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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