NMR ofQuadrupolarNuclei inSolid Materials

EMR HandbooksBased on the Encyclopedia of Magnetic Resonance (EMR), this monograph series focuses on hot topics and majordevelopments in modern magnetic resonance and its many applications. Each volume in the series will have a specificfocus in either general NMR or MRI, with coverage of applications in the key scientific disciplines of physics, chemistry,biology or medicine. All the material published in this series, plus additional content, will be available in the online versionof EMR, although in a slightly different format.

Previous EMR HandbooksNMR CrystallographyEdited by Robin K. Harris, Roderick E. Wasylishen, Melinda J. DuerISBN 978-0-470-69961-4

Multidimensional NMR Methods for the Solution StateEdited by Gareth A. Morris, James W. EmsleyISBN 978-0-470-77075-7

Solid-State NMR Studies of BiopolymersEdited by Ann E. McDermott, Tatyana PolenovaISBN 978-0-470-72122-3

Forthcoming EMR HandbooksRF Coils for MRIEdited by J. Thomas Vaughan and John R. GriffithsISBN 978-0-470-77076-4

UTE ImagingEdited by Graeme M. Bydder, Gary Fullerton and Ian R. YoungISBN 978-0-470-68835-9

Encyclopedia of Magnetic ResonanceEdited by Robin K. Harris, Roderick E. Wasylishen, Edwin D. Becker, John R. Griffiths, Vivian S. Lee, Ian R. Young, AnnE. McDermott, Tatyana Polenova, James W. Emsley, George A. Gray, Gareth A. Morris, Melinda J. Duer and Bernard C.Gerstein.

The Encyclopedia of Magnetic Resonance (EMR) is based on the original printed Encyclopedia of Nuclear MagneticResonance, which was first published in 1996 with an update volume added in 2000. EMR was launched online in 2007with all the material that had previously appeared in print. New updates have since been and will be added on a regular basisthroughout the year to keep the content up to date with current developments. Nuclear was dropped from the title to reflectthe increasing prominence of MRI and other medical applications. This allows the editors to expand beyond the traditionalborders of NMR to MRI and MRS, as well as to EPR and other modalities. EMR covers all aspects of magnetic resonance,with articles on the fundamental principles, the techniques and their applications in all areas of physics, chemistry, biologyand medicine for both general NMR and MRI. Additionally, articles on the history of the subject are included.

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NMR ofQuadrupolarNuclei inSolid MaterialsEditors

Roderick E. WasylishenUniversity of Alberta, Edmonton, Canada

Sharon E. AshbrookUniversity of St Andrews, St Andrews, UK

Stephen WimperisUniversity of Glasgow, UK

A John Wiley and Sons, Ltd., Publication

This edition first published 2012© 2012 John Wiley & Sons Ltd

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Library of Congress Cataloging-in-Publication Data

NMR of quadrupolar nuclei in solid materials / editors, Roderick E. Wasylishen,Sharon E. Ashbrook, Stephen Wimperis.

p. cm.Includes Index.ISBN 978-0-470-97398-l (cloth)

1. Nulclear quadrupole resonance spectroscopy. 2. Nuclear spin. 3. Solids–Analysis.I. Wasylishen, Roderick E. II. Ashbrook, Sharon E. III. Wimperis, Stephen.

QD96.N84N57 2012538’.362–dc23

2012002021

A catalogue record for this book is available from the British Library.

ISBN-13: 978-0-470-97398-1

Set in 9.5/11.5 pt Times by Laserwords (Private) Limited, Chennai, IndiaPrinted and bound in Singapore by Markono Print Media Pte Ltd

Encyclopedia of Magnetic Resonance

Editorial Board

Editors-in-ChiefRobin K. HarrisUniversity of DurhamDurhamUK

Roderick E. WasylishenUniversity of AlbertaEdmonton, AlbertaCanada

Section EditorsSOLID-STATE NMR & PHYSICS

Melinda J. DuerUniversity of CambridgeCambridgeUK

Bernard C. GersteinAmes, IAUSA

SOLUTION-STATE NMR & CHEMISTRY

James W. EmsleyUniversity of SouthamptonSouthamptonUK

George A. GrayVarian Inc.Palo Alto, CAUSA

Gareth A. MorrisUniversity of ManchesterManchesterUK

BIOCHEMICAL NMR

Ann E. McDermottColumbia UniversityNew York, NYUSA

Tatyana PolenovaUniversity of DelawareNewark, DEUSA

MRI & MRS

John R. GriffithsCancer Research UK

Cambridge ResearchInstitute

CambridgeUK

Vivian S. LeeNYU Langone Medical

CenterNew York, NYUSA

Ian R. YoungImperial CollegeLondonUK

HISTORICAL PERSPECTIVES

Edwin D. BeckerNational Institutes of HealthBethesda, MDUSA

vi Encyclopedia of Magnetic Resonance

International Advisory Board

David M. Grant (Chairman)University of UtahSalt Lake City, UTUSA

Isao AndoTokyo Institute

of TechnologyTokyoJapan

Adriaan BaxNational Institutes of HealthBethesda, MDUSA

Chris BoeschUniversity of BernBernSwitzerland

Paul A. BottomleyJohns Hopkins UniversityBaltimore, MDUSA

William G. BradleyUCSD Medical CenterSan Diego, CAUSA

Graeme M. BydderUCSD Medical CenterSan Diego, CAUSA

Paul T. CallaghanVictoria University

of WellingtonWellingtonNew Zealand

Richard R. ErnstEidgenossische Technische

Hochschule (ETH)ZurichSwitzerland

Ray FreemanUniversity of CambridgeCambridgeUK

Lucio FrydmanWeizmann Institute

of ScienceRehovotIsrael

Maurice GoldmanVillebon sur YvetteFrance

Harald GuntherUniversitat SiegenSiegenGermany

Herbert Y. KresselHarvard Medical SchoolBoston, MAUSA

C. Leon PartainVanderbilt University Medical

CenterNashville, TNUSA

Alexander PinesUniversity of California

at BerkeleyBerkeley, CAUSA

George K. RaddaUniversity of OxfordOxfordUK

Hans Wolfgang SpiessMax-Planck Institute

of Polymer ResearchMainzGermany

Charles P. SlichterUniversity of Illinois

at Urbana-ChampaignUrbana, ILUSA

John S. WaughMassachusetts Institute

of Technology (MIT)Cambridge, MAUSA

Bernd WrackmeyerUniversitat BayreuthBayreuthGermany

Kurt WuthrichThe Scripps Research

InstituteLa Jolla, CAUSAandETH ZurichZurichSwitzerland

Contents

Contributors ix

Series Preface xiii

Volume Preface xv

Part A: Basic Principles 1

1 Quadrupolar InteractionsPascal P. Man 3

2 Quadrupolar Nuclei in SolidsAlexander J. Vega 17

3 Quadrupolar Coupling: An Introduction and Crystallographic AspectsSharon E. Ashbrook, Stephen Wimperis 45

4 Quadrupolar Nuclei in Solids: Influence of Different Interactions on SpectraDavid L. Bryce, Roderick E. Wasylishen 63

Part B: Advanced Techniques 75

5 Acquisition of Wideline Solid-State NMR Spectra of Quadrupolar NucleiRobert W. Schurko 77

6 Sensitivity and Resolution Enhancement of Half-Integer Quadrupolar Nuclei in Solid-State NMRThomas T. Nakashima, Roderick E. Wasylishen 95

7 Quadrupolar Nutation SpectroscopyArno P.M. Kentgens 107

8 Dynamic Angle SpinningPhilip J. Grandinetti 121

9 Double Rotation (DOR) NMRRay Dupree 133

10 MQMAS NMR: Experimental StrategiesJean-Paul Amoureux, Marek Pruski 143

11 STMAS NMR: Experimental AdvancesSharon E. Ashbrook, Stephen Wimperis 163

12 Correlation Experiments Involving Half-Integer Quadrupolar NucleiMichael Deschamps, Dominique Massiot 179

13 Computing Electric Field Gradient TensorsJosef W. Zwanziger 199

viii Contents

Part C: Applications 211

14 Quadrupolar NMR to Investigate Dynamics in Solid MaterialsLuke A. O’Dell, Christopher I. Ratcliffe 213

15 Alkali Metal NMR of Biological MoleculesGang Wu 233

16 Nitrogen-14 NMR Studies of Biological SystemsLuminita Duma 255

17 Oxygen-17 NMR Studies of Organic and Biological MoleculesGang Wu 273

18 Oxygen-17 NMR of Inorganic MaterialsSharon E. Ashbrook, Mark E. Smith 291

19 Chlorine, Bromine, and Iodine Solid-State NMRDavid L. Bryce, Cory M. Widdifield, Rebecca P. Chapman, Robert J. Attrell 321

20 Quadrupolar NMR of Ionic Conductors, Batteries, and other Energy-Related MaterialsFrederic Blanc, Leigh Spencer, Gillian R. Goward 349

21 Quadrupolar NMR of Nanoporous MaterialsMohamed Haouas, Charlotte Martineau, Francis Taulelle 371

22 Quadrupolar NMR in the Earth SciencesJonathan F. Stebbins 387

23 Quadrupolar NMR of SuperconductorsNicholas J. Curro 401

24 Quadrupolar NMR of SemiconductorsJames P. Yesinowski 417

25 Quadrupolar NMR of Metal Nuclides in Biological MaterialsTatyana Polenova, Andrew S. Lipton, Paul D. Ellis 439

26 Nuclear Waste Glasses: Insights from Solid-State NMRScott Kroeker 453

27 Quadrupolar Metal NMR of Oxide Materials Including CatalystsOlga B. Lapina, Victor V. Terskikh 467

28 Quadrupolar NMR of Intermetallic CompoundsFrank Haarmann 495

Index 511

Contributors

Jean-Paul Amoureux Universite de Lille, Villeneuve d’Ascq 59650, FranceChapter 10: MQMAS NMR: Experimental Strategies

Sharon E. Ashbrook School of Chemistry and EaStCHEM, University of St Andrews,St Andrews KY16 9ST, UKChapter 3: Quadrupolar Coupling: An Introduction andCrystallographic AspectsChapter 11: STMAS NMR: Experimental AdvancesChapter 18: Oxygen-17 NMR of Inorganic Materials

Robert J. Attrell Department of Chemistry and Centre for Catalysis Research andInnovation, University of Ottawa, 10 Marie Curie Private, Ottawa,ON K1N 6N5, CanadaChapter 19: Chlorine, Bromine, and Iodine Solid-State NMR

Frederic Blanc Department of Chemistry, University of Cambridge, Lensfield Road,Cambridge CB2 1EW, UKChapter 20: Quadrupolar NMR of Ionic Conductors, Batteries, andother Energy-Related Materials

David L. Bryce Department of Chemistry and Centre for Catalysis Research andInnovation, University of Ottawa, 10 Marie Curie Private, Ottawa,ON K1N 6N5, CanadaChapter 4: Quadrupolar Nuclei in Solids: Influence of DifferentInteractions on SpectraChapter 19: Chlorine, Bromine, and Iodine Solid-State NMR

Rebecca P. Chapman Department of Chemistry and Centre for Catalysis Research andInnovation, University of Ottawa, 10 Marie Curie Private, Ottawa,ON K1N 6N5, CanadaChapter 19: Chlorine, Bromine, and Iodine Solid-State NMR

Nicholas J. Curro Department of Physics, University of California, Davis, CA 95616, USAChapter 23: Quadrupolar NMR of Superconductors

Michael Deschamps Departement de Chimie, Universite d’Orleans, BP 6759, 1 Rue deChartres, 45067 Orleans cedex 2, FranceChapter 12: Correlation Experiments Involving Half-IntegerQuadrupolar Nuclei

x Contributors

Luminita Duma Ecole Normale Superieure, Department de Chimie, Laboratoire desBioMolecules, UMR 7203 CNRS-ENS-UPMC, 24 rue Lhomond, 75005Paris, FranceUniversite Pierre et Marie Curie Paris 6, 4 Place Jussieu, 75005 Paris,FranceChapter 16: Nitrogen-14 NMR Studies of Biological Systems

Ray Dupree Department of Physics, University of Warwick, Coventry CV4 7AL, UKChapter 9: Double Rotation (DOR) NMR

Paul D. Ellis Biological Sciences Division, K8-98, Pacific Northwest National Labora-tory, Richland, WA 99352, USAChapter 25: Quadrupolar NMR of Metal Nuclides in BiologicalMaterials

Gillian R. Goward Department of Chemistry and Brockhouse Institute for MaterialsResearch, McMaster University, Hamilton, ON L8S 4M1, CanadaChapter 20: Quadrupolar NMR of Ionic Conductors, Batteries, andother Energy-Related Materials

Philip J. Grandinetti Department of Chemistry, The Ohio State University, Columbus, OH43210-1185, USAChapter 8: Dynamic Angle Spinning

Frank Haarmann Institute of Inorganic Chemistry, RWTH Aachen University, AachenD-52074, GermanyChapter 28: Quadrupolar NMR of Intermetallic Compounds

Mohamed Haouas Tectospin, Institut Lavoisier de Versailles, Universite de Versailles-St. Quentin en Yvelines, 78035 Versailles, FranceChapter 21: Quadrupolar NMR of Nanoporous Materials

Arno P.M. Kentgens Radboud University Nijmegen, Institute for Molecules and Materials,Heyendaalseweg 135, 6525 AJ Nijmegen, The NetherlandsChapter 7: Quadrupolar Nutation Spectroscopy

Scott Kroeker Department of Chemistry, University of Manitoba, Winnipeg, MB R3T2N2, CanadaChapter 26: Nuclear Waste Glasses: Insights from Solid-State NMR

Olga B. Lapina Boreskov Institute of Catalysis, Russian Academy of Sciences, ProspectLavrent’eva 5, Novosibirsk 630090, RussiaChapter 27: Quadrupolar Metal NMR of Oxide Materials IncludingCatalysts

Andrew S. Lipton Biological Sciences Division, K8-98, Pacific Northwest National Labora-tory, Richland, WA 99352, USAChapter 25: Quadrupolar NMR of Metal Nuclides in BiologicalMaterials

Contributors xi

Pascal P. Man Universite Pierre et Marie Curie, Paris 94200, FranceChapter 1: Quadrupolar Interactions

Charlotte Martineau Tectospin, Institut Lavoisier de Versailles, Universite de Versailles-St. Quentin en Yvelines, 78035 Versailles, FranceChapter 21: Quadrupolar NMR of Nanoporous Materials

Dominique Massiot CNRS-CEMHTI, Site Hautes Temperatures, 1D Avenue de la RechercheScientifique, 45071 Orleans cedex 2, FranceChapter 12: Correlation Experiments Involving Half-IntegerQuadrupolar Nuclei

Thomas T. Nakashima Department of Chemistry, University of Alberta, Edmonton, AB T6G 2G2,CanadaChapter 6: Sensitivity and Resolution Enhancement of Half-IntegerQuadrupolar Nuclei in Solid-State NMR

Luke A. O’Dell Steacie Institute for Molecular Sciences, National Research Council ofCanada, 100 Sussex Drive, Ottawa, ON K1A 0R6, CanadaChapter 14: Quadrupolar NMR to Investigate Dynamics in SolidMaterials

Tatyana Polenova Department of Chemistry and Biochemistry, 036 Brown Laboratories,University of Delaware, Newark, DE 19716, USAChapter 25: Quadrupolar NMR of Metal Nuclides in BiologicalMaterials

Marek Pruski Department of Chemistry, Ames Laboratory, Iowa State University, Ames,IA 50011, USAChapter 10: MQMAS NMR: Experimental Strategies

Christopher I. Ratcliffe Steacie Institute for Molecular Sciences, National Research Council ofCanada, 100 Sussex Drive, Ottawa, ON K1A 0R6, CanadaChapter 14: Quadrupolar NMR to Investigate Dynamics in SolidMaterials

Robert W. Schurko University of Windsor, Department of Chemistry and Biochemistry,Windsor, ON N9B 3P4, CanadaChapter 5: Acquisition of Wideline Solid-State NMR Spectra ofQuadrupolar Nuclei

Mark E. Smith Department of Physics, University of Warwick, Coventry CV4 7AL, UKChapter 18: Oxygen-17 NMR of Inorganic Materials

Leigh Spencer Department of Chemistry and Brockhouse Institute for MaterialsResearch, McMaster University, Hamilton, ON L8S 4M1, CanadaChapter 20: Quadrupolar NMR of Ionic Conductors, Batteries, andother Energy-Related Materials

xii Contributors

Jonathan F. Stebbins Department of Geological and Environmental Sciences, Stanford Univer-sity, Stanford, CA 94305, USAChapter 22: Quadrupolar NMR in the Earth Sciences

Francis Taulelle Tectospin, Institut Lavoisier de Versailles, Universite de Versailles-St. Quentin en Yvelines, 78035 Versailles, FranceChapter 21: Quadrupolar NMR of Nanoporous Materials

Victor V. Terskikh Steacie Institute for Molecular Sciences, National Research CouncilCanada, Ottawa, ON K1A 0R6, CanadaChapter 27: Quadrupolar Metal NMR of Oxide Materials IncludingCatalysts

Alexander J. Vega Department of Chemistry and Biochemistry, University of Delaware,Newark, DE 19716, USAChapter 2: Quadrupolar Nuclei in Solids

Roderick E. Wasylishen Department of Chemistry, University of Alberta, Edmonton, AB T6G 2G2,CanadaChapter 4: Quadrupolar Nuclei in Solids: Influence of DifferentInteractions on SpectraChapter 6: Sensitivity and Resolution Enhancement of Half-IntegerQuadrupolar Nuclei in Solid-State NMR

Cory M. Widdifield Department of Chemistry and Centre for Catalysis Research and Inno-vation, University of Ottawa, 10 Marie Curie Private, Ottawa, ON K1N6N5, CanadaChapter 19: Chlorine, Bromine, and Iodine Solid-State NMR

Stephen Wimperis School of Chemistry and WestCHEM, University of Glasgow, GlasgowG12 8QQ, UKChapter 3: Quadrupolar Coupling: An Introduction andCrystallographic AspectsChapter 11: STMAS NMR: Experimental Advances

Gang Wu Department of Chemistry, Queen’s University, Kingston, ON K7L 3N6,CanadaChapter 15: Alkali Metal NMR of Biological MoleculesChapter 17: Oxygen-17 NMR Studies of Organic and BiologicalMolecules

James P. Yesinowski Chemistry Division, Naval Research Laboratory, Washington, DC 20375-5342, USAChapter 24: Quadrupolar NMR of Semiconductors

Josef W. Zwanziger Department of Chemistry, Dalhousie University, Halifax, NS B3H 4J3,CanadaChapter 13: Computing Electric Field Gradient Tensors

Series Preface

The Encyclopedia of Nuclear Magnetic Resonancewas published in eight volumes in 1996, in part tocelebrate the fiftieth anniversary of the first publica-tions in NMR in January 1946. Volume 1 containedan historical overview and ca. 200 short personalarticles by prominent NMR practitioners, while theremaining seven volumes comprise ca. 500 articleson a wide variety of topics in NMR (including MRI).Two “spin-off” volumes incorporating the articles onMRI and MRS (together with some new ones) werepublished in 2000 and a ninth volume was broughtout in 2002. In 2006, the decision was taken to pub-lish all the articles electronically (i.e. on the WorldWide Web) and this was carried out in 2007. Sincethen, new articles have been placed on the web everythree months and a number of the original articleshave been updated. This process is continuing. Theoverall title has been changed to the Encyclopedia ofMagnetic Resonance to allow for future articles onEPR and to accommodate the sensitivities of medicalapplications.

The existence of this large number of articles, writ-ten by experts in various fields, is enabling a new

concept to be implemented, namely the publicationof a series of printed handbooks on specific areasof NMR and MRI. The chapters of each of thesehandbooks will comprise a carefully chosen selec-tion of Encyclopedia articles relevant to the area inquestion. In consultation with the Editorial Board,the handbooks are coherently planned in advance byspecially selected editors. New articles are writtenand existing articles are updated to give appropriatecomplete coverage of the total area. The handbooksare intended to be of value and interest to researchstudents, postdoctoral fellows, and other researcherslearning about the topic in question and undertak-ing relevant experiments, whether in academia orindustry.

Robin K. HarrisUniversity of Durham, Durham, UK

Roderick E. WasylishenUniversity of Alberta, Edmonton, Alberta, Canada

November 2009

Volume Preface

In August 1950, the classic paper by R. V. Pound,“Nuclear Electric Quadrupolar Interactions in Crys-tals”, appeared in Physical Review and opened thedoor for NMR studies of quadrupolar nuclei in solids.Looking back at this 18-page masterpiece (Phys. Rev.,79, 685–702) one is struck by the numerous theoret-ical and experimental insights provided by Profes-sor Pound. Apart from discussing the 7Li, 23Na, and27Al NMR spectra of single crystals of Li2SO4•H2O,NaNO3, and Al2O3, respectively, powder lineshapesand relaxation effects (including the results of satu-rating satellite transitions) are provided. Many out-standing papers quickly followed and a 1957 reviewby M. H. Cohen and F. Reif, published in SolidState Physics—Advances and Applications, summa-rized the early quadrupolar NMR literature.

Over the next 15 years, many papers dealt withthe analysis of quadrupolar NMR powder lineshapescomplicated by anisotropic magnetic shielding, dipo-lar interactions, and so on. A comprehensive reviewof magnetic resonance lineshapes in polycrystallinesolids appeared in 1975 (P. C. Taylor, J. F. Baugherand H. M. Kritz, Chem. Rev., 1975, 75, 203–240).It is interesting to mention that in the section ofthis review discussing spinning techniques, the fol-lowing statement appears: “Little has been donewith nuclei possessing quadrupole moments. Becausethe quadrupolar interaction tensor is traceless, spin-ning the sample will eliminate first-order quadrupolareffects. Second-order effects will remain in a modi-fied form, however, and these could be studied inthe absence of dipolar broadening” (two referencesto E. R. Andrew and coworkers follow). We thinkit is fair to say that up to this time almost all con-tributions in the field of quadrupolar NMR of solidshad been made by physicists working largely with“home-built equipment”. In the early 1980s, therewas an explosion of activity involving NMR stud-ies of quadrupolar nuclei with nonintegral spins insolids using magic angle spinning techniques. Many

chemists and chemical physicists (E. Oldfield, A.Samoson, E. Lippmaa, R. K. Harris, C. A. Fyfe andothers) were responsible for demonstrating the ad-vantages of magic angle spinning (MAS) for inves-tigating noninteger quadrupolar nuclei in solids. Atthe same time, the potential of quadrupolar nuclei,in particular 2H, for studying dynamics in solids wasdemonstrated by several research groups.

It is not the purpose of this preface to present a re-view of NMR activity involving quadrupolar nucleiin solids; however, it is important to recognize thatthe pace at which techniques and applications in thisarea of research has developed in recent years has notsubsided. In fact, the number of relevant publicationscontinues to expand. There are several reasons for thistrend, including the following: First, the availabilityof high magnetic field strengths has had an enor-mous impact on the nature of the problems that onecan tackle. In particular, because the second-orderquadrupolar interaction scales as the inverse Larmorfrequency, the quadrupolar perturbation of the cen-tral NMR transition, mI = 1

2 ↔ mI = − 12 , decreases

at higher fields. Second, there have been considerabletechnological improvements in producing rotors forsample spinning experiments that spin rapidly, stably,and reliably. As outlined in this handbook, this hasled to the development and use of several techniques(e.g., double-rotation (DOR), dynamic-angle spin-ning (DAS), multiple-quantum magic-angle spinning(MQMAS), satellite transition magic angle spinning(STMAS)). At the same time, commercial vendorsof NMR equipment have made available double- andtriple-resonance probes capable of MAS in standard-bore magnets. Moreover, improvements in spectrom-eter hardware and software have provided experi-mentalists with unprecedented flexibility in designingpulse sequences, and so on. Finally, computers andquantum mechanical techniques for computing elec-tric field gradient tensors, magnetic shielding tensors,and so on for nuclei embedded in crystal lattices as

xvi Volume Preface

well as in “isolated” molecules are making importantcontributions to science in this area.

The purpose of the present handbook is to pro-vide under a single cover the fundamental prin-ciples, techniques, and applications of quadrupolarNMR as it pertains to solid materials. The chaptersherein have been taken from or will appear as in-dividual articles in the Encyclopedia of MagneticResonance (both the online and printed versions).Each chapter has been prepared by an expert whohas made significant contributions to our understand-ing and appreciation of the importance of NMRstudies of quadrupolar nuclei in solids. The text isdivided into three sections: (i) Basic Principles, (ii)Advanced Techniques, and (iii) Applications. Thefirst section provides the reader with the backgroundnecessary to appreciate the challenges in acquiringand interpreting NMR spectra of quadrupolar nu-clei in solids. The second section presents cutting-edge techniques and methodology for employingthese techniques to investigate quadrupolar nucleiin solids. The final section explores applications ofsolid-state NMR studies of solids ranging from inves-tigations of dynamics, characterizations of biologicalsamples, organic and inorganic materials, porous ma-terials, glasses, catalysts, semiconductors, and high-temperature superconductors.

As mentioned above, the articles can alsobe found with minimal differences in format inthe online Encyclopedia of Magnetic Resonance,at www.wileyonlinelibrary.com/ref/emr. The online

versions also include brief autobiographies of theauthors, a list of related encyclopedia articles, and insome cases, acknowledgements by the authors. Theyalso have cross-references to encyclopedia articlesthat are not part of this handbook. Additionally,article abstracts and keywords can be found online.

We hope that readers find the contributions here in-structive and that the knowledge acquired advancestheir research. Also, the related handbooks, NMRCrystallography and Solid-State NMR of Biopolymersshould provide complementary information aboutNMR of solids. Finally, we wish to thank all authorsfor their contributions to this handbook, and Profes-sor Robin K. Harris for many helpful suggestions.We also thank people at Wiley, particularly StaceyWoods, Elizabeth Grainge and Rosanna Curran, fortheir efforts, patience in assembling author contribu-tions as well as Martin Rothlisberger for his supportand leadership.

Roderick E. WasylishenUniversity of Alberta, Edmonton, Canada

Sharon E. AshbrookUniversity of St Andrews, St Andrews, UK

Stephen WimperisUniversity of Glasgow, UK

June 2012

PART ABasic Principles

Chapter 1Quadrupolar Interactions

Pascal P. ManUniversite Pierre et Marie Curie, Paris 94200, France

1.1 Introduction 31.2 Quadrupolar Hamiltonian in a Uniform

Space 41.3 Spherical Tensor Representation for the

Quadrupolar Hamiltonian 51.4 Quadrupolar Interaction as a Perturbation of

Zeeman Interaction 61.5 Energy Levels and the Spectrum of a Single

Crystal 71.6 Powder Spectrum 101.7 Appendix 12

References 15

1.1 INTRODUCTION

Nuclei are characterized by an atomic number Z, amass number A, and a nuclear spin I. The value ofI depends on those of A and Z (Table 1.1). Nucleiwith spin I > 1/2 are multiple energy level systemsand are called quadrupolar nuclei. They representmore than 70% of those in the Periodic Table.However, they are not as frequently investigated inNMR as other elements, because of their quadrupolemoments Q, which interact with the electric field

NMR of Quadrupolar Nuclei in Solid MaterialsEdited by Roderick E. Wasylishen, Sharon E. Ashbrook andStephen Wimperis© 2012 John Wiley & Sons, Ltd. ISBN: 978-0-470-97398-1

gradient (EFG) generated by their surroundings.This coupling, called the quadrupolar interactionand denoted by HQ, may be much stronger than theamplitude of the rf excitation pulse. As a result, itaffects the line intensity and alters the lineshape.These effects make the interpretation of spectramore difficult. Usually, only the first two expansionterms of HQ are considered: the first-order, (H [1]

Q ),

and second-order, (H [2]Q ), quadrupolar interac-

tions, in the vocabulary of standard perturbationtheory. H [1]

Q splits the spectrum of a half-integerquadrupole spin system in a single crystal into2I − 1 satellite lines, but the central line remainsat the Larmor frequency ω0. The additional effectof H [2]

Q is to shift further all the lines, including thecentral line.

When the sample is in powder form, as it usuallyis, it is mainly the central line that is observed. More-over, its lineshape becomes nonsymmetrical whenH

[2]Q is large. In favorable cases, the powder pattern

of the satellite spinning sidebands is detected us-ing the popular MAS technique. The powder patternof the central line is characterized by three parame-ters: the quadrupolar coupling constant χ = e2qQ/h,which is the product of a nuclear property (eQ) anda crystal property (eq), the asymmetry parameter ηand the center of gravity of the experimental line,δ

expCG (in ppm). χ is a measure of the strength of the

quadrupolar interaction and η a measure of the de-viation of the EFG from axial symmetry. The truechemical shift δCS of the central line is related to

4 Basic Principles

Table 1.1. Value of nuclear spin I as a function of atomicnumber Z and mass number A

Z

A Odd Even

Odd Half-integer I Half-integer IEven Integer I I = 0

these three parameters:1,2

δCS = δexpCG + 1

30

(1 + 1

3η2)

×[I (I + 1)− 3

4

] [3χ

2I (2I − 1)ω0

]2

(1.1)

A precise determination of δCS is required if its valuehas to be correlated with bond lengths and bondangles. Several methods are available for determiningχ and η. They can be grouped into two categories:

1. there is a series of techniques, especially themechanical spinning of the sample,1 – 5 basedon the frequency domain response of the spinsystem (see Chapter 9);

2. the second series deals with the time domainresponse of the spin system to rf excitation6 – 8

(see Chapters 7, 10 and 11).

The experimental center of gravity δexpCG is determined

by spectral simulation. However, spectra acquiredwith DAS or DOR probes provide this value directly5

(see Chapter 8). Books dealing with these moderntechniques are available.9 – 11

In the present chapter, we focus on the frequencydomain response of half-integer quadrupolar spinlarger than 1. (Jellison and co-workers12 calculatedperturbation terms up to third order for integer spinsI = 1 and 3.) The first part is devoted to a deriva-tion of the Hamiltonians corresponding to first- andsecond-order perturbations, with the emphasis on thedifferent conventions used in the literature, namely,the asymmetry parameter, the components of spheri-cal tensors in their principal axis system, the Larmorfrequency, transitions, and the transition frequency.With this in mind, the Magnus expansion is appliedinstead of standard perturbation theory. For simplic-ity, Hamiltonians are expressed in angular velocityunits and relaxation phenomena are not taken intoaccount. In the second part, NMR parameters re-lated to single crystal spectra and powder patterns

in static and MAS measurements are presented (seeChapters 19 and 22), in particular, the second-orderquadrupolar shift, the critical points and the line-shapes of the powder patterns for various values ofη, and the second-order quadrupolar shift for the cen-ter of gravity of a powder pattern. In the appendix,the commonly used Euler angles as well as thoseused by Baugher and co-workers13 – 15 are given ingraphical form. The Wigner rotation matrix, express-ing the components of the same spherical tensor intwo different coordinate frames, is also given.

1.2 QUADRUPOLAR HAMILTONIAN IN AUNIFORM SPACE

Slichter16 and others17,18 introduce the quadrupolarinteraction from the classical concept of the chargedensity for a nucleus in a space where the three co-ordinate axes x, y, and z are equivalent. Then, thequantum mechanical form of this interaction is ob-tained using operators. Thanks to the Wigner–Eckarttheorem, the Hamiltonian representing the quadrupo-lar interaction independently of the Cartesian coordi-nate frame is defined:

hHQ = eQ

6I (2I − 1)

∑α,β=x,y,z

Vαβ [ 32 (IαIβ + Iβ Iα)

−δαβI (I + 1)] (1.2a)

with

Vαβ = ∂2U

∂α ∂β

∣∣∣∣r=0

(1.2b)

δαβ is the Kronecker delta symbol, U is the elec-trostatic potential at the origin (inside the nucleus)generated by external charges, and Vαβ are the Carte-sian components of the EFG at the origin, V, whichis a second-rank symmetrical tensor. In the principalaxis system ΣPAS of the EFG, V is diagonal:

V =⎡⎣ VXX 0 0

0 VYY 00 0 VZZ

⎤⎦ (1.3)

with the convention |VZZ| ≥ |VYY| ≥ |VXX|. Further-more, the Laplace equation VXX + VYY + VZZ = 0holds for V. Thus, only two independent parametersare required:

eq = VZZ (1.4a)

η = VXX − VYY

VZZ(1.4b)

Quadrupolar Interactions 5

the largest component and the asymmetry parameter,respectively, with 1 ≥ η ≥ 0.

In the coordinate frame ΣPAS, the Cartesiantensor representation of the quadrupolar interaction[equation (1.2a)] takes the form

hHQ = e2qQ

4I (2I − 1)[3I 2

Z − I (I + 1)+ η(I 2X − I 2

Y )]

(1.5a)

In terms of the operators

I+ = IX + iIY , I− = IX − iIY (1.5b)

equation (1.5a) becomes

hHQ = e2qQ

4I (2I − 1)

×[3I 2Z − I (I + 1)+ 1

2η(I2+ + I 2−)]

(1.5c)

Sometimes, the opposite convention is adopted for η:

η = VYY − VXX

VZZ(1.6)

which is associated with the condition |VZZ|≥ |VXX| ≥ |VYY|.19,20 As a result, a negativesign appears in front of η in equations (1.5a)and (1.5c) and in subsequent expressions con-taining η.

1.3 SPHERICAL TENSORREPRESENTATION FOR THEQUADRUPOLAR HAMILTONIAN

The passage from one coordinate frame to anotheris more conveniently realized if the quadrupolar in-teraction is expressed as a second-rank irreduciblespherical tensor, according to Mehring:21

hHQ = eQ

2I (2I − 1)

2∑q=−2

(−1)qV (2,q)T (2,−q)

= eQ

2I (2I − 1)

2∑q=−2

(−1)qV (2,−q)T (2,q)

(1.7)In any Cartesian coordinate frame Σ , the spherical

tensor and Cartesian tensor components of V are

related by

V (2,0) ≡ V0 = 3√

16Vzz

V (2,1) ≡ V1 = −Vxz − iVyzV (2,−1) ≡ V−1 = Vxz − iVyz

V (2,2) ≡ V2 = 12 (Vxx − Vyy)+ iVxy

V (2,−2) ≡ V−2 = 12 (Vxx − Vyy)− iVxy

⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭

(1.8)

and those of T as

T (2,0) = 16

√6[3I 2

z − I (I + 1)]

T (2,1) = − 12 (IzI+ + I+Iz)

T (2,−1) = 12 (IzI− + I−Iz)

T (2,2) = 12 I+I+

T (2,−2) = 12 I−I−

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

(1.9)

with I+ = Ix + iIy and I− = Ix − iIy. These twooperators are different from those of equation (1.5b)despite the same notation. It is worth noting thatthe numerical factors in the components of V andT [equations (1.8) and (1.9)] differ from author toauthor.

Using equations (1.7)–(1.9), the spherical tensorrepresentation of the quadrupolar interaction in thecoordinate frame Σ becomes

hHQ = eQ

4I (2I − 1){ 1

3

√6[3I 2

z − I (I + 1)]V0

+ (IzI+ + I+Iz)V−1

− (IzI− + I−Iz)V1 + I 2+V−2 + I 2

−V2}(1.10)

Slichter16 uses nearly the same relationship, apartfrom a negative sign due to another choice of V1.From equations (1.4a), (1.4b), and (1.8), the sphericaltensor components of V in ΣPAS are obtained:

V PAS0 =

√32eq, V PAS

1 = V PAS−1 = 0,

V PAS2 = V PAS

−2 = 12eqη (1.11a)

If the other convention for η, namely, equation (1.6)is used then the spherical tensor components of V inΣPAS are20

V PAS0 =

√32eq, V PAS

1 = V PAS−1 = 0,

V PAS2 = V PAS

−2 = − 12eqη (1.11b)

6 Basic Principles

1.4 QUADRUPOLAR INTERACTION AS APERTURBATION OF ZEEMANINTERACTION

A nuclear spin possesses a magnetic moment μ andan angular momentum hI , which are related by thegyromagnetic ratio γ :

μ = γ hI (1.12)

In the laboratory frame Σ lab, the direction of thestrong static magnetic field B0 is taken as the z axis.The coupling of the magnetic moment with B0 is theZeeman interaction HZ:

hHZ = −μ · B0 = −hω0Iz (1.13a)

ω0 = γB0 (1.13b)

where ω0/2π is the Larmor frequency. Sometimesthis frequency is defined as ω0/2π = −γB0/2π. As aresult, the Zeeman interaction takes the form hHZ =hω0Iz. As with η, the choice of ω0 changes the signof some expressions below.

We deal with the case where HQ can be treated asa weak perturbation of the Zeeman interaction. It isthen more convenient to express interactions in theframe Σobs rotating relative to Σ lab with an angularvelocity ω0 so that the spherical tensor representationof the quadrupolar interaction expressed by equation(1.10) becomes time-dependent:22

hHQ(t) = exp(iHZt)hHQ exp(−iHZt)

= eQ

4I (2I − 1){ 1

3

√6[3I 2

z − I (I + 1)]V0

+ I+(2Iz + 1)V−1 exp(−iω0t)

− I−(2Iz − 1)V1 exp(iω0t)

+ I 2+V−2 exp(−i2ω0t)+ I 2

−V2 exp(i2ω0t)}(1.14)

However, the first term in the curly brackets (i.e.,the secular term) remains time-independent. In or-der to make the quadrupolar interaction completelytime-independent, HQ(t) is averaged over one Lar-mor period 2π/ω0 up to first-order, using the Magnusexpansion:23

〈HQ(t)〉 = ω0

2π

∫ 2π/ω0

0dt HQ(t)

− iω0

4π

∫ 2π/ω0

0dt∫ t

0

× dt ′ [HQ(t), HQ(t′)] = H

(0)Q +H

(1)Q

(1.15)

with

H(0)Q = eQ

4I (2I − 1)h

√6

3[3I 2

z − I (I + 1)]V0

(1.16)

H(1)Q = − 1

ω0

[eQ

4I (2I − 1)h

]2

× {√

6V0V−1I+(2Iz + 1)2

−√

6V0V1I−(2Iz − 1)2

+ 2√

6V0V−2I2+(Iz + 1)

+ 2√

6V0V2I2−(Iz − 1)

+ 2V−1V1Iz[4I (I + 1)− 8I 2z − 1]

+ 2V−2V2Iz[2I (I + 1)− 2I 2z − 1]}

(1.17)Usually, only the secular terms that commute with

Iz (i.e., the last two terms in the curly brackets ofH

(1)Q are considered. With this simplification, H(0)

Q

and H(1)Q are equivalent to the first-order, H [1]

Q , and

second-order, H [2]Q , terms in standard perturbation

theory,24 i.e.,

H[1]Q = H

(0)Q = eQ

4I (2I − 1)h

√6

3[3I 2

z − I (I + 1)]V0

(1.18)

H[2]Q = H

(1)Q = − 1

ω0

[eQ

4I (2I − 1)h

]2

× {2V−1V1Iz[4I (I + 1)− 8I 2z − 1]

+ 2V−2V2Iz[2I (I + 1)− 2I 2z − 1]}

(1.19)respectively. Equations (1.18) and (1.19), derived inthe rotating frame Σobs, are unchanged in the lab-oratory frame Σ lab. This is because they commutewith the Zeeman interaction. In other words, theycommute with the operator Iz. From now on, weshall use the language of standard perturbation theory.The first-order quadrupolar interaction H

[1]Q is inde-

pendent of ω0, whereas the second-order quadrupo-lar interaction H

[2]Q is inversely proportional to ω0.

Therefore, a strong static magnetic field is requiredto reduce the effects of H [2]

Q .

Quadrupolar Interactions 7

1.5 ENERGY LEVELS AND THESPECTRUM OF A SINGLE CRYSTAL

When a free spin I is introduced into a strongstatic magnetic field, the Zeeman interaction splits its2I + 1 energy levels |m〉, whose energy is defined by

〈m|HZ|m〉 = −mω0 (1.20)

and the difference between two consecutive energylevels (m − 1, m), expressed in angular velocity units,is

ω(Z)m−1,m = 〈m− 1|HZ|m− 1〉 − 〈m|HZ|m〉 = ω0

(1.21a)We choose the same convention as Abragam18 forthe pair (m − 1, m) and equation (1.21a) to rep-resent the transition and the transition frequency,respectively, but other authors choose (m, m − 1),(m, m + 1), (m + 1, m), equation (1.21a) or its nega-tive

ω(Z)m−1,m = 〈m|HZ|m〉 − 〈m− 1|HZ|m− 1〉 (1.21b)

Of course, these choices affect some later relation-ships dealing with transitions and transition frequen-cies. Equation (1.21a) implies that the energy levels|m〉 of a free spin in a strong static magnetic fieldB0 are equally spaced. The separation between twoadjacent levels is ω0. In the spectrum, a single line islocated at ω0. However, these energy levels may beshifted by other interactions, including the quadrupo-lar interaction discussed in this chapter.

The first-order quadrupolar interaction H[1]Q shifts

the energy levels |m〉 by an amount

〈m|H [1]Q |m〉 = eQ

4I (2I − 1)h

√6

3[3m2 − I (I + 1)]V0

(1.22)and in the spectrum, its contribution to the lineposition, i.e., the first-order quadrupolar shift ω(1)m−1,mof the line position associated with the transition(m − 1, m), is

ω(1)m−1,m = 〈m− 1|H [1]

Q |m− 1〉 − 〈m|H [1]Q |m〉

= 3eQ

4I (2I − 1)h

√6

3(1 − 2m)V0 (1.23)

The spectrum consists of 2I lines, the central one ofwhich, associated with the transition (−1/2, 1/2), isstill located at ω0. The other 2I − 1 lines are calledsatellite lines.

When the second-order quadrupolar interactionH

[2]Q is taken into account, the energy levels |m〉 are

shifted further:25

〈m|H [2]Q |m〉 = − 1

ω0

[eQ

4I (2I − 1)h

]2

× {2V−1V1m[4I (I + 1)− 8m2 − 1]

+ 2V−2V2m[2I (I + 1)− 2m2 − 1]}(1.24)

and its contribution ω(2)m−1,m to the line position, i.e.,

the second-order quadrupolar shift of the line, is26

ω(2)m−1,m = 〈m− 1|H [2]

Q |m− 1〉 − 〈m|H [2]Q |m〉

= − 2

ω0

[eQ

4I (2I − 1)h

]2

× {V−1V1[24m(m− 1)− 4I (I + 1)+ 9]

+ 12V−2V2[12m(m− 1)− 4I (I + 1)+ 6]}

(1.25)Therefore, the line associated with the transition(m − 1, m) is located in the spectrum at

ωm−1,m = ω0 + ω(1)m−1,m + ω

(2)m−1,m (1.26)

In the following two subsections, we apply equation(1.26) to two experiments, in which the single crystalis either static or is spinning at the magic angle.

1.5.1 Spectrum of a Static Single Crystal

We have to express V0 in equation (1.23), and V1,V−1, V2, and V−2 in equation (1.25) in terms of thecomponents of V in ΣPAS, equation (1.11a). For thispurpose, the following relationship is used:

Vi =2∑

j=−2

D(2)j,i (α, β, γ )V

PASj (1.27)

where the Euler angles α, β, and γ describe thedirection of the strong static magnetic field in ΣPAS

(Figure 1.1) and D(2)j,i (α, β, γ ) is the Wigner rotation

matrix defined in the appendix. For example,

V0 =√

32 eq

[ 12 (3 cos2 β − 1)+ 1

2η sin2 β cos 2α]

(1.28)Its substitution into equation (1.18) yields

H[1]Q = 1

3ωQ[3I 2z − I (I + 1)] (1.29)

8 Basic Principles

Z

b

g

Y

Xa

B0

SPAS

Figure 1.1. Euler angles defining the direction of B0 inthe principal axis system ΣPAS of the EFG during a staticexperiment.

with

ωQ = 3χ

4I (2I − 1)× [ 1

2 (3 cos2 β − 1)+ 12η sin2 β cos 2α

](1.30)

A negative sign will appear in front of η if the otherconvention for η, equation (1.6), is chosen or the Eu-ler angles used by Baugher and co-workers13 – 15 areused. The definitions of H [1]

Q by equation (1.29) orof ωQ by equation (1.30) are not unique. Other defi-nitions can be found in the literature. The first-orderquadrupolar shift of the lines (m − 1, m), equation(1.23), becomes

ω(1)staticm−1,m = (1 − 2m)ωQ (1.31)

The lines in the spectrum are separated by the samequantity 2ωQ, but the central line is not shifted.

The other two factors V1V−1 and V2V−2 in equation(1.25) are

2V1V−1

= − 32e

2q2[(− 13η

2 cos2 2α + 2η cos 2α − 3) cos4 β

+ ( 23η

2 cos2 2α − 2η cos 2α − 13η

2 + 3) cos2 β

+ 13η

2(1 − cos2 2α)] (1.32a)

V2V−2

= 32e

2q2[( 124η

2 cos2 2α − 14η cos 2α + 3

8 ) cos4 β

+ (− 112η

2 cos2 2α + 16η

2 − 34 ) cos2 β

+ 124η

2 cos2 2α + 14η cos 2α + 3

8 ] (1.32b)

The second-order quadrupolar shift of the central line,using equation (1.25), is given by

ω(2)static−1/2,1/2

= − 1

6ω0

[3χ

2I (2I − 1)

]2

[I (I + 1)− 34 ]

× [A(α, η) cos4β +B(α, η) cos2β +C(α, η)](1.33)

with

A(α, η) = − 278 + 9

4η cos 2α − 38 (η cos 2α)2

B(α, η) = 308 − 1

2η2 − 2η cos 2α + 3

4 (η cos 2α)2

C(α, η) = − 38 + 1

3η2 − 1

4η cos 2α − 38 (η cos 2α)2

⎫⎪⎬⎪⎭

(1.34)

When the EFG has axial symmetry (η = 0), equation(1.33) becomes simply

ω(2)static−1/2,1/2 = − 1

16ω0

[3χ

2I (2I − 1)

]2

[I (I + 1)− 34 ]

× (1 − cos2 β)(9 cos2 β − 1) (1.35)

It is worth noting that the third Euler angle γ does notappear in equations (1.30), (1.33), and (1.34); this isbecause B0 is a symmetry axis for the spins. Our re-sults are identical to those of Narita and co-workers27

[note that their paper contains a typographical errorconcerning the expression of cos 2α in C(α, η)]. Sub-sequently, Baugher and co-workers15 obtained ex-pressions similar to equation (1.34), except that theirterms containing η have the opposite sign. Their com-ment 23, concerning the sign in front of all the termsin cos 2α, is explained in our appendix using the Eulerangles (Figure 1.10) defined by Goldstein.14 Anotherway to obtain the same results as those of Baugherand co-workers15 is to employ the usual Euler angles(Figure 1.9) and to replace η by −η (the other con-vention for η). This point is confirmed by Hirshingerand co-workers28 and by Chu and Gerstein.29 Wolfand co-workers30 have determined the third-orderperturbation term, and shown that it is proportionalto (2m− 1)/ω2

0. Therefore, the position of the centralline is not shifted further by this new term.

1.5.2 Spectrum of a Rotating Single Crystal

First of all, the expressions for V0, V1, V−1, V2, andV−2 must be expressed in terms of the componentsof V in the coordinate frame ΣMAS of the rotor.To do this, the Wigner rotation matrix is applied

Quadrupolar Interactions 9

z

y

x

B0

wrt

qm

SMAS

Figure 1.2. Euler angles defining the direction of B0 inthe rotor coordinate frame ΣMAS during a MAS experiment.In ΣMAS, B0 rotates around the rotor with the angularvelocity ωr; θm is the magic angle; the third angle is γ

= 0. B0 performs a right-hand, positive rotation in∑MAS.

Therefore, the rotor performs a right-hand, negative rotationin∑lab.

once more:

Vi =2∑

j=−2

D(2)j,i (ωrt, θm, 0)VMAS

j (1.36)

where ωr is the angular velocity of the rotor andθm = 54.73◦ is the magic angle (Figure 1.2). Then,VMASj must be expressed in terms of the components

of V in ΣPAS:

VMASj =

2∑k=−2

D(2)k,j (α, β, γ )V

PASk (1.37)

where the Euler angles α, β, and γ describe thedirection of the rotor in ΣPAS (Figure 1.3).

The first step, equation (1.36), yields

ω(1)MASm−1,m =

√23 (1 − 2m)

3eQ

8I (2I − 1)h

× VMAS0 (3 cos2 θm − 1) (1.38)

The second step, equation (1.37), yields the first-orderquadrupolar shift:

ω(1)MASm−1,m = 1

2 (1 − 2m)ωQ(3 cos2 θm − 1) (1.39)

This shift is zero when the crystal rotates at the magicangle. In other words, all the energy levels becomeequally spaced. Therefore, a single line instead of 2Ilines appears in the spectrum at ω0.

Z

YX a

b

SPAS

g

Figure 1.3. Euler angles defining the direction of the rotorin the principal axis system ΣPAS of the EFG during aMAS experiment. In ΣPAS, the rotor containing the sampleappears static.

For the second-order quadrupolar shift, the firststep, equation (1.36), yields

ω(2)MASm−1,m = − 2

ω0

[eQ

4I (2I − 1)h

]2

× {− 16V

MAS2 VMAS

−2 [50m(m− 1)

− 6I (I + 1)+ 17]

+ 13V

MAS1 VMAS

−1 [8m(m− 1)+ 2]

− 12V

MAS0 VMAS

0

× [14m(m− 1)− 2I (I + 1)+ 5]}(1.40)

The second step, equation (1.37), yields, in the fastrotation regime,1

ω(2)MASm−1,m = − 3

32ω0

[χ

I (2I − 1)

]2

(1 + 13η

2)

× [2I (I + 1)− 14m(m− 1)− 5]

+ 3

128ω0

[χ

I (2I − 1)

]2

× [6I (I + 1)− 34m(m− 1)− 13]

× g(α, β, η) (1.41)

with

g(α, β, η) = 12 (1 + 6 cos2 β − 7 cos4 β)

+ 13η(1 − 8 cos2 β + 7 cos4 β) cos 2α

+ 118η

2[−7(1 − cos2 β)2

× cos2 2α + 8 − 4 cos2 β] (1.42)

10 Basic Principles

For the central line, the second-order quadrupolarshift is31

ω(2)MAS−1/2,1/2 = − 1

6ω0

[3χ

2I (2I − 1)

]2

[I (I + 1)− 34 ]

× [D(α, η) cos4 β

+E(α, η) cos2 β + F(α, η)]

(1.43)

with

D(α, η) = 2116 − 7

8η cos 2α + 748 (η cos 2α)2

E(α, η) = − 98 + 1

12η2 + η cos 2α − 7

24 (η cos 2α)2

F(α, η) = 516 − 1

8η cos 2α + 748 (η cos 2α)2

⎫⎪⎬⎪⎭

(1.44)

As in the case of a static sample, the Euler angle γdoes not appear in equations (1.43) and (1.44). Thisis because the experimental conditions correspond tothe fast rotation regime. However, this angle doesappear in the intermediate regime where the angularvelocity of the rotor is of the same order of magnitudeas the linewidth. As a result, spinning sidebandsappear in the spectrum. Samoson and co-workers23

established a general expression for Ω(2)MASm−1,m that

clearly shows the presence of modulations due to therotation of the rotor:

Ω(2)MASm−1,m = ω

(2)MASm−1,m +

4∑n=1

[An cos(nωrt)

+Bn sin(nωrt)] (1.45)

Two typographical errors appear in the annexe of theirpaper:23 the expressions for Aλ

1 and Aλ2 should be

Aλ1 = − 1

2

√2 sin 2βλρλ20

+ 13

√3 cos 2αλ sin 2βλρλ22

Aλ2 = 1

2 sin2 βλρλ20+ 1

6

√6 cos 2αλ(1 + cos2 βλ)ρλ22

⎫⎪⎪⎪⎬⎪⎪⎪⎭

(1.46)

Equation (1.45) allows us to investigate the spinningsidebands.

1.6 POWDER SPECTRUM

In most cases, the sample is in powder form, be-cause the growth of single crystals of significantsize is not always possible. As a result, only thecentral line is detected in NMR. However, satellitelines can be detected without spinning the sample ifχ /2π <300 kHz; for example, for 23Na (I = 3/2) in

NaNO3 or 7Li (I = 3/2) in LiNbO3. When the MAStechnique is applied, the spinning sidebands of thesatellite lines are detected; for example, for iodine127I (I = 5/2) in KI or the two isotopes of bromine(I = 3/2) in KBr. These two compounds are usedfor setting the magic angle of the MAS probe in thevicinity of 29Si and 27Al frequencies, respectively.

In a powder sample, the principal axes of the EFGassociated with each crystallite are randomly orientedwith respect to B0. The transition frequencies arenot unique, but depend on the distribution of theEuler angles α and β describing the direction ofthe rotor in the coordinate frame ΣPAS in a MASexperiment at high spinning rate, or the direction ofB0 in the coordinate frame ΣPAS in an experimentwithout spinning.

The resonance condition ω(α, cos β) represents asurface in a 3D space described by the parameters ω,α, and cos β. The critical points of this surface definedivergences and shoulders in the spectrum. They areroots of the two coupled equations12,13

∂∂αω(α, cosβ)

∣∣∣∣α=r,cosβ=s

= 0

∂∂ cosβ ω(α, cosβ)

∣∣∣∣α=r,cosβ=s

= 0

⎫⎪⎪⎬⎪⎪⎭ (1.47)

The nature of the critical point (r, s) is related to thesign of the Wronskian determinant DW:

DW =[(

∂2ω

∂α ∂ cosβ

)2

−(∂2ω

∂α2

)(∂2ω

∂2 cosβ

)]α=r,cosβ=s

(1.48)

If DW is positive then the critical point (r, s) repre-sents a divergence; if DW is negative then the crit-ical point represents a shoulder. Unfortunately, thismethod does not allow us to determine the lineshape.Therefore, numerical calculations are required.

1.6.1 Powder Pattern for a Pair of SatelliteLines

For static samples,27 numerical calculations are basedon the summation of signals for the direction of eachcrystallite. The two Euler angles are redefined by α =2πp/300 and cosβ = p/300, where p = 0, . . . , 300.

Quadrupolar Interactions 11

– (1 + h) – (1 – h) 0 2

w – w0

b

Figure 1.4. Critical points13 of a satellite powder patternassociated with the transition (m − 1, m); b = 3χ (1 −2m)/8I(2I − 1).

For each value of η, the summation is performed on

ω(1)staticm−1,m

/3χ(1 − 2m)

8I (2I − 1)= 3 cos2 β − 1

+ η sin2 β cos 2α(1.49)

A satellite lineshape is shown in Figure 1.4. Thepositions of the shoulder and divergence13 are derivedfrom equations (1.47) and (1.48). The powder patternof a pair of satellite lines for several values of η isplotted in Figure 1.5.

1.6.2 Powder Pattern for the Central Line

The critical points in the powder pattern of thecentral line have been determined for the twoexperiments. For a static sample, they have beendetermined by Stauss.32 Two patterns (Figure 1.6)appear according to the value of η compared with1/3; for example, for 39K in inorganic potassiumsalts33 or 139La in lanthanum salts.34 It is worthnoting that these critical points depend neither on theconvention for η nor on the choice of Euler angles:changing η to −η yields the same set of criticalpoints. For a rotating sample, these points have beendetermined by Muller.31 As previously, two powderpatterns (Figure 1.7) appear, depending on the valueof η compared with 3/7. These patterns are clearlyobserved in 27Al compounds.35

The powder pattern of the central line is obtainedas in the previous case using equation (1.33) for astatic sample or equation (1.43) for a rapidly rotat-ing sample. To facilitate the comparison, the two setsof spectra are superposed as shown in Figure 1.8. We

00.2

0.4

0.7

1

Figure 1.5. Simulated powder pattern of a pair of satellitelines for increasing values of the asymmetry parameter ηfrom 0 to 1 in steps of 0.1.

can see that the linewidth is reduced by a factor rang-ing from 2 to 4, depending on the value of η when aMAS experiment is performed. From a practical pointof view, the asymmetry parameter of an experimentalspectrum can be estimated by comparing the line-shape with those in Figure 1.8. Then the quadrupolarcoupling constant can be calculated using the posi-tions of the experimental critical points and thoserepresented in Figures 1.6 and 1.7. Other simple pro-cedures are also defined for extracting quadrupolarparameters from the spectra of static36 or rotating37

samples.The second-order quadrupolar shift of the center

of gravity ωiso(m − 1, m) of the powder pattern isdetermined as follows:1

ωiso(m− 1,m) = 1

4π

∫ π

0dβ sinβ

∫ 2π

0dα ω(2)MAS

m−1,m

= − 3

40ω0

[χ

I (2I − 1)

]2

(1 + 13η

2)

× [I (I + 1)− 9m(m− 1)− 3](1.50)

12 Basic Principles

w – w0

a

w – w0

a

h > 13

124

– (3 + h)2 13

– (1 – h2) 0 23

(1 + h)

124

– (3 – h)2 16

– h2 23

(1 – h)

h < 13

Figure 1.6. Critical points determined by Stauss.32 Theyare associated with the powder pattern of the central line ina static experiment;

a = − 1

6ω0

[3χ

2I (2I − 1)

]2 [I (I + 1)− 3

4

]

From a practical point of view, the experimentalchemical shift of the center of gravity associated withthe transition (m − 1, m), δexp

CG (m− 1,m), consists oftwo terms: the true chemical shift δCS(m − 1, m) andthe contribution from the second-order quadrupolarshift ωiso(m − 1, m)/ω0. Thus,

δCS(m− 1,m) = δexpCG (m− 1,m)− ωiso(m− 1,m)/ω0 (1.51)

For the central line, equation (1.51) becomes equation(1.1).

1.7 APPENDIX

The Euler angles are extensively used in the study ofthe quadrupolar interaction, especially in MAS, VAS,DAS, and DOR. They are defined as three successivepositive angles of rotation for the coordinate frame(c.f.) as described in Figure 1.9. First (Figure 1.9a),the starting c.f. (x, y, z), called the old c.f., in which

114

w – w0

h < 37

w – w0

h > 37

0 (1 + h)2 27

12 (1 + h2)

148 (15 + 6h + 7h2)

148 (15 – 6h + 7h2)

114

(1 – h)2

16

a

a

Figure 1.7. Corrected critical points9,11 determined byMuller.31 They are associated with the powder pattern ofthe central line in a MAS experiment;

a = − 1

6ω0

[3χ

2I (2I − 1)

]2 [I (I + 1)− 3

4

]

we know the components V OLDk of the spherical

tensor V, is rotated counterclockwise around the zaxis by an angle α. This rotation generates a new c.f.(X, Y, z). Then (Figure 1.9b) the counterclockwiserotation of this intermediate c.f. around the Y axis byan angle β generates a second intermediate c.f. (X′,Y, z′). Finally (Figure 1.9c), this second intermediatec.f. is rotated counterclockwise by an angle γ aroundthe z′ axis, resulting in a c.f. (x′, y′, z′), called thenew c.f., in which we wish to know the componentsV NEWj of the spherical tensor V. It is worth noting

that, in this definition of the Euler angles, β and α

also represent the polar angles of the z′ axis in theold coordinate frame.

As explained by Spiess,20 the mathematical toolfor expressing the components of the same sphericaltensor V in two different coordinate frames, where thenew c.f. is obtained by three positive angles of rota-tion (α, β, γ ) of the old one, is the Wigner rotationmatrix D(2)

p,q (α, β, γ ) reported in Table 1.2. This