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The Lecture Notes in Physics The series Lecture Notes in Physics (LNP), founded in 1969, reports new developments in physics research and teaching – quickly and informally, but with a high quality and the explicit aim to summarize and communicate current knowledge in an accessible way. Books published in this series are conceived as bridging material between advanced graduate textbooks and the forefront of research to serve the following purposes: • to be a compact and modern up-to-date source of reference on a well-defined topic; • to serve as an accessible introduction to the field to postgraduate students and nonspecialist researchers from related areas; • to be a source of advanced teaching material for specialized seminars, courses and schools. Both monographs and multi-author volumes will be considered for publication. Edited volumes should, however, consist of a very limited number of contributions only. Proceedings will not be considered for LNP. Volumes published in LNP are disseminated both in print and in electronic formats, the electronic archive is available at springerlink.com. The series content is indexed, abstracted and referenced by many abstracting and information services, bibliographic networks, subscription agencies, library networks, and consortia. Proposals should be sent to a member of the Editorial Board, or directly to the managing editor at Springer: Dr. Christian Caron Springer Heidelberg Physics Editorial Department I Tiergartenstrasse 17 69121 Heidelberg/Germany [emailprotected]

J. Dolinšek M. Vilfan S. Žumer (Eds.)

Novel NMR and EPR Techniques

ABC

Editors Professor Dr. Janez Dolinšek+∗ Professor Dr. Marija Vilfan∗ Professor Dr. Slobodan Žumer+∗

E-mails: [emailprotected] [emailprotected] [emailprotected]

+ Physics

Department Faculty of Mathematics and Physics University Ljubljana Jadranska 19 1000 Ljubljana Slovenia

∗ Jozef Stefan Institute Jamova 39 1000 Ljubljana Slovenia

J. Dolinšek et al., Novel NMR and EPR Techniques, Lect. Notes Phys. 684 (Springer, Berlin Heidelberg 2006), DOI 10.1007/b11540830

Library of Congress Control Number: 2006921047 ISSN 0075-8450 ISBN-10 3-540-32626-X Springer Berlin Heidelberg New York ISBN-13 978-3-540-32626-7 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2006 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: by the authors and techbooks using a Springer LATEX macro package Cover design: design & production GmbH, Heidelberg Printed on acid-free paper

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This book is dedicated to Professor Robert Blinc, with respect and kind regards

Professor Robert Blinc

Preface

This book is a collection of scientiﬁc articles on current developments of NMR and ESR techniques and their applications in physics and chemistry. It is dedicated to Professor Robert Blinc, on the occasion of his seventieth birthday, in appreciation of his remarkable scientiﬁc accomplishments in the NMR of condensed matter. He is a physicist commanding deep respect and aﬀection from those who had the opportunity to work with him. Robert Blinc was born on October 31, 1933, in Ljubljana, Slovenia. He graduated in 1958 and completed his Ph.D in 1959 in physics at the University of Ljubljana. His doctoral research on proton tunneling in ferroelectrics with short hydrogen bonds was supervised by Professor Duˇsan Hadˇzi. After a postdoctoral year spent in the group of Professor John Waugh at M.I.T., Cambridge, Mass., Robert Blinc was appointed as a professor of physics at the University of Ljubljana at a time when there was scarce research in the ﬁeld of condensed matter in Slovenia. With his far-sighted mind, Robert Blinc, together with Ivan Zupanˇciˇc, started the NMR laboratory at the Jozef Stefan Institute in Ljubljana. He immediately realized the enormous potential of NMR methods in the research of structure, dynamics, and phase transitions in solids. In the subsequent years he made signiﬁcant contributions in applying magnetic resonance to the research of ice, ferroelectric materials, liquid crystals, incommensurate systems, spin glasses, relaxors, fullerenes, and fullerene nanomagnets. His work led to the detailed understanding of the microscopic nature and properties of those materials. To mention only a few: Robert Blinc and coworkers elucidated the isotopic eﬀect in ferroelectric crystals, predicted the Goldstone mode in ferroelectric liquid crystals, studied the impact of collective orientational ﬂuctuations on spin relaxation, detected solitons and phasons in incommensurate systems using NMR, and determined the Edwards-Anderson order parameter in glasses and relaxors. He also pioneered the application of NMR to the nondestructive oil-content measurements in seeds and the development of the NMR measurements of the self-diﬀusion coeﬃcient in broad-line materials. In the early stage of the double resonance technique, he succeeded in obtaining the ﬁrst nitrogen NMR

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Preface

spectra in nucleic acids and peptides. An important achievement of Professor Robert Blinc, which attracted considerable attention in the broad scientiﬁc community, is the book Soft Modes in Ferroelectrics and Antiferroelectrics ˇ s in 1974. The book was (North Holland), written by him and Boˇstjan Zekˇ translated into Russian (1975) and Chinese (1982) and belongs to the 600 most-cited scientiﬁc books in the world. Another of his books, written toˇ s, The Physics of Ferroelectric and gether with Igor Muˇseviˇc and Boˇstjan Zekˇ Antiferroelectric Liquid Crystals, was published by World Scientiﬁc in 2000. Apart from being professor of physics at the University of Ljubljana, the head of the Condensed Matter Physics Department at Joˇzef Stefan Institute and a member (and vice-president in the years 1980–1999) of Academy of Science and Arts of Slovenia, Robert Blinc maintained a wide range of contacts with scientists worldwide. There is an amazingly long list of his international scientiﬁc activities. To mention only a few of them, he was a visiting professor at the University of Washington in Seattle; ETH Zurich; Federal University of Minas Gerais in Belo Horizonte, Brazil; University of Vienna in Austria; University of Utah in Salt Lake City; Kent State University in Ohio, Argon National Laboratory, and several others. In the years 1988–1994 he was the president of the Groupement AMPERE (Atomes et Molecules Par Etudes Radio Electrique), and president of the European Steering Committee on Ferroelectrics (1990–1999). He is a member of seven foreign Academies of Sciences and has received several national and international scientiﬁc prizes and medals. As the head of the Condensed Matter Physics Department at the Joˇzef Stefan Institute for more than 40 years, Robert Blinc promoted, in addition to NMR, other experimental techniques: ESR, dielectric measurements, optical spectroscopy. He also atomic force microscopy. He also took active part in solving theoretical problems related to the systems under study. He was the supervisor of 67 diploma works and 35 Ph. D. theses in the ﬁeld of condensed matter physics in Ljubljana. He can therefore be recognized as the founder and tireless promoter of the condensed matter physics research in Slovenia. Most of Robert Blinc’s research is tightly related to nuclear magnetic resonance. Therefore we invited a number of prominent researchers in this ﬁeld to write chapters on the recent condensed matter physics research based on new NMR and ESR techniques. The book covers: • Adiabatic and nonadiabatic magnetization caused by rotation of solids with dipole-dipole coupled spins. • Magnetic resonance techniques for studying spin-to-spin pair correlation in multi-spin systems. • Studies of selectively deuterated semisolid materials and anisotropic liquids by deuterium NMR. • Initial steps toward quantum computing with electron and nuclear spins in crystalline solids.

Preface

• Laser radiation-induced increase of the spin polarization in various magnetic resonance experiments. • Multiple-photon processes in cw and pulse electron paramagnetice resonance spectroscopy. • NMR and EPR for the determination of ion localization and charge transfer in metallo-endofullerenes. • NMR shifts in metal nanoparticles of silver, platinum, and rhodium. • NMR relaxation studies of diﬀerent superconducting systems. • Investigations of static and dynamic properties of low dimensional magnetic systems by NMR. • NMR-NQR relaxation studies of spin ﬂuctuations in two-dimensional quantum Heisenberg antiferromagnets. • The dynamics of the deuteron glass in KDP type crystals studied by various one-dimensional and two-dimensional NMR techniques. • Nuclear Magnetic Resonance cryoporometry based on depression of the melting temperature of liquids conﬁned in pores. We congratulate Professor Robert Blinc on his great scientiﬁc achievements and also express our deep gratitude for his continuous eﬀorts in stimulating and supporting the NMR and condensed matter physics community.

Ljubljana June 2005

Janez Dolinˇsek Marija Vilfan ˇ Slobodan Zumer

Contents

Nuclear Spin Analogues of Gyromagnetism: Case of the Zero-Field Barnett Eﬀect E.L. Hahn, B.K. Tenn, M.P. Augustine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Personal Tribute . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Parameter Rules for Interpretation of Gyromagnetic Experiments . . 4 Nuclear Spin Analogues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 hom*onuclear Dipole-Dipole Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Adiabatic Demagnetization and Remagnetization . . . . . . . . . . . . . . . . . 7 Sample Spinning Non-axial with DC Field . . . . . . . . . . . . . . . . . . . . . . . 8 Lattice Structure Dependence of the Barnett Eﬀect . . . . . . . . . . . . . . . 9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 3 5 5 14 15 18 18 19

Distance Measurements in Solid-State NMR and EPR Spectroscopy G. Jeschke, H.W. Spiess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Dipole-Dipole Interaction in a Two-Spin System . . . . . . . . . . . . . . . . . . 3 Measurement Techniques for Isolated Spin Pairs in Solids . . . . . . . . . . 4 Complications in Multi-Spin Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Application Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21 21 24 33 44 49 58 59 60

NMR Studies of Disordered Solids J. Villanueva-Garibay, K. M¨ uller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Simulation Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65 65 66 75

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4 Model Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Applications for Guest-Host Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75 80 85 85

En Route to Solid State Spin Quantum Computing M. Mehring, J. Mende, W. Scherer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 1 Brief Introduction to Quantum Algorithms . . . . . . . . . . . . . . . . . . . . . . 87 2 Combined Electron Nuclear Spin States in Solids . . . . . . . . . . . . . . . . . 92 3 Entanglement of an Electron and a Nuclear Spin 12 . . . . . . . . . . . . . . . 94 4 Entangling an Electron Spin 32 with a Nuclear Spin 12 . . . . . . . . . . . . . 100 5 The S-Bus Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Laser-Assisted Magnetic Resonance: Principles and Applications D. Suter, J. Gutschank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 2 Optical Polarization of Spin Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 3 Optical Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 4 Applications to NMR and NQR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5 EPR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Multiple-Photon Transitions in EPR Spectroscopy M. K¨ alin, M. Fedin, I. Gromov, A. Schweiger . . . . . . . . . . . . . . . . . . . . . . . 143 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 2 Diﬀerent Types of Multiple-Photon Transitions in EPR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 3 Eﬀects of Oscillating Longitudinal Field . . . . . . . . . . . . . . . . . . . . . . . . . 158 A Second Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Multi-Frequency EPR Study of Metallo-Endofullerenes K.-P. Dinse, T. Kato . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 2 La@C82 – a Case Study of Ion Localization . . . . . . . . . . . . . . . . . . . . . . 189 3 La2 @C− 80 Radical Anion – Evidence for Reduction of the Encased Cluster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 4 Gd@C82 – Determination of Exchange Coupling Between Ion and Cage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

Contents

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Beyond Electrons in a Box: Nanoparticles of Silver, Platinum and Rhodium J.J. van der Klink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 2 Bulk Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 3 Small Silver Particles: Bardeen-Friedel Oscillations . . . . . . . . . . . . . . . 218 4 Small Platinum Particles: Exponential Healing . . . . . . . . . . . . . . . . . . . 223 5 Small Rhodium Particles: Incipient Antiferromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 The Study of Mechanisms of Superconductivity by NMR Relaxation D.F. Smith, C.P. Slichter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 2 Normal Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 3 Development and Veriﬁcation of the BCS Theory . . . . . . . . . . . . . . . . . 246 4 Type I and II Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 5 The Alkali Fullerides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 6 The Cuprate Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 7 The Organic Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 NMR in Magnetic Molecular Rings and Clusters F. Borsa, A. Lascialfari, Y. Furukawa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 2 Challenges of NMR in Molecular Nanomagnets . . . . . . . . . . . . . . . . . . . 300 3 NMR at High Temperature (kB T J) . . . . . . . . . . . . . . . . . . . . . . . . . 302 4 NMR at Intermediate Temperatures (kB T ≈ J) . . . . . . . . . . . . . . . . . . 306 5 NMR at Low Temperatures (kB T J) . . . . . . . . . . . . . . . . . . . . . . . . . 318 6 Miscellaneous NMR Studies of Molecular Clusters: Fe2, Fe4, Fe30, Ferritin Core, Cr4, Cu6, V6, V15 . . . . . . . . . . . . . . . . . 338 7 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 Correlated Spin Dynamics and Phase Transitions in Pure and in Disordered 2D S = 1/2 Antiferromagnets: Insights from NMR-NQR A. Rigamonti, P. Carretta, N. Papinutto . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 1 Introduction and Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 2 The Phase Diagram of 2DQHAF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 3 Basic Aspects of the Experimental Approach . . . . . . . . . . . . . . . . . . . . . 356 4 NMR-NQR Relaxation Rates: Amplitude and Decay Rates of Spin Fluctuations and Critical Behaviour in 2D Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

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Pure 2DQHAF: Temperature Dependence of the Correlation Length (in La2 CuO4 and in CFTD, within Scaling Arguments) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 6 Spin and Charge Lightly-Doped La2 CuO4 : Eﬀects on the Correlation Length and on the Spin Stiﬀness . . . . . . . . 363 7 Spin and Charge Doped La2 CuO4 Near the AF Percolation Thresholds: Spin Stiﬀness, Correlation Length at the Transition and Staggered Magnetic Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 8 The Cluster Spin-Glass Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 9 The Quantum Critical Point in an Itinerant 2DAF – Eﬀect of Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 10 Summarizing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

Two-Dimensional Exchange NMR and Relaxation Study of the Takagi Group Dynamics in Deuteron Glasses R. Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 2 Model of the Glass Phase Dynamics in DRADP-50 . . . . . . . . . . . . . . . 385 3 87 Rb 2D Exchange-Diﬀerence NMR Reveals a Correlated Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 4 Distinction of the Six Slater Conﬁgurations by the Anisotropic 31 P Chemical Shift Tensor . . . . . . . . . . . . . . . . . . . . 391 5 Slow Polarization Fluctuations of the PO4 Groups Observed by 31 P 2D Exchange NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 6 Interpretation of the 87 Rb T1 Measurements . . . . . . . . . . . . . . . . . . . . . 399 7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 Characterising Porous Media J.H. Strange, J. Mitchell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 2 Measurement of Liquid Fraction Using NMR . . . . . . . . . . . . . . . . . . . . . 410 3 The NMR Cryoporometry Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 411 4 Determining the Melting Point Depression Constant . . . . . . . . . . . . . . 413 5 Cryoporometry Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 6 Applications of NMR Cryoporometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431

List of Contributors

M.P. Augustine Department of Chemistry One Shields Avenue University of California Davis, CA 95616 [emailprotected] F. Borsa Dipartimento di Fisica “A.Volta” e Unita’ INFM, Universita’ di Pavia, 27100 Pavia, Italy and Department of Physics and Astronomy and Ames Laboratory Iowa State University Ames, IA 50011 [emailprotected] P. Carretta Department of Physics “A.Volta” and Unit` a INFM University of Pavia Via Bassi n◦ 6 I-27100, Pavia (Italy) K.-P. Dinse Physical Chemistry III Darmstadt University of Technology Petersenstrasse 20 D-64287 Darmstadt, Germany [emailprotected]

M. Fedin Laboratory for Physical Chemistry ETH Zurich, CH-8093 Zurich Switzerland Y. Furukawa Division of Physics Graduate School of Science Hokkaido University Sapporo 060-0810, Japan I. Gromov Laboratory for Physical Chemistry ETH Zurich, CH-8093 Zurich Switzerland J. Gutschank Universit¨at Dortmund Fachbereich Physik 44221 Dortmund, Germany E.L. Hahn Department of Physics University of California Berkeley, CA 94720 [emailprotected] G. Jeschke Max Planck Institute for Polymer Research Postfach 3148

XVI

List of Contributors

55021 Mainz, Germany [emailprotected] T. Kato Institute for Molecular Science Myodaiji Okazaki 444-8585, Japan present address: Department of Chemistry Josai University, 1-1 Keyakidai Sakado 350-0295, Japan [emailprotected] M. K¨ alin Laboratory for Physical Chemistry ETH Zurich, CH-8093 Zurich Switzerland R. Kind Institute of Quantum Electronics ETH-Hoenggerberg, CH-8093 Zurich Switzerland [emailprotected] J.J. van der Klink Institut de Physique des Nanostructures EPFL, CH-1015 Lausanne, Switzerland [emailprotected] A. Lascialfari Dipartimento di Fisica “A.Volta” e Unita’ INFM, Universita’ di Pavia, 27100 Pavia, Italy lascialfari@fisicavolta. unipv.it M. Mehring Physikalisches Institut University Stuttgart D-70550 Stuttgart, Germany [emailprotected]

J. Mende Physikalisches Institut University Stuttgart D-70550 Stuttgart, Germany J. Mitchell Department of Physics University of Surrey Surrey, UK, GU2 7XH [emailprotected] K. M¨ uller Institut f¨ ur Physikalische Chemie Universit¨at Stuttgart Pfaﬀenwaldring 55 D-70569 Stuttgart, Germany [emailprotected] N. Papinutto Department of Physics “A.Volta” and Unit` a INFM University of Pavia Via Bassi n◦ 6 I-27100, Pavia (Italy) A. Rigamonti Department of Physics “A.Volta” and Unit` a INFM University of Pavia Via Bassi n◦ 6 I-27100, Pavia (Italy) [emailprotected] W. Scherer Physikalisches Institut University Stuttgart D-70550 Stuttgart, Germany A. Schweiger Laboratory for Physical Chemistry ETH Zurich, CH-8093 Zurich Switzerland [emailprotected]

List of Contributors

C.P. Slichter Department of Physics and Frederick Seitz Materials Research Laboratory University of Illinois Urbana, IL 61801 [emailprotected] D.F. Smith Department of Physics and Frederick Seitz Materials Research Laboratory University of Illinois Urbana, IL 61801 H.W. Spiess Max Planck Institute for Polymer Research Postfach 3148 55021 Mainz, Germany [emailprotected] J.H. Strange School of Physical Sciences University of Kent Canterbury, Kent UK, CT2 7NR, [emailprotected]

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D. Suter Universit¨at Dortmund Fachbereich Physik 44221 Dortmund, Germany [emailprotected] B.K. Tenn Department of Chemistry One Shields Avenue University of California Davis, CA 95616 J. Villanueva-Garibay Institut f¨ ur Physikalische Chemie Universit¨at Stuttgart Pfaﬀenwaldring 55 D-70569 Stuttgart, Germany

Nuclear Spin Analogues of Gyromagnetism: Case of the Zero-Field Barnett Eﬀect E.L. Hahn1 , B.K. Tenn2 and M.P. Augustine2 1

Department of Physics, University of California, Berkeley, CA 94720 [emailprotected] Department of Chemistry, One Shields Avenue, University of California, Davis CA 95616 [emailprotected]

Abstract. A short review of the history and elementary principles of gyromagnetic eﬀects is presented. The Barnett eﬀect is considered as a mechanism for inducing nuclear spin magnetization in solids by sample spinning in zero and low ﬁeld. Simulations of rotation induced adiabatic and non-adiabatic magnetization derived from initial dipolar order in hom*onuclear dipole-dipole coupled spins are carried out. Aspects of the converse Einstein-de Haas eﬀect are included.

1 Personal Tribute We thank the editors for the invitation to write in honor of Robert Blinc and to celebrate his 70th birthday. Over his many years of international research collaborations and leadership of NMR research groups at the Josef Stefan Institute, he and his colleagues have generated a body of comprehensive experimental data leading to new concepts and clariﬁcations concerning unusual solid state structures. These have involved topics such as ferroelectrics, disordered systems, and liquid crystals, often connected with phase transitions and modal behavior. While continually devoting himself to his group as a pioneering research physicist working out new interpretations of experiments, Professor Blinc also served as a leader of Slovenian science, maintaining many personal contacts as a virtual “Science Ambassador” in Europe. During the Cold War and through its waning years, because of Robert’s international connections in the East, he was able to arrange contacts with people from both the East and the West to attend conferences in Slovenia and on the Dalmation Coast. He has made it possible for many NMR research people from the Eastern block to interact with those of us from the West. His international inﬂuence has been unique in making the world a better place for scientiﬁc cooperation. Personally I can testify how much we have enjoyed Robert’s hospitality, friendship, and E.L. Hahn et al.: Nuclear Spin Analogues of Gyromagnetism: Case of the Zero-ﬁeld Barnett Eﬀect, Lect. Notes Phys. 684, 1–19 (2006) c Springer-Verlag Berlin Heidelberg 2006 www.springerlink.com

E.L. Hahn et al.

stimulation provided by visitations to interact with his NMR research group at the Jozef Stefan Institute for which we are grateful. We wish him and his spouse many happy and fruitful years and that he should remain active and not really retire.

2 Introduction The motivation of this work is the possibility of the ultra-sensitive detection of pico and femto-Tesla ﬁelds from the nuclear spin polarization induced in spinning solids at zero ﬁeld caused by the Barnett eﬀect. Both the Superconducting Quantum Interference Device (SQUID) [1] and non-linear optical Faraday rotation methods [2] of measuring magnetic ﬁelds with an ultimate sensitivity of about one femto-Tesla Hz−1/2 or 10−11 G Hz−1/2 promise to detect these small ﬁelds generated by diamagnetic solids. A brief review of well known elementary principles of gyromagnetic experiments [3, 4] sets the stage for discussion of coupling mechanisms in the Barnett eﬀect that may account for momentum transfer from a mechanically spinning macroscopic body to microscopic nuclear spins within the body. Only a minute fraction of the total mechanical angular momentum of the spinning sample is transferred to the oriented macroscopic magnetic spin angular momentum, thus conserving the total angular momentum of the system. In the absence of diamagnetic eﬀects, the ratio of the change of the macroscopic magnetism of a rotating body to this corresponding change in angular momentum of the body is well known as the eﬀective gyromagnetic ratio (q/2mc) g where q is the fundamental charge, m is the electron mass, c is the speed of light, and g is the empirical g factor. The Barnett eﬀect was ﬁrst observed in 1914 [4] by detecting the magnetism due to the polarization of electron spins caused by rotation of a cylinder of soft unmagnetized iron. Although the Barnett eﬀect is looked upon today as an archaic experiment, it was an important experiment of the old physics era. Today many people are not aware of the Barnett eﬀect because it is referred to so little in the literature. It is interesting that even though the concept of electron spin did not exist at that time, the original Barnett experiment provided the ﬁrst evidence that the electron had an anomalous magnetic moment with a g factor of 2. Today it is well known that the electron spin g factor can also diﬀer signiﬁcantly from the value of 2 (ignoring the small radiative correction) because of spin orbit coupling. Barnett concluded in 1914 that he measured the gyromagnetic ratio of classical rotating charges q to be q/mc, an anomalous value twice the classical value he expected. In 1915 Einstein and de Haas [5] carried out the converse of the Barnett experiment. The reversal of an initially known magnetization M0 , or the growth of M0 from zero, of an iron cylinder produces a small mechanical rotation of the cylinder. In contrast to Barnett’s experiment, there is an apparent transfer of spin angular momentum from M0 into mechanical rotation. Curiously

Nuclear Spin Analogues of Gyromagnetism

Einstein and de Haas reported g = 1 from their measurements, apparently rejecting data that deviated from the expected classical value of g = 1 to account for the orbital magnetism. In 1820 Ampere established that the current due to a charge q and mass m rotating in a circle of radius r, multiplied by circle area πr2 , expresses the classical orbital magnetic moment. This picture supported the idea of hidden classical Amperian currents in permanent magnets, a view that held sway into the early years of the 20th century. But this view of classical magnetism, and ultimately the Einstein and de Haas g = 1 experimental interpretation, was ﬁrst challenged by a theorem formulated by Miss van Leeuwen [6] and Bohr, namely, that any conﬁned conﬁguration of free charges obeying classical laws of motion and precessing in any magnetic ﬁeld must yield zero magnetic susceptibility. Finally the advent of momentum quantization and the concept of magnetic spin made possible a break away from the invalid classical picture of magnetism. The classical Amperian magnetic moment was replaced by the non-classical entity of magnetism, the Bohr magneton, e = γ . (1) µ= 2mc

3 Parameter Rules for Interpretation of Gyromagnetic Experiments Let the ratio nµ/n = M/Ω = γ = e/2mc be deﬁned from (1), where n is the number of polarized spins, or circulating charges in the old picture, lined up to deﬁne a macroscopic magnetic moment M = nµ. The corresponding angular momentum is given by Ω = n. By itself this ratio is a trivial identity, given that the Bohr magneton of every particle with L = 1 is µ = e/2mc. The terms n and contained in γ always cancel, implying in ﬁrst order that the macroscopic body must display the same γ as a single spin would, and provide a measure of e/2mc multiplied by any anomalous g factor. However this argument deserves a better physical justiﬁcation, relating phenomenologically and still somewhat obscurely to the response of a gyroscope to torque. Sample rotation at a given frequency ωr may be viewed as equivalent to a Larmor precession caused by a magnetic ﬁeld H. As shown in Fig. 1, the imposed torque due to H tends to line up the spins. Changes in M and Ω evolve coaxially. They precess about H independent of the angle between H and M or Ω. A real magnetic ﬁeld H causes spin precession of M about the direction of H while M develops and ﬁnally reaches equilibrium because of spin-lattice relaxation. However, if the sample is not left to rotate freely as in the Einstein-de Haas experiment, there can be no direct evidence of any mechanical exchange of momentum Ω no matter how minute. Except for certain special circ*mstances of macroscopic radiation damping, theories of spin relaxation keep track of energy degrees of freedom but not of elusive internal mechanisms of spin lattice momentum transfer.

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Fig. 1. Relationship of the magnetization M , rotational angular momentum Ω, and static magnetic ﬁeld H used to discuss magnetomechanical rotation experiments

Before the development of gyromagnetic experiments, in 1861 Maxwell perceived Amperian currents as hidden gyroscopic sources of permanent magnetism. He tried to detect the precession of a permanent magnet in response to an outside torque, but the eﬀect is too small to detect. His attempt relates to Fig. 1. In place of a mechanical torque, the magnetic ﬁeld H subjects the magnetization M to the torque T = M sin θ H = Ω sin θ ωr . Including the empirical factor g the ratio e ωr M = = g Ω H 2mc

(2)

deﬁnes values of M and Ω as ﬁnal values representing changes from zero. As a gyromagnetic rule, M/Ω should be written as the ratio of changes ∆M/∆Ω at any time in the evolution of the spin alignment. The Einstein-de Haas experiment measures the ratio of any imposed M change to the resulting sample rotation angular momentum which is observed. The Barnett experiment measures the ratio of ωr to a calibrated H ﬁeld that produces the same M caused by sample rotation at the frequency ωr . Generation of a calibrated H in the pico to femto-Tesla range from a stable current source would be an extremely diﬃcult requirement. No real ﬁeld is present when the Barnett eﬀect takes place. Instead, the ﬁeld H in (2) acts like a “ghost” ﬁeld Hghost , having the same eﬀect as a real ﬁeld. Its deﬁnition relates to Larmor’s theorem, where Hghost = ωr /γ is deﬁned as an equivalent ﬁeld. Equation (2) must follow from energy conservation arguments. A sample rotating at the rate ωr and with moment of inertia I is endowed with rotational energy U = Iωr2 /2. Any small momentum transfer ∆Ω = I∆ωr that might take place to the spins would require a corresponding increase in spin energy ∆M H. The total energy transferred between spin and rotation is then ∆U = Iωr ∆ωr = ∆Ωωr = ∆M H, a relation that immediately rearranges to express the gyroscopic rule given by (2).

Nuclear Spin Analogues of Gyromagnetism

4 Nuclear Spin Analogues Prior to 1940, gyromagnetic experiments served as a measure of g values of electron spin systems in ferromagnetic and paramagnetic substances [4]. The Barnett magnetization, although at least a thousand times or more greater than nuclear spin magnetization in samples of comparable size, is very small, diﬃcult to measure, and easily obscured by instrumental instabilities and stray ﬁelds. Even large magnetic ﬁelds in those days could not be measured to better than a fraction of a percent by rotating pick up coils. These methods are now obsolete, superseded by the application of magnetic resonance detection methods [7, 8, 9]. As a physical mechanism, the nuclear Barnett eﬀect was invoked later by Purcell [10] to account for the observation of weakly polarized starlight. In that account, Purcell discusses the mechanism of diﬀerential light scattering from fast “suprathermal” rotating grains in interstellar space. Because of their rotation it is postulated that these grains become magnetized due to the nuclear Barnett eﬀect. The common directivity and polarization of light scattering by the grains occurs over vast distances because the polarized grains in turn precess about the direction of weak interstellar magnetic ﬁelds. Rather than discuss parameters of this very special unearthly case [11] of Barnett polarization, consider a more representative and yet marginal case on Earth. Here a 1 cm3 sample of N = 1022 nuclear Bohr magnetons where µB = (9.27/1840) × 10−21 erg G−1 is rotated at the rate ωr /2π = 4 kHz in zero applied magnetic ﬁeld at T = 300 K. Assume that the sample acquires an equilibrium magnetization M0 = N µB (ωr /kT ) because of spin lattice relaxation in the ghost ﬁeld Hghost = ωr /γ. The resulting polarization ﬁeld in the sample is about 4πM0 ≈ 10−10 G or about 1–10 femto-Tesla. Clearly rotation at higher speeds and at lower temperature could provide M0 values 10 to 100 times larger, providing an extra margin for weak ﬁeld detection [1, 2].

5 hom*onuclear Dipole-Dipole Coupling The crude estimate of the ﬁeld due to a Barnett induced magnetization mentioned above assumes that the spins polarize in a ghost ﬁeld Hghost = ωr /γ during a spin-lattice relaxation process as though it were a real ﬁeld. However, the complexity of many momentum transfer relaxation mechanisms between spin and lattice thermal reservoirs is too diﬃcult to handle. Some understanding can be gained from a speciﬁc example of momentum conservation by simulating the eﬀect of sample spinning on dipolar interactions among nuclear spins. A rigid lattice ﬁrmament of spins is assigned only a spin temperature with lattice coordinates θ, r and φ independent of time in the absence of sample spinning. Since there is no lattice thermal reservoir in this picture, the source of magnetization is obtained from previously prepared dipolar order in zero ﬁeld that is converted to magnetization by sample rotation. A simple starting point considers two identical spins I1 and I2

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separated by the distance r and coupled via their dipolar ﬁelds. A real DC magnetic ﬁeld H0 , or sample spinning at the rate ωr may be applied separately or simultaneously, axially or non-axially. The dipolar coupling is deﬁned as 2 (2) (2) HD = ωD q=−2 (−1)q Tq (I1 , I2 )R−q (θ, φ) where ωD = γ1 γ2 /r3 and the (2)

(2)

irreducible components Tq (I1 , I2 ) and Rq (θ, φ) are given by (2)

T0 (I1 , I2 ) = (2)

T±1 (I1 , I2 ) = (2)

√1 (3Iz,1 Iz,2 − I1 · I2 ) 6 √1 (I±,1 Iz,2 + Iz,1 I±,2 ) 2

T±2 (I1 , I2 ) = I±,1 I±,2

(2)

R0 (θ, φ) =

√

6 2 (1

− 3 cos2 θ) ,

(2) R±1 (θ, φ) = ±3 sin θ cos θe±iφ , (3) (2)

R±2 (θ, φ) =

3 2

sin2 θe±2iφ .

When written in this way it should be clear that HD not only couples the spins (2) I1 and I2 to each other with the Tq (I1 , I2 ) operators, but also to the lat(2) tice with the Rq (θ, φ) coeﬃcients. The lattice degrees of freedom with large heat capacity are usually assumed to be at constant temperature in diagonal states of the density matrix, while the spin temperature may change due to relaxation. For this reason the understanding of NMR experiments in solids considers only the eﬀect of the lattice on the spins, while eﬀects of the spins on the lattice are usually neglected. However gyromagnetic experiments conﬁrm that the lattice hooked to the rotor does exchange angular momentum with the spins, showing that the spins have a momentum eﬀect on the lattice. In connection with nuclear spin diﬀusion in zero ﬁeld, Sodickson and Waugh [12] introduced the interesting and related question about momentum exchange and conservation between nuclear spins and the lattice in zero ﬁeld. Here the components of the angular momentum operators L = r × p are given by ∂ ∂ + cot θ cos φ , Lx = i sin φ ∂θ ∂φ ∂ ∂ + cot θ sin φ , (4) Ly = i − cos φ ∂θ ∂φ ∂ , Lz = −i ∂φ which in combination with Ehrenfest’s theorem can be used to show that the time derivative of the expectation value of the total spin angular momentum J = I 1 + I 2 is given by d d J = I 1 × H D,2 + I 2 × H D,1 = i[HD , J ] = − L , dt dt

(5)

where H D,1 and H D,2 are the dipolar ﬁelds from spin 1 and 2. This reformulation of Bloch’s equation shows that the total angular momentum, spin plus lattice, is conserved. This angular momentum conservation relationship is at the heart of recovering Zeeman order from dipolar order by sample rotation and can be used to determine the eﬀect of the spins on the lattice. As an example consider an ensemble of identical dipole-dipole coupled two

Nuclear Spin Analogues of Gyromagnetism

spin systems at thermal equilibrium in zero applied magnetic ﬁeld. The thermal equilibrium density operator corresponding to this situation is given by ρeq = exp(−HD /kT ) ≈ 1 − HD /kT where k is the Boltzmann constant and the high temperature approximation has been applied. Provided that there are no real magnetic ﬁelds present, the expectation value J = T r{ρeq J } is zero at all times. In order to verify (5) or equivalently show that L = 0 in this example, a rigorous quantum mechanical treatment of the expectation value of the lattice angular momentum L is needed. This treatment requires that the rotational motion of the lattice be quantized in direct analogy to the rotational level structure in molecular H2 gas [7] or in tunneling methyl groups [13]. Adopting this approach requires some knowledge of the partition function corresponding to rotation in addition to the calculation of (2) (L) the matrix elements of the Rq (θ, φ) coeﬃcients from the |L, m = Ym (θ, φ) spherical harmonic basis functions. In the case of H2 gas in a molecular beam where the moment of inertia I is small, molecular rotation is fast, and the temperature is low, it is safe to truncate the basis set to a ﬁnite number of rotational energy levels. However, in the case of a real macroscopic sample at room temperature, I is large and the sample rotation is slow. This means that an untractably large number of very closely spaced energy levels will be populated at thermal equilibrium. The inability to deﬁne the density matrix of the rotor in this case can be circumvented by realizing that the expectation value for the lattice angular momentum L must reduce to the classical result Iωr for a macroscopic object. In this way in zero ﬁeld (5) reduces to d d (Iωr ) = − J , dt dt

(6)

therefore in the absence of sample rotation, ωr = 0 and L = Iωr = 0. Equation (6) contains a very important result, namely, any change in sample rotation rate ωr will lead to a corresponding change in sample magnetization. If a stationary sample is initially in zero magnetic ﬁeld, (6) indicates that a jump in rotational frequency from zero to a ﬁnal value ωr will lead to a corresponding change in sample magnetization from zero to a ﬁnal value. In certain special cases like the ensemble of identical two spin systems mentioned above, (6) can be used in combination with the solution to the Liouville-von Neumann equation during sample spinning along the +z direction to determine the dynamics of both the formation of magnetization J and the change in the spin rate ωr due to this magnetization. Here the static dipolar coupling HD mentioned above becomes (2) (2) time dependent in φ(t) as HD (t) = ωD (−1)q Tq (I1 , I2 )R−q (θ, φ)eiqωr t since the internuclear vector r now precesses about the rotation direction taken along +z. This particular time dependent form for the laboratory frame Hamiltonian Hlab = HD (t) is not convenient for practical calculations and does not oﬀer much insight into how magnetization develops due (2) (2) to sample spinning. The symmetry between the Tq (I1 , I2 ) and Rq (θ, φ)

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products in HD (t) permit the exp(iqωr t) time dependence to be grouped to(2) gether with the spin operators as Tq (I1 , I2 ) exp(iqωr t) instead of with the (2) Rq (θ, φ) spatial terms that describe the orientation of r as a function of time. In this way it is clear that transformation to a rotating frame in spin space by rotation operator U (ωr t) = eiqIz ωr t about the laboratory +z axis as (2) (2) U (ωr t) Tq (I1 , I2 ) U † (ωr t) = Tq (I1 , I2 ) exp(iqωr t) yields the time independent rotating frame Hamiltonian (2) Hrot = ωr (Iz,1 + Iz,2 ) + ωD (7) (−1)q Tq(2) (I1 , I2 )R−q (θ, φ) , where the ghost ﬁeld Hghost = ωr /γ appears. As expectation values of the total spin magnetization J are independent of unitary transformations in the trace, the ﬁctitious term in the rotating frame generates the same magnetization that a real ﬁeld H0 = Hghost would develop in the laboratory frame. Furthermore, since (5) and (6) relate traces of spin and lattice momentum operators, these relations are also frame independent. A combination of the solution to the Liouville-von Neumann equation in the rotating frame using Hrot in (7) with the angular momentum conservation rule in (6) allows the dynamics of the magnetization and the rotor frequency to be determined starting from a stationary sample in zero magnetic ﬁeld. Figure 2 shows the eﬀect of instantaneously switching on an ωr /2π = 10 kHz sample rotation to an ensemble of dipolar coupled two spin systems with ωD /2π = 10 kHz. The plots in Fig. 2(a) are appropriate for an ensemble of identical two-spin systems with θ = π/2 and φ = 0 while the plots in Fig. 2(b) describe a somewhat more realistic case involving an isotropic distribution of θ and φ values. In both of these plots the feedback due to angular momentum conservation requires that the alternating magnetization will modulate the ωr /2π = 10 kHz applied sample spin rate. In addition the phase of the periodicity introduced into the spin rate is 180◦ out of phase with the periodicity in Jz , consistent with the negative sign in (6). The same two spin system used in Fig. 2(a) was used to determine the peak z magnetization as a function of applied rotation rate in Fig. 2(c). This plot demonstrates that the largest Barnett magnetization can be obtained in zero ﬁeld when ωr ≈ ωD . It is natural to ask if an analogue of the above experiments can be obtained by causing an ensemble of two-spin systems at thermal equilibrium in zero ﬁeld to rotate by instantaneously jumping a real DC magnetic ﬁeld. Introduction of a real DC magnetic ﬁeld H0 along the +z direction in the laboratory frame where θ = 0 adds a Zeeman term Hz = γH0 (Iz,1 + Iz,2 ) to the dipolar coupling Hamiltonian HD yielding the full laboratory frame Hamiltonian as Hlab = Hz + HD . One consequence of the ﬁeld H0 is to add an additional term to the zero ﬁeld angular momentum conservation relation shown in (5) since dL/dt = i[Hlab , L] = i[HD , L] = −i[HD , J ] and dJ /dt = i[Hlab , J ] = i[Hz , J ] + i[HD , J ]. Rearranging these equations, noting that L commutes with Hz , and using the fact that L = Iωr in the classical limit recasts (5) as

Nuclear Spin Analogues of Gyromagnetism

Fig. 2. Simulation of the dynamics of the two spin system rotating at the frequency ωr /2π with ωD /2π = 10 kHz and ω0 /2π = 0. In each case (a)–(c) the t < 0 thermal equilibrium condition corresponds to an ensemble of two spin systems in zero magnetic ﬁeld. At the time t = 0 the spin rate is instantaneously increased from zero to ωr /2π = 10 kHz in (a) and (b) and a variable level in (c). The eﬀects of the sample rotation are to create magnetization (solid line) as shown by the left ordinate in (a) and (b) for the θ = π/2 orientation and the isotropic powder respectively. The polarization Jz = Iz,1 + Iz,2 is scaled by the initial density operator so that a left ordinate value of 1 corresponds to a polarization of ωD /2kT . The right ordinate shows how the magnetization feeds back via angular momentum conservation to provide only mHz changes in the spin rate (dashed line) because the ratio N /I is only 10−3 s−1 for a real sample containing N spins. The plot in (c) suggests that the maximum transfer of rotational angular momentum into Zeeman polarization occurs when ωr = ωD

E.L. Hahn et al.

d d (Iωr ) = − J + i [Hz , J ] = −i [HD , J ] . dt dt

(8)

The expectation values are the same regardless of reference frame since the trace is independent of unitary transformation. Comparison of (8) to (6) shows that the interaction of the spin angular momentum J with the applied DC ﬁeld H0 represents an additional source of angular momentum that indirectly inﬂuences the momentum conservation. The rotor only exchanges angular momentum with the spins through HD , the dipolar interaction – not through the direct coupling with Hz . The external Hz or any other applied ﬁeld represents an energy-momentum source from a solenoid or an oscillator not directly coupled to the rotor. When initial angular momentum ﬂows into Jz by jumping the rotation frequency from zero to ωr , the built in feedback in (6) and demonstrated by simulation in Fig. 2 gives rise to the ghost ﬁeld Hghost = ωr /γ. This ﬁeld can not couple to the solenoid. On the other hand, when an applied ﬁeld is jumped from zero to H0 the source of angular momentum ﬂowing into Jz is the solenoid. The solenoid plays the role now of the rotor because of the application of an electromotive force which launches a current and therefore momentum into the applied H0 ﬁeld. The resulting exchange of order between the dipolar and Zeeman reservoirs is that of the Strombotne and Hahn experiment [14]. Therefore, Hz has only an indirect eﬀect on the rotor but remains to deﬁne the Zeeman Bloch equations when ωr = 0. Equation (8) also explains why neither the Barnett nor the Einstein-de Haas eﬀects have been noticed in routine NMR experiments in solids performed at high ﬁeld. At high magnetic ﬁeld the nuclear Zeeman interaction scales the non-secular q = 0 part of the 2 /ω0 ωD ω0 . To zeroth order in perdipole-dipole coupling to roughly ωD turbation theory the NMR spectrum is governed by the Zeeman interaction Hz and the right hand side of (8) is identically zero thus removing any coupling between the nuclear spin and mechanical angular momenta. Extending this argument to ﬁrst order in perturbation theory also does not yield any useful coupling between ωr and J when the applied DC ﬁeld is parallel to (2) (2) the sample spinning axis because the spatial term T0 (I1 , I2 )R0 (θ, φ) does not have any ωr rotational dependence. It is the time dependence imparted on the q = 0 parts of HD (t) that translate into the generation of magnetization in zero magnetic ﬁeld or the onset of sample rotation in a magnetic ﬁeld. This can be appreciated by considering the Hamiltonian for the two-spin system with a magnetic ﬁeld applied parallel to the sample spinning axis along the +z direction. This time dependent Hamiltonian Hlab = Hz + HD (t) is best considered in the rotating frame at the frequency ωr . Since this transformation involves a rotation about the z axis, the nuclear Zeeman interaction Hz is simply added to the Hamiltonian in (7) to give (2) (−1)q Tq(2) (I1 , I2 )R−q (θ, φ) , (9) Hrot = (ωr + ω0 )(Iz,1 + Iz,2 ) + ωD a result suggesting that sample rotation can either add to or subtract from the apparent Larmor frequency ω0 . Judging from (9) the largest eﬀect of magnetic

Nuclear Spin Analogues of Gyromagnetism

ﬁeld on sample rotation would be in low-ﬁeld where ωr is comparable to ω0 , and given (8) combined with the zero ﬁeld results mentioned above, the largest eﬀect should be observed when ωr is comparable to ωD . In direct analogy to the zero ﬁeld case mentioned in Fig. 2, a combination of the solution to the Liouville-von Neumann equation in the rotating frame using Hrot in (9) with the angular momentum conservation rule in (8) permits the dynamics of the magnetization and the rotor frequency to be determined starting from a stationary sample in zero magnetic ﬁeld. Figure 3 demonstrates the eﬀect of instantaneously switching on an H0 = 2.35 G DC magnetic ﬁeld or equivalently an ω0 /2π = 10 kHz 1 H Larmor frequency to an ensemble of dipole-dipole coupled two spin systems with ωD /π = 10 kHz. The plots in Fig. 3(a) correspond to the same ensemble of identical two-spin systems with θ = π/2 and φ = 0 shown in Fig. 2(a) while the plots in Fig. 3(b) show results for the same isotropic distribution of θ and φ values used in Fig. 2(b). In both of these plots the feedback due to angular momentum conservation requires that the magnetization will generate a periodic sample rotation. Since the commutator in (8) is always zero for the case of a ﬁeld applied parallel to the maximum moment of inertia, the negative sign in (8) causes the phase of the periodicity in the spin rate to be 180◦ out of phase with the periodicity in Jz . The same two-spin system used in Fig. 3(a) was used to determine the peak spin rate ωr as a function of applied DC ﬁeld and hence Larmor frequency ω0 in Fig. 3(c). This plot demonstrates that the largest induced sample rotation rate starting in zero ﬁeld can be obtained when the applied magnetic ﬁeld H0 is comparable to the inherent dipolar ﬁeld i.e. when H0 ≈ ωD /γ or ω0 ≈ ωD . The eﬀects predicted by the admittedly crude two-spin system in Figs. 2 and 3 continue to manifest themselves in larger more realistic cases [14]. Figure 4 shows the Barnett initially induced magnetization in (a) and the DC ﬁeld jump induced rotation in (b) for an eight spin system. In these examples the eight spins are positioned on the corners of a stationary cube in zero magnetic ﬁeld that occupies the normal x, y, and z cartesian axis system while the gated sample rotation and magnetic ﬁeld directions are along the +z axis. The side length of the cube is 2.3 ˚ A giving an ωD /2π = 10 kHz dipolar coupling for protons with the smallest separation. In this example thermal equilibrium is appropriate for an ensemble of identical eight spin cubes held at the temperature T . To remain consistent with Figs. 2 and 3, the rotation speed in Fig. 4(a) is jumped from zero to ωr /2π = 10 kHz while the ﬁeld is jumped from zero to H0 = ω0 /2πγ = 2.35 G in Fig. 4(b). Comparison of Fig. 4 to Figs. 2 and 3 suggest that the dynamics of the eight spin system are similar to that for the two spin system. The major diﬀerence is that the presence of three diﬀerent dipolar coupling values corresponding to spins separated on the edge, face diagonal, and body diagonal of the cube, have the net eﬀect of smearing the Zeeman-dipolar oscillations in Figs. 2 and 3, an averaging that makes the eight spin simulation closely resemble oscillations observed in real ﬁeld cycled experiments [14]. Admittedly, Figs. 2(a) and 3(a), Figs. 2(b) and 3(b), Figs. 2(c) and 3(c), and Fig. 4(a) and Fig. 4(b) look

E.L. Hahn et al.

Fig. 3. Simulation of the dynamics of the two spin system with applied DC ﬁeld at frequency ω0 /2π with ωD /2π = 10 kHz and ωr /2π = 0. In each case (a)–(c) the t < 0 thermal equilibrium condition corresponds to an ensemble of two spin systems in zero magnetic ﬁeld. At the time t = 0 the DC magnetic ﬁeld is instantaneously increased from zero to γH0 = ω0 /2π = 10 kHz in (a) and (b) and a variable level in (c). The eﬀects of this ﬁeld are to create magnetization (solid line) as shown by the left ordinate in (a) and (b) for the θ = π/2 orientation and the isotropic powder respectively. The polarization Jz is scaled by the initial density operator so that a left ordinate value corresponds to a polarization of ωD /2kT . The right ordinate shows how the magnetization feeds back via angular momentum conservation so that a small fraction of N spins in the sample becomes polarized, where the acquired sample rotation rate ωr is typically N /I = 10−3 s−1 . The plot in (c) suggests that the maximum production of Zeeman polarization occurs when ω0 = ωD

Nuclear Spin Analogues of Gyromagnetism

Fig. 4. Simulation of the dynamics of an eight spin system with the spins centered on the corners of a cube. Taking the spins as protons, the ωD /2π = 10 kHz dipolar coupling corresponds to the strongest coupling or the shortest distance between nuclei on the 2.3 ˚ A long cube side. The plot in (a) corresponds to jumping of an ωr /2π = 10 kHz sample rotation rate while in (b) similar results are explored by turning on an ω0 /2π = 10 kHz Larmor frequency. In both cases the t < 0 thermal equilibrium situation reﬂects an ensemble of dipolar coupled eight spin systems. Here the value of the magnetization on the left ordinate is scaled to the thermal equilibrium density operator for the eight spin system while the right ordinate pertains to changes in the sample rotation frequency due to angular momentum conservation

identical. This similarity is intended as the ghost ﬁeld due to sample rotation in Figs. 2 and 4(a) yield the same dynamics by independent calculations as the real ﬁeld in Figs. 3 and 4(b). In Figs. 2–4 one may interpret in the case of jumping the rotor frequency from zero to ωr , that when momentum ﬂows from Jz to the rotor, the process is Einstein-de Haas. If the momentum ﬂows from the rotor to Jz the process is Barnett. Hence there is an oscillatory display of both eﬀects. These oscillations due to angular momentum conservation can be connected with the oscillations in energy due to population exchange between Zeeman and dipolar order.

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6 Adiabatic Demagnetization and Remagnetization Because of spin entropy conservation in a hom*onuclear dipolar coupled spin system [15, 16] in the absence of relaxation, the adiabatic demagnetization of a spin system converts Zeeman order to dipolar order after an initial polarizing Zeeman ﬁeld Hi is reduced adiabatically to zero. During this process, the initial spin temperature is reduced from Ti to a ﬁnal dipolar spin temperature Tf = (Hloc /Hi ) Ti assuming Hi Hloc , where Hloc ∝ ωD /2πγ is the local average dipolar ﬁeld. Suppose that the thermodynamic relation dU = −M dH governing the adiabatic process of demagnetization in the presence of a real ﬁeld H applies also to the ghost ﬁeld Hghost . More accurately, this means that the spin temperature will be a function of the ghost ﬁeld Hghost as if it were a real ﬁeld. On this basis any polarization induced by sample spinning will be Barnett in character. When a real ﬁeld Hi is reduced during demagnetization, the adiabatic reduction in Zeeman energy −M ∆H requires that the spins must do work on the solenoid. Conversely the solenoid must do work on the spins to restore M ∆H in the process of remagnetization. With the case of a Barnett polarization mediated by sample spinning, it is necessary to show experimentally that the ghost ﬁeld Hghost can bring about an analagous adiabatic response. This would seem to follow by virtue of the identity dU = −(M/γ)(γdHghost ) = −Ωdωr , if in this case positive and negative work must be done by the rotor instead of the solenoid. If this picture is true, the outside work by the rotor must be diﬀerent and not related to the microscopic changes in rotor energy and momentum mentioned in connection with spin lattice relaxation. There is an ambiguity here because the same terms that apply to either situation have been invoked by assuming that the same spin temperature applies in both cases. A negative or positive Hghost added parallel to H by sample spinning in opposite directions would require both of these mechanisms to operate simultaneously. In the limit that the ghost ﬁeld behaves just like a real ﬁeld, a sizeable portion of initially polarized magnetization Mi = Cspin Hi /Ti may be recovered and sustained after adiabatic demagnetization by turning on a small ghost ﬁeld Hghost . Here the initial ﬁeld Hi is sizeable and Cspin is the nuclear spin Curie constant. After adiabatic remagnetization in the ghost ﬁeld Hghost , a good fraction of Mi given by Mf = Mi

Hghost 2 Hghost

2 Hloc

= Mi

ωr ωr2

2 + ωD

(10)

should be recovered due to the sample spinning. Since the adiabatically recovered magnetization in (10) is not directly proportional to Hghost or equivalently ωr as it is in the genuine Barnett eﬀect, the ﬁnal magnetization Mf should be considered as a re-magnetized “pseudo–Barnett” polarization suspended in the absence of real ﬁelds. Beginning at room temperature without demagnetization one would obtain a much smaller magnetization Mi = Cspin Hghost /Ti .

Nuclear Spin Analogues of Gyromagnetism

The dynamics of the “pseudo-barnett” polarization in a small ghost ﬁeld can be easily tested using the ensemble of eight spin systems described above. Here the initial thermal equilibrium polarization is taken to be in the high ﬁeld high temperature limit ρeq = 1 − Hz /kT − HD /kT where both the Zeeman Hz and dipolar coupling HD Hamiltonians are appropriate for eight spins situated at the corners of a cube. The Zeeman and dipolar temperatures are taken to be the same, and the ﬁeld Hi is parallel to the sample spinning direction in addition to deﬁning the +z axis. Figure 5(a) shows the timing of the real magnetic ﬁeld and sample rotation ramps during the adiabatic demagnetization and remagnetization process while Figs. 5(b)–(d) show how the total magnetization Jz , dipolar order HD , and spin rate ωr evolve in time respectively. Comparison of Figs. 5(b) and (c) indicate that the demagnetization process from a ﬁeld of Hi = 2, 350 G to zero completely transfers proton Zeeman order into dipolar order at the end of the ramp at t = 100 µ s. After persisting as dipolar order for t = 250 µ s, the adiabatically ramped sample spinning from zero to a ﬁnal value of ωr /2π = 30 kHz causes the “pseudoBarnett” magnetization to appear. The dashed line in Fig. 5(b) corresponds to the ratio Mf /Mi anticipated on the basis of (10) with ωr /2π = 30 kHz and 2 1/2 = 16 kHz for the eight spin problem. Figures 5(b)–(d) all display what ωD appears to be a large amount of noise at low ﬁeld. Closer inspection of these plots reveals that the noise is periodic and that the source of the periodicity is the angular momentum feedback due to the Barnett eﬀect.

7 Sample Spinning Non-axial with DC Field In all of the previous cases with the DC ﬁeld and sample rotation applied in the +z direction, Jx = Jy = 0 and any feedback contributions due to oﬀ axis sample rotation are discarded in (6) and (8). The more useful case applying to narrow NMR lines in solids involves rapid sample spinning at some angle with respect to an applied DC ﬁeld. New time dependent terms in the Hamiltonian are generated from rotation transformations of only the secular terms. In a small magnetic ﬁeld comparable to the dipolar ﬁeld, inclusion of the transformation of the non-secular terms show that any static ﬁeld component normal to the axis of spin rotation results in saturation of any polarization M0 that may be acquired by the Barnett eﬀect. The coupling of the sample spinning to the magnetization in the case when a real DC ﬁeld is applied at some initial angle that is not parallel to the direction of sample rotation can be understood by writing the dipolar coupling interaction in the principal axis frame of the moment of inertia tensor I. In this frame the z-axis corresponds to the direction of the maximum moment of inertia. Realizing that the applied static ﬁeld can be at any orientation with respect to this frame recasts the Zeeman interaction as Hz = ωx (t)Jx +ωy (t)Jy +ωz (t)Jz where it is understood that the Jx , Jy , and Jz operators pertain to the total spin angular momentum in the x, y, and z directions in the inertial frame

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Fig. 5. Eight spin simulation of the eﬀects of adiabatic demagnetization from a high Hi = ω0 /γ = 2, 350 G magnetic ﬁeld to zero ﬁeld and subsequent remagnetization in an eﬀective spinning ﬁeld of Hghost = ωr /γ = 7.05 G. The timing of the magnetic ﬁeld and sample spinning is shown in (a) while the change in magnetization Jz , dipolar order HD , and sample spin rate ωr are included in (b)–(d) respectively

Nuclear Spin Analogues of Gyromagnetism

while ωx (t), ωy (t), and ωz (t) correspond to the time dependent orientation of the ﬁeld H = ω(t)/γ = (ωx (t)/γ)i + (ωy (t)/γ)j + (ωz (t)/γ)k in the inertia frame as the sample rotates. The three principal components of the inertia tensor Ixx , Iyy , and Izz can be used along with the angular momentum conservation relation in (8) to develop a similar conservation rule for any general orientation of the DC ﬁeld with respect to the sample rotation direction as         0 −ωz (t) ωy (t) Jx I ω (t) J d  x   d  xx r,x  0 −ωx (t)   Jy  , Iyy ωr,y (t) = − Jy + ωz (t) dt dt 0 Izz ωr,z (t) Jz −ωy (t) ωx (t) Jz (11) where ωi (t) = ωi (t) + ωr,i (t), i = x, y, z and ωr,i (t) corresponds to the frequency of rotation of the object along the principal axes of I in the inertial frame. The expectation values of the total magnetization in (11) in the presence of a DC ﬁeld are most easily obtained by expressing the Zeeman and hom*onuclear dipolar interactions in a rotating frame at the sample rotation frequency as Hrot =

(ωr,i + ωi (t))Ii,j +

i∈{x,y,z} j=1

ωD (i, j)

(12)

i=1 j=i (2)

(−1)q Tq(2) (Ii , Ij )R−q (θij , φij ) ,

q=−2

where the sum over all of the spins considered in the sample is included for 3 clarity, ωD (i, j) = γi γj /ri,j , and the polar angles θi,j and φi,j are included in the ﬁnal term to specify that the internuclear direction r i,j between each spin pair in the sample might have a diﬀerent orientation with respect to the moment of inertia frame. In the special case of Barnett induced magnetization, the discussion in the previous sections suggests that any changes due to the feedback predicted in (11) in practical spin rates of several kHz along the +z direction are small and most likely negligible. Taking ωr,x (t) = ωr,y (t) ≈ 0 and ωr,z (t) = ωr suggests that the only time dependence remaining in (12) in the rotating inertia frame is due to the precession of the applied DC ﬁeld around the sample. The transverse ωx (t) and ωy (t) components will modulate at ωr or equivalently, are at exact resonance with M0 and will thus cause saturation transitions while the ωz term is time independent and will add a detuning eﬀect and thus a resonance oﬀset. It is important to note that a similar eﬀect occurs in standard magnetic resonance experiments when a DC ﬁeld perpendicular to the Zeeman polarizing ﬁeld is turned on in the lab frame. An initially polarized value of M0 would disappear as it precesses around the eﬀective ﬁeld at a rate depending on the size of the perpendicular ﬁeld.

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8 Lattice Structure Dependence of the Barnett Eﬀect In a liquid the dipole-dipole interaction between spins is cut oﬀ because of random ﬂuctuations of the lattice coordinates. The competition between mutually interacting spins precessing in the local dipolar ﬁeld and the apparent precession caused by a small ghost ﬁeld Hghost disappears if the local ﬁeld averages to zero. To understand this averaging assume ﬁrst that the dipolar coupling between two spins is zero. The two spins of course would precess about any applied DC ﬁeld, but for the moment let them remain pointed in ﬁxed directions in space like gyroscopes. One can easily see for example that rotation of the internuclear vector r by some precession angle about a perpendicular axis will alter the orientation angles of both dipoles relative to one another. Now turn on the dipole-dipole interaction. Comparison of the interaction energy at the two diﬀerent orientations reveals that it is in fact diﬀerent. If instead of two discrete angles a continuous set of angles is explored through sample rotation one ﬁnds that if the rotation rate ωr is comparable to or exceeds the dipole-dipole coupling strength, the spins appear to be precessing coherently about the rotation axis perpendicular to r in the rotating frame. It is in this case that an apparent torque can be attributed to the ﬁctitious ﬁeld Hghost = ωr /γ. As long as r retains the integrity of the lattice structure, the spins can sense Fourier components characteristic of the lattice rotation appearing in that frame and ultimately polarize by spin-lattice relaxation. But as stated above, if the lattice structure disappears because of coordinate ﬂuctuations as in a liquid due to motional narrowing, the Barnett eﬀect will be quenched. Basically the lattice environment becomes isotropic, looking virtually the same to the spins regardless of the rotational aspects of the liquid. If a torque is to be attributed to Hghost in a rotating sample, the spins must mutually interact over many cycles of rotation in a reference frame that connects them to a ﬁxed lattice structure.

9 Conclusion One motivation for this work was the lure of polarizing nuclear spins in solids in conventional high ﬁeld applications by transferring the massive angular momentum Iωr from a rotating macroscopic object. The arguments provided in the above sections suggest that a small fraction of this massive sample angular momentum can indeed be transferred but only in low ﬁeld situations where the sample rotation rate ωr and Larmor frequency ω0 is comparable to the dipolar coupling strength ωD . At present conventional high ﬁeld applications of this method to hom*onuclear dipole-dipole coupled spin systems are limited because the high magnetic ﬁeld quenches the spin-lattice coupling manifest in the non-secular dipole-dipole terms that drive the eﬀect. Before identifying possible uses of the Barnett eﬀect in nuclear spin systems, experiments must be completed to verify whether or not the ghost ﬁeld does in fact behave like

Nuclear Spin Analogues of Gyromagnetism

a real DC ﬁeld. Instead of relying only on conservation laws to account for the transfer of angular momentum between the spin system and the rotor, the challenge remains to devise a more rigorous two reservoir formalism similar to that applied to cross relaxation between two spin species.

Acknowledgements In particular, ELH is grateful to Dietmar Stehlik for stimulating discussions and initiating interest in the Einstein-de Haas and Barnett eﬀects. We also gratefully acknowledge useful discussions with Alex Pines, Maurice Goldman, John Waugh, Jean Jeener, Eugene Commins, Carlos Meriles, Demitrius Sakellari, Andreas Trabesinger and Jamie Walls. MPA is a David and Lucile Packard and Alfred P. Sloan foundation fellow.

References 1. R. Mc Dermott, A.D. Trabesinger, M. Muck, E.L. Hahn, A. Pines, J. Clarke: Science 295, 2247–2249 (2002) 2. D. Budker, D.F. Kimball, V.V. Yashchuk, M. Zolotorev: Phys. Rev. A 65, 55403 (2002) 3. L.F. Bates: Modern Magnetism (University Press, Cambridge 1951) 4. (a) S.J. Barnett: Phys. Rev. 6, 239–270 (1915) (b) S.J. Barnett: Rev. Mod. Phys. 7, 129–166 (1935) 5. (a) A. Einstein, W.J. de Haas: Verhandl. Deut. Phsik. Ges. 17, 152–170 (1915) (b) A. Einstein, W.J. de Haas: Verhandl. Deut. Phsik. Ges. 18, 173–177 (1916) 6. J.H. Van Vleck: The Theory of Electric and Magnetic Susceptibility (Clarendon, Oxford 1932) pp 94–97 7. J.M.B. Kellogg, I.I. Rabi, N.F. Ramsey, J.R. Zacharias: Phys. Rev. 56, 728–743 (1939) 8. (a) F. Bloch, W.W. Hansen, M. Packard: Phys. Rev. 70, 474–485 (1946) (b) E.M. Purcell, H.C. Torrey, R.V. Pound: Phys. Rev. 69, 37–38 (1946) 9. A. Abragam, B. Bleaney: Electron Paramagnetic Resonance of Transition Ions (Clarendon, Oxford 1970) 10. E.M. Purcell: Astrophys. J. 231, 404–416 (1979) 11. A. Lazarian, B.T. Draine: Astrophys. J. 520, L67–70 (1999) 12. D.K. Sodickson, J.H. Waugh: Phys. Rev. B 52, 6467–6469 (1995) 13. B. Black, B. Majer, A. Pines: Chem. Phys. Lett. 201, 550–554 (1993) 14. R.L. Strombotne, E.L. Hahn: Phys. Rev. 133, A1616–A1629 (1964) 15. A. Abragam: Principles of Nuclear Magnetism (Clarendon, Oxford 1961) 16. M. Goldman: Spin Temperature and Nuclear Magnetic Resonance in Solids (Clarendon, Oxford 1970)

Distance Measurements in Solid-State NMR and EPR Spectroscopy G. Jeschke and H.W. Spiess Max Planck Institute for Polymer Research, Postfach 3148, 55021 Mainz, Germany [emailprotected] [emailprotected] Abstract. Magnetic resonance techniques for the measurement of dipole-dipole couplings between spins are discussed with special emphasis on the underlying concepts and on their relation to site-speciﬁc distance determination in complex materials. Special care is taken to reveal the approximations involved in data interpretation and to examine the range of their validity. Recent advances in the understanding of measurements on multi-spin systems and in the extraction of spin-to-spin pair correlation functions from dipolar evolution functions are highlighted and demonstrated by selected experimental examples from the literature.

1 Introduction The majority of our knowledge on the geometric structure of matter has been obtained by scattering techniques; mainly by X-ray, electron, and neutron diffraction. Diﬀraction methods are the natural choice for systems with translational symmetry and are very useful for all repetitive structures, in particular, structures with at least some degree of long-range order. Scattering techniques in general can reveal the size and shape of certain objects in a system, provided that the distribution of sizes and shapes is reasonably narrow and that suﬃcient contrast between the objects and their environment can be achieved. A limitation of scattering techniques results from the fact that destructive interference cancels any signals due to non-repetitive features of the structure. This limits the complexity of the structures that can be understood on the basis of scattering data. The structural picture of a system obtained by magnetic resonance techniques, such as nuclear magnetic resonance (NMR) and electron paramagnetic resonance (EPR), is to a large extent complementary to the one obtained by scattering techniques. Nuclear and electron spins can be considered as local probes of the structure. Signals due to diverse environments of the observed spins may overlap, but they do not cancel. Indeed, analysis of NMR spectra provides detailed information about the degree of disorder, e.g., in incommensurate systems or spin glasses [1, 2]. Moreover, interactions of spins with G. Jeschke and H.W. Spiess: Distance Measurements in Solid-State NMR and EPR Spectroscopy, Lect. Notes Phys. 684, 21–63 (2006) c Springer-Verlag Berlin Heidelberg 2006 www.springerlink.com

G. Jeschke and H.W. Spiess

their environment are mostly so weak that external perturbations can compete with them. It is therefore possible to manipulate the Hamiltonian during the experiment almost at will [3, 4, 5]. By such manipulations it is possible to separate interactions and thus to reduce signal overlap, for instance by disentangling the spectrum into two or more dimensions. Complexity of the spectra can also be reduced by intentionally and selectively suppressing signal contributions of spins in more mobile or more rigid environments [4]. Finally, only the structural features of interest for a given problem can be addressed by site-selective isotopic labelling in NMR or site-selective spin-labelling in EPR. With this arsenal of approaches for obtaining just the information relevant in a certain context, magnetic resonance techniques are well suited for studying complex structures. Limitations arise mainly from the comparatively low sensitivity of magnetic resonance experiments, which is caused by the low energy of spin transitions. These limitations often dictate the choice between NMR and EPR. Except for systems featuring native paramagnetic centers, NMR is the ﬁrst choice, as it can be applied to the system as it is or after a mere isotope substitution that usually does not inﬂuence structure and dynamics significantly. However, if the structural features of interest correspond to only a small fraction of the material, EPR spin labelling techniques may be required to obtain suﬃciently strong signals. Examples are chain ends in polymers or single residues in proteins with molecular weights larger than 20 kDa. As the paramagnetic moiety of the common nitroxide spin probes has a size of approximately 0.5 nm, such spin-labelling approaches are restricted to structural features that are larger than 1 nm, and it must be ascertained that the labelling does not change the structure or function of the system. In the past, most method development and applications work in NMR and EPR has been devoted to elucidation of the chemical structure of diamagnetic and paramagnetic molecules, respectively. Starting with proteins in solution [6, 7, 8], this has changed recently, and obtaining information on geometric structure and on structural dynamics has now become the main goal of method development. Geometric structure is derived by a molecular modelling approach that takes into account general knowledge on bond lengths, bond angles, and dihedral angles in molecules [9] as well as constraints on spinto-spin distances (for application of such an approach to a weakly ordered system, see [10]). The distance constraints in turn result from the magnetic resonance spectra. Constraints on the relative orientation of groups (angular constraints) may also be useful. As the experience with protein structures has shown, such an approach works quite well even if only part of the structure is well deﬁned. In fact, for many problems in biomolecular and materials science knowledge about part of the structure already provides valuable insight into the relationships between structure, properties, and function. Determination of the full geometric structure requires a large number of constraints. Moreover, this number increases strongly with increasing size and complexity of the system. Nevertheless, protein structure determination by

Distance Measurements by NMR and EPR

high-resolution NMR in solution successfully implements such an approach [6, 7]. Usually precision of the constraints derived from high-resolution NMR spectra is rather limited, but for proteins with molecular weights up to approximately 20 kDa it is often possible to obtain signiﬁcantly more constraints than would be strictly required to solve the structure. Overdetermination of the problem then compensates for lack of precision of the individual measurements. In this chapter we explore the basics of the alternative approach in which only part of the structure is solved by a smaller number of more precise constraints. Such precise information may be required for understanding self-organization phenomena in supramolecular systems, for ﬁne-tuning the properties of materials, and for understanding how the function of biomacromolecules is optimized. In the approach discussed here, which is applicable mainly to solid materials and soft matter, information on spin-to-spin distances is obtained by measurements of the dipole-dipole coupling, which scales with the inverse cube of the distance. The strength of this coupling depends only on the distance, fundamental constants, and the angle between the spin-to-spin vector and the quantization axis of the two spins. It is not inﬂuenced by the medium in between the spins, so that such measurements are potentially very precise. Furthermore, information on the spin-to-spin pair correlation function can be obtained even in cases where the distances of the spins under consideration are broadly distributed. To fully utilize these advantages, the most appropriate experiment for a given problem has to be selected. This requires an overview of the existing techniques and, in particular, of the concepts on which they are based. In these Notes we attempt for the ﬁrst time to provide such an overview. Our concentration on concepts implies that only a limited selection of experiments can be treated. Further useful experiments are discussed in NMR and EPR monographs [4, 5, 11] and in a number of reviews [12, 13, 14, 15, 16, 17]. This chapter is organized as follows. In Sect. 2 we introduce the dipoledipole Hamiltonian and discuss its truncation for several cases of interest as well as its spectrum. We also examine under which conditions isotropic spinspin couplings, such as J couplings, are signiﬁcant and how they inﬂuence the spectra. Eﬀects of molecular motion and spin delocalization on the average dipole-dipole interaction are considered. In Sect. 3 we introduce the principal approaches for measuring the spectrum of the dipole-dipole Hamiltonian and discuss them with special emphasis on the range of their applicability. Complications in the measurement of the distance between two spins that are caused by couplings to further spins of the same kind are considered in Sect. 4. In this Section we also examine how dipolar time evolution functions are related to the spin-spin pair correlation function and under which conditions the pair correlation function can be determined. Where applicable, the approaches based on sample spinning are compared with experiments on static samples.

G. Jeschke and H.W. Spiess

2 Dipole-Dipole Interaction in a Two-Spin System Coupling between magnetic dipoles is a pairwise interaction. We may thus ﬁrst discuss a two-spin system and consider later the complications that arise in multi-spin systems. The dipole-dipole coupling is the energy of the magnetic dipole moment µ1 in the ﬁeld induced by dipole moment µ2 and vice versa: E = −µ1 · B 2 (r 12 , µ2 ) = −µ2 · B 1 (r 12 , µ1 ) ,

(1)

where r 12 is the distance vector connecting the two moments. Hence, the coupling depends only on the distance and the relative orientation of the two dipole moments with respect to each other. With the expression for the ﬁeld induced by the dipole,

3 µ0 1 (2) µ − 2 (µ · r) r , B (r, µ) = − 4 π r3 r we have E=−

3 µ0 1 · µ − (µ · r ) (µ · r ) . µ 1 2 1 12 2 12 3 2 4 π r12 r12

(3)

2.1 Hamiltonian of the Dipole-Dipole Interaction For two magnetic moments associated with spins S and I, the Hamiltonian for the dipole-dipole coupling can be derived from (3) by the correspondence principle. Writing the Hamiltonian in units of angular frequencies we ﬁnd

ˆ dd = 1 µ0 γS γI S ˆ · r SI Iˆ · r SI ˆ · Iˆ − 3 1 S H , (4) 3 4π 2 rSI rSI where the γS,I = gS,I µS,I /

(5)

are the magnetogyric ratios of the spins with the appropriate magnetons µS,I (Bohr magneton µB for electron spins, nuclear magneton µn for nuclear spins) and the appropriate g values. For historical reasons, the electron g value is taken as positive in most literature. In these notes we conform to the convention in which the electron g value is negative [18] to ensure that (5) is valid for both nuclear and electron spins. In all the experimental situations that we shall discuss the dipole-dipole coupling is small compared to the Zeeman interaction of at least one of the two spins. Indeed, in magnetic ﬁelds above 1 T the coupling is small compared to the Zeeman interactions of both spins, except for hyperﬁne couplings in EPR. With the magnetic ﬁeld axis chosen as the z axis, it is therefore convenient to consider the dipole-dipole Hamiltonian in a basis spanned by the eigenstates |αS αI , |αS βI , |βS αI , and |βS βI of the operator Fˆz = Sˆz + Iˆz . The energy level schemes for the cases of comparable and strongly diﬀerent Zeeman

Distance Measurements by NMR and EPR

Fig. 1. Energy level schemes of two-spins systems for the cases where the Zeeman interactions of both spins (a) or of spin S (b) are much larger than the dipole-dipole coupling. Labelling of the states corresponds to the NMR case (positive magnetogyric ratio) in (a) and to the EPR case (negative magnetogyric ratio) in (b). Solid lines correspond to allowed transitions, dotted lines to forbidden transitions. Operators designate the assignment of terms in the dipolar alphabet to the transitions

energies are shown in Fig. 1. Considerable simpliﬁcations are possible if the quantization axes of the two spins are parallel to each other. Experiments are usually performed at magnetic ﬁelds B0 that are suﬃcient to align the quantization axes with the magnetic ﬁeld axis. Exceptions are nuclear spins S, I > 1/2 with substantial quadrupole couplings, where the required ﬁelds may be technically inaccessible, and electron spins S of transition metal or rare earth metal ions with large g anisotropy, where alignment of the quantization axis with the external ﬁeld generally cannot be achieved. For all other cases we may deﬁne a common frame for the two spins. As the z axis we choose the quantization axis, the x and y axes are for the moment left unspeciﬁed. Introducing polar coordinates and the shift operators Sˆ+ = Sˆx + iSˆy , Sˆ− = Sˆx − iSˆy , Iˆ+ = Iˆx + iIˆy , Iˆ− = Iˆx − iIˆy , we thus obtain a representation in terms of the dipolar alphabet: ˆ dd = 1 µ0 γS γI Aˆ + B ˆ + Cˆ + D ˆ +E ˆ + Fˆ H 3 4π rSI

(6)

(7)

with Aˆ = Sˆz Iˆz 1 − 3 cos2 θ , ˆ = − 1 Sˆ+ Iˆ− + Sˆ− Iˆ+ 1 − 3 cos2 θ , B 4 3 ˆ C = − Sˆ+ Iˆz + Sˆz Iˆ+ sin θ cos θe−iφ , 2

(8) (9) (10)

G. Jeschke and H.W. Spiess

ˆ = − 3 Sˆ− Iˆz + Sˆz Iˆ− sin θ cos θeiφ , D 2 3 ˆ+ ˆ+ 2 −2iφ ˆ E = − S I sin θe , 4 3 Fˆ = − Sˆ− Iˆ− sin2 θe−2iφ . 4

(11) (12) (13)

Which terms are relevant depends on the relative magnitude of the dipolar frequency 1 µ0 γS γI , (14) ωdd (rSI ) = 3 rSI 4 π given here in angular frequency units, with respect to the splittings between the levels in the absence of dipole-dipole coupling. These splittings can be expressed in terms of the Larmor frequencies ωS and ωI in the absence of coupling. The secular term Aˆ is signiﬁcant in all experimental situations and corresponds to a ﬁrst-order correction of the Zeeman energies. It causes a splitting of both the S and I spin transitions in doublets. The double-quantum ˆ and Fˆ do not signiﬁcantly inﬂuence the eigenvalues and eigenvectors terms E of the Hamiltonian if the Larmor frequency of at least one spin is much larger than the dipolar frequency, which is the case for all experiments discussed in these Notes. Hence these terms can safely be neglected. For like spins, ωdd ∆ωSI = |ωS − ωI | ,

(15)

ˆ causes signiﬁcant mixing of the |αS βI and |βS αI states. This the term B situation is usually encountered in hom*onuclear NMR experiments, where S and I are spins corresponding to the same isotope, and may be encountered in pulse EPR experiments when both the S and I spins are excited by pulses ˆ term is signiﬁcant in such EPR at the same microwave frequency. The B experiments unless both the width of the EPR spectrum and the excitation bandwidth of the microwave pulses are much larger than ωdd . In pulse electron electron double resonance (ELDOR) experiments this term can generally be neglected. In the following we denote experiments where both the S and I ˆ term is thus spins are excited by the same irradiation frequency – and the B ˆ term signiﬁcant – as experiments on like spins, and experiments where the B can be neglected as experiments on unlike spins. ˆ are negligible if The terms Cˆ and D ωdd ωS , ωI .

(16)

This inequality is fulﬁlled except for dipolar hyperﬁne couplings in electron nuclear double resonance (ENDOR) and electron spin echo envelope modulation (ESEEM) experiments, where the dipole-dipole coupling may be comparable to the nuclear Zeeman frequency. In the latter situation only the terms ˆ are non-secular and thus negligible. Neglecting Sˆ+ Iˆz and Sˆ− Iˆz in Cˆ and D these terms and now choosing the x axis so that φ = 0, we ﬁnd the truncated dipole-dipole Hamiltonian for the electron-nuclear two-spin system

Distance Measurements by NMR and EPR

ˆ dd = ωdd H

1 − 3 cos2 θ Sˆz Iˆz − 3 sin θ cos θSˆz Iˆx .

(17)

For like spins (most hom*onuclear NMR experiments and single-frequency pulse EPR experiments under certain conditions) the truncated dipole-dipole Hamiltonian is given by

ˆ dd = ωdd 1 − 3 cos2 θ Sˆz Iˆz − 1 Sˆ+ Iˆ− + Sˆ− Iˆ+ H , (18) 4 and, ﬁnally, for unlike spins (ELDOR and heteronuclear NMR experiments), it is given by ˆ dd = ωdd 1 − 3 cos2 θ Sˆz Iˆz . H (19) In either of the three cases, the dipolar frequency ωdd and thus the interspin ˆ dd if the distribution distance rSI is uniquely determined by the spectrum of H of angle θ between the spin-spin vector and the common quantization axis is known. In particular, for macroscopically disordered systems orientations with a given angle θ are realized with probability sin θ. Note also that the dipoledipole coupling is purely anisotropic – if we express it in tensorial form as ˆ I, ˆ we ﬁnd that the tensor D is traceless. Usually the spectrum ˆ dd = SD H ˆ dd is not directly accessible by analyzing the lineshape of NMR or EPR of H spectra, as ωS and ωI are broadly distributed due to anisotropy of the Zeeman interaction or due to interactions with other spins in the sample. Separation of interactions as described in Sect. 3 is then required to measure the spectrum ˆ dd . As we shall see, it is usually possible to eliminate the contributions of of H all other interactions by applying appropriate external perturbations and it is often even possible to factor out relaxational broadening. However, J coupling between the S and I spins is described by the same product operators and is thus inseparable from the dipole-dipole coupling for fundamental reasons. It is therefore necessary to discuss in which situations J coupling may interfer with the measurement of ωdd and how reliable interspin distances can be obtained in such a situation. 2.2 J Coupling and Isotropic Hyperﬁne Coupling Coupling between two spins may arise not only due to the dipole-dipole interaction through space, but may also be mediated by the electron cloud. Overlap of the singly occupied molecular orbitals of two unpaired electrons leads to Heisenberg exchange between the two electron spins. Exchange coupling between electron spins may also proceed through orbitals of neighbor molecules, as for instance solvent molecules. This superexchange relies on a correlation of the spin states of the unpaired electron and electrons of the solvent molecules in spatial regions where the corresponding orbitals overlap. Similarly, the nuclear spin state is correlated to the spin states of electrons in s orbitals at this nucleus, and this correlation can be transported through a chain of overlapping orbitals to another nucleus. With the exception of the hyperﬁne coupling

G. Jeschke and H.W. Spiess

between the nucleus and an unpaired electron in its s orbitals (Fermi contact coupling), all these coupling mechanisms are usually denoted as J couplings and are often referred to as through-bond couplings. Note that the latter term may be misleading, as through-solvent or through-space J couplings can be substantial in certain situations [19]. In general, J couplings may have both an isotropic and an anisotropic contribution [20, 21]. However, the anisotropic contribution, which is also called a pseudodipolar contribution, is usually negligible in NMR experiments on elements in the ﬁrst and second row of the periodic system and in EPR experiments on spin pairs with a distance exceeding 0.5 nm. In most cases of interest, we are thus left with a situation where the J coupling is purely isotropic and the dipole-dipole coupling is purely anisotropic. The two interactions can then in principle be separated by sample rotation. For macroscopically oriented systems such as crystals or liquid-crystals such a separation is feasible in both NMR and EPR spectroscopy by studying the orientation dependence of the spectra. For macroscopically disordered systems fast sample reorientation during an NMR experiment (see Sect. 2.5) can be used to average the dipole-dipole coupling, so that the J coupling can be measured separately. In the next Section we shall see that analysis of the lineshape corresponding to the total spin-spin coupling (dipole-dipole and isotropic J coupling) also provides unique values for ωdd and J. In many cases, J coupling can be neglected altogether, as it is much smaller than dipole-dipole coupling. This situation is usually encountered in solid-state NMR if only ﬁrst-row and second-row elements are involved and in EPR when rSI is longer than 1.5 nm and the two spins are separated by an insulating matrix. Note however that J coupling in solid-state NMR may also be useful for detecting through-bond correlations by the two-dimensional INADEQUATE experiment [22, 23, 24]. Where necessary, we shall use the Hamiltonian

1 ˆ+ ˆ− ˆ− ˆ+ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ S I +S I HJ = J S I = J Sx Ix + Sy Iy + Sz Iz = J Sz Iz + , 2 (20) for isotropic J coupling in both the NMR and EPR cases. For the EPR case, this deﬁnition of J is the one adopted in most modern textbooks [5, 25, 26]. ˆ Iˆ [27] and H ˆ Iˆ [28] are ˆ J = −J S ˆ J = −2J S Note that two other deﬁnitions H also widely used in EPR literature, so that care must be taken when comparing reported values of J. 2.3 The Pake Pattern We are now in a position to discuss spectral patterns that result from coupling of two spins. For negligible J coupling, the Hamiltonians in (18) and (19) both give rise to the Pake pattern [29] shown in Fig. 2a. Note that for like spins the splitting ω⊥ is 1.5 times as large as for unlike spins, since in the former

Distance Measurements by NMR and EPR

Fig. 2. Spectral patterns arising from two-spin coupling (simulations). (a) Pake pattern. For like spins, ω⊥ = 3ωdd /2, for unlike spins, ω⊥ = ωdd . (b) Pattern in the presence of J coupling for unlike spins, J = 0.275ωdd . (c) Orthorhombic dipolar pattern for unlike spins as it may be observed when the point-dipole approximation is violated, here η = 0.2

ˆ contributes to the splitting. case the pseudo-secular zero-quantum term B Furthermore, for like spins, J coupling does not inﬂuence the spectral pattern. For unlike spins in the presence of signiﬁcant J coupling, the singularities and outer inﬂection points are found at ω⊥ = ± |ωdd + J| , ω = ± |−2ωdd + J| ,

(21)

respectively (Fig. 2b). If the whole pattern can be measured the dipolar frequency can thus always be extracted. 2.4 The Point-Dipole Approximation So far we have assumed that both spins are strictly localized in space, i.e., that they can be considered as point dipoles. This point-dipole approximation is certainly well justiﬁed for nuclear spins, as nuclear radii are of the order of only a few femtometers. For electron spins of some paramagnetic species, notably nitroxide radicals and many transition metal complexes, the point-dipole approximation is valid at distances of 2 nm and longer, as the distribution of distances implied by the conformational freedom of the molecules is much broader than implied by the spatial distribution of the electron spin. If the unpaired electron is delocalized on the length scale of the measured distance, this delocalization has to be taken into account explicitly. This case is usually encountered when a distance between an electron spin S and a nuclear spin IN is determined from the dipolar contribution to the hyperﬁne coupling. Assuming that we know all signiﬁcant spin densities ρk at the other nuclei as well as the distances Rk > 0.25 nm of these nuclei from the nucleus with index N , we may compute a dipole-dipole coupling tensor D by the electron-nuclear point-dipole formula

G. Jeschke and H.W. Spiess

ρk 3nk nT µ0 k −1 γS γI D= , 4π Rk3

(22)

k=N

where the nk are unit vectors denoting the direction cosines of Rk and superscript T denotes the transpose. If the spatial distribution of the electron spin ρS (r) is known from a quantum-chemical computation, we have r SI r T 1 µ0 SI γS γI ρS (r) 3 5 − 3 D= dr , (23) 4π rSI rSI where r SI depends on integration variable r. For a system of two distributed electron spins we ﬁnd r SI r T 1 µ0 γS γI ρS (r S ) ρI (r I ) 3 5 SI − 3 (24) dr I dr S , D= 4π rSI rSI where r SI depends on both integration variables r S and r I . Using these formulas it is possible to check whether a quantum-chemical computation is consistent with experimental ﬁndings or to obtain estimates for the deviation between a computed and an experimental geometry of the system. It has been demonstrated that such approaches can distinguish between possible alternatives for the inner electronic structure of strongly coupled clusters of paramagnetic ions [30, 31]. If the point-dipole approximation is not valid, the dipole-dipole coupling tensor does not in general have axial symmetry. It is thus characterized by three principal values, which can be expressed as Dxx = (1 + η) ωdd , Dyy = (1 − η) ωdd , and Dzz = −2ωdd with the average dipolar frequency ωdd and the asymmetry η. The approximate spin-to-spin distance obtained by substituting the average dipolar frequency for the dipolar frequency in (14) is usually shorter than the distance between the centers of gravity of the 3 . Note howspatial distributions of the two spins, as the averaging is over 1/rSI ever that this is only a rule of thumb – for strong delocalization contributions from diﬀerent spatial regions may cancel each other due to the orientation dependence of the sign of the coupling. This may lead to a smaller average dipolar frequency than expected and thus to an overestimate of the distance. For such an orthorhombic dipole-dipole coupling tensor the orientation dependence of the Hamiltonian can be written as ˆ dd (θ, φ) = ωdd Psec Sˆz Iˆz + PZQ Sˆ+ Iˆ− + Sˆ− Iˆ+ + PSQ Sˆz Iˆx , (25) H where the orientation dependence of the secular, zero-quantum, and singlequantum contributions is given by Psec (θ, φ) = 1 − 3 cos2 θ + η sin2 θ cos 2φ , 1 PZQ (θ, φ) = − Psec , 4 PSQ (θ, φ) = − sin θ cos θ (3 + η cos 2φ) . The corresponding spectral pattern is shown in Fig. 2c.

(26)

Distance Measurements by NMR and EPR

2.5 Averaging of the Dipole-Dipole Interaction by Internal or External Motion The dependence of the dipole-dipole interaction on the relative orientation of the spin-to-spin vector r SI with respect to the external ﬁeld (angle θ) leads to partial or complete averaging when the spin pair reorients on the time scale of the experiment. In soft matter such reorientation occurs due to local dynamics. For fast reorientation of a spin-to-spin vector r SI of constant length on a cone, the Pake pattern is simply scaled by a local dynamic order parameter. Such an eﬀect can be recognized if the distance rSI is known a priori or if a second measurement can be performed at temperatures that are suﬃciently low to obtain the static dipolar spectrum. With ωdd,stat and ωdd,dyn being the dipolar frequencies observed under static and dynamic conditions, respectively, the local dynamic order parameter is deﬁned by [32] ωdd,stat , (27) SSI = 1 2 2 (3 cos α − 1) ωdd,dyn where 2α is the opening angle of the cone. Note that for fast anisotropic motion of a spin-to-spin vector of constant length, the average dipole-dipole coupling is in general not axially symmetric [4, 33]. For the case of isotropic dynamics during which both length and orientation of the spin-to-spin vector change, the Pake pattern is preserved if and only if the dynamics is much faster than the time scale of the experiment and the changes in length and orientation of the vector are uncorrelated. In this situation an average dipolar frequency can be computed by averaging (14) over rSI (t) and the pattern is furthermore scaled by a local dynamic order parameter analogous to the one deﬁned in (25). If the changes of length and orientation are correlated, the spectral pattern has to be computed by aver3 , where P2 (α) = (3 cos2 α − 1)/2 is the second Legendre aging over P2 (α) /rSI polynomial. If dynamics proceeds on the time scale of the experiment spectral patterns can be computed numerically by the general approach introduced in [34]. Intentional reorientation of the sample with respect to the external magnetic ﬁeld can be used to separate isotropic from anisotropic interactions. Such an approach is most easily realized by sample rotation. As magnetic resonance linewidths in solids are usually dominated by the anisotropy of interactions, sample rotation leads to motional narrowing [35, 36]. For a spin-spin coupling with both an isotropic component J and a purely anisotropic dipolar component with axial symmetry, we ﬁnd for the time dependence of the secular part of the coupling Hamiltonian ˆ SI = H =

ˆJ + H ˆ dd (t) H Sˆz Iˆz [C0 + C1 cos (ωrot t + γ) + C2 cos (2ωrot t + 2γ)] ,

(28)

G. Jeschke and H.W. Spiess

where ωrot is the angular frequency of sample rotation and γ is an Euler angle relating the principal axis frame of the dipolar tensor D to the rotor-ﬁxed frame. The coeﬃcients are ωdd 3 cos2 θrot − 1 (1 + 3 cos 2β) , C0 = J − 4 3ωdd sin 2θrot sin 2β , C1 = 2 3ωdd sin2 θrot 1 − cos2 β , (29) C2 = − 2 where β is another Euler angle and θrot is the angle between the rotation axis and the magnetic ﬁeld axis. The time dependence simpliﬁes for two choices of θrot , the magic angle θrot = 54.74◦ , and the right angle, θrot = 90◦ . At the magic angle the timeindependent part (coeﬃcient C0 ) is purely isotropic. The spectrum then consists of a series of sidebands that are spaced by the rotation frequency and whose amplitudes roughly trace the Pake pattern (Fig. 3). For signiﬁcant J coupling, the centerband and each sideband become doublets with splitting J. At moderate rotation frequencies, ωrot < ωdd /5 the dipolar frequency ωdd can be determined by ﬁtting the sideband pattern. Generally, the eﬀect of magic angle spinning (MAS) can be considered as a refocusing of the anisotropy of interactions. As a result, rotational echoes are observed at integer multiples of the rotor period. At high rotation frequencies, ωrot ωdd , the dipole-dipole coupling of an isolated spin pair is averaged completely, at least if chemical shift anisotropy is negligible. For hom*onuclear spin pairs with coinciding chemical shift, the dipole-dipole coupling is not fully averaged unless the chemical shift anisotropy is also fully averaged [37, 38]. While MAS is widely applied in solid-state NMR spectroscopy [12, 39], it is not directly applicable to EPR spectroscopy as technically feasible sample rotation frequencies are much smaller than the anisotropy of the electron Zeeman and hyperﬁne interaction. This limitation can in principle be overcome

Fig. 3. Dipolar patterns during sample rotation. Dashed lines are the static Pake patterns. (a) Moderately fast rotation at the magic angle θrot = 54.74◦ , J = 0, ωrot = 0.1875ωdd . (b) Fast rotation at the right angle θrot = 90◦ , J = 0, ωrot = 5ωdd

Distance Measurements by NMR and EPR

by applying the magic angle turning experiment [40]. In this experiment, the Larmor frequency is averaged over only three discrete orientations, which is suﬃcient to fully suppress anisotropic broadening. Using fast sample rotation with a frequency of 20 kHz or higher the required 240◦ turn can be completed in a time shorter than the longitudinal relaxation time of the electron spins [41]. The experiment is, however, still limited by the requirement that the excitation bandwidth of the pulses is at least comparable to the total width of the EPR spectrum, which cannot yet be achieved for most samples [42]. At the right angle the term with coeﬃcient C1 vanishes. Due to the term with coeﬃcient C2 there is still anisotropic broadening, however, the width of the dipolar pattern is scaled by a factor of 1/2. Such right-angle spinning has been demonstrated to provide resolution enhancement in EPR spectroscopy for a broad range of samples [43, 44]. Routine use of this technique is hampered by the problem that suﬃciently stable sample rotation is hard to achieve at temperatures below 200 K, where many EPR experiments have to be performed. An additional wiggling magnetic ﬁeld perpendicular to the static magnetic ﬁeld provides a similar relative motion of sample and magnetic ﬁeld as sample rotation and can thus be used to overcome this technical problem [45].

3 Measurement Techniques for Isolated Spin Pairs in Solids As we have pointed out in Sect. 2 the spin-spin distance rSI in a spin pair can be determined or at least estimated from dipolar patterns or dipolar MAS sideband patterns of one of the spins. In practice, magnetic resonance spectra of solids are always spectra of multi-spin systems. Dipolar spectra of such multi-spin systems are more complicated to analyse and will be discussed in Sect. 4. However, in many cases an appropriate combination of experimental techniques and data analysis procedures can provide dipolar spectra that are dominated by the interaction of a single pair of spins. Such approaches are discussed in the following. 3.1 Lineshape Analysis If the dipole-dipole interaction in the spin pair under consideration is of a similar magnitude as all the other interactions of spin S or even larger, distance information can be extracted from ordinary NMR or EPR spectra. This situation may occur when the distance rSI is much shorter than all distances of spin S to other spins that are of the same kind as spin I. In NMR spectroscopy, this is the usual case for rare spins I or if spin I has been introduced by isotope labelling. In EPR spectroscopy, the situation is commonplace for distances up to approximately 2 nm between nitroxide radicals introduced by site-directed spin-labelling [46, 47]. Usually the other interactions in the spin

G. Jeschke and H.W. Spiess

Hamiltonian cannot be neglected, thus lineshape analysis requires that the spectrum of spin S in the absence of dipole-dipole coupling is known. If S and I are like spins both spectra have to be known. Spectrum Simulation Dipolar splittings are resolved in the spectra if the distance is well deﬁned and the dipole-dipole interaction is larger than the width of the most narrow peaks in the NMR or EPR spectrum. In this situation the spectrum may contain suﬃcient information to extract not only the spin-spin distance rSI but also the orientation of vector r SI with respect to the molecular frame deﬁned by the chemical shift anisotropy tensor or g tensor [48]. The reliability of this kind of lineshape analysis is much enhanced when a global ﬁt of several spectra obtained at diﬀerent magnetic ﬁelds is performed, as this corresponds to a variation of the ratio of dipolar to Zeeman anisotropy [49]. Deconvolution and Convolution Approaches In some applications the orientation of vector r SI with respect to the molecular frames of spins S and I is random or it can be assumed that eﬀects of orientation correlation on the lineshape are negligible. Thus, the absorption lineshape A (ω) is the convolution of the absorption lineshape A0 (ω) in the absence of dipole-dipole coupling with the dipolar pattern S (ω). The dipolar pattern can then be extracted even for cases where the dipole-dipole interaction merely causes broadening of the lineshape rather than resolved splittings. Furthermore, a precise analysis is possible even if the spectrum in the absence of dipole-dipole coupling cannot be simulated, provided that this spectrum is experimentally accessible. According to the convolution theorem of Fourier transformation, a convolution of the spectrum with the dipolar pattern corresponds to the product of the Fourier transforms F of the spectrum and the dipolar pattern (30) F {A (ω)} = F {A0 (ω)} F {S (ω)} , so that the dipolar pattern can be obtained by [50] F {A (ω)} −1 S (ω) = F , F {A0 (ω)}

(31)

where F −1 denotes the inverse Fourier transformation. Alternatively, the distance or a distribution of distances can be ﬁtted by computing the corresponding dipolar pattern and simulating the spectrum A (ω) by inverse Fourier transformation of (30) [46].

Distance Measurements by NMR and EPR

3.2 Double-Resonance Techniques In many cases dipolar frequencies are signiﬁcantly smaller than the width of the most narrow peaks in the inhom*ogeneously broadened static NMR or EPR spectrum, but still larger than or comparable to the hom*ogeneous linewidth Γhom = 2/T2 , where T2 is the transverse relaxation time. Dipolar broadening of the ordinary NMR or EPR lineshape is then negligible, however, the dipolar frequency can still be measured if the inﬂuence of all the other broadening mechanisms is suppressed. Such suppression can be achieved by refocusing the other anisotropic interactions in echo experiments or by averaging them by MAS. In both cases the dipole-dipole interaction is usually also suppressed, exceptions being the coupling between like spins in the Hahn echo experiment [51] and the coupling between unlike spins in the solid-echo experiment [52]. In the following Sections we shall discuss ways to recouple exclusively the desired dipole-dipole interaction. For the case of unlike spins this can be achieved by double resonance experiments. Static Solid-State NMR In a two-pulse echo sequence 90◦x −τ −180◦x −τ −echo applied to spin S the 180◦ pulse with phase x refocuses all contributions to the spin Hamiltonian that are linear in the Sˆz operator. In particular, this includes the Zeeman interaction of spin S and the coupling to spin I, provided that the pulse does not excite spin I. Refocusing is based on inversion of the spin magnetic moment with respect to the external ﬁeld and to the local dipolar ﬁeld generated by spin I. The coupling to spin I can thus be selectively reintroduced by inverting only the local dipolar ﬁeld, which can in turn be achieved by applying a 180◦ -pulse to spin I (Fig. 4). In this spin echo double resonance (SEDOR) experiment [53] the variation of the echo amplitude with time τ is described by Vdip = cos ωdd τ 1 − 3 cos2 θ + J . (32) For a macroscopically disordered system, the dipolar time evolution function Vdip (τ ) is the Fourier transform of the dipolar pattern. In the usual case where J coupling can be neglected, it is the Fourier transform of the Pake pattern (Fig. 4c). Due to tranverse relaxation, the dipolar time evolution function is damped, V (τ ) = Vdip (τ ) exp (−2τ /T2 ). In another double-resonance approach, the heteronuclear dipole-dipole coupling drives the equilibration of spin temperatures between baths of S and I spins, i.e., it determines the dynamics of cross polarization. Precise measurements are possible if the spin temperature of the I spins is inverted halfway through the polarization transfer and hom*onuclear couplings among the I spins are suppressed by Lee-Goldburg decoupling. In such Lee-Goldburg decoupling the I spins are spin-locked along the magic angle. This polarization inversion spin-exchange at the magic angle (PISEMA) experiment is a

G. Jeschke and H.W. Spiess

Fig. 4. SEDOR experiment. (a) Pulse sequence. (b) Eﬀect of the 180◦ -pulse applied to the I spin on the local ﬁeld at the S spin. (c) Variation of the echo amplitude S (τ ) for a macroscopically disordered system with well deﬁned distance rSI (simulation)

two-dimensional experiment with dipolar evolution in the indirect dimension and acquisition of a free induction decay corresponding to the ordinary decoupled NMR spectrum in the direct dimension [54]. Compared to its more simple predecessor, separated local ﬁeld spectroscopy [55], the introduction of Lee-Goldburg decoupling in PISEMA strongly improves resolution in the indirect dimension and thus the precision of distance measurements. Solid-State MAS NMR Sample rotation at the magic angle with ωrot ωdd averages dipole-dipole coupling to zero. This is because the local dipolar ﬁeld imposed by spin I at spin S changes its sign during rotation and the average over a full rotor period vanishes. Recoupling is again possible by inverting the local ﬁeld by 180◦ pulses applied to the I spin. In the basic version of the experiment [56, 57], two 180◦ -pulses are applied during each rotor period, one after the ﬁrst half and one at the end of the rotor period. To obtain a pure dipolar evolution, at least one 180◦ -pulse must be applied to the S spins in the center of the total evolution period, as MAS does not refocus resonance oﬀsets due to the isotropic chemical shifts. The dipolar evolution is traced by the amplitudes of rotational echoes. Usually the basic unit of the experiment consists of several rotor periods trot . In this rotational echo double resonance (REDOR) experiment, the dipolar evolution is also damped by transverse relaxation. For longer distances, the dipolar oscillation may be overdamped, so that it becomes necessary to factor out relaxational decay. This can be done by measuring the rotational echo amplitudes S0 (ntrot ) in the absence of the recoupling pulses for the I spins and the amplitudes S (ntrot ) in their presence. The diﬀerence ∆S = S0 − S

Distance Measurements by NMR and EPR

Fig. 5. REDOR data in diﬀerent domains (experiment on 10% selectively 2−13 C−15 N- labelled glycine diluted into unlabelled gylcine). (a) Time-domain data (dipolar evolution function). (b) Frequency-domain data obtained by complex Fourier transform. (c) Distribution of dipolar frequencies obtained by the REDOR transform. (d) Distribution of dipolar frequencies obtained by REDOR asymptotic rescaling. (e) Distribution of dipolar frequencies obtained by Tikhonov regularization. Reproduced with permission from [87]

corresponds to the variation of the rotational echo amplitude caused by both dipole-dipole interaction and transverse relaxation, while the normalized signal VREDOR = 1 − ∆S/S0 corresponds to the REDOR dipolar evolution function (Fig. 5a) [58]: √ √ √ 2π J1/4 2ωdd t J−1/4 2ωdd t , (33) VREDOR (ωdd , t) = 4 where J1/4 and J−1/4 are Bessel functions of the ﬁrst kind. The diﬀerence with respect to the static dipolar evolution function, as observed in the SEDOR experiment (32), results since rotational averaging is only partially oﬀset by

G. Jeschke and H.W. Spiess

the recoupling pulses. Accordingly, the Fourier transform of this function does not correspond to a Pake pattern (Fig. 5b). Numerous variants of the basic REDOR experiment have been proposed that correct for imperfections encountered in certain situations [13, 14]. For instance, it is usually impossible to fully invert quadrupole spins I > 1/2 by a 180◦ -pulse. The rotational echo adiabatic passage double resonance (REAPDOR) experiment circumvents this problem by applying a prolonged pulse to the I spins that inverts these spins with higher eﬃciency than a 180◦ pulse [59]. The prolonged pulse achieves inversion since during rotation the resonance frequency of the I spins adiabatically passes the frequency of the applied pulse. The PISEMA experiment introduced in the previous section can also be applied under MAS conditions [60]. In contrast to REDOR there is no need to synchronize the pulses with sample rotation. Pulse EPR In pulse EPR spectroscopy, excitation bandwidths are often signiﬁcantly smaller than the width of the whole spectrum of a single paramagnetic species. For example, spectra of nitroxide spin labels have a width of ∼180 MHz at X-band frequencies of ∼ 9.6 GHz, which is mainly due to nitrogen hyperﬁne anisotropy. As experiments with good sensitivity can be performed with pulse lengths of 32 ns, corresponding to excitation bandwidths of ∼30 MHz, two microwave frequencies can be placed in the spectrum that excite nitroxide radicals with diﬀerent orientations or with diﬀerent magnetic spin quantum numbers of the nitrogen nucleus. As pulses at one frequency excite exclusively S spins and pulses at the other frequency excite exclusively I spins, such ELDOR experiments can be considered as experiments on unlike spins, although both spins are electron spins of nitroxide radicals. The situation thus corresponds to a heteronuclear experiment in NMR. Accordingly, the principle of SEDOR can also be applied in EPR spectroscopy. Because of a more unfavourable ratio between typical dipolar frequencies (0.1–10 MHz) and typical transverse relaxation times (1 µs), it is necessary to factor out relaxation. This can be done by using a constant interpulse delay τ , and varying the delay of the 180◦ -pulse for the I spins with respect to the 90◦ -pulse for the S spins, which results in the pulse sequence (90◦ )S − t − (180◦ )I − (τ − t) − (180◦ )S − τ − echo. The local ﬁeld at the S spin is thus changed at time t during the defocusing period of length τ , which corresponds to an interchange of magnetization between the two transitions of the dipolar doublet. Hence, the magnetization vector precesses 2 1 − 3 cos ± ω θ /2 during time t and with frequency with frequency ω S dd ωS ∓ ωdd 1 − 3 cos2 θ /2 during time τ − t as well as in the refocusing period of duration τ . The echo amplitude as a function of t is then modulated with frequency ωdd 1 − 3 cos2 θ +J. It does not vary due to relaxation as the total

Distance Measurements by NMR and EPR

duration of the experiment is constant. The choice of the ﬁxed interpulse delay τ in this pulse ELDOR (PELDOR) or double electron electron resonance (DEER) experiment [61, 62] thus involves a tradeoﬀ between sensitivity and resolution, as the signal amplitude decreases with increasing τ while resolution of the dipolar frequency increases with increasing maximum observation time tmax ≤ τ . In EPR typical instrumental deadtimes td of a few ten nanoseconds after a pulse compare unfavourably to the period of dipolar oscillations, at least for distances up to 2.5 nm. With standard equipment t > td has to be chosen to avoid signal distortion. This problem can be circumvented either by using a bimodal microwave cavity and separate high-power ampliﬁers for the two frequencies [62] or by inserting another 180◦ -pulse for refocusing the two-pulse echo. In this four-pulse DEER experiment [63, 64] t = 0 for the dipolar decay corresponds to the unobserved ﬁrst echo (see Fig. 6a), so that the complete time evolution can be observed when choosing τ1 ≥ td . Selective excitation by the microwave pulses implies that for a given S spin the corresponding I spin is ﬂipped by the 180◦ -pulse with probability λ < 1. Thus, for an isolated spin pair only a fraction λ (θ) of the echo is modulated: VDEER (ωdd , t) = 1 −

π/2

λ (θ) 1 − cos ωdd t 1 − 3 cos2 θ sin θdθ . (34)

Fig. 6. Four pulse DEER experiment. (a) Pulse sequence. (b) Experimental data set for a well deﬁned distance rSI ≈ 2.8 nm The inset shows the structure of the biradical. (c) Experimental data set for a broad distribution of distances in a [2]catenane [82]. (d) Pair correlation functions obtained from the data in (c) by direct transformation (solid line) and by ﬁtting a simpliﬁed geometric model of the structure (dotted line)

G. Jeschke and H.W. Spiess

If correlation between the molecular frames of the two spins and the orientation of vector r SI is negligible, λ does not depend on angle θ and can be pulled in front of the integral. A fraction 1 − λ of the echo is then unmodulated. In practice, spin pairs are never completely isolated and the signal is thus also aﬀected by coupling of spin S to remote spins. For a hom*ogeneous spatial distribution of remote spins with concentration c, the eﬀect is an exponential damping of the signal by a factor 2πgS gI µ2B µ0 NA √ λct , (35) Vhom (c, t) = exp − 9 3 where NA is the Avogadro constant, µB is the Bohr magneton, and the concentration is given in units of mmol L−1 . A typical experimental data set for a rigid biradical with rSI = 2.8 nm diluted into a matrix at a concentration of 2 mmol L−1 is shown in Fig. 6b. 3.3 Rotational Resonance Per deﬁnition double resonance methods are not applicable to like spins. However, in solid-state MAS NMR hom*onuclear recoupling can be achieved by a technique that is similar in spirit to the PISEMA experiment. In the PISEMA experiment, double resonance irradiation establishes a degeneracy of levels of the two coupled spins in the doubly rotating frame, so that the dipole-dipole coupling can drive a magnetization exchange. In other words, an external perturbation oﬀsets the diﬀerence in the resonance frequencies, and as a result, dipole-dipole coupling causes strong mixing. In hom*onuclear MAS NMR, diﬀerences in resonance frequencies of the two spins are usually of the same order of magnitude as the frequency of sample rotation ωrot . By matching an integer multiple of the rotation frequency to this diﬀerence [65, 66], (36) nωrot = ωS − ωI , rotational resonance occurs, at which a mixing of spatial and spin-dependent contributions to the Hamiltonian takes place. The phenomenon can best be discussed by Floquet theory, as it corresponds to a degeneracy of levels in Floquet space [67]. Such Floquet states correspond to spin states in the presence of a periodic external perturbation that couples to the spins. In the case at hand this perturbation is sample rotation. Thus, degeneracy of states in Floquet space means that, in the presence of this perturbation, the states can be mixed by a small oﬀ-diagonal element of the Hamiltonian. Here this ˆ in the dipolar alphabet, (9). State oﬀ-diagonal element is the ﬂip-ﬂop term B mixing in the vicinity of the level anti-crossing, (36), causes line broadening from which the dipolar frequency can be estimated. If the MAS NMR spectrum contains lines from multiple spins, this rotational resonance condition can usually be established selectively for each pair of spins. Site-selective distance measurements by this method thus do not require acquisition of a complete two-dimensional data set.

Distance Measurements by NMR and EPR

3.4 Pulse-Induced Recoupling During MAS in hom*onuclear Spin Systems In Sect. 3.2 we have seen that radiofrequency pulses resonant with one of the spins in the spin pair can partially oﬀset MAS averaging of the dipoledipole coupling. For hom*onuclear spin systems pulses excite both spins simultaneously. To achieve similar recoupling, the 180◦ -pulse applied in REDOR halfway through the rotor period has to be replaced by a pair of 90◦ -pulses with phases x and −x that are separated by an interpulse delay τ [69]. The pulse pair is placed symmetrically with respect to the centre of the rotor period. To ensure averaging of the chemical shift anisotropy and of resonance oﬀsets, 180◦ -pulses with alternating phases x and −x are applied at the end of each second rotor period, leading to a basic cycle that extends over four rotor periods. In this dipolar recovery at the magic angle (DRAMA) experiment, the recoupling eﬃciency and shape of the dipolar pattern depend on the ratio τ /trot , where trot = 2π/ωrot is the duration of the rotor period. The recoupling is achieved as a 90◦x -pulse applied to both spins interconverts Sˆz Iˆz and Sˆy Iˆy terms. Several alternative experiments for hom*onuclear recoupling during MAS have been developed, some of which can be performed with narrowband excitation and combined with the rotational resonance experiment [70]. A systematic treatment of the combined inﬂuence of radiofrequency ﬁelds and sample rotation based on symmetry considerations can provide optimized pulse schemes [71]. 3.5 Build-Up of Double-Quantum Coherence In the low-temperature limit, kB T / ωS , ωI , only the lowest level of the four-level system (see Fig. 1a) of a spin pair is populated in thermal equilibrium. A 90◦ -pulse applied to this initial state excites coherence at all six transitions. However, usually magnetic resonance experiments are performed in the high-temperature limit, kB T / ωS , ωI , where population diﬀerences between the levels are small and all four single-quantum transitions are equally polarized in thermal equilibrium. In this situation a 90◦ -pulse excites coherence exclusively at the four single-quantum transitions. The zeroquantum and double-quantum (DQ) transition are therefore sometimes called forbidden transitions. Indeed, for irradiation in the linear regime their excitation is forbidden by selection rules at any temperature. However, even in the high-temperature limit, coherence can be excited on forbidden transitions by a combination of pulses and free evolution [72, 73, 74]. The basic building block of such experiments consists of a 90◦ -pulse that excites coherence on all single-quantum transitions, an interpulse delay τ during which the coherences of the two transitions of each spin acquire a phase diﬀerence ∆φ = ωdd t 1 − 3 cos2 θ /2, and another 90◦ -pulse that converts a fraction sin ∆φ of the single-quantum coherence to DQ coherence. At a later stage of

G. Jeschke and H.W. Spiess

the experiment, a third 90◦ -pulse transfers the DQ coherence back to singlequantum coherence in antiphase (Sˆy Iˆz and Sˆz Iˆy ). Single-quantum coherence in antiphase does not give rise to a signal, but is reconverted to observable ˆ dd . By appropriate phase cysingle-quantum coherence by evolution under H cling all signals can be eliminated that stem from other coherence transfer pathways [3], so that the build-up of DQ coherence can be monitored as a function of time τ . hom*onuclear MAS NMR This basic scheme for excitation of DQ coherence cannot simply be apˆ dd is averaged. However, the scheme plied under MAS conditions, where H can be combined with DRAMA recoupling or with other recoupling techniques that use rotor-synchronized pulses [75]. In a DRAMA-type pulse cycle 90◦y − τ1 − 90◦x − τ2 − 90◦−x − τ1 − 90◦−y with 2τ1 + τ2 = trot , the doublequantum contribution to the dipolar Hamiltonian under MAS [76] is eﬀective only during interpulse delay τ2 . This contribution is thus not averaged over the rotor cycle and DQ coherence is generated. The double-quantum contriˆ dd can also be reintroduced by the back-to-back (BABA) pulse bution of H ◦ cycle 90x − trot − 90◦−x 90◦y − trot − 90◦−y [77]. Information on the dipolar frequency can be obtained either from the DQ build-up curves or from DQ sideband patterns. To obtain the patterns, a variable delay t1 with increment ∆t1 < trot , corresponding to the indirect dimension of a two-dimensional experiment, is introduced in between the DQ excitation and reconversion subsequences. Each subsequence consists of one or several of the recoupling pulse cycles introduced above. Fourier transformation along the t1 dimension yields DQ spinning-sideband patterns whose frequency dispersion results from rotor-encoding of the DQ Hamiltonian. In other words, sample reorientation between excitation and reconversion rather than the evolution of DQ coherence determines the width of the patterns and the intensity distribution among the sidebands. This has two consequences. First, sidebands are also observed for rotation frequencies that signiﬁcantly exceed the dipolar frequency ωdd . This results from recoupling. Second, the pattern depends on the number of rotor cycles (pulse cycles) used for excitation and reconversion of the DQ coherence. For a given rotation frequency, the experiment can thus be adapted to the expected magnitude of the dipolar frequency by applying an appropriate number of recoupling pulse cycles. This approach is restricted by relaxation which imposes an upper limit on the number of recoupling pulse cycles that can be used. For small couplings, ωdd < ωrot /10, one may thus be conﬁned to a regime where only the two ﬁrstorder sidebands can be observed and where only the total intensity rather than the shape of the pattern contains information on ωdd . In this situation, the information can be extracted from build-up curves, i.e., from the dependence of the total intensity of the DQ sideband pattern on the number of recoupling pulse cycles. The fact that DQ NMR can be eﬃciently applied at high MAS

Distance Measurements by NMR and EPR

frequencies makes this experiment particularly suitable for the measurement of proton-proton distances [12]. Pulse EPR In EPR spectroscopy, the basic scheme for excitation and reconversion of DQ coherences can be applied, but must be supplemented with 180◦ -pulses for refocusing the dispersion of frequencies ωS and ωI caused by g and hyperﬁne anisotropy and by unresolved isotropic hyperﬁne couplings. The resulting excitation subsequence, 90◦ −τ1 /2−180◦ −τ1 /2−90◦ , and reconversion subsequence, 90◦ − τ2 /2 − 180◦ − τ2 /2 − echo, sandwich a period t1 /2 − 180◦ − t1 /2 during which the DQ coherence evolves. To factor out relaxation, this DQ EPR experiment is performed with ﬁxed t1 and a constant sum τ1 + τ2 of the other interpulse delays, varying the diﬀerence τ1 − τ 2 [78, 79]. Assuming excitation of the whole spectrum, the signal for an isolated spin pair is then given by [5] 1 cos ωdd (τ1 − τ2 ) 1 − 3 cos2 θ /2 2 1 + cos ωdd (τ1 + τ2 ) 1 − 3 cos2 θ /2 , 2

VDQ (τ1 , τ2 ) =

(37)

where we have assumed that the total width of the EPR spectrum is much ˆ term in the dipolar alphabet can be neglected. larger than ωdd , so that the B Note that the second term on the right-hand side of (37) is constant as τ1 +τ2 is kept constant. Fourier transformation of the ﬁrst term gives the Pake pattern. 3.6 Solid-Echo Techniques The solid-echo sequence 90◦x − τ − 90◦y − τ − echo refocuses the dipole-dipole coupling of like spins but does not refocus the one of unlike spins [80]. In solidstate NMR, this sequence can thus be used to separate heteronuclear from hom*onuclear couplings [52]. An echo decay that is purely due to heteronuclear couplings and transverse relaxation is obtained if resonance oﬀsets, chemical shift anisotropies and hom*onuclear multi-spin eﬀects can be neglected. By supplementing the experiment with additional 180◦ -pulses halfway through the evolution periods, an echo experiment is obtained that refocuses resonance oﬀsets, couplings between unlike spins, and couplings in isolated pairs of like spins. By introducing a diﬀerence τ1 −τ2 between the defocusing and refocusing time, 90◦x − τ1 /2 − 180◦x − τ1 /2 − 90◦y − τ2 /2 − 180◦x − τ2 /2 − echo, we obtain an experiment in which variation of the echo amplitude as a function of τ1 − τ2 for constant τ1 + τ2 is solely due to couplings between like spins [81]. In EPR spectroscopy, this single-frequency technique for refocusing dipole-dipole couplings (SIFTER) allows for observing dipolar evolution for longer times than alternative experiments. This is because the contribution of remote like spins to the echo decay is smaller for the solid echo than for the Hahn echo.

G. Jeschke and H.W. Spiess

4 Complications in Multi-Spin Systems Considering isolated spin pairs is a useful approximation for a measurement on spin S when the distance rSI(k) to spin Ik under consideration is much shorter than the distance to any other I spin. The signal contribution due to the latter, remote spins can then be neglected at short dipolar evolution times or can be accounted for in a summary way as indicated in Sect. 3.2 for the DEER experiment. For many problems of interest such a situation can be generated by choosing the appropriate experiment, e.g., an experiment with selective perturbation of I spins, or by isotope or spin labelling. However, in a sizeable number of cases the intrinsic structure of the material precludes this approximation or a selective labelling approach would be too tedious. Experiments and data analysis procedures for multi-spin systems are thus required. While experiments on isolated spin pairs correspond to two-body problems, which can in principle be solved exactly, any experiment on a multi-spin system corresponds to a many-body problem for which there is no general solution. Depending on the topology of the system exact solutions may or may not exist. If no exact solution exists, it is often possible to ﬁnd a regime where approximations are suﬃciently precise, or the ambiguities of the many-body problem can be overcome by analyzing the data in terms of a preconceived structural model. Finally, one may resort to calibration, i.e., to systematic comparison of the experimental data to data obtained on a similar system with known structure. 4.1 Dipolar Broadening and Moment Analysis Consider the resonance line of S spins that is broadened by dipole-dipole coupling to a large number of like spins, in this case also denoted as S spins, or to a large number of unlike spins, denoted as I spins. As the number of coupled spins roughly scales with r2 , the rS(j)S(k) (or rS(j)I(k) ) are in eﬀect continuously distributed at long distances, so that no a priori limit can be put on the number of spins that have to be considered in computation of the lineshape. Thus, the problem cannot be solved by explicit computation of resonance frequencies and amplitudes of all transitions of the multi- spin system. However, the lineshape can be analyzed by the method of moments, in which characteristics of the absorption line A (ω) are deﬁned that can be computed analytically for any given spatial distribution of spins and can be obtained easily from the experimental lineshape [83, 84]. The ﬁrst moment of the lineshape, ∞ ωA (ω) dω , (38) ω = 0 ∞ A (ω) dω 0 is the average frequency of the resonance line, which is not inﬂuenced by the dipole-dipole coupling. Higher moments are deﬁned by

Distance Measurements by NMR and EPR

∞

(ω − ω) A (ω) dω ∞ . (39) A (ω) dω 0 Analysis is often restricted to the second moment, ∆ω 2 , which is of the order of the square of the linewidth. In general, experimental precision decreases with increasing order n of the moment, as the low-amplitude wings of the lineshape with lower signal-to-noise ratio contribute increasingly. The second moment can be computed without diagonalizing the Hamiltonian of ˆ terms of the the system [83]. For the case of like spins, where the Aˆ and B dipolar alphabet have to be included, it is given by 2 3 µ0 2 4 2 1 1 − 3 cos2 θjk γS S (S + 1) , (40) ∆ω SS = 6 4 4π N rjk ∆ω n =

j,k

where N is the number of spins included in the summation. This sum can be computed for a given crystal lattice. For an isotropic system where all spins S are similarly situated, angular correlations can be neglected, and we have [83] 2 3 µ0 2 4 2 1 −6 ∆ω SS = γS S (S + 1) rjk , (41) 5 4π N k

where the sum does no longer depend on j, as it is the same for any given spin S. ˆ term has to be neglected, which leads For the case of unlike spins, the B to scaling of the second moment by a factor of (2/3)2 : 2 1 µ0 2 2 2 2 1 1 − 3 cos2 θjk ∆ω SI = γS γI I (I + 1) . (42) 6 3 4π N rjk j,k

Equation (41) changes accordingly. Note that the width of the Pake pattern ˆ term, but for an isolated spin pair scales by a factor 2/3 on neglecting the B that formulas for the nth moment cannot generally be converted by scaling by (2/3)n . 4.2 Relation of the Dipolar Pattern to the Spin-Spin Pair Correlation Function For an isotropic system with similarly situated observer spins Sj , the second moment is fully deﬁned by the distribution of distances rjk to coupled spins Sk or Ik as can be seen in (41). It can thus be expressed in terms of the spin-spin pair correlation function G (r), which gives the probability to ﬁnd a coupled spin at distance r from the observer spin. In fact, if angular correlations between spin pairs can be neglected, this applies to any moment, and as the lineshape (or dipolar pattern of the multi-spin system) is fully determined by a series expansion into its moments, the dipolar pattern itself is fully determined by G (r). Because of the unique mapping between distances rSI and

G. Jeschke and H.W. Spiess

dipolar frequencies ωdd the dipolar pattern or the dipolar evolution function in time domain can be converted to the pair correlation function G (r). Direct integral transformations of experimental data to the pair correlation function or the distribution of dipolar frequencies have been proposed for the REDOR [58], see Fig. 5c,d, and DEER [85], see Fig. 6d, experiments. Tikhonov regularization with an adaptive choice of the regularization parameter [86] is generally applicable for this task if an analytical expression for the dipolar evolution function is available. As demonstrated in Fig. 5, Tikhonov regularization is advantageous for noisy data in cases where the pair correlation function consists of narrow peaks [87]. Distance distributions with broad peaks are harder to analyze, as transformation of the dipolar evolution function to a distance distribution corresponds to an ill-posed problem. A systematic comparison of diﬀerent procedures for data analysis in this situation can be found in [88]. Whether or not G (r) can be obtained from magnetic resonance data by one of the procedures mentioned above depends on the availability of an analytical expression for the dipolar time evolution function. For isolated spin pairs such expressions are available for most experiments. In the following we shall see that under certain conditions, the expression for a multi-spin system can be obtained from the expression for an isolated spin pair. 4.3 Eﬀective Topology of Spin Systems Consider a system consisting of an observer spin S with dipole-dipole couplings to several spins Ik . In general, the I spins will also be coupled among themselves, so that a true multi-body problem arises (Fig. 7a). In that situation the dipolar evolution function for the multi-body system cannot be expressed in terms of the functions of the pairs SIk . On the other hand, if couplings among the I spins can be neglected due to the design of the experiment or if they are much smaller than the couplings in pairs SIk (Fig. 7b), the

Fig. 7. Eﬀective topology of multi-spin systems. (a) All couplings are signiﬁcant during the experiment. Factorization into pair contributions is impossible. (b) Only couplings between spin S and spins Ik are signiﬁcant. The signal is a product of pair contributions

Distance Measurements by NMR and EPR

Hamiltonian is block diagonal with the blocks corresponding to the spins Ik . The Hamiltonian and density operator of the spin system can then be factorized into pair contributions, which can be treated separately (see, for instance, [89]). As a result, the signal of the multi-spin system can be expressed as a product of the signals of the pairs VSI(k) . (43) Vmulti = k

This situation is typical for electron-nuclear spin systems, in which usually one electron spin S is hyperﬁne-coupled to several nuclear spins Ik with the hyperﬁne coupling being by several orders of magnitude larger than the couplings among the nuclear spins. For all the other cases, couplings among I spins are generally comparable to couplings between S and I spins. Nevertheless, an eﬀective topology allowing for factorization can be achieved when the I spins are excited by only one pulse with a duration that is short compared to the inverse of the dipolar linewidth of these spins. The couplings among I spins can then be neglected during the pulse. They do inﬂuence the evolution of I spin coherence after the pulse, but as there is no subsequent mixing between S and I spins, this inﬂuence does not extend to the signal observed on S spin transitions. Thus, (43) applies to SEDOR and DEER experiments. For experiments with prolonged irradiation of I spins or experiments in which several pulses are applied to these spins, it has to be checked explicitely whether or not (43) is a good approximation for the signal of the multi-spin system. Note that simpliﬁcation of the eﬀective topology can also be achieved by hom*onuclear decoupling of the I spins during the dipolar evolution period of the experiment. In the PISEMA experiment [54] this is done by Lee-Goldburg decoupling. Eﬀective topology of the spin system is also the key to understanding why the build-up of I spin multiple-quantum coherences in heteronuclear NMR experiments can be used to count I spins in the vicinity of the S spin [90]. Such spin counting relies on the fact that the maximum coherence order which can be excited in a given evolution time is limited by the number of spins in a cluster. The cluster consists of spins coupled among themselves by dipoledipole interactions that are comparable to the inverse of the evolution time [74]. In the heteronuclear case, the build-up of I spin multiple-quantum coherence is due to the coupling of all the I spins to a single S spin. As the 2 while the coupling decreases with number of I spins increases only as ∼ rSI −3 rSI , the build-up curve converges with time [90, 91]. 4.4 Multi-Spin Eﬀects in MAS NMR As long as the rotation frequency ωrot does not strongly exceed the total anisotropy of the static NMR spectrum, the extent of line narrowing under MAS depends on the commutation properties of the Hamiltonian at diﬀerent

G. Jeschke and H.W. Spiess

ˆ 0 that times t1 and t2 during the rotor cycle [37]. Consider the Hamiltonian H ˆ 0 (t2 )] = 0 for ˆ 0 (t1 ) , H includes all anisotropic interactions of the S spins. If [H ˆ 0 is inhom*ogeneous in the sense of Maricq and Waugh all times t1 and t2 , H [37]. For this case MAS leads to complete refocusing of the transverse S spin magnetization to a rotational echo after a full rotor cycle. Already at low MAS frequencies, the anisotropic spectrum of the S spins is then resolved into a sideband pattern, with the widths of the centerband and all individual sidebands being determined by the transverse relaxation time T2 . If, on the ˆ 0 (t2 )] = 0 at least for some combinations of t1 and ˆ 0 (t1 ) , H other hand, [H ˆ 0 is called hom*ogeneous t2 , such complete refocusing does not occur and H ˆ 0 , the width of the in the sense of Maricq and Waugh. For a hom*ogeneous H sidebands depends on sideband order and on ωrot . Complete narrowing to the limit of the relaxational linewidth is only achieved for ωrot that are much larger than the total anisotropy of the static spectrum. In many cases, such high rotation frequencies may be technically inaccessible. Any combination of chemical shift anisotropy terms and Aˆ terms of the dipolar alphabet is inhom*ogeneous, as [Sˆz , Sˆz Iˆkz ] = 0. Thus, high resolution can be achieved at moderate ωrot as long as only heteronuclear dipole-dipole couplings are involved. This is true irrespective of whether an isolated spin pair or a multi-spin system is considered, since Aˆ terms of diﬀerent spins also comˆ terms of the dipolar alphabet are mute. For the hom*onuclear case, where the B ˆ signiﬁcant, H0 is still inhom*ogeneous for an isolated spin pair in the absence of signiﬁcant chemical shift anisotropy, as [Sˆz Iˆz , Sˆx Iˆx ] = [Sˆz Iˆz , Sˆy Iˆy ] = 0. In ˆ 0 becomes hom*ogeneous, as Sˆz the presence of chemical shift anisotropy, H ˆ term. For an isolated hom*onuclear spin pair does not commute with the B with coinciding isotropic shifts [37] or for small clusters of nuclei of the same isotope [38] the centreband and each sideband are then broadened into charˆ 0 is acteristic dipolar patterns. More signiﬁcantly for the problem at hand, H ˆ terms of diﬀeralso hom*ogeneous for multi-spin systems of hom*onuclei, as B ˆ term of one spin does not ent spins do not commute and furthermore the B commute with the Aˆ term of another spin. ˆ 0 can be obDetailed insight into sideband broadening for hom*ogeneous H tained by applying Floquet theory to MAS NMR on multi-spin systems [92]. Although quantitative analysis requires numerical computations for systems consisting of a small ﬁnite number of nuclei and moment analysis for a large or inﬁnite number of nuclei, analytical expressions reveal how multi-spin correlations inﬂuence the patterns [12, 93]. It is found that the inﬂuence of coupling −k . Thus, the multi-spin terms that involve correlations of k spins scales with ωrot character of the signal decreases with increasing rotation frequency. 4.5 Multi-Spin Eﬀects in Double-Quantum Sideband Patterns The DQ sideband pattern of an isolated spin pair with distance rSI consists of only odd-order sidebands, spaced by (2n + 1) ωrot from the center of the pattern, where n is an integer number. The eﬀect of a third spin on such a

Distance Measurements by NMR and EPR

pattern can be described in terms of a dimensionless perturbation parameter [12] pert 2πωdd ξ= . (44) ωrot pert where the perturbing dipole-dipole coupling ωdd is deﬁned as the coupling to the spin of the original pair that is closer to it. Generally, in such a multi-spin system even-order sidebands spaced by 2nωrot from the center of the pattern appear and their intensity increases with increasing ξ. While the intensity of the additional sidebands can provide an estimate of the overall inﬂuence of perturbing spins, the detailed intensity pattern and the occurence of broadening of the sidebands depends on the spatial arrangement of the three spins. In a linear conﬁguration, no additional broadening is observed irrespective of the magnitude of ξ, i.e., the linear three-spin system behaves inhom*ogeneously. Furthermore, the centreband is the most intense of the new bands for this geometry. In a ∆ arrangement, where the perturbing spin has the same distance from both spins of the original pair, the second-order sideband is the most intense of the new bands, and signiﬁcant broadening of the original odd-order sidebands is observed. The three-spin system in a ∆ arrangement thus behaves hom*ogeneously. In contrast to DQ build-up curves, DQ sideband patterns thus exhibit a dependence on the geometrical arrangment of the spins in the multi-spin system that can provide additional information in favorable cases where the total number of spins is small or further constraints are available.

5 Application Examples How many distances have to be measured and which precision has to be achieved strongly depends on the system of interest and on the information that is required to understand its properties or function. In highly disordered systems it may be suﬃcient to prove spatial proximity of certain substructures. On the other hand, full determination of the well-deﬁned structure of a biomacromolecule may require a sizeable number of constraints on distances and angles. For soft matter, a static structure often has to be supplemented by some information on dynamics to understand the function of the system. In the following, we illustrate these issues on a number of application examples and model studies. 5.1 Detecting

P-31 P-Spatial Proximity in Phosphate Glasses

Glasses are characterized by a rather low degree of order beyond the trivial constraints on bond lengths, bond angles, and dihedral angles. Nevertheless, it has been found that some heterogeneity, and hence some order, is present over a wider range of length scales. For any particular glass it is of interest

G. Jeschke and H.W. Spiess

what types of order do persist, as such insight may show ways of tailoring the material to certain applications. Inorganic silicate and phosphate glasses can be described in terms of elementary building blocks in which the central silicon or phosphorous atom is coordinated by four oxygen atoms in a roughly tetrahedral geometry. Such a building block with n ≤ 4 oxygen atoms that bridge to a neighbour building block is called a Q(n) group. The distribution of Q(n) groups, and thus the structure of the glass, can be inﬂuenced by varying the ratio between the network formers SiO2 or P4 O6 and network modiﬁers, which may be akali or earth alkali oxides. This distribution can be determined by lineshape analysis of 29 Si or 31 P MAS NMR spectra, as the diﬀerent Q(n) groups are characterized by diﬀerent chemical shift ranges. However, the connectivities among the Q(n) groups, which are a characteristic for the hom*ogeneity or heterogeneity of the structure, are not accessible from such spectra. These connectivities can be detected by hom*onuclear 31 P DQ NMR spectroscopy [94]. Two-dimensional DQ spectra of two phosphate glasses containing 58 and 35 mol % of the network modiﬁer Na2 O are displayed in Fig. 8. The glass with the higher content of Na2 O is expected to form chain-like networks consisting mainly of Q1 and Q2 groups. As seen by the crosspeaks 1 − 2 and 2 − 1 in Fig. 8a, signiﬁcant fractions of the Q1 and Q2 groups are close enough for build-up of DQ coherence within the excitation time of 640 µs used in the experiments. This excludes structures consisting mainly of rings of Q2 groups and isolated Q1 groups and is consistent with the presence of relatively short chains of Q2 groups that are end-capped by Q1 groups. The larger mean

Fig. 8. 31 P-31 P-double quantum NMR on two phosphate glasses. Connectivities between structural units Q(n) and Q(n ) are indicated by n − n (a) Glass containing 58 mol % Na2 O. (b) Glass containing 35 mol % Na2 O. Reproduced with permission from [94]

Distance Measurements by NMR and EPR

isotropic chemical shift of −16 ppm for the 2 − 1 crosspeak compared to the shift of −18 ppm for the 2 − 2 autopeak strongly indicates that the type of the adjacent building blocks has a stronger inﬂuence on chemical shift than varaiations of bond lengths and bond angles caused by strain in the glass. Such strain is stronger in the glass with lower content of the network modiﬁer (Fig. 8b), which is expected to form a three-dimensional network consisting mainly of Q2 and Q3 groups. Connectivity between Q2 and Q3 groups is again manifest by crosspeaks 2 − 3 and 3 − 2, and again the dominating contribution to chemical shift variation appears to come from the type of neighboring Q(n) group. 5.2 Measurement of 1 H-1 H Distances in Bilirubin One of the principal limitations of x-ray crystallography lies in the diﬃculty to detect the position of hydrogen atoms, which is caused by the small electron density on these atoms. Hydrogen atoms involved only in conventional chemical bonds do not present too much of a problem, as their position can usually be predicted from the known positions of the heavier atoms by reyling on the known potentials for bond lengths, bond angles, and dihedral angles. However, the strength of hydrogen bonds and hence the position of hydrogen atoms involved in a hydrogen bond cannot be predicted easily and precisely with the currently available computational approaches. Furthermore, structures that are dominated or strongly inﬂuenced by hydrogen bonding often have to be studied in the solid state, as the hydrogen bonds are broken on dissolving the material. Recently, it has been demonstrated in a number of cases that hydrogen-bonded structures can be conveniently studied by DQ MAS NMR. As an example, consider the yellow-orange pigment bilirubin (for the structure, see Fig. 9e), which is a product in the metabolism of hemoglobin. Strong hydrogen bonding renders this compound insoluble under physiological conditions and thus unexcretable, unless it is enzymatically conjugated with glucoronic acid. This process is usually performed in the liver and its failure causes the yellow discolouration of the skin associated with hepatitis. The hydrogen bonds involve three protons in each of the two pseudo-symmetryrelated moieties of the molecules. The triangle made up by these protons is fully characterized by two distances and one angle, which can be determined from DQ MAS measurements [95]. As discussed in Sect. 4.5, intensities in the sideband pattern of such a three-spin system are dominated by the largest dipole-dipole coupling corresponding to the shortest proton-proton distance and modiﬁed by the perturbation due to the third proton. Hence, in the case at hand the shortest distance of 0.186 nm between the lactam and pyrrole NH protons can be determined most precisely (error of 0.002 nm), while the distance of 0.230 nm between the lactam NH and carboxylic acid OH protons is less certain (error of 0.008 nm). Likewise, the H-H-H angle of 122◦ is uncertain by 4◦ . Note however, that even the precision of these less well deﬁned values

G. Jeschke and H.W. Spiess

Fig. 9. 1 H (700.1 MHz) DQ MAS NMR on bilirubin. (a) Spinning- sideband pattern for a BABA excitation period 2trot . (b) Best-ﬁt simulation of the pattern in (a) assuming an isolated three- spin system. (c) Spinning-sideband pattern for a BABA excitation period 3trot . (d) Best-ﬁt simulation of the pattern in (c) assuming an isolated three-spin system. (e) Structure of bilirubin. Hydrogen bonds and proton chemical shifts (in ppm) are indicated in the left-hand moiety, distances and angles determined from the best ﬁts are indicated in the right-hand moiety. Adapted with permission from [95]

is still much better than the precision of proton positions derived from x-ray data. 5.3 Site-Selective Measurement of Distances Between Paramagnetic Centers Supramolecular assemblies that extend over several nanometers can be designed on the basis of metal ions, multidentate ligands, which provide a well deﬁned coordination geometry at the metal, and rigid spacers between these ligands [96]. The known structures of such assemblies have been derived by xray crystallography. However, crystallizing these materials may become more

Distance Measurements by NMR and EPR

and more diﬃcult if one increases the complexity of the structure. An alternative way of characterizing the assemblies in frozen solution would be the measurement of distances between the metal centers, between a metal center and selected sites on the linkers, or between two sites on the linkers. As typical distances between such sites exceed 2 nm, pulse EPR is the techniques of choice. This applies particularly to assemblies that contain paramagnetic transition metals such as copper(II). The potential of this kind of structure determination has been demonstrated on a model complex of copper(II) with two ligands, each of which consists of a terpyridine coordinating unit, a rigid spacer, and a nitroxide spin label as an endgroup (Fig. 10a) [97]. As the EPR spectra of the copper centre and the nitroxide labels overlap only slightly (Fig. 10b), it is possible to measure the end-to-centre and end-to-end distance separately by two DEER experiments with diﬀerent choices of the observer and pump frequencies. Pumping at position A in the nitroxide spectrum and observing at position B in the copper spectrum provides a dipolar evolution function that is solely due to the copper-nitroxide pair (Fig. 10c). The distance of nm obtained by ﬁtting this function is in nice agreement with the distance of 2.43 nm predicted by molecular modelling. Pumping again at position A but observing now at position C also in the nitroxide spectrum provides a dipolar evolution function that is solely due to the nitroxide- nitroxide pair (Fig. 10d). In this case the ﬁt value of 5.2 nm overestimates the expected distance by 0.34 nm. A more precise measurement of this long distance would require observation of the dipolar evolution function for a longer time, which is precluded here by enhanced relaxation of the nitroxide due to the nearby copper centre. However, the precision that could be achieved should be suﬃcient to elucidate the principal structure of a supramolecular assembly. 5.4 Averaging of Dipole-Dipole Interactions by Dynamics in the Discotic Phase of a Hexabenzocoronene Some materials are applied not in the solid state but as a liquid or in a liquid-crystalline state. In cases like this, a static structure is insuﬃcient to explain the properties. However, once a static structure at lower temperature is known, it is often possible to derive information on structural dynamics at higher temperatures by an analysis of motional averaging of the dipole-dipole interaction (see Sect. 2.5). The interest in polycyclic aromatic materials, which form liquid-crystalline columnar mesophases derives from their ability to form vectorial charge transport layers. Such transport layers may be suitable for applications in xerography, electrophotography, or in molecular electronic devices. Hexaalkylsubstituted hexa-peri -benzocoronenes (for a structure, see Fig. 11c) are a class of such materials with exceptionally high one-dimensional charge carrier mobility. The solid-state structure of the columns (stacks) of HBC-C12 molecules could be elucidated by analyzing ring-current eﬀects on the chemical shift of

G. Jeschke and H.W. Spiess

Fig. 10. Spectral selection in DEER measurements on a model compound for coordination polymers. (a) Structure of the model complex with end-to-centre and end-toend distances predicted by molecular modelling. (b) EPR spectrum of the complex and excitation positions for DEER. (c) DEER data obtained for observer (S spin) frequency B and pump (I spin) frequency A, corresponding to the copper-nitroxide distance. The ﬁt (dashed line) corresponds to a distance of 2.43 nm. (d) DEER data obtained for observer (S spin) frequency C and pump (I spin) frequency A, corresponding to the nitroxide-nitroxide distance. The ﬁt (dashed ) line corresponds to a distance of 5.20 nm

a given polycycle due to the adjacent polycycles in the stack. Furthermore, spatial proximities of certain types of aromatic protons were determined from cross-peaks in two-dimensional DQ MAS NMR spectra [98] in the same way as discussed above for the phosphate glasses. The structure of the columns is characterized by a herring-bone packing as it was also found in an x-ray structure of unsubstituted hexabenzocorenene. This regular packing is lost in the discotic phase as the polycycles begin to rotate independently about an axis parallel to the long axis of the stack and perpendicular to the aromatic plane. This leads to reorientation of the spin-to-spin vector in the proton pairs marked by ellipses Fig. 11c. As a result, the dipole-dipole interaction is partially averaged. By analysing the DQ MAS NMR sideband pattern, a dipole-dipole coupling of 15.0 kHz is found for these pairs at a temperature of 333 K in the solid state. In the liquid crystalline state (T = 386 K), the same analysis yields a coupling of only 6 kHz, corresponding to a reduction by a factor of 0.4. Fast axial rotation about an axis perpendicular to the polycycle and passing through its centre of symmetry would result in a reduction by a factor of 0.5. This indicates that there is either a signiﬁcant out-of-plane motion of

Distance Measurements by NMR and EPR

Fig. 11. Motional averaging of the proton-proton dipole-dipole coupling in the hexabenzocoronene HBC-C12 . (a) DQ MAS NMR sideband pattern of a proton pair in the solid state (333 K) measured at an MAS frequency of 35 kHz. (b) Sideband pattern measured in the discotic phase (386 K) at an MAS frequency of 10 kHz. (c) Structure of HBC-C12 . Isolated proton pairs are marked by dashed ellipses. Reproduced with permission from [98]

the C-H bonds for these protons or, maybe more likely, an out-of-plane motion of the polycycles themselves. 5.5 Structure Determination of a Peptide by Solid-State NMR Full structure determination of proteins or peptides by solid-state NMR relies on sequence-speciﬁc assignment of backbone carbon and nitrogen resonances in 13 C and 15 N NMR spectra. Furthermore, distances have to be measured for a suﬃcient number of pairs of backbone nuclei and sidechain nuclei. The number of required distance constraints can be drastically reduced if also torsion angles can be determined. By correlating spectra of two dipole-dipole coupled spin pairs in a three-dimensional NMR experiment, the orientation dependence of the dipole-dipole coupling can be utilized to obtain such constraints [99]. This method is based on the fact that the bond length and hence the magnitude of the dipole-dipole coupling for pairs of directly bonded nuclei are known. In a three-dimensional experiment with two chemical shift dimensions and one dimension corresponding to dipolar evolution, the frequency in the dipolar dimension thus depends on the relative orientation of the two dipolar tensors. For sensitivity reasons, application of such approaches to proteins requires 13 C and 15 N isotope labelling. In many model studies, selective labelling has been used for the distance measurements, so that the approximation of isolated spin pairs could be applied. However, broad application of such methodology may depend on techniques that can solve the structure of uniformly

G. Jeschke and H.W. Spiess

labelled proteins, as this signiﬁcantly reduces the eﬀort required in the labelling process. For the chemotactic peptide N -formyl-L-Met–L- Leu–L-Phe-OH a full structure determination based on uniform isotope labelling was demonstrated by applying a frequency-selective version of the REDOR experiment [99]. In this frequency-selective REDOR technique, broadband recoupling is combined with chemical shift refocusing by weak Gaussian-shaped pulses that are resonant only with one 13 C and one 15 N nucleus [100]. As in site-selective DEER it is thus possible to obtain dipolar evolution functions that are solely due to one selected spin pair (Fig. 12b-d). With this technique, 16 long-range 13 C-15 N-distances between 0.3 and 0.6 nm were measured. Furthermore, 18 torsion angle constraints on 10 angles could be obtained with four diﬀerent three-dimensional experiments. Sequence-speciﬁc shift assignment was achieved from a three-dimensional shift-correlation experiment that is also based on spatial proximity. The complete data set allowed for constructing a structural model of the peptide (Fig. 12a) by simulated annealing techniques or full search of the conformational space. In the latter procedure, conforma-

Fig. 12. Frequency-selective REDOR measurements of carbon-nitrogen distances in a uniformly 13 C-15 N-labelled peptide diluted into the unlabelled peptide. (a) Structural model of the peptide. (b-d) Experimental REDOR curves for selected spin pairs together with ﬁts and ﬁt residuals. Reproduced with permission from [99]

Distance Measurements by NMR and EPR

tional space is factored into subspaces and then reduced by excluding all those subspaces in which at least one experimental constraint is violated. In the case at hand, the structure could be fully determined except for the orientations of the phenyl group and of the C terminus, for which no constraints were available. 5.6 Constraints on Long Distances in a Protein by EPR Structure determination by high-resolution NMR is restricted to soluble proteins, while structure determination by x-ray crystallography requires that the protein can be crystallized. For most membrane proteins, which are notoriously hard to crystallize and may not fold into their functional structures in solution, neither of the two approaches is applicable. Solid-state NMR spectroscopy as well as EPR spectroscopy on singly and doubly spin-labeled mutants can provide at least partial information on structure and structural dynamics for this important class of proteins. Determining the fold of a protein or at least recognizing that a protein belongs to a known class of folds may be possible even if only a few distances can be measured. For this purpose, long distances exceeding 2 nm provide particularly valuable constraints as they contain information on the relative arrangement of secondary structure elements such as α-helices and β-sheets. Pulse EPR methods such as the DEER experiment or DQ EPR can be used to measure distances between spin labels in the range between 2 and 5 nm, and in favourable cases up to 8 nm [11, 17, 64, 79]. That such methods can be applied to protein structure determination in the solid state has been demonstrated on the soluble protein T4 lysozyme consisting of 164 amino acid residues [101]. Distances ranging between 2.1 and 4.7 nm could be measured by DQ EPR for eight selected pairs of spin-labelled residues in shock-frozen solutions of this protein. Experimental data for three pairs are shown in the left column of Fig. 13a together with ﬁts by distance distribution consisting of a single Gaussian peak. Broadening due to a distribution of distances with a typical widths of 0.2 nm is also apparent in the dipolar spectra shown in the left column of this ﬁgure. The determined distances are by 0.3–1.0 nm larger than the distances between the respective α- or β-protons of the residues in the crystal structure. Without relying on this structure, it is possible to construct a rough three-dimensional model from the measured distances by the triangulation approach that is illustrated in Fig. 13b. It turned out that the number of eight distance constraints was too small to fully specify the relative positions of all the residues which had been spin-labelled. However, as is also shown in Fig. 13b a suggestion for an additional label site could be derived which should complete the triangulation.

G. Jeschke and H.W. Spiess

Fig. 13. DQ EPR distance measurements on spin-labelled double mutants of T4 lysozyme. (a) DQ coherence temporal envelopes and their ﬁts (dashed lines) for three double mutants (left column) and dipolar spectra obtained by Fourier transformation (right column). Simulations of dipolar spectra for double mutants 65/86 und 61/135 are shown as dashed lines, the simulation for double mutant 65/135 (ii) is shifted downward with respect to the experimental spectrum (i). (b) Experimental distances for several double mutants showning “triangulation” in progress. Grey spheres correspond to average widths of distributions obtained by ﬁtting the experimental data. The position of residue 86 cannot be ﬁxed for lack of a suﬃcient number of distance constraints. The black sphere depicts an additional label site which would complete the triangulation. Reproduced with permission from [101]

6 Conclusion Precision and sensitivity of distance measurements by magnetic resonance methods depend substantially on the choice of technique, in particular, on the elimination of signal contributions due to other interactions and due to relaxation. Adapting spectral selectivity of the measurement to the problem at hand is also important. Generally, better deﬁned structures are studied with higher site selectivity than less well deﬁned structures. Both precision and sensitivity may also be strongly inﬂuenced by the data analysis procedures used in the interpretation of experimental raw data. Consideration of experimental imperfections, in particular of imperfect suppression of other interactions or of the unavoidable inﬂuence of isotropic spin-spin couplings (J couplings) may be crucial. It should be borne in mind that deriving spin-spin pair correlation functions from dipolar evolution functions is an ill-posed mathematical problem, so that noise may inﬂuence the results in a diﬀerent way than in the more familiar Fourier transformation techniques. Choosing a numerically stable procedure and cross-checking the results by model computations is necessary to ensure reliability of the results [88]. In general, the reliability of structural models derived from NMR and EPR distance measurements can

Distance Measurements by NMR and EPR

be better estimated if they are compared to the results of molecular modelling techniques. NMR crystallography is based on such an approach [102]. Quantum-chemical computation of magnetic resonance parameters may help to resolve ambiguities and keep the number of adjustable parameters and their ranges in the ﬁtting process to an absolute minimum. A study of a solid-state structure by magnetic resonance techniques thus amounts to a complex task, for which no simple set of rules can be given. This is not expected to change in the future, as the complexity of the process derives from the complexity of the structures that are studied by magnetic resonance techniques. It is exactly the strong variability of magnetic resonance experiments and the possibility to adapt them to the problem under investigation that allows for a study of such structures. The question which technique is optimum for given types of spins thus cannot be answered in general. The history of method development and subsequent application or non-application of the methods suggests that Einstein’s dictum is valid also in this ﬁeld: Make things as simple as possible – but no simpler.

Acknowledgment We thank I. Schnell for helpful discussions. Financial support from the Deutsche Forschungsgemeinschaft is gratefully acknowledged.

A Appendix BABA back-to-back: an NMR recoupling pulse sequence for broadband excitation of multiple-quantum coherences during fast MAS DEER d ouble electron electron r esonance: acronym for a spin-echo double resonance experiment in EPR spectroscopy; used synonymously with PELDOR DQ d ouble-quantum: designates transitions of spin S that involve a change ∆mS = 2; build-up of coherence on such transitions depends on the dipoledipole coupling DRAMA d ipolar r ecovery at the magic angle: an NMR recoupling pulse sequence that reintroduces hom*onuclear dipole-dipole coupling during MAS ELDOR el ectron electron double r esonance: a collective term including continuous-wave and pulse EPR experiments in which two microwave frequencies or a magnetic ﬁeld step are applied to obtain information on dynamics or on couplings between electron spins ENDOR electron nuclear double resonance: indirect detection of the NMR spectrum of nuclei that are hyperﬁne coupled to an electron spin by observation on electron spin transitions to increase sensitivity and bandwidth ESEEM electron spin echo envelope modulation: a modulation in the decay of the primary (Hahn) or stimulated echo that is caused by coherence

G. Jeschke and H.W. Spiess

transfer echoes; can be used for indirect detection of an NMR spectrum of hyperﬁne coupled nuclei if the normally forbidden electron-nuclear zeroand double-quantum transitions are slightly allowed INADEQUATE i ncredible natural abundance d ouble quantum transfer experiment: signals from pairs of rare S spins are selectively detected in the presence of much more abundant isolated S spins by applying a double-quantum ﬁlter MAS magic angle spinning: fast rotation of the sample about an axis that includes an angle of 54.74◦ with the magnetic ﬁeld axis; increases resolution if broadening in the spectra is caused by anisotropy of interactions PELDOR pulse ELDOR: an abbreviation used mostly for spin echo double resonance experiments in EPR; synonymous with DEER PISEMA polarization i nversion exchange at the magic angle: an NMR double resonance technique in which I spin magnetization is spin-locked at an angle of 54.74◦ with respect to the magnetic ﬁeld axis and inverted halfway through the evolution period; polarization transfer between I and S spins driven by the heteronuclear dipole-dipole couplings can then be observed over a longer time, thus enhancing resolution REAPDOR r otational echo adiabatic passage double r esonance: broadband recoupling technique for heteronuclear couplings of spins I > 1/2 that relies on long pulses and on the change of the resonance frequency during MAS REDOR r otational echo double r esonance: recoupling of heteronuclear dipole-dipole coupling during MAS by applying 180◦ pulses to one of the spins, so that rotational averaging is disturbed SEDOR spin echo double r esonance: the heteronuclear dipole-dipole coupling is reintroduced into the nuclear primary (Hahn) echo decay of S spins by applying an additional 180◦ pulse to the I spins SIFTER si ngle f requency technique for r efocusing: by using a combination of primary (Hahn) echo and solid-echo refocusing, the coupling of like spins is separated from Zeeman anisotropy and hyperﬁne couplings

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75. R. Graf, D.E. Demco, J. Gottwald, S. Hafner, H.W. Spiess: J. Chem. Phys. 106, 885 (1997) 76. D.E. Demco, S. Hafner, H.W. Spiess: J. Magn. Reson. A 116, 36 (1995) 77. M. Feike, D.E. Demco, R. Graf, J. Gottwald, S. Hafner, H.W. Spiess: J. Magn. Reson. A 122, 214 (1996) 78. P.P. Borbat, J.H. Freed: Chem. Phys. Lett. 313, 145 (1999) 79. P.P. Borbat, J.H. Freed: ‘Double-Quantum ESR and Distance Measurements’. In: Biological Magnetic Resonance, Vol. 19, ed. by L.J. Berliner, G.R. Eaton, S.S. Eaton (Kluwer, New York 2000) pp. 383–459 80. J.G. Powles, P. Mansﬁeld: Phys. Lett. 2, 58 (1962) 81. G. Jeschke, M. Pannier, A. Godt, H.W. Spiess: Chem. Phys. Lett. 331, 243 (2000) 82. G. Jeschke, A. Godt: ChemPhysChem 4, 100 (2003) 83. J.H. van Vleck: Phys. Rev. 74, 1168 (1948) 84. C.P. Slichter: Principles of Magnetic Resonance (Springer, Berlin 1990) 85. G. Jeschke, A. Koch, U. Jonas, A. Godt: J. Magn. Reson. 155, 72 (2001) 86. J. Weese: Comput. Phys. Commun. 69, 99 (1992) 87. F.G. Vogt, D.J. Aurentz, K.T. Mueller: Mol. Phys. 95, 907 (1998) 88. G. Jeschke, G. Panek, A. Godt, A. Bender, H. Paulsen: Appl. Magn. Reson. 26, 223 (2004) 89. W.B. Mims: Phys. Rev. B 5, 2409 (1972) 90. K. Saalw¨ achter, H.W. Spiess: J. Chem. Phys. 114, 5707 (2001) 91. H.W. Spiess: ‘Double-Quantum NMR Spectroscopy of Dipolar Coupled Spins Under Fast Magic-Angle Spinning’. In: Encyclopedia of Nuclear Magnetic Resonance, Vol. 9, ed. by D.M. Grant, R.K. Harris (John Wiley & Sons, Chichester 2002) pp. 44-58 92. C. Filip, X. Filip, D.E. Demco, S. Hafner: Mol. Phys. 92, 757 (1997) 93. C. Filip, S. Hafner, I. Schnell, D.E. Demco, H.W. Spiess: J. Chem. Phys. 110, 423 (1999) 94. M. Feike, C. J¨ ager, H.W. Spiess: J. Non-Cryst. Solids. 223, 200 (1998) 95. S.P. Brown, X.X. Zhu, K. Saalw¨ achter, H.W. Spiess: J. Am. Chem. Soc. 123, 4275 (2001) 96. J.M. Lehn: Angew. Chem. Int. Ed. Engl. 29, 1304 (1990) 97. E. Narr, A. Godt, G. Jeschke: Angew. Chem. Int. Ed. 41, 3907 (2002) 98. S.P. Brown, I. Schnell, J.D. Brand, K. M¨ ullen, H.W. Spiess: J. Am. Chem. Soc. 121, 6712 (1999) 99. C.M. Rienstra, L. Tucker-Kellog, C.P. Jaroniec, M. Hohwy, B. Reif, M.T. McMahon, B. Tidor, T. Lozano-P´erez, R.G. Griﬃn: Proc. Natl. Acad. Sci. USA 99, 10260 (2002) 100. C.P. Jaroniec, B.A. Tounge, J. Herzfeld, R.G. Griﬃn: J. Am. Chem. Soc. 123, 3507 (2001) 101. P.P. Borbat, H.S. Mchaourab, J.H. Freed: J. Am. Chem. Soc. 124 5304 (2002) 102. F. Taulelle: Curr. Opin. Sol. State Mater. Sci. 5 397 (2001)

NMR Studies of Disordered Solids J. Villanueva-Garibay and K. M¨ uller Institut f¨ ur Physikalische Chemie, Universit¨ at Stuttgart, Pfaﬀenwaldring 55 70569 Stuttgart, Germany [emailprotected]

Abstract. In this contribution an introduction to dynamic solid state NMR spectroscopy is presented. The main emphasis is given to dynamic 2 H NMR techniques, since these methods – in combination with selectively or partially deuterated compounds – have demonstrated a particular suitability for studying the molecular properties (i.e. order and dynamics) of solid, semisolid materials as well as anisotropic liquids. A general overview about the theoretical background of dynamic NMR spectroscopy is provided in the ﬁrst part, which also includes the description of the main experimental methods in dynamic 2 H NMR spectroscopy. In the second part representative results from model simulations are given, considering various types of motional processes which are frequently discussed in disordered materials. Applications of dynamic 2 H NMR techniques during the study of inclusion compounds are shown in the last section.

1 Introduction Dynamic NMR spectroscopy is a well established technique for the evaluation of molecular dynamics in condensed media. Apart from the frequent application of such techniques in the ﬁeld of high-resolution (liquid) NMR spectroscopy [1, 2], dynamic NMR methods were also applied successfully on quite diﬀerent types of anisotropic materials [3], such as polymers [4, 5], thermotropic liquid crystals [6, 7], (lyotropic) lipid bilayers and biological membranes [8, 9], guest-host systems (clathrates, zeolites) [10], molecular and plastic crystals [11], etc. These latter studies have clearly demonstrated that dynamic solid state NMR spectroscopy represents a powerful method for the determination of the dynamic and structural features of even complex systems. Dynamic NMR spectroscopy thus can be used to probe the ordering characteristics in terms of conformational, orientational and positional order. Likewise, a comprehensive analysis of such experiments gives access to the inherent motional contributions, comprising conformational, reorientational and lateral motions. In favourable cases, dynamic NMR spectroscopy is able J. Villanueva-Garibay and K. M¨ uller: NMR Studies of Disordered Solids, Lect. Notes Phys. 684, 65–86 (2006) c Springer-Verlag Berlin Heidelberg 2006 www.springerlink.com

J. Villanueva-Garibay and K. M¨ uller

to follow such motions over a very broad time-scale, ranging from the sub-kHz to the GHz region [12, 13]. In this contribution we will provide a brief introduction to dynamic solid state NMR methods. In particular, we will focus on dynamic 2 H NMR techniques, since these methods in combination with selectively or partially deuterated compounds have demonstrated their particular suitability for studying the aforementioned solid and semisolid materials as well as anisotropic liquids [13, 14, 15]. In the ﬁrst part of this contribution we brieﬂy describe the theoretical background of dynamic NMR which also includes the description of the main experimental methods. In the second part some representative model simulations are provided. Results from the application of dynamic NMR techniques during the study of guest-host systems are shown in the last section.

2 Theoretical Background 2.1 General Theory The description of NMR experiments in general is done by considering the time evolution of the spin-density operator ρ(t). In the absence of molecular motions the time evolution of the spin-density operator ρ(t) is given by the Liouville-von Neumann equation [16] i dρ(t) = [ρ(t), H] . dt

(1)

H is the time-independent Hamiltonian of dimension n (n: dimension of Hilbert space) which includes various terms for magnetic interactions of the nuclei with their local surrounding as well as terms for the r.f. pulses H = Hrf + HCS + HD + HQ + . . . .

(2)

The formal solution of (1) is given by ρ(t) = U (t) ρ(0) U (t)−1 ,

(3)

where the propagator U (t) is deﬁned as U (t) = e−(i/)Ht .

(4)

The NMR experiment is furthermore subdivided in time intervals τ1 , τ2 . . . τn that possess a constant Hamiltonian, e.g. with and without r.f. pulses and/or particular magnetic interactions. The density matrix operator ρ(t) at a particular time t can then be easily calculated via the summation of the intervals with constant Hamiltonian [11] (5) ρ(t) = ρ τi .

NMR Studies of Disordered Solids

In the presence of molecular motions the starting point is the Stochastic Liouville equation, which in general is given by [16, 17] dρ dρ(t) i = [ρ(t), H] + . (6) dt dt dyn Here, the second term on the right side accounts for the contribution due to dynamic processes, such as molecular motion or chemical exchange. Equation (6) can be solved after rewriting in the form i dρ(t) = L ρ(t) dt

(7)

to yield ρ(t) [11] i

ρ(t) = ρ(0) e Lt .

(8)

As before, the full time evolution of the density matrix is then described by dividing the experiment in intervals with constant Hamiltonian. The Liouville superoperator L in (7) and (8) is derived from the Hamiltonian by the prescription [12] (9) L = H ⊗ E − E ⊗ H , where E is the identity operator in Hilbert space. The matrix L is further expanded by the part that accounts for molecular motion to give the matrix L. The ﬁnal dimension of L is given to n2 N , where N is the number of exchanging sites [11, 18]. In the most general case the following terms are included in the spin Hamiltonian [19] (10) H = Hrf + HCS + HD + HQ . They refer to the radio frequency part Hrf i Hrf = ωrf (t) (Iix cos Φi + Iiy sin Φi )

(11)

and the contributions from several magnetic interactions, namely (i) chemical shift interaction (HCS ) i ωCS,0 (t) Iiz , HCS =

(12)

(ii) dipole-dipole interaction (HD ) between spins i and k HD =

i,k

1 i ωD,0 (t) √ (3Iiz Ikz − I i · I k ) , 6

(13)

J. Villanueva-Garibay and K. M¨ uller

(iii) and quadrupolar interaction (HQ ) with ﬁrst and second order contributions 1 i ωQ,0 (t) √ 3Iiz 2 − I 2i HQ = 6 i (14) 2 i i 2 ωQ,−2 (t) ωQ,+2 (t) 2I i − 2Iiz − 1 Iiz + 1 . + i i 2 − 1) I 2ω0i ωQ,−1 (t) ωQ,+1 (t) (4I 2i − 8Iiz iz In (11) to (14) ω0 and ωα,i denote the Larmor frequency and the corresponding interaction constants, which can be found elsewhere [19]. The angular dependence (anisotropy) of the various magnetic interactions is obtained by the transformation of the respective magnetic interaction from its own principle axis system (PAS) into the laboratory (LAB) frame using second-rank rotation matrices [20]. In the presence of molecular motions a minimum of two transformations is required, namely (i) from the PAS to an intermediate axis system (IAS), deﬁned by the symmetry axis of the motional process, and from the IAS to the LAB frame. If several motions are superimposed, additional transformations (through intermediate axis systems IAS-1 to IAS-n , n = number of superimposed motions) are required [11, 14, 21]. The spin-part of the Hamiltonian, i.e., the spin operators, is given in its matrix representation. For a general spin system the basis of the Hamiltonian is obtained via the direct product of the relevant single spin operators Iiα (i = 1 . . . n, α = x, y, z), e.g. I1x I2y = Ix ⊗ Iy ⊗ 1 ⊗ . . . ⊗ 1(n) ; I3x = 1 ⊗ 1 ⊗ Ix ⊗ . . . ⊗ 1(n) ; etc. (15) 1 is the single spin unity operator. The single spin operators are obtained by the relations I, m|Iz |I, m = m , (16) I, m ± 1|I± |I, m = I(I + 1) − m(m ± 1) . (17) On this basis, the time evolution of the density matrix, as given by (3) or (8), can then be calculated. In the most general case this requires diagonalization of the spin Hamiltonian or the matrix L within the various time intervals. The dimension of these matrices is given by n or by n2 N in the absence and presence of molecular motions, respectively [11, 18]. 2.2 Dynamic 2 H NMR Spectroscopy In the following we will restrict ourselves to the case of dynamic 2 H NMR spectroscopy on static samples, i.e., broadline NMR conditions. In 2 H NMR spectroscopy we have a very particular situation, since the spin Hamiltonian is dominated by the quadrupolar interaction with a coupling constant ωQ /2π = e2 qQ/h between about 165 kHz (aliphatic deuterons) and 185 kHz

NMR Studies of Disordered Solids

Fig. 1. Sensitive time-scales of diﬀerent types of dynamic NMR experiments

(aromatic deuterons). In dynamic NMR spectroscopy various sensitive timescales (see Fig. 1) and NMR experiments can be distinguished, where – instead of the most general formalism (see Sect. 2.1) – suitable approaches can be employed for the theoretical description of these experiments. In the following we thus distinguish among (i) the fast (rate constant k ∼ ω0 , k ωQ ), (ii) intermediate (k ∼ ωQ ), and (iii) ultraslow motional region (k ωQ ) [13, 22]. From the experimental point of view (see Fig. 2), the quadrupole echo sequence is used for the detection of 2 H NMR line shapes and spin-spin (T2 ) relaxation. The time-scale of such NMR line shape studies and T2 eﬀects is given by the strength of the (motionally) modulated magnetic interaction – here the quadrupolar interaction – which is in the MHz range [11, 22, 23, 24, 25, 26]. Spin-lattice (T1 ) relaxation is probed with a modiﬁed inversion recovery experiment from which fast molecular motions in the vicinity of the Larmor frequency, i.e., in the MHz to GHz-region, are accessible [11, 14, 24]. Finally, 2D exchange and related NMR experiments can be used to study ultra-slow motional processes in the Hz- and sub-Hz range. Here, the accessible

Fig. 2. Basic pulse experiments in dynamic 2 H NMR spectroscopy, top: quadrupole echo sequence, middle: inversion recovery sequence, bottom: 2D exchange sequence

J. Villanueva-Garibay and K. M¨ uller

time-scale is determined by the length of the exchange interval τm , and is limited by spin-lattice (lower limit) and spin-spin relaxation (upper limit) [12, 13, 27]. 2

H NMR Spectra in the Rigid Limit

If the motional processes are slow on the NMR time-scale, and the sample refers to a polycrystalline material, then the observed 2 H NMR powder spectrum (“Pake” pattern) is the sum over all (static) orientations of the crystallites with respect to the external magnetic ﬁeld (see Fig. 3). Thus, the powder spectrum is the weighted sum of individual pairs of lines whose frequencies ωq are given by [13] 3 ωq = ± ωQ 3 cos2 θ − 1 + η sin2 θ cos 2φ . 8

(18)

φ and θ are the spherical polar angles which specify each crystallite orientation, and η is the asymmetry parameter (which for aliphatic deuterons normally is close to 0). The powder spectrum is obtained via Fourier transformation of the free induction decay (FID) signal

sin θdθ

S(t) = N 0

2π

" ! dφ e−iωq (φ,θ)t ,

(19)

where N is a normalization constant. It should be noted that the singularities in the 2 H NMR powder spectra of rigid samples directly reﬂect the three main AS ) in its principle axis components of the quadrupolar interaction tensor (QP ii system (see Fig. 3) [4, 22, 24]. For the case of an axially symmetric quadrupolar interaction tensor (η = 0), a splitting of 3/4 ωQ between the perpendicular singularities is registered. 2

H NMR Spectra in the Fast Exchange Region

If the molecular motions are in the fast exchange region, then the inspection of the experimental line shapes already gives an indication about the symmetry (or type) of the underlying motional process (see Fig. 4). In fact, the description of the fast exchange NMR line shapes does not require a complex line shape simulation. Rather, it is just necessary to transform the quadrupolar interaction tensor QP AS from its principle axis system to the IAS coordinate system, which is deﬁned by the particular motional process (i.e., motional symmetry axis), using appropriate transformation matrices R(α, β, γ) (α, β, γ = Euler angles specifying the coordinate transformation) QIAS (αβγ) = R(αβγ)QP AS R−1 (αβγ) with

(20)

NMR Studies of Disordered Solids

Fig. 3. Relationship between crystal orientation and 2 H NMR powder pattern (upper spectrum: η = 0; lower spectrum: η = 0)

QP AS

  1+η 0 0 3 e2 qQ  0 1−η 0  . = 4 h 0 0 −2

(21)

The averaged tensor components are then calculated by using the equilibrium population of the relevant molecular orientations Peq (αk , βk , γk ) which are necessary for the particular motion under consideration. ¯ ij = Q Peq (αk , βk , γk )QIAS (22) ij (αk , βk , γk ) . k

J. Villanueva-Garibay and K. M¨ uller

Fig. 4. Theoretical fast exchange 2 H NMR spectra based on diﬀerent motional models

The fast exchange 2 H NMR line shapes are determined by the residual prin¯ P AS . These quantities, which again can be taken ciple tensor components Q ii directly from the spectral singularities, are obtained via diagonalization of the averaged tensor matrix, according to [24] P AS ¯ ¯ (23) Qij − δij Qij = 0 . j The procedure for the analysis of the fast exchange spectra is thus very simple. The most convenient way is the implementation of the above procedure into standard mathematical software packages [28], such as MathcadTM [29], MatlabTM [30], etc. 2

H NMR Line Shape and T2 Eﬀects

If the molecular motions occur in the intermediate time-scale (k ∼ ωQ ), then H NMR line shapes from the quadrupolar echo experiment can be adequately described via (24), where inﬁnitesimally sharp δ-pulses are assumed, and ﬁnite pulse eﬀects thus are neglected [22, 23, 24, 25, 26]. 2

∗

S(t, 2τe ) = 1 eAt eAτe eAτe Peq (0) .

(24)

Here, the evolution of the magnetization (i.e., spin-spin relaxation) is explicitly considered during the intervals τe between the r.f. pulses. S(t, 2τe ) is the FID starting at the top of the quadrupole echo, and the vector Peq (0) denotes the fractional populations of the N exchanging sites in thermal equilibrium. A is a complex matrix of size N with A = iΩ + K .

(25)

NMR Studies of Disordered Solids

The imaginary part of A is given by the diagonal matrix Ω whose elements ωii (= ωq,i ) describe the frequencies of the exchanging sites. The real part corresponds to a kinetic matrix K. Here, the non-diagonal elements kij are the jump rates from site j to site i, while the diagonal elements kii represent the sums of the jump rates for leaving site i; they also contain the residual line widths in terms of 1/T20 .   ωq,1   ωq,2   Ω= ; ..   . ωq,N    K= 

−

ki1 + k21 .. . kN 1

0 1/T2,1

−

k12 0 k + 1/T2,2 i2 i .. . kN 2

... ... .. . ... −

k1N k2N .. .



(26)

  . 

0 kiN + 1/T2,N

It should be noted that (26) also accounts for the general case that – depending on the complexity of the system studied – several (superimposed) internal and intermolecular processes might be present at the same time. Equation (24) is solved numerically using standard diagonalization routines, from which 2 H NMR line shapes, partially relaxed spectra and spin-spin relaxation times T2 are derived. The inﬂuence of ﬁnite pulse eﬀects can be taken into account after calculation of the FID by making use of analytical expressions, as shown in [31]. 2

H NMR Line Shape and T1 Eﬀects

The simulation of partially relaxed 2 H NMR spectra from the modiﬁed inversion recovery experiment is feasible with the help of (27) [24] S(t, 2τe , τr ) = 1 − 2 e−τr /T1 S(t, 2τe ) . (27) S(t, 2τe ) is the FID signal obtained by (24) or by tensor averaging in the fast motional limit, as also described earlier. The delay τr refers to the relaxation interval between the inversion pulse and the quadrupole echo sequence used for signal detection. In order to calculate spin-lattice relaxation eﬀects, second order perturbation theory [32, 33, 34] is employed which, however, is not only restricted to the fast motional limit. Rather, the condition must hold that the spin-lattice relaxation rate is slower than the motional rate k responsible for spin relaxation (1/T1 k) [11]. Again, if only the quadrupolar interaction determines spin-lattice relaxation, then the 2 H spin-lattice relaxation time T1 is given by

J. Villanueva-Garibay and K. M¨ uller

1 3 = T1 16

e2 qQ h

2 [J1 (ω) + 4J2 (2ω)] .

(28)

The spectral densities Jm can be derived by solving the following equation for a general N-site exchange [35, 36] Jm (ω) = 2

d(2) ma (θ )dma (θ ) (2)

a,a =−2

Xl Xl Xj Xj d0a (θl )d0a (θj ) (0)

(n)

(0)

(n) (2)

(2)

n,l,j=1

λn × cos(aφl − a φj ) 2 λn + ω 2 with φi = φi − φ .

(29) (30)

Here, X (n) and λn are the corresponding eigenvectors and eigenvalues of the symmetrized rate matrix K (see (27) without 1/T20 terms). The angles θ and φ are the spherical polar angles between the PAS and an IAS (determined by the motional process), whereas the angles θ and φ connect the IAS and (2) the LAB system. dab (θ) are elements of the reduced Wigner rotation matrix. If there is a superposition of several motional modes, then the transformation from the PAS to the LAB frame is subdivided into several steps according to the number of motional contributions (see above). 2D Exchange 2 H NMR Spectra In order to describe the 2D exchange NMR experiments for the detection of ultraslow motions, motional eﬀects normally are only considered during the exchange interval τm . With the assumption of inﬁnitesimally sharp r.f. pulses, the quadrupolar order (SQ ) and Zeeman order (SZ ) signals are calculated using the following equations [13, 27, 37] j i sin(ωqi t1 ) e−t1 /T2 Pij (τm ) sin(ωqj t2 ) e−t2 /T2 , SQ (t1 , t2 ; τm ) = C ij

SZ (t1 , t2 ; τm ) = C

cos(ωqi t1 ) e−t1 /T2 Pij (τm ) cos(ωqj t2 ) e−t2 /T2 (31) i

with Pij (τm ) = Pi (0) eKτm ij . Pij and Pi (0) denote the conditional probability that a nucleus jumps from site j to site i during τm and the equilibrium population of site i, respectively. K is the exchange matrix, already introduced above (26). In (31) spin-lattice relaxation contributions during τm were neglected. The processing of the data sets Sz (t1 , t2 ; τm ) and SQ (t1 , t2 ; τm ) in order to obtain the pure absorption 2D exchange NMR spectrum is described elsewhere [27, 38].

NMR Studies of Disordered Solids

3 Simulation Programs The theoretical NMR spectra and relaxation eﬀects for I = 1 spin systems were obtained by employing appropriate FORTRAN programs [39, 40, 41] that are based on the theoretical approaches and assumptions, as outlined in Sect. 2.2. In general, a numerical diagonalization is required in order to calculate the theoretical line shapes and relaxation times, which is achieved by employing appropriate routines [42]. All simulations were performed on personal computers (Windows and LINUX platforms) or SUN workstations (UNIX platform) [43]. The parameters which enter in the simulation programs are the quadrupolar coupling constant, the transformation angles between the PAS of the quadrupolar interaction, i.e., the C-2 H bond direction, and an internal coordinate system that is deﬁned by the motional process. In the case of superimposed motions further transformation angles are necessary. Furthermore, pulse intervals, motional correlation times, equilibrium populations of the various jump sites are required along with a residual line width, the sweep width, the number of acquired data points, and the number of crystallite orientations in order to calculate the NMR powder spectrum.

4 Model Simulations In the following, a few representative results from model simulations are provided that demonstrate the impact of various molecular or simulation parameters on the 2 H NMR line shapes and relaxation data. It should be emphasized that these examples are closely related to cases that are frequently encountered during the study of various types of disordered solids. To begin with, we recall the eﬀect of diﬀerent molecular motions on the fast exchange 2 H NMR line shapes, which is demonstrated in Fig. 4. As can be seen, diﬀerent types of overall motions (tetrahedral jumps, methyl group rotation, 180◦ ﬂips) give rise to quite diﬀerent fast exchange line shapes, which at the same time – as outlined earlier – reﬂect the symmetry of the underlying molecular motions. Model calculations showing the inﬂuence of motional processes in the intermediate time-scale are given in Figs. 5 to 7. The two series of 2 H NMR line shapes in Fig. 5 were obtained with the assumption of a 3-fold jump motion around a motional symmetry axis which is perpendicular to the C-2 H bond direction. As can be seen, the variation of the equilibrium populations of the jump sites p1 , p2 and p3 (with p1 = 1 − p2 − p3 ; p2 = p3 ) has a signiﬁcant inﬂuence on such 2 H NMR line shapes. Likewise, Fig. 6 depicts theoretical 2 H NMR line shapes that are obtained by considering two superimposed overall rotations – modelled by degenerate 3-site jump motions (i.e., with equally populated jump sites) – about two motional symmetry axes that are oriented perpendicular on each other. It is quite obvious that the actual rate constants of both motions have a considerable

J. Villanueva-Garibay and K. M¨ uller

Fig. 5. Theoretical 2 H NMR line shapes (quadrupole echo, τe = 20 µs) for a nondegenerate 3-fold jump process with diﬀerent populations p1 , as indicated, and at correlation times of 3 × 10−7 s (left) and 1 × 10−6 s (right). C-2 H bonds are oriented perpendicular with respect to motional symmetry axis

impact on the overall appearance of these 2 H NMR spectra. At the same time, it is found that the relative orientation of the two motional symmetry axes as well as the equilibrium populations of the jump sites also play a signiﬁcant role on such NMR line shapes (spectra not shown). That is, both the types of motion and the motional correlation times can be determined with a high precision, since the change of these molecular quantities is directly reﬂected by the alterations in the 2 H NMR spectra. In Fig. 7 three series of partially relaxed 2 H NMR spectra – calculated for the quadrupole echo sequence – are shown. These spectra were obtained on the basis of a degenerate 3-site jump motion, where three diﬀerent angles between the motional symmetry axis and the C-2 H bond direction have been chosen. Here, the observed changes in the partially relaxed 2 H NMR spectra, as a function of the pulse spacing τe , are a direct measure of the angular dependence of T2 (or T2 anisotropy) [11, 22], which strongly depends on the particular model assumptions. The present examples clearly demonstrate the inﬂuence of diﬀerent opening angles between the motional symmetry axis

NMR Studies of Disordered Solids

Fig. 6. Theoretical 2 H NMR line shapes (quadrupole echo, τe = 20 µs) for two degenerate 3-fold jump processes (both motional symmetry axes are perpendicular on each other, C-2 H bond oriented perpendicular with respect to ﬁrst motional symmetry axis) for the correlation times given in the ﬁgure (rows: correlation times of outer rotation are varied; columns: correlation times of inner rotation are varied)

and the C-2 H bond direction. In quite the same way, the T2 anisotropy also is strongly aﬀected by the actual equilibrium populations or the motional correlation time (data not shown). In Figs. 8 and 9 model simulations are given that refer to partially relaxed 2 H NMR spectra from the modiﬁed inversion recovery sequence. The ﬁrst example in Fig. 8 refers to a 3-fold jump motion in the fast exchange limit assuming a perpendicular orientation of the motional symmetry axis with respect to the C-2 H bond direction. The three series of spectra demonstrate the inﬂuence of the actual equilibrium population p1 on the spin-lattice relaxation. As reﬂected by the characteristic changes of these spectra as a function of the relaxation period τr after the inversion pulse, the relaxation rate is not

J. Villanueva-Garibay and K. M¨ uller

Fig. 7. Theoretical 2 H NMR line shapes (quadrupole echo, partially relaxed spectra) for a degenerate 3-fold jump process at diﬀerent angles between the C-2 H bond direction and the motional symmetry axis. The motional correlation time is 1×10−6 s

Fig. 8. Theoretical 2 H NMR line shapes (inversion recovery, partially relaxed spectra) for a 3-fold jump process and diﬀerent equilibrium populations p1 . The angle between the C-2 H bond direction and the motional symmetry axis is 90◦ . The motional correlation time is 1 × 10−11 s

NMR Studies of Disordered Solids

Fig. 9. Theoretical 2 H NMR line shapes (inversion recovery, ∆ = 20 µs) for two degenerate 3-fold jump processes (both motional symmetry axes are perpendicular on each other, C-2 H bond oriented perpendicular with respect to ﬁrst motional symmetry axis). The correlation times are left: 1×10−13 s (inner rotation), 8×10−7 s (outer rotation), and right: 5 × 10−9 s (inner rotation), 8 × 10−7 s (outer rotation)

identical across the 2 H NMR spectrum, which is a direct consequence of the angular dependence of T1 , i.e., T1 anisotropy [11, 14, 36]. The second set of partially relaxed spectra, shown in Fig. 9, were calculated on the basis of two superimposed degenerate 3-fold jump motions with motional symmetry axes that are perpendicular on each other. The interesting point is that one motion occurs in the intermediate motional regime, giving rise to line shape (or T2 ) eﬀects along with a characteristic line broadening. The second motion, however, takes place in the fast motional limit, and is responsible for the characteristic spin-lattice relaxation eﬀects [24]. Figure 10 shows a 2D exchange 2 H NMR spectrum. The simulation was performed with the assumption of a mutual exchange between two sites which are distinguished by their quadrupolar coupling constants. In fact, such a situation is encountered if six-membered ring hydrocarbons exhibit a fast overall rotation – leading to a diﬀerent motional averaging of the quadrupolar coupling constants for the axial and equatorial deuterons – along with ultraslow ring inversion, the latter of which determines the 2D exchange pattern [40].

J. Villanueva-Garibay and K. M¨ uller

Fig. 10. Theoretical 2D exchange 2 H NMR spectrum assuming chemical exchange between axial and equatorial deuterons in six-membered ring hydrocarbons. Due to fast rotation around the molecular C3 -axis, the ratio of quadrupolar coupling constants is ωax : ωeq = 1 : (−1/3). In addition, the condition of (1/τc )τm 1 (τc : correlation time for chemical exchange) holds

An analysis of the height of these exchange ridges provides the actual rate constants for the chemical exchange process. If the 2D exchange spectra are dominated by ultraslow reorientational motions, then the analysis of such exchange pattern provides valuable information about the underlying motional process (jump angle, distribution of correlation times, etc.) [13, 27, 37].

5 Applications for Guest-Host Systems In the following we report on the application of dynamic 2 H NMR methods for the characterization of guest species in cyclophosphazene (CPZ) inclusion compounds [40, 44, 45]. The basic structure of the host matrix is given by parallel, hexagonal channels (see Fig. 11), in which various types of guest molecules can be incorporated. The examples discussed in the following refer to NMR studies on CPZ inclusion compounds with benzene-d6 or pyridine-d5 as guest molecules. In general, it could be shown that these guest molecules exhibit a high mobility, which even holds for low temperatures (around 100 K). From a thorough data analysis it could be shown that the benzene guests undergo two motional processes, namely a fast rotation around the molecular C6 symmetry axis (degenerate 6-fold jump process, C6 -axes perpendicular to channel axis), and a second rotation around the channel long axis (3-fold jump process with unequally populated jump sites). The 2 H NMR line shapes and partially relaxed spectra (quadrupole echo experiment) in Figs. 12 and 13 thus are dominated by the latter motional

NMR Studies of Disordered Solids

Fig. 11. Host structure of CPZ inclusion compounds

Fig. 12. Experimental and theoretical 2 H NMR spectra (quadrupole echo, τe = 20 µs) for benzene-d6 /CPZ at diﬀerent temperatures. The relevant simulation parameters are given in the Figure and in the text

contribution around the CPZ channel long axis, which occurs on a slower timescale. The rotation around the molecular C6 axis is signiﬁcantly faster and dominates spin-lattice relaxation, as can be derived from the partially relaxed 2 H NMR spectra (inversion recovery experiment), depicted in Fig. 14. The general very good agreement between the experimental and theoretical data sets in Figs. 12 to 14 strongly supports the chosen model assumptions. The Arrhenius plots for the derived motional correlation times are given in Fig. 15, which yielded very low activation energies of 2.1 kJ mol−1 and 4.6 kJ mol−1 for the rotation around C6 -axis and the rotation around channel axis, respectively. The pyridine guests again turned out to be highly mobile. However, due to their diﬀerent chemical structure and lower symmetry, the motional behaviour is diﬀerent from that discussed for the benzene guests. The experimental 2 H

J. Villanueva-Garibay and K. M¨ uller

Fig. 13. Experimental and theoretical 2 H NMR spectra (quadrupole echo, partially relaxed spectra, T = 140 K) for benzene-d6 /CPZ. The correlation time for rotation around channel axis τCH is 8.5 × 10−7 s. Other parameters are given in the text

Fig. 14. Experimental and theoretical 2 H NMR spectra (inversion recovery, partially relaxed spectra, T = 60 K) for benzene-d6 /CPZ. The correlation time for rotation around channel axis τC6 is 2.2 × 10−10 s. Other parameters are given in the text

NMR Studies of Disordered Solids

Fig. 15. Arrhenius plot for the correlation times of C6 rotation (τC6 , circles) and rotation around the CPZ channel axis (τCH , squares) in benzene-d6 /CPZ

Fig. 16. Experimental and theoretical 2 H NMR spectra (quadrupole echo, τe = 20 µs) for pyridine-d5 /CPZ at diﬀerent temperatures. The relevant simulation parameters are given in the Figure and in the text

NMR spectra of the pyridine guests as well as the corresponding partially relaxed spectra (inversion recovery experiment), given in Figs. 16 and 17, can thus be reproduced by the assumption that the molecules undergo a fast rotation on a cone with on opening angle between 59◦ and 73◦ (see Fig. 18).

J. Villanueva-Garibay and K. M¨ uller

Fig. 17. Experimental and theoretical 2 H NMR spectra (inversion recovery, partially relaxed spectra, T = 210 K) for pyridine-d5 /CPZ. The correlation time for rotation on a cone τCH is 1.1 × 10−12 s. Other parameters are given in the text

Fig. 18. Arrhenius plot for the derived correlation times τCH for rotation on a cone of pyridine-d5 in the CPZ channels

As a result, the 2 H NMR spectra can be understood as a superposition of three subspectra due to magnetically non-equivalent deuterons (i.e., subspectra from deuterons 1, 5; deuterons 2, 4, and deuteron 3). The analysis of the spin-lattice relaxation data provided the correlation times for this motional process that are summarized in Fig. 18. As before, a relatively low

NMR Studies of Disordered Solids

activation energy of 8.7 kJ mol−1 is found for the overall pyridine rotation, which, however, is consistent with the above results for the benzene guests and the published data on related compounds [39, 40, 46, 47].

6 Conclusions In the present work the basics of dynamic NMR spectroscopy were brieﬂy reviewed. For the case of dynamic 2 H NMR spectroscopy it has been shown that such techniques are very sensitive to motional processes that can occur on quite diﬀerent time-scales. A comprehensive analysis of the experimental data on the basis of appropriate simulation programs can provide a very detailed picture about the motional characteristics and ordering features of quite diﬀerent (motionally) disordered solids. As an example, results from the application of dynamic 2 H NMR spectroscopy during the characterization of the guest species in cyclophosphazene inclusion compounds were reported.

Acknowledgement The authors would like to thank the Deutsche Forschungsgemeinschaft and the Fonds der Chemischen Industrie (FCI) for ﬁnancial support.

References 1. J.I. Kaplan, G. Fraenkel: NMR of Chemically Exchanging Systems (Academic Press, New York 1980) 2. J. Sandstr¨ om: Dynamic NMR Spectroscopy (Academic Press, London 1982) 3. R. Tycko (ed.): Nuclear Magnetic Resonance Probes of Molecular Dynamics (Kluwer, Dordrecht 1994) 4. H.W. Spiess: Adv. Polym. Sci., 1985, 66, 23 5. K. M¨ uller, K.-H. Wassmer, G. Kothe: Adv. Polym. Sci. 95, 1, (1990) 6. G.R. Luckhurst, C.A. Veracini: The Molecular Dynamics of Liquid Crystals (Kluwer, Dordrecht 1989) 7. R. Dong: Nuclear Magnetic Resonance of Liquid Crystals (Springer, Berlin 1994) 8. R.G. Griﬃn: Methods Enzymol. 72, 108, (1981) 9. J.H. Davis: Biochim. Biophys. Acta 737, 117, (1983) 10. J. Ripmeester in Inclusion Compounds, Eds. J.L. Atwood, J.E.D. Davies, D.D. MacNicol: Oxford University Press, 1991; Vol.5, p 37 11. R.R. Vold: in NMR Probes of Molecular Dynamics, Ed. R. Tycko, Kluwer, Dordrecht, 1994, p 27 12. R.R. Ernst, G. Bodenhausen, A. Wokaun: Principles of Nuclear Magnetic Resonance in One and Two Dimensions (Clarendon, Oxford 1987) 13. K. Schmidt-Rohr, H.W. Spiess: Multidimensional Solid-State NMR and Polymers (Academic Press, London 1994)

86 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47.

J. Villanueva-Garibay and K. M¨ uller R.R. Vold, R.L. Vold: Adv. Magn. Opt. Res. 16 (1991) 85 C.A. Fyfe: Solid State NMR for Chemists (CFC Press, Guelph 1983) J.I. Kaplan: J. Chem. Phys. 28, 278, (1958); 29, 462, (1958) S. Alexander: J. Chem. Phys. 37, 967, (1962) J. Jeener: Adv. Magn. Reson. 10, 1, (1982) M. Bak, J.T. Rasmussen, N.C. Nielsen: J. Magn. Reson. 147, 296, (2000) D.M. Brink, G.R. Satchler: Angular Momentum (Clarendon, Oxford 1975) H.W. Spiess in NMR, Basic, Principles and Progress, Eds. P. Diehl, E. Fluck, R. Kosfeld, Springer-Verlag, Berlin, 1978, Vol. 15, p 55 K. M¨ uller, P. Meier, G. Kothe: Progr. Nucl. Magn. Reson. Spectrosc. 17, 211, (1985) H.W. Spiess, H. Sillescu: J. Magn. Reson. 42, 381 (1981) R.J. Wittebort, E.T. Olejniczak, R.G. Griﬃn: J. Chem. Phys. 86, 5411, (1987) A.J. Vega, Z. Luz: J. Chem. Phys. 86, 1803, (1987) M.S. Greenﬁeld, A.D. Ronemus, R.L. Vold, R.R. Vold, P.D. Ellis, T.E. Raidy: J. Magn. Reson. 72, 89, (1987) C. Schmidt, B. Bl¨ umich, H.W. Spiess: J. Magn. Reson. 79, 269, (1988) Sample ﬁles are available from the authors MathcadTM , Mathsoft Engineering & Education, Inc., Cambridge, MA MATLABTM , The MathWorks, Inc., Natick, MA M. Bloom, J.H. Davis, A.L. MacKay: Chem. Phys. Lett. 80, 198, (1981) R.K. Wangsness, F. Bloch: Phys. Rev. 89, 728, (1953) A.G. Redﬁeld: Adv. Magn. Reson. 1, 1, (1965) A.G. Redﬁeld: IBM J. Res. Develop. 1, 19, (1953) R.J. Wittebort, A. Szabo: J. Chem. Phys. 69, 1722, (1978) D.A. Torchia, A. Szabo: J. Magn. Reson. 42, 107, (1982) C. Boeﬀel, Z. Luz, R. Poupko, A.J. Vega: Isr. J. Chem. 28, 283, (1988) B. Bl¨ umich, H.W. Spiess: Angew. Chem. 100, 1716, (1988) J. Schmider, K. M¨ uller: J. Phys. Chem. A 102, 1181, (1998) A. Liebelt, A. Detken, K. M¨ uller: J. Phys. Chem. B 106, 7781, (2002) K. M¨ uller: Phys. Chem. Chem. Phys. 4, 5515, (2002) B.T. Smith, J.M Boyle, B.S. Garbow, Y. Ikebe, V.C. Klema, C.B. Moler: Matrix Eigensystem Routines – EISPACK Guide (Springer, Berlin 1976) Further information about the simulation programs are available from the authors H.R. Allco*ck, in Inclusion Compounds, (J.L. Atwood, J.E.D. Davies, D.D. MacNicol, Eds.) Academic Press, New York (1984), Vol. 1, p 351 E. Meirovitch, S.B. Rananavare, J.H. Freed: J. Phys. Chem. 91, 5014, (1987) A. Liebelt, K. M¨ uller: Mol. Cryst. Liq. Cryst. 313, l45, (1998) J. Villanueva-Garibay, K. M¨ uller: J. Phys. Chem. B 108, 15057, (2004)

En Route to Solid State Spin Quantum Computing M. Mehring, J. Mende and W. Scherer Physikalisches Institut, University Stuttgart, 70550 Stuttgart, Germany [emailprotected]

Abstract. We present routes to quantum information processing in solids. An introduction to electron and nuclear spins as quantum bits (qubits) is given and basic quantum algorithms are discussed. In particular we focus on the preparation of pseudo pure states and pseudo entangled states in solid systems of combined electron and nuclear spins. As an example we demonstrate the Deutsch algorithm of quantum computing in an S-bus system with one electron spin coupled to a many 19 F nuclear spins.

1 Brief Introduction to Quantum Algorithms It was Feynman [1] who suggested more than twenty years ago to use quantum algorithms to simulate physical phenomena. A few years later Deutsch proposed the concept of a quantum computer [2, 3]. This initiated some exciting new ideas leading to quantum cryptography [4, 5] and quantum teleportation [6, 7]. This new area of science was stimulated enormously by the proposal of fast searching algorithms by Grover [8] and its NMR implementation by Chuang et al. [9] and more so by the quantum factoring algorithm proposed by Shor [10] which was implemented recently with liquid state NMR by Vandersypen et al. [11]. These quantum algorithms demonstrated the impressive parallelism of quantum computation which could speed up calculations tremendously well beyond classical computing. A number of NMR experiments in the liquid state have demonstrated the concept of spin quantum computing [12, 13, 14, 15, 16, 17]. For an introduction to these concepts see [18]. After these initial experiments a number of other liquid NMR realizations have been published. It is beyond this overview to reference all these. A critical account on the quantum nature of these experiments can be found in [19]. In the following sections we want to summarize our initial steps for performing quantum computing with electron and nuclear spins in crystalline solids. The early proposal to perform solid state spin quantum computing by M. Mehring et al.: En Route to Solid State Spin Quantum Computing, Lect. Notes Phys. 684, 87–113 (2006) c Springer-Verlag Berlin Heidelberg 2006 www.springerlink.com

M. Mehring et al.

Kane [20] is based on the nuclear spin of phosphorous in silicon. Here we utilize the combined states of the electron spin and nuclei in solids. We brieﬂy introduce the concept of quantum gates and quantum algorithms. Next we consider the quantum states of an electron spin S = 1/2 coupled to a nuclear spin I = 1/2 and how basic quantum gates can be realized with such a spin system. As a speciality we treat the case of an electron spin S = 3/2 coupled to a nuclear spin I = 1/2. Finally we discuss the new S-Bus Concept, where an electron spin S = 1/2 couples to many nuclear spins Ij = 1/2 which allows to create multi spin correlated and entangled states. Moreover we present experiments with a qubyte+1 nuclear spin system in CaF2 :Ce in context of the S-Bus concept. 1.1 Basic Quantum Gates Quantum gates are the building blocks for quantum computation. A quantum bit (qubit) is represented not only by the two binary states 0 and 1 like a classical bit, but by the whole two-dimensional Hilbert space representing the wavefunction (1) ψ = c1 |0 + c2 |1 , with complex numbers c1 and c2 obeying the condition c1 c∗1 + c2 c∗2 = 1. Here we have used the qubit basis states |0 and |1. Considering a spin 1/2 as the ideal qubit, we will often use the notation | ↑ = |+ = |0 and | ↓ = |− = |1 here. The one bit gate which can easily be implemented as a quantum gate, is the NOT gate (see Fig. 1).

Fig. 1. NOT gate

It simply inverts an arbitrary bit a. The corresponding unitary transformation can be represented in matrix form as 01 1 0 UNOT = with basis states |0 = and |1 = . (2) 10 0 1 The bit ﬂip operation is readily implemented in a qubit system by applying the unitary transformation Py (π) = e−iπIy which performs essentially the same operation as UNOT . The bit ﬂip can be considered as a classical operation. An important aspect of qubits is the fact that the superposition of quantum states can be exploited. This is achieved by the Hadamard transformation 1 1 1 H=√ . (3) 2 1 −1

Spin Quantum Computing

If applied to the √ it transforms it to the superposition √ state |+ = |0 state (|+ + |−)/ 2 = (|0 + |1)/ 2. This corresponds to the transverse components in the xy-plane in magnetic resonance. In magnetic resonance this is usually achieved by a π/2-pulse. However, the unitary transformation Py (π/2) does not correspond exactly to the Hadamard transform, but can be used instead if one obeys the fact that the Hadamard transform corresponds to its own inverse, whereas Py (−π/2) must be applied if the inverse operation is required. For completeness we mention that the Hadamard transform can be implemented by the composite pulse Py (−π/4)Px (π)Py (π/4). A little more advanced is the two qubit CNOT (Controlled NOT) operation, sketched as a block diagram in Fig. 2.

Fig. 2. CNOT gate (see text)

It requires two qubits, namely qubit a which is called the control bit and qubit b which is the target bit. The target bit b is inverted only if the control bit a is in a particular state. The most common case is where qubit b is inverted if the control bit a = 1. The control bit a stays unchanged in this operation. The matrix representation of the CNOT operation is given by   1000 0 1 0 0  (4) UCNOT =  0 0 0 1 . 0010 These gates are reversible. In order to demonstrate the application of these quantum gates we discuss the preparation √ of the fundamental √ EinsteinPodolsky-Rosen (E-P-R) state (| + − − | − +)/ 2 = (|01 − |10)/ 2 which is at the heart of quantum mechanics, by starting from the two qubit product state | − − = |11 [21]. The E-P-R state results by applying a Hadamard transformation to the ﬁrst qubit followed by a CNOT gate (Fig. 3): 1 H CNOT 1 |11 −→ √ (|01 − |11) −→ √ (|01 − |10) . 2 2

(5)

It corresponds to a superposition state which intimately involves both qubits and cannot be expressed as a product state of qubits one and two. The state of each individual qubit is undetermined. All four possible entangled states of a two qubit system are called Bell states:

M. Mehring et al. 1

1 2

   

01 − 10

   

Hadamard

UCNOT

Fig. 3. Quantum gate for creating an entangled state

1 1 Ψ ± = √ (|01 ± |10) and Φ± = √ (|00 ± |11) . 2 2

(6)

Superposition and entanglement of qubits are the essential ingredients of quantum computing. They can be obtained from simple product states by the unitary transformations corresponding to the quantum gates NOT, H and CNOT and their combinations.

The basic block diagram for quantum computing, as sketched in Fig. 4, comprises the preparation of a particular initial quantum state, a series of unitary transformations, representing the quantum algorithm and ﬁnally the detection of the outcome. We will exemplify this in the following for the Deutsch-Jozsa algorithm. For a summary on quantum gates and liquid state NMR applications see [18, 22]. 1.2 The Deutsch-Jozsa Algorithm The Deutsch-Jozsa (DJ) algorithm is a quantum algorithm which evaluates a binary function and decides if the function is constant or balanced [2]. Let us consider a function f (ab) of two qubits a and b where the function returns only a single bit state. The following table lists some of the 24 = 16 possible functions. Note that the ﬁrst two functions are constant. Their value is either 0 or 1 independent of the variable ab. The other functions are called balanced, because their values represent an equal number of 0 and 1. There are six of those. Since the total number of possible functions is 16 there must be

Spin Quantum Computing

eight more functions which are neither constant nor balanced. Their values correspond to an odd number of 0 or 1. In the context of the Deutsch-Jozsa algorithm one is only interested in distinguishing the constant and balanced functions. In order to implement the DJ algorithm one needs to represent the functions f (ab) by unitary transformations. In Table 1 we have therefore included the label of the corresponding unitary transformations. They are constructed such that on the diagonal every 0 is represented by 1 and every 1 by −1. An example for U0101 is given in (7)   1 0 0 0  0 −1 0 0   U0101 =  (7) 0 0 1 0  . 0 0 0 −1

Table 1. Function f (ab) f (ab)

U0000 U1111 U0101 U0011 U1001 U1010 U1100 U0110

0 1 0 0 1 1 1 0

0 1 1 0 0 0 1 1

0 1 0 1 0 1 0 1

0 1 1 1 1 0 0 0

constant constant balanced balanced balanced balanced balanced balanced

The block diagram of the DJ algorithm is presented in Fig. 5. Note that the output of qubit 1 can be 0 or 1. The DJ algorithm is constructed such that a constant function gives 0 and a balanced function gives 1. This quantum algorithm evaluates the functions in a single computational step, whereas the classical computer must evaluate each function separately in order to decide if the function is constant or balanced. Other quantum algorithms have been formulated and were demonstrated partially by NMR quantum computing. Space does not allow to go into details here.

Fig. 5. Block diagram for the DJ algorithm

M. Mehring et al.

In Sect. 5.6 we present the implementation of this algorithm within the S-Bus Concept in a single crystal of CaF2 :Ce.

2 Combined Electron Nuclear Spin States in Solids Here we consider the situation, where a number of nuclei Ij are connected via hyperﬁne interaction to an electron spin S. In the context of quantum computing we have labelled this an S-Bus system to be discussed in more detail in Sect. 5. 2.1 Quantum States The total Hamiltonian of the S-Bus system can be expressed as Htot = ωS Sz + ωI Iz + Sz

N j=1

aj Izj +

Djk Izj Izk ,

(8)

j=k

with the Larmor frequencies ωS = gµB B0 / and ωI = −γI B0 . In order to keep only the diagonal terms of the Hamiltonian we applied the approximation ωS ωI > aj Djk , where the absolute values of these parameters are considered in these inequalities. Oﬀ-diagonal terms may exist, but will be rather small in most cases. The energy spectrum is schematically sketched in Fig. 6. Representative electron spin resonance (ESR) transitions (∆mS = ±1, ∆mI = 0) are indicated by solid lines and some nuclear spin transitions |mS〉

|mS, mI1, mI2〉

|+½〉

|+ + +〉 |+ + −〉 |+ − +〉 |+ − −〉

|−½〉

|− + +〉 |− + −〉 |− − +〉 |− − −〉

Fig. 6. Schematic energy level scheme of two nuclear spins I1,2 = 1/2 coupled to an electron spin S = 1/2. Solid lines: ESR transitions. Dashed lines: ENDOR transitions. Dotted lines: Entangled states (see text))

Spin Quantum Computing

(∆mS = 0, ∆mI = ±1) are drawn as dashed lines. Experiments are performed by applying microwave pulses at the electron spin transitions combined with pulsed irradiation in the radio frequency spectrum at the diﬀerent nuclear transitions. Direct detection of the nuclear spin transitions is exceedingly difﬁcult because of the small gyromagnetic ratio and most of all because of the low concentration and the low thermal polarisation. The nuclear transitions and coherences can, however, be observed indirectly by monitoring the electron spin signal while irradiating at the NMR transition. This type of double resonance is called Electron Nuclear Double Resonance (ENDOR). 2.2 Equilibrium Density Matrix The initial state of a real physical spin system will correspond to an equilibrium state which is usually assumed to be the Boltzmann state. The Boltzmann equilibrium density matrix for the S-Bus system is deﬁned as ρB =

e−βHtot , Tr(e−βHtot )

(9)

where β = /kB T and Tr(A) implies taking the trace (sum oﬀ diagonal elements) of the corresponding matrix A. This is the typical Boltzmann density matrix of an ensemble (large number) of spin clusters consisting of a single S spin coupled to N nuclear spins I as expressed by the Hamiltonian according to (8). Because of the dominance of ωS over all other interactions in the electron nuclear spin system discussed here, we assume for simplicity that the Boltzmann density matrix is represented by the electron Zeeman term ωS Sz which leads for an S spin 1/2 and arbitrary spins I to 1 1 βωS , (10) ρBS = (I0 − 2KB Sz ) where KB = tanh 2(2I + 1)N 2 with 0 ≤ KB ≤ 1 and where I0 is the 2(2I + 1)N × 2(2I + 1)N identity matrix. We note that under this approximation there is no nuclear spin polarization or correlation whatsoever. Since we have no control over the identity matrix in the density matrix expression of (10) we ignore it usually in magnetic reso nance and deal with the truncated equilibrium density matrix ρB = −KB Sz . In the high temperature low ﬁeld approximation KB is rather small. This is the usual case in magnetic resonance (MR) experiments. In order to prepare for the concept of pseudopure states, ﬁrst introduced by Cory and co-workers [12, 13], we use some freedom to rearrange the expression for the Boltzmann density matrix (10) in the following way. 1 KB KB ρPB , (11) 1− ρBS = I0 + N 2(2I + 1) K K which deﬁnes the pseudo Boltzmann density matrix

M. Mehring et al.

ρPB =

1 (I0 − 2KSz ) . 2(2I + 1)N

(12)

Note that the Boltzmann density matrix as expressed by (12) is still exact with arbitrary parameters K > 0. This allows us to express the pseudo Boltzmann density matrix ρPB at will. The idea behind this is to manipulate the operator part 2KSz in ρPB in such a way that it is converted into a density matrix which has the same operator structure as a pure state. Let us consider the simplest possible case, namely spin I = 0 and K = 1. This would convert ρPB into 00 ρ1 = , (13) 01 which clearly would represent a density matrix of the pure state |1. Nevertheless, it is still nothing but the Boltzmann density matrix at temperature T > 0 and as such represents a mixed state. The usefulness of the pseudo pure density matrix becomes more obvious for the case, when we manage to convert, by some means of manipulations, −KSz into Sz + Iz + 2Sz Iz for two spins S and I. This leads to the pseudopure density matrix   1000 0 0 0 0 1 1  ρPB → ρ00 = |0000| = I0 + (Sz + Iz ) + Sz Iz =  (14) 0 0 0 0 . 4 2 0000 This and related types of pseudopure density matrices we will use in the following as initial states. We note that such a pseudopure density matrix requires to introduce a correlation between the spins, represented by the operator product Sz Iz . The relevance of the spin correlation and pseudopure states for liquid state NMR quantum computing was discussed by Warren et al. [19] and Cory et al.[12, 13, 23].

3 Entanglement of an Electron and a Nuclear Spin

1 2

Typically one discusses the entanglement between spins 1/2 of the same type, like either electrons or protons. Although quantum algorithms have been formulated independent of the type of qubit system, the entanglement of an electron and a nucleus is somewhat exotic because of their very diﬀerent properties like coupling to external ﬁelds and their strong hyperﬁne interaction. In this section we will demonstrate the pseudo entanglement between and electron spin 1/2 and a nucleus with spin 1/2 namely a proton and in a separate section between an electron spin 3/2 and a 15 N nucleus with spin 1/2. What we are aiming at are Bell states introduced in (6) and represented here by spin symbols, where the ﬁrst spin represents the electron spin 1 Ψ ± = √ (| ↑↓ ± | ↓↑) 2

and

1 Φ± = √ (| ↑↑ ± | ↓↓) . 2

(15)

Spin Quantum Computing

They correspond to a superposition of the states in Fig. 8 connected by dotted arrows. 3.1 An Electron Nuclear Spin Pair in a Single Crystal This section is based on a recent publication observing pseudo entanglement between an electron spin 1/2 and a nuclear spin 1/2 in a crystalline solid [24]. Let us consider the intensively studied malonic acid radical in a single crystal. The basic molecular unit is sketched in Fig. 7. The electron spin density extends over whole molecular unit and beyond. The strongest hyperﬁne coupling is to the adjacent proton to the central carbon which carries most of the spin density. The hyperﬁne interaction is highly anisotropic. We consider here a special orientation of the magnetic ﬁeld where we can to a good approximation describe the Hamiltonian and the corresponding energy levels of this spin pair by (8) with a < 0. The corresponding energy level scheme is sketched in Fig. 8.

Fig. 7. The malonic acid radical

+−

energy

−+ −−

Fig. 8. Four level scheme of a two qubit system

We have doubly labelled quantum states in Fig. 8 according to their spin orientation and the qubit terminology. We have also included the phase rotation property of each individual state under the operation e−iφ1 Sz e−iφ2 Iz . This will turn out to be useful when we discuss the tomography of the entangled states. The corresponding spectrum consists of four allowed transitions two at the ESR frequency (∆mS = ±1, ∆mI = 0) and two at NMR frequencies

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Fig. 9. Complete pulse sequence for creating an entangled state of a two qubit system by transition selective excitation

(∆mS = 0, ∆mI = ±1). All four transitions are well resolved in the spectra of the malonic acid radical. The complete pulse sequence for the preparation of the pseudopure initial states, the creation of entanglement which applies a CNOT (controlled NOT) operation and the density matrix tomography is shown in Fig. 9. The sequence ends with the detection of the spin echo sequence for the electron spin. The diﬀerent building blocks of this sequence will be discussed in the following sections. 3.2 Creating Pseudopure States by Selective Excitation Due to the well resolved transitions we can apply transition selective excitations in contrast to most liquid state NMR quantum computing experiments, where spin selective excitations have been performed. In order to prepare the pseudopure state   0000 0 0 0 0 1 1 1  ρ10 = |1010| = I0 − Sz + Iz − Sz Iz =  (16) 0 0 1 0 4 2 2 0000 we need to convert −KSz in (12) into −Sz + Iz − 2Sz Iz which is readily achieved by the pulse sequence ∆

∆

1 2 Py(12) (π/2) −→ ρ10 , Py(24) (arccos(−1/3)) −→

(17)

with K = 3/4. This pulse sequence is sketched in Fig. 9 in the prepara(jk) tion segment. Here we use the notation Py (β) for a β-pulse in y-direction at the transition j ↔ k which corresponds to a unitary transformation (jk) (jk) Uy (β) = exp(−iβIy ). We will use this deﬁnition for x- and y-pulses throughout this article. The whole process of creating the pseudopure state

Spin Quantum Computing

is, however, not a unitary process, because we eliminate after each pulse the oﬀ-diagonal components of the density matrix by allowing for decoherence times ∆1 and ∆2 after pulses. In a similar way we prepare the pseudopure density matrix   0000 0 0 0 0 1 1  ρ11 = |1111| = I0 − (Sz + Iz ) + Sz Iz =  (18) 0 0 0 0 . 4 2 0001 3.3 Tomography of the Pseudopure States An elegant way of performing a spin density matrix tomography was introduced by Madi et al. [25] by utilizing the concept of two-dimensional NMR spectroscopy. A more recent account of density matrix tomography applied after a quantum algorithm in liquid state NMR quantum computing can be found in [26, 27]. Here we apply a diﬀerent and simpler approach because ESR lines in solids are rather broad (inhom*ogeneous broadening) and moreover we can address every transition selectively. In order to prove that we have prepared the wanted pseudopure state we need to perform a density matrix tomography. Because of the decoherence times we need to determine only the four diagonal elements of the density matrix. Due to the normalized trace of the density matrix we need to determine only three diﬀerent parameters. This is readily obtained by measuring the population diﬀerences p1 −p2 , p1 −p3 and p3 −p4 . This we could obtain by measuring the amplitudes of the Rabi precession of the corresponding transitions. This requires a proper calibration in particular since ESR and NMR transitions are to be compared. By virtue of relating all Rabi precessions to the change in the electron spin echo amplitude we were able to determine all parameters with reasonable precision. A matrix representation of the experimentally determined ρ10 gives   0.01 0 0 0  0 −0.06 0 0   . ρ10 =  (19)  0 0 1.02 0  0 0 0 0.03 Similar results were obtained for ρ11 . 3.4 Entangled States For this spin system we have prepared all of the four Bell states according to (6, 15). The pulse sequence for creating the entangled states from the prepared pseudopure state comprises a π/2-pulse at a selective NMR transition,

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replacing the standard Hadamard transformation, followed immediately by πpulse at a selective ESR transition which corresponds to a CNOT operation as is included in the pulse sequence Fig. 9. This corresponds exactly to the quantum algorithm for creating entangled states as presented in Sect. 1.1. The question arises how do we know that we have indeed created an entangled state. In order to prove this we need to perform a density matrix tomography. Before we discuss this procedure and the results obtained we ﬁrst take a look at the phase dependence of the entangled states. When applying the phase rotation operator exp(−i(φ1 Sz + φ2 Iz )) to the E-P-R state, which corresponds to a phase rotation about the z-axis, one observes the relation 1 1 1 ρΨ − (φ1 , φ2 ) = √ (e−i 2 (φ1 −φ2 ) | + − − ei 2 (φ1 −φ2 ) | − +) , 2 1 1 −i 1 (φ1 +φ2 ) ρΦ+ (φ1 , φ2 ) = √ (e 2 | + + + ei 2 (φ1 +φ2 ) | − −) , 2

(20) (21)

which identiﬁes this entangled state through its phase diﬀerence. By incrementing both phases φ1 and φ2 in steps phase interferograms like the one shown in Fig. 10 are obtained. The phase increments can be related to a frequency which is deﬁned as ∆φj = 2πνj ∆t. The phase interferograms shown in Fig. 10 therefore correspond to the sum φ1 + φ2 (Fig. 10 (bottom)) and diﬀerence φ1 − φ2 (Fig. 10 (top)) of the applied phases. After Fourier transformation of the phase interferograms a spectrum is displayed as is shown in Fig. 11, with lines appearing at particular frequencies. Here we used the following individual frequencies: spin 1: ν1 = 2.0 MHz; spin 2: ν2 = 1.5 MHz.

Fig. 10. Phase interferograms, top: Ψ − state, bottom: Φ+ state

Spin Quantum Computing

Fig. 11. Fourier transform of the phase interferograms of Fig. 10

We expect to see a line at these frequencies for the superposition states of the individual spins. For the entangled states we expect to see lines at ν1 ± ν2 depending on which of the Bell states were created. One clearly observes these features in Fig. 11. The upper spectrum represents the ρΨ − state, whereas the lower spectrum corresponds to the state ρΦ+ . The characteristic frequencies ν1 ± ν2 = 2.0 ± 1.5 MHz are clearly identiﬁed as the dominant lines. The lines appearing at ν1 and ν2 are clearly contaminations of unwanted superposition states. Density Matrix Tomography The tomography of the entangled states requires to determine the diagonal part and the oﬀ-diagonal parts of the density matrix. The diagonal part is obtained in a similar way as already discussed in the case of the pseudopure density matrix by measuring the Rabi precession of the diﬀerent allowed transitions and obtain from their amplitudes the diagonal elements. The oﬀdiagonal elements are obtained from the phase rotation and the corresponding spectral amplitudes discussed in the previous section. The numerical values obtained by the tomography procedure are   0.49 0.00 0.00 0.49  0.00 −0.03 0.00 0   ρΦ+ =  (22)  0.00 0.00 0.02 0.00  , 0.49 0 0.00 0.52 where the oﬀ-diagonal elements labelled 0.00 correspond to values 0.00 ± 0.05. The label 0 refers to values which could not be measured, but are expected to be small, because no excitation was performed at that transition. In a similar way we obtained ρΨ − . The following data were obtained for the density matrix

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

−0.02  0.00 =  0 0.00

ρΨ −

 0.00 0 0.00 0.55 −0.47 0.00   . −0.47 0.50 0.00  0.00 0.00 −0.03

(23)

More details on this subject can be found in [28].

4 Entangling an Electron Spin

3 2

with a Nuclear Spin

1 2

ESR and ENDOR spectra of the molecule N@C60 were ﬁrst reported in [29]. The electron spin of the nitrogen atom is S = 3/2, whereas the 15 N nucleus has spin I = 1/2. There have been a number of proposals how one could use 15 N@C60 in order to perform quantum computing [30, 31, 32]. However, these proposals neither addressed the problem of what is the relevant qubit in this system and how, realistically a quantum algorithm could be performed. In a recent publication we have deﬁned the relevant qubit and performed a CNOT operation leading to entanglement [33]. We have also performed a density matrix tomography in order to evaluate the entangled state. The corresponding Hamiltonian of 15 N@C60 can be expressed in ﬁrst order as H = (ωS Sz + ωI Iz + aSz Iz ) ,

(24)

with energy eigenvalues E(mS , mI ) = (ωS mS + ωI mI + amS mI ) .

(25)

According to the negative g-factor and the negative γ of N ωS ωI > 0. Furthermore the approximation |a|, ωI ωS has been applied in (24). The eigenstates |mS mI are labelled here + + + 3 1 3 1 3 1 {|1, |2, · · · |8} = , , ,− , ··· − ,− . (26) 2 2 2 2 2 2 15

The energy levels including the quantum states are shown in Fig. 12. The energy level structure can be considered as separated into two four level systems, each corresponding to an electron spin 3/2, where the levels |1, |3, |5, |7 correspond to the nuclear spin quantum number mI = +1/2, whereas the levels |2, |4, |6, |8 correspond to the nuclear spin quantum number mI = −1/2. We introduce two ﬁctitious spins 3/2 by deﬁning S (mI ) as S (±) for mI = ±1/2. In the following we will concentrate on the ﬁctitious two qubit subsystem + + + + 3 1 3 1 3 1 3 1 {|ab} = , , , − , − , + , − , − . (27) 2 2 2 2 2 2 2 2 Although ESR pulses are applied at the two ﬁctitious spin 3/2 systems, the pulse sequences are tailored such that the wanted state of the ﬁctitious two qubit subsystem is reached.

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Fig. 12. Schematic energy level scheme of .

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N@C60

4.1 Pseudopure States The preparation of the pseudopure density matrix ρ11 of the ﬁctitious two qubit system as deﬁned before is performed by a similar pulse sequence as in the electron spin 1/2 case as ρBP

Py(+) (β0 )

−→

∆t1

Py(12) (π/2)

−→

∆t2 =⇒ ρP11 .

(28)

First a selective β0 -pulse, with β0 = arccos(−1/3), is applied at the ﬁctitious spin 3/2 subsystem corresponding to mI = +1/2 followed by the decay time ∆t1 which allows to decohere the oﬀ-diagonal states. After this a π/2pulse is applied to the 1 ↔ 2 transition, followed again by the decay time ∆t2 . The resulting density matrix is given by   00000 000 0 0 0 0 0 0 0 0   0 0 1 0 0 0 0 0 3   0 0 0 0 0 0 0 0  (29) ρP 11 =  0 0 0 0 1 0 0 0 , 6   0 0 0 0 0 1 0 0 2   0 0 0 0 0 0 0 0 00000 001 where we have marked the ﬁctitious two qubit sublevel system as bold face. Extracting from this the density matrix for the ﬁctitious two qubit subsystem leads obviously to

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

ρ11

0 0 = 0 0

0 0 0 0

 0 0  . 0 1

(30)

The preparation of the pseudopure density matrix ρ10 proceeds in a similar way as described for ρ11 . These are used as initial matrices for creating the entangled states of the ﬁctitious two qubit subsystem. 4.2 Entangled States We consider as entangled states of the ﬁctitious two qubit subsystem the density matrices 1 (±) (31) ρPΨ = |2 ± 72 ± 7| 2 and 1 (±) ρPΦ = |1 ± 81 ± 8| (32) 2 in analogy to the two qubit Bell states (6, 15). Preparation of these entangled (27) (−) (78) states is achieved by the unitary transformations U± = PSy (∓π) PIy (π/2) (18)

(+)

(78)

and U± = PSy (±π) PIy (π/2). The following results are obtained. When extracting the corresponding density matrices of the ﬁctitious two qubit subsystem, as discussed before, we obtain   0 0 0 0 0 1 ±1 0 2 2  (33) ρΨ =  0 ±1 1 0 2 2 0 0 0 0 and



1 2

0  0 0 ρΦ =   0 0 ± 12 0

 0 ± 12 0 0   , 0 0  0 12

(34)

which correspond to the Bell states. 4.3 Density Matrix Tomography The tomography of the diagonal part of the diﬀerent density matrices is performed by measuring the amplitude of the Rabi precessions when particular transitions are excited as was discussed in the electron spin 1/2 case. These amplitudes are proportional to the diﬀerence of the corresponding diagonal elements. A typical Rabi precession for two diﬀerent transitions of the ρ10 state is shown in Fig. 13. As expected for this state, the population diﬀerence at the

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time (µs) Fig. 13. Rabi precession of the transitions 1 ↔ 2 (top) and 7 ↔ 8 (bottom) for the ρ10 state in 15 N@C60

1 ↔ 2 transition is close to zero, whereas the large population diﬀerence at the 7 ↔ 8 transition gives rise to a large amplitude as is expected for the ρ10 state. The tomography of the entangled states proceeds via a detection sequence which basically converts the entangled state to an observable state. This will be in general a product state. In order to demonstrate the entanglement the characteristic phase dependence of the particular entangled state should appear in the detection signal. As was already discussed in the case of the entangled spins S = 1/2 and I = 1/2 in Sect. 3.4 phase interferograms were obtained as is shown for the E-P-R state ρΨ − in Fig. 14. We apply a similar pulse sequence as in the case of the electron spin 1/2, where ﬁrst an ESR π-pulse is applied at the mI = −1/2 sublevel system (−) represented by the unitary transformation PSx (−π, φ1 ). Immediately after this a selective ENDOR π/2-pulse at the 7 ↔ 8 transition follows represented (78) by unitary transformation PIx (π/2, φ2 ). Phase rotation is applied here with frequencies ν1 = 2.0 MHz and ν2 = 1.5 MHz. It is interesting to note that due to the electron spin S = 3/2 we now expect to see a tripled frequency for the ESR transition which is indeed the case as is seen in the interferogram (Fig. 14 middle) as well as in the correponding spectrum. The entangled state, however, of ρPΨ ± is expected to result in the detection signal Ψ (φ1 , φ2 ) = const. ± S±

3 cos(3φ1 − φ2 ) . 20

(35)

The phase diﬀerence leads to the diﬀerence frequency 3ν1 − ν2 as seen in Fig. 15 (bottom). In a similar way all the other Bell states were analyzed. More details on this subject can be found in [28, 33].

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Fig. 14. Phase interferogram obtained with frequencies ν1 = 2.0 MHz and ν2 = 1.5 MHz. Top: Variation of φ2 (φ1 = 0). Middle: Variation of φ1 (φ2 = 0). Bottom: φ1 = 0 and φ2 = 0

Fig. 15. Spectrum corresponding to the phase interferogram of Fig. 14 with frequencies ν1 = 2.0 MHz and ν2 = 1.5 MHz

5 The S-Bus Concept The S-Bus concept for spin quantum computing was ﬁrst presented at the ISMAR conference (Rhodos, Greece 2001) and is derived from multiple quantum ENDOR (MQE) [34] and was ﬁrst published elsewhere [22, 35, 36]. It basically consists of a central spin, called S spin, which acts as a sort of server

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which is coupled to a network of nuclear spins, labelled I spins. Only the S spin is observed similar to a Turing machine, where the head moves along a tape and only state change of the head is observed. Details will be laid out in the following sections. In this contribution the S spin is always an electron spin whereas the I spins are nuclear spins. In this case the advantage of this concept is very pronounced, because the high spin polarization of the electron spin can be used to reach highly polarized and correlated states of the nuclear spins. The principle is, however, rather general and the S and I spins could be any other spins. 5.1 S-Bus Structure A typical topology of the S-Bus is presented in Fig. 16. The dominant coupling considered here is the hyperﬁne coupling aj between the electron spin S and the diﬀerent nuclear spins Ij . The coupling constants aj will in general be diﬀerent for the diﬀerent nuclei. The internuclear interaction will be dipoledipole interaction in solid samples (considered here) or else scalar couplings in liquid samples. In any case the internuclear interactions are considered weak compared with the hyperﬁne interactions which are orders of magnitude larger, i.e. aj , ak Djk with Djk being the internuclear coupling. We will show in the following that one can prepare highly correlated nuclear spins, just through their interaction with the electron spins S, even in the limit where Djk = 0 and the initial correlation among the nuclear spins is zero. The topology in Fig. 16 should not suggest that all coupling constants aj are equal. In general the coupling constants will be diﬀerent and the distance between the nuclei also varies considerably. 5.2 A Multi Qubit Solid State S-Bus System In the preceding section we have investigated the entanglement between an electron spin and a single nuclear spin. The S-Bus system in its genuine sense

S I1 I2 I3

…I

Fig. 16. Basic S-Bus topology

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−

Fig. 17. A qubyte, consisting of eight with lattice constant a = 0.546 nm

F nuclei surrounding a Ce3+ ion in CaF2

implies a large number of nuclear spins coupled to a single electron spin. As an example we present here a Ce3+ with eﬀective electron spin 1/2 replacing a Ca2+ ion in CaF2 as displayed in Fig. 17. The Ce+ ions represents a ﬁctitious electron spin S = 1/2 with large ganisotropy due to spin-orbit interaction. The combined orbital and spin states have been discussed in detail in the literature and will not be dwelled on here. This S spin together with the hyperﬁne coupled eight near neighbor 19 F nuclear spins I comprises our qubyte S-Bus system. There is also the charge compensating F− ion which also shows appreciable hyperﬁne interaction and can be considered as another qubit. This and the weaker hyperﬁne couplings to the further distant nuclei will not be considered here. We note that this center possesses C4 symmetry and can be viewed as consisting of two layers of four ﬂuorines, one near the F− ion (layer 1) and another layer 2 opposite to the F− ion. Their isotropic hyperﬁne interaction appears to be diﬀerent, thus rendering the two layers inequivalent. Still all four ﬂuorines of each layer would be magnetically equivalent without anisotropic hyperﬁne interaction. Due to this anisotropic interaction all eight ﬂuorines become nonequivalent for certain orientations of the magnetic ﬁeld. This allows to address any of the ﬂuorines individually as is obvious from the related ENDOR spectrum displayed in Fig. 18. The general pulse sequence for performing quantum computing in the SBus system is depicted in Fig. 19. It consists of a series of pulses applied to the electron spin S and individually to each nuclear spin Ij which takes part in the quantum algorithm. Detection of the ﬁnal quantum state is through the electron spin echo. The preparation segment is responsible for creating highly correlated nuclear spin states. The excitation at nuclear spin transitions creates pseudopure initial density matrices, performs quantum algorithms and applies density matrix tomography.

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Fig. 18. ENDOR spectrum of the qubyte consisting of the eight near neighbours to the Ce3+ S spin

πy

2 y

S (ESR)

I1 (NMR)

manipulation I1

I2 (NMR)

manipulation I2

preparation PS qc algorithm Aj

detection DS

Fig. 19. General Pulse sequence for S-Bus quantum computing

5.3 Creating Multi Nuclear Correlations Multi nuclear correlations are readily correlated out of a totally uncorrelated Boltzmann state as represented just by the electron spin Zeeman interaction (10). Simply the application of two π/2-pulses separated by a free evolution time τ as is used in the Mims type pulsed ENDOR [22, 37] suﬃces to create highly correlated nuclear spin states. More elaborate sequences which ﬁlter out certain hyperﬁne interactions are possible. In the two pulse sequence one usually applies the same phase (e.g. yy) for the π/2-pulse pair. If we let the residual coherences after the second pulse decay, the diagonal part of the density matrix can be expressed as ρ0 (τ ) = (ab) −Sz ρI (τ ), where ab ∈ {yy, yx} and   N (yy) ρI (τ ) = Re  Pj  with Pj = eiaj τ Izj (36) j=1

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and where aj is the hyperﬁne interaction of nucleus j [22, 36]. Note that only even products of spin operators appear in this expression when one expands the factors. In order to get odd numbered operator products one could apply a yx phase for the initial pulse pair in the sequence. In this case the imaginary part of (36) applies   N (yx) (37) Pj  . ρI (τ ) = Im  j=1

In the special case of I = 1/2 one arrives at   N (yy) ρI (τ ) = Re  (cj I0 + i2Izj sj )

(38)

j=1



and (yx)

ρI

(τ ) = Im 

 (cj I0 + i2Izj sj ) ,

(39)

j=1

where I0 is the one qubit identity matrix and we have used the abbreviations cj = cos( 12 aj τ ) and sj = sin( 12 aj τ ). By this technique an arbitrary degree of nuclear spin correlations can be obtained depending on the hyperﬁne interactions and the delay time τ [22, 36]. (ab) We further note that ρI (τ ) represents the S-Bus or sublevel density matrix which refers either to the mS = +1/2 or mS = −1/2 electron spin state. In general we want to extract a submatrix of a certain number n of spins in an N spin S-Bus system. As an example we extract the two qubit part for spin I1 and I2 which can be expressed as [35, 36] (yy)

ρI2

(2)

= C0 I0 − 2(C1 Iz1 + C2 Iz2 ) − 4C12 Iz1 Iz2 ,

(40)

where C0 = KR c1 c2 , C1 = KI s1 c2 , C2 = KI c1 s2 and C12 = KR s1 s2 and with     N N 1 1 2 = 1 + cos(aj τ ) and KI2 = 1 − cos(aj τ ) . (41) KR 2 2 j=3 j=3 This procedure is readily extended to an arbitrary number of sub-spins as is shown elsewhere [35, 36]. 5.4 Multiple Quantum ENDOR In order to detect a nuclear spin correlated state we apply Multiple Quantum ENDOR (MQE) as ﬁrst introduced in [34]. In its simplest version two π/2pulses are applied to every nuclear spin transition under consideration. For this discussion we set the delay time between the pulses to zero and introduce

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a phase shift π + φ to the second pulse. As an example we consider the case N = 2 (40). After the ﬁrst pulse all z-components of the nuclear spin operators are converted into x-operators. For φ = 0 the second −y-pulse would just reconvert the x-operators back to z. In order to see how the density matrix ρI changes for φ = 0 we perform the corresponding transformation and obtain (yy)

ρI2

= C0 I0 − 2(C1 cos φ1 Iz1 + C2 cos φ2 Iz2 )− 4C12 cos φ1 cos φ2 Iz1 Iz2 , (42)

where we have assumed that all oﬀ-diagonal components have decayed after the second pulse. This state is reached after some delay time. The generalization to an arbitrary number N of nuclear spins can readily be written down [35, 36]. An early example of a multiple quantum ENDOR spectrum with nonselective excitation where all phase angles φ were varied in increments ∆φ = 2πνφ ∆t which deﬁnes the phase frequency νφ and the virtual time increment ∆t was already published in [34]. After Fourier transform of the phase interferogram a spectrum is obtained with multiple quantum lines appearing at integers of the base frequency νφ . In Fig. 20 we demonstrate the eﬀect of applying individual phase rotation frequencies ν1 = 0.9 MHz and ν2 = 1.1 MHz to two diﬀerent spins I1 and I2 . Note the appearance of single quantum (1Q) lines at ν1 and ν2 due to the linear spin components in the density matrix and additional lines at ν1 ± ν2 (zero (0Q) and double quantum (2Q) lines) due to the bilinear component. Extension to a larger number of correlated spins is straightforward [35, 36].

Fig. 20. MQE-interferogram and MQE-spectrum of a two spin correlated state with single spin frequencies ν1 = 0.9 MHz and ν2 = 1.1 MHz

5.5 Creating Pseudopure Nuclear Spin States The S-Bus density matrix represented by (36) or (37) depends on the hyperﬁne interactions aj and the delay time τ . In general one will choose an optimum value of τ for a given distribution of hyperﬁne interactions. Except for fortuitous cases this will not correspond to a pure or pseudopure state. We

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therefore need to prepare a pseudopure state. This can be achieved simply by applying a Py (β) pulse to each addressable nuclear spin after the S spin preparation sequence PS . The prefactors in front of the spin operators can also be modiﬁed at will by the MQE sequence discussed in Sect. 5.4. After the decay of transient components the corresponding nuclear spin transition is scaled by cos(β). For this we ignore the identity matrix which will be added appropriately later. Suppose we want to prepare the following truncated pseudopure density matrix 1 1 (12) (43) ρ00 = const. I0 + Iz1 + Iz2 + Iz1 Iz2 . 2 2 We simply need to modify the prefactors appropriately in order to prepare the pseudopure state. Examples of some pseudopure states are shown in Fig. 21. Note that the sign of the linear spin components of the density matrix is directly reﬂected in the sign of the 1Q signal, whereas the sign of the two spin correlated state depends on both spins. It can be read oﬀ from the MQE-spectrum directly from the 0Q and 2Q components. By evaluating the intensity of the MQE lines in reference to the as prepared intensities one can evaluate the density matrix components. Details of this procedure are published elsewhere [35, 36].

ρ00

0.5

1.5

ρ10

0.5

1.5

frequency [MHz] Fig. 21. MQE-spectra of the density matrices ρ00 and ρ10 with phase frequencies ν1 = 0.85 MHz and ν2 = 1.15 MHz

5.6 Deutsch-Jozsa Algorithm in the S-Bus The Deutsch-Jozsa algorithm was introduced in Sect. 1.2. It was already implemented in liquid state NMR [38, 39, 40, 41]. In order to implement it in the S-Bus system, we need to perform unitary transformations according to the block diagram Fig. 5. The ﬁrst Hadamard transform was realized by Py (π/2) pulses at two nuclear spins and the second Hadamard transform by the inverse pulse. The unitary transformations representing the diﬀerent functions f (ab) according to Table 1 were implemented by phase shifts [35, 36]. The two

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balanced functions correspond to identity matrices and therefore represent a NOP (no operation). As an example for the balanced functions we consider the transformation   10 0 0 0 1 0 0   U0011 = eiπIz2 =  (44)  0 0 −1 0  . 0 0 0 −1 In a similar way all other transformations can be implemented by applying a π rotation to the other spin or both spins together. As an example we display in Fig. 22 the result of the operations U0000 and U0011 on the initial density matrix ρ00 . More examples are presented elsewhere [35, 36]. Under the U0000 transformation the initial density matrix ρ00 is unchanged as expected since it represents a constant function. Note, however, that the DJ algorithm changes the initial state if the transformation represents a balanced function. This fulﬁlls the requirement of the DJ algorithm, namely that a balanced function leads to an output diﬀerent from ρ00 . The tomography of the ﬁnal state density matrices when applying the balanced transformations U0101 , U0011 , U1001 results in fact in ρ01 , ρ10 and ρ11 . The other three balanced functions are equivalent in the sense that they just involve a sign change of the unitary transformations which has no eﬀect on the outcome. We note that no entangled states are involved in the two qubit DJ algorithm as demonstrated here. In a three qubit DJ algorithm, however, some of the balanced functions involve entangled states. We have also implemented entangled nuclear spin states within the S-Bus system which will be discussed elsewhere. ν1 ν2 − ν1

ν2

ν1 ν2 + ν1 ν2 − ν1

ν2

ν2 + ν1

Fig. 22. MQE-spectra of the density matrices after applying the DJ algorithm with the constant function U(0000) (left) and the balanced function U(0011) (right). ν1 and ν2 are the single spin frequencies

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Acknowledgements We gratefully acknowledge ﬁnancial support by the BMBF, the Landesstiftung Baden W¨ urttemberg and the Fond der Chemischen Industrie.

References 1. 2. 3. 4. 5.

6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

24. 25. 26. 27.

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Laser-Assisted Magnetic Resonance: Principles and Applications D. Suter and J. Gutschank Universit¨ at Dortmund, Fachbereich Physik, 44221 Dortmund, Germany [emailprotected]

Abstract. Laser radiation can be used in various magnetic resonance experiments. This chapter discusses a number of cases, where laser light either improves the information content of conventional experiments or makes new types of experiments possible, which could not be performed with conventional means. Sensitivity is often the main reason for using light, but it also allows one to become more selective, e.g. by selecting signals only from small parts of the sample. Examples are given for NMR, NQR, and EPR spectra that use were taken with the help of coherent optical radiation.

1 Introduction The interest in the ﬁeld of magnetic resonance spectroscopy is based largely on the huge potential for applications: spins can serve as probes for their environment because they are weakly coupled to other degrees of freedom. In most magnetic resonance experiments, these couplings are used to monitor the environment of the nuclei, like spatial structures or molecular dynamics. While the direct excitation of spin transitions requires radio frequency or microwave irradiation, it is often possible to use light for polarizing the spin system or for observing its dynamics. This possibility arises from the coupling of spins with the electronic degrees of freedom: optical photons excite transitions between states that diﬀer both in electronic excitation energy as well as in their angular momentum states. 1.1 Motivation Some motivations for using light in magnetic resonance experiments include • Sensitivity: In many cases, the possible sensitivity gains are the primary reason for using optical methods. Compared to conventional NMR, sensitivity gains of more than 10 orders of magnitude are possible. The ultimate D. Suter and J. Gutschank: Laser-Assisted Magnetic Resonance: Principles and Applications, Lect. Notes Phys. 684, 115–141 (2006) c Springer-Verlag Berlin Heidelberg 2006 www.springerlink.com

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limit in terms of sensitivity was reached in 1993, when two groups showed that it is possible to observe EPR transitions in single molecules [1, 2]. The same technique was later used to observe also NMR transitions in a single molecule [3]. • Selectivity: Lasers can be used to selectively observe signals from speciﬁc parts of the sample, like surfaces, at certain times which may be deﬁned by laser pulses with a resolution of 10−14 s, or from a particular chemical environment deﬁned, e.g., by the chromophore of a molecule or the quantum conﬁned electrons in a semiconductor. • Speed: Magnetic resonance requires the presence of a population diﬀerence between spin states to excite transitions between them. In conventional magnetic resonance, this population diﬀerence is established by thermal relaxation through coupling with the lattice, i.e. the spatial degrees of freedom of the system. At low temperatures, this coupling process may be too slow for magnetic resonance experiments. In the case of optical excitation, the population diﬀerences are established by the polarizing laser light. Depending on the coupling mechanism, this polarization process can be orders of magnitude faster than the thermal polarization process, independent of temperature. • Electronically excited states: If information about an electronically excited state is desired that is not populated in thermal equilibrium, it may be necessary to use light to populate this state. It is then advantageous to populate the diﬀerent spin states unequally to obtain at the same time the polarization diﬀerences that are needed to excite and observe spin transitions. 1.2 What Can Lasers Do? Light can support magnetic resonance experiments in diﬀerent ways. They can, e.g., initiate a chemical reaction that one wishes to observe, like in photosynthetic processes. These light-induced modiﬁcations of the sample will not be considered here; instead we concentrate on the use of light for the magnetic resonance experiment, where light aﬀects directly the spin degrees of freedom, rather than spatial coordinates. Typically, the laser is then used either to increase or to detect the spin polarization of nuclear or electronic spins. These two approaches are largely independent of each other: It is, e.g., possible to use optical pumping to enhance the spin polarization and observe the transitions with a conventional NMR coil; conversely, optical detection can be used with or without increasing the population diﬀerence with laser light. In many cases, however, it is advantageous to combine both approaches. In some cases, a single laser beam may provide an increase of the spin polarization and an optical signal that can be related to a component of the magnetization. In others, a pump-probe setup separates the excitation and detection paths.

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In addition to these applications of lasers, light can also be used to drive the dynamics of spin systems, e.g., through Raman transitions [4]. For this review, however, we will concentrate on the issues of increasing the spin polarization and on optical detection.

2 Optical Polarization of Spin Systems Magnetic resonance spectroscopy requires a spin polarization inside the medium. In conventional magnetic resonance experiments, this polarization is established by thermal contact of the spins with the lattice. This process is relatively slow, especially at low temperatures, where relaxation times can be many hours, and it leads to polarizations that are limited by the Boltzmann factor. Photon angular momentum, in contrast, can be created in arbitrary quantities with a polarization that can be arbitrarily close to unity. If it is possible to transfer this polarization to nuclear or electronic spins, their polarization can increase by many orders of magnitude. A number of diﬀerent approaches have been used to achieve this goal. The oldest and best known approach is known as optical pumping [5]; it was originally demonstrated on atomic vapors [6] and later applied to condensed matter. While optical pumping allows one to create very high spin polarization in atomic vapors, it is less suitable for applications to anisotropic systems such as low symmetry solids. Other techniques were therefore developed, which can still be used in such an environment. While optical pumping was originally implemented with conventional light sources, most of the other approaches require the use of coherent optical radiation, i.e. laser light.

Fig. 1. Four ways for optically increasing the spin polarization

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2.1 Optical Pumping The possibility to use optical radiation for exciting and detecting spin polarization can be traced back to the angular momentum of the photon. Photons as the carriers of the electromagnetic interaction carry one unit () of angular momentum, which is oriented either parallel or antiparallel to the direction of propagation of the light. In an isotropic environment, angular momentum is a conserved quantity. When a photon is absorbed by an atom or molecule, its angular momentum must therefore be transferred to the atom. The resulting angular momentum of the atom is equal to the vector sum of its initial angular momentum plus the angular momentum of the absorbed photon. The use of angular momentum conservation for increasing the population diﬀerence between spin states was ﬁrst suggested by Alfred Kastler [7, 8, 5]. If an atom is irradiated by circularly polarized light, the photons have a spin quantum number ms = +1. Since the absorption of a photon is possible only if both, the energy and the angular momentum of the system are conserved, the atoms can only absorb light by simultaneously changing their angular momentum state by one unit. After the atom has absorbed a photon it will reemit one, decaying back into the ground state. Spontaneous emission can occur in an arbitrary direction in space and is therefore not limited by the same selection rules as the excitation process with a laser beam of deﬁnite direction of propagation. The spontaneously emitted photons carry away angular momentum with diﬀerent orientations and the atom can therefore return to a ground state whose angular momentum state diﬀers by ∆m = 0, ±1. The net eﬀect of the absorption and emission processes is therefore a transfer of population from one spin state to the other and thereby a polarization of the atomic system. 2.2 Spin Exchange Spin polarization can be transferred between diﬀerent reservoirs not only within one atomic species, but also between diﬀerent particles. This was ﬁrst demonstrated by Dehmelt who used transfer to free electrons to polarize them [9]. Another frequently used transfer process uses optical pumping of alkali atoms, in particular Rb and Cs and transfer of their spin polarization to noble gas atoms like Xe. These atoms cannot be optically pumped from their electronic ground state (although He can be pumped in the metastable state [10, 11]); spin exchange allows one to optically pump an alkali gas (typically rubidium) and transfer the spin polarization from there to the Xe nuclear spin. This method was pioneered by Happer [12], applied to the study of surfaces [13, 14], and used in a number of medical applications [15, 16, 11]. The transfer from alkali to noble gas atoms is relatively eﬃcient when the two species form van der Waals complexes. During the lifetime of this quasi-molecule, the two spins couple, mainly by dipole-dipole interaction. This

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coupling allows simultaneous spin ﬂips of the two species which transfer polarization from the Rb atoms to the Xe nuclear spin. Typical cross-polarization times are on the order of minutes, but the long lifetime of the Xe polarization permits to reach polarizations close to unity. The spin polarization survives freezing [17] and can be transferred to other spins by thermal mixing [18]. 2.3 Excited Triplet States In many classical optically detected magnetic resonance experiments, absorption of light excites the system into a singlet state that can, through nonradiative processes, decay into a triplet state, whose energy is below the excited singlet state. This intersystem conversion process as well as the decay of the triplet state can be spin-dependent, therefore creating a signiﬁcant spin polarization of the triplet state. In many systems, these processes are quite eﬃcient, even for unpolarized light, generating a high degree of spin polarization in the triplet state. Under certain conditions, this polarization of the electron spin can also lead to a polarization of the nuclear spin, which survives when the molecule returns to its ground state. 2.4 Spectral Holeburning When the spin is located in a host material with low symmetry, the electronic angular momentum is quenched. Figure 2 shows the situation schematically: While angular momentum states with total angular momentum J are 2J + 1 fold degenerate in free space, the Coulomb interaction of the atom or ion with neighboring charges (electrons and nuclei) lifts this degeneracy. The resulting states are usually no longer angular momentum eigenstates. While this argument applies directly only to orbital angular momentum, the spin-orbit interaction often is strong enough to also quench the electron spin. If the angular momentum is quenched, optical pumping with circularly polarized light becomes ineﬃcient for excitation of spin polarization. In these systems, other approaches may increase spin polarization. One possibility exists when the diﬀerent spin states can be distinguished in frequency space, Crystal field Free atom

degenerate angular momentum states

degeneracy lifted

Fig. 2. Quenching of angular momentum by interaction with the crystal ﬁeld

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i.e. when the energy diﬀerence between them is larger than the hom*ogeneous width of a suitable optical transition. The situation is shown schematically in the lower right of Fig. 1. The laser only excites those ground state atoms whose spin is in the ↑ state. Since the excited state can decay to both ground states, the population accumulates in the |g↓ state. This allows one to use a laser to selectively depopulate one of the spin states, while increasing the population of the other states. Since the inhom*ogeneous width of the optical transitions is usually large compared to the energy of magnetic resonance transitions, it is rarely possible to address only a single spin state. The laser frequency selects then a subset of all the spins, for which the resonance condition is fulﬁlled; only for those systems, the spin polarization will be increased. This situation is known as spectral holeburning, since the depopulation of speciﬁc spin states reduces the absorption of light at the frequency of the pump laser beam. Additional details are discussed in the context of optical detection.

3 Optical Detection Any magnetic resonance experiment includes a scheme for detection of timedependent components of the spin polarization, usually as a macroscopic magnetization. In NMR, the precessing transverse magnetization changes the magnetic ﬂux through the radio frequency (rf) coil. According to Faraday’s law, the time derivative of the ﬂux induces a voltage over the coil, which is detected as the free induction decay (in pulsed experiments) or as a change in the impedance of the coil (in continuous wave experiments). The optical detection schemes that we discuss here can sometimes replace this inductive detection. They can be used together with optical polarization or they can be combined with conventional excitation schemes. In suitable systems, optical detection provides a number of advantages over the conventional method: First, optical radiation introduces an additional resonance condition, which can be used to distinguish diﬀerent signal components and thereby separate the target signal from backgrounds such as impurities. Second, optical radiation can be detected with single photon sensitivity (in contrast to microwave or radio frequency radiation). This has made detection of single spins possible in suitable systems. A third possible use of the optical radiation is that the laser beam breaks the symmetry of isotropic samples, such as powders or frozen solutions. As we discuss in Sect. 5.3, this allows one to derive the orientation of tensorial interactions, such as electron g-tensors or optical anisotropy tensors from non-oriented samples. 3.1 Circular Dichroism An early suggestion that magnetic resonance transitions should be observable in optical experiments is due to Bitter [19]. The physical process used in

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|e>

|g> Fig. 3. Optical detection through circular dichroism

such experiments may be considered as the complement of optical pumping: the spin angular momentum is transferred to the photons and a polarization selective detection measures the photon angular momentum. Figure 3 illustrates this for the same model system that we considered for optical pumping. Light with a given circular polarization interacts only with one of the ground state sublevels. Since the absorption of the medium is directly proportional to the number of atoms that interact with the light, a comparison of the absorption of the medium for the two opposite circular polarizations yields directly the population diﬀerence between the two spin states. This population diﬀerence is directly proportional to the component of the magnetisation parallel to the laser beam. Early experimental implementations of these techniques were demonstrated in atomic vapors [20, 21, 5], where angular momentum conservation is exact and the principle is directly applicable. Similar considerations hold also for solid materials [22], although, as we discussed above, angular momentum is not always a conserved quantity in such systems. It depends therefore on the symmetry of the material if absorptive detection is possible [23]. Nevertheless, even small optical anisotropies can be measured; changes in these parameters upon saturation of the spins provide a clear signature of magnetic resonance transitions [24]. While most implementations measure the longitudinal spin component by propagating a laser beam parallel to the static magnetic ﬁeld, it is also possible to observe precessing magnetization with a laser beam perpendicular to the static ﬁeld [25]. The two approaches provide complementary information [26] and a combination of longitudinal and transverse measurements is therefore often helpful for the interpretation of the spectra. 3.2 Photoluminescence Photoluminescence is another important tool for measuring spin polarization. Depending on the system, the intensity or the polarization of spontaneously emitted photons can be a measure of the spin polarization in the ground – or in an electronically excited state. In free atoms, angular momentum conservation imposes correlations between the direction and polarization of the spontaneously emitted photons which depend on the angular momentum state of the excited atom. Photoluminescence has therefore long been used to measure spin polarization in electronically excited states [27, 28].

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Hanle effect

Polarization of luminescence Sz

B S(τ) ωL

0 ∆B

Fig. 4. Hanle eﬀect: A magnetic ﬁeld perpendicular to a circularly polarized excitation laser forces Larmor precession of the electron spins. The spin polarization of the excited electrons decreases therefore with increasing magnetic ﬁeld strength

Spin polarization in the electronic ground state also aﬀects the photoluminescence, since the absorption of polarized light depends on the spin state. If the spin orientation prevents absorption of light, the intensity of the photoluminescence decreases correspondingly. The intensity and polarization of the photoluminescence can therefore serve for detecting ground state spin polarization and, e.g., by saturation with a resonant rf ﬁeld, for detecting magnetic resonance transitions [29, 30]. The eﬀect of Larmor precession on the spin polarization of excited states has been observed as early as 1924 by Hanle [31]. He noticed that the polarization of the photoluminescence decreases if a magnetic ﬁeld is applied perpendicular to the direction of the spin polarization (Fig. 4). The observed polarization of the photoluminescence changes with the ﬁeld B0 as Sz =

∆B 2 , ∆B 2 + B02

(1)

where the width ∆B = (Γr + Γs )/γ is determined by the gyromagnetic ratio γ and the relaxation rates Γs and Γr of the spin and excited state population. The Hanle eﬀect can also be observed in four-wave mixing experiments [32] in atomic vapors as well as in crystals [33]; in this case, signiﬁcant polarization of the photoluminescence is only obtained if the crystal has high enough symmetry and mechanical strain is small enough to avoid depolarization. It is particularly suitable for measuring spin polarization in semiconductors with a direct band gap, such as GaAs [34]. 3.3 Coherent Raman Scattering Raman processes are optical scattering processes in which the frequency (and therefore the wavelength) of the scattered light diﬀers from that of the incident light [35]. The energy diﬀerence between the incident and the scattered photon is absorbed (or emitted) by excitations of the material in which the scattering occurs. While this excitation of the material is often a vibration, it can also be

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|3> Laser Laser

|1>

|2> Microwaves

Fig. 5. Coherent Raman scattering from a three-level system. Laser excitation creates a coherence between levels |1 and |3 and microwaves between |1 and |2. The resulting non-linear polarisation in the third transition creates a Raman wave

associated with spin degrees of freedom, in which case the scattering process can be used to detect magnetic resonance transitions. Figure 5 shows the relevant process for the simplest possible case: The two states |1 and |2 represent two spin states of the electronic ground state, while |3 is an electronically excited state. If a microwave ﬁeld (rf in the case of nuclear spin transitions) resonantly excites the transition between states |1 and |2, it creates a coherence between the two spin states. Similarly, the laser excites an optical coherence in the electronic transition |1 ↔ |3. Since the two transitions share state |1, the two ﬁelds create a superposition of all three states, which contains coherences not only in the two transitions that are driven by the external ﬁelds, but also in the third transition |2 ↔ |3. If this transition has a non-vanishing electric dipole moment, this coherence is the source of a secondary optical wave, the Raman ﬁeld. As the ﬁgure shows, the frequency of this wave diﬀers from that of the incident wave by the frequency of the microwave ﬁeld. It has the same spatial dependence as the incident laser ﬁeld and therefore propagates in the same direction. If the two optical ﬁelds are detected on a usual photodetector (photodiode or photomultiplier), they interfere to create a beat signal at the microwave frequency. The type of scattering process used for magnetic resonance detection is referred to as “coherent” Raman scattering [36] since the Raman ﬁeld is phase-coherent with the microwave as well as with the incident laser ﬁeld. This is an important prerequisite for the detection process: If the laser frequency drifts, the frequency of the incident ﬁeld as well as that of the Raman ﬁeld are shifted by the same amount. As a result, the diﬀerence frequency is not aﬀected and the resolution of the measurement is not aﬀected by laser frequency jitter or broad optical resonance lines [37]. Coherent Raman processes provide therefore a combination of high resolution with high sensitivity. Like in conventional magnetic resonance experiments, the excitation of the magnetic resonance transition indicated in Fig. 5 can be performed either in a continuous (cw) [38] or pulsed [39, 40] mode. Furthermore, the microwave or rf

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Laser frequency

Magnetic field

Fig. 6. Two-dimensional optically detected EPR (ODEPR) spectrum as a function of the laser frequency and the magnetic ﬁeld strength. The result is a complete microwave resonance spectrum for each laser wavelength. The projections on the axes represent the conventional EPR and absorption spectra

ﬁeld can be replaced by optical ﬁelds, applied to the two electronic transitions, that can excite the spin coherence by another Raman process [41, 42, 43]. Since the coherence that generates the signal is excited by two resonant ﬁelds, it depends on the frequencies of both ﬁelds. As shown in Fig. 6, the resulting signal is doubly resonant and contains therefore information about the optical as well as the magnetic resonance transition. As with other twodimensional experiments, it allows one to correlate information from the two frequency dimensions. Examples that demonstrate this feature will be discussed in Sect. 5. While we have discussed the process here as involving magnetic resonance transitions in the ground state, equivalent processes are also possible that relate to spins of electronically excited states. 3.4 Spectral Holeburning In Sect. 2.4, we discussed how narrowband lasers that cause spectral holeburning can increase the polarization of spins, in analogy to optical pumping. In most such experiments, a second laser beam, whose frequency can be swept around the frequency of the pump beam, is used to monitor the changes in the populations. The resulting spectra are known as holeburning spectra [44]. As shown in Fig. 7, holeburning requires a pump and a probe laser beam. The pump laser modiﬁes the population of those atoms for which the laser frequency matches an electronic transition frequency. When the probe laser hits the same transition, the absorption is reduced in line with the smaller population of the relevant ground state. The population that has been removed from this state is accumulated in the other spin state. When the probe laser frequency is tuned to the transition from this ground state to an electronically excited state, it ﬁnds increased absorption, which is referred to

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Absorption

Antihole: increased absorption Spectral hole: reduced absorption

νP νT

0 Fig. 7. Schematic representation of a holeburning spectrum. The hole represents reduced absorption, the “antiholes” increased absorption. The separation between hole and antihole is equal to the transition frequency of the two spin states

as an “antihole”. The separation between the hole and antihole matches the energy diﬀerence between the two spin states and the hole burning spectrum can therefore measure magnetic resonance transition frequencies [45]. While this discussion has centered on spin transitions between electronic ground states, the procedure also allows one to measure energy diﬀerences between spin states of electronically excited states [46]. The optical detection techniques discussed here were chosen to represent the most frequently used approaches. There are several additional techniques which cannot be discussed, which include purely optical techniques like photon echo modulation [47, 48].

4 Applications to NMR and NQR 4.1 Rare Earth Ions Ions of rare earth elements have been studied extensively with high resolution optical spectroscopy [49, 50]. The relevant optical transitions are between f electron states and have relatively small hom*ogeneous and inhom*ogeneous broadening. Figure 8 shows the relevant energy levels for the 141 Pr ion doped into the host material YAlO3 . The electronic ground state as well as the electronically excited states are split into substates that diﬀer with respect to their nuclear spin coordinates. The separation between these spin states is due to nuclear quadrupole coupling and second order hyperﬁne coupling, which combine into an eﬀective quadrupole interaction [51, 52]. One approach to measure the NQR transition frequencies is by holeburning spectroscopy, as shown in Fig. 9. Since pump and probe laser beam can each

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0.92 MHz 1.57 MHz

Electronic transition

141

Pr quadrupole splittings (I=5/2) 7.06 MHz

3H 4

14.11 MHz

Fig. 8. Relevant level scheme for

141

Pr doped into YAlO3 Excited state splittings

Ground state quadrupole splittings

Theory

Commercial laser

Stabilized laser

-20

-10

Laser frequency difference [MHz] Fig. 9. Holeburning spectrum of Pr: YAlO3

be resonant with nine diﬀerent transitions (from three ground- to three excited states), the holeburning spectrum, which depends on the diﬀerence between the pump and probe laser frequencies, has a total of 81 resonances. A number of these resonances have identical frequencies (e.g., νP − νT = 0), resulting in a total of 49 distinguishable frequencies. The width of each resonance line increases with the laser frequency jitter. As the comparison of the middle and lower traces shows, it is therefore important to use a narrowband laser for measuring these spectra. In this example, the laser linewidth was ≈ 30 kHz; the width of the observed resonance lines was therefore close to the hom*ogeneous width of the optical transition [46]. The amplitudes of the individual resonance lines depend on the optical transition matrix elements and are proportional to the overlap integral χg |χe of the ground and excited state nuclear spin states. While the nuclear spin is not involved in the electronic transition, the electronically excited state can have diﬀerent quantization axes than the ground state if, as in this example, the eﬀective quadrupole interaction changes with the electronic excitation. The precise measurement of the holeburning spectrum allows one then to determine the relative orientation of the principal axis system between the ground- and excited states [46].

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Pr3+:YAlO3 1D

| 1/2> | 3/2> | 5/2>

λ = 611 nm

| 1/2> | 3/2>

3H 4

| 5/2>

frequency (MHz) Fig. 10. Raman-heterodyne detected NQR spectra of Pr:YAlO3 . All six resonances can be detected in a single wide frequency scan, but the low frequency part, which is associated with the NQR transitions in the electronically excited state has been expanded in the upper part of the ﬁgure

The spectrum can be simpliﬁed considerably if the Raman scattering experiment is used instead. In this case, every nuclear spin transition gives rise to a single resonance at the transition frequency. Figure 10 shows an example of a coherent Raman spectrum of Pr:YAlO3 in zero magnetic ﬁeld, which was recorded by irradiating the optical transition at 611 nm with a laser beam and measuring the Raman-heterodyne signal while sweeping the rf frequency. In the example shown here, the three NQR transitions at 7, 14, and 21 MHz occur within the electronic ground state, while the three low-frequency transitions (< 3 MHz) belong to the electronically excited state. 4.2 Sign Information An interesting case of additional information that is unavailable with conventional techniques is the sign of the nuclear quadrupole interaction. As is well known [53], conventional magnetic resonance experiments cannot provide the sign of the quadrupole coupling. In the simplest case of axial symmetry, the Hamiltonian HQ of the nuclear quadrupole interaction is given by a coupling constant D times the square of the nuclear spin operator Iz , HQ = DIz2 . The coupling constant D is determined by the size of the nuclear quadrupole moment and the electric ﬁeld gradient. It can be measured either in the absence of a magnetic ﬁeld, which corresponds to the case of pure quadrupole coupling, or in a high magnetic ﬁeld, which corresponds to the case of highﬁeld NMR. Figure 11 shows schematically the NQR (zero magnetic ﬁeld) and NMR spectra of a spin I = 5/2 with axial quadrupole interaction. The spectra for the positive and negative coupling constant D are identical, unless the spin

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Levels (B=0)

NQR

NMR

HR optical

m = 5/2

D>0

3/2 1/2

Frequency

m = 1/2 3/2

D 1, we use the term multiple-photon transition throughout this work. During a multiple-quantum transition the radiation frequency matches the energy diﬀerence between the two levels involved in the transition so that, like in a single-quantum transition, only one photon is absorbed or emitted. On the other hand, during a multiple-photon transition several photons are absorbed and/or emitted.

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century [8]. Also in magnetic resonance multiple-photon phenomena have been known for a long time, but up to now extensive use of them as a spectroscopic tool has not yet been made. In any transition between two energy levels energy and angular momentum are conserved. As a consequence, during a multiple-photon process in magnetic resonance the energy and the polarization of the photons involved in the process must correspond to the total change in energy and angular momentum in the spin system. Electrons have an intrinsic angular momentum characterized by the spin quantum number S = 1/2. When an electron spin is subjected to a static magnetic ﬁeld B 0 oriented along the laboratory z-axis, the projection of the angular momentum along this axis is either /2 or −/2, and the energy levels corresponding to the magnetic quantum numbers mS = 1/2 and mS = −1/2, are split owing to the Zeeman interaction. Photons have an intrinsic angular momentum , characterized by the spin quantum number J = 1. Photons in state mJ = −1 have angular momentum − relative to the direction of propagation and are called σ − photons (negative helicity), whereas photons in state mJ = 1 have angular momentum and are called σ + photons (positive helicity) [3]. Left and right circular ﬁelds perpendicular to B 0 can be associated with σ − and σ + photons propagating along B 0 . The state with projection mJ = 0 is forbidden, since a photon has a vanishing mass. However, a linear ﬁeld oriented parallel to B 0 can be associated with so-called π photons√propagating √ perpendicular to B 0 and described by the wave function π = ( 2/2)σ + + ( 2/2)σ − . The eigenstate of this function thus corresponds to mJ = 0, and absorption or emission of π photons does not change the total angular momentum of the spin system. In magnetic resonance radio frequency (rf) or microwave (mw) radiation ﬁelds are applied. A radiation ﬁeld with only one frequency is called a monochromatic ﬁeld. Correspondingly, we call a radiation ﬁeld with two frequencies a bichromatic ﬁeld, independent on whether the frequencies are very close to each other or diﬀer by orders of magnitudes. During a multiple-photon process, several photons are simultaneously absorbed and/or emitted from the radiation ﬁeld. The sum of the angular momenta of all absorbed σ photons must equal the change in angular momentum of the electron spin system expressed by the change in mS . There is no restriction to the number of absorbed or emitted π photons, since they do not add to the angular momentum. Several theoretical approaches have been developed to describe multiplephoton processes [9]. In magnetic resonance the Bloch equations [10, 11], or semiclassical theories, with the radiation ﬁeld treated as a classical observable, are frequently used. However, to understand the physics behind a multiplephoton process second quantization, where not only the spins but also the radiation ﬁelds are quantized [12, 13, 14], and Floquet theory [15, 16, 17] has

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to be applied. A brief introduction to second quantization and Floquet theory is given in the Appendix. In this Lecture Notes we discuss diﬀerent types of multiple-photon processes that can be observed in electron paramagnetic resonance (EPR) spectroscopy. Completeness in the cited literature is not aimed at. In pulse EPR spectroscopy multiple-photon resonances are also attractive from the point of view of spin dynamics and potential applications. In Sect. 2 we give a brief overview of the diﬀerent types of multiple-photon transitions that can be observed in EPR with monochromatic or bichromatic mw ﬁelds. Section 3 is devoted to multiple-photon transitions induced by a bichromatic radiation ﬁeld consisting of a transverse mw ﬁeld and a longitudinal rf ﬁeld. In Sect. 3.1 we describe how these processes manifest in continuous wave (cw) EPR. Although this type of multiple-photon transitions is omnipresent in cw EPR, most spectroscopists are not aware of the fact that they are dealing with them in their daily routine work. In Sect. 3.2 we demonstrate how multiple-photon transitions can be observed in pulse EPR. Finally, in Sect. 3.3 a new type of multiple-photon transitions induced by a bichromatic ﬁeld is introduced. We particularly stress on the phenomenon of π-photon-induced transparency and describe the characteristics of these transitions and their potential applications.

2 Diﬀerent Types of Multiple-Photon Transitions in EPR The energies of states of a paramagnetic species with an electron spin S and nuclei with spins I subject to a static magnetic ﬁeld B 0 are described by the static spin Hamiltonian H0 . The Hamiltonian consists of terms which describe the interactions between the electron spin and B 0 , as well as the interaction between electron spins and between electron and nuclear spins. The coupling between an electron spin and a linear radiation ﬁeld Bmw (t) = 2B1 cos(ωmw t) with amplitude 2B1 and mw frequency ωmw is expressed by the perturbation Hamiltonian H1 (t) [18]. In this Lecture Notes energies and amplitudes of radiation ﬁelds are usually given in angular frequency units, i.e., = E/ and ω1 = −γe B1 . In the great majority of EPR studies, the linear response of a spin system to a weak radiation ﬁeld is recorded. The transition amplitudes are then determined by the matrix elements of the perturbation operator H1 (t). The transition amplitude for a single-photon transition between the two eigenstates with energies 1 (labeled by 1) and 2 (labeled by 2) of the static Hamiltonian having wave functions φ1 and φ2 is given by b12 = φ1 |H1 |φ2 ,

(1)

where H1 is the time-independent perturbation Hamiltonian in a frame that rotates with the mw frequency ωmw (rotating frame). The computation of b12

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is most convenient in the eigenbasis of H0 , where b12 is the matrix element of H1 connecting the states 1 and 2. Due to state mixing, the magnetic quantum numbers of the electron and nuclear spins may no longer be good quantum numbers, and forbidden EPR transitions of the type (∆mS = ±1; ∆mI = ±1, ±2, . . .), or (∆mS = ±2, ±3, . . .) become weakly allowed. Since for these forbidden transitions, |∆mS | > 1 or |∆mI | > 0, they are often called multiple-quantum transitions. Note, however, that such a transition, where only one photon is absorbed or emitted, is still a single-photon process. During a single-photon transition, the spin system absorbs or emits one photon of a radiation ﬁeld perpendicular to the quantization direction of the spin, so that the angular momentum is conserved. During an electron spin transition with a positive g value or a nuclear spin transition with a negative gn value, the absorbed or emitted photon is a σ + photon (Fig. 1a). In the rare case of a negative g value [19], or for a nucleus with positive gn , the absorbed or emitted photons are σ − photons (Fig. 1b). (a)

σ+ 1

(b) σ+

σ−

Fig. 1. Energy level diagram for a single-photon transition in a two-level system, fulﬁlling energy and angular momentum conservation (curved arrows symbolize the polarization of the absorbed or emitted photon). (a) (mS = 1/2) > (mS = −1/2): One σ+ photon is absorbed or emitted. (b) (mS = −1/2) > (mS = 1/2): Order of the energy levels is reversed, one σ− photon is absorbed or emitted

In magnetic resonance linear radiation ﬁelds are used with a very few exceptions [19, 20, 21]. Such a linear ﬁeld consists of both σ − and σ + photons, since it can be considered as a superposition of a left and a right circular ﬁeld. The generation of a circular ﬁeld is experimentally more demanding, but has the advantage that the power needed to generate the same circular ﬁeld amplitude ω1 is reduced to one quarter compared to a linear ﬁeld, and that the Bloch-Siegert shift (see below) is absent. Note, however, that for many of the multiple-photon processes described in these Lecture Notes, both σ − and σ + photons are required, and transitions can not be induced by a circular radiation ﬁeld.

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2.1 Multiple-Photon Transitions in Spin Systems with Unequally Spaced Energy Levels In EPR spectroscopy multiple-photon transitions, where the magnetic quantum number changes by ∆mS > 1, have been observed in various spin systems with unequally spaced energy levels (see for example [22, 23, 24, 25, 26]). In this kind of multiple-photon transitions, which requires real intermediate energy levels, the ﬁnal state is reached without directly altering the populations of the intermediate states. For the description of these multiple-photon transitions, higher-order timedependent perturbation theory is usually applied. It has been found in cw EPR (and cw NMR), that (a) the unsaturated linewidth of an n-photon transition is 1/n of the linewidth of the single-photon transition, (b) in the absence of saturation eﬀects the intensity of the signal for an n-photon transition is proportional to ω12n−1 , (c) the intensity of an n-photon transition between levels with energy a and b depends on the position of the intermediate energy levels with respect to the (n − 1) equally spaced virtual levels between a and b [27]. One Radiation Field Perpendicular to B 0 In this type of multiple-photon transitions, two or more σ + photons of the same frequency are absorbed when the energy levels are almost equally spaced. In EPR these transitions can be observed either with a linear or a right circular radiation ﬁeld perpendicular to B 0 (Fig. 2 a). z

(a)

(b) 3 σ+

∆

2' y

σ+

B1 x

Fig. 2. (a) Field conﬁguration in the laboratory frame and, (b) energy level diagram for a two-photon transition induced in a three-level system with unequally spaced energy levels by one right circular radiation ﬁeld perpendicular to B 0 (σ + photons)

As an example, we consider the simple situation of a right circular radiation ﬁeld applied to a spin system consisting of three energy levels labeled by 1, 2, and 3 (Fig. 2b). In addition to the two single-photon transitions with ∆mS = ±1 between level 1 and 2, and level 2 and 3, a two-photon transition can be induced between level 1 and 3. If the states are not mixed the transition

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amplitude of the single-photon transition with ∆mS = ±2 is zero. For the twophoton transition, where two σ + photons of the same frequency ωmw = ω (13) /2 are absorbed, the transition amplitude is given by [28] b13 ∝ ω12

φ1 |S + |φ2 φ2 |S + |φ3 , ω (32) − ω (31) /2

(2)

with ω (32) = 3 − 2 , ω (31) = 3 − 1 , and the raising operator S + = Sx + iSy . Such a two-photon transition can be considered as a forbidden transition between level 1 and 3 with ∆mS = ±2, carried out in two allowed steps, ﬁrst from level 1 to an intermediate (virtual) level 2’, and then from level 2’ to level 3. The transition amplitude b13 in (2) is proportional to the square of the mw ﬁeld amplitude ω1 and inversely proportional to ∆ = ω (32) − ω (31) /2, where ∆ is the energy diﬀerence between level 2 and the virtual level 2’. Thus, to observe a two-photon transition with suﬃcient intensity, the amplitude of the pumping ﬁeld has to be large and ∆ has to be small. Corresponding formulas hold for transitions where three or more photons are absorbed. For multiplephoton transitions in spin systems with unequally spaced energy levels, the number of absorbed σ + photons is equal to the total change in mS of the electron spin, so that angular momentum is conserved. Two examples of multiple-photon transitions observed in electron spin systems with unequally spaced energy levels are given in Fig. 3. The spectrum of a Ni2+ -doped MgO single crystal with S = 1 is shown in Fig. 3a. When the spectrum is recorded with high mw power a broad single-photon transition (a)

(b)

2 B0 3 B0 Fig. 3. Multiple-photon transitions in spin systems with unequally spaced energy levels. (a) Two-photon transition (narrow center line) of Ni+2 in a MgO single crystal (adapted from [29]). (b) Multiple-photon transitions of atomic oxygen. 1: mw power 0.6 mw; 2: 4.5 mW (arrows mark two-photon transitions); 3: 42 mW (adapted from [30])

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superimposed by a narrow double-photon line is observed [29]. The broadening of the single-photon line is caused by lattice strains, which shift the state |0 by diﬀerent amounts relative to the states |−1 and |1. Since for the twophoton transition state |0 plays only the role of an intermediate level, the line remains narrow. The second example (Fig. 3b) shows the spectral changes observed in atomic oxygen in the gas phase as a function of the mw power [30]. At low power (0.6 mW) only single-photon transitions are observed. But already at a power of 4.5 mW the spectrum is dominated by three two-photon transitions (marked by arrows). At a power of 42 mW the two three-photon transitions are the most intense lines, while the single-photon transitions are buried in the noise. In all these EPR experiments where the multiple-photon transitions are directly observed, the line intensity is proportional to ω12n−1 . The situation is diﬀerent when the multiple-photon transitions are indirectly detected via the change in polarization of another transition. This is the case, for example, in cw electron-nuclear double resonance (ENDOR) experiments where nuclear frequencies are detected via desaturation of an EPR line. For a non-saturating rf ﬁeld the intensity of an n-photon transition is then proportional to ω12n . Figure 4 shows two examples for the observation of two-photon ENDOR transitions (marked by arrows) in a copper complex with two magnetically equivalent nitrogen ligands [28]. The intensity of the two-photon transitions depends on the parameter ∆ which can be varied via the orientation-dependent nuclear quadrupole interaction by rotating the crystal.

νrf / MHz

21.5

νrf / MHz

Fig. 4. Two-photon cw ENDOR transitions observed in bis(salicylaldoximato) copper(II), Cu(sal)2 , diluted into a Ni(sal)2 single crystal. Two diﬀerent orientations of the crystal are shown in the left and right panel. Bottom: low rf power, no two-photon transitions are observed; top: high rf power, arrows denote two-photon transitions (adapted from [28])

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Two Radiation Fields Perpendicular to B 0 The restriction ω (32) ≈ (3 − 1 )/2 for the intermediate energy level may be dropped by using a bichromatic ﬁeld perpendicular to B 0 with frequencies ωmw,1 and ωmw,2 and amplitudes ω1 = γe B1 and ω2 = −γe B2 , as is shown in Fig. 5a. z

(a)

(b) 3

σ+ ∆

σ+

B2 x

Fig. 5. (a) Field conﬁguration in the laboratory frame and, (b) energy level diagram for a two-photon transition induced in a three-level system with unequally spaced energy levels by two right circular radiation ﬁelds perpendicular to B 0 (σ + photons)

In this scheme, ωmw,1 and ωmw,2 fulﬁlling the resonance condition ωmw,1 + ωmw,2 = ω (31) , may be chosen such that ∆ = ω (21) − ωmw,1 is small for any value of 2 . The transition amplitude for this type of multiple-photon transitions is then given by b13 ∝ ω1 ω2

φ1 |S + |φ2 φ2 |S + |φ3 . ω (21) − ωmw,1

(3)

Again corresponding formulas hold for transitions where three or more photons are absorbed. An example for a two-photon transition induced by two mw frequencies is shown in Fig. 6 [31]. The system is again a Ni2+ -doped MgO single crystal. Line (a) or (c) is observed when the absorption of mw with frequency ωmw,1 or ωmw,2 is recorded. Line (b) in the center of the spectrum is attributed to the two-photon transition, where one photon of each radiation ﬁeld is absorbed. 2.2 Multiple-Photon Transitions in Two-Level Systems Multiple-photon transitions can also be induced between two energy levels when no real intermediate level exist. In a two-level system, or a multi-level system where only two levels are involved in the experiment, one has to distinguish between multiple-photon processes where either an odd or an even number of photons is absorbed.

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c +2

Fig. 6. Two-photon transitions of Ni in a MgO single crystal induced by two mw ﬁelds of diﬀerent frequencies (adapted from [31]). Top: ωmw,1 is swept, bottom: ωmw,2 is swept. (a), (c) indicate single-photon transitions, (b) indicates two-photon transition

One Linear Radiation Field Consider a two-level system with electron spin S = 1/2 interacting with a linear radiation ﬁeld with frequency ωmw . For an angle θ between this ﬁeld and the static ﬁeld B 0 , the laboratory frame Hamiltonian is given by H(t) = ωS Sz + 2ω1 cos(ωmw t) [cos(θ)Sz + sin(θ)Sx ] ,

(4)

with the electron Zeeman frequency ωS = −γe B0 . The two eigenstates of the unperturbed spin system are denoted by α and β, with energies α = ωS /2 and β = −ωS /2. To follow the evolution of the spin system under the Hamiltonian in (4) the Schr¨ odinger equation for the corresponding evolution operator has to be solved. To eliminate the time dependence of the Hamiltonian, Shirley implemented the Floquet theorem [15] as is outlined in the Appendix. Following this approach the transition probability between the two eigenstates of the time-dependent Hamiltonian H(t) can be found by using the corresponding Floquet Hamiltonian with matrix elements n−m + nωmw δαβ δnm , αn|HF |βm = Hαβ

(5)

where αn| represent the Floquet states, α describes the state and n spin n inωmw t e . The denotes the component in the Fourier expansion H(t) = n Hαβ Hamiltonian in (5) has non-zero oﬀ-diagonal elements, which couple states that diﬀer in the number of photons by ∆n = 1 and in the spin projections by ∆mS = 0, ±1. The possible transitions can be revealed by checking the behavior of the energy levels in the vicinity of the (n−m)-resonance conditions α − β = (n − m)ωmw . The corresponding energy levels αn = α + nωmw and βm = β + mωmw are degenerated when the oscillating ﬁeld is zero. For ω1 = 0, the level crossings αn = βm are lifted. The existence of multiplephoton transitions thus implies a level anticrossing, and vice versa.

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Fig. 7. Eigenvalues of the Floquet Hamiltonian calculated as a function of the normalized photon energy ωmw /ωS for ω1 /ωS = 0.2 and two angles θ between B 0 and B 1 . (a) Eigenvalues for θ = 90◦ . (b) Eigenvalues for θ = 45◦

Fig. 8. Time-averaged transition probabilities as a function of the normalized Larmor frequency ωS /ωmw for ω1 /ωS = 0.25. (a) Radiation ﬁeld perpendicular to B 0 . (b) Direction of the radiation ﬁeld tilted by θ = 45◦ (adapted from [32])

To demonstrate the existence of anticrossings, the eigenvalues of HF have been calculated as a function of ωmw /ωS with θ = 90◦ and 45◦ , and ω1 /ωmw = 0.25 (Figs. 7a and b). Anticrossings are found in the vicinity of ωmw /ωS = 1/n. For an oscillating ﬁeld perpendicular to B 0 an anticrossing is observed at ωmw /ωS = 0.38, corresponding to the position of a three-photon resonance (Fig. 7a). For θ = 45◦ , anticrossings occur at ωmw /ωS =0.55, 0.37 and 0.26; i.e., at two-, three- and four-photon resonances (Fig. 7b). A radiation ﬁeld at an angle of 45◦ to B 0 may thus induce both odd and even resonances. Numerical computations of the time-averaged transition probability performed for HF as a function of the normalized Larmor frequency ωS /ωmw are plotted in Figs. 8a and b for θ = 0◦ and 45◦ . For a radiation ﬁeld with two frequencies, the analysis is similar.

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• One Linear Radiation Field Perpendicular to B 0 In a two-level system with electron spin S = 1/2 and energy levels 1 and 2, multiple-photon transitions with an odd number of photons can be induced by a linear radiation ﬁeld perpendicular to B 0 (Fig. 9a). In a (2n + 1)-photon transition, (n + 1) σ + photons and n σ − photons are absorbed, so that the angular momentum is conserved. z

(a)

(b) 2

σ+

B0 σ− y

σ+

Fig. 9. (a) Field conﬁguration in the laboratory frame and, (b) energy level diagram for a three-photon transition induced between the two states of an S = 1/2 spin system by a linear radiation ﬁeld perpendicular to B 0 (σ + and σ − photons)

This type of multiple-photon transitions can only be induced by a linear ﬁeld which consists of both σ + and σ − photons. The energy level scheme for a three-photon transition σ+ + σ− + σ+ with ωmw = ωS /3, where two σ+ photons and one σ− photon are absorbed, is given in Fig. 9b. For this process the transition amplitude is found to be (3)

(3)

b12 ∝ ωeﬀ = ω13 /2ωmw 2 .

(6)

The observation of such multiple-photon resonances has been reported for Cr+ -doped gallium phosphide [33]. Figure 10 shows the X-band cw EPR spectrum induced by red light. In addition to the single-photon transition at 328 mT, also the two-photon transition at 656 mT and the three-photon transition at 984 mT (arrow) are induced. At high mw power multiple-photon transitions with n up to 7 could be observed. It is assumed that the very long relaxation time of this material allows the observation of the multiple-photon transitions. In this experiment with the mw ﬁeld oriented perpendicular to B 0 , not only odd resonances as predicted by theory, but also even resonances could be observed (see below). • One Linear Radiation Field at Arbitrary Orientation A linear radiation ﬁeld, which includes an angle θ < 90◦ with the static ﬁeld B 0 , can be decomposed into a linear ﬁeld component perpendicular to B 0 consisting of σ − and σ + photons, and a ﬁeld component parallel to B 0

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0.1

0.5

1 B0 / T

Fig. 10. Light-induced EPR spectrum of chromium-doped gallium phosphide showing a single-photon, a two-photon, and a three-photon transition (marked by arrow) (adapted from [33])

(a)

(b)

B0 θ

σ+

1 1

Fig. 11. (a) Field conﬁguration in the laboratory frame and, (b) energy level diagram for a two-photon and a three-photon transition induced between the two states of an S = 1/2 spin system by a linear radiation ﬁeld which includes an angle θ < 90◦ with B 0 (σ + , σ − , and π photons)

consisting of π photons (Fig. 11a) [34]. Consequently, multiple-photon transitions with both an even and an odd number of absorbed photons can be observed, as is shown in Fig. 11b for the transitions σ+ + π and σ+ + 2 × π. For a two-photon transition induced by a radiation ﬁeld with frequency ωmw = ωS /2, the transition amplitude given by b12 ∝

ω12 sin(2θ) 2ωS

(7)

is maximum for θ = 45◦ . First observations of multiple-photon transitions of the type σ+ + k × π have been reported by Winter in an optical pumping experiment [7]. The observation of even resonances in the spectrum of Cr+ -doped gallium phosphide mentioned above (Fig. 10) is explained by a distortion of the mw ﬁeld caused by the high dielectric constant of the sample, which results in a component of the radiation ﬁeld along B 0 .

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The so-called second-harmonic-detected EPR experiments described by Bosca-ino and coworkers [35, 36, 37, 38, 39] are also based on two-photon transitions of the type σ+ + π. In this approach a bimodal cavity with two resonant modes in the frequency range 2–6 GHz is used. In the ﬁrst mode, a linear radiation ﬁeld with frequency ωmw ≈ ωS /2, which includes with B 0 an angle of θ = 45◦ , induces two-photon transitions in an S = 1/2 spin system. The resulting second-harmonic signal at the Larmor frequency ωS is detected via the second mode perpendicular to B 0 . This scheme has the advantage that the signal at frequency ωS can be recorded during excitation, since the radiation ﬁeld at half this frequency is far oﬀ-resonant from the observation frequency. The experimental scheme has been applied in various investigations of two-level systems (see for example [40]). Very recently, second-harmonic detection has been used to study the decay properties of stimulated nutation echoes [41]. The two-pulse sequence used to generate this type of echoes is shown in Fig. 12a. Both pulses with frequency ωmw ≈ ωS /2 drive the two-photon transition σ+ + π. The ﬁrst pulse of length τ1 burns a polarization pattern into an inhom*ogeneous line [18], which is recalled by the second pulse after a waiting time τ τ1 . The stimulated nutation echo is formed at time t = τ1 after the onset of the second pulse (Fig. 12b [41]). Two or More Radiation Fields Perpendicular to B0 Multiple-photon processes become more complex when transitions are induced in a two-level system by linear bichromatic or tetrachromatic radiation ﬁelds perpendicular to B 0 . Multiple-photon transitions with an odd number n of photons can be induced by two linear radiation ﬁelds with frequencies ωmw,1 and ωmw,2 and amplitudes ω1 = ω2 (Fig. 13a), when the resonance condition

(a)

τ2

Signal / dB

τ1

∼ ∼

(b)

-10

-20

-30 0

t / µs Fig. 12. Stimulated nutation echo induced by a linear radiation ﬁeld (θ = 45◦ and recorded via the second-harmonic signal. (a) Pulse sequence, arrow marks echo maximum. (b) Stimulated nutation echo in γ-irradiated quartz glass (adapted from [41])

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(a)

(b) 2

σ+

σ−

Fig. 13. (a) Field conﬁguration in the laboratory frame and, (b) energy level diagram for a three-photon transition induced between the two states of an S = 1/2 spin system by two linear radiation ﬁelds perpendicular to B 0 (σ + and σ − photons)

ωS = ωmw,1 ± [n2 (ωmw,2 − ωmw,1 )2 − ω12 ]1/2

(8)

is fulﬁlled [42]. A three-photon transition of this type is shown in Fig. 13b. Recently, Hyde and co-workers introduced a multiple-photon EPR experiment, where the spin system is excited by two (or four) cw radiation ﬁelds with closely spaced mw frequencies [43, 44, 45, 46, 47]. In the case of two radiation ﬁelds with frequencies ωmw,1 = ωmw + 12 ∆ω and ωmw,2 = ωmw − 12 ∆ω and amplitudes ω1 , ω2 , the multiple-photon process creates coherences that manifest as sidebands at frequencies ωmw ± (k + 12 )∆ω. Corresponding experiments have also been reported in pulse NMR [48, 49] and cw NMR [12, 50, 51], and in the optic regime [52]. Each individual sideband can be down-converted with respect to the reference frequency ωmw and detected with a phase-sensitive detector. The same detection scheme has also been used in cw electron-electron double resonance [53] and cw ENDOR [54] experiments. This approach can be used to measure pure absorption EPR spectra [55]. As an example, in Fig. 14 the conventional cw EPR spectrum of nitrous oxide reductase is compared with the three-photon EPR spectrum recorded without ﬁeld modulation, and the corresponding ﬁrst-harmonic spectrum obtained by pseudo-modulation [56]. The diﬀerence between the two spectra (marked by an arrow) is traced back to the enhanced suppression of the forbidden transitions in the multiple-photon spectrum. This type of multiple-photon process can also be studied with pulse excitation. Recently, an inversion recovery experiment with the pulse sequence shown in Fig. 15a and a tetrachromatic radiation ﬁeld has been carried out on γ-irradiated quartz glass. The tetrachromatic pulses were created by mixing the mw frequency ωmw with two radio frequencies ωrf,1 and ωrf,2 . An odd number of photons is absorbed from the tetrachromatic radiation ﬁeld, which induces updown transitions and creates polarization at frequency ωS , corresponding to the center frequency ωmw of the inversion pulse. The resonance condition, ωS = ωmw + n1 ωrf,1 + n2 ωrf,2 , is fulﬁlled for example for the three-photon

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(a)

(b) 10 mT

(c) Fig. 14. EPR on nitrous oxide reductase substituted with 63 Cu, temperature 20 K. (a) Cw EPR spectrum. (b) First harmonic of the spectrum in (c), using pseudomodulation. (c) Three-photon EPR spectrum (adapted from [55])

(a)

(b)

νrf / MHz π

π/2

8 10 12

ωmw ± ωrf,1 ± ωrf,2

      

ωmw

16 18 20

3ωrf,1-2ωrf,2 10

2ωrf,1- ωrf,2 40

Ωs /2π / MHz Fig. 15. Three- and ﬁve-photon resonances in γ-irradiated quartz glass at Xband frequency. (a) Inversion recovery pulse sequence with a tetrachromatic inversion pulse. (b) Multiple-photon resonances as a function of ωrf,2 at diﬀerent ﬁxed frequencies ωrf,1 of the tetrachromatic pulse. Three-photon resonances are observed for 2ωrf,1 − ωrf,2 = 0 (dashed line). Very weak ﬁve-photon resonances with 3ωrf,1 − 2ωrf,2 = 0 (dashed line) can be observed at the two top traces. For convenience, the spectra are inverted

transition with n1 = 2 and n2 = −1, and the ﬁve-photon transition with n1 = 3 and n2 = −2. The integrated echo intensities as a function of ωrf,2 at diﬀerent ﬁxed frequencies ωrf,1 are plotted in Fig. 15b. In addition to the three-photon resonances, weak ﬁve-photon resonances become visible with radio frequencies of 8 and 10 MHz.

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3 Eﬀects of Oscillating Longitudinal Field In this section we describe multiple-photon transitions induced by a bichromatic radiation ﬁeld consisting of a linear or right circular mw ﬁeld in the gigahertz range, perpendicular to B 0 , and an rf ﬁeld in the megahertz range, parallel to B 0 (Fig. 16a). Such a bichromatic ﬁeld induces multiple-photon transitions of the type σ+ mw + k × πrf . In this section we understand by a bichromatic ﬁeld always this kind of radiation ﬁeld, and for the multiplephoton transitions we use the abbreviated form σ+ + k × π. z

(a)

(b) π B0

σ+

B2 y

π σ+

σ+

B1 x

Fig. 16. (a) Field conﬁguration in the laboratory frame and, (b) energy level diagram for the experiments with one mw ﬁeld perpendicular to B 0 and one rf ﬁeld parallel to B 0 , showing two-photon resonances with ωS = ωmw ± ωrf

Two-photon transitions σ+ + π of this type (Fig. 16b) have ﬁrst been reported by Winter in 1958 [34] on the free radical DPPH, and three years later by Burget and coworkers and Hashi on DPPH and other two-level systems [57, 58]. Recently, such two-photon transitions has also been observed in electronnuclear multi-level systems, where they appear in cw ENDOR spectra at the corresponding hyperﬁne frequencies [59]. 3.1 The Field Modulation Used in cw EPR as a Multiple-Photon Process In this section we demonstrate that ﬁeld modulation commonly used in cw EPR spectroscopy, together with the mw ﬁeld, represents a bichromatic radiation ﬁeld which gives reason to complex multiple-photon processes. Since the dawn of cw EPR spectroscopy, modulation of the static magnetic ﬁeld B0 with subsequent phase-sensitive detection is used to improve signal-to-noise. As a consequence of this ﬁeld modulation the derivative of the absorption signal is observed. When the modulation amplitude or the modulation frequency are larger than the width of a particular line, the line shape is distorted. The former situation is known as modulation broadening due to overmodulation, and in the latter case sidebands appear in the spectrum [60].

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A ﬁrst theoretical description of modulation eﬀects was given by Karplus [61] for mw rotational spectroscopy, where either the excitation frequency or the energy levels of the molecule are modulated. In magnetic resonance this theory was used by Smaller [62], and was further developed by several authors to obtain quasi steady-state solutions of the modiﬁed Bloch equations [63, 64, 65, 66, 67]. These equations [67] give a correct description for any harmonics of the cw EPR signal, including distortions due to overmodulation. In the classical approach the modulation of B 0 with frequency ωrf , leads to the time-dependent ﬁeld B(t) = B 0 + 2B2 cos(ωrf t). The corresponding signal can be expanded to a Taylor series ∞ 1 dn A(B0 ) S(t) ∝ A(B0 ) + [2B2 cos(ωrf t)]n , n n! dB 0 n=1

(9)

where A(B0 ) is the absorption spectrum (Fig. 17a). For modulation amplitudes small compared to the linewidth, only the ﬁrst-order term with n = 1 has to be considered. The ﬁrst-harmonic absorption spectrum, the part of the absorption signal oscillating with frequency ωrf , then consists of lines which are the ﬁrst derivative dA(B0 )/dB0 of the absorption lines (Fig. 17b). (a)

(b)

(c)

Fig. 17. (a) Lorentzian absorption line with width Γ . (b) First-harmonic line, modulation amplitude Γ/20 (ﬁrst derivative of the absorption line). (c) First-harmonic line with strong overmodulation, modulation amplitude 6Γ

With increasing modulation amplitude higher-order terms of the Taylor series in (9) become relevant (Fig. 17c) [68, 69, 70]. For modulation amplitudes large compared to the linewidth, the spacing between the positive and negative peak maximum of the distorted line corresponds to the modulation amplitude 2B2 . This eﬀect is routinely used to calibrate the modulation coils of EPR spectrometers. We now discuss two semi-classical approaches, which in contrast to the modiﬁed Bloch equations can also be applied to pulse experiments.

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Multiply Tilted Rotating Frames The inﬂuence of an rf ﬁeld parallel to B 0 on a spin system can be described by a series of transformations to tilted rotating frames [71, 11, 32]. This approach is restricted to cases where higher-order eﬀects caused by strong radiation ﬁelds can be neglected. An S = 1/2 electron spin system subject to an external ﬁeld B 0 and a bichromatic radiation ﬁeld can be described by the Hamiltonian Hlab (t) = wS Sz + 2ω1 cos(ωmw t)Sx + 2ω2 cos(ωrf t)Sz ,

(10)

where ω2 = −γe B2 is the amplitude of the radio frequency in angular frequency units. In a ﬁrst step the Hamiltonian in (10) is transformed to a frame rotating with frequency ωmw . Omitting the counter-rotating component of the mw ﬁeld we ﬁnd HSRF = ΩS Sz + ω1 Sx + 2ω2 cos(ωrf t)Sz ,

(11)

with the resonance oﬀset ΩS = ωS − ωmw . The Hamiltonian in this singly rotating frame is shown graphically in Fig. 18a. (a)

(b)

(c)

y rf

rf rf rf

Fig. 18. Pictorial representation of the cw EPR Hamiltonian. (a) Singly rotating frame. (b) First tilted rotating frame, tilted from the z-axis by angle α and rotating with ωrf (β is the tilt angle for the next transformation). (c) Toggling frame with k = 0 and z = 2ω2 /ωrf

The eﬀective ﬁeld vector ω eﬀ , and the z-axis include an angle α = arctan(ω1 /ΩS ). The frame is then tilted around the y-axis by this angle, resulting in a new z -axis parallel to ω eﬀ . Finally, this frame is transformed to a frame rotating with frequency ωrf (Fig. 18b). In this doubly rotating frame the equilibrium magnetization lies on a cone with apex angle 2α. For small angles α, corresponding to ω1 ΩS , the magnetization is approximatively oriented along to the z -axis. After omitting the counter-rotating term and the remaining linear component, the Hamiltonian is given by HTRF,1 = ΩS,1 Sz + ω1,1 Sx ,

(12)

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with the new resonance oﬀset ΩS,1 = ω12 + Ω2S − ωrf ≈ ΩS − ωrf . For the eﬀective ﬁeld amplitude ω1,1 in the ﬁrst tilted rotating frame we ﬁnd ω1,1 = −ω2 sin(α). Near resonance (ΩS,1 ωrf ), ω1,1 can be simpliﬁed to b12 ∝ ω1,1 ≈

−ω1 ω2 , ωrf

(13)

leading to the Hamiltonian HTRF,1 = (ΩS − ωrf )Sz − ω1,1 Sx ω1 ω2 = (ΩS − ωrf )Sz + Sx . ωrf

(14)

Equation (14) describes a two-photon transition, resonant with ωS = ωmw + ωrf . By repeating this procedure, a Hamiltonian for any multiple-photon transition with frequency ωS = ωmw + kωrf can be derived. For the three-photon transition, with frequency ωS = ωmw + 2ωrf , for example, the Hamiltonian in the triply rotating frame is found to be HTRF,2 = (ΩS − 2ωrf )Sz +

ω1 ω22 Sx . 2ωrf 2

(15)

The tilted frame approach delivers only correct results for rf amplitudes much weaker than the ﬁelds usually used in EPR experiments. Toggling Frames The toggling frame approach does not suﬀer from this drawback [72, 73]. Starting with the Hamiltonian in the singly rotating frame (11), the timedependent longitudinal component of the Hamiltonian is replaced by timedependent transverse components using the rotation operator 2ω i kωrf t+ ω 2 sin(ωrf t) Sz

R(t) = e

(16)

We then arrive at the toggling frame Hamiltonian HTF,k = R(t)HSRF R−1 (t) − kωrf Sz − 2ω2 cos(ωrf t)Sz = (ΩS − kωrf )Sz + ω1 J−k(z) Sx ∞ + ω1 Jn(z) ei(k+n)ωrf tSz Sx e−i(k+n)ωrf tSz ,

(17)

n=−∞ n=−k

with the normalized rf amplitude z = ω2 /ωrf , and the Bessel function of the ﬁrst kind Jn(z). HTF,k contains the resonance oﬀset (ΩS − kωrf ), a timeindependent eﬀective ﬁeld amplitude ω1 J−k(z), and the sum of the remaining time-dependent transverse perturbations. For a better understanding HTF is

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visualized in Fig. 18c for k = 0. The component ω1 J0(z) points along the xaxis, while all the other transverse components rotate around the z-axis with frequencies that are multiples of ωrf . For a properly chosen value of k, the remaining resonance oﬀset is minimum and the time-dependent terms of the Hamiltonian can be neglected in ﬁrst-order. A multiple-photon transition of the type σ+ + k × π has thus the transition amplitude b12 ∝ ω1,k = ω1 J−k(z) .

(18)

Although the toggling frame is a convenient tool for the understanding of multiple-photon processes, it remains a semiclassical description. Fully Quantum Mechanical Description For a fully quantized description of the cw EPR experiment with ﬁeld modulation, only two modes of the radiation ﬁeld, corresponding to the two frequencies, are of importance (see Appendix). All other frequencies are either strongly damped or far oﬀ-resonant. The fully quantized Hamiltonian is then given by 1 1 T T H = ωS Sz + ωmw amw amw + 1 + ωrf arf arf + 1 2 2 ω1 T ω2 T amw + amw Sx + √ arf + arf Sz . + √ (19) Nmw Nrf The function space is spanned by |mS , n, m = |mS ⊗ |n ⊗ |m, where n and m are the numbers of mw and rf photons. Due to the low frequency of the mw and rf ﬁelds, H can be replaced by a Hamiltonian in a two-mode Floquet space, as is shown in Appendix A.1–A.3. This approach provides a simpler mathematical treatment of the problem and is used for further evaluation. However, it is important to mention that Fourier indices of the Floquet states correspond to the photon state occupation numbers of the fully quantized Hamiltonian in (19). This correspondence proves that the modulation sidebands discussed later are caused by multiple-photon transitions. The resonance condition for a multiple-photon transition of the type σ+ + k × π between the two levels α, 0, 0| and β, 1, k| is fulﬁlled when α, 0, 0|HF |α, 0, 0 ≈ β, 1, k|HF |β, 1, k, or ωS ≈ ωmw + k ωrf . Coupling between the degenerate levels will then lead to a coherent transition in the form of Rabi oscillations. Figure 19 a shows a segment of the matrix representation of the two-mode Floquet Hamiltonian. The two levels |α, 0, 0 and |β, 1, 1 are degenerate, corresponding to the two-photon transition σ+ + π. Since there is no direct coupling between the elements, only second- and higher-order eﬀects are relevant. The two pathways for the two-photon transition are indicated by arrows and

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(a)

Fig. 19. (a) Segment of the two-mode Floquet space Hamiltonian HF . Full rectangles: Degenerate levels |α, 0, 0 and |β, 1, 1. Dashed rectangles: Coupling elements and intermediate levels of the two-photon transition. The two pathways I and II are indicated by arrows. (b) Energy level scheme with transitions of the two two-photon pathways I and II and the four four-photon pathways III to VI. Black arrows: σ+ mw photons, dashed arrows: σ− mw photons, grey arrows: πrf photons. (c) Segment of the single-mode Floquet Hamiltonian HF,TF,0 (adapted from [72])

correspond to the transitions I and II in Fig. 19b. The next possible transition pathways are those of the four-photon transitions, −σ+ + π + 2σ+ , 2σ+ + π − σ+ , −π + σ+ + 2π, and 2π + σ+ − π (transitions III to VI in Fig. 19b). Using a perturbation approach [74], the eﬀective Hamiltonian for any two coupled Floquet state can be derived. For two-photon transitions, I and II in

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Fig. 19, the Hamiltonian is given by

ω2 ω12 Heﬀ = ΩS − ωrf + 1 + Sz + ω1,1 Sx , 2ωrf 4ωmw

(20)

where ∆BS = ω12 /(4ωmw ) is the Bloch-Siegert shift [75]. The eﬀective ﬁeld amplitude ω1,1 is approximately (−ω1 ω2 /ωrf ), a result which is also obtained with the tilted frame approach (13). The perturbation approach or a numerical evaluation of the two-mode Floquet Hamiltonian provide solutions for all possible transitions and transition amplitudes. However, it is better to reduce the number of modes in the Floquet space to get a more compact but still complete presentation of related states and transitions. This can be done by using a toggling frame approach, which as we have seen is a convenient choice for a semiclassical description of σ+ + k × π transitions. The parameter k of the toggling frame may be chosen arbitrarily and does not have to fulﬁll any kind of resonance condition, since the resonant transition is selected later. We simply set k = 0 and get, according to (17), the toggling frame Hamiltonian HTF,0 = (ΩS + ∆BS ) Sz +

∞

ω1 Jn(z) einωrf tSz Sx e−inωrf tSz .

(21)

n=−∞

In contrast to the semiclassical approach in (17), the time-independent singlemode Floquet Hamiltonian HF,TF is derived from (21). A small segment of the matrix representation is shown in Fig. 19c. The Hamiltonian HF,TF has a direct coupling element between two degenerate levels, taking into account all pathways where one mw σ+ photon is absorbed. The coherence is described in the function space of HF,TF . The back-transformation to the singly rotating frame splits this single coherence into a set of coherences corresponding to all involved virtual levels. For the eﬀective Hamiltonian of a resonant σ+ + k × π multiple-photon transition calculated from HF,TF (see Fig. 19c), we ﬁnd [72, 73] Heﬀ = ([ωS + ∆BS + ∆k ] − [ωmw + kωrf ]) Sz + ω1,k Sx , with a Bloch-Siegert-like resonance shift (in second order) 2 2 2ω1 Jl(z) ∆k = , ωrf (k − l)

(22)

(23)

l=k

and the eﬀective ﬁeld amplitude2 (0) (3) ω1,k = ω1 ck + ck   2 J (z) J (z) J (z) ω1 −l m−l m−k . = ω1 J−k(z) + 2ωrf (l − k)m l=k m=0

(3)

Note that the expression for the term ck

in [72] contains a sign error.

(24)

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For ω1 ωrf , ∆k , the higher-order terms in (24) and the Bloch-Siegert shift ∆BS can be neglected, resulting in Heﬀ = ΩS − kωrf Sz + ω1 J−k(z) Sx . (25) This expression is equivalent to the Hamiltonian derived from the semiclassical toggling frame approach in (17) after omitting the time-dependent perturbations. Description of Sidebands The cw EPR absorption signal is proportional to the transverse component of the steady-state magnetization out-of-phase with the mw ﬁeld. The steadystate solution is most easily obtained from the eﬀective Hamiltonian given in (25) with the assumption that each σ+ + k × π transition can be treated separately. The density operator at thermal equilibrium, σSRF = −Sz , is transformed to the kth toggling frame by σTF,k (t) = R(t)σSRF (t)R−1 (t). Since the detection operator D = Sy is expressed in the singly rotating frame, the density operator has to be transformed back to this frame, σSRF (t) = R−1 (t)σTF,k (t)R(t) ∞ = Jn(z) e−i(k+n)ωrf tSz σTF,k (t)ei(k+n)ωrf tSz .

(26)

n=−∞

The time-independent density operator in the toggling frame is then transformed to a series of terms in the singly rotating frame, oscillating with multiples of the modulation frequency. Summation over all possible multiple-photon resonances results in the general solution for the cw EPR absorption signal Sy = tr[σSRF Sy ]

∞ ∞

ω1 Jn(z) J−k(z)

n=∞ k=∞

T2 cos([k + n]ωrf t) − (ΩS − kωrf ) T22 sin([k + n]ωrf t) . 1 + ω12 J2k(z) T1 T2 + (ΩS − kωrf )2 T22

(27)

Apart from the saturation term ω12 J2k(z) T1 T2 in the denominator, (27) is identical to the formula derived from the Bloch equations [67]. Using (27), the absorption spectrum for any harmonic of the modulation frequency can be calculated. For the zeroth-harmonic spectrum, only the constant terms are relevant Sy 0 =

∞ k=−∞

ω12

T2 ω1 J2k(z) 2 Jk(z) T1 T2 + (ΩS

− kωrf )2 T22

In cw EPR the ﬁrst-harmonic absorption spectrum, given by

(28)

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T2 ω1 J−k(z) J1−k(z) + J−1−k(z) Sy 1 = cos(ωrf t) 1 + ω12 J2k(z) T1 T2 + (ΩS − kωrf )2 T22 k=−∞ (ΩS − kωrf ) T22 ω1 J−k(z) J1−k(z) − J−1−k(z) − sin(ωrf t) , (29) 1 + ω12 J2k(z) T1 T2 + (ΩS − kωrf )2 T22 ∞

is usually measured. These equations becomes more clear from the plots shown in Fig. 20, which represent the ﬁeld-swept EPR absorption spectra of the zeroth-harmonic and the ﬁrst- and second-harmonic signal with the components in-phase and out-of-phase with respect to the rf ﬁeld. (a) π

σ+

(b)

(c)

(d)

(e)

(f)

Fig. 20. Simulated ﬁeld-swept cw EPR spectra for diﬀerent harmonics of the modulation frequency. k indicates the number of absorbed rf π photons. (a) Multiplephoton transitions. (b) Zeroth-harmonic spectrum. (c) First-harmonic spectrum, in-phase with the radio frequency. (d) Corresponding out-of-phase spectrum. (e) Second-harmonic spectrum, in-phase with the radio frequency. (f ) Corresponding out-of-phase spectrum (adapted from [72])

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The zeroth-harmonic spectrum is symmetric (Fig. 20b) and consists of a center line (single-photon transition) and sidebands (multiple-photon transitions) with spacings ∆B = ωrf /γe . All lines have an absorptive shape, with the intensity of the kth sideband proportional to J2k(z). The ﬁrst-harmonic spectrum consists of in-phase components with absorptive lineshapes (Fig. 20c), and out-of-phase components with dispersive lineshapes (Fig. 20d). The spectra are anti-symmetric with respect to the center line. The second-harmonic spectra shown in Figs. 20e,f are again symmetric. In cw EPR spectroscopy the ﬁrst-harmonic absorption signal in phase with the modulation frequency is usually measured (Fig. 20c). The derivative-like lineshape is then a superposition of a large number of overlapping sidebands separated by ωrf . The two multiple-photon transitions with |k| of approximately 2ω2 /ωrf give the maximum contributions to the signal. For a hom*ogeneous line and 2ω2 1/T2 , (29) reduces to the commonly observed derivative lineshape ω1 T2 d Sy 1 ≈ 2ω2 . (30) dΩS 1 + Ω2S T22 The signal amplitude is proportional to the modulation amplitude, and does not depend on the modulation frequency. Equation (30) can be extended to the more general case of an inhom*ogeneous broadened line, which consists of a distribution of unresolved hom*ogeneous lines. As long as the modulation amplitude does not exceed the inhom*ogeneous linewidth, again the derivative of the absorption line is observed. It is noteworthy that in the ﬁrst-harmonic in-phase cw EPR spectrum the observed derivative lines consist of a large number of multiple-photon transitions, and that the signal intensity is zero at the position of the single-photon transition. For example, in a standard cw EPR experiment with 100 kHz ﬁeld modulation and a modulation amplitude of 0.1 mT (2ω2 /2π ≈ 3 MHz), the two multiple-photon transitions constituting the peak maximum and the peak minimum are σ+ ± 28 × π transitions, where 28 rf photons are absorbed and emitted. The ﬁnding that the lineshapes observed in cw EPR experiments are caused by multiple-photon transitions was veriﬁed by experiments on a single crystal of lithium phthalocyanine with a linewidth of 4–8 µT. Two series of ﬁrst-harmonic in-phase EPR spectra with diﬀerent modulation amplitudes are shown in Fig. 21. The spectra recorded with a 1 MHz ﬁeld modulation (Fig. 21a) show the patterns predicted by (29). When a 100 kHz ﬁeld modulation is used (Fig. 21b), the individual multiple-photon transitions are no longer resolved. The out-of-phase signal (not shown) is averaged to zero, since the individual multiple-photon transitions with derivative lineshape (see Fig. 20d) compensate each other. Although it was not realized during many years since the beginning of EPR, in the traditional cw experiment we actually observe multiple-photon

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(a)

(b)

0.3V0

1V0 2

4 2 6 4

12 6

-1

-0.4

0.4

∆B / mT Fig. 21. First-harmonic EPR spectra of lithium phthalocyanine recorded at Qband with diﬀerent modulation amplitudes in phase with the modulation frequency. Numbers are proportional to the signal voltage. (a) Modulation frequency 1 MHz. (b) Modulation frequency 100 kHz (adapted from [72])

transitions. This recent ﬁnding gives a deeper insight into the origin of the observed phenomena and the interpretation of the results. 3.2 Pulse EPR Experiments We now demonstrate that multiple-photon transitions created by a bichromatic radiation ﬁeld can also be observed in pulse EPR experiments. In particular, the creation of multiple-photon echoes is discussed. The pulse sequence for the generation of multiple-photon echoes is shown in Fig. 22a. The mw part consists of the conventional two-pulse sequence [18]. The sequence of rf pulses with the ﬁeld along B 0 is applied synchronously with the mw pulses. The ﬁrst bichromatic pulse creates coherence at ωmw and ωmw + kωrf . For simplicity we consider lowest (two-photon) transitions only ωmw ± ωrf . During the free evolution period of time τ between the ﬁrst and the second pulse the coherence dephases. The second bichromatic pulse inverts the directions of spin precession, resulting in the formation an echo at time τ after the second pulse. The echo appears again at ωmw and at ωmw ± ωrf . By varying the mw and rf amplitude, as well as by choosing a proper B 0 value, the echoes observed at diﬀerent frequencies can be optimized or separated from each other, as is demonstrated in next paragraph [32, 73].

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Fig. 22. Multiple-photon pulse EPR with bichromatic pulses. (a) Pulse sequence for a multiple-photon echo. (b) Observed time-domain echo signals. 1: Single-photon echo, ωS = ωmw , ω1 /2π = 2.5 MHz. 2-5: Two-photon echoes measured at ΩS −ωmw ≈ ωrf , with ΩS /2π (in MHz) 40 (2), 60 (3), 80 (4), and 100 (5) (adapted from [32])

Two-photon echoes on γ-irradiated quartz glass are recorded at Q-band frequencies [32]. The lengths of the bichromatic pulses are 100 ns and 200 ns and the mw and rf amplitudes are both approximately 10 MHz. With these pulse sequences the signal of the conventional echo is maximum for an mw amplitude of about 2.5 MHz (Fig. 22b, trace 1). The echo traces 2–5 in Fig. 22b are obtained at B0 ﬁelds corresponding to resonance oﬀsets of 37.8, 58.2, 78.4 and 97.4 MHz, which correspond to the radio frequencies used in these experiments. The echo-detected two-photon EPR spectra obtained by Fourier transformation of the time-domain traces and recorded with radio frequencies of 40 and 80 MHz are shown in Figs. 23b,c. As a reference, the single-photon EPR spectrum is measured with the same sequence. The two-photon EPR spectra are symmetrically placed to the position of the single-photon EPR spectrum, with shapes that are exact replicates of the latter. The observed peak-to-peak splittings of the two-photon EPR spectra are approximately twice the radio frequency. 3.3 π-Photon-Induced Transparency In this section we describe a new transparency phenomenon which can be observed in spin systems. For a two-level system that is exposed to a bichromatic radiation ﬁeld consisting of an mw and an rf ﬁeld, where the two-level system is resonant with the mw frequency (ωmw = ωS ), energy and angular momentum are not only conserved for the single-photon transition but also for all multiple-photon transitions of the type −m × π + σ+ + m × π, with 1 < m < ∞ (Fig. 24a).

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Fig. 23. Echo-detected single- and two-photon EPR spectra of γ-irradiated quartz glass. (a) Single-photon spectrum, ω1 /2π = 2.5 MHz. (b) Two-photon spectrum with ωrf /2π = 40 MHz, ω1 /2π = ω2 /2π = 10 MHz. (c) Two-photon spectrum with ωrf /2π = 80 MHz, ω1 /2π = ω2 /2π = 15 MHz (adapted from [32])

(a)

(b)

π σ+

σ+ 0

-1 0

z Fig. 24. (a) Multiple-photon transitions of the type −m × π + σ + + m × π. (1) Single-photon process σ + ; (2) three-photon processes, π+σ + −π and −π+σ + +π; (3) ﬁve-photon processes 2π + σ + − 2π and −2π + σ + + 2π. (b) Normalized eﬀective ﬁeld ω1,0 = ω1 J0(z) for the centre band at ωmw = ωS , as a function of z = 2ω2 /ωrf (bold solid line). The contributions of the processes (1), (1)+(2) and (1)+(2)+(3) are also given (dashed lines) Adapted from [73]

In the single-photon process (1), one mw σ+ photon is absorbed, whereas in the two three-photon processes (2) with m = 1, one mw σ+ photon is absorbed, and one rf π photon is absorbed and one is emitted. Also ﬁve-photon transitions (3) with m = 2 and transitions of higher order are induced.

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For properly chosen experimental parameters of the rf ﬁeld, the spin system becomes transparent owing to destructive interference of the single-photon and the multiple-photon processes [76]. Since this phenomenon is caused by rf π photons, we call it π-photon-induced transparency. For the description of this type of multiple-photon transitions the phases ϕmw and ϕrf of the mw and rf ﬁeld have to be taken into account. Starting from the laboratory frame Hamiltonian Hlab (t) = ωS Sz + 2ω1 cos(ωmw t + ϕmw )Sx + 2ω2 cos(ωrf t + ϕrf )Sz ,

(31)

one arrives at the eﬀective spin Hamiltonian in the toggling frame HTF,k = ([ωS + ∆k ] − [ωmw + k ωrf ]) Sz + ω1,k e−iϕk Sz Sx eiϕk Sz ,

(32)

where ϕk = ϕmw + kϕrf is the phase of the eﬀective ﬁeld. The Hamiltonian in (32) essentially describes the spin system exposed to an eﬀective radiation ﬁeld with frequency ωmw + kωrf , phase ϕk , and amplitude ω1,k . The eﬀective transition amplitudes for the single-photon transition and the contributions of the processes (1), (1) + (2) and (1) + (2) + (3) are plotted in Fig. 24b as a function of the normalized rf amplitude z = ω2 /ωrf . The eﬀective transition amplitude of the sum of all multiple-photon processes is then given by 2m ∞ (−1)m z2 = ω1 J0(z) . (33) ω1,0 = ω1 (m!)2 m=0 For the ﬂip angle of a bichromatic pulse we ﬁnd βeﬀ = ω1,0 tp = ω1 J0(z) tp .

(34)

Corresponding expressions ω1,k can be derived for each of the sidebands. Of special interest is the ﬁnding that the eﬀective transition amplitudes ω1,k can be zero; the two-level system then becomes transparent. For the center band with ωS = ωmw , the eﬀective transition amplitude ω1,0 is zero at the zero-crossings of the Bessel function J0 (z). The ﬁrst zero-crossing is at z = j0,1 ≈ 2.4048, the second one at z = j0,2 ≈ 5.5201, etc. Figure 25 describes the motion of the tip of the magnetization vector M in the rotating frame during bichromatic radiation with an mw ﬁeld of amplitude ω1 along the x-axis and an rf ﬁeld 2ω2 sin(ωrf t), with ωrf /2π MHz, along the z-axis. The situation for a weak mw ﬁeld with amplitude ω1 /2π = 0.01 MHz, typically used in cw EPR experiments and z = 2.4 (ﬁrst transparency condition) is shown in Fig. 25a. When the radiation ﬁeld is turned on (t = 0), the eﬀective ﬁeld vector is given by ω eﬀ = ω 1 and M , which is oriented along the z-axis, starts to nutate around x. Since ω2 ω1 , the angle between the eﬀective ﬁeld ω eﬀ and the z-axis gets rapidly very small, so that ω eﬀ ≈ 2ω 2 . After one quarter of the rf period the magnetization is in the xz-plane, with a

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(c)

(b)

(a)

y z

z x

Fig. 25. Trajectories of the magnetization vector M under a bichromatic ﬁeld, starting from thermal equilibrium (M along the z-axis). (a) Weak mw ﬁeld, ω1 /2π = 0.01 MHz, resonant with the single-photon transition. Transparency condition fulﬁlled, z = 2.4048. Projection onto the xy-plane. (b) Transparency condition not fulﬁlled, z = 1. The trajectory is shown for the ﬁrst ﬁve rf periods. The scaling of the axes is reduced by the factor 20 compared to (a). (c) Strong on-resonant mw ﬁeld, ω1 /2π = 5 MHz. The transparency condition z = 2.3306 is fulﬁlled (adapted from [73])

deviation from the z-axis of only 0.07◦ . At half of the rf period (t = π/ωrf ), M has moved back to the z-axis and the eﬀective ﬁeld is again ω eﬀ = ω 1 . During the second half of the rf period, M describes a corresponding trajectory with negative x-values, so that after the full rf period M is again oriented along the z-axis. Thus, the tip of the magnetization vector M describes a ﬁgure of eight in the very close vicinity of the z-axis. If the transparency condition is not fulﬁlled, the trajectory is no longer a closed curve, and the magnetization moves on a toggling path towards the xy-plane, as is shown in Fig. 25b for the ﬁrst ﬁve rf periods and z = 1. For strong mw ﬁelds, as used in pulse EPR experiments, the third-order (3) contribution c0 (24) to the eﬀective ﬁeld amplitude can no longer be neglected. This reduces the value of z for which transparency is observed. For example, for an mw ﬁeld with amplitude ω1 /2π = 5 MHz, corresponding to a π pulse with a length of 100 ns, transparency occurs at z = 2.3306 instead of z = 2.4048. The tip of M describes again a ﬁgure of eight, but with larger deviations from the z-axis (Fig. 25c). Bichromatic pulses can be used as a tool to experimentally control the transition amplitude. This also includes the full suppression of the interaction of the mw ﬁeld with the spin system. For example, during a resonant mw pulse a free evolution period can artiﬁcially be created by an rf pulse with a ﬁeld amplitude that fulﬁlls the ﬁrst transparency condition. In this way an electron spin echo can be observed using a single mw pulse [76]. The pulse sequence for such a one-pulse echo experiment is shown in Fig. 26a. During the mw pulse of length tmw , the rf pulse of length τ1 is turned on at time tp1 and turned oﬀ at time tmw − tp2 , with 2tp1 = tp2 and

Multiple-Photon Transitions in EPR Spectroscopy

(a)

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(b)

Fig. 26. One-pulse echo experiment with a π-photon-induced free evolution period on a coal sample, ωmw /2π = 9.626 GHz, ωrf /2π = 15 MHz, B0 = 341.1 mT. (a) Pulse sequence with tp1 = 300 ns, tp2 ‘0 ns, tmw = 1.3–3.1 µs, τ1 = 0.4–2.2 µs. The rf pulse applied during time τ1 fulﬁlls the ﬁrst transparency condition for the single-photon transition. (b) Experimental time traces showing FIDs and echoes for diﬀerent τ1 values (adapted from [73])

τ1 = tmw − tp1 − tp2 . The mw ﬁeld amplitude is chosen such that the nominal ﬂip angle is β1 = π/2 during tp1 and β2 = π during tp2 . The pulse scheme thus corresponds to a two-pulse echo sequence, (π/2) − τ1 − (π) − τ2 − (echo), with a π-photon-induced free evolution period of time τ1 , and a usual free evolution period of time τ2 = τ1 . During time τ1 the spin coherence evolves as it would do during a free evolution period, apart from a phase shift that depends on z, the rf phase, and the length of the bichromatic pulse. Experimental time traces of such one-pulse echo experiments are shown in Fig. 26b for diﬀerent τ1 values. The signal that follows the mw pulse consists of an FID and an electron spin echo at time τ2 = τ1 , as in a conventional two-pulse echo experiment. The Bichromatic Pulse as a Substitute for a Second mw Frequency The π-photon-induced transparency phenomenon can be applied, for example, in pulse EPR experiments that require two mw frequencies ωmw,1 and ωmw,2 . A bichromatic pulse, which is transparent for the mw frequency ωmw = ωmw,1 , may then be used as a substitute for the second mw frequency ωmw,2 [77]. When the transparency condition is fulﬁlled for the center band at ωmw , the bichromatic pulse only drives eﬃciently two- and three-photon transitions with resonance frequencies ωmw ± ωrf and ωmw ± 2ωrf .

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(a)

(b)

-40

-20

0 20 Ω s / MHz

Fig. 27. Simulation of the polarization patterns of an S = 1/2 spin system with an inhom*ogeneous linewidth Γinh = ∞ after a bichromatic pulse with a length of 500 ns and a radio frequency of 10 MHz, as a function of ΩS . (a) Transparency condition fulﬁlled, z = 2.4. The intensity of the center band is zero. (b) Transparency condition not fulﬁlled, z = 1.5 (adapted from [77])

This is illustrated in Fig. 27, which shows simulations of the Mz magnetization of an S = 1/2 spin system with an inhom*ogeneous linewidth Γinh = ∞ after a bichromatic π pulse of length 500 ns, as a function of the resonance oﬀ-set ΩS . If the transparency condition is fulﬁlled (z = 2.4), only spins with resonance frequencies ωmw ± ωrf , ωmw ± 2ωrf , and ωmw ± 3ωrf are excited (Fig. 27a). The contribution of higher-order multiple-photon transitions can be neglected. A bichromatic pulse of proper length, fulﬁlling the transparency condition, can thus eﬃciently be used for broadband excitation with frequencies diﬀerent from the mw frequency. If the transparency condition is not fulﬁlled, transitions at ωmw and ωmw + kωrf are simultaneously excited. This is shown in Fig. 27b for z = 1.5. For the transverse magnetizations Mx and My an analogous behavior is found, reﬂecting the excitation at frequencies ωmw + kωrf by the bichromatic pulse. • Stimulated Soft ESEEM Stimulated Soft Electron Spin Echo Envelope Modulation (SS-ESEEM) is a pulse EPR experiment that is used to determine weak hyperﬁne and nuclear quadrupole interactions in paramagnetic species in solids [78]. The pulse sequence of SS-ESEEM shown in Fig. 28a makes use of two mw frequencies. The ﬁrst pulse with ﬂip angle π/2 and frequency ωmw,1 creates electron coherence on an allowed (forbidden) transition. The second pulse with ﬂip angle π and frequency ωmw,2 is resonant with a forbidden (allowed) transition and transfers the electron coherence to nuclear coherence, which then evolves

Multiple-Photon Transitions in EPR Spectroscopy

(a)

175

(c)

mw1

mw2

1 2

(b) mw1

3 rf 8

νESEEM / MHz

Fig. 28. Stimulated soft ESEEM. (a, b) Pulse sequences for SS-ESEEM (a), and bichromatic-pulse SS-ESEEM (b). (c) Comparison of three-pulse ESEEM and bichromatic-pulse SS-ESEEM experiments on a single crystal of 63 Cu(II)doped Zn(picolinate)2 , arbitrary orientation, T K. 1: Three-pulse ESEEM spectrum obtained with mw pulses with a length of 20 ns. 2: Corresponding spectrum obtained with mw pulses with a length of 400 ns. 3: Bichromatic-pulse SS-ESEEM spectra obtained with pulses with a length of 400 ns and a radio frequency of 14 MHz. Plots (2) and (3) use the same scale (adapted from [77]).

during the following free evolution period of time T . The third pulse with ﬂip angle π and frequency ωmw,1 transfers the nuclear coherence back to forbidden (allowed) electron coherence, which refocuses to a coherence-transfer echo. In this experiment, time T is incremented, and the modulation of the echo amplitude is monitored. SS-ESEEM experiments provide the same information as three-pulse ESEEM. They have, however, the advantages to be free of blind spots, and to require an mw power which is about two orders of magnitudes lower than for the standard three-pulse ESEEM experiment. In the bichromatic version of SS-ESEEM, the second pulse with frequency ωmw,2 is replaced by a bichromatic pulse (Fig. 28b) [77]. The general features of SS-ESEEM are discussed elsewhere [78], here we concentrate on the characteristics of the experiment with a bichromatic pulse. The potential of bichromatic-pulse SS-ESEEM is demonstrated on a single crystal of 63 Cu(II)-doped Zn(picolinate)2 . All the spectra are measured at a temperature of 20 K with an arbitrary crystal orientation. The three-pulse ESEEM spectrum shown in Fig. 28c, trace 1, is obtained with short nonselective mw pulses with a length of 20 ns, and consists of several proton peaks close to the proton Larmor frequency of 12.7 MHz. Trace 2 and 3, demonstrate the superiority of SS-ESEEM over three-pulse ESEEM, when long low-power π/2 and π pulses with a length of 400 ns are used. Under this condition, three-pulse ESEEM is no longer operative, but SS-ESEEM still is. Trace 2 shows the three-pulse ESEEM spectrum. The excitation bandwidth of a pulse with a length of 400 ns is by far too small

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to simultaneously excite allowed and forbidden transitions. Trace 3 shows the bichromatic-pulse SS-ESEEM spectrum measured with two mw pulses and a bichromatic pulse with a radio frequency of 14 MHz. The improvement in the signal/noise ratio is tremendous. The line positions in three-pulse ESEEM and the bichromatic-pulse SSESEEM spectrum are the same. Since SS-ESEEM does not suﬀer from blind spots, some of the lines are more intense in the bichromatic-pulse experiment than in three-pulse ESEEM. The line intensities are diﬀerent in the two spectra, because the selective bichromatic pulse excites a narrow band of frequencies close to ωmw + kωrf . Bichromatic-pulse SS-ESEEM is easy to implement and does not require a precise tuning of the bichromatic pulse to the transparency condition. • Double Electron-Electron Resonance Another experiment that makes use of two mw frequencies is double electron-electron resonance (DEER) [79], which has recently been extended to four pulses [80]. DEER may be used for the determination of distances between two spins A and B in solids by measuring the dipole coupling between two unpaired electrons. The pulse sequence for four-pulse DEER is shown in Fig. 29a. It consists of three pulses with frequency ωmw,1 and one pulse with frequency ωmw,2 . The ﬁrst pulse with ﬂip angle π/2 and frequency ωmw,1 creates electron coherence of the A-spins. The second pulse with ﬂip angle π and frequency ωmw,1 refocuses this coherence to an electron spin echo, which is formed at time τ1 after

Fig. 29. Four-pulse DEER. (a, b) Pulse sequence for DEER (a), and bichromaticpulse DEER (b). (c) Comparison of DEER time-traces of the nitroxide biradical CAS 312624-83-21 shown on top. 1: Standard four-pulse DEER with two mw frequencies, (ωmw,1 −ωmw,2 )/2π MHz. 2: Four-pulse DEER with a bichromatic pulse, ωrf /2π MHz and z ≈ 2.4 (adapted from [77])

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the pulse, and the third pulse with ﬂip angle π and frequency ωmw,1 refocuses the coherence to a refocused echo. At time t after the second pulse an mw pulse with ﬂip angle π and frequency ωmw,2 is applied to the B-spins. The eﬀect of this pulse on the refocused echo intensity is recorded by varying time t between 0 and 2τ1 (τ1 < τ2 ). The pulse with frequency ωmw,2 can again be replaced by a bichromatic pulse with mw frequency ωmw = ωmw,1 and radio frequency ωrf (Fig. 29b). Figure 29c shows experimental DEER time traces obtained with the twofrequency experiment (trace 1) and with bichromatic-pulse DEER (trace 2). The time traces are very similar, and the dipolar coupling obtained from the corresponding spectra agrees well with results reported earlier [80]. In the case of DEER, it is important that the bichromatic pulse fulﬁlls the transparency condition. Otherwise the residual excitation of A-spins at frequency ωmw destroys the relevant signal, since A-spins and B-spins with resonance frequencies ωmw and ωmw + kωrf are both ﬂipped simultaneously. A bichromatic pulse of at least one period of the radio frequency has to be used in bichromatic-pulse DEER. This leads to a loss in signal intensity compared to standard four-pulse DEER. On the other hand, in bichromatic-pulse DEER a larger number of spins with frequencies ωmw +kωrf are simultaneously excited, which again improves the signal intensity. The examples of SS-ESEEM and DEER demonstrate that bichromatic pulses can successfully be used to substitute pulses with a second mw frequency. In this Chapter we described the diﬀerent types of multiple-photon transitions that can be observed and used in EPR, from both a theoretical and an experimental point of view. We especially emphasized on the multiple-photon transitions induced by an additional longitudinal oscillating ﬁeld. These transitions play a major role in traditional ﬁeld-modulated cw EPR and are responsible for its derivative-like lineshapes. Moreover, π-photon-induced transparency based on this type of transitions has proved to be a very interesting and useful phenomenon. One area of applications refers to two-frequency pulse EPR experiments such as DEER, Soft ESEEM and others. In the future, we will study the multiple-photon phenomena in more detail and screen their potential for practical applications.

A Second Quantization A.1 Hamiltonian of the Photon Field For a fully quantum-mechanical description of the interaction between matter and an electro-magnetic radiation ﬁeld the oscillating magnetic ﬁelds have to be replaced by a set of quantized harmonic oscillators [3, 81], this is called second quantization. The Hamiltonian of a spin system in a radiation ﬁeld is then given by

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H = HS + HR + Hi ,

(35)

where HS is the Hamiltonian of the undisturbed system, HR describes the radiation ﬁeld, and Hi describes the magnetic dipole coupling between spin system and radiation ﬁeld. The radiation ﬁeld is represented by a superposition of an inﬁnite number of transverse electro-magnetic waves, propagating in all possible directions. The diﬀerent modes, distinguished by their propagation vectors k with |k| = ωk c, are orthogonal and do not interact with each other. Every mode has two independent orthogonal polarizations λ. The polarization of the magnetic ﬁeld component of a mode is given by the vector βkλ . For a circular ﬁeld it is more convenient to describe the polarization with the unit vectors√β +1 = √ −(β x + iβ y )/ 2 (right circular polarization), β −1 = (β x − iβ y )/ 2 (left circular polarization) and β 0 = β z (linear polarization), instead of using the basis system of Cartesian unit vectors, {β x , β y , β z }. Using the correspondence principle the Hamiltonian of the radiation ﬁeld is found to be † 1 (36) ωk akλ akλ + 1 , HR = 2 kλ

with the photon annihilation and creation operators a and a† . The combined operator Nkλ = kλ† akλ is called the photon number operator, since it gives the number nkλ of photon quanta ωk present in an eigenstate |nkλ kλ of mode kλ, (37) Nkλ |nkλ kλ = nkλ |nkλ kλ . Consequently, the eigenstates of the operator Nkλ are called number states. The polarization vector of a radiation mode does enter in the interaction Hamiltonian Hi , which describes the coupling between the magnetic ﬁeld and the magnetic dipole of the spin, 1 1 T Hi = − B T µ = − B kλ µ .

(38)

kλ

The operator of the quantized magnetic ﬁeld is given by . ωk µ0 B(r) = i akλ eik·r − a†kλ e−ik·r . β kλ 2V

(39)

kλ

For low frequencies and correspondingly long wavelengths, as used in NMR and EPR spectroscopy, one ﬁnds kT r 1, so that in good approximation exp(ik · r) = 1, over all positions r within the sample volume. In this dipolar approximation the interaction between the radiation ﬁeld and the magnetic dipole is reduced to the energy of a dipole in a dipolar ﬁeld. The interaction Hamiltonian of a certain mode kλ can then be written as . ωk µ0 T (40) i β kλ akλ + a†kλ S , Hi = −γ 2V

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179

with the square root describing the magnetic ﬁeld produced by a single photon with frequency ωk in a cavity of volume V . For a linear ﬁeld mode oriented parallel to the j-axis (j = x, y, z) the interaction Hamiltonian is given by Hi = λ akλ + a†kλ Sj , (41) with an eﬀective √ coupling factor λ. For a right circular ﬁeld mode, β kλ = iβ +1 = (i, −1, 0)/ 2, we ﬁnd for the corresponding Hamiltonian (42) Hi = λ akλ S + + a†kλ S − . Since these coupling elements connect spin states with ∆m = 1, a right circular ﬁeld consists of photons with helicity mJ = 1, i.e., σ+ -photons. The ﬁrst term in (42) with the photon annihilation operator akλ and the raising operator S + connects an initial state (|β state + photon) with a ﬁnal state (|α state, photon absorbed (annihilated)). The second term a†kλ S − , on the other hand, describes the emission (creation) of a photon. For a left circular ﬁeld, the situation is reversed. A.2 Coherent States The quantum-electrodynamical equivalents to classical radiation ﬁelds are coherent states [82, 83, 2]. A coherent state |α is an eigenfunction of the annihilation operator with a complex eigenvalue α, (a|α = α|α), leading to the deﬁnition ∞ n 2 α 1 |α = (43) e− 2 |α| √ |n . n! n=0 The populations of the diﬀerent number states follow a Poisson distribution |n|α|2 = e|α|

|α|2n n!

(44)

with an average photon number N = |α|2 . A typical semiclassical interaction Hamiltonian of a linear radiation ﬁeld, oriented perpendicular to the static magnetic ﬁeld, is Hi,semicl = 2ω1 cos(ωt)Sx .

(45)

By deﬁnition it is equal to the corresponding quantized interaction Hamiltonian in (41), Hi,quant = Hi,semicl , which leads to a coupling factor of λ=

ω1 . N 1/2

The fully quantized Hamiltonian ﬁnally results in

(46)

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H = HS + HR + Hi,quant ω1 1 † = ωS Sz + ω a a + 1 + (a† + a)Sx , 2 N 1/2

(47)

with a function space spanned by |mS , n = |mS ⊗ |n. Physically, this can be understood as spin states being dressed by a certain number of photons. The resulting states are thus called dressed states [84]. For low frequencies and high power as used in magnetic resonance experiments N is extremely large, which allows further simpliﬁcations, as is shown in the following section. A.3 Floquet Theory For very high average photon numbers N the Poisson distribution in (43) approximates a Gaussian distribution with very narrow relative linewidth. Almost exclusively states with n ≈ N are populated. For these the coupling terms mS , n|Hi,quant |mS − 1, n + 1 in (47) are in very good approximation equal to 12 ω1 . The quantized ﬁeld Hamiltonian can then be replaced by a Hamiltonian in the semiclassical Floquet space [15, 74, 85], 1 (48) HF ≈ H − N + ω1 . 2 The advantages of the Floquet Hamiltonian are the reduced number of considered photons (only diﬀerences of photon numbers are of interest) and the simpler mathematical treatment due to the periodicity of eigenfunctions and eigenvalues. Floquet theory was introduced in magnetic resonance by Shirley [15] to describe transition probabilities of nuclear spins in an atomic beam, subject to a linear rf ﬁeld. Later it was applied to analyze saturation-transfer EPR spectra, by calculating the steady-state solution of a stochastic Liouville equation under the inﬂuence of radiation and modulation ﬁelds [86, 87, 88, 89]. Nowadays Floquet theory is used, for example, for the description of multiple-photon processes in NMR [16] and EPR [46, 17, 32], and for explaining experimental observations in magic angle spinning (MAS) NMR [74, 85]. Mathematically, the Floquet formalism is based on a solution of linear differential equations with periodic coeﬃcients [90]. A semiclassical periodically time-dependent Hamiltonian H(t) = H0 + H1 (t) in the eigenbasis of H0 , with fundamental frequency ω/2π and period tp = 2π/ω, is expanded to a Fourier series, ∞ p|hn |qeinωt , (49) p|H(t)|q = n=−∞

where h is the nth Fourier component of H(t). A Floquet Hamiltonian is deﬁned as a Hermitian matrix of inﬁnite dimension in Floquet space n

Multiple-Photon Transitions in EPR Spectroscopy

p, n|HF |q, m = p|hn−m |q + nωδnm δpq ,

181

(50)

where |p, n represents a spin function |p dressed by n quanta with frequency ω. The evolution of the density operator in Floquet space is described in analogy to the Hilbert space by

with the propagator

σF (t) = UF (t; t0 )σF (t0 )UF−1 (t; t0 ) ,

(51)

UF (t; t0 ) = e−iHF (t−t0 ) .

(52)

For the calculation of the evolution of the spin system in Hilbert space either the density operator or the propagator has to be transformed back to Hilbert space. For time-dependent Hamiltonians with more than one basic frequency the Floquet theory is expanded to the many-mode Floquet theory [91, 92]. When transient eﬀects are investigated the evolution of the density operator in Hilbert space can be calculated numerically [93].

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Multi-Frequency EPR Study of Metallo-Endofullerenes Klaus-Peter Dinse1 and Tatsuhisa Kato2 1

Physical Chemistry III, Darmstadt University of Technology, Petersenstrasse 20 64287 Darmstadt, Germany [emailprotected] Institute for Molecular Science, Myodaiji, Okazaki 444-8585, Japan, (present address: Department of Chemistry, Josai University, 1-1 Keyakidai, Sakado 350-0295, Japan) [emailprotected]

Abstract. Immediately after the observation that fullerenes can act as cages for persistent trapping of ions or atoms, Electron Paramagnetic Resonance (EPR) was used for the investigation of these compounds. In contrast to group 15 atoms entrapped in C60 or C70 , which are nearly perfectly decoupled from the cage and which at time average are found at the center of the cage, it was observed that single lanthanide ions are localized at speciﬁc binding sites of the internal carbon surface. 6+ , which Apparently this is no longer true for encased ion clusters like La6+ 2 or Sc3 N are exploring the full inner space of the fullerenes in such a way that the inherent symmetry of the cage is retained. Although simple 2-pulse sequences have been used to elucidate ion and cage motions in the single ion case, more advanced techniques like pulsed electron-nuclear double resonance (ENDOR) as well as two-dimensional EPR had to be invoked for the study of the cluster compounds. Performing experiments at 95 GHz, one is beneﬁting from superior spectrometer sensitivity as well as from the inherent capability for improved orientation selection. In this contribution − results about Sc3 N@C− 80 , La@C82 , and La2 @C80 compounds are compiled. It was the purpose of our investigation to obtain information about the amount of spin and charge transfer in these cage compounds. Using DFT methods, a reasonable description of the observed spin Hamilton parameters could be obtained.

1 Introduction Since trace amounts of metallo-endofullerenes (MEF) were detected more than a decade ago in 1991, scientiﬁc interest was focused on two problems, i.e., the possibility of ion localization at a speciﬁc binding site and the amount of charge transfer from the encased metal ion to the cage [1, 2]. To clarify both points would be a prerequisite for a deeper understanding of electronic properties of MEF. With respect to localization, major progress was made after pure substances in macroscopic quantities became available several years K.-P. Dinse and T. Kato: Multi-Frequency EPR Study of Metallo-Endofullerenes, Lect. Notes Phys. 684, 185–207 (2006) c Springer-Verlag Berlin Heidelberg 2006 www.springerlink.com

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later. This enabled direct structure determination via synchrotron radiation powder diﬀraction techniques, the analysis not only yielding the binding site but also allowing to identify the cage topology [3]. Furthermore, under favorable conditions, the oxidation state of the ion (or cluster) could also be determined by performing a combined Rietveld/Maximum Entropy analysis of the X-ray data. Compared to X-ray diﬀraction, application of less direct magnetic resonance methods like NMR and EPR for structure determination nevertheless has advantages given by the fact that such investigations can be performed using highly diluted material. In most cases, a very wide temperature range can also be investigated, thus probing the binding potential and testing the inﬂuence of solvents and solid matrices. Even more important is the fact that magnetic resonance methods are uniquely capable to determine the spin multiplicity of the compound, thus enabling to explore details of the molecular wave function of the compound. Knowledge of the eﬀective electronic spin for instance is of utmost importance for a complete understanding of the weakly coupled electronic system, generated by attaching the metal ion to the carbon cage. Envisioning the MEF as examples of internal charge transfer complexes, sign and size of the resulting exchange coupling between cage and ion will determine the properties of the ground state of the coupled system. It should be noted, however, that the assumption of a purely ionic interaction is too simplistic, because photoelectron spectroscopy of La@C82 for instance has revealed that there is a ﬁnite electron density in 5d levels of La [4]. Finally, for magnetic resonance, an additional degree of freedom is given by the possibility to study dilute solutions of compounds that can be reduced or oxidized by chemical or electrochemical methods, thus allowing to probe charge redistribution in the status of electron excess or deﬁciency. The localization problem is best stated by describing the potential of the inner fullerene surface seen by the encased atom or ion after intra-molecular charge transfer. Depending on the topology of this potential, either localization or quasi-free motion is possible. Clearly, the binding potential will depend on fullerene topology (i.e., carbon number and cage symmetry) as well as on the identity of the metal ion, implying that both extremes of localized particle and free motion can be realized depending on temperature. In order to distinguish between both cases, magnetic resonance techniques can be invoked. Using magnetic resonance, static as well as dynamic terms of the eﬀective spin Hamiltonian can in principle be determined from line positions and line widths data, respectively. For this purpose, a line shape analysis of individual spectral transitions is performed using a model of thermally activated hopping between diﬀerent binding sites. If the correlation time of this process is fast on the time scale of typical frequency diﬀerences in the spectrum, averaging of spectroscopic properties of diﬀerent sites occurs. Molecular tumbling on the other hand leads to characteristic line broadening, which can be analyzed in terms of various second-rank tensor interactions. Changing the temperature, the correlation time as well as the variance of the time-modulated interactions can be deduced. By this method, it is in principle

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possible to distinguish between internal hopping of the ion and a tumbling motion of the whole molecule with the ion rigidly attached to its docking position [5]. Although quite powerful for the fast correlation limit, magnetic resonance investigations based on such a line shape analysis are limited to a temperature range for which liquid solvents are available. Even more restrictive is the condition that a fully resolved and assigned spectrum must be available. Meeting these conditions, a qualitative picture emerges from a ﬁt of individual lines using Lorentzians of varying widths, thus revealing the concerted action of the anisotropic g-matrix, the electron-nuclear magnetic dipole-dipole, and the nuclear quadrupole interactions. Additionally the principal elements of these second-rank tensors can quantitatively be deduced, although an axes assignment is generally not possible. At lower temperatures, for which localization is expected, spectral analysis of the rigid-limit spectrum allows to determine these tensor elements directly, thus providing information about the binding site of the localized ion. Combining results from a high temperature dynamic study with rigid limit data, it is possible to detect the anticipated transition from a temperature-activated large-scale internal motion to rigid attachment from an analysis of the hyperﬁne (hf) tensor elements. The ground state of a coupled spin system is advantageously deﬁned by its spin multiplicity. In particular, if MEF are considered to consist of encased Lanthanide ions with an empty valence shell after charge transfer and a partially ﬁlled cage molecular orbital (MO), overlap of 4f orbitals with carbon centered orbitals will be small even at short distance. As a result, the exchange part of the Coulomb interaction will probably be small, leading to close-lying spin multiplets. Information about the eﬀective electronic spin Sef f of the compound can be determined from the analysis of the EPR spectrum. If Sef f ≥ 1, second and higher rank electronic spin-spin tensor interactions generally dominate the EPR spectrum. Evaluation of these Zero-Field-Splitting (ZFS) terms usually allows a qualitative characterization. Multi-frequency EPR data are in most cases required to discriminate between integer and half-integer spin systems. A simple method to distinguish between diﬀerent Sef f ≥ 2 spin systems with additional fourth-rank tensor components is unfortunately not available for disordered samples. In neutral MEF, a subtle balance of electron aﬃnity of the cage and ionization potential of the encaged atom governs charge distribution between ion and cage. Depending on the energy level structure of both subcomponents, accommodation of an additional electron, for instance provided by reduction to the mono-negative ion, cannot be predicted unambiguously. The question arose, whether the oxidation state of the ion could be “switched” by reduction, thus allowing changing its eﬀective spin multiplicity. Such reversal of charge transfer would be counter-intuitive, however, if modelling the MEF as Faraday cage system. Very early, the surprisingly small isotropic hyperﬁne interactions (hﬁ) of encased ions observed in Sc@C82 [7, 6], Y@C82 [9, 8], and

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La@C82 [2] were taken as evidence for a complete charge (and spin) transfer of all three metal valence electrons to the cage. A convenient way to probe a possible spin redistribution under reduction or oxidation is therefore given by probing the isotropic Fermi contact interaction of the ion. In the following, we present recent results obtained by multi-frequency EPR investigations of three diﬀerent MEF La@C82 , La2 @C80 , and Gd@C82 . These compounds nicely exemplify the above stated problems, viz. localization, exchange coupling, and charge redistribution under reduction of the compound. The system La@C82 (I) was chosen for this study because of conﬂicting conclusions published by various groups. X-ray studies by the group of Takata indicated large amplitude motion around a well-deﬁned site (quasi-localization) at room temperature [10, 11]. An early theoretical study of Andreoni and Curioni [12] came to the conclusion that the La ion could be localized at two diﬀerent positions, one of them allowing large amplitude motion. As was shown later [10, 13], the C82 topoisomer investigated in their ﬁrst study had the wrong cage symmetry (C 2 instead of C 2v ). In a later study by the same authors [14], the proper C 2v topoisomer was also investigated, apparently exhibiting a unique binding site with strong localization in addition to a second local minimum. The hypothesis of diﬀerent binding sites was in contradiction to our ﬁnding that even at room temperature molecular tumbling controls the correlation time of hf interactions seen by the ion [5]. The observation of resolved 13 C satellites in the EPR spectrum conﬁrmed our model but would still allow large amplitude motion about a deﬁned equilibrium position, which would be in agreement with Takata’s room temperature result. Apart from C60 , only C80 can be used to realize conﬁnement with the highest possible symmetry I h . As an example, diamagnetic La2 @C80 was studied by NMR in solution, indicating that the encased cluster is not restricted to any speciﬁc binding site, thus preserving the I h symmetry on time average as seen by 13 C nuclei of the cage [15]. Using qualitative arguments derived from MO theory, six electrons can be accommodated by the 4-fold degenerate Highest Occupied Molecular Orbital (hom*o) of C80 , suggesting an oxidation state of +6 for the cluster. This hypothesis was recently conﬁrmed by powder diﬀraction data [16]. In case of the mono-negative ion the ground state of the 7− 5+ 6− compound could either be represented either as La6+ 2 :C80 or as La2 :C80 , resulting in a drastic change of the La hﬁ. In the former case, the “spin-less” conﬁguration with small hﬁ of the cluster would be preserved under reduction, whereas the latter conﬁguration would be characterized by a large La hﬁ, because the additional electron would have to occupy 5d or 6s orbitals.

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2 La@C82 – a Case Study of Ion Localization 2.1 Experimental Conventional arc vaporization of a rod composed of graphite and La2 O3 in a helium atmosphere was used to prepare a fullerene soot containing La@C82 . After soxhlet extraction, the major topoisomer La@C82 (I) was separated using multi-stage HPLC techniques as described in [17]. Diluted solutions of the pure ≥98 percent compound were degassed on a high vacuum line and subsequently sealed in 4 mm or 0.9 mm o.d. quartz tubes appropriate for EPR investigations at 9.5 and 94 GHz, respectively. EPR spectra were taken with a BRUKER ELEXSYS 680 spectrometer at 9.4 GHz (X-band) and 94 GHz (W-band). W-band ENDOR experiments were performed using a home-built ENDOR probe head [18]. The microwave pulse power of 5 mW provided by the commercial microwave bridge was boosted to a level of 100 mW using a multi-stage microwave ampliﬁer [19]. The microwave ﬁeld amplitude at the sample was approximately 0.25 mT, as determined by the π/2 pulse length of 35 ns. 2.2 Results and Discussion EPR Spectra Caused by the rather low C 2v symmetry of the C82 cage of the major topoisomer of La@C82 (I), the spin Hamiltonian parameters of the paramagnetic compound will exhibit the full possible anisotropy of second-rank tensor interactions. For this reason, the g-matrix as well as electron-nuclear dipolar hﬁ and electric ﬁeld gradient (EFG) at the La site will be characterized by three diﬀerent components in their respective eigen frames. Isotropic averaging by molecular tumbling let one recover the trace of these interactions. Data thus derived can be used for a convenient identiﬁcation of the various compounds which is important to discriminate, for instance, incorporation of La in C82 cages of diﬀerent topology [20]. Fluctuating values of spin Hamiltonian parameters induced either by molecular tumbling or by internal mobility of the encased ion will in general lead to characteristic relaxation processes. Under favorable conditions, an analysis of relaxation-induced line broadening of resolved hf components (hfc) of the EPR spectra can be utilized to discriminate between various possible relaxation processes and this information can be used to derive a detailed model of cage and ion mobility. Solution spectra of La@C82 are well resolved, not only showing the dominant hﬁ of the La nucleus (I = 7/2) but also showing hﬁ with 13 C nuclei of the cage (in natural abundance). In frozen solution, however, no hf information can be extracted neither from X- nor from W-band EPR powder spectra, probably because g-matrix and hﬁ tensor eigen frames are not co-linear. Nevertheless, the g-matrix

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magnetic field (mT)

Fig. 1. 95 GHz (W-band) echo-detected EPR spectra of La@C82 (I) as a function of temperature

anisotropy is suﬃciently large to lead to a powder pattern in W-band, which is characteristic for a rhombohedral g-matrix as expected. The individual subcomponent line width masked at low temperatures by inhom*ogeneous spectral broadening can be recovered in principle by simple 2-pulse echo experiment. In the typical case of a paramagnetic sample of high dilution, the de-phasing time T2 of a particular EPR transition is of the order of a few microseconds resulting in a hom*ogeneous line width of less than 1 MHz. Because this value is much less than the spectral range of the powder spectrum, selective excitation by microwave pulse sequences is possible. Considering that the eﬀective spectral width of microwave pulses used for W-band ENDOR is less than 5 MHz, orientation selection is possible. In Fig. 1, a series of low temperature EPR spectra is depicted. As can be noted, spectral features are still changing even below 80 K. Similar observations have already been reported in [21]. Because spectra are obtained using a simple 2-pulse echo sequence with ﬁxed pulse delay, apparent intensity variations could result from an orientation-dependent spin de-phasing time T2 . This could be taken as evidence for residual cage mobility. In addition, the total spectral width is reduced noticeably comparing 80 K with 10 K spectra as is shown in Fig. 2. This change in g matrix anisotropy probably indicates structural changes, which occur also in this temperature interval. Note that in Fig. 2, ﬁeld modulation was used to emphasize canonical positions in the spectrum. A completely diﬀerent EPR spectrum is obtained when observing La@C82 in liquid solution. Hﬁ with 139 La nuclei (100 percent natural abundance) is

191

field-modulated EPR signal intensity (arb. units)

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magnetic field (mT)

Fig. 2. Continuous wave (cw) EPR spectra of La@C82 (I) at 10 K and 80 K

apparent from 8 equidistant lines. Because of very narrow EPR lines, it is also possible to resolve additionally various hf coupling constants (hfcc) with 13 C nuclei of the cage (see Fig. 3). Assuming equal spin de-phasing rates for all La hfc, spin statistics predict 8 major lines of equal intensity. The observed characteristic deviation from equal intensities can be evaluated and used to determine the dominant EPR relaxation mechanisms. A consistent picture had emerged when ﬁtting the widths of individual La hfc in the liquid phase temperature range of 200 to 300 K. A detailed analysis was performed by two research groups, showing that ﬂuctuations of the EFG at the La site in combination with electron-nuclear dipole-dipole interaction, both caused by molecular tumbling, are the major sources of electron spin relaxation. Considering that 13 C hﬁ could also be observed over a wide temperature range, it could be concluded that the ion is rigidly attached to the inner surface of the carbon cage. Any additional internal hopping of the ion to a diﬀerent attachment site would drastically inﬂuence the 13 C hﬁ, because the spin density distribution on the cage would change if the ion would be attached to another site, thus modifying the 13 C hfcc. As a result, the widths of these 13 C satellites would drastically diﬀer from those of the main components. In addition to the isotropic part of La hﬁ (aiso (139 La) = 3.22 MHz), which can be directly deduced form the EPR spectrum, the line width analysis resulted in an estimate of the principal component of the traceless dipole-dipole interaction Az z (139 La) = 6.5 MHz in its eigen frame denoted by z’. For this analysis, axial symmetry had to be assumed, and no axis assignment with respect to the molecular frame is possible. An estimate of the nuclear quadruple

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magnetic field (mT)

Fig. 3. Room temperature EPR spectrum of La@C82 (I) in toluene

coupling constant e2 Qq/(2I(2I − 1)h) = 0.85 MHz was also obtained. Because nuclear quadrupole interaction (nqi) is traceless, this interaction does not lead to ﬁrst order line splittings in the EPR spectrum. Any attempt to conﬁrm this analysis therefore had to invoke electron-nuclear double resonance techniques. ENDOR Spectra Apart from simplifying crowded EPR spectra, ENDOR gives the opportunity to detect EFG using the electric quadrupole moment Q of I > 1/2 nuclei as local sensor. This can be seen by calculating ENDOR frequencies from eigenvalues of the spin Hamiltonian H/h = νe − νn + SAI + IQI ,

(1)

under the condition of ∆ms = 0 and ∆mI ± 1. Here, in usual notation, the electron and nuclear Larmor frequencies are indicated as νe and νn , respectively. The dipolar hf coupling tensor (including its isotropic part) is denoted as A, the traceless quadrupole coupling tensor as Q (all values given in frequency units). Its principal value Qz z is related to the quadrupole coupling constant e2 Qq/h by Qz z = e2 qQ/(2I(2I − 1)h). The presence of a nonvanishing nqi leads to two characteristic multiplets of 2I lines, each centered at |νn + Azz ms |. ∆νmI +1 ,mI = |νn + Azz (Θ, Φ)ms + 3/2Qzz (Θ, Φ)(2mI + 1)| .

(2)

In (2), the nuclear spin quantum number mI is ranging from −|I| to |I| − 1. For simplicity only allowed NMR transitions are considered which might be

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too restrictive if eﬃcient nuclear spin mixing results from non collinear nqi. In both electron spin sublevels, the same characteristic quadrupolar splitting of 3Qzz between adjacent lines is predicted. The orientational dependence of all interactions is denoted by the angles Θ and Φ, relating the eigen axes of the interactions (given as “primed” coordinates) to the laboratory z axis, which in turn is deﬁned by the magnetic ﬁeld direction. The orientation dependence of both hf interactions causes a “powder-like” ENDOR line shape, in general precluding the determination of the coupling constants. In contrast, if eﬃcient orientation selection can be obtained by frequency-selective excitation of spin packets in the EPR spectrum, a resolved “single crystal-like” ENDOR spectrum is predicted. Suﬃcient orientation selection can for instance be obtained if electron Zeeman anisotropy dominates the EPR spectrum, thus leading to an unique set of Θ and Φ values for a given ﬁeld setting. In our case, because of C2v symmetry of the compound, which is preserved as site symmetry at the ion, there will be a common eigen axes system of the g matrix and La hﬁ and nqi tensors. Selective excitation at the edge of the EPR absorption spectrum is thus predicted to yield a narrow-line ENDOR spectrum, which can be analyzed in terms of one of the eigenvalues of the hﬁ and nqi tensors. In case of La@C82 (I), electron spin de-phasing times of approximately 5 µs were observed, nearly independent of temperature. Spin-lattice relaxation rates were strongly temperature dependent and reached 5 ms at 10 K. Under these conditions, pulsed ENDOR spectra could be recorded using a π-T-π/2τ -π sequence (tπ/2 = 40 ns, T = 180 µs, τ = 1 µs), nuclear spin transitions being excited during interval T after the inverting π pulse. ENDOR signals were extremely weak and could only be detected after more than 10 hours accumulation time. The resulting spectrum is depicted in Fig. 4. In agreement with (2), equidistant transitions are observed, centered about the nuclear Zeeman frequency νn (139 La) = 20.4 MHz, the low-frequency multiplet, however, being barely detectable. As is indicated by the simulation, the expected seven-line pattern of both ms multiplets, separated by Az z , overlap partially. Clear identiﬁcation of transitions of the high-frequency multiplet nevertheless allows the unambiguous determination of all relevant parameters, viz. |Az z | = 5.9(2) MHz, aiso = 4.1(2) MHz, and |Qz z | = 0.44(2) MHz. Comparing these values with the data set determined in liquid solution, viz. |Az z | = 5.8(2) MHz, and aiso = 3.2(2) MHz, it can be stated that the dipolar hﬁ is nearly invariant over the full temperature range. Evaluation of the nqi data is more complicated because no axis assignment of the solution value e2 qQ/2I(2I − 1)h = 0.85(1) MHz can be made. Assuming that z is also the principal axis of the nqi, the z orientation-selected low temperature quadrupole coupling |Qz z | = 0.44 MHz corresponds to the solution value 0.85 MHz, thus indicating a signiﬁcant reduction. Experiments performed under diﬀerent orientation selection conditions indicate, however, a larger quadruple splitting, better in agreement with the solution value [22]. Further pulsed experiments have to be performed at still higher Larmor frequencies with inherent better orientation selection to clarify this point.

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Fig. 4. (a) W-band pulsed ENDOR spectrum of La@C82 (I) measured at 10 K at B0 = 3371.6 mT. The spectrum was accumulated when exciting the high-ﬁeld edge of the EPR spectrum. (b) Simulated spectrum used to identify the individual line positions

HYSCORE Spectra As was observed quite early, hﬁ can lead to pronounced deviations from an exponential decay of 2-pulse electron spin echo decays. The so called electron spin echo envelope modulation (ESEEM) results from the interference of electron spin coherences generated by the microwave pulses and can be used to extract hﬁ data otherwise only available by double resonance techniques like ENDOR [23]. Later an elegant 2-dimensional version termed Hyperﬁne Sublevel Correlation Spectroscopy (HYSCORE) was proposed, by which nuclear spin dependent frequency diﬀerences in diﬀerent electron spin sublevels can be correlated [24]. This technique is now widely applied in the study of disordered samples and is particularly useful as the method complementary to ENDOR. This 2-dimensional technique is particularly useful in disentangling spectral features in ENDOR spectra originating from diﬀerent nuclear spins, a condition frequently met in X-band ENDOR. The observation of deep ESEEM eﬀects using a 13 C-labelled sample of La@C82 (I) prompted us to use HYSCORE to obtain additional information about La nqi. In contrast to the attempt to use eﬃcient orientation selection to obtain frequency resolved transitions within the nuclear spin manifold as described above, nearly perfect sampling of all orientations was attempted here. The reason for this strategy is as follows: Powder broadening of nuclear spin transitions occurs because of a varying orientation of the magnetic ﬁeld

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“seen” by the nuclear spin with respect to its eigenframe, for instance, deﬁned by the local EFG. If the external ﬁeld is compensated by the hyperﬁne ﬁeld, the resulting nuclear spin transitions resemble a pure quadrupole resonance spectrum (“zero ﬁeld NQR”) [25]. The possibility to vary the external ﬁeld using multi-frequency EPR and thus to “tune” into the perfect cancellation condition is advantageously used to determine 14 N nqi in biological samples. Clearly, compensation is only possible in one of the electron spin manifolds because of the dependence of the hyperﬁne ﬁeld on the electron spin magnetic quantum number mS . A minimum number of narrow lines therefore is only obtained if averaging over the full orientation sphere can be performed, thus wiping out transitions originating from one of the electron spin manifolds. This requires exciting the full spectrum. Being limited in frequency width of the microwave excitation pulses, this condition cannot be met in X-band. Using 13 C labelling, however, the spectral broadening by strong 13 C hﬁ leads to eﬀective orientation averaging even under semi-selective excitation conditions. A representative HYSCORE spectrum of the major isomer obtained at 80 K is shown in Fig. 5. Projections of the 2D plot on both frequency axes are also shown. The resulting plots correspond to spectra which would be obtained performing a one-dimensional ENDOR experiment. Because of spectral overlap in the range from 1 to 6 MHz originating from 13 C and 139 La ENDOR transitions, no clear assignment would be possible if only ENDOR data are available. In the 2D plot, however, three major features can be noted, one is a diagonal peak at 14.7 MHz, another is a pair of ridges which runs perpendicular to the frequency diagonal with a crossing point at 3.8 MHz, the other is a strong diagonal peak at 2.9 MHz. The ﬁrst peak at 14.7 MHz corresponds to the free proton Zeeman frequency and it arises from “distant” protons of spurious solvent molecules. The observed ridges, crossing the diagonal at 3.8 MHz, can be assigned to arise from 13 C dipolar hﬁ with various non-equivalent nuclei of the cage. The prominent diagonal peak at 2.9 MHz is identical with the line obtained by Fourier transformation of the deep ESEEM modulation observed in both 2-pulsed and 3-pulsed spin echo decays. Although the nuclear Zeeman frequency of 139 La at the chosen ﬁeld of 2.1 MHz is close to this value, a transition at the La nuclear Zeeman frequency cannot be observed unless lanthanum would be situated at a site of at least cubic symmetry, which is not available in our case. We therefore have to assign this prominent narrow line feature to a quadrupole transition in eﬀective zero ﬁeld between La nuclear spin levels, which could be classiﬁed in ﬁrst order (neglecting mixing of mI states by non-axial EFG components) as mI = ±1/2 ←→ ±3/2, ±3/2 ←→ ±5/2, or ±5/2 ←→ ±7/2, respectively. It should be noted, that this unambiguous assignment is possible only because in the 2D frequency plot of HYSCORE the partially overlapping ENDOR transitions from 13 C nuclei are separated and clearly identiﬁed. In the axial approximation, transitions at 3, 6, and 9 in units of e2 qQ/42h are predicted. Unfortunately, because of limited signal-to-noise and overlap with 13 C lines we were not successful in identifying unambiguously the full set

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Fig. 5. X-band HYSCORE spectrum of La@C82 at 80 K

of three quadrupole transitions, which would have enabled the determination of the asymmetry parameter also. Lacking this additional information, only an order of magnitude interpretation can be given: Assuming that the observed prominent peak at 2.9 MHz corresponds to one of the transitions given above, quadrupole coupling constants e2 qQ/2I(2I − 1)h would be obtained as 0.96, 0.64, and 0.32 MHz, respectively. These values are well in the range of |Qz z | = 0.44 MHz obtained from the ENDOR experiment. 2.3 La@C82 – Resum´ e The analysis of EPR and ENDOR data obtained at diﬀerent electron Larmor frequencies and in diﬀerent phases leads to the conclusion that no signiﬁcant change occurs with respect to the binding site of the encased ion. Apparently, the room temperature large-scale motion of the encased ion observed freezes at low temperatures, and because of averaging of hﬁ on the time scale of the EPR experiment no drastic change of dipolar and quadrupolar hﬁ is observed. These interactions are sensitive measures of the structure of the binding site because

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they are determined by the local spin and charge distribution. The detection of hﬁ in disordered samples was possible only by using orientation selection in the 94 GHz EPR spectrum, which is expanded with respect to magnetic ﬁeld by g-anisotropy. It would be a challenge to reproduce these results of spin and charge density distribution at the Lanthanum ion by quantum-chemical modelling. This will be essential for a better understanding of the electronic structure of the complex.

3 La2 @C− 80 Radical Anion – Evidence for Reduction of the Encased Cluster 3.1 Experimental La2 @C80 was prepared and separated by the method reported in [15, 26]. Mono anions of La2 @C80 were produced by chemical reduction using 1, 5diazabicycloundecene (DBU) as well as by electrochemical reduction. The electrochemical reduction was performed in well-dried o-dichlorobenzene (ODCB) with electrochemical grade tetra-n-butylammonium perchlorate (TBAP). EPR spectra of La2 @C− 80 mono anions obtained by these two reduction processes were recorded by EPR using a microwave frequency of 95 GHz (W-band) and of 9.5 GHz (X-band). Anions prepared by both methods gave identical X-band EPR spectra in the temperature range from 3 K to 295 K. 3.2 Results and Discussion EPR Analysis Having shown by NMR that neutral La2 @C80 has an electronic singlet spin ground state [15], the question arose how the mono anion would accommodate an additional electron, provided by chemical or electrochemical reduction. As stated above, the ground state of the negative ion could either be represented 7− 5+ 6− in ﬁrst approximation as [La6+ 2 :C80 ] or as [La2 :C80 ]. In this notation, localization of the additional electron either on the cage or at the conﬁned cluster is evident. In the former case, a very small La hﬁ is predicted because of vacant La valence orbitals and a completely empty 4f shell. In contrast, the latter conﬁguration would be characterized by a large La hﬁ, because the additional electron would have to occupy La 5d or 6s valence orbitals. First evidence of the possibility of “in cage reduction” was provided by the EPR spectrum of the mono-anion of Sc3 N@C80 , characterized by Sc hﬁ, one order of magnitude larger [27] than observed for Sc3 @C84 [6]. Spin polarization was proposed as reason for the small Sc hﬁ in Sc3 @C84 . The increase in hﬁ of the mono anion was explained by assuming electron localization at the cluster in reduced Sc3 @C84 , synonymous for “in cage reduction”.

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magnetic field (mT) Fig. 6. X-band EPR spectrum of La2 @C80 mono anion in toluene at 235 K

In Fig. 6, an EPR spectrum of La2 @C80 mono anions in solution is depicted. In ﬁrst order perturbation theory, a simple equidistant 15 line spectrum would be expected for this spin system consisting of S = 1/2 coupled to two equivalent I = 7/2 nuclei. The observed spectrum clearly cannot be explained even qualitatively within this scheme. The reason being twofold: First, the La hﬁ is so large that the usual ﬁrst-order truncated spin Hamiltonian H/h = νe Sz − νn Iz + aiso Sz Iz ,

(3)

obtained by omitting non-secular terms aiso (S+ I− + S− I+ ) cannot be used. These terms are lifting the frequency degeneracy of allowed EPR transitions being characterized by a total value of MI . The magnitude of frequency shifts of the center of each of these MI multiplets is of the order of a2iso /νe which amounts to 4 mT in our case (vide infra). Second, nuclear spin dependent line width variations might broaden some transitions beyond detectability. For spectral simulation, the spectrum of two equivalent nuclear spins can easily be constructed by using a coupled representation of total nuclear spin, in our case ranging from Itot = 14 to 0. Only within these nuclear spin subspaces, EPR transitions are allowed, giving a total of 64 transitions. Because of the large hﬁ constant, the spectrum extends over a very wide ﬁeld range thus varying the frequency shifts considerably. As a result, even in the simulated spectrum no simple feature evolves. However, as is shown in Fig. 7, the high ﬁeld edge of the spectrum is least disturbed, allowing to extract aiso = 38.6 mT from the separation of outmost hf components by direct comparison with the experimental data.

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E P R spectrum (arb. units)

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magnetic field (mT) Fig. 7. Simulated X-band EPR spectrum of La2 @C80 mono anion (g = 1.984, aiso = 38.6 mT, νe = 9.4 GHz.)

Modelling La hﬁ The measured isotropic La hﬁ constant of 38.6 mT corresponding to 1.03 GHz clearly cannot be rationalized as arising from spin polarization eﬀects. In LaO, a large La hf constant of 3.89 GHz has been observed, the absence of a noticeable anisotropy being taken as evidence for dominant spin density in the 6s orbital of the metal [28]. This assumption is consistent with the observation of a hfcc of 8.1 GHz of the free La2+ ion. Taking the free ion value as reference for La hﬁ arising from a singly occupied 6s orbital, in ﬁrst approximation a local 6s spin density of approximately 0.25 can be deduced for both metal atoms of the internal La5+ 2 cluster. Apart from this qualitative argument, Density Functional Theory (DFT)type calculations were performed to support the hypothesis of internal reduction. For the calculation a ﬁxed geometry had to be assumed, although at room temperature the internal cluster is highly mobile, preserving the Ih symmetry of the cage on the time scale of EPR. Preserving the inversion symmetry, the La2 cluster was placed along a C3 axis through the center of 6-membered carbon rings. The results of the DFT calculations, which were performed using the B3LYP parametrization and 6-31G* and Hay/Wadt basis sets for carbon and lanthanum, respectively, are best compiled showing the change in charge density of the compound under reduction. In agreement with the qualitative argument given above, charge density is piled up at the cluster as seen in Fig. 8. Correspondingly, the spin density is also concentrated at La. Calculated La hfcc are much smaller than the observed value, however, which

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Fig. 8. Calculated charge diﬀerence between neutral La2 @C80 and its mono anion. For details of the calculation see text

is not too surprising considering the ﬁxed core basis set used for La, which does not allow spin polarization of the inner shell orbitals to occur. e La2 @C− 80 Radical Anion – Resum´ The analysis of the EPR spectrum of mono anions of La2 @C80 gives evidence for the rare case of “in cage reduction” for this compound. In analogy to mono anions of Sc3 N@C80 , the additional electron cannot be accommodated in the closed 4-fold degenerate hom*o of the carbon cage but rather resides on the internal cluster. The observed very large La hfcc of 1.03 GHz indicates that 6s orbitals of the metal cluster are mainly populated. The predicted small anisotropy of the hﬁ was conﬁrmed by the observation of a well resolved EPR spectrum in solid solution, which could be fully analyzed invoking spectra taken at 94 GHz as well as at 9.4 GHz [29].

4 Gd@C82 – Determination of Exchange Coupling Between Ion and Cage 4.1 Experimental A highly puriﬁed sample of the major isomer of Gd@C82 (I)(C2v ) was prepared by applying various HPLC steps as described above [20, 30]. Liquid solution ESR spectra using CS2 or tri-chlorobenzene (TCB) as solvent as well as powder spectra of 100 percent Gd@C82 were obtained using a Bruker E500 (X-band) and a E680 (W-band) spectrometer. Temperature was controlled by a helium ﬂow cryostat set at 4.0 K for the X-band ESR measurement and at 20 K and 4.0 K for W-band experiments.

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4.2 Results and Discussion EPR Analysis Having identiﬁed lanthanides as class of atoms which can easily be encapsulated in fullerenes it was obvious to search for eﬀects originating from diﬀerent orbital and spin momenta resulting from varying number of 4f electrons. Lanthanum, gadolinium and lutetium are special because of their vanishing orbital momentum. In contrast to lanthanum and lutetium, each of spin multiplicity one, gadolinium with its half ﬁlled 4f shell has an octet spin multiplicity. Here we assume complete charge transfer of three valence electrons for each of the elements. As a result, La@C82 and Lu@C82 exhibit EPR properties characteristic of eﬀective electronic spin Sef f = 1/2. In case of Gd@C82 , however, the spin multiplicity of the ground state will depend on the details of coupling between ion and cage. Taking Scage = 1/2 and Sion = 7/2, Sef f = 4 or 3 can be realized. The energy diﬀerence between both states can be described by an eﬀective spin Hamiltonian (4) H = H1 + H2 − 2JS1 S2 . In (4) contributions from the isolated constituents (viz., ion and cage) are denoted by H1 and H2 , respectively, the coupling being approximated by assuming an isotropic exchange term. In this description the analogy to singlet/triplet splittings in organic molecules and charge transfer complexes is evident. For a rough estimate of J in our case the following arguments can be used: In typical small organic molecules the energy diﬀerence between the ﬁrst excited singlet and the lowest triplet state is of the order of 1 eV, in charge transfer (CT) complexes this value is reduced by at least one order of magnitude, giving values for J ranging from +4000 cm−1 to +100 cm−1 . The positive sign of J indicates that the triplet state is situated below the corresponding singlet level as is usually found in small organic molecules. The decrease in absolute value of J under CT conditions can be attributed to charge localization at diﬀerent centers with resulting decrease in orbital overlap. For weakly coupled CT complexes even both signs are observed. In MEF like Gd@C82 with large charge transfer and localization of the metal ion at the inner surface, a rather large electronic exchange term could result. However, because of an empty valence shell after transfer of three electrons, only the charge distribution of compact 4f electrons has to be considered, which might lead to a drastic reduction of J. In brief, highly localized atomic orbitals (AO) in combination with small charge separation distances prevailing in MEF preclude a reliable prediction of sign and size of J in this case. Determination of the ground state spin multiplicity would be suﬃcient to deduce the sign of J. Although it is rather straightforward to discriminate between integer and non integer spin states by EPR, the predicted diﬀerence in EPR spectra of Sef f = 3 and Sef f = 4 spin systems is not large if for instance the same ﬁne structure splitting parameter is used. In Fig. 9 ﬁctitious

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magnetic field (mT) Fig. 9. Simulated X-band EPR spectra of ﬁctitious S = 3 and S = 4 spin systems

X-band powder spectra are depicted, calculated with an identical ZFS tensor. Here, for simplicity we have not included higher forth-rank terms in the spin hamiltonian which could contribute for both spin systems. Diagonal elements Dii = [−1.56−2.64 4.20] GHz and gii = [1.99 1.99 2.00] were chosen, which are in the range of those determined for the Gd@C82 compound (see below). The spectra are calculated by summing over all transitions between eigenstates of a truncated spin Hamiltonian H = µe B T gS + S T DS .

(5)

In (5) the index T indicates transposition. The situation is slightly better if spectra are taken at W-band, i.e., by increasing the Zeeman term by one order of magnitude. Simulated spectra using the same parameter set as in Fig. 9 are shown in Fig. 10. Obviously, if the spin system exhibits a non-axial g matrix and ZFS tensor, both cases cannot be discriminated unambiguously without further knowledge. It should be noted, however, that the assumption of an identical ZFS tensor for both Sef f = 3 and Sef f = 4 multiplets is not justiﬁed. A small exchange term J implies a weakly coupled spin system consisting of S = 7/2 ˜ under and S = 1/2 constituents. Denoting the principal element of D as D, ˜ can easily be calculated in the coupled weak coupling condition the resulting D representation by invoking the relevant Clebsch-Gordan coeﬃcients giving ˜ (S=3) = 5/4D ˜ (Gd) − 1/8D ˜ cage , D (S=4) (Gd) ˜ ˜ cage . ˜ = 3/4D + 1/8D D

(6) (7)

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magnetic field (mT) Fig. 10. Simulated W-band EPR spectra of ﬁctitious Sef f = 3 and Sef f = 4 spin systems

In this approximation one ﬁnds an exclusive contribution from the metal ˜ of the S = 1/2 subcompoion to the ZFS tensor D because of a vanishing D ˜ values of both spin multiplets are predicted to scale nent. The respective D in the ratio 5 : 3. Anticipating that J might be rather small and in the range of a few wavenumbers only, EPR spectra taken at low temperature can in principal then be used to determine the ground state spin multiplicity and the energy diﬀerence to the upper electronic spin multiplet by observing the temperature dependence of line intensities and the appearance of additional lines because of increasing population of the upper levels in thermal equilibrium. ˜ incipient thermal populaBecause of the predicted rather large change in D, tion of the “excited” spin manifold will lead to much larger eﬀects in the EPR spectrum than shown in Figs. 9, 10. In case of close-lying spin multiplets the interpretation of EPR spectra becomes also complicated because no a priori information about relaxation rates neither between states of diﬀerent eﬀective spin nor within a spin multiplet is provided. It can be argued, however, that spin selection rules will restrict matrix-induced transitions between the upper and lower spin sublevels compared to transitions between states of identical eﬀective spin. In this case no dominant lifetime broadening of excited state levels is expected and therefore no diﬀerent line shape parameters have to be assumed for spectral simulations of transitions within both spin multiplets. Anticipating that line broadening by orientational disorder prevents resolution of individual spin packets, an

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magnetic field (mT) Fig. 11. W-band EPR spectra of Gd@C82 diluted in TCB taken at 4 K and 20 K, respectively

identical line width parameter can also be used for transitions connecting both spin multiplets. Spectra taken in the temperature range of 4 K and 20 K showed a noticeable change of spectral features, as shown in Fig. 11. The appearance of additional lines cannot be easily explained unless thermal population of a closelying electronic state occurs. As was reported elsewhere [31], this observation can be explained by assuming a negative exchange term J = −1.8 cm−1 , corresponding to Sef f = 3 as ground state, situated 14.4 cm−1 below the Sef f = 4 excited state. Furthermore, there was evidence that the exchange coupling is not completely isotropic. Such a traceless contribution could either arise from dipole-dipole interaction of electronic spin distributions at cage and ion or from anisotropic overlap between subcomponent orbitals. There is no way to discriminate between both possibilities by EPR. In Fig. 12 simulated spectra are shown derived from the eigenvalues of the following spin Hamiltonian H = µe B T g1 S1 + µe B T g2 S2 + S1T D1 S1 − 2JS1T S2 + S1T D12 S2 .

(8)

In (8), hﬁ contributions are omitted because no data could be extracted from the spectra. Components attributed to the metal ion and the cage are indexed as 1 and 2, respectively. Only for the S = 7/2 subcomponent corresponding to the metal ion a ZFS term D has to be included. Numerical values for diagonal elements of all matrices were obtained by a spectral ﬁt and are compiled in Table 1. Although no very good agreement between experimental and simulated spectra was obtained, the general trend is well reproduced.

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magnetic field (mT) Fig. 12. Temperature dependence of W-band EPR spectra of Gd@C82 (simulated) Table 1. ZFS (in GHz), g-matrix elements, and isotropic coupling constant J = −54 GHz used for the simulation of the EPR spectrum of Gd@C82 (1)

(1)

Dxx

Dyy

Dzz

gxx

gyy

gzz

−2.364

−2.784

+5.151

2.0090

2.0100

1.9775

(12) Dxx

(12) Dyy

(12) Dzz

(2) gxx

(2) gyy

gzz

+6.471

+8.031

−14.499

2.1050

2.0970

2.0570

(2)

Most important, not even qualitative agreement can be obtained if the sign of the exchange coupling constant J is reversed. Exchange Interaction in Gd@C82 Unravelled Apparently the compound Gd@C82 is an interesting model system for a class of compounds exhibiting weak Coulomb exchange. In such a case Coulomb interaction between spin-separated subcomponents results in energy separation of spin multiplets with the eﬀect that thermal population of “excited” levels not only leads to noticeable spectral changes but also in a non-standard temperature dependence of the magnetic susceptibility. If the compound is investigated as “pure” substance, i.e., if Coulomb interaction is also admitted between adjacent molecules for instance in closely packed crystalline topology, the resulting eﬀective spin will also be inﬂuenced by the electronic

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band structure. “Anti-ferromagnetic” coupling could lead to complete spin compensation of cage spins, leaving the remaining “shielded” S = 7/2 spins of the metal ions which form a super paramagnet. Switching from an integer spin system to a dipolar coupled half-integer situation by studying pure Gd@C82 instead of a dilute solid solution was indeed observed [31].

5 Conclusion The examples presented give an impression about the rich variety of problems related to ground state properties of these new molecular structures with varying degree of electronic interaction between well deﬁned subcomponents. Application of advanced EPR methods like pulsed EPR and pulsed ENDOR as well as the capability to use electronic Zeeman energies comparable with ZFS interaction is mandatory for successful characterization of MEF. A clear understanding of the transition between localization observed in Gd@C82 and La@C82 and quasi-free internal rotation as seen in La2 @C80 anions probably requires development of new parametrization for DFT calculations. Reliable predictions of redox potentials, and more general about chemical reactivity also await improvement of current quantum chemical methods.

Acknowledgement In this report results obtained by various graduate students (P. Schweitzer, M. R¨ ubsam, P. Jakes, S. Okubo) and post-docs (Dr. N. Weiden, Dr. H. Matsuoka, Dr. K. Furukawa) are presented. Without their dedicated work this study could not have been completed. Fruitful and long lasting collaborations with Prof. H. Shinohara and Prof. T. Akasaka were essential for this interdisciplinary research. Financial support by the Deutsche Forschungsgemeinschaft under various grants is gratefully acknowledged. German/Japanese collaboration was supported by a visiting professor fellowship and travel grants of the Institute for Molecular Science, Okazaki. We thank Dr. S. Stoll (ETH Z¨ urich) for providing the EasySpin software package.

References 1. Y. Chai, T. Guo, C. Lin, R. Hauﬂer, L.P.F. Chibante, J. Fure, L. Wang, J.M. Alford, R.J. E. Smalley: Phys. Chem. 95, 7564 (1991) 2. R.D. Johnson, M.S. de Vries, J.R. Salem, D.S. Bethune, C.S. Yannoni: Nature 355, 239 (1992) 3. For a recent review see: Endofullerenes, A New Family of Carbon Clusters, Akasaka, T. and Nagase, S., Eds. (Kluwer Academic Publishers, Dordrecht 2002) 4. B. Kessler, A. Bringer, S. Cramm, C. Schlebusch, W. Eberhardt, S. Suzuki, Y. Achiba, F. Esch, M. Barnaba, D. Cocco: Phys. Rev. Lett. 79, 2289 (1997)

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5. M. R¨ ubsam, P. Schweitzer, K.-P. Dinse: J. Phys. Chem. 100, 19310 (1996) 6. H. Shinohara, H. Sato, M. Ohkohchi, Y. Ando, T. Kodama, T. Shida, T. Kato, Y. Saito: Nature 357, 52 (1992) 7. C.S. Yannoni, M. Holnkis, M.S. de Vries, D. Bethune, J.R. Salem, M.S. Crowder, R.D. Johnson: Science 256, 1191 (1992) 8. H. Shinohara, H. Sato, Y. Saito, M. Ohkohchi, Y. Ando: J. Phys. Chem. 96, 3571 (1992) 9. J.H. Weaver, Y. Chai, G.H. Kroll, C. Jin, T.R. Ohno, R.E. Hauﬂer, T. Guo, J.M. Alford, J. Conceicao, L.P.F. Chibante, A. Jain, G. Palmer, R.E. Smalley: Chem. Phys. Lett. 190, 460 (1992) 10. E. Nishibori, M. Takata, M. Sakata, H. Tanaka, M. Hasegawa, H. Shinohara: Chem. Phys. Lett. 330, 497 (2000) 11. M. Takata, E. Nishibori, M. Sakata: Endofullerenes, A New Family of Carbon Clusters, T. Akasaka and S. Nagase, Eds. (Kluwer Academic Publishers, Dordrecht 2002) 12. W. Andreoni, A. Curioni: Phys. Rev. Lett. 77, 9606 (1996) 13. T. Akasaka, T. Wakahara, S. Nagase, K. Kobayashi, M. W¨ alchli, K. Yamamoto, M. Kondo, S. Shirakura, S. Okubo, Y. Maeda, T. Kato, M. Kako, Y. Nakadaira, R. Nagahata, X. Gao, E. van Caemelbecke, K.M.J. Kadish: J. Am. Chem. Soc. 122, 9316 (2000) 14. W. Andreoni and A. Curioni: Appl. Phys. A 66, 299 (1998) 15. T. Akasaka, S. Nagase, K. Kobayashi, M. W¨ alchli, K. Yamamoto, H. Funasaka, M. Kako, T. Hoshino, T. Erata: Angew. Chem. Int. Ed. Engl. 36, 1643 (1997) 16. E. Nishibori, M. Takata, M. Sakata, A. Taninaka, H. Shinohara: Angew. Chem. Int. Ed. 40, 2998 (2001) 17. K. Yamamoto, H. Funasaka, T. Takahashi, T. Akasaka: J. Phys. Chem. 98, 2008 (1994) 18. N. Weiden, B. Goedde, H. K¨ aß, K.-P. Dinse, M. Rohrer: Phys. Rev. Lett. 85, 1544 (2000) 19. The add-on unit was produced by W. Krymov. 20. S. Okubo, T. Kato, M. Inakuma, H. Shinohara: New Diamond and Frontier Carbon Technology 11, 285 (2001) 21. S. Knorr, A. Grupp, M. Mehring, U. Kirbach, A. Bartl, L. Dunsch: Appl. Phys. A 66, 257 (1998) 22. N. Weiden, T. Kato, K.-P. Dinse: J. Phys. Chem. B 108, 9469 (2004) 23. W.B. Mims: Phys. Rev. B 5, 2409 (1972) 24. P. H¨ ofer: J. Magn. Reson. A 144, 77 (1994) 25. H.L. Flanagan, D. Singel: J. Chem. Phys. 87, 5606 (1987) 26. T. Suzuki, Y. Maruyama, T. Kato, K. Kikuchi, Y. Achiba, K. Kobayashi, S. Nagase: Angew. Chem. Int. Ed. Engl. 34, 1094 (1995) 27. P. Jakes and K.-P. Dinse: J. Amer. Chem. Soc. 123, 8854 (2001) 28. W. Weltner, Jr., D. McLeod, P.H. Kasai: J. Chem. Phys. 46, 3172 (1967) 29. T. Kato, S. Okubo, H. Matsuoka, K.-P. Dinse in: Fullerenes and Nanotubes: The Building Blocks of Next Generation Nanodevices, Proc. of the Intern. Symp. on Fullerenes, Nanotubes, and Carbon Nanoclusters (Paris, France 2003); D.M. Guldi, P.V. Kamat, F. D’Souza, Eds: Vol. 13, The Electrochemical Society (Pennington, NJ 2003) 30. S. Okubo, T. Kato: Appl. Magn. Reson. 23, 481 (2003) 31. K. Furukawa, S. Okubo, H. Kato, H. Shinohara, T. Kato: J. Phys. Chem. A 107, 10933 (2003)

Beyond Electrons in a Box: Nanoparticles of Silver, Platinum and Rhodium J.J. van der Klink Institut de Physique des Nanostructures, EPFL, 1015 Lausanne, Switzerland [emailprotected]

Abstract. This chapter presents a review of NMR in small particles of three elemental metals: the “odd electron” Ag and Rh, and the “even electron” Pt. These experiments on elements close to each other in the periodic system demonstrate the versatility of the NMR technique, detecting several properties that are not captured by the simplest “electrons in a box” approximation. The NMR of Ag particles shows Bardeen-Friedel oscillations in the spatial density of Fermi-level electrons. The case of Pt is dominated by a strong spatial variation in the local density of states of d-like electrons, and Rh shows at low temperatures signs of incipient antiferromagnetism. As an introduction, a summary of relevant NMR theory is presented, and its application to the NMR of bulk Na metal.

1 Introduction The framework provided by the early explanation of the NMR shift in metals in terms of a hyperﬁne coupling and a Pauli-type of susceptibility [1], is still solidly upright today. But although the estimates for these quantities at the time were good enough to show the essential correctness of the theory, it was only much later, with the advent of methods developed by Kohn, Hohenberg and Sham, that a suﬃcient theoretical and numerical power became available to yield nearly quantitative comparisons between theory and experiment. It is somewhat unfortunate that in the meantime a vast body of experimental data had been accumulated (for a compilation, see [2]), without an adequate theoretical framework for their interpretation. While the last extensive treatment of metal NMR in book form appeared in 1970 [3], the ﬁrst modern collection of “Calculated Electronic Properties of Metals” was published only eight years later [4], and the ﬁrst review paper that attempted to discuss metal NMR somewhat comprehensively in the light of such theoretical results dates from 2000 [5]. As we will see, some of these calculated results are astonishingly close to what are now considered as the best experimental data for the simple metals. For transition metals the available experimental data are not complete enough to provide a full test of the numerical results, although a J.J. van der Klink: Beyond Electrons in a Box: Nanoparticles of Silver, Platinum and Rhodium, Lect. Notes Phys. 684, 209–241 (2006) c Springer-Verlag Berlin Heidelberg 2006 www.springerlink.com

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1983 calculation speciﬁc to Pt [6] has given an amazing agreement with the experimental value for the so-called core polarization hyperﬁne ﬁeld, a quantity ﬁrst discussed, both experimentally and theoretically, by Yafet, Jaccarino and colleagues around 1964 [7]. This hyperﬁne ﬁeld is one of three phenomena not covered in the very ﬁrst descriptions of metal NMR; in fact it was expected at the time [1, p. 115] that no shift would be associated with the paramagnetism from d electrons. The two others are the Stoner enhancement of the static uniform susceptibility χ, as compared to the value predicted by simple Pauli theory, and the Moriya desenhancement of the spin-lattice relaxation rate T1−1 with respect to its Korringa value. All three phenomena are consequences of the exchange enhancement of the nonlocal complex paramagnetic susceptibility. Our present ideas on this point [8] are based on theories concerning the inhom*ogeneous electron gas published around 1975 [9, 10]. Formally, this susceptibility is a zero-ﬁeld groundstate property that can be calculated in a linear response theory; in practice the core polarization ﬁelds are more easily obtained by explicitly including a magnetic ﬁeld in the Hamiltonian. In most metals the temperature dependence of the susceptibility is only weak, and therefore χ can be calculated as a zero-temperature property. However, some strongly enhanced metals, of which Pd is the archetypical example [11], show a marked (and not necessarily monotonic) temperature dependence of χ. This phenomenon can be described by spin ﬂuctuation theory [12]. The basic idea is that a thermal ﬂuctuation sets up an “internal ﬁeld”, similar to that in Stoner theory, which couples to the other ﬂuctuation modes. In this way, there is a temperature-dependent magnetic energy in zero applied ﬁeld, associated with the thermal ﬂuctuations of the local spin density around a zero average. This theory is purely thermodynamical and considers the hom*ogeneous electron gas: it yields expressions for the low-frequency wavevectordependent susceptibility, but not for the corresponding hyperﬁne ﬁelds. The ﬁrst NMR experiments on nonbulk metal used thin ﬁlaments of lead, embedded in porous glass [13]. The results were interpreted in terms of surface eﬀects. That discussion appeared just a year after an inﬂuential theoretical paper predicting a quantum size eﬀect for the susceptibility of small metal particles [14]. From simple free electron theory for N electrons in a small “box”, one expects the spacing between energy levels to be of the order Ef /N , where Ef is the Fermi energy of the corresponding bulk solid. At temperatures kT Ef /N , the susceptibility is expected to deviate from its bulk value. Furthermore, for an element with an odd number of electrons per atom there should appear a Curie-like susceptibility for those “boxes” in the sample (supposed to consist of many particles of similar, but not identical, size) that contain an odd number of atoms. Several groups have searched for such eﬀects in samples of small copper particles, with moderate success. A possible reason is that at temperatures low enough for the discrete energy levels to be felt, the usual T1 mechanism (electron spin ﬂip–ﬂops) disappears

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because the requirement of energy conservation cannot be satisﬁed, so that the signal is very easily saturated. This paper presents a review of the NMR in small particles of three elements: the “odd electron” Ag and Rh, and the “even electron” Pt. As a preliminary, the next Sect. 2 introduces the notation and mentions results for some bulk metals. The NMR of Ag particles shows the Bardeen-Friedel oscillations in the spatial density of Fermi-level electrons, and their “washing-out” at higher temperatures. The case of Pt is dominated by a strong variation in the local density of states of d-like electrons (the “exponential-healing” phenomenon) and Rh shows at low temperatures signs of incipient antiferromagnetism. These experiments on elements close to each other in the periodic system demonstrate the versatility of NMR as a technique, detecting several properties that are not captured by an “electrons in a box” approximation. Perhaps the most evident of such properties is the size-dependent structural change detected in small particles of Tc by 99 Tc NMR [15]. That nucleus has spin 9/2, and therefore the hexagonal structure of the bulk leads to a NMR spectrum of about a MHz wide (in a 7 T ﬁeld), whereas small particles (average diameter 2.3 nm, but with a rather wide size distribution) give a single narrow line, about 1–5 ”kHz wide. This clearly shows that the small Tc particles have a cubic structure.

2 Bulk Metals 2.1 Theory The theory of the NMR of paramagnetic metals [5, 8] is based on the local density approximation of the density functional theory for the low-frequency limit of the complex nonlocal spin susceptibility of the inhom*ogeneous electron gas χ(r, r ; ω), which gives the response in r to a magnetic ﬁeld with frequency ω applied in point r [9, 10, 16, 17, 18]. That susceptibility can be written in the form of an integral equation: χ(ρ, ρ + Rα ; ω) = χP (ρ, ρ + Rα ; ω) N χP (ρ, ρ1 + Rβ ; ω)ν(n(ρ1 + Rβ ))χ + β=1

cell

(ρ1 + Rβ , ρ + Rα ; ω) dρ1 ,

(1)

where ν(n(r)) is related to a second derivative of the exchange-correlation energy, and is (in the local-density approximation) a function only of the charge density n at r. The quantity χP (ρ, ρ + Rα ; ω) is the “noninteracting” Pauli susceptibility. The vectors ρ and ρ1 are in the unit cell at the origin, and Rα and Rβ are lattice vectors. N is the number of unit cells.

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The dependence of χ on N lattice vectors in real space can be replaced by a dependence on N vectors q α in the reciprocal lattice through the Bloch Fourier transform: χ(ρ, ˜ ρ ; q α ; ω) =

e iqα ·Rβ χ(ρ, ρ + Rβ ; ω) .

(2)

β=1

The ν(n(r)) that appears in (1) creates simultaneously the Stoner enhancement of the susceptibility, the core-polarization hyperﬁne ﬁelds, and the desenhancement factor in the spin-lattice relaxation rate [8]. In terms of χ ˜ (the real part of χ) ˜ the static uniform susceptibility is χ = Ω −1 (3) χ ˜ (ρ, ρ ; 0; 0) dρdρ , cell

where Ω is the volume of the unit cell. For some metals, the spin-only static susceptibility has been measured from the intensity of the ESR absorption. The ESR signal can be calibrated by using the same setup for an NMR measurement [1] on the same sample, since the nuclear susceptibility can easily be calculated. Such experiments were performed by Schumacher and Slichter [19]. The (isotropic part of the) Knight shift K is 2 χ ˜ (ρ, ρ ; 0; 0) dρ , K(ρ) = (4) 3 where the nucleus under consideration has a relative position ρ in the unit cell. By deﬁnition, the hyperﬁne ﬁeld Bh is simply related to the ratio of the Knight shift and the uniform susceptibility through the dimensionless quantity ΩBh (ρ) K(ρ) = , µ0 µB χ

(5)

but we will see later, in (9), that this simple deﬁnition of “the” hyperﬁne ﬁeld is not always physically meaningful, and that it can be more judicious e.g. to attribute diﬀerent hyperﬁne ﬁelds to s-like and to d-like electrons. The general expression for the relaxation rate of a nucleus in ρ is −1

S(T1 (ρ)T )

µ0 = 4π

4µB 3

N 2 χ ˜ (ρ, ρ; q α ; ωS − ωI ) , N α=1 (ωS − ωI )

(6)

where S = (2µB )2 /(4πkγ 2 ), and ωS and ωI are the electronic and nuclear Larmor frequencies. The imaginary part of the susceptibility χ is an odd function of frequency, and linear for small values of ω. The right-hand side of (6) is then frequency-independent and can be evaluated in the limit of

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vanishing frequencies. For metals χ is (nearly) temperature-independent, in which case T1−1 ∝ T , as was ﬁrst found by Heitler and Teller in 1936 [20]. For completeness we mention the expression for the scalar coupling between two nuclei, at positions ρ1 and ρ2 in the unit cell, mediated by the electron spin susceptibility: J(ρ1 , ρ2 ) = µ0 (2/3)2 γ1 γ2 N −1

χ ˜ (ρ1 , ρ2 ; q α ; 0) .

(7)

α=1

This coupling was experimentally detected and theoretically explained by Bloembergen and Rowland [21]; it has also been discussed by Ruderman and Kittel [22]. The origin of the Knight shift is the contact interaction, which is “symmetric” in nuclear and electronic spins. The corresponding shift of the ESR line, the Overhauser shift O, is proportional to the thermal average of the nuclear spin magnetization γI I = χn H 0 , with χn the nuclear magnetic susceptibility, just as the Knight shift is proportional to the electron spin susceptibility. If the (5) is meaningful, we will therefore have that O = (χn / χ )K. The interesting thing is that at practical temperatures the nuclei form a magnetically non-interacting system with an easily computed (Langevin) susceptibility, so that a measurement of O can be interpreted as a measurement of Bhf . 2.2 Practical Equations Starting from (1), the static susceptibility, (3), the Knight shift, (4), and the relaxation rate, (6), for transition metals can be decomposed approximately into sums of s- and d-like contributions [8, 23], to which an additional orbital term [24] must be added: Ds (Ef ) Dd (Ef ) χ = µ0 µ2B Ω −1 + (8) + χorb = χs + χd + χorb 1 − αs 1 − αd Ω K= χs Bhf,s + χd Bhf,d + χorb Bhf,orb = Ks + Kd + Korb (9) µ0 µB S(T1 T )−1 = ks Ks2 + kd Kd2 Rd + (µB Dd Bhf,orb )2 Rorb , (10) where Ω is the atomic volume; Ds and Dd are densities of states at the Fermi energy Ef ; αs and αd Stoner enhancement factors that are related to exchange integrals Is,d = αs,d /Ds,d (Ef ); the Bhf eﬀective hyperﬁne ﬁelds (including core polarization); ks and kd Moriya desenhancement factors; and Rd and Rorb reduction factors related to the decomposition of Dd into contributions of diﬀerent symmetry Dt2g and Dte . The three densities of states Ds (Ef ), Dt2g (Ef ) and Deg (Ef ) can be found from band structure calculations, and the d-like hyperﬁne ﬁeld can sometimes be determined by experiment. The αl (l = s, d) are treated as ﬁttable parameters. It is usually assumed that kl can be calculated from some l-independent function of the Stoner parameter

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J.J. van der Klink

k(α), thus kl = k(αl ). We have often used the Shaw–Warren result [25], that can be ﬁtted to (11) kSW (α) = (1 − α)(1 + 1/4 α) , while the k(α) relation of the original Moriya equation [26, 27] can be approximately represented as kM (α) = (1 − α)(1 + 5/3 α2 ).

(12)

For strongly enhanced paramagnets, K and/or T1 T may become temperature dependent through spin ﬂuctuations. In Sect. 5.2 we summarize how these temperature dependences are expected to diﬀer for ferromagnetic and antiferromagnetic enhancements. For noninteracting electrons α = 0, and for the case of s-electron metals we retrieve the Korringa relation [28] S(T1 T )−1 = K 2 ,

(13)

that has been independently derived by Slichter [1]. As we will see shortly, even for Na metal the relation (13) is only very approximately fulﬁlled. 2.3 Na Metal The properties of the alkali metals are often discussed with reference to a free electron gas with the same electron density. If the volume per electron (and therefore also per atom) is Ω and the electron density parameter rs is deﬁned through Ω 43 πrs3 then the density of states at the Fermi energy (number of states per atom and per Joule) in the free electron gas (feg) is Dfeg (Ef ) = 1.869 × 1017 (rs /a0 )2 ,

(14)

where a0 is the Bohr radius. The susceptibility is χfeg = 32.65 × 10−6 (rs /a0 )−1

(15)

and has no dimension, although it is often referred to as the “volume” susceptibility. The corresponding cgs value is obtained by dividing the right-hand side by 4π. The cgs susceptibility in emu mol−1 results from a further multiplication by the molar volume Vm = NA Ω, and therefore has the dimension of a volume, and units cm3 . In the early papers [1] the Knight shift Kfeg was written as the product of a susceptibility as obtained above and the intensity at the nucleus of a Fermienergy single-electron wavefunction, |ψ(0)|2 , normalized in the volume Ω χ 8π feg Ω|ψ(0)|2 Kfeg = 4π 3 χ 4π ΩB feg hf = (16) 4π µ0 µB = µB Dfeg (Ef )Bhf ,

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where the second line introduces the equivalent hyperﬁne ﬁeld Bhf , with units Tesla (T). The corresponding cgs equations are obtained by replacing the factors χfeg /4π by χcgs /NA Ω, where χcgs is the susceptibility in emu mol−1 , and further replacing µ0 by 4π. The resulting hyperﬁne ﬁeld is in Gauss (G), and 1 T = 104 G. In the case of Na, the “best” experimental value for the spin susceptibility at low temperature (4.2 K) is thought to be (13.63 ± 0.14) × 10−6 [29], and the value found for the hyperﬁne ﬁeld by Overhauser shift measurements is Bhf = 24.76 ± 1.2 T [30]. The molar volume of sodium at low temperatures is Vm = 22.72 cm3 . These ESR-based values of susceptibility and hyperﬁne ﬁeld predict a value for the Knight shift K = (1.09 ± 0.05) × 10−3 , in good agreement with the accepted value K = 1.07 × 10−3 at 10 K. The value of Bhf corresponds to |ψ(0)|2 = 120 Ω −1 , which shows clearly how the wavefunctions that are important for the contact shift pile up near the nucleus. For rs /a0 = 3.93, appropriate for Na, (14) gives a density of states Dfeg = 2.89 × 1018 J−1 , or 6.3 states per atom and per Rydberg (1 Ry = 2.1795 × 10−18 J) in good agreement with the value 6.12 Ry−1 from band structure calculations [4]. Therefore χfeg = 8.31 × 10−6 is close to the value from the band structure calculation in the Pauli approximation, but both are clearly diﬀerent from the experimental χobserved = 13.63 × 10−6 . The Stoner exchange integral I, as determined from the χobserved and the calculated D(Ef ) is I = 66.5 mRy, whereas the directly calculated value is I = 67.8 mRy. The calculated value of the direct hyperﬁne ﬁeld (without core polarization) is Bhf = 31 T; the agreement with experiment (24.8 T) is not perfect, but nevertheless rather impressive. From the calculated density of states, exchange integral and hyperﬁne ﬁeld we ﬁnd K = 1.38×10−3 , which is 30% higher than the experimental value. All these theoretical results are from the collection of calculated properties of metals [4]. Associated with the Stoner enhancement of the uniform susceptibility is a disagreement between the T1 T value calculated from the Korringa relation (13), 3.29 s K, and the experimental value 5.1 ± 0.3 s K.The Shaw–Warren model, (11), gives k(0.407) ≈ 0.653 , close to the value of 0.64 observed in Na. All this shows that susceptibility enhancement is important even in such a “simple” metal as sodium, and that computational methods as already available in 1978 [4] can yield good agreement with experiment. The free electron gas model has a very smooth density of states around the Fermi energy, and the only temperature eﬀect is thermal expansion, which changes rs /a0 . The same parameter describes pressure eﬀects. The volume dependences of the susceptibility, (15), and of the Knight shift, (17), are given by d ln χfeg = −1/3 d ln Ω d ln Kfeg d ln χfeg d ln |ψ(0)|2 = +1+ . d ln Ω d ln Ω d ln Ω

(17) (18)

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J.J. van der Klink

χ / χ0

1.03 1.02 1.01 1 1.02 1

0.95

0.9 23

K/K0

0.99 0.98 0.97 1.02 1

0.95 V(p)/ V0

0.9

Fig. 1. Volume dependence of the spin susceptibility in Na metal, measured by the CESR method and of the 23 Na Knight shift. All quantities relative to room temperature and atmospheric pressure. [After Kushida et al., [34] and Benedek et al., c [31]. 1976, 1958 by the American Physical Society]

Clearly, changes of volume can aﬀect both the uniform susceptibility and the hyperﬁne ﬁeld, and both variations have been observed for Na, see Fig. 1. The change of susceptibility with volume at constant temperature (4.2 K) has been measured by the CESR method [31] for volume reductions to Ω(p)/Ω(0) = 0.9, and the quantity in (17) has been found as −0.34 ± 0.03. The 23 Na pressure-dependent NMR data [31, 32, 33] in the same range of relative volume give a slope of +0.13 ± 0.02 for (18), so that d ln |ψ(0)|2 /d ln Ω ≈ −0.54. If the changes in volume scale at the level of the wavefunctions, such that Ω|ψ(0)|2 remains constant, then d ln |ψ(0)|2 /d ln Ω = −1. As a kind of opposite case, the changes in volume might aﬀect the wavefunction far away from the nucleus, while |ψ(0)|2 stays constant: d ln |ψ(0)|2 /d ln Ω = 0. In the free electron gas approximation we may thus expect for the volume dependence of the Knight shift − 1/3 ≤

d ln Kfeg ≤ 2/3 . d ln Ω

(19)

Beyond Electrons in a Box

Pressure (GPa) 1 2 4

K(p) / K 0

1.01

217

1.00 0.99 0.98 1.0

0.9

0.8 V(p)/ V0

0.7

0.6

Fig. 2. Volume dependence of the relative 23 Na Knight shift in Na metal measured at 295 K in a diamond anvil cell, achieving very large volume reductions. Two sets c of data, taken in slightly diﬀerent cells. [After Kluthe et al., [33]. 1996 by the American Physical Society]

In sodium metal, this slope of the Knight shift changes sign at a volume reduction V (p) ≈ 0.8V (0), see Fig. 2, and becomes approximately −0.1. There are no corresponding data for the susceptibility, but if we continue to use the value of −0.34, then d ln |ψ(0)|2 /d ln Ω ≈ −0.76, closer to the value −1 expected for Ω|ψ(0)|2 constant. In this interpretation of the NMR data, initially the value of the wavefunction far away from the nucleus changes more than the value close to it; but at higher pressure the “compression” becomes more uniform. For Li, the best value for the spin susceptibility is (27.4 ± 0.1) × 10−6 [29], and the hyperﬁne ﬁeld from the Overhauser shift is 5.39 ± 0.2 T [35]. The molar volume is 12.8 cm3 . The Knight shift calculated from these data is (2.69 ± 0.1) × 10−4 , while the experimental value is 2.6 × 10−4 . The calculated direct hyperﬁne ﬁeld (without core polarization) is 6.9 T and the calculated K = 4.33 × 10−4 . These satisfactory agreements hide in fact a conceptual problem concerning the meaning of the hyperﬁne ﬁelds. Several bandstructure calculations that project out the partial densities of states of s-, p- and dsymmetries, see e.g. [36], agree that for Li about 3/4 of the density of states at the Fermi energy corresponds to p-states, that have zero amplitude at the nucleus; for Na the value is about 2/5. A CESR experiment, and therefore the Overhauser experiment also, sees all electrons at the Fermi energy, as shown by the good agreement between measured and calculated susceptibilities: the measured hyperﬁne ﬁeld is an average value over all these electrons. We now face a delicate question on the principle of a Knight shift calculation, and even more so for the relaxation rates. Should we consider only the s-like susceptibility, its enhancement, and the (high) purely s-like hyperﬁne

218

J.J. van der Klink

ﬁeld, or rather the average susceptibility, its enhancement and the (lower) average hyperﬁne ﬁeld? The reasonable agreement between calculated properties and the observed χ and K for two alkali metals, Li and Na, with rather different band structures speaks in favor “average sp”-approach over the “pure s-type” models. A disadvantage of the average sp-approach is that its hyperﬁne ﬁelds cannot really be considered as atomic properties, and can, strictly speaking, not be compared easily between unit cells in diﬀerent environments. Nevertheless, we will assume the existence of two site-independent hyperﬁne ﬁelds when using (9) for the interpretation of small-particle 195 Pt NMR. The expression for the Knight shift is a sum of terms linear in the densities of states and hyperﬁne ﬁelds, and can be interpreted as representing an average, weighted by lifetime. The relaxation rate contains a sum of squares, and its interpretation as an average requires that on some relevant time scale the d electrons are distinguishable from the s (or rather sp) electrons. The relevant time scale is of the order of the time it takes for an electron to ﬂy through the Wigner-Seitz cell [37]. Formally, the requirement for the separation into these two contributions is that cross-enhancement between d- and sp-susceptibilities be negligible [8].

3 Small Silver Particles: Bardeen-Friedel Oscillations For bulk silver metal, Vm = 10.27 cm3 , K = 5.22 × 10−3 and (T1 T )−1 = 0.111 (s K)−1 . Using (11) and the calculated D(Ef ) = 3.67 Ry−1 yields I = 153 mRy and χ = 24.3 × 10−6 (in dimensionless SI units). Between room temperature and the liquid phase at 1360 K the molar volume increases by nearly 13%. The Knight shift of 109 Ag plotted as a function of molar volume is continuous across the melting transition, Fig. 3, but the desenhancement factor k(α) shows a discontinuous increase in relaxation rate in the liquid, Fig. 4 [38]. Over this range d ln K/d ln Ω = 0.873 ± 0.004; and additional data down to 25 K also fall very well on this plot [39]. This value is large compared to that for Na, and falls outside the expected range in the free electron gas, (19). The apparent jump in k(α), Fig. 4, might be due to additional contributions to the relaxation rate in the liquid phase, e.g. through atomic motion. On the other hand, there might be a real increase in the relaxation rate, still described by (6), but not correctly parameterized by (11). It has been argued [40] that disorder enhances the dynamic susceptibility (related to the relaxation rate), while it does not aﬀect the static susceptibility (connected to the shift) by much. The mechanism can be understood by considering the relaxation process as an example of Abragam’s [37] “scalar relaxation of the second kind” [5]. This is a form of BPP theory in the extreme narrowing limit, so that the relaxation rate is proportional to a mean-squared interaction multiplied by a relevant correlation time. It is plausible to take this time proportional to the hopping time for electronic conduction: increasing the disorder tends to

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219

1.12 109

K / K RT

liquid

1.08

solid

1.04

m.p.(solid)

1.00 1.00

1.04

m.p.(liquid)

1.08 V(T)/ V0

1.12

Fig. 3. Relative shift of 109 Ag in Ag metal as a function of volume. Here the experimental parameter is temperature (rather than pressure). The plot is continuous c across the melting transition. [After El-Hanany et al., [38]. 1974 by the American Physical Society]

(S / T1T) / K

0.80

0.70

0.60

0.50 1.00

m.p.(solid)

m.p.(liquid)

1.05 1.10 V(T)/ V0

1.15

Fig. 4. The Korringa ratio (S/T1 T )/K 2 for 109 Ag in Ag metal as a function of volume. The relaxation rate shows a discontinuous increase in the liquid phase. c [After El-Hanany et al., [38]. 1974 by the American Physical Society]

localize the electron, increases the hopping time, and therefore the spin-lattice relaxation rate. The mean-squared interaction, proportional to K 2 , remains unaﬀected by the disorder. The number of NMR studies on small silver particles is quite limited, which is rather surprising when one thinks of the advantage that 109 Ag has over 63 Cu in being a spin 1/2, thereby avoiding possible problems with quadrupole broadening of the signals. In bulk silver, the full linewidth at half maximum (FWHM) ∆H is of the order of 0.75 G, which in a 7 T ﬁeld gives ∆H/H = 11 ppm, compared to K = 5200 ppm. For copper the corresponding ﬁgures are ∆H/H = 120 ppm, K = 2500 ppm. It therefore seemed that studies on small

220

J.J. van der Klink

silver particles might resolve some of the ambiguities left in attempts to detect the quantum size eﬀect [14] by 63 Cu NMR [41, 42, 43, 44]. Three groups have reported 109 Ag NMR [45, 46, 47] in systems of supported particles: but two of them found it impossible to detect signals in samples where all particles were smaller than 50 nm. In practice, it is not easy to avoid the presence of such large particles: the reactivity of silver (think e.g. of its use in photographic processes) makes the preparation of small particles of this metal rather diﬃcult, and often leads to asymmetrical or even bimodal size distributions [45]. We have prepared some samples by the colloidal route, using surfactants as protecting agents, and deposing the particles on alumina, titania or silica carriers. The sample with the smallest particles [47] had an average particle diameter determined from X-ray line broadening analysis (XLBA) of 13 nm. We have very little information on the particle size distributions, because of contrast problems in transmission electron microscopy (TEM). The available TEM pictures suggest that nearly all samples contained at least some large particles of the order of 50 nm. The 109 Ag NMR in all our samples showed a symmetric line, detectable by Fourier transform methods but clearly broadened with respect to a bulk signal. The line position and the spin-lattice relaxation time were those of bulk silver. Most experiments were performed at 20 K in 7 or 8 T ﬁelds, but occasionally temperatures down to 15 and up to 400 K were used. For quantitative analysis of the line broadening, the observed signals were ﬁtted to a sum of two Lorentzians, centered at the same frequency, but with diﬀerent width. While this was done simply for convenience, it was also found this way that the lineshape for diﬀerent samples could be very diﬀerent. This is thought to be related to details in the particle size distributions (e.g. bimodality), and makes it very diﬃcult to characterize the linewidth by a single number. Two diﬀerent measures were used: the ﬁrst simply the FWHM of the ﬁtted line; the other the ratio (M23 /M4 )1/2 , with M2 and M4 the second and fourth moment for a truncated Lorentzian (it is well known that the moments of a full Lorentzian diverge). If the two measures gave a very diﬀerent ranking for the same spectrum in the particle diameter vs. linewidth ordering, the data on that sample were discarded. Nevertheless, there remained considerable scatter in the plot of linewidth vs. particle diameter d, although it is pretty sure that the variation is somewhere between proportional to d−1 and d−2 , see Fig. 5. Given the particle sizes and the temperature at which the experiments are performed, the quantum size eﬀects can be excluded as the reason for the particle-size dependent line broadening. The proposed explanation is related to one of the ﬁrst discussions of size eﬀects in the NMR of metals, where the broadening of the 207 Pb NMR line in ﬁlaments of lead, obtained by the impregnation of porous glass, was ascribed [13] to surface-induced spatial variations in the density of Fermi-level electrons. The associated charge oscillations in the free electron gas have been mentioned by Bardeen [48], and go asymptotically as (2kf x)−2 cos(2kf x) where x is the distance from the surface

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221

∆H /K [%]

8 6 4 3 2

1 8 6 4

5 6 7 8 100

Diameter [nm]

Fig. 5. Linewidths (fwhm) in units of the bulk metal shift for 109 Ag in small silver particles as a function of particle diameter. The lines are simple guides to the eye, showing slopes d−1 (solid) and d−2 (dashed ). Data at 20 K. [After Bercier et al., c [47]. 1993 by the Institute of Physics]

and kf the Fermi wavevector. The asymptotic variation of the susceptibility goes as sin(2kf x) for x kf−1 , χ(x) ∝ (20) 2kf x and therefore decays slower than the charge perturbation [49]. The free electron gas has no crystal lattice, and therefore it is not easy to compare its characteristic distance, the inverse of the Fermi wavevector, with a distance in a crystal lattice. Anyway, it has been proposed [47] that these BardeenFriedel oscillations imply that in the surface region some sites have higher and other sites have lower local susceptibilities than the average, thereby leading to NMR line broadening. Strictly speaking of course the decaying oscillation cannot give a perfectly symmetric broadening, but there remains a clear difference with the “exponential-healing” model to be discussed in Sect. 4 that results in a one-sided broadening. A very coarse estimate of the broadening can be made on basis of the free electron model. The calculation of the position-dependent susceptibility [49] (with the free electron gas parameter set for silver) indicates that the relative shift of the surface atoms with respect to the bulk might be as large as ±0.15, of the order of the linewidth of the smallest sample at low temperatures. It is then calculated that the local susceptibility oscillations decay to an amplitude that corresponds to the bulk linewidth over a distance of ten atomic layers. In the largest-size sample studied, this region contains about one tenth of all atoms; in the smallest-size sample it is about half of all atoms. This model suggests that for samples with average particle diameters above 5.7 nm (twenty

222

J.J. van der Klink 2

10 8

∆H /K [%]

6 5 4 3 2

1 8 6 5 4 3

3 4 5 6 7 8100

3 4

Temperature [K] Fig. 6. Linewidths (FWHM) in units of the bulk metal shift for 109 Ag in small silver particles as a function of temperature. Typical particle diameter 1.3 nm (top) c and 8.0 nm (bottom). [After Bercier et al., [47]. 1993 by the Institute of Physics]

atomic layers) the linewidth should decrease as d−1 . The scatter in the data of Fig. 5 is then attributed to diﬀerences in the particle size distributions. There is a remarkable variation of the observed linewidth with temperature, Fig. 6. To a good approximation, the lineshapes observed at 20 K and at 80 K can be brought into coincidence by scaling of the shift axis (and, of course, of the amplitude). The scaling factor varies somewhat from one sample to the next, but is about 1.6, clearly diﬀerent from the factor 4 that one would expect for broadening by paramagnetic impurities in the dilute limit. (The lineshapes scale linearly with applied ﬁeld between 4 T and 8 T). When the temperature is increased further, the linewidth decreases more rapidly. The temperature dependence does not have the characteristics of thermally activated motional narrowing, e.g. by self-diﬀusion. The proposed explanation is based on the vibrational motion of the surface. This can be an important eﬀect in small particles, where increased vibration amplitudes are seen in the Debye-Waller factor [50] and in a lowering of the melting temperature [51]. The motion diminishes the sharpness of the boundary of the electron gas, and thus diminishes the amplitude of the Bardeen oscillations in the model of [49]. A more dynamic image of this process can

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be obtained by considering the local hyperﬁne ﬁeld acting on some arbitrarily selected nucleus inside the particle. When the surface vibrates, the Bardeen oscillations and therefore the local ﬁeld on this nucleus vary in time. When the variation is rapid compared with the total range of NMR frequencies of the nucleus during the course of one vibration period, there will be “motional narrowing” of the NMR signal, but this time not due to motion of the nucleus itself, but of the surface of the particle. The phenomenon of size- and temperature-dependent magnetic broadening of NMR lines also seen by others in copper [42, 43], and perhaps in lead and tin [52, 53] (but not in d-band metals like Pt) might well arise from intrinsic properties of small particles, although it has been usual to ascribe it to, often poorly deﬁned, extrinsic sources.

4 Small Platinum Particles: Exponential Healing The NMR parameters for bulk Pt, according to (8) to (10), are given in Table 1 [8, 23]. The values of Bhf,s , Is and Id have simply been ﬁtted. Bhf,d can be measured from the correlation between Knight shift and susceptibility, with temperature as parameter [23]. The values of Bhf,orb , χorb and χdia are estimated [23]. There is no term in the Knight shift or the relaxation that corresponds to the diamagnetic part of the susceptibility. The shift associated with diamagnetism of core electrons is included in the deﬁnition of the zero of the shift scale. The Landau diamagnetism creates a demagnetizing ﬁeld, leading to overall shifts of a few tens of ppm (the numerical value of the Landau diamagnetic susceptibility). For small particle NMR, we keep all parameters ﬁxed at the bulk value, except the densities of state. We neglect the loss of cubic symmetry near surface sites (where the R-factors can be expected to be diﬀerent) as well as the site to site variation of the orbital Knight shift (assuming that such variations are small compared to variations in the spin Knight shift). Table 1. Fitted values of partial contributions to the susceptibility, the Knight shift and the relaxation rate of 195 Pt in platinum metal. The next four rows give the parameters used: hyperﬁne ﬁelds Bh , reduction factors R, exchange integrals I and densities of states D(Ef ). Vm = 9.10 cm3 From [5], with corrections

χ K (T1 T )−1 Bhf R I D(Ef )

orb

dia

sum

exp.

22.2 7.8 7.2 270

289 −44.2 19.9 −118 0.208 37.7 20.4

13.8 2.0 6.35 110 0.389

−41

284 −34.4 33.4

98 4.08

×10−6 ×10−3 s−1 K−1 T mRy Ry−1

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J.J. van der Klink

While the Ag lineshape described in Sect. 3 could be described with concepts from the free electron gas, this is clearly not the case for the d-band metal Pt, as shown e.g. by an interesting series of NMR experiments on 195 Pt in bulk platinum alloyed with very small quantities of other transition metals [54]. In these spectra “satellite” lines appear (see Fig. 7), due to 195 Pt in sites close to an alloying impurity. The diﬀerence in shift between a satellite line and the bulk NMR line was taken as a direct measure for a diﬀerence in local susceptibility, the hyperﬁne ﬁelds being considered as site-independent. This spatial distribution of the susceptibility on Pt sites around the solute is related to the change in the bulk susceptibility of similar (more concentrated) alloys. To ﬁt the data, an exponential decay of the change in susceptibility with distance from the impurity, rather than the damped sinusoid of (20), was assumed. This assumption is empirical, it has no fundamental justiﬁcation. The characteristic length of the decay is between 0.5 and 0.6 lattice constants (0.392 nm), depending on solute. The spatial decay of the local susceptibility corresponds to a decay in the local density of states at the Fermi energy: the experimental data show that a measurable change in this quantity can extend as far as the third neighbor Pt shell.

Pt V

Pt Cr

Pt Nb 1

Pt Re 1

Pt Rh 1

Pt Mo 3

-3.5

-3.0 -2.5 K (%)

-2.0

Fig. 7. Sketches of the main (near K = −0.035) and satellite 195 Pt lines in Pt metal slightly alloyed with other transition metals. The assignments to ﬁrst, second and third neighbors (numbered arrows) are based on simultaneous analysis of all data. c [After Inoue et al., [54]. 1978 by the Physical Society of Japan]

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A similar exponential decay has been assumed to explain the 195 Pt linewidth in small metal particles [55], with the particle surface now playing the part of the impurity in the so-called layer model. In this model, the atoms in a small-particle sample are divided into groups belonging to diﬀerent atomic layers: the surface layer, the subsurface layer and so on. To ﬁnd the fraction of atoms in each group, the particle size histograms obtained by electron microscopy are interpreted in terms of fcc cubooctahedra. We assume that the diﬀerent sites in a given layer, Fig. 8, of a cubooctahedral particle are suﬃciently similar that the resonance frequencies of all nuclei in the same layer are relatively close to each other on the scale of the total spectrum width. The superposition of NMR signals from all nuclei in a given layer is supposed to result in a Gaussian, completely characterized by the position of its maximum in the spectrum, by its width (of the order of a MHz) and by its integral. The integral must be proportional to the relative number of atoms in the corresponding layer, given in Fig. 8. For the position of the maximum as a function of layer number, we impose an exponential decay (see (21) below), similar to the behavior in the vicinity of an impurity in very dilute alloys, Fig. 7. Finally, the width of the Gaussian is considered a freely ﬁttable parameter, but for not too diﬀerent samples (like the case of Fig. 9) it is assumed to be sample-independent. The maximum of the Gaussian peak corresponding to the nth layer is assumed to occur at a Knight shift Kn (K0 is the Knight shift on the surface, K∞ that in the inﬁnite solid) obeying the relation: (21) Kn − K∞ = (K0 − K∞ ) e−n/m , where the dimensionless constant m represents the “healing length” for the Knight shift, expressed in units of a layer thickness (0.23 nm). According to 60

(a)

frêquency (%)

20 10 0 0 20

(b)

3 Pt-60

10 0 0

(c)

Pt-36

2 4 diameter (nm)

fraction of atoms (%)

Pt-36 Pt-60

40 30 20 10 0

5 layer number

Fig. 8. Particle size distributions and layer statistics. (a,b) size histograms, giving the percentage of particles in diﬀerent diameter classes, for two samples of Pt particles on titania TiO2 . (c) distribution of atoms over the layers of the NMR layer c model. Layer 0 is the surface. [After Bucher et al., [56]. 1989 by Elsevier Science]

226

J.J. van der Klink Pt-36 "clean"

(a)

(c)

(b)

Pt-60 "clean"

(d) n=0 1

1.08 1.09 1.10 1.11 1.12 1.13 1.14 1.15 B0 /ν0 (G/kHz)

Fig. 9. 195 Pt NMR spectrum and layer statistics. (a,b) point-by-point spectra under clean-surface conditions for the samples in Fig. 8. (c,d) ﬁts by a superposition of Gaussians that represent the NMR line of a given layer. [After Bucher et al., [56]. c 1989 by Elsevier Science]

this assumption and to the data in Fig. 8c, the NMR spectrum of Fig. 9b should consist mainly of a superposition of three Gaussian peaks with relative areas 0.60, 0.29 and 0.09. The spectrum of Fig. 9a contains these same Gaussians (having the same positions in the spectrum and the same widths), but now with relative areas 0.36, 0.25 and 0.17, and several more Gaussian peaks. Fits according to this principle are shown in Fig. 9c and Fig. 9d. They correspond to K0 = 0 and m = 1.35. The agreement between ﬁtted and experimental spectra is not perfect, but suﬃcient to demonstrate the usefulness of the NMR layer model. The ﬁtted subsurface (n = 1) peak of the clean sample falls approximately halfway between the surface and bulk resonances. This is in very good agreement with a ﬁve-layer slab calculation [6] and shows that more than half of the spectrum contains information from the surface region. The NMR layer model simply considers the layer-to-layer variation of the NMR shift. It might perhaps be more reasonable to look instead at the local density of states at Ef (LDOS). The necessary additional experimental information can be obtained from the spin-lattice relaxation data in the following way. It is easily seen from (9) that at a single resonance frequency (ﬁxed value of K) one might ﬁnd signals from nuclei with many diﬀerent combinations of s- and d-like LDOS. Each such combination would give rise to the same K, but a diﬀerent T1 . Therefore, generally, the spin-lattice relaxation curves measured at a certain resonance frequency could be nonexponential. The spin-lattice relaxation mechanism by conduction electrons requires that such nonexponential decay curves obey “time-temperature scaling”, because

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the spin-lattice relaxation rate T1−1 for each individual nucleus is proportional to temperature T , so that T1 T = C [20]. In a relaxation experiment at a given resonance position and temperature, we measure a series of recovered signal amplitudes Ai as a function of the relaxation interval τi between the initial saturation pulses and the inspection pulse(s). When the Ai are normalized by the fully relaxed amplitude, the Bloch equation for a single site is 1 − Ai = e−τi /T1

(22)

and using the Heitler–Teller relation, one has 1 − Ai = e−τi T /C .

(23)

If the relaxation curve is a sum of N diﬀerent exponentials (corresponding to diﬀerent sites), there are N diﬀerent constants C; but a collection of curves taken at diﬀerent values of T will collapse onto a single curve when plotted as normalized Ai vs. τi T , as shown in Fig. 10 [57]. It is usually impossible to determine a value of N from the experimental data. We ﬁnd that most relaxation curves can be described by a sum of two exponentials, with temperatureindependent amplitude ratios. While it is clear that in such a case nuclei in at least two diﬀerent environments resonate at the frequency under consideration, it is of course impossible to demonstrate that there are not more than two environments. Therefore one should be very cautious in interpreting the amplitude ratio of the two exponential decays as a ratio of “site occupations”. The time-temperature scaling in Fig. 10 is seen to persist down to at least 22 K. These particles have (on the average) 60% of their atoms exposed on the surface. For comparison, a cubooctahedron of 309 atoms (2.1 nm equivalent diameter) has 162 atoms in the surface, 92 in the ﬁrst subsurface layer, and 42, 12 and 1 atoms in the next layers. It is therefore somewhat astonishing that no hint of discreteness of the electron energy spectrum is visible in Fig. 10. However, such a “loss of metallic character” has been found for platinum in zeolite of similar or slightly higher dispersion. The platinum particles are inside the (sometimes partially damaged) supercages of the zeolite matrix. Samples of this dispersion show virtually no signal at the bulk resonance position, since in terms of the NMR layer model less than 1% of the atoms are below layer 2 (the sub-subsurface layer). In three Pt/zeolite systems the 195 Pt spin-lattice relaxation at the position 1.10 G kHz−1 was studied between 22 and 250 K, see Fig. 11. In the following discussion we focus mainly on the sample “64-clean”. From 250 K down to 80 K the relaxation curves can very well be normalized by time/temperature scaling. Below 80 K only a temperature-dependent fraction of the experimental relaxation curve can still be normalized this way. At 22 K, only about 1/3 of the nuclei resonating at this position still show the “metallic” spinlattice relaxation. The scaling behavior of the relaxation could be studied below 83 K at one spectral position, 1.091 G kHz−1 , where only surface atoms should resonate (the layer model supposes that the maximum of the surface

228

J.J. van der Klink

1.0

(a) Pt-60-H10 22 K 30 K 40 K 80 K 100 K 120 K 140 K

NMR amplitude

0.5

0.0 1.0

(b) Pt-60-H50 22 K 40 K 80 K 100 K 120 K 140 K

0.5

0.0 1.0

0.5

0.0 0.01

0.1

100

τT (sK) Fig. 10. Time-temperature scaling for nonexponential relaxation curves in a series of 195 Pt saturation-recovery experiments. The time points τ are multiplied by the temperature of the experiment (see key to symbols) and the individual equilibrium signal amplitudes are scaled to the same value. The sample is Pt/TiO2 of dispersion 0.6 under several hydrogen coverages (0.1, 0.5 and 1.0 monolayer). The squares in (c) show data at 110 K for another Pt/TiO2 catalyst of dispersion 0.36. [After Tong c et al., [57]. 1994 by the American Chemical Society]

peak is at 1.10 G kHz−1 , but there is some signal from the subsurface layer at this frequency), and another, 1.110 G kHz−1 , where the contribution from the subsurface layer should be more important. At both frequencies, roughly the same 1/3 of the signal has metallic relaxation at 22 K. This suggests that there are two classes of particles: those that have at 22 K metallic relaxation on all sites, and those that have no metallic relaxation. It is then reasonable to think that the metallic particles must be the larger ones, and from the size distribution it is calculated that their diameter is 1.6 nm or more.

Beyond Electrons in a Box

NMR Amplitude

1.0

(a) 80 K

0.5 0.0 1.0

229

(b) 60 K

(d) 22 K

0.5 0.0 0.01

0.1

τ T(s K)

100

0.01 0.1

100

τ T(s K)

Fig. 11. Time-temperature-scaled relaxation data for three clean-surface Pt/zeolite samples, taken at spectral position 1.100 G Hz−1 . The full curve in all panels is c a double exponential ﬁt to the 80 K data [After Tong et al., [58]. 1995 by the American Physical Society]

The diﬀerence between relaxation behavior in oxide-supported and in zeolite-encaged particles may be related to the fact that particles sit on the surface of the oxide, but are enclosed in the matrix of the zeolite. The latter probably ﬁt rather snugly inside the cavity created by the local partial collapse of the aluminosilicate network, while the surfaces of the oxide-supported particles are relatively unconstrained. Free particles can execute “breathing” vibrations, whereby their volume varies in time. This is a phonon-like motion, and the electron distribution can adjust very rapidly on this timescale, so as to be at any moment in equilibrium with the instantaneous conﬁguration of the nuclei. As a simpliﬁcation we consider the particle as an elastic continuum to describe the vibrations, and as a free electron gas inside this vibrating membrane to calculate an eﬀective density of states. When under inﬂuence of the vibration the radius of the particle varies between R+ 12 ∆R and R− 12 ∆R, the electron density parameter will vary between rs + 12 ∆rs and rs − 12 ∆rs , with ∆rs /rs = ∆R/R and R = N 1/3 rs . The variation of Dfeg (Ef ) is given by (14). The usual estimate of the level splitting in a particle containing N electrons is δ = 2/N D(Ef ). Washing-out of the level structure will then occur if the variation in Fermi energy that corresponds to the variation in rs is larger than the level splitting δ: 1 1 (24) Ef rs − ∆rs − Ef rs + ∆rs ≥ 2/N D(Ef ) , 2 2 which leads to the condition ∆R/rs ≥ (2/3)N −2/3 . For N = 100, the requirement is that the amplitude of the surface motion is about 3% of an atomic radius; in bulk gold the atomic rms displacements are estimated to satisfy the requirement above 100 K; and it is well known that the amplitude of thermal motion in a surface is larger than that in the bulk.

230

J.J. van der Klink

Such a vibration mechanism could therefore explain the absence of discretelevel eﬀects above 20 K in particles with essentially “free” surfaces, such as those anchored on oxides. On the other hand, the “encaged” particles in zeolites are mechanically clamped, so that higher temperature is needed to have suﬃcient amplitude of these breathing modes to average out the discrete level structure.

5 Small Rhodium Particles: Incipient Antiferromagnetism In pure bulk metals, magnetism is found only in the 3d row of the periodic system, from Cr to Ni. Non-bulk forms of 4d and 5d metals can also be magnetic, either due to surface eﬀects as in ﬁlms [59] or because of size eﬀects, as in atomic clusters [60]. As shown in Sect. 4, the local susceptibility in the surface layer of platinum is lower than that of the bulk, while the bulk value is retrieved about 3 atomic layers deep. At least qualitatively, this agrees with calculations for a ﬁve-layer platinum ﬁlm [6]. The calculated Stoner enhancement for bulk rhodium is rather small [16], but the calculated surface-LDOS is larger than the bulk value [61, 62]. The experimental situation is not quite clear [63], but suggests that the Rh(100) surface shows superparamagnetism or some extremely unstable two-dimensionsional ferromagnetic order. In rhodium clusters of up to roughly a hundred atoms, size-dependent magnetism has been found both experimentally [64] and in calculations [62]. The incipient magnetic structure of a paramagnetic system is best described by its wavevector dependent susceptibility χ(q). ˜ In this respect, calculations give distinct results for bulk Pd and bulk Rh [65, 66, 67]. The χ(q) ˜ in Pd peaks strongly at zero wavevector and decays monotonously for larger q, which is characteristic for an incipient ferromagnet. The calculated q = 0 enhancement in Rh is clearly weaker than that in Pd; but there is a rather strong secondary maximum that corresponds to a tendency towards antiferromagnetic ordering. We have used NMR of 103 Rh of small supported rhodium particles to study some of these issues [68]. As a prerequisite for the data analysis, we give the bulk 103 Rh NMR parameters and present a brief review of spin ﬂuctuation theory as applied to NMR. 5.1 NMR of Bulk Rh The Knight shift of bulk Rh is nearly temperature independent up to 100 K and then decreases (becomes less positive) going to room temperature. We ﬁnd the metal resonance at low temperatures at 1.34373(5) MHz T−1 , in excellent agreement with the literature value of 1.34374(3) MHz T−1 [69]. But because we use another reference frequency [5], we give the corresponding Knight shift

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231

Table 2. Fitted values of partial contributions to the susceptibility, the Knight shift and the relaxation rate of 103 Rh in rhodium metal. The next four rows give the parameters used: hyperﬁne ﬁelds Bhf , reduction factors R, exchange integrals I and densities of states D(Ef ). The molar volume Vm = 8.30 cm3 . For χcgs in µemu mole−1 multiply the table body entries by Vm /(4π)

χ K (T1 T )−1 Bhf R I D(Ef )

orb

dia

sum

exp.

2.5 0.60 1.3 200

130.6 −5.25 11.6 −34 0.20 28 18.0

71.2 8.42 96.9 100 0.44

−59.7

144.6 3.76 109.8

143.8 3.75 111.5

0 0.7

18.7

×10−6 ×10−3 ×10−3 s−1 K−1 T mRy Ry−1

as K = 3.75 × 10−3 , instead of 4.3 × 10−3 . At 280 K, we have K = 3.57 × 10−3 . Between 15 and 200 K, we ﬁnd the relaxation rate linear in temperature, with T1 T = 8.97(5) s K, in agreement with the earlier result 9 s K at helium temperatures [69]. These low-temperature NMR data for bulk Rh can be ﬁtted by the usual (8) to (11), see Table 2. In this ﬁt, the diamagnetic susceptibility of Rh was set equal to that of Ag (−39.4 µemu mole−1 in cgs units), as calculated from the total experimental susceptibility of silver (−19.5 µemu mole−1 ) and its spin susceptibility as determined by 109 Ag NMR [5] (19.9 µemu mole−1 , see Sect. 3). The reduction factors R were set to equal occupancy for all ﬁve types of d-orbitals, and a term 1/25 was added to Rorb to account for dipolar relaxation [24]. Since Ds (Ef ) is low anyway, the corresponding exchange integral Is = αs /Ds (Ef ) was set to zero; for the same reason the s-like hyperﬁne ﬁeld was simply set to a plausible value, the ﬁtted value for Pd [5]. The d-like hyperﬁne ﬁeld was constrained to be between zero and the value determined experimentally for Pd [11]. The orbital susceptibility and hyperﬁne ﬁeld were constrained to be comparable to calculated values [70]. The d-like exchange integral Id was constrained to yield a moderate susceptibility enhancement. The ﬁtted enhancement factor is 2.02, in reasonable agreement with the calculated value of 1.79 [16]. The parameter values reproduced in Table 2 [68] are somewhat diﬀerent from those in the preliminary analysis proposed in [71]. The latter unfortunately contains numerical errors and is therefore inconsistent. 5.2 Spin Fluctuations, K and T1 For strongly enhanced paramagnets, K and/or T1 T may become temperature dependent through spin ﬂuctuations. Below we summarize how these temperature dependences are expected to diﬀer for ferromagnetic and antiferromagnetic enhancements. These theories have been developed for the hom*ogeneous electron gas [12, Ch. 5], and we will give the relevant quantities an index h.

232

J.J. van der Klink

Paramagnets with Ferromagnetic Spin Fluctuations These are systems that show a strong temperature dependence of the uniform static susceptibility, but that nevertheless remain paramagnetic. The paramagnetic phase of low-TC ferromagnets can be described by the same theory. In addition to an exchange parameter νh , spin ﬂuctuation theory introduces a function δ(T ), such that the temperature dependent susceptibilities in a ferromagnetically enhanced paramagnet are given by χ ˜h (q, T ) =

χ ˜P,h (q) 1 . νh (δ(T ) + 1) χ ˜P,h (0) − χ ˜P,h (q)

(25)

The Stoner enhancement factor is related to the exchange parameter and the q = 0 static susceptibility through αh = νh χ ˜P,h (0; ω = 0)

(26)

and to the low-temperature limit of δ(T ) by lim δ(T ) =

T →0

1 − αh . αh

(27)

The inverse of the static uniform susceptibility, as in (8), is −1 χ−1 νh δ(T ) . h (T ) = Ω

(28)

At low temperatures the usual Stoner enhanced susceptibility is retrieved because of (27). At higher temperatures the experimental Curie-Weiss behaviour implies that (29) δ(T ) ∝ (T − TC ) and the cases of interest to us are close to the limit TC → 0. According to (28) the Knight shift becomes temperature dependent, and from (6) and (25) the spin-lattice relaxation rate is given by: 2(ΩBhf )2 Ω πµ0 (2π)3 2 χ ˜P,h (q; ω) δ(T ) + 1 1 dq ω αh δ(T ) + 1 − F (q) / +2 1 δ(T ) + 1 ≈ S(T1 T )−1 P,h αh δ(T ) + 1 − F (q) FS

S(T1 T )−1 h =

≈ S(T1 T )−1 P,h

1 + 53 (δ(T ) + 1)−2 χh (T ) , χP,h

(30) (31) (32)

where F (q) is the Lindhard function. The angular brackets in (31) indicate an average over vectors q that connect two points on the spherical Fermi

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233

surface. The expression in the numerator of (32) comes from the approximation in (12). It is usually assumed that the main temperature dependence of T1 T is contained in the factor χh (T ), which for an incipient ferromagnet is ∝ 1/T . The result is a T1 independent of temperature, and a Knight shift K ∝ χh (T ) ∝ 1/T . The low-temperature limit of (27) introduces the Moriya desenhancement factor of (12) into (32): −1 (T1 T )−1 h = (T1 T )P,h kM (αh ) .

(33)

Paramagnets with Antiferromagnetic Spin Fluctuations In antiferromagnetic systems, the static paramagnetic susceptibility has its maximum at an ordering vector Q = 0. A moderate tendency towards antiferromagnetism has been found in calculations for bulk rhodium, where a secondary maximum in χ(q) ˜ appears at a nonzero wavevector [66], the absolute maximum remaining at q = 0. In the NMR of the paramagnetic state ˜ (Q), of antiferromagnets a situation can arise where T1 is dominated by χ whereas the Knight shift is determined by χ ˜ (0). In that case, there exists no equivalent of the k(α) relation as in (11) or (12), since the α in K refers to q = 0, and that in T1 to Q. There is no closed expression, analogous to (30), for the spin-lattice relaxation in antiferromagnetically enhanced paramagnets. Still the temperature dependence of T1 T can be estimated in the following way. The starting assumption is that, because of the enhancement, χ ˜ is strongest around a wavevector Q diﬀerent from zero, and that around that wavevector it can be expanded as C χ ˜ (Q + q; ω) = , (34) ω νQ (δQ (T ) + A q 2 )2 where C and A are coeﬃcients of the expansion, and νQ and δQ (T ) are analogous to the quantities in ferromagnetic ﬂuctuations theory. The sum in (6) is converted into an integral over a spherical volume equivalent to a 3 = 6π2/Ω, so that (T1 T )−1 Brillouin zone centered at Q, of radius qB with qB is proportional to qB q=0

4πq 2 π2 dq = 1/2 νQ (δQ (T ) + A q 2 )2 A νQ (A δQ (T )) 3 = 2πqB νQ χ ˜2 P (Q)

for δQ → 0

(35)

for νQ → 0 .

(36)

In the weak-enhancement limit, νQ → 0, the T1 T is independent of T , as expected. In the limit of a strong antiferromagnet, δQ → 0, we have T1 T ∝ δ(T ) ∝ T − TN (37) and for an incipient antiferromagnet, TN → 0, the resulting relaxation rate is proportional to the square root of temperature.

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J.J. van der Klink

The detection of antiferromagnetic enhancement by NMR is usually based on qualitative arguments. At low temperature the Knight shift is temperature independent, whereas T1 T is not. The experimental relaxation rate T1−1 (T ) is ﬁtted to (38) T1−1 = aT + bT 1/2 . The term ∝ T is attributed to the term ∝ Bhf,orb in (10). It is usually impossible to give a quantitative interpretation of the value of b. 5.3

103

Rh Spectra of Small Rhodium Particles

We have studied four samples of small Rh particles by NMR. A ﬁrst sample, denoted Rh/TiO2 , has an average diameter d = 3.6 nm, dispersion 26%. A second one, Rh/PVP, has d = 3.0 nm, dispersion 37%. A third sample was prepared on RP3 alumina (Rhˆ one-Poulenc) with d = 2.6 nm and dispersion 40%. The fourth sample was on GSF400 alumina (Rhˆ one-Poulenc) with 4% loading. Its size distribution is diﬃcult to measure by TEM; we have used small-angle X-ray scattering to estimate the average diameter as 1.5 nm and the dispersion as 54%. The 103 Rh spectra for these four small-particle samples are shown in Fig. 12. They are all approximately centered at the bulk (a)

7.35

7.40 7.45 7.50 7.55 7.35 7.40 7.45 7.50 7.55 Field / frequency (G/kHz) Field / frequency (G/kHz)

(c)

7.35

(b)

(d)

7.40 7.45 7.50 7.55 7.35 7.40 7.45 7.50 7.55 Field / frequency (G/kHz) Field / frequency (G/kHz)

Fig. 12. Point by point 103 Rh NMR spectra for clean surface Rh particles of diﬀerent sizes on diﬀerent supports, taken at 80 K. The spectra are normalized to the same area. The samples are (a) Rh/TiO2 , (b) Rh/PVP, (c) Rh/RP3, (d) Rh/GSF400. The dotted line in (c) shows the eﬀect of chemisorption of a monolayer of hydrogen. On this scale, the width of the bulk Rh resonance is comparable to a line thickness. c [After Burnet et al., [68]. 2002 by the Institute of Physics]

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235

resonance position, and broaden rather symmetrically when the particle size decreases. Chemisorption of hydrogen has a measurable, but rather small effect, Fig. 12(c). These spectral characteristics are markedly diﬀerent from those of the small platinum particles in Sect. 4. There are two reasons for this diﬀerent behaviour: one in the NMR of Rh metal, the other in the electronic structure of the Rh surface. According to Table 2, the Knight shift and the relaxation rate of bulk Rh are dominated by the orbital parts, with additional contributions from the spins of the d-like electrons, and negligible s-like parts. This is very diﬀerent from Pt (compare Table 1), where the d-like spin part dominates, and the orbital parts are nearly negligible. In small particles of Pt the magnetic behaviour can be reasonably well described by considering only the site to site variation of the d-like spin part, and as it happens this part of the susceptibility is measurably smaller on surface sites than in the bulk. From calculations for a rhodium slab [61] it is expected that the site to site variation of the 103 Rh spin Knight shift is comparatively small: therefore the spectral shapes in Fig. 12 arise from competition between (positive) orbital shifts and (negative) spin shifts. On some surface sites the net result is negative, on others positive; therefore no distinct surface signal can be found in the spectra. From calculations for the fcc(111) surfaces for Pd and Rh [72] it is found that hydrogen has markedly less inﬂuence on the D(Ef ) at surface sites on rhodium than on palladium. It is believed that in this respect Pt behaves as Pd, which would explain that the 195 Pt spectral shape shows large qualitative changes upon hydrogen adsorption, whereas the 103 Rh spectrum in Fig. 12(c) only shows a shift (by about 14 kHz or 740 ppm, upﬁeld). 5.4 Spin-Lattice Relaxation of

103

Rh in Small Particles

In 195 Pt NMR of clean surface particles the relaxation rate at a given resonance position is independent of support or particle size [73] (with the exception of zeolite carriers). Therefore Pt nuclei in diﬀerent samples that resonate at the same spectral position are in very similar environments. Such a simple identiﬁcation of atomic environment with spectral position cannot be made for Rh particles: at the same resonance position, we can ﬁnd very diﬀerent relaxation rates. In the sample with the largest particles, Rh/TiO2 , the value of T1 T is roughly the same at 80 K and 20 K for several spectral positions, see Fig. 13(a,b). Assume for the sake of argument that the drawn line in Fig. 13(b) correctly represents the spectral variation of the T1 T product. It is nowhere larger than the bulk value and, the contribution of Ds (Ef ) to the relaxation being negligible, (10) says that there can be no atomic sites with a local value for the density of states Dd (Ef ) smaller than the bulk value. The symmetry of the NMR spectrum and of the sketched T1 T curve imply that there must also be a site to site variation of the orbital susceptibility χorb . The parts (c) and (d) of Fig. 13 show the relative changes across the spectrum of Dd (Ef ) and χorb , assuming all other parameters in Table 2 to be constant.

236

J.J. van der Klink

0.8

K (%) 0.4 0

-0.4

(a)

7.40 7.42 7.44 7.46 7.48 7.50 0.8 8 T1 T (sK) 6

0.4

-0.4

(b)

4 7.40 7.42 7.44 7.46 7.48 7.50 26 Dd (Ry-1)

(c)

22 18 7.40 7.42 7.44 7.46 7.48 7.50 130

(d)

χorb 110 (10-6) 90 70 7.40 7.42 7.44 7.46 7.48 7.50 Field / frequency (G/kHz)

Fig. 13. The NMR spectrum (a) and value of T1 T for several spectral positions (b) in sample Rh/TiO2 , at 20 K (full circles) and 80 K (open circles). The line drawn in (b) is meant to suggest that T1 T is independent of temperature, and that it is close to the bulk value (straight dashed lines) in the centre of the spectrum. The drawn line in (b) can be represented by the variations of Dd (Ef ) and of χorb shown in (c) c and (d). [After Burnet et al., [68]. 2002 by the Institute of Physics]

At the low ﬁeld end of the spectrum for somewhat smaller particles, Rh/PVP, the relaxation rate is less than it is in the bulk, see Fig. 14(a,b). Since the orbital relaxation rate should be unaﬀected by magnetic ﬂuctuations, this immediately says that at least in part of this sample the LDOS is lower than the D(Ef )bulk . There is a clear low-temperature enhancement of

Beyond Electrons in a Box

1.2

0.8

K (%) 0.4

-0.4

(a)

-0.5

(c)

7.40 7.42 7.44 7.46 7.48 7.50 1.2

K (%) 0.5

237

0.8

0.4

7.36

-0.4

(b)

1 12

T1 T 8 (sK)

0 7.40 7.42 7.44 7.46 7.48 7.50 Field/frequency (G/kHz)

7.4

7.44 0.5

7.48 0

7.52 -0.5

(d)

0 7.36

7.4 7.44 7.48 Field/frequency (G/kHz)

7.52

Fig. 14. The NMR spectrum (a) and value of T1 T for several spectral positions (b) in sample Rh/PVP, at 20 K (full circles) and 80 K (open circles). The lines in (b) are drawn only to connect points at the same temperature. The temperature eﬀect on the relaxation (but not on the spectrum) is much larger than in Fig. 13. At the low ﬁeld end in (b) the relaxation is slower than in bulk Rh (straight dashed lines). Similar data for sample Rh/RP3 are shown in (c) and (d). Here the site to site variation is rather small, but the temperature eﬀect is large. [After Burnet et al., c [68]. 2002 by the Institute of Physics]

the relaxation rate over most of the spectral width, while the spectral shape in Fig. 14(a) hardly varies. For still smaller particles, the spectral shape remains nearly independent of temperature, but the T1 T products vary strongly, see Fig. 14(c,d). At ﬁxed temperature, the site to site variation of the relaxation is relatively small, since T1 is nearly constant across the spectrum. From an interpolation at the bulk resonance position we have T12 T = 0.63 ± 0.03 s2 K for the two temperatures. This is a strong indication for relaxation dominated by the bT 1/2 term in (38). The relative unimportance of the aT term implies a drop in D(Ef ) with respect to the bulk value, as has already been seen at the low ﬁeld end in Fig. 14. Since the spectrum remains centered at the bulk position, there must be a drop in χorb as well.

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The smallest particle sample in Fig. 12(d) contains only a small weight of rhodium, and its signal is spread over a large spectral width, which makes it very diﬃcult to perform the relaxation measurements. Some results (not shown) have been obtained at 80 K. Their large error bars range from the 80 K values in Fig. 14(b) to those in Fig. 14(d). Very qualitatively the results agree with those in the other samples, but the large error range and the lack of low-temperature data prevent further discussion. 5.5 Surface and Size Eﬀects The 103 Rh NMR of the 3.6 nm particles, Rh/TiO2 , is mainly determined by surface eﬀects. Calculations for slabs exhibiting diﬀerent Rh surfaces [61, 62] indeed show enhanced values of D(Ef ) at the surface, as suggested by the curve in Fig. 13(c). From comparison with Fig. 13(d), it follows that on sites where Dd (Ef ) increases, χorb increases also. In particles of 3.0 nm, Rh/PVP, the T1 values at 80 K show that at least in a part of the sample the Dd (Ef ) is less than the bulk value. The spectrum stays symmetric and centered at Kbulk , which suggests that where Dd (Ef ) diminishes, χorb diminishes also. We believe that this is a size- rather than a surface eﬀect, that somehow accompanies the onset of the size-dependent antiferromagnetic relaxation enhancement. This enhancement is more clearly seen at 20 K; at 80 K it is decreased by spin ﬂuctuations. For the particles in this sample the enhancement probably is in between the two limits of (36), so that the overall temperature dependence of T1 T does not have the structure of (38). The very similar values of T12 T at the two temperatures for the bulk resonance position in Fig. 14(d) suggest that here (38) might be applicable. However, the considerable amount of experimental time that would be required has kept us from gathering enough data to do a T1−1 = aT + b(T − TN )1/2 analysis, for which an interpretation of a and b would be diﬃcult anyway. To illustrate this point, assume that in Fig. 14(d) the a term has half the value of the bulk relaxation, ≈0.05 (s K)−1 , and furthermore that we are in the δQ → 0 limit of (36). In that case, our two temperature points could be ﬁtted by assuming TN ≈5–10 K, and χorb ≈0.7χorb,bulk . The only modest variation of T1 across the spectrum in Fig. 14(d) suggests that the relaxation mechanism acts more or less in the same way on all atomic sites: it is related to a size- rather than a surface eﬀect. Magnetism has been found in Rh clusters of up to ≈80 atoms, both experimentally [64] and theoretically [62]. Our particles in Rh/PVP and Rh/RP3 are larger (at least several hundreds of atoms), but the tendency towards magnetism now expresses itself through an enhancement of the antiferromagnetic χ(Q) ˜ found in calculations for the bulk [66]. From the 103 Rh NMR we conclude that rhodium particles of approximately 2.6 nm diameter show incipient antiferromagnetism below 80 K. This incipient antiferromagnetism is accompanied by a lowering

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of the density of states at the Fermi level and of the orbital susceptibility with respect to their bulk values.

Acknowledgements The small-particle NMR shown here has been taken from the PhD theses of Jean-Pierre Bucher, Jean-Jacques Bercier, YuYe Tong and S´everine Burnet. Their work was supported by the Swiss National Science Foundation, recently under grant 20-53637.98.

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30. Ch. Ryter: Phys. Lett. 4, 69 (1963) 31. G.B. Benedek, T. Kushida, J. Phys. Chem. Solids 5, 241 (1958) 32. R. Bertani, M. Mali, J. Roos, D. Brinkmann: J. Phys.: Condens. Matter 2, 7911 (1990) 33. S. Kluthe, R. Markendorf, M. Mali, J. Roos, D. Brinkmann: Phys. Rev. B 53, 11369 (1996) 34. T. Kushida, J.C. Murphy, M. Hanabusa: Phys. Rev. B 13, 5136 (1976) 35. Ch. Ryter: Phys. Rev. Lett. 5, 10 (1960) 36. D.A. Papaconstantopoulos: Handbook of the Bandstructure of Elemental Solids (Plenum Press, New York 1986) 37. A. Abragam: Principles of Nuclear Magnetism (Oxford University Press, Oxford 1961/1986) 38. U. El-Hanany, M. Shaham, D. Zamir: Phys. Rev. B 10, 2343 (1974) 39. J.-J. Bercier: Thesis EPFL, Lausanne 1993 40. B.S. Shastry, E. Abrahams: Phys. Rev. Lett. 72, 1933 (1994) 41. S. Kobayashi, T. Takahashi, W. Sasaki: J. Phys. Soc. Jpn. 32, 1234 (1972) 42. M. Ido, R. Hoshino: J. Phys. Soc. Jpn. 36, 1325 (1974) 43. P. Yee, W.D. Knight: Phys. Rev. B 11, 3261 (1975) 44. M.J. Williams, P.P. Edwards, D.P. Tunstall: Faraday Discuss. 92, 199 (1991) 45. J.K. Plischke, A.J. Benesi, M.A. Vannice: J. Phys. Chem. 96, 3799 (1992) 46. V.M. Mastikhin, I.L. Mudrakovsky, S.N. Goncharova, B.S.Balzhinimaev, S.P. Noskova, V.I. Zaikovsky: React. Kinet. Catal. Lett. 48, 425 (1992) 47. J.J. Bercier, M. Jirousek, M. Graetzel, J.J. van der Klink: J. Phys.: Condens. Matter 5, L571 (1993) 48. J. Bardeen: Phys. Rev. 49, 653 (1936) 49. R.L. Kautz, B.B. Schwartz: Phys. Rev. B 14, 2017 (1976) 50. C. Solliard: Solid State Commun. 51, 947 (1984) 51. Ph. Buﬀat, J.P. Borel: Phys. Rev. A 13, 2287 (1976) 52. F. Wright Jr: Phys. Rev. 163, 420 (1967) 53. W.A. Hines, W.D. Knight: Phys. Rev. B 4, 893 (1971) 54. N. Inoue, T. Sugawara: J. Phys. Soc. Jpn. 45, 450 (1978) 55. H.E. Rhodes, P.-K. Wang, H.T. Stokes, C.P. Slichter, J.H. Sinfelt: Phys. Rev. B 26, 3559 (1982) 56. J.P. Bucher, J. Buttet, J.J. van der Klink, M. Graetzel: Surf. Sci. 214, 347 (1989) 57. Y.Y. Tong, J.J. van der Klink: J. Phys. Chem. 98, 11011 (1994) 58. Y.Y. Tong, D. Laub, G. Schulz-Ekloﬀ, A.J. Renouprez, J.J. van der Klink: Phys. Rev. B 52, 8407 (1995) 59. H. Dreyss´e, C. Demangeat: Surf. Sci. Rep. 28, 65 (1997) 60. J.P. Bucher, L.A. Bloomﬁeld: Int. J. Mod. Phys. B 7, 1097 (1993) 61. A. Eichler, J. Hafner, J. Furthm¨ uller, G. Kresse: Surf. Sci 346, 300 (1996) 62. C. Barreteau, R. Guirado-L´ opez, D. Spanjaard, M.C. Desjonqu`eres, A.M. Ol´es: Phys. Rev. B 61, 7781–7794 (2000) 63. A. Goldoni, A. Baraldi, M. Barnaba, G. Comelli, S. Lizzit, G. Paolucci: Surf. Sci. 454–456, 925–929 (2000) 64. A.J. Cox, J.G. Louderback, S.E. Apsel, L.A. Bloomﬁeld: Phys. Rev. B 49, 12295 (1994) 65. H. Winter, E. Stenzel, Z. Szotek, W.M. Temmerman: J. Phys. F. Met. Phys. 18, 485–500 (1988)

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66. L.M. Sandratskii, J. K¨ ubler: J. Phys.: Condens. Matter 4, 6927–6942 (1992) 67. J.B. Staunton, J. Poulter, B. Ginatempo, E. Bruno, D.D. Johnson: Phys. Rev. B 62, 1075–1082 (2000) 68. S. Burnet, T. Yonezawa, J.J. van der Klink: J. Phys.: Condens. Matter 14, 7135 (2002) 69. A. Narath, H.T. Weaver: Phys. Rev. B 3, 616 (1971) 70. H. Ebert, M. Battocletti, M. Deng, H. Freyer, J. Voitl¨ ander: J. Computational Chemistry, 20, 1246 (1999) 71. P.A. Vuissoz, T. Yonezawa, D. Yang, J. Kiwi, J.J. van der Klink: Chem. Phys. Lett. 264, 366 (1997) 72. R. L¨ ober, D. Hennig: Phys. Rev. B 55, 4761 (1997) 73. J.J. van der Klink: Adv. Cat. 44, 1 (2000)

The Study of Mechanisms of Superconductivity by NMR Relaxation Dylan F. Smith and Charles P. Slichter Department of Physics and Frederick Seitz Materials Research Laboratory University of Illinois, Urbana, IL 61801 [emailprotected] Abstract. This chapter describes the use of NMR to study superconductivity. The focus is on the use of NMR measurements of spin-lattice relaxation (T1 ) to provide insight into the possible mechanisms which give rise to superconductivity. The chapter begins with a review of NMR T1 in the normal state of metals, followed by an explanation of how, according to the theory of Bardeen, Cooper, and Schrieﬀer (BCS), the situation is modiﬁed when the metal becomes superconducting. These ideas are then applied in a discussion of several recent superconducting systems. Data presented for alkali fullerenes show that the fullerides are conventional BCS superconductors, while data for cuprate superconductors suggest a much different type of superconducting pairing. The authors conclude with a discussion of some organic materials that exhibit superconductivity or, with slight modiﬁcation in structure, antiferromagnetism. In the studies of the antiferromagnetism, it is found that NMR can detect the electron spin-spin interaction discovered by Dzialoshinskii and Moriya.

1 Introduction Since the discovery of magnetic resonance, consideration of the interaction of the nucleus with its surroundings has been a subject of great importance. Indeed, such considerations precede the discovery of NMR since the ﬁrst discussion of the interaction was motivated by an investigation of the possibility of using the nuclear magnetism of a metal to produce cooling of the metal by adiabatic demagnetization. In a classic paper, Heitler and Teller [1] calculated the spin-lattice relaxation time, T1 , of paramagnetic ions and of nuclei. They concluded that in diamagnetic insulators, the nuclear relaxation would be of order 106 years, whereas for a metal it would be about 1 second at 0.1 K. They also showed that T1 would vary inversely with temperature in a metal. In their paper announcing the discovery of “Resonance Absorption by Nuclear Magnetic Moments in a Solid”, Purcell, Torrey, and Pound [2] point out that “a crucial question concerns the time required for the establishment of thermal equilibrium between spins and lattice”. The ﬁrst experiments of the D.F. Smith and C.P. Slichter: The Study of Mechanisms of Superconductivity by NMR Relaxation, Lect. Notes Phys. 684, 243–295 (2006) c Springer-Verlag Berlin Heidelberg 2006 www.springerlink.com

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Purcell group after the discovery of NMR in 1945 were studies of relaxation times of protons in liquids and were reported in the classic paper known to all students of magnetic resonance as BPP [3]. It turns out that the “solid” of their ﬁrst paper, paraﬃn, was not all that solid because had it been so the T1 would have been too long for the experiment to have been successful! So from the start it was evident that nuclear magnetic resonance could be used to study the material in which the nuclei were imbedded. In the subsequent history of NMR, one of the most important contributions of NMR was its use in the discovery of the superﬂuid phase of 3 He by Osheroﬀ, Richardson, and Lee [4]. In this paper, we describe the use of NMR to study superconductivity. This ﬁeld is enormous, so we focus on a particular aspect, the use of NMR measurements of T1 to give insight into the possible mechanisms which give rise to superconductivity, illustrating with examples of several superconducting systems. We ﬁrst review the situation in the normal state of metals, then explain how, according to the theory of Bardeen, Cooper, and Schrieﬀer (BCS) [5, 6], the situation is modiﬁed when the metal becomes superconducting. We show how NMR conﬁrms the essential idea of their theory. Then we discuss several other systems (the alkali fullerides and the cuprates). We conclude with a discussion of some organic materials that exhibit superconductivity or, with slight modiﬁcation in structure, antiferromagnetism. In the studies of antiferromagnetism, we encountered an NMR mystery. We describe it and give its explanation, showing that NMR can detect the electron spin-spin interaction discovered by Dzialoshinskii and Moriya.

2 Normal Metals In 1949, Knight [7] discovered the famous shift, which bears his name, of resonance frequency experienced by nuclei in metals. A year later, Korringa published his paper [8] on the theory of NMR relaxation and shifts in metals which related T1 to the Knight shift, a formula which is known as the Korringa relation, γ2 , (1) T1 T K2 = e2 γn 4πkB where γe and γn are the electron and nuclear gyromagnetic ratios, K is the Knight shift, and kB is the Boltzmann constant. A second relationship, the Heitler-Teller equation, says that 1 ∝T . T1

(2)

These relationships were studied extensively and veriﬁed, for example by Norberg and Slichter [9] and by Holcomb and Norberg [10] in the alkali metals. Pines [11] pointed out that the Korringa relationship is modiﬁed if one

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takes into account electron-electron interactions. If these are on the whole antiferromagnetic in nature, the right hand side is reduced, if they are ferromagnetic, the right hand side is increased. The left hand side of (1) is often called the “Korringa product”. It is interesting that if one can ignore the electron-electron eﬀects, the right hand side depends only on fundamental constants and is independent of the material in which the nucleus sits. The constant of proportionality of (2) is given in [1] and [8]. For our purposes, we may sketch its derivation as follows. Nuclear relaxation in a metal may be viewed as a scattering process in which an electron experiences a force from the nuclear magnetic moment and is scattered, ﬂipping the nucleus in the process. If one thinks of the electron-nuclear magnetic interaction as arising from the hyperﬁne coupling, one may imagine that as an electron comes close to the nucleus, the two spins form a resultant total spin angular momentum vector about which they precess. When the collision is over, both spins have changed their directions in space. If the collision takes place in the presence of an external magnetic ﬁeld, the spins will have changed their Zeeman energies and, since their γ’s diﬀer, the electron must also changed its kinetic energy. If the hyperﬁne coupling is the Fermi contact term1 , and we treat the electrons as a free electron gas, the scattering is independent of the direction of the electron velocity. Then we can make a simple quantum mechanical theory of the T1 as follows. We think of the nucleus as initially in a spin up state and the electron as initially in a spin down state with wave vector k. The scattering leaves the nucleus with spin down, the electron with spin up and a new wave vector k . We describe the scattering using Fermi’s Golden Rule: the probability per second of a transition of an electron from the initial state, i, to the ﬁnal state, f , is Wif =

2π | i| Vˆ |f |2 δ(Ef − Ei − ∆E) ,

(3)

where Vˆ is the hyperﬁne coupling energy operator (the Fermi contact interaction), ∆E is the change in nuclear spin Zeeman energy, and Ei and Ef are the energies of the initial and ﬁnal states of the electron. For the electrons, the initial state must be occupied and the ﬁnal state empty. The probability that the initial state is occupied is F (Ei ), where F (E) is the Fermi function of energy E. The probability that the ﬁnal state is empty is 1 − F (Ef ). We denote the number of states within dE at energy E by (E)dE. Then, summing over initial and ﬁnal states, the total probability per second, W , of ﬂips of the nuclear spin up and the electron spin down is 1

Interaction energy of a nucleus at the origin with an electron at position r: Vˆ = 8π γ γ 2 δ(r) 3 e n

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2π | i| Vˆ |f |2 δ(Ef − Ei − ∆E)F (Ei )(1 − F (Ef )) (Ei )dEi (Ef )dEf 2π = | i| Vˆ |f |2 F (Ei )(1 − F (Ei + ∆E))(Ei )(Ei + ∆E)dEi . (4)

W =

Now the energy diﬀerence between the initial and ﬁnal electron states is just the change in the nuclear Zeeman energy. This change is very small compared to kB T (unless one is so cold that there is gigantic nuclear polarization!). Consequently we neglect the energy diﬀerence, integrate over the ﬁnal energy, and get 2π (5) W = | i| Vˆ |f |2 F (Ei )(1 − F (Ei ))ρ(Ei )2 dEi . It is easy to show that F (E)(1 − F (E)) = −

∂F (E) . ∂E

(6)

If then we can neglect any changes of | i| Vˆ |f | and ρ(E) over energies comparable to kB T , we get W =

2π [| i| Vˆ |f |EF ]2 (Ei )2 kB T ,

(7)

where the subscript EF means evaluation for electrons at the Fermi energy. Equation (7) is essentially the Heitler-Teller result. The linear dependence on T represents the fact that, as a result of the exclusion principle, only electrons within about kB T of the Fermi surface can be scattered by the nuclear moment. The number of such electrons is a fraction of the total number of electrons of the order of (kB T /EF ). The linear dependence on temperature is displayed dramatically in Fig. 1 that shows data from a paper by Spokas and Slichter [12] on nuclear relaxation in aluminum. The measurements extend from 1.1 K to 1000 K and include data also from Anderson and Redﬁeld [13] as well as Hebel and Slichter for temperatures below 4.2 K.

3 Development and Veriﬁcation of the BCS Theory 3.1 Bardeen’s Early Gap Concept Although superconductivity was discovered in 1911 [14], its explanation resisted the eﬀorts of many outstanding physicists for many years. In 1950 Maxwell [15], and independently Reynolds, Serin, Wright, and Nesbitt [16] discovered that the superconducting transition temperature varied with the

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Fig. 1. Thermal relaxation time in aluminum versus the reciprocal temperature. The line describes (T1 T ) = 1.85 s K. Reproduced with permission from [12]. Copyright 1959 by the American Physical Society

isotopic content of the metal, showing that in all likelihood lattice vibrations played an important role in the mechanism of superconductivity. In 1951, John Bardeen came to the University of Illinois from Bell Laboratories. Bardeen had for many years been interested in superconductivity. The experimental discovery of the isotope eﬀect led him to focus on the role of the lattice vibrations. Bardeen and his postdoc, David Pines, worked out the theory of how the response of the ions to the presence of an electron changed the interaction of an electron with a second one in its vicinity [17]. Although many people associate superconductivity with the vanishing of electrical resistance, in fact the most fundamental property of a superconductor is its perfect diamagnetism: superconductors exclude magnetic ﬁelds, a property called the Meissner eﬀect. In 1954, Bardeen [18] proposed a very simple model that he argued could give the Meissner eﬀect and thus explain superconductivity. His idea was that in a superconductor there was a gap in the density of states right at the Fermi energy. The states below EF were the same as in the normal state. States which had been above the Fermi energy were all shifted to a higher energy by an amount EG so that the density of states vanishes between EF and EF + EG . This density of states is shown in the middle of Fig. 2. He argued that the size of EG would be approximately kB Tc , where Tc is the temperature of the superconducting transition. One of the authors of this article, CPS, ﬁrst learned of these ideas in 1954 from a

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Fig. 2. The energy dependence of the density of states for normal metals (top), the Bardeen model (middle), and the BCS model (bottom)

talk by Bardeen. Since the electrons on which Bardeen was focused were just those which give rise to the NMR T1 of a metal, the transition to superconductivity should produce a big eﬀect on the nuclear T1 . CPS resolved to try to measure the nuclear T1 , but there was a problem: as a result of the Meissner eﬀect, superconductors exclude the magnetic ﬁelds needed to do NMR (this event predates discovery of Type II superconductors which do not exclude the magnetic ﬁeld completely). However, another consequence of the Meissner eﬀect is that application of a suﬃciently strong ﬁeld suppresses the superconductivity. This critical ﬁeld, Hc , is 98.4 gauss for Al metal, for which Tc is 1.172 K. The solution to measuring T1 in the superconducting state was to cycle the magnetic ﬁeld. The cycle used by Hebel and Slichter [19, 20] is shown in Fig. 3. One sits initially at a ﬁeld H0 , greater than Hc , so that the metal is in the normal state, the magnetic ﬁelds can penetrate, and a net nuclear magnetization can be established. Then one reduces the magnetic ﬁeld to zero, cooling the nuclear spins (an adiabatic demagnetization), and converting the metal to its superconducting state. Since the nuclear spin temperature is now much lower than the lattice temperature, the nuclear spins warm towards

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Fig. 3. The ﬁeld cycling experiment expressed as magnetic ﬁeld vs. time. The time dependence is shaped to satisfy the conditions of adiabaticity

the lattice temperature via the spin-lattice relaxation processes. After some time, ts , in the superconducting state, the magnet is switched on again to H0 , an adiabatic remagnetization. The nuclear resonance apparatus, operating in the steady state mode, is tuned to a resonance frequency somewhat less than γn H0 , so that as the magnetic ﬁeld returns to H0 the ﬁeld passes through the resonance condition, producing an NMR signal, S(ts ). The experiment is repeated for varying times ts . The form of the signal S(ts ) is, to a good approximation, (8) S(ts ) = S0 exp (−ts /T1s ) , where T1s is the spin-lattice relaxation time in the superconducting state, and S0 a normalization constant. Thus, one can deduce the relaxation time in the superconducting state by measurements of dependence of S on ts . For this scheme to work the turn-oﬀ and turn-on of the magnetic ﬁeld must satisfy the two conditions of adiabaticity. The ﬁrst requirement is that the process be slow enough to be reversible. This requirement is equivalent to saying that at all times the spin system must be describable by a spin temperature θs , so that the internal parts of the system must be able to exchange energy suﬃciently eﬀectively that internal temperature gradients do not arise. In practice this means that at those parts of the magnetization cycle where the Zeeman and dipolar heat capacities are comparable, those two energy reservoirs must be able to exchange energy rapidly (A discussion of the general situation, distinguishing between slow and sudden turn-oﬀs can be found in the textbook Principles of Magnetic Resonance [21]). The second requirement

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is that heat transfer occurs only when the system is in the superconducting state. The time to turn the magnet oﬀ must be substantially shorter than the spin-lattice relaxation time in either the normal or superconducting states. This condition ruled out many materials with high values of Tc . For the experiments of Hebel and Slichter, the second condition led them to choose as their superconductor the metal aluminum, which has a long relaxation time of about 1 second in the liquid helium range. Even with aluminum, it was necessary to make a special magnet to achieve a fast enough switching time. They constructed the magnet using laminated iron that could be switched from 500 gauss to zero in a millisecond. The major diﬃculty of aluminum was that its transition temperature was so low. The normal way of cooling was to use liquid He, pumping on it to obtain temperatures below 4.2 K. It is very hard owing to the Rollin ﬁlm to get below 1.2 K by this method. By use of a triple Dewar system, to combat the Rollin ﬁlm, Hebel and Slichter succeeded in reaching 0.94 K. They observed the resonance at a ﬁeld of 360 gauss. By the fall of 1956, they had data which are shown in Fig. 4.

Fig. 4. Relaxation rate in a superconductor, Rs , relative to the zero-ﬁeld value extrapolated from the normal state, Rn (0), versus reduced temperature θ/θc . Three theoretical curves from the BCS theory are also shown. Reproduced with permission from [20]. Copyright 1959 by the American Physical Society

These data show that the transition to the superconducting state causes the nuclear relaxation to speed up. This result was a great surprise. One of the important models of the superconducting state was the two-ﬂuid model of Gorter and Casimir [22, 23]. The electrons were divided into a superﬂuid

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component and normal component. As one cooled below Tc the fraction of superﬂuid component grew from zero to 100%. One would expect that only the “normal” component could relax nuclei, hence T1 should lengthen as one cooled. If one viewed the problem in terms of the existence of an energy gap in the density of electron states, one would suppose that at low temperatures there would be few excitations to states above the gap, leading to a strong inhibition of the relaxation process. In examining their data, Hebel and Slichter thought of one possible explanation. Equation (5) shows that 1/T1 involves the square of the density of states. If the creation of a gap caused the density of states to pile up on the edges of the gap, this eﬀect might enhance W as long as the gap was not too large. They showed their data to Bardeen together with this possible explanation but of course lacking a true theory of the superconducting state, one could not judge the validity of this explanation. As we see below, this is part of the explanation of the enhanced relaxation rate, but the other part, dealing with the scattering matrix element in the superconducting state, is more important since it is the crucial manifestation of the pairing of electrons, the central feature of the later successful theory by Bardeen, Cooper, and Schrieﬀer. 3.2 The Bardeen, Cooper, Schrieﬀer Theory In the spring of 1956, Leon Cooper, who succeeded David Pines as Bardeen’s post doctoral associate, made an important discovery. In thinking about metals, he considered a pair of electrons with energies above a ﬁlled Fermi sea. He found that if they had an attractive interaction, they could form a bound state. These so-called Cooper pairs demonstrated the essential role of degeneracy and its resolution. He published these results in the fall of 1956, just as Hebel and Slichter were getting their ﬁrst experimental results. Bardeen, Cooper, and Bardeen’s graduate student Bob Schrieﬀer were deeply immersed in an eﬀort to generalize Cooper’s result to include all the electrons, not just a pair above the Fermi sea. In his Nobel Prize lecture, Schrieﬀer describes how the solution to the zero temperature wave function came to him while away on a trip (actually while riding the New York City subway). It was wave a function made up entirely of Cooper pairs in which each pair state (k, spin up; −k, spin down) was occupied fractionally. Utilizing this wave function, Bardeen, Cooper, and Schrieﬀer [5, 6] solved the Hamiltonian at absolute zero and then rapidly extended the analysis to temperatures up to the transition temperature. They found that there was an energy gap, 2∆(T ), that varied with temperature, which signaled the transition to the normal state at the temperature at which it vanished. (Note that the quantity ∆ is deﬁned by BCS as the diﬀerence in energy from the gap edge to the Fermi energy. Thus the diﬀerence in energy between the upper and lower gap edges is 2∆). The relationship between Tc and the gap is Tc ≈

2∆(0) . 3.52

(9)

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Fig. 5. Nuclear spin-lattice relaxation rate in the superconducting state of Al. The circles represent data of Hebel and Slichter, the crosses data of Redﬁeld and Anderson, and the solid line the BCS theory. Reproduced with permission from [25]. Copyright 1965 by the American Physical Society

They were able to calculate the results of many experiments with excellent agreement, all with only one parameter, the size of the energy gap at absolute zero. BCS showed Hebel and Slichter their theory, and the latter used it to analyze their data [19, 20]. (Details of their calculation of T1 in the normal and superconducting states can be found in the appendix of Reference [20]). They found that theory accounts for the increase in 1/T1 just below Tc and predicts a fall-oﬀ at lower temperatures where it says 1 ∝ e−∆(T )/kB T . T1

(10)

Redﬁeld independently thought of the same experiment as Hebel and Slichter, but started work somewhat after they began. He likewise studied Al, but to cool his sample he obtained some 3 He, thus avoiding the diﬃculties of the Rollin ﬁlm which arose with 4 He [24]. Figure 5 shows the data of Redﬁeld and Anderson combined with that of Hebel and Slichter (used in the Nobel Prize Lecture of Leon Cooper) [25]. Figure 6 shows data of Morse and Bohm [26] on the absorption rate of ultrasound, likewise from Cooper’s Nobel Prize Lecture. The temperature dependence is strikingly diﬀerent from that of the NMR relaxation rate. For ultrasound, the rate drops precipitously just below Tc . How can this be? The absorption of sound waves is much like nuclear spin-lattice relaxation. Both are low energy scattering events. For NMR, the nucleus ﬂips, ﬂipping the electron and scattering its wave vector. In sound absorption, a sound quantum (phonon) is absorbed, scattering the electron’s wave vector (the electron spin is not ﬂipped). For both processes, the energy exchange is much smaller than kB T . One would expect that in a one-electron theory of metals, both rate

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processes would be described by an equation such as (4), so they should have the same temperature dependence in the superconducting state. One of the major triumphs of the BCS theory is that it explains this diﬀerence. Indeed, the explanation is considered to be one of the best proofs of the correctness of the essential idea of the BCS theory, the pairing concept. One can in fact see the origin of the diﬀerence in detail by going back to (5), 2π | i| Vˆ |f |2 F (Ei )(1 − F (Ei )(Ei )2 dEi . (11) W = As we have remarked, nuclear relaxation and sound absorption both involve electron scattering, but in the former case the electron spin is ﬂipped whereas it is not ﬂipped in the latter case. The BCS wave function is made up of electron pairs in which an electron with spin up and wave vector k is paired with another electron of wave vector −k and spin down. As a consequence, there are two matrix elements that join any initial state to the same ﬁnal state. Denoting these as Vˆ1 and Vˆ2 , we then replace Vˆ by (Vˆ1 + Vˆ2 ) for NMR and (Vˆ1 − Vˆ2 ) for ultrasound, giving 2π W = | i| Vˆ1 ±Vˆ2 |f |2 F (Ei )(1−F (Ei +∆E))(Ei )(Ei +∆E)dEi , (12) where the plus sign applies to NMR and the minus sign to sound absorption. According to the BCS theory, the density of states s (E) in the superconductor obeys the equation

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s (E) =

n (E)E E 2 − ∆2

1/2

(13)

where n (E) is the density of states in the normal metal, and where E is measured from the Fermi energy and has a minimum absolute value of |∆|. Thus, at the edges of the gap where E approaches ∆, s (E) blows up, as shown at the bottom of Fig. 2. This increase of the density of states close to the edge of the energy gap satisﬁes the concept that Hebel and Slichter had postulated as to why the relaxation was faster in the superconducting state than in the normal state. But to explain the contrast between the NMR and the sound absorption one also needs the fact that the scattering matrix elements have diﬀerent energy dependence. For NMR the square of the matrix element between states at E and E goes as (EE + ∆2 )/EE , whereas for sound absorption it goes as (EE − ∆2 )/EE . Therefore if one neglects the diﬀerence in energy between E and E , the energy dependence of the matrix element cancels the singularity in the square of the density of states for sound waves, but not for NMR. This is directly a consequence of the pair nature of the wave functions. There is a beautiful and detailed explanation of all this in Cooper’s Nobel Prize Lecture [27].2 The details of the T1 calculation can also be found in the appendix of [20]. One consequence of the BCS form of the density of states is that the expression for the rate, W , such as that of (5) has a logarithmic inﬁnity. One can then return to (4), including the nuclear energy change, but this still predicts too large a peak. Another approach is to introduce a level broadening function. This was the approach of Hebel and Slichter [20, 29]. One can also consider anisotropy of the gap arising from lattice eﬀects. In a beautiful set of experiments, Masuda and Redﬁeld [30] perfected the ﬁeld cycling technique and the 3 He cooling apparatus. Their data ﬁt the BCS theory at both high and low values of T /Tc with a gap equal to 3.2 kB Tc , slightly less than the BCS value, and a 10% broadening of the temperature dependent energy gap. So, together the NMR T1 and the ultrasonic absorption were among the ﬁrst evidence that gave strong conﬁrmation of the central concept of the BCS theory, correlated electron pairs. 3.3 The General Pairing Conditions of the BCS Theory The original BCS paper explained superconductivity as resulting from the eﬀect of the lattice degrees of freedom on the interactions of electrons.3 There is an energy gap “parameter” ∆k that is the solution of an integral equation 2

These lectures are also in the appendix of the book by J.R. Schrieﬀer [28]. We caution the reader that there appears to be a typographical error in Cooper’s formula for the matrix element for NMR. The expression (1 + E 2 /∆2 ) should be (1 + ∆2 /E 2 ). Excellent accounts of the BCS theory can be found in the books by Schrieﬀer [28] and Tinkham [31].

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called the gap equation. As basis states, we use free electron Bloch states with energy measured with respect to the Fermi energy denoted as k . In the superconducting state, the interaction between the electrons Vkk changes the energy to Ek where (14) Ek = (2k + ∆2k )1/2 and ∆k is the solution of the integral equation (we give the equation at absolute zero for simplicity) Ek = −

V k

kk ∆k

2Ek

(15)

BCS approximated ∆k as a single number, ∆0 , for electrons within a given cutoﬀ energy of the Fermi energy, and zero outside this range. The energy gap parameter was independent of the direction in k-space, depending only on energy. One then gets that Ek = (2k + ∆20 )1/2

(16)

for electrons within the cutoﬀ range from the Fermi energy. In fact, the theory is more general. Depending on the details of the interaction, one may get other pairings. In the state of zero current ﬂow, all pairs have zero momentum. However, the pairs may have orbital angular momentum about their mass centers with quantum numbers, L, of 0, 1, 2, 3, etc. Owing to the exclusion principle, the spin, S, of the pair will be 0 if L is an even number and 1 if L is an odd number. The standard BCS state is (L = 0, S = 0). But other possibilities are (L = 1, S = 1) (so called p-wave superconductors); (L = 2, S = 0) (d-wave superconductors) and so on. Such pairings are often called “unconventional”. An example of unconventional pairing is liquid 3 He. The superﬂuid state of 4 He was a challenging problem which appeared to have some similarity to superconductivity, but the 4 He atoms obey Bose-Einstein statistics, making the parallel unlikely. However, 3 He atoms obey Fermi-Dirac statistics, which lead theorists to speculate in the 1960’s that there should be a superﬂuid state described in terms of a BCS-like theory. Early estimates of the transition temperature were overly optimistic. Experimenters looked unsuccessfully, pushing the possible temperature lower and lower. Because at short distances the interaction between 3 He atoms is repulsive from the overlap of the electron clouds, it had been suspected that the most favorable pairing state might not be L = 0. Then, in 1972, D. D. Osheroﬀ, R. C. Richardson, and D. M. Lee [4, 32] found the transition at 2.7 mK using NMR to study 3 He. These data were explained by A. J. Leggett [33] as arising from a BCS-like phase in which (L = 1, S = 1, Sz = ±1). Osheroﬀ, Richardson, and Lee shared the Nobel Prize in 1996 for their work, and Leggett received the Nobel Prize in 2003 for his. In solving the Hamiltonian for interactions that give other pairings, one ﬁnds that ∆k varies with direction in k-space as well as with the energy. The

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conventional BCS state has an isotropic gap so the gap function is simply a single number. More generally, it is a function of wave vector, and may even change sign as one moves around in k-space. In fact, one must think of ∆k as being a function that has an algebraic sign. The sign has signiﬁcance and can indeed be directly measured in certain tunnelling experiments [34]. Superconductivity was discovered in elemental metals (Hg and Pb). At ﬁrst it was thought to be a highly unusual phenomenon, however, in subsequent times it has been found to have very widespread occurrence. Materials such as Nb3 Sn and Nb3 Ge had much higher transition temperatures than Hg, Pb, or Al, getting above 20 K for Nb3 Ge. Some of the so-called heavy-fermion metals also exhibited superconductivity. More recently the cuprates, the alkali fullerides, and several classes of organic crystals have been shown to be superconductors. A major issue has become what mechanism explains the gigantic leap in Tc represented by the cuprates. The discovery of the cuprates as superconductors by George Bednorz and Alex M¨ uller resulted from careful and imaginative thought about methods of enhancing the electron-phonon interaction, thinking about electron-lattice coupling from a background that beneﬁted from deep experience with topics such as the role of the Jahn-Teller eﬀect, especially by Alex M¨ uller. This approach led to a revolutionary result, long sought after by experimenters and theorists alike. However, other mechanisms have been proposed, such as mechanisms involving the electron spins. As we discuss later, any given mechanism tends to favor a particular pairing state. In a given theory, only certain pairings can result, so one can rule out such a theory if one ﬁnds a diﬀerent pairing state. Thus, much eﬀort has gone into trying to determine the pairing states for the various materials that are superconductors. We now turn to several examples and show what NMR can say about the pairing state.

4 Type I and II Superconductors One of the earliest theories of superconductivity was that of the London Brothers [35]. A natural outcome of their theory was the concept of the penetration depth, λ, which described the penetration of a magnetic ﬁeld into a superconductor. Their theory expressed the fact that the diamagnetic shielding described by Meissner took place over a non-zero distance, λ, and gave a formula for it, mc2 , (17) λ2 = 4πns e2 where m is the electron mass, e its charge, and ns the number density of superconducting electrons. At absolute zero, ns is expected to be of the order of the number density of electrons. In 1953, A. B. Pippard [36] discovered that an additional length was needed to describe experiments on microwave electrical conductivity. This length,

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which he called the coherence length, ξ, describes the fact that sometimes one must think of the properties of a material in terms of non-local equations. Thus, when the microwave penetration depth gets less than the electron mean free path, Ohm’s Law is no longer accurate at a point in space since the electric ﬁeld varies so much between electron collisions. He argued that a similar thing should apply in superconductors, and estimated the length by an uncertainty principle argument as avF , (18) ξ= kB Tc where a is a constant of order unity, vF the velocity of electrons at the Fermi energy, and Tc the superconducting transition temperature. A similar concept had arisen earlier in the theory of V. L. Ginzburg and L. D. Landau [37]. One may think of ξ as describing a stiﬀness of the superconducting wave function: if one subjects a superconductor to a spatially varying disturbance, it is very costly in energy for the superconducting wave function to respond over lengths shorter than ξ. In the conventional superconductors such as Al, the penetration depth is much shorter than the coherence length: λ 1. ξ

(19)

In 1957, A. A. Abrikosov published a paper [38] in which he considered what would happen to the Ginzburg-Landau theory if instead λ 1. ξ

(20)

The result is quite remarkable. Consider an interface between normal and superconducting regions of a metal. In the presence of a magnetic ﬁeld, (19) corresponds to a positive surface energy (a cost in energy to have such an interface), whereas (20) corresponds to a negative surface energy, hence an energy gain by having such an interface. Abrikosov called those materials obeying (19) Type I superconductors, those obeying (20) Type II superconductors. If an external magnetic ﬁeld is applied to a Type I superconductor, the material excludes the magnetic ﬁeld until it reaches a critical value Hc above which the material becomes a normal metal and the magnetic ﬁeld can penetrate. If one applies a magnetic ﬁeld to Type II superconductor, the magnetic ﬁeld is excluded until a ﬁeld Hc1 is reached. Then the magnetic ﬁeld begins to penetrate, but not uniformly. Rather it consists of ﬂux in tubes, the core of which is much like a normal metal. These tubes, called ﬂuxoids, increase in spatial density as the applied ﬁeld increases until a ﬁeld Hc2 is reached at which the ﬂuxoid separations are of the order of the coherence length. Above that ﬁeld, the material is no longer superconducting. For these theoretical contributions, Ginzburg and Abrikosov were awarded the Nobel Prize in 2003. Landau had been awarded the prize in 1962. From the point of view of magnetic resonance, the discovery of these two types of

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superconductors has great importance since the ﬁeld penetration in a Type II superconductor makes possible direct observation of NMR in a superconductor in the presence of a magnetic ﬁeld. From (17) and (18), one expects that Type II superconductors arise from materials with high transition temperatures and low carrier densities. The alkali fullerides, the cuprates, and many organic superconductors are Type II superconductors.

5 The Alkali Fullerides 5.1 Introduction The discovery of the C60 molecule opened many new avenues of science. 13 C NMR made possible interesting studies of solid C60 [39]. The addition of alkali atoms, A, to form materials such as Ax C60 , where x ranges from 0 to 3, added another dimension. The A3 C60 materials are superconductors with remarkably high transition temperatures. For example, some transition temperatures for various A3 ’s are 19.5 K for K3 , 29.5 K for Rb3 , and 31 K for Rb2 Cs. All of these materials are ideal for NMR study since all the alkalis give strong NMR signals, and there are strong 13 C signals from natural abundance 13 C. Studies have been made for a variety of A3 ’s including Rb3 , Rb2 K, Rb2 Cs, RbK2 , RbCs2 , and K3 [39, 40]. 5.2 NMR in the Normal State The NMR properties of the alkali fullerides in the normal state are rather similar to conventional metals. Figure 7 shows log(1/T1 T ) vs. T for 87 Rb, 13 C, and 133 Cs nuclei for Rb2 CsC60 from Stenger et al. [40]. The curves at all three sites are uniformly displaced from one another on the log scale, showing that the ratio of the T1 ’s is independent of temperature. There is, however, a marked deviation from the Heitler-Teller law, 1/T1 T = constant. In fact, Stenger reports that there is an 80% increase from 30 K to 292 K in Rb2 CsC60 . For K3 C60 the increase is 45% and for Cs2 RbC60 it is 140%. Tycko et al. observed a similar phenomenon in Rb3 C60 [41]. Figure 8a shows their data. The normal state plots of 1/T1 T vs T show that T1 T is nearly independent of T in the normal state, but with a slight upward drift as the temperature rises. 5.3 T1 in the Superconducting State Tycko et al. [41] studied 13 C T1 in K3 C60 and Rb3 C60 in both the normal and superconducting states. Figures 8 and 9 show their data below Tc . The plot of log T1 vs 1/T is a straight line, obeying the equation for the BCS (L = 0, S = 0) pairing

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Fig. 7. Spin-lattice relaxation (1/T1 T ) versus temperature for 87 Rb, 13 C, and 133 Cs in Rb2 CsC60 at an applied ﬁeld of 8.8 T. Uncertainty is estimated as ±5%. Reproduced with permission from [40]. Copyright 1995 by the American Physical Society

Fig. 8. Temperature dependence of 13 C NMR spin-lattice relaxation in Rb3 C60 : (a) plotted as (T1 T )−1 vs. T ; (b) plotted as log T1 vs. T −1 . The dashed line is a ﬁt to the BCS (L = 0, S = 0) equation from 8 to 12 K. Reproduced with permission from [41]. Copyright 1991 by the American Physical Society

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D.F. Smith and C.P. Slichter

Fig. 9. Temperature dependence of 13 C NMR spin-lattice relaxation in K3 C60 : (a) plotted as (T1 T )−1 vs. T ; (b) plotted as log T1 vs. T −1 . The dashed line is a ﬁt to the BCS (L = 0, S = 0) equation below 9 K. Reproduced with permission from [41]. Copyright 1992 by the American Physical Society

1 ∝ e−∆/kB T . T1

(21)

The ﬁt gives ∆ as 21.3 K and 46.8 K for K3 C60 and Rb3 C60 , respectively. These numbers correspond to 2∆/kB TcNMR of 3.0 and 4.1 respectively, compared to the BCS weak coupling value of 3.5. Thus, these data support the BCS (L = 0, S = 0) state found for electron-phonon coupling mechanism of superconductivity. As we shall see, the same is not true for NMR data in the superconducting states of the cuprates or the organics. There is, however, one diﬀerence from the data on conventional superconductors. There is no sign of the coherence peak in 1/T1 . It is known from the studies of 51 V3 Sn by Y. Masuda and N. Okuba [42, 43] that a static magnetic ﬁeld may suppress the coherence peak. Accordingly, Stenger et. al. studied the magnetic ﬁeld dependence of the 13 C T1 in Rb2 CsC60 . Their results are shown in Fig. 10, which displays the temperature dependence of the ratio of

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the relaxation rate in the superconductor, Rs , to the rate in the normal state, Rn . We see that a “coherence peak” is present for suﬃciently low magnetic ﬁelds. 5.4 The Knight Shift Although this article is largely focused on the role of spin-lattice relaxation, useful information can also be obtained from measurements of the Knight shift. We turn to that subject here. Electron spins in an S = 0 state do not couple to an applied magnetic ﬁeld. This law is quite rigorous. A Zeeman term in the spin Hamiltonian has no non-zero matrix elements to other states. Consequently, the Knight shift in a superconductor with spin singlet pairing should vanish at T = 0. In fact, the Knight shift did not appear to vanish in early experiments on colloidal Hg [44, 45] and platelets of Sn [46]. The situation turned out to be rather complicated (eﬀects of impurities, of small particle sizes, and of changes in the chemical shift arising from narrow metal bands) and is beyond the scope of this article to discuss. K. Yosida [47] worked out the theory of the Knight shift, Ks , for BCS spin-singlet superconductors. He found that the ratio of Ks to the value Kn in the normal state is ∞ Ks (T ) −∞ = ∞ Kn (Tc ) −∞

∂F s (E) ∂E dE ∂F n (E) ∂E dE

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D.F. Smith and C.P. Slichter

∞ = −∞

E E 2 − ∆2

1/2

∂F dE , ∂E

(22)

where F is the Fermi function and E is given by (16). Measurement of the Knight shift involves several complicating factors. They can be seen by studying (23) which gives the expression for the resonance angular frequency ω in an magnetic ﬁeld B resulting from an applied ﬁeld B0 : ω = γn (1 + K S + K L )B ,

(23)

where K S is the Knight shift and K L is the chemical shift. The former arises from polarizing electron spins, the latter from polarizing electron orbital magnetism. B is the magnetic induction in the sample. In most magnetic resonance B = B0 . However, the Meissner eﬀect makes B diﬀerent from B0 . We write B = B0 + ∆B .

(24)

To a good approximation for Type II superconductors, ∆B/B0 is small, enabling us to write ∆B ω = γn 1 + K S + K L + (25) B0 . B0 Equation (25) is equivalent to saying that there is a shift tensor, K, given by K = KS + KL +

∆B . B0

(26)

Suppose one has more than one nuclear species in the sample. We label the species by a number such as 13 for C and 87 for Rb. Then 13

K = 13 K S + 13 K L +

∆B B0

(27)

and

∆B . B0 If we write the electron spin-nuclear spin Hamiltonian as 87

K = 87 K S + 87 K L +

H = γn H0 Iz + γe H0 Sz − γn Iz Aγe Sz ,

(28)

(29)

where we have made explicit the dependence of the hyperﬁne coupling on the nuclear and electron magnetic moments, then the Knight shift is α

K(T ) = α Aχs (T ) ,

(30)

where χs (T ) is the electron spin susceptibility and α denotes the nuclear species. The α A’s vary with species because they depend on the local wave function but are independent of T .

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We can get rid of the unknown quantity ∆B/B0 by subtracting (27) from (28). This is the procedure used by Stenger et al. [48]. Thus, they get 87 K(T ) − 13 K(T ) = 87 A − 13 A χs (T ) + constant . (31) For a BCS superconductor with (L = 0, S = 0), χs (T ) vanishes at T = 0. Unfortunately we do not have an independent method of determining the chemical shifts in these materials, hence we can not prove from the shift measurements that χs (0) vanishes. However, if we assume it does, we can use (31) to test the temperature dependence. Stenger et al. used (31) to test the relationship between the energy gap and Tc . Their data are shown in Fig. 11. They ﬁnd a good ﬁt for 2∆0 = 3.52kB Tc , the weak-coupling BCS value.

Fig. 11. NMR shift diﬀerences (K87 − K13 ) for Rb3 C60 (ﬁlled circles) and (K133 − K87 ) for Rb2 CsC60 (ﬁlled squares) versus T /Tc . Also shown are BCS spin susceptibility predictions for 2∆/kB Tc = 3.52 (solid line), 4 (dotted line), and 5 (dashed line). Reproduced with permission from [48]. Copyright 1993 by the American Physical Society

5.5 Conclusions for the Alkali Fullerides In conclusion, the exponential behavior of T1 at low temperatures, the existence of a peak in the relaxation rate just below Tc , the temperature dependence of the Knight shift, the fact that the temperature dependence of both

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T1 and Knight shift ﬁt the BCS gap-Tc relationship, all lead to the conclusion that these materials are conventional (L = 0, S = 0) BCS superconductors.

6 The Cuprate Superconductors 6.1 Background The discovery of superconductivity in lanthanum barium copper oxide at a temperature close to 40 K by J. D. Bednorz and K. A. M¨ uller in 1986 [49] opened a vast array of exciting new physics. Shortly thereafter, they found similar results in lanthanum strontium copper oxide. La2 CuO4 is an antiferromagnetic insulator. If one thinks of it as made up of La3+ , Cu2+ , and O2− ions, one can view this formula as saying that the La atoms give up 3 electrons and the Cu atoms give up 2 electrons in order for the O atoms to form closed p shells, becoming O2− ions. Since the ionic form of Sr is Sr2+ , substitution of a Sr atom for a La means that there is one less electron available to contribute to the O atoms. Thus every Sr substituted for a La eﬀectively adds a hole to the closed shells of the O2− ions. One speaks of the Sr as producing hole doping. The crystal structure of this material, often abbreviated as LSCO, is shown in Fig. 12. For low doping, the material remains an insulator, but the antiferromagnetic transition temperature drops with doping. Writing the formula as La2−x Srx CuO4 , one can make a phase diagram, shown in Fig. 13. Above x ≈ 0.05, the material is no longer an antiferromagnet but becomes both a conductor and a superconductor.

Fig. 12. Crystal structure of La2 CuO4

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Temperature (K)

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Strontium content, x Fig. 13. Phase diagram of La2−x Srx CuO4

It is customary to say that La2 CuO4 is the parent compound of this family of materials. Not long after the discovery of the systems based on La2 CuO4 , a material with superconducting transition temperature near 94 K, YBa2 Cu3 O7 , was discovered by C.W. Chu, M.K. Wu and collaborators [50]. Their material has a parent compound, YBa2 Cu3 O6 , which is also an antiferromagnetic insulator. If one writes the formula as YBa2 Cu3 O6+x , the phase diagram is that of Fig. 14 [51]. The crystal structure is shown in Fig. 15 [52].

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According to band theory, both parent compounds should be conductors. It is thought that they are instead Mott insulators, materials in which electronelectron correlations interfere with electron ﬂow because they make it very unfavorable to have two charge carriers simultaneously on the same atom. A Cu2+ ion has one hole in the d-shell, resulting in its having a net electron spin of 1/2. It might be natural to think of the doped materials as having a conduction band formed from holes on the O atoms with Cu spin-1/2 ions imbedded. The system might be much like Cu metal with magnetic impurities, the system which gives rise to the famous Kondo eﬀect. But this is not a correct description and NMR gives some of the best proof. The situation of an isolated hole has been analyzed by Zhang and Rice [53]. They conclude that “Cu-O hybridization strongly binds a hole on each square of O atoms to the central Cu2+ ion to form a local singlet. This moves through the lattice in a similar way as a hole in the single band eﬀective Hamiltonian of the strongly interacting Hubbard model.” This viewpoint explains why one observes in NMR that both the Cu and the O nuclei experience the transition to superconductivity at the same temperature and are not separable into two independent systems. From the earliest experiments it became apparent that the cuprate conductors are not ordinary metals. In the normal state of a normal metal, the spin (Knight) shift is independent of temperature, 1/T1 is proportional to temperature, and T2 is independent of temperature. None of these statements holds true in the electrically conducting cuprates. For example, we show data from N. J. Curro et al. for the 81 K superconductor YBa2 Cu4 O8 [54]. Figure 16 shows the normal state Knight shift of the planar 63 Cu nuclei. We see it is strongly temperature dependent, reaching a maximum Knight shift at about 500 K. Figure 17 shows 1/T1 versus T and T1 T versus T . At high temperatures T1 becomes independent of T . In fact,

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Fig. 16. The measured 63 Cu(2) Knight shift in the perpendicular direction as a function of temperature when the static magnetic ﬁeld is perpendicular to the crystal c axis. A temperature-independent orbital contribution has been subtracted oﬀ. Reproduced with permission from [54]. Copyright 1997 by the American Physical Society

Fig. 17. (a) 63 Cu(2) spin-lattice relaxation rate 1/T1 as a function of temperature. (b) T1 T as a function of temperature. The open circles are the data [55] of Corey et al. Reproduced with permission from [54] Copyright 1997 by the American Physical Society

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the plot of T1 T shows that above about 130 K it is linear in T . Figure 18 shows the quantity T2G , a measure of the transverse relaxation time, versus T , again showing a strong T dependence, in contrast to the usual situation in a metal. The basic problem in understanding these data is that one does not have a theory of the cuprates even for the normal state. The situation is analogous to that which prevailed for the superconducting state of conventional superconductors prior to the BCS theory. At that time, one had quite a good theory of the normal state of metals. Using it one could understand the NMR shifts and relaxation times, things such as the Korringa relation and how it was modiﬁed by electron-electron interactions. However, one could not understand experimental results in the superconducting state since there was no microscopic theory of superconductivity. It was only after one had the BCS theory that one could explain results such as the diﬀerent temperature dependence of the NMR T1 and the ultrasonic absorption rate. For the cuprates, the goal of NMR studies is to help in the development of a theory of the normal state as well as the superconducting state. We give some indication below of what progress has been made. As we have remarked, using NMR we are certainly able to say that these are not conventional metals.

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6.2 The Spin Hamiltonian The cuprates are rich in nuclear moments: 63 Cu, 65 Cu, 89 Y, 137 Ba, 139 La, and if one does isotopic substitution, 17 O. Thus NMR can probe throughout the unit cells of these materials. In general, the shifts and the magnetic susceptibility vary with doping (i.e. with x in the chemical formulas above) and temperature. Alloul et al. [56] studied the 89 Y shifts in YBCO, to use a shorthand notation that covers all dopings, and found that the shifts were a linear function of the magnetic susceptibility as he varied it as a function of temperature and of O dopings. From this he concluded that the 89 Y shifts were determined by the Cu electron spin magnetization. Subsequently Takigawa et al. [57] studied the 17 O and 63 Cu shifts for nuclei in the CuO planes of YBa2 Cu3 O6.63 and found out that not only were both those shifts linear functions of the spin magnetic susceptibility, but in fact were all proportional to one another since at low temperatures in the superconducting state they went to values given by the chemical shifts only. Since one expects that the spin susceptibility arises largely from the spin magnetism of the Cu atoms, the conclusion is that there must be transferred hyperﬁne coupling between the Cu electron spins and the 17 O and 89 Y nuclei. It is also found that the isotropic component of the 63 Cu spin shift relative to its value at T = 0 is positive, showing that it is not due to core polarization, which is always negative. Mila and Rice [58] showed that this could be explained by postulating there was transferred hyperﬁne coupling from one Cu atom to its four nearest neighbors. They showed theoretically that this mechanism could account for the size of the observations. These results can be summarized in a spin Hamiltonian for the Cu nuclei: , 63 H = γ63 63 Iˆ · 1 + 63 KL · H 0 + 63 Iˆ · A · S 0 + B · S n + 63 HQ , (32) n

where H 0 is the applied magnetic ﬁeld, 63 KL is the orbital (chemical) shift tensor, A the on-site hyperﬁne tensor, B the transferred hyperﬁne tensor, S 0 the on-site electron spin, S n the electron spin of the four nearest neighbor Cu atoms, and 63 HQ the electric quadrupole coupling to the on-site Cu nuclear spin. The Hamiltonian for a 17 O nucleus is 17 H = γ17 17 Iˆ · 1 + 17 KL · H 0 + 17 Iˆ · C · Sn , (33) n

where 17 KL is the oxygen chemical shift, C the transferred hyperﬁne coupling constant, and S n the spin of the two nearest neighbor Cu spins. Note that neither (32) nor (33) makes mention of an electron spin on the O atom. It is assumed, in the spirit of the Zhang-Rice singlet, that the spins are so tightly coupled that all eﬀects are encompassed in terms of the Cu spins and the hyperﬁne and transferred hyperﬁne coupling coeﬃcients.

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In calculating the electric quadrupole coupling it is useful to consider small clusters of atoms. Then if the z axis is perpendicular to the Cu–O planes, and the x and y axes are oriented along Cu–O bonds, to a ﬁrst approximation Cu2+ has a single hole in the x2 − y 2 orbital. The holes are then in the highest energy orbital (since the electrons will be in the lowest energy arrangement), that is, Cu–O antibonding σ orbitals. The presence of the Cu electron spins means that it is useful to think of a spatial Fourier spectrum of electron spins. Thus at a given spatial position, r, we write S q eiq·r . (34) S(r) = q

The ground state in the absence of an applied static magnetic ﬁeld is unmagnetized. Application of a static ﬁeld will cause magnetization. There will also be thermal excitation of magnetization. Using (34) we may think of the wavelength, λ, of the excitations or the q vector. The two are related by q=

2π . λ

(35)

It is found experimentally that the T1 ’s of the O and the Cu nuclei have very diﬀerent temperature dependences. Shastry [59] and Hammel et al. [60] proposed that the relaxation arose from spin ﬂuctuations, and that the different species responded to diﬀerent wavelengths. Thus, for long wavelengths (q ≈ 0), both species respond, but for ﬂuctuations close to the antiferromagnetic wavelength (qx , qy ) = (π/a, π/a), where a is the lattice constant, the Cu responds, but the O does not since the spins ﬂuctuation on its two neighboring Cu neighbors are opposite to each other and thus, by (34), cancel each other. This point of view can be expressed formally quite elegantly using a formula discovered by Moriya [61] for nuclei that are relaxed by coupling to electron spins, kB T 1 = 2 |Aβ (q)|2 χ (q, ωn )/ωn , T1α 2µB q

(36)

β=α

where α denotes the direction of the applied static magnetic ﬁeld, β the two perpendicular directions, Aβ (q) the β component of the hyperﬁne coupling tensor of wave vector q, and χ (q, ωn ) the imaginary part of the electron spin susceptibility at wave vector q and angular frequency ωn , the nuclear Larmor frequency. Equation (36) has a simple interpretation. A nucleus precessing at angular frequency ωn exerts an alternating magnetic ﬁeld on the electron spin system. The strength of the alternating ﬁeld is given by the transverse hyperﬁne coupling Aβ . The electron spin system absorbs energy at a rate given by χ (q, ωn ). The quantities |Aβ (q)|2 can be obtained from the spin Hamiltonians. For example,

The Study of Mechanisms of Superconductivity by NMR Relaxation 63

Ax (q) = 63 Ay (q) = Axx + 2B [cos(qx a) + cos(qy a)]

Az (q) = Azz + 2B [cos(qx a) + cos(qy a)] ,

271

(37)

where Axx = Ayy and Azz are components of the on-site Cu hyperﬁne tensor of (32). Finding a theory of the nuclear T1 ’s therefore can be considered a search for theoretical expressions for χ (q, ωn ). Or phrased alternatively, NMR T1 data measure a weighted average of the imaginary part of the low frequency electron spin susceptibility. Pennington and Slichter [62] discovered that the T2 in the cuprates was dominated by an indirect coupling mechanism through the electron spin system so that measurements of T2 give information about the real part of the electron spin susceptibility. Thus, the interaction of the z components of nuclear spin of a nucleus at position r 1 with a second at position r 2 had the form of a contribution H12 in the nuclear spin Hamiltonian given by G(r 1 , r)χ (r − r )G(r , r 2 )Iˆz2 , (38) H12 = Iˆz1 r,r

where the G’s are form factors such as G(0, r) = Azz + B

δr,r+ρ

(39)

and χ (r − r ) is the static magnetism produced at position r by a spatial delta function magnetic ﬁeld of unit amplitude at position r . χ (q, 0) is the spatial Fourier transform of χ (r − r ). They calculated the contribution to the second moment, denoting it as (1/T2G )2 . A more detailed discussion is in the article by Slichter et al. [63]. Thus we see that NMR can provide data about both the real and the imaginary part of the electron spin susceptibility. We have already seen NMR data for one of the cuprates, YBa2 Cu4 O8 . In Fig. 19 we show T1 data by Imai et al. [64] for La2−x Srx CuO4 . The striking feature is that at high temperatures the T1 curves for both the undoped and the doped samples all converge to the same value and become independent of temperature. These data were analyzed by Chubukov and Sachdev [66] using scaling concepts based on the two-dimensional Heisenberg antiferromagnet. They found indeed that at high temperatures the T1 should become independent of temperature. Figure 20 shows 1/T1 T from Takigawa et al. [67] for YBa2 Cu3 O7−y with O6.63 (y = 0.37) and O7 (y = 0). Shown are data for both 63 Cu and 17 O. From (36) it is evident that the two nuclei must see diﬀerent regions of q-space. The form factors in fact insulate the 17 O nuclei from magnetic ﬂuctuations at the antiferromagnetic wave vector, whereas the Cu nucleus sees them fully. We discuss this topic in greater detail below in Sect. 6.4.

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Fig. 19. Temperature dependence of 1/63 T1 measured by NQR for La2−x Srx CuO4 (•, x = 0; ◦, x = 0.04; , x = 0.075; , x = 0.15). Reproduced with permission from [64]. Copyright 1993 by the American Physical Society

Fig. 20. Temperature dependence of 1/T1 T in YBa2 Cu3 O7−y with y = 0.37 (open circles) and y = 0 (closed circles). On the left are data for the O(2,3) sites, on the right are data for the Cu(2) site. Reproduced with permission from [67]. Copyright 1991 by the American Physical Society

Millis, Monien, and Pines (MMP) [68] have proposed an explicit model which has these features. They assume the spin susceptibility is given by the expression αξ 2 , (40) χ(q, ω) = 1 + (q − Q)2 ξ 2 − i ωωsf

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where α is a constant, ξ a temperature-dependent correlation length, Q the point at which the susceptibility peaks, and ωsf the inverse of a relaxation time. For Q they pick the antiferromagnetic value (Qx , Qy ) = (±π/a, ±π/a). They assume that (40) is accurate near Q, but add a second term, independent of q, to represent the form near the origin. A series of papers (Sokol and Pines [69], Barzykin and Pines [70], and Zha, Barzykin and Pines [71]) using concepts of scaling as in that of Chubukov and Sachdev led to the realization that at high temperatures T1 T should be a linear function of T , as is shown in Fig. 17. Figure 21 shows T1 T versus T for optimally doped LSCO from the paper by Zha et al. [71] showing the linear character at high temperatures.

Fig. 21. The interpolated 63 T1c for La1.86 Sr0.14 CuO4 is shown together with the measured values of 63 T1c T for La1.87 Sr0.13 CuO4 and La1.85 Sr0.15 CuO4 of Ohsugi et al. [72]. Shown on the righthand side is the scale for ωsf (T ) ∝ 63 T1c T for La1.86 Sr0.14 CuO4 inferred from the ﬁt to the neutron-scattering experiments. Reproduced with permission from [71]. Copyright 1996 by the American Physical Society

The upturn in T1 T at low temperatures is a signature of a phenomenon called the spin gap or spin pseudogap. It is reminiscent of the situation one would have for a nucleus relaxed by a pair of coupled electron spins with a ground state singlet and excited state triplet. The ground state electron spin singlet does not couple to the nuclear spin, hence does not contribute to nuclear relaxation. At high temperatures, the spin-triplet state becomes occupied. Since it is magnetic, it could produce relaxation of the nucleus. But as the temperature drops, the occupancy of the triplet state drops, freezing

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out the relaxation when kB T becomes comparable to the singlet-triplet energy splitting. 6.3 The Superconducting State Relaxation time measurements for YBCO in the superconducting state were ﬁrst observed by Imai et al. [73] and by Kitaoka et al. [74]. Figure 22 shows data from Imai et al. plotted on a log-log scale. The low temperature behavior seems to ﬁt the power law 1 ∝ T3 . (41) T1 Figure 23 displays low ﬁeld 63 Cu and 17 O data of Martindale et al. [75] for YBa2 Cu3 O7 showing again the temperature dependence of (41). Figure 24 shows the data of Martindale et al. in a plot of log(63 T1c ) vs. 1/T , both axes normalized to their values at Tc for YBa2 Cu3 O7−y with y close to 1. The conventional BCS form in such a plot, once one is below the coherence peak, is a straight line with a slope proportional to the energy gap, ∆. The straight line shows the BCS slope using (9) and (10). We note the data have no coherence peak and clearly do not ﬁt an exponential in 1/T . For an L = 0 orbital state, ∆ is uniform in k space and grows in size as one goes down in T , levelling oﬀ at low T . This clearly does not ﬁt the data.

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Fig. 23. W1c from sample 2 measured in a 0.67 T ﬁeld vs. temperature on a log-log scale. The circles are 63 Cu and the squares are 17 O. The vertical line is at Tc = 91.2 K for this ﬁeld strength and orientation. The lines through the data show the powerlaw behavior of (41). Reproduced with permission from [75]. Copyright 1993 by the American Physical Society

T1(Tc)/T1(T)

0.1

0.01

BCS S-wave

1E-3 1

Tc /T Fig. 24. Data of Martindale et al. [75] for YBa2 Cu3 O7−y (y ≈ 1), contrasting with the straight line appropriate for the BCS (L = 0, S = 0) state prediction at low temperatures

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D.F. Smith and C.P. Slichter

The slope of the data appears to be continuously changing with temperatures, getting smaller at lower temperatures. As one goes to lower T , the gap appears to be shrinking. Such a situation might arise if the gap function ∆ were not uniform in k space. At higher T there are excitations across all parts of the gap, but at low T excitations occur only where the gap is small. Thus the data suggest a non-uniform gap. Since the Knight shift measurements by Barrett et al. [76] and by Takigawa et al. [67] showed that the electron spins were in the S = 0 (spin-singlet) state below Tc , we expect the orbital pairing to be one of L = 0, 2, 4, etc. Having ruled out L = 0, we turn to L = 2. The fact that the cuprate parent compounds are antiferromagnets suggested that possibly the superconducting mechanism involved electron spins. Several theorists proposed such models [77, 78, 79]. They found that the pairing would be L = 2 (d-wave). The gap function would have nodes along the lines |kx | = |ky | corresponding to the form ∆(k) ∝ kx2 − ky2 .

(42)

Moreover, at low temperatures thermal excitations exist only near the nodes and are expected to lead to a T 3 rate for relaxation processes, consistent with the data of Fig. 23. Thus the NMR data rule out L = 0 pairing (unless for some reason crystal asymmetry produces an enormous asymmetry in the L = 0 gap function) and strongly suggest d-wave pairing. Historically, NMR gave some of the earliest evidence for d-state pairing, the state now generally accepted as correct. 6.4 T1 Anisotropy In the normal state of the cuprates, the T1 is anisotropic as a result of the anisotropy of the hyperﬁne coupling (Axx = Ayy = Azz ). However, although the T1 ’s are temperature dependent the anisotropy is independent of temperature. The temperature dependence comes from the T dependence of χ . Figure 25 shows data by Barrett et al. [80]. The inset gives the ratio of the rate 1/T1 for a static ﬁeld parallel to that perpendicular to the Cu–O planes. In the superconducting state they found that the ratio became T -dependent. Martindale et al. [75] explored this further. They discovered that the relaxation rate varied with the strength of the applied static ﬁeld, and attributed this variation to the presence of the ﬂuxoids. They reasoned that the cores of the ﬂuxoids, which might be expected to be much like the normal material, might be a source of spin-lattice relaxation. The lower the applied ﬁeld, the lower the ﬂuxoid density, hence the less the ﬂuxoid cores should contribute to relaxation. The crystal c axis is deﬁned as the normal to the Cu–O planes. We can think of T1c , the relaxation time when the static ﬁeld is along the c axis, as arising from ﬂuctuating magnetic ﬁelds in the a and b directions. Likewise, we

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277

Fig. 25. The 63 Cu(2) spin-lattice relaxation rate 63 W1a vs temperature for H 0 a (♦, sample 1; , sample 4) and for H 0 c (◦, sample 1; , sample 4). The vertical solid line is at 92 K. Inset: The normal-state ratio W1a /W1c vs. temperature for sample 4 (•). The horizontal line is at W1a /W1c = 3.73, and the vertical line is at 92 K. Reproduced with permission from [80]. Copyright 1991 by the American Physical Society

can think of the T1a as arising from ﬂuctuating ﬁelds in the b and c directions. The c direction is also the axis of quantization for the 63 Cu nuclei when the applied ﬁeld is zero, as in an NQR experiment. Thus, if one has an applied ﬁeld along the c direction, and then lowers it towards zero, the resulting T1c should go over smoothly to the NQR value, measuring the eﬀects of the x and y components of ﬂuctuating ﬁeld. There is no way that one can measure the z components of ﬂuctuating ﬁelds from T1 measurements in zero ﬁeld. But a weak applied ﬁeld transverse to the c axis makes this possible. Accordingly, Martindale et al. [75] extended the measurements of Barrett et al. [80] to low ﬁelds as did Takigawa, Smith, and Hults [57]. Their results are shown in Fig. 26, along with theoretical results from Bulut and Scalapino [81] that we discuss below. As we have remarked, the anisotropy of the 63 Cu T1 can be analyzed in terms of hyperﬁne coupling terms of (37). These terms depend on the components of q. It is convenient to deﬁne the quantities F below which are the square of the terms in (37): 2 2 2 2 1 63 63 F = 63 Axx (q) = 63 Ayy (q) = Axx (q) + 63 Ayy (q) , 2 2 63 2 1 63 63 F⊥ = Axx (q) + Azz (q) , 2 1 17 2 F = 2 C 1 + (cos(qx a) + cos(qy a)) . (43) 2 It is convenient for some calculations to deﬁne a quantity

F⊥eﬀ as

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D.F. Smith and C.P. Slichter

Fig. 26. Temperature dependence of the Cu(2) anisotropy (T1−1 )ab /(T1−1 )c . The experimental data are from Martindale et al. [82] (squares) and Takigawa, Smith and Hults [57] (solid and open circles). The curves are the theoretical results of Bulut and Scalapino [81]. Reproduced with permission from [81]. Copyright 1992 by the American Physical Society

2 F⊥eﬀ = 63 Azz (q) .

(44)

F⊥eﬀ = 263 F⊥ − 63 F .

(45)

We note that 63 63

F comes into the calculation of 63 T1c , and 63 F⊥ comes into the calculation of 63 T1b = 63 T1a . These functions are represented in q-space in Fig. 27 taken from Thelen, Pines, and Lu [83]. The ﬁgure shows the variation along the line from (0, 0) to (π/a, 0), then along the line from (π/a, 0) to (π/a, π/a), and ﬁnally from (π/a, π/a) back to (0, 0). The ﬁrst Brillouin zone is shown in Fig. 28 from the same paper. We see that the form factors F are very large at some parts of the Brillouin zone, very small at others. If B were zero, the Cu F ’s would all have the same shape in q-space (independent of q), but they would still diﬀer from the oxygen F . The important point is that there are three NMR measurements of T1 (63 T1c , 63 T1a , 17 T1 ), each of which probes a diﬀerent part of q-space. Thus NMR not only probes real space, being able to look at the various atomic sites (Cu sites in the planes or chains, O sites in either the planes or chains, Y sites, La sites etc), but also it probes q-space. This second capability is a result of the existence of transferred hyperﬁne coupling in addition to on-site hyperﬁne coupling. The MMP expression (40) peaks at q = Q. For YBa2 Cu3 O7 this is the antiferromagnetic wave vector, (π/a, π/a). At low temperatures, all excitations are near to the Fermi surface. From Fig. 28 we see how the MMP

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279

Fig. 27. Form factors as a function of momentum for oxygen 17 F in units of C 2 and for copper sites 63 F⊥eﬀ and 63 F in units of 4B 2 . Reproduced with permission from [83]. Copyright 1993 by the American Physical Society

Fig. 28. Fermi surface for t = −0.45t. The + symbols indicate nodes on the Fermi surface for a ds2 −y2 superconductor, connected by momenta (0.742π/a, 0.742π/a). Reproduced with permission from [83]. Copyright 1993 by the American Physical Society

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D.F. Smith and C.P. Slichter

Q ﬁts the Fermi surface. At low temperatures, the gap nodes are the only places where excitations can occur. These are evidently excited by q = 0 or q = 0.742(π/a, π/a). We can see from Fig. 27 that this will change the ratios of the various F ’s and therefore change the anisotropy of the relaxation times. This is in essence the explanation of Bulut and Scalapino and of Thelen, Pines, and Lu for the T -dependent anisotropy of the 63 Cu T1 ’s in the superconducting state. The several theoretical curves show that the anisotropy of T1 strongly favors d-wave orbital pairing over s-wave, in agreement with the conclusions reached from Fig. 26. Note further that ﬁtting the NMR ratios at low T tells one the location on the Fermi surface of the nodes.

7 The Organic Superconductors 7.1 Introduction Soon after the discovery of the cuprate superconductors came the discovery in 1988 of κ−(BEDT−TTF)2 Cu(NCS)2 [84], the ﬁrst in the family of organic superconductors of the form κ−(BEDT−TTF)2 X , where BEDT-TTF (further abbreviated ET) represents the organic molecule bis – (ethylenedithiatetrathiafulvalene), X represents one of a number of polymeric anions, and κ represents the particular packing phase of this family. While superconductivity had ﬁrst been found in organic compounds in 1980 [85, 86], the κ−(ET)2 X family of superconductors distinguished itself by presenting the highest Tc of any organic compound at that date and by demonstrating similarities to the cuprates. Among the characteristics that the κ−(ET)2 X family shares with the cuprates are the following • Highly anisotropic properties, quasi two-dimensional in nature. • A similar phase diagram if one equates doping in the cuprates with pressure in the κ−(ET)2 X family [87]. In particular, antiferromagnetic and superconducting phases are found to be adjacent in the phase diagram, as seen in Fig. 29. • Unconventional superconductivity, with d-wave being the most likely pairing state. NMR has played a major role in developing an understanding of the physics of the κ−(ET)2 X systems and demonstrating the similarities to the cuprates. After a brief discussion of the structure of the κ−(ET)2 X family, we will discuss the contributions of NMR in the normal, superconducting, and antiferromagnetic states of the family. 7.2 Structure The κ−(ET)2 X compounds are composed of conducting layers of ET molecules alternating with insulating layers of polymers, as shown in Fig. 30. We

The Study of Mechanisms of Superconductivity by NMR Relaxation

paramagnetic

antiferromagnetic

Temperature (K)

281

superconducting ~300

Pressure (atm) Fig. 29. Low-temperature phase diagram of the κ−(BEDT−TTF)2 Cu[N(CN)2 ]X family. The vertical lines indicate the ambient pressure characteristics of the particular compounds

Fig. 30. Structure of κ−(BEDT−TTF)2 Cu[N(CN)2 ]X . Adjacent dimers have different shading. In order to aid viewing, the 1 H in the terminal ethylene groups of the molecules have not been shown

will focus on the compounds with the polymer Cu[N(CN)2 ]Y , where Y = Cl, Br or I, which we will hereafter simply refer to as κ−(ET)2 Cl, κ−(ET)2 Br and κ−(ET)2 I [88, 89, 90, 91]. The unit cell of these compounds is orthorhombic with a and c lying in the plane of the ET layers and b oriented in the interlayer direction. The polymers run along a. Within the conducting layer the

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D.F. Smith and C.P. Slichter

Fig. 31. An ET dimer with the central

C sites indicated

ET molecules pair up into dimers, as shown in Fig. 31. There are two ET layers in the unit cell with two dimers in each layer, for a total of four dimers per unit cell. The highest occupied molecular orbitals (hom*o’s) of the molecule are composed primarily of atomic p orbitals. In a dimer the hom*o’s of the two molecules interfere, producing bonding and antibonding orbitals of the dimer. Each dimer transfers one electron from an antibonding orbital (the higher in energy of the two dimer orbitals) to the polymer layer, resulting in a half-ﬁlled conduction band formed of overlapping ET orbitals. As seen in the phase diagram of Fig. 29, κ−(ET)2 Cl is an antiferromagnet at ambient pressure with TN = 27 K. At pressure of 300 atm this compound becomes a superconductor with TC = 13 K. κ−(ET)2 Cl might be said to be analogous to the parent compounds in the cuprates, La2 CuO4 and YBa2 Cu3 O6 , in that it is located at the region of lowest pressure (doping) in the phase diagram and is believed to be a Mott insulator. κ−(ET)2 Br is a superconductor at ambient pressure (TC = 12 K) and κ−(ET)2 I is an insulator showing neither magnetism nor superconductivity. There are several nuclei in the ET molecule that have been studied by NMR. We will discuss the research involving molecules labelled with 13 C at the two central carbon sites, as shown in Fig. 31. This particular doping is denoted κ−(13 C2−ET)2 X . As the electronic density is highest at the center of the molecules, the central carbons are ideally located for probing the nature of the electronic interactions. Since the molecules pair up into dimers, the two central carbons on a given molecule exist in slightly diﬀerent chemical environments, which we refer to as the inner and outer sites, shown in Fig. 31. 7.3 The Normal State This section concerns the behavior of the materials at temperatures above the superconducting (κ−(ET)2 Br) and antiferromagnetic (κ−(ET)2 Cl) transition temperatures. One of the ﬁrst tasks of NMR in these systems was to assign the observed resonance frequencies to their corresponding nuclear sites. It was found that for an arbitrary orientation of the applied ﬁeld, the 13 C NMR spectrum of κ−(13 C2−ET)2 Cl will contain 16 resonance lines in a single crystal

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283

sample. A factor of 8 in the number of lines is due to the presence of 8 inequivalent central carbon sites in the compound: a factor 4 for the four inequivalent dimers in a unit cell and a factor of 2 from the inner/outer site distinction (Sect. 7.2). There is an additional factor of 2 (bringing the total number of lines to 16) from the strong nuclear-nuclear dipolar coupling between the two central 13 C nuclei of a given molecule. From data on the dependence of the room temperature resonance frequencies on the orientation of the external ﬁeld, De Soto et al. [92] were able to assign the resonance frequencies to the nuclear sites. An example of the orientation data is shown in Fig. 32. The ﬁt comes from a spin Hamiltonian of the form (46) Hi = −γH 0 · (1 + KLi + KSi ) · Iˆi , which includes Zeeman (1), orbital (KL ) and spin (KS ) contributions. The subscript i denotes the particular site of the nucleus, corresponding to one of the 8 physical sites discussed above. The spin (Knight shift) tensor is found by subtracting out the orbital contribution, which is known from data on pure ET molecules [93]. De Soto et al. found that the two Knight shift tensors per dimer, which must be attributed to the inner and outer sites, had principle axes that varied slightly from the principle axes of the molecule, one tensor showing signiﬁcantly more deviation than the other. Based on the geometry of the dimer, it is expected that the p orbitals of the inner sites will have more overlap with the hom*o of the adjacent molecule than do the orbitals

−

Fig. 32. An orientation study of the central 13 C NMR resonance frequencies (expressed as frequency shift from a reference in parts per million) as function of external ﬁeld orientation. The data were taken at room temperature with an applied ﬁeld of 8.3 T oriented in the a-b plane. The ﬁlled points are from the outer site, the open points are from the inner site, and the solid lines are ﬁts using (46)

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D.F. Smith and C.P. Slichter

of the outer sites. Noting that the angle of overlap of the inner site orbitals corresponds to the angle of deviation shown by the tensor that deviated more from the molecular plane, De Soto et al. assigned the tensor with greater deviation to the inner site of the dimer. The components of the tensors are KSinner = (−57, −135, 423) ± 25 ppm and KLouter = (55, −33, 728) ± 25 ppm. Both tensors are dominated by the z component, indicating their p-like nature. Note that the outer site has a larger tensor, suggesting a larger value of the electron spin density at that site. This is somewhat surprising, as it is generally known that the electronic density is larger near the center of the molecules and so one might also expect the electron density to be greatest at the center of the dimer. M.-H. Whangbo and H.-J. Koo have provided an explanation for this result (private communication) by considering the π-type hom*o’s of the ET molecules in the ET dimer, which has only three electrons to occupy the resulting two orbitals. Their calculations indicate that there is indeed greater charge density at the inner sites for the bonding orbital of the dimer. However, the reverse is true for the antibonding orbital, in which the unpaired electron is located. This explains the higher spin density at the outer site. The temperature dependence of the Knight shift for κ−(13 C2−ET)2 Br is shown in Fig. 33(a), along with relaxation data. Above 150 K, T1 T ≈ constant while the Knight shift K S varies, so that the Korringa relation (1) is not followed. The reason for this can be seen in the Korringa factor, shown in Fig. 33(b). As discussed in Sect. 2, this factor is less than one when antiferromagnetic correlations are present, and this is seen to be the case in this temperature range. The presence of the antiferromagnetic ﬂuctuations are seen in T1 T , which decreases with decreasing temperature until 50 K. Below 50 K, T1 T increases and K S decreases, apparently caused by a freezing out of the spin susceptibility, similar to the pseudogap in the cuprates. The inﬂuence of antiferromagnetic ﬂuctuations on the nuclear relaxation in the normal state can be seen more clearly by comparing the spin-lattice relaxation of κ−(ET)2 Br to that of κ−(ET)2 Cl, shown in Fig. 34. The spinlattice relaxation peaks near TN in κ−(ET)2 Cl, consistent with the slowing down of the antiferromagnetic ﬂuctuations near the magnetic transition. The relaxation in κ−(ET)2 Br follows that of κ−(ET)2 Cl down to 50 K, suggesting that antiferromagnetic ﬂuctuations are present in κ−(ET)2 Br. Below 50 K, T1 drops oﬀ in κ−(ET)2 Br, showing the pseudogap-like behavior. 7.4 The Superconducting State As with the cuprates (Sect. 6.4), the analysis of spin-lattice relaxation in κ−(ET)2 Br is complicated by the presence of ﬂuxoids. Again, the eﬀect of ﬂuxoids can be minimized by taking data in smaller ﬁelds. Figure 35 shows the superconducting state spin-lattice relaxation data of De Soto et al. As expected, the data in the lower ﬁeld (0.6 T) show a lower relaxation rate than the data in the higher ﬁeld (8.3 T), a diﬀerence attributable to the extra

240

210

180

(a)

150

120

0.3

1.5

0.2

1.0 0.5

0.1 (b)

0.0

100

150

200

250

K (Outer) / K (Inner)

K (Outer)

285

Ks [PPM]

T1T [Ks]

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0.0 300

Temperature [K] Fig. 33. (a) The temperature dependence at H0 = 8.3 T and H 0 a of: (a) the Korringa product, T1 T (circles) and the Knight shift (triangles), for the outer site (b) the Korringa factor Kouter (squares) and the ratio Kouter /Kinner (diamonds). Reproduced with permission from [92]. Copyright 1995 by the American Physical Society

Fig. 34. The temperature dependence of the 13 C NMR spin-lattice relaxation rate in the κ−(ET)2 Br (squares) and κ−(ET)2 Cl (circles) compounds

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100

T1(Tc) / T1(T)

-1

1/T1 (s )

-1

-2

~T3

-3

-2

-3

0.1

T/Tc

Tc / T

Fig. 35. The temperature dependence of the spin-lattice relaxation rate in the superconducting state, with H 0 a, and H0 = 8.3 T (squares) and H0 = 0.6 T (circles). A “zero-ﬁeld” extrapolation point (triangle) is shown. The solid lines indicate ﬁts to the weak-coupling BCS expression (10), T1−1 ∝ exp(−∆(T )/kB T ), treating ∆ as a parameter. Near TC the dotted line shows the ﬁt ∆ = 3.0, and at low T the dotted line shows the ﬁt ∆ = 0.3. The solid line shows the BCS result of ∆(0)/kB TC = 1.76. The inset: the same data shown on a log-log plot, now ﬁt to the d-wave power law, T1−1 ∝ T 3 . Reproduced with permission from [92]. Copyright 1995 by the American Physical Society

relaxation provided by the ﬂuxoids in the higher ﬁeld. For this reason, the analysis treats only the data in the 0.6 T ﬁeld. No spin-coherence peak is present in the data of Fig. 35, and the data are clearly not ﬁt by a single exponential, as shown by the diﬀerent values of ∆ required to ﬁt the data to the BCS expression (10) at diﬀerent temperatures. Near TC , ∆ = 3 is found, while at lower temperatures, ∆ = 0.3, suggesting an anisotropy of the superconducting gap of at least 3.0/0.3 = 10. The inset shows a reasonable ﬁt of the relaxation data to the T1−1 ∝ T 3 power law of (41), consistent with superconductivity mediated by magnetic correlations. We conclude this section with a direct comparison of the κ−(ET)2 Br relaxation data to that of the cuprates, provided in Fig. 36. The cuprate data are those of Fig. 24, now combined with the relaxation data of De Soto et al. The similarity between the scaled relaxation rates of the two materials is striking, and it is again demonstrated that neither the cuprates nor the organics have relaxation behavior that is described by the BCS (L = 0, S = 0) expression (10). 7.5 The Antiferromagnetic State Just as with the cuprates, theorists have developed theories of superconductivity mediated by antiferromagnetic correlations [94, 95, 96, 97], spurred largely

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κ -(ET)2Cu[N(CN)2]Br

C NMR, H0 = 6kG

T1(Tc)/T1(T)

YBa2Cu3O7 63

Cu(2) NQR

0.1

0.01

BCS S-wave 1E-3

Tc/T

Fig. 36. A comparison of the temperature dependence of the spin-lattice relaxation rates of κ−(ET)2 Br [92] to that of YBa2 Cu3 O7 [75]. The rates have been scaled to their values at TC . The solid line indicates the BCS (L = 0, S = 0) result. Reproduced with permission from [92]. Copyright 1995 by the American Physical Society

by the proximity of the antiferromagnet state to the superconducting state. Here we review what NMR has revealed about the antiferromagnetic ordering. The ﬁrst NMR study of the magnetic state of κ−(ET)2 Cl was that of Miyagawa et al. [98]. Their NMR data was taken on the 1 H nuclei found in the terminal ethylenes of the ET molecules. The spin-lattice relaxation showed a peak in (T1 T )−1 near 27 K, providing a value for TN . They also found that the 1 H line split into three discrete lines below the magnetic transition, suggesting that the magnetic ordering was commensurate with the crystal lattice, and allowing them to estimate the magnetic moment as (0.4 − 1.0)µB /dimer. The knowledge that the ordering is commensurate strongly suggests that magnetism is a result of a Mott insulator state, rather than a spin-density wave, which would lead to incommensurate ordering. Magnetometry further identiﬁed both an easy-axis along b and the presence of weak ferromagnetic canting. The work of our group [99] on κ−(ET)2 Cl began with the study of the central 13 C resonance frequencies in the magnetic state. The data for the outer site are shown in Fig. 37, and the data for the inner site are qualitatively similar. The data show the resonance lines developing large orientation-dependent shifts in the antiferromagnetic state. We learned several things from these data. First, the large magnitude of the shifts in the antiferromagnetic state suggests a magnetic moment of approximately 0.5 µB per dimer, consistent with the ﬁndings of Miyagawa et al. Second, that there is no additional splitting of the lines in the antiferromagnetic state conﬁrms the commensurate ordering and puts constraints on the possible orientation of the electron spins.

288

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Shift (MHz)

0.5 0.0 −0.5

H0 || a H0 || b H0 || c

−1.0 −1.5 −2.0 0

Temperature (K) Fig. 37. The temperature dependence of 13 C NMR outer site frequency shift in κ−(ET)2 Cl, for several orientations of the applied ﬁeld. Reproduced with permission from [99]. Copyright 2003 by the American Physical Society

It is quite surprising that for each orientation of applied ﬁeld the antiferromagnetic ordering gives rise to only one line. We mention two reasons why one might expect otherwise. The ﬁrst concerns the potential for domains of diﬀerent electron spin orientation. Reversing the orientation of all electron spins leaves unchanged the energy of a system with isotropic exchange coupling. Therefore, one might expect two conﬁgurations of spin ordering (differing solely by spin reversals) which represent the ground state, and both conﬁgurations would be found in a given sample. If one conﬁguration shifts the NMR frequency up, the other shifts it down. Thus all NMR lines would occur as pairs of opposite shift. The fact that we see a single line means that there cannot be multiple domains of spin orientation in the sample. We show later that the argument for multiple domains is invalidated because of the presence of non-isotropic elements of the exchange interaction, speciﬁcally the Dzialoshinskii-Moriya interaction. A second argument for why more than one line might be expected arises from the distinct hyperﬁne tensors of the various dimer sites. As discussed in Sect. 7.3, there are four hyperﬁne tensors per unit cell. While in the normal (paramagnetic) state these four tensors provide degenerate resonance frequencies when the external ﬁeld is applied along one of the crystal axes (points of highest symmetry), it is surprising that in the antiferromagnetic state these distinct hyperﬁne tensors do not lead to distinct resonance frequencies. To see this, we consider a basic nuclear spin Hamiltonian with a Zeeman and hyperﬁne interaction, H i = HZeeman,i + Hspin,i

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ˆi = −γn Iˆi · H 0 + Iˆi · Ai · S , ˆi · S A i = −γn Iˆi · H 0 − γn ˆ eﬀ,i , ≡ −γn Iˆi · H

(47)

ˆi are respectively the nuclear and electron spin operators at where Iˆi and S ˆ eﬀ,i to be the eﬀective ﬁeld perceived by the nuclear the site i. We deﬁne H spin at site i. If the Zeeman term dominates the hyperﬁne interaction with the ﬁeld applied along the axis α, then to ﬁrst order the nuclear spin Hamiltonian is 1 a αb b αc c (Aαa S + A S + A S ) H i = −γn Iˆiα H0 − i i i i γn i i α ≡ −γn Iˆiα Hef (48) f,i , ˆ with the thermal average where we have replaced the electron spin operator, S expectation value, S. In the paramagnetic state the electron spin is also polarized along the direction α, so that the Hamiltonian reduces to 1 αα α α ˆ A S H i = −γn Ii H0 − , (49) γn i i so that now only a single diagonal component (Aαα i ) of the hyperﬁne tensor is sampled. Since the diagonal components of the tensors are the same for all four dimer sites, the degeneracy in the paramagnetic state when the ﬁeld is along a crystal axis is explained. However, in the magnetic state the electron spins are no longer aligned along the direction of applied ﬁeld. In fact, a simple antiferromagnet with just the Zeeman and isotropic exchange interactions will have the electrons aligned perpendicular to the ﬁeld to maximize the Zeeman interaction while keeping the nearest neighbor spins mostly antiparallel (this assumes that the Zeeman interaction is much smaller than the isotropic exchange interaction, a condition true for all of the ﬁelds we used in our study). Let’s assume that the electron spins align along b when the ﬁeld is along a. Then if we label the two dimer sites in the A layer as A1 and A2, we ﬁnd an eﬀective ﬁeld at A1 of 1 ab b A S , (50) Heﬀ,A1 = H0 − γn A1 A1 and an eﬀective ﬁeld at A2 of Heﬀ,A2 = H0 −

1 ab b A S . γn A2 A2

(51)

ab From symmetry arguments we know that Aab A2 = AA1 , and antiferromagnetic b B , which leads to ordering of nearest neighbor spins implies SA2 = −SA1

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Heﬀ,A2 = H0 +

1 γn

b Aab A1 SA1 .

(52)

b Thus, for this conﬁguration, Heﬀ,A1 and Heﬀ,A2 diﬀer by γn2 Aab A1 SA1 , and two resonance lines result. We can therefore rule out the possibility that the electron spins align along b when the ﬁeld is along a. From this demonstration we see that the condition imposed by our data that every site see the same eﬀective ﬁeld does in fact reveal constraints on the electronic ordering. Further constraints result due to the oﬀ-diagonal components of the hyperﬁne tensor changing sign between sites, unlike the diagonal components. Following through with the analysis, we deduced the electronic spin orderings shown in Fig. 38. These diagrams overlook the slight canting of the spins toward the applied ﬁeld due to the Zeeman interaction. Note that the type of the interlayer ordering (between A1 and B1, for instance) changes as the direction of the ﬁeld changes, indicating that the interlayer exchange interaction is not playing a signiﬁcant role in the ordering. Also, no ordering is shown for H 0 c because in that case we do not have enough constraints to make an assignment.

Fig. 38. Electron spin ordering of the AF state of κ−(ET)2 Cl for ﬁelds along the a and b directions, as deduced from NMR data. The dashed lines indicate glide planes of symmetry. Reproduced with permission from [99]. Copyright 2003 by the American Physical Society

To explain our NMR results along with the magnetometry data of our group and other groups, we developed an electron spin Hamiltonian that includes an isotropic exchange interaction (to provide for antiferromagnetic alignment of nearest neighbor spins), an antisymmetric or DzialoshinskiiMoriya (DM) interaction [100, 101] (to provide for the weak ferromagnetism), an anisotropic exchange interaction (to provide for the b easy-axis), and the Zeeman interaction (to couple the spins to the applied ﬁeld):

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H spin = H iso + H DM + H anis + H Zeeman |J| ˆ ˆ 1 ˆn × S ˆm ) Sn · Sm + = D nm · (S 2 n,m 2 n,m +

∆J ˆb ˆb ˆn . Sn Sm + gµB H · S 2 n,m n

(53)

Here n and m represent nearest neighbor (intralayer) dimer sites. The DM interaction leads to weak ferromagnetism through the cross product in the DM expression. Consider two neighboring spins, S A1 and S A2 , which are antiferromagnetically aligned along the c axis, as shown in Fig. 39. If the DM vector between the spins (D A1,A2 ) is along b, and the ﬁeld is along a, then as the spins cant toward a to minimize the Zeeman interaction, they will develop a cross product that points along -b to minimize the DM interaction. Note that if one switches the spins S A1 and S A2 the cross product points in the opposite direction, and the DM energy is not minimized. Thus, the presence of the DM interaction explains why the argument for multiple domains of spin orientation does not hold up.

Fig. 39. A diagram showing the eﬀect of the DM interaction on two neighboring spins. The spins are drawn with the same origin to aid viewing

To determine the electron spins orientations that minimize the DM interaction in κ−(ET)2 Cl for particular orientation of the applied ﬁeld, it was necessary to know something about the orientation of the DM vectors (one for each layer). We were able to identify several constraints on the DM vectors through a symmetry analysis in the spirit of the analysis carried out by Coffey et al. [102] for La2 CuO4 . With the information gained from the symmetry analysis, we learned that the electronic ordering implied by the DM vectors was in fact the ordering determined through analysis of our NMR data. Furthermore, additional magnetometry measurements, when interpreted with our model, enabled us to determine values for both the isotropic exchange and DM interaction.

8 Conclusion From the classic BCS superconductors to the more exotic cuprate and organic superconductors, NMR has been an important part of the extensive body of

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research on these intriguing materials. NMR work on aluminum was central to the conﬁrmation of the BCS theory of superconductors. Later relaxation studies of both the normal and superconducting states of other superconductors have informed theoretical work on the pairing state, leading us closer to an understanding of the fundamental mechanisms of superconductivity. This work was supported by the U.S. Department of Energy, Division of Materials Sciences under Award No. DEFG02-91ER45439, through the Frederick Seitz Materials Research Laboratory at the University of Illinois at Urbana-Champaign.

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NMR in Magnetic Molecular Rings and Clusters F. Borsa1,2 , A. Lascialfari1 and Y. Furukawa3 1

Dipartimento di Fisica “A.Volta” e Unita’ INFM, Universita’ di Pavia, 27100 Pavia, Italy [emailprotected] Department of Physics and Astronomy and Ames Laboratory, Iowa State University, Ames, IA 50011 [emailprotected] Division of Physics, Graduate School of Science, Hokkaido University, Sapporo 060-0810, Japan

Abstract. Molecular nanomagnets (MNM) are magnetic molecular clusters containing a limited number of transition ions in a highly symmetric conﬁguration and coupled by strong exchange interaction (either ferromagnetic (FM) or more often antiferromagnetic (AFM)). The magnetic intermolecular interaction is very weak and thus the clusters behave as single nanosize units. NMR has proved to be an excellent probe to investigate the static magnetic properties and the spin dynamics of this new fascinating class of magnetic materials. The chapter contains a comprehensive review of the work performed in the last few years by the present authors with only a brief reference to work performed by other researchers. Most of the NMR measurements were performed on protons but important results were obtained also using other nuclei like 55 Mn, 57 Fe, 7 Li, 23 Na, 63 Cu, 19 F. In some cases the NMR was observed at low temperature in zero external ﬁeld. Some novel NMR phenomena speciﬁc of the systems investigated were discovered and explained. For example in the anisotropic ferrimagnetic clusters Mn12 and Fe8, the ground state is a high total spin S = 10 state whereby the crystal ﬁeld anisotropy generates an energy barrier typical of superparamagnets. It is shown how NMR and relaxation measurements can detect the microscopic local spin conﬁguration in the ground state and the dynamics of quantum tunnelling of the magnetization (QMT). Another example is the case of the AFM rings, Fe10, Fe6 and Cr8, in which the ground state is a singlet, S = 0, separated from the ﬁrst triplet excited state by an energy gap of about 5–10 K. By applying a magnetic ﬁeld one can observe level crossing eﬀects. These eﬀects were studied by proton NMR and relaxation measurements vs ﬁeld at low temperature (1.5–3 K). Finally, the nuclear relaxation rate as a function of temperature in the above mentioned AFM rings displays a ﬁeld dependent peak at a temperature of the order of the exchange constant J, which can be ﬁtted with a general scaling law. From these data, the lifetime broadening of the energy levels can be determined.

F. Borsa et al.: NMR in Magnetic Molecular Rings and Clusters, Lect. Notes Phys. 684, 297– 349 (2006) c Springer-Verlag Berlin Heidelberg 2006 www.springerlink.com

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1 Introduction In the last years there has been a great interest in magnetic systems formed by a cluster of transition metal ions covalently bonded via superexchange bridges, embedded in a large organic molecule [1, 2, 3, 4, 5]. Following the synthesis and the structural and magnetic characterization of these magnetic molecules by chemists, the physicists realized the great interest of these systems as a practical realization of zero-dimensional model magnetic systems. In fact the magnetic molecules can be synthesized in crystalline form whereby each molecule is magnetically independent since the intramolecular exchange interaction among the transition metal ions is dominant over the weak intermolecular, usually dipolar, magnetic interaction. Magnetic molecules (see Figs. 1 (a)–(d) for some example systems) can be prepared nowadays with an unmatched variety of parameters: (i) the size of the magnetic spin can be varied, spanning from high “classical” spins to low “quantum” spins by using diﬀerent transition metal ions i.e., Fe3+ , Mn2+ (s = 5/2); Mn3+ (s = 2); Cr3+ , Mn4+ (s = 3/2); Cu2+ , V4+ (s = 1/2); (ii) the exchange interaction can go from antiferromagnetic (AFM) to ferromagnetic (FM) with values of the exchange constant J ranging from a few K to more than 1000 K; (iii) the geometrical arrangement of the magnetic core of the molecule can vary from simple coplanar regular ring of magnetic ions as found in many Fe and Cr rings to totally asymmetric three dimensional clusters such as [Fe8 (N3 C6 H15 )6 O2 (OH)12 ]8+ × [Br8 ·9H 2 O]8− (in short Fe8); (iv) the symmetry of the magnetic Hamiltonian can go from isotropic Heisenberg type as in most cases to easy axis or easy plane. Choosing from this variety of model systems one can investigate fundamental problems in magnetism taking advantage of the fascinating simplicity of zero-dimensional systems. Examples of issues of interest are the transition from classical to quantum behavior, the eﬀect of geometrical frustration, the form of the spectral density of the magnetic ﬂuctuations, the spectrum of the low-lying excitations with the connected problem of quantum spin dynamics and tunneling. NMR has proved to be a powerful tool to investigate both static and dynamic properties of magnetic systems. In particular, it has been very successful in addressing some special features in low dimensional magnetic systems. For example, in one dimensional magnetic Heisenberg chains the long-time persistence of spin correlation has dramatic consequences on the ﬁeld dependence of the nuclear spin-lattice relaxation rate T1−1 which directly probes the lowfrequency spectral weight of spin ﬂuctuations. From T1 and T2 measurements one can detect the crossover of spin dimensionality from Heisenberg to XY to Ising as a function of temperature in both one- and two-dimensional systems. One ﬁnal example among the many is the study of the gap in energy in Haldane s = 1 spin chains. With the above scenario in mind we have undertaken a systematic NMR investigation of molecular nanomagnets since back in 1996. The present review tries to give an account of the main results obtained so far and of the many

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Fig. 1. top left: Structure of Fe30 [2]. Light gray: iron, dark gray: molybdenum, small gray spheres: oxygen. Hydrogen atoms are not shown for simplicity. top right: Structure of Fe8 [1]. The hydrogen atoms are not shown for simplicity. bottom left: Structure of Fe6(X) (X = alkaline ion). Dark gray: iron, black: oxygen, light gray: carbon, central sphere: alkaline atom (Li, Na). Hydrogen atoms omitted for clarity. bottom right: Structure of Cr8. Main atoms: black: chromium, dark gray: ﬂuorine, light gray: oxygen

exciting projects that still lie ahead. The work was done through a continuous very fruitful collaboration among three NMR laboratories: at the University of Pavia, Italy, at Iowa State University and Ames Laboratory, Ames, IA, USA, and at Hokkaido University, Sapporo, Japan with occasional very useful collaborations with the high ﬁeld NMR laboratory in Grenoble, France. None of the work could have been done without the precious collaboration and help of our colleagues in chemistry at the University of Florence and of Modena, Italy, and at Ames Laboratory in USA who synthesized and characterized the samples used in the NMR work.

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In Sect. 2 we will mention some of the problems encountered in doing NMR in molecular nanomagnets. These are related for example to the presence of many non-equivalent nulclei, to very broad and structured resonance lines, to very short relaxation rates, to vanishingly weak signals. In the three following Sects. (3, 4, 5) we present and discuss the experimental data. To organize the considerable amount of data we chose somewhat arbitrarily to divide them according to the temperature range. In fact, the physical issues regarding the magnetic properties and the spin dynamics of the molecular nanomagnets depend on the relative ratio of the thermal energy kB T and the magnetic exchange energy J. At high T the individual magnetic ions in the molecule behave as weakly correlated paramagnetic ions; at very low T the individual spins are locked into a collective quantum state of total spin S; at intermediate T the interacting spins develop strong correlations in a way similar to what happens in magnetic phase transitions in three dimensional systems. In the illustration of the physical issues encountered in the diﬀerent temperature ranges we utilize the most representative results for diﬀerent kind of molecules. The magnetic molecules which were investigated by NMR but whose results are not mentioned in Sects. 3, 4 and 5 are reviewed separately in Sect. 6.

2 Challenges of NMR in Molecular Nanomagnets Molecular nanomagnets oﬀer a wide variety of nuclei which can be used to probe the magnetic properties and the spin dynamics. Most of the measurements were done on proton NMR. In this case the signal is very strong but the width of the spectrum and the presence of many inequivalent protons in the molecule require some special attention in the analysis of the results. Due to the above reasons the recovery of the nuclear magnetization was found in many cases to be strongly non-exponential. There are two sources for nonexponential recovery. The ﬁrst is due to incomplete saturation of the broad NMR spectrum which leads to an initial fast recovery of the nuclear magnetization due to spectral diﬀusion. If the spectrum is not too wide (at most twice the spectral width of the rf pulse) one can still saturate the whole line by using a sequence of rf pulses provided that the condition of T1 much longer and T2 much shorter than the pulse spacing can be met. In nanomagnets with magnetic ground state like [Mn12 O12 (CH3 COO)16 (H2 O)4 ] (in short Mn12) and Fe8 [1] the proton spectrum can be as wide as 4 MHz and structured at low temperature. In this case the spectrum has to be acquired by sweeping the ﬁeld and/or the frequency and plotting the amplitude of the echo signal after proper correction for changes of irradiation conditions. In this case the relaxation can only be measured at given points of the spectrum and the spectral diﬀusion eﬀect cannot be avoided. One can try to establish the percentage of the fast initial recovery which is aﬀected by spectral diﬀusion and exclude that from the measurement but a large systematic error can still be unavoidable. The second source of non-exponential recovery is due to the presence in the

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molecule of protons having a diﬀerent environment of magnetic ions and thus having a diﬀerent relaxation rate. If T2 is fast compared to T1 then a common spin temperature is achieved during the relaxation process and the recovery is exponential with a single T1−1 which is a weighted average of the rates of the inequivalent protons in the molecule. In the opposite limit encountered when T1 is comparable to T2 , each nucleus or group of nuclei relaxes independently with its own spin temperature and the recovery of the nuclear magnetization results in the sum of exponentials: n(t) =

M (∞) − M (t) −t/T1i = pi e . M (∞) i

(1)

If there is a continuous distribution of T1 ’s the recovery follows a stretched exponential function exp(−(t/T1∗ )β ) where β < 1 is the smaller the wider is the distribution and T1∗ is a relaxation parameter related to the distribution of T1 ’s in a non-trivial manner. When the recovery is non-exponential it is best to measure the T1 parameter from the recovery of the nuclear magnetization at short times. In fact the slope at t→ 0 of the semilog plot of n(t) vs t yields an average relaxation rate T1−1 = i pi exp(−t/T1i ). Unfortunately, in most cases the situation is intermediate between the two above limiting cases. In this circ*mstance there is no simple way that one can deﬁne a spin-lattice relaxation parameter. Since in many instances one is interested in the relative changes vs T and H, one can simply deﬁne an eﬀective relaxation parameter R by taking the time at which the recovery curve n(t) reduces to 1/e of the initial value. Other isotopes which have been utilized for NMR studies of magnetic molecules include 2 H, 13 C, 7 Li, 23 Na, 63,65 Cu. The disadvantage of a weaker signal in 13 C is in part compensated by the advantage of having a nucleus with strong hyperﬁne coupling to the magnetic ions and with less number of inequivalent sites with respect to protons. For the remaining quadrupole nuclei there is the additional information obtained by the quadrupole coupling with the electric ﬁeld gradient. When the quadrupole interaction is suﬃciently strong to remove the satellite transitions from the central line the non exponential decay of the nuclear magnetization becomes very diﬃcult to analyze because besides the non-equivalent sites one has to take into account the intrinsic non-exponential decay due to unequal separation of the Zeeman levels. The 63,65 Cu case is the only one where, to our knowledge, a pure NQR experiment has been performed in molecular clusters (i.e., [Cu8 (dmpz)8 (OH)8 ] × 2C6 H5 NO2 , in short Cu8). The NQR spectrum was found to contain several lines in the frequency range 16–21 MHz. Very useful information was obtained from the 55 Mn and 57 Fe NMR in Mn12 and Fe8 clusters, respectively. The NMR of the above nuclei can be observed only at low temperature (T < 4 K) since with increasing temperature the relaxation times T1 and T2 become too short. The 55 Mn and 57 Fe (in isotopically enriched sample) NMR was detected both in zero ﬁeld and

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in an externally applied ﬁeld. The zero ﬁeld 55 Mn NMR spectrum in Mn12 consists of three quadrupole broadened lines (i.e., each several MHz wide) in the frequency range 230–370 MHz while the 57 Fe NMR spectrum in Fe8 is made of eight diﬀerent rather narrow lines (i.e., 100 kHz) in the frequency range 63–73 MHz. Both Mn12 and Fe8 are ferrimagnetic molecules at low temperature. However, since there are no domain walls and the anisotropy is very high no signal enhancement due to the rf enhancement in domain walls and/or domains is present contrary to normal ferro- or ferri-magnetic long range ordered systems. As a consequence, the NMR signal intensity in zero external ﬁeld is small (particularly in Fe8) even at low temperature since the frequency range of the overall spectrum is quite broad.

3 NMR at High Temperature (kB T J ) Most of the magnetic molecular clusters investigated are characterized by exchange constants J which are well below the room temperature energy value kB T . Exceptions to this are the Cu8 ring and to a certain extent also Mn12 and Fe8 clusters. If kB T J, the magnetic moments in the cluster are weakly correlated and the system behaves like a paramagnet at high temperature. In this case the nuclear spin lattice relaxation due to the coupling to the paramagnetic ions should be ﬁeld independent as indeed found in paramagnets such as MnF2 . On the other hand, the T1 in molecular nanomagnets is strongly ﬁeld dependent as shown for a number of systems in Figs. 2 (a)–(f). All data reported here refer to proton NMR. The ﬁeld dependence of T1 is a characteristic feature of the zero dimensionality of the magnetic system. A similar ﬁeld dependence is well known to occur in one dimensional magnetic chains and, to a lesser extent, in two dimensional paramagnets. The fundamental reason for this is that in Heisenberg isotropic paramagnets the time dependence of the spin correlation function has a long time persistence in low dimensions. We will review brieﬂy this result in the following. In the weak-collision approach T1−1 can be expressed as [6]: ij αij J± (ωe ) + βij Jzij (ωL ) , (2) T1−1 ∝ ij

where i,j number the electronic spins, ωe and ωL are the Larmor frequencies of the electron and of the nucleus, respectively, αij and βij are geometrical ij are the transverse and longitudinal spectral densities of the factors and J±,z ij spin ﬂuctuations. In (2), J±,z (ω) can be expressed by the Fourier transform (FT) of the spin correlation function (CF): ij iωt (ω) = Gα dt . (3) J±,z ij (r, t)e An approximate expression for the correlation function can be obtained for an inﬁnite Heisenberg classical chain at high temperature by matching

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(a)

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(d)

(b)

(e)

(c)

(f)

Fig. 2. The external ﬁeld dependence of 1 H-1/T1 at room temperature. (a) Cr8; (b) Fe6 and Fe10; (c) Cu6; (d) V12; (e) V6; (f ) V15. The solid lines in the ﬁgure are calculated results using (7) with a set of parameters listed in Table 1. All samples are in powder form. The two points at very low ﬁeld in (d) for V12 refer to spin-lattice relaxation rates in the rotating frame, T1ρ , measured at 4.7 T

the short time expansion to the long time diﬀusive behavior due to the conservation of the total spin and of its component in the direction of the applied ﬁeld [7]. For temperatures T J/kB the conservation property can be incorporated for spins on a ring by means of a discretized diﬀusion equation to which cyclic boundary conditions are applied. For this model it is found [8, 10] that the auto-correlation function (CF) decays rapidly at short times until it reaches a constant value which depends on the number of spins in the cluster. −1 where ωD The plateau in the CF is reached after a time of the order of 10ωD is the exchange frequency given at the simplest level of approximation by [6]:

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ωD =

1 2πJ [S(S + 1)] 2 , h

(4)

with J the exchange constant between nearest neighbor spins S. The same result is obtained for the CF by using a one-dimensional hopping model on a closed loop [9] or by calculating the spin correlation function with a modecoupling approach [16]. The leveling-oﬀ of the time dependence of the CF at a value approximately given by 1/N with N the number of spins in the cluster is the result of the conservation of the total spin component for an isotropic spin-spin interaction. In practice the anisotropic terms in the spin Hamiltonian will produce a decay of the CF via energy exchange with the “lattice”. A sketch of the time decay of the CF and of the corresponding spectral density is shown in Fig. 3. The decay at long time of the CF has a cut-oﬀ at a time ΓA−1 due to the anisotropic terms in the spin hamiltonian [11, 16]. In the following we will discuss the magnetic ﬁeld dependence of the nuclear relaxation rate at room temperature in terms of a simpliﬁed model which incorporates the theoretical understanding of the spin dynamics in clusters as described above. On the basis of the time dependence of the CF discussed above and sketched in Fig. 3, we model the spectral function in (2) as the sum of two components [12]:

Fig. 3. Sketch of the decay in time of the autocorrelation function of the spins −1 while on magnetic ions. The initial fast decay is characterized by the constant ΓD −1 the slow decay at long time is characterized by the constant ΓA . In the inset the behavior of the Fourier transform i.e. the spectral density is shown

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J ± (ω) = J z (ω) = J‘(ω) +

ω2

ΓA ΓD ΓA = 2 + 2 , 2 2 + ΓA ω + ΓD ω + ΓA2

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(5)

where we assume the same CF for the decay of the transverse (±) and longitudinal (z) components of the spins. The ﬁrst term in (5) represents the Fourier transform (FT) of the initial fast decay of the CF while the second term represents the FT of the decay at long time of the CF due to anisotropic terms in the spin Hamiltonian and we model this second part with a Lorentzian function of width ΓA . From (4) one can estimate that the exchange frequency ωD is of the order of 1013 Hz for typical values of J/kB (10–20 K) and spin values S (1/2–5/2). The spectral function J‘(ω) in (5) reaches a plateau and becomes almost frequency independent for ω < ωD /10. For the magnetic ﬁeld strength used in the experiment (see Fig. 3) both ωn and ωL are smaller than ωD /10. Thus we −1 in (5) where the characteristic frequency will assume J‘(ωn ) = J‘(ωe ) = ΓD ΓD is of the same order of magnitude as ωD /10. Finally, by assuming ωL ΓA in (5), (2) can be rewritten as:

1 ± ΓA 1 1 1 A± 1 z A 2 =K + + A + , (6) T1 2 ωe + ΓA2 2 ΓD ΓD ΓA where the constants A± and Az are averages over all protons in the molecule of the products of the hyperﬁne dipolar tensor components αij and βij , respectively (see (2)). The constant K which has been factored out from the 2 n γe ) = 1.94 × 10−32 (s−2 cm6 ). dipolar tensor coeﬃcients is given by K = (hγ4π The width ΓA of the narrow component in the spectral function represents the frequency which characterizes the exponential time decay of the spin CF in the cluster due to anisotropic terms in the spin Hamiltonian. The experimental data in Figs. 2 (a)–(f) were ﬁtted by using an expression of the form 1 A = + C (ms−1 ) , (7) T1 1 + (H/B)2 where the magnetic ﬁeld H is expressed in Tesla and B = ΓA /γe (Tesla). The ﬁtting parameters for the diﬀerent rings and clusters are summarized in Table 1. The most signiﬁcant parameter in Table 1 is B which measures the cut-oﬀ frequency ΓA of the electronic spin-spin correlation function. Except for the Cr8 case (complete formula: [Cr8 F8 Piv16 ], HPiv=pivalic acid) B is around 1 T corresponding to ΓA ≈ 1011 rad s−1 or hΓA /kB ≈ 1 K. The cut-oﬀ eﬀect is provided, in principle, by any magnetic interaction which does not conserve the total spin components. In practice, such small terms stem from a variety of mechanisms including intracluster dipolar and anisotropic exchange interaction, single ion anisotropies, inter-ring dipolar or exchange interactions etc. [11]. A detailed calculation for Fe6 based on intraring dipolar interaction yielded ΓA = 1.5 × 1011 s−1 [16]. A similar estimate for Cu6 based on known

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Table 1. The ﬁtting parameters in (7) for the diﬀerent molecular rings and clusters (see text for complete chemical formula), in powders Single Molecular Magnet

A (ms−1 )

B (T)

C (ms−1 )

Cr8 (AFM ring – s = 3/2) Fe6 (AFM ring – s = 5/2) Fe10 (AFM ring – s = 5/2) Cu6 (FM ring – s = 1/2) V12 (AFM square – s = 1/2) V6 (AFM triangle – s = 1/2) V15 (AFM ring – s = 1/2)

2.7 2.7 2 0.65 8.5 17 0.13

5 1.5 0.5 0.5 1.6 1 2

1.7 2.6 3.6 0.18 ∼0 2.5 0.28

anisotropic nearest neighbor (exchange and dipolar) contributions to nearest neighbor interactions yielded 1.4×1011 s−1 [16]. Both these results can account very well for the experimental ﬁndings in Table 1. From the comparison of (6) and (7) one has A = KA± /2ΓA and C ≈ KAz /ΓA (since ΓD ΓA ). Thus the order of magnitude of the hyperﬁne constants is A± ≈ Az ≈ 1÷10×1046 cm−6 . Since A± , Az are the product of two dipolar interaction tensor components they are of order of r−6 where r is the distance between a 1 H nucleus and a transition metal local moment. For most of the rings the value of the hyperﬁne constants is consistent with a purely nuclear-electron dipolar interaction. V For V12 (complete formula: (NHEt)3 [VIV 8 V4 As8 O40 (H2 O)]×H2 O) and V6 (complete formula of one variant: Na6 [H4 (V3 L)2 P4 O4 ]×18H2 O), the A± ( Az ) hyperﬁne constant is one order of magnitude higher indicating the presence of an additional contribution probably due to a contact interaction due to the admixing of the hydrogen s wave function with the d wave function of the Vanadium ions. An alternative way to explain the anomalous values (A C) for V6 and particularly for V12 in Table 1 is to go back to (7) and assume that B = ΓA /γN instead of B = ΓA /γe . This implies that the cut-oﬀ frequency ΓA is much less than in other clusters, namely of order of the nuclear Larmor frequency in (6). In this case the value of the constant C in (7) is close to zero in agreement with the experiments as can be seen easily by modifying in the appropriate way the approximate expression (6). It is, however, diﬃcult to justify such small value for the cut-oﬀ frequency in V12 [17].

4 NMR at Intermediate Temperatures (kB T ≈ J ) As the temperature is lowered and it becomes comparable to the magnetic exchange interaction J strong correlations in the ﬂuctuations of the magnetic moments of the molecule start building up. The situation is analogous to macroscopic three-dimensional magnetic systems when the temperature approaches the critical temperature for the transition to long range magnetic

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order. In molecular magnets, as a result of the ﬁnite size of the system the low lying magnetic states are well separated among themselves. Therefore the correlation of the magnetic moments at low temperature has to be viewed as the result of the progressive population of the collective low lying quantum total spin states of the magnetic molecule without any phase transition. In this intermediate temperature range two very interesting situations arise which can be investigated by NMR and relaxation. On one hand, one can follow the evolution of the electronic spin correlation function as the system crosses over from an uncorrelated ﬁnite size paramagnet to a total spin S collective quantum state. On the other hand, one can investigate the nature of the ﬂuctuations of the local electronic spin while the system is in its ground state but at a temperature for which excitations to higher states are important. In the following we will consider examples of the two situations. For the evolution of the spin correlation function we will refer to simple antiferromagnetic rings having a total spin S = 0 ground state. For the thermal ﬂuctuations in the ground state we will refer to ferromagnetic clusters having a total high spin S = 10 ground state. In order to describe the nuclear spin-lattice relaxation in a magnetic system in presence of a correlated spin dynamics it is more convenient to express the nuclear T1 in terms of the q components of the electronic spins [6, 18, 19]: 1 (hγn γe )2 ± = (t) dt cos(ωn t) dq 1/4A± (q)Sq± (t)S−q T1 4π z (t) (8) +Az (q) Sqz (t)S−q or in terms of the response functions by using the ﬂuctuation-dissipation theorem [20]: 1 (hγn γe )2 = kB T 1/4 A± (q)χ± (q)fq± (ωe ) 2 2 T1 4πg µB q Az (q)χz (q)fqz (ωn ) , +

(9)

where γn and γe are the gyromagnetic ratios of the nucleus and of the free electron, respectively, g is Lande’s factor, µB is the Bohr magneton, kB is the Boltzmann constant. The coeﬃcients A± (q) and Az (q) are the Fourier transforms of the spherical components of the product of two dipole-interaction tensors [describing the hyperﬁne coupling of a given proton to the magnetic moments] whereby the symbols ± and z refer to the transverse and longitudinal components of the electron spins with respect to the quantization direction which is here the external magnetic ﬁeld. The collective q-dependent spin correlation function is written as the product of the static response function and the normalized relaxation function fq±,z (ω). At high temperature (kB T J) one can neglect in (9) the q-dependence of the generalized susceptibility χα (q) and of the spectral density function

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fqα (ω). If one assumes an isotropic response function 1/2χ± (q) = χz (q) = χ(q = 0) and one takes a q-independent average value for the dipolar hyperﬁne interaction of the protons with the local moment of the electronic spins, i.e., A± (q) = A⊥ , Az (q) = Az in units of cm−6 , then (9) reduces in this high temperature limit to: 1 (hγn γe )2 = kB T χ(q = 0) 1/2A± J ± (ωe ) + Az J z (ωn ) . 2 2 T1 4πg µB

(10)

If one further assumes for the spectral density of the spin correlation the expressions (5), then (10) reduces to (6) used in the previous paragraph to analyze the ﬁeld dependence of 1/T1 at room temperature. By decreasing the temperature to values such that kB T becomes comparable to J one expects that the nuclear spin-lattice relaxation rate displays a characteristic temperature dependence related to the correlated spin dynamics according to (8) and (9). 4.1 Thermal Fluctuations in AFM Rings Measurements of proton T1 as a function of temperature in a number of antiferromagnetic molecular rings has shown a surprisingly large enhancement of the relaxation rate at low temperatures, resulting in a ﬁeld dependent peak of T1−1 centered at a temperature of the order of the magnetic exchange constant J/kB (see e.g., [12], [21] and [22]). The results are shown for three diﬀerent rings in Fig. 4 (a), in Fig. 4 (b), and in Fig. 4 (c). The systems investigated are: Cr8 (s = 5/2, J ∼ 17.2 K) [15]; Fe6(Na) (s = 5/2, J ∼ 28.2 K) [5]; Fe10 (s = 5/2, J ∼ 13.8 K) [5, 21] (for the last two compounds the respective complete formulas are: [NaFe6 (µ2 -OMe)12 (dbm)6 ]Cl and [Fe(OMe)2 (O2 CCH2 Cl)]10 ). In all three samples the ground state is nonmagnetic with total spin ST otal = 0 and the energies of the lowest lying exchange multiplets can be described to ﬁrst approximation by Lande’s interval rule ES = 2JS(S + 1)/N , where N is the number of spins in the ring [5]. The main feature in the temperature dependence of T1−1 is the strong (a)

(b)

(c)

Fig. 4. Temperature dependence of 1 H-1/T1 under various external magnetic ﬁelds. (a) Cr8, (b) Fe6(Na), and (c) Fe10. The solid line in (a) shows the temperature dependence of χT in arbitrary unit

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Fig. 5. 1/(T1 χ T ) as a function of temperature in log-log plot. (a) Cr8; (b) Fe6(Na); (c) Fe10. The solid lines in the ﬁgures are theoretical calculation (see text). All samples are in powders

enhancement at low T and the presence of a maximum at a temperature T0 for each of the samples investigated. For T < T0 , T1−1 decreases approaching at low T , an exponential drop due to the “condensation” into the Stotal = 0 singlet ground state as discussed later on. It should be noted that the behavior of the relaxation rate is diﬀerent than the behavior of the uniform magnetic susceptibility. The latter, when plotted as χT , shows a continuous decrease with an exponential drop at very low temperature, consistent with what is expected for an AFM system with a singlet ground state. When the T1−1 is plotted together with χT as shown in Fig. 4 (a) for Cr8, one can see that the two quantities are approximately proportional in the whole temperature range except for the region where the peak in T1−1 occurs. In order to emphasize the critical enhancement it is more eﬃcient to plot the relaxation rate divided by χT as shown in Fig. 5 (a), Fig. 5 (b) and Fig. 5 (c) for the cases of Cr8, Fe6 and Fe10, respectively. As shown in these same ﬁgures, the peak in the relaxation rate is depressed by the application of an external magnetic ﬁeld and the position of its maximum moves to higher temperature on increasing the external ﬁeld. One should also note the ﬁeld dependence of the intensity of the maximum which decreases as the ﬁeld is increased. In Fig. 6 we plot (T1−1 )/(χT ) vs. temperature for Fe6(Li), obtained by using two diﬀerent nuclei, i.e., 1 H and 7 Li. The qualitative temperature behavior for temperatures below the peak is very similar for the two nuclei. Finally, in Fig. 7 we plot the renormalized (T1−1 )/(χT ) data as a function of reduced temperature t = T /T0 where T0 is the temperature of the maximum for diﬀerent samples and diﬀerent ﬁelds. The sets of data overlap suggesting that the occurrence of a maximum is a universal eﬀect of antiferromagnetically coupled rings with ST otal = 0 ground state. Recently, it has been shown [23] that the relaxation rate data around the peak for Cr8 can be ﬁtted very well by the simple expression for T1 (7) used in the high temperature approximation, provided that the constant B is assumed to depend on temperature as B = DT n with the exponent n close to the value of three. The same kind of ﬁt appears to work reasonably well also for Fe6(Na) and Fe10. This simple result is very surprising because it implies that the expression (10) which is obtained in the limit of high temperature

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Fig. 6. Plot of renormalized 1/T1 χT as a function of T for Fe6(Li), in two diﬀerent magnetic ﬁelds for two diﬀerent nuclei

Fig. 7. Plot of renormalized 1/T1 χT as a function of T /T0 for diﬀerent magnetic ﬁelds and systems. All samples are in powders

remains valid even at low temperature provided that the spectral density of the ﬂuctuations in (5) is allowed to narrow down and to become peaked at very low frequency. Thus from (10) and (5) one can write down an expression for the nuclear relaxation of the form: 1 B = A + C (ms−1 ) , T1 T χ 1 + (H/B)2

(11)

which can be used to ﬁt the data around the temperature of the peak. The constant C in (11) groups together the terms in (10) which are weakly T dependent, particularly in the region of the peak. The quality of the ﬁt obtained using (11) is exceptionally good for Cr8 as shown in Fig. 5 (a) and moderately good for Fe6(Na) and Fe10 as shown in Figs. 5 (b) and 5 (c) (Note that only the ﬁrst term in (11) in shown in Fig. 5). The constant B in (11) can be identiﬁed either with ΓD /γe or with ΓA /γn . In the ﬁrst case the peak would be due to the slowing down of ΓD as T 3 from a value at room temperature much higher than ωe to a value of the order of ωe in the region of the peak. In the second case the peak would be due to the slowing down of ΓA from a value of the order of ωe at room temperature (see previous paragraph) to a value of the order of ωn in the region of the peak. Measurements

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of proton T1 alone cannot distinguish between these situations. On the other hand, measurements of relaxation on two nuclear species in the same molecular cluster can give us the answer. These measurements have been performed in the AFM ring Fe6(Li), in all identical to Fe6(Na) except for the replacement of Na with Li which induces a change of J from J = 28 K for Fe6(Na) to J = 23 K for Fe6(Li). The results are shown in Fig. 6. It is quite clear that the maxima in T1−1 overlap when the resonance frequency is the same and not the magnetic ﬁeld. This is a direct proof that the constant B in (11) has to be identiﬁed with ΓA /γn , namely that the width of the Lorenzian in (5) becomes of order of ωN at the temperature of the peak. Thus it appears that the peak arises from the spectral density of the longitudinal ﬂuctuations J z (ωn ) in (10). As a direct consequence of the ﬁtting formula (11) (where C can be neglected in the region of the peak) one ﬁnds that the renormalized plot of 1/(T1 T χ) vs t = T /T0 shown in Fig. 7 has the simple form 1/(T1 T χ) 2tn = . (1/(T1 T χ))max 1 + t2n

(12)

In Fig. 8, we show the renormalized plot compared to the function in (12). The ﬁt is remarkably good with n = 3 and no adjustable parameters [23].

Fig. 8. Log-log plot of renormalized 1/(T1 χT ) as a function of T /T0 . The solid line is the theoretical curve (12) with n = 3. The samples are in form of powders

At present there is no theoretical description which explains the critical enhancement manifested in the peak of T1−1 for AFM rings. One possible scenario is that the eﬀect arises from the increase of the antiferromagnetic correlation which generates an enhancement and a slowing down of the spin ﬂuctuations at the staggered wave vector q = π. To see this, we rewrite (9) in an approximate form where we neglect any diﬀerence between the transverse (±) and longitudinal (z) terms and we divide the response functions in a non-critical part which is described by the q = 0 term and in a critical part described by the q = π term:

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0 1 (hγn γe )2 A(π) fπ (ωe ) χ(π) = kB T χ(0) A(0) f0 (ωe ) 1 + 2 2 T1 4πg µB A(0) f0 (ωe ) χ(0) 1 A(π) fπ (ωn ) χ(π) + A(0)f0 (ωn ) 1 + . A(0) f0 (ωn ) χ(0)

(13)

By comparing (13) with (10) which ﬁts so well the experimental data one would deduce that the critical term fπ (ωe )/f0 (ωe ), times χ(π)/χ(0), has a simple Lorenzian form with a correlation frequency ΓD = γe B which displays a critical behavior as T n . This result is diﬃcult to justify on the basis of dynamical scaling arguments similar to the ones used in phase transitions [20]. First principles theoretical calculations of the spin correlation function could give the answer. An alternative scenario is one in which the magnetic critical slowing down plays no relevant role in these ﬁnite size systems. In this case the peak in the nuclear relaxation rate could be simply related to a decrease of the cut-oﬀ frequency ΓA which reﬂects the anisotropy terms in the magnetic Hamiltonian which do not commute with the Heisenberg Haniltonian. These terms determines the electronic spin lattice relaxation via spin-phonon coupling. An important clue for the understanding of the problem is the dependence of the position in temperature of the peak of 1/T1 from the exchange interaction J in the AFM ring. The ﬁts of the experimental data with (11) yields B = DT 3 with a value of D diﬀerent for the three AFM rings. The values of the constant D are plotted in Fig. 9 (a) vs the exchange constant J of the three AFM rings in a log-log plot. In Fig. 9 (b) the same values are plotted vs the energy gap ∆ separating the ground state ST = 0 from the ST = 1 excited state according to Lande’s rule ∆ = 4J/N . In both cases there appears to be a negative power dependence i.e., D ∝ J −α and D ∝ ∆−β with α = 4 ± 0.5 and β = 2.3 ± 0.5.

Fig. 9. (a) A relation between D and J. (b) A relation between D and ∆. The data refer to Cr8, Fe6(Na) and Fe10

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4.2 Thermal Fluctuations of the Magnetization in Nanomagnets: Mn12, Fe8 The cluster containing twelve Mn ions, Mn12, has a high total spin (S = 10) ground state in spite of the large antiferromagnetic coupling between the local Mn moments [14]. This circ*mstance together with the large magneto crystalline anisotropy generates an easy axis nanoferrimagnet with spectacular superparamagnetic eﬀects [24]. Fe8 is similar to Mn12 with regards to the high spin ground state (S = 10) but it has a more complex anisotropy and a lower barrier for the reorientation of the magnetization (about 27 K against 67 K for Mn12) [25].The interest in molecular magnets is largely related to the possibility of observing quantum resonant tunneling of the magnetization (QTM) [26]. For the proper description of the quantum dynamical eﬀects in the high spin ground state one has to take into account the environmental eﬀects represented by spin-phonon coupling, intermolecular magnetic interactions and hyperﬁne interactions with the nuclei. NMR is a very suitable local microscopic probe to investigate the above-mentioned environmental effects. The NMR spectrum at low temperature shows a multiplicity of spectral lines, which yield directly the local hyperﬁne ﬁeld at the nuclear site due to the coupling with the local moment of the magnetic ions in the high total spin ground state. Furthermore, the nuclear spin-lattice relaxation is driven by the ﬂuctuations of the orientation of the total magnetization of the molecular cluster and thus yields information about spin-phonon coupling which limits the lifetime of the m components (i.e., along the easy axis) of the total spin S. At very low temperature (≤ 4 K) both Mn12 and Fe8 are in their magnetic ground state and the magnetization of the molecule is frozen in the time scale of an NMR experiment. Thus the nuclei experience a large static internal ﬁeld. This allows to detect 55 Mn NMR in zero external ﬁeld in Mn12 and 57 Fe NMR in zero external ﬁeld in Fe8. For protons the internal ﬁeld is small, being generated by the proton-Mn (Fe) moment dipolar interaction and thus the NMR in zero ﬁeld is weak and can be observed only at low frequency over a broad frequency interval (2–4 MHz) due to the presence of many inequivalent proton sites in the molecule. On the other hand, the proton NMR in an external magnetic ﬁeld shows a broad structured spectrum with a ﬁeld independent shift of the lines of order of the internal ﬁeld. For the analysis of the hyperﬁne ﬁeld at the proton site and at the deuteron site in deuterated Fe8, as well as of the temperature dependence of the spectrum, we refer to [27]. At intermediate temperatures (4–30 K) the magnetization of the Mn12 and Fe8 molecules are subject to large and fast ﬂuctuations due to the thermal populations of the higher energy magnetic quantum states m = ±9, ±8, ±7... separated by crystal ﬁeld anisotropy within the total spin ground state S = 10. As a result the zero ﬁeld NMR lines progressively disappear on increasing the temperature and the spin-lattice relaxation becomes very fast.

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We have utilized this circ*mstance to investigate the thermal ﬂuctuations of the magnetization by measuring the temperature and ﬁeld dependence of T1−1 . The study was possible for proton NMR [27] and Muon Spin Rotation (µSR) [28] in Fe8 over the whole temperature range. In Mn12 the proton relaxation becomes too fast in the temperature range (4–20 K) and we had to measure the muon longitudinal relaxation by µSR technique [29]. For 55 Mn NMR in Mn12 the signal can be detected only up to about 4 K and for 57 Fe NMR the signal is lost at about 1.8 K because of a very short T2 . The proton T1 vs temperature for Fe8 is shown in Fig. 10. The proton T1 in Mn12 is shown in Fig. 11 vs magnetic ﬁeld at low temperature below 4.2 K. Regarding the temperature dependence in the range 4–30 K we refer to the µSR relaxation rate since the proton NMR cannot be detected due to the short T1 . The results for the longitudinal muon relaxation rate (a parameter analogous to 1/T1 ) are shown in Fig. 14 for diﬀerent applied longitudinal ﬁelds in Mn12 powders. The 55 Mn T1 as a function of temperature is shown in Fig. 12 in the narrow temperature range in which the signal is observable in zero external ﬁeld. Finally, the ﬁeld dependence of the 55 Mn T1 is shown in Fig. 13.

Fig. 10. Temperature dependence of 1 H-1/T1 in Fe8 “oriented” powders at diﬀerent magnetic ﬁelds, parallel to the easy-axis. The lines are theoretical estimation calculated by (18) with a set of parameters described in the text

Fig. 11. The external ﬁeld (parallel to the easy-axis) dependence of 1 H-1/T1 in Mn12 “oriented” powders measured at T = 4.2 and 3 K. The curves are ﬁtting results obtained by (18)

All the relaxation rate data shown here can be interpreted in terms of a simple model of nuclear spin-lattice relaxation via a direct process driven by the ﬂuctuations of the molecular magnetization. Since the ﬂuctuations of the magnetization are a consequence of the ﬁnite lifetime of the m magnetic substates due to spin-phonon interactions, one can obtain from the ﬁt of the NMR data the spin-phonon coupling constant, a quantity not easily derived by other techniques. We give in the following a brief account of the model.

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Fig. 12. Temperature dependence of 55 Mn-1/T1 for each Mn site in Mn12 oriented powders. Solid curves are ﬁtting curves according to (18). The inset shows the temperature dependence of 1/T1 for the three peaks renormalized at the same value at T = 1.4 K

Fig. 13. External ﬁeld (parallel to the easy-axis) dependence of 55 Mn-1/T1 for each Mn site in Mn12 oriented powders measured at T = 1.4 K. The solid lines are calculated results by (18)

Fig. 14. Muon longitudinal relaxation rate for diﬀerent applied longitudinal ﬁelds in Mn12 powders, as a function of temperature. The lines are ﬁts to the data following (18)

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We start from a semi-classical approach by expressing the nuclear spinlattice relaxation rate (NSLR) in terms of the correlation function of the transverse component h± (t) of the time dependent transverse hyperﬁne ﬁeld at the proton site: 1 1 2 = γN (14) h± (t) h± (0)eiωL t dt , T1 2 with γN the nuclear gyromagnetic ratio and ωL the Larmor frequency. The fundamental assumption in this model is that, although the hyperﬁne ﬁeld h± (t) is due to the interaction of the protons with the local moments of the Fe3+ (Mn) ions, its time dependence is the same as the time dependence of the orientation of the total magnetization of the molecular cluster. Thus one can use for the correlation function of h± (t) an exponential function with the correlation time τm determined by the lifetime broadening of the m sublevels: h± (t) h± (0) =

−10

∆h2± m=+10

t − τm

− kEmT B

(15)

where Z is the partition function. The term ∆h2± is the average square of the change of the hyperﬁne ﬁeld when the magnetization of the molecule changes orientation (i.e., ∆m = ±1, ±2). For sake of simplicity we assume an average value independent of the ∆m transition considered. The assumption should be acceptable since in the temperature range investigated most of the ﬂuctuations occur between few m states close to the m = 10 ground state. The lifetime τm for each individual m state is determined by the probability of a transition from m to m ± 1, Wm,m±1 , plus the probability for a transition with ∆m = ±2, Wm,m±2 , i.e, 1 = Wm→m+1 + Wm→m−1 + Wm→m+2 + Wm→m−2 . τm

(16)

The transition probabilities are due to spin-phonon interaction and can be expressed in terms of the energy level diﬀerences as [30, 31]: Wm→m±1 = W±1 = C s±1

(Em±1 − Em ) , eβ(Em±1 −Em ) − 1 (17)

Wm→m±2 = W±2 = 1.06 C s±2

(Em±2 − Em ) , eβ(Em±2 −Em ) − 1

where s±1 = (s ∓ m)(s ± m + 1)(2m ± 1)2 , s±2 = (s ∓ m)(s ± m + 1)(s ∓ m − 1)(s ± m + 2) .

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The spin-phonon parameter C in (17) is given by C = D2 /(12πρv 5 h4 ) with ρ the mass density and v the sound velocity and D a constant related to the crystal ﬁeld anisotropy. Finally we can write for the NSLR: −10 − m A τm e k B T 1 = 2 τ2 , T1 Z m=+10 1 + ωL m E

(18)

2 where A = γN ∆h2± . The energy levels Em in the above equations can be obtained from the Hamiltonian of the molecules expressed in terms of the total spin S:

H = −DSz2 − BSz4 + E(Sx2 − Sy2 ) + gµB Sz H .

(19)

For Mn12 one has D = 0.55 K, B = 1.2 × 10−3 K, and E = 0 while for Fe8 one has D = 0.27 K, B = 0 and E = 0.046 K. The experimental data both as a function of temperature at diﬀerent ﬁelds (Figs. 10, 14 and 12) and as a function of ﬁeld at a ﬁxed temperature (Figs. 11 and 13) were ﬁtted to (18) by using (17) and (19) and treating A and C as adjustable parameters. From the ﬁt of the proton relaxation in Fe8 one obtains the parameters: C = 31 Hz K−3 and A = 1.02 × 1012 (rad2 s−2 ). The ﬁt of the proton relaxation data in Mn12 (as well as the µSR relaxation data, see Fig. 10) was obtained with a somewhat larger value of C and a smaller value of A = 0.45 × 1012 (rad2 s−2 ). The coupling constant A represents the average hyperﬁne interaction squared between protons and transition metal magnetic moments. The value found for Fe8 is larger than the value obtained in Mn12 indicating that in Fe8 the protons are subject to non-negligible hyperﬁne interaction due to contact terms in addition to the dipolar interaction. From the knowledge of the spin-phonon coupling parameter C one can estimate the lifetime of the m sublevels by using (16) and (17). For a detailed discussion of the ﬁtting parameters we refer to the original papers. Finally, the 55 Mn relaxation data were ﬁtted with reasonable values of C and A, although for this case the two parameters cannot be determined independently. It should be mentioned that 55 Mn relaxation data have also been reported by Goto et al. [32] and analyzed with a model based on a two-state pulse ﬂuctuation corresponding to the hyperﬁne ﬁelds in the two lowest energy levels m = ±10, ±9. This model is indeed a better model for the very low temperature region (< 4.2 K) while our model has the advantage to be applicable even at higher temperatures where the higher energy levels become populated. A diﬀerent approach has been proposed by Yamamoto and collaborator [33], based on a spin-wave approximation for the description of the energy levels and a Raman two-magnon scattering mechanism to describe the 55 Mn T1−1 data. Since both the phenomenological models based on a direct relaxation process [49, 32] and the spin-wave model [33] appear to ﬁt the low temperature 55 Mn relaxation data, a connection between the two approaches should be established.

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5 NMR at Low Temperatures (kB T J ) When the temperature is much lower than the exchange interaction among magnetic moments in the molecule, i.e., T J/kB , the system is mostly in its collective quantum ground state characterized by a total spin S. We have to distinguish the two cases of singlet ground state S = 0 and of high spin ground state S > 0. In the ﬁrst case which pertains to AFM rings the residual weak magnetism of the molecule at low temperature is due to the thermal population of the ﬁrst excited state which is normally a triplet S = 1 state. In the second case the molecule at low temperature behaves like a nanomagnet with a spontaneous magnetization proportional to the value of the ground state spin S. If there is no anisotropy the molecule acts like a soft nanomagnet with a magnetization which can be aligned by an external magnetic ﬁeld with no hysteresis in the magnetization cycle. This is the case of Cu6 FM ring which is discussed in Sect. 6. In presence of an anisotropy the molecule behaves as a hard nanomagnet with hysteresis in the magnetization cycle. However, since there is no long range order each molecule acts as a superparamagnetic particle. At temperatures much lower than the anisotropy barrier the relaxation of the magnetization can be dominated by quantum tunneling. NMR can give interesting information in this low temperature regime. We will review the main results treating separately the case of a non-magnetic ground state (AFM rings) and the case of a magnetic ground state (Mn12 and Fe8). 5.1 Energy Gap of AFM Rings in the Magnetic Ground State AFM rings such as Fe10, Fe6, Cr8 already discussed in Sect. 3, are characterized by a single nearest neighbor exchange interaction J which generates a singlet ground state of total spin S = 0 separated by an energy gap ∆ from the ﬁrst excited triplet state S = 1. From simple Lande’s interval rule one has E(S) = 2J/N S(S + 1). Thus in absence of crystal ﬁeld anisotropy the gap is ∆ = 4J/N where N is the number of magnetic moments in the ring. In presence of crystal ﬁeld anisotropy with axial symmetry characterized by the parameter D (see (19)) the gap between S = 0 and S = 1, M = ±1 is 4J/N + D/3 for the case of positive axial anisotropy. In Table 2 we summarize the magnetic parameters for the above mentioned three rings, for the Cu8 ring and for the cluster V12 which can be assimilated to a square of V4+ magnetic ions. From the inspection of the gap value in Table 2 it appears that below liquid helium temperature the molecular magnets will be mostly in the non-magnetic ground state. In Cu8 the gap is so big that the ring is in its singlet ground state even up to room temperature. As a result the magnetic susceptibility goes to zero and the nuclear relaxation rate becomes also very small. Measurements of T1−1 in this low temperature range can yield interesting information about the energy gap ∆ and about the quantum ﬂuctuations in the S = 0 ground state.

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Table 2. Comparison of single ion spin values, exchange coupling constant (J) and energy gap ∆ (K) between the ground state and ﬁrst excited state at H = 0 for diﬀerent AFM rings AFM Ring

Magnetic Ion Spin

J (K)

∆ (K) – No Anisotropy

5/2 5/2 3/2 1/2 1/2

13.8 28 17 ≈ 1000 17.2

5.5 18.7 8.5 ≈ 500 17.2

Fe10 Fe6(Na) Trigonal Cr8 Cu8 V12

The electronic spin correlation function entering the expression of T1−1 (see (2) and (3)) is deﬁned as: n|Siα |l e−βEn +iEn t/−iEl t/ l|Sjα |n , (20) Gα ij (r, t) = n

where n,l number the eigenstates, En , El are the energy eigenvalues, β = α are the spin operators of the ith (j th ) spin and α = x, y, z. For a 1/kB T , Si(j) ﬁnite system the energy diﬀerence between eigenstates is very large compared to the Larmor Zeeman energy. Therefore for a direct process (see (3)) only the matrix elements with n = l in (20) need to be considered and a broadening of the energy levels has to be introduced in order to fulﬁll energy conservation (i.e., in order to have some spectral density of the ﬂuctuations at ωe and ωL in (2)). It should be noted that an alternative approach is to describe the nuclear relaxation in terms of a Raman process [33]. Even in this case one needs to have a broadening of the levels or a spin wave band. We have not explored this possibility since the direct relaxation process appears to be able to explain the experimental data. As a consequence of the presence of the Boltzmann factors in (20) the NSLR at very low temperature will be simply proportional to the population of the excited states. For temperatures less than the energy gap ∆ one has approximately: e−∆/kB T , (21) T1−1 = A 1 + 3e−∆/kB T where A is a ﬁtting constant which contains the hyperﬁne coupling constants and the matrix elements in (20). The gap ∆ depends on the applied magnetic ﬁeld as a result of the Zeeman splitting of the excited triplet state of the AFM rings. The validity of the simple prediction of (21) was tested in the AFM rings described in Table 2. The dependences of T1−1 vs T in Cr8, Fe6 and V12 are shown in Figs. 15 (a), 15 (b) and 15 (c), respectively. For these systems the condition T ∆/kB can be met in the temperature range (1.4–4.2 K). For Fe10 one has to perform the measurements at lower temperature since the gap is smaller [34].

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(a)

(b)

(c) Fig. 15. Semilog plot of 1 H-1/T1 as a function of 1/T (Cr8, (a)) and 1000/T (Fe6(Na), (b), and V12, (c)). The solid lines are calculated results using (21). All samples are in the form of powders

As seen in the ﬁgures the temperature dependence of T1−1 appears to be ﬁtted well by (21) in the low T limit for Cr8 and Fe6 with values of the gap ∆ in qualitative agreement with the values of the gap in Table 2. Also the decrease of the measured gap with increasing ﬁeld in Cr8 is in agreement with the notion that the application of an external ﬁeld should close the gap. It should be noted that in presence of crystal ﬁeld anisotropy the value of the gap is diﬀerent from the one in Table 2 and, in presence of a magnetic ﬁeld, the gap depends on the angle between H and the symmetry axis of the molecule. Thus measurements in a powder sample cannot be expected to give better quantitative agreement than found here. The case of V12 is particularly interesting. In fact the low temperature proton relaxation data (see Fig. 15 (c)) can be ﬁtted by the sum of a term described by (18) with a gap value in agreement with Table 2 and of a constant term. In other words, it appears that the relaxation rate saturates at low T instead of decreasing as expected for the “condensation” of the AFM ring in the singlet ground state. There are several reasons to exclude the possibility that the constant T1−1 at low T in V12 is due to paramagnetic impurities [35]. Then one can speculate that the low T contribution to T1−1 is due to quantum ﬂuctuations in the singlet ground state for the quantum s = 1/2 system (V12), ﬂuctuations not evident for the classical s = 3/2 (Cr8) and s = 5/2 (Fe6) rings.

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The case of Cu8 appears to be the ideal case for the study of the energy gap and the singlet ground state since the gap is larger than room temperature. The Cu8 ring has been investigated both by proton NMR and 63,65 Cu NMR and NQR [36]. The 63,65 Cu spectrum is complicated by the presence of four non equivalent Cu sites. The 63,65 Cu NQR spectra are composed of four separate lines for each isotope plus an additional line (probably due to impurities), spanning over the frequency range 16–22 MHz. The 63,65 Cu NMR spectrum in high ﬁeld displays the powder pattern of a central line transition broadened by second order quadrupole eﬀects (diﬀerent for each of the four inequivalent sites) and anisotropic paramagnetic shift. On the other hand, the proton NMR line is narrow (≤ 40 KHz) and ﬁeld independent. The nuclear spin-lattice relaxation rate (NSLR) decreases exponentially on decreasing temperature for all nuclei investigated as shown in Fig. 16 and Fig. 17. In both ﬁgures we have drawn the curves corresponding to (21) with the gap value given in Table 2 and derived from susceptibility measurements. One can see that both for 1 H NMR and 63,65 Cu NQR-NMR the experimental NSLR deviates from the exponential behavior at relatively high temperature in a manner similar to the case of V12 discussed above. However, in the case of Cu8 it appears more diﬃcult to rule out the eﬀect of paramagnetic defects and thus it is still to be proved that the NSLR in rings of spin 1/2 in the singlet ground state is driven by quantum ﬂuctuations.

Fig. 16. Semilog plot of 63 Cu-1/T1 as a function of 1000/T in Cu8 powders

Fig. 17. Semilog plot of 1 H-1/T1 as a function of 1000/T in Cu8 powders

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5.2 Level Crossing in AFM Rings In AFM ring of the kind discussed above with S = 0 singlet ground state and S = 1 triplet excited state, an external magnetic ﬁeld H removes the residual Kramers degeneracy of the triplet state and induces multiple level-crossings (LCs) at speciﬁc magnetic ﬁeld values Hci , whereupon the ground state of the molecule changes from S = 0 to S = 1 (Hc1 ), from S = 1 to S = 2 (Hc2 ), and so on. Because of magnetic anisotropy, the values of Hci depend on the angle between H and the molecular axis z [15, 37]. The situation of near-degeneracy of the magnetic levels near the LC ﬁelds, and particularly the situation of level repulsion or anticrossing (LAC), is favorable to the observation of quantum tunneling and quantum coherence [38]. A situation of LC is also found between singlet and triplet states in 1D gapped quantum magnets [39], but the physical context and the continuum of excited states make the situation not comparable to that in ﬁnite-size magnets. For the spin dynamics near LC, a crucial issue is the role played by the coupling between magnetic molecular levels and the environment such as phonons and/or nuclear spins [40]. Essential information on this problem is accessed through measurements of the nuclear spin-lattice relaxation rate 1/T1 since the nuclei probe the ﬂuctuations of the local ﬁeld induced at the nuclear site by the magnetic moments of the transition ions. The spin dynamics at LC was studied by means of 1 H NMR in Fe10 powder sample [41], Fe6:Li(BPh4) single crystal [42] and Cr8 single crystal [43] close to Hc1 and Hc2 . We will review in the following the main results and conclusions derived in the above mentioned studies. One crucial issue in the study of LC in AFM rings is the structure of the magnetic levels at the critical ﬁeld where the S = 0 state becomes degenerate with the S = 1, M = 1 state (ﬁrst level crossing), and similarly for other LC’s. If the magnetic Hamiltonian does not contain terms that admix the two degenerate levels, one can have in principle a true LC. If on the other hand the Hamiltonian contains terms which strongly admix the levels, one expects a gap at the critical ﬁeld resulting in what we call level anticrossing (LAC). The spin Hamiltonian describing an N -membered ring-like spin topology can be written as: si · si+1 + U (si ) + Ui,j (si, sj ) + gµB H si , (22) H=J i

i>j

where i and j run from 1 to N . The ﬁrst term is the nearest-neighbor Heisenberg exchange interaction, the second and third terms represent the crystal-ﬁeld anisotropies and the anisotropic spin-spin interactions (including eventually a Dzyaloshinski-Moriya (DM) term Cij si ⊗ sj ), respectively, and the last one is the Zeeman term. In (22), sN +1 = s1 and the exchange interactions are supposed to be limited to nearest neighbors. Magnetic torque and speciﬁc heat measurements have been performed as a function of magnetic ﬁeld in molecular rings [15, 37, 42]. The former is a powerful tool to locate level crossings, which lead to abrupt variations of the torque signal

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at low temperature, and to determine the ZFS parameters (DS ) of excited spin states. Speciﬁc heat measurements, when performed at suﬃciently low temperature, are in principle able to distinguish a true LC from a LAC [42]. Particularly interesting is the possibility to extract the value of the energy gap at the anticrossing ﬁeld, which is directly proportional to the matrix element connecting the two states involved. Measurements of proton spin-lattice relaxation as a function of the magnetic ﬁeld at ﬁxed T are shown in Figs. 18, 19 and 20 for the three AFM rings Fe10, Fe6(Li) and Cr8. In all three cases a strong enhancement of T1−1 is observed in correspondence to critical ﬁeld values for LC (or LAC). The enhancement is clearly related either to cross relaxation eﬀects or to magnetization ﬂuctuations due to the almost degeneracy of the crossing magnetic levels as will be discussed further on. In Fe10 the peaks of T1−1 are at 4.7, 9.6 and 14 T. The ﬁeld dependence of the gap in Fe10 leading to the ﬁrst LC for H parallel to the z axis should be ∆(H) = ∆(0) + D1 /3 − gµB H, where the anisotropy parameter D1 = 3.2 K and ∆(0) = 5.5 K from Table 2. The gap closes for H = 6.3 K/1.33 K/T ≈ 4.7 T in excellent agreement with the NMR data. Since the measurements in Fe10 were performed in a powder sample

Fig. 18. Magnetic ﬁeld dependence of 1 H-1/T1 in Fe10 powders measured at low temperature T ≈ 1.5 K

Fig. 19. Magnetic ﬁeld dependence of 1 H-1/T1 in Fe6(Li) single crystal measured at T = 1.5 and 3 K, for θ ≈ 20◦ (θ = angle between H and the molecular axis)

Fig. 20. Magnetic ﬁeld dependence of 1 H-1/T1 in Cr8 single crystal measured at T = 1.5 K, for two slightly diﬀerent values of the angle θ between H and the molecular axis

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there is a distribution of LC ﬁelds which generate a broadening of the T1−1 vs H curve. In Fe6(Li) single crystal the maximum of T1−1 can be observed at 11.7 T in excellent agreement with torque measurements for the same angle θ ∼ 25◦ between the z axis and H. In Cr8 two peaks in T1−1 could be observed at 6.85 T and 13.95 T. The measurements were performed in single crystal but the orientation of the crystal is not deﬁned. From the knowledge of D1 = 2.3 K and ∆(0) = 8.5 in Table 2 one can deduce that the NMR experiments were performed at θ ∼ 65◦ (1st crossing) and θ ∼ 50◦ (2nd crossing). Having established that the peaks in the proton T1−1 vs H curve are indeed evidence for LC (or LAC) it remains to be seen what kind of information one can obtain about the spin dynamics at level crossing from NMR measurements. The processes involved in nuclear spin-lattice relaxation close to LC or LAC are currently not well understood. Ultimately, the challenge is to be able to distinguish between thermal and quantum spin ﬂuctuations in the molecule. In the presence of true LC or very small LAC, there are two magnetic ﬁeld values for which the level separation ∆1 = gµB |Hci − H| matches the 1 H Zeeman energy hνL . At these two ﬁeld values, one expects a large enhancement of 1/T1 due to a direct exchange of energy between electronic and nuclear reservoirs, i.e., a cross-relaxation [18, 41]. The cross-relaxation depends on the relative values of the nuclear and electron spin-lattice relaxation rates T1 and T1el respectively, which limit the energy transfer to the “lattice”, and on the rate of internal energy exchange between the nuclear Zeeman reservoir and the molecular magnet reservoir. If T1el is much shorter than the nuclear T1 and the exchange time between reservoirs is very short, one expects a large T -independent double-peak of T1−1 close to LC. On the other hand, if T1el ≥ T1 , the cross-relaxation may not be eﬀective. However, one should keep in mind that in practice this situation can only occur in the case of ultra small LACs (as small as the nuclear Zeeman energy, i.e., of the order of mK) and that the separation between the two peaks of 1/T1 (H) would also be as small as the nuclear Zeeman energy. Furthermore, to be experimentally observable as two separate peaks with ﬁnite width, this mechanism would require some broadening of the levels. If the LAC is large (with respect to nuclear energies), the nuclear relaxation should have dominant contributions from indirect processes (e.g., Raman-like, Orbach-like [18, 40]). The width of the peak and its T -dependence may be inﬂuenced by many parameters, and should certainly be diﬀerent from the above-cited case of cross-relaxation. To obtain a phenomenological expression for 1/T1 near the LC ﬁelds, we suppose that the ﬂuctuations of the magnetization between two adjacent magnetic states will drive the nuclear relaxation, that can be written as: 1 = A2 J(ωL ) , T1

(23)

where J(ωL ) is the spectral density of spin ﬂuctuations at the nuclear Larmor frequency ωL and A is an average hyperﬁne coupling constant. In a phenomenological model where the spin-spin correlation function is supposed to decay

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exponentially with time, we assume 1 A2 Γ = 2 , 2 T1 (T, H) Γ + (hωL − ∆(H))

(24)

where Γ is a temperature-dependent damping factor associated with level broadening. In the case of nuclear relaxation induced by a purely quantum process, one does not expect any T dependence of T1 . The relaxation can still be described by an expression similar to (24), with a T -independent Γ , now having a diﬀerent meaning. Near the LC ﬁelds we can write 1/2 2 , ∆(H) = [gµB (Hci − H)] + ∆2i

(25)

where ∆i is the temperature independent “gap” at the anticrossing. The use of (24) near the LC condition is justiﬁed by the fact that the next excited state is at least several Kelvin higher in energy. Three diﬀerent cases regarding the peak of T1−1 vs H can be distinguished: a) for ∆i Γ (T ), the width of the peak is determined by ∆i while the intensity depends mainly on Γ (T ). Qualitatively, for H = Hci one has 1/T1 (T ) ∝ Γ (T ). This situation corresponds to a well-identiﬁed LAC. b) for ∆i Γ (T ), both the width and the intensity of the peak are determined by Γ (T ), and for H = Hci one has 1/T1 (T ) ∝ 1/Γ (T ). In this situation, it is not possible to identify a LAC, either because there is a true LC or because ∆i is too small to inﬂuence NMR relaxation. c) for ∆i ∼ Γ (T ), the width of the peak is due to both ∆i and Γ (T ). Note that in general a dependence T1−1 (H = Hci ) vs. T shows a maximum if Γ (T ) = |∆i − hωL |. Let us now focus on the experimental results. As previously noted, in Fe10 three broad peaks were observed in the powder sample at the ﬁrst three LC ﬁelds (see Fig. 18). In the range 1.5 < T < 4.2 K, it was shown that 1/T1 (T ) = const., thus suggesting that Γ is temperature independent. Out of the crossing ﬁeld values, 1/T1 has an exponential behavior [41] (as also shown by the authors of [34]). Since Fe10 data can be ﬁtted by (24)) assuming ∆1 = 0, the physical situation could correspond to the case discussed above at point b), i.e., a true LC or small LAC. First principles T1−1 calculations conﬁrmed this scenario [44]. In the case of Fe6(Li) the peak of the dependence 1/T1 vs H (see Fig. 19) is quite broad in spite of the fact that the measurements are done in single crystal. The NMR data appear to conﬁrm the speciﬁc heat results which are suggestive of a LAC at Hc1 = 11.8 T with an anticrossing gap ∆1 /kB = 0.86 K [42]. In fact, the proton relaxation data can be ﬁtted well near the maximum by using (24) with the above values for the parameters and with A = 8.5 × 107 rad s−1 . The two curves in Fig. 19 at 1.5 K and 3 K are well reproduced by assuming a damping factor varying quadratically with temperature Γ ∼ T 2

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[42]. We note that the T -dependent broadening Γ of the magnetic levels is small (Γ = 0.26 T at 3 K) and thus it aﬀects only the magnitude of 1/T1 , while the width of the 1/T1 peak is determined by the anticrossing gap ∆1 . This corresponds to the situation of the above-cited case (a). The observation of a large LAC has raised the problem of explaining its occurrence in terms of antisymmetric interaction in (22) [42, 45, 46]. In Cr8, two LCs are observed in the NMR (see Fig. 20). Both peaks show a high ﬁeld shoulder whose origin is not yet understood and will be disregarded in the discussion which follows. The peaks of 1/T1 are narrower than in Fe6:Li(BPh4) by a factor ∼ 2.5 for the ﬁrst crossing and by a factor ∼ 4 for the second crossing. Actually, the dipolar or hyperﬁne interaction extracted from 1 H NMR spectra were estimated to yield a broadening of about 0.1÷0.2 T, not far from the measured width of the peak of 1/T1 on the second crossing. This suggests that LAC eﬀects are smaller in Cr8 than in Fe6, or that the damping factor in Cr8 is comparable to or greater than the energy gap at the anticrossing. This experimental observation seems to be in qualitative agreement with speciﬁc heat data [47] that indicate anticrossing gap values much smaller than in Fe6(Li). However, from C(H) data it was not possible to obtain more than an upper limit for ∆1 (∼ 0.2 K). As a consequence, one has to determine all three parameters A, ∆1 and Γ in (24) from T1 data alone. This requires extensive T1 measurements as a function of H and T , currently not available. We note that the above discussion in terms of the phenomenological expression (24), is strictly valid only in the very vicinity of the LC critical ﬁelds. The ﬁeld dependence at a ﬁxed very low temperature value over a wide range of H on both sides of Hci ’s follows a behavior of the type 1/T1 = A(H, T ) exp(−∆(H)/kB T ). The exponential dependence of 1/T1 with a ﬁeld-dependent gap parameter ∆(H) reﬂects the same nuclear spin-lattice direct relaxation process described by (21) in the previous section. 5.3 Local Spin Conﬁguration in the Ground State of Nanomagnets Mn12, Fe8 As mentioned before, at very low temperature (≤ 4 K ) both Mn12 and Fe8 are in their magnetic ground state and the magnetization of the molecule is frozen in the time scale of an NMR experiment leading to the possibility of observing zero ﬁeld NMR. The 55 Mn NMR spectrum in zero external ﬁeld in Mn12 was ﬁrst observed by Goto et al. [48]. A detailed study was later reported by us [49]. The 55 Mn NMR spectrum is shown in Fig. 21. The narrow low frequency line originates from the Mn4+ ions while the two broader lines are from the Mn3+ ions whereby the broadening of the latter two lines is of quadrupole origin [49, 50]. The 57 Fe NMR signal in zero external ﬁeld was detected in isotopically enriched Fe8 [51, 52] and the complete spectrum is shown in Fig. 22. The eight narrow lines correspond to the eight non equivalent Fe sites in the molecule. The identiﬁcation of the eight lines with the corresponding Fe sites was possible from measurements as a function of the

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(a)

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Intensity (arb. unit)

Spin echo intensity (arb. unit)

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(c)

225

250 275 300 325 350

375

Frequency (MHz)

0.7 T

400 62

Frequency (MHz)

Fig. 21. 55 Mn-NMR spectra in Mn12 oriented powders measured at T = 1.5 K under zero magnetic ﬁeld

Fig. 22. 57 Fe-NMR spectra on Fe8 powders measured at T = 1.5 K. (a) H = 0.5 and 0.7 T parallel to the easy axis; (b) H = 0; (c) H = 0.7 T perpendicular to the easy-axis

applied magnetic ﬁeld [51, 52]. For protons the internal ﬁeld is small and the NMR in zero ﬁeld is weak and can be observed only at low frequency over a broad frequency interval (2–4 MHz) due to the presence of many inequivalent proton sites in the molecule. We have observed the signal in zero ﬁeld but the complete spectrum has never been published. On the other hand the proton NMR in an external magnetic ﬁeld can be observed easily. It shows a broad structured spectrum with a ﬁeld independent shift of the lines of order of the internal ﬁeld. The proton spectrum in Fe8 at 1.5 K is shown in Fig. 23 [27, 53]. The proton spectrum in Mn12 at low temperature is shown in Fig. 24 [60]. Further details about the zero ﬁeld NMR and about the analysis of the hyperﬁne ﬁeld at the diﬀerent nuclear sites are given in [49] and [50]. One important issue in molecular nanomagnets is the knowledge of the local spin conﬁguration corresponding to the collective quantum state described by the total spin i.e., S = 10 for both Mn12 and Fe8. This issue has been addressed very successfully by NMR and we will summarize the results for both Mn12 and Fe8 in the following. Mn12 Figure 25 shows the external magnetic ﬁeld dependence of the 55 Mn resonance frequencies of the three signals in the spectrum, with the magnetic ﬁeld applied along the easy-axis in Mn12. With increasing parallel ﬁeld, P1 shifts to higher frequency while the other two peaks P2 and P3 (pertaining to Mn3+ ions) shift to lower frequency. Since the resonance frequency is proportional to the

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Fig. 23. 1 H-NMR spectrum of Fe8 oriented powders at 36.8 MHz (Hres = 0.864 T) and T = 1.4 K. In the inset the shifts of the lines at diﬀerent ﬁelds H1...4,p as a function of Hres are shown to be constant. Here H is parallel to the easy-axis

Fig. 25. Parallel ﬁeld dependence of resonance frequency for each 55 MnNMR peak at T = 1.5 K

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Mn12 H-NMR 160.5 MHz

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Fig. 24. 1 H-NMR spectra in Mn12 oriented powders measured at 160.5 MHz for three diﬀerent temperatures. Here H is parallel to the easy-axis

Fig. 26. Schematic structure of the magnetic core of Mn12 and orientation of the Mn moments in the ground state

vector sum of the internal ﬁeld (H int ) and the external ﬁeld (H ext ) i.e., ωR = γN |H int + H ext |, this result indicates that the direction of the internal ﬁeld at the Mn sites for Mn3+ ions is opposite to that for Mn4+ ions. Since H int originates mainly from the core-polarization [49], H int is negative and the direction of the internal ﬁelds at nuclear sites is opposite to that of the Mn spin moment. Thus one can conclude that spin direction of Mn4+ ions is antiparallel to the external ﬁeld, while that of Mn3+ ions is parallel to the external ﬁeld, corresponding to the standard spin structure of magnetic core of Mn12 cluster (see Fig. 26) [54]. In the case when the external magnetic ﬁeld is applied perpendicular to the easy axis (which is the common axis of the oriented powder), the ﬁeld dependence of the resonance frequencies is the one shown in Fig. 27. As described

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Frequency (MHz)

400 350 300 250 200 150 100 50 0.0

2.5

5.0

7.5

10.0 12.5 15.0 17.5

Transverse field (T)

Fig. 27. Transverse ﬁeld dependence of resonance frequency for each peak at T = 1.5 K

Mn-NMR

above, the resonance frequency is proportional to the eﬀective internal ﬁeld at the nuclear site, which is the vector sum of H int due to spin moments and H ext due to the external ﬁeld i.e., |H eff | = |H int + H ext |. Thus the opposite ﬁeld dependence of |H eff | for Mn4+ and Mn3+ ions indicates that the direction of Mn4+ spin moments is antiparallel to that of Mn3+ spin moments. This leads to the conclusion that the individual spin moments of both Mn4+ and Mn3+ ions do not cant independently along the direction of the transverse ﬁeld but rather rotate rigidly maintaining the same relative spin conﬁguration. For a detailed quantitative analysis of the results we refer to the original paper [55]. Fe8

Frequency (MHz)

A similar study has been performed in Fe8 at 1.5 K where the nanomagnet is in its ground state. The ﬁeld dependence of the eight 57 Fe resonance frequencies as a function of a magnetic ﬁeld applied along the main easy axis is shown in Fig. 28. An analysis analogous to the one performed in Mn12 leads to the conclusion that the internal spin conﬁguration of the ground state is as

72 68 64 60 56 0

Parallel field (T)

Fig. 28. Parallel ﬁeld dependence of resonance frequency for each at T = 1.5 K

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Frequency (MHz)

H//medium axis

57Fe-NMR

T=1.5 K

Transverse field (T) Fig. 29. Structure of Fe8 and orientation of the Fe moments in the ground state

Fig. 30. Transverse ﬁeld dependence of resonance frequency for each 57 FeNMR peak at T = 1.5 K

depicted in Fig. 29 [56]. When the magnetic ﬁeld is applied perpendicular to the main easy axis and parallel to the medium axis in the xy hard plane, the ﬁeld dependence of the resonance frequencies is quite diﬀerent, as shown in Fig. 30. Again, a quantitative analysis of the results leads to the same conclusion as for Mn12 i.e., that the local spin conﬁguration remains intact up to at least 4 T transverse ﬁeld whereby the eﬀect of the ﬁeld is the one of rotating rigidly the eight Fe3+ moments. For details see [52]. 5.4 Spin Dynamics and Quantum Tunneling of the Magnetization in Nanomagnets: Mn12, Fe8 At very low temperature, when the nanomagnet occupies mostly the magnetic ground state, the thermal ﬂuctuations of the magnetization become vanishingly small. In this temperature range the spin dynamics is dominated by quantum ﬂuctuations. In this paragraph we will concentrate on the issue of what information NMR can give about the phenomenon of quantum tunneling of the magnetization (QTM). Let us ﬁrst brieﬂy summarize the QTM phenomenon. The ground state of both Mn12 and Fe8 clusters is a high total spin ground state i.e., S = 10. The S = 10 ground state is split into eleven sublevels by a strong easy axis anisotropy [13, 14, 57]. The remaining Kramers degeneracy is removed by an external magnetic ﬁeld directed along the z easy axis. The energy levels for H // z are obtained from Hamiltonian (19) as: Em = −Dm2 − Bm4 + gµB Hm .

(26)

Assuming g = 2 one has gµB H = 1.33H (K) for H in Tesla. For Mn12 one has D = 0.55 K, B = 1.2 × 10−3 K; for Fe8 one has D = 0.27 K, B = 0 and in the total Hamiltonian H = −DSz2 − BSz4 + gBH · S, the rhombic term E(Sx2 − Sy2 ), with E = 0.046 K, must be added. This term modiﬁes the (26) of the energy levels in a non-simple way.

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Below liquid helium temperature the clusters occupy mostly the m = ±10 states and the reorientation of the magnetization between these two states becomes extremely long (about one day for Mn12 at 2.4 K) due to the anisotropy barrier giving rise to a pronounced superparamagnetic behavior [13, 14, 57]. When the relaxation rate of the magnetization is measured in response to a varying magnetic ﬁeld Hz along the easy axis, peaks are observed which have been interpreted as a manifestation of resonant tunneling of the magnetization [26, 57]. The qualitative explanation is that the relaxation rate of the magnetization is maximum at zero ﬁeld and at ﬁeld values where the total spin states become pairwise degenerate again. The longitudinal ﬁeld at which this occurs can be easily calculated from (26) with the parameters given for Mn12 and Fe8, respectively. It is this degeneracy which increases the tunneling probability and thus shortens the relaxation time. The size of the eﬀect depends on terms not shown in the Hamiltonian (19), i.e., terms which couple the pairwise degenerate states. In particular, a transverse magnetic ﬁeld component can greatly enhance the tunneling splitting of the degenerate levels and thus the QTM. On the other hand, the QTM is reduced by the smearing out of the energy levels of the spin states due to spin-phonon coupling, intermolecular interactions, and/or hyperﬁne interactions with the nuclei [58]. NMR can detect the QTM in two quite diﬀerent ways which will be brieﬂy described in the following. Measurement of the Relaxation Rate of the Magnetization by Monitoring the NMR Signal Intensity in Oﬀ-Equilibrium State The idea is quite simple. We have seen above that in the NMR spectrum at low temperature in Mn12 and Fe8 the position of the resonance lines depend upon the internal ﬁeld due to the magnetization of the molecule. Thus if an external ﬁeld is also applied the position of the line depends on the vector sum of the external ﬁeld and the internal ﬁeld, the latter being directed along the magnetization of the molecule. If the direction of the external ﬁeld is suddenly reversed (or the sample is ﬂipped by 180◦ ) the position of the NMR line changes in the new oﬀ-equilibrium situation. The intensity of that particular line starts from zero and grows back to the full intensity as the magnetization of the molecule relaxes back to equilibrium along the applied ﬁeld. The method was ﬁrst described for proton NMR in Mn12 by looking at the echo intensity at the Larmor frequency when the external ﬁeld is turned on adiabatically at low temperature [59]. A more straightforward implementation was later applied to the signal intensity of shifted proton lines in Mn12 as described below. The detailed results for proton NMR in Mn12 are discussed in [60]. The magnetization of the Mn12 clusters is initially prepared in equilibrium conditions with the magnetic ﬁeld along the easy c-crystal axis. By inverting the magnetic ﬁeld, or better, by ﬂipping the oriented powder by 180◦ one creates an oﬀ-equilibrium condition whereby the magnetization of each molecule wants to realign along the external ﬁeld (m = −10 to m = +10 transition). At

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Echo intensity (a. u.)

332

0.2

0.4

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H (T) Fig. 31. Time evolution of 1 H-NMR spectrum of Mn12 oriented powders measured at 30.801 MHz and T = 2.4 K, for H parallel to the easy-axis. The spectrum at the bottom corresponds to the thermal equilibrium state. The second lowest spectrum corresponds to the oﬀ-equilibrium situation following the ﬁeld inversion. The remaining spectra from bottom up are taken at diﬀerent times after the ﬁeld inversion

low temperature and in magnetic ﬁelds less than 1 T this process is prevented by the crystal ﬁeld anisotropy and proceeds very slowly via spin tunneling and phonon assisted relaxation [57, 58]. Figure 31 shows the experimental results. The spectrum at the bottom of Fig. 31 corresponds to the thermal equilibrium state before the inversion, where the easy axis of the clusters is along the magnetic ﬁeld. Just after the inversion of direction of the sample, the observed spectrum changes drastically as shown in the second spectrum from the bottom of Fig. 31. In the ﬁgure, the time evolution is from the bottom up. Since the spectra were obtained by sweeping the magnetic ﬁeld, a process, which takes about 30 minutes for each spectrum, the spectra do not correspond to a precise oﬀ-equilibrium state. However, since the overall process of relaxation of the magnetization at this temperature takes a two or three hundred minutes, the diﬀerent spectra give a qualitative idea of the evolution of the NMR spectrum with time. The signals of the shifted peaks with positive hyperﬁne ﬁelds disappear, while new signals can be observed at magnetic ﬁelds higher than the Larmor ﬁeld H0 , where no signal could be detected before the inversion. After a long time (for example, 400 minutes in this case), the spectrum becomes independent of time and recovers the initial shape before the ﬁeld inversion. In order to investigate the eﬀect quantitatively one can sit at ﬁxed ﬁeld on one of the shifted lines (see Fig. 24) and follow its amplitude as a function of time without need of recording the full spectrum. The signal intensity for each shifted peak (P2 to P5 shown in Fig. 24) in the spectrum at thermal equilibrium corresponds to the total number of clusters occupying the magnetic m = −10 ground state. Immediately after the 180◦ -rotation of the sample the state m = −10 becomes m = +10. Then the growth of the signal intensity for each peak after the inversion is proportional to the increase of the number of clusters which return to the m = −10 new ground state. Therefore,

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we can measure a relaxation time of magnetization by monitoring the echo intensity as a function of time. Figure 32 shows a typical time dependence of the echo intensity h(t) measured at 0.4212 T (at the position of the P2 peak) and at T = 2.4 K. The experimental results can be ﬁtted tentatively by the expression (27) h(t) = a (1 − e−t/τ (H) ) + b , where τ (H) is a relaxation time and a + b is the echo intensity for the thermal equilibrium state, hT.E. . As can be seen in the ﬁgure, the growth of the signal intensity is well ﬁtted by a single exponential function (27) except for an initial fast growth, which accounts for about 30% of the signal. From the slope of 1 − h(t)/hT.E. on semi log plot, we can estimate τ (H). In Fig. 33 it is shown the comparison of the ﬁeld dependence of the relaxation time measured with NMR and the one measured directly by monitoring the magnetization with a SQUID. Although the two sets of data refer to two diﬀerent ways of extracting the relaxation time from the recovery curves, one can conclude that the results from both methods are in good agreement. In particular, in both cases one sees minima of τ (H) at the level crossing ﬁelds i.e., H = 0 (only for the magnetization), H = 0.45 T, and H = 0.9 T. Except for the minima, indicated by the arrows in Fig. 33, the H-dependence of τ (H) follows a thermally activated law τ (H) = τ0 e(67−13.3H)/kB T with τ0 ∼ 10−6 s which is consistent with background thermal excitations over the barrier due to the 100000

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t (min) Fig. 32. Time dependence of the echo intensity in Mn12 oriented powders taken at H = 0.4212 T and T = 2.4 K following the sudden inversion of the orientation of the easy axis of the sample with respect to the applied magnetic ﬁeld. Solid and open circles represent h(t)/hT.E . and 1 − h(t)/hT.E , respectively

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10 1 0.0

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H (T) Fig. 33. Field dependence of the relaxation time for Mn12 (oriented powders) magnetization measured at T = 2.4 K by 1 H-NMR (closed circles) and SQUID (open circles). The solid line represents τ (H) = τ0 e(67−13.3 H)/kB T discussed in the text. H is parallel to the easy-axis

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anisotropy as modiﬁed by the applied magnetic ﬁeld. The ﬁeld values corresponding to the minima in the relaxation time agree with critical ﬁelds where the magnetic level crossing occurs. It should be pointed out that the NMR method of monitoring the relaxation of the magnetization of the molecule is not entirely equivalent to the thermodynamic method since the NMR signal is a local probe of the magnetization. This subtle diﬀerence suggests that in the NMR method there is a microscopic information about the mechanism of reversal of the magnetization. The method has been conﬁrmed [50, 61, 62] by measuring the 55 Mn NMR spectrum in Mn12, with results very similar to the ones obtained with proton NMR [60], and it has been extended to very low temperature to observe the avalanche eﬀect of the spin reversal in the magnetization recovery in Mn12 [63]. Also it has been applied to proton NMR in Fe8 at very low temperature to obtain information about Landau-Zener transition as the ﬁeld is swept through a level crossing condition [64]. Nuclear Spin-Lattice Relaxation Induced by Quantum Tunneling of the Magnetization in Molecular Nanomagnets: Fe8, Mn12 Quantum ﬂuctuations of the magnetization are expected to have a distinctive eﬀect on nuclear spin-lattice relaxation (NSLR). For example, in the case of Fe8 the proton NSLR was observed to decrease very fast down to 400 mK as the result of slowing down of the thermal ﬂuctuations but then to become temperature independent below 300 mK [64]. This leveling-oﬀ of proton T1 was taken as indication of a crossover to a regime of quantum ﬂuctuations of the magnetization. A more direct way to detect the eﬀect of quantum tunneling on NSLR is obtained from the ﬁeld dependence of T1 as illustrated in the following. The occurrence of QTM in zero external magnetic ﬁeld or in an external ﬁeld applied along the anisotropy axis is related to the splitting of the pairwise degenerate magnetic levels by an amount ∆T that is due to oﬀ-diagonal terms in the magnetic Hamiltonian arising from anisotropy in the xy plane, intermolecular dipolar interactions, and hyperﬁne interactions. Normally the tunnel splitting ∆T in the Mn12 and Fe8 clusters is much smaller than the level broadening so that measurements of ∆T are diﬃcult. The eﬀect of QTM as a function of the longitudinal magnetic ﬁeld can be seen as dips in the relaxation time of the magnetization, in correspondence with the critical ﬁelds for level crossing as shown in Fig. 33. However, under applied parallel ﬁeld no clear eﬀect could be observed in the proton T1 . This can be seen in Fig. 34 where we show the longitudinal ﬁeld dependence of T1 in both Mn12 and Fe8. The critical ﬁelds corresponding to the ﬁrst level crossing in the ground state (i.e., m = −10 to m = 9) is estimated to be 0.5 T for Mn12 and 0.2 T for Fe8, respectively, from (23). As seen in Fig. 34 the data can be ﬁtted well by the simple model of thermal ﬂuctuations of the magnetization described in Paragraph 4.2 without any marked anomaly at the level crossing ﬁelds. Thus we may conclude tentatively that for H//z no eﬀects of quantum tunneling

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H (T)

Fig. 34. Parallel ﬁeld dependence of 1 H-1/T1 in both Mn12 (T = 3 K) and Fe8 (T = 2.1 K), oriented powders. The arrows indicate the critical ﬁeld values for level crossing across the energy barrier

can be observed on 1/T1 since the tunneling dynamics is too slow for longitudinal applied ﬁelds. As a word of caution we may add that the data in Fig. 34 indicate some enhancement of the NSLR in Mn12 around the level crossing ﬁeld and in Fe8 the data are incomplete since the ﬁrst level crossing cannot be reached as the result of the experimental diﬃculties at low ﬁelds thus leaving open the possibility that a small eﬀect of QTM may be observed for longitudinal ﬁelds. On the other hand, a recent study of 55 Mn NMR in Mn12 at very low temperature (down to 20 mK) has revealed the presence of (temperature independent) tunneling ﬂuctuations [64]. On the other hand, by applying a magnetic ﬁeld perpendicular to the easy axis (transverse ﬁeld), one can increase ∆T of all levels while leaving the symmetry of the double-well potential intact [58, 65]. An increase of the tunneling splitting corresponds to an increase of the tunneling frequency or of the tunneling probability. For H 1 T, in Fe8, the relaxation (ﬂuctuation) of the magnetization driven by tunneling (coherent and/or incoherent) becomes so fast that it falls within the characteristic frequency domain (MHz) of a NMR experiment. Therefore, when the magnetic ﬁeld is applied perpendicular to the main easy axis z (transverse ﬁeld) a pronounced peak in the spin-lattice relaxation rate, 1/T1 , of protons in a single crystal of Fe8 as a function of external magnetic ﬁeld can be observed at 1.5 K, as shown in Fig. 35. The eﬀect is well explained by considering that by increasing the transverse ﬁeld the incoherent tunneling probability becomes suﬃciently high as to match the proton Larmor frequency. When the applied ﬁeld goes through this condition the ﬂuctuation rate of the magnetization is most eﬀective in driving the nuclear relaxation and a maximum appears in 1/T1 . The peak disappears when a parallel ﬁeld component is introduced in addition to the transverse ﬁeld, by tilting the single crystal about 5 degrees in yz plane (see Fig. 35). Since the parallel ﬁeld component removes the degeneracy of the ±m magnetic states and consequently the possibility of tunneling, it is clear that the peak in 1/T1 must be related to a contribution to the nuclear relaxation rate

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H-NMR single crystal Fe8 T=1.5 K

1/T1 (sec)-1

10000 7500 5000 2500 0

H (T)

Fig. 35. Transverse ﬁeld dependence of 1 H-1/T1 at T = 1.5 K in Fe8 single crystal as a function of the ﬁeld along the medium axis (closed circles). Open squares are the results obtained when the single crystal is tilted so that the applied ﬁeld is 5 degrees oﬀ the xy-plane

from the tunneling dynamics. A quantitative interpretation of the eﬀect and further details can be found in [66]. Measurements of NSLR as a function of transverse ﬁeld were also performed in oriented powder of Mn12 both on proton NMR and on 55 Mn NMR [55]. However, in this case we failed to observe a peak of the NSLR such as for Fe8 although the relaxation rate was found to be faster than expected on the basis of purely thermal ﬂuctuations of the magnetization. The results are shown in Fig. 36 where we plot the transverse magnetic ﬁeld dependence of 55 Mn-1/T1 (P1; Mn4+ ions). With increasing the transverse magnetic ﬁeld, 1/T1 increases rapidly by about two decades and shows a broad maximum around 5.5 T. We have calculated the expected ﬁeld dependence obtained from the simple thermal ﬂuctuation model described in Paragraph 4.2, a model which works well to describe the longitudinal ﬁeld dependence. As seen from the comparison of the dotted line and the experimental points in Fig. 36, there appears to be an additional contribution to NSLR of 55 Mn in transverse ﬁeld. This additional contribution was explained [55] by a phenomenological model which considers the eﬀect of canting of the magnetization in presence of a transverse ﬁeld. In fact, the quantization axis of the nuclear spin does not coincide with that of the internal magnetic ﬁeld due to the electron spins. Therefore the transverse ﬁeld generates components of the local ﬁeld perpendicular to quantization axis of the nuclear spin, whose ﬂuctuation can be very eﬀective in producing nuclear relaxation. For the details of the model which generates the ﬁtting curve in the ﬁgure we refer to [55]. The transverse ﬁeld dependence of 1/T1 for protons at 1.5 K in Mn12 is also shown in the inset of Fig. 36. Contrary to 55 Mn, the proton 1/T1 has a rapid initial decrease with increasing ﬁeld, roughly as H −2 , which can be explained very well by (18) in the limit of “slow motion” (ωN τ 1), where 2 ). The key to understand the diﬀerent T1−1 can be approximated with A/(ωN 55 1 behavior of 1/T1 for Mn and for H is the fact that the internal magnetic

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ﬁelds at proton sites are very small (at most 0.4 T in comparison with the case of Mn nuclei, ∼ 22 T for Mn4+ ions) and thus ωN ∝ Hext for protons while ωN depends very little on the external ﬁeld for manganese. At higher ﬁelds the proton NSLR levels oﬀ to an almost constant value and no peak can be observed contrary to Fe8 (see inset of Fig. 36). The peak due to the matching of the Larmor frequency with the tunneling probability was expected in Mn12 in transverse ﬁeld around 6–7 T. We do not have as yet an explanation for the diﬀerent behavior of Mn12 and Fe8 except for the fact that the data in Mn12 were taken in oriented powder since no single crystal large enough for NMR work is available for Mn12. The leveling-oﬀ of 1/T1 for protons at high ﬁelds is due to the same decrease of the lifetime τ which is the dominant eﬀect for 55 2 Mn. This can be seen by extracting τ fom the equation of T1−1 = A/(ωN ) 55 1 and the experimental points for both Mn and H. The two sets of data coincide within experimental error as shown in Fig. 37. In the ﬁgure, we show also for comparison the transverse ﬁeld dependence of τ −1 calculated from the spin phonon interaction. The diﬀerence of the experimental points from the solid line represents the contribution to the ﬂuctuation of the magnetization which cannot be related to spin-phonon interaction. 1000000

100000

1/τ (sec)-1

10000

Transverse field dependence 55

1/τMn from Mn-T1 1

1/τH from H-T1 Spin-phonon interaction

1000

120 -1

spin-phonon int. + canting comp. 2 4

H-NMR T=1.5 K

100

1/T1 (sec)

1/T1 (s-1)

Mn4+ 100 T=1.5 K

100

Parallel field dependence Mn4+ (P1)

Mn3+ (P2)

Mn3+ (P3) Spin-phonon interaction

Transverse field (T)

Fig. 36. Transverse ﬁeld dependence of 1/T1 for 55 Mn-NMR at the M n4+ site (P1) measured at T = 1.5 K in Mn12. The inset shows the transverse ﬁeld dependence of 1/T1 of 1 H-NMR

external magnetic field (T)

Fig. 37. External magnetic ﬁeld dependence of 1/τ in Mn12 oriented powders, estimated from 2 τ) a relation of T1−1 = A/(ωN for both 57 Mn and 1 H NMR. Solid and dotted lines are transverse and parallel ﬁeld dependence of τ −1 calculated from the spinphonon interaction

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6 Miscellaneous NMR Studies of Molecular Clusters: Fe2, Fe4, Fe30, Ferritin Core, Cr4, Cu6, V6, V15 In this paragraph we will brieﬂy review the NMR work done in a number of magnetic molecular clusters that were not included in the previous paragraphs because they present speciﬁc features which do not ﬁt entirely into the general issues discussed above. 6.1 Iron Clusters: Fe2, Fe4, Fe30 and Ferritin Core The Iron(III) S = 5/2 dimer, [Fe(OMe)(dbm)2 ]2 (in short Fe2), has a nonmagnetic S = 0 ground state. The separation between the singlet ground state and the ﬁrst excited (triplet) state was determined by susceptibility measurements to be about 22 K; proton NMR measurements were performed on Fe2 [67]. The nuclear spin-lattice relaxation rate (NSLR) was studied as a function of temperature at 31 and 67 MHz and as a function of the resonance frequency (10–67 MHz) at T = 295 K. The results are shown in Fig. 38. At room temperature the 1 H NSLR is independent of frequency (see inset in Fig. 38) contrary to the strong ﬁeld dependence found in AFM rings as discussed in Paragraph 3. The temperature dependence of the proton NSLR shows a monotonous decrease on lowering the temperature without the peak characteristic of the AFM as described in Paragraph 4.1. 1/T1 is approximately proportional to χT where χ is the uniform susceptibility measured at H = 1 T. The cluster of four Iron(III) ions, Fe4 (OCH3 )6 (dpm)6 (in short Fe4), is characterized by a total spin ground state ST = 5 and Ising anisotropy [68]. The cluster behaves at low temperature like a superparamagnet just like Mn12 and Fe8 except that the anisotropy barrier is much lower. The D = 0.29 K in (26) corresponds to an energy barrier between the lowest MS = ±5 state and

Fig. 38. Temperature dependence of 1 H-1/T1 in Fe2 powders. The inset shows magnetic ﬁeld dependence of 1/T1 at room temperature

Fig. 39. Temperature dependence of 1 H-1/T1 in Fe4 “oriented” powders

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the highest MS = 0 state of only 7.25 K. The Fe4 cluster has been investigated by 1 H NMR as a function of temperature (0.5–295 K) and external magnetic ﬁeld (0.3–7.2 T) [69]. The results of NSLR are shown in Fig. 39. At very low T (0.5 K) the spectrum becomes very broad indicating the freezing of the Fe3+ moments in a superparamagnetic state. The temperature dependence of T1−1 is characterized by a ﬁeld dependent maximum as shown in Fig. 39. For low magnetic ﬁelds the peak of 1/T1 becomes so high that the proton NMR signal cannot be detected in the temperature region of the peak. The ﬁeld dependent maximum as well as the loss of NMR signal in the region of the peak at low magnetic ﬁelds (see Fig. 39) is qualitatively similar to the one observed in Mn12 [70]. In Mn12, 13 C [71] and 2 D NMR [72, 73] were also performed giving similar qualitative experimental results. In [73] an isotope eﬀect on NMR data was also suggested. It should be noted that the ﬁeld dependent peaks observed in Fe4 and in Mn12 with maxima located at temperatures of the order of the exchange constant J have many similarities with the peaks discussed for AFM rings in Sect. 4. A systematic analysis of the nuclear relaxation data in ferrimagnetic molecular clusters is in progress to establish the relaxation mechanism at these intermediate temperatures. The cluster {Mo72 Fe30 O252 (Mo2 O7 (H2 O))2 (Mo2 O8 H2 (H2 O))–(CH3 COO)12 (H2 O)91 } × 150H2 O [2], Fe30 in brief, has 30 Fe(III) ions occupying the 30 vertices of an icosidodecahedron. The magnetic properties are characterized by a ground state with total spin state ST = 0 due to O-Mo-O bridges mediating antiferromagnetic (AF) coupling with the exchange coupling constant J = 1.57 K between nearest-neighbor Fe ions. An accurate description of the magnetic properties of the cluster has been based on classical Heisenberg model of spins on the vertices of an icosidodecahedron for arbitrary magnetic ﬁelds [2, 74]. 1 H nuclear magnetic resonance (NMR) and relaxation measurements have been performed in the Keplerate species Mo72Fe30 [75]. The 1 H NMR linewidth increases gradually with decreasing T and it saturates below about 4 K, as expected for a non-magnetic ground state with ST = 0. The results for the magnetic ﬁeld and temperature dependence of T1−1 are shown in Figs. 40 (a) and (b). The magnetic ﬁeld dependence of T1−1 at room temperature (see Fig 40 (a)) can be ﬁtted by (7) used to ﬁt the results in AFM rings. The best ﬁt constants A = 5 ms−1 , B = 0.1 T and C = 2.2 ms−1 compare well with the constants in Table 1. The small value of B, corresponding to ΓA = γe B = 1.7 × 1010 (rad s−1 ), is consistent with a highly isotropic Heisenberg system. As shown in Fig. 40 (b) a strong enhancement of T1−1 with a peak around T = 2 K is observed, a feature similar to the one observed in antiferromagnetically (AFM) coupled rings (see Paragraph 4.1). It must be mentioned that the ﬁrst direct experimental conﬁrmation of the transition from ST = 0 to ST = 1 for standard Fe30, was from µSR experiments where, in correspondence to the ﬁrst crossing ﬁeld Hc ∼ 0.24 T, the width of the distribution of the local ﬁelds at the muon site tends to a plateau [76]. An antiferromagnetic molecular cluster containing a large number of Fe(III) ions (a variable number of the order of 5000) is the biomolecule ferritin.

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(a) Fe30 T=300 K

3 2 1 0 0.1

(b)

1/T1 (ms-1)

H=0.75 T H=4.7 T

4 3 2 1

0 0

H (T)

T (K) 1

Fig. 40. (a) Magnetic ﬁeld dependence of H-1/T1 at 300 K in Fe30 powders. (b) Temperature dependence of 1/T1 measured at H = 0.75 K (open squares) and H = 4.7 T (closed circles)

It is commonly reported [77] that ferritin becomes antiferromagnetic (AF) although the evidence is largely indirect and the Neel temperature reported varies widely in the range 50 K < TN < 240 K. Moreover, for T < 30 K, magnetization measurements [78] give evidence of superparamagnetic relaxation and of spin freezing of the total magnetic moment associated with the uncompensated spins at the boundary of the cluster of iron ions. Recently the interest in the magnetic behavior of ferritin has been revived by the conﬁrmation [79, 80] of an early report [81] of the occurrence of macroscopic quantum coherence phenomena at low temperature. Since the naturally occuring ferritin biomolecule contains the largest number of protons in the outer protein shell and thus are not coupled to the iron ions in the core, one has to strip the protein shell in order to use the NMR of the protons inside the core to probe the magnetic behavior of the Fe3+ ions. Proton NMR was performed in ferritin core as a function of temperature from 220 K down to 4.2 K at 4.7 T [82]. The proton spectrum is inhom*ogeneously broadened and the NSLR depends on the position in the spectrum where the measurements are performed making the analysis of the results difﬁcult and uncertain. The relevant result of the investigation is represented by the maximum in the NSLR observed at a temperature of about 100 K which is a clear indication of the occurrence of an antiferromagnetic transition. The study has to be viewed as preliminary. More experiments are needed to investigate the critical behavior of the NSLR around the ordering temperature and the eﬀect of the superparamagnetic ﬂuctuations and quantum tunneling. 6.2 Ferromagnetic (FM) Ring (Cu6) and Cluster (Cr4) Both [(PhSiO2 )6 Cu6 (O2 SiPh)6 ] (in short Cu6) and [Cr4 S(O2 CCH3 )8 (H2 O)4 ] (NO3 )2 ×H2 O (in short Cr4) are molecular nanomagnets with nearest neighbor ferromagnetic exchange interaction. Cu6 is a planar ring of six spin 1/2 Cu2+

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ions with J = 61 K and a ground state with ST = 3 separated by an energy gap of 30 K from the ﬁrst excited state ST = 2 [83, 84]. The ground state is split by an anisotropy D = 0.435 K. Thus in presence of an external magnetic ﬁeld H, the energy levels of the 7 magnetic substates m in ST = 3 state are given by: Em = 0.435 m2 − 1.33 m H (Kelvin, with H in Tesla) . kB

(28)

On the other hand, Cr4 is a cluster formed by four spin 3/2 Cr3+ ions on the vertices of almost regular tetrahedron [85]. The exchange constant J = 28 K and the ground state has ST = 6. The crystal ﬁeld anisotropy is negligibly small. Proton NMR measurements have been performed in both systems in the low temperature range where the magnetic clusters are in their total ST ground state [84, 85]. It is noted that contrary to Mn12 and Fe8 no superparamagnetic eﬀects are observable since the anisotropy for Cu6 is positive and for Cr4 is negligible. Thus no energy barrier is present for the reorientation of the magnetization even at low temperature. These FM clusters could be thought of as soft nanomagnets in contrast to Mn12 and Fe8 which are hard nanomagnets. Therefore in the temperature range of our measurements (1.4– 4.2 K) the thermal ﬂuctuations of the magnetization are still faster than the NMR frequency and no static local ﬁeld is present at the proton site (i.e., no spin freezing). The systems behave like an assembly of very weakly interacting paramagnetic spins of size ST = 3 and ST = 6 for Cu6 and Cr4, respectively. The results for the proton NSLR are shown in Fig. 41 and Fig. 42 for Cu6 and Cr4, respectively. The NSLR results can be explained very well by a simple model of spinlattice relaxation due to the ﬂuctuations of the total magnetization of the cluster. One can assume that the transverse hyperﬁne ﬁeld at the proton site due to the interaction with the diﬀerent Cu (Cr) magnetic moments is

(a)

(b)

Fig. 41. 1 H-1/T1 in Cu6 powders. (a) 1/T1 plotted as a function of temperature for assorted values of the magnetic ﬁeld. (b) 1/T1 plotted as a function of gµB H/kB T

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Fig. 42. 1 H-1/T1 in Cr4 powders. (a) 1/T1 plotted as a function of temperature for assorted values of the magnetic ﬁeld. (b) 1/T1 plotted as a function of gµB H/kB T

proportional to the component of the magnetization of the cluster in the direction z of the applied ﬁeld i.e., h± (t) = C Mz (T ) = C (Mz (T ) + δMz (T )). Then from (14) it follows for the NSLR: 1 1 2 2 Γ = γ C (29) δM (0)2 e−Γ t e−iωL t dt = A δM (0)2 2 2 , T1 2 Γ + ωL where we dropped the subscript z for simplicity. The characteristic frequency Γ can be viewed classically as a probability per unit time for the total spin ST = 3 (6) of the Cu6 (Cr4) cluster to change its orientation along the external magnetic ﬁeld or, quantum mechanically, as the broadening of the corresponding magnetic eigenstate. The average of the square of the ﬂuctuation can be calculated exactly as: m2 e−Em /kT kB T − δM (0)2 = χ(H, T ) = (gµb )2 Z m

, -2 me−Em /kT m

, (30)

where Em is given by (28). The data in Figs. 41 (a) and 42 (a) have been replotted in Figs. 41 (b) and 42 (b) as a function of gµB H/kB T . In Cr4 the NSLR data at diﬀerent ﬁelds and temperatures rescale well with gµB H/kB T indicating that Γ ωL in (29). The data are reproduced very well by a theoretical curve obtained from (29) and (30) assuming Γ to be H- and T independent and using the energy levels, (28), with negligible anisotropy. On the other hand, the data in Cu6 shown in Fig. 41 (b) do not scale so well vs gµB H/kB T . This is attributed to the non-negligible anisotropy term in (25) and the H and T dependence of Γ [84]. In conclusion, the above results of proton NSLR demonstrate that in the FM molecular clusters Cu6 and Cr4 the low temperature spin dynamics can be described as a simple thermal ﬂuctuation of the magnetization of the cluster in its collective quantum state of total spin ST . The simple model for the NSLR leads to the same result as obtained from the ﬁrst-principles calculation of the equilibrium two-ion time

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correlation function for the given set of magnetic eigenstates of the molecule whereby the characteristic frequency Γ for the ﬂuctuations of the total spin of the molecule corresponds to the broadening of the magnetic eigenstates [85]. 6.3 Vanadium Clusters: V6 and V15 V15 cluster (complete formula: K6 [V15 As6 O42 (H2 O)] × 8H2 O) contains 15 paramagnetic vanadium ions V4+ , each of which has spin s = 1/2. The V15 cluster is a very peculiar Heisenberg spin-triangles system which can be viewed as a triangle formed by three groups of ﬁve V4+ ions [1]. As a result of frustration the ground state is formed by two ST = 1/2 doubly degenerate states separated by a small gap of about 0.2 K. One expects that the magnetic properties and the spin correlations are dominated by interlayer coupling rather than by intralayer coupling. This situation can be compared with another spin-triangles system, namely V6 [86]. In V6 the ions are arranged in two almost independent trinuclear units, each forming a strongly antiferromagnetic frustrated triangle. The low temperature ground state for each of these triangles can be characterized by a doubly degenerate ST = 1/2, a doubly degenerate ﬁrst excited state with ST = 1/2 separated by a gap of about 60 K, and a fourfold degenerate second excited state with ST = 3/2 separated by 85 K [86]. 1 H NMR and relaxation measurements were performed in both clusters [87, 88], in form of powders. For V6 the T -dependence of proton T1−1 is approximately proportional to χT , where χ is the uniform susceptibility. On the other hand, in V15 the T -dependence of T1−1 is quite diﬀerent from the one of χT . This was taken as indication of a critical T -dependence of Γ , a parameter which measures the width of the V4+ spin ﬂuctuation spectral density. For details in these two frustrated spin triangles we refer to the [87, 88]. A proton NMR study has been performed also in a frustrated triangular system of Mn spins [Mn3 O(O2 CCH3 )6 (C5 H5 N)3 ] × C5 H5 N, in brief Mn3 [89], powders. The Mn system is similar in many respects to the frustrated spintriangle system V6 although the three Mn ions in the triangle are diﬀerent, with one Mn2+ (s = 5/2) ion and two Mn3+ (s = 2) ions. Also the coupling constant J between the two Mn3+ spins is three times as large as the coupling J between the Mn3+ and the Mn2+ spins. The most remarkable NMR result is a strong enhancement of 1/T1 with a peak at T0 ≈ J just like in the AFM rings (see Paragraph 4.1). Another frustrated spin-triangle system is (NH4 )[Fe3 (µ3 -OH)(H2 L)3 (HL)3 ] (in short Fe3, L = neutral ligand) [90]. Here again one observes a peak in T1−1 at T ∼ J, analogously to Mn3.

7 Summary and Conclusions In the present review we tried to illustrate in some details a large body of NMR data obtained by our groups starting back in 1996 in magnetic mole-

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cular clusters (or molecular nanomagnets or single molecule magnets). We tried to quote brieﬂy also work done by other NMR groups although some reports may have escaped our attention and we apologize for this. As a conclusion of the review we may say that the main characteristics of NMR in these new interesting magnetic systems have been established and the ﬁeld is now mature for a deeper investigation and a better understanding of some remarkable eﬀects reported here. In fact, the theoretical analysis of the results in this review has been often rather qualitative and based on simple models to describe the NMR and relaxation eﬀects observed. We will summarize in the following some of the major issues emerging from our investigation with the focus on what remains to be done. We have observed a magnetic ﬁeld dependence at room temperature of proton T1 in all nanomagnets investigated. Since at room temperature molecular clusters have to be viewed as nanosize paramagnets, the ﬁeld dependence is a clear signature of the zero dimensionality of the system. From the ﬁeld dependence of T1 one can obtain the “cut-oﬀ” frequency for the time decay of the electron spin correlation function. The detailed origin of the “cut-oﬀ” frequency for the diﬀerent molecules remains to be established. At intermediate temperatures, namely when the electron spin in the molecular cluster becomes strongly correlated (kB T ≈ J), we have observed a T - and H-behavior of T1−1 which can be explained satisfactorily with simple phenomenological models based on a direct relaxation process. The models are based on the assumption that the relaxation process is a direct process due to the hyperﬁne ﬁeld ﬂuctuations related to the lifetime of the magnetic molecular states. In particular, for the cases of Mn12 and Fe8 the low temperature nuclear relaxation can be described in terms of ﬂuctuations of the magnetization in the total spin S = 10 ground state manifold due to spinphonon interaction, and a value for the spin-phonon coupling constant could be derived for these two nanomagnets. The fact that the nuclear T1 can be explained in such a simple way came as a surprise since normally in a highly correlated spin system such as ferro and antiferromagnetc bulk system close to the phase transition one has to consider the interplay of ﬂuctuations of diﬀerent symmetry i.e., diﬀerent q-vectors. In nanosize molecular magnets it appears that the discreteness of the magnetic energy levels plays an important role whereby only the lifetime of the total spin quantum state of the molecule is all it matters in determining the nuclear spin-lattice relaxation. One issue which remains to be clariﬁed is the possibility of introducing a spin-wave description of the excited states in these nanomagnets and consequently the relation between a description of the relaxation in terms of a direct process and the description in terms of one- and two-magnons scattering. At even lower temperature (kB T J) we have observed dramatic eﬀects on the nuclear relaxation rate due to level crossing eﬀects and quantum tunneling of the magnetization. It appears that NMR can be useful to distinguish between level crossing (LC) and level anticrossing (LAC) in AFM rings but a general theory to describe the observed enhancement of T1−1 at the critical

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ﬁeld is still lacking. The same is true for quantum tunneling of the magnetization. The phenomenon can be detected by NMR in diﬀerent ways but a detailed description of the coupling between the nuclear Zeeman reservoir and the tunneling reservoir has to be ﬁrmly established in order to extract information from NMR measurements. Finally, it should be mentioned that the NMR spectra at low temperature give important information about hyperﬁne interactions. In particular, the zero ﬁeld NMR and the evolution of the spectrum as a funtion of an applied ﬁeld in single crystals or oriented powders is a unique method which enables one to determine the size and orientation of the local magnetic moments of the molecular cluster in its ground state conﬁguration.

Acknowledgements We thank all collaborators who made possible this investigation and whose names appear in the many papers quoted in the review. A special thank to our colleagues chemists who introduced us to this ﬁeld and provided the many samples and helped us to understand their properties. Their names also appear in most of our joint publications. We remark that all the ﬁgures presented by the authors of the current work were drawn using data taken from references listed in the bibliography and referred to along the main text.

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Correlated Spin Dynamics and Phase Transitions in Pure and in Disordered 2D S = 1/2 Antiferromagnets: Insights from NMR-NQR A. Rigamonti, P. Carretta and N. Papinutto Department of Physics “A.Volta” and Unit` a INFM, University of Pavia Via Bassi n◦ 6, I-27100, Pavia (Italy) [emailprotected] Abstract. A recall of the phase diagram for two-dimensional quantum Heisenberg antiferromagnets (2DQHAF) and of the main issues involving phase transitions and spin dynamics in these systems is ﬁrst given. After a pedagogical description of the basic aspects for the NMR-NQR relaxation rates in terms of amplitudes and decay rates of spin ﬂuctuations, the problem of the temperature dependence of the correlation length in prototype, pure 2DQHAF (CFTD and La2 CuO4 ) is addressed. Then spin- and charge-doped systems (Zn/Mg for Cu and Sr for La substitutions in La2 CuO4 ) are considered and the eﬀects on the spin stiﬀness, on the correlation length and on the staggered Cu2+ magnetic moment are reported, in particular near the percolation threshold. A critical outline of the properties of the cluster spin glass phase in Sr- doped La2 CuO4 is given. Finally the results of a 63 Cu NQR-NMR relaxation study around the quantum critical point (in CeCu6−x Aux ) are presented. It is pointed out how the 2D response function with anomalous exponent and energy/temperature scaling indicated by neutron scattering is basically conﬁrmed, while the 63 Cu relaxation measurements reveal novel eﬀects involving low-energy spin excitations and the role of an external magnetic ﬁeld.

1 Introduction and Contents Since the discovery that La2 CuO4 , the parent of high temperature superconductors, was also the experimental realization of the model two-dimensional (2D), quantum (S = 1/2), Heisenberg antiferromagnet (AF), a great deal of interest was triggered towards low-dimensional quantum magnetism. The system we are going to deal with is basically a planar array of S = 1/2 Cu2+ magnetic ions onto a square lattice, as sketched in Fig. 1, in antiferromagnetic interaction, namely described by the magnetic Hamiltonian

A. Rigamonti et al.: Correlated Spin Dynamics and Phase Transitions in 2DQHAF, Lect. Notes Phys. 684, 351–382 (2006) c Springer-Verlag Berlin Heidelberg 2006 www.springerlink.com

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Fig. 1. Pictorial sketch of planar arrays of S = 1/2 magnetic moments, in antiferromagnetic (AF) super-exchange interaction onto a square lattice, a model system rather well describing La2 CuO4 and Copper formiate tetrahydrate (CFTD). The spin dilution is achieved, in La2 CuO4 , by S = 0 Zn2+ or Mg2+ for S = 1/2 Cu2+ substitutions, while charge doping, namely injection of holes in the plane, is obtained by substituting the La3+ ions (out of plane) by Sr2+

J Si · Sj 2 i,j

J >0,

(1)

while the summation is often limited to the ﬁrst nearest neighbours spin operators. We shall devote our attention to a variety of aspects involving static and dynamical properties of that 2D array: the temperature dependence of the inplane magnetic correlation length ξ2D entering in the equal-time correlation function S i (0) · S j (0); the critical spin dynamics driving the system towards the long-range ordered state (at T = 0 in pure 2DQHAF in the absence of interplanar interaction J⊥ ); the validity of the dynamical scaling, where ξ2D controls the relaxation rate Γ of the order parameter according to a law of −z , with a given critical exponent z. We shall also discuss the form Γ ∝ ξ2D the modiﬁcations induced by spin dilution (or spin doping) namely when part of the S = 1/2 magnetic ions are substituted by non-magnetic S = 0 ions, as well as the eﬀects related to charge doping, namely the injection (for instance by heterovalent substitutions) of S = 1/2 holes creating local singlets which can itinerate onto the plane, locally destroying the magnetic order and inducing novel spin excitations. These aspects are of particular interest in the vicinity of the percolation thresholds, where the AF order is about to be hampered at any ﬁnite temperature. This can be considered a situation similar to a quantum critical point, where no more the temperature but rather Hamiltonian parameters can drive the transition.

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In principle the problems involving the aforementioned aspects could be studied by spectroscopic techniques relaying on the electron’s response. However electron paramagnetic resonance is often prevented by very broad lines, possibly due to the strong electron-electron correlation. Neutron scattering is a powerful technique but it could suﬀer of resolution limitation for low-energy excitations or for short correlation length. Thus the nuclei have represented in recent years one of the best tools to investigate the properties of 2DQHAF, by resorting to the hyperﬁne interactions and to their time-dependence, as explored by NMR-NQR spectra and relaxation. The chapter is organized as follows. In Sect. 2 we brieﬂy discuss the rich phase diagram of 2DQHAF, as it results from a variety of experimental studies and theoretical eﬀorts (that will not be described in detail). Then (Sect. 3) the basic principles of the experimental approach using the nuclei as local probes are recalled, describing the sources of the hyperﬁne ﬁelds and giving the expressions for the relaxation rates W in terms of the amplitudes |Sq |2 and decay rates Γq of the spin ﬂuctuations, with an illustrative example for a 2D Ising-like system approaching the ordering transition (Sect. 4). It is shown, in particular, how the temperature dependence of the critical decay rate is extracted. In Sect. 5 the results obtained in two 2DQHAF prototypes, namely La2 CuO4 and CFTD, are described, showing how quantitative estimate of ξ2D is derived and concluding that in a wide temperature range the predictions of the renormalized classical (RC) regime are rather well followed. The role of the “ﬁltering factor” Aq which weights in diﬀerent way diﬀerent regions of the Brillouin zone, is illustrated by comparing the results in La2 CuO4 and CFTD. Then (Sect. 6) the case of charge and spin doped 2DQHAF is treated and the insights derived from 63 Cu and 139 La NQR in Sr-doped and in Zn (or Mg) -doped La2 CuO4 are illustrated. It is shown, in particular for spin doping, how the “dilution model” namely the simple correction in Hamiltonian (1) by the probability of presence pi,j of the magnetic ion, explains rather well the behaviour of ξ2D (x, T ), of the spin stiﬀness ρs (x) and of the ordering temperature TN (x), when the doping amount x is far from the percolation threshold. The modiﬁcation occurring when the threshold is approached are then discussed in Sect. 7, by comparing the NQR results with the ones obtained by neutron scattering. The eﬀects on the staggered magnetic moment are also addressed. The main properties of the cluster spin glass phase, occurring when the charge doping is above the percolation threshold in Sr-doped La2 CuO4 , are then brieﬂy reviewed (Sect. 8). Finally, in Sect. 9 we present and discuss the results of a recent NQR-NMR study carried out in a system (CeCu6−x Aux ) around the quantum critical point (QCP) (x = 0.1) separating Fermi liquid and AF phases, with a ﬁeld-dependent magnetic generalized susceptibility of 2D character, with anomalous exponent and energy/temperature scaling. While the main results obtained by other authors by means of inelastic neutron scattering are conﬁrmed also by our measurements based on low-energy

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susceptibility over all the Brillouin zone, novel aspects are pointed out in regards of the role of the magnetic ﬁeld. Summarizing remarks are collected in Sect. 10.

2 The Phase Diagram of 2DQHAF In Fig. 2 the phase diagram for a planar array of S = 1/2 spin in square lattice and AF interaction is shown. The diagram results from a variety of experimental studies in synergistic interplay with theoretical descriptions, that will not be recalled in detail (for a nice introduction and an exhaustive review of the studies of the magnetic properties of 2DQHAF, see [1]). In the Figure g is a dimensionless parameter measuring the strength of quantum ﬂuctuations and it can be related to spin wave velocity csw and to the spin stiﬀness ρs : √ csw 2π , (2) g= kB ρs a (a lattice parameter). The spin stiﬀness ρs has been introduced in order to describe the properties of disordered AF’s and it measures the increase in the ground state energy for a rotation by an angle θ of the sublattice magnetization (∆E = ρs kB θ2 /2). It can be written ρs ∝ (c2sw /χ⊥ ), χ⊥ being the transverse spin susceptibility. For classical Heisenberg spins, from usual spin wave theories one has

Fig. 2. Phase diagram argued to describe the occurrence in 2DQHAF of various regimes, as a function of temperature and of the parameter g measuring the strength of quantum ﬂuctuations (see discussion in the text)

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ρs = Zρ (S)JS 2 , 1/2

csw = Zsw (S) (8)

355

(3) kB JSa/h,

with Zρ (S) 1 − 0.235/2S and Zsw (S) 1 + 0.158/2S. The parameter g is expected to increase upon doping and disorder. Starting from the seminal paper by Chakravarty et al. [2] (who mapped the Hamiltonian (1) onto the so-called non-linear σ model, that for T → 0 is the simplest continuum model with the same symmetry and the same spectrum of excitations) the diagram in Fig. 2 has emerged. Below a given value gc the ground state, at T = 0, is the Neel AF state, which is somewhat extended to a ﬁnite temperature TN because of the interplane interaction J⊥ J. The percolation threshold for the AF state at T = 0 as a function of g, is at gc . For g < gc , upon increasing temperature above TN one enters in the renormalized classical (RC) regime. Here the eﬀect of the quantum ﬂuctuations is to renormalize ρs and csw with respect to the mean ﬁeld values for the “classical” 2D Heisenberg paramagnet. Thus the in-plane correlation length goes as ξ2D ∝ exp [2πρs /T ]. The prefactor (and a correction term) have been more recently evaluated [3] so that 2 T T 1.15J/T ξ2D = 0.493 e 1 − 0.43 + O , (4) J J (hereafter ξ2D √ = ξ is expressed in lattice units a) and 1.15J = 2πρs , while csw = 1.18 2JkB a/. Weakly damped spin waves exist for wave vectors q ≥ ξ −1 while for longer wave lengths only diﬀusive spin excitations of hydrodynamic character are present. For T ≥ J/2 ≈ 2ρs instead of entering into the classical limit (that would be reached for T J) the planar QHAF should cross, according to the above mentioned descriptions [1–3] to the quantum critical (QC) regime. In this phase, typical of 2D and 1D quantum magnetic systems, the only energy scale is set by temperature and ξ = J/T . On increasing g, according to proposals by Anderson [4] and to quantum phase transitions theories [5] the increase in quantum ﬂuctuations can inhibit an ordered state even at T = 0. The system is then in the quantum disordered (QD) regime, the correlation length being short and temperature independent. In the spectrum of spin excitations a gap of the order of csw /ξ opens up. The somewhat speculative phase diagram illustrated in Fig. 2 is still under debate. In particular the validity of the non-linear σ model is questioned at large T and/or for large g regions. For instance, as we shall see later on, some experimental studies do not provide evidence of crossovers from RC to QC or QD regimes. Also the occurrence of micro-segregation and/or a phase separation in hole-depleted and hole-rich stripes, as well as the real nature of the low-energy excitations are still open questions. We conclude the discussion of the diagram in Fig. 2 just mentioning the following. The cluster-spin-glass (SG) phase is the one for g > gc in which the experiments (primarily NQR spectra and relaxation and magnetic susceptibility) indicate the presence of mesoscopic “islands” of AF character separated

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by domains walls, with eﬀective magnetic moments undergoing collective spin freezing without long range order even at temperature close to zero. Above a given amount of charge doping (e.g. hole injection as in La2−y Sry CuO4 ) the systems become superconductors (SC phase), with the so-called underdoped and overdoped regimes characterized by a transition temperatures Tc < Tcmax (the one pertaining to the optimal doping), as discovered by M¨ uller and Bednorz. In SG and SC underdoped phases a gap in the spin excitations at the AF wave vector q AF = (π/a, π/a) has been experimentally observed to arise at a given temperature T ∗ . The spin gap (and charge pseudo-gap) region has possibly to be related to superconducting ﬂuctuations of “anomalous” (i.e. non-Ginzburg-Landau) character or to AF ﬂuctuations locally creating a “tendency” towards a mesoscopic Mott insulator. Exotic excitations of various nature have been considered to occur in the regions of high g’s. In this Chapter we shall not go into detail involving these aspects, which are still under debate and less settled than the ones for low g, namely for the doped non-superconducting 2DQHAF. Finally we shall present and discuss the results around the quantum critical point in CeCu6−x Aux that could be placed at the critical value gc separating the AF phase from the Fermi liquid region.

3 Basic Aspects of the Experimental Approach In the 2DQHAF considered here the electron paramagnetic resonance of Cu2+ ions can hardly be carried out. Therefore the nuclei are used as microscopic probes to detect the Cu2+ S = 1/2 spin dynamics through the time-dependent hyperﬁne interaction driving the NQR-NMR relaxation processes. In NQR experiments instead of the external magnetic ﬁeld as in NMR is the Vzz component of the electric ﬁeld gradient at the Cu site (perpendicular to the CuO2 planar lattice) that acts as quantization axis. The recovery towards the thermal equilibrium, after the usual sequences of RF pulses, is driven by the time dependence of the ﬁctitious hyperﬁne ﬁeld h(t) due to Cu2+ S = 1/2 electrons and the relaxation rate is γ2 (5) 2W = h+ (0) h− (t) e−iωm t dt , 2 where ωm (ωQ or ωL ) is the measuring frequency in the MHz range and the correlation function ..... involves the transverse components of the ﬁeld

Si 3(r i · S i ) r i li 8π S i δ(r i ) . γe 3 − − 3− (6) h(t) = ri ri5 ri 3 i The spin dynamics is reﬂected in the spin operators S i (t), with collective spin components

Correlated Spin Dynamics and Phase Transitions in 2DQHAF

1 −iqri Sq = √ e S i (t) , N i

357

(7)

yielding the eﬀective ﬁeld in the form 1 Aq S q (t) , h(t) = √ N q

(8)

where Aq is the Fourier transform of the lattice functions specifying the “positions” of the magnetic ions. For the systems we are going to discuss, h(t) originates from the on-site contribution (namely from the 3d electrons of the magnetic ion) and from transferred hyperﬁne scalar and dipolar contributions. From (5) and (8) one has γ2 2 α Aq (t) e−iωm t dt , (9) 2W = Sqα (0) S−q 2N q ⊥ where {...}⊥ indicates the dynamical structure factor components S αα (q, ω) at ωm involved in the ﬁeld components perpendicular to the quantization axis. From the ﬂuctuation-dissipation theorem S αα (q, ω) = kB T χ

αα

(q, ω)/ω ,

(10)

(for details see [6] and references therein).

4 NMR-NQR Relaxation Rates: Amplitude and Decay Rates of Spin Fluctuations and Critical Behaviour in 2D Systems In the cases we shall discuss later on, the decay rates of the spin ﬂuctuations will remain greater than ωm and (9) can be written γ 2 2 |Sqαα |2 Aq , (11) 2W = 2N q Γq ⊥

αα 2 αα 2 where αα |Sq | are the amplitudes of the collective ﬂuctuations, namely |Sq | ≡ S (q, ω) dω and Γq the correspondent decay rates. It is noted that in correspondence to the critical wave vector q c in practice the amplitude and the decay rate of the order parameter are involved. In (11) A2q has the relevant role of weighting particular regions of the Brillouin zone (BZ), or, in other words, a particular symmetry of the spin ﬂuctuations. In case of dominant on-site contribution to the eﬀective ﬁeld h(t), in practice only the auto-correlation function S i (0) S i (t) is involved in (5) and thus A2q is q-independent:

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Fig. 3. Spin lattice relaxation rate in p-terphenil on approaching the transition temperature Tc = 194.5 K (a). No dependence from the measuring frequency ωm was detected. In part b) the temperature behaviour of the critical frequency (see text) is reported and compared with the few results from high resolution neutron back-scattering in the deuterated crystal ( ) (see [7])

2 2W = ωint

1 |Sqαα |2 , 2N q Γq

(12)

ωint = γ A being an eﬀective strength of the nucleus-electrons interaction. Direct access to the spin dynamics is thus obtained. A long-lasting 2D illustrative example, for an Ising-like pseudo-spin system, is oﬀered by the relaxation measurement in the p-terphenil crystal [7]. The relaxation data on approaching the order-disorder transition (Fig. 3) can be analysed as follows. The pseudo-spin variable Sz (t) = (+, −)1 describes the time dependence of the dipole-dipole nuclear Hamiltonian and the correspondent dynamical structure factor is written ω S(q, ω) = | S q |2 Γq −1 F ( ) , Γq where F is a sharp function with width of the order of unit and unit area, that can be taken F = 1 when no frequency dependence of the relaxation rates is detected (see Fig. 3). The amplitudes and the decay rates of the spin ﬂuctuations scale with the correlation length as follows: | S q |2 = ξ 2−η f (qξ) , Γq = Γqc g(qξ), where f and g are hom*ogeneous functions of qξ, which correspond to scale the generalized susceptibility involved in (10) in the form

Correlated Spin Dynamics and Phase Transitions in 2DQHAF

χ(q, ω) = χ0 ξ z f (qξ, ω/ξ z ) , with χ0 = S(S + 1)/3 kB T . From the ﬁrst order expansion of f and g one has |Sq |2 ξ 2−η 1 = D−2 dq q (D−1) f g −1 Γq π Γqc q and

2W ∝

q (D−1)

dq Γ −1 ∝ εν(D−z)−γ , (1 + q 2 ξ 2 )2 qc

359

(13)

(14)

(15)

with Γqc ∝ ενz , ε = (T −Tc )/Tc , ν critical exponent for ξ, γ = (2−η)ν critical exponent for |Sqc |2 and z dynamical critical exponent (while in 2D η 0). Equation (12) is reduced to 2 −1 Γqc , 2W = ωint

justifying the temperature behaviour of the relaxation rates and yielding the critical frequency Γqc when ωint can be obtained from the spectra or from numerical estimate (see Fig. 3 b)). It should be remarked that the 2D scaling is represented by Γqc ∝ ξ −1 , namely by the dynamical critical exponent z = 1.

5 Pure 2DQHAF: Temperature Dependence of the Correlation Length (in La2 CuO4 and in CFTD, within Scaling Arguments) Real systems that, to a good approximation, obey to the Hamiltonian in (1) (neglecting for the moment the interplanar coupling) are La2 CuO4 (the parent of high-temperature superconductors) and CFTD (copper formiate tetrahydrate deuterated). The former is characterized by the in-plane superexchange constant J 1500 K. Due to the weak (J⊥ 10−5 J) interplanar coupling La2 CuO4 orders 3D at TN = 315 K. Still a wide temperature range exists where the in-plane correlation length is much larger than the lattice constant, while no 3D long range AF order sets in. However the main reason to deal with La2 CuO4 is that charge and spin doping can easily be obtained by Sr2+ for La3+ substitutions (thus injecting itinerant holes in the CuO2 plane) or by Zn2+ (or Mg2+ ) S = 0 for Cu2+ substitutions, which induce spin dilution. Instead CFTD is a good prototype of 2DQHAF (with J⊥ /J 10−4 ), with rather small J 80 K. Thus a large (T /J) range can experimentally be explored, searching for possible crossover to QC regime, for instance (see Fig. 2). In the light of (11) one realizes that a quantitative estimate of the correlation length by means of NMR-NQR relaxation requires a reliable form of the

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Fig. 4. Sketch of the simpliﬁed magnetic array of the Cu2+ ions in La2 CuO4 assumed in order to relate the 63 Cu NQR relaxation rates to the correlation length

A2q factor. For La2 CuO4 the ﬁctitious ﬁeld at the Cu nucleus can be written (see Fig. 4) 4 h = A S0 + B Si (16) i=1

and the values A⊥ = 80 kOe and B = 83 kOe have been estimated ([6] and references therein). Then for external ﬁeld H0 parallel to the c-axis or in NQR, one has 2 (17) A2q = [A − 2B(cos qx a + cos qy a)] , with q starting from q AF = (π/a, π/a). At the 139 La site more complicate expression holds: A2q has maxima at the center and at the borders of the Brillouin zone and an average value is approximately A2q (1 kOe)2 . In CFTD, at the hydrogen atom lying in the Cu plane one has A2q = A2 + B 2 (cos qx a + cos qy a)

(18)

and from the rotation pattern of the paramagnetic shift [8] one derives A = 2.31 kOe and B = 1.46 kOe. The form factor for CFTD given in (18) is depicted in the 2D BZ in Fig. 5. It is noted that the form factor is peaked at the center of the BZ. Thus the “critical” AF ﬂuctuation is ﬁltered out and a kind of anti-divergence of 2W as a function of temperature can qualitatively be expected in correspondence to the enhancement and slowing down of the order parameter ﬂuctuations. The experimental results for 63 Cu 2W in La2 CuO4 and for 1 H 2W ≡ T1−1 in CFTD are reported in Fig. 6. Now we outline how the temperature behaviour of ξ is extracted from the experimental data. Since ξ 1, scaling arguments for |Sq|2 and Γq should hold. Then, in the light of (10)–(13) and (14) we write

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Fig. 5. Contour plot of A2q for CFTD, as derived from the rotation pattern for the paramagnetic shift ∆K of the H NMR signal (from [8])

Fig. 6. (a) 63 Cu NQR relaxation rate in La2 CuO4 as a function of temperature. The solid line is the theoretical behaviour correspondent to (21), for correlation length in the RC regime (from [9]; (b) proton-spin lattice relaxation rate as a function of temperature in CFTD. The dashed line is the theoretical mode-coupling calculation while the full line is the Monte Carlo numerical estimate with the correlation length derived in the framework of the pure-quantum self-consistent harmonic approximation, in substantial agreement with (4) (From [10])

|Sq |2 = |SqAF |2 f (qξ) = p and

S(S + 1) 2 ξ f (qξ) 3

(19)

2ωE (20) Γq = ΓqAF g(qξ) = √ ξ −z g(qξ) , 2π where p < 1 accounts for the zero-temperature reduction due to quantum eﬀects, ωE is the Heisenberg exchange frequency describing the uncorrelated spin ﬂuctuations for T → ∞. A normalizing factor preserving the sum rule

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2 |S α q| = pN

S(S + 1) 3

has to be introduced in f = g −1 = β(1 + q 2 ξ 2 )−1 (see (15)). Thus one writes √ 2 2π 2 S(S + 1) z+2 β 2W = γ pξ 3 ωE 2 a2 [A⊥ − 2 B (cos qx a + cos qy b)] dq . 4π 2 (1 + q 2 ξ 2 )2 BZ

(21)

An approximate expression is obtained by averaging A2q (as given by (17)). For 63 Cu in La2 CuO4 the approximate form reads 2W ≈ [(4.2 × 103 )/(ln qm ξ)2 ] ξ z ,

(22) √ with qm = 2 π. However for the quantitative evaluation of the correlation length the complete q-dependence, and therefore (21), has to be taken into account. As shown in Fig. 7 the temperature behaviours of ξ deduced from the experimental results in La2 CuO4 and in CFTD are consistent with (4), namely with the RC regime and dynamical scaling with z = 1.

Fig. 7. (a) In plane magnetic correlation length as a function of temperature in the paramagnetic phase of La2 CuO4 as extracted from the inversion of (21) in the text (for ε the value ε = 0.3 was used, according to theoretical estimates and neutron scattering data). The solid line tracks (4). For comparison some representative results from neutron scattering are also shown (for complete references see [9]). (b) In plane magnetic correlation length for CFTD as a function of T /J and comparison with the theoretical estimates. In the inset the lack of a possible crossover to the QC regime, in the high temperature range, is evidenced (the dotted line is the theoretical behaviour within the mode-coupling theory and Kubo relaxation function; for details see [10])

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363

The conclusions that can be drawn from the studies in pure 2DQHAF are the following. The NMR-NQR relaxation rates allow one to derive reliable estimates of the absolute value and of the temperature dependence of the magnetic correlation length, particularly when the proper form of the factor A2q is taken into account in the inversion yielding ξ from the experimental results. In La2 CuO4 and in CFTD the RC regime appears to hold, up to a temperature of the order of 1.5 J and no evidence of crossover to QC or QD regimes is observed. The in-plane magnetic correlation length follows rather well the theoretical expression given by (4).

6 Spin and Charge Lightly-Doped La2 CuO4 : Eﬀects on the Correlation Length and on the Spin Stiﬀness As already mentioned, in La2 CuO4 charge doping is obtained by heterovalent Sr2+ for La3+ substitutions, while spin dilution is obtained by S = 0 Zn2+ (or Mg2+ ) for S = 1/2 Cu2+ substitutions. In La2−y Sry CuO4 the Neel temperature drops very fast with the Sr content. Analogous eﬀect, at a lower rate, has the spin dilution in La2 Znx Cu1−x O4 , in terms of the Zn (or Mg) content x. (See Fig. 8). Upon spin dilution the spin stiﬀness is expected to change and we shall derive insights on the doping dependence of the correlation length. At this aim we start by analysing ﬁrst the experimental results for 63 Cu 2W , again using the general theoretical framework leading to (21), in the limit of weak doping, when the dilution model should hold. The dilution model amounts to modify the Hamiltonian (1) simply by considering the probability pi that a given site is spin-empty: 2 pi pj S i · S j = J(0) [1 − x] Si · Sj . (23) H=J i,j

i,j

Then the spin stiﬀness should depend on doping according to ρ(x) = 2 ρ(0) [1 − x] , the correlation length becoming (see (4)) ξ(x, T ) = ξ(0, T ) e−(2−x) x 1.15J/T .

(24)

Equation (21) is then rewritten 2W = C ξ z+2

1 ωE

dq BZ

A2q , (1 + q 2 ξ 2 )2

(25)

where C is a constant including all the quantities that do not change while 2 Aq = A⊥ −2(1−x)B [cos qx a + cos qy b] and ωE (x) = ωE (0) [1 − x] . From the 63 experimental results for Cu NQR relaxation rates, the correlation lengths

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Fig. 8. Drop of the Neel temperature TN in La2 CuO4 upon charge or spin doping, as obtained from a combination of measurements (most µSR, NQR spectra and SQUID magnetization). The percolation threshold is around x = 0.02 for Sr doping (experimental observation) while for Zn or Mg doping is expected around y = 0.41, according to quantum Monte Carlo simulation and to experimental ﬁndings (see later on). The solid line for the Sr-doped compound is a guide for the eye, while for Zn-doped La2 CuO4 the line tracks the initial suppression rate (–3.2) expected from theoretical treatments and experimental data in analogous compounds

reported in Fig. 9 have been derived. The echo signal is lost for temperature below about 450 K, due to the shortening of the dephasing time T2 . An indication for the value of the correlation length at TN can be obtained from the mean ﬁeld argument according to which TN (x) = ξ 2 (x, TN ) J⊥ (x) .

(26)

Then for the correlation length one has ξ(x, TN ) = ξ(0, TN ) [1 − 4x]1/2 /(1 − x) .

(27)

The solid lines in Fig. 9 are the theoretical behaviours of ξ(x, T ) according to (24). In the inset the values estimated for the correlation lengths at TN from (27) are also included. As it appears from the comparison with the experimental data it can be concluded that for doping x ≤ 0.1 and T ≤ 800 K

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365

Fig. 9. Temperature dependence of the in-plane magnetic correlation length in spin doped La2 CuO4 , in the weak doping regime, in the assumption that the Hamiltonian (1) and the related quantities are modiﬁed according to the dilution model. The inset shows the extrapolation at TN of ξ(x, T ) given by (24) and the comparison with the values derived according (27), with the doping dependence of TN evidenced in Fig. 8 for x → 0

the system remains in the RC regime, with reduction of the spin stiﬀness according to the dilution. No evidence is achieved of a crossover to QD regime, as it is also conﬁrmed by the Gaussian contribution to the 63 Cu echo dephasing, which was experimentally found [9] to follow the temperature dependence −1 ∝ (ξ/T ), as it is derived from the scaling conditions applied to the qT2G integrated static generalized susceptibility q χ(q, 0) which controls the echo dephasing rate. Now we are going to discuss how the charge doping aﬀects the properties of 2DQHAF by referring to Sr doped La2 CuO4 . The eﬀect of itinerant holes is more complicate than the one due to spin vacancies, essentially because of the occurrence of charge inhom*ogeneities: the holes tend to segregate along stripes, leaving randomly distributed eﬀective magnetic moments inside AF domains, which in turn are long-range interacting. Cooperative spinfreezing is expected to occur on cooling, with consequent rise of low-energy

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Fig. 10. Temperature dependence of the correlation length in Sr-doped La2 CuO4 for doping amount smaller (a) and larger (b) than the “critical” value y = 0.02 (see Figs. 2 and 8), as deduced from 63 Cu NQR relaxation. In part b) of the Figure some representative data from neutron scattering [14] and from Imai et al. [15] are also reported. The solid lines are the best ﬁt behaviours according to (4), with the spin stiﬀness as adjustable parameter

spin excitations (cluster-spin glass phase, somewhat extending into the SC phase, see Fig. 2 and discussion later on). Let us ﬁrst consider the problem of the spin dynamics in the paramagnetic phase well above TN , where the 63 Cu dephasing rate is not so large to wipe-out the NQR signal [13]. From 63 Cu NQR relaxation measurements in La2−y Sry CuO4 , with a procedure similar to the one outlined above for spin-doped compounds, the correlation length has been extracted. The experimental data for ξ are compared in Fig. 10 with the theoretical behaviours according the RC regime (4), having left the spin stiﬀness ρ(y) as adjustable parameter. The conclusions that can be drawn from the Cu NQR relaxation are the following. For doping smaller than the percolation threshold the RC regime is still obeyed, although the spin stiﬀness decreases with the Sr content less than

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367

expected for itinerant holes (for which a dependence of the form ρ(y) ∝ y −1 has been derived [16]). For Sr content above x = 0.02, while in the high temperature region one has a tendency of the correlation length to follow the RC behaviour, in the low temperature range one has ξ(y, T → 0) ∝ 1/ny (with n in between 1 and 2). This dependence is diﬀerent from the one related to √ the average distance between holes, namely 1/ y, and therefore is compatible with the hypothesis of charge segregation in stripes, altering the topology from 2D to 1D. It is also remarked that from the sub-lattice magnetization data diﬀerent spin dynamics seems to drive the relaxation process in diﬀerent temperature ranges. Well above TN , where ξ is smaller than the average stripe distance ds , the stripes are mobile. On the contrary, in the low temperature range where ξ > ds the stripes appear quasi-static, their hopping rate being less that the NQR line width, i.e. about 100 kHz. We mention here that in the ordered state, in spin-diluted and in chargedoped La2 CuO4 , spin excitations diﬀerent from the paramagnon spin dynamics considered until now arise, related to eﬀective magnetic moments and their cooperative freezing. As regards instead the doping and temperature dependences of the ordered parameter, i.e. the sub-lattice magnetization, namely of the expectation value of the Cu2+ magnetic ion, we postpone the discussion, in order to include the eﬀects over all the doping range.

7 Spin and Charge Doped La2 CuO4 Near the AF Percolation Thresholds: Spin Stiﬀness, Correlation Length at the Transition and Staggered Magnetic Moment Now we extend the analysis of the NQR relaxation rates to the spin doping region where the dilution model, and therefore (23), is evidently a too crude assumption. In practice only 139 La NQR relaxation have been measured in the temperature range of interest. By a procedure similar the one outlined in the previous Section, the correlation length has been extracted. In Fig. 11 the absolute values and the temperature dependences derived in the strong dilution condition are compared with the data for light doping and with the theoretical behaviours described by (4), by leaving the spin stiﬀness as adjustable parameter. Again one is led to the conclusion that the RC regime does hold also for strong dilution. It is noted that also the values for ξ(x, TN ) deduced from (26), still follow the temperature trend of the NQR relaxation data. The unexpected reliability of the RC description in spin diluted AF has been independently conﬁrmed by Vajk et al. [18] by means of neutron scattering, up to x = 0.35. Only close to the percolation threshold (x = 0.41) for the nearest-neighbour square AF lattice and in the high temperature range, ξ(x, T ) seems to cross over from exponential behaviour to a power law (see Figure 3B in [18]).

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Fig. 11. In-plane magnetic correlation length in spin-doped La2 CuO4 (x: Zn or Mg amount substituting Cu) derived from nuclear relaxation rates (close symbols) and from TN (x) (open symbols). The solid lines track the RC behaviour, according to (4). The dotted line is the value ξ(x, T → TN ), for x 1. Data from [17]

It should be remarked, however, that in the strong dilution regime the reduction of the spin stiﬀness dramatically departs from the one predicted by the dilution model, as it is evidenced in Fig. 12. Another relevant observation involves the absolute value of the in plane correlation length at the transition temperature. In spite of the drastic reduction in the spin stiﬀness and of the Neel temperature, still the transition to the AF state occurs when the correlation length reaches an in-plane value around 150 lattice steps, as in pure or lightly doped systems. Analogous conclusion is obtained in regards of the charge-doped La2 CuO4 . In Fig. 13 we report the temperature behaviour of ξ(y, T ) deduced from a combination of 63 Cu and 139 La NQR relaxation measurements in La2−y Sry CuO4 , for Sr amount y = 0.016, close to the percolation threshold (see Fig. 8). It has to be observed that the doping amount y = 0.02, coinciding with the percolation threshold, would not provide more pertinent results. In fact, as it has been conﬁrmed by a detailed analysis of inelastic relaxation [19] for y close to yc a microscopic phase separation in AF and cluster-spin-glass phase is noticed. The Sr concentration y = 0.016 was found the closer one to the percolation threshold still providing information on the charge doped La2 CuO4 without detectable “contamination” from phase separation. Again the solid line in Fig. 13 is the theoretical behaviour according (4), in correspondence to a value of the spin stiﬀness reduced to almost one third of the one in pure 2DQHAF. Still one notes that the maximum of ξ is reached around

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Fig. 12. Spin stiﬀness ρ(x) in spin-diluted La2 CuO4 and comparison with the dependence expected within the dilution model (23)

TN = 150 K and corresponds to about 150 lattice constant, similarly to the spin doped compound. Another quantity of interest for the quantum eﬀects in disordered 2DHQAF is the zero-temperature staggered magnetic moment < µCu (x, T → 0) > along the local quantization axis , namely the dependence of the sublattice magnetization on spin dilution. The staggered magnetic moment is diﬀerent from the classical S = 1/2 value because of the quantum ﬂuctuations, that in turn are expected to increase with spin dilution. The quantity R(x, T = 0) =

< µCu (x, 0) > < µCu (0, 0) >

has been obtained to a good accuracy from the magnetic perturbation due to the local hyperﬁne ﬁeld on 139 La NQR spectra or from µSR precessional frequencies [21, 17] and recently evaluated also close to the percolation threshold from neutron diﬀraction in a single crystal of Zn-Mg doped La2 CuO4 [18]. In Fig. 14 we report the x-dependence of R as it results from a combination of NQR [21, 17] and neutron diﬀraction data [18]. While the classical doping dependence [24] (for S → ∞) as well as the one predicted by the quantum non-linear σ model [22] are not supported by the experimental ﬁndings, the data in 14 indicate a doping dependence of the form R = (xc − x)β , with critical exponent β = 0.45, close to the behaviour deduced from spin wave theory and T-matrix approach [23]. The non-classical critical exponent β is in substantial agreement with ﬁnite -size scaling analysis [25].

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Fig. 13. In plane magnetic correlation length in Sr doped La2 CuO4 , for Sr amount y= 0.016, close to the percolation threshold, as a function of the inverse temperature. The values are obtained along the procedure outlined in the text, from 63 Cu NQR and from 139 La NQR relaxation rates (data from [20]). The solid line tracks the behaviour according to (4) and yields a spin stiﬀness ρ = 107 K. Such a reduction implies a small growth, on cooling, of the correlation length and therefore a reduction in TN , the transition to the ordered state still occurring when ξ is about the same as in pure system

Fig. 14. Zero-temperature normalized staggered magnetic moment R in La2 CuO4 as a function of spin dilution, from neutron diﬀraction data [18] and from 139 La Zeeman perturbed NQR spectra or from µSR precessional frequencies [21,17] and comparison with the behaviours expected from some theoretical approaches [22-24] (see also [18] and references therein)

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As regards the temperature dependence of R, it appears that both in the light doping regime (where it has been derived from the eﬀect of the AF ﬁeld on the 139 La NQR spectra [21]) as well as for strong dilution (from elastic neutron diﬀraction [18]) an universal law of the form R(x = const, T ) ∝ [TN (x) − T ]n holds, with a small critical exponent n that appears to be around 0.2 for light doping while on approaching the percolation n increases to 0.3 (Fig. 15).

Fig. 15. Temperature dependence of the staggered magnetic moment in spin diluted La2 CuO4 . Up to a dilution amount x ≤ 0.12 a second order transition is detected and the critical exponent turns out about β = 0.2 (solid line, data from [21]). These data can hardly be ﬁtted by an exponent β = 0.3 (dotted line) that instead appears to justify the results for strong dilution, according to the ﬁndings from neutron scattering reported in [18]

We do not discuss in detail the complicate and still open issue of the magnetic excitations in the ordered state, namely for T < TN (x, y), in spin and/or charge disordered La2 CuO4 and only mention the following. Around the S = 0 impurity eﬀective magnetic moments µe have been envisaged, long-range interacting through the AF matrix and yielding anomalous lowenergy excitations superimposed to almost unaltered (at least for light doping) magnons. The cooperative freezing, on cooling, of the µe is evidenced by a peak in the 139 La NQR relaxation rates, occurring at the temperature Tf where the ﬂuctuation frequency of the site-dependent magnetic ﬁeld at the La nucleus becomes of the order of the quadrupole coupling constant. The “freezing”

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temperature Tf increases linearly with the Zn content x. For strong doping the situation in unclear. Even more complicate picture holds for the magnetic excitations in the ordered state of Sr-doped La2 CuO4 . Below TN (y) the holes itinerate along “rivers” (the stripes) separating small domains of almost unaltered AF. The stripes should “evaporate” when the holes tend to localize and their lowfrequency “diﬀusion” along the stripe can be expected to imply a strong relaxation mechanism. Below the localization temperature a region of magnetic perturbation around the localized hole is induced, with a spin-texture in the AF layer, equivalent to the extra-magnetic moments. Their ﬂuctuations cause maxima in the relaxation rates, with recovery laws characterized by a distribution of relaxation times, and a continuous freezing transition similar to the one occurring when the spin vacancies are induced by Zn doping. Again, the freezing temperature Tf increases linearly with the Sr amount. For some more details, see [6] and references therein.

8 The Cluster Spin-Glass Phase Above the Sr concentration yc = 0.02 in La2−y Sry CuO4 the AF ordered state is no longer attained at any temperature. The typical experimental observation regarding the 139 La NQR relaxation is reported in Fig. 16, for y = 0.03. One should remark that in this range of charge doping the recovery law does no longer keep the simple exponential form but it is rather given by a stretched exponential, with characteristic time τe . In discussing the experimental data in Fig. 16, ﬁrst of all one has to realize that in this temperature and doping regime, according to Sect. 6 and Fig. 10 b), the in-plane correlation length is drastically reduced (around 10–15 lattice steps) with respect to the theoretical RC value and practically temperature independent. Thus the marked peaks in the relaxation rates observed on cooling have to be attributed to a relaxation mechanism diﬀerent from the critical paramagnons imbedded in the behaviour of a divergent correlation length. It turns out that a good ﬁt of the data (particularly above about 6 K, see inset in Fig. 16) is obtained according to a law of the form 2 ), τe−1 ∝ τ /(1 + τ 2 ωm

(28)

with τ ∝ exp(E/T ). The peak in τe−1 occurs at a temperature Tg where the characteris−1 tic correlation time τ reaches ωm , the inverse of the measuring frequency (ωm = 2ωQ = 2π 12.4 MHz, with ωQ 139 La quadrupole coupling frequency, for the data in Fig. 16). Remarkably, Tg was found to decrease with y, approximately in the form Tg ∝ y −1 . Since a long ago Cho et al. [27] interpreted Tg as the freezing temperature into a cluster spin glass phase. It was believed that the localization of the holes along the “stripes” generates 2D ﬁnite size

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Fig. 16. Temperature dependence of the characteristic 139 La NQR relaxation rate τe−1 in La1.97 Sr0.03 CuO4 . In the inset the activated temperature behaviour is emphasized, the departure below Tg being the consequence of the distribution of relaxation rates involved in the stretched exponential recovery (data from [26])

AF domains, including an odd number of Cu2+ and therefore an eﬀective magnetic moment with enhanced ﬂuctuations. The walls of the domains are mobile, somewhat corresponding to charge density waves. The doping dependence of Tg is compatible with a mean ﬁeld argument whereby the magnetic coupling among clusters is proportional to the cluster size L2 ∝ ξ 2 ∝ 1/y. An alternative interpretation of the NQR relaxation data typically indicated by Fig. 16 could be given in terms of “ﬂuctuating stripes” . In this case the inverse of the correlation time in (28) represents the average frequency ωstr for the motion of the stripes. For T < Tg the stripes are “frozen” in the time scale of the NQR experiment. Although the stripes are deﬁned as “static” in neutron scattering experiment when ωstr decreases below about 1011 rad s−1 , they are actually not-static down to 350 mK [28]. A ﬁne conﬁrmation of the spin-glass features of the low temperature phase in Sr doped La2 CuO4 above yc = 0.02 has been accomplished by Wakimoto et al [29], by means of high resolution magnetic susceptibility measurements in single crystals. Diﬀerence between FC and ZFC magnetization were found below Tg = 6 K, in the crystal at y = 0.03. Furthermore a canonical spinglass order parameter was detected, displaying temperature and magnetic ﬁeld dependence in accordance to universal scaling relationships expected for spin glasses. The Curie constants evaluated from the in-plane susceptibility [29] resulted independent of the Sr concentration, a feature that might reﬂect an

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inhom*ogeneous distribution of the holes in the CuO2 plane, thus compatible with the stripe structure. Finally we should mention that in the light of µSR [30] and NMR-NQR experiments [31], the cluster spin glass phase was suggested to extend into the underdoped phase of superconducting LSCO, for hole content larger than about y = 0.05. On cooling from about the superconducting transition temperature, divergent behaviours of the relaxation rates were noticed, reminiscent of the one reported in Fig. 16 for the spin-glass phase, both in LSCO and in Ca-doped YBCO. Thus one is led to conclude that coexistence of superconductivity and of the spin freezing process typical of the cluster spin glass phase occurs in the underdoped phases of cuprate superconductors. In reality, it seems that in those phases a variety of low-energy magnetic excitations can actually be present, for instance sliding motions of orbital currents implying ﬂuctuating magnetic ﬁelds at the La or at the Y sites [32]. The µSR and NQR experimental observations, in particular the strengths, the distribution and the temperature dependences of the ﬂuctuating ﬁelds, are as well compatible with the freezing processes of sliding orbital vortex-anti vortex currents [32, 33]. The highly inhom*ogeneous charge distribution in the CuO2 plane in underdoped superconducting cuprates and in the cluster spin glass phase is proved by the phenomenon called “wipe out eﬀect” , namely the fact that on cooling some Cu nuclei no longer contribute to the NQR signal because their echo dephasing times become too short to allow one to detect them (see Julien et al. [31], and references therein).

9 The Quantum Critical Point in an Itinerant 2DAF – Eﬀect of Magnetic Field Going back to the schematic phase diagram depicted in Fig. 2 a very interesting issue involves the point at T = 0 and g = gc . This should be considered a quantum critical point (QCP). In fact, when for T → 0 the parameter g can be tuned, for instance by applying pressure, external magnetic ﬁeld or chemical composition, a quantum phase transition [34] can be induced. In strongly correlated electron systems, transition of quantum character are rather ubiquitous and of prominent interest is the problem of the spin dynamics accompanying the transition and of the related low-energy excitations [5]. In charge-doped typical 2DQHAF, as in Sr-doped La2 CuO4 , it appears diﬃcult to locate QCP’s, primarily because of the simultaneous tendency to phase separation, so that in general a multiphase system occurs, with AF and cluster-spin-glass or superconducting phases. Thus the experimental investigation of the spin-dynamics driving the phase transition could hardly be carried out in the 2DQHAF leading to high temperature superconductors. In

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recent times groups of related strongly correlated metals have been discovered where tuning between the AF and the paramagnetic phases can easily be obtained by controlling the atomic composition in intermetallic alloys. Au-doped CeCu6 can be considered the prototype of these systems, the magnetic response being of 2D character and the electronic properties of the Cu2+ ions being involved, as in the more conventional 2DQHAF. The low-temperature phase diagram for CeCu6−z Auz around the Au content zc = 0.1 is shown in Fig. 17. CuCe6 , namely in correspondence to z = 0, is a heavy fermion system, that below the Kondo temperature TK 6 K, where the 4f Ce electrons are delocalized into the Fermi sea, is a good example of Fermi liquid (FL), with itinerant pseudo-particles of fermionic character, speciﬁc heat going linearly with T and resistivity quadratic in temperature. For z > zc = 0.1 one has an itinerant AF metal with transition to the 3D AF state at TN = 2.3 K for z = 1. The Neel temperature TN decreases on decreasing z towards zc . On cooling along the line at z = zc one approaches the QCP, which is believed to result from a competition between Kondo mechanism, which tends to screen the Ce magnetic moments, and the long-range RKKY interaction favouring an ordered magnetic state. Also an external magnetic ﬁeld can be used to tune the system around the QCP, by suppressing the AF state for z > 0.1. Around the quantum criticality, inelastic neutron scattering revealed a generalized susceptibility of the form [35, 35] χ(q, ω) = 1/[f (q) + A ω α ] ,

(29)

Fig. 17. Schematic phase diagram (z, T ) for CeCu6−z Auz , as derived from a combination of experimental data ([37]) and speculative theoretical considerations

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with f (q) going to zero along lines of the Brillouin zone, thus corresponding to 2D magnetic ﬂuctuations in real space. Anomalous critical exponent α and (ω/T ) scaling have been also pointed out. By means of 63 Cu NMR-NQR relaxation the low frequency generalized spin susceptibility over all the Brillouin zone, hardly detectable by neutron scattering, can be accessed. Here we discuss the spin-lattice relaxation measurements in CeCu6−z Auz for z = 0, z = 0.1 and z = 0.8, for external magnetic ﬁeld ranging from zero (NQR) up to 110 kOe [38]. The 63 Cu NQR spin-lattice relaxation rates are reported in Fig. 18, in selected temperature ranges. As it appears from the Figure, the temperature behaviour of 2W for z = 0.8 is apparently similar to the one typical of itinerant metal, with carriers in AF interaction [39]. However, for T TN one observes a weak temperature dependence of 2W which contrasts with the critical behaviour that one would derive from (9)–(11) for a generalized susceptibility of a nearly AF metal [39], that would imply (2W/T ) ∼ (T − TN )−0.5 . On the contrary the data for 2W for T ≥ 30 K are rather accounted for by the weak logarithmic divergence expected for dipolar interactions, with anisotropic response function. Below about 20 K the almost constant value of 2W is likely to reﬂect the screening of the magnetic moments for T → TK . For z = 0 (pure CeCu6 ) the FL behaviour 2W ∝ T is obeyed, as expected for T TK .

1400

z=0 z=0.1 z=0.8

1200

-1

2W (s )

1000 800 600 400

α=0.7, ξ=3/T

1/2

α=1

200 0 0.1

100

T (K) Fig. 18. 63 Cu NQR relaxation rates in CeCu6−z Auz (for the resonance line at frequency around 11.3 MHz). The low-temperature data (T ≤ 1 K) for z = 0 and z = 0.1 are from other authors (see [38] and references therein). The solid lines correspond to (31) in the text, for α = 1 and α = 0.7

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From inelastic neutron scattering [40, 41], in combination to heuristic arguments [36], the magnetic response function, of 2D character, can be envisaged in the form

(T − iω/a)α α + f (k, ω, T ) = k (k, T ) , (30) χ−1 B 2D c with anomalous exponent α = 1, (ω/T ) scaling and renormalized Curie Weiss constant c. Let us ﬁrst discuss the NQR relaxation data, namely the case of zero external magnetic ﬁeld. From (30) one can derive the form of the generalized susceptibility by expanding f α (k, T ) in even powers of q starting from the critical AF wave vector under the condition that the excitation frequencies Γq ≥ ΓAF remain larger than the measuring frequency (See (11)–(15)). Then −2 BZ γ2 c 1 + (q ξ)2α 2 2W = kB T A , (31) 2N aα kB T α+1 2D

where (aT ) = ΓAF is the critical frequency and ξ(T, H = 0) is the correlation length (again in lattice units). In the case that the critical exponent α would be 1, from the 2D integration in (31) one has 2W =

2 qD γ 2 2 S(S + 1) A 2 , 8π 2 aT 1 + ξ 2 qD

(32)

√ with the correlation length given by [42, 5] ξ(T, H = 0) = (1/ T ) ln(TK /T ) and qD Debye-like wave vector. In Fig. 18 the solid lines correspond to (32) for a = 1010 rad s−1 K−1 and A = 3.8 kOe (according to [43]). It is evident that the experimental results for T → 0 indicate α = 1, as already noticed from neutron scattering. Numerical integration is required to derive 2W from (31) for α = 1. The line in Figure 18 has been obtained in correspondence to α = 0.7 and ΓAF = (1.5 × 1010 T ) rad s−1 and for ξ = 3 at T = 1 K. This value of the correlation length is close to the one derived from neutron scattering, where it was found [40] ξ 2 = (ω0 /kB T q02 ) with A2 , yielding ξ = 10 ˚ A at T = 1 K. One could remark (ω0 /q02 ) = 10 meV ˚ that below about 1 K the critical frequency ΓAF slows down to less than 1011 rad s−1 and therefore hardly detectable by neutron scattering because of resolution limits. Now we discuss the eﬀect of the magnetic ﬁeld. Since close to the quantum criticality, for both energy and ﬁeld dependence, the only scale in the response function is temperature, one can reformulate the derivation leading to (31) and (32) by simply substituting T with an eﬀective temperature 1/2 2 1/2 ≡ T + (Tmag )2 . (33) Tef f (H) = T 2 + (g µB H/kB )2 By considering ﬁrst, for simplicity of discussion, the case of α = 1, in the presence of the ﬁeld (32) is modiﬁed in

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−1 2 (2W/T ) ∝ (T 2 + Tmag ) ξ 2 (T, H) ,

(34)

showing that for T Tmag no ﬁeld dependence should be expected. This is in agreement with the trend of the data in Fig. 19. For strong ﬁeld so that Tmag is dominating, (34) predicts (W/T ) temperature independent and going as H −1 , as approximately indicated by the experimental results in Fig. 19, in the temperature range 0.5 − 2 K. The solid lines reported in (Fig. 19 b) are the theoretical behaviours obtained by means of numerical integration, having used in (31) for T , including in the correlation length, the eﬀective temperature given by (33). As it appears from the Figure, above a given temperature T ∗ (H), the experimental results for representative strengths of the ﬁeld justify rather well the theoretical behaviours. Below T ∗ (H) a drastic departure of the data from the theoretical trends is noticed, with a sudden decrease of the scaled relaxation rates on cooling. For T < T ∗ (H), W takes a temperature dependence approximately of the form W ∝ exp[−∆(H)/T ], typical of a system with a gap in the spin excitations. To give an order of magnitude, one ﬁnds ∆(H = 6.7 T) 0.6 K. Recent SQUID magnetization and 63 Cu NMR measurements in high ﬁelds [44] indicate that the gap in the magnetic excitations results from the saturation of the magnetization, implying the quenching of the spin ﬂuctuations. Summarizing, one can state that 63 Cu NQR-NMR spin lattice relaxation measurements in CeCu5.9 Au0.1 provide interesting new insights around QCP. On one side the k-integrated response function at low energy conﬁrm the 2D character of the magnetic ﬂuctuations, the anomalous critical exponent and the energy/temperature scaling. Furthermore, in the low temperature range where the critical frequency is below the resolution limit in neutron scattering new aspects involving the role of an external magnetic ﬁeld and unconventional scaling [44] are pointed out.

10 Summarizing Remarks In this Chapter it has been shown how NMR-NQR relaxation can be a valuable tool in order to study the correlated spin dynamics and the phase transitions in systems that in the last decade have called strong interest as models for quantum magnetism and as parents of high temperature superconductors, the square planar arrays of S = 1/2 magnetic moments in antiferromagnetic interaction (2DQHAF). Their rich phase diagram as a function of temperature and of spin dilution or hole injection, can be explored by means of NQR-NMR spin-lattice relaxation measurements. Quantitative estimate of the quantumﬂuctuations-aﬀected correlation length, spin stiﬀness and order parameter can be derived by resorting to the integration of the generalized susceptibility in the Brillouin zone, once that the wave vector dependence of the electron-nuclei hyperﬁne interaction is properly taken into account. First it has been shown that in pure, non-disordered, 2DQHAF the inplane correlation length ξ(0, T ) can actually be obtained from the relaxation

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Fig. 19. a) Temperature dependence of the 63 Cu relaxation rates in CeCu5.9 Au0.1 for H = 0 ( ), H = 6.7 Tesla (), H = 9.8 T ( ) and H = 11.2 T ( ). For H = 0 the relaxation rates have been extracted from the recovery laws for Zeeman perturbed NQR or for quadrupole perturbed NMR. Some modiﬁcations in the hyperﬁne coupling term can be expected (see text). In part b) of the Figure the relaxation rates have been scaled by T and the solid lines represent (31) in the text, with eﬀective temperatures according to (33). The arrows indicate the temperature T ∗ (H) at which the data depart from the theoretical forms. The dashed-dotted line tracks the behaviour of W in a magnetic ﬁeld H = 6.7 Tesla expected in the presence of a gap in the excitations. The departure from the behaviour predicted on the basis of (33) could be due to the breakdown for strong ﬁeld of that scaling law (that is feasible to hold only up to Tef f TK ) (see [44])

◦

•

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rates. By working in the prototype CFTD, where the exchange constant J is rather small, it has been possible to prove that the classical regime for ξ, with renormalization of the spin stiﬀness and of the spin wave velocity due to the quantum ﬂuctuations, holds in a wide (T /J) range. No evidence of the crossover to quantum critical regime, expected on the basis of an extension of the non-linear σ model, is observed. Similar conclusion holds also for La2 CuO4 , although only a more limited (T /J) range can be explored in this compounds, where J = 1500 K. On the other hand La2 CuO4 allows one to perform spin dilution by Zn2+ S = 0 (or Mg) for Cu2+ S = 1/2 substitutions or charge doping by Sr2+ for La3+ substitutions, which corresponds to injecting itinerant holes in the CuO2 plane. The disordered 2DQHAF is thus created and a variety of interesting eﬀects is observed. For moderate spin doping the dilution model is found to lead to a reliable description: the spin stiﬀness and therefore the correlation length are still the ones pertaining to the renormalized classical regime, once that the probability of a spin vacancy is taken into account in the AF Hamiltonian. For strong dilution the model is evidently inadequate, the spin stiﬀness decreasing with increasing the Zn or Mg content x with a x-dependence much stronger when the percolation threshold is approached. Still the spin doped La2 CuO4 was found to remain in the RC regime, ξ(x, T ) displaying a temperature dependence similar to the one for ξ(0, T ), once that the spin stiﬀness is renormalized to the corrected value. The transition to the 3D ordered state occurs at the temperature where ξ(x, T ) reaches about the same value as in the pure 2DQHAF, namely about 150 lattice steps. Similar results have been found also in charge doped La2 CuO4 , for a Sr content y = 0.016, not far from the percolation value y = 0.02. Again the temperature behaviour of ξ(y, T ) appears almost the same as in the pure compound, although with a strongly reduced spin stiﬀness, and ξ(y = 0.016) at TN turns out around 150 lattice steps. The reduction of the expectation value of the Cu2+ magnetic ion as a function of the spin dilution has also been derived. In accordance to neutron diﬀraction data it has been found that the x-dependence turns out of the form (x − xc )β with β = 0.45, close to the one expected from spin wave theories and T matrix description and in agreement with ﬁnite-size scaling. The temperature dependence of the staggered magnetic moment seems to follow a rather universal law in terms of the x-dependent Neel temperature, with an abrupt but continuous phase transition and a critical exponent close to 0.2 for small x, possibly increasing to 0.3 for large doping amounts. A system where a nice study of the spin dynamical properties at the disorder conditions corresponding to a quantum critical point has been possible, is CeCu5.9 Au0.1 . The NQR-NMR 63 Cu relaxation rates have provided enlightening insights on the magnetic response function, particularly when the critical frequency slows down below the resolution limit of neutron scattering and in regards of the role of an external magnetic ﬁeld. On one side a 2D response, with a critical exponent diﬀerent from 1 and the energy/temperature

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scaling have been conﬁrmed from the relaxation measurements involving the k-integrated dynamical susceptibility at low energy. On the other hand it has been shown that below a certain ﬁeld-dependent temperature the system crosses over to a phase of gapped spin excitations, with the quenching for T → 0 of the magnetization ﬂuctuations and unconventional magnetic ﬁeld scaling. On the whole it has been shown that the nuclei can be used as useful tools in the attempt to unravel the many static and dynamical phenomena occurring in 2DQHAF upon charge and spin doping. The powerfulness of the NQR-NMR measurements, when accompanied by a suitable analysis, has been illustrated vis a vis to inelastic neutron scattering. While in some cases ﬁne conﬁrmations of the data found by this technique have been obtained, in other cases the information obtained from NQR-NMR relaxation in regards of low-energy spin excitations have turned out even more subtle and novel aspects have been pointed out, in turn stimulating new scientiﬁc work.

Acknowledgments In this Chapter some results obtained in works carried out in cooperation with the authors indicated in the References have been used. In particular F. Borsa, R. Cantelli, F. Cordero, M. Corti, M. Eremin, M.J. Graf, A. Lascialfari, M. Julien, J. Spalek, V. Tognetti and A. Varlamov are gratefully thanked for their contributions and for useful discussions.

References 1. D.C. Johnston, in Handbook of Magnetic Materials Vol. 10, Ed. K.H.J. Buschow (Elsevier 1997) Chapter 1 2. S. Chakravarty, B.I. Halperin, D.R. Nelson: Phys.Rev. B 39, 2344 (1989) 3. P. Hasenfratz, F. Niedermayer: Phys. Lett. B 268, 231 (1991); Z. Phys. B 92, 91 (1993) 4. P.W. Anderson: Frontiers and Borderlines in Many-particle Physics Eds. R.A. Broglia and J.R. Schriﬀer (North Holland 1988) and references therein. 5. S. Sachdev: Quantum Phase Transitions (University Press, Cambridge 1999) 6. A. Rigamonti, F. Borsa, P. Carretta: Rep.Prog. Phys. 61, 1367 (1998) 7. M. Conradi, T. Guillion, A. Rigamonti: Phys. Rev. 31, 4388 (1985) 8. P. Carretta, T. Ciabattoni, A. Cuccoli, A. Rigamonti, V. Tognetti, P. Verrucchi: J. Appl. Magn. Resonance 19, 391 (2000) 9. P. Carretta, A. Rigamonti, R. Sala: Phys. Rev. B 55, 3734 (1997) 10. P. Carretta, T. Ciabattoni, A. Cuccoli, E.R. Mognaschi, A. Rigamonti, V. Tognetti, P. Verrucchi: Phys. Rev. Lett. 84, 366 (2000) 11. F.C. Chou et al: Phys. Rev. Lett. 70, 222 (1993) 12. H.H. Klauss et al: Phys. Rev. Lett. 85, 4590 (2000) 13. P. Carretta, F. Tedoldi, A. Rigamonti, F. Galli, F. Borsa, J.H. Cho and D.C. Johnston: Eur. Phys. J. B 10, 233 (1999)

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14. B. Keimer et al: Phys. Rev. B 46, 14034 (1992) 15. T. Imai et al: Phys. Rev. Lett. 70, 1002 (1993) 16. M. Acquarone: Physica B 259-261, 509 (1999), Proceedings of the SCES Conference, Paris (1998) 17. P. Carretta, A. Rigamonti, E. Todeschini, L. Malavasi: Acta Physica Polonica B 34 (2003), Proceedings of the SCES Conference 2002 (Kracow) 18. O.P. Vajk, P.K. Mang, M. Greven, P.M. Gehring, J.W. Lynn: Science 295, 1691 (2002) 19. A. Paolone, F. Cordero, R. Cantelli, M. Ferretti: Phys. Rev. B 66, 094503 (2002) 20. A. Paolone, R. Cantelli, F. Cordero, M. Corti, A. Rigamonti, M.Ferretti: Inter. J. Modern. Phys. B 17, 512 (2003) 21. M. Corti, A. Rigamonti, F. Tabak, P. Carretta, F. Licci, L. Raﬀo Phys.Rev.B 52, 4226 (1995) 22. Y.C. Cheng, A.H. Castro Neto: Phys. Rev. B 61, R3782 (2000) 23. A.L. Cheryshev, Y.C. Chen, A.H.Castro Neto: Phys. Rev B 65, 104407 (2002) 24. A.W. Sandvik: Phys. Rev. B 66, 024418 (2002) 25. K. Kato et al: Phys. Rev. Lett. 84, 4204 (2000) 26. A. Campana, M. Corti, A. Rigamonti. F. Cordero R. Cantelli: Europ. J. Phys. B 18, 49 (2000) 27. J.H. Cho, F. Borsa, D.C. Johnston, D.R. Torgeson: Phys. Rev. B 46, 3179 (1992) 28. P.M. Singer, A.W. Hunt, A.F. Cederstrom, T. Imai: cond-mat /0302077 (2003) 29. S. Wakimoto, S. Ueki, Y. Endoh, K. Yamada: Phys. Rev. B 62, 3547 (2000) 30. Ch. Niedermayer et al: Phys. Rev. Lett. 80, 3843 (1998) 31. M.H. Julien et al: Phys. Rev. B 63, 144508 (2001) See Ref.6 and references therein for early data. 32. See M. Eremin and A. Rigamonti, Phys. Rev. Lett. 88, 037002 (2002) and references therein. 33. A. Rigamonti, M. Eremin, A. Campana, P. Carretta, M. Corti, A. Lascialfari, P. Tedesco: Intern. J. Modern Physics B 17, 861 (2003) 34. J.A. Hertz: Phys. Rev. B 14, 1165 (1976) 35. Q. Si et al: Nature 413, 804 (2001) 36. A. Schroder et al: Nature 407, 351 (2000) 37. H. v.L¨ ohneysen et al: Physica B 223-224, 471 (1996); H. v.L¨ ohneysen: J. Phys. Condens. Matter 8, 9689 (1996) 38. P. Carretta, M. Giovannini, M. Horvatic, N. Papinutto, A. Rigamonti: Phys. Rev. B 68, 220404 (2003). See also N. Papinutto, M. J. Graf, P. Carretta, M. Giovannini and A. Rigamonti: Physica B 359, 89 (2005) 39. T. Moriya: Spin Fluctuations in Itinerant Electron Magnetism, Vol. 56 (Springer, Berlin 1985) 40. O. Stockert et al: Phys. Rev. Lett. 80, 5627 (1998) 41. A. Schroder et al: Phys. Rev. Lett. 80, 5623 (1998) 42. A.J. Millis: Phys. Rev. B 48, 7183 (1993) 43. M. Winkelmann, G. Fisher, B. Pilawa, E. Dormann: Eur. Phys. J. B 26, 199 (2002) 44. N. Papinutto, M.J. Graf, P. Carretta, A. Rigamonti, M. Giovannini, K. Sullivan: Proceedings of SCES’05 meeting (Vienna) to be published on Physica B

Two-Dimensional Exchange NMR and Relaxation Study of the Takagi Group Dynamics in Deuteron Glasses R. Kind Institute of Quantum Electronics, ETH-Hoenggerberg, 8093 Zurich Switzerland [emailprotected] Abstract. The dynamics of the deuteron glass Rb1−x (ND4 )x D2 PO4 has been studied with various one dimensional and two dimensional NMR techniques. Each of these techniques provides a low dimensional hom*omorphous mapping of the true situation, corresponding to a small piece in a huge puzzle. To create a model which is as close as possible to the true situation it is therefore necessary to combine all available macroscopic and microscopic information on the system. In this combination the NMR-techniques play a dominant role because of the broad spectral window extending from the mHz (2D NMR) to the GHz (1D NMR) region. In this contribution we show how the combination of various NMR techniques, together with symmetry relations, geometrical constraints and model calculations is leading to a consistent model of the deuteron glass dynamics. This model is based on a random Slater lattice with a certain amount of defects i.e., Takagi pairs and unpaired Takagi groups that can propagate through the lattice analogous to OH− and OH+ 3 ions in hexagonal ice.

1 Introduction In contrast to most contributions of this book, where a single investigation method is described and it’s application is demonstrated on various samples, we present here the application of diﬀerent complementary methods on a single class of mono-crystalline samples, the substitutionally disordered pseudo-spin glass system Rb1−x (ND4 )x D2 PO4 (D-RADP-x). It will be shown, that for the construction of a consistent microscopic model of the glass transition, a combination of NMR methods with diﬀerent spectral windows had to be applied and analyzed. The dynamics of the freezing transition in spin glasses and their dielectric analogues, namely, proton and deuteron glasses, has remained one of the important problems of condensed matter physics. In the latter category, the mixed ferroelectric-antiferroelectric (FE-AFE) solid solution Rb1−x (ND4 )x D2 R. Kind: Two-Dimensional Exchange NMR and Relaxation Study of the Takagi Group Dynamics in Deuteron Glasses, Lect. Notes Phys. 684, 383–405 (2006) c Springer-Verlag Berlin Heidelberg 2006 www.springerlink.com

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PO4 (DRADP-x) has probably been investigated more thoroughly than any other glassy system [1, 2, 3]. A real breakthrough in the understanding was achieved when the random-bond-random-ﬁeld model of Pirc et al. [4, 5] was introduced to explain the NMR line shapes of the acid deuterons in DRADP44 [6], as well as the Tl2+ ESR line shapes of Tl-doped RADP-70 [7]. This analysis revealed unambiguously the glassy character of the material, and conﬁrmed the thermally activated dynamics observed in earlier T1 measurements [8, 9]. A further proof of the glassy character was the observation of a bifurcation between the ﬁeld-cooled and zero-ﬁeld-cooled static dielectric susceptibility in DRADP-60 at 61 K [10]. The Edwards-Anderson order parameter qEA deﬁned in the models of [4, 5], which can be written as the second moment of the probability distribution function of the time averaged local polarization p as qEA

+1 = W (p) p2 dp, −1

1 where p = τ

p (t) dt, −1 < p < +1 ,

(1)

cannot, by itself, distinguish between a random freeze-out and a locally correlated freeze-out of the protons or deuterons, respectively, in biased hydrogen bonds. The length over which the deuterons are coherently frozen in one out of the six Slater conﬁgurations (H2 PO4 ) [11], has been derived from the diffuse diﬀracted x-ray intensity in RADP. It is 2.0 nm at low temperatures [12]. This is longer than the average deuteron-deuteron distance, but considerably shorter than the typical size of a conventional domain. Similar results were obtained from a 87 Rb NMR line shape analysis [13]. In [14] it was shown that in the KDP framework a great variety of pure Slater lattices (lattices containing only Slater groups) can be realized, with a ratio of FE to AFE conﬁgurations ranging continuously from zero to one. Using the proper ratio the low temperature 87 Rb NMR line shape can be reproduced. However, the picture presented above has two major drawbacks: First, it is assumed that the system is static at low temperatures, i.e. that the long time average of each local polarization pi is constant and diﬀers from zero, and second, it is based on symmetric hydrogen bonds in the paraelectric PE phase state. The ﬁrst assumption was disproved by the deuteron two-dimensional exchange NMR measurements at low temperatures in DRADA-32 of Dolinsek et al. [15], which revealed that all local polarizations vanish in the long-time (10–200 s) average, i.e., qEA = 0 in this time scale, whereas qEA diﬀers from zero in the time scale of one-dimensional NMR (10−3 s). The second assumption is conﬁrmed by the deuteron NMR measurements Bjorkstam [16] revealing symmetric potentials for the hydrogen bonds above Tc and asymmetric potentials below Tc . This is in apparent contradiction with the Slater ice rules [11], which predict only asymmetric hydrogen bond potentials. As 1D-NMR is performing a gliding time average with an integration time of the order of T2 the asymmetric potentials must be reversed stochastically or periodically with a correlation time much shorter than T2 to overcome the contradiction.

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As known since the early work of Slater [11] and Takagi [17], the corresponding bias ﬂuctuations, which include the exchange between FE and AFE Slater conﬁgurations, can take place at low temperatures exclusively via diﬀusion of unpaired Takagi groups (HPO4 and H3 PO4 ). It seems that this mechanism is dominant also in the PE high temperature phase. In the following sections we will ﬁrst present model calculations showing how the motion of unpaired Takagi groups aﬀects the order in a KDP lattice and then some 1D and 2D NMR experiments which prove the presence of this motion and the corresponding exchange of FE and AFE Slater groups.

2 Model of the Glass Phase Dynamics in DRADP-50 Since the pioneering work of J.C. Slater 1941 on the ferroelectric FE transition in KH2 PO4 (KDP) [11] it is known that due to the so called Pauling ice-rules only two out of the four protons linking adjacent PO4 groups via O-H...O bonds are close to each PO4 ion, see Fig. 1. The double well potentials of all O-H...O bonds are here asymmetric since any intra-bond proton transfer from one well to the other creates a Takagi [17] pair (HPO4 -H3 PO4 ) which has a higher energy than the Slater pair (H2 PO4 -H2 PO4 ) it is originating from. Nevertheless, for tractability reasons most order disorder model calculations for the KDP family, including the pseudo-spin model, are working with symmetric hydrogen bonds just using an FE or AFE order parameter eigenvector leading to the corresponding long range ordered Slater lattices. However, as mentioned in the introduction, the two FE and the four AFE domains are not the only possible Slater lattices. There exists a whole variety of Slater lattices which could all serve as equivalent ground states of

D 0

D 1/2

1/4

D 0

3/4

D D X-bonds

D 0

Y-bonds

1/4

3/4

D D

D PO4

c D

D D a

Fig. 1. PO4 -tetrahedra with hydrogen bonding network of the KDP-family

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R. Kind

an intra-bond proton or deuteron ordering. The crucial question is whether these ground states are stable or whether there exists a mechanism allowing transitions among these states. In analogy to hexagonal ice where isolated ion states HO− and H3 O+ can diﬀuse through the lattice without changing the internal energy, in the KDP family the diﬀusion of unpaired Takagi groups HPO4 (T 1) and H3 PO4 (T 3) could transform an initial Slater lattice into another. An unpaired Takagi group in an otherwise perfect Slater lattice has three symmetric and one strongly asymmetric hydrogen bond. Moving the proton on one of the symmetric bonds to the opposite side corresponds to moving the T 1 or T 3 along this bond to the adjacent PO4 group leaving a Slater group behind, see Fig. 2. However, moving the proton on the asymmetric bond to the opposite side would create a T 0 (PO4 ) or T 4 (H4 PO4 ) state which is highly improbable.

Fig. 2. Schematic illustration of the bond bias inversion in a random Slater lattice by the passage of an unpaired Takagi defect H3 PO4 (T 3). The left side shows the conﬁguration evolution in four steps (a)–(d). Corresponding double well potentials of the deuteron (indicated by the open circle) are shown on the right side

To check the eﬀect of the random walk of an unpaired Takagi group, we have used a rigid lattice model of 6 × 6 × 8 lattice constants in the x,y,z directions containing 1152 PO4 groups. As initial state for all calculations a long range ordered FE Slater lattice was used. In this lattice a Takagi T 1–T 3 pair was created and separated at an arbitrary position. Then the unpaired

Two-Dimensional Exchange NMR

387

Takagi groups could start their random walk through the lattice, while the state of every PO4 group was monitored. Periodic boundary conditions were used so when a T 3 or T 1 accidentally left the lattice on side it was reentered at the opposite side of the prism. Upon accidental recombination of a pair a new one was created again at an arbitrary position. After a large number (≈ 107 ) of individual steps a state is reached, which we call a random Slater lattice, where all six Slater conﬁgurations are about equally populated. This state, which corresponds to the situation in hexagonal ice, was used as initial state for any further calculations. The fact that such a state can be reached from an initially long range ordered state by the diﬀusion of unpaired Takagi groups clearly demonstrates the power of the mechanism. However, the ice-rule barrier is not the only short range interaction in the system especially when we deal with a random substitution of K or Rb ions by NH4 or ND4 groups as e.g., in the solid solution Rb1−x (ND4 )x D2 PO4 (DRADP-x). Depending on the value of x either the FE or the AFE Slater conﬁgurations become more probable than the random walk model predicts. To take care of this fact we have introduced a biased random walk of unpaired Takagi groups. The corresponding bias energy corresponds to an average Slater energy (2εav ). Starting with a random Slater lattice and depending on this average bias we can reach any ratio from 100% FE to 100% AFE Slater conﬁgurations. For symmetry reasons the two FE conﬁgurations are equally populated, as well as the four AFE conﬁgurations appear with equal probabilities so that the macroscopic polarization is always zero and the average tetragonal structure is established. Locally, however, the situation is more complicated. For a given “symmetric” O...D-O bridge of a T 3 or T 1 the Slater bias energy εjd depends in ﬁrst order on whether a short, or a long, or no N-D...O bridge is leading to one of the oxigens of the bridge, i.e., we have distribution of εjd that might be even quasi-continuous if also second order contributions are taken into account. The inner energy of two Slater lattices connected by a Takagi step diﬀer thus by εjd and so sort of a fractal energy landscape is established. For none of these Slater lattices the local symmetry, is tetragonal and non-polar, but as the time evolves the local time averaged symmetry develops these properties, as the 2D-NMR experiments revealed. To reach the observed local symmetry, however, the diﬀusion of unpaired Takagi groups is not suﬃcient, since a static distribution of Slater energies εjd biasing the random walk of the T 1 or T 3 always leads to static local deviations from the average symmetry. The only possibility to reach also locally the observed average symmetry, is a time dependent distribution of εd , where each local εjd averages to zero in the long time average. In the random-bond random-ﬁeld model this would correspond to a time dependent random ﬁeld with constant variance. The origin of the εjd is the random Rb-ND4 distribution but it is highly improbable that the symmetry is reached by a chemical exchange of Rb and ND4 groups, which would, of course, do the job. A more probable mechanisms based on the fact that each ND4 groups forms four N-D...O hydrogen bridges to PO4 oxygens,

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two of them being short, the other two long. This leads to four sites for the nitrogen deviating slightly from A site of the lattice. An exchange between these four sites, combined with the corresponding “short-long” exchange of the N-D...O bonds would provide the requested time dependence of the random ﬁeld. At high temperatures reorientational motion of the ND4 -groups can be observed (ND4 deuteron T1 minimum at 160 K) [8]. For geometrical reasons these reorientations are restricted to the symmetry operations of the tetrahedron, i.e., ±120◦ reorientations around the four trigonal axes, and 180◦ ﬂips around the three twofold axes. The latter are less probable, since all four N-D...O bonds have to be opened and reformed, whereas for the 120◦ reorientations only three bonds are aﬀected. Anyhow, during the reformation of the bonds, the ND4 -groups have the occasion to adjust their positions to the state of lowest potential energy, so that all four nitrogen positions are reached during successive reorientations. At low temperatures the reorientations are supposed to be frozen-in. Nevertheless, when the Takagi motion results in local polarizations that correspond to a high energy state for the actual ND4 position, there is a non-vanishing probability that the ND4 group reorients to reach a site with lower energy, thus changing the εd for the four O-D...O bonds involved. The next Takagi group visiting one or some of these bonds will ﬁnd thus changed biases.

3 87 Rb 2D Exchange-Diﬀerence NMR Reveals a Correlated Motion The aim of these experiments was to observe polarization ﬂuctuations during the glass ordering process to understand the way the order is established. The gradual freeze-out predicted by the pseudo-spin random-bond random-ﬁeld model should lead to a smooth static (but temperature dependent) probability distribution w(p) of local polarizations pi where all values between −1 and +1 are allowed [4, 5]. This can only be the case for the time averaged value of pi , as the instantaneous values are either −1 or +1, i.e., there must be a smooth static probability distribution of bond bias energies. This picture is conﬁrmed by the 87 Rb 1D NMR measurements indicating a smooth distribution of local polarizations for intra-bond hopping frequencies well above the hom*ogeneous NMR line width. On the other hand this picture is in conﬂict with the icerules as long as we do not allow a second mechanism at the same time namely the fast inversion of the ice-rule bias. Otherwise there would be a gap in w(p) around p = 0. Thus the ice-rule dilemma aﬀects not only the PE phase of the KDP family but the whole “freezing” process of the glass phase e.g., in DRADP-50. Assuming that the ice-bias ﬂuctuations are much less frequent than the individual intra-bond hopping of the deuterons in the asymmetric potential

Two-Dimensional Exchange NMR

389

wells, we have performed 87 Rb 2D-exchange NMR measurements at low temperatures for B 0 c. This orientation of the external magnetic ﬁeld is in so far special as the signal intensity above the Larmor frequency νL comes from Rb spins in a predominantly FE surrounding, whereas the signal intensity below νL from spins in a predominantly AFE environment [14]. Thus polarization ﬂuctuations should become visible in 2D-experiment provided that the ice-bias ﬂuctuations become slower than the hom*ogeneous line width. The 2D-exchange diﬀerence NMR data were recorded using the two pulse sequences one having a long mixing time, the other a short one (shorter than the expected exchange time) are shown in Fig. 3. The idea is to take the diﬀerence of the two resulting 2D exchange spectra (one with exchange, the other without exchange) to resolve also oﬀ-diagonal intensity close to the diagonal. To compensate for T1 eﬀects both pulse sequences have the same length by introducing a T1 weighting time which makes up for the diﬀerence in the two mixing times [18]. T1 -weighting

2D exchange

90x 90x τ tw1

90x 90x

90x 90x τ P8

90x

∆

tmix1 P4

90x 90x tw2

t1 P9

P10

∆

90x

180x

∆

tmix2 P11

Acq. t2

180x

s1 (t1 ,t2 ) t1 Acq. t2

∆ P12

s2 (t1 ,t2 ) t1

Fig. 3. Pulse sequences used for the 87Rb 2D-exchange-diﬀerence NMR measurements

The spectra show indeed some pronounced oﬀ-diagonal intensity which consists, however, of two rather narrow rims close to the diagonal, exhibiting frequency diﬀerences ∆ν between initial and ﬁnal state of less than 5 kHz, see Fig 4a. At this point the suspicion arose that the observed oﬀ-diagonal intensity could be the result of spectral spin diﬀusion and not of chemical exchange. To check this a separate study [19] was performed in the FE phase state of a RbH2 PO4 single crystal well below the phase transition temperature where the system is completely frozen-out. Any oﬀ-diagonal intensity can here be only the result of spectral spin diﬀusion. The time constant determined for this process was about 600 ms for ∆ν = 2 kHz, so that for the case of D-RADP0.5 a spin-diﬀusion time of 2.4 s can be expected in view of the 50% reduction of Rb atoms due to the substitution with ND4 . This time increases rapidly with ∆ν so that e.g., for ∆ν = 6 kHz we get 8 s. Thus for the mixing times τm of less than 1 s used in our measurements the spin diﬀusion contribution can safely be neglected.

390 -5

R. Kind 20

-10

c) 10

-15

4 -20 -25

-16

-30 -35 -35 -30 -25 -20 -15 -10

-6 -10

-26 -5

-26 -16 -6

14 24 34

-20 -20

-10

Fig. 4. Comparison of measured 87 Rb 2D exchange-diﬀerence NMR spectrum at 45 K (a) with model calculations: Uncorrelated bias inversions (b) and motion of unpaired Takagi groups in a Slater lattice (c)

This leaves us with the task to understand the narrow oﬀ-diagonal exchange rims. For this we have used the rigid lattice model described in the previous section but this time by taking only the eight closest deuterons to a given 87 Rb into account. The EFG-tensor and the corresponding NMR frequencies were calculated for all 28 possible deuteron conﬁgurations like in [14]. For symmetry reasons only 35 diﬀerent distinct frequencies were obtained, the distribution of which corresponding roughly to the measured 1D signal. This knowledge opens the path for a computer simulation of the 2D-exchange. Any of the 256 states can serve as initial state with frequency νi or after a mixing time (represented by a certain amount of deuteron intra-bond jumps selected by a random generator) as ﬁnal state with frequency νf , respectively. The resulting coordinates (νi ,νf ) are then accumulated in a 2D histogram for about 105 repetitions. Figure 4b shows the resulting 2D frequency pattern for eight steps between initial and ﬁnal state. It clearly displays frequency diﬀerences ∆ν of up to 40 kHz and is entirely diﬀerent from the measured rim structure. This clearly indicates, that independent ice-bias inversions do not explain the observed pattern [20]. On looking for a correlated mechanism with the required properties the idea arose to try out the diﬀusion of unpaired Takagi groups. As shown in Fig. 2 they have the property of inverting the ice bias on every hydrogen bond they pass. To check this mechanism we have used the rigid lattice model described in Sect. 2, but instead of monitoring the states of the PO4 groups, the positions of the sixteen closest deuterons to a given 87 Rb were observed and the corresponding NMR frequencies calculated. Starting every set with a freshly calculated random Slater lattice and the related initial frequency νi a certain number of random Takagi steps (16 to 4064) were calculated to reach the ﬁnal state with νf . Again the resulting coordinates (νi ,νf ) were accumulated in a 2D histogram for about 107 sets. The result is shown in Fig. 4c as 2D contour plot. At least for a small number of Takagi steps the pattern looks similar to the measurements. A better impression can be gained by looking at the cross sections through the model patterns, see Fig. 5. Here we clearly

Two-Dimensional Exchange NMR

16 steps

0.06 intensity (a.u.)

391

4096 steps 0.04 0.02 0.00 -6 -5 -4 -3 -2 -1 0

ν [kHz] Fig. 5. Cross-Section through the calculated 87 Rb 2D NMR exchange spectrum for two diﬀerent numbers of Takagi steps between initial and ﬁnal state. The intensity versus frequency shift curve clearly shows that most of the intensity is in the two rims close to the diagonal of Fig. 4c

ﬁnd most of the oﬀ-diagonal intensity close to the diagonal as in the experiment. This strongly supports the Takagi diﬀusion as responsible mechanism. However, for a ﬁnal proof it must be shown experimentally that the polarization ﬂuctuations of the PO4 groups (between the six Slater conﬁgurations) associated with the Takagi diﬀusion are really taking place.

4 Distinction of the Six Slater Conﬁgurations by the Anisotropic 31 P Chemical Shift Tensor The ﬁrst 31 P NMR measurements in the KDP family were performed to investigate the PE-FE phase transition in KH2 PO4 [21]. The authors have determined the anisotropic part σ of the 31 P chemical shift (CS) tensor in both the PE and FE phase states. Since the phosphorus atoms are located in the center of the PO4 tetrahedra the symmetry of the 31 P CS-tensor reﬂects the symmetry of the B-site in the KDP lattice. Because of the symmetry element 4z of this site in the PE phase the CS-tensor is diagonal with σxx = σyy = − 21 σzz :   −9 0 0 σPE [ppm] =  0 −9 0  , (2) 0 0 18 whereas the remaining symmetry element 2z in the FE phase requires the elements σxz and σyz to vanish:     30 ∓52 0 −15 ±52 0 σFE1,4 [ppm] =  ∓52 −15 0  , σFE2,3 [ppm] =  ±52 30 0  . 0 0 −15 0 0 −15 (3)

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The two sets are related by the diagonal glide planes of the FE space group Fdd2, i.e., x ⇒ y, y ⇒ −x and thus exchanging σxx and σyy while σxy changes sign. Within the set the tensors are related by the lost symmetry elements 2x and 2y relating the two FE domains with electric polarizations up or down, respectively. The CS tensor elements were obtained 31 P NMR from rotation patterns where the single crystal is rotated in the external magnetic ﬁeld B 0 around its a or c-axis, respectively. This corresponds to a rotation of the CS tensor . and the anisotropic part of the CS is given by the transformed element σzz σ = RσR−1 ,

∆ν = (σiso + σzz ) νL .

(4)

This yields the following angular dependencies for the a and c-rotations: a-rotation c-rotation

σzz = − 12 σxx + 12 (σzz − σyy ) cos (2β) + σyz sin (2β) , σzz = − 12 σzz + 12 (σxx − σyy ) cos (2β) + σxy sin (2β) .

(5)

For the FE CS tensor the sin(2β) vanishes for the a-rotation and the base line is determined by σiso − 12 σxx . From (5) it is evident that that both rotation patterns (a and c) are needed to determine the whole FE CS tensor. To determine the AFE CS tensors we have measured 31 P rotation patterns in a single crystal of DRADP-95 which undergoes an AFE ordering. The AFE phase state has the space group P21 21 21 , i.e. there are no elements left transforming the PO4 groups into themselves. Therefore there are no symmetry restrictions for the AFE CS tensor. Nevertheless, for unknown reasons the c-rotation does not show any angular dependence, meaning that (σxx − σyy ) = σxy = 0. The a-rotation is shown in Fig. 6 yielding for the four AFE CS tensors:   16.5 0 ±16.3 16.5 ∓51.8  , σAFE1.2 [ppm] =  0 ±16.3 ∓51.8 −33.0 (6)   16.5 0 ±51.8 16.5 ∓16.3  . σAFE3,4 [ppm] =  0 ±51.8 ∓16.3 −33.0 Though (5) allow only the determination of (σzz −σyy ) and σyz , the remaining elements can be obtained from the same rotation pattern because of the four AFE domains present in the lattice, which are related by the lost symmetry elements of the space group I42d of the PE high temperature phase. The four lines shown in Fig. 6 (right side) correspond to the four AFE domains, or in other words, to the four AFE Slater conﬁgurations. For unknown reasons the two inner lines have much less intensity than the two outer lines, though the number of spins is the same for all four lines. If the crystal is rotated around the a -axis, the inner and outer lines are exchanged, but again the inner lines have much less intensity. Thus it is dangerous to draw conclusions

angle [deg]

Two-Dimensional Exchange NMR

180

150

120

0 -10

-5

0 ν [kHz]

-10

-5

393

ν [kHz]

Fig. 6. Left: Measured 31 P NMR rotation pattern of DRADP-95 around the a-axis in the AFE phase state at T = 195 K. The two inner lines are only barely visible in this scale. Right: calculated angular dependence with ﬁtted chemical shift tensor

alone on the basis of signal intensities, especially if known symmetries are apparently violated. We know now the CS tensors of the six Slater conﬁgurations for the long range ordered systems. It is clear, that we cannot expect that their size is the same in DRADP-50, but the symmetry relations in the glass phase are very likely the same as for the pure systems since the average tetragonal non-polar symmetry of the PE phase is maintained in the glass phase. In DRADP-50 the chemical shift tensors of the 31 P give rise to an NMR spectrum with twelve separated lines for general orientation of the crystal in the external magnetic ﬁeld B 0 . For symmetry reasons one can ﬁnd orientations where sets of FE or AFE NMR lines merge to a single line. For instance if B 0 is either perpendicular to the c or to the a-axis of the crystal, the lines of corresponding Slater groups of the two physically non-equivalent 31 P sites in the primitive unit cell merge and the set of lines is reduced from twelve to six. Due to the inhom*ogeneous line widths of about 2 kHz the spectra are not well resolved, see Figs. 7, 8. Nevertheless, it was possible to determine the CS tensors from the low temperature rotation patterns by using the symmetry relations presented above and by ﬁtting the orientation dependence of the corresponding set of Gaussians simultaneously to the whole rotation pattern. We can again distinguish the six CS tensors for the two diﬀerent FE and the four AFE Slater conﬁgurations. In the c-rotation pattern only the FE CS tensors exhibit an angular dependence (satellites), while similar to the AFE phase state above, the lines originating from AFE Slater groups are all part of the central line.

394

R. Kind

angle [deg]

T = 57 K 180

180

150

120

0 -10

-5

0 ν [kHz]

0 -10

-5

ν [kHz]

Fig. 7. Left: Measured 31 P NMR rotation pattern of DRADP-50 around an axis in the a-b plane accidentally only close to the a-axis (oﬀset 7.3◦ ) at 57 K. Right: Corresponding calculated angular dependence exhibiting all twelve lines as obtained from the ﬁt. The full lines belong to AFE Slater conﬁgurations, the dotted to the FE ones

angle [deg]

T = 45 K 180

180

150

120

0 -10

-5

0 5 ν [kHz]

0 -10

-5

ν [kHz]

Fig. 8. Left: Measured 31 P NMR rotation pattern of DRADP-50 around the c-axis at 50 K. Right: Corresponding calculated angular dependence, where the full line belongs to the four AFE Slater groups, the dotted to the FE ones



 25.0 ∓49.7 0 σFE1,4 [ppm] =  ∓49.7 −14.1 0  , 0 0 −10.9 

 −25.0 ±49.7 0 0  . σFE2,3 [ppm] =  ±49.7 25 0 0 −10.9

(7)

Two-Dimensional Exchange NMR

 σAFE1.2 [ppm] = 

395



12.7 0 ±18.2 0 12.7 ∓51.3  , ±18.2 ∓51.3 −25.4 (8)



 12.7 0 ±51.3 12.7 ∓18.2  . σAFE3,4 [ppm] =  0 ±51.3 ∓18.2 −25.4

However, it was not possible to identify CS tensors belonging to Takagi conﬁgurations. To illustrate the strong temperature dependence of the spectra during the glassy ordering we have measured the a and c-rotation patterns also at 168 K, Fig. 9. The a-rotation pattern shows the angular dependence of the PE phase, with (σzz − σyy ) cos(2β), whereas the c-rotation pattern exhibits only a modulation of the line width, indicating the onset of the glassy ordering. In Fig. 9 a set of spectra with temperature as parameter is shown for a special orientation of the external magnetic ﬁeld B 0 , where for symmetry reasons only three lines are observed. It corresponds to an orientation in the c-rotation pattern where the lines of the two physically inequivalent FE Slater groups (of the primitive unit cell) merge. The two satellites correspond to the two FE polarizations up or down, respectively, whereas the central line is a superposition of all AFE-lines. Though the temperature dependence of the spectra (Fig. 10) looks like a dynamic line shape transition, it is in fact the result of the pseudo-static glass ordering. It should be noted, that because of the intensity problems mentioned above, the intensity ratio R = 1.86 of the AFE central line and one FE satellite is very likely not equal to the corresponding ratio of AFE and FE up, or FE down Slater groups. Thus one should not draw corresponding conclusions from the ratio R.

T = 168 K

180 150 120 90

0 -10

-5

0 ν [kHz]

T = 168 K

c-rot angle [deg]

angle [deg]

a-rot

0 -10

-5

0 5 ν [kHz]

Fig. 9. Measured 31 P NMR rotation patterns of DRADP-50 around the a-axis (left) and the c-axis (right) at 168 K

396

R. Kind

c⊥ B 0

T [K]

∠(a,B 0) = 45o

200 180 160 140 120 100 80 60 40

-10

-5

0 5 ν [kHz]

Fig. 10. Measured 31 P NMR spectra at various temperatures for a special orientation in the external magnetic ﬁeld where for symmetry reasons only three lines are visible. The satellites correspond to the two FE polarizations up or down, respectively, whereas the central line is a superposition of the four AFE contributions

5 Slow Polarization Fluctuations of the PO4 Groups Observed by 31 P 2D Exchange NMR Since an unambiguous assignment of the 31 P NMR lines to the corresponding Slater conﬁgurations is possible a 31 P 2D-exchange NMR experiment can reveal transitions between the diﬀerent Slater states. Thus if the random walk mechanism of unpaired T 3 or T 1 Takagi groups described in the previous section takes place it should be observable in the 31 P 2D-exchange NMR spectrum. Clearly there are no transitions possible between Slater states of diﬀerent 31 P sites and therefore there is no loss in generality if one performs the experiment with a set of six lines only. Following Ernst et al. [22] we start with a set of six 1D NMR lines with frequencies ω1 , ω2 , . . . , ω6 and measured integral intensities (zero’th moments) denoted by Mj0 (j = 1, 2, . . . , 6) among which chemical exchange is supposed to take place. The 2D intensities recorded after a mixing time τm are then given by (9) Iij (τmix ) = aij (τmix ) Mj0 and aij (τmix ) = [exp {Lτmix }]ij .

(10)

Neglecting relaxation, the transition matrix L relating the six Slater States has for symmetry reasons the form    L= 

−2 (K1 + K2) K1 0 K1 K2 K2 K1 −2 (K1 + K2) K1 0 K2 K2  0 K1 −2 (K1 + K2) K1 K2 K2  K1 0 K1 −2 (K1 + K2) K2 K2  K3 K3 K3 K3 −4K3 0 K3 K3 K3 K3 0 −4K3

, (11)

Two-Dimensional Exchange NMR

397

where the indices of L denote 1 = AFE1, 2 = AFE2, 3 = AFE3, 4 = AFE4, 5 = FE1, 2 = FE2. The Lij are transitions rates from state j to state i and the zeros indicate the transitions that are not directly possible for geometrical reasons, i.e., there are two successive visits of a T 3 or T 1 needed to perform such a transition. Provided that there is any chemical exchange taking place in the system, the intensities of the signals in the 1D spectrum are the continuous or equilibrium result of that mechanism. Therefore the transition rates Lij (except for the forbidden transitions) are related to Mj0 in the following way: Lij /Lji = Mi0 /Mj0 . The observed average tetragonal symmetry of the system indicates that the four AFE Slater conﬁgurations are equally populated and thus there is only a single transition rate K1 relating these states. Since the intensity of the two equivalent FE signals diﬀers from the one of the four AFE signals (M50 = M60 = M10 = M20 = M30 = M40 ) the rate K2 of the FEAFE transitions diﬀers from the rate K3 of the AFE-FE transitions. The six eigenvalues L∗jj of L are the time constants of the system and the backtransform of exp(L∗ τmix ) yields the time dependence of all signal intensities in the 2D spectrum. For a crystal orientation with B 0 perpendicular to the c-axis the four AFE lines merge to a single line and the 1D spectrum consists now of an AFE central line and two FE satellites corresponding to the two anti-parallel FE polarizations. This allows the use of a reduced transition matrix, as well as the analytical calculation of the 2D intensities. Provided that the 1D spectrum is given by M 0 = (1, R, 1)/(2 + R) the transition matrix becomes   −K K 0 L =  R K −2R K R K  , (12) 0 K −K where the indices of L denote 1 = FE ↑, 2 = AFE, 3 = FE ↓. From the measured 1D spectrum we have determined R = 1.86. The eigenvalues of L are [0, −(2 + R)K, −RK] and for the normalized intensities one obtains 1+0.5[(2+R) e(−KRτm ) +R e(−(2+R)Kτm ) ] (2+R)2 R2 +2R e(−(2+R)Kτm ) (2+R)2 (−(2+R)Kτm ) I21 = I32 = I23 = R−R e(2+R)2 1−0.5[(2+R) e(−KRτm ) −R e(−(2+R)Kτm ) ] I31 = , (2+R)2

I11 = I33 = I22 = I12 = I13 =

(13)

which for τm = 0 (no exchange) and for very long τm (exchange saturation) become     1 0 0 1 R 1 I =  0 R 0  (2 + R)−1 and I =  R R2 R  (2 + R),−2 respectively. (14) 1 R 1 0 0 1 If the above 2D exchange NMR saturation pattern is observed in the experiment, then we have the proof that all spins contributing to the 1D NMR

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ν1 [kHz] 15

ν2 [kHz]

-15 -20

τm = 800 ms

-25 -30

ν2 [kHz]

-15 -20

τm = 4 s

-25 -30

ν2 [kHz]

-15 -20

τm = 8 s

-25 -30

T = 45 K 31

Fig. 11. Contour plots of the P 2D-exchange NMR spectra in DRADP-55 at 45 K for three diﬀerent mixing times (τm )

signal are involved in the chemical exchange mechanism, i.e. every PO4 group takes on all six Slater conﬁgurations within a certain time span, so that the local time average is also reﬂecting the average non-polar tetragonal symmetry. Figure 11 shows the contour plots of the 2D exchange spectra for three diﬀerent mixing times illustrating the evolution of the cross-peaks. In Fig. 12 the evolution of the measured intensity ratios I12 /I11 and I13 /I11 is shown as a function of the mixing time τm for T = 45 K. The measurements clearly show that the 2D saturation pattern is reached after about 15 s. The ﬁtted value for the transition rate is K = 0.26 s−1 . This clearly shows the existence of slow

Two-Dimensional Exchange NMR

399

Intensity ratios

1.5

IAFE - FE↑

/ IFE↑ - FE↑

IFE↑ - FE↓

/ IFE↑ - FE↑

0.5

0 0

Mixing time (s)

Fig. 12. Evolution of the intensity ratios between cross-peaks of Fig. 11 and one diagonal FE peak as a function of the mixing time (τm )

PO4 polarization ﬂuctuations in D-RADP-50. Together with the results of the previous section we have the unambiguous proof that these polarization ﬂuctuations are resulting from the biased random walk of unpaired Takagi groups. These results were presented in [23, 24].

6 Interpretation of the

Rb T1 Measurements

According to second order time dependent perturbation theory the spectral density responsible for the relaxation rate is the Fourier transform of the auto-correlation function of the Rb-EFG-tensor ﬂuctuations. A rigid lattice point-charge model revealed that the intra-bond motions of the protons or deuterons on the O..D-O bonds yield the suﬃciently high ﬂuctuation amplitudes of the EFG tensor elements to explain the observed relaxation rate at the T1 minimum. The dynamics of this motion is coming from two sources: The motion of unpaired Takagi groups and the intra-bond motion of the protons or deuterons in the asymmetric potential wells. The autocorrelation function of the latter mechanism is well known. It decays exponentially with an autocorrelation time τc = τ + τ − /(τ + + τ − ), where τ + and τ − are the mean dwell times of the deuterons in the two potential wells. The amplitude is reduced by the so called depopulation factor 1 − p2 , with p =

τ+ − τ− = tanh (Eb /kB T ) . τ+ + τ−

(15)

where 2Eb is the energy needed to create a Takagi pair. For symmetric bonds p = 0 and τc = τ + /2 = τ0 , where τ0 is given by the Arrhenius law for symmetric bonds τ0 = τ∞ exp (Ea /kb T ).

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(b) p(t)

p(t)

(a) 0

-1 6.10

-1 6.15 time (a.u.)

6.20

time (a.u.)

Fig. 13. Calculated time dependence of the local polarization p(t) for p = 0.95 using an exponential distribution for the dwell times τ + and τ − . Superposed is a slower mechanism inverting the bias to p = −0.95, corresponding to the passage of an unpaired Takagi group (a). Gliding time average p(t) of p(t) exhibiting the bias inversions by successive passages of unpaired Takagi groups (b). The arrow indicates the bias inversion shown in Fig. 13a

In Fig. 13a an example for the function p(t) for a given hydrogen bond is shown for p = ±0.95, where p is the gliding short time average of p(t). The sign of p changes when an unpaired Takagi group is passing the bond. Such a change is indicated by the vertical dashed line. In Fig. 13b the ﬁltered function p(t) is shown, i.e., the fast asymmetric motion is ﬁltered out and only the eﬀect of the Takagi group motion remains. It is calculated to reach a long-time average of zero for p(t), thus symmetrizing the bond. The arrow indicates the Takagi passing shown in Fig. 13a. The autocorrelation function of p(t) is exponentially decaying with a pre-factor p2 and a time constant given by the average number of Takagi steps between two bias inversions nBI times the autocorrelation time τ0 . Since there are twice as many bonds than PO4 -groups, nBI is related to the number of unpaired Takagi groups NT u by: nBI =

1 1 ≈ . − ln (1 − 2NT u ) 2NT u

(16)

where 1/NT u ≈ 52 (1 + exp (2Eb /kB T )). The well known Fourier transform of the two autocorrelation functions yields for the normalized spectral density 2τ0 1 − p2 2nBI τ0 2 J (ωL ) = 1 − p + p2 . (17) 1 + (1 − p2 ) (ωτ0 )2 1 + (nBI ωτ0 )2 In contrast to the usual result of the BPP theory, τ0 is not the only temperature dependent contribution in both terms of (17), so that the apparent activation energy Eapp that is usually obtained from the slope of T1 diﬀers

Two-Dimensional Exchange NMR

300 150

401

40 T (K)

107

Inverse relaxation rates (µs)

106

105 1/W2 104 1/W1 103

102 0

1000/T (K–1) Fig. 14. Inverse 87 Rb spin-lattice relaxation rates 1/W1 and 1/W2 versus inverse temperature. The solid lines are a ﬁt of (17) and (18) to the data. The dashed line is calculated for a lower ice-rule barrier, see text

considerably from the activation energy Ea of τ0 , e.g., for the product nBI τ0 we have Eapp ≈ 2Eb + Ea . For Eb > 15 meV the second term is dominant down to 65 K and the observed Eapp of 80 meV consists mainly of this sum. The ﬁrst 87 Rb spin-lattice relaxation measurements on D-RADP-50, where the non-exponential magnetization recovery curves were systematically analyzed (to discriminate not only the mean relaxation rates at ωL and 2ωL , W1 and W2 , respectively, but also their probability distribution functions) were performed by N. Korner [25] and ﬁrst published in [26]. These measurements are shown in Fig. 14. In order to account for the reduced slope in the slow motion regime and for the distribution of the auto-correlation time, N. Korner used a Havrilak-Negami spectral density [27] for ﬁtting the theory to the experiment.

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The Havrilak-Negami spectral density is given by:

(ωτc )δ sin(δπ/2) 2 JHN (ω, τc , δ, ε) = sin ε arctan ω 1 + (ωτc )δ cos(δπ/2) −ε/2 . × 1 + 2(ωτc )δ cos(δπ/2) + (ωτc )2δ

(18)

Four our calculations we have used this spectral density for the second term in (17) with τc = τ0 NBI and ω = ωL or 2ωL , respectively. For the calculation of the solid lines the following parameters were used: Eapp = 80 meV, 2Eb = 37.6 meV, Ea = 42.4 meV, τ∞ = 0.7 × 10−13 s, νl = 98.163 MHz, δ = 1.0 and ε = 0.85. For the ﬁt we need also the mean square ﬂuctuation amplitudes A1 = 0.60 × 1013 s−2 and A2 = 0.23 × 1013 s−2 for the transitions ∆m = ±1 and ∆m = ±2, respectively. The ﬁt is excellent except for room temperature, i.e., where due to their high density the Takagi motion suﬀers from mutual hindering and where probably also the existence of T 0 and T 4 Takagi groups cannot be neglected anymore. To show the eﬀect of the ﬁrst term of (17) we have calculated the relaxation rates for 2Eb = 30 meV (dashed lines). Such a deviation clearly is not observed and whether the bending of the inverse relaxation rates at 40 K is due to this term was questionable without further low temperature data. However, such measurements exist in the literature [28] and there this eﬀect should be visible if it exists at all. This is in fact the case: In Fig. 2 of this reference where the T1 of the acid deuterons in DRADA-32 is displayed versus the inverse temperature, a systematic deviation from the calculated T1 is observed that has exactly the shape we are looking for. The same feature is observed (though less pronounced because of the scale and the number of data points) in Fig. 1 for the 87 Rb T1 in DRADP-50. These facts support our theory and allow for the ﬁrst time to determine the ice barrier in DRADP-50: Eb = 37.6 meV. Since the apparent activation energy Eapp = 80 meV, the activation energy for the symmetric bonds is Ea = 42.4 meV. This clearly shows that also for the asymmetric bonds (subdued to the ice-bias) in DRADP-50 there are two localized states for the deuterons.

7 Discussion The 87 Rb T1 results show that the time scale of the Takagi visits is much faster than expected from the 31 P 2D-exchange NMR. The Takagi visits are in fact the mechanism that establishes the glass order. It is therefore not astonishing that the oﬀ-diagonal rims in the 87 Rb 2D-exchange diﬀerence spectra become visible already for mixing times as short as 10 ms. These rims can even be observed at room temperature for short mixing times [29]. The exchange time determined from 31 P 2D-exchange NMR is thus not the time between two Takagi visits to a given PO4 group as erroneously stated in [25] but reﬂects the presence of a much slower motion, namely the reorientation of

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Fig. 15. ND4 hydrogen bonding scheme in DRAD-P-50 with two short and two long N-D...O bonds. The average local tetragonal symmetry can only be restored by a stochastic reorientation of the bonding scheme in ±90◦ steps. The 31P 2D exchange NMR measurements revealed a correlation time for this motion of the order of seconds at 45 K

the ND4 groups. A frozen-in conﬁguration of all ND4 groups (Fig. 15) acts via the corresponding Slater energies εj like a maze for the Takagi group motions leading to a certain set of glassy states. With the ND4 reorientation this maze is slowly changing resulting in a corresponding change of the glassy states. Within a long enough time span enough possible states are taken on so that the non-polar tetragonal tetragonal symmetry is established also in the local time average. To conclude we want to resume all known experimental facts concerning the dynamics in the deuteron glass DRADP-50. The following facts are deduced from combining the results of all experiments performed on the system so far. They include all macroscopic techniques, as well as, x-ray and neutron scattering results, and, of course, NMR and ESR measurements. 1. Down to at least 40 K the system relaxes at any temperature to a dynamic equilibrium state. Though the responsible mechanisms are slowing down, they are far from being frozen-in. 2. This dynamic equilibrium state consists of an inﬁnite sequence of diﬀerent Slater lattices, so that the non-polar tetragonal symmetry, as well as the speciﬁc ratio of FE and AFE Slater groups is established in the long time average for every single PO4 -group. 3. Furthermore, there exists an intermediate dynamic equilibrium state, that consists of a ﬁnite sequence of diﬀerent Slater lattices. It is observed when a gliding time average of ﬁnite length is applied to all local ﬂuctuations of the system before taking the space average. The gliding time average can be achieved by a typical 1D quadrupole perturbed NMR experiment where the integration time is of the order of T2 and where the deviation from the average local symmetry is changing the EFG-tensor of the nucleus under investigation linearly. This intermediate dynamic equilibrium

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state is clearly not the result of a slowing down, but is resulting from a pseudo static glass ordering. This means, it can be described by a randombond random-ﬁeld model with “static” random interactions and a “static” random ﬁeld. Though the local gliding time-average is time dependent on a longer time scale, its spatial distribution function (and its moments) is always time independent. This is reﬂected in the fact that, e.g., the NMR line shape does not depend on the absolute time. 4. We are thus faced with three mechanisms with diﬀerent characteristic times. A: The uncorrelated deuteron intra-bond jumps in the ice-rule biased double well potentials. This motion creates or annihilates the (T 1-T 3) Takagi pairs with the loss or gain of the energy 2Eb . B: Once the Takagi pairs are formed there is a non-vanishing probability that they separate and move independently through the Slater lattice as unpaired Takagi groups, hindered or aided by the Slater energies according to their signs, until they recombine with the original or another partner. At low temperatures there is also a non vanishing probability of getting trapped by a lattice defect. Such Takagi groups are then out of the game until they become loose again at higher temperatures. The motion of unpaired Takagi groups is neutralizing the ice biases of the system. However, this motion alone cannot restore the high temperature symmetry. C: The “static” distribution of Slater energies εd always leads to a “static” distribution of local polarizations corresponding to the results of the RB-RF model. Note that the εd are resulting not only from the random ﬁeld but also from the random interactions. As mentioned in Sect. 2 this requires the motion of the nitrogens among the four possible sites (corresponding to the N-D...O hydrogen bonding scheme) induced by the reorientational motion of the ND4 -groups. This motion must be considerably slower than the averaging performed by the Takagi group motion, otherwise the continuous glass ordering could not be observed at all and the glass order parameter qEA determined from 1D-NMR would never diﬀer from zero. In a recent 14 N T1 study [30] the interpretation of the results by the authors diﬀer considerably from our model. However, a closer analysis of the data revealed that they can as well be explained with our model.

Acknowledgments The author would like to thank his former PhD students P.M. Cereghetti, C. Jeitziner, T. Koenig and N. Korner for performing the measurements and some of the model calculations presented here. Furthermore he is very grateful to Professor Robert Blinc – to whom this contribution is dedicated – for the stimulating scientiﬁc cooperation and his repeated hospitality over almost 40 years, as well as for his friendship.

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References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

21. 22. 23. 24. 25.

26. 27. 28. 29. 30.

E. Courtens: J. Phys. (Paris) Lett. 43, L199 (1982) E. Courtens: Ferroelectrics 72, 229 (1987) U.T. H¨ ochli, K. Knorr, A. Loidl: Adv. Phys. 39, 405 (1990) R. Pirc, B. Tadic, R. Blinc: Phys. Rev. B 36, 405 (1990) R. Pirc, B. Tadic, R. Blinc, R. Kind: Phys. Rev. B 43, 2501 (1991) R. Blinc, J. Dolinsek, R. Pirc, B. Tadic, B. Zalar, R. Kind, O. Liechti: Phys. Rev. Lett. 63, 2284 (1989) R. Kind, R. Blinc, J. Dolinsek, N. Korner, B. Zalar, P. Cevc, N.S. Dalal, J. Delooze: Phys. Rev. B 43, 2511 (1991) R. Blinc, D.C. Ailion, B. G¨ unther, S. Zumer: Phys. Rev. Lett. 57, 2826 (1986) J. Slak, R. Kind, R. Blinc, E. Courtens, S. Zumer: Phys. Rev. B 30, 85 (1984) A. Levstik, C. Filipic, Z. Kutnjak, I. Levstik, R. Pirc, B. Tadic, R. Blinc: Phys. Rev. Lett. 66, 2368 (1991) J.C. Slater: J. Chem. Phys. 9, 16 (1941) R.A. Cowley, T.W. Ryan, E. Courtens: Z. Phys. B 65, 181 (1986) N. Korner, R. Kind: Phys. Rev. B 49, 5918 (1994) R. Kind, N. Korner, T. Koenig, C. Jeitziner: J. Korean Phys. Soc. 32 S799 (1998) J. Dolinsek, B. Zalar, R. Blinc: Phys. Rev. B 50, 805 (1994) J.L. Bjorkstam: Phys. Rev. 153, 599 (1967) Y. Takagi: J. Phys. Soc. Jpn. 3, 273 (1948) J. Dolinsek, G. Papavassiliu: Phys. Rev B 55, 8755 (1997) P.M. Cereghetti, R. Kind: J. Magn. Resonance 138, 12 (1999) C. Jeitziner: NMR studies of the low-temperature structure and dynamics of the pseudo-spin glass D-RADP-X. PhD. Thesis, ETH, Diss. Nr. 13257, Zurich (1999) R. Blinc, M. Burgar, V. Rutar, J. Seliger, I. Zupancic: Phys. Rev. Lett. 38, 92, (1977) R.R. Ernst, G. Bodenhausen, A. Wokaun: Principles of nuclear magnetic resonance in one and two dimensions (Clarendon Press, Oxford 1987) P.M. Cereghetti: On the dynamics of glassy phase states: An NMR investigation. PhD. Thesis, ETH, Diss. Nr. 13806, Zurich (2000) R. Kind, P.M. Cerghetti, Ch.A. Jeitziner, B. Zalar, J. Dolinsek, R. Blinc: Phys. Rev. Lett. 88, 195501-1 (2002) N. Korner: From long range order to glass order: Static and dynamic properties of the solid solution Rb1−x (ND4 )x D2 PO4 . PhD. Thesis, ETH, Diss. No. 9952, Zurich (1993) N. Korner, Ch. Pfammatter, R. Kind: Phys. Rev. Lett. 70, 1283, (1993) P.A. Beckmann: Physics Reports 171, 85–128 (1988) J. Dolinsek, D. Arcon, B. Zalar, R. Pirc, R. Blinc, R. Kind: Phys. Rev. B 54, R6811 (1996) Th.J. Koenig: Cluster dynamics in the solid solution D-RADP-x investigated by 87 Rb NMR. PhD. Thesis, ETH, Diss. No. 12027, Zurich (1997) A. Gregorovic, B. Zalar, R. Blinc, D. Ailion: Phys. Rev. B 60, 76 (1999)

Characterising Porous Media J.H. Strange1 and J. Mitchell2 1

School of Physical Sciences, University of Kent, Canterbury, Kent, UK, CT2 7NR [emailprotected] Department of Physics, University of Surrey, Surrey, UK, GU2 7XH [emailprotected]

Abstract. The method of Nuclear Magnetic Resonance cryoporometry has gained popularity since its inception in 1993 as a non-destructive technique for measuring pore size distributions in the nano-scale range. NMR cryoporometry is a secondary method of measuring pore sizes by observation of the depressed melting point of a conﬁned liquid. The melting point depression constant of the absorbate has to be determined empirically although this constant is only a function of the absorbed liquid and its associated solid, not the porous matrix. Cryoporometry has the major advantage of oﬀering, with care, directly calibrated measurements of pore volume as a function of pore diameter, of non-destructive pore measurement, structural resolution of spatially dependent pore size distributions, and behavioural information about the conﬁned liquid. This chapter focuses on the history of NMR cryoporometry, the basic equipment required to run an experiment, and highlights some of the major results that have been achieved by various research groups around the world using this technique.

1 Introduction Porous media are prevalent in the natural world and are widely used for industrial applications. Porous materials exist in a vast range of forms; everything from biological cells to rocks, drying agents and catalysts in chemical reactors. It is important to be able to characterise the many properties of these systems, such as porosity, pore size, permeability, and surface morphology. Only under exceptional circ*mstances will porous media contain pores of a single shape and size. Normally the pores come in a range of sizes. Knowing the distribution of pore sizes in a material is particularly important when using porous media in technical applications. For example, the pore geometry can inﬂuence the use of porous catalysts where the accessible surface area (related to the open pore volume) will determine the reaction rate. There is great interest in techniques that can non-destructively, easily, and inexpensively measure pore size distributions. The technique of Nuclear Magnetic Resonance (NMR) J.H. Strange and J. Mitchell: Characterising Porous Media, Lect. Notes Phys. 684, 407–430 (2006) c Springer-Verlag Berlin Heidelberg 2006 www.springerlink.com

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cryoporometry meets these requirements. It is a method for determining calibrated pore size distributions using the depression in melting point of a liquid conﬁned in pores compared to the bulk melting point of the same liquid. It is known that the melting point of small crystals is depressed relative to the bulk crystal melting point. The melting temperature is directly related to the size of the crystal. Therefore, when the crystal size is limited by a conﬁning geometry, the melting point depression is related to the size of the conﬁning space. NMR cryoporometry relies on the established Gibbs-Thomson equation (1) [1, 2, 3, 4, 5, 6]. The equation relates the depression in melting point (∆T) of a conﬁned liquid to the diameter of the conﬁning pore (x) by a constant (k). It must be assumed in order to determine a calibrated pore volume that the density of the absorbed liquid remains constant throughout the experiment, and that the absorbate remains pure. ∆T =

k . x

(1)

The Gibbs-Thomson equation (1), derived in terms of the absorbate’s properties, was veriﬁed by Jackson and McKenna [7] in a study of the thermal properties of organic crystals in porous media using Diﬀerential Scanning Calorimetry (DSC). This was exploited in conjunction with variable temperature NMR measurements by Strange et al. in 1993 [8] to measure pore size distributions, and the method named ‘NMR cryoporometry’. Earlier NMR studies demonstrated that the molecular behaviour of absorbates conﬁned in porous media was changed considerably from that of the bulk material over a wide temperature range [9, 10, 11]. The mechanisms behind the altered relaxation times were discussed in terms of surface interactions. The presence of a Non-Frozen Surface Layer (NFSL) on the pore wall was ﬁrst observed around 1973 [11]. Later molecular motion studies conﬁrmed the presence of a NFSL in a variety of systems and explored the general modiﬁcations in molecular mobility of conﬁned materials [12, 13, 14]. In general it was found that crystalline materials conﬁned in porous media form a twophase system below their depressed melting point. A core lattice is normally present of similar structure and properties to the bulk solid. A more disordered surface layer also appears, possibly being a plastic crystal phase or even a glassy state, extending for approximately two molecular layers in depth and having considerable molecular mobility. The ﬁrst cryoporometry experiments were conducted by simply observing the change in NMR signal intensity of a frozen sample as it melted, with very simple temperature control. A sample was frozen, placed in the spectrometer and the liquid signal amplitude observed as it slowly warmed in a thermally insulated container. An obvious problem was the lack of control over the sample warming rate. Without this control, the pore size could not be reliably determined using equation (1). Subsequent versions of the cryoporometry apparatus included programmable temperature control and warming rates of the NMR sample. Various methods of attaining temperature control were used, oﬀering

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diﬀerent advantages. Gas ﬂow temperature control was most frequently employed, although this does have limitations. Improvements to the temperature control will be vital to the continued advancement of cryoporometry. At the same time as the introduction of cryoporometry by Strange et al. [8], a paper was published on a related topic by Overloop et al. [15] using a high-resolution spectrometer. This group performed freezing studies of water in porous media, noting the hysterisis occurring due to super cooling. Although additional information on the behaviour of the conﬁned material can be gleaned from cooling measurements (it is good practise to observe both the freezing and melting sections of each cycle), pore size distributions should be derived only from data collected on a warming run. Over small temperature changes the phase transition of a material melting in a pore can involve meta-stable states due to super-cooling eﬀects. NMR cryoporometry is a secondary method of measuring pore sizes because the melting point depression constant of the absorbate (k) has to be initially calibrated using samples with known pore diameters. This constant is only a function of the absorbed liquid and its associated solid, not the porous matrix. It is commonly evaluated using model samples with narrow pore size distributions using the BJH method of (nitrogen) gas adsorption to obtain the pore surface-to-volume ratio [16]. The NMR cryoporometry experiment provides directly calibrated measurements of pore volume as a function of pore diameter. Two distinct methods of obtaining NMR cryoporometry data have been devised: scanning measurements (using a continuous temperature ramp) [17] and static measurements (using discontinuous temperature steps) [18]. In both cases the NMR cryoporometry measurement records the number of 1 H protons in molecules in a liquid state as a function of temperature. This is possible because NMR can be used to distinguish easily between a solid and liquid due to the very diﬀerent nuclear magnetic relaxation of the protons in the two states. This technique therefore provides a measure of the total number of molecules that have undergone a solid-to-liquid phase transition at any given temperature. This is taken to be proportional to the total liquid volume, vL , as a function of temperature and can be used to determine the volume distribution, dv/dx, of pores with diameter x; see (2) [8]. dv dvL k = . dx dT x2

(2)

NMR cryoporometry can be used for studying more than just pore size distributions. In this chapter we will discuss a number of applications of cryoporometry. Combined with complimentary NMR techniques, it can be used to categorise a range of pore properties other than the geometry, including surface wetting, surface aﬃnity and absorbate interactions. NMR cryoporometry has a unique place among the methods of porosimetry in that it can be combined with Magnetic Resonance Imaging (MRI) to provide spatially resolved pore characterisation [19, 20, 21].

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2 Measurement of Liquid Fraction Using NMR For basic NMR cryoporometry measurements, we are interested in recording the total volume of liquid in a sample as a function of temperature. The transverse relaxation time (T2 ) [22] of a liquid is easily distinguishable from that of a solid. Liquids have characteristically long T2 relaxation times, ranging from milliseconds to seconds, whereas solids usually have T2 relaxation times in the order of microseconds. This generally allows the NMR signal from the liquid fraction of the absorbate to be measured independently from the solid fraction. Water or hydrogenous organic liquids are usually used as the absorbates in cryoporometry experiments, and are easily detected since they exhibit a strong NMR proton signal. The observed proton magnetisation is a measure of the number of hydrogen nuclei in the liquid. If the liquid density remains constant, the signal will be linearly proportional to the liquid volume [8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. In cryoporometry experiments we are concerned with the precise quantity of liquid in the sample. The height of an NMR spin-echo [23] can be used as a measure of the liquid volume. This is usually the most appropriate NMR measurement of relative magnetisation since minimal post-measurement data extrapolation is required to determine the signal at zero time. In an ideal system the Free Induction Decay (FID) [22] (signal following a 90◦ pulse) directly provides a measure of the transverse magnetisation, I(0); see (3). The signal intensity is often represented as an exponentially decaying function of time as shown in (3). However, in a typical real system the FID will depend on magnetic susceptibility changes throughout the sample and inhom*ogeneities in the applied magnetic ﬁeld and as well as nuclear magnetic dipolar interactions. These eﬀects cause the spins to dephase more rapidly than they would in a hom*ogeneous static ﬁeld. Unless T2 is short (< 300 µs for low-resolution spectrometers) the time constant taken from a real FID signal will be strongly dependent on the spread in Larmor frequencies of the spins due to B0 inhom*ogeneities; see (4). The measured transverse decay time T2∗ is often a property of the spectrometer magnet and not the sample. It comes about due to a loss in coherence of the spins in the measuring frame of reference. The FID could be extrapolated back to zero time to provide the full liquid signal intensity but since the shape of the decay would be determined by the unknown magnet system and sample geometry, assumptions would have to be made. I(τ ) = I(0) e−τ /T2 .

(3)

γ∆B0 1 1 . ≈ + T2∗ T2 2

(4)

So for cryoporometry measurements, where the liquid component is of interest and has a T2 greater than 10 ms, direct FID measurements are of limited value. To overcome this problem a spin echo can be used and thus observe the liquid signal, ideally without seeing any residual solid signal. By using a

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Fig. 1. Producing a spin-echo: The nuclear spins start parallel to the z-axis. (a) A 90◦ pulse along the x-axis shifts the spins onto the y-axis. (b) The spins dephase due to ﬁeld inhom*ogeneities. (c) The 180◦ pulse rotates the spins about the y-axis and so they move back into phase (d) forming an echo

90◦x − τ − 180◦y − τ -echo pulse sequence, the loss in coherence of the spins due to magnetic inhom*ogeneities can be overcome. The 90◦x pulse shifts the spins onto the y-axis where they start to recover but dephase during the time of the experiment. After a time τ (where τ > T2∗ ) they have dephased in the x-y plane. Applying an 180◦y pulse rotates the spins about the y-axis, causing them to rephase. A maximum signal will be measured at 2τ ; see Fig. 1. This is known as a spin echo and the height of the echo is directly dependent on the volume of liquid in the sample. The echo amplitude is attenuated by transverse relaxation according to (3). Further to this we can capture extra information by repeating the sequence to form an echo train, called the Meiboom-Gill modiﬁed Carr-Purcell (CPMG) sequence [24, 25]. A closely spaced train of echoes with short τ can overcome attenuation of the echo amplitude due to diﬀusion in magnetic ﬁeld gradients. By applying the 180◦ pulse π/2 out of phase with the excitation pulse and taking only the even echoes we can negate signal loss due to inaccurate 180◦ pulses. The envelope of the echo train gives a good approximation to the FID that would be observed if all the spins were in an hom*ogeneous magnetic ﬁeld, and hence provides the true T2 ; see Fig. 2. This then allows precise estimation of the signal intensity arising from liquid in the sample so quantifying the liquid content.

3 The NMR Cryoporometry Experiment In a standard cryoporometry pore volume measurement the samples are imbibed with liquid to just overﬁll the pores. The extra bulk liquid between the grains provides the bulk melting point as a source of reference. Care must

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Fig. 2. To ﬁnd the T2 of a liquid, multiple spin-echoes can be plotted on the same graph. The maxima describe the transverse relaxation curve. In the CPMG measurement, this is performed in a single scan by repeating the 180◦ refocusing pulse

be taken to choose a suitable absorbate to ﬁll the pores. Ideally, something should be known of the porous sample before the experiment is conducted. Water, for example, is more suited to very small pore diameters, whereas an organic liquid like cyclohexane would have to be used if the sample were hydrophobic. Also some organic absorbates are totally unsuitable because the solid sublimes easily or the liquid is very volatile. This would be revealed by a decrease of integral liquid volume during the cryoporometry experiment. Samples used in cryoporometry are normally sealed in glass tubes under vacuum. The absorbate is then completely frozen, e.g., by cooling in liquid nitrogen. For relaxation measurements the liquid should be pumped (frozen, evacuated, thawed and the process repeated several times) to remove any dissolved oxygen. Any paramagnetic oxygen dissolved in the sample will reduce the T2 time and will therefore aﬀect the relaxation measurement. Under-ﬁlled samples are made in the same way, with the exception that the pore volume and density of the sample has to be predetermined to allow the correct quantity of absorbate to be added. A cryoporometer can be used either in a scanning mode, or stepped in temperature. The scanning cryoporometer uses a continuous heating ramp. The rate of warming is simultaneously controlled and measured, providing a scale against which the pore size distribution can be calibrated. All cryoporometry experiments rely on capturing as many points as possible on the melting curve to maximise the resolution of the ﬁnal pore size distribution. Improved resolution has been achieved by ramping the temperature as slowly as possible and using a logarithmic warming ramp to measure melting point depressions of less than 1 K. This increases the resolvable pore diameter since the amount of data recorded between the bulk and depressed melting points of the absorbate increases. Since the signal-to-noise ratio may be low on each

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point, ﬁltering processes can be applied to allow clear pore size distributions to be determined. A greater number of data points provide a better trend for the ﬁltering functions to resolve, producing a more accurate picture of the pore volume distribution. A recent version of the scanning cryoporometer automatically controls the warming ramp, records the height of a single spin-echo, and determines the pore size distribution without user input. The stepped cryoporometer permits the simultaneous measurement of relaxation times as well as the liquid volume fraction as a function of temperature. The stepped cryoporometer sets the temperature but allows it to stabilise before either a single spin-echo or a CPMG echo train is recorded. Recording an echo train provides information not only on the signal amplitude but also signal decay (T2 ) at each temperature. The precision of the pore size distribution determination may be improved by combining the cryoporometry results with the relaxation data. The transverse decay shown by CPMG may be ﬁtted using an exponential decay function, such as (3). More frequently a multiple exponential function is required, (5), where the subscript n represents the relaxation time of the nth component. −τ I= In (0) e T2n . (5) n

The ﬁtted curves may then be back projected to provide the liquid signal amplitude at zero time. This allows for any residual signal due to the solid or any signal attenuation eﬀects to be overcome. Although the method does not provide as many points on the cryoporometry curve as the scanning technique, the data produced have a very high signal-to-noise ratio and require little, if any, ﬁltering. Such stepped cryoporometry data sets are suitable for determining mean pore diameters and comparing samples, although ﬁne structure would be indistinguishable. The temperature steps allow other NMR parameters to be measured during the heating cycle, including T1 , T1ρ (spin-lattice relaxation and spin-lattice relaxation in the rotating frame) and diﬀusion rates.

4 Determining the Melting Point Depression Constant The melting point depression of the absorbate is inversely related to the diameter of the conﬁning pore, if it can be assumed that all the other terms in the full Gibbs-Thomson equation can be replaced by a constant k giving (1). To determine the k constant for a particular absorbate, a series of cryoporometry measurements are conducted on standard samples. The absorbate is added to just overﬁll the pores of a set of test porous silicas with well-deﬁned pore size distributions. Each samples is then cooled to well below the depressed melting point of the absorbate to overcome any super-cooling eﬀects. This is best performed in situ, the NMR signal being used to determine when the entire sample has frozen. The cooling should be slow to prevent the formation

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of metastable states that could potentially occur if the sample were quenched. The cryoporometry experiment can then be conducted as detailed above and the data used to determine the average pore melting temperature from the melting curve; see Fig. 3. The average pore melting points are plotted against the inverse of the median pore diameter of the test samples as determined by gas adsorption. The resulting data is ﬁtted linearly to determine the k value. Careful analysis of the melting point depression data shows that in some cases it is not a linear relationship; rather a curve that is ﬁtted precisely by the modiﬁed Gibbs-Thomson equation of the form: ∆T =

k . x − 2Sl

(6)

Introducing a surface layer thickness, Sl, allows the curvature to be quantiﬁed. There is a factor of 2 multiplying the surface layer thickness to take into account that x is the pore diameter. This disappears if x is expressed as pore radius. The surface layer was ﬁrst observed [11] and studied in detail with

Fig. 3. (a) Shows an ideal cryoporometry curve with four main features: (1) the pore melting step; (2) the total pore volume plateau; (3) the bulk melting step, and (4) the total liquid volume plateau. 4(b) shows the pore distribution expected from such a cryoporometry curve conducted on an artiﬁcial porous silica sample. To calibrate the pore size scale, the predetermined k value is used. Sample mass ratios are used to calibrate the pore volume (vertical) scale

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water [12, 26], and also noted in organic liquids [27]. It was suggested by Stapf and Kimmich [28] that the surface layer may interfere with the method of cryoporometry. The surface layer will reduce the observed pore radius since the material at the pore wall does not undergo a sharp solid-liquid transition as deﬁned by the Gibbs-Thomson equation. However, studies suggest the surface layer is independent of pore structure or geometry, allowing it to be included in the k constant. The fraction of non-frozen surface material can contribute to the NMR liquid signal. Normally the T2 decay time of the surface layer is much less than the rest of the liquid due to enhanced surface relaxation. In some cases therefore, the signal component can be ignored or overcome, particularly when measuring CPMG data on a static cryoporometer. Problems arise when the molecular size of the absorbate is large compared to the pore diameters, such that the surface layer occupies a signiﬁcant volume of the pores. In this case it has been observed that organic crystals do not truly freeze, making a cryoporometry experiment impossible. Rather, they enter an amorphous state at low temperatures, similar in properties to a glassy phase. This phenomenon has been observed for cyclohexane in pores of diameter less than 40 ˚ A [17, 29]. For naphthalene, this occurs around 50–60 ˚ A [14, 18]. Water, on the other hand, has been successfully used to measure pore diameters below 20 ˚ A [17, 30]. Water is perhaps the most obvious absorbate to be employed in cryoporometry experiments. It has a k value of 573 K ˚ A [30] determined using the scanning cryoporometer without including the surface layer factor, and noted to be independent of τ . Cyclohexane is more widely used since it has a number of advantages: its higher k value means larger pore diameters can be resolved and its soft plastic crystal does not damage the porous matrix, allowing the sample to be reproducibly re-measured.

5 Cryoporometry Hardware Simple NMR apparatus is required for NMR cryoporometry experiments, regardless of the technique used or modiﬁcations to the method. The arrangement consists of a magnet (either permanent or super-conducting), an NMR spectrometer (computer) and a temperature control system. Complexity arises when considering the implementation of accurate temperature control. As previously mentioned, the temperature control needs to be as precise and as stable as possible. Temperature regulated gas ﬂow is now the most commonly used method. Assuming the gas ﬂows at a high rate over the sample and that the sample has a suitable thermal mass, the temperature can be extremely stable. The two cryoporometers discussed in this section are given as examples of temperature-regulated NMR systems and were constructed using low-resolution (broad-line) spectrometers. The same principles of temperature control apply to high-resolution spectroscopy and imaging systems.

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In early cryoporometry experiments cooling was achieved using the direct injection of liquid nitrogen into a ‘splash-pot’ in the probe (the Lindacot system) [31]. A heater in a Dewar forced droplets of liquid nitrogen into the probe. Thus the full latent heat of evaporation of the nitrogen was available to cool the probe and sample. The thermal mass of the splash-pot smoothed out suddenly temperature ﬂuctuations due to the evaporation of individual liquid nitrogen droplets. A resistive heater was built directly into the probe to enable it to be warmed above room temperature. Later this probe was used eﬀectively with gas ﬂow cooling [17]. Instead of forcing droplets of liquid nitrogen into the probe, a dried air supply was cooled in liquid nitrogen and passed into the splash-pot. This provided a much more uniform warming ramp than the Lindacot system, although it did consume more liquid nitrogen. An air heater was added to the gas ﬂow just prior to entering the probe. This was used to control the temperature of the gas passing into the splash-pot. A thermocouple inside the splash-pot was used to monitor the temperature and provide feedback for the temperature control unit. Another thermocouple was soldered to an earthed copper shield in thermal contact with the sample to provide an accurate sample temperature for the cryoporometry measurement. A schematic of this probe, which is placed inside a Dewar between the poles of a permanent magnet, can be seen in Fig. 4. This probe is used for scanning cryoporometry measurements over a temperature range −100◦ C to 80◦ C, and had a repeatable temperature resolution of 1 mK. Measuring the temperature of the sample is a fundamental problem in cryoporometry. Ideally a thermocouple should be placed in direct thermal contact with the sample. However, a thermocouple placed inside the NMR coil will conduct RF noise into the system, thus ruining any result. Electrically insulating and earthing of the thermocouple can overcome this [17], although great care is required to ensure that no RF interference reaches the NMR coil. An alternative method, not yet fully explored, is the use of an optical temperature sensor to remotely monitor the sample temperature. Current optical temperature sensors are limited in their ability to measure temperatures below ambient, but as the technology improves this may become possible. Current optical thermometers would be suitable for super-ambient cryoporometry measurements. The stepped cryoporometer had a more modern variable temperature NMR probe (see Fig. 5) that was used in a commercial spectrometer system [18, 32]. This probe relies on the thermal insulation properties of PTFE to provide a sample chamber with a temperature range of −70◦ C to +180◦ C without risk to the magnet. This probe is also temperature regulated by a gas ﬂow system, although in this case the gas ﬂows directly over the sample and does not utilise the thermal mass of a splash-pot. As with the scanning probe (Fig. 4), dried air was passed through a Dewar of liquid nitrogen and an air process heater used to regulate the temperature. To warm the sample above room temperature, the compressed air was passed directly to the heater without the use of liquid nitrogen cooling. A thermocouple was placed at the base

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Fig. 4. Gas-ﬂow cooled NMR probe; custom design built by Webber et al. [17] for use in the scanning cryoporometer

of the probe where the gas enters, to simultaneously regulate the temperature and monitor the sample. It is important in all cryoporometry experiments that the sample temperature does not ﬂuctuate above the set temperature; Fig. 6(top). Due to super-cooling eﬀects, the melting of the absorbate is a non-reversible process over small temperature increments. Therefore the measured liquid volume may not coincide with the measured sample temperature under such ﬂuctuation conditions. In the cryoporometry experiment the recorded temperature should always be the maximum temperature to which the sample is exposed, see Fig. 6(bottom).

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Fig. 5. Design of the variable temperature sample chamber used in the stepped cryoporometer [18]. This probe was installed and used successfully for cryoporometry measurements in a commercial low-resolution bench-top spectrometer system

Fig. 6. Using a PID temperature control routine usually results in a temperature change that oscillates around the SET temperature before stabilising (top). In cryoporometry experiments the maximum temperature seen by the sample must be the SET, or recorded temperature, resulting in the necessity for a smooth temperature step with no overshoot (bottom)

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6 Applications of NMR Cryoporometry The methodology of cryoporometry has been expanded to include high temperature measurements and has also been combined with other NMR techniques such as imaging [19, 20, 21], relaxometry [33], diﬀusion [34] and NMR spectroscopy [15, 27]. By combining techniques, more information can be gained about the nature of the porous sample and the interactions at the solid/liquid interface [32]. Testing partially ﬁlled samples [29] or samples ﬁlled with more than one absorbate [35, 36] has been shown to provide information on surface interactions between the liquid and the pore wall. From these studies, the way in which liquids ﬁll pores was deduced. A great deal of interest in cryoporometry has come from cement manufacturer. This technique is ideal for the study of the formation of open microstructure during the hydration of cement pastes. Two examples of the use of cryoporometry in this ﬁeld have been taken from papers by Jehng, Halperin et al. [26, 37]. In Fig. 7 the results from NMR cryoporometry and Mercury Intrusion Porometry (MIP) have been compared for a cement paste during drying. Signiﬁcant diﬀerences in the distributions were observed: the MIP data only contained information on the larger porous structures, whilst the NMR contained information on small pores as well. The authors explained that the mercury was unable to penetrate the small structures. Therefore the cryoporometry data yielded extra information. A similar comparison has been made (see Fig. 8) between NMR cryoporometry and NMR relaxometry. The

Fig. 7. Pore radius distributions of cement paste at two stages during the drying process. Both Mercury Intrusion Porometry (MIP) and NMR cryoporometry (histogram) have been used. Graphs reproduced with permission from [37]

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Fig. 8. Pore distributions in cement powder. The data compare results from NMR cryoporometry (◦) and NMR relaxometry (•). Good agreement is shown at small pore sizes but not in the large capillary pores. In both techniques the absorbate used was water. Graphs reproduced with permission from [26]

two techniques diverge for the pore distributions in the large capillary pores, although cryoporometry measurements with water are limited to about 1–2 microns. For the smaller pores the two techniques agreed extremely well. Cryoporometry has been used to study the eﬀect of adding a surfactant to the surface of Ultra High Pore Volume (UHPV) silicas [38]. The results from these tests were quite dramatic and can be seen in Fig. 9. The surfactant coated sample B2 had quite obviously lost its smallest pores and a signiﬁcant quantity of pore volume where the space had been occupied by the surfactant. This was interpreted as a coating 25 ˚ A deep. Sample C appeared to have lost nearly all its pore volume on coating. However, it must be stressed that the ﬂuorinated surfactant used was potentially hydrophobic, preventing water uptake. Cryoporometry is independent of absorbate/pore wall interactions, making it a perfect choice for these types of experiments, as long as the absorbate can enter the pores. Sample C was fully categorised later by using cyclohexane as the absorbate and the surface coating thickness successfully deduced. Experiments conducted on partially ﬁlled systems [29, 39, 40] have been used to determine the behaviour of liquids in pores. Cryoporometry experiments were conducted on porous sol-gel silicas partially ﬁlled with varying proportions of cyclohexane, Fig. 10 (a), or water, Fig. 10 (b). As the volume of cyclohexane was increased, the modal pore size in the measured distributions was seen to increase, but the minimum observed pore diameter was never less than the smallest pores in the silica. This suggested that the cyclohexane

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Fig. 9. Four samples of UPHV silica: sample A – unmodiﬁed silica, nominal pore diameter 150 ˚ A. Sample B2 – silica modiﬁed with standard surfactant. Sample C – silica modiﬁed with ﬂuorinated surfactant. Sample B2 was tested both fully and partially (50%) ﬁlled. Water was the absorbate used in all cases. Graphs previously presented in [38]

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Fig. 10. Puddle size distributions for (a) cyclohexane and (b) water in 60 ˚ A silica at various ﬁlling fractions. Graphs reproduced with permission from ref [29]

preferentially ﬁlled the smallest pores with the highest surface-to-volume ratio. Similar experiments conducted with water showed a signiﬁcant shift in the entire pore size distributions as the volume of water was increased. The smallest ‘pore’ diameters observed (at the lowest ﬁlling factor) were far less than the smallest pores in the sample. It was deduced, with the aid of T2 relaxation analysis, that water forms puddles together with physisorbed layers that partially coat the silica pore walls. As the volume of water was increased, the puddles increased and joined together until the pores were eventually ﬁlled. Whilst cryoporometry on overﬁlled samples is independent of surface interactions, combining the technique with partial ﬁlling can provide a lot of information on absorbate behaviour and pore morphology. The stepped cryoporometer allowed NMR cryoporometry and NMR relaxometry experiments to be combined, as demonstrated by Valckenborg, Pel and Kopinga [33]. Two silicas provided by diﬀerent manufacturers, both having nominal pore diameters of 60 ˚ A ± 2˚ A, were compared by this technique. The silica samples were prepared by slightly over-ﬁlling with cyclohexane. A standard cryoporometry measurement was conducted as detailed above. The CPMG data sets collected at each temperature step were inverted to provide distributions of relaxation times, as in the technique of relaxometry

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Fig. 11. Combined cryoporometry and relaxometry of cyclohexane in two 60 ˚ A nominal pore diameter silicas from diﬀerent manufacturers. The surface interaction dominates over the conﬁnement in the relaxometry measurement [18]

[41, 42, 43]. The data sets were constructed into three-dimensional graphs, plotting the temperature against T2 relaxation time and showing the component amplitude at each point. The results from the two silica gels can be seen in Fig. 11 (a) and (b). Ideally both the signal amplitude (gray scale) and T2 relaxation time (x-axis) should exhibit the same temperature dependence in both samples. It can easily be seen that the relaxation time as a function of temperature for the cyclohexane in the two silicas varied greatly. Although the nominal pore diameter of the two silicas was quoted to be 60 ˚ A, a slight deviation was observed in gas adsorption measurements (±2 ˚ A). However, such a deviation would not result in the diﬀerent relaxation times seen in the two samples. The concentration of paramagnetic ions in the silicas was suggested as a possible cause of the variation in relaxation times. Since the

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cyclohexane in silica (a) exhibited shorter relaxation times, silica (a) was expected to contain more paramagnetic impurities than silica (b). Silicon NMR relaxation studies and XFD measurements were also conducted on the same silica gels. These conﬁrmed that silica (a) contained more paramagnetic impurities than the silica (b), as expected [18]. Such impurities can be introduced at the time of manufacture and may vary with the production method or the batch, depending on the source of the trace components. By combining the NMR cryoporometry and relaxometry experiments it was clear that surface relaxation times for liquids conﬁned in porous media can be modiﬁed by more than just the pore geometry. Although this review has concentrated mainly on low-resolution NMR cryoporometry, high-resolution measurements can provide alternative information. Valiullin and Fur´ o have used this technique to study a dual phase absorbate [35]. A nitrobenzene/hexane mixture was absorbed into porous glass and studied using high-resolution cryoporometry to resolve the two phases. An example of the results obtained can be seen in Fig. 12. From these results it was determined that the nitrobenzene coalesces into small droplets surrounded by hexane when in the pores.

Fig. 12. The chemical shift of nitrobenzene relative to hexane provides cryoporometry information on the two absorbates. Graph reproduced with permission from [35]

Various applications of Nuclear Magnetic Resonance (NMR) have been used to study the behaviour of oil and water in natural rocks [44, 45, 46, 47, 48]. This has been a source of considerable interest to those studying the removal of oil from oil-bearing rocks by ﬂushing the systems with salt water [49]. Normally the natural rock samples have extremely broad pore size distributions, ranging from microns to millimetres [50], limiting the utility of NMR cryoporometry for studying the behaviour of the conﬁned liquids [17]. The most frequently used NMR technique has been relaxometry [44]. Although

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this method can provide estimates of the pore size distributions in the rocks [51], it is highly sensitive to surface interactions [32] and can be diﬃcult to interpret accurately [52]. These studies all used model immiscible liquid mixtures in real rock systems. In a recent study cryoporometry has been employed to analyse binary mixtures of immiscible liquids, water and decane, in a model system of porous sol-gel silicas [36]. The pore geometry had already been characterised by gas adsorption and standard NMR cryoporometry measurements [17], allowing the interactions between the water, decane and pore surface to be investigated. The experiments were conducted in 100 ˚ A nominal pore diameter silica. The samples were made by ﬁrst slightly over ﬁlling the silica with decane. The decane was allowed to soak into the pores without assistance. Water was then added incrementally to the top of the sample and a cryoporometry measurement made without delay. The addition of known amounts of water, a little at a time, was conducted until a bulk water melting step was observed in the cryoporometry measurement. The cryoporometry melting curves for the binary mixtures can be seen in Fig. 13. The added water entered the pores and no bulk water signal was detectable until half of the absorbate mix (by volume) was water. As the water entered the pores, the decane shifted from the pores into the bulk. This can be seen by a shift in the ratio of the pore melting step and the bulk melting step for the decane. The total decane signal remained approximately constant. No more water was added once 50% of the absorbate mix was water. The sample was re-measured after twenty-four hours where no signal could be observed for conﬁned decane. Instead, the

˚ Fig. 13. The cryoporometry melting curves of decane and water mixtures in 100 A nominal pore diameter silica. The decane melting region is clearly separate from the water melting region. As the fraction of water increases the decane shifts from the pores into the bulk

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Fig. 14. The pore volume distributions for decane (top) and water (bottom) generated from the cryoporometry melting curves in Fig. 13. It can be seen that water preferentially displaces decane from the smallest pores

water occupied all of the open pore volume, indicating that the water could displace the decane completely over time. The pore size distributions taken from the cryoporometry results in Fig. 13 can be seen in Fig. 14. Whereas the cryoporometry melting curve only shows that the water displaces the decane, the pore size distributions show where the water goes. The water preferentially displaced the decane from the smallest pores. For a mix containing 10% water, pores with a nominal diameter of 80 ˚ A were occupied by water. When the quantity of water was increased to 50% of the mixture, the nominal pore diameter of pores occupied by the water increased to 95 ˚ A. At the same time the width of the pore size distribution occupied by the water increased dramatically. By contrast the decane occupied pores of a nominal diameter of 97 ˚ A when 10% of the mixture was water; this increased to 160 ˚ A when 50% of the mixture was water. That meant the remaining decane only resided in the largest pores. The results of this study clearly show that water can displace decane from porous sol-gel silicas, given suitable time for the displacement to occur. It has been shown elsewhere that water has a far stronger aﬃnity to the surface in these silicas than non-polar organic liquids [32, 53, 54]. This is because the silica surface contains hydroxyl groups to which the water molecules can form hydrogen bonds. It was therefore reasonable to expect the water to preferentially enter the smallest pores (with the greatest surface to volume ratio) ﬁrst, displacing the decane. The last application of cryoporometry discussed here is the most visually impressive. By conducting a cryoporometry experiment in a spectrometer

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Fig. 15. Cryoporometry combined with 3D NMR imaging as used to study reservoir rock core samples at (a) 268 K and (b) 295 K. Imagine intensity is related to porosity: black for high porosity down to light grey for low porosity. Images previously displayed in Report to CPMT (2000) and presented in [38]

ﬁtted with imaging gradients, spatially resolved pore size distributions were obtained. To demonstrate this method, a special silica sample was constructed, consisting of a 60 ˚ A nominal pore diameter silica column ‘wearing’ a collar of 500 ˚ A nominal pore diameter silica [21]. The intended sample shape and the actual 3D rendered volumes can be seen in Fig. 16. This technique has since been used practically to image oil reservoir rock samples [38]; Fig. 15. The results, combined with diﬀusion measurements, illustrated the interconnectedness of the pores in the rock. This type of research oﬀers great potential for petrology, catalyst and construction materials industries.

7 Conclusion In this chapter we have attempted to cover the important practical aspects involved with accurate NMR cryoporometry measurements. It has been shown that quantitative pore volume distributions can be obtained easily from a simple NMR spectrometer system. Although cryoporometry was born on a custom NMR system, it has spread to be used on a wide range of equipment and for many purposes. It has been shown to produce results equal to, and in same cases better, than those from well-established pore volume measurement methods. Cryoporometry has the major advantage of oﬀering direct volumetric measurements that few other techniques can match and variations of the technique permit the detailed study of ﬂuid-surface interactions. Useful in its own right, cryoporometry has been combined with other NMR techniques (relaxometry, diﬀusion, imaging, and spectroscopy) to provide a

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Fig. 16. Spatially resolved silica test sample via combination of NMR cryoporometry and 3D NMR imaging. (a) The intended shapes of the silica sample. (b) Horizontal view of the rendered volumes. (c) Vertical view of the 500 ˚ A silica volume, clearly showing the hollow centre. Images reproduced with permission from [21]

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toolbox that can fully characterise almost any porous system. The technique can be used to characterise pore surfaces using partially ﬁlled samples, and compare surface aﬃnities of absorbates using binary liquids. Now used for the measurement of porous rocks, cements, biological cells, and in many other research areas, NMR cryoporometry is proving to be invaluable in the ﬁeld of porous media exploration.

References 1. J. W. Gibbs: The Scientiﬁc Papers of J. Willard Gibbs, volume 1:Thermodynamics, new dover edition (Dover Publications Inc., Constable and Co., New York, London 1906) 2. J. W. Gibbs: Collected Works (Longmans, Green and Co., New York 1928) 3. J. Thomson: Trans. Roy. Soc. xi, (1849) 4. J. Thomson: Proc. Roy. Soc. xi, (1862) 5. J. J. Thomson: Application of Dynamics to Physics and Chemistry (Macmillan & Co., London 1888) 6. W. Thomson: Phil. Mag., Ser. 4 42, 282 (1871) 7. C. L. Jackson, G. B. McKenna: J. Chem. Phys. 93, 12 (1990) 8. J. H. Strange, M. Rahman, E. G. Smith: Phys. Rev. Lett. 71, 21 (1993) 9. H. A. Resing, J. K. Thompson, J. J. Krebs: J. Phys. Chem. 68, 7 (1964) 10. H. A. Resing: J. Chem. Phys. 43, 2 (1965) 11. R. T. Pearson, W. Derbyshire: J. Colloid Interf. Sci. 46, 2 (1973) 12. E. W. Hansen, M. Stocker, R. Schmidt: J. Chem. Phys. 100, 6 (1996) 13. H. F. Booth, J. H. Strange: Molec. Phys. 93, 2 (1997) 14. J. Mitchell, J. H. Strange: Molec. Phys. In Press, (2004) 15. K. Overloop, L. v. Gervan: J. Magn. Reson. Ser. A 101, (1993) 16. E. P. Barrett, L. G. Joyner, P. P. Halenda: J. Am. Chem. Soc. 73, (1951) 17. J. B. W. Webber: Characterising Porous Media. PhD Thesis Thesis, University of Kent, Canterbury (2000). 18. J. Mitchell: A Study of the Modiﬁed Behaviour of Organic Macromolecules in Conﬁned Geometry. PhD Thesis, University of Kent, Canterbury (2003). 19. J. H. Strange, J. B. W. Webber, S. D. Schmidt: Magn. Reson. Imaging 14, 7/8 (1996) 20. J. H. Strange, J. B. W. Webber: Appl. Magn. Reson. 12, 2-3 (1997) 21. J. H. Strange, J. B. W. Webber: Meas. Sci. Technol. 8, 1-7 (1997) 22. E. L. Hahn: Physics Today 4, November (1953) 23. E. L. Hahn: Phys. Rev. 80, 4 (1956) 24. H. Y. Carr, E. M. Purcell: Phys. Rev. 94, (1954) 25. S. Meiboom, D. Gill: Rev. Sci. Instrum. 29, (1958) 26. J. Y. Jehng, D. T. Sprague, W. P. Halperin: Magn. Reson. Imaging 14, 7/8 (1996) 27. E. W. Hansen, R. Schmidt, M. St¨ ocker: J. Phys. Chem. 100, (1996) 28. S. Stapf, R. Kimmich: J. Chem. Phys. 103, 6 (1995) 29. S. G. Allen, P. C. L. Stephenson, J. H. Strange: J. Chem. Phys. 108, 19 (1998) 30. J. B. W. Webber, J. H. Strange, J. C. Dore: Magn. Reson. Imaging 19, 3-4 (2001)

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31. M. O. Norris, J. H. Strange: J. Phys. E 2, 2 (1969) 32. J. H. Strange, J. Mitchell, J. B. W. Webber: Magn. Reson. Imaging 21, 3-4 (2003) 33. R. M. E. Valckenborg, L. Pel, K. Kopinga: J. Phys. D: Appl. Phys. 35, (2002) 34. A. V. Filippov, V. D. Skirda: Colloid Journal 62, 6 (2000) 35. R. Valiullin, I. Furo: J. Chem. Phys. 116, 3 (2002) 36. S. M. Alnaimi, J. Mitchell, J. H. Strange et al.: J. Chem. Phys. 120, 2075 (2004) 37. S. Bhattacharja, M. Moukwa, F. D’Orazio et al.: Advanced Cement Based Materials 1, (1993) 38. J. H. Strange, L. Betteridge, M. J. D. Mallett: Characterisation of Porous Media by NMR. In: NATO ASI series II: Mathematics, Physics and Chemistry, vol Magnetic Resonance in Colloid and Interface Science, ed by J. Fraissard (Kluwer Academic Publishers, Dordrecht 2002) 39. S. M. Alnaimi, J. H. Strange, E. G. Smith: Magn. Reson. Imaging 12, 2 (1994) 40. S. G. Allen, P. C. L. Stephenson, J. H. Strange: J. Chem. Phys. 106, 18 (1997) 41. J. R. Zimmerman, W. E. Brittin: J. Phys. Chem. 61, (1957) 42. K. R. Brownstein, C. E. Tarr: J. Magn. Reson. 26, (1977) 43. K. R. Brownstein, C. E. Tarr: Phys. Rev. A 19, 6 (1979) 44. R. J. S. Brown, I. Fatt: Petroleum T. AIME 207, (1956) 45. J. J. Tessier, K. J. Packer, J. F. Thovert et al.: AICHE J. 43, 7 (1997) 46. S. Godfrey, J.-P. Korb, M. Fleury et al.: Magn. Reson. Imaging 19, (2001) 47. G. C. Borgia, E. Mesini, P. Fantazzini: J. Appl. Phys. 70, 12 (1991) 48. P. J. Barrie: Ann. R. NMR S. 41, (2000) 49. W. G. Anderson: J. Petrol. Technol. 38, 11 (1986) 50. F. A. L. Dullien: Porous Media. Fluid Transport and Pore Structure (Academic Press, London 1979) 51. M. H. Cohen, K. S. Mendelson: J. Appl. Phys. 53, 2 (1982) 52. R. J. S. Brown, G. C. Borgia, P. Fantazzini et al.: Magn. Reson. Imaging 9, (1991) 53. T. Zavada, S. Stapf, U. Beginn et al.: Magn. Reson. Imaging 16, 5/6 (1998) 54. J.-P. Korb, M. Whaley-Hodges, T. Gobron et al.: Phys. Rev. E 60, 3 (1999)

Index

180◦ -pulse 36 90◦ -pulse 38, 41 2D Heisenberg paramagnet 355 in-plane correlation length 355, 368, 372 2D exchange NMR 74, 389, 402 2 H spectrum 80 spectra 389 2DQHAF (two-dimensional quantum Heisenberg antiferromagnet) 351 phase diagram 354 absorption lines 34, 159 activation energy 400 adiabatic demagnetization and remagnetization 14 AFM or AF (antiferromagnetic) 298 alkali fullerides 258 anisotropic exchange 305 anisotropic spin-spin interactions 322 annihilation operator 178 antiferromagnetic state 245, 286 correlation length 273, 311 ﬂuctuations 356 incipient 230 insulator 264 percolation thresholds 352, 367, 368 ring 307 transition 264 AO (atomic orbitals) 201 Arrhenius law 399 Arrhenius plot 81 asymmetry parameter 30 atomic vapors 117

azurin

134

B3LYP parametrization 199 BABA (back-to-back) pulse cycle 42, 59 Bardeen-Friedel oscillations 218 Barnett eﬀect 2, 15 BCS (Bardeen, Cooper, and Schrieﬀer) theory 244, 246 Bell states 94 bichromatic ﬁeld 150, 172 bichromatic pulse 171, 173 bilirubin 51 biomacromolecule 49 biradical 40 bit ﬂip 88 Bloch-Siegert shift 146, 164 Bohr radius 214 Boltzmann density matrix 93 Bose-Einstein statistics 255 BPP relaxation 218, 243, 400 Brillouin zone 357 broadening lifetime 316 13

C 55 C4 symmetry 106 C60 100, 188, 258 C6 symmetry 80 C80 188 CaF2 :Ce 88 carbon cage 186 internal hopping 187 reduction 200

432

Index

CeCu6−z Auz 374 cement 419 CESR method 216 CF (auto-correlation function) 303 electronic spin-spin 305 spin correlation function 307 CFTD (copper formiate tetrahydrate) 352 deuterated 359 charge carrier mobility 53 charge doping 352 charge segregation 367 chemical exchange 67 chemical shift 32 interaction 67 chemisorption 235 circular dichroism 120 circular polarization 121 circular radiation ﬁeld 146 classical bit 88 Clebsch-Gordan coeﬃcients 202 cluster spin-glass phase 372 CNOT (Controlled NOT) 89 coherence-transfer echo 175 coherent optical radiation 117 coherent optics 143 coherent Raman scattering 122, 131 coherent states 179 commensurate ordering 287 complex of copper 53 conformational space 56 continuous freezing transition 372 convolution approaches 34 Cooper pairs 251 Cooperative spin-freezing 365 correlation length 359 CPMG (Meiboom-Gill modiﬁed Carr-Purcell) sequence 411 CPZ (cyclophosphazene) 80 creation operator 178 critical exponent 369 CS (chemical shift) tensor 391 CS2 200 CT (charge transfer) 185, 201 complexes 186, 201 Cu-O planes 269 Curie Weiss constant 377 cw ENDOR 149, 156 cyclohexane 415

DBU (1, 5-diazabicycloundecene) 197 deadtimes 39 Debye-Waller factor 222 decane 425, 426 decoherence 97 deconvolution approaches 34 DEER (double electron resonance) pulse technique, also PELDOR 39 DEER (double electron electron resonance) pulse technique, also PELDOR 59, 176 bichromatic-pulse DEER 177 demagnetizing ﬁeld 223 density matrix tomography 97, 99, 102 density operator 7, 165 matrix 93 DFT (density functional theory) 199, 211 diamagnetism 247 dilution model 353 dimer 282 dipolar broadening 35 frequency 26, 30 interactions 17 order 15 reservoirs 10 dipolar approximation 178 dipolar evolution function 37 dipole moment 24 dipole-dipole interaction 24, 67 coupling 40, 43, 187 dipolar alphabet 25 ﬂip-ﬂop term 40 heteronuclear 35, 38, 43 hom*onuclear 5, 41, 43 direct relaxation process 319 discotic phase 53 DJ (Deutsch-Jozsa) algorithm 90, 110 DM (Dzialoshinskii-Moriya) interaction 290 doped gallium phosphide 154 doped MgO 148, 150 double resonance 35 electron-electron 26, 59 electron-nuclear 26 DQ (double-quantum) 41, 59 coherence 41

Index sideband patterns 48 transition 41 two-dimensional spectra 50 DRAMA (dipolar recovery at the magic angle) 41, 59 dressed states 180 DSC (diﬀerential scanning calorimetry) 408 dynamical scaling 352 E-P-R (Einstein-Podolsky-Rosen) 89 echo sequence 35 echo traces 169 Edwards-Anderson order parameter 384 eﬀective electronic spin 187 eﬀective gyromagnetic ratio 2 eﬀective topology 47 EFG (electric ﬁeld gradient) 127, 189 ﬂuctuations 399 Ehrenfest’s theorem 6 Einstein-de Haas experiment 3 ELDOR (electron electron double resonance) 26, 59 electron coherence 176 electron spin 144, 145 electron spin susceptibility 271 electron-electron interactions 245 ENDOR (electron nuclear double resonance) 59, 93, 103, 189 spectra 192 W-band 189, 190 X-band 194 energy gap 318 entangled states 97, 102 entanglement 90 EPR (electron paramagnetic resonance), also ESR 21, 92, 103, 142 cw Hamiltonian 160 light-induced 154 multi-frequency 187 pulse 168 spectroscopy 57 ESEEM (electron spin echo envelope modulation) 26, 59, 194 ESR (electron spin resonance), also EPR 21, 92, 103, 142 ET (ethylenedithia-tetrathiafulvalene) 280

even resonances 152 exchange coupling 200 exchange interactions 305 exchange-correlation energy exponential damping 40 external ﬁeld 328

433

211

Faraday cage 187 fast exchange region 70 FE-AFE (ferroelectric-antiferroelectric) solid 383 Fe8 ([Fe8 (N3 C6 H15 )6 O2 (OH)12 ]8+ ×[Br8 · 9H2 O]8− ) 298 Fermi contact coupling 28, 245 Fermi function 245 Fermi sea 251 energy gap 254 Fermi surface 279 Fermi’s Golden Rule 245 Fermi-Dirac statistics 255 ferritin core 338 ferro- or ferri-magnetic 302 FID (free induction decay) 70, 410 ﬁeld-dependent gap 326 ﬁeld-sweep mode 132 FL (Fermi liquid) 375 Floquet theory many-mode theory 181 states 40, 151, 162 theorem 151 ﬂuctuation-dissipation theorem 357 ﬂuxoids 284 d-state pairing 276 FM (ferromagnet) 298 incipient 230 forbidden transitions 146 Fourier transformation 34 free electron gas 218, 245 free radical DPPH 158 fullerene 186 g-matrix (anisotropic) 187 elements 205 GaAs 129 gas adsorption 414 BJH method 409 Gaussian-shaped pulses 56 Gd@C82 200, 205

434

Index

ghost ﬁeld 18 Gibbs-Thomson equation 408 Ginzburg-Landau theory 257 glassy state 403 1

H-1 H distances 51 Hadamard transformation 88 Hahn echo 35, 43 Hamiltonian dipole-dipole interaction 8, 24 eﬀective spin 201 ﬁrst-order truncated spin 198 Floquet 180 magnetic 312, 351 non-secular terms 198 rotating frame 8 semiclassical interaction 179 spin system 67, 177, 269, 271, 283 time-dependent 151 time-independent 66 time-independent perturbation 145 toggling frame 164 total 92 Hanle eﬀect 122, 129 Havrilak-Negami spectral density 401 3 He 255 4 He 255 healing length 225 heating ramp 412 Heisenberg model 298 antiferromagnet 351 chains 298 exchange interaction 27, 322 Heitler–Teller equation 227, 244 helicity of photons 144, 179 herring-bone packing 54 hexabenzocoronene 53 hﬁ (hyperﬁne interactions) 187 hf tensor 187 hfcc (hf coupling constants) 191 high-resolution NMR 23 high-temperature limit 41 Hilbert space 66, 88 hom*o (highest occupied molecular orbital) 188, 282 hom*ogeneous linewidth 35 hom*onuclear MAS NMR 42 hopping ion 191

HPLC technique 189 hydrogen (O-H...O) bond 384, 426 hyperﬁne ﬁelds 212 tensor 269 HYSCORE (hyperﬁne sublevel correlation spectroscopy) 194 IAS (intermediate axis system) 68 INADEQUATE (incredible natural abundance double quantum transfer experiment) 60 interpulse delay 38, 41 indirect processes (Raman-like and Orbach-like) 324 internal ﬁeld 328 intra-molecular 186 ion localization 185 Ising model 298 isolated spin pair 33, 48 isotropic exchange interaction 289 isotropic hyperﬁne coupling 27, 43 J coupling

KDP (KH2 PO4 ) 385 Knight shift 213, 244, 284 Kondo eﬀect 266 Kondo temperature 375 Korringa relation 214, 244 Kramers degeneracy 330 La2 CuO4 264, 351, 359 La2 @C80 188 La2 O3 189 La@C82 186, 189 LAB (laboratory) frame 68 laboratory frame 8, 153 LAC (level repulsion or anticrossing) 322 Landau diamagnetism 223 Landau-Zener transition 334 Lande’s rule 312 Lanthanide ions 187 Larmor frequency 11, 68 Larmor precession 3 laser light 117 laser-assisted magnetic resonance 114 latent heat of evaporation 416 LCs (multiple level-crossings) 322

Index Lee-Goldburg decoupling 35 level anticrossing 151 lifetime broadening 203 ligand 53 Light induced EPR spectrum ﬁrst-harmonic 167 second harmonic 155 linear radiation ﬁeld 145, 151 linear response 145 Liouville superoperator 67 Liouville-von Neumann equation 7, 66 liquid-crystalline state 53 lithium phthalocyanine 168 local density of states 226 local dipolar ﬁeld 35 localization temperature 372 locally correlated freeze-out 384 longitudinal ﬁeld 158 Lorentzian absorption line 159, 305 magic angle 36 magnetic ﬁeld gradient 411 ground state 300, 318 ions 357 molecules 301 susceptibility 309 magnetic resonance spectrum eﬀect of surfaces 130 ﬁrst moment 44 moment analysis 44 second moment 45 spectrum simulation 34 magnetization ﬂuctuations quantum ﬂuctuations 320, 334 thermal 313 magnetogyric ratios 24 magneton Bohr 3, 24 nuclear 24 magnets molecular 298, 307 nanomagnets 313, 326, 330, 334 malonic acid radical 95 MAS (magic angle spinning) 32, 36, 42, 60 complete refocusing 48 fast sample rotation 33 odd-order sidebands 48

435

sidebands 42 MCD (magnetic circular dichroism) 131 MEF (metallo-endofullerenes) 185 Meissner eﬀect 247 metalloproteins 134 metastable states 414 depression 408 microwave 144 bimodal 39 pulse 190 MIP (Mercury Intrusion Porometry) 419 Mn12 ([Mn12 O12 (CH3 COO)16 (H2 O)4 ]) 300 MO (molecular orbital) 187 mode-coupling theory 362 molecular clusters 301 molecular motion 67 Moriya equation 214 motional averaging dipole-dipole interaction 31 Mott insulator 266, 282, 356 MQE (multiple quantum ENDOR) 108 MRI (magnetic resonance imaging) 409 multi nuclear correlations 107 multi qubit 105 multi-body system 46 multi-spin eﬀects 47 system 44, 46 multiple-photon echoes 168 µSR (muon spin rotation) 314 15

N 55 nanoparticles 208 ND4 387 Neel temperature 363 neutron scattering 367 NFSL (non-frozen surface layer) 408 NH4 387 nitroxide radical 33 NMR (nuclear magnetic resonance) 21, 66, 125 coil 416 cryoporometry 408, 411, 419 dynamic 2 H spectroscopy 68

436

Index

solid state 35 non-exponential recovery 300 non-uniform gap 276 NOT gate 88 nqi (nuclear quadrupole interaction) 187, 192 NQR (nuclear quadrupole resonance) zero ﬁeld 195 NSLR (nuclear spin-lattice relaxation rate) 298, 316, 331, 334, 341 nuclear magnetic relaxation spin-lattice in the rotating frame 413 transverse 409 nuclear spin 145 nuclear Zeeman energy 246 ODCB (o-dichlorobenzene) 197 odd resonances 152 ODEPR (optically detected EPR) 131, 132 lineshapes 133 oﬀ-equilibrium magnetization 331 optical coherence 123 optical detection 120 optical polarization 117 optical pumping 118 spin polarization 130 Xe nuclear spin 130 optical transitions band structure 136 orbital pairing 280 orbital shifts 235 order parameter ﬂuctuations 360 order-disorder transition 358 organic liquids 426 Overhauser shift 213 overmodulation 158 oxidation state 187 31

P 391 P-31 P distances 49 p orbital 283 Pake pattern 28 paramagnet ﬁnite size 307 paramagnetic center 52 ions 30, 423 state 289 31

paramagnets 232, 233 metals 211 paramagnon 367 PAS (principle axis system) 68 Pauli susceptibility 211 PELDOR (pulse ELDOR) 39, 60 Peptide 55 phase frequency 109 phenyl group 57 phosphate glasses 49 photoluminescence 121 photon number 178 π-photon-induced transparency 169, 177 PISEMA (polarization inversion spin-exchange at the magic angle) 35, 60 plastic crystal 415 platinum particles 223 PO4 ion 385 point-dipole approximation 29 Poisson distribution 179 polarization ﬂuctuations 396 population diﬀerences 41 pore characterisation 409 diameter 414 morphology 422 size distribution 412 porous glass 220 porous media 406 ﬁlling factor 422 powder spectrum (“Pake” pattern) 70 protein 55 α- or β-protons 57 proton magnetisation 410 proton-proton distances 43 pseudo-spin glass 383 pseudo-spin variable 358 pseudopure states 93, 101, 109 Pt/zeolite systems 227 pulse EPR 38, 43 pulse sequence 36, 38, 168, 172, 174 QD (quantum disordered) regime 355 QTM (quantum tunneling of the magnetization) 313, 330, 334 quadrupolar interaction 68 echo 72, 73, 76

Index order 74 spin-lattice relaxation 73 quadrupole coupling 25 coupling constant 127 interaction 127 moment 127 nuclei 301 quantization axes 25 quantum algorithm 90 quantum computing 87 quantum critical point 352, 374 criticality 377 quantum gates 88 quantum numbers 146 quantum tunneling 318 quantum well materials 128 quantum-chemical computation 30 qubit (quantum bit) 88 Rabi precession 97 radiation ﬁeld 144 radical anion 197 Raman ﬁeld 123 Raman process two-magnon scattering 317 Raman processes 122 random freeze-out 384 random-bond random-ﬁeld model 404 rare earth 125 87 Rb 388 Rb1−x (ND4 )x D2 PO4 (D-RADP-x) 383 REAPDOR (rotational echo adiabatic passage double resonance) 60 reconversion subsequence 43 recoupling pulse 42 REDOR (rotational echo double resonance) 36, 60 relaxation spin-lattice 70 spin-spin 70 relaxation processes 356 NQR 356, 360 NQR-NMR 356, 357 RF noise 416 103 Rh 235 rhodium particles 234 RKKY 375

rotating frame 8, 10 double 160 multiply tilted 160 rotational echoes 32 rotational resonance 40 rotor-ﬁxed frame 32 rotor-synchronized pulses rubredoxin 137

437

S-Bus 88, 92, 104 system 105 topology 105 sample spinning 15 Sc3 @C84 197 Sc3 N@C80 197 Sc@C82 187 second Legendre polynomial 31 second quantization 177 SEDOR (spin echo double resonance) 35, 60 semiconductors 128 SG (spin-glass) 355, 373 Shaw–Warren model 215 shift operators 25 sidebands 158, 165 SIFTER (single frequency technique for refocusing) 43, 60 signal/noise ratio 176 silica gels 423 singlet ground state 321 Slater lattice 384, 386 conﬁgurations 384 ice rule 384, 404 solid echo 43 techniques 43 solid-state MAS NMR 36 solid-state NMR 35, 57 space group Fdd2 392 I42d 392 spatial proximity 49 spectral densities transverse and longitudinal 302 spectral holeburning 119, 124 spectrum zeroth-harmonic 165, 167 spin coherence 173 dilution 352

438

Index

dynamics 330 exchange 118 gap and pseudogap 273 label 53 multiplicity 186, 187 polarization 118, 199 polarized electron-hole pairs 128 precession 3 selective excitations 96 shifts 235 stiﬀness 354 topology 322 spin echo 411 sequence 96 spin ﬂuctuations 231, 357 quantum 324 thermal 324 spin temperature 14, 35 dipolar 14, 15 Zeeman 15 spin transitions multiple-photon 143, 162 multiple-quantum 143 single-photon 145 three-photon 149 two-photon 147 spin-density operator 66 spin-phonon coupling 313, 316 spin-spin pair correlation function 45 spin-wave velocity 354 spinning-sideband pattern 52 SQUID (superconducting quantum interference device) 2 SS-ESEEM (stimulated soft electron spin echo envelope modulation) 174 staggered magnetic moment 353, 367, 369 static dipolar spectrum 31 static magnetic ﬁeld 145 Stochastic Liouville equation 67 Stoner exchange integral 215 strong delocalization 30 sub-lattice magnetization 367 supercages 227 superconductors cuprate 264 d-wave 255 organic 280

p-wave 255 type I 256 type II 248, 256 superﬂuid state 255 superparamagnetism 230, 338 eﬀects 313 particle 318 superposition 90 supramolecular assembly 53 surface interactions 425 Takagi diﬀusion 391 Takagi groups 385 TBAP (tetra-n-butylammonium perchlorate) 197 TCB (tri-chlorobenzene) 200 TEM (transmission electron microscopy) 220 Tikhonov regularization 37, 46 toggling frame 161 tomography 97 topoisomer 188 total spin states 307 transferred hyperﬁne tensor 269 transformation matrices 70 transition matrix 396 transition selective excitations 96 transparency condition 174 transverse decay time T2∗ 410 transverse hyperﬁne ﬁeld 316 transverse relaxation time (decay time) 35, 48, 410 triplet excited state 322 tumbling motion 187 Turing machine 105 Two-dimensional DQ spectra crosspeaks 50 two-dimensional Heisenberg antiferromagnet 271 two-spin system 24 UHPV (ultra high pore volume) silicas 420 ultraslow motion 74 unitary transformation 88 V V12 ((NHEt)3 [VIV 8 V4 As8 O40 (H2 O)]×H2 O) 306 vanadium clusters 343

Index vortex-anti vortex currents

374

wavefunction 88 weak Coulomb exchange 205 weak-collision approach 302 Wigner rotation matrix 74 X-band 38 XLBA (X-ray line broadening analysis) 220 XY model 298 Y@C82 187 YBa2 Cu3 O6 265

Zeeman anisotropy 34 interaction 10, 15, 289 levels 301 order 15, 74 reservoirs 10 Zeeman-dipolar oscillations 11 zeolite matrix 227 Zeolites 130 zero-quantum transition 41 ZFS (zero-ﬁeld-splitting) 187 tensor 203

439

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