PERGAMON MATERIALS SERIES VOLUME 6
Multinuclear Solid-State NMR of Inorganic Materials
PERGAMON MATERIALS SERIES Series Editor: Robert W. Cahn VRS Department of Materials Science and Metallurgy, University of Cambridge, Cambridge, UK
Vol. 1 Vol. 2 Vol. 3 Vol. 4 Vol. 5 Vol. 6
C ALPHAD by N. Saunders and A. P. Miodownik Non-Equilibrium Processing of Materials edited by C. Suryanarayana Wettability at High Temperatures by N. Eustathopoulos, M. G. Nicholas and B. Drevet Structural Biological Materials edited by M. Elices The Coming of Materials Science by R. W. Cahn Multinuclear Solid-State NMR of Inorganic Materials by K. J. D. MacKenzie and M. E. Smith
A selection offorthcoming titles in this series: Underneath the Bragg Peaks: Structural Analysis of Complex Materials by T. Egami and S. L. J. B illinge Phase Transformations in Titanium- and Zirconium-Based Alloys by S. Banerjee and P. Mukhopadhyay Thermally Activated Mechanisms in Crystal Plasticity by D. Caillard and J.-L. Martin Nucleation by A. L. Greer and K. F. Kelton Non-Equilibrium Solidification of Metastable Materials from Undercooled Melts by D. M. Herlach and B. Wei The Local Chemical Analysis of Materials by J. W. Martin Synthesis of Metal Extractants by C. K. Gupta Structure of Materials by T. B. Massalski and D. E. Laughlin Intermetallic Chemistry by R. Ferro and A. Saccone
P E R G A M O N MATERIALS SERIES
Multinuclear Solid-State NMR of Inorganic Materials by Kenneth J.D. MacKenzie School of Chemical and Physical Sciences, Victoria University of Wellington, and N e w Zealand Institute for Industrial Research and D e v e l o p m e n t
Mark E. Smith Department of Physics, University of Warwick, U K
2002
PERGAMON An Imprint of E l s e v i e r S c i e n c e A m s t e r d a m - Boston - London - New Y o r k - Oxford - P a r i s S a n D i e g o - S a n F r a n c i s c o - Singapore - Sydney - Tokyo
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Preface Although the concept of solid state nuclear magnetic resonance with magic angle spinning (MAS NMR) has been with us for many years, it is only over the last two decades that the technique has become increasingly used in other disciplines. The advantages of being able to probe the atomic environment of poorly crystalline materials were quickly appreciated by mineralogists and zeolite chemists, who exploited the fact that the two nuclides of major interest to mineral studies (29Si and 27A1) are readily amenable to MAS NMR study. Other subject areas, including ceramics, glasses and cements, have now followed this lead, and today MAS NMR is being applied to most of the major topics in the area of inorganic materials research. In the early stages of its development, MAS NMR tended to be the preserve of specialist spectroscopists and theoreticians, but, as its possibilities have become better known, it has become very much a technique used by engineers and scientists to solve real problems. The literature, which was once found mainly in fundamental and highly technical texts, is now just as likely to be found in applied science journals, as increasing numbers of non-specialist researchers turn to MAS NMR to solve problems in real and often non-ideal systems. Arising from questions directed to us by students and colleagues wishing to use MAS NMR in just such studies, especially of inorganic materials, for some time we have been conscious of the need for a book addressed to non-specialist researchers. Ideally, such a book would provide accessible answers to the most common questions about the theory and practice of MAS NMR asked by novices, and it should also provide a more specialised and up-to-date treatment of the most important areas of inorganic materials research to which MAS NMR has application. Such a book would bring together all the theory necessary to appreciate the scope (and limitations) of the technique, together with practical details and hints which are often not published elsewhere because they seem too self-evident and mundane to established practitioners. In short, the book we have envisioned is precisely that which we would have found invaluable when we first began to use MAS NMR to study inorganic materials, and into which we would continue to delve for background information as the need arose to branch out into the study of new systems. This, then, is the philosophy behind the present book, which we hope above all will prove useful to MAS NMR users whatever their level of expertise, and whatever inorganic materials they wish to study. MAS NMR is constantly being extended to a more diverse range of materials, pressing into service an ever-expanding range of nuclides including some previously considered too intractable to provide usable results. At the
vi
Preface
same time, new developments in both hardware and software are being introduced and refined. In this book we have tried to cover the most important of these new developments without spreading ourselves too thinly over the subject matter. For this reason, the book does not include any of the extensive literature on carbon-based polymers which is already catered for by comprehensive texts. Materials science is very much an interdisciplinary exercise, and its practitioners are likely to come from a variety of scientific backgrounds. It is expected that the readers of this book will be at least slightly acquainted with some aspects of physics, chemistry, mineralogy or engineering, and some may be deeply knowledgeable in one or more aspects of these subjects. Because of the diverse nature and range of inorganic materials which have been subjected to NMR investigation (from cements, catalysts and glasses to superconductors and metals), it is beyond the scope of this book to treat in depth the most advanced chemical and physical aspects of the various materials. We have therefore confined ourselves to discussion and illustration of how solid state NMR techniques have been applied to a wide range of systems, and have provided an extensive bibliography to allow a reader with particular interests a convenient entry into the literature to gain more advanced information on any specific topic. The plan of the book is straightforward; the second chapter lays the theoretical groundwork for an understanding of the NMR experiment, including the factors which must be taken into account both when establishing the appropriate experimental conditions and when interpreting the data. Particular attention has been given to the theoretical basis of new experimental techniques such as multiple-quantum MAS NMR, which holds considerable promise for a wide range of inorganic solids. The third chapter covers the practical aspects of solid state NMR spectroscopy, while in the following chapters, the foregoing principles are applied to specific nuclides, with particular emphasis on applications of the technique to solving problems in inorganic materials science. A range of illustrative examples has been included, together with tabular data summarising the present state of the literature on the various nuclides in inorganic materials. Finally we should note that this book is possible only because of the ingenuity of the NMR researchers who have contributed to the rapid development of the subject. We are especially grateful to the many NMR colleagues who have shared their enthusiasm and expertise with us, and to members of our research groups both past and present, whose efforts have been very important. The copyright holders who gave their permission to use figures are gratefully acknowledged. MES is grateful for the support of funding bodies such as EPSRC, the Royal Society and the Leverhulme Trust, and particularly thanks Susan Holmes for her support and understanding during the preparation of this book. KM is indebted to the Royal Society of New Zealand for a James Cook Research Fellowship allowing two years to be spent in Oxford during which time the book became a reality, and to the Materials Department of Oxford
Preface
vii
University for being gracious hosts and providing the necessary facilities for this undertaking. We also wish to record our gratitude to the Series Editor, Prof. Robert Cahn, and the Editorial staff of Elsevier Science for their assistance in bringing this book to fruition. KENNETH MACKENZIE Email: kenneth.mackenzie@ vuw. ac.nz MARK SMITH Email: m.e.smith, [email protected] Oxford and Warwick, October 2001
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Contents Preface CHAPTER 1 INTRODUCTION 1.1. Methodology of Materials Characterisation by NMR 1.2. Historical Aspects of NMR Spectroscopy 1.3. Brief Description of the NMR Experiment 1.3.1 General Principles 1.3.2 Overcoming NMR Spectral Broadening in Solids by MAS 1.3.3 Other NMR Experiments used with Solids 1.3.3.1 Decoupling 1.3.3.2 Cross-Polarisation (CP) 1.3.3.3 Spin-Echo Experiments 1.3.3.4 Two-Dimensional Experiments 1.3.4 Nuclei Suitable for NMR Spectroscopy 1.4. Further Reading References
3 6 7 7 10 11 12 12 12 12 13 17 18
CHAPTER 2 PHYSICAL BACKGROUND 2.1. Fundamental Interaction with External Magnetic Fields 2.1.1 A Quantum Mechanical Description of the Zeeman Interaction 2.1.2 Bulk Magnetisation 2.1.3 The Rotating Frame and the Application of RF Pulses 2.1.4 Observation of the NMR Signal 2.2. Internal Interactions 2.2.1 The Dipolar Interaction 2.2.2 Scalar Coupling 2.2.3 Paramagnetic Coupling 2.2.4 Chemical Shielding 2.2.5 Knight Shift 2.2.6 Quadrupole Interaction 2.2.7 Nature of Interactions
23 25 26 29 34 35 37 40 43 44 48 50 57
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2.3.
58 59 59 61 63 64
One Dimensional Methods for Improving Resolution 2.3.1 Magic Angle Spinning and First-Order Effects 2.3.1.1 Physical Principles 2.3.1.2 Formation of Spinning Sidebands 2.3.2 Magic Angle Spinning and Higher-Order Effects 2.3.2.1 MAS of Second-Order Quadrupole Effects 2.3.2.2 Residual Coupling Effects due to Quadrupolar Nuclei in MAS Spectra 2.3.2.3 Nonequivalent Homonuclear Spins 2.3.3 Variable Angle Spinning 2.3.4 Double Angle Spinning 2.3.5 Multiple Quantum Transitions 2.3.6 Ultrasonically-Induced Narrowing 2.4. Dipolar Decoupling 2.4.1 Heteronuclear Dipolar Decoupling 2.4.2 Homonuclear Dipolar Decoupling 2.5. Spin-locking 2.6. Cross-Polarisation 2.7. Two-Dimensional Methods 2.7.1 Dynamic Angle Spinning 2.7.2 2D MQMAS 2.8. NMR Relaxation 2.8.1 Introduction to Relaxation 2.8.2 Mechanism for Relaxation Processes References
71 74 74 75 77 78 78 78 79 83 85 90 92 93 98 98 101 105
CHAPTER 3 EXPERIMENTAL APPROACHES 3.1. Basic Experimental Principles of FT NMR 3.2. Instrumentation 3.2.1 Overview of a Pulsed FT NMR Spectrometer 3.2.2 Magnets 3.2.3 Shimming 3.2.4 Transmitters 3.2.5 Probes 3.2.6 Connection of the Probe 3.2.7 Signal Detection 3.2.8 Additional Equipment
111 112 112 113 115 116 120 122 124 127
Contents
3.3.
Practical Acquisition of NMR Spectra 3.3.1 Processing the FID to Produce a Spectrum 3.3.1.1 Window Functions 3.3.1.2 Shifting of the Time Origin and Linear Back Prediction 3.3.1.3 Zero Filling 3.3.1.4 Phase Correction 3.3.1.5 Baseline Correction 3.3.2 Complications in Recording Spectra 3.4. Static Broad Line Experiments 3.4.1 Pulsed Echo Experiments 3.4.2 Stepped Experiments 3.5. One-Dimensional High Resolution Techniques 3.5.1 Magic Angle Spinning (MAS) 3.5.2 Extraction of Parameters from MAS NMR Spectra 3.5.3 Suppression of Spinning Sidebands 3.5.4 Special Considerations for MAS of Quadrupolar Nuclei 3.5.5 Magic Angle Spinning Observation of Satellite Transitions 3.5.6 Double Angle Rotation of Quadrupolar Nuclei 3.5.7 Practical Implementation of CRAMPS 3.6. Two-Dimensional Experiments 3.6.1 Nutation NMR 3.6.2 Off-Resonance Nutation 3.6.3 Order-Resolved Sideband Spectra 3.6.4 Dynamic Angle Spinning (DAS) 3.6.5 Two-Dimensional Sequences Developed from Solution NMR 3.6.6 Multiple Quantum Experiments in Dipolar Coupled Systems 3.6.7 Multiple Quantum NMR Experiments of Non-Integer Spin Quadrupolar Nuclei 3.6.8 2D XY Correlation Methods 3.6.9 Correlation of Tensor Information- Separated Local Field Experiments 3.7. Summary of Approaches for Examining Quadrupole Nuclei 3.8. Multiple Resonance 3.8.1 Cross-Polarisation (CP) 3.8.2 SEDOR, REDOR and TEDOR 3.8.3 TRAPDOR and REAPDOR 3.9. Techniques for Determining Relaxation Times and Motional Parameters 3.9.1 Measurement of T1
xi 127 128 128 129 129 130 130 130 133 133 136 138 138 143 143 144 149 150 152 153 153 154 155 156 157 160 161 168 170 172 172 173 178 182 183 183
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3.9.2 Other Spin-Lattice Relaxation Times (TI~, T1D) 3.9.3 Transverse Relaxation Times (Y2) 3.9.4 Molecular Motion 3.9.5 Diffusion Measurements 3.10. NMR Under Varying Physical Conditions 3.10.1 Variable Temperature NMR 3.10.2 High Pressure Experiments References
CHAPTER 4 298I NMR 4.1. General Considerations 4.1.1 Broadening Effects in 298i Spectra 4.1.2 Relaxation Effects in 298i Spectra 4.1.3 Effect of Structure on 298i Spectra 4.2. Si-O Compounds 4.2.1 Relationships between 29Si NMR Spectra and Structure/Bonding 4.2.2 Four-Coordinated Si-O-Compounds 4.2.3 Tetrahedral 298i Chemical Shifts in Silicates 4.2.4 29Si Chemical Shifts in Aluminosilicates 4.2.5 Effects of Other Nearest Neighbours on the 298i Shift 4.3. Order-Disorder Effects in Minerals 4.4. Identification of Silicate Minerals 4.5. Thermal Decomposition of Silicate Minerals 4.6. Relationships between 298i Chemical Shift (d) and Structure 4.6.1 Relationships between 6 and the Si-O Bond Length 4.6.2 Relationships between 6 and the Si-O-Si Bond Angle 4.6.3 More Complex Relationships between 6 and the Structure 4.7. Five and Six-Coordinated Si-O Compounds 4.8. Cross-Polarisation (CPMAS) Experiments 4.8.1 Cross-Polarisation between ~H and 298i 4.8.2 Cross-Polarisation between 19F and 298i 4.8.3 Other Cross-Polarisation Experiments with 298i 4.9. Glasses, Gels and Other Amorphous Materials 4.9.1 Silicate Glasses 4.9.2 Deconvolution of 298i NMR Spectra 4.9.3 Connectivities in Glass 4.9.4 Chalcogenide Glasses
184 185 186 187 187 187 189 190
201 201 202 204 205 205 205 205 206 208 208 212 214 217 218 219 223 225 227 227 229 229 230 231 235 236 238
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4.9.5 Gels 4.9.6 Other Amorphous Materials 4.10. Si-N and Si-N-O Compounds 4.11. Si-A1-O-N Compounds 4.11.1 fl-Sialon, Si6_zAlzOzNs_z 4.11.2 O-Sialon, Si2_xAlxOl+xNz_x 4.11.3 X-Sialon, nominally Si12AllsO39Ns 4.11.4 Polytypoid Sialons, (Si,A1)m(O,N)m+l 4.11.5 oL-Sialons,MxSilz_(m+n)Alm+nOnN16-n 4.12. Other Metal Silicon Nitrides and Oxynitrides 4.13. Si-C, Si-C-O and Si-C-N Compounds 4.13.1 Silicon Oxycarbide Species 4.13.2 Silicon Carbonitride Species 4.14. Other Materials 4.14.1 Biologically Compatible Glasses 4.14.2 Cements 4.14.3 Inorganic Polymers References
240 242 244 247 247 25O 251 253 253 253 255 256 257 257 257 257 259 260
CHAPTER 5 27AL NMR 5.1. General Considerations 5.2. Chemical Shifts in 27A1 Spectra 5.2.1 27A1Chemical Shifts in A1-O Environments 5.2.2 27A1Chemical Shifts in Aluminosilicates 5.2.3 Relationships between 27A1 Chemical Shift (giso) and Structure 5.3. Five-Coordinated A1-O 5.3.1 A1(v) in Well-Defined (Crystalline) Environments 5.3.2 A1(v) in Non-Crystalline Environments 5.3.3 A1(v) in Zeolites 5.4. Aluminium Oxides 5.5. Amorphous Aluminium Compounds 5.5.1 Aluminate Gels 5.5.2 Glasses 5.5.3 Other Amorphous Systems 5.6. Aluminophosphates 5.7. Aluminium Borate and Molybdate 5.7.1 Aluminium Borate 5.7.2 Aluminium Molybdate
271 272 273 274 279 281 281 283 287 291 294 294 299 303 304 307 307 307
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Aluminium Fluorides 5.9. Thermal Decomposition Reactions 5.10. Cements 5.11. Nitride and Oxynitride Compounds 5.12. Sialon Compounds 5.12.1 Polytypoid Sialons 5.12.2 [3-Sialons 5.12.3 O-Sialons 5.12.4 X-Sialons 5.12.5 oL-Sialons 5.12.6 Sialon Glasses References
308 310 313 316 317 317 318 320 321 322 323 324
CHAPTER 6 170 NMR 6.1. Introduction 6.2. Background 6.2.1 Enrichment Schemes 6.2.2 Experimental NMR Methodology 6.2.3 Relationships between NMR Parameters and Structure 6.3. Binary Oxides 6.3.1 Crystalline Materials 6.3.2 Sol-Gel Produced Samples 6.4. Crystalline Ternary Ionic Systems 6.5. Silicates and Germanates 6.5.1 Crystalline Materials 6.5.1.1 Silica and Germania 6.5.1.2 Ternary Silicates 6.5.1.3 Silicates and Germanates of Zirconium and Titanium 6.5.2 Amorphous Materials 6.5.2.1 Silica and Germania 6.5.2.2 Metal Silicate and Germanate Glasses 6.5.2.3 Gel-Based Silicates 6.6. Aluminium- and Gallium-Containing Systems 6.6.1 Alumina and Aluminates 6.6.2 Crystalline Alumino- and Gallosilicates 6.6.3 Amorphous Aluminosilicates 6.7. Boron-Containing Systems 6.7.1 Borates
333 334 334 337 346 349 349 352 355 359 359 359 361 365 366 366 367 369 372 372 375 379 381 381
5.8.
Contents
6.7.2 Ternary and Quaternary Systems 6.8. Other Systems 6.9. Hydrogen-Containing Samples 6.9.1 Crystalline Hydroxides and Other Hydrogen-Containing Materials 6.9.2 Hydrous Gels and Glasses 6.10. High Temperature Ceramic Superconductors References
CHAPTER 7 NMR OF OTHER COMMONLY STUDIED NUCLEI 7.1. 23NaNMR 7.1.1 General Considerations 7.1.2 23NaNMR Spectra of Sodium Compounds 7.1.3 Relationships between the 23Na Chemical Shift and Structural Parameters 7.1.4 23NaNMR of Crystalline Materials 7.1.5 23NaNMR Studies of Thermal Reactions 7.1.6 23NaNMR of Glasses 7.1.6.1 Silicate and Aluminosilicate Glasses 7.1.6.2 Sodium Borosilicate Glasses 7.1.6.3 Sodium Borate, Germanate and Tellurite Glasses and Melts 7.1.6.4 Phosphate Glasses 7.1.6.5 Miscellaneous Glass Studies 7.1.7 23NaNMR of Zeolites 7.2. llBNMR 7.2.1 General Considerations 7.2.2 11B NMR of Crystalline Compounds 7.2.3 I~B NMR of Glasses 7.2.4 ~B NMR of Zeolites 7.3. 31PNMR 7.3.1 Relationships between 31p NMR Parameters and Structure 7.3.2 31p NMR of Glasses 7.3.2.1 Binary Phosphate Glasses 7.3.2.2 Phosphosilicate Glasses 7.3.2.3 Alkali Borophosphate Glasses 7.3.2.4 Borosilicophosphate Glasses 7.3.2.5 Phosphoaluminosilicate Glasses 7.3.2.6 Alkali Phosphoaluminoborosilicate Glasses
XV
382 384 386 386 387 388 390
399 399 399 403 406 412 413 413 414 415 415 416 418 420 420 421 424 431 432 438 441 441 443 445 445 446 447
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7.3.3 7.3.4 References
Contents
7.3.2.7 Phosphorus Chalcogenide Glasses 31p NMR of A1PO4 Molecular Sieves 31p NMR of Biomaterials
CHAPTER 8 NMR OF LOW-~/NUCLIDES 8.1. General Considerations 8.1.1 Problems Associated with Low-y Nuclei 8.2. NMR of Spin-1/2 Nuclei 8.2.1 89y NMR 8.2.2 l~ and l~ NMR 8.2.3 183WNMR 8.3. Quadrupolar Nuclei 8.3.1 laN NMR 8.3.2 25Mg NMR 8.3.3 33S NMR 8.3.4 35C1and 37C1 NMR 8.3.5 39K NMR 8.3.6 43CaNMR 8.3.7 47Ti and 49Ti NMR 8.3.8 67ZnNMR 8.3.9 91ZrNMR 8.3.10 95Mo and 97Mo NMR 8.3.11 135Baand 137Ba NMR 8.3.12 Other Miscellaneous Low-y Nuclei References
CHAPTER 9 NMR OF OTHER SPIN-l/2 NUCLEI 9.1. Introduction 9.2. Abundant High-~/Nuclei 9.2.1 ~H NMR 9.2.1.1 Background to Proton Studies in Inorganic Materials 9.2.1.2 Studies of Stoichiometric Protons in Crystalline Materials 9.2.1.3 Non-Stoichiometric Proton Environments in Crystalline and Glassy Materials
447 448 450 452
461 461 462 462 469 473 475 475 479 488 491 495 502 505 511 514 516 522 525 526
535 536 536 536 539 542
Contents
9.3.
9.2.1.4 1H NMR of Hydrous Glasses 9.2.1.5 Biomineral-Related Materials 9.2.2 19F NMR 9.2.2.1 Introduction 9.2.2.2 Simple Inorganic Fluorides 9.2.2.3 More Complex Fluorides 9.2.2.4 Applications to Fluoroapatite Studies 9.2.2.5 Fluorine in Aluminosilicate Minerals and Related Materials 9.2.2.6 Surface Interaction of Fluorine with Silica- and Alumina-Based Materials 9.2.2.7 Fluorine in Alumino- and Gallophosphates 9.2.2.8 Fluorine in Oxygen-Containing Glasses 9.2.2.9 Fluoride Glasses 9.2.2.10 Fluorine in Other Materials 9.2.2.11 Fluorine as a Source of Cross-Polarisation 9.2.2.12 Summary of 19F Shift Trends and Other NMR Properties Dilute or Medium-~/Nuclei 9.3.1 13C NMR 9.3.1.1 13C NMR of Elemental Carbon 9.3.1.2 Silicon Carbide 9.3.1.3 Other Binary Carbides 9.3.1.4 Ternary and Quaternary Carbides 9.3.1.5 Carbonates 9.3.2 15NNMR 9.3.2.1 Nitrides 9.3.2.2 Silicon Aluminium Oxynitride Ceramics and Glasses 9.3.2.3 Nitride Ceramics from Polymeric Precursors 9.3.2.4 Nitrates and Nitrites 9.3.3 77Se NMR 9.3.4 111Cd and 113CdNMR 9.3.5 ll5Sn, ll7Sn and ll9Sn NMR
9.3.6
9.3.5.1 Crystalline Oxygen-Containing Materials 9.3.5.2 Oxide Solid Solutions and Glasses 9.3.5.3 Non-oxide Materials 123Te and 125TeNMR 9.3.6.1 Crystalline Tellurides 9.3.6.2 Crystalline Tellurites and Tellurates 9.3.6.3 Glassy Tellurium-Containing Materials
xvii 545 550 550 550 551 554 555 556 557 559 559 560 562 562 562 563 563 563 568 570 572 572 574 575 576 579 582 583 587 591 591 594 595 598 598 599 601
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9.3.7 129Xe NMR 9.3.8 195pt NMR 9.3.9 199Hg NMR 9.3.10 2~ and 2~ NMR NMR 9.3.11 2~ 9.3.11.1 Correlations between 2~ Chemical Shifts and Structure 9.3.11.2 2~ NMR of Crystalline Lead Compounds 9.3.11.3 2~ NMR of Lead-Containing Glasses 9.3.11.4 2~ in Sol-Gel Prepared Ceramics References
CHAPTER 10 NMR OF OTHER QUADRUPOLAR NUCLEI 10.1. 6Li and 7Li NMR 10.1.1 General Considerations 10.1.2 6'7LiNMR of Crystalline Solids 10.1.3 Relation between 6Li Chemical Shifts and Structure 10.1.4 6'7LiNMR of Fast Lithium Ion Conductors 10.1.5 6'7LiNMR of Glasses 10.2. 9Be NMR 10.3. 5iV NMR 10.3.1 General Considerations 10.3.2 5iV NMR of Vanadium Oxides and the Vanadates 10.3.3 5~V NMR of Zeolites and Catalysts 10.4. 63CHand 65CHNMR 10.4.1 63CHNMR of Superconductors and Superfast Ionic Conductors 10.5.
10.6.
10.7. 10.8.
69Ga and 71Ga NMR 10.5.1 General Considerations 10.5.2 69'71GaNMR of Crystalline Compounds 10.5.3 69'71GaNMR of Other Compounds 87Rb NMR 10.6.1 General Considerations 10.6.2 87RbNMR of Crystalline Compounds 10.6.3 87RbNMR of Rubidium Fullerides 93Nb NMR 133Cs NMR 10.8.1 General Considerations
601 603 604 604 607 607 609 613 615 616
629 629 630 634 636 638 639 642 642 642 646 649 650 653 653 655 657 658 658 658 661 662 665 665
Contents
10.8.2 10.8.3 10.8.4
133CsNMR of Crystalline Caesium Compounds
133CsNMR of Minerals and Zeolites 133CsNMR of Fullerides, Superionic Conductors and Semiconductors 10.9. 139LaNMR References
xix 666
669 673 674 678
CHAPTER 11 SOLID STATE NMR OF METALS AND ALLOYS 11.1. Introduction 11.2. Experimental Approaches 11.3. Metallic Elements 11.4. Intermetallic Alloys 11.5. Phase Transformations, Ordering and Defect Sites 11.6. Phase Composition and Precipitation 11.7. Atomic Motion References
687 689 691 693 696 698 700 701
SUBJECT INDEX MINERAL INDEX
703 721
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Chapter 1
Introduction 1.1. Methodology of Materials Characterisation by NMR 1.2. Historical Aspects of NMR Spectroscopy 1.3. Brief Description of the NMR Experiment 1.3.1 General Principles 1.3.2 Overcoming NMR Spectral Broadening in Solids by MAS 1.3.3 Other NMR Experiments used with Solids 1.3.3.1 Decoupling 1.3.3.2 Cross-Polarisation (CP) 1.3.3.3 Spin-Echo Experiments 1.3.3.4 Two-Dimensional Experiments 1.3.4 Nuclei Suitable for NMR Spectroscopy 1.4. Further Reading References
3 6 7 7 10 11 12 12 12 12 13 17 18
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Chapter 1
Introduction Solid state NMR spectroscopy is a powerful technique capable of providing information both about the structure of materials and about the dynamics of processes occurring within those materials. The focus of this book is on the application of NMR spectroscopy to structural studies of materials rather than their dynamics. NMR spectroscopy has been applied to the study of a wide range of materials including polymers, organic compounds, organometallics and foodstuffs, but these are outside the scope of this book, which has been restricted to the range of diverse inorganic materials amenable to multinuclear NMR spectroscopy in its various forms.
1.1. METHODOLOGY OF MATERIALS CHARACTERISATION BY NMR
A number of experimental techniques exist for investigating the structures of materials (Figure 1.1). On the largest scale, materials are characterised by measurements of their bulk properties including density, surface area, chemical composition and thermal properties. At the next level, the macroscopic and mesoscopic properties can be probed by optical and electron microscopy, with small angle X-ray and neutron scattering (SAXS, SANS) also used to probe the mesoscopic properties. In the medium range of atomic ordering, structures can be probed by conventional X-ray, neutron or electron diffraction. Diffraction of amorphous materials yields the radial distribution function which is being increasingly used to characterise such materials. Structures are investigated on the atomic scale by vibrational spectroscopies (infrared, ultraviolet-visible, Raman), by X-ray absorption (EXAFS, XANES) or by the various multinuclear NMR techniques. Materials can exist in an almost infinite variety of states of structural disorder and heterogeneity (Figure 1.2). Unlike techniques such as X-ray diffraction which conventionally require the presence of long-range atomic order, NMR spectroscopy is almost equally useful for probing the atomic environments of the most disordered (melts and colloid gels) as for the most ordered single crystal systems. NMR is thus able to monitor the changes occurring in the atomic environment when materials change from one state of structural disorder or heterogeneity to another and has proved to be a powerful technique for studying the transition from glasses to melts, nucleation of polycrystals from glasses and the formation and growth of crystalline phases from colloids or gels. NMR is also an ideal method for studying the intermediate phases formed when minerals react or transform to other phases by heating or
Multinuclear Solid-State NMR of lnorganic Materials
Atomic scale probes
Bulk measurements
Multinuclear NMR:
MAS, CP, MQ
i
X-ray absorption: EXAFS, XANES
>
Density Surface area Chemical composition Thermal analysis
I
Vibrational spectroscopy: IR, UV-visible, Raman
Macroscopic and mesoscopic probes
Medium range probes X-ray diffraction Neutron diffraction Electron diffraction Radial distribution function
Electron microscopy Optical microscopy
Mesoscopic probes Small angle X-ray scattering Neutron scattering
Figure 1.1. Relationship between the various methods for investigating the structures of materials.
Melt
Colloid (Gel) /
I Glass transition
/
Agglomeration
O oN
Glass
Nanocrystal /
,4~
Nucleation ~
~//Crystal growth
Polycrystai
i / ~ Single Crystal
Composite
Structural heterogeneity Figure 1.2. Schematic relationship between disorder and heterogeneity of complex materials. mechanical grinding, which often lack the long-range order necessary for conventional diffraction studies. An important element in the use of NMR techniques to determine the structural characteristics of materials is to establish relationships between the experimental
Introduction
5
NMR spectrum and recurring structural motifs. In its simplest form this is done by a "fingerprinting" procedure in which an extensive database of NMR spectra is established for related materials of known composition and structure, and the characteristics of an unknown material are deduced by comparison, (shown schematically in Figure 1.3, pathway 3). The parameter most readily determined from an NMR spectrum is the position of the resonance peak, and although this may not necessarily represent the true value of the isotropic chemical shift ~iso (see below), empirical relationships have been developed for a limited range of similar compounds between the resonance position and some characteristic feature of the structural unit (e.g. the mean bond angle or bond length). To put these empirical relationships on a sounder theoretical basis and make them applicable to a wider range of compounds, other spectral parameters such as the nuclear quadrupolar coupling constant (• or the chemical shift anisotropy (CSA) may have to be invoked; these are deduced from a more in-depth analysis of the NMR spectrum, in some cases making measurements at more than one magnetic field and using specialised techniques to improve the resolution of the spectral lineshapes. Another approach to deducing the structures of materials from their solid state NMR spectra is to make an ab initio theoretical calculation of the electric field gradient at the nucleus of an atom in a known crystal structure environment (Figure 1.3, pathway 1), from which the nuclear quadrupolar parameters and hence the expected NMR spectrum can be calculated and compared with the experimental spectrum (Figure 1.3. pathway 2). The difficulties inherent in making such ab initio calculations are such that relatively few real compounds have so far been solved completely, but developments in ab initio theory, and mathematical algorithms and computing techniques should make this approach much more widely accessible in future.
I
' Sam le 'i P I Structure I
(~
~- Nuclear l, Spin / Hamiltoniau
.... J
] xperlmeut INMR ~ Spectrum J Figure1.3. Relationship between experimental and theoretical approaches for establishing the structural characteristics of materials by NMR spectroscopy.
Multinuclear Solid-State NMR of lnorganic Materials
1.2. HISTORICAL ASPECTS OF NMR SPECTROSCOPY This book is about an experimental technique which has been a mainstay of solution chemistry for many decades. Since very well resolved and structurally informative 13C NMR spectra can readily be obtained from solutions at modest magnetic field strengths, organic chemists have traditionally been the principal beneficiaries. In earlier times inorganic chemists attempting to apply the technique to solid samples did not fare well, as these spectra were too broad to provide much useful information. Furthermore, the interesting nuclei are often quadrupolar (e.g. 27A1), ideally requiring much higher magnetic fields. The need for higher field strengths was eventually satisfied by the development of superconducting magnets, the field of which could not, however, readily be varied. Thus the traditional technique of irradiating the sample with a continuous source of radiofrequency radiation while scanning over a range of the magnetic field had to be modified; the field was now kept constant and the radiofrequency was applied as a sequence of pulses. An important breakthrough in NMR spectroscopy of solids came when spectroscopists in the UK (Andrew et al. 1959) and America (Lowe 1959) deduced that some of the factors causing broadening in these materials could be minimised by rapidly rotating the sample at a particular angle to the magnetic field axis (the magic angle, 54.73~ However there were still difficulties associated with observing materials of interest such as ~3C in organic polymers as the sensitvity of ~3C was low and the relaxation time was long. In addition the ~H-~3C dipolar coupling was too strong to be narrowed by MAS at the spinning speeds then available. A number of other experimental breakthroughs were necessary. The concept of polarisation transfer from an abundant to an insensitive nucleus was demonstrated by Hartman and Hahn (1962). This was combined successfully with decoupling for ~H-13C (Pines, Gibby and Waugh 1973). The final piece of the jigsaw was to combine cross-polarisation, decoupling and MAS (Schaefer and Stejskal 1976) to reveal the fine structure in ~3C NMR spectra of solid materials. These developments opened the way for commmercial probe development to allow usefully narrow solid state NMR spectra to be obtained from nuclei such as 298i and 27A1, a fact rapidly exploited by mineralogists and materials scientists working on zeolite catalysts. The study of the latter received considerable impetus and funding from the oil crisis of the 1970s, resulting in a great deal of detailed work being done to establish practical relationships between the structural units of the silicates and zeolites and their 298i NMR spectra (summarised in Engelhardt and Michel 1987). Although 298i NMR has historically received the greatest amount of attention for solid state NMR of inorganic materials because of its ubiquity and its technical importance in materials science, other nuclei (especially quadrupolar nuclei such as 27A1, 23Na and ~B) have become increasingly accessible with the availability of higher magnetic fields, faster spinning speeds and new methodologies for improving resolution and
Introduction
7
offsetting the effects of quadrupolar broadening (double rotation, double angle spinning and multiple quantum techniques). All these aspects are treated in depth in the following chapters.
1.3. B R I E F D E S C R I P T I O N O F T H E N M R E X P E R I M E N T
1.3.1 General principles Many nuclei possess a quantised property called nuclear spin which can be adequately described only by quantum mechanics (see Chapter 2) but can usefully be thought of as being caused by the physical spinning of the nucleus. Nuclei with even mass number and even charge (e.g. 12C, 160) have zero spin and are therefore of no use for NMR spectroscopy. The angular momentum of a spinning nucleus is a function of its spin quantum number I which can have either integer or half-integer values. When such a nucleus is placed in a strong magnetic field, the energy levels between the various spin states are split (the Zeeman interaction). The differences between the various energy levels are small compared with spectroscopies involving electronic energy states, and transitions are only possible between adjacent energy levels, with the absorption or emission of a photon in the radiofrequency (rf) range. It is the frequency of this rf radiation which is measured in an NMR experiment. The nuclei in different structural environments in a solid may experience slightly different magnetic fields because they are shielded by the surrounding electrons and consequently absorb photons of slightly different frequency. Since the absolute NMR frequencies are difficult to measure with sufficient accuracy, the resonance frequencies are normally reported as chemical shifts (~) relative to an external standard compound. Unless the nucleus is in a very symmetrical environment, it experiences anisotropic shielding which will be reflected in its ~ value. A more useful parameter for comparison purposes is the isotropic chemical shift ~iso, the average shift which is experienced by the nucleus. The goal of all NMR experiments is to determine the change in the separation of the energy levels for different environments. In its simplest form, an NMR experiment consists of three parts, the preparation of the nuclear spin system by placing it in an external magnetic field, its perturbation by applying a pulse of rf radiation, and the detection of phenomena accompanying its return to the initial state when the perturbation is removed.
Preparation. When the nuclear spin system is placed in the external magnetic field Bo, its magnetic moment precesses about the magnetic field axis similar to the manner in which a gyroscope precesses about the orientation of a gravitational field (Figure 1.4A). In the NMR experiment we are dealing with many spins simultaneously and the observed response is an average of the behaviour of the individual spins. The
Multinuclear Solid-State NMR of Inorganic Materials
Z
Z' M
"~Xx
A. Preparation
B. Perturbation
C. Detection
(evolution)
N
/ \
/ \
I \
/ /
N N
/
Figure 1.4. Schematic representation of the three stages of a simple NMR experiment. In the preparation stage the magnetic moment of the nucleus precesses about the principal axis of the applied magnetic field Bo. In the evolution stage the sample is irradiated with plane-polarised rf radiation at the Larmor frequency containing a magnetic field component Bo which causes the net magnetisation M to incline, becoming oriented perpendicular to Bo in the case of a -rr/2 pulse. In the detection stage the spin system coherently precesses in the transverse plane inducing a voltage in the coil producing a signal which is detected (the FID). frequency of the precession is called the Larmor frequency and is a characteristic of the particular nucleus (since each nuclide has its own characteristic Larmor frequency, it follows that in any particular experiment the technique acquires information only about the nuclide for which the system is specifically tuned). The magnetic moments of all the nuclei in a particular sample can be visualised as forming a cone about the z-axis. yielding a net magnetisation M parallel to the Bo axis. When viewed in a frame rotating about the z-axis at the Larmor frequency (the so-called rotating frame), the spin system appears to be stationary. When thermal equilibrium is attained between the nuclear spin moments and their surrounding lattice, an equilibrium magnetisation is obtained, which is, however, only a very small fraction of the maximum possible value if complete alignment of the nuclear moments with Bo was achieved.
Perturbation. The sample is now irradiated with a pulse of plane-polarised rf radiation at the Larmor frequency. Plane polarised electromagnetic radiation consists of electric and magnetic fields oscillating in fixed planes perpendicular to each other and to the direction of the radiation. The nuclear spin packets which are already in thermal
Introduction
9
equilibrium with the static external magnetic field interact with the magnetic field component of the radiation, causing the equilibrium magnetisation to incline with respect to the applied magnetic field (Figure 1.4B). In the simplest experiment, the angle of inclination is 90 ~ and provided the pulse is of short duration compared with the time taken for the spin packet to spiral into the 'new' field direction, the precession can be assumed to remain fixed at 90 ~ Such a pulse, which is long enough to rotate the spin packet through 90 ~ is described as a 7r/2 pulse (a 7r pulse is one which rotates the spin packet through 180~
Detection. When the rf field is turned off at the end of the simple 1-pulse experiment,
the spin system dephases in the x-y plane with a characteristic relaxation time T2, and returns to thermal equilibrium along the z-axis with a characteristic relaxation time T1. The transverse magnetisation after the pulse induces a voltage in the coil which is recorded as a function of time and is called a Free Induction Decay (FID) (Figure 1.5) containing frequency information. This can be extracted mathematically by a process called Fourier Transformation (FT) giving a plot of amplitude vs. frequency (Figure 1.5) showing which frequencies would have to be added together with what relative amplitudes to reproduce the time-dependent shape of the FID. Thus, Fourier transformation of a "rr/2 single-pulse NMR experiment gives exactly the same information as would be obtained from a continuous wave spectrum, allowing the entire frequency range to be covered in the time taken to acquire a single FID. Usually the sample is irradiated with many pulses and the resulting signals added together to give an improved signal/noise ratio. A clear and simple description of the processes involved in pulsed NMR has been given by Schwartz (1988). Many of the advances in the application of NMR to solids have involved the development of more complex pulse sequences for manipulating the spin systems in order to improve the spectral resolution or extract other information (bond lengths, atom connectivities, etc.)
~
,~
FT
>
t
v
FID
NMR spectrum
Figure 1.5. Schematic representation of Fourier transformation of a FID in the time domain to an NMR spectrum in the frequency domain.
10
Multinuclear Solid-State NMR of lnorganic Materials
1.3.2 Overcoming NMR spectral broadening in solids by MAS The NMR spectra of solids suffer from broadening due to various interactions between the dipole moments of the nuclei, between the quadrupole moments of quadrupolar nuclei and the electric field gradient (EFG) at the nucleus, and by anisotropy of the electronic shielding at different sites in the structure. These interactions are dealt with in detail in Chapter 2. These broadening effects do not arise in liquids because their atomic motion is faster than the interaction frequency, allowing all the nuclei in a particular atomic environment to experience the same average magnetic field, thus producing extremely narrow NMR spectra. A number of the interactions giving rise to line broadening in solids can be cancelled out or at least reduced by spinning the sample very rapidly (typically 10-15 kHz or 10,000-15000 revolutions per second) at an angle of 54.74 ~ to the axis of the applied magnetic field (the so-called magic angle, Figure 1.6A). Broadening in spin I = 1/2 nuclei arises mainly from interactions between the magnetic dipoles of adjacent nuclei which are anisotropic in powder samples. In quadmpolar nuclei where the spin I > 1/2 (a category containing more than two-thirds of all the NMR-active nuclei) the non-zero nuclear electric quadrupole moment results from the fact that the charge distribution is non-spherically symmetric. Interaction of this quadrupole moment with the electric field gradient at the nucleus causes the broadening associated with these nuclei. As discussed in detail in Chapter 2, some of these interactions contain terms in (3cos 2 0-1) the second-order Legendre polynomial (P2(cos0)), where 0 is the angle between the axis of the applied magnetic field and the principal axis of the interaction. At the magic angle, cos0 = 1/~/3 and the term P2(cos0), becomes zero. Magic angle spinning (MAS) thus removes the dipoledipole and chemical shift anisotropy (CSA) interactions, as well as the first-order quadrupolar interactions, narrowing the resonance lines from both spin I - 1/2 and quadrupolar nuclei to reveal spectral details (Figure 1.6B). Lines equispaced on each side of the central resonance (spinning side bands) are caused by modulation of the interaction by the physical act of spinning the sample, and, since they are harmonics of the spinning speed, they can be distinguished from tree peaks because their position changes with the spinning speed, moving away from the central peak at higher speeds. It should be noted that although the well-resolved NMR spectra of solids obtained by MAS may resemble solution spectra, they are normally broader due to distributions of the isotropic chemical shifts in solids. There is also another important difference between liquid and solid MAS NMR spectra. Unlike the random molecular motion in liquids, magic angle spinning is a coherent averaging process, introducing the possibility of synchronising the pulse sequences with the rotor speed to selectively reintroduce anisotropic interactions and allowing the measurement of internuclear distances. MAS removes the broadening of the spectra of quadrupolar nuclei resulting from first-order quadrupolar interactions, but not broadening due to higher-order interac-
11
Introduction
A
~;o
|
~;o
!
i
o
!
!
-lOO
!
-~'oo
~7AIshift (ppm) w.r.t AI(H20)~3+
Figure 1.6. A. A typical magic angle spinning probe illustrating the orientation at the magic angle of the sample within the stator/coil assembly with respect to the axis of the applied magnetic field B0. B. Typical solid state 27A1 NMR spectra of a sintered mixture of A1203 and Y203. Upper spectrum unspun, lower spectrum spun at the magic angle. The peaks marked with asterisks are spinning side bands.
tions which have angular dependences other than the (3 C O S 2 0 - 1) term eliminated by MAS (see Chapter 2). In complex spectra of samples containing several overlapping sites, the removal of higher-order broadening may be important, and can be accomplished by spinning at more than one angle either simultaneously as in double rotation (DOR) or sequentially as in double angle spinning (DAS) (see Chapter 3). A more recent and extremely promising method which does not require complex and fragile probe hardware involves the excitation of higher order quantum coherences, exploiting differences in their associated evolution under different interactions. This method, called multiple quantum or MQMAS NMR, is described in detail in Chapter 2.
1.3.3 Other NMR experiments used with solids Although MAS NMR with a simple pulse sequence is the technique most generally used to study solid materials, there are a number of other experiments which may be used with or without MAS to provide specific information or improve the quality of the spectrum. Since these are described in Chapters 2 and 3 and will be encountered in many of the practical examples given in the subsequent chapters, they are only treated briefly here.
12
Multinuclear Solid-State NMR of Inorganic Materials
1.3.3.1 Decoupling. In some cases the broadening arising from interactions between the dipole moments of two different nuclei (heteronuclear broadening) is too large to be removed completely by MAS. Typically the second nucleus may be 1H, in which case the protons in the system are continuously irradiated at their Larmor frequency, which continuously modulates their spin state, cancelling the effect of their dipole moment. The spectrum of the other nucleus is collected during this decoupling period and is thus freed of heteronuclear broadening.
1.3.3.2 Cross-Polarisation (CP). In systems containing two nuclei, one with more abundant spins than the other, magnetisation can be transferred from the more abundant (often 1H) to the less abundant nucleus by irradiating both nuclei at their correct Larmor frequencies in fulfillment of the Hartmann-Hahn condition (Hartmann and Hahn 1962) (see Chapter 2). The spectrum of the less abundant nucleus can then be collected at a higher magnetisation level determined by the protons, and, since the relaxation time T~ is now that of the protons, a greater number of scans can be collected in a given time, giving a better signal/noise ratio. Further, since the signal from the less abundant nuclei located in structural sites in closest proximity to the protons is preferentially enhanced, additional structural information can be gained by comparing CP and non-CP spectra of the same sample.
1.3.3.3 Spin-echo experiments. Spectra containing broad resonances can suffer from artifacts and distortion introduced during the first few microseconds of instrumental dead time at the beginning of the FID. To overcome this, the spin system can be given an initial pulse and is then subject to another pulse (or pulse sequence) which causes the dephasing spin system to rephase (the refocussing pulse) and is sometimes termed a spin-echo (Hahn 1950). The FID is then collected and Fourier transformed in the usual way. Spin-echo experiments are also often used for determining relaxation times. 1.3.3.4 Two-dimensional experiments. A wide range of experiments are now available in which a series of FIDs (and hence spectra) are collected, each with a different value of some time-dependent variable (e.g. the evolution period in a multiple-pulse experiment) that modulates the FID. This has been termed two-dimensional spectroscopy and was first suggested by Jeneer (1971) and implemented practically a few years later (Aue et al. 1976). The spectra are first Fourier-transformed in the usual way, then again with respect to the second time variable, and may be plotted as a contour with the chemical shift along one axis and the effects related to the evolution period along the other. 2D experiments can be used to provide a range of structural information, including the identification of sites which are directly connected, and are discussed in detail in Chapter 2.
Introduction
13
1.3.4 Nuclei suitable for NMR spectroscopy To produce an N M R spectrum, a nucleus must possess a nuclear spin. Nuclei with odd mass numbers (e.g. 29Si, 27A1) have half-integer spins and are of most interest for solid state NMR. Nuclei with even mass numbers and odd charge (e.g. 2H, 14N) have integer spins, and although subject to difficulties they can still be useful N M R nuclei. Of the 120 nuclei suitable for NMR, 9 have spin I - 1, 31 are spin I - 1/2, 32 are spin I = 3/2, 22 are spin I = 5/2, 18 are spin I = 7/2 and 8 are spin I = 9/2. One factor influencing the usefulness of an N M R nucleus is its natural abundance, which can range from 100 percent down to the vanishingly small. In the latter case, it may only be possible to acquire an N M R spectrum if the sample has been artificially isotopically enriched, as is normally done for 15N and 170 NMR. The natural abundances of the most useful spin I = 1/2 nuclei are listed in Table 1.1 and those of the quadrupolar nuclei in Table 1.2. The standard substances against which the chemical shifts of the various nuclides are quoted are listed for the spin I -- 1/2 nuclei in Table 1.1 and for the quadrupolar nuclei in Table 1.3. Tables 1.1 and 1.2 also provide information about the N M R sensitivities of the various nuclei, listed as their receptivity relative to Si, with the absolute sensitivity defined by Harris (1984) as @COO + 1)) where C is the natural abundance of the nucleus and ~/is its gyromagnetic ratio. The relative receptivities of the quadrupolar nuclei in Table 1.2 have been adjusted by taking into account the fractional contribution of the central transition.
Table 1.1. NMR properties of the spin I = 1/2 nuclei. Nucleus
Natural abundance (%)
Vo at 7.05 T (MHz)
1H 3He 13C 15N
99.985 1.3 • 10 - 4 1.108 0.37 100 4.70
300.00 228.546 75.435 30.408 282.282 59.601
2.71 • 1.56 • 4.77 • 1.04 ;< 2.26 • 1
100 2.19
121.440 9.714
1.80 • 102 2.01 • 10 -3
7.58 100 100 51.82 48.18 12.75 12.26 0.35
57.213 14.751 9.516 12.141 13.569 63.645 66.579 98.580
1.47 3.22 • 10 -1 8.56 • 10 -2 9.43 • 10 -2 1.33 X 10 -1 3.33 3.66 3.36 • 10 -1
19F 29Si 31p 57Fe
77Se 89y
l~ 1~ l~ 111Cd 113Cd 115Sn
Relative receptivity* 103 10 -3 10 -1 10 -2
103
Standard substance TMS TMS MeNO2 CC13F TMS H3PO4 Fe(CO)5
Me2Se Y(NO3)33aq mer-[RhC13(Sme2)3] Ag+aq Ag+aq CdMe2 CdMe2 Me4Sn
Multinuclear Solid-State NMR of lnorganic Materials
14
Table 1.1. (Continued) Nucleus
Natural abundance (%)
Vo at 7.05 T (MHz)
Relative receptivity*
Standard substance
ll7Sn l l9Sn
7.61 8.58 6.99 26.44 100 14.31 14.40 1.64 33.8 16.84 29.50 70.50 22.6
106.896 111.870 94.647 83.430 24.810 52.830 12.483 6.846 64.242 53.733 173.100 172.899 62.760
9.46 1.22 • 10 6.07 1.54 X 10 1.53 2.12 2.87 X 10 .2 5.42 X 104 9.19 2.66 1.54 X 102 3.79 X 102 5.45
Me4Sn Me4Sn Me2Te XeOF4
125Te ~29Xe 169Tm 171yb
183W 1870S 195pt 199Hg 2~ 2~ 2~
WF6 OsO4 [Pt(CN)6] 2MezHg T1NO3aq T1NO3aq Me4Pb
* relative receptivity normalised to 298i
Table 1.2. NMR properties of the quadrupolar and integer-spin nuclei. Nucleus
I
Natural abundance
Vo (MHz) at 7.05 T
(%) 2H 6Li 7Li 9Be l~ liB 14N
170 21Ne 23Na 2SMg 27A1 33S 35C1 37C1
39K 41K 43Ca
45Sc 47Ti 49Ti 51V
1 1 3/2 3/2 3 3/2 1 5/2 3/2 3/2 5/2 5/2 3/2 3/2 3/2 3/2 3/2 7/2 7/2 5/2 7/2 7/2
1.5 x 10 -2 7.42 92.5 100 19.58 80.1 99.63 0.037 0.27 100 10.0 100 0.75 75.77 24.23 93.26 6.73 0.135 100 7.28 5.51 99.75
46.051 44.146 16.67 42.20 32.239 96.32 21.671 40.71 23.71 79.44 18.39 78.27 23.06 29.44 24.51 14.02 7.70 20.23 73.03 16.95 16.95 79.05
Quadrupole Relative moment receptivity* (mb)
Quadrupole Stemheimer broadening antishielding factor** factor (~)
7.35 x 102 3.76 x 10
-0.808 -40.1 52.9
2.28 X 10-1 1.10
0.2 0.2
3.59 x 10 2 5.22 2.93 x 10 -2 1.80 x 10 -2 2.51 x 10 2 7.29 x 10 -1 5.61 x 102 4.62 x 10 -2 9.71 1.79 1.29 1.46 x 10 -2 2.35 x 10 -2 8.19 x 102 4.16 x 10 -I 5.61 x 10-1 1.04 x 103
40.6 20.44 -25.6 101.6 104 199.4 146.6 -67.8 -81.7 -64.6 58.5 71.0 -49.0 -220 302 247 -52
2.83 x 10 -1 NA 6.40 x 10 -2 7.21 2.64 8.60 1 3.30 3.75 2.80 4.20 11.3 1.39 X 10 -1 1.12 21.40 6.08 5.78 X 10 -2
0.19 -6.7 -13.8 -9.5 -5.5 -4.1 -3.6 -52.2 -42.0 -42.0 -21.8 -21.8 -18.8 -23.1 -9.0 -9.0 -7.6
15
Introduction Table 1.2. (Continued) Nucleus
I
Natural abundance
Vo (MHz) at 7.05 T
Relative receptivity*
(%) 53Cr 55Mn
59Co 61Ni
63Cu 65Cu 67Zn 69Ga 71Ga 73Ge 75As 79Br 81Br 83Kr 85Rb 87Rb 878r 91Zr 93Nb 95Mo
97Mo 99Ru lOlRu lO5pd 113in 115in
121Sb 123Sb 127I 131Xe
133Cs 135Ba 137Ba 139La 141pr 143Nd 145Nd 1478m 1498m 151Eu 153Eu 155Gd
157Gd
3/2 5/2 7/2
3/2 3/2 3/2 5/2 3/2 3/2 9/2
3/2 3/2 3/2 9/2
5/2 3/2 9/2 5/2 9/2 5/2 5/2 5/2 5/2 5/2 9/2 9/2
5/2 7/2 5/2 3/2 7/2
3/2 3/2 7/2 5/2 7/2 7/2 7/2 7/2 5/2 5/2 3/2
3/2
9.50 100 100 1.14 60.11 39.89 4.11 60.1 39.9 7.73 100 50.69 49.31 11.5 72.16 27.84 7.00 11.22 100 15.92 9.55 12.7 17.0 22.33 4.3 95.7 57.36 42.64 100 21.2 100 6.59 11.23 99.91 100 12.18 8.30 15.0 13.8 47.8 52.2 14.8 15.65
17.00 74.56 71.04 26.87 79.65 85.32 18.82 72.24 91.79 10.50 51.57 75.46 81.34 11.59 29.08 98.56 13.06 28.02 73069 19.65 20.07 13.78 15.45 13.80 66.02 66.17 72.30 39.15 60.47 24.79 39.64 30.02 33.58 42.73 91.90 16.35 10.07 12.51 10.31 74.63 32.95 9.28 12.18
2.34 x 10 -1 4.85 x 102 7.57 x 102 1.11 x 10 -1 1.75 X 102 9.59 x 10 3.19 x 10 -1 1.14 X 102 1.55 x 102 2.96 X 10 -1 6.90 X 10 1.09 X 102 1.33 x 102 5.95 x 10-1 2.88 x 10 1.34 x 102 5.16 x 10 -1 2.88 1.32 x 103 1.41 9.03 x 10 -1 3.89 x 10-1 7.35 x 10 -1 6.84 x 10-1 4.09 x 10 9.20 x 102 2.54 x 102 5.39 x 10 2.59 x 102 1.62 1.31 X 102 8.92 X 10 -1 2.13 1.64 x 102 9.09 X 102 1.12 1.78 x 10 -1 6.17 x 10 -1 3.18 X 10 -1 2.32 X 102 2.19 x 10 6.00 • 10 -2 1.40 x 10 -1
Quadrupole moment (mb) -150 330 420 162 -220 -204 150 171 107 - 196 314 313 261.5 259 276 133.5 335 -176 - 320 -22.0 255 79.0 457 660 799 810 -360 -490 -616 -114 -3.4 160 245 200 -59.0 -630 -330 -260 75.0 903 2410 1270 1350
Quadrupole Sternheimer broadening antishielding factor**
factor (Y)
21.9 5.81 4.20 16.18 10.06 8.08 4.76 6.70 2.06 3.36 31.67 21.50 13.92 5.32 10.42 3.00 7.91 6.02 1.28 0.98 12.88 1.80 53.75 125.5 8.9 9.13 8.98 10.36 24.95 8.68 4.93 x 10 -4 14.12 29.60 1.58 1.51 X 10 -1 41.02 18.28 9.13 0.92 43.45 700.9 2878 2478
-6.6 -5.8 -4.7 -25.2 -25.2 -21.9 - 17.0 -17.0 -8.7 -80 -80 - 85.5 -52.8 -52.8 -47.8 -26.6 - 23.0 -20 -20 -28 -28 - 162 -110 -110 -110 - 71 -
Multinuclear Solid-State NMR of lnorganic Materials
16
T a b l e 1.2. (Continued) Nucleus
I
Natural abundance
Vo (MHz) at 7.05 T
Relative receptivity*
Quadrupole moment (rob)
72.12 10.33 14.46 64.07 8.66 14.61 34.28 12.18 7.65 36.40 68.50 69.21 23.64 5.4 5.87 5.22 20.07 49.09 5.85
1.88 X 102 2.46 X 10 -1 8.75 X 10 -1 5.54 X 102 3.11 X 10 -1 5.89 X 10 - l 8.25 X 10 7.07 X 10 -1 2.03 X 10- l 1.02 X 102 1.41 X 102 2.43 X 10 2 1.07 3.04 X 10 -2 6.79 X 10 -2 8.13 X 10 -2 5.32 X 10 - l 3.92 X 102 2.93 X 10 -3
1432 2507 2650 3580 3570 2800 4970 3360 3790 3170 2180 2070 856 816 751 547 386 -516 4936
(%) 159Tb 161Dy 163Dy 165Ho 167Er 173yb 175Lu 177Hf 179Hf 18~Ta 185Re ~87Re ~89Os 191Ir 193Ir 197Au 2~ 2~ 235U
3/2 5/2 5/2 7/2 7/2 5/2 7/2 7/2 9/2 7/2 5/2 5/2 3/2 3/2 3/2 3/2 3/2 9/2 7/2
100 18.9 24.9 100 22.95 16.12 97.41 18.606 13.629 99.988 37.40 62.60 16.1 37.3 62.7 100 13.18 100 0.72
Quadrupole Sternheimer broadening antishielding factor** factor (~/) 470.9 2419 1931 338 2487 2134 1218 1566 1728 466.5 275.9 246.1 513.3 2042 1591 949 122.9 4.99 7038
--73.8 -
* relative receptivity normalised to 298i ** quadrupole broadening normalised to 27A1 Data from Smith and van Eck (1999) and references therein, and Pyykk6 (1992, 2001)
T a b l e 1.3. Standard reference substances for the quadrupolar and integer-spin nuclei, with compounds commonly used as secondary references and their shift relative to the primary reference. Nucleus
Reference substance
Secondary reference compound
2H 6'7Li 9Be l~ 14N 170 23Na 25Mg 27A1
TMS LiCl(aq) BeO(s) BF3.Et20 CH3NO2 H20 NaCl(aq) MgSO4(aq) Al(H20)63+
NaBH4 NH4C1 NaCl(s) MgO(s) Y3A15OI2 (octahedral peak)
33S
CS 2
NH4AI(SO4)2.12H20
35'37C1
NaCl(aq)
CaS KCl(aq)
Shift from the primary compound (ppm)
3.2 342.4 7.2 26 0.7 333 -28.5 3.07
17
Introduction
Table 1.3. (Continued) Nucleus
Reference substance
39'41K 43Ca
KCl(aq) CaC12(aq) 47'49Ti TIC14 51V VOC13 63'65Cu CuC1 67Zn Zn(NO3)2(aq) 69,71Ga Ga(H20)63+ 85'87Rb RbCl(aq) 91Zr Cp2ZrBr2 93Nb NbC15in wet acetonitrile 95'97Mo Na2MoO4(aq) 131Xe Xe(g) at zero pressure 133Cs CsCl(aq) 135Ba BaClz(aq) , 139La LaC13(aq) 2~ (CH3)zHg
Secondary reference compound
Shift from the primary compound (ppm)
CaO(s) SrTiO3 0.16M NaVO3(aq) ZnSe ' _ BaZrO3 Nb2Os(s) Mo(CO)6 BaZrO3 -
128 - 843 -574.4 276 _ 208.1 - 1240 - 1854 279 -
A further factor to be taken into account when assessing the relative ease with which a quadrupolar nucleus may be observed is the width of the line which is determined for a given electric field gradient by the quadrupole moment (also listed in Table 1.2). Often only the central transition is observed, for which the linewidth is determined by its second-order quadrupolar broadening and is proportional to [Q2(I(I + 1)-3/4)2]/ [ ( 2 I - 1)2]. These quadrupolar broadening factors listed in Table 1.2 have been normalised to 27A1. The actual linewidth will be given by this factor multiplied by the electric field gradient and although this property is not easily defined for a particular nucleus, it is amplified by the Sternheimer antishielding factor (Table 1.2) which must therefore also be taken into account when comparing nuclei. Hence to get a feeling for the observability of the central transition of a given quadrupolar nucleus, the secondorder broadening factor should be multiplied by the square of the Sternheimer antishielding factor. Values of the latter are given in Table 1.2 only for cases where meaningful calculations have been carried out for ions with a closed shell structure, e.g. A13+, C1-. Where possible, these values are quoted for a lattice (Schmidt et al. 1980), otherwise they pertain to the free ions.
1.4. F U R T H E R R E A D I N G
This book is intended to be a self-contained introduction to the background of solidstate NMR, its experimental techniques and applications to inorganic materials.
18
Multinuclear Solid-State NMR of lnorganic Materials
Nevertheless there are a number of excellent books dealing with different aspects of NMR spectroscopy and solid state NMR with a different emphasis and perspective. Some of these are listed below, and the interested reader is encouraged to consult them. The background to the physics of NMR can be found in the excellent books by Abragam (1983), Slichter (1990) and Munowitz (1988), the latter providing a very readable quantum mechanical description of solid state NMR experiments. Harris (1983) provides a nice introduction to the chemical aspects of NMR techniques, and the physical background of multidimensional NMR techniques is developed in detail by Ernst et al. (1988). High-resolution solid state NMR is dealt with by Haeberlen (1976) and Mehring (1983). A book devoted to pulsed solid state NMR has been written by Gerstein and Dybowski (1985). Although 13C NMR of organic materials is not dealt with in the present book, there is ample literature on this subject, including the books by Stejskal and Memory (1994) and Fyfe (1983), the latter providing numerous applications of 13C NMR. Schmidt-Rohr and Spiess (1996) present a comprehensive background to modem NMR techniques for characterising solid polymeric materials. Applications to silicates and zeolites were dealt with comprehensively by Engelhardt and Michel (1987). On practical matters, the authors find it useful to have copies of the books by Fukushima and Roeder (1981) and Derome (1987) in the laboratory. Last but not least is the Encyclopaedia of NMR, published in 1996 by Wiley.
REFERENCES
Abragam, A. (1983) Principles of Nuclear Magnetism, Oxford University Press, Oxford. Andrew, E.R., Bradbury, A. & Eades, R.G. (1959) Nature 183, 1802. Aue, W.P., Bartholdi, E. & Ernst, R.R. (1976) J. Chem. Phys. 64, 2229. Derome, A.E. (1987) Modern NMR Techniques for Chemistry Research, Pergamon, Oxford. Engelhardt, G. & Michel, D. (1987) High-Resolution Solid State NMR of Silicates and Zeolites, Wiley, Chichester. Ernst, R.R., Bodenhausen, G. & Wokaun, A. (1988) Principles of Nuclear Magnetic Resonance in One and Two Dimensions, Oxford University Press, Oxford. Fukushima, E. & Roeder, S.W.B. ( 1981) Experimental NMR Spectroscopy, Addison-Wesley, Reading, Mass. Fyfe, C.A. Solid State NMR for Chemists, CRC Press, Guelph. Gerstein, B.C. & Dybowski, C.R. (1985) Transient Techniques in NMR of Solids, Academic Press, Orlando, Florida. Haeberlen, U. Advances in Magnetic Resonance, Supplement 1, Academic Press, San Diego. Hahn, E.L. (1950) Phys. Rev. 80, 580. Harris, R.K. (1984) NMR Spectroscopy, (Pitman, London). Hartmann S.R. & Hahn, E.L. (1962) Phys. Rev. 128, 2042. Jenner, J. (1971) Amp6re International Summer School, Basko Polje, Yugoslavia.
Introduction
19
Lowe, I.J. (1959) Phys. Rev. Lett. 2, 285. Mehring, M. (1983) High Resolution NMR of Solids, Springer-Verlag, Berlin. Munowitz, M. (1988) Coherence and NMR, Wiley, New York. Pines, A., Gibby, M.G. & Waugh, J.S. (1973) J. Chem. Phys. 59, 569. Pyykk6, P. (1992) Z. Naturforschung 47a, 189. Pyykk6, P. (2001) Mol. Phys. 99, 1617. Schaefer, J. & Stejskal, E.O.J. (1976) J. Amer. Chem. Soc. 98, 1031. Schmidt, P.C., Sen, K.D., Das, T.P. & Weiss, A. (1980) Phys. Rev. B 22, 4167. Schmidt-Rohr, K. & Speiss, H.W. (1996) Multidimensional Solid-State NMR and Polymers, Academic Press, San Diego. Schwartz, L.J. (1988)J. Chem. Educ. 65, 752. Slichter, C.P. (1990) Principles of Magnetic Resonance, Springer-Verlag, Berlin. Smith, M.E. & van Eck, E.R.H. (1999) Prog. Nucl. Magn. Resort. Spectrosc., 34, 159. Stejskal, E.O. & Memory, J.D. (1994) High Resolution Solid State, NMR Oxford University Press, New York.
This Page Intentionally Left Blank
Chapter 2
Physical Background 2.1.
Fundamental Interaction with External Magnetic Fields 2.1.1 A Quantum Mechanical Description of the Zeeman Interaction 2.1.2 Bulk Magnetisation 2.1.3 The Rotating Frame and the Application of RF Pulses 2.1.4 Observation of the NMR Signal 2.2. Internal Interactions 2.2.1 The Dipolar Interaction 2.2.2 Scalar Coupling 2.2.3 Paramagnetic Coupling 2.2.4 Chemical Shielding 2.2.5 Knight Shift 2.2.6 Quadrupole Interaction 2.2.7 Nature of Interactions 2.3. One Dimensional Methods for Improving Resolution 2.3.1 Magic Angle Spinning and First-Order Effects 2.3.1.1 Physical Principles 2.3.1.2 Formation of Spinning Sidebands 2.3.2 Magic Angle Spinning and Higher-Order Effects 2.3.2.1 MAS of Second-Order Quadrupole Effects 2.3.2.2 Residual Coupling Effects due to Quadrupolar Nuclei in MAS Spectra 2.3.2.3 Nonequivalent Homonuclear Spins 2.3.3 Variable Angle Spinning 2.3.4 Double Angle Spinning 2.3.5 Multiple Quantum Transitions 2.3.6 Ultrasonically-Induced Narrowing 2.4. Dipolar Decoupling 2.4.1 Heteronuclear Dipolar Decoupling 2.4.2 Homonuclear Dipolar Decoupling 2,5, Spin-locking 2.6. Cross-Polarisation 2.7. Two-Dimensional Methods 2.7.1 Dynamic Angle Spinning 2.7.2 2D MQMAS 2.8. NMR Relaxation
23 25 26 29 34 35 37 40 43 44 48 50 57 58 59 59 61 63 64 71 74 74 75 77 78 78 78 79 83 85 90 92 93 98
2.8.1 2.8.2 References
Introduction to Relaxation Mechanism for Relaxation Processes
98 101 105
Chapter 2
Physical Background 2.1. FUNDAMENTAL INTERACTION WITH EXTERNAL MAGNETIC FIELDS
The nuclei of interest to the NMR experiment possess a nuclear spin angular momentum ] and consequently a nuclear magnetic moment ~. A classical description of this magnetic moment placed in a static magnetic field B shows a torque z being exerted of
z-ExB
(2.1)
The magnetic dipole moment ~ is related to the spin angular momentum via
= ?'hi and I~[- 17[h4I(I + 1)
(2.2)
where ~/is the nuclear gyromagnetic ratio (units of rads-lT-1). The torque in Eq. 2.1 causes the magnetic moment to change direction such that
dt
=7~xB
(2.3)
If the moment was inclined at 0 to B this would cause a precession of the moment in a conical path at constant 0 about B at an angular frequency O~ogiven by O)~ -?'[B I
(2.4)
COois known as the Larmor or resonance frequency. Strictly this is the angular resonance frequency but the literature uses this term interchangeably with the direct frequency which is of more immediate experimental relevance, Vo = eOo/2~r. This means NMR is a radiofrequency spectroscopy so for protons at B = 1 T, Vo - 42 MHz. ~/can be either positive or negative, although the consequences of this are usually ignored. Levitt (1997) has explored this issue and the consequences it has for phase shifting and detecting the signal. Classically, a magnetic dipole moment in a magnetic field has an energy of
E--~.B~ o --Th[.B
o
(2.5)
which not only depends on ~/, h and the magnitude of B o but also on the relative orientation of ~ and B o. In NMR spectroscopy the energy difference between different spin 23
24
Multinuclear Solid-State NMR of Inorganic Materials
levels (i.e. effectively different orientations of I) is of interest. This is the well known Zeeman interaction and is the basis of magnetic resonance spectroscopy. Although the magnetic moment has been treated classically, and indeed many NMR effects can be conveniently described classically, the spin is quantised. This means that its values and components can only take discrete values. The energy levels can be calculated using the spin Hamiltonian which is an operator giving the energy of the system. In principle all energy levels in a system (nucleus, system of coupled nuclei, molecules) are given by the time-dependent Schr6dinger equation: ih ~)l/t _ HI// (2.6) ~)t where H = Hamiltonian and qJ is the wave function. In NMR the way the system is interrogated (e.g. with an rf pulse) is assumed not to alter the energy levels of the system so that it can be assumed that the Hamiltonian is time-independent. Then the wave function that solves Eq. 2.6 will be of the form
= qg. exp(-iEt / h)
(2.7)
where ~ is time-independent. If these two equations are combined then q~ will satisfy
Hq~= Eq~
(2.8)
where q~ is the eigenfunction of the Hamiltonian H with eigenvalue (energy) E. The Hamiltonian gives all possible energy levels of a system, including rotations, vibrations, bonding etc. However the NMR energy splittings are --~ 1.24 X 10 -6 eV (= 300 MHz = 2 x 10 -25 J = 0.01 cm-~ in various spectroscopic units) which is sufficiently different from other interactions that it is possible to split the Hamiltonian, and hence the resulting wavefunctions, into those associated with the nuclear spin and those with the other interactions (e.g. vibrations). Wave functions that represent the interactions of the nuclear spin alone will give an equivalent equation to 2.8 but relating only to the nuclear spin. This removes all the interactions that do not influence NMR experiments. The Hamiltonian H will henceforth in this book be taken to be the nuclear spin Hamiltonian and the eigenfunction q~refers to this Hamiltonian. The eigenfunctions of the Hamiltonian are defined in terms of a complete set of orthogonal spin functions q~n. A general eigenfunction q~ can now always be written as a linear combination of the spin functions q~n where the coefficients Cn vary for different Hamiltonians (interactions). q9 - Z C,~0n B
(2.9)
25
Physical B a c k g r o u n d
Table 2.1. Nuclear spin operators. Operator
Description
Operator II,m > =
I
Nuclear Spin
~/I(I + l)h l I, m >
12 ---Fx -/-Fy -/-Fz
Square of the total nuclear spin
I ( I + 1)h 2 I I, m >
Iz
z-component of nuclear spin
mhlI, m>
I+ --Ix + ily
Raising Operator
~/I(I + 1 ) - m ( m + 1)h l I, m + 1 >
1_ : Ix- ily
Lowering Operator
~/I(I + 1)- m(m-
1)h l I, m -
1>
In NMR the q~n are characterised by the quantum numbers I and m where I is the spin quantum number (0, 1/2, 1, 3/2,...) and m is the z-component of I with values of m = - I, - I + 1. . . . . I - 1, I. The spins of nuclei are given in Tables 1.1 and 1.2. The state of a nucleus would be described by the pair of quantum numbers I and m and is usually written I I, m > in Dirac bra and ket notation. This text is not intended as a primer in quantum mechanical operators but to understand papers that interpret interactions and develop new pulse sequences it is worth noting some of the most important results. It is immediately apparent from Table 2.1 that II, m > is an eigenfunction of both I 2 and Iz so that II, m > is unchanged. In contrast the raising and lowering operators, I+ and I_, change the spin function so that I I , m > is not an eigenfunction of I+ or I_. The raising and lowering operators describe changes in the z-component of the spin state corresponding to a transition of a spin between energy levels.
2.1.1 A quantum mechanical description of the Zeeman interaction The classical Zeeman interaction between a magnetic moment and an applied magnetic field was defined above. In a quantum mechanical description the operators for the quantities need to be used, such as the nuclear spin I. With an externally applied magnetic field B o the Zeeman Hamiltonian is given by (2.10)
H z =-yh[.B_ o
The usual convention is to let the direction of the externally applied static magnetic field B o define the z-direction in the laboratory frame. Then the Zeeman Hamiltonian can be written as
H z - -~,'h Iy
9
- _~,hBoI z
Bo
(2.11)
26
Multinuclear Solid-State N M R of Inorganic Materials Zero field Bo = 0
Bo finite E = YhBO 2
(,,
./,.,-' """ x\
AE= y Bo
E - -TfiBO
~
I~-~>t I~.~) J
Figure 2.1. Energy level diagram of a spin-l/2 nucleus showing the Zeeman interaction. The Zeeman Hamiltonian can be used to calculate the energy difference between the nuclear spin states. For a nucleus the energy eigenvalues can be obtained by taking the Hamiltonian operator given in Eq. 2.11 so that H_ I I, m >= E m I I, m >= -COol_ I I, m >= - m h ~ o I I, m >
For I
=
1/2, m can be
_+ 1/2
E% - - ~ h
(2.12)
which gives the eigenvalues Em of
COo(m - ~ ) and E_,/: = ~ h COo(m - - ~ )
(2.13)
The separation (Figure 2.1) of m - _+ 1/2 levels is AE = E - l / - El/ - h O~o which is 2 2 . the energy of the m - _+1/2 transition and matches the classical expression given in Eq. 2.4.
2.1.2 Bulk magnetisation A sample on which an NMR experiment is carried out will usually contain a large number of nuclei. They will all precess around the direction of B o but the phase of precession of each individual spin will vary (the so-called random phase approximation). Consequently, in a unit volume the total nuclear magnetisation M is obtained by summing the individual contributions. The components of magnetic moment perpendicular (transverse) to B o cancel leaving M pointing parallel to the applied magnetic field. M acquires a non-zero equilibrium value Mo in this longitudinal direction since the applied field B o tends to align the magnetic moments preferentially in the B o direction through a normal susceptibility mechanism. It is pertinent at this point to reflect on the different ways of describing magnetic fields. They can be characterised by the magnetic induction (or flux density) designated by B and measured in Tesla (T) or by the magnetic field intensity H measured in Am-1. The difference between these two vector fields is that H is a measure of the field from free current alone whereas B depends on the material. B is the more fundamental quantity and is what is meant when the term "magnetic field" is used in this book. The magnetisation (M) results from the
27
Physical Background
additional internal field provided by the sample itself and is also measured in Am-~. The three vector fields are related by B
H = ,~ - M ~
~0
(2.14)
~
Where ~o = 4w x 10 -7 Hm -~ is the permeability of a vacuum. The amount of magnetisation generated by a sample when placed in magnetic field is characterised by the susceptibility Xv. The literature on magnetic susceptibility is littered with confusing units and careful attention needs to be paid to the units being used. Magnetisation is effectively the amount of magnetic moment per unit volume so the most directly useful measure is the volume susceptibility thus
Zv -
~M
(2.15)
~H
In a linear isotropic material B = / z H where p~ is the permeability of the material. The relative permeability ~r = P~/~o or = 1 + • is a measure of how much magnetisation a material will generate. The molar and specific susceptibilities are also quoted in the literature. The equilibrium bulk magnetisation comes from the net moment on summation of contributions from individual spins. A sample containing spin-1/2 nuclei has two states (energy levels) with the nuclear spins distributed between these according to the Boltzmann distribution which gives the probability of occupation of the different states. N1/2
AE = e kT
~-. 9~,~
N-1/2
The total number of spins is given by N = N1/ + N_ and the difference in population is n = N1/z-N 1/2. Then N1/2 -- (N-n)/2 aZnd N_ 1/2' (N+n)/2. Substituting into 1/2 - Eq. 2.16 gives AE
N-n
~ - - e
AE
kT
1 - e kv n - N ~
(2.17)
N+n
l+e
k~
which in the high temperature limit (nearly always a good approximation in NMR) AE AE gives ~ << 1 meaning that the population difference is n - N . From Eq. 2.13 kT 2kT AE = ~/hBo. The z-component of the magnetic moment is given by P~z = ~/hm and
28
Multinuclear Solid-State NMR of Inorganic Materials
hence per spin the average z-component of the magnetic moment is (/xz) --/xz (m) P(m) where P(m) is the probability of m, given by the Boltzmann distribution. Then
(~: ) - ThlB~( ThlB~"
(2.18)
where B~ is the Brillioun function. Then
M~176
)
(2.19)
In the high temperature approximation
Bt
ThlBo ) _ ThlB o I(I + 1) kT kT 31
(2.20)
so that the equilibrium magnetisation is
Mo = N ,v 2Bo~ 2I,,I t +1) 3kT
(2.21)
and the volume susceptibility is
OM Z~, = 11o OB
~oN~'2h2I(l + 1) 3kT
(2.22)
which is termed Curie-like. It can be seen immediately from the form of the magnetisation that the strongest signal will be obtained from samples with many spins (large N), at high magnetic field and at low temperatures. The Boltzmann factor also gives a clue as to what lies at the core of one of the major weaknesses of NMR, namely sensitivity. The population difference, which is the source the magnetisation, depends on the energy separation AE compared to the thermal energy (kT). Hence the NMR signal is produced by a very small fraction of the total number of nuclei in the sample, so that NMR is unfortunately a relatively insensitive technique. For example at 9.4 T at room temperature for 1H only 3.2 • 10-3% of the nuclei contribute. This gets very much worse for "low-~/" nuclei with small magnetic moments such as 89y, where this fraction is reduced to only 1.6 • 10-4%. In spectroscopy involving electromagnetic radiation both spontaneous and stimulated events can occur. NMR is a relatively low frequency electromagnetic spectroscopy so that spontaneous events are very unlikely (-~ 10-22s - l ) and stimulated events therefore dominate. Hence NMR is termed a coherent spectroscopy meaning that the excess population can be made to work in a concerted way. This is achieved via the application of additional radiofrequency (rf)
29
Physical Background
pulses. To understand how the magnetisation is manipulated it is often convenient to observe the effects in the so called rotating frame rather than the laboratory frame and this is introduced next. This provides a classical description of what happens to the magnetisation.
2.1.3 The rotating f r a m e a n d the application o f RF pulses At any instant the total magnetisation M is given by
M-
(2.23)
M x i+My ~+M z k
The change of the magnetisation with time is then
- = dt
dt
i+ -
j+ dt -
/
+ M X - +My - +M z dt dt
dt
(2.24)
In a rotating frame of reference the magnitude of the unit vectors remains the same but not the vector directions di - = ~x/, dt
dJ - = cox j, dt - -
dk - = cox _k, dt -
(2.25)
so that dM
d M- I
dt
dt
lab
+ cox (Mxi + My j+ M~k) - -dM_ ~
rot
dM
~
=
dt rot
~
-
~
- e x M = y M x B + M x
+ (ox M
(2.26)
rot
(2.27)
lab
CO
= yM • (B+ ~ ) - y M • B~ff Y
(2.28)
CO
where Beff= B+ - is the effective magnetic field in the rotating frame. If the frequency Y and direction of the rotating frame are chosen to coincide with those of the magnetic moments
30
Multinuclear Solid-State NMR of Inorganic Materials - ~0 - - T B 0
(2.29)
so that
B~eff-
-7B o ~B~q 7 - B0 "[- ~ 2 , =0
(2.30)
- 7M • B~ - 0 (co - COo)
(2.31)
dM dt
rot
then M is stationary in a frame rotating at the Larmor frequency. Now consider what happens if a second, smaller magnetic field B1 is introduced which is perpendicular to the Bo field and is rotating at a frequency ~ ( - to k). This frequency need not be equal to the Larmor precession frequency tOo, The B1 field is given by B - B e -;~o, ~1
(2.32)
1
In a rotating frame with the same frequency as the 01 field, the 01 field will appear stationary and the effective field in that frame will be given by (Figure 2.2) o)
%---- 00+ -~- + B1
(2.33)
As B1 is usually applied perpendicular to Bo the strength of the effective field is given by
_
[
Laboratory. frame
2 +B 7
2
(2.34)
.1 Rotatin~ frame
z)
Z
B.0 +z
y
x/"
coo
-B1
y,
xi,
Figure 2.2. Representation of fields in the laboratory and rotating frames.
Physical Background
31
The change of the magnetisation with time is still
dm dt
- ?'M x By,
(2.35)
rot
so that M precesses about B eff in the rotating frame. If the frequency to of rotation of the B~-field (and hence of the rotating frame) is the same as the Larmor frequency too, then %
(2.36)
#er 0+7 +
Now the effective field lies in the xy plane (say along the x'-axis for convenience) in the rotating frame. The magnetisation will now precess around the x'-axis with a frequency of ~1 = -3/B 1
(2.37)
If M = M k at t = 0 then M is tipped towards the - y ' axis through an angle, termed the tip angle 0 = eOlt p - -'yBltp. In almost all NMR experiments the rf field is applied close enough to the Larmor frequency that it is effectively on resonance. Also all values of tip angle are used but there are a few special cases of the duration tp that are most commonly used. These are 0 = 90 ~ which tips M onto the - y ' axis, and 0 = 180 ~ when the magnetisation ends up along - z ' - a x i s (Figure 2.3A, B). The above has adopted a classical picture for describing the interaction between an rf field and the magnetisation. However, just as for the interaction with the main static applied magnetic field there is an analogous quantum mechanical description. The interaction between the rf magnetic field and the nuclear spins is simply another Zeeman interaction. The difference for this field is that it is time-dependent. In practice, the sample is irradiated with a linearly polarised if-field of strength 2B1, frequency tOrf and phase oL. A
iiiii I
B
Z'
,.'""--
M ~
i/
Z'
1""'" y?
Figure 2.3. The precession of the magnetisation in the rotating frames for 90~ and 180~pulses (assuming that ~ is positive).
32
Multinuclear Solid-State NMR of Inorganic Materials
(2.38)
B(t) = 2B 1cos(co,ft + a)
Then the Hamiltonian for an if-field applied along the x-axis is
/ H,u = 2hooll x cos(co,it + ~)
(2.39)
(2.40)
This Hamiltonian contains the spin operator Ix and in Table 2.1 above this can be rewritten in terms of the raising and lowering operators as
H,f - -?'fiB12 (t+e" -i~o~,, + I e i~
I~ - - ~1 [ i + + i _ ] so that (2.41)
These operators induce transitions between different spin states so that by applying an if-field to nuclear spins in the presence of a large static magnetic field close to the Larmor frequency, the spin distribution between the energy levels is perturbed away from thermodynamic equilibrium. In pulsed NMR experiments the spin system is excited with a short rf pulse near resonance and the system is measured afterwards. The total external Hamiltonian from the applied fields is then H = H z + H,f = hOgol.- + 2ho9~I x cos(og,ft + a)
(2.42)
The induced transitions change the populations of the different energy levels. After a 90 ~ pulse the populations are equalised, and after a 180 ~ pulse the populations are inverted so that for a spin-l/2 nucleus the + 1/2 state now contains the population excess. To keep track of the populations of the different levels and how pulses affect the magnetisation, the most complete and powerful formalism is to use the density operator or density matrix (Farrar 1990, 1990a). This approach will not be adopted here but ideas that have come out of this can be extremely informative, allowing an understanding of some of the more complex pulse sequences adopted today. A central idea is that of coherences and coherence order (Munowitz 1988, Ernst, Bodenhausen and Wokaun 1988, Keeler 1990). Transverse magnetisation is a concept that can be readily visualised and can be represented by a classical vector. As the rf pulses are applied as
Physical Background
33
described above, the spins change their magnetic quantum number m by + 1 and are termed single quantum coherences because the change corresponds to absorption (or emission) of one quantum. It is only this coherence that gives rise to a directly observable signal and the sign convention adopted is to regard Am = - 1 as the observable magnetisation. However it is possible that other coherences are generated. For example for nuclei with spin higher than 1/2 it is possible to excite transitions that cause spins to undergo a change Am > 1, which is also possible for systems of coupled spins which flip in unison. The higher order coherences cannot be observed directly but if a series of pulses is applied the higher order coherences will influence the final transverse magnetisation. It is worth noting that a zero-order quantum coherence corresponds to a net change Am -- 0 but that longitudinal magnetisation is not a coherence. Then in a two-dimensional (2D) NMR experiment this influence is recorded as a function of a second time period. Such ideas have been long used in 2D sequences applied to liquids. 2D sequences are now more commonly used in solids, and multiple quantum techniques are at the heart of generating higher resolution NMR spectra from noninteger spin quadrupolar nuclei. To follow the effects of a pulse sequence a coherence level diagram can be helpful. The coherence values are shown and the effects of the pulses of a sequence are then shown by the coherence pathway line (Figure 2.4). Such a sequence starts with the longitudinal magnetisation at m = 0. The changes in the mvalue occur when there are pulses present. Often when a pulse is applied there is no clean generation of a single change of coherence order but different coherence changes occur together with differing probabilities. However the desired coherence pathway can be selected by the appropriate phase cycling of the pulses. In Eq. 2.40 there was an arbitrary phase oLwhich, in the rotating frame, corresponds to the direction in which the magnetisation points. Spectrometers are capable of generating such phases. This phase
2 t2
I" "1 3
,/
10 Longitudina magnetisati~
3Qcoherence /
-1 -2
~~
Observable magnetisation
,,
-3 Figure 2.4. An example of a coherence level diagram for a 3Q experiment with symmetrised pathways and a z-filter pulse.
34
Multinuclear Solid-State NMR of lnorganic Materials
often needs to be rapidly set (--~ Ixs). A decade ago spectrometers for solids came with four hard-wired channels with the four phases + x and _+ y always available. Modem spectrometers are capable of setting the phases for the pulses to 0.09 ~ in less than 200 ns. The phases of the pulses can be used to discriminate between the coherence orders and hence pathways since if the phase of a pulse is changed by ot then a coherence order change Am will experience a phase change of otAm (Ernst et al. 1988).
2.1.4 Observation of the NMR signal The magnitude of the transverse magnetisation produced by the pulse is proportional to Bo (from the Boltzmann factor) and ~2, with one ~/arising from the Boltzmann factor and the other from the magnetic moment. It is also possible to detect magnetisation directly by using a SQUID (superconducting quantum interference device) and development of NMR probes based on such an approach is on-going. However the vast majority of NMR experiments use a coil to apply the perturbing rf pulse and then to record the signal. The maximum signal is generated after all the initially longitudinal magnetisation becomes transverse (i.e. after a 90 ~ pulse). The pulse is turned off and then in the laboratory frame (where the NMR coil is located) this transverse magnetisation precesses through the torque exerted by the Bo field as M ~ • B o so that the precession is proportional to ~/Bo. Through Faraday's Law of electromagnetic induction the induced voltage is proportional to the rate of change (~ ~/Bo) of magnetisation (~ ~/2Bo), producing an induced signal S given by 3
2
S - NVc~' B~
+ l)
(2.43)
3kT where Vc is the volume of the sample. Since there are now no rf fields perturbing the system this is called the free induction decay (FID) signal. Typically induced voltages are --~ ~V although there is significant variation with ~/. The influence of these factors in determining the signal intensity can be seen by comparing the intrinsic sensitivity (per magnetic nucleus) of ~3C which is 133 times greater than 89y (this assumes that the sensitivity is proportional to ~/3 which is generally accepted although some analysis has shown that the variation is proportional to @/2). Even though 89y is 100% naturally abundant compared to only 1.1% for 13C, the ~/-factor results in 89y displaying a relative sensitivity of only 69.6% of 13C. The relative receptivity (R) (Harris 1983) takes into account the nuclear factors, including natural abundance (C), on the sensitivity via R = ]yl 3 CI(I + 1). Equation 2.43 indicates that other factors can be used to enhance signal intensity. A larger sample increases Vc, the experiment can be carried out at lower temperature and at higher applied magnetic field. Also, if the natural abundance of an isotope is low, isotopic enrichment will help sensitivity. Papers should be read carefully as solid state NMR reports especially of low abundance
Physical Background
35
nuclei often use enriched samples, making the experiment much more tractable than at natural abundance. The most significant change in experimental technology is the wider availability of higher magnetic fields (i.e. >- 11.7 T). Detailed analysis indicates that the signal increases with applied magnetic field as Bo3/2-Bo 7/4. The detail is not important, but as sensitivity rises rapidly with field the study of many nuclei in solids is now much more feasible than even just a few years ago.
2.2. INTERNAL INTERACTIONS The total interaction energy of the nucleus is the sum all the individual Hamiltonians it experiences. The external magnetic fields applied allow the spectroscopist to manipulate the spin. The information about the atomic scale surroundings of a nucleus is contained within the interactions between the nucleus and those surroundings. These interactions are summarised in the internal Hamiltonian H~nt -- H D + Hcs + H K + Hj + Hp + HQ ~1) + HQ ~2) + - . .
(2.44)
In diamagnetic insulating solids spin-1/2 nuclei experience a range of interactions that include magnetic dipolar (HD), interaction through space with nearby nuclear magnetic moments, chemical shielding (Hcs), modification of the magnetic field at the nucleus due to the surrounding electrons and indirect spin-spin coupling (Hj), interaction of nuclear magnetic moments mediated by intermediate electron spins. In materials that contain paramagnetic centres the unpaired electrons can interact strongly with the nuclei (Hp) and possibly cause very large shifts and severe broadening of the NMR signal. The fluctuating magnetic fields produced by the electron spins can produce very efficient relaxation. Hence for solids where the nuclei are slowly relaxing and which will dissolve paramagnetic ions, small amounts (--~0.1 mol%) are added to aid relaxation. In materials containing conduction electrons these can also interact strongly with the nuclear spin via a contact interaction HI,: that produces both relaxation and a change in resonance position termed the Knight shift, which provide important information on the nature of the density of states at the Fermi surface. Nuclei with spin I > 1/2 are also affected by the electric quadrupole interaction (HQ), an interaction between the nuclear electric quadrupole moment (eQ) and the gradient in the electric field at the nucleus. Although this is an electrical interaction it depends on the magnetic quantum number and so affects the NMR spectrum. Of the NMR-active nuclei around three quarters have I >_ 1. The quadrupole interaction can cause a significant splitting relative to the Zeeman effect.
Multinuclear Solid-State NMR of lnorganic Materials
36
All these internal interactions that affect the NMR spectra can be represented by a general expression involving tensors Hm = kmliTmij.Amj
(2.45)
where Hm is one of the component Hamiltonians in Eq. (2.44). For each interaction there is a constant k, a 3 • 3 second-rank tensor T and another vector quantity, either a field or a spin with which the spin (I) interacts. The interactions are summarised in Table 2.2 along with the interactions with the external magnetic fields. Three numbers are needed to describe a 3 • 3 tensor relating two vectors and these are usually the isotropic value, the anisotropy and the asymmetry (Haeberlen 1976). Their exact definition can vary even though there are conventions that are normally adopted, but any paper should be examined carefully to see how the quantities are being defined. In a typical isotropic powder the random distribution of particle orientations means the principal axes systems (where the tensor only has diagonal elements) will be randomly distributed. In the presence of a large magnetic field, this random distribution gives rise to broadening of the NMR spectrum since the exact resonance frequency of each crystallite will depend on its orientation relative to the main magnetic field. Fortunately, to first-order, all these interactions have similar angular dependencies of (3cos20 - 1 + xlsin20cos2q~) where ~1 is the asymmetry parameter of the interaction tensor (xl = 0 for axial symmetry). Lineshapes can provide very important information constraining the local symmetry of the interaction which can often be related to some local structural symmetry.
Table 2.2. Summary of NMR Interactions.
Hm
Interaction
Tij
Ai
Typical Size (Hz)
Comments
Hz HRF HD
Zeeman RF Dipolar
Unitary Unitary D
Bo B~ I
107-109
Hcs
cr
Bo
102-105
Hj
Chemical Shielding Indirect Spin
J
I
1-103
He
Paramagnetic
P
S
102-105
HK
Knight Shielding Quadrupolar
K
S
102-105
eq
I
103-107
Interaction with main magnetic field Interaction with rf field Through space spin-spin interaction, axially symmetric traceless tensor Alteration of the magnetic field by the electrons Spin-spin interaction mediated via the bonding electrons through the contact interaction Interaction with isolated unpaired electrons Interaction with conduction electrons via the contact interaction Interaction of nuclear quadrupolar moment with the electric field gradient (q), effectively an 12 interaction
HQ
103-105 103-104
Physical Background
37
2.2.1. The dipolar interaction The dipole interaction arises from the coupling between two magnetic dipoles. Classically the energy of two interacting dipoles ~1 and tx2, a distance r apart, is given by
ED
~
r3
r5
(2.46)
The quantum mechanical Hamiltonian can be derived directly by substitution of ~1 = 3/lh/1 and ~2 = y2h/-2 which leads to Ho=/'to 4re
~/l~t2h2[ r3
1-1 "1"2-
3(/1" r)(I2"r) / r2-
(2.47)
or in Cartesian coordinates
Ii. r - I1xx + IlyY + Ilzz
(2.48)
This can be usefully rewritten in terms of polar coordinates as x = r sin0 cosq~,
y = r sin0sin q~, z = r cos0
(2.49)
This leads to the important "alphabet" expression for the dipolar Hamiltonian ]'to
H D = -~
~/1~/2h2 r3
(A + B + C + D + E + F)
(2.50)
The terms A - F in this Hamiltonian are
A - IlzI2z(3 COS20-1) B--1[I1+I2_+I1_I2+1(3cos20-1) 4 C - -3[I1+12z + 11zI2+](sin0 cos0) exp(-iq~) 2
(2.51)
D - --~[I1_I2z 3 + IlzI2_l(sinO cos0) exp(+iqg) E - - 3 [I1+I2+] sin20 exp(-2iq~) 4 F = - 3 [I1_I2_ ] sin20 exp(+2iq~) 4 Only terms A and B commute with Iz and this is the Hamiltonian HS~,~the secular part of HD which is independent of time.
Multinuclear Solid-State NMR of Inorganic Materials
38
__(A+B)=t.to 8iv ~/1~'2 r 3j~2 I 2Ilzi2z --21 [I1+12- + I1-I2+ ]1(3
H sec _ 1"/o ~/1~'2 h 2 D -3
47c
r
COS20-
1) (2.52)
Here 0 is the angle between the internuclear vector and the magnetic field Bo as the magnetic field lies along the z-axis. The terms C - F are called the non-secular part H 79~ sec of the dipolar Hamiltonian liD. These (C-F) are time-dependent at frequencies tOo and 2tOo. Since the frequency of the time dependency of H~ ~ sec is much higher than the frequency/energy of H~,C the time average of the non-secular part of the dipolar Hamiltonian is essentially zero and the non-secular part can thus be neglected for all spectroscopic purposes. The terms C-F still play a role however as they are responsible for relaxation of spins to Boltzmann equilibrium through the raising and lowering operators. For the secular term it is important to distinguish the effect of the B-term for homonuclear and heteronuclear cases. The spin part of term B is called the flip-flop operator because it can cause transitions between different spin states of two coupled spins. For two coupled spin-1/2 it can be envisaged that as one spin changes from m - 1/2 to m = -1/2 the coupled spin goes from m - -1/2 to m = 1/2. The probability of this flip-flop process occurring is highest when this process is energy conserving, i.e. no external energy needs to be supplied. Whether or not term B is energy conserving has implications as to whether it can be really included as a secular term, and hence for the effect of the dipolar interaction on the spectrum. The energy difference (AED) between the two states of the coupled spins is to a first approximation (2.53)
AED = - ( 7 , - ~ ' 2 ) h B o
So unless Yl - Y2 an energy difference between these states exists. If the coupled spins are different elements, the B-term in the dipolar Hamiltonian vanishes since the flipflop process is no longer energy-conserving. Hence the secular dipolar Hamiltonian will change for a set of like-spins Ii and Ij
H~ _ lao
E
l y'2h 2 2li_ij___~
I';+'; +'
cos2
-1)
(2.54)
whereas the secular heteronuclear dipolar Hamiltonian for spin I and spin S is given by
H ,oS = ~o ~~'I ~I S pl 2 21_S_(3cos 20is -1) 8Jr r/3 " "
(2.55)
Both the heteronuclear and the homonuclear dipolar coupling depend on the orientation of the inter-nuclear vector and the inverse cube of the distance, and often give rise to broad lines in powders. However the characteristic lineshape has features, particularly singularities. For an isolated pair of spin-l/2 nuclei a pair of strong singularities
Physical Background
39
are formed in a so-called Pake pattern (Figure 2.5A, Pake 1948). In single crystals relatively sharp lines are obtained but they could occur at positions covering a wide frequency range within the powder envelope. For a coupled spin-1/2 pair, a pair of lines is observed with the separation depending on the orientation and also the distance. The behaviour is similar for homonuclear and heteronuclear coupled spins but there is a scale factor of 1.5 for the homonuclear case compared to the heteronuclear one (this difference arises from the change from the nonsecular to the secular form of the B term in the dipolar H a m i l t o n i a n - in addition note the replacement of "~i for ~/s). The dipolar interaction is very important in NMR since by measuring the magnitude of the dipolar interaction between 2 nuclei the distance between these nuclei can be determined. An example of this dipolar coupling is the water molecule in NaA1SizO6.H20 (Figure 2.5B). The dipolar coupling is also central to many schemes for magnetisation transfer between nuclei using double resonance techniques such as cross-polarisation. The data become more difficult to unravel when a nucleus experiences numerous different dipolar interactions at the same time. Even an increase to just three spins causes the lineshape
D n - 3/~o72_____h h 32/r2r3
~ _ _ ~ _
zs _/ZoYzYsh - 16~.2r3
!
!
D H,Zs
,,
[
D n.Zs
D n aS
_D U,~'s
_
2
300
200
100
2
0
-100 -200
-300
1H shift in p p m
Figure 2.5. A. Schematic powder pattern of a spin-l/2 nucleus dipolar coupled to a second spin-l/2 nucleus. B. Example of a 1H Pake powder pattern from NaA1Si206.H20 from Yesinowski, Eckert and Rossman (1988) with permission of the American Chemical Society.
40
Multinuclear Solid-State NMR of Inorganic Materials
to become much more complex and to depend on the relative orientation of the spins. As the number of interacting spins increases, the lineshape rapidly loses distinctive features, and the observed distribution of intensity can be described by the method of moments. The nth moment for a lineshape L(v) is defined by oo
[. (V- Vo )" L(v)dv
M
Il
(2.56)
- - - -.-oo c~
I L(,v)dv The second moment is the most used description of the intensity distribution but this alone cannot unambiguously define the lineshape and higher order moments have to be examined. Often the lineshape in solids can be described by a Gaussian which has the distribution
l((V-Vo G ( v ) - 2 r ~ - A exp -
)2 )
(2.57)
A2
where A is the standard deviation of the line which is related to the more directly measurable full width half maximum (FWHM) by A = 2.355FWHM. The second moment can be related to the dipolar interaction. For the homonuclear interaction the second moment is given by
__ 3 (lUo ")2 t4];12i(i -]-1)Z
J
M211 -4 \ 41r J
(1-
3cos2Oij)2 ,)6
(2.58)
For a powder an average value of the factor (1 - 3cos20ij) 2 can be taken as 5/4, resulting in the expressions for M2
M2tt - -5 \ 4zc J ~'t h I(I + 1) 9
Mzls -
4.
)'1Ys2 h S(S + 1)
--T 1)j
(2.59)
r-6
(2.60)
If the lattice type is known then the lattice sum of rij -6 can be performed, and conversely from the second moment, information can be deduced about the distribution of nuclear spins.
2.2.2 Scalar coupling The scalar coupling (also called spin-spin, indirect or J-coupling) is a quantum mechanical-based indirect interaction of nuclear moments mediated via the electrons
Physical Background
41
in the chemical bonds between the nuclei. The first nucleus perturbs the surrounding electrons and this perturbation of the electrons produces an additional magnetic field at the second nucleus. In contrast to dipolar coupling, which is a through-space interaction, scalar coupling is a through-bond interaction. Hence, it is most important in covalent materials. The Hamiltonian for the scalar coupling is given by Hj - hJ12/l"/_2,
(2.61)
where J is the scalar coupling constant in Hz. The source of the scalar coupling can be spin-orbital, spin-dipolar and Fermi contact interactions. The Fermi contact interaction is probably the most important. The sources of these terms are discussed in detail by Jameson (1987) and Facelli (1996). From Eq. 2.61 there is no Bo dependence of this interaction. In reality the scalar coupling has to be represented by a tensor as it is orientation-dependent. However, since the tensor character is only apparent in solids where it is usually a relatively small perturbation compared to the other spin interactions it is generally ignored. The scalar coupling is mainly observed in liquids since, as it is usually small, it is often obscured in solids by other interactions. When there are several coupled spins, the total scalar Hamiltonian is Hj - Z hJijmimj i,j
(2.62)
Hence the pattern produced by the coupling gives important information about the local spin network and has proved very useful in liquids for identifying the local fragments of molecules, especially in large molecules such as proteins. If a spin-I is coupled to several spins S then including the Zeeman interaction the Hamiltonian is H = H z + H j = h VoI z + JlshlzSlz + JishlzS2z
(2.63)
With only the Zeeman interaction a single resonance is observed, but the scalar coupling now splits the resonance with the pattern depending on the number and type of coupled spins. The number and intensity ratios follow a binomial distribution. If the observed spin I is coupled to a single spin-1/2 nucleus then a pair of resonances split by the J-coupling value is observed, whereas if it is coupled to two identical spin-1/2 nuclei then the resonance is split in three with an intensity ratio of 1:2:1, again separated by the scalar coupling constant J (Figure 2.6A). The magnitude of the scalar coupling depends on the degree of covalence of the bonds, the number of bonds between the nuclei, the gyromagnetic ratio, and the degree of orbital overlap. Heavy nuclei have more extensive orbitals leading generally to larger scalar couplings. The value of J also depends directly on the product of the gyromagnetic ratios of the two nuclei involved, and often, reduced coupling constants normalised by this product are quoted to remove
42
Multinuclear Solid-State NMR of lnorganic Materials
A
B
I
C
J
'" "i
,
-640 .....
i
-680
-720
~19Sn shift (ppm) w.r.t. (CH3)4Sn Figure 2.6. Coupling multiplets of a spin which is coupled to A. one and B. two spin-l/2 nuclei, and C. example of l l9Sn in BaSnO3 showing the J-coupling with crystallographically equivalent but magnetically inequivalent ~17Sn, from Clayden, Dobson and Fern (1989) with permission of the Royal Society of Chemistry.
this factor (Jameson 1987, Sanders and Schrobilgen 1990). This reduced factor illustrates how other influences play a role, with one-bond coupling increasing for 1H from --- 40 to carbon to 940 for lead, and increasing for 19F from 195 for nitrogen up to 2100 for bismuth. There is no reason why this coupling should be purely isotropic and anisotropy in the J tensor is possible. The J-coupling is usually taken to be directed along the internuclear vector about which the tensor is probably axially symmetric. J is then described by an anisotropy AJ = J - J• The complete J-Hamiltonian is then ,, ,, Z~ r Hj - JI.S. +
[3]:j: (S. j) -
IzSz ]
(2.64)
where jz is a projection of the unique axis of the J tensor onto the z-direction. The anisotropic component interaction is identical in functional form to the direct throughspace dipolar coupling and hence it is very difficult to separate these contributions. Often a pseudo-dipolar Hamiltonian is written to describe the bilinear coupling between two spins I and S, which is the same as Eq. 2.64 except that the dipolar coupling is replaced by an effective dipolar coupling constant D ' = P~ 8Jr 2ri3
ALl 3
(2.65)
Then the coupling between the two spins I and S including these two effects gives a frequency shift for the (m, m - 1) transition of Av m - - m J + mD'(3cos 2 0 - 1 )
(2.66)
where 0 describes the direction of the internuclear vector in space. Reports of J-coupling are quite infrequent in solids, as the effects are often small compared to other effects
Physical Background
43
which mask the J-coupling. Figure 2.6B shows an example of a J-split spectrum of 119Sn in CaSnO3. Reports of anisostropy in J are even rarer.
2.2.3 Paramagnetic coupling If a material contains a localised unpaired electron (e.g. on an ion) this electron can cause an additional magnetic field at the nucleus. The interaction Hamiltonian between the nucleus (I) and the electron (S) can be generally written as H I, = ~ h I . P . S
(2.67)
The coupling tensor (P) to the paramagnetic ion produces two main sources of additional field. One is the Fermi contact or hyperfine interaction. The general form of the contact interaction produces a Hamiltonian Hcon = -12 - -o- 8re y , ye I. S a(r)
4re 3
(2.68)
where 8(r) is the Dirac delta function. Unpaired electron spin density, through spin polarisation, will induce unpaired spin polarisation on the s-orbitals of the nucleus, which has a finite value at the nucleus. This s-polarisation interacts directly with the nuclear magnetic moment to give a frequency shift of Avc - A y B ~ (Sz) h
(2.69)
where (A/h) is the electron-nucleus hyperfine coupling constant in Hz and (Sz) is the expectation value of Sz, the average component of the electron spin along the magnetic field. The rapid thermal flipping of the electron spin can greatly reduce this expectation from its full value, and is in general given by (Sz) _ _ ].tBJ(J + 1 ) g j ( g j -- 1)B o 3kT
(2.70)
where IXB is the Bohr magneton, J is the total angular momentum quantum number for the electron state and gj is the Land6 g factor given by gj =
3 J ( J + 1) - L ( L + 1) + S(S + 1)
(2.71)
2 J ( J + 1)
The mean electron moment given by Eq. 2.69 will only be parallel to Bo if g is isotropic. If the nucleus being observed is not the ion with the initial unpaired electron, spin s-orbital polarisation can still be induced through effects transferred via the chemical bond, in which case it is called the transferred hyperfine interaction. This effect can still be strong although it will decrease rapidly as the number of intermediate bonds
44
Multinuclear Solid-State N M R of Inorganic Materials
increases. For example in the series Y2-xRExSn207 each yttrium (ignoring oxygen) has a shell of tin, and further out a shell of yttrium. If a rare-earth (RE) substitutes for the central yttrium the shift effects on the tin shell are an order of magnitude greater than the next yttrium shell (Grey et al. 1989, 1990). In paramagnetic systems the Fermi contact term is expected to dominate. However in addition there is also the possibility of a through-space dipolar effect. The shift caused by the trace of this tensor interaction is often called the pseudocontact effect, which for a lanthanide ion is given by 2
2
AVpc = _ g j l J s J ( J + 1)(2J - 1)(2J + 3 ) F ) ' B o
1207r(kT) 2 r 3
(2.72)
where F is a function that depends on the angle between the principal magnetic axis and the lanthanide-resonating nucleus vector, the crystal field parameter of the lanthanide site and a numerical coefficient. In Y2-xRExSn207 for the tin sites the Fermi contact term dominates but for yttrium site contact and pseudocontact terms make comparable contributions, so it is important to take both effects into account (Grey et al. 1989, 1990). The Fermi contact interaction is isotropic but the pseudocontact interaction is a full anisotropic tensor interaction. The angular dependence of these terms is of the form A0 (3cos 20 - 1) + Azsin 20 cos2q~
(2.73)
2.2.4 C h e m i c a l s h i e l d i n g
The factor which turned NMR from simply an interesting physical observation of the nuclear magnetic moment into a widely useful technique in analytical sciences was the realisation that the exact resonance depends on the chemical environment of the nucleus (Proctor and Wu 1950, Dickinson 1950). This was termed the chemical shielding and means that the exact resonance frequency is an extremely sensitive discriminator of the chemical environment as it depends on the electron density and hence reflects the chemical bonding. The chemical shielding arises from the change in the actual magnetic field at the nucleus since the applied magnetic field induces changes in electrons about the nucleus with generate local fields. To a first-order approximation the chemical shielding is linear with the applied field strength (Grant 1996). This effect can either add to or subtract from the main applied magnetic field (termed paramagnetic or diamagnetic respectively). Hence the energy of interaction is E - -yhI
. B o - ~ It~
H ~~t~ ? ' h / ( l - o r - Z ) H o 27r )'h/~.(M_ + _ o ) - 2n: .__ .~
(2.74)
This equation splits the contributions related to chemical shielding and susceptibility effects. This distinction is somewhat arbitrary as they are both a result of the response
Physical Background
45
of the electrons to the applied magnetic field giving rise to additional internal magnetic fields. X is usually only significant if the susceptibility of the solvent in liquid state NMR is very different, or if the shape of a solid sample (including the grains of a powder) is very different, giving rise to very different demagnetising effects. The three-dimensional nature of the electron density in a material means that the chemical shielding is represented by a second-rank tensor (if) and the corresponding Hamiltonian is
Hcs
=
7~h I_.q. B_o
(2.75)
~
The orientational dependence is present in the chemical shielding tensor which in its principal axis system (PAS) is diagonal with principal components ~rxx, ~ryy and (rzz. In a crystalline powder sample all orientations of the PAS system of the shielding tensor are present and for most orientations the PAS does not coincide with the laboratory (LAB) frame where the z-axis is aligned with the direction of the applied magnetic field Bo. Hence the tensor representation gives in the PAS q P A S - - I ~ axx
0 ayy 0
00
I
(2.76)
r
which when transformed into the laboratory flame becomes O'xx
O'xy
(Txz
~= -- (Tyx
O'yy
(Tyz
O'zx
(Y zy
($ zz
(2.77)
The tensor components are distinguished by capital subscripts in the PAS and lower case subscripts in the laboratory frame. The two frames are related by rotation of the axes using matrices with the Euler angles defining the rotation to give G=Lab -- R ( a ~ ) q
(2.78)
PAS R -1 (aft'J/)
Then the field at the nucleus, including both the main field and the shielding, gives
1-a~ -a~y -a~z ~ I-a~z Bo B-(1-q).B o-
-Cryx -az~
1-Cryy -ay z -azy
1-azz
Bo
-ay z Bo ~,(1-azz)Bo
(2.79)
46
Multinuclear Solid-State N M R o f Inorganic Materials
It is only the zz-component that is of interest since after this transformation this is the component that is aligned with Bo, Oxz and (ryz make only second-order contributions. Hence the Hamiltonian representing the chemical shielding is now given by Hcs - ?'t hi_. o'__- B o
(2.80)
Crzzis obtained by performing the appropriate set of rotations from the PAS to the LAB frame and this tensor element is given by Ao" [(3cos=O-1)+ 77( sin=0 cos 20)]
(~zz -- l~iso + T
where Act - o'-: - ~riso is sometimes called the shielding anisotropy and r / -
(2.81) I~ XX -- l ~ y y
A~y
is the asymmetry. The angles 0 and + are the polar angles of the LAB frame in the PAS of the tensor. The variation of different components of the shielding tensor often reflects the local symmetry around the nucleus and is characterised by three parameters. Rather than the three principal components, combinations of these are quoted, namely the isotropic value, the anisotropy and the asymmetry. Chemical shielding (or) is normalised by looking at the difference of the resonance frequency between the sample (Psample) and a reference Larmor frequency (PLarmor)for that nucleus (i.e. the bare nuclear frequency). It is then normalised by (PLarmor) and quoted in parts per million (ppm)
O'sample =
VLarm~ -- Vsample (X 10 6 ).
(2.82)
VLarmor
This normalisation means that measurements at different magnetic fields can be directly compared. A lower resonance frequency indicates a higher shielding value. Rather than the chemical shielding the chemical shift is quoted, where the sample frequency (Psample) is referenced to the resonance frequency of a standard reference sample (Vref) instead of the Larmor frequency. It also is normalised, with units of ppm, and its symbol is 5. Chemical shifts are, in fact, used far more often than chemical shielding since this is the parameter experimentalists measure. 5sample -- Vsample -- Vref (• Vref
6)
(2.83)
For the chemical shift a higher resonance frequency means a higher chemical shift value. The chemical shielding and chemical shift are related by 5 -- l~ref -- l~Ysample 1 -- (~ref
(2.84)
Physical Background
47
which to a good approximation for IO'refl < < 1 gives ~sample -- O'ref -- O'sample
(2.85)
The difference in sign between g and cr arises because g is defined in terms of frequency whereas cr is defined in terms of the fractional change of the magnetic field. There are no inconsistencies with these approaches, which have arisen through the historical development of the technique in which sweeping the magnetic field has been largely superseded by pulsed Fourier transform methods. The different terminology can also be quite confusing when it is first encountered. A more positive value for indicates a higher frequency (and thus downfield) whereas for shielding this would mean a lower frequency. An interesting consequence of the definition of g has been pointed out by Neue (1996). If different references are used in different studies then to relate the two studies, the values are compared by making a simple additive correction of the two references. If the references are a long way apart then more correctly the shift on the second scale is I~2 -- (--I~R1,R2 nt- ~1 ) V~
(2.86)
Vo2
This uses the convention in Eq. 2.84 for the definition of the chemical shift where ~1 and ~2 are the shifts of a material with respect to references R1 and R2, and gm,r~2 is the shift of reference 1 measured against reference 2. The references 1 and 2 have absolute frequencies of Vo1,2 respectively. Hence it can be seen that for many compounds this correction factor will make very little difference, but for elements with big shift ranges or using different nuclei to obtain a reference means that this factor must be taken into account if high accuracy is required. The tensor elements of the chemical shift also need to be defined and in the literature there have been a number of approaches adopted to define parameters describing the tensor. Naturally the three principal components of the tensor and the spatial orientation of these components on the molecular frame will define the tensor. However, papers often do not define the individual tensor components and the parameters adopted by each paper to define the tensor need to be carefully considered. A recent suggestion (Mason 1993) for standardising the definitions is to have the three elements in the PAS designated ~11, ~22 and ~33, with ~11 corresponding to the highest frequency. Then the isotropic value is still the average of the sum of these three terms with the span defined as (2.87) which will always be -> 0, and the skew
48
Multinuclear Solid-State NMR of Inorganic Materials --0"22) = 3(~22 -- (~iso ) 0"33 --O'll (~11 --I~33
tr = 3(r
(2.88)
which will vary between - 1 and 1. The tensor elements are necessarily ordered for the above definitions as 033 -> 0"2= -> 0"33 and consequently 811 -> ~22 ~ ~33. Particular special cases are (Figure 2.7):
811=~'~2
833 811
822
833 811
822--q~33
Figure 2.7. Example of static CSA powder patterns showing different skews. spherically symmetric where 811 = ~22 = ~33, ~ -- K -- 0 axially symmetric (prolate) where 811 = ~22 5/:~33, ~'~ finite, K= 1. axially symmetric (oblate) where 81~ 4= ~22 = ~33, , [-~ finite, K = -- 1.
2.2.5 Knight shift In a conducting solid the nucleus will couple to electrons of the chemical bonds in the same manner as above for diamagnetic insulating solids, but there is the additional effect of the conduction electrons. Unlike paramagnetic and diamagnetic solids, the electrons are delocalised in conductors and a given nucleus will experience the magnetic field from a range of electrons. The effect of this collective influence must be described. The conduction electrons occupy a Fermi distribution within the electronic states of the material. These states are occupied in a pairwise fashion due to the Pauli exclusion principle (spin up and spin down) from the lowest energy up to the Fermi energy. At Bo = 0 the pairing will mean that there is effectively no net magnetic moment from these electrons. However, under a magnetic field there is a shift in energy of the spin-up and down states. The resulting imbalance in the number of spin-up and down states leads to a net magnetisation of the material and hence an associated susceptibility termed the Pauli susceptibility (Xp). Then, if the change in populations are n + 1/, M = / l e ( n ~ - n_,/2)
(2.89)
if the electron distribution is described by fiE) so that
M - lu2Bf nf(E)dE For a free electron gas, in the limit that Ef > > kT then
(2.90)
49
Physical Background M - IaBBon(E f ) ~ Zp - ~ t o l 2 B 2 n ( E f )
(2.91)
The shift in the field thus produced can be determined by calculating the field caused by the electrons producing the susceptibility. This can be expressed in terms of the hyperfine field produced by the conduction electrons, a contact type of interaction,
4re 3
F
This depends on the value of the wavefunction at the nucleus (r = 0) for electrons averaged over the Fermi surface, where 12 is the volume per electron. This shift of frequency in a metal was first observed in copper in 1949 by W.D. Knight and was subsequently termed the Knight shift (Townes, Herring and Knight 1950, Knight 1956), given by K - AB
2
(2.93)
Since K depends on the wavefunction density at the nucleus, the effect is dominated by s-electrons which is certainly true in metals with unpaired s-electrons. If the Pauli susceptibility and electron density can be independently measured then the Knight shift will give an independent measure of the s-component of the conduction electron spin density. These shifts are positive and are much larger than chemical shift effects, some typical values being L i - 0.025%, A g - 0.52% and H g - 2.5%. In other metals the situation is more complicated when the s-electrons are paired but there are other electrons (e.g. p but especially d). As only s-electrons have significant density at the nucleus the effects of these other electrons are much smaller. The hyperfine fields of these electrons induce polarisation in the s-electrons that subsequently produce a shift, termed core polarisation.
R-orb + (y
Total s h i f t - K + c r - K s + Kcp + "~d
(2.94)
A major problem is that it is the net value of the shift which is measured in the experiment. Two points emerge from this. The zero of the scale needs to be known so that the contribution of the chemical shielding has to be taken into account. Also, in more complicated metals the different terms have different signs, with Ks and Kd ~ positive whereas Kcp is negative. If the symmetry of the site is lower than cubic the full tensor form of the electronnucleus interaction needs to be used, so that in addition to an isotropic term there is an anisotropic contribution. If in the PAS of the Knight shift tensor the components of the tensor are Kx, Ky and Kz, then in the laboratory frame with its orientation in the frame defined by Bo described by the Euler angles 0 and +,
50
Multinuclear Solid-State NMR of lnorganic Materials B - B o { K z cos20+Kx sin2Ocos2~+Kr sin2Osin2~}
(2.95)
In many materials the tensor is axially symmetric so that Kx = Ky - - Kz/2 and the anisotropy is usually axially symmetric Kax. The frequency shift is then given by
AV-
YBo (Kis o + Kax(3 2re
COS 2
O-- 1))
(2.96)
2.2.6 Quadrupole interaction It can be shown that for nuclei with I > 1/2, the electrical charge distribution of the nucleus is non-spherical, giving rise to an electric quadrupole moment. The background to the quadrupole interaction is given in the classic article by Cohen and Reif (1957) and a comprehensive recent introduction has been given by Man (2000). The quadrupole moment is designated eQ and can be prolate (eQ > 0) or oblate (eQ < 0). The energy of a charge distribution in an electrostatic potential is V(r)
E - J
p(r). V ( r ) d r
(2.97)
In the volume surrounding the nucleus V(r) can be expanded as a Taylor series as
V(r)- V(O)+
Z
i--~t
i=x,y,z
+~
ij
r=0
" "
aiaj[
+
(2.98)
r=0
which can be substituted into Eq. (2.97) to give 1
i
(2.99)
t,y
where ~V
Vi - - ~ / . and
V~, -
~2V
Oi---0-]
(2.100)
Vii is a second rank symmetric tensor which is diagonal in its PAS so that Vq - ~ij . Vq
(2.101)
The zero-order term in Eq. 2.100 represents the electrostatic energy which is the same for all orientations. This term will have no influence on the spectrum and can be ignored. The first-order term represents the electric dipole moment and from the fact that the nuclear wavefunction is symmetric (i.e. r(r) 5 r (2 r)), the product rr(r) is antisymmetric and this term will be identically zero for all nuclei. The second-order term is the electric quadrupole moment and this is the most important in giving an
Physical Background
51
orientational dependence to the effect of this interaction on the NMR spectrum. The variation will be determined from the deviation of the charge distribution from spherical symmetry and is defined as
eQ - I P(r)(3z2 - r2 )
(2.102)
For the potential Laplace's equation states VZv = 0, from which it follows that for the electric gradient tensor, s
-0
(2.103)
i
This property means that to fully define V only two of its three components need to be known. It is usually chosen to define these as the largest component
Vzz = eq
(2.104)
and the asymmetry parameter
(Henceforth the term xI will be taken to refer exclusively to the asymmetry parameter in the quadrupole interaction). The energy equation for the quadrupole interaction can be transformed into a form that makes it compatible with the other Hamiltonians above by substituting spatial operators with spin operators using the Wigner-Eckart theorem (Slichter 1990) which after some manipulation gives the quadrupole Hamiltonian in the PAS of this interaction
)]-
3# -
+
(i+ +
where XQ is the quadrupole coupling constant defined as
ZQ=
eeqQ h
(2.107)
The electric field gradient (efg) is set up by the charge distribution outside the ion (e.g. A13+) but the initially spherical charge distribution of inner shells of electrons will become polarised to lower their energy in this efg. This polarisation of the inner electrons produces an additional efg at the nucleus so that eq, = eq (1 - ~/~) where ( 1 - ~/~) is the Sternheimer antishielding factor (Sternheimer 1954). This factor is a measure of the magnification of eq due to distortion of the inner electrons close to a nucleus. Full energy band structure calculations of efgs have improved markedly in recent years with developments of code such as WIEN97 (Blaha, Schwartz and
52
Multinuclear Solid-State NMR of Inorganic Materials
Dederichs 1988, Blaha, Schwartz and Luitz 1999). These show the importance of the contribution of the electrons on the ion itself compared to the lattice. These direct calculations remove the need to apply this correction factor but it is helpful to make it clear why some nuclei experience much larger quadrupole interactions than others. To obtain the quadrupole Hamiltonian of a spin in a magnetic field the Hamiltonian needs to be transformed from the PAS to the LAB frame, keeping only those terms that commute with Iz. This is called truncation of a Hamiltonian and is only valid when HQ < < Hz (the high field approximation). To perform the transformation it is much more convenient if second-rank irreducible spherical tensors are used. The Cartesian and spherical tensor elements (T) can be related (see Schmidt-Rohr and Spiess 1994 and Eq. 8, in Man 2000), with two of the more common elements being (2.108)
~o - i= and "~/-6/~2o- 3~2 _ 12 In these operators, the quadrupole Hamiltonian is
where Vi are the elements of the electric field gradient in the PAS so that
V0 = ,1~ eq, V+~ - 0 , ~2 -
V+_2=
eqJ~
(2.1 10)
2
In the high field limit, where the quadrupole interaction acts as a perturbation of the Zeeman states, the terms of this Hamiltonian which commute with Iz lead to the perturbation of first-order
4(1)_ o
eQ
4 1 ( ~ - 1)
~[3i2 i2]V0
(2.111)
and the second-order term is
(
H~2) _
2
/2
{VIV liz (412 -- 8 # - 1 ) +
eQ
V0 4 I ( 2 1 - I )
VzV2I~( 2 I
2 - 2 I 2 - 1)} ( 2 . 1 1 2 )
These perturbations then lead to a shift in the separation of the Zeeman states Em.m- 1 - E m - E m - 1 which to first-order is E(l) _
3eQ
- 41(21 - 1)
(1 - 2m)V0
(2.113)
PhysicalBackground
53
However second-order effects are also present and for the central (1/2, -- 1/2) transition as there is no first-order energy shift (m = 1/2), these become particularly important. This shift is E(2) =
2[
eQ ]2~V-~Vl(24m(m-1)-4I(I+l)+9)+} I V-2V2
v0 4 I ( 2 I - 1 )
(2.114)
(lZm(m-1)-41(I+l)+6)
The Vi then need to be transformed from the PAS to the Laboratory frame. If the Euler angles (or, [3, ~/) describe the orientation of Bo with respect to the PAS system the tensor elements can be transformed by Wigner rotation matrices Dij (n) (Haeberlen 1976) via
2 (2.115)
_ Z
j=-2
Substituting these values and operating on the states with the spin operators gives the perturbations and the orientation dependence
E(1)=
Zoh 8I(2I-1)
[ cos
0 -1+
0
-
+
The Zeeman splitting gives a set of equally-spaced energy levels, shown for a spin-5/2 system in Figure 2.8. To first-order, the quadrupole splitting gives a set of symmetric transitions with the magnitude of the effect depending on XQ and a characteristic shape determined by 0. The first-order term splits the spectrum into 2I components of which the single quantum intensity is detected. This intensity depends on I<mlIxlm+ 1>12 (proportional to I(I+ 1 ) - m ( m + 1 ) = oL) at frequency V(1)m, m -- 1 away. The intensity distribution between the different transitions has to be carefully accounted for when accurate quantification is required from NMR spectra. The splitting between different transitions also causes a change in the pulse response. If XQ is very small so that the energy levels remain effectively evenly spaced, the response to the if-pulse is as would be expected for a spin-l/2 nucleus with the same ~/(Schmidt 1971, Fenzke 1984, Man 1988). This is called non-selective excitation and the magnetisation precesses as sino~rft. When XQ becomes significant, the separation between adjacent levels is quite different and at each set of energy levels the nucleus behaves as an effective spin1/2 nucleus but with a different ~/. This limit is termed selective excitation. This ability to consider transitions in pairs has become known as the fictitious spin-b2 formalism with each different transition showing its own nutation rate, and its pulse response now being sinoLtOrft. This change of response to the rf pulse means that the effective 90 ~ pulse
et al.
et al.
54
Multinuclear Solid-State NMR of lnorganic Materials B0 only Zeeman interaction
-hv0m
First-order quadrupolar
Second-order quadrupolar
~0~(3cos20_1)
9hz~ 6400vo
5
5(2 sin 2 20+ sin 4 O)
/
-512
-3/2
_~
-1
.3(sin 4 8 - 2sin 228)
-1/2
N
-4
2(sin' O- 2sin' 20)
112
~
-4
/ 2(2sin' 20- sin' O)
3/2
_...
-1
/
3(2sin2 20-sin4 O) 5 5/2
-5(2sin 220+ sin4 O)
/
Figure 2.8. Energy level diagram of a spin-5/2 system showing the Zeeman interaction and the first- and second-order quadrupole perturbation of the energy levels.
Table 2.3. Changes in the 90 ~ pulse and the intensity distribution for different transitions for quadrupole nuclei. I 3/2 5/2
7/2
9/2
mz
1/2 3/2 1/2 3/2 5/2 1/2 3/2 5/2 7/2
1/2 3/2 5/2 7/2 9/2
90~ 0.500 0.578 0.333 0.354 0.447 0.250 0.258 0.389 0.378 0.200 0.204 0.218 0.250 0.333
Int (%) 40.0 60.0 25.7 45.6 28.6 19.0 35.8 28.6 16.6 15.2 29.0 25.4 19.4 11.0
t _ For large • comparedto the case where there is no quadrupole splitting.
55
Physical Background
changes, as shown in Table 2.3. The change of response to the rf pulse with XQ is the basis of the 2D quadrupole nutation technique (Samoson and Lippmaa 1983, Guerts et al. 1985). The differences in the pulse response and the observed intensity for the selective and non-selective cases need to be considered. For non-selective excitation,
2 lm+l,m _ am+l,m "nonsel -- I-1
sin mrf t
Z 2 am+l,m -I
(2.117)
and the equivalent expression for selective excitation is m+l,m i sel
_
am+l,m I-1
Z -I
sin a~o~ft
(2.118)
2
am+l,m
so that in the small pulse angle limit (sin0 --- 0) these two expressions become the same and intensity between sites with very different XQ can be compared directly (Schmidt 1971, Lippmaa, Samoson and Magi 1986). The frequency shift can cause the non-central transitions (i.e. m # 1/2) to be shifted sufficiently far from the Larmor frequency that these transitions become difficult to observe with conventional pulse techniques. Equation 2.116 also shows that there is an orientational dependence of the frequency so that very broad lines occur in a powder. This is important for spin-1 nuclei as there is no central transition and all transitions are broadened to first-order. For the satellite transitions the shape extent will depend on XQ and the shape will depend on ~ (Figure 2.9A). It should be noted that for m <--->- m transitions there is no first-order effect. For the central transition of non-integer quadrupolar nuclei, v(]~ _ i/ - 0 , and "2' 2 . the dominant perturbation is second-order. The rotation of the tensor elements in the second-order equation gives _
1//2
(--~
V1V_1 - - ~ - 3
COS2 20~ + 2 r / c o s 2 a - 3)cos 4 fl
e2q2 q-(~-3/72cos 2 2 0 : - 2 r / c o s 2 a -
1//2 --b3)cos 213
3
(2.119)
1//2 (1 - cos 2 20:) and 1
2
2
1
( - ~ r / cos 2 c t - - r / c o s 2 a 4
V2V_2 -
_3_.e2q2
+(--12//2 COS2
2
1.2
_
3
8 )cOS4fl
2a_ 1 2
77 - 3 )cos
1
3
+ - ,t cos 2 2 a + - 7/cos 2 a + 3 4 8
(2.120)
Multinuclear Solid-State NMR of Inorganic Materials
56 A
B
rt=l
.n=l
n =0.
n =0.9
n =0.
n =0.8 n = 0.7
n---0.4 r--J
~_~~--a
n--~
~ , ~
.... H
.... ,___ '--~ .....
I
-2
,__
n-0.3
....
n =0 I
n-0.4
n =0 I
-2VQ
I
l
-VQ
I
I
I
0
I
I
I
I
2VQ
VQ
I
I
1
I
0.5
0
I
-0.5
I
-1
I
-1.5
I
-2
A
Figure 2.9. The quadrupole perturbed powder patterns for an I = 3/2 nucleus. A. The outer satellite transitions to first-order. B. The second-order quadrupolar broadening of the central transition with A = (I(I+ 1) - 3/4)p~/I10. The second-order line can then be calculated for any transition so that for the (1/2, -- 1/2) transition ~,(2) _ 1 I 3ZQ ~-~ - - 6v----~ 2 I ( 2 I - 1)
[A(a,~)cosa fl+B(a,~)cosZ fl+C(a,~)]
I(I+1)-
(2.121) where 27 9 A ( a , rl) = - - - + - 0 c o s 2 a - 8 4
B(a, rl) =
30
/72
8
2
3 ~2
cos 2 2 a
(2.122)
8
277 cos 2 a +
3 -4~
2
cos2 2 a
(2.123)
Physical Background
57
2
C(a, 7/) - - ~3 + __f_0- - ~-r/cosZct--8 1 3 r/2 cos 2 2o~
(2.124)
To obtain the lineshape of the central transition the second-order perturbation of the quadrupolar interaction must be calculated on the (1/2,-1/2)transition. These give distinctive lineshapes for the centreband (Figure 2.9B) which provide information about the local symmetry of the electrical charge about the nuclear site. For the special axially symmetric case where rl = 0,
,
(2)
"~-~
_
1
3XQ
I ( I + 1)-
(1- cos 2 fl)(9 cos 2 f l - 1)
(2.125)
16v o 2 I ( 2 I - 1)
If the total width of the central transition is Av, an estimate of the quadrupolar interaction is
xQ~r/~ + 22r/Q + 25 - 8 I ( 2 I - 1)l
v~ 3 (I(I + 1 ) - ~)
(2.126)
2.2.7 Nature of interactions All these interactions are quite similar in form, containing an anisotropic part so that in powders the lines are usually significantly broadened. The first-order perturbation shows an angular variation of the form A((3cos213 - 1) + "q sin2[3cos2o0 which is similar for all interactions. Sometimes it is necessary to distinguish between homogeneous and inhomogeneous interactions. The distinction can affect the way their averaging must be approached and also the way the interaction responds to refocusing, e.g. in echo experiments. An inhomogeneous interaction means that a broad spectral line can be regarded as being made up of many individual contributions, with each spin, (e.g. depending on its orientation) contributing intensity to a specific part of the line. For a homogeneous line, a spin via its interactions with other spins (especially via spin diffusion caused by the flip-flop term of the dipolar Hamiltonian) contributes intensity effectively to the whole line. This distinction gives rise to the well known phenomenon in laser and optical spectroscopy of hole burning. If a resonance is strongly irradiated by narrow band radiation at a specific frequency within a broad spectral line, a homogeneous line will show a decrease in intensity across the whole line whereas an inhomogeneous line will show a decrease in intensity only in the vicinity of the irradiation. Most interactions are inhomogeneous e.g. shielding effects, the quadrupolar interaction and Bo inhomogeneity, whereas homonuclear dipolar coupling can be homogeneous. A more rigorous distinction between these two interactions
58
Multinuclear Solid-State NMR of lnorganic Materials
is that given by average Hamiltonian theory. If the Hamiltonian operator is taken as a function of time and the terms commute at different times, the interaction is inhomogeneous. There is also possible intermediate behaviour where irradiation at a specific point affects other parts of the line but not the complete line, for which the term heterogeneous has been coined, although this can be regarded as largely inhomogeneous. For these interactions there is also the concept of coherent and incoherent homogeneous interactions. T2 relaxation is incoherent due to the independently fluctuating local fields whereas coherent effects would be generated by multiparticle interactions such as the dipolar couplings between spins. This is discussed in some detail by SchmidtRohr and Spiess (1994). The inhomogeneous interactions such as shift anisotropy and dispersion, two-spin dipolar interactions and the quadrupolar interaction can be refocused by spin echoes (Hahn 1950) whereas coherent multispin interactions can only be refocused by sequences such as the solid echo (Powles and Strange 1965) and magic sandwich echo. Incoherent effects are essentially irreversible.
2.3. ONE DIMENSIONAL METHODS FOR IMPROVING RESOLUTION
The anisotropic part of the interaction can often provide insight into structure, but is usually regarded as the poor relation of the isotropic information. In materials with relatively few sites static NMR spectroscopy is often worth considering. Its disadvantage is that the anisotropic part causes broadening which can often be very significant so that there is strong overlap between different sites, meaning that different sites cannot really be resolved. In liquids the interactions responsible for line broadening are averaged by the continuous, random tumbling and translational motions of the molecules. This isotropic averaging of the second-rank tensor interactions produces high resolution spectra. However, for solids it is often necessary to improve resolution by deliberately averaging the anisotropic parts of the interactions, thereby obtaining line narrowing. The Hamiltonians representing these interactions have all been seen to contain a spatially-dependent part and a spin-dependent part. To average these interactions, either (or both) of these parts of the Hamiltonian must be manipulated which usually means making them time-dependent. By far the most common approach is to make the spatial part time-dependent, for example by applying magic angle spinning (MAS). Sometimes this is extended to other angles and is then called variable angle spinning (VAS). There are other more complex time dependencies imposed such as double angle rotation (DOR) and dynamic angle spinning (DAS). The spatial part can also be varied by diluting the spins (i.e. increasing r) which reduces the dipolar coupling since the magnitude of the dipolar coupling is proportional to the inverse cube of the distance between spins. By making that distance large, the dipolar coupling can be made small. This situation occurs naturally for 13C where the low natural abundance ensures that carbon-carbon dipolar couplings are not
Physical Background
59
observed in J3C spectra. For nuclei such as protons where the dipolar coupling between protons can be very strong, diluting the protons can be extremely helpful. This can be done by deuterating a sample, which is acceptable provided it does not change the chemical nature of the sample (which might not be true, for example, for hydrogen-bonding). For species that are naturally rare e.g. 13C, aSN, selective isotopic enrichment selectively introduces the dipolar coupling which can be used to determine bond lengths. Techniques for averaging the spin part include decoupling, both for heteronuclear and, by multiple pulse techniques, for homonuclear dipolar coupling cases. Averaging techniques can be used in combination with each other, leading to even higher resolution enhancement. Included in such approaches are the combination of multiple pulse homonuclear dipolar coupling averaging with MAS which is then termed CRAMPS (Combined Rotation and Multiple Pulse Sequence), and more recently for non-integer spin quadrupolar nuclei, multiple quantum excitation has been combined with MAS in MQMAS sequences. For multinuclear studies of solid materials all of these techniques should ideally be available so that high resolution solid-state NMR spectra can be obtained from a range of solids.
2.3.1 Magic angle spinning and first-order effects 2.3.1.1 Physical principles. Magic Angle Spinning is probably the most widely used technique to enhance spectral resolution in solid state NMR spectroscopy (Andrew 1971, 1981). The solid sample, usually a powder, is loaded into a container called a rotor which is inclined at a fixed angle to the magnetic field and rapidly spun about its symmetry axis. The Hamiltonian thus becomes time-dependent and to find its new form the tensor elements have to be rotated, now not directly from the PAS to the Laboratory (LAB) frame, as for a static powder, but first to the rotor frame (ROT). With the Euler angles describing the relative orientations of these frames shown in Figure 2.10 this can be readily achieved by using the Wigner rotation matrices to give in the Laboratory frame
H L~b(t) - CI~AjToo + CIA Z "mO r) (2) (0, O, (_Ort+ q~)Z D(2) m l m (CZ,i~, ~') m
(2.127)
m1
This then leads to a Hamiltonian that can be split up into modified static and timedependent parts
H Lab (t)
= H + H * (t)
(2.128)
Multinuclear Solid-State NMR of Inorganic Materials
60
ZL ZROT~~
io xL~
.....
n ""--~
x Yt~ YROT
X~S
Figure 2.10. The relative orientations of the principal axis system (PAS), rotor (ROT) and laboratory (LAB) frames.
The part which now appears static in the Laboratory frame is
H - Cl_AjToo + ~/-~CI:Aj (3 c~ "
0 - 1 ) A A [(3cos 2 / 3 - 1 ) + r/sin 2/3cos2y]
(2.129)
2
This Hamiltonian is very similar in form to the anisotropic parts of the Hamiltonians found above. Thus, in a powder where [3 and ~/are different for different crystallites, the same broadening effect would occur. However, every crystallite orientation has now acquired a modulation factor 1/2(3cos20 - 1) (= P2(cos0), the second-order Legendre polynomial). If 0 is set to 54 ~ 44' 8", the value of Pz(cos0) is zero and the anisotropic part disappears leaving only the isotropic part as would appear in solution; this is termed the magic angle. It should be emphasised that this is strictly true if only first-order broadening effects are present. An alternative way to envisage the effect of MAS on spatially-dependent Hamiltonians is to consider the dipolar Hamiltonian. For a fixed internuclear vector it is the orientation of this vector that determines the magnitude of the interaction. If the sample is rotated about an axis, for any given internuclear vector its "average" orientation will become the rotation axis (Figure 2.11) which can be set such that
(3C0S 2 0 - 1)- ~-1 (3C0S2 0 m --1)(3 cos 2 1 3 - 1 ) - 0
(2.130)
Physical Background
_
J
61
A>f
2o,.rp.,r
Figure 2.11. Schematic reorientation of a dipolar pair under MAS.
2.3.1.2 Formation of spinning sidebands. The above expression for the effect of MAS has neglected the H(t) part of Eq. 2.128. This is a good approximation if the spinning speed is very high compared to the magnitude of the interaction, or the Hamiltonian is sampled after complete cycles only (termed rotor synchronised acquisition). This is not usually the case so that the full signal produced by the modulation process is obtained. Without losing generality the phase factor + in Eq. 2.127 can be set to zero which gives
H(t)- ./27 CIzAiAA[-cos2~cos20cos(.Ort + sin 2 flsin 20cos2(-Ort ] ~32
L
(2.131)
This time modulation produces spinning sidebands which are signals separated from the isotropic line by integer multiples of the spinning frequency yr. They will extend out to cover a range comparable with the frequency extent of the static anisotropy. The spinning speed necessary to break a static powder pattern up into a set of spinning sidebands must be carefully considered. Sideband formation is discussed in a classic article by Maficq and Waugh (1979). For sample spinning to have any averaging effect, a causality argument can be invoked. If an interaction produces a line that spans a frequency range Av, in the time domain this magnetisation remains coherent for --~1/Av. Any imposed modulation must be occurring faster than this to have any effect. This would apparently limit the applicability of MAS since, even with speeds of 50 kHz becoming available, many lines encountered in solid state NMR are much broader. Fortunately a distinction can be drawn between homogeneous and inhomogeneous interactions. For homogeneous interactions (e.g. strong dipolar coupling between groups of nuclei such as protons) this condition does indeed have to be approached before MAS will have any significant effect since the spins lose complete coherence on the timescale 1lAy. However, for an inhomogeneous line, the transverse magnetisation dephases after a pulse, but this is because the individual spins lose coherence with one another. A 180 ~pulse would refocus this magnetisation and a spin echo would
62
Multinuclear Solid-State NMR of lnorganic Materials
form. This means that the lifetime associated with an individual spin in the transverse plane is quite long, i.e. the intrinsic linewidth associated with any particular spin is small. It is this intrinsic width associated with a given spin that the spinning speed must exceed. In terms of average Hamiltonian theory for an inhomogeneous interaction, the Hamiltonian commutes at different times, so precise averaging will occur at rotation rates small compared to Av. Following an rf pulse, because the tensor orientations of each crystallite are different, the resonant frequency for each crystallite is different and the magnetisation rapidly dephases. This can be envisaged pictorially from the chemical shift interaction. In the static powder pattern the frequency axis could be read as an orientation axis. Then in Figure 2.12 the two sets of spins starting off at A and B have different initial precession rates. The azimuthal phase angle picked up by each of these orientations after a time t is t
(2.132)
qgi(t)- OgoCrisot+ coo H i (t')dt" 0
The time domain signal (ignoring relaxation effects) is then t
gE (t) - - Z exp(i q~i(t)) - exp(i coo~Yisot )exp Z i coi ~ H;(t')dt" i
i
(2.133)
0
f~
"0
TR
Figure 2.12. Schematic variation of the frequency of two spin packets under a chemical shift interaction showing that the frequency varies during the MAS period (TR) and that the same range of frequencies are experienced by the two spin packets.
Physical Background
63
The purely oscillatory nature of H*(t) (Eq. 2.132) means that H*(t) will disappear at times Tr-- P/vr, where P is integer. Thus after one complete revolution the particles will have regained their phase coherence and an echo (a rotational spin-echo) will form. This can be seen since as the particles change orientation due to the spinning their instantaneous precession frequency, which depends on orientation, will change. However, over a complete cycle all spin packets will have experienced the same set of orientations so that at integer multiples of the spinning frequency all spins will have undergone the same average precession governed by the isotropic shielding. Hence in the time domain the signal will be formed by a series of these rotational echoes and it is the number of these rotational echoes, and hence how long the magnetisation lasts in the time domain, that governs the resolution of an MAS NMR spectrum. Spinning sidebands are formed through the accumulated phase with the FID represented by
g(t)- e i~r176176
(2.134)
which can be used to calculate the intensity of the spinning sidebands. The intensities of the spinning sidebands formed provide information about the anisotropic parts of the interactions. Detailed calculations have been made of the sidebands for different interactions, including the chemical shift interaction, by Maricq and Waugh (1979), and Herzfeld and Berger (1980). Nayeem and Yesinowski (1988) have carried out similar calculations for sideband formation in paramagnetic solids.
2.3.2 Magic angle spinning and higher-order effects In many cases it is not sufficient to consider only the first-order broadening terms. It has already been seen that the second-order quadrupole effects often dominate the lineshape of the static spectra of the central transition of non-integer quadrupolar nuclei. It can be anticipated from the more complex angular variation of this interaction that MAS is unlikely to achieve complete averaging of this term. Quadrupole effects often result in more complex spectra than from first-order effects alone. Also, in a system that experiences multiple interactions there is coupling between these interactions and the cross terms of the interactions (the second-order quadrupole interaction can be regarded as the cross term of the quadrupole interaction with itself). These terms can produce both additional isotropic effects and residual anisotropic effects under MAS. Stuart (1994) has dealt in detail with the case of magnetically dilute nuclei that can simultaneously experience the quadrupole effect, shielding anisotropy and heteronuclear spin coupling. There are two specific cases that are more commonly observed
64
Multinuclear Solid-State NMR of Inorganic Materials
than other combinations and will be examined here viz. (i) direct second-order quadrupole effects and (ii) the distorting effects of quadrupole mixing of spin states on spin coupling interactions.
2.3.2.1 MAS of second-order quadrupole effects. The same type of rotations using Wigner rotation matrices can be applied to the second-order effects as for the firstorder interaction which leads to a second-order quadrupole energy of interaction such that
2 Co (l,m)Fo (71)+ II3ZQIc2(I,m)Pz(cosO)F2([3,)',~)+ E~2)lm, m- 1 ) - V--o 21(21-1) C4(I,m)P4(COS 0 ) F 4 (/~, ~',/7)
(2.135)
1 P4(COS0) - ~ (35 cos 4 0 - 3 0 c o s 2 0 + 3)
(2.136)
where
772
F0(r/) - 1 + ~
(2.137)
3
1-y
(cos/3)7/sin2 t3 cos 27'
(2.138)
F4(/3,r, rt,)- 1+ i 8 P4(cos/3)+ 57/ 96 ( 7 c o s 4 f l - 4 c o s 2 f l - 3)cos 2)' + 35rl 2
~(cos4/3 1152
(2.139)
- 4 cos 2/3 + 3) cos 4~'
It is of central importance to realise that this second-order quadrupole broadening has an isotropic part as well as anisotropic terms that are proportional to the second-order and fourth-order Legendre polynomials. The variations of these two functions are sketched in Figure 2.13. It is apparent that there is no single angle which can simultaneously satisfy that P2(cos0) = 0 and P4(cos0) = 0. Spinning around a single axis can
Physical Background
65
T a b l e 2.4. The coefficients Cn(I,m) for second-order
quadrupole effects. I 3/2
M
Co(I,m)
C2(I,m)
C4(I,m)
1/2
2/5 6/5 -- 16/15 - 4/5 20/3 -- 30/15 - 54/15 30/15 294/15 - 48/15 - 108/15 - 60/15 168/15 648/15
- 8/7 0 - 64/21 - 40/7 40/21 - 120/21 - 96/7 - 240/21 168/21 - 192/21 - 168/7 - 600/21 - 336/21 432/21
54/35 - 6/5 144/35 228/35 - 60/7 270/35 606/35 330/35 - 966/35 432/35 1092/35 1140/35 168/35 - 2332/35
30.6 ~
54.7 ~
--
3/2 5/2
1/2
3/2 5/2 7/2
9/2
1/2
3/2 5/2 7/2 1/2 3/2 5/2 7/2 9/2
,... \
1
\ 0.5
X
9 ' x
""
X
I I "[
I"
P4(cosO)\\
I I I
.P2(cosO).
I I
\
70.1 ~
I
I
"'..
I / " I / I/ /
"'.
\1\
" "..
0
Ix I I i
".
/I /"'/ ...
\ \
'/ ~
"-
I
-0.5
I -I
'
0
10
20
30
40
50
60
70
" ""
80
90
0
Figure 2.13. The Legendre functions P2(cos0) and P4(cos0).
c o m p l e t e l y r e m o v e e i t h e r o n e or the o t h e r o f t h e s e t e r m s b u t w i l l t h e n o n l y p a r t i a l l y a v e r a g e the other. T h e a p p r o a c h o f v a r i a b l e a n g l e s p i n n i n g b a l a n c e s this p a r t i a l avera g e to a c h i e v e the b e s t r e s o l u t i o n (see b e l o w ) . B y far the m o s t c o m m o n a p p r o a c h is to spin at the c o n v e n t i o n a l m a g i c a n g l e o f 54.7 ~ as this e n s u r e s that
all
the first-order
effects are r e m o v e d a n d to live w i t h the c o n s e q u e n c e s o f o n l y p a r t i a l l y a v e r a g i n g the P4(cos0) t e r m . T h e n the f r e q u e n c y w i l l be g i v e n b y
Multinuclear Solid-State NMR of lnorganic Materials
66
3[xQ 3[xQ
Q(2), Vm,m+ l -- _
(I(I + 1 ) - 9m(m + 1 ) - 3) 1 +
40v o 1(21-1)
2 (61(1 + 1)- 34m(m + 1)- 13)
80v o 1(21-1)
(2.140)
X [F4 (0~, F])cos 4 ~-k- F2 (a, ~)cos 2 ~--I- Fo (a, F])] where 105 16
35 35,2 8 r / c o s 2 a +~48 ,i cos 2 2 a
(2.141)
F2(a, r/) - - ~45 + 57/2 + 57/cos 2 a - 35r/2 cos 2 2 a 8 12
(2.142)
F4 (a, rl)
9
~+ F~(a' r/) - 16
7/2
5 r/cos 2 a -k-~ 35 7/2 COS2 2 a
.... 3
8
48
(2.143)
For the normally observed central transition then, this frequency in the fast spinning limit is (2)MAS
~'-~
1 [
I(I + 1)-
3XQ
6v o 21(21-1)
F~(a' r/) c~
fl +
(2.144)
F2 (a, q) cos 213+ F4 (a, r/)
where F0 (a, r/) -
~__m21 7 16
8
7
r/cos 2 a + ~ (r/cos2o~) 2 48
F2 ( a, tl ) -- ----9 + ~712 7/COS2 a -- ~(7/COS 7 20;) 2 8
12
(2.145)
(2.146)
24
5 1 7 F4 (a, r / ) - 1---~- ~ r / c o s Z a +-4--~(rlcosZa) 2
(2.147)
The centrebands produced under this limit will also have characteristic lineshapes even though the shapes will be very different from the static ones. The scale of the
67
Physical Background
narrowing can be gauged from the change in ~M2 under MAS which varies from 3.6 ('q = 0) to 2.4 (~q - 1) (Behrens and Schnabel 1981). The ratio of the overall linewidth static to MAS is generally given by the factor 7(1/2 + 2277 + 25) 2(6+//) 2
(2.148)
Again, if the singularities can be observed the quadrupole parameters can be obtained from spectral simulation and the lineshape is a strong function of xl (Figure 2.14). These lineshapes were seen in early calculations of the effect of MAS on the central transition (Kundla et al. 1981, Samoson et al. 1982, Muller 1982). In the case where 11 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 .......
,,
4t)
,
.
20
.
.
.
0
,
-2t) kl-l[z
.2 4;0.... '
20'
I) . . . .
,
,
-40
,
-r
\ -20
'-4'0
-60
8~o.~ ~ Figure 2.14. The MAS centreband of a second-order quadrupole perturbed powder pattern as a function of the asymmetry parameter xl for an I = 3/2 nucleus with XQ = 3.2 MHz at Uo = 95 MHz.
Multinuclear Solid-State NMR of Inorganic Materials
68
the spinning speed is not in the "infinite" speed limit, spinning sidebands appear. There will then be a dependence on the third Euler angle ~/. An interesting and complicating consequence of this is that the lineshape, even from the same transition, will vary according to the order of the spinning sideband. Hence a complex set of spinning sidebands will result which can make interpretation difficult, especially if a sample contains multiple sites with similar chemical shifts. Another consequence of the residual second-order term is that the centre of gravity of the lineshape does not coincide with the isotropic chemical shift. There is an isotropic second-order quadrupolar shift which is given generally by v~2) ( i , m ) _ -
Q,iso
3Z~
40Voi~i~i_ l)2
[i(i+l)_9m(m_l)_3]ll+~_l(2.149)
When only a featureless, usually asymmetric line is observed the peak position is often quoted and this can be significantly different from the isotropic chemical shift. This effect can be so large that a peak is shifted into an isotropic chemical shift range usually associated with another structural unit. There are numerous examples in the literature of mis-assignments based on this effect. Often when a material is not sufficiently crystalline for the singularities to be observed the resulting line is not just simply a featureless symmetric peak but is asymmetric. This asymmetry results from structural disorder producing a range of electric field gradients. Hence, if the sites had very similar isotropic chemical shifts (as expected if the sites are chemically very similar) then the lineshapes start from approximately the same position but the differing values of • stretch the lineshapes to different extents towards lower frequency (Eq. 2.149 is negative for m = 1/2). Hence the range of • stretches the overall lineshape to negative frequency giving the distinctive tail observed in the MAS powder spectra of disordered solids for nuclei such as 23Na and 27A1 (Figure 2.15). The inverse dependence of this effect on the applied magnetic field means that the lines become more symmetric as the magnetic field is increased and the actual peak position becomes closer to the true isotropic chemical shift. It is emphasised that most papers concerning such quadrupole nuclei when a featureless lie is observed quote the peak position. This position is, of course, a characteristic of the peak but must be quoted in the context of the applied magnetic field used since it is not a universal characteristic. Hence for the (1/2, - 1/2) transition the effect of the residual second-order isotropic shift is to produce a shift of the centre of gravity of ~cg, which means the overall position is given by 2'Q ~cg-~c~iso-3F(1)~2 1+ ~' 40 vo
(2.150)
69
Physical Background
A
B
D
i
i
1
'
'i
i
0
'i
i
-1
i
i
-2
i
-3
A Figure 2.15. Effect of a distribution of interactions on the second-order quadrupole powder lineshape of the central transition in detail with the mean interaction XQ = 2 MHz and ~q -- 0 at a Larmor frequency of 80 MHz showing A. no distribution, B. a Gaussian distribution of isotropic chemical shifts with FWHM = 0.17A, C. a Gaussian distribution of the quadrupole interaction of 340A and D. both the chemical shift and quadrupole interactions distributed, (A = 2344Hz). where F(I) is a spin-dependent factor given by Table 2.5. Spin dependent factors of the central transition.
I F(I)
3/2 1/3
5/2 2/25
7/2 5/147
9/2 1/54
The residual second-order effect produces a linewidth which is a function of I and m and then 2
Av(eZ~(i,m)~
ZQ [6I(I+l)-34m(m-1)-13] Voi2(2I-1) 2
For the central transition m = 1/2 a width can be associated with
(2.151)
Multinuclear Solid-State NMR of Inorganic Materials
70
2
2
Av~2) - f (l) ZQ(6 + ~) 224v o
(2.152)
This represents the speed that must be exceeded for MAS to cause even partial averaging of the second-order quadrupolar interaction, but fulfilling this condition does not mean that the fast spinning limit has been achieved. Although it is the central transition that is usually observed, the non-central or satellite transitions (i.e. different values of m) can also be observed. The same situation where MAS removes the Pz(cos0) term, but only partially averages the P4(cos0) term, remains. However it can be noted from Eq. 2.140 that the coefficient is a function of I and m (Samoson 1985). This means that most importantly the residual second-order quadrupole effects under MAS will be different for each transition and each spin. The factors for the relative isotropic chemical shift and residual widths normalised to the central transition are given in Table 2.6. The removal of the second-order quadrupolar width to improve resolution is the most important consideration. The best combinations of I and m for reducing these residual effects are the
(
+ 3 2 9 transitions of I = 2
tranitionsof I
n
~
5 2
nuclei with
and
/ +-
m
nuclei with only 5.6% of the width compared to the central
transition. This dependence was first noted by Samoson (1985) and was then developed extensively by the groups of Jager (see Jiiger 1994 for a summary) and Skibsted (1991), with the technique becoming known as satellite transition spectroscopy (SATRAS). For the above transitions not only are second-order broadening effects reduced, so that resolution is better, but the second-order shift away from the isotropic chemical shift position is smaller. Satellite transition spectroscopy has been most widely reported from I - 5/2 spins because of its applicability to nuclei such as 27A1 and 1 7 0 . The different factors for the different transitions have effects similar to determining the spectrum of thecentral transition at different applied magnetic fields. The m-dependence of the shift for a particular value of I also produces a different net isotropic shift, and this allows the isotropic chemical shift to be determined from a single spectrum. Hence simulation of all transitions provides an internal check on the NMR parameters. For amorphous solids the extent of the sideband manifold can be used to estimate the Table 2.6. The second-order isotropic shifts 8 (Q)isoand residual w i d t h s AV(2)Qin MAS spectra for satellite transitions (I, m) normalised to the central transition for the same spin. x/2, y/2 ~(2)Oiso
Avr
3,3 5,3 5,5 7,3 7,5 7,7 9,3 9,5 9,7 9,9 2.000 - 0.125 - 3.500 0.400 - 1.400 - 4.400 0.625 - 0.500 - 2.375 - 5.000 -0.889 0.292 - 1.833 0.622 -0.511 -2.400 0.764 0.056 - 1.125 -2.778
--
Physical Background
71
AV77
o ,~ I 1 ' 1
+
,~l++ltlN~itlllllill]l~l!lll[[tll)!lll I II ,,t t'1
-
,
,
c
(*10Snz) 8
,
,
4
.....
1'
,
9
0
,
20
.-
0
(*10Snz)
Figure 2.16. Effect of a changing distribution in the main component of the electric field gradientVzz comparing the effect on A. the satellite transition and B. the centreband with a mean quadrupole frequency of 1 MHz and then a distribution of quadrupole frequencies as shown. Only one half of the satellite transition is shown. The Larmor frequency is 104.26 MHz and a spinning speed of 11.5 kHz has been taken. From J~iger et al. 1993 with permission of the copyright owner.
average value of • and the shape of the envelope can be used to estimate the distribution. 27A1 SATRAS has been applied to a range of compounds, including ordered crystalline materials, atomically disordered solids and amorphous solids. A general problem with 27A1 NMR spectra from highly disordered solids is that the centreband peaks tend to be broad and asymmetric, often with extensive overlap of resonances that make the complete lineshape difficult to simulate. However, the inner satellite transition sidebands for I = 5/2 are better resolved and, just as importantly, are significantly more symmetric since the second-order quadrupolar broadening is reduced relative to the chemical shift dispersion broadening, allowing more straightforward deconvolution of the spectra. The effects of distributions on the shape of the centreband and the envelope of the spinning sideband from the inner transition have been calculated (J~iger et al. 1993). The sidebands take on a more monotonic decrease as the distribution becomes broader relative to the mean interaction (Figure 2.16). Even for nuclei where the satellite transitions are broader than the central transition (Table 2.6), if their second-order quadrupolar structure can be observed, they act as an independent check of quadrupolar parameters deduced from the central transition. The larger shift of the satellite transitions for I = 3/2 has been extremely useful in identifying different boron species from 11B NMR of borate glasses.
2.3.2.2 Residual coupling effects due to quadrupolar nuclei in M A S spectra. It was seen that for both dipolar and spin-spin coupling between nuclei the first-order anisotropic effects are proportional to Pz(cos0) so that under MAS if a nucleus experiences both these effects, only the isotropic effects of J remain. However, the
72
Multinuclear Solid-State NMR of lnorganic Materials
case is complicated if one of the coupled spins is quadrupolar. Suppose a spin-l/2 nucleus (I) is being observed which is coupled to a quadrupole nucleus (S). If the quadrupole interaction becomes strong enough, the eigenstates m of the quadrupole spin are not pure Zeeman states and mixing of spin states occurs, leading again to second-order perturbation terms of the Hamiltonian (Bohm et al. 1983, Harris and Olivieri 1992). This leads to a second-order frequency shift of
E
AV~Q2)- 3D'zo_ S ( S + I ) - 3 m 8Vos S ( 2 S - 1)
21
sin20sin/3Icos/3cos~O(3-r/cos2a) ]-r/sinOsin2a
]
(2.153)
This term is anisotropic and produces a powder pattern. It has been derived under the assumptions that first-order perturbation of the S-states is sufficient, that the J tensor is axially symmetric and that the unique axis of J is aligned with the internuclear vector. Under MAS this term will be scaled but, as it is not proportional to P2(cos0), it cannot be completely removed. Hence the MAS spectrum will still have some residual width, but the most profound effect is to leave an isotropic term which can be calculated by averaging the powder lineshape. Hence for a J-coupled system with an axially symmetric quadrupole interaction, the spectrum is shifted from the isotropic chemical shift by: Av- -mJ+
(2.154)
10Vos
S(2S - 1)
Hence the basic multiplet structure remains, although at low field the additional perturbation can strongly disrupt the coupling pattern. The implications of this frequency shift are that for non-integer spin-S the innermost and outermost lines will be shifted in opposite directions. Also the centre of gravity of the entire multiplet pattern is invariant under second-order effects, so that the chemical shift can be estimated from the centre of gravity of the spectrum. For a spin-l/2 nucleus coupled to a non-integer spin quadrupolar nucleus the spacings of the different transitions have been given (see Tables 2 and 3 in Harris and Olivieri 1992). The variation of the spectrum as the parameter ZD' varies is shown schematically in Figure 2.17A. Vos This effect was first reported for 13C-14N(Opella et al. 1979, Frey and Opella 1980, Groombridge et al. 1980) and was apparent as an asymmetric 1:2 doublet. Since that time these residual coupling effects have been observed for a wide range of spin pairs including 119Sn-35'37C1, 1H-14N, 13C-2H, 13C-59C0, 13C-75As, 295i-14N, 29Si-27A1, 3~p_55Mn ' 63,65Cu' 95,97M0" An example of its application is given by compounds of the family [Cu(PPh3)2(4-RC6HaNCN)]2 which have a spin fragment of the type P2CuN2 with the copper in tetrahedral coordination. This provides a cluster of three coupled spins of P2Cu in which the two phosphorus atoms have slightly different
PhysicalBackground
73 Magnetic field (T) total
A
11.7
--~&~-~-,~'~" ,~-#"".~-"~-r- ~sc u ......... ! ..... . ...........
~..!
2000
0
-2000
(Hz)
f
total
w
9.4
-%.
___s- / 1~_s ,~,-u~ , - ~ _ 6SCu 1 ""20~)0....... 6 ...... -2o0o
total ~
~
~
~
~ 2000
~
6
,
S
~
C
0
7.05
63Ci 1
u
-2000
2.11
~
,
total 63Cu
6SCu
"'20'r ...... 6...... -20'00 (nz)
Figure 2,17, A. The effect on a multiplet due to a spin-S/2 nucleus of increasing parameter
(XQD'/voS) with B. an example of 31p multiplets from coupling to copper. Note that there are two inequivalent phosphorus nuclei and that there are 2 copper isotopes (63'65Cu) both with spin-S/2 but with different moments and quadrupole effects that cause distortion. From Hanna et al. (1992) with permission of the American Chemical Society.
chemical shifts. The coupling pattern will be 2 sets of multiplets for the chemically distinct phosphorus (1,2) which are each split into a quartet by the J-coupling to the S = 3/2 copper. An additional complication is that copper has 2 isotopes both with a spin of 3/2 but with different nuclear properties. The scalar couplings will differ by a factor of 0.934 and the quadrupolar perturbation is proportional to the quadrupole moment so the asymmetry will differ by a factor 1.07. The coupling patterns are then weighted by the natural abundances of 69.09% for 63Cu and 30.91% for 65Cu. An example of this is shown in Figure 2.17B for [Cu(PPh3)z(4-PhNCN)]2. Data were recorded at 4 magnetic fields from 2.11 to 11.7 T. From observation of the multiplet coupling the asymmetry can be deduced, which depends on the both the parameter
ZQD"and the relative VoS
orientation of the quadrupolar and coupling tensors.
74
Multinuclear Solid-State NMR of lnorganic Materials
Thus, to obtain definitive information from these measurements requires other input such as a determination of the orientation of the quadrupole tensor relative to the internuclear vector, a knowledge of the internuclear vector or a value of XQ from other determinations. Given these constraints these coupling patterns have provided very useful information about the local structure around metal centres. To calculate these effects average Hamiltonian theory has been applied. There have also been full calculations of these effects, (Menger and Veeman 1982) which supersede average Hamiltonian theory when the Hamiltonian cannot be averaged by MAS. These authors applied an adiabatic variation of the eigenstates. The calculation by Menger and Veeman also showed that the isotropic J-coupling will be affected by the quadrupole interaction but this is a higher order effect and most experiments are carried out at sufficiently high magnetic field that J can be regarded as an invariant. These effects are not common in MAS spectra but are very interesting spin effects that can provide additional structural information. 2.3.2.3 Nonequivalent homonuclear spins. If two homonuclear spins that are not equivalent are coupled, then when other interactions are present the MAS lineshape will depend on the MAS rate (Wu and Wasylishen 1993). In the case of ~SN in cisazobenzene dioxide large changes in the lineshape were observed and simulations showed that the lineshape was dependent on the relative orientation of two chemical shift tensors and their orientation with respect to the internuclear vector.
2.3.3 Variable angle spinning Spinning a sample rapidly around any angle will introduce a scaling of the NMR inhomogeneous lineshape for the central transition. If the quadrupolar interaction is dominant so that effectively only second-order quadrupolar broadening is present in the central transition, there are angles that are more efficient at reducing second-order effects than 54.7 ~ This approach has been termed variable angle spinning (VAS) (Ganapathy et al. 1982). When dipolar and chemical shielding effects can be neglected, VAS can be effective, but as these interactions are not normally negligible, VAS is useful only in a limited range of applications. However, it should be noted that when the second-order perturbation is modulated there is no single angle that can remove both the P2(cos0) and P4(cos0) terms simultaneously. The effect on the lineshape when second-order effects dominate is shown in Figure 2.18. Spinning at 0 ~ produces the same lineshape as the static case. Spinning at the conventional magic angle produces the familiar lineshape, narrowed by comparison with the static case. The total width of the centreband of the central transition has its minimum in the range 60 ~ - 70 ~ depending on xI, the line being approximately twice as narrow as under MAS (Lefebvre et al. 1987, Amoureux et al. 1990). At 43.5 ~ the linewidth is independent of rl. For xl = 0,
75
Physical Background
q -->
0.0
0.3
0.6
N~ 4,
1.0
90
A____A__
___3_2___
- - - - - - - - ~ ----~------ -----.,~
~
70
__h__A___A_ A_ _A___A._ ~ - - - ~
__k._ __/.A_ l,
10
I
I
~
i--
_~
_
9
I
I
,
so
~
A____A
a _ ,
i_
80
40
L_
0
0 -1010 0 -10 10 0 -10 10 0 -10
(v- v,) (kHz) Figure 2.18. Variable angle spinning spectra for different angles as a function of different ~q, from Ganapathy, Schramm and Oldfield (1982) with permission of the copyright owner.
spinning at 36 ~ and 75 ~ gives a line consisting of an extremely intense narrow component with a broader component, so that although the total linewidth is not at its narrowest at this point, in a complex material the resolution of the intense component may be an advantage. This advantage becomes increasingly less evident for higher ~q. Calculations have also been performed for the sidebands of the central transition formed by spinning off the magic angle (Ganapathy et al. 1990).
2.3.4 Double angle spinning Although the quadrupolar information is itself useful, often it is desirable to produce better resolution by complete removal of second-order quadrupolar effects. As spinning about a single axis is unable to remove the P2(cos0) and P4(cos0) effects simultaneously, a more complex time dependence needs to be imposed upon the sample. The most direct solution to this problem is to make the spinning axis a continually varying function of time (Llor and Viflet 1988, Samoson et al. 1988, Zwanziger and Chmelka 1994). A scheme that achieves this uses a spinning rotor (termed the inner rotor) which moves bodily as a function of time by being enclosed in a spinning outer rotor so that the axis of the inner rotor describes a complicated but continuous trajectory as a function of time (Figure 2.19A). This is termed double angle rotation (DOR) and the second-order quadrupolar frequency of the central transition experiences a double modulation in the laboratory flame (similar to the single modulation of MAS). The effect of this modulation can be calculated by using Wigner rotation matrices, but now there are three
Multinuclear Solid-State NMR of Inorganic Materials
76
B ~ z ~ Bo
~ z,o~.~
~_~.~ ~~01 ~
/
Static
-"z,o,
_54,7o 02=30.6~
L
I ....
1 , , ~ , 1 , , , , I , ~ , , I
1 O0
....
0
I,,,,I
....
!
- 1 O0
Frequency(ppm)
Figure 2.19. A. The relative orientations and the motion of the rotors in a DOR experiment. B. The increase in resolution possible for 170 in wollastonite (CaSiO3) comparing static, MAS and DOR spectra from Wu, Sun and Pines (1990) with permission of the copyright owner. consecutive transformations, from the PAS of a given crystallite to the inner rotor axes which are defined by the Euler angles [3 and o~, then from the inner rotor to the outer rotor and, finally, from the outer rotor to the laboratory frame. The angle between the outer rotor axis and the applied magnetic field is 0~ and the angle between the two rotor axes is 02. If the angular velocity of the inner rotor is o~ and +1 describes the azimuthal orientation of the outer rotor in the laboratory, then in the laboratory frame the secondorder quadrupolar Hamiltonian becomes
~ (I,m)Fo(~)
I
+A 2 (I,m)P: (cosP 1)P2 (c~ 02 )F2 (fl, a , 77)
(2.155)
H ~ ) ( t ) - 2 V o ( 2 1 ( 2 1 _ 1)) 2 +A4 (l,m)P4 (c~ 01 )P4 (c~ 02 )/74(fl, a, 77) +terms o~ cos(t~ + ~1 )
where 01 and 02 c a n be chosen so that P2(cos0]) = 0 (01 -- 54.74 ~ and P4(COS02) -- 0 (02 = 30.56 ~ or 70.15~ DOR works well if the quadrupolar interaction is dominant and the sample is highly crystalline, in which case extremely impressive gains in resolution are achieved.
Physical Background
77
Simulation of the complete DOR spectrum (centreband plus the spinning sidebands) will yield the NMR interaction parameters (Sun et al. 1992, Cochon and Amoureux 1993, Amoureux and Cochon 1993). However, it is most usual to perform the experiment to give improved resolution and simply quote the measured peak position which appears at the sum of the isotropic chemical and second-order quadrupole shifts. DOR experiments at more than one applied magnetic field will allow these different contributions to be separated and hence provide an estimate of the quadrupole interaction via the combined quadrupole effect parameter PQ
(2.156)
/72
Po - Z O (1+ T )
This approach is similar to the use of the field variation of the centre of gravity of the MAS centreband, but has the advantage that the narrower, more symmetric line makes determination of the correct position of the centre of gravity more precise. For experiments carried out at two magnetic fields where the Larmor frequencies are vol and Vo2 for the measured DOR peak positions (in ppm) at the two magnetic fields of ~dorl,2 then
(~iso,cs =
(2.157)
V21(~dorl __ V022(~dor2 2 2 V01 -- V02
and I 3(I(I + 1 ) - 3 ) 1 772 4012(21 1)2 Z~(1 + - T ) --
2
2
Vo1Vo2((~d~ --(~dor2)2
(2.158)
V01 -- V02
This means that measurements at multiple magnetic fields will constrain the interactions. Detailed examples given in Chapter 6.
2.3.5 Multiple quantum transitions It has been pointed out that the central (1/2,-- 1/2) transition does not experience any first-order quadrupole interaction. The absence of first-order broadening effects is a general property of symmetric ( m , - m) transitions. There are cases where this can be a distinct advantage, the most direct instance being for integer spin nuclei (e.g. 2D and 14N, both I = 1) where there is no (1/2,- 1/2) transition. The main problem is to excite and detect such higher-order transitions, for which there are two separate approaches. The sample may either be irradiated and detected at the multiple quantum frequency (called overtone spectroscopy) or the MQ transition can be excited arid a 2D sequence used to detect the effect on the observable magnetisation.
78
Multinuclear Solid-State NMR of Inorganic Materials
Overtone spectroscopy developed for 14N irradiates the sample at approximately twice the Larmor frequency (Tycko and Opella 1987). If the quadrupole interaction is sufficiently large that second-order quadrupole effects are significant, the ( - 1 <--->1) transition becomes weakly allowed. In powders the spectrum is still structured, allowing the interactions to be deduced, but is narrowed by a factor of 8Vo/XQ. Symmetric transitions remove first-order quadrupole effects and the coefficients of the anisotropic terms of the second-order broadening depend on I and m. Amoureux (1993) pointed out that for the 3Q transition of I = 3/2 spins the coefficient of the P2(cos0) term is zero. Hence spinning at either 30.56 ~ or 70.12 ~ will remove the P4(cos0) term. The other first-order interactions still have the P2(cos0) variation so that spinning away from 54.7 ~ will only partially scale these. Comparing the widths of these interactions at the roots of P4(cos0) for the MQ transition with the static width of the (1/2, - 1/2) transition shows a scaling of 1.84 at 30.56 ~ and 0.98 at 70.12~ hence the latter is usually preferred. The other combination where there may be an advantage is the m = 7/2 transition for I = 9/2. Although calculations have been carried out there are no experimental data using this approach in the literature to date.
2.3.6 Ultrasonically-induced narrowing Narrowing of NMR lines has been observed in colloidal suspensions of ultrafine particles, presumably by Brownian motion. A suggestion related to this principle is to take small solid particles into a liquid medium and induce sufficient reorientational motion of these particles to produce narrowing (Satoh and Kimura 1990). Ultrasound induces translational motion and collisions of the particles produce rotation. This sonically induced narrowing has been observed for 27A1 in aluminium sulphate, with the advantage that no spinning sidebands were observed (Homer et al. 1991). The experiment depends on the frequency and the power of the ultrasound and the liquid medium. Although the first reports of this approach appeared in 1989 there have been few developments in this field.
2.4. DIPOLAR DECOUPLING
2.4.1 Heteronuclear dipolar decoupling Heteronuclear dipolar coupling can be a major broadening mechanism, especially as in a number of materials protons are present which often have dipolar couplings to other nuclei in excess of 50 kHz. This means that MAS is usually not sufficient to remove the effect of the dipolar coupling in the spectra. ~3C NMR spectra are, for instance, often obscured by dipolar coupling to the protons. In order to remove the carbon-proton coupling the protons are irradiated with an rf field while, at the same time, the carbon spectrum is measured. Consider the CH system where the heteronuclear dipolar
Physical Background
79
interaction is proportional to the factor Sz(carbon)Iz(proton). Initially, the proton magnetisation will lie parallel to the z-axis (Iz). The rf field rotates the proton magnetisation to the xy-plane and then to t h e - z-axis so that the proton magnetisation is then represented b y - Iz. Hence the spin part of the Hamiltonian is manipulated. Representing this schematically, the proton spin will sequentially vary as Iz --a -Iz --~ Iz ~ -Iz ~ Iz ~ -Iz. If this rotation is rapid compared to the timescale of the experiment, the time-average of Iz will tend to zero. Hence the heteronuclear dipolar Hamiltonian oscillates between + IzSz so that the average of the spin operator will be zero. This is an example of averaging interactions by manipulating the spin part of the Hamiltonian. In order for this to work, the rf field strength has to be greater than the strength of the dipolar interaction. This technique is often used in combination with MAS, resulting in almost liquid-like resolution for solids. In recent years there has been a significant amount of theoretical and experimental work on schemes to improve on standard CW decoupling, and some of these are discussed in Chapter 3.
2.4.2 Homonuclear dipolar decoupling Solid-state spectra of many materials containing protons (and fluorine) are usually dominated by the homonuclear proton-proton coupling (or fluorine-fluorine, and in cases where the phosphorus density is high, phosphorus-phosphorus). This is especially true when the nuclei are physically close together so that r is small, making the dipolar coupling strong. The proton spectra are usually featureless due to the fact that protons have a high natural abundance and a high gyromagnetic ratio causing even distant protons (--~10.~) to contribute to the broadening of the spectrum. To remove homonuclear dipolar coupling the lifetime of the magnetisation must be prolonged in the transverse plane (Powles and Mansfield 1962). Several multiple pulse schemes (Mansfield 1971) have been developed to accomplish this, including WHH4 (Waugh et al. 1968), MREV-8 (Mansfield et al. 1973, Rhim et al. 1974), BR24 (Burum and Rhim 1979) and BLEW-12 (Caravatti et al. 1982, 1983). These sequences involve long trains of pulses that cause coherent motion of the magnetisation. A distinction must be drawn as to whether the sequence is being used simply for decoupling and the heteronucleus is being observed, or whether the same nucleus is being observed. In cases where observations of the same nucleus are being made, a so called "windowed" sequence is necessary so that the magnetisation can be recorded. These sequences are quite unlike all those encountered to date as the multiple pulse sequences sample the magnetisation stroboscopically. This means that there are certain times within the pulse sequence, which often involve repeated short cycles where the magnetisation is in the correct state. At each point a single data point is recorded. Just as using the rotating frame oftensimplifies the understanding of many NMR experiments, to understand these multiple pulse sequences it
80
Multinuclear Solid-State NMR of Inorganic Materials
is easier to observe the magnetisation in a flame in which it remains stationary so that the flame's motion compensates for the effect of the rf pulses. In the literature this is termed the toggling flame. Note that to properly understand such pulse sequences, second-averaging and Magnus expansion techniques should be applied (Gerstein and Dybowski 1985). Consider the simplest of these sequences, WHH4, where the cycle has four pulses and five time-intervals as shown in Figure 2.20. If the dipolar coupling is dominant then, in the toggling flame, the magnetisation is rotated successively to different directions, the average dipolar interaction being "" tl
)-
(3Ilxl2x
--/-1" ~/2 )'k-
(3Ilyl2y - 11" [2 )+ (311zl2z - 1-1"1-2)
(2.159)
The factor of 1/3 arises from the fact that the spin spends an equal amount of time along each of the Cartesian directions. It is clear from this expression that the sequence successfully removes the homonuclear dipolar coupling. However, it is important to realise that such manipulation by pulses has the effect of scaling other operators and not completely removing them. The most important is Iz, its average value for the above sequence being
(1 eft) -- 89 x + Iy + lz) ]
(2.160)
This means that there is an effective field along the unit vector 1L/3(1,1,1) direction in the Cartesian basis, producing an effective scaling of the frequency of 1/~/3. This basic sequence was developed into longer sequences such as MREV-8 and BR-24 (Table 2.7).
90~
Magnetisation along
Z
90~
Y
90%
x
90~
Y
z
Figure 2.20. The periods of a WHH4 multipulse experiment for narrowing homonuclear dipolar coupling.
Physical Background
81
Table 2.7. Comparison of some of the more commonly used homonuclear decoupling sequences with a normal one pulse experiment. Sequence
Pulse cycles (phase)
Cycle time
Scale factor for Izeff
One pulse
x
"r
1
WHH-4
x- yy- x
6'r
MREV-8
x - yy - x - x - yyx
12'r
BR-24
xy-y-x -xy-yx yx-x-y -yx-xy xy-x-y -yx-xy
36'r
45
2
As the cycles become longer they have better self-compensation and second-averaging of the interactions which improves the resolution of the spectrum, and they are more robust to frequency offset effects. The downside is that the cycle time increases markedly. The causality argument applied to MAS can also be invoked here in that the cycle time which is doing the averaging must be fast compared with the timescale of the modulation of the magnetisation caused by the interaction to be averaged. Hence this condition is much more easily met when the sequence has a shorter cycle time. When the number of pulses increases, the only way to decrease the cycle time is to decrease the n- spacing, which is governed by the rate of recovery of the receiver system after a pulse and the speed of the electronics. The 90 ~ pulse time can also be reduced which is a stringent test of the power handling capabilities of the probe. When it is simply decoupling that is required (as opposed to observation as well) an off-resonance proton field can be applied such that the proton magnetisation is spin-locked at the magic angle. This is termed Lee-Goldburg decoupling (Lee and Goldburg 1965). A variant of this sequence currently favoured is Frequency Switched Lee-Goldburg decoupling (FSLG) (Bielecki et al. 1990). The ability of NMR hardware to abruptly change the frequency while keeping the phase has meant that this approach is now possible. This approach does not require the use of hard, resonant 90 ~ pulses and is therefore relatively more efficient at decoupling I-spins over a greater frequency. Such considerations have become more important even for 1H as very high magnetic fields are applied. The basic principle of the Lee-Goldburg decoupling concept is that it causes coherent precession of the magnetisation about an effective field inclined to Bo. With B1 (say) along the x-axis, this effective field will be inclined at 0 to Bo, defined by
82
Multinuclear Solid-State NMR of Inorganic Materials
tanO =
BI Bo
~o,
(2.161)
If the effective field is directed along the magic angle (i.e. directed along the direction (111)), 0 = 54.7 ~ This will remove the dipolar coupling since the effective spin direction is inclined at the magic angle to Bo. The offset frequency ALG = tOo - tO1.Rather than continuously irradiate at this single frequency, it has been found to be much more efficient to flip the frequency by _+ ALG with precession for a time "rm corresponding to 2-rr. Between each period the phase is also successfully changed by ~r. The FSLG decoupling has been shown to be one of the most effective decoupling schemes. It will also scale other spin operators, with a factor of (~/3)-1 for the chemical shift. FSLG has been used for the observation of abundant spins as an alternative to the multiple pulse sequences based on hard 90 ~ pulses (Levitt et al. 1993). These sequences have the benefit of short cycle times and high scaling factors but are sensitive to rf homogeneity. These pulse sequences completely remove only the homonuclear dipolar coupling, which means that interactions such as shielding are scaled but still present. The anisotropy can, of course, provide useful information but to achieve the best resolution this should also be removed. This is readily achieved by MAS and the combination of multiple pulse and MAS averaging is termed CRAMPS (Combined Rotation And Multiple Pulse Sequence) (Ryan et al. 1980, Jackson and Harris 1988, Maciel et al. 1990). When such multiple pulse sequences and MAS are combined care must be taken to ensure that the independent averaging processes do not interfere with each another. Hence it has been assumed that there should be at least five multiple pulse cycles per rotor period for this to be approximately true, so that modest spinning speeds (i.e. --~2 kHz) were typically used. In strongly dipolar coupled proton systems CRAMPS gives much better resolution than MAS (Figure 2.21). For Lee-Goldburg decoupling, similar time constraints are necessary so that 2"rLC should be short compared to the period of sample rotation. More recently, consideration has been given to combining multiple pulse sequences with fast MAS. The timing of the fast MAS with a sequence means that symmetry of the sequences allows their combination. The sequences proposed have largely been windowless so that one data point is accumulated at the end of the complete sequence and the sequence is then incremented (Hafner and Spiess 1998, Filip et al. 1993). If indirect detection is being used, as in many 2D sequences, this is not a drawback. The technique has been developed still further to use semiwindowless decoupling so that at points in the sequence quasi-static conditions apply, even under fast MAS. These approaches may offer new impetus to high resolution proton spectroscopy even in systems with a high proton density.
83
Physical Background
\
15
10
5
0
-5
\
PPM
/ \
/ \
r \
/ \
/ \
/ ~
/ /
-~~
~
A
10.7 kHz
j
~
9.2 kI-Iz
5.0 kHz
40
20
0
-20
-40
kHz
Figure 2.21. Example of ~H spectrum comparing static, MAS and CRAMPS from citric acid from Dec et al. (1989) with permission of the copyright owner.
2.5. SPIN-LOCKING A useful concept is the creation of spin-locked transverse magnetisation after a pulse, with the phase of the rf then flipped by 90 ~ This means that in the rotating frame after the phase shift of the rf field, the transverse magnetisation and the B1 field are aligned. The transverse magnetisation will not then dephase under interactions such as chemical shift and dipolar coupling, as would normally be the case. The transverse magnetisation will decay by genuine relaxation processes in the rotating frame, characterised by the relaxation time TI~. In this situation the magnetisation is said to be spin-locked, since the transverse magnetisation lasts much longer than under normal T2 dephasing. For quadrupole nuclei the situation is more complex and it is well known that
84
Multinuclear Solid-State NMR of Inorganic Materials
rotationally induced spin-locked magnetisation can appear when an if-field is applied to a quadrupolar nucleus in conjunction with MAS. The spin-locking behaviour of quadrupolar nuclei on resonance can be described by considering the first-order quadrupolar interaction only, so the Hamiltonian in the rotating frame will be given by the first-order quadrupolar interaction (Eq. 2.111) plus the rf contribution VlIx. Diagonalising this Hamiltonian gives the eigenfunctions (Vega 1981, 1992), which, in the case of the quadrupolar interaction being much larger than the if-field for an I = 3/2 spin, are the single (c+) and triple (t_) coherences, defined as
ic+) = [ ~ ) + - I - ~ ) and It+)= 13/2)+1-3/2) -
(2.162)
-
The energy level diagram for these eigenstates is shown in Figure 2.22 which demonstrates that the eigenvalues change smoothly as the magnitude of the quadrupole interaction changes relative to the rf field. When IVQI > > Vl the states are spin-locked. However, when VQ is comparable to v~, c+ and t+ are no longer the eigenstates but linear combinations are (Vega 1992). In a spinning sample the orientation of the quadrupole nucleus changes continuously. Hence, the splitting between the energy levels which depends on the crystallite orientation through the anisotropic term will vary, and level crossings between the different spin levels will occur (Vega 1992a, Grey et al. 1993, Grey and Vega 1995). During one rotor period, spin density is transferred from the outermost spin levels to a spin-locked state and back again, continuing as the rotation continues. Depending on the orientation of the quadrupole tensor with respect to the spinning axis, either two or four level crossings occur per rotor period. There are three spin-lock regimes depending on the so-called adiabaticity parameter = v~2/VQVr (Vega 1992). For e~ > > 1 the level crossings are adiabatic and spin density will be transferred from one level to another. For e~ < < 1 the change is sudden and no transfer of population occurs, and in this case the normal spin-lock behaviour is expected. In the intermediate regime the transfer of spin density is not efficient but still occurs, giving rise to very short effective spin-lock times. The situation is more complex when there are additional levels for higher spins, but the same reasoning can be used. For 27A1 when • -- 2 MHz, spinning at 5 kHz and an rf field-strength of 60 kHz gives oL = 2.4, which means it is in the adiabatic regime. So for effective spin-locking the system should be well into the adiabatic or sudden regime. The rotationallyinduced level crossings adversely affect the performance of various NMR experiments such as cross-polarisation (Sec. 2.6) and nutation NMR (Sec. 3.6.1). It also lies behind the TRAPDOR effect (Sec. 3.8.3). A more detailed and thorough discussion of MAS and spin-locking of quadrupole nuclei has been provided by Vega (1992a).
Physical Background
85
Spin I=3/2 _
~Eigenvalue (v0 c+
4
t+
Spin I=5/2
Figure 2.22. The variation in the energy level eigenvalues of a quadrupole nucleus as a function of the ratio VQ/V~which is changed by sample rotation since VQ is a function of orientation. The c+ and t+ states are as given in Eq. 2.162 and the diagrams follow ideas from Vega (1992a), and Grey and Vega (1995).
2.6. C R O S S - P O L A R I S A T I O N
Even with the line-narrowing techniques described earlier, NMR experiments on solids with dilute spin-1/2 nuclei are still relatively unattractive on two principal counts. One is the lack of sensitivity due to their low net polarisation and the other is the relatively long spin-lattice relaxation time that is often encountered. In solids where both abundant (I) and dilute (S) nuclei coexist, polarisation transfer techniques can usually be used to overcome both these problems. There are many schemes to effect such a transfer but the most common technique is to create and then spin-lock transverse I-magnetisation. This experiment is best understood using ideas from spin thermodynamics. The magnetisation is given by Curie's Law (Eq. 2.21) and the temperature in
86
Multinuclear Solid-State NMR of Inorganic Materials IH Channel 90, ~ i |
(Spin-lock)y Decoupling
,, i
X Channel F1D
CP
H-spins
X-spins
Lab
Rotating Frame
Bo ~ ~
Bm Blx Hartmann-Hahn matched
Lab Be
C
,
0
~
5
....
, ....
, ....
, ....
10 15 20 25 30 35
msec
Figure 2.23. A. Schematic representation of the CP sequence for ~H --~ X. B. the changes of the energy levels from the laboratory frame and in the rotating frame showing the Hartmann-Hahn match (separations are not to scale). C. The build-up of magnetisation for the 13C signals in the 2 carbons of glycine as a function of contact times with an inset of short contact times showing the effect of different Tis for each carbon.
this equation is the thermodynamic lattice temperature. The pulse sequence has three main steps (see Figure 2.23A), which in terms of the spin thermodynamics are" 1. Cooling down the abundant spin system, 2. Contact between the I and S spins to allow polarisation transfer, 3. Observation of the dilute spins.
Physical Background
87
The transverse I-magnetisation is created and then spin-locked. In the spin-locking frame (which is equivalent to examining the magnetisation in the rotating frame as in Eqs. 2.36 and 2.37) the effective field is only B1. In the spin-locking frame, immediately after the 90 ~ pulse, the I-magnetisation is still MI with the same degree of order, but the energy levels are now much closer (i.e. ~lhB1 as opposed to ~/h Bo). With the new field B1 the system can now be assigned with an effective spin temperature Tpl defined by M1=
NlyZh2I(I + 1)Bo
-
N1YZh2I(I + 1)BlI
(2.163)
As B li < < Bo and the degree of order amongst the spins remains the same (i.e. M~ is constant) it follows that Bli
Tpl - -~o Tc
(2.164)
Hence the I-spins are effectively very cold. The S-spins, in contrast, start off with no transverse magnetisation so, in terms of thermodynamics, are very hot. There is a thermodynamic driving force for the transfer of magnetisation. However, the spin systems have to be allowed to communicate efficiently and this is achieved by applying a second B1 field, this time to the S-spins. If the two spins to be brought in contact are spin-1/2 then the condition the two fields must meet are the Hartmann-Hahn condition (Hartmann and Hahn 1962) given by (Figure 2.23B). ~/IBII -
(2.165)
YsBls
When this term is present the dipole flip-flop terms (e.g. I+S_) are energy conserving so that order can be transferred between the I and the S spins. Thermodynamics means that the transfer of order occurs, tending to give the two systems a common spin temperature. Magnetic energy is conserved, given by
E - N?2h2I(I + 1)B2 3kT
(2.166)
Conservation of total energy amongst the spin system from these flip-flop terms can be invoked on a timescale that is less than the spin-lattice relaxation times so that
N, yZh:I(I + 1)B2I
NiY~h2I(I + 1)B2I + Nsy2h2S(S + 1)B2s
3k 1
3k 2
(2.167)
Using the Hartmann-Hahn condition and rearranging leads to a new temperature of
Multinuclear Solid-State NMR of lnorganic Materials
88
(2.168)
The implication of this equation is that the spin temperature for the I-spin system changes very little, so that I-magnetisation essentially remains unchanged as a result of the contact, and the S-magnetisation created is
Ms=
NsT2h2S(S + l) T1 1 NsT~h2S(S + I)B o = 3kTp2 ~"s 1 + A 3kTL
(2.169)
where A
n
NsS(S+I) NII(I+I)
(2.170) ~I I
Hence in a single contact experiment the gain in signal intensity is approximately
7s compared to a single 90 ~ pulse on the S-spins (which for I - 1H and S - 29Si produces a factor of--~ 5). This produces transverse S-magnetisation. Although this describes the driving mechanism for the transfer of the magnetisation between I and S, it gives no clue as to the dynamics of the process. Mehring has carried out an extensive analysis of the CP process (Mehring 1983), leading to
M ce(t) - )'i Mo exp -~/~ 3 - e x p Ts (P+-P-)
(2.171)
Tis )]
where
,+ [l+I/1 ]
(2.172)
with
x--
1/ 1+a+~7-+~-3--
2
T~p
T~p
andy-
1+
T~p~,
+
s
T~So) Tip
(2.173)
where ot is a factor depending on the number of nuclei of each type in contact with one another. It has been shown that for many CP experiments where there are many more I spins than S and when Tis < < TS~p, the following dynamics approximately hold
Physical Background
exp -
89
-exp -
M ce (t) - MCoe
(2.174) 1
Tip
In this equation T~s determines the rate of polarisation transfer and hence the build up of signal, while Tlo is the relaxation time of the spin-locked I-magnetisation in the rotating frame determining the time scale of the decay of the reservoir of magnetisation. The rate of transfer of the magnetisation is found to depend on the second moments (M2) of the I-S and I-I spin systems as
1
M ,s
Tzs = Czs ~/MzII
(2.175)
C~s is a constant. In cases where the CP occurs between relatively isolated spins (e.g. pairs) the spin thermodynamic description cannot be applied. Rather than a smooth, continuous build-up of magnetisation, the CP curve shows structure, termed dipolar oscillations (Muller et al. 1974). The magnetisation oscillates between the proton and the X-nucleus. In the case where the system is exactly Hartmann-Hahn matched, the magnetisation of the S-spin is given by (Levitt et al. 1986)
M s ( t ) - Mso ~I [I_cos(_P2(cosO)t)],
(2.176)
where Mso is the equilibrium Zeeman magnetisation of the S-spin system. An example is shown in Figure 6.7 for 1H-170 from Mg(OH)x(OCH3)z-x allowing the distance between these nuclei to be determined. As the contact time increases, destructive interference occurs and the homonuclear proton dipolar coupling begins to dominate the CP curve. There is also an interesting contrast between Mg(OH)x(OCH3)z-x and Mg(OH)2 where the more dilute proton bath in the former leads to weaker proton-proton coupling and hence more pronounced oscillations. The above analysis is usually applied to spin-1/2 systems but there is no reason why magnetisation transfer cannot be applied to quadrupole systems. If XQ is small the theory is identical to that given above, but if • becomes significant the HartmannHahn condition is modified to
aiTiBli = aS~tsB1s
(2.177)
90
Multinuclear Solid-State N M R o f Inorganic Materials
where oL- 1 for spin-l/2 systems and a - ~ / [ S ( S + 1 ) - m ( m - 1 ) ] for the ( m , m - 1 ) transition of a spin-S quadrupole nucleus. The enhancement of the signal depends on the heat capacities of the two spin systems, which depend in turn on the gyromagnetic ratios of the two spins and the number of spins in contact. Thus, the CP magnetisation (Mcr,) is related to the normal Zeeman magnetisation (Woessner 1987, Walter et al. 1988) Mcp = M o
7t N s [S(S + 1)- m ( m - 1)]1 NI
I(I + 1)
J
(2.178)
All of the above remains only strictly true for a static sample, but CP is usually combined with MAS. If hvr < < HII, the effect of sample spinning on the cross-polarisation process may be ignored. Once MAS begins efficiently to average the I-I dipolar coupling, CP may be weakened as magnetisation can no longer diffuse freely through the I-spin reservoir. This often leads to modulation in the Hartmann-Hahn-match condition across the spectrum causing distortions in the spectrum. This effect has become more noticeable as faster-spinning probes have been used to overcome problems caused by spinning sidebands arising from CSA as ever higher applied magnetic fields become available. Quantification is often required, and, in CP, signal generation depends on the heteronuclear dipolar coupling which determines the CP rate via Tis. At slow MAS rates a broad matching condition is usually obtained around the central Hartmann-Hahn condition and any mismatch is relatively unimportant since the strong homonuclear dipolar coupling compensates. As the spinning speed increases, the matching profile breaks up into a series of narrowing matching bands separated by yr. The CP rate at the central matching condition is slower than at some of the sidebands (Metz et al. 1994, 1996). The ability to maintain efficient matching under fast MAS has attracted much attention. Sequences have been developed based on either phase and/or amplitude modulation. Ramping the amplitude across one of the matching sidebands with the ramp centred on one of the matching sidebands has been found to improve greatly the CP efficiency and broaden the matching condition. This approach is straightforward to implement on modern spectrometers. An example of the improvements brought by this approach is that of N-t-Boc-alanine (Metz et al. 1996), where there are several different carbons with different dipolar couplings, together with the influence of different match conditions (Figure 2.24).
2.7. TWO-DIMENSIONAL METHODS
Most of the methods described to this point have simply excited single quantum magnetisation and manipulated the Hamiltonian by making it time-dependent. They are all
Physical Background
91
90~ CP
L
Decoupling
,,...f
SACP Centreband
B
SACP Sideband
RAMP-CP Sideband
i~l
e ~
0
1---
-
9
0
-
-
9
-
9
9
9
9
- ,
. . . .
,
~
10
1
,
0
.
.
.
.
.
.
.
,
9
-
9
9
9
9
9
9
9
,
~
1
10
0
10
Contact Time (ms) Figure 2.24. A. One of the pulse sequences for modulating the Hartmann-Hahn match condition by ramping the X-nucleus transmitter amplitude. B. Comparison of 1H --+ 13C CP for conventional CP (SACP) matched on the centreband and a sideband with RAMP-CP on a sideband of N-t-Boc-alanine showing the very much improved quantitative data for the six equally populated carbon sites. Taken from Metz, Ziliox and Smith (1996) with permission of the copyright owners. essentially one-dimensional techniques, the only time domain associated with the experiment being that of the normal FID. A whole range of further experiments becomes available when a second time domain is introduced. There are many experiments that use this concept of two-dimensional (2D) NMR; some of those more commonly used are described in Section 3.2. The concept of the experiment is that the Hamiltonian is manipulated in some way so that the system is prepared in a particular state. This state is then allowed to evolve for a time (tl) and the system is interrogated by applying a pulse that creates observable magnetisation. The FID signal accumulated in this time is designated t2 (Figure 2.25, Ernst et al. 1988). On the basis of one experiment nothing can really be said about the effect of the evolution in tl. However, a whole series of FIDs are accumulated in which the time tl is varied. Fourier transforming this set of FIDs will produce spectra which are modulated in some way by the evolution of the system during the preceding tl-period, and a second FT will produce a second frequency dimension containing information about this modulation, and hence about the interaction causing it. This 2D concept is important in some of the approaches to be described next.
92
Multinuclear Solid-State NMR of Inorganic Materials tvl
t2 v
Evolution Interrogationi
Figure 2.25. Time intervals in a basic two-dimensional pulse sequence.
2.7.1 Dynamic angle spinning Dynamic Angle Spinning (DAS) is a 2D experiment in which the sample is spun about different axes sequentially during different periods (Eastman et al. 1992, Zwanziger and Chmelka 1994). During the first evolution time tl the sample is spun at an angle of 01 degrees. The magnetisation is then stored along the z-axis and the angle of the spinning axis is changed to 02. After the rotor has stabilised (--~ tens of milliseconds) the magnetisation is then brought into the xy-plane again and a signal is acquired. The secondorder quadrupole frequency of an individual crystallite depends on the angle of the spinning axis, so during tl the quadrupole interaction will produce a frequency in the crystallite of v~, and 1)2 during t2. If v2 is of opposite sign to vl the signal from the crystallite will be at its starting position again at some time later during t2. The angles can be chosen in such a way that the signals from each individual crystallite will be at the starting position at exactly the same time so that an echo forms with the secondorder quadrupolar broadening removed at the peak of the echo. The two angles should fulfil the following equations simultaneously P2(cos 01 ) = -kP 2(cos 02 ) and P4 (cos 01 ) --
-ke 4(cos 0 2 )
(2.179)
where k is the scaling factor. There is a continuous set of solutions for 01 and 02, the so-called DAS complementary angles, and each set has a different scaling factor. For these solutions, the second-order quadrupole powder pattern at 01 is exactly the scaled mirror image of the pattern at 02, and an echo will form at t2 = ktl. For the combination 01 = 30.56 ~ 02 = 70.12 ~ the P4(cos0) terms are zero and the scaling factor k = 1.87. For the combination 01 = 37.38 ~ 02 = 79.19 ~ the scaling factor k = 1, the spectra are exact mirror images and an echo will form at t l = t2. Finally, the combination 01 = 0 ~ 02 = 63.43 ~ (k = 5) is special because it allows efficient CP for signal enhancement prior to the DAS sequence. There are several ways in which the DAS spectra can be acquired; one can acquire the entire echo, which means that the resulting 2D spectrum will be sheared. Some additional processing is then required to obtain an isotropic spectrum in F1. The advantage of acquiring the entire echo is that the second-order information is retained. Hence the 2D data set has improved the resolution whilst providing secondorder quadrupolar information as well. Alternatively, the acquisition could also start at
Physical Background
93
the position of the echo so that an isotropic spectrum in F1 is obtained directly. This has the disadvantage that the S/N ratio is less than if the complete echo is acquired. A third possibility is to carry out the experiment as a pseudo-1D experiment where only the top of the echo is acquired as a function of t l. In this case the isotropic spectrum is acquired directly but there is no shortening of the duration of the experiment. It is, however, the best way to combine DAS with 2D heteronuclear correlation. The DAS sidebands can be analysed to deduce interactions (Sun et al. 1992).
2.7.2 2D MQMAS From the DAS experiment it is known that in different time intervals the sample is spun at different angles so there is a point (where the echo forms) where anisotropic effects are refocused. Also it is known that there are no first-order quadrupole effects in symmetric transitions. Using these ideas, in 1995 a new experiment emerged from the group of Frydman that has had an enormous impact on solid state NMR spectroscopy of noninteger quadrupolar nuclei (Frydman and Harwood 1995, Medek et al. 1995). The 2D Multiple Quantum Magic Angle Spinning (2D MQMAS) experiment greatly enhances resolution of the spectra of half-integer spin quadrupolar nuclei. Under MAS for a symmetric transition, as the P2(cos0) term is removed the frequency evolves under the quadrupole interaction as
I
(2.180)
2I(2I-1)
+ v o [C4(i,m)P4(54.7O)F4(fl,)/,r D
where F4(~,'y,'q) is defined in Eq. 2.139. The experiment correlates ( m , - m) (the multiple quantum ( p - 2m) transition) to the central (~/2,-1/2) transition. The resolution enhancement stems from the fact that the quadrupole frequencies for both transitions are correlated. It can be seen from the coefficients in Table 2.8 that both the isotropic and anisotropic parts show different time evolution for the different coherences. The experiment is shown schematically in Figure 2.4, together with the coherence level diagram. The 3Q experiment has so far been the most common since, of the higher coherence transitions, 3Q is the easiest to generate. Initially the 3Q transition is generated and the Hamiltonian evolves for a period h. Another pulse converts the 3Q signal to an observable ( - 1) single quantum coherence (in Figure 2.4 a three pulse sequence where the coherence changes is 0 ~ +_ 3 --~ 0 ~ - 1) so that at t2 = ktl the anisotropic F2 term in Eq. 2.180 is refocused when
k - C2(I'3/2) C2(I,1/2)
(2.181)
94
Multinuclear Solid-State NMR of Inorganic Materials Table 2.8. Coefficients for the second-order quadrupolar interaction and the ratio for the echo periods. I
3/2
m
p
Co
C2
C4
k
1/2
1 3 1 3 5 1 3 5 7 1 3 5 7 9
3 - 9 8 6 - 50 15 27 - 15 - 147 24 54 30 - 84 - 324
- 12 0 - 32 - 60 20 - 60 - 144 - 120 168 -- 96 - 252 - 300 - 168 216
- 27 21 - 72 - 114 150 - 135 - 303 - 165 483 -- 216 - 546 - 570 - 84 116
NA - 7/9 NA 19/12 - 25/12 NA 101/45 11/9 - 161/45 NA 91/36 95/36 7/18 - 31/6
3/2 5/2
7/2
9/2
1/2 3/2 5/2 l/2 3/2 5/2 7/2
1/2 3/2 5/2 7/2 9/2
NA - not applicable.
At this point an e c h o forms but the coefficients of the isotropic terms are different and so are not r e f o c u s e d at this point. This has i m p o r t a n t c o n s e q u e n c e s for the e x p e r i m e n t . T h e isotropic part VoQ of the q u a d r u p o l e f r e q u e n c y is
vQ - - _
(2.182)
,~'~(3 if-/7 2 )
4012 (21 - 1)2V0 and the anisotropic part vaQ([3,~/) is g i v e n by
7
4
i - ~ ( 3 - ~ c o s 2 ~ ' ) 2 sin fl 9Z~ 44812 (21 - 1)2Vo
/7 2
+ 2 ( r / c o s 2 y - 2 - --~-) sin 2 fl
(2.183)
2//2
4 +-45 5
As the 2D e x p e r i m e n t is carried out at different e v o l u t i o n times the echo m a x i m u m will always be free from anisotropy but will s h o w m o d u l a t i o n due to isotropic effects. A double F T will then contain both isotropic and anisotropic information. After a 2D F T the signals s h o w up as ridges lying along the Q u a d r u p o l e A n i s o t r o p y (QA) axis w h i c h has a gradient d e p e n d i n g on k (Figure 2.26). T h e isotropic s p e c t r u m can be obtained by projection of the entire 2D s p e c t r u m on a line t h r o u g h the origin p e r p e n d i c u l a r to
95
Physical Background Site
1
2
3
vl ~ p
~ F1
Co(P) Co(l) -'"
ganiso--"
C4(P )
C2(1---~ F2
Figure 2.26. A schematic 2D MQ MAS data set of a sample containing 3 sites- 1 (large XQ, no distribution), 2 (moderate XQbut with some distribution) and 3 (very small XQ and no distribution). 3 lies on line Vl = PP2, 2 and 3 displaced from this line along a line with gradient giso,Qand 3 showing a lineshape along a line with gradient ganiso.
the QA-axis (vide infra). Cross-sections along the ridge will retain the second-order quadrupolar lineshape. The sign of k is varied for different coherence orders and has important consequences for the way the experiment will be carried out. If the value is negative, anisotropic broadening is refocused at a positive value of t2 and the correlation must be between - p and - 1 (the observable coherence). However, for positive k, to get refocusing at positive t2 the correlation has to be between the + p and - 1 coherences. A scheme that gives refocusing at positive t2 values is termed an echo pathway whereas refocusing at negative values is an antiecho pathway. Under MAS the terms involving P2(cos0) (for both the quadrupole and chemical shift) and P4(cos0) are refocused along the ridge but there are isotropic terms. The isotropic parts of both the chemical shift Aviso and the isotropic second-order quadrupole effects v Q, are scaled. These scale factors (SF) are for the chemical shift
SFcs(i,m)_
p+ k
l+lkl
(2.184)
96
Multinuclear Solid-State NMR of lnorganic Materials
and for the quadrupolar term
C~
-k
SFQ(I,m) - CO(I,- 1/2)
(2.185)
l+lkl The MQ experiment involves excitation of higher order coherences and selection of particular pathways by phase cycling. These effects lead to considerable reduction in intensity compared to normal single quantum experiments. Hence, much attention must be paid to efficient generation of the MQ coherences and the efficient reconversion of the MQ signal to observable single quantum magnetisation. If the quadrupole interaction is significant, single pulse excitation is a comparatively efficient way of generating MQ coherences. For samples where there is no quadrupole interaction there can be no MQ signal generated, and these sites are missing from the MQ spectra. There have been a number of investigations to optimise these experiments. Generation of MQ transitions depends upon both the ratio r -- VQ/Vl and the flip angle of the pulse. The optimum pulse tip angles are almost independent of the ratio r but depend on the coherence order to be maximised. However, the efficiency of the generation of the MQ transition is a peaked function (Figure. 2.27), having an optimum value of r. The key is that with typical • values there is usually a gain from using higher rf fields. This is increasingly true as the order of the coherence to be generated increases. This has meant that most work has concentrated on the lower coherence orders. Most of the calculations have assumed a static sample but MAS imposes a time dependence on the Hamiltonian. As spinning speeds increase the generation of the MQ coherence decreases, but even at the highest speed available (--~45 kHz) the intensity is reduced by a factor of only --~3 and the optimum pulse angle is not much changed. Numerical calculations have been carried out on the effect of the angle on the conversion efficiency. The work of Amoureux and Fernandez (1998) is summarised in Table 2.9. One of the key observations is that the efficiency of the flip angles shows no real dependence on r. However, the conversion is quite an inefficient process so that the overall sensitivity of the experiment is poor relative to one-pulse experiments and much effort has gone into improving the excitation and reconversion efficiency. As I increases, the optimum flip angle for maximum excitation decreases. Since both excitation and conversion are dependent on r, MQ spectra are normally not directly quantitative, so great care has to be taken to interpret the spectra. The situation will generally improve as the applied rf field increases. Since the 2D data set contains the anisotropic information, lineshapes can be analysed and the quadrupolar parameters deduced. The lineshape will show distortion as there will be crystallites whose orientation is such that their quadrupolar frequency is zero and will not contribute. Apart from the excitation and the conversion, the actual coherence pathway
97
Physical Background 60./'~3a i\ '
-
40
20"
3Q
/=\
$
/ /i i '............ . ' ~ '"~'" '~\" ''\
/;. ,o7~ \:\
0"~
'
I
'
I
,
! ~
,I
'
I
30-
20
""
5Q
i~,- ,t2
.. ,,..
~aS!s
/
20
il\i
"~'~
10-
fl
._n
-4
-2
0
2
4
6
VQffS(S+I)-3/~4/V1 =2" F i g u r e 2.27. The overall efficiency of a 2-pulse MQ pulse sequence for different excitation (3Q, 5Q, 7Q, 9Q) for different spins with the quadrupole frequency scaled by the spin factor to allow direct comparison of the different spins with the optimum pulse angle as given in Table 2.9, after Amoureux and Fernandez (1998).
Table 2.9. Numerically calculated optimum flip angles and the relative efficiency of the sequences after Amoureux and Fernandez (1998) for VQ/Vl = 1.25. I
3/2 5/2 7/2 9/2
Pulse
1 2 1 2 1 2 1 2
Ipl = 3
Ipl = 5
Ipl = 7
Ipl = 9
0~
Eft
0~
Eft
0~
Eft
0~
Eft
240 55 180 60 120 45 90 35
1.50 0.30 1.50 0.30 1.50 0.30 1.50 0.26
170 72 120 65 100 50
0.32 0.05 0.49 0.05 0.64 0.07
220 80 150 72
0.21 0.03 0.29 0.03
250 86
0.14 0.02
The relative efficiency (Eft) is one for the central transition observedafter a perfectly selective90~pulse.
98
Multinuclear Solid-State N M R of Inorganic Materials
will affect the lineshape, so, for example, the coherence pathway 0 --~ p --~ - 1 will produce an echo but there will be absorptive and dispersive components. Ideally the sequence 0 --~ _+ p --> - 1, (especially if both pathways are equal) will markedly improve the lineshape. For I = 3/2, echo and anti-echo pathways can be equalised but this is not possible for higher spins. Triple quantum experiments are the most popular because they are most readily generated, but with higher spins there is the possibility of generating higher order coherences. The sensitivity of such experiments decreases rapidly for the higher order coherences. There has been much work to improve the efficiency of such experiments. Instead of single pulses both composite pulses (Marinelli et al. 1998) and shaped pulses (Ding and McDowell 1997) have been investigated. The group of Kentgens has used amplitude modulation of the rf field which amounts to a sweep of the frequency (DFS) and causes an adiabatic transfer between coherences. Through DFS good enhancement of the signal and excitation of an improved range of XQ were obtained (Kentgens and Verhagen 1999, Iuga et al. 2000, Schiifer et al. 2001, Iuga and Kentgens 2001). A more readily implemented scheme is the fast amplitude-modulated conversion pulses of Madhu et al. (1999, 2000) (FAM) which again produce coherence transfer through adiabatic passages. Wu et al. (1996) showed that as for spin-locking, rotation of the samples produces interconversion between 1Q and 3Q coherences, this approach being termed RIACT. The conversion of the single quantum to triple quantum coherence by the first long pulse and its reconversion by the second pulse depends on two main factors. The situation is most closely described for I = 3/2. High power rf and relatively slow MAS are required to ensure the conditions are as close as possible to adiabatic. Under adiabatic conditions, for spins to contribute, the system must undergo an odd number of passages or zero-crossings during the period of the spin-locking pulse. An adiabatic zero-crossing will the convert central-transition coherence (1Q) to 3Q, but after either two or four passages, the system will return to its initial state. The RIACT(II) sequence in which the excitation and conversion of the coherences is achieved by a spin-lock of duration Tr/4 instead of a hard pulse, has the advantage of more uniform excitation and conversion, which is essential in obtaining quantitative information. Schemes for sensitivity enhancement based on the selective inversion of the satellite transitions have been proposed (Haase et al. 1994). This can be combined into the MQ sequence as has been successfully demonstrated for 23Na and 87Rb (Gan 2000, Yao et al. 2000).
2.8. NMR RELAXATION
2.8.1 Introduction to relaxation
The spectroscopy described above is effectively the static part of the interaction, but there are dynamical processes that determine how rapidly the spins lose coherence and
99
Physical Background
how rapidly thermodynamic equilibrium is achieved between the spin states. This aspect of N M R is termed relaxation. Relaxation can provide information on the processes that cause transitions. First, a description of the process is necessary and a classical approach is to adopt the phenomenological B loch equations for an ensemble of spins in an external magnetic field. This model describes the behaviour of the magnetisation through the torque as in Eq. 2.3. Parallel to the main magnetic field the time dependence of the magnetisation is given by (2.186)
dMz = M~ - Mz dt
T1
where T1 is the longitudinal or spin-lattice relaxation time. If the sample is placed in the Bo field and initially has no magnetisation, the magnetisation along the direction of the magnetic field builds up as
exp/;//
(2.187)
The transverse components of the magnetisation decay as dM x = -M x dt
(2.188)
dMy _ -My
T2
dt
T2
where T2 is the transverse or spin-spin relaxation time. Then, in the laboratory frame the overall equation of motion of the magnetisation in an applied magnetic field B will be given by dM
---=-- = ?'M_x B dt
-
M x i + Myj
- -
M~ - M o k
T2
T1
(2.189)
-
which can be rewritten in the rotating frame as
dM
_ ) ' M x B_eff
MX'i + My,j
(2.190)
Mz' - M~ k
dt
T1 CO
In the rotating frame at v - Vo if there is no rf field present then Beff= B + -o = 0 and the B loch equations can be written as -o ), dM~, =
dt
M X,
r2'
dMy______.~=, _ My________[, dMz._____~,= M o - M~,
dt
dt
The solutions for the transverse (x',y') magnetisation are
T,
(2.191)
100
Multinuclear Solid-State NMR of Inorganic Materials M,(t)- M:,,(O)e-t/7"2 and My,(t)- My,(O)e-t/T2
(2.192)
Hence any magnetisation in the x'y'-plane (and xy) decays exponentially to zero. The transverse relaxation time is a result of the dephasing of the magnetisation in the xy-plane of the magnetic moments precessing about Bo. In principle, if each magnetic moment of the same nuclear species precessed with the same Larmor frequency about Bo no dephasing would occur. However, the moments do not experience the same magnetic field due to interactions and inhomogeneities in the static magnetic field Bo. The result of these effects is that each magnetic moment precesses with a different frequency which will result in the dephasing of the magnetisation in the xy-plane. These processes can be coherent or incoherent so that the T2* seen through the decay of the magnetisation in the FID after a pulse can be regarded as being made up of different contributions and written as 1
1
1
---r = - - - t - ~
(2.193)
where T2 is the decay of the magnetisation due to the interactions, and T'2 is due to the the effect of inhomogeneities in the field. The real physical interest lies in T2, since it can provide information about the interactions and the nature of the motion that modulates the interactions and causes relaxation. Spin echo techniques allow T2 to be measured rather than T2*. In this technique a sequence of two rf pulses is used as shown in Figure 2.28. The first pulse is a 90 ~ pulse and following this the individual nuclear moments spread out around the transverse plane due to their different precession rates. After a time "r a second pulse of 180 ~ is applied. All the spins are thus flipped about B1 so that those moments which had been precessing faster than average prior to --~2
~
3 ----~ 4
J
i ' v
AA~
'
5
L,
I I" 5 ,~..___m
Figure 2.28. The basic spin-echo pulse sequence and effect on spin packets showing the formation of a spin-echo.
Physical Background
101
application of the second pulse, and had thus got ahead of the other moments now find themselves, after the second pulse, behind the others. They are, however, still precessing faster than average and so, after a further time "r, they catch up with the slower moments. The same argument applies in reverse to the slower moments, the nett effect being to restore phase coherence to the precession at a time 2"r after the first pulse, thus forming a spin echo. The echo shape itself decays away with the characteristic time Y2* as the moments continue to precess at different rates. During the time 2T irreversible T2 processes will have been going on so that the re-phased magnetisation is less than Mo. Consequently the echo amplitude is reduced and varies with 2"r according to the B loch equation (Mx,(0) = Mz,(O) = O)
dMy, _
My,
My, (2z')- My,(O)e -2"c/T2
dt
(2.194)
By measuring the amplitude of the signal as a function of the echo time 2-r, T2 may be determined. This method assumes that each nucleus remains in the same field for the duration 2"r of the experiment. If, however, the nuclei move into a different field e.g. through diffusion, the echo amplitude will be further reduced because the precession frequency of the individual nuclei will have changed during the echo as they move from a region with one particular value of the magnetic field to another region. The signal amplitude at t = 2-r is now
2"c Mxy(2"c) = M ~ ( 0 ) e x p ( - T 2
2y2G2D'c 3 ) 3
(2.195)
where G is the magnetic field gradient in the z-direction and D the diffusion coefficient. To measure T2 in the presence of diffusion in an inhomogeneous field the Carr-Purcell spin-echo technique (a train of 180 ~pulses) can be applied. Carrying out the experiment with the echo spacing constant means that field gradient effects are much less significant than in a experiment with varying "r delays, but the disadvantage is that each 180 ~ pulse has to be very accurate. Diffusion measurements can also be made by deliberately applying a pulsed field gradient during both -r periods of a spin echo, and measuring the signal as a function of the gradient strength.
2.8.2 Mechanism for relaxation processes Transitions between Zeeman energy levels, and hence nuclear magnetic relaxation, are caused by fluctuations (time variations) in the local interactions at the nucleus that can cause transitions. The transition rates between these energy levels that cause nuclear spin relaxation depend on two factors: Rate of relaxation - (Strength of interaction) • (Number of fluctuations at Vo)
(2.196)
Multinuclear Solid-State NMR of lnorganic Materials
102
The fluctuations are often caused by atomic motion e.g. Brownian motion in liquids, ionic hopping, molecular rotations, librations and atomic vibrations. These motions are often complex and it is the range of frequencies that are present in the motions that determine relaxation. The spectral density function describes the relative intensities of different frequencies in the motions and can be used to calculate relaxation rates. Let y(t) be some function of time, such as the orientation of the internuclear vector and some other function f (e.g. the dipolar interaction) that depends on y. It is possible to define a probability function p(y,t) which is the probability that at a time t, the internuclear vector has some orientation y. Then (2.197)
f (t) - ~ p(y,t)f (y)dy
This assumes that y(t) is a stochastic function (i.e. y(t) varies completely randomly) but often y(t) shows some correlation. It is useful to define the probability function P(y~,t~;y2,t2) which means that if y = y~ at t = t~ this value is the probability p of y = Y2 at t -- t2. Then the probability that y = Y2 at t -- t2 and y = y~ at t = t~ and y = y~ at t -- t~ is p(y~,t~)P(y~,t~;y2,t2). The correlation function G(t~,t2) is
G(tl ,t2 ) - f (tl ) f * (t2) - ~ P(Yl ,tl )P(Yl ,tl ;Y2,t2 ) f (Yl ) f * (Y2)dYldY2
(2.198)
Since G is the correlation of the function with itself it is termed the auto-correlation function of f(y). If the function y(t) is stationary then it is invariant to changes of the time origin and G depends only on the difference, a" -- t2 - t~, so that G(r)
~ P(Y~ )P(Y~,Y2, v)f(y~ ) f * (Y2)dyldy2
(2.199)
The spectral density function is defined to be the temporal Fourier transform of G('r) so that ~x~
oo
--~
0
J(m) represents the "strength" (or energy) of fluctuations in f(y) at the frequency m and, as an example, the dipole interaction can be considered using the alphabet expression of Eq. 2.52. In Eq. 2.52 the terms A, B, C, D, E and F can be considered as having the form (Geometric term i.e. depends on the polar angles (0,~) fly)) • (Spin-operator term). The spectral density functions are
J(P)(m)- f f where
Y2p(Oa, ~a ) 3
J~,
Y2p(Ob'g~ 1~,3
,r b , m) dr_ dr b
(2.201)
Physical B a c k g r o u n d
103
oo
P(r~a ' ~ b , co) - FT[P(r~a ' r_~, r)] - I P(ra'~ rb , oo)e -iarcdr
(2.202)
--oo
with Y2p(Oi, (~i) the spherical harmonic functions. As an example for homonuclear dipolar coupling
1
-~1
l
3
3,2
-
-~
= -~'4h2I(I
(//0) 2
[jO) (CO~ )+ j(2) (2r ~)],
(2.203)
[J(~ (2(.01) + 10J(1) ((.0o) + J(2) (2(.0o)],
(2.204)
h 2 I ( I + 1) G
+ 1)
(/./O ~2 \4~J
and
l = 3 /4}~2i(I -t-1)(1"/o ~2[J(~ T2 8 \4,cJ
- 10j(1)((.Oo)--kJ(2)(2(.0o)].
(2.205)
The functional form of J(m) in turn depends on the form of P(G, G, ~) which will depend on the details of the atomic motion. The function P can often either be derived analytically or can be calculated numerically. The key is to understand that these expressions show that the different relaxation times probe different frequencies. The spectral density function is a measure of the distribution of fluctuations. For relaxation to be efficient there needs to be significant spectral density at the frequencies in Eqs. 2.203-2.205. Often a simplified functional form can be assumed, of which the most common is the BPP approximation (Bloembergen, Purcell and Pound 1948). This model assumes that the autocorrelation function is exponential
exp( /
206,
where % is a correlation time and is related to the jump rate. Then J(P) ((.o) - 2 G (p) (0)
vc
1 + (o)~"C)2
(2.207)
The relaxation rates are dominated by the behaviour of the spectral density function at frequencies related to the spectral densities at ~o (T1), o~1 (T~p) and 0 (T2). The correlation time % follows an Arrhenius-type activation so that r c - r o exp I Ea ]
(2.208)
104
Multinuclear Solid-State NMR of lnorganic Materials
where Ea is an activation energy for the motion and % is the inverse of the jump attempt frequency. If an experiment is carried out at constant Bo (and hence too) while the temperature (T) is changed then the form of the spectral density will change. This is shown schematically in Figure 2.29 so that as the temperature is increased the spectral density is spread out over a wider frequency range. There is a temperature where there will be a maximum amount of spectral density at COo. The correlation time can be represented by
Ea
ln(r C) - ln(~') + ~ kT
(2.209)
Therefore, a plot of In(relaxation time) against T - 1 gives a characteristic curve (Figure 2.30) with the position of T1 being minimum at ~Oo'rc - 1 since this is the maximum of j(1) (COo). Similarly, the minimum of Tlo occurs at tol"rc = 1/2. The slopes at high and low temperatures for T1 can be used to estimate Ea of the motion since
j(~)' Tx '
Tv
COO
O~
Figure 2.29. Schematic representation of the spectral density functions J(~o) at three temperatures where Tx < Tu < Tz.
\
/B01
Rigid Lattice Limit -T2
1/T Figure 2.30. Plots of the relaxation times In T1, T~p and T2 as a function of inverse temperature T-1
Physical Background
105
/
ln(T1) ~ in 1 + 0) 2 2 Tc
(2.210)
then
T --->,,o :cor << 1 : ln(T1) ~ : - I n ( z ) ~:
-E a
(2.211)
kT
and T ---->0"coy >> 1" ln(T1) ~: -ln(CO2T) ,~ ln(CO2) + E. kT
(2.212)
This symmetry is lost if there is a range of activation energies, as in a glassy material (Brinkman 1992). Hence relaxation measurements can provide significant insight into atomic scale motion. T2 has the same slope at high temperature as T1, but at low temperature the J(0) term leads to a constant value termed the rigid lattice limit. The use of relaxation time measurements can have important effects on constraining local motion in materials (Strange 1987, Brinkmann 1992).
REFERENCES Amoureux, J.P., Fernandez, C. & Lefebvre, F. (1990) Mag. Resort. Chem., 28, 5. Amoureux, J.P. (1993) Solid State Nucl. Mag. Reson., 2 83. Amoureux, J.P. & Cochon, E. (1993) Solid State Nucl. Mag. Reson., 2, 223. Amoureux, J.P. & Fernandez, C. (1998) Solid State Nucl. Mag. Reson., 10, 211. Andrew, E.R. (1971) Progress Nucl. Mag. Reson. Spectrosc., 8, 1. Andrew, E.R. (1981) Int. Rev. Phys. Chem., 1, 195. Behrens, H.S. & Schnabel, B. (1981) Physica B, 114, 185. Bielecki, A., Kolbert, A.C., de Groot, H.J.M., Griffin, R.G. & Levitt, M.H. (1990) Advances in Mag. Reson., 14, 111. Blaha, P., Schwartz, K. & Dederichs, P.H. (1988) Phys. Rev. B, 37, 2792. Blaha, P., Schwartz, K. & Luitz, J. (1999) Technical University of Vienna, Austria, ISBN 3-95010031-0-4. Bloembergen, N., Purcell, E.M. & Pound, R.V. (1948) Phys. Rev., 73, 679. Brinkmann, D. (1992) Progress Nucl. Mag. Reson. Spectrosc., 24, 527. Bohm, J., Fenkze, D. & Pfeifer, H. (1983) J. Mag. Reson., 55, 197. Burum, D.P. & Rhim, W.-K. (1979) J. Chem. Phys., 71,944. Caravatti, P., Bodenhausen, G. & Ernst, R.R. (1982) Chem. Phys. Lett., 89, 363. Caravatti, P., Braunschweiler., L. & Ernst, R.R. (1983) Chem. Phys. Lett., 100, 305. Clayden, N.J., Dobson, C.M. & Fern, A. (1989) J. Chem. Soc., Dalton Trans., 843.
106
Multinuclear Solid-State NMR of Inorganic Materials
Cochon, E. & Amoureux, J.P. (1993) Solid State Nucl. Mag. Reson., 2, 205. Cohen, M.H. & Reif, F. (1957) Solid State Phys., 5, 321. Dec, S.F., Bronnimann, C.E., Wind, R.A. & Maciel, G.E. (1989) J. Mag. Reson., 82, 454. Dickinson, W.C. (1950) Phys. Rev., 77, 736. Ding, S. & McDowell, C.A. (1997) Chem. Phys. Lett., 270, 81. Eastman, M.A., Grandinetti, P.J., Lee, Y.K. & Pines, A. (1992) J. Mag. Reson., 98, 333. Ernst, R.R., Bodenhausen, G. & Wokaun, A. (1988) Principles of Nuclear Magnetic Resonance in One and Two Dimensions, Oxford University Press, Oxford. Facelli, J.C. (1996) Encyclopedia ofNMR, Eds. Grant D.M. & Harris, R.K., John Wiley and Sons, Chichester, p. 2576. Farrar, T.C. (1990) Concepts in Mag. Reson., 2, 1. Farrar, T.C. (1990a) Concepts in Mag. Reson., 2, 55. Fenkze, D., Freude, D., Frohlich, T. & Haase, J. (1984) Chem. Phys. Lett., 111, 171. Fernandez, C. & Amoureux, J.P. (1998) Solid State Nucl. Mag. Reson., 10, 211. Filip, C., Hafner, S., Schnell, I., Demco, D.E. & Spiess, H.W. (1999) J. Chem. Phys., 110,423. Frey, M.H. & Opella, S.J.J. Chem. Soc., Chem. Commun., 474. Frydman, L. & Harwood, J.S. (1995) J. Amer. Chem. Soc., 117, 5367. Gan, Z. (2000) J. Amer. Chem. Soc., 122, 3242. Ganapathy, S., Schramm, S. & Oldfield, E. (1982) J. Chem. Phys., 77, 4360. Ganapathy, S., Shore, J. & Oldfield, E. (1990) Chem. Phys. Lett., 169, 301 Geurts, F.M.M., Kentgens, A.P.M. & Veeman, W.S. (1985) Chem. Phys. Lett., 120, 206. Grant, D.M. (1996) Encyclopedia ofNMR, Eds Grant D.M. & Harris, R.K., John Wiley and Sons, Chichester, p. 1298. Grey, C.P., Dobson, C.M., Cheetham, A.K. & Jakeman, R.J.B. (1989) J. Amer. Chem. Soc., 111,505. Grey, C.P., Smith, M.E., Cheetham, A.K., Dobson, C.M. & Dupree, R. (1990) J. Amer. Chem. Soc., 112, 4670. Grey, C.P., Veeman, W.S. & Vega, A.J. (1993)J. Chem. Phys., 98, 7711. Grey, C.P. & Vega, A.J. (1995) J. Amer. Chem. Soc., 117, 8232. Groombridge, C.L., Harris, R.K., Packer, K.J., Say, B.J. & Tanner, S.F. (1980) J. Chem. Soc., Chem. Commun., 174. Haase, J., Conradi, M.S., Grey, C.P. & Vega, A.J. (1994) J. Mag. Resort. A, 109, 90. Haeberlen, U. (1976) High Resolution NMR in Solids: Selective Averaging, Advances in Magnetic Resonance, Academic Press, New York. Hafner, S. & Spiess, H.W. (1998) Concepts in Mag. Reson., 10, 99. Hahn, E.L. (1950) Phys. Rev., 80, 580. Hanna, J.V., Smith, M.E., Stuart, S.N. & Healy, P.C. (1992) J. Phys. Chem., 96, 7560. Harris, R.K. (1983) Nuclear Magnetic Resonance Spectroscopy, Pitman Books Limited, London. Harris, R.K. & Olivieri, A.C. (1992) Progress Nucl. Mag. Reson. Spectrosc., 24, 435. Hartmann, S.R. & Hahn, E.L. (1962) Phys. Rev., 128, 1042. Herzfeld, J. & Berger, A.E. (1980)J. Chem. Phys., 73, 6021. Homer, J., McKeown, P., McWhinnie, W.R., Patel, S.U. & Tilstone, G.J. (1991) J. Chem. Soc., Faraday Trans., 87, 2253.
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Iuga, D., Sch~ifer, H., Verhagen, R. & Kentgens, A.P.M. (2000) J. Mag. Reson., 147, 192. Iuga, D. & Kentgens, A.P.M. (2001) Chem. Phys. Lett., 343, 556. Jackson, P. & Harris, R.K. (1988) Mag. Reson. in Chem., 26, 1003. J~iger, C., Kunath, G., Losso, P. & Scheler, G. (1993) Solid State Nucl. Mag. Reson., 2, 73. J~iger, C. (1994) inNMR Basic Principles and Progress, vol. 31, Eds. Blumich, B. & Kosfeld, R., Springer-Verlag, Berlin, p. 135. Jameson, C.J. (1989) in Multinuclear NMR, Ed. Mason, J., Plenum Press, New York, p. 89. Keeler, J. (1990) in Multinuclear Magnetic Resonance in Liquids and Solids - Chemical Applications, vol. 322, Eds. Granger, P. & Harris, R.K., NATO ASI, p. 103. Kentgens, A.P.M. & Verhagen, R. (1999) Chem. Phys. Lett., 300, 435. Knight, W.D. (1956) Solid State Phys., 2, 93. Kundla, E., Samoson, A. & Lippmaa, E. (1981) Chem. Phys. Lett., 83, 229. Lee, M. & Goldburg, W.I. (1965) Phys. Rev. A, 140, 1261. Lefebvre, F., Amoureux, J.P., Fernandez, C. & Derouane, E.G. (1987) J. Chem. Phys., 86, 6070. Levitt, M.H., Suter, D. & Ernst, R.R. (1986) J. Chem. Phys., 84, 4243. Levitt, M.H., Kolbert, A.C., Bielecki, A. & Ruben, D.J. (1993) Solid State Nucl. Mag. Reson., 2, 151. Levitt, M.H. (1997) J. Mag. Reson., 126, 164. Llor, A. & Virlet, J. (1988) Chem. Phys. Lett., 152, 248. Maciel, G.E., Bronnimann, C.E. & Hawkins, B.L. (1990) Advances in Mag.Reson., 14, 125 Madhu, P.K., Goldburt, A., Frydman, L. & Vega, S. (1999) Chem. Phys. Lett., 307, 41. Madhu, P.K., Goldburt, A., Frydman, L. & Vega, S. (2000) J. Chem. Phys., 112, 2377. Man, P.P. (2000) in Encyclopedia of Analytical Chemistry, Ed. Meyers, R.A., John Wiley and Sons, Chichester, p. 12224. Man, P.P., Klinowski, Trokiner, A., Zanni, H. & Papon, P. (1988) Chem. Phys. Lett., 151, 143. Mansfield, P. (1971) Progress Nucl. Mag. Reson. Spectrosc., 8, 41. Mansfield, P., Orchard, M.J., Stalker, D.C. & Richards, K.H.B. (1973) Phys. Rev. B, 7, 90. Maricq, M.M. & Waugh, J.S. (1979) J. Chem. Phys., 70, 3300. Marinelli, L., Medek, A. & Frydman, L. (1998) J. Mag. Reson. A, 132, 88. Mason, J. (1993) Solid State Nucl. Mag. Reson., 2, 285. Medek, A., Harwood, J.S. & Frydman, L. (1995) J. Amer. Chem. Soc., 117, 12779. Mehring, M. (1983) Principles of High Resolution NMR in Solids, Springer-Verlag, Berlin. Menger, E.M. & Veeman, W.S. (1982) J. Mag. Reson., 46, 257. Metz, G., Wu, X. & Smith, S.O. (1994)J. Mag. Reson. A, 110, 219. Metz, G., Ziliox, M. & Smith, S.O. (1996) Solid State Nucl. Mag. Resort., 7, 155. Mtiller, D. (1982) Annales der Physike, 39, 451. Mfiller, L., Kumar, A., Baumann, T. & Ernst, R.R. (1974) Phys. Rev. Lett., 32, 1402 Munowitz, M. (1988) Coherence and NMR, John Wiley and Sons, New York. Nayeem, A. & Yesinowski, J.P. (1988) J. Chem. Phys., 89, 4600. Neue, G., Dybowski, C., Smith, M.L., Hepp, M.A. & Perry, D.L. (1996) Solid State Nucl. Mag. Reson., 6, 241, Opella, S.J., Frey, M.H. & Cross, T.A. (1979) J. Amer. Chem. Soc., 101, 5856. Pake, G.E. (1948) J. Chem. Phys., 16, 327. Powles, J.G. & Mansfield, P. (1962) Phys. Lett., 2, 58
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Powles, J.G. & Strange, J.H. (1963) Proc. Phys. Soc., 82, 6. Proctor, W.G. & Yu, F.C. (1950) Phys. Rev., 77, 717. Rhim, W.-K., Elleman, D.D. & Vaughan, R.W. (1973) J. Chem. Phys., 58, 1772. Ryan, L.M., Taylor, R.E., Paff, A.J. & Gerstein, B.C. (1980) J. Chem. Phys., 72, 508. Samoson, A., Kundla, E. & Lippmaa, E. (1982) J. Mag. Reson., 49, 350. Samoson, A. & Lippmaa, E. (1983) Phys. Rev. B, 28, 6567. Samoson, A. (1985) Chem. Phys. Lett., 119, 29. Samoson, A., Lippmaa E. & Pines, A. (1988) Molecular Phys., 65, 1013. Sanders, J.C.P. & Schrobilgen, G.J. (1990) in Multinuclear Magnetic Resonance in Liquids and Solids - Chemical Applications, vol. 322, Eds. Granger, P. & Harris, R.K., NATO ASI, p. 157. Sato, N. & Kimura, K. (1990) J. Amer. Chem. Soc., 112, 4688. Sch~ifer, H., Iuga, D., Verhagen, R. & Kentgens, A.P.M. (2001) J. Chem. Phys., 114, 3073. Schmidt, V.H. (1971) Proceedings of the Ampere International Summer School//, p. 75. Schmidt-Rohr, K. & Spiess, H.W. (1994) Multidimensional Solid State NMR and Polymers, Academic Press, London. Skibsted, J., Nielsen, N.C., BildsCe, H. & Jakobsen, H.J. (1991) J. Mag. Reson., 95, 88. Slichter, C.P. (1990) Principles of Magnetic Resonance, Springer-Verlag, Berlin. Sternheimer, R.M. (1954) Phys. Rev., 95, 738. Strange, J.H. (1987) Cryst. Lattice Defects and Amorphous Mater., 14, 183. Stuart, S.N. (1994) Solid State Nucl. Mag. Reson., 3, 199. Sun, B.Q., Baltisberger, J.H., Wu, Y., Samoson, A. & Pines, A. (1992) Solid State Nucl. Mag. Reson. 1,267. Townes, C.H., Herring, C. & Knight, W.D. (1950) Phys. Rev., 77, 852 Tycko, R. & Opella, S.J. (1987) J. Chem. Phys., 86, 1761. Vega, A.J. (1992) J. Mag. Reson., 96, 50. Vega, A.J. (1992a) Solid State Nucl. Mag. Reson., 1, 17. Vega, S. (1981) Phys. Rev. A, 23, 3152. Walter, T.H., Turner, G.L. & Oldfield, E. (1988) J. Mag. Reson., 76, 106. Waugh, J.S., Huber, L.M. & Haeberlen, U. (1968) Phys. Rev. Lett., 20, 180. Woessner, D.E. (1987) Z. Phys. Chem. Neue Folge, 152, 309. Wu, G. & Wasylishen R.E. (1993) J. Mag. Reson. A, 102, 183. Wu, G., Rovnyak, D., Sun, B. & Griffin, R.G. (1995) Chem. Phys. Lett., 249, 210. Wu, G., Rovnyak, D. & Griffin, R.G. (1996) J. Amer. Chem. Soc., 118, 9326. Wu, Y., Sun, B.Q. & Pines (1990) J. Mag. Reson., 89, 297. Yao, Z., Kwak, H.-T., Sakellariou, D., Emsley, L. & Grandinetti, P. (2000) Chem. Phys. Lett., 327, 85. Yesinowski, J.P., Eckert, H. & Rossman, G.R. (1988) J. Amer. Chem. Soc., 110, 1367. Zwanziger, J.W. & Chmelka, B.F. (1994) in NMR Basic Principles and Progress, vol. 31, Eds. Blumich, B. & Kosfeld, R., Springer-Verlag, Berlin, p. 202.
Chapter 3
Experimental Approaches Basic Experimental Principles of FT NMR Instrumentation 3.2.1 Overview of a Pulsed FT NMR Spectrometer 3.2.2 Magnets 3.2.3 Shimming 3.2.4 Transmitters 3.2.5 Probes 3.2.6 Connection of the Probe 3.2.7 Signal Detection 3.2.8 Additional Equipment 3.3. Practical Acquisition of NMR Spectra 3.3.1 Processing the FID to Produce a Spectrum 3.3.1.1 Window Functions 3.3.1.2 Shifting of the Time Origin and Linear BackPrediction 3.3.1.3 Zero Filling 3.3.1.4 Phase Correction 3.3.1.5 Baseline Correction 3.3.2 Complications in Recording Spectra 3.4. Static Broad Line Experiments 3.4.1 Pulsed Echo Experiments 3.4.2 Stepped Experiments 3.5. One-Dimensional High Resolution Techniques 3.5.1 Magic Angle Spinning (MAS) 3.5.2 Extraction of Parameters from MAS NMR Spectra 3.5.3 Suppression of Spinning Sidebands 3.5.4 Special Considerations for MAS of Quadrupolar Nuclei 3.5.5 Magic Angle Spinning Observation of Satellite Transitions 3.5.6 Double Angle Rotation of Quadrupolar Nuclei 3.5.7 Practical Implementation of CRAMPS 3.6. Two-Dimensional Experiments 3.6.1 Nutation NMR 3.6.2 Off-Resonance Nutation 3.6.3 Order-Resolved Sideband Spectra 3.6.4 Dynamic Angle Spinning (DAS)
3.1. 3.2.
111 112 112 113 115 116 120 122 124 127 127 128 128 129 129 130 130 130 133 133 136 138 138 143 143 144 149 150 152 153 153 154 155 156
3.6.5 3.6.6 3.6.7
Two-Dimensional Sequences Developed from Solution NMR Multiple Quantum Experiments in Dipolar Coupled Systems Multiple Quantum NMR Experiments of Non-Integer Spin Quadrupolar Nuclei 3.6.8 2D XY Correlation Methods 3.6.9 Correlation of Tensor Information- Separated Local Field Experiments 3.7. Summary of Approaches for Examining Quadrupole Nuclei 3.8. Multiple Resonance 3.8.1 Cross-Polarisation (CP) 3.8.2 SEDOR, REDOR and TEDOR 3.8.3 TRAPDOR and REAPDOR 3.9. Techniques for Determining Relaxation Times and Motional Parameters 3.9.1 Measurement of T~ 3.9.2 Other Spin-Lattice Relaxation Times (Tip, T1D) 3.9.3 Transverse Relaxation Times (T2) 3.9.4 Molecular Motion 3.9.5 Diffusion Measurements 3.10. NMR Under Varying Physical Conditions 3.10.1 Variable Temperature NMR 3.10.2 High Pressure Experiments References
157 160 161 168 170 172 172 173 178 182 183 183 184 185 186 187 187 187 189 190
Chapter 3
Experimental Approaches 3.1. BASIC EXPERIMENTAL PRINCIPLES OF FT NMR
In principle the NMR experiment is straightforward in that it has relatively few requirements. The equilibrium magnetisation must be produced by the application of a high static magnetic field which creates the Zeeman states lifting the degeneracy of the nuclear spin energy levels. The central task of an NMR spectroscopy experiment is to determine the separation of these energy levels. This measurement is carried out by the application of a usually much smaller magnetic field oscillating at a frequency that can induce transitions between the energy levels. This condition dictates that with magnetic fields in the range 4.7 to 18.8 T typically available today, the corresponding resonance frequencies are in the range 10-800 MHz, and it will not be long before 1H NMR is carried out at 1 GHz. The original method employed to detect a signal was to scan either the frequency of the exciting oscillator or the applied magnetic field until resonant absorption occurred. However, compared to simultaneous excitation of a wide range of frequencies by an rf pulse, scanning either the frequency or Bo is a very time-inefficient way of recording the spectrum. In the linear approximation there is a direct Fourier relationship between the time domain signal after a pulse (the FID) and the spectrum so that the spectrum is produced by Fourier transformation (FT) of the FID. Hence, with the advent of computers that can be dedicated to spectrometers, and efficient Fourier transform algorithms pulsed FT has become the normal mode of operation in NMR experiments (Ernst and Anderson 1966). Operating at constant field and frequency also produces big advantages in terms of the stability of the whole experiment. With the development of persistent superconducting magnets very stable applied magnetic fields are now used. In an FT NMR experiment a pulse of duration Tp is applied close to resonance. The discussion of the rotating frame in Chapter 2 shows that the pulse causes a coherent motion of the magnetisation, tipping the magnetisation through an angle 0p. The signal observed in the NMR coil by electromagnetic induction is termed the FID. To improve the signal/noise ratio (S/N), a number of FIDs (n) are usually coherently added, increasing the signal, with the S/N improving as ~n. It is a tacit assumption that everything behaves in a linear fashion, for example, that the excitation (or effective rf field) is uniform across the entire spectrum. In many cases this situation is closely approximated but distortions can occur for the broad lines that may be encountered. The frequency spectrum A(v) of a pulse of duration Tp applied at Vo is given by a sinc function sin ~Tp (V o - V) A(v) -
rC(Vo - v ) T p
111
(3. l)
Multinuclear Solid-State NMR of Inorganic Materials
112
A(v) rf
/'~
T Tp Fouriertransform,. field[~ ] ]Envelopeof ' ~
sincoet Vo 1
Frequene3 1
Figure 3.1. Fourier relationship between an rf pulse of duration Tpand the amplitude distribution A(v) of the frequency components present.
The sinc function is simply the FT of a rectangular pulse. Hence, from the bandwidth theorem, as the duration of the pulse decreases the frequency range covered increases. In Figure 3.1 the central lobe of this sinc function increases its frequency coverage as the pulselength decreases. To obtain an undistorted spectrum it should ideally be confined to the central portion of this lobe so that the irradiation is then uniform across the spectrum. Unfortunately there are examples in the solid state NMR literature of wide lines where the resonances are clearly much broader than the central lobe so that effectively only the irradiation envelope is reported and not the true spectral lineshape. The sinc 2 function describes the best possible case, with often a much stronger frequency dependence of power output delivered at the probehead. (It should be noted here that alternative excitation schemes to pulses are possible such as adiabatic passage (Kentgens 1991) and stochastic excitation (Yang et al. 1998) but to date these are only very infrequently applied).
3.2. INSTRUMENTATION 3.2.1 Overview of a pulsed F T N M R spectrometer The basic components of a pulse FT NMR spectrometer are shown schematically in Figure 3.2. There is a high field magnet which these days is nearly always a superconducting solenoid magnet. The probe circuit containing the sample is placed at the centre of this magnetic field. The probe is connected to the transmitter. All parts of the spectrometer are coupled together via transmission lines whereby the rf power/signal are transferred. Typically the coaxial cable which acts as the transmission line has a characteristic impedance determined by its capacitance and inductance per unit length. In NMR experiments, the convention is to standardise on an impedance of 50 l). The transmitter consists of a synthesiser to produce the central frequency, which is gated to form the pulses. These pulses must be amplified so they can be made sufficiently short to cause broad enough frequency excitation for the frequency width to be observed.
Experimental Approaches
ii ii
@
I
113
Magnet
Synthesiser r
j Ampl~'~r
Figure 3.2. Basic components of FT NMR spectrometer.
The probe is also connected to the receiver, requiring careful design to ensure that the receiver that is sensitive to IxV does not see any of the large excitation voltages produced by the transmitter. The induced voltage detected in the receiver must be digitised for storage in a computer where it can be manipulated and converted into a spectrum. The relatively simple concept of both the experiment and the instrument belies the extensive research and development effort that has gone into NMR spectrometers.
3.2.2 Magnets NMR demands ever higher magnetic fields, the highest currently commercially available for solid state NMR being 18.8 T. Standard instruments are now considered to be 4.7-9.4 T. The drive for higher fields is based on the increased chemical shift dispersion (in Hz) and the increase in sensitivity via both the Boltzmann factor and higher frequency of operation. Superconducting solenoids dominate the magnets sold and are usually based on Nb3Sn or NbTi multifilament wire maintained at liquid helium temperatures. However, fields and current densities now used are close to the critical limits of these materials, demanding improved materials technology. The demand for ever higher magnetic fields has seen the extension of the superconductor performance by operating at lower temperature. This is achieved by pumping on part of the helium cryostat which reduces the boiling point of helium. The principle of operation is very simple; a high current is circulated through a long (several km) coil of wire at a typical current of at least 50 A. This means that the magnet stores significant amounts of energy (up to 10 MJ) in its field (= 1/2LI2, where L is the solenoid inductance and I is the current flowing). A superconducting magnet consists of a cryostat, main coil, superconducting shim set and a means for attaching the current supply to the main coil (Figure 3.3). The
114
Multinuclear Solid-State NMR of lnorganic Materials
Longitudinal cut through a vertical cryostat ~ ] A - liquid helium (-269~ B - superconducting coil C - l i q u i d nitrogen (-196~
D - vacuum space E - room temperature bore
Figure 3.3. Schematic view of a high field superconducting magnet.
cryostat consists of two vessels for the liquid cryogens, the inner one for helium and the outer one for nitrogen. These are insulated by several vacuum jackets with a radiation shield. The aim is to reduce heat leakage to the inner chamber to conserve helium. NMR superconducting magnets are usually operated in persistent mode, which means that after a current is introduced, the two ends of the main coil are effectively connected so that the current follows a continuous path within the superconductor, allowing the power supply then to be disconnected. To attach and detach the coils (main and shim) to the power supply, the circuits within the cryostat have a superconducting switch. The coil circuits are also designed to cope with a sudden irreversible loss of superconductivity, termed a quench. There are resistors present (called dump resistors) to disperse the heating effect and prevent damage to the main coil when a quench occurs (Laukien and Tschopp 1993).
Experimental Approaches
115
Magnet technology is continually advancing, even for "standard" 7.05 T solid-state NMR magnets which have been available for --~20 years. In early versions the field was typically generated by a current of 50A, whereas modem versions run at nearer 90A since the inductance of the main coil in newer magnets is much lower. This brings the advantage that modem magnets can be charged much more rapidly. The complexity and size of the associated cryostat is greater than in the more conventional magnets. Although higher magnetic fields are available than those used in NMR, the NMR experiment imposes additional constraints on the field. An NMR spectroscopy experiment also demands homogeneity and stability of the magnetic field. Long term stability is aided by persistent-mode operation and the drift should be < 2 X 10 - 7 per day. Homogeneity requirements for solid state NMR experiments are typically 2 X 10 - 9 over a volume of--~ 1 cm s. The main coil alone is unable to produce this level of homogeneity so in the main cryostat there is a set of smaller superconducting coils called cry0shims. The number of these cryoshims depends on the design and also the purpose of the magnet (e.g. solid state NMR, high resolution NMR, imaging), but typically varies between three and eight. For many solid state NMR experiments the homogeneity produced by the cryoshims is sufficient. However, most commercial spectrometers also have a room-temperature shim set which further improves the homogeneity of the magnetic field. A final consideration for the magnet is the accessible room-temperature bore size of the magnet. A standard magnet has a bore of 52 mm diameter, but most solid state NMR spectroscopists prefer an 89 mm bore as this gives much more room for the probe, allowing the use of larger, more robust high-power electrical components, and accommodating some of the more specialised probe designs (e.g. double angle rotation, dynamic angle spinning etc.). Superwide bore magnets also exist, with an accessible diameter of 150 mm. For imaging experiments, even wider-bore magnets are made, and often rather than the bore being in a vertical orientation, as is favoured in spectroscopy, the bore is horizontal.
3.2.3 Shimming For most solid-state NMR spectroscopists, shimming is a relatively minor consideration. For many of the studies discussed later in this book the resolution provided by a well cryoshimmed magnet is more than sufficient. Nevertheless, there will be occasions when shimming is important. The resolution required for high-resolution solution experiments is typically --~0.1 Hz. At a resonance frequency of 400 MHz this corresponds to a magnetic field homogeneity of one part in 2.5 X 10 ~~ this resolution ideally achieved over a cylindrical volume typically 20 mm high and 10 mm in diameter. Most commercial spectrometers are bought with a room temperature shim set. Shimming to this level of precision can be difficult, as the process amounts to trying to find a global minimum in the linewidth determined by the inhomogeneity in the
116
Multinuclear Solid-State NMR of Inorganic Materials
magnetic field which is a multidimensional function. For an ideal shim set all shims would be orthogonal with each gradient having no interaction with other gradients. No shim set is perfect but often local minima are only relatively shallow, although this is not always so. Each shim gradient (the direction and shape of gradient that results when a current is applied to that specific coil) has a shim centre. In a cryomagnet there will be quite a strong field dependence (i.e. a gradient) along the z-axis. If a shim current is used to apply an additional field to cancel out this gradient there should be a change in the value of the field at all points except one; this unchanged point is termed the shim centre for that gradient. The determination of this point for the z direction is made in the factory and is recorded in the magnet handbook as a distance from the magnet top and bottom flanges. Different magnet manufacturers supply different cryoshims, some making only x, y and z shims available but others offering xy, xZ--y2, XZ and yz as well. Cryoshimming is carried out during installation so that most operators never come into contact with this procedure, which needs not be repeated unless the magnet is moved to a significantly changed environment (e.g. another magnet is located close by), or a cryoshim quenches. The authors' experience is that even after a full quench the old cryoshim currents are usually satisfactory. Cryoshimming should ideally be carried out on the largest sample volume likely to be used. Most NMR spectrometers have 12 to 18 shim controls (Churmny and Hoult 1990). Each user will adopt their own procedure but the aim is to produce the minimum linewidth consistent with a good lineshape. In practice, some shims are much more significant than others and for particular probes different shims will be important. For solid-state operation, shimming usually needs to be carried out relatively infrequently. One possible procedure for probes tuned to ~H is to crudely shim on H20. If there is no proton channel most multinuclear probes will tune to 2D, so D20 can be used. For CPMAS probes that tune to 13C, adamantane is a useful compound which should be shimmed under spinning and 1H decoupling conditions. A typical resolution for 13C in admantane of 3-4 Hz at 7.05 T and --~10 Hz at 11.7 T should be achieveable. In any shimming procedure the gradients should be varied systematically (Churmny and Hoult 1990) and it should be borne in mind that even with very good shim sets the gradients will be interdependent. The speed of the Fourier transformation in modem spectrometers allows the spectrum itself to be used for shimming, rather than the FID. There will always be a compromise between the very narrowest lines that can be produced and a good lineshape (free of shoulders and humps).
3.2.4 Transmitters The frequency required to observe a particular nucleus is the first consideration. This frequency was formerly synthesised by analogue methods. Frequencies below 200 MHz
ExperimentalApproaches
117
were synthesised directly, whereas higher frequencies were synthesised via frequency mixing. Synthesising the frequency in a stable and accurate way is very important. The technology of synthesisers has advanced, with digital synthesisers now available. In some experiments the frequency of the synthesiser needs to be switched rapidly. There are two alternative schemes for switching the frequency. One possibility is that it is phase-continuous, whereby, when the frequency changes, the phase continues instantaneously with the same value or phase coherently (i.e. the phase of the second frequency is as if it had started at the same point in time in phase with the other frequency). This frequency is normally continuously synthesised and then gated to form the pulse. Solid-state NMR experiments often call for short pulses and rapid phase switching, and in earlier spectrometers the most commonly-used quadrature phases (0 ~ 90 ~ 180~ 270 ~ or x, y , - x , - y) were simultaneously present in a hardwired modulator. Each of the four channels had separate amplitude and phase controls, and the rf was generated simply by taking the appropriate output. This set-up requires maintenance to correct for changing conditions and the frequency dependence of the operation means that different settings will be necessary for the different channels at different frequencies. Such a set-up produces the maximum switching speed for the phases, but in recent years digital synthesis has become more common. Here, the computer memory holds a sine wave in digital form. The required frequency is then produced by the speed at which the computer memory is read out, the rf being generated by a digital-to-analog converter. The phase shifts are generated by simply varying the memory address from which the signal read-out commences. This means that pulse phases can be generated continuously with typical settability currently of--~0.1 ~ Hence, non-90 ~ phase shifts are readily generated, an important consideration since an increasing number of experiments routinely demand such phase differences (e.g. MQ-MAS of quadrupole nuclei). The pulse from the modulator, which is typically at the --~V level, is then amplified to a level that depends on the type of experiment. The power of a transmitter can be determined by examining the pulsed voltage on an oscilloscope (but NEVER plug a high power transmitter directly into an oscilloscope). Attenuation is measured in decibels (dB) and in terms of voltage:
10
(3.2)
Hence 3 dB represents a factor of 1.4 in voltage while 20 dB is a factor of 10. If Vpp is the measured peak-to-peak voltage after taking into account the attenuation, the power is
P(watts) = 200-,~
(3.3)
118
Multinuclear Solid-State NMR of lnorganic Materials
Amplification is achieved via solid-state transistors or vacuum tube amplifiers. Transistors have always been used for power of less than 100 W, but for amplification up to the 1 kW level the philosophy has changed in recent years. Originally, tube amplifiers were the only real choice for high power rf applications. However, solid-state transistor technology has advanced so that up to 200 MHz 1 kW transistor amplifiers are now a serious option. At higher frequencies, tube or cavity amplifiers are still the only real possibility. For more demanding high-power applications, tube amplifiers are preferred since, if the duty cycle increases, producing a significant heating effect, transistors are susceptible to changes in their characteristics, whereas tube amplifiers already run hot. Tube amplifiers are usually part of an active tuned circuit so that as the frequency of operation is changed the amplifier circuit has to be retuned. By contrast, a transistor amplifier is broadbanded, requiting only the new frequency to be set. This makes the transistor design increasingly popular and the spectrometer easier to use. The behaviour of rf amplifiers falls into one of several classes (normally termed A, AB, B or C) depending on the relationship between the input and the output voltage. Figure 3.4 gives a diagrammatic representation of this relationship. Class A is termed linear, the output being directly proportional to the input. This relationship can make resetting of the CP condition at different power settings relatively straightforward and is necessary for the application of high-power shaped pulses in solid-state NMR. The disadvantage of linear amplifiers is that without careful circuit design the output can be less stable (e.g. due to heating effects). Also, as the circuit operates in the active region it generates significant noise even in the quiescent state, and so needs to be actively blanked by a gating circuit. For some applications (e.g. CRAMPS), the speed of the Anode current
Anode current
Class A
Class C
. . . .
Output
0 ,---i .....
o Grid voltage
i
i i
Grid voltage
i
'Input Input Figure 3.4. Schematic representation of the input-output relations of Class A and C amplifiers with
O the operating point of the grid voltage.
Experimental Approaches
119
Gate
TR i-,,
. v~
I -~-...................... i................................. j ............ [ ,,/~ "--,, i]~Droop i," ~ A-""-^-.....~--~- ....... ~.........................
i/'
.
.
.
.
.
.
.
Ripple
Figure 3.5. Schematic representation of pulse imperfections with TR the rise time and TF the fall time. blanking circuit becomes an important consideration. Class C operation is highly nonlinear but has several advantages. In its quiescent state, the amplifier is biased hard off so that there is very little noise generated by the transmitter between pulses. It then turns on at a fixed level with relatively little change in the output with input level, making the transmitter very stable. Often class AB operation is preferred which, at the extremes, shows the advantages of class C behaviour while nevertheless having a linear region. The pulses from the transmitter are often far from ideal. To check their non-ideality they should be observed on an oscilloscope via a broadband 50 ~ impedance, termed a dummy load. Typical characteristics are the rise and fall times, the on-off ratio and the amounts of tipple and droop. These are shown schematically in Figure 3.5. The turn-on time is usually defined as the time to go from 10 to 90% of the maximum power. The on-off ratio is especially important for determining the noise generated by the pulse during the FID. The effect of droop becomes more noticeable when longer pulses are applied, as for spin-locking. All these effects which alter the shapes of pulses mean that the irradiation profile is often much more complicated than the sinc function of Eq. 3.1. Also, for a pulse incident on a narrowed-banded mismatched load (e.g. a probe), the effects and the irradiation are more complicated still. Even if the bandwidth of the pulse is high, any narrow bandwidth components in the spectrometer sequence will cut this down and will also give rise to transient oscillations.
120
Multinuclear Solid-State NMR of lnorganic Materials
3.2.5 Probes
The probe is the heart of the NMR experiment. It is essentially a tuned resonant circuit with the sample contained within the main inductance (the NMR coil). The inductance of the coil is proportional to p~l,Zon21A where n is the number of turns per unit length on the coil (i.e. N (total turns)/1 (coil length), IX~o is the permeability of the sample and A is the cross-sectional area of the coil. The details of the circuits are often complex, and a typical circuit is shown (Figure 3.6). The circuits often use inductive coupling so that different parts of the circuit need not be physically linked (Kuhns et al. 1988). Both series and parallel circuit designs are used (Fukushima and Roeder 1981, Stejskal and Memory 1994) but they all consist of capacitances and inductances which have frequency-dependent reactive impedances. Usually a parallel-tuned circuit is preferred, and in simplified form the inductance (L) and capacitance (C) can be related to the required resonant frequency (Vo) via Vo - 1/(2-rrL ~ ) . The resonance frequency is the most important parameter, but the input impedance, which should be 50 12, also has to be satisfied. Furthermore, the quality factor Q, (a measure of the sharpness of the resonance of the probe circuit, one definition being the resonance frequency/half width of the resonance response of the circuit, also = tooL/R, R is the coil resistance), must also be considered. The probe must satisfy competing requirements which can be characterised in terms of Q, as summarised in Table 3.1. It can be immediately seen from Table 3.1 that a probe cannot be designed with all the properties optimised because of their differing Q-dependencies. This means that most probes are a compromise, focussing on the most important aspects for a specific application. Several probe designs exist which have advantages for different applications. 74 nH
10 nH 22 pf . . . 1 _
1-10 pf 13C <-",
.,
22pf + '
-r-
/l't-
'lgpf I
13 nH 1.5 pf
1-10 pf
----) I
'P
t"~ /
11. l i Tt
J'
400 MHz 100 MHz
_ J 1-10 pf
:~
1-10 pf
l r -
7nH~ ,
10.,n _
,,
~
1 ~r
- :
~
"
33 nH 1
Figure 3.6. Typical circuit of a doubly resonant probe from Doty et al. (1988) with permission of the copyright owner.
121
Experimental Approaches
Table 3.1. Summary of the dependence on various properties of the NMR probe on the quality factor (Q). Property
Q-dependence
Length of a 90~pulse at constant transmitter power Probe deadtime Voltage developed across the circuit Frequency response bandwidth Sensitivity per nucleus
1/]Q Q Q I!Q
Various choices of coil are also possible (Hoult and Richards 1976), the optimum design again depending on the specific application. The traditional coil design, particularly applicable at lower frequencies and for solids, is the conventional solenoid. For large samples loaded externally to the magnet, Helmholtz or saddle coils are used widely in high resolution liquid state NMR studies, but in MRI birdcage coils are finding increasing use. The resistance of the coil (wire) increases with the frequency of the experiment through an electromagnetic phenomenon known as the skin effect. This effect produces an ac current in a layer determined by the skin depth d d - ~[
1
(3.4)
rCfloOV o
As frequency increases, d decreases, causing the wire to present a smaller cross-sectional area. This increases the ac resistance and consequently lowers the circuit Q. NMR coils are often made of flat wire rather than normal round wire, the former also having the advantage of better rf homogeneity. The magnitude of the induced B1 field is related to the power applied, Q, and the coil volume, V, by B 1 - J t't~ 7Wv o
(3.5)
Coils operate better if they do not have any sharp comers, since there is less tendency for the magnetic field to bunch, and also voltage breakdown is more likely to occur at sharp points. The capacitors are also important in probe design since they need to be able to withstand high voltages. The maximum voltage developed in a probe circuit is given approximately by Winput X Q Vmax --
5O
(3.6)
For a 1 kW transmitter, the peak-to-peak input voltage is --- 600 V, and with a Q of 250, which is common in commercial CP-MAS probes, 3 kV can be generated. It is thus important for the capacitors to be able to withstand such voltages and for their dielectric
122
Multinuclear Solid-State NMR of Inorganic Materials
insulation to have a high breakdown voltage. High power rf components must be handled with great care so that grease and dirt are not introduced as these will severely degrade performance; once the performance has begun to deteriorate it is very difficult to improve the situation without complete probe overhaul. The leads between the coil and the probe tuning, and matching capacitors and inductors also play a central role in the overall performance of the probe. A simple consideration is the length of the leads from the coil; the longer the leads, the farther other probe components can be removed from the active area of interest. Removing components minimises their possible disturbance of the magnetic field, although this is often not a significant consideration for solid-state NMR. The downside is that the leads add to the inductance and capacitance of the probe circuit. Stray capacitance can be a significant problem especially if one is interested in higher frequencies, since the main tuning capacitor may only be a small part of the total circuit capacitance, giving a small tuning range and limiting the minimum capacitance that can be achieved. The leads also add inductance to the circuit and since they do not contain sample, they reduce the filling factor of the circuit. Maximum sensitivity occurs when all the circuit inductance is filled with sample. One way to minimise the effect of the leads is to get as much of the tuning capacitance near the active part of the coil as possible. Linear analysis is often applied to determine the electrical response of a probe circuit, but in probes designed for solids (and therefore for high-power operation), nonlinear effects can occur (Doty 1996). In multiple-resonant probes, for decoupling and cross-polarisation, even small voltages generated at the second resonant frequency are unacceptable since these would swamp NMR signal generated in the coil which is part of more than one resonant circuit. A detailed consideration of the design criteria for doubly-tuned CP-MAS probes has been given (Doty et al. 1988). All probes have background signals. This is more of a problem for static wideline work, since the background signals are usually themselves broad. Wide background signals would not have much effect on the narrower spectral windows used in high resolution work. Some common background signals are given in Table 3.2. To check that a signal comes from a sample, a dataset should be acquired under identical conditions but without the sample. Some of the interfering nuclei listed in Table 3.2 are very important for solid-state NMR and probes can always be constructed to minimise the background from a particular nucleus, to allow the observation of that nucleus in the sample.
3.2.6 Connection of the probe It is usual to have the probe physically connected at all times to both the transmitter and the receiver. The electrical circuit acts effectively to couple the probe to the transmitter and receiver when the pulse is respectively on and off (termed duplexing). In solid-state NMR operation the duplexing circuit should withstand high power rf and
123
Experimental Approaches Table 3.2. Nuclei observed as background signals. Source of background
Nucleus 1H
~B 17 0
19F 23Na 27A1 29Si 35,37C1 63,65Cu lO7,1O9Ag
Moisture, polymer parts Ceramic and glass parts As the background is unenriched not normally a problem but for weak MAS signals ZrO2rotors can be seen at "-~ 383 ppm Ceramic and polymer parts, PTFE tape, Teflon of the coaxial cable Glass and ceramic parts, be careful with salt off the fingers Ceramic parts, capacitors and from aluminium metal itself which is widely used in sample manufacture, A1N impurities in Si3N4rotors Ceramic and glass parts, Si3N4rotors Some fluorinated polymers give a strong chlorine signal Coil wire Coil wire
Crosseddiodes
[ Tx L
(~l
Z/4 ___~ Probe [
~4 Crosseddiodes< >1
] Preamp ,l
TX on /
/
Open
Closed .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
ti
Preamp ,I
Figure 3.7. Lowe-Tarr circuit design for connecting the NMR probe to both the transmitter and the receiver with crossed diodes and M4 cables. remain stable throughout the sequence of pulses. A common approach to this is the Lowe-Tarr circuit (Lowe and Tarr 1968) which contains a series of diodes and path lengths related to the wavelength of the rf radiation (Figure 3.7). Pairs of diodes in parallel but with their conducting direction oppositely oriented (termed crossed) act as active switches. When the pulse is on, the two switches are closed. Hence the probe is
124
Multinuclear Solid-State NMR of Inorganic Materials
connected to the transmitter so that with the impedance transformation of the h/4 cable, the far side of the probe appears earthed and the receiver is protected, allowing the maximum voltage to be delivered to the probe. Then when the pulse is off the transmitter is effectively disconnected from the probe which is itself connected to the preamplifier. The path lengths may be physically produced by cables of the correct lengths (the simplest approach). The drawback is that the h/4 length of cable is only efficient over a relatively narrow bandwidth of frequencies. Reactive components provide alternatives to cables; several circuit designs from the literature are extensively discussed by Fukushima and Roeder (1981). More sophisticated active networks have been suggested, such as pin-diode switches.
3.2.7 Signal detection The microvolt NMR signal generated in the coil requires amplification prior to detection and digitisation. The first stage is typically 20-30 dB amplification, using a preamplifier whose most important characteristic is the noise figure (NF). This is essentially a measure of the noise added to the signal by the amplifier, and is defined by,
I V2nRs NF - 10 loglo Vth2R~ ) - G
(3.7)
where W n is the output noise voltage, R~ is the load resistance, Rs is the input resistance of the preamplifier, Vth is the thermal or Johnson noise and G is the power gain of the preamp. Careful consideration is necessary for the production of a low noise figure and rapid recovery from saturation. Another way of regarding the noise figure is the ratio of the minimum detectable signal plus the thermal noise to the minimum detectable signal. The nearer this ratio is to one, the less noise is added in the preamplifier. Here again some compromise is necessary, with recovery of < 2 Ixs normally sufficient. The preamplifier should also have extremely good linearity and rapid recovery from saturation. The initially amplified signal is then usually detected in the receiver using a threeport device acting as a double-balanced mixer. Three port devices, which can be used as gates, mixers or phase-sensitive detectors, consist of a series of diodes and transformer coils. The phase-sensitive detector has two input signals, the NMR signal (COo) and the frequency of the synthesiser ((Oref). The output is proportional to COS((O)o- O)ref) -- (~) -[- COS(((.Oo -'['- (.Oref) Jr- (~).
(3.8)
The sum frequency is much larger than the total bandwidth of the spectrometer and so is lost, leaving the difference frequency, which is the frequency of the observed FID. This lower frequency is usually further amplified by audio-amplifiers. It can be seen that phase-sensitive detection is equivalent to examining the NMR signal in a reference
ExperimentalApproaches
125
flame rotating at (.Oref. If the frequencies too and O)re f are equal and relaxation effects are ignored, a constant output is obtained that depends on the phase difference (+), hence the term phase-sensitive detector. Most modern spectrometers employ two phase-sensitive detectors having reference signals that differ in phase by 90 ~, hence termed quadrature phase-sensitive detection. Quadrature detection (Traficante 1990) has several advantages compared to single channel detection. The most important is that quadrature detection can distinguish whether a signal is above or below the reference frequency. The positioning of the transmitter frequency in the spectral window being examined is then less critical. This ability to place the transmitter frequency near the centre of the range of interest improves both the pulse power efficiency and the signal-to-noise ratio. However, the system needs to be set to remove imbalance in the gains of the two channels and to correct for any deviation from 90 ~ of the phase shift between the two channels. The two channels are termed the real and imaginary channels since, if the signal is Fourier-transformed, the channels will give the absorptive and dispersive spectral components. If these factors are not corrected for, spurious signals termed quadrature images will appear. Any net DC offset, either between the channels or overall, will also cause spectral artefacts, most obviously as a zero-frequency spike. Careful set-up of the spectrometer can effectively remove such effects. However, phase cycling will also efficiently cancel quadrature images. The most common sequences include the use of four phases 90 ~ apart. The CYCLOPS sequence is widely used to remove quadrature artefacts (Hoult and Richards 1975). This pulse sequence has four steps in the phase cycle. The phase cycling and routing of the signal, shown in Figure 3.8, eliminate DC offset, amplitude imbalance and any deviation from 90 ~ of the phase between the two channels. It should be emphasised that this phase cycle only removes artefacts by cancellation not reconstruction so that S/N is poorer than if the system is set up perfectly. To improve S/N, the high frequency noise from outside the spectral range of interest should be cut down. As the signal is digitised, the Nyquist sampling theorem for a discretely sampled time domain signal (with a spacing of DW = the dwell time) shows that the highest frequency that can be unambiguously determined is 2/DW. Higher frequencies thus appear at lower frequencies, folding higher-frequency noise into the spectral range. Audio filters are used to cut out this high frequency noise. An optimum value is 1.2-1.3 of the spectral width of interest and the spectrometer manufacturers usually build this automatically into their software as the default setting. There are various designs of analogue filter available. The simplest are RC circuits, where the time constants determine the upper frequency response. A number of more complex filter circuits exist, two commonly used in NMR being the Bessel and Butterworth designs (Fukushima and Roeder 1981). Filters do not have infinitely sharp cut-offs and these designs show differing fall-off characteristics. Older designs of NMR spectrometer used Bessel or Butterworth filters for spectral ranges below 100-120 kHz, whereas
126
Multinuclear Solid-State NMR of Inorganic Materials Reference Frequency
Cyclops Computer memory
r
] Filter+l 1~
PS
"[ ADC
Real
Tx Phase X Y -X -Y
Signal sent to Real 1 2 -1 -2
Imaginary 2 -1 -2 1
7-"
Signal I
I
900 Phase shift
-'9 PSD2 [i "-i
Imaginary
] Filter + 2_ l .
7 ADC
Figure 3.8. Quadrature phase sensitive detection along with the phase cycling and signal routing used in CYCLOPS for eliminating quadrature artefacts.
above this range RC filters were used. A compromise must be made in improving the S/N by filtering, since the filter introduces distortion (e.g. deadtime, overshoot) to the start of the FID that can distort the lineshape of broad resonances. In determining the optimum filtering, the philosophy normally adopted is to use a filter value slightly larger than the spectral range of interest, but in certain cases this is not necessarily the best approach. If a broad line is of interest, the signal can be digitised very rapidly so that the FID is recorded accurately but the filter is set significantly smaller than spectral range. This improves the S/N, but the noise will show a variation in the spectral range. This is not a problem as long as it does not coincide with the spectral position. The signal is then digitised for storage in the computer memory. There are two ways in which the data can be captured when the two quadrature channels are being used. A data point for each channel can be captured at the same time (called simultaneous acquisition) or, alternatively, one data point is taken at each dwell time but routed to the different channels of the phase-sensitive detector (called sequential acquisition). The relationship of the spectral width and DW is slightly different for the two approaches: Simultaneous acquisition.'DW =
1
SW
2
,and sequential acquisition: DW = ~ SW
(3.9)
The number of time domain data points (TD) means that the time over which data are collected (termed the acquisition time (AQ)) is given by AQ - TD • DW. Digitisers are characterised by the number of bits (determining the dynamic range with which the
Experimental Approaches
127
voltage is recorded), the rate of digitisation (determining the dwell time) and the size of memory capable of storing the data points. Until very recently the ability to record narrow spectral objects over a broad range of frequencies was limited, usually by the on-board computer memory, but commercial spectrometers have now addressed this problem. Another possible approach to filtering that is becoming increasingly popular is to filter the digitised data (termed digital filtering).
3.2.8 Additional equipment Although spectrometers are self-contained, it is often very useful to have some additional hardware for optimising the operation of a spectrometer, especially for those who want to build components (e.g. probes) themselves and develop new approaches. A good oscilloscope is essential with a high rf bandwidth and sweep rates allowing very high frequencies to be detected. The triggering and calibration should be reliable. For normal phosphor screen instruments, intense displays are necessary to observe shortlived events. More recently oscilloscopes with liquid crystal displays have become available, allowing operation much closer to the magnetic field. Attenuators are essential so that pulses can be observed either on oscilloscopes or with the spectrometer receiver without causing damage. Although NMR pulses can be of high power, they are usually on for a relatively short period of time so the CW power rating need not be too high. The attenuator should have a broad bandwidth. An oscilloscope and attenuators are probably the minimum additional requirement for a properly equipped NMR laboratory. Directional couplers are also useful since the power reflected from the probe can be continuously monitored, thereby allowing tuning and monitoring for breakdown, which is sometimes important in high-power experiments. For probe tuning, a magic T with a sweep generator is a worthwhile asset, especially for development and modification of probes. There is increasing sophistication in this area and self-contained instruments that simply monitor circuit response are now available. A vector impedance meter is the top of the range instrument for examining probe circuits. 3.3. PRACTICAL ACQUISITION OF NMR SPECTRA
In single pulse acquisition the following simple steps need to be followed. (1) Choose an appropriate frequency for observation of the nucleus of interest and tune the probe for the experiment. (2) Start with an appropriate set-up compound that allows the 90 ~ pulse to be determined for given settings of the transmitter and the chosen probe. This set-up compound should also allow the spectral reference to be determined. (3) Make sure that the irradiation bandwidth and frequency range being digitised
128
(4)
(5) (6) (7)
Multinuclear Solid-State NMR of lnorganic Materials (usually termed the spectral or sweep range), are appropriate for the width of the resonance being observed. Choose a proper dead time delay between the detection pulse and the beginning of the acquisition. An optimum choice is to set this duration to approximately 0.75DW. This represents the very best that can be done and has to be set to avoid deadtime and tinging effects. Set AQ sufficiently long time that the signal is not truncated. Choose an appropriate gain in the receiver (pre-amplifier) so that the signal does not overload the receiver. Use a recycle time that means all the sites are fully relaxed in the spectrum.
3.3.1 Processing the FID to produce a spectrum The FID contains all the information of the NMR signal from the sample. All the spins have been excited simultaneously, and the FID is a record of all the information about the nuclei (e.g. the distribution of Larmor frequencies, interactions etc.) Spectroscopists find it somewhat difficult to interpret the time domain data but Fourier transformation to a spectrum greatly aids understanding. Usually a full discrete complex Fourier Transform is carried out using a numerical algorithm. The Cooley-Tukey (1954) (or the FFT (fast FT)) algorithm carried this field forward. The efficiency of this algorithm results from a repetitive pairwise sorting of data points, for data sets with a power of 2 number of data points. For direct transformation, the consumed time increases with N 2 of data points while it increases with 2NlogN for the FFT algorithm. Much development has occurred in numerical FT algorithms but they are largely based on the original algorithm. T2 relaxation processes mean that the information is not equally distributed throughout the time domain data. The T2 decay of the signal means that the contribution from the signal relative to the noise decreases as the time increases. Therefore weighting functions can be applied to the time domain data points in order to emphasise the parts of the FID where the signal preferentially resides. 3.3.1.1 Window functions. The experimental signal S(t) is multiplied by the window function in a process termed apodisation to produce a new weighted signal A(t) which is Fourier transformed. Useful window functions commonly used are: (i) Exponential multiplication A(t) = S(t) X e x p ( - 7rLBt)
(3.10)
This function signal improves S/N as it cuts down the high frequency noise contribution. It causes Lorentzian linebroadening, LB being the broadening (in Hz) produced by the function, and it reduces effects caused by truncation of the FID.
Experimental Approaches
129
(ii) Gaussian multiplication A(t) - S(t) x exp - ~ L B ( t - (2GB.AQ) )
(3.1~)
This can be used to improve the S/N by using Gaussian linebroadening. If the constant is set to a negative value, S/N will be degraded but the line is narrower (i.e. resolution enhancement). Such a process is often used in solution since it requires good S/N to be effective.
(iii) Trapezoidal multiplication S(t) can be multiplied by a trapezoidal shape. If a FID ends well before the end of AQ this can be used to cut down the high frequency noise. It can also be used to define a region around an echo to cut out deadtime effects.
(iv) Squared sine bell multiplication
[E AI
A(t) - S(t) x sin ~: -
AQ
A
(3.12)
The multiplication here is a good approximation to exponential multiplication (for A = 2) and to Gaussian multiplication (for A ~> 3). For zero shift, the first data point will be nulled and hence dispersive contributions to the FID are eliminated. An advantage of this function over the commonly used real EM or GM is the fact that the sine functions decay to zero, eliminating truncation oscillations.
3.3.1.2 Shifting of the time origin and linear back prediction. Often because of various deadtime effects (Sec. 3.3.2) the data cannot be recorded accurately and sometimes result in the appearance of large spurious induced voltages in the FT as spectrum-like features. These features can be removed by moving the time domain zero point in a process called left shifting. This is done by simply specifying the number of data points to be removed. The opposite process can move the data set back towards the real zero time domain point. However, this raises the question of what to insert into this region to avoid sinc oscillations. Linear back prediction has become standard in commercial spectrometer software, but because of the complex lineshapes that often occur, such an approach should be used with caution in the NMR of solids. The process works by looking at the data points and then predicting data points at earlier times using a relatively simple function.
3.3.1.3 Zero filling. To increase the digital resolution of the spectrum, the number of points can be increased before FT. The FID then consists of the acquired number
130
Multinuclear Solid-State NMR of lnorganic Materials
of data points plus additional data points containing zeroes. This simulates longer acquisition of the FID, but if the FID has already decayed, no noise is accumulated. However, there is no advantage in zero-filling more than three times (i.e. 2 3 = 8 times more data points).
3.3.1.4 Phase correction. Phase-sensitive acquisition of the FID with quadrature detection means that the pure absorption lineshape can be formed after FT but it may require some manipulation of the data to achieve this. If the real and the imaginary parts of the signal are AR and A~, the actual absorption spectrum A(oJ) can be formed as A(co) - pA R cos(to) + ~/1 - p2 AI sin(og)
(3.13)
A phase difference between the carrier frequency and the pulse leads to a phase shift which is almost the same for all resonance frequencies (o~). This effect is compensated for by the so-called zero-order phase correction, which produces a linear combination of the real and imaginary parts in the above equation with p - Po. The finite length of the excitation pulse and the unavoidable delay before the start of the acquisition (dead time delay) leads to a phase error varying linearly with frequency. This effect can be compensated for by the frequency-dependent, first-order phase correction p - Po + p~(~o - COo),where the factor p is frequency dependent. Electronic filters may also lead to phase errors which are also almost linearly frequency-dependent. 3.3.1.5 Baseline correction. The baseline of a spectrum may be distorted for a number of reasons, but often for reliable evaluation of the spectral information it is important that the baseline be flat. Commercial spectrometers are equipped with software packages containing a number of useful correction algorithms. Different algorithms can be used, depending on the nature of the baseline distortion. These include automatic linear baseline correction where a straight line is subtracted from the spectrum, which is calculated from a fit to the first and last N data points of the spectrum. Similarly, there are higher-order automatic polynomial corrections, a sinc curve or a set of data points can be defined manually and written into a file. These data points define the baseline of the spectrum and are used with a function fit between them. The availability of similar automatic routines is especially important for baseline correction of 2D NMR spectra when a large number of rows or columns need to be corrected.
3.3.2 Complications in recording spectra The best possible case for recording an undistorted spectrum when the pulse is incident on the probe circuit is given by the sinc function for the power distribution of the pulse. Complications beyond this include the frequency response of the probe in detecting the
Experimental Approaches
131
signal intensity. Thus, a broad line will not only experience non-uniform irradiation, but the intensity detected per spin at different frequency offsets will depend on this probe response, which depends on Q. The width of the frequency response decreases as Q increases, so typically for a Q of 100, the halfwidth of the frequency response is about 1.5 MHz. If sensitivity can be sacrificed, broad lines can be recorded by deliberately degrading or "spoiling" the Q by putting extra resistance in series with coil. Direct FT-pulse observation of broad spectral lines becomes impractical with pulse techniques for linewidths greater than --~200 kHz. Broader spectral lines can be reproduced provided corrections are made for the rf-irradiation and probe responses. Other potential pitfalls are experimental effects which corrupt the start of the time domain signal; these can be usefully split into deadtime and probe ringing. Deadtime is a general term used to describe effects causing the spectrometer system to be unable to record the NMR signal. Deadtime comes from several sources, the two principal ones being (i) the probe and (ii) other electronic spectrometer components including filters, preamplifier, receiver and even the cables. The pulses applied to the probe, which is essentially a tuned rf circuit, are usually rectangular (i.e. with very rapid rise and fall times compared to the duration of the pulse). However, the time at which the voltage in the circuit actually falls to zero depends on a time constant which is determined by the circuit (Hoult et al. 1983). The deadtime is the time constant of the exponential decay at the probehead and is given by
tdead =
Q (3.14) /~'V o
For a Q of 100 at 20 MHz the minimum value of tdead is 1.6 tXS. Sometimes a more practical definition of deadtime is the time for any induced voltage to drop to --~ 3 times the thermal noise. Hence the voltage will be too large in the coil for the sensitive receiver to record the tiny NMR signal for --~10 txs, and this represents the best possible case. Again the situation could be improved by reducing Q. Decreasing Q will also reduce the rf field at the probehead (for a given power), increasing the 90 ~pulse length. To obtain the most uniform irradiation over a broad range when observing broad lines, a short 90 ~ pulse length is necessary. The effect of the loss of the start of the FID (f(t)) can be considered most easily by writing an idealised time domain representation of the FID: D(t) • f(t), where D(t) = 0 when 0 --< t --< tdead, and D(t) = 1 when t >- tdead (3.15) Fourier transforming this gives the FT of the FID (i.e. the expected spectrum) convoluted with sinc(o~- tOo)t~ead, the latter manifested as a rolling of the baseline. An example is shown in Figure 3.9A. The effect of this deadtime-induced baseline roll is not a significant problem if the spectral lines are narrow compared to the frequency of the
132
Multinuclear Solid-State NMR of lnorganic Materials
A
Baseline = A
6000
2000
4000
0
sink(v-I/o) k(v- Vo)
-2000 -4000 -6000
Shift ppm B line
....
I
I
-800
I
I
I
-1000 -1200 -1400 -1600
sgy chemical shift in ppm Figure 3.9. A. The spectrum over a 1.66 MHz spectral range at 130.32 MHz for 27A1 with the initial deadtime producing a sinc roll of the baseline (taken from Alemany et al. 1991 with permission of the copyright owner). B. At 17.64 MHz the more complex baseline produced by ringing effects at the low frequency with the 89y just visible (taken from Smith and van Eck (1999) with permission of the copyright owner). baseline roll. The practical situation is often more complex than this analytic function, and an example is shown in Figure 3.9B. If the resonance was much broader, the lineshape would be significantly distorted by the baseline roll and it would be difficult to know what the true lineshape was. Working at higher frequency decreases tdead. However, practice often shows that the advantage of removing distortion by working at higher frequency is not as marked as might be anticipated, usually because the start of the FID is corrupted by ringing. Ringing is an all-embracing term to describe effects that induce FID-like voltages in the
Experimental Approaches
133
coil. Typical sources of this effect are the electromagnetic effects of the pulse interacting with the surroundings (e.g. the probe body) so that either mechanical and/or re-radiation effects occur. Acousto-mechanical effects can also occur in the coil and can continue after the pulse is turned off, producing a short-lived pseudo-FID. The overall efficiency (Ec) of this process is given by
klB~
EC =
(3.16)
d v s 1+
4
Vs
where kl and k2 are constants, e is the coil resistivity, d is the bulk density and Vs is the acoustic wave velocity (Fukushima and Roeder 1979). Eq. 3.16 shows that for a constant Bo the situation is worse at lower frequency, and at a constant frequency it is worse at higher Bo. These short-lived voltages can look like broad resonances or baseline distortion. Careful probe design can reduce ringing effects and the considerable work put into reducing these effects has been extensively reviewed by Geronthanassis (1987). In addition to the effects from the probe there is the electronic deadtime, including pulse ringdown (100 ns), preamplifier recovery (800 ns), filter overdrive recovery (1 txs) and ADC conversion droop (200 ns) (Hoult 1979). Magnetoacoustic ringing (up to 200 ~s) can be very significant if careful probe design (e.g. coil wire) is not considered. Samples that exhibit peizoelectric behaviour can lead to very long response times of up to 10 ms.
3.4. STATIC BROAD LINE EXPERIMENTS
3.4.1 Pulsed echo experiments Static powder patterns offer one way of characterising a material, and if spectral features can be observed and the line simulated, accurate determination of the NMR interaction parameters is possible. A simple consideration not to be overlooked is that the theoretical powder lineshape assumes the powder sample to be randomly oriented. This means that the powder should be finely ground. Coarse powders show structure in the lineshape which can complicate the lineshape analysis. Despite the effects outlined above, one-pulse acquisition experiments allow relatively narrow spectral features such as singularities to be recorded over a broad spectral range (Figure 11.3A). Detailed analysis of the singularity positions of the different transitions has allowed the different interactions (e.g. quadrupolar, shift) to be estimated (Baugher et al. 1969). Spectral intensity between the singularities is mostly lost so that true lineshape is not observed.
134
Multinuclear Solid-State NMR of Inorganic Materials
A common way of overcoming deadtime problems is to form a signal with an effective time-zero point outside the deadtime, i.e. an echo. There are many methods for forming such echoes. Most involve two-pulse sequences with the classic spin-echo (Sec. 2.8). In general, the loss of transverse magnetisation is incoherent, but under some circumstances this phase loss can be reversed and echoes observed in solids. If this dephasing is due to effects such as inhomogeneity of the applied magnetic field, chemical shift effects or heteronuclear dipolar effects, all these interactions are proportional to the operator iz. A 180 ~ pulse reverses the direction of iz and hence the magnetisation refocuses. In the other important cases of homonuclear dipolar and quadrupolar interactions echo phenomena cannot really be described classically and the loss of phase coherence is properly described by quantum mechanics (Solomon 1958) by which a complete density matrix description of the system is developed. The lineshape will only be accurately reproduced with the spin-echo if iz is accurately refocused. This will certainly be true at short refocusing times. Since the dipolar coupling and chemical shift do not commute, under some circumstances if there are dipolar couplings, the echo will not necessarily be correctly refocused. The spin-echo for an I spin will also not refocus well if the S-spin undergoes spin fluctuations on the timescale of the experiment. Such interactions will occur for strong homonuclear dipolar coupling or if S nuclei show short T~. Hence if there is strong homonuclear dipolar coupling or short S T1, the I spin is not fully refocused by a spin echo via the heteronuclear dipolar coupling. The efficiency of signal recovery in the echo is greatly reduced as a- is increased and the optimum echo signal amplitude will be observed when -r is equal to the spectrometer dead-time following the 90 ~ rf pulse. The echo decay shape is a good replica of the original FID and its observation can be used to obtain more reliable and quantitative information about solids. Echo methods can be used to observe just the central transition of quadrupolar nuclei, allowing the determination of sites or nuclei with larger quadrupole interactions, or the whole satellite transition manifold can be determined. In forming echoes it is convenient to consider two cases depending on the spacing a- relative to the length of the FID (Tf). A so-called Solomon echo forms if a- < < Tf, whereas a Hahn type of echo forms if a- -> Tf. Early calculations of this behaviour assumed a "hard" pulse regime in which the rf nutation frequency is much greater than the effective frequency of the interaction, so evolution during the pulse is solely governed by the if-pulse and the effect of the interaction can be ignored. With the advent of computers, numerical calculations of echo formation have become possible. These effects have been most widely calculated for the quadrupole interaction (initially the second-order quadrupole interaction) and can be readily extended to the regime where the first-order quadrupole interaction is significant, so evolution during the pulse occurs under both the rf and first-order quadrupole interactions. Calculations have been made for a variety of two-pulse sequences and for varying if-field strengths (Haase and Oldfield 1993, 1993a, Man 1993, 1995). These calculations have been
Experimental Approaches
135
extended to include other interactions as well (Bodart et al. 2000, Man 2000). The general conclusion of this work is that echoes will generally be formed but that to obtain quantitatively reliable information great caution has to be exercised. For quadrupole nuclei it is useful to note that VQ can be deduced from the intensity variation as a function of the length of the second pulse (Haase and Oldfield 1993a, Man 1995). In practice, hard if-pulses are used for uniform excitation of broad lines. Our own work has tended to use an echo sequence with the phase cycling first proposed by Kunwar, Turner and Oldfield (1986) which combines quadrature phase cycling with further cycling designed to cancel direct magnetisation (the remaining FID) and ringing effects: Phase pulse 1: Phase pulse 2: Receiver phase:
xxxxyyyy - x- x- x- x- y- y - y- y xy- x- yxy - x- yxy- x- yxy- x- y - yy - yy - xx - xxy - yy - yx- xx- x
Other phase cycling to prevent distortion due to mis-set pulse lengths (Rance and Byrd 1983) has pointed out that response of a spin system covering a wide frequency range is not simply a reflection of the power distribution in the pulse, but that there are spectral distortion effects with some spins remaining in the same state before and after the pulse. The rotation produced by the second pulse in the two-pulse echo experiment is not critical. In practice, the best choice is to make the second pulse twice the length of the first, the actual length being a trade-off between sensitivity and uniformity of the irradiation. The uniformity can be checked by decreasing the pulselength and seeing if the lineshape changes. The excitation has only sufficient bandwidth to allow detection of the first two singularities of the satellite transitions, but compared to the one-pulse experiment, it provides a better record of the satellite transitions between the singularities, although still below the theoretical expectation. An important practical consideration in recording echoes is that the reason for applying the echo is to move the effective t -- 0 position for the FID outside the region where the signal is corrupted. However, so that phasing problems do not re-emerge, the data sampling rate should be sufficient to allow this point to be accurately defined. If T2 is sufficiently short that an echo time can be used that allows the whole echo (both before and after the maximum) to be accurately recorded without an unacceptably large loss of intensity, there is no need accurately to define the new t = 0 position. Fourier transformation of the whole echo (which effectively amounts to integration between +_w), followed by magnitude calculation removes phasing errors and produces a pure absorption lineshape with a signal-to-noise ~/2 larger than that obtained by transforming from the echo maximum. Rather than using a single echo it has recently been suggested that a train of refocusing w-pulses (selective on the central transition), similar in form to the Carr-PurcellMeiboom-Gill (CPMG) sequence, could have significant advantages. Early work,
136
Multinuclear Solid-State NMR of Inorganic Materials
termed quadrupolar CPMG or spikelet spectroscopy, was developed in Aarhus (Larsen et al. 1997, 1998, 2000). The w-pulses refocus the quadrupolar-induced dephasing. At the start of the sequence an optimised quadrupole echo is used. Hence, the refocused magnetisation decays relatively slowly compared to the timescale of the decay of an individual echo. A string of echoes is recorded and the complete echo train is Fouriertransformed to give a series of sharp bands that follow closely the static lineshape. The inhomogeneous interaction is refocused at the echo maxima. FT of the full echo contains both the homogeneous and inhomogeneous interactions. The homogeneous interaction affects the lineshape of the individual bands and the envelope provides information about the inhomogeneous part of the interaction. An extensive phase cycle is applied for the 90~ - 180~ -- [ 180~ ]n -REC+rec sequence (Larsen et al. 1997).
+2
+3 +rec
xy-x-yxy-x-yxy-x-yxy-x-y yxyxy- x- y- xyxyxy- x- y- x yxyx - y- x - y- x - y- x- y- xyxyx - x- yxy - x- yxy- x- yxy- x- yxy
This sequence provides significant signal enhancement, since the intensity from the powder lineshape is concentrated into a series of sharp bands. This gain is especially true if T~ is long compared to the timescale of the echo train. For example, if T~ required a recycle time of 16 s but 60 echoes could be collected without significant magnetisation decay, an experiment using single echoes would take --~30 minutes compared to a few ms using an echo train. The interpulse delay ('ra) determines the frequency spacing of the sharp lines (1/a'a) that make up the spectrum. The delays must be adjusted so that baseline artefacts are minimised, to give significant improvement in S/N but also to provide enough lines to define accurately the powder lineshape allowing its simulation. Splitting the spectra into a series of narrow bands also has the advantage that low-intensity second sites can be observed, and even if baseline artefacts are not completely removed, functions can be used to correct any residual problems (Section 3.3.2) because the baseline can be observed.
3.4.2 Stepped experiments Several alternative approaches have been developed for recording broad spectral lines, based on the philosophy that although a line is broad it can be recorded stepwise using a series of narrow-banded experiments to overcome the distortions introduced into a broad spectrum. One of these approaches is to carry out a spin-echo experiment using relatively weak rf pulses, recording only the intensity of the on-resonance magnetisation and repeating the experiment at many frequencies to map out the lineshape. The system has to be retuned at each new frequency so that the lineshape is mapped out
Experimental Approaches
137
point by point. This is an extremely laborious process, but the lineshapes are accurately recorded, as for high temperature superconductors and 91Zr in ZrO2 (Bastow and Smith 1992). The titanium signal from YzTi207, with a lineshape 250 kHz wide recorded by this method is shown in Figure 3.10, which also shows a pulse echo experiment for comparison. In the direct experiment, the singularities of the powder lineshape can be readily seen but bandwidth effects (excitation and response) result in the centre of the spectrum having much enhanced intensity compared to the edges, distorting the lineshape. Although these static experiments which step the frequency produce accurate lineshapes, they are relatively time inefficient. The time efficiency can be improved by slowly rotating the sample (--~ 1 rpm) to bring different crystallites into resonance. This approach, termed ROTISSERIE, has been shown to give large savings in the time necessary to produce 14N spectra from ceramics (Yesinowski and Hill 1999). The modulation of the echo amplitude produced by the rotation itself contains
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138
Multinuclear Solid-State NMR of lnorganic Materials
information about the interaction and has been used in an experiment termed STEAMER (Yesinwoski and Hill 1999). Another approach is to use low power, frequency selective pulses (Sindorf and Bartuska 1989) with phase-coherent sampling. A long (1 ms) pulse at --~10 mW acquires a single data point, with each point shifted by a frequency of 1 kHz. The spectrum is obtained directly without the need for Fourier transformation but this approach has not been widely applied. An alternative is to sweep the main magnetic field. There are several examples of solid spectra obtained in this way dating from the earliest days of NMR, but only a limited number of reports have used superconducting magnets. Much of the work to date has used relatively low-resolution magnets. Examples of studies made using superconducting magnets include work on model alumina-containing compounds (Xu et al. 1994), studies of the aluminium sites in catalyst-related materials and on high temperature ceramic superconductors. There is no doubt that this approach will work in all cases. It is possible for a single NMR spectrometer to be capable of both conventional high resolution spectroscopy and also field-sweep operation at relatively little extra cost. The field sweep in the 7.05 T instrument described by Poplett and Smith (1998) was limited to + 0.5 T, sufficient to cover many broad lines. Control of the field is completely automated and integrated with the pulse programme. As with the stepped-frequency experiment, relatively soft pulses are applied, and although strictly the on-resonance part of the magnetisation should be used, direct use of the spin-echo intensity accurately reproduces the lineshape. 27A1 spectra of ot-A1203 obtained by one pulse, spin-echo and field-sweep methods are shown in Figure 3.11. The intensity between the singularities in the field sweep spectrum is much higher and much more closely matches the theoretical expectation. There is also essentially no bandwidth limit on the field sweep experiment so that the outer singularities are easily recorded. The total frequency width of this lineshape is --~1.4 MHz but the approach can be applied to much broader lines, as demonstrated for 27A1 in A13Zr where the frequency width is well beyond 3 MHz (Figure 11.4A). A field-sweep approach could be extremely useful as an alternative method for examining low-y nuclei, where narrowing techniques do not yet offer widespread opportunities for improved resolution.
3.5. ONE-DIMENSIONAL HIGH RESOLUTION TECHNIQUES
3.5.1 Magic angle spinning (MAS) The most widely used technique for solid state NMR observation is MAS. The early design of MAS rotors by Andrew (1981) had a single bearing/drive surface over which compressed gas was forced (Figure 3.12A). This provided a very low friction gas bearing allowing rapid rotation. These rotors were typically --~ 1 cm diameter, and, using compressed air, could be spun to 3-5 kHz. On spinning at 5 kHz the peripheral
ExperimentalApproaches
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acceleration is - 5 • 105g, so only very strong materials are employed. For inorganic solids these rotors were typically made of plastics such as Delrin and Torlon. Commercially available systems for rapid spinning have tended towards variations of the double bearing (DB) design (Doty and Ellis 1981, Figure 3.12B). A large range of diameters of rotor is available commercially (2.5 to 14 mm) with 1.8 mm diameter rotors currently providing MAS of - 50 kHz. The driving force is provided by compressed gas, usually air or nitrogen, forced through a series of nozzles on to vanes, usually on the rotor cap. Very careful design of the optimum nozzle diameter and angle of incidence on to the vanes is needed. The very tight clearance also requires accurate
140
Multinuclear Solid-State NMR of lnorganic Materials
Figure 3.12. MAS rotor design showing A. A single beating Andrew-Beams (Andrew 1981a) and B. a double bearing (Doty and Ellis 1981) with permission of the copyright owner.
machining of tough ceramics. For example, the optimum gap to the beating in a 12 mm rotor is 0.027 mm (Doty 1994). The nature of the gas supply is important, the minimum requirement being that it is dry and free from oil. Nitrogen boil-off is undoubtedly the best drive gas but is much more expensive than using compressed air for this purpose. The very high spinning rates create considerable pressures across the wall from the resulting centripetal force, and material strength is often the limiting factor for safe spinning speed. Careful note needs to be taken of the sample density as this will determine the fastest speed that can safely be used. Polymeric materials have now largely given way to advanced engineering ceramics such as Si3N4 and stabilised ZrO2. For many sequences it is often better to have a uniform rf field over the sample. This can be achieved by decreasing the volume occupied by the sample within the rotor, and spacers can be used. In CRAMPS experiments spacers with spherical cavities often give
Experimental Approaches
141
the best results. Other more specialised designs have been suggested, some for spinning air-sensitive samples (Gay 1984, Lee et al. 1984, Carpenter 1986, Merwin et al. 1989). Even though air bearings are used, allowing cooling by the gas stream to occur, there has been considerable discussion of the magnitude of the frictional heating effects in such systems. In addition to frictional heating, rapid expansion of the gas through the nozzles in the stator brings about the Joule-Thompson effect. There is no doubt that for spectroscopy at accurately known temperature, careful calibration is required. Elements with large shift ranges (such as 2~ or paramagnetic compounds with a large Curie temperature dependence of shift such as 31p in (VO2)2P207(Pan and Gerstein 1991) and ~3C in samarium acetate (Haw 1988) are ideal for this purpose. Lead in PbNO3 has been widely used to determine the temperature in MAS experiments (Bielecki and Burum 1995, Mildner et al. 1995). The 119Sn signal from Sm2Sn207 is also very sensitive to temperature changes (van Moorsel et al. 1995, Grimmer et al. 1997, Langer et al. 1999). It is apparent that the temperature is a strong function of Yr. For 7 mm rotor the temperature rises by --~ 4 K as the spinning rate is increased from 2-5 kHz (Bielecki and Burum 1995) and for a 2.5 mm rotor spinning at 35 kHz the temperature rise is 30-40 K (Langer et al. 1999). Mildner et al. (1995) have modelled the temperature difference as 2
Tsample - Tinle t
--
3~2FlsR2 Vr
(3.17)
where "qs is the viscosity of the gas, R is the radius of the spinner and X is the thermal conductivity of the gas. In addition to the average rise there will also be temperature gradients which must be well characterised. The temperature rise can change the observed chemical shift tensor and the gradient can limit resolution (Brus 2000). To achieve efficient narrowing by MAS, the magic angle has to be accurately set and fine adjustments can be achieved by the quadrupole satellite technique (Frye and Maciel 1982). This technique is based on materials which contain quadrupole nuclei in a nominally cubic local environment, some of which are distorted, causing first-order quadrupole broadening of the non-central transitions at some sites. As the narrowing factor is ~(3COS2[~ -- 1) for a deviation of d[3, the residual broadening factor is 3cos[3sin~3d[3. Hence, the first-order quadrupolar broadening in a compound to be used to set the magic angle should not be too severe, to enable the spectrum to be narrowed into a visible set of sidebands over a limited but not too restrictive a range of angles about the magic value. Far away from the angle, only the sidebands from second-order broadening of the central transition are observable. Then, as the magic angle is approached, the sidebands from the satellite transitions are observed, and become progressively narrower. The bromine resonance of KBr is typically used for this purpose, but with increasing demand for more accurate setting of the magic angle, other compounds are now often used. Other commonly used angle references are 23Na in
142
Multinuclear Solid-State NMR of Inorganic Materials
NaNO2, and a favourite of the authors is 27A1 in Y3A15012, in which the A106 site with a small XQ value can be used for initial setting of the angle (as good as KBr and the carbonyl carbon in glycine). Fine adjustments are made using the A104 site with a much larger XQ. Figure 3.13A shows the changes in the 27A1 FID as the magic angle is approached. Initially, away from the magic angle, only a few sharp rotational echoes are seen. As the magic angle is approached these become sharper, and more importantly they can be seen at longer times. Typically with 24 scans, using a 4 mm rotor, these should be visible out to --~ 5 ms if the angle is well set. In the corresponding spectra (Figure 3.13B) as the correct angle is approached, the satellite transition sidebands become much sharper. For lower frequency probes that will not tune to the region of
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Experimental Approaches
143
23Na, 27A1 and 79'81Br, 85Rb in RbC1 is ideal. Alternative nuclei with small magnetic moments that can be used to set the magic angle include 14N in compounds where the resonance can be narrowed into a set of sidebands over a range that is not too limited. The molybdenum resonance of Mo(CO)6 has also been suggested as a candidate for angle setting since it provides a strong signal with sidebands whose widths are sensitive to angle but can nevertheless be observed a long way off the magic angle. A number of factors result in residual broadening of MAS NMR spectra (Andrew 1972): (a) Instrumental- Bo, B1 inhomogeneity, inaccurate setting of the magic angle, too slow spinning, bulk susceptibility effects, temperature gradients. (b) Residual interactions- residual dipolar coupling, chemical shift distribution, antisymmetric tensor-terms, quadrupole effects. (c) Motional and relaxation effects- spin-lattice and cross relaxation, slow internal motions.
3.5.2 Extraction of parameters from MAS NMR spectra Once the NMR spectrum has been obtained, the NMR interaction parameters must be extracted. The use of the sideband intensity to extract CSA parameters has always been popular. The relative sideband intensities depend on the asymmetry and the anisotropy; Herzfeld and Berger (1980) produced extensive graphical plots allowing the CSA parameters to be deduced from this relative intensity. Now a variety of software packages are available such as WINFIT (Massiot et al. 1994) and SIMPSON (Bak et al. 2000) to simulate solid-state NMR spectra. Some packages assume that when centrebands (e.g. of quadmpole nuclei) are being simulated, one is in the "infinite" spinning speed limit. Alternatively, the low-order moments of spectra can be used to deduce the interactions (Maricq and Waugh 1979, Herreros et al. 2000).
3.5.3 Suppression of spinning sidebands Where there is a combination of different sites, each of which produces a significant number of sidebands, simplification of the spectrum by removal of the spinning sidebands would be extremely helpful. Dixon (1982) introduced the concept of breaking up the rotation period into intervals separated by 180~ pulses. The phase of each spin is inverted by the pulses and the total phase accumulated by each spin can be calculated. Then for all the sidebands to be removed, the sum of the phase for spins contributing to the spinning sidebands needs to vanish, independent of the orientation of the particular crystallite. The basic TOSS sequence has been developed to compensate for imperfections by extension of the applied phase cycles and by paying careful
144
Multinuclear Solid-State NMR of lnorganic Materials
attention to the timings in the pulse sequence. The sequence is sensitive to the accuracy of the 180~ pulses and works by effectively cancelling the magnetisation in the sidebands, although it does not regain the magnetisation in the centreband. Hence, the recorded intensities from a sample containing sites with differing anisotropies will not reflect the site distribution. In addition, when the anisotropy becomes large compared to both the spinning speed and the irradiation width of the 180~ pulses, complete cancellation of the sidebands cannot be achieved. Careful selection of the applied magnetic field can often greatly aid in minimising the sidebands, for example, making it advantageous to accumulate l l9Sn and 2~ spectra at lower applied magnetic fields. For 13C, TOSS has been combined with the sequences for suppression of protonated species (Carduner 1987) described in Section 3.7.1. The TOSS sequence has also been demonstrated for 27A1 (Carduner 1989). This greatly clarified the spectrum when spinning speeds of only a few kHz were available, but has largely been superseded by the use of very much faster spinning speeds. Since sidebands form an important part of MAS spectra, there has been continued development of TOSS. More recently, new shorter sequences have been proposed to optimise TOSS sequences, including one employing 5 almost evenly spaced 180 ~ pulses (Samoson and Tegenfeldt 1995). Antzutkin (1994) has extensively reviewed sequences for manipulating sidebands. An alternative scheme for sideband suppression is to interrupt the evolution of the magnetisation for fractions of the rotor period by using pairs of rf pulses with reversed phases. The interruption means that destructive interference occurs for the magnetisation producing the sidebands (Hong and Harbison 1993). The sequence, termed SELTICS, prevents the sideband magnetisation from refocusing, rather than manipulating the magnetisation to cause cancellation.
3.5.4 Special considerations for MAS of quadrupolar nuclei The technique to be applied depends on the magnitude of VQ (= 3XQ/(2I(2I-1))). For XQ < 0.7 MHz, one pulse acquisition of the entire spectrum is possible (e.g. 27A1 in LaA103). If features of the centreband lineshape can be observed, the quadrupole parameters can be deduced by simulation from the MAS centreband. As the spinning speed and applied magnetic field are increased, sites with larger • values will narrow. Accurate 27A1 lineshapes have thus been recorded for kyanite (XQ up to 10 MHz), andalusite (• up to 15 MHz) and zoisite (XQ up to 10 MHz) (Alemany et al. 1991, 1999, 2000). However, the spectra show distortions from the expected powder pattern until the spinning speed well exceeds the residual second-order quadrupole width. This is shown in Figure 3.14 for the 23Na MAS spectrum of crystalline albite acquired at speeds from 0-10 kHz. The true "infinite" spinning speed lineshape of this material is not obtained until --- 15 kHz at 7.05 T. The spin-echo sequence developed for improving the sensitivity of the static lineshapes from low-~/nuclei (Larsen 1998, 1998a) has
Experimental Approaches
145
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Figure 3.14. 23Na spectra at 7.05 T from crystalline albite (NaA1Si3Os) as the spinning speed is increased from 0 to 10 kHz showing the distortion at low spinning speed from Smith and van Eck (1999) with permission of the copyright owner.
also been used to increase the S/N ratio of the MAS centrebands from sites with large XQ values. In this case, the sequence must be set up so that the spin-echo and rotationalecho effects form coincidentally. When no distinct second-order quadrupole lineshape can be observed, the quadrupole product Po - (Xo +~/2/3)1/2 and the ~cs,isovalue can be estimated by recording the peak position as a function of the applied magnetic field. Eq. 2.150 shows that the field-dependence will give a linear plot (Figure 3.15). Good spectra can also be obtained, often at relatively low spinning speeds, from amorphous materials with a distribution of interactions, provided care is taken to ensure that narrowing of all the sites in the sample occurs and that the spectrum is representative of the whole sample. Eq. 2.152 shows there is a spinning speed that must be exceeded for narrowing to occur. In glassy samples, more sites will narrow as the spinning speed increases, and since these sites have different second-order quadrupole shifts, the composite line will apparently change so that at higher spinning speeds the peak becomes less positive. This shift is accompanied by broadening (Figure 3.16) as Vr increases, since the residual secondorder quadrupole width of the narrowed sites is greater. Quantification of MAS NMR spectra of quadrupolar nuclei posed many problems 10-15 years ago, with a number of 27A1 NMR studies on both crystalline and glassy powders noting that the integrated signal intensity failed to agree both with the known
146
Multinuclear Solid-State NMR of Inorganic Materials Shift of Centre of Gravity (ppm)
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aluminium distribution in different sites, and with the total aluminium content. However, the understanding and available experimental technology have improved markedly over the last few years. Use of the correct experimental conditions (e.g. small tip angles) with higher magnetic fields and faster MAS rates has removed many of the difficulties that gave rise to the idea of "NMR invisible" aluminium. By using small flip angle pulses so that work is performed in the linear regime, sites with widely differing XQ values can produce the same intensity response (a minor problem with this is that typical values of B1 of 75 kHz for 27A1 would require only a 0.4 Ixs pulse to be used!). This is illustrated by the two sites in Y3AlsOl2, whose response is the same at small angles (Figure 3.17). Even after this precaution is taken, quantitative interpretation of quadrupole-perturbed NMR spectra needs to be approached cautiously. One of the sources of error is that it is often only the prominent centreband that can be recorded accurately. The intensity will be distributed between the centreband and the spinning sidebands; although these could be integrated as well, with more than one transition for quadrupole nuclei, partitioning this magnetisation can be problematic. Full-scale simulation of the complete manifold can be time consuming and requires a good S/N ratio of the spinning sidebands. A more pragmatic approach is to record the centreband and
Experimental Approaches -15
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20
spinning speed (kHz) Figure 3.16. The effect of spinning speed on the peak position and the linewidth of 23Na from albite glass at 7.05 T from (Kohn et al. 1998 with permission of the copyright owner).
to correct the observed intensity by a factor corresponding to the amount of magnetisation that appears in the centreband. For the (1/2,-1/2) transition, this depends on the parameter v~/VoVr, while for the satellite transitions it is a function of 11-2 m IvQ/Vr, both being only weak functions of rlQ (Figure 3.18A,B, Massiot et al. 1990). Once these factors are taken into account, the measured integrated intensity can be corrected to give the actual distribution in compounds to a high degree of accuracy, even for sites with widely differing XQ values. An interesting case study is provided by Y3A15012, which has A104 and A106 sites in the structure in the ratio of 3:2. Integration of the
148
Multinuclear Solid-State NMR of Inorganic Materials (1/2,-1/2) intensity per 27A1 nucleus (Arbitrary units) 25
20
15
AI(H20)6] 3+ in aqueous
/
/
\
1
--~o, YAG vo = 900 kHz
I \.. -5
Tp (~ts)
regime
Figure 3.17. The pulse response of 27A1 comparing a solution where XQ = 0 and the two resonances in Y3A15012 with, for the AIO6, XQ = 0.6 MHz and, for the AIO4, XQ = 6 MHz.
1.0
1.0
0.75
0.75
0.50
o.so
B
C
]
Cat a
L @ ~ ~ i a t i o
JL
n
AIO,0/Z,1/2) AIO6(:~3/2, •
0.2:5
0.25
11=0 _
AIO6(:~-5/2,:t:3/2)
rl 0
1
2
3
VQ2/VOVr
4
0
]"-A, 20
40
60
ll-2mlvQ/Vr
80
200
0
AIO4(1/2,-1/2) -200
Z7Al c h e m i c a l shift in p p m
Figure 3.18. The dependence of the intensity of the centreband of an MAS spectrum for the A. central and B. satellite transition as a function of the spinning speed Vr, the quadrupole frequency VQ and the Larmor frequency Vo along with C. the 27A1 MAS NMR spectrum of Y3A15012 at 7.05 T at a spinning speed of 7 kHz. From Massiot et al. (1990) with permission of copyright owner.
149
Experimental Approaches
Table 3.3. Correction of integrated centreband intensities to derive the actual aluminium site distribution in Y3A15012 derived from the MAS spectrum accumulated at 7.05 T spinning at 7 kHz shown in Figure 3.18C.
Factor
A104 signal
A106signal
])Q (kHz) Experimentally integrated intensities of the centreband ( m , - 1/2) transition
900 1
90 1
vQZ/voVr
1.48 1 0.78 0.78
0.015 1 1 1
257 1.78 0 0
25.7 1.78 0.045 0.08
514
51.4
1.11 0 0 0.78 1.28 1.44
1.11 0.304 0.039 1.12 0.89 1
Inherent relative intensity of the transition* Fraction of this transition contributing to the centreband Intensity this transition contributes to the centreband (_+ 3/2, + 1/2)transition [1-2 m[vQ/vr Inherent relative intensity of the transition Fraction of this transition contributing to the centreband Intensity this transition contributes to the centreband (_+ 5/2, + 3/2)transition I1- 2 mlvQ/vr Inherent relative intensity of the transition Fraction of this transition contributing to the centreband Intensity this transition contributes to the centreband Total relative intensity contributed to the centreband from all transitions Corrected centreband intensity to give the actual site distribution Actual intensity normalised to A106 resonance
* See Table 2.3 where the intensity column has been used and the contributions are normalised to that from the (1/2,-l/z) transition.
centrebands of the 7.05 T, 7 kHz 27A1NMR spectrum gives a ratio of 1:1 (Figure 3.18C). In this case, the centreband of the A104 site effectively contains only intensity from the central transition, with 78% of this central transition intensity present and no contribution from the outer transitions (Table 3.3). However, for the A106 site, all the central transition intensity is in the centreband, which also contains a significant contribution from the outer transitions (Figure 3.18C, Table 3.3). Once these factors are corrected for, the centreband intensity turns out to be 1.44:1, close to the value of 1.5:1 expected from the crystal structure. The advantages of fast spinning and high fields are also apparent in other materials. The observed signal of 27A1 MAS N M R spectra of certain zeolites increased from 65 to 95% of that expected from the aluminium content when the magnetic field was increased from 9.4 to 18.8 T; it should also be noted that these were samples in which the maximum • was only 6.3 MHz (Fyfe et al. 2001).
3.5.5 Magic angle spinning observation of satellite transitions Although the majority of N M R studies of non-integer spin quadrupolar nuclei have concentrated on the central transition, observation of the satellite transitions under
150
Multinuclear Solid-State NMR of Inorganic Materials
MAS has distinct advantages. If the quadrupolar interaction is small but non-zero, the second-order quadrupolar structure of the central transition can be obscured by other broadening mechanisms. However, if the powder pattern of the outer transitions can be recorded, XQ and -q can be deduced. As detailed above, this can be carried out in static experiments, but with fast, accurate (--- 0.005 ~ MAS this pattern splits up into a set of sharp sidebands that closely follow the contour of the powder pattern envelope. The MAS approach is less susceptible to initial deadtime problems (Section 3.3.2) and provides sensitivity advantages. The sidebands can be simulated to deduce • and xI, and using conventional pulse spectrometers for solids, this method is applicable to materials with values of XQ up to 2 MHz and 3.3 MHz for spins of 3/2 and 5/2 respectively (Skibsted et al. 1991). In amorphous solids, the change in the extent of the sideband manifold and the shape of the envelope can be used to estimate the average XQ value and distribution for each site. The main practical requirements are for fast, stable spinning precisely at the magic angle and accurate probe tuning. Any variation of the spinning speed or mis-setting of the angle will degrade the resolution of the sidebands. In accumulating the data, the corrections made necessary by if-irradiation and the probe can be achieved by semi-theoretical modification (for samples with smaller values of XQ), or by measuring the spectrum at several frequencies and combining these after applying correction factors (for samples with larger values of XQ). Since single-pulse excitation is used there are also often baseline distortions present, but provided the spinning sidebands are separated so that intermediate baseline can be observed, normal spline fitting using the spectrometer software is adequate. There is no reason why the field sweep method outlined above for accumulating broad static lines cannot be used for such MAS satellite spectra, conveniently making the step size in the field equal to Yr.
3.5.6 Double angle rotation o f quadrupolar nuclei Spinning a sample about two axes simultaneously can be achieved in specialised probes (Figure 3.19, Samoson et al. 1988). From a mechanical point of view, DOR is intriguing since the angular momentum of the inner rotor will tend to inhibit rotation of the outer rotor. However the torque due to the inner rotor may be eliminated (Wu et al. 1990) if the rotation rates of the two rotors satisfy
v,
V2
-- COS ~ l
(,,r / la.,.
(3.~8)
where [31 is determined from the NMR conditions. Since Itr and Iax are the transverse and axial moments of inertia of the inner rotor, the ratio of spinning rates is typically around 5. The stability of this system with respect to changes in rotation rate is best when Vl/V2 is close to, but slightly higher than the value given by the above equation,
Experimental Approaches
151
10' a'. .
9 ;.
13'
2'
4 14
--.-.
Figure 3.19. An example of a DOR rotor from Wu et al. (1990) with permission of the copyright owner. since the system will then self-adjust for small variations of the inner rotor. The magic angle must be set as for MAS experiments by using Na2C204 with a pulse that selectively excites the central transition, the sidebands narrowing as the magic angle is approached. The intensities and phases of the sidebands depend on the relative phases of the two rotors and the ratio of the spinning speeds of the two rotors. The dependence on the phase angle Y2, the instantaneous orientation of the outer rotor, means that by triggering the signal acquisition synchronously at ~2 = 0 and ~2 = q ' r , the inversion symmetry of the odd-order sidebands eliminates the odd-order sidebands upon co-addition of the signals, effectively doubling the outer rotor speed (Samoson and Lippmaa 1989). If the quadrupolar interaction is dominant and the sample is highly crystalline, extremely impressive gains in resolution are observed. One of the major limitations is the relatively slow rotation speed of the large outer rotor. The maximum actual spinning speed that can be routinely obtained from the latest system with active computer control of the gas pressures has been reported in the literature to be --~1800 Hz, and
152
Multinuclear Solid-State NMR of lnorganic Materials
undoubtedly the technology associated with the technique will continue to improve, leading to increased spinning speeds and expanding the application of the technique. Higher spinning speeds not only mean fewer spinning sidebands but will broaden the applicability of the technique to amorphous materials, since the outer rotor speed must exceed the residual width, produced, for example, by a dispersion of parameters such as the chemical shift. An example of DOR narrowing from an amorphous solid has been reported (MacKenzie et al. 2001). The other main problem with DOR is that the rf coil encloses the whole system, giving a small filling factor which leads to low sensitivity. The large coil size also means that the rf generated is quite low. Double tuning for cross-polarisation is also difficult, although such an experiment has been performed (Wu et al. 1992). Simulation of the complete DOR spectrum (centreband plus the spinning sidebands) yields the NMR interaction parameters. However, it is most usual to perform the experiment to give improved resolution and simply quote the measured peak position which appears as the sum of the isotropic chemical and second-order quadrupole shifts. DOR experiments at more than one applied magnetic field allow these different contributions to be separated and hence provide an estimate of PQ (see Section 6.5.1.2). This approach is similar to that using the field variation of the centre-of-gravity of the MAS centreband, but has the advantage that the narrower, more symmetric resonance makes determination of the correct position of the centre of gravity more precise.
3.5.7 Practical implementation o f C R A M P S
The drawback to CRAMPS is that the long pulse trains must be set very accurately with stable pulse lengths and phases. This has discouraged the wider use of the approach, but improved reproducibility of digitally-held information in the most modem spectrometers (rather than analog control) should result in CRAMPS being more routinely used. Accurate control of the phase and amplitude of each channel is necessary. Practical considerations have been detailed (e.g. Jackson and Harris 1988, Maciel et al. 1990, Burum 1990) but a key for success is short (1-2 Ixs) homogeneous 90 ~pulses, so that the sample is often constrained to a spherical volume at the centre of the coil. The smaller volume and the reduction of susceptibility by using a spherical sample results in more homogeneous rf across the sample. The recovery of the probe from the pulse should be rapid, so that the sampling windows can be short, keeping the cycle time down. Since recovery is important, CRAMPS probes tend to have lower Q than conventional CP-MAS probes. The multiple pulse sequence also scales the chemical shift by effectively scaling i: (Section 2.4.2), and this must be carefully calibrated, depending on the pulselength and the cycle time (Haeberlen 1976, Gerstein and Dybowski 1985). The averaging efficiency and frequency range covered is improved if quadrature rather than single-phase detection is employed (Burum et al. 1993). This also
Experimental Approaches
153
means that 19F CRAMPS becomes more amenable, since its much wider chemical shift range often resulted in folding in singly-detected CRAMPS experiments. This is not a problem in CRAMPS with quadrature detection.
3.6. TWO-DIMENSIONAL EXPERIMENTS
3.6.1 Nutation NMR Quadrupole parameters can often be obtained by measurement of the static or MAS spectra of powdered samples as outlined above. Unfortunately, the distinct features of a quadrupolar lineshape are often obscured by the existence of a spread in chemical shifts, residual dipolar broadening and/or the quadrupole interaction. The idea behind nutation NMR is to study the quadrupole interaction in an if-field in the rotating frame, whereby the influence of the chemical shifts is dramatically reduced since they scale linearly with the applied field. At the same time, there is the sensitivity benefit of working in a high static magnetic field. The 2D nutation experiment consists of monitoring the magnetisation as a function of the nutation pulse (h). During tl the spin system is governed in the rotating frame by the Hamiltonian.
H1
__VIIx + Vo(O,c~)(3iZz
_ I2)
(3.19)
with VQ(O,~))- VQ(3COS2 0 - 1 + r/sin 2 0 cos 2q~)
(3.20)
To calculate the nutation spectrum the density matrix formalism is used, in which the Hamiltonian is diagonalised either numerically (Kentgens et al. 1987) or analytically (Pandey et al. 1986, Janssen and Veeman 1988). In Section 2.2.6, the nutation frequency of a quadrupolar nucleus was seen to vary from (I + l/2)v 1 when vl < < 1)Q to vl when 1)1 > > I)Q. The nutation spectra display distinct features in the region where vl --~VQ. Simulation of nutation behaviour has shown that the useful range of rf field-strengths lies with VQ/Vl in the range 0.05-1. The distinct features of nutation spectra allow the quadrupole parameters to be obtained. This means that only quadrupole frequencies up to a certain limit are accessible with currently available rf-fields of 500 kHz in specially dedicated probeheads, and in fields of around 100 kHz in most commercially available systems. This constraint especially limits the investigation of spin I = 3/2 nuclei such as 23Na. In cases where more than one resonance could be observed, one would like to increase the resolution in the F2-dimension by applying MAS. However, as was discussed previously, rotationally-induced spin-locked magnetisation can appear when an
154
Multinuclear Solid-State NMR of lnorganic Materials
rf-field is applied in conjunction with spinning. This process severely distorts the nutation spectrum as it gives rise to strong dispersive components in the 2D spectrum and a large signal at zero frequency. The spinning speed should therefore be kept low (2-4 kHz) so that the nutation signal has decayed within a quarter rotor period at most. It is possible however to simulate the nutation behaviour of quadrupole nuclei under the influence of MAS, from which the quadrupole parameters can be extracted (Nielsen et al. 1992). A 27A1 quadrupole nutation experiment on A1PO4-8 under fast MAS revealed higher resolution in the cross-section at 3v~f compared to that at Vrf. This was attributed to a smaller distribution of orientations contributing to the 3Vrf cross-section (Rocha et al. 1992).
3.6.2 Off-resonance nutation Off-resonance nutation has been developed (Kentgens 1993) to increase the upper limit of quadrupole couplings that are accessible by nutation NMR. In this experiment the nutation behaviour of the spins is monitored in the effective field rather than in the rffield, the effective field being the vector sum of the if-field and the resonance offset. This greatly increases the upper limit of accessible • values since the resonance offsets can be freely chosen and are limited only by the Q of the probe. For increased sensitivity and for the reduction of the zero frequency signal it is beneficial for the magnetisation to nutate perpendicularly to the effective field. Therefore, one can either apply a frequency-stepped adiabatic half passage (Veenedaal et al. 1998) or a soft 90 ~ pulse to bring the magnetisation into the xy-plane; the former option gives the best results. The if-phase is then shifted 90 ~ and the frequency is switched to the required resonance offset. The useful range of resonance offsets is between one to eight times v~, depending on the ratio VQ/V~. After a time t~ the signal is detected on resonance. The phase-coherent frequency switching (switching times of around 200 ns) of the if-field required in this experiment is within the capabilities of most modem spectrometers. An extensive review has dealt with many of the details of off-resonance nutation spectroscopy (Kentgens 1998). Whereas the on-resonance nutation spectra are amplitude-modulated and can be properly phased in the F1 dimension, the off-resonance nutation spectra are phasemodulated and can no longer be phased properly. Therefore, a magnitude calculation is performed after Fourier transformation. Furthermore, because the magnetisation evolves during t~ around the effective field, the sine and cosine components of the modulation are different and the amplitudes of positive and negative F1 signals are unequal. Figure 3.20 shows an example of the strength of the off-resonance nutation technique applied to various samples of ~-alumina impregnated with phosphorus and/or molybdenum. The nutation spectrum of the tetrahedral resonance is very similar to the nutation spectrum of "7-A1203. The octahedral resonance however shows a nutation
155
Experimental Approaches
A
B
-400
C
0
400
Nutationfrequency(kHz) j
~~~~~
|
-400
|
i
'l
!
0
j
u
i
!
400
Nutationfrequency(kHz) Figure3.20. Off-resonance nutation experiments on Mo(12)p(2)/~/-A1203 with an rf field strength of 41 kHz. The normal 1D MAS spectrum A. is shown and the nutation spectra of the tetrahedral B. and octahedral region C. with the tetrahedral region made up of only ~/-A1203 but the octahedral region requiring a superposition of ~/-A1203 and Alz(MoO4)3 from Krauss et. al. (1996) with permission of the American Chemical Society.
spectrum that can only be simulated using two subspectra, one the nutation spectrum of ~/-A1203 and the second with quadrupole parameters similar to those of A12(MoO4)3. This led to the conclusion that impregnation of ~/-A1203 with molybdenum results in an interaction with the 7-A1203 surface to form A12(MoO4)3 (Kraus et al. 1996).
3.6.3 Order-resolved sideband spectra Techniques have been developed for modulating the phase of sidebands according to their order through a series of pulses. These 2D experiments separate the centreband and sidebands in the second dimension even if they strongly overlap in the normal 1D spectrum. In these sequences, sidebands can often be summed by using either hopping
15