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Magnetic resonance line-shape and relaxation time studies of rotational diffusion in liquids Phillips, Paul Stewart 1985

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MAGNETIC RESONANCE LINE-SHAPE AND RELAXATION TIME STUDIES OF ROTATIONAL DIFFUSION IN LIQUIDS by P.S.PHILLIPS B.Sc. The University of Sussex, 1974 M.Sc. The University of B r i t i s h Columbia, 1978 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES DEPT. OF CHEMISTRY We accept th i s thesis as conforming to the^equ^red standard THE UNIVERSITY OF BRITISH COLUMBIA JUNE 1985 © P.S.PHILLIPS, 1985 In presenting t h i s thesis in p a r t i a l fulfilment of the requirements for an advanced degree at the The University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his or her representatives. It i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. DEPT. OF CHEMISTRY The University of B r i t i s h Columbia 2036 Main Mall, Vancouver, Canada V6T 1Y6 Date: JUNE 20th 1985 A b s t r a c t A new n u m e r i c a l a n a l y s i s method, d i s p e r s i o n vs. a b s o r p t i o n p l o t s (DISPA), has been d e v e l o p e d f o r ESR. T h i s method may be used f o r s e m i - q u a n t i t a t i v e l i n e - s h a p e s t u d i e s and i s u s e f u l b o t h as a d i a g n o s t i c and a n a l y t i c a l t o o l . In a d d i t i o n i t p r o v i d e s a method of a u t o m a t i c p h a s i n g f o r magnetic resonance s p e c t r a . Numerous examples of i t s a p p l i c a t i o n s , b oth s i m u l a t e d and e x p e r i m e n t a l a r e p r e s e n t e d , w i t h emphasis on s p i n - p r o b e s t u d i e s . The d i g i t a l a c q u i s i t i o n and p r o c e s s i n g methods used f o r t h e s e s t u d i e s a r e a l s o b r i e f l y d i s c u s s e d . ESR and NMR r e l a x a t i o n time s t u d i e s of the b i s ( d i a l k y l - N - c a r b o d i t h i o a t e ) m e t a l ( I I ) c l a s s of s p i n - p r o b e s have been performed. The T,'s of 1 3 C and 2H e n r i c h e d n i c k e l complex were measured by NMR. The l i n e - w i d t h s of 6 3 C u complex were measured by ESR and a n a l y s e d by R e d f i e l d t h e o r y . The two s e t s of r e s u l t s were combined t o g i v e the p r i n c i p a l elements of the r o t a t i o n a l d i f f u s i o n t e n s o r f o r t h e p y r o l l i d i n e d e r i v a t i v e i n t o l u e n e . T h i s i s t h e f i r s t t ime t h a t ESR and NMR s t u d i e s have been combined t o measure a d i f f u s i o n t e n s o r . A g e n e r a l s t r a t e g y f o r t h i s approach i s p r e s e n t e d . ESR d a t a from p r e v i o u s work has been r e - a n a l y s e d i n t h e l i g h t of the new r e s u l t s . The a n a l y s i s shows t h a t t h e commonly used assumption of i s o t r o p i c d i f f u s i o n i s e x t r e m e l y m i s l e a d i n g . i i T a b l e of C o n t e n t s A b s t r a c t i i L i s t of T a b l e s x i i L i s t of F i g u r e s x i v Acknowledgements x i x PART 1. DISPERSION VS. ABSORPTION PLOTS: DISPA 1 1 . INTRODUCTION 2 1.1 A B r i e f H i s t o r y 3 2. THE BASIC THEORY OF DISPA 5 2.1 The DISPA C i r c l e 5 2.2 L o r e n t z i a n L i n e s D i s t r i b u t e d i n A m p l i t u d e 8 2.3 The E f f e c t of S a t u r a t i o n on L o r e n t z i a n L i n e s ..8 2.4 D i s t r i b u t i o n i n L i n e w i d t h s of L o r e n t z i a n L i n e s 10 2.5 D i s t r i b u t i o n i n Resonant Frequency of L o r e n t z i a n L i n e s 11 2.6 D i s t r i b u t i o n i n Resonant Frequency and A m p l i t u d e 13 2.7 M o d u l a t i o n B r o a d e n i n g 14 2.8 The E f f e c t of D i s p e r s i o n 15 2.9 The Dysonian L i n e 16 3. THE HILBERT TRANSFORM AND DATA PRESENTATION 18 3.1 G e n e r a t i n g t h e D i s p e r s i o n Spectrum 18 3.2 P r e - P r o c e s s i n g of the Spectrum f o r the FFT ...21 3.3 The D i f f e r e n c e P l o t 23 3.3.1 The Index D i f f e r e n c e P l o t 26 3.3.2 The P o l a r D i f f e r e n c e P l o t 29 i i i 3.3.3 The Absorption Difference Plot 30 3.4 The Gaussian Difference Plot 33 4. INSTRUMENTAL DIAGNOSTICS AND APPLICATIONS 34 4.1 Time Constant 34 4.2 Noise 35 4.3 Baseline Artefacts 36 4.4 Amplifier Phasing 37 4.5 Microwave-Bridge Phasing 38 4.6 Saturation 38 4.7 Modulation 39 4.8 Line Truncation and Padding 40 5. THE AUTOMATIC PHASING OF SPECTRA 43 5.1 Basic Theory of Phase Correction 43 5.2 Use of DISPA plots for Phase Correction 45 6. APPLICATIONS TO LINE SHAPE ANALYSIS IN LIQUIDS 52 6.1 C l a s s i f i c a t i o n of Lobes 54 6.2 Notes on the Simulations and Plots 56 6.3 Detecting Two Superimposed Lorentzian Lines ..57 6.4 Detecting Two Superimposed Gaussian Lines ....59 6.5 Detecting Two Overlapping Lorentzian Lines ...62 6.6 Detecting Two Overlapping Gaussian Lines 65 6.7 Detecting Combinations of Lorentzian and Gaussian Lines 67 6.8 Detecting and Measuring Unresolved Hyperfine Couplings 69 6.9 Applications to Line-Shape Analysis of Solids 71 7. EXPERIMENTAL EXAMPLES 73 iv 7.1 Temperature Dependence of Unresolved Hyperfine 73 7.2 Mixtures of Spin Probes 75 7.3 Unresolved Hyperfine Coupling Constants 76 7.4 Using DISPA plots to Detect S a t e l l i t e s 77 7.5 The Detection of Chemical Exchange. Solvation Effects 78 7.6 The Spectrum of Grey Pitch 79 7.7 Graphite Spectra 80 7.8 Coal Spectra 81 7.9 Wood Spectra 84 7.10 Nitroxides in the Slow-Motional Regime and Powder Spectra 86 8. CONCLUSIONS 88 8.1 Summary of Results 88 8.2 Rules-of-Thumb 88 8.3 Conclusions 89 PART 2. RELAXATION STUDIES BY MAGNETIC RESONANCE 91 9. INTRODUCTION TO THE MOTIONAL STUDIES 92 9.1 Introduction 92 9.2 Choice of Spin Probe 98 9.3 Choice of Probe Substituents 101 9.4 Choice of Central Metal 102 v 9.4.1 C e n t r a l M e t a l f o r ESR Exp e r i m e n t s ....102 9.4.2 C e n t r a l M e t a l f o r NMR Exp e r i m e n t s ....103 10. GENERAL THEORY 104 10.1 I n t r o d u c t i o n t o R e d f i e l d Theory 105 10.2 On S p e c t r a l D e n s i t i e s 111 10.3 Choice of the A x i s System 113 10.4 Hydrodynamic Models f o r R o t a t i o n a l D i f f u s i o n 116 11 . GENERAL EXPERIMENTAL 119 11.1 P r e p a r a t i o n of Sodium D i t h i o c a r b a m a t e s and C a r b o d i t h i o a t e s 119 11.2 T r a n s i t i o n M e t a l D i t h i o c a r b a m a t e s 121 11.3 P r e p a r a t i o n of S o l u t i o n s 122 PART 3. ELECTRON SPIN RESONANCE STUDIES 124 12. ESR THEORY 1 25 12.1 The I s o t r o p i c ESR Spectrum 125 12.2 The ESR Problem: Development of the R e d f i e l d E q u a t i o n 129 12.2.1 The T r a n s i t i o n F r e q u e n c i e s 131 12.3 The F i n a l E q u a t i o n 133 12.4 The Debye D i f f u s i o n Model f o r an Asymmetric Ro t o r 140 12.5 S p i n R o t a t i o n a l R e l a x a t i o n 141 13. ESR EXPERIMENTAL 144 v i 13.1 P r e p a r a t i o n of 6 3 C o p p e r ( I I ) C h l o r i d e 144 13.2 P r e p a r a t i o n of C o p p e r ( I I ) D i t h i o c a r b a m a t e Complexes 144 13.3 P r e p a r a t i o n of C o p p e r - f r e e N i c k e l Complexes f o r ESR M a t r i x E x p e r i m e n t s 145 13.4 P o l y c r y s t a l l i n e ESR S p e c t r a 146 13.5 P r e p a r a t i o n of the s o l u t i o n s f o r ESR 146 13.6 ESR Sample Tubes 146 13.7 R e c o r d i n g ESR S p e c t r a 147 13.8 Temperature Measurement i n ESR ex p e r i m e n t s ..150 13.9 F i e l d C a l i b r a t i o n of ESR S p e c t r a 150 13.10 C o l l e c t i o n and A n a l y s i s of ESR S p e c t r a 151 14. ESR ERROR DISCUSSION 152 14.1 The A x i a l Symmetry A p p r o x i m a t i o n f o r the S p i n H a m i l t o n i a n 152 14.2 On A p p r o x i m a t i n g S p e c t r a l D e n s i t i e s .154 14.2.1 The ( U 0 T C ) Z « \ A p p r o x i m a t i o n 155 14.2.2 The {(jiaTc)2«\ A p p r o x i m a t i o n 156 14.2.3 The w a « c j 0 A p p r o x i m a t i o n 156 14.3 C o n t r i b u t i o n s From the N u c l e a r Zeeman Term ..157 14.4 The F i r s t and Second Order C o n t r i b u t i o n 158 14.5 The R e s i d u a l L i n e w i d t h 158 14.5.1 D i p o l a r B r o a d e n i n g 158 14.5.2 Paramagnetic Broadening 159 14.5.3 S o l v e n t C o o r d i n a t i o n 159 14.5.4 I n t e r n a l M o t i o n 159 14.5.5 U n r e s o l v e d H y p e r f i n e 160 14.5.6 Magnetic F i e l d Inhomogeneity 161 14.5.7 S p e c t r o m e t e r P h a s i n g 161 v i i 14.5.8 Time Constant and M o d u l a t i o n 162 14.6 Temperature Inhomogeneity 162 14.7 F i t t i n g A r t e f a c t s and N o i s e 163 14.8 F i e l d C a l i b r a t i o n and C a v i t y S h i f t 163 ESR RESULTS AND DISCUSSION 165 15.1 ESR R e s u l t s 1 65 15.2 Approximate Methods f o r Data A n a l y s i s 166 15.2.1 S i m u l a t i o n s 167 15.2.2 The I s o t r o p i c Assumption 167 15.2.3 The F a s t M o t i o n a l A p p r o x i m a t i o n 168 15.2.4 The A x i a l A p p r o x i m a t i o n 168 15.3 U s i n g the A p p r o x i m a t i o n s 169 15.4 I n v e r s i o n of Data w i t h the A x i a l A p p r o x i m a t i o n 170 15.5 D i r e c t I n v e r s i o n U s i n g The I s o t r o p i c Assumption 173 15.6 I n t e r p r e t i n g Data from I s o t r o p i c I n v e r s i o n s .174 15.7 C o n c l u s i o n s 178 PART 4. NMR STUDIES 180 NMR THEORY 1 8 1 16.1 Chemical S h i f t A n i s o t r o p y (CSA) * 182 16.1.1 I s o l a t i n g the CSA Term 183 16.2 Quadrupolar R e l a x a t i o n 184 v i i i 16.3 S p i n R o t a t i o n a l R e l a x a t i o n 186 16.4 C h o i c e of T, Experiment 187 16.4.1 The I n v e r s i o n Recovery Experiment ....187 16.4.2 The S a t u r a t i o n Recovery Experiment ...188 16.4.3 I n v e r s i o n Recovery v s . S a t u r a t i o n Recovery 189 17. NMR EXPERIMENTAL 190 17.1 P r e p a r a t i o n of the s o l u t i o n s f o r NMR 190 17.2 NMR sample tubes 190 17.3 Powder S p e c t r a 190 17.4 T, Measurements 191 17.5 A n a l y s i s of NMR Data 193 18. NMR ERROR DISCUSSION 195 18.1 On A p p r o x i m a t i n g the S p e c t r a l D e n s i t i e s 195 18.2 R e s i d u a l C o n t r i b u t i o n s t o R e l a x a t i o n 195 18.2.1 I n t e r m o l e c u l a r D i p o l a r R e l a x a t i o n ....196 18.2.2 I n t r a m o l e c u l a r D i p o l a r R e l a x a t i o n ....197 18.2.3 F l u c t u a t i o n s , i n t h e S c a l a r C o u p l i n g s .197 18.2.4 I n t e r n a l M o t i o n ..198 18.3 E r r o r s from Data A n a l y s i s 199 18.4 E r r o r s i n T, Measurements 199 19. NMR RESULTS AND DISCUSSION 201 19.1 1 3 C R e s u l t s 201 19.2 Deuterium T, R e s u l t s 202 19.3 D i s c u s s i o n 203 ix PART 5. COMMENTS ON THE COMBINED NMR-ESR STUDIES 204 20. COMBINED ESR AND NMR RESULTS AND DISCUSSION 205 20.1 I n t r o d u c t i o n 205 20.2 Comments on Data I n v e r s i o n 205 20.3 The D i f f u s i o n Tensor 206 20.4 The Hydrodynamic Model 208 20.5 Summary of the R e s u l t s 211 20.6 A S t r a t e g y f o r Measurement of D i f f u s i o n Tensors 212 20.7 F i n a l Remarks 217 PART 6. NOTES ON THE DIGITAL ACQUISITION OF ESR SPECTRA 219 21. THE DIGITAL ACQUISITION OF ESR SPECTRA 220 21.1 I n t r o d u c t i o n 220 21.2 The Hardware 222 21.3 The B a s i c Problems i n A c q u i r i n g ESR S p e c t r a .225 21.4 ADC R e s o l u t i o n 226 21.5 No. of P o i n t s C o l l e c t e d . The N y q u i s t C r i t e r i o n 227 21.6 F i l t e r i n g Methods 228 21.7 I n t e r p o l a t i o n 231 x 21.8 Box-Car I n t e r p o l a t i o n and F i l t e r i n g 232 21.9 Peak S e a r c h i n g and F i t t i n g 233 21.10 B a s e l i n e F i t t i n g and F l a t t e n i n g 234 21.11 I n t e g r a t i o n of S p e c t r a 237 21.12 A d d i t i o n and S u b t r a c t i o n of S p e c t r a 240 21.13 S h i f t i n g S p e c t r a 243 APPENDICES 247 22. APPENDICES 248 22.1 Nomenclature 248 22.2 The 2H NMR Spectrum of P y r o l l i d i n e 249 22.3 NMR S p e c t r a l Parameters 250 22.4 ESR S p e c t r a l Parameters 252 22.5 Comparison of R e d f i e l d and Other T h e o r i e s ...253 22.6 H a m i l t o n i a n i n a S p h e r i c a l B a s i s 254 22.7 Notes on U n i t s f o r ESR 255 22.8 On P y r o l l i d i n e R i n g Pucker 257 22.9 The F a s t - m o t i o n a l L i m i t 258 22.10 ESR L i n e - w i d t h Data 260 22.11 NMR R e l a x a t i o n D ata. Deuterium 261 22.12 NMR R e l a x a t i o n D ata. 1 3 C - 264 R e f e r e n c e s 267 x i LIST OF TABLES 7.1 Temperature dependence of the u n r e s o l v e d h y p e r f i n e 74 7.2 I d e n t i f i c a t i o n of and notes on the c o a l samples 84 8.1 Summary of r e s u l t s f o r s i m p l e DISPA p l o t s 88 11.1 M i c r o a n a l y s e s f o r D i t h i o c a r b a m a t e s 122 11.2 S o l u b i l i t i e s of m e t a l dtc's 123 12.1 S p e c t r a l d e n s i t y F r e q u e n c i e s 133 12.2 The M a t r i x Elements f o r R e d f i e l d Theory 139 13.1 Spectrum R e c o r d i n g C o n d i t i o n s 148 14.1 E f f e c t of h y p e r f i n e on c o r r e l a t i o n t i m e s 161 14.2 E f f e c t of Phase on Observed L i n e - w i d t h s 162 15.1 S p e c t r a l d e n s i t i e s 166 15.2 ESR d a t a i n v e r t e d w i t h a p p r o x i m a t i o n s .....170 15.3 A x i a l a p p r o x i m a t i o n used w i t h CuPydtc i n t o l u e n e 171 15.4 A x i a l a p p r o x i m a t i o n used w i t h CuMeOddtc 172 15.5 A x i a l a p p r o x i m a t i o n used w i t h CuMeOddtc 172 15.6 Comparison of T C ' S 174 15.7 R e l a t i v e r e l a x a t i o n c o n t r i b u t i o n s 178 19.1 1 3 C T,'s 201 19.2 Deuterium T / s 202 19.3 The d i f f u s i o n t e n s o r from 1 3 C and 2H d a t a 203 20.1 The d i f f u s i o n t e n s o r 207 20.2 F r i c t i o n c o e f f i c i e n t s f o r the probe 210 20.3 P r e d i c t e d d i f f u s i o n c o e f f i c i e n t s ..210 x i i 22.1 2H relaxation times for neat d 9 p y r o l l i d i n e 249 22.2 Line-width data for CuPydtc in chloroform ...260 22.3 T=3 1 OK. 61 . 4MHz 261 22.4 T=323K. 61.4MHz 261 22.5 T=333K. 61.4MHz 262 22.6 T=310K. 30.7MHz 262 22.7 T=310K. 30.7MHz ...262 22.8 T=323K. 30.7MHz 263 22.9 T=310K. 50.3MHz 264 22.10 T=310K. 100.7MHz ..264 22.11 T=323K. 50.3MHz 264 22.12 T=323K. 100.7MHz 265 22.13 T=333K. 50.3MHz 265 22.14 T=333K. 50.3MHz ' 265 22.15 T=333K. 100.7MHz 266 xi i i LIST OF FIGURES 2.1 Typical Cole-Cole format DISPA plot 8 2.2 DISPA for superimposed Lorentzian Lines 10 2.3 DISPA c i r c l e in polar coordinates 12 2.4 DISPA for overlapping Lorentzian lines 13 2.5 DISPA plot for a poorly phased l i n e . 0=10° 16 2.6 The Dysonian Line-shape 17 3.1 Flow-chart for generating the dispersion data 23 3.2 Diagrammatic d e f i n i t i o n of a difference plot 25 3.3 Diagrammatic d e f i n i t i o n of the lobe parameters 26 3.4 Linear and square root indexed difference plots 27 3.5 Logarithmic, LN, and Lorentzian, LZ, indexed difference plots 28 3.6 Logarithmic indexed plot as a function line-width/sweep-width. 29 3.7 Polar difference plot 30 3.8 The absorption difference plot 31 3.9 Typical Cole-Cole and difference DISPA plot 32 4.1 Eff e c t of a large PSD f i l t e r on a DISPA plot 35 4.2 E f f e c t of noise on a DISPA plot 36 4.3 DISPA plot showing the e f f e c t of baseline artefacts 37 4.4 DISPA plot for a mis-phased microwave bridge 38 4.5 DISPA plot for an overmodulated l i n e 40 4.6 DISPA plot for a truncated Lorentzian l i n e 40 4.7 DISPA plot for a truncated l i n e with unresolved hyperfine 41 xiv 4.8 Effect of various padding schemes on the DISPA plot 42 5.1 Radial difference plots for various phase angles 46 5.2 Difference plot lobe asymmetry as a function of phase angle 46 5.3 Flow chart for the automatic phase correction of spectra 48 5.4 Phase error as a function of l i n e - p o s i t i o n 49 5.5 The center l i n e for Fremies salt before ( l i g h t l i ne) and after (heavy li n e ) automatic phase correction 50 5.6 The r a d i a l difference plot corresponding to the diagram above 50 5.7 An unidentified r a d i c a l before ( l i g h t l i n e ) and after (heavy l i n e ) automatic phase correction 51 6.1 The influence of integration on resolution 54 6.2 C l a s s i f i c a t i o n of difference plots 55 6.3 Miscellaneous c l a s s i f i c a t i o n of difference plots 56 6.4 DISPA plots for superimposed Lorentzian l i n e s 58 6.5 DISPA plots for superimposed Lorentzian l i n e s 59 6.6 DISPA plot for superimposed Gaussian l i n e s 60 6.7 DISPA plot for superimposed Gaussian l i n e s 60 6.8 DISPA plot for superimposed Gaussian l i n e s 61 6.9 DISPA plot for superimposed Gaussian l i n e s 61 6.10 DISPA plot for superimposed Gaussian l i n e s 62 6.11 DISPA plot for superimposed Gaussian l i n e s 62 6.12 DISPA plot for overlapping Lorentzian l i n e s 63 xv 6.13 DISPA p l o t f o r o v e r l a p p i n g L o r e n t z i a n l i n e s 64 6.14 DISPA p l o t f o r o v e r l a p p i n g L o r e n t z i a n l i n e s 64 6.15 DISPA p l o t f o r o v e r l a p p i n g L o r e n t z i a n l i n e s 65 6.16 DISPA p l o t f o r o v e r l a p p i n g G a u s s i a n l i n e s 66 6.17 DISPA p l o t f o r o v e r l a p p i n g G a u s s i a n l i n e s 66 6.18 DISPA p l o t f o r a m i x t u r e of a L o r e n t z i a n and Ga u s s i a n l i n e s 67 6.19 DISPA p l o t f o r a m i x t u r e of a L o r e n t z i a n and Ga u s s i a n l i n e s 68 6.20 DISPA p l o t f o r a m i x t u r e of a L o r e n t z i a n and Ga u s s i a n l i n e s 68 \ 6.21 DISPA p l o t f o r a m i x t u r e of a L o r e n t z i a n and Ga u s s i a n l i n e s 69 6.22 DISPA p l o t f o r u n r e s o l v e d h y p e r f i n e 70 6.23 C a l i b r a t i o n c h a r t f o r u n r e s o l v e d h y p e r f i n e 71 7.1 D i f f e r e n c e p l o t s f o r TEMPO and TEMPONE 74 7.2 The DISPA p l o t f o r two superimposed s p i n l a b e l s 75 7.3 The DISPA p l o t f o r an a m p h i p a t h i c s p i n - p r o b e 76 7.4 The DISPA p l o t f o r 6 5 C u M e 2 d t c i n t o l u e n e 77 7.5 A S q u a r e - r o o t d i f f e r e n c e p l o t showing s a t e l l i t e s 78 7.6 DISPA p l o t showing Chemical Exchange 79 7.7 The DISPA p l o t f o r g r e y - p i t c h 80 7.8 DISPA p l o t f o r SP1 g r a p h i t e 81 7.9 DISPA p l o t s f o r v a r i o u s c o a l samples 82 7.10 DISPA p l o t s f o r v a r i o u s c o a l samples 83 7.11 DISPA p l o t f o r n a t u r a l decayed wood 85 x v i 7.12 DISPA p l o t f o r decayed wood a f t e r i r r a d i a t i o n 85 7.13 DISPA p l o t f o r decayed wood a f t e r i r r a d i a t i o n and r e l a x a t i o n 86 7.14 DISPA p l o t f o r a powder spectrum 87 7.15 DISPA p l o t f o r a n i t r o x i d e i n a membrane 87 9.1 The g e n e r a l s t r a t e g y 98 9.2 T y p i c a l m etal d i t h i o c a r b a m a t e 101 10.1 A x i s system f o r t e n s o r s 115 12.1 T y p i c a l m e t a l d i t h i o c a r b a m a t e spectrum 128 12.2 T r a n s i t i o n diagram 132 13.1 The Spectrometer 149 14.1 S p e c t r a l d e n s i t i e s vs. f r e q u e n c y 156 14.2 L i n e - w i d t h E r r o r s f o r the S p e c t r a l D e n s i t y A p p r o x i m a t i o n s 1 57 15.1 P y r o l l i d i n e dt c l i n e - w i d t h d a t a 165 15.2 The e f f e c t of a n i s o t r o p y on TJ /T p l o t s 176 15.3 A 77/T p l o t from p r e v i o u s work 177 17.1 Schematic of the p u l s e sequence 191 17.2 T y p i c a l IR d a t a s e t 192 17.3 T y p i c a l SR d a t a s e t 193 20.1 The probe as an e l l i p s o i d 209 21.1 B l o c k diagram of the a c q u i s i t i o n system 223 21.2 F l o w - c h a r t f o r the s o f t w a r e of the a c q u i s i t i o n system 224 21.3 Flow c h a r t f o r i n t e r a c t i v e b a s e l i n e f l a t t e n i n g 237 x v i i 21.4 The e f f e c t of low d a t a d e n s i t y on i n t e g r a t i o n 238 21.5 I n t e g r a t i o n e r r o r s v s . d a t a d e n s i t y 239 21.6 I n t e g r a t i o n e r r o r s v s . d a t a d e n s i t y 240 21.7 I n t e g r a t i o n e r r o r s v s . d a t a d e n s i t y 240 21.8 Spectrum of the f r e e s p i n - p r o b e 242 21.9 Spectrum of f r e e and bound s p i n - p r o b e 243 21.10 Spectrum of a bound s p i n - p r o b e 243 22.1 Chemical s h i f t v a l u e f o r n i c k e l dtc's 251 22.2 C o u p l i n g parameters (Hz) f o r n i c k e l dtc's 252 22.3 The powder spectrum f o r N i E t 2 d t c 252 xvi i i Acknowledgements I w i s h t o s i n c e r e l y thank Dr. F.G. H e r r i n g f o r h i s h e l p , g u i d a n c e and p a t i e n c e throughout my r e s e a r c h . I would a l s o l i k e t o thank D r . K . A . M i t c h e l l and D r . R . F . S n i d e r f o r u s e f u l comments on the t h e s i s , D r . A . S t o r r f o r the e i g h t year l o a n of h i s m o l e c u l a r models and Kam S u k u l , Mike H a t t o n and o t h e r members of the e l e c t r o n i c shop f o r t h e i r never e n d i n g m a i n t a i n e n c e of the equipment. I a l s o w i s h t o acknowledge the C h e m i s t r y Department of the U n i v e r s i t y of B r i t i s h Columbia f o r p r o v i d i n g f i n a n c i a l s u p p o r t , d e s p i t e an u n f a v o u r a b l e economic c l i m a t e , d u r i n g the c o u r s e of t h i s work. x i x "Onwards always Onwards, In Si I e nee and in Gl oom. . . " 1 1 "Fungus the Bogeyman", Raymond B r i g g s , Hamish H a m i l t o n , London (1977) xx PART 1. DISPERSION VS. ABSORPTION PLOTS: DISPA 1 1. INTRODUCTION Any l i n e i n a spectrum may be c h a r a c t e r i s e d by i t s a m p l i t u d e , p o s i t i o n and shape. The a m p l i t u d e i s u s u a l l y a f u n c t i o n of t h e c o n c e n t r a t i o n of the s p e c i e s a s s o c i a t e d w i t h the l i n e . The p o s i t i o n g i v e s s t r u c t u r a l i n f o r m a t i o n . The shape i s u s u a l l y assumed t o be L o r e n t z i a n and the w i d t h of the l i n e i s then used t o complete the c h a r a c t e r i s a t i o n . The w i d t h of t h e l i n e can g i v e i n f o r m a t i o n about the s p e c i e s ' dynamics. In r e c e n t y e a r s t h e r e has been g r e a t i n t e r e s t i n e x t r a c t i n g m o t i o n a l i n f o r m a t i o n from ESR l i n e - w i d t h s . I t i s v e r y i m p o r t a n t , f o r these s t u d i e s , t o be sure t h a t t h e l i n e i s i n f a c t L o r e n t z i a n , or the l i n e - w i d t h w i l l not a c c u r a t e l y r e f l e c t the r e l a x a t i o n time and hence the motion of t h e s p e c i e s . A q u i c k and easy method f o r a s s e s s i n g d e v i a t i o n s from the L o r e n t z i a n l i n e - s h a p e would be of g r e a t v a l u e t o such s t u d i e s . To t h i s end we have d e v e l o p e d a method of a n a l y s i n g the l i n e - s h a p e of ESR s p e c t r a . The method i s based on C o l e - C o l e p l o t s (1) and not o n l y d e t e c t s d e v i a t i o n s from L o r e n t z i a n b e h a v i o u r , but a l s o i d e n t i f i e s the cause of t h e d e v i a t i o n . I t i s thus u s e f u l as both a d i a g n o s t i c and a n a l y t i c a l t o o l . A p l o t of an a b s o r p t i o n s i g n a l vs. i t s c o r r e s p o n d i n g d i s p e r s i o n s i g n a l w i l l produce an a p p r o x i m a t e l y c i r c u l a r graph. Such a p l o t i s known (here) as a DISPA p l o t . I f t h e a b s o r p t i o n s i g n a l i s L o r e n t z i a n then th e DISPA p l o t i s a c i r c l e . I f t h e s i g n a l i s d i s t o r t e d by, f o r i n s t a n c e , a 2 3 second l i n e , then a d i s t o r t e d e l l i p s e i s produced. The graph's d e v i a t i o n from a c i r c l e i s c h a r a c t e r i s t i c ( o f t e n u n i q u e l y so) of the d i s t o r t i n g mechanism. DISPA p l o t s t h u s p r o v i d e a method of q u a n t i t a t i n g the shape of a l i n e and f o r d e t e c t i n g u n r e s o l v e d f e a t u r e s . However, i t s h o u l d be emphasised t h a t the u t i l i t y of DISPA l i e s not o n l y w i t h the i n f o r m a t i o n o b t a i n e d , but a l s o w i t h the ease w i t h which i t i s o b t a i n e d . A DISPA p l o t r e q u i r e s no s p e c i a l experiment and o n l y t a k e s a few minutes a t most. S i m u l a t i o n s , f o r example, can t a k e days. 1.1 A BRIEF HISTORY D i s p e r s i o n vs. a b s o r p t i o n p l o t s have been l o n g used i n d i e l e c t r i c s t u d i e s (as C o l e - C o l e p l o t s ) t o a n a l y s e r e l a x a t i o n mechanisms (1). There have been b r i e f a l l u s i o n s t o DISPA p l o t s i n magnetic resonance l i t e r a t u r e (2)(3)(4), but i t s f u l l p o t e n t i a l as a t o o l i n l i n e - s h a p e a n a l y s i s was not r e a l i s e d u n t i l d e v e l o p e d f o r NMR by M a r s h a l l (5)(6)(7)(8). I t s a p p l i c a t i o n t o ESR was i n i t i a l l y hampered by the d i f f i c u l t i e s of p r o d u c i n g ESR d i s p e r s i o n s p e c t r a . T h i s problem was c i r c u m v e n t e d by u s i n g the K r a m e r s - K r o n i g r e l a t i o n s (9) t o g e n e r a t e d i s p e r s i o n s p e c t r a from a b s o r p t i o n s p e c t r a . However, t h i s method proved i m p r a c t i c a l u n t i l a more e f f i c i e n t a l g o r i t h m u t i l i s i n g the f a s t F o u r i e r t r a n s f o r m (FFT) was i n t r o d u c e d (10). I t r a p i d l y became app a r e n t t h a t d a t a p r o c e s s i n g f o r ESR s p e c t r a was c o n s i d e r a b l y d i f f e r e n t (and more d i f f i c u l t ) than i t was f o r 4 NMR s p e c t r a and the development of DISPA f o r t h e s e two t e c h n i q u e s d i v e r g e d . M a r s h a l l c o n t i n u e d t o r e f i n e DISPA f o r NMR (11)(12)(13) and H e r r i n g and P h i l l i p s (10)(14) d e v e l o p e d the method f o r use w i t h ESR. Only the l a t t e r r e s e a r c h w i l l be d i s c u s s e d h e r e . 2 2 Much of t h i s work i s a p p l i c a b l e t o NMR and some o v e r l a p w i t h M a r s h a l l ' s work i s i n e v i t a b l e . 2. THE BASIC THEORY OF DISPA A l i n e i s c h a r a c t e r i s e d by i t s w i d t h , h e i g h t , a r e a , p o s i t i o n and shape. A t h e o r e t i c a l t r e a t m e n t of DISPA p l o t s s h o u l d take a l l of these v a r i a b l e s i n t o a c c c o u n t . U n f o r t u n a t e l y , g e n e r a l a n a l y t i c a l e x p r e s s i o n s f o r most of the cases do not e x i s t so the b a s i c t h e o r y i s r e s t r i c t e d t o s i m p l e d i s t r i b u t i o n s of L o r e n t z i a n l i n e s . More complex cases have t o be d e a l t w i t h e m p i r i c a l l y . 2.1 THE DISPA CIRCLE In magnetic resonance the a b s o r p t i o n (A(CJ)) and d i s p e r s i o n (D(o))) s i g n a l s f o r a s i n g l e L o r e n t z i a n l i n e a r e g i v e n by (15) al \ M 0 7 B i T 2 * \ C 0 > = 1 + T 2 (l0-cjQ) Z + 7 2 B 2 T i T 2 D ( a , ) = 1+TlU-o> 0H +7 2BiT 1T 2 ( 2 ' 1 ) The e x p e r i m e n t a l l y o b s e r v e d s i g n a l s , v f o r a b s o r p t i o n , u f o r d i s p e r s i o n , a r e 3 3 S t r i c t l y s p e a k i n g , f o r ESR, the s i g n a l s a r e the d i f f e r e n t i a l s of thes e f u n c t i o n s . The a b s o r p t i o n s i g n a l i s always i n t e g r a t e d b e f o r e use so t h i s i s of l i t t l e immediate consequence. 5 6 Ab v ~ 1 + (bp72+S „ - Apb 2 " " 1 + (bpF+S ( 2 * 2 ) where A i s the amplitude of v at resonance (~M0yBy), b = T 2, p = (CJ-O) 0) and S i s the saturation term ( 7 2 B 2 T 1 T 2 ) . If we note that v=bpu we can eliminate the parameter p from Eqn.2.2 and after rearranging we get ( i f there i s no saturation, i.e., S<<1) u2 + v 2 = bAv or u 2 + (v-ibA) 2 = ( i b A ) 2 (2.3) i f we l e t £bA=R then we get u2 + (v-R) 2 = R2 (2.4) This i s a c i r c l e of radius R displaced along the v axis by R (Fig.2.1). R i s proportional to the spectrum line-width and amplitude. A plot of u vs. v superimposed on a reference c i r c l e (a c i r c l e of radius equal to the maximum of v, R, in th i s case and displaced along the v axis by R) i s c a l l e d a DISPA plot (in the Cole-Cole format). Such a plot i s only a c i r c l e for 7 a pure L o r e n t z i a n l i n e . G e n e r a l l y a d i s t o r t e d e l l i p s e i s ob s e r v e d (see F i g . 2 . 1 ) . The d i s t o r t i o n i s c h a r a c t e r i s t i c f o r the mechanism c a u s i n g d e v i a t i o n s from L o r e n t z i a n b e h a v i o u r . Note from Eqn.2.4 t h a t the fre q u e n c y dependence (the p term) has been e l i m i n a t e d so t h a t DISPA p l o t s a r e independent of l i n e p o s i t i o n . A l s o , i f the spectrum i s s c a l e d t o a f i x e d a m p l i t u d e b e f o r e p l o t t i n g , the l i n e - w i d t h and a m p l i t u d e dependence of DISPA p l o t s a r e removed. In magnetic resonance the s p e c t r a l l i n e s can o f t e n be c o n s i d e r e d as c o m p o s i t e s of,many L o r e n t z i a n l i n e s d i s t r i b u t e d i n a m p l i t u d e , f r e q u e n c y or w i d t h (and c o m b i n a t i o n s t h e r e o f ) . The v a r i o u s c a s e s a r e d i s c u s s e d below. (AR) Figure 2.1. T y p i c a l C o l e - C o l e format DISPA p l o t f o r a s i n g l e l i n e . The s o l i d l i n e i s the r e f e r e n c e c i r c l e . The diamonds a r e t h e d a t a . 8 2.2 LORENTZIAN LINES DISTRIBUTED IN AMPLITUDE The a m p l i t u d e , A, i n Eqn.2.2 i s s i m p l y a h e i g h t s c a l i n g f a c t o r . So p r o v i d i n g t h a t the l i n e s have the same w i d t h and p o s i t i o n a d d i n g L o r e n t z i a n l i n e s t o g e t h e r j u s t produces a DISPA c i r c l e of l a r g e r r a d i u s . As the s p e c t r a a r e always s c a l e d t o a c o n s t a n t h e i g h t f o r DISPA s t u d i e s t h i s e f f e c t i s not o b s e r v e d . The c o r o l l a r y t o t h i s i s t h a t a n y t h i n g t h a t changes o n l y the spectrum a m p l i t u d e ( s m a l l power and m o d u l a t i o n changes, a m p l i f i e r g a i n etc.) does not a f f e c t the DISPA p l o t f o r t h a t l i n e . T h i s i s t r u e even i f the l i n e i s n o n - L o r e n t z i a n . Another p o i n t i s t h a t , f o r a s i n g l e L o r e n t z i a n l i n e , a change i n l i n e - w i d t h o n l y a f f e c t s the a m p l i t u d e and so i s not o b s e r v e d i n DISPA p l o t s . These r a t h e r t r i v i a l r e s u l t s l e a d t o the i m p o r t a n t c o n c l u s i o n t h a t , f o r a s i n g l e L o r e n t z i a n l i n e , homogeneous b r o a d e n i n g mechanisms do not a f f e c t the DISPA p l o t . The most commonly e n c o u n t e r e d homogeneous b r o a d e n i n g mechanisms a r e ; l i f e t i m e b r o a d e n i n g due t o exchange or m o t i o n , d i p o l a r b r o a d e n i n g and microwave s a t u r a t i o n . 2.3 THE EFFECT OF SATURATION ON LORENTZIAN LINES I f we e l i m i n a t e p from our e q u a t i o n s as b e f o r e , but r e t a i n t h e s a t u r a t i o n term S we get v2(1+S) + u2 = Abv (2.5) 9 i f we l e t q=v/(1+S) then we o b t a i n u2 + ( v q - b A / q ) 2 = ( i b A / q ) 2 (2.6) As p o i n t e d out by Abragam (4) and shown n u m e r i c a l l y by M a r s h a l l (11), t h i s i s an e l l i p s e whose major a x i s i s p a r a l l e l t o the u a x i s The l e n g t h of which depends on the microwave power. The p o s i t i o n and w i d t h of the e l l i p s e a r e a l s o a f u n c t i o n of microwave power, but t h i s e f f e c t d i s a p p e a r s because the a b s o r p t i o n s p e c t r a a r e s c a l e d t o a c o n s t a n t a m p l i t u d e , vide supra. The a b s o r p t i o n l i n e i n the presence of s a t u r a t i o n i s s t i l l L o r e n t z i a n , o n l y i t s w i d t h and a m p l i t u d e change. S i m i l a r l y f o r the d i s p e r s i o n l i n e . The e l l i p t i c a l b e h a v i o u r a r i s e s because the a m p l i t u d e dependence on microwave power of a s a t u r a t e d d i s p e r s i o n l i n e i s d i f f e r e n t from t h a t of a s a t u r a t e d a b s o r p t i o n l i n e (16). However, the d i s p e r s i o n l i n e i s g e n e r a t e d by a H i l b e r t t r a n s f o r m * of the a b s o r p t i o n l i n e . The computer program j u s t has a L o r e n t z i a n d a t a s e t of a g i v e n w i d t h and h e i g h t and g e n e r a t e s the c o r r e s p o n d i n g ( i n w i d t h and h e i g h t ) d i s p e r s i o n l i n e . The s a t u r a t i o n term does not appear e x p l i c i t l y so the e l l i p t i c a l b e h a v i o u r d i s a p p e a r s from the DISPA p l o t . 9 The H i l b e r t t r a n s f o r m i s o n l y v a l i d f o r l i n e a r , i . e . , u n s a t u r a t e d , systems. (17) 10 2.4 DISTRIBUTION I_N LINEWIDTHS OF LORENTZIAN LINES I f we assume the l i n e i s composed of s e v e r a l L o r e n t z i a n l i n e s of the same r e s o n a n t f r e q u e n c y , but d i f f e r e n t r e l a x a t i o n times then we have t o s u b s t i t u t e Ab=ZA.b. i n t o i i i Eqn.2.3, 5 hence we get / c i r c l e s If ut)2 + [f ( v / - i A / b / ) 3 2 = [f i A . b , ] 2 ] (2.7) where u. = -/ -/ etc. (2.8) 1 1+(bp /) 2+S Eqn.2.7 r e p r e s e n t s a s e t of c i r c l e s each of r a d i u s b./2 and d i s p l a c e d by b^/2, which, when summed, w i l l form a d i s t o r t e d o b l a t e e l l i p s e ( F i g . 2 . 2 ) . D ID Figure 2.2. DISPA f o r superimposed L o r e n t z i a n L i n e s . 5 A and b a r e l i n k e d by the a r e a of the l i n e , w h i c h i n t u r n i s p r o p o r t i o n a l t o t h e c o n c e n t r a t i o n of the s p e c i e s . However, i t i s sometimes c o n v e n i e n t t o r e g a r d t h e s e p a r a m e t e r s as independent v a r i a b l e s . The r a d i u s of the i ' t h c i r c l e depends on both the l i n e - w i d t h and c o n c e n t r a t i o n of s p e c i e s i so i t i s not p o s s i b l e t o make u s e f u l q u a n t i t a t i v e s t a t e m e n t s about the d i s t o r t i o n e x cept maybe f o r the case of i=2. The d i s t r i b u t i o n f o r i>1 i s v e r y common i n d i e l e c t r i c s t u d i e s . I t s c h a r a c t e r i s a t i o n has been d i s c u s s e d by C o l e & Davidson (18). 2.5 DISTRIBUTION IN RESONANT FREQUENCY OF LORENTZIAN LINES In t h i s case we s u b s t i t u t e p= I p. i n t o Eqn.2.3, where so t h a t A^ . r e p r e s e n t s the s h i f t of the i ' t h l i n e from the c e n t e r of the spectrum, w 0. Eqn.2.3 i s i n an i n c o n v e n i e n t form f o r t h i s s u b s t i t u t i o n , but can be r e c a s t i n t o a c o n v e n i e n t form u s i n g p o l a r c o o r d i n a t e s . A c i r c l e i n p o l a r c o o r d i n a t e s i s g i v e n by r=2Rcos0=Abcos0. See F i g . 2 . 3 f o r a d e f i n i t i o n of terms. 1 2 F i g u r e 2.3. DISPA c i r c l e i n p o l a r c o o r d i n a t e s . The a n g l e , 8, i s g i v e n by 6 = a r c t a n { D ( c j ) / A ( c j ) } = a r c t a n { b p } ( 2 . 9 ) So f o r a d i s t r i b u t i o n o f r e s o n a n t f r e q u e n c y we g e t I r . = A b c o s { a r c t a n Z [ b ( p + A . ) ] } / ' i 1 ( 2 . 1 0 ) N o t e t h a t t h i s e q u a t i o n r e p r e s e n t s one c i r c l e . T he p o s i t i o n o f a n y g i v e n p o i n t on t h e c i r c l e , f o r a g i v e n f r e q u e n c y , p, d e p e n d s on A.. T h e c o m p o s i t e f i g u r e i n t h i s c a s e i s d i f f i c u l t t o v i s u a l i s e , b u t i t w i l l be a d i s t o r t e d p r o l a t e e l l i p s e ( F i g . 2 . 4 ) . 1 3 ID Figure 2.4. DISPA for overlapping Lorentzian l i n e s . 2.6 DISTRIBUTION IN RESONANT FREQUENCY AND AMPLITUDE This i s an important d i s t r i b u t i o n as i t includes the case of unresolved hyperfine structure. The DISPA plot i s simply given by Zr. = bIA.cos{arctanZ[(b.p+A.)]} (2.11) / i The exact shape of the DISPA plot w i l l depend on A. and how the amplitudes are d i s t r i b u t e d , but i t w i l l be, as above, a distorted prolate e l l i p s e . If i s constant and Af. follow the binomial d i s t r i b u t i o n , then v represents a l i n e with unresolved hyperfine coupling (approximately the Voigt line-shape). In the l i m i t of A —> 0 Eqn.2.11 gives the result for a Gaussian l i n e (A. d i s t r i b u t e d normally) or a 14 V o i g t l i n e (19) (A^ d i s t r i b u t e d as a c o n v o l u t i o n of a L o r e n t z i a n and a G a u s s i a n ) . 2.7 MODULATION BROADENING The a b s o r p t i o n and d i s p e r s i o n s i g n a l s m o d i f i e d t o a l l o w f o r m o d u l a t i o n a re (10)(20) v _ y AbF(k) k=0 1+b 2(p+kw ) 2 m _ A b 2 F ( k ) (p+kcj ) " = £ o H F T p + k w F " 1 -m where F ( k ) = J? {7B /CJ (2.12) K m m 70 m J i s a B e s s e l f u n c t i o n of the f i r s t k i n d , co i s the m o d u l a t i o n f r e q u e n c y , B^ i s the m o d u l a t i o n a m p l i t u d e and 7 i s the e l e c t r o n gyromagnetic r a t i o . I f we combine A and F ( k ) ( t o g i v e A.) and note the s i m i l a r i t y of kw and A. i n ^ k m i Eqn.2.11 we get I r . = lA^cosUrctantlb^p+kcjp ]} (2.13) /' k k which i s the same as Eqn.2.11, a s u p e r p o s i t i o n of L o r e n t z i a n l i n e s d i s t r i b u t e d i n f r e q u e n c y (the harmonics of u^) and a m p l i t u d e (a f u n c t i o n of the m o d u l a t i o n a m p l i t u d e and f r e q u e n c y ) . The shape of the DISPA p l o t i s d i f f e r e n t from 1 5 t h a t c a u s e d by u n r e s o l v e d h y p e r f i n e c o u p l i n g a s g e n e r a l l y CJ «A. . ( a t 100kHz, u> =30mG) a n d A. i s d i s t r i b u t e d a c c o r d i n g m i m i t h e B e s s e l f u n c t i o n s a s o p p o s e d t o a P a s c a l t r i a n g l e . A l t h o u g h i n p r a c t i c e t h e d i f f e r e n c e s a r e o f t e n o b s c u r e d by n o i s e . 2.8 THE E F F E C T OF DISPERSION F o r a p o o r l y p h a s e d s p e c t r o m e t e r t h e o b s e r v e d s i g n a l O(w) an d i t s q u a d r a t u r e Q(CJ) a r e g i v e n by (21) 0(u>) = A ( c j ) c o s ( e ) + D ( c j ) s i n ( 0 ) Qioj) = D ( a ) ) c o s ( 0 ) - A ( w ) s i n ( 0 ) ( 2 . 1 4 ) where A(w) a n d D ( u ) a r e t h e a b s o r p t i o n a n d d i s p e r s i o n s i g n a l s r e s p e c t i v e l y . The a n g l e 6 i s c a l l e d t h e ' p h a s e a n g l e ' o r ' p h a s e ' . F r o m C a r t e s i a n g e o m e t r y f o r t h e r o t a t i o n o f c o o r d i n a t e s , t h e l o c u s o f ( 0 ( w ) , Q ( w ) ) ( t h e DISPA p l o t ) i s j u s t t h e l o c u s o f ( A ( W ) , D ( C J ) ) r o t a t e d by 6. I f A(w) i s L o r e n t z i a n t h e n d i s p e r s i o n l e a k a g e s i m p l y g i v e s a r o t a t e d c i r c l e ( F i g . 2 . 5 ) . F o r o t h e r f u n c t i o n a l f o r m s o f A ( w ) , t h e e f f e c t i s t h e same, t h e DISPA p l o t s a r e j u s t r o t a t e d by 6. T h i s i s t h e o n l y m e c h a n i s m t h a t c a u s e s a s y m m e t r i c DISPA p l o t s f o r s p e c t r a w i t h s y m m e t r i c a b s o r p t i o n l i n e s ( f r e e f r o m b a s e l i n e a n d f i l t e r a r t e f a c t s ) . A more d e t a i l e d d i s c u s s i o n o f t h e e f f e c t o f s p e c t r o m e t e r p h a s e c a n be f o u n d i n (22) a n d 16 Sect.5.2. DM F i g u r e 2.5. DISPA plot for a poorly phased l i n e . 8=10°.Corresponding spectrum on the l e f t . 2.9 THE DYSONIAN LINE The Dysonian line-shape arises from paramagnetic centers in a conductor (23). The exact form of the line-shape depends on the system; conductor thickness, d i f f u s i o n rate of the center etc. , but the spectrum i s , in a l l cases, similar to (and in some cases ac t u a l l y is) an admixture of absorption and dispersion signals. A simulated spectrum i s shown in Fig.2.6. The DISPA plot i s similar to Fig.2.5, except that the c i r c l e i s rotated =*45°. F i g u r e 2.6. The D y s o n i a n L i n e - s h a p e . 3. THE HILBERT TRANSFORM AND DATA PRESENTATION 3.1 GENERATING THE DISPERSION SPECTRUM ESR dispersion spectra are very d i f f i c u l t to produce d i r e c t l y . Also i t i s not currently possible to produce the absorption and dispersion spectra simultaneously, which presents very serious scaling problems. The dispersion spectrum can, however, be obtained via Kramers-Kronig relations (9). A ( u ' ) = + 7 T - 'Pf D ( O J ) — , D(w') = - * - ' P £ A ( u ) ^ , (3.1) The P denotes use of the Cauchy p r i n c i p l e value. This is more usually known as a Hilbert transform and i s completely general for a l l spectroscopy except non-linear systems. 6 Eqn.3.1 can be solved numerically as i t stands. However, the computation i s very slow and the pole (at o)0=u) can cause problems. The transform can be done very e f f i c i e n t l y once i t is recognised as a convolution integral (24) so that 6 A H i l b e r t transform can be done with any spectroscopic data set. However, the quadrature spectrum so obtained may not be the true quadrature signal (see Sect.2.3 on saturation). This generally does not a f f e c t DISPA's use as a diagnostic t o o l , but care should be taken when making comparisons with NMR data, where the true quadrature signal i s always available. 18 19 F-Hll D ( w ) Z P o > ' } = F - 1 { D ( W ) } F - M C J - M (3.2) where F " 1 i s t h e r e v e r s e F o u r i e r t r a n s f o r m . 7 8 b u t F " ' ! " " 1 } = i . rr. sgn (CJ) s o F" 1{D(C J)} = i .TTF- 1 {A(CJ) } . s g n ( c j ) The p r o b l e m i s t h u s r e a d i l y s o l u b l e by u s i n g t h e f a s t - F o u r i e r - t r a n s f o r m ( F F T ) a l g o r i t h m (25)(26). A few wor d s o f c a u t i o n a r e n e e d e d a s t h e F F T a l g o r i t h m i s n o t e x a c t l y e q u i v a l e n t t o t h e a n a l y t i c t r a n s f o r m . F i r s t , t h e ESR s p e c t r u m i s c o l l e c t e d a s a f u n c t i o n o f t i m e i n u n i t s p r o p o r t i o n a l t o G a u s s . I t i s n e v e r t h e l e s s a s p e c t r u m i n t h e f r e q u e n c y d o m a i n . T h i s means t h a t , i f one w i s h e s t o m a i n t a i n an a n a l o g y w i t h FT-NMR, one h a s t o u s e t h e i n v e r s e t r a n s f o r m t o g e n e r a t e t h e ESR e q u i v a l e n t o f t h e f r e e i n d u c t i o n d e c a y ( F I D ) ; m u l t i p l y by t h e s g n f u n c t i o n ; t h e n do a f o r w a r d t r a n s f o r m t o r e c o v e r t h e d i s p e r s i o n s p e c t r u m . S e c o n d l y t h e d i s c o n t i n u i t y o f t h e s g n f u n c t i o n s h o u l d l i e a t t h e o r i g i n o f t h e d a t a s e t . F o r a d a t a s e t g e n e r a t e d by an F F T t h e o r d i n a t e i s n o t d e f i n e d ( i t s an a r r a y i n d e x ) , t h e o r i g i n , a s 7 The s g n f u n c t i o n i s d e f i n e d a s -1 f o r x<0 a n d +1 f o r x>0, i.e., i t s i m p l y a s i g n c h a n g e c e n t e r e d a t t h e o r i g i n . 8 T h e t r a n s f o r m f r o m t h e t i m e d o m a i n t o t h e f r e q u e n c y d o m a i n i s c a l l e d t h e f o r w a r d t r a n s f o r m a n d i s d e f i n e d a s S(w) = F { G ( t ) } = J G ( t ) e - / w ' d t The r e v e r s e t r a n s f o r m ( f r e q u e n c y t o t i m e ) i s d e f i n e d a s G ( t ) = F - 1 { S ( c o ) } = $*- , J , S ( t ) e + / w ' d t 20 such l i e s a t the N / 2 t h d a t a p o i n t ( f o r N d a t a p t s . ) . I f N i s even, then s t r i c t l y s p e a k i n g the N/2 p o i n t w i l l be u n d e f i n e d a f t e r m u l t i p l i c a t i o n by the sgn f u n c t i o n as i t - l i e s on a d i s c o n t i n u i t y . In p r a c t i c e i t can be m u l t i p l i e d by ±1 as c o n v e n i e n t . ( T h i s i s e q u i v a l e n t t o s e t t i n g the p o l e of Eqn.3.4 e q u a l t o one of the a d j a c e n t p o i n t s ) . F i n a l l y the FFT a l g o r i t h m uses independent r e a l and i m a g i n a r y p a r t s . The a b s o r p t i o n spectrum i s u s u a l l y r e g a r d e d as t h e i m a g i n a r y p a r t so s h o u l d be l o a d e d t o the i m a g i n a r y p a r t of the data a r r a y f o r the t r a n s f o r m . The r e s u l t a n t t r a n s f o r m w i l l a l s o be i m a g i n a r y . T h i s w i l l be then m u l t i p l i e d by the sgn f u n c t i o n and the p r o d u c t f o r w a r d t r a n s f o r m e d t o g i v e the d i s p e r s i o n spectrum which w i l l be i n the r e a l p a r t of the a r r a y . The c h o i c e of which p a r t of the spectrum i s r e a l and i m a g i n a r y i s a matter of c o n v e n t i o n so one can t a k e l i b e r t i e s as t o the d i r e c t i o n of the t r a n s f o r m and which p a r t of the a r r a y i s used t o s t o r e the d a t a . However, i t i s i m p o r t a n t t o keep t r a c k of where the d a t a a r e a t each stage of t h e t r a n s f o r m . T h i s i s e s p e c i a l l y t r u e i f t h e a b s o r p t i o n spectrum's maximum does not l i e a t the o r i g i n (N/2) as the time domain spectrum (FID) w i l l c o n t a i n both r e a l and i m a g i n a r y components (and b o t h p a r t s have t o be m u l t i p l i e d by t h e sgn f u n c t i o n ) . T h i s can c r e a t e problems w i t h m i n i c o m p u t e r s as they o f t e n o n l y p e r m i t r e a l d a t a t o be used f o r t h e f o r w a r d t r a n s f o r m . F u r t h e r m o r e , the d a t a must c o n s i s t of 2n (where n i s i n t e g r a l ) e q u a l l y spaced d a t a p o i n t s . A l s o the a b s o r p t i o n spectrum i s r e q u i r e d , not the 21 d e r i v a t i v e , as i s produced i n ESR. The l a t t e r two problems a r e d i s c u s s e d i n the next s e c t i o n . 3.2 PRE-PROCESSING OF THE SPECTRUM FOR THE FFT The d e r i v a t i v e spectrum can be r e a d i l y i n t e g r a t e d u s i n g Simpsons Rule or the t r a p e z o i d a l r u l e . , but c a r e must be taken t o a v o i d low d a t a d e n s i t i e s (27). B a s e l i n e o f f s e t and d r i f t p r e s e n t problems. B a s e l i n e c o r r e c t i o n methods a r e d i s c u s s e d i n Sect.21.10. The e f f e c t s of poor b a s e l i n e c o r r e c t i o n on DISPA p l o t s a r e o u t l i n e d i n S e c t . 4 . 3 . U n l i k e NMR, ESR d a t a a r e not u s u a l l y e q u a l l y spaced. A l g o r i t h m s f o r F o u r i e r t r a n s f o r m i n g u n e q u a l l y spaced d a t a are a v a i l a b l e , but a r e much sl o w e r t o r u n , e.g. (28). I t i s more c o n v e n i e n t t o i n t e r p o l a t e t h e d a t a t o o b t a i n an e q u a l l y spaced d a t a s e t . T h i s i s d i s c u s s e d i n S e c t . 2 1 . 7 . A d a t a s e t of 2n d a t a p o i n t s i s e a s i l y produced by adding e x t r a d a t a p o i n t s ; p a d d i n g . The p a d d i n g s h o u l d be done s y m m e t r i c a l l y (so t h a t any a r t e f a c t s a r e symmetric) i n such a manner t h a t t h e r e a r e no d i s c o n t i n u i t i e s i n the spectrum or a t the ends of the spectrum. (The ends of the data s e t s h o u l d be a t z e r o ) . The e f f e c t s of padding a r e d i s c u s s e d i n S e c t . 4 . 8 . The optimum method seems t o be t o pad t h e d a t a by i n t e r p o l a t i o n t o 1024 p t s ; reduce the d a t a s e t back t o 512 p o i n t s by b o x - c a r r i n g ; s y m m e t r i c a l l y pad t h i s d a t a s e t t o 1024 p o i n t s u s i n g a l i n e a r ramp; do the H i l b e r t t r a n s f o r m , then r e c o v e r o n l y t h e p a r t of the d i s p e r s i o n spectrum 22 c o r r e s p o n d i n g t o the 512 p o i n t a b s o r p t i o n spectrum. A f l o w c h a r t f o r the b a s i c p r o c e d u r e i s g i v e n i n F i g . 3 . 1 . ( T h i s i s f o r the LSI-11 computer, f o r a l a r g e r computer i t would be b e t t e r t o double the number of p o i n t s i.e. , do a 2048 p o i n t t r a n s f o r m ) . F L A T T E N D E R I V A T I V E D A T A I N T E G R A T E D E R I V A T I V E D A T A A B S O R P T I O N D A T A P A D & C E N T E R A B S O R P T I O N D A T A R E V E R S E F F T M U L T I P L Y B Y S G N A T C E N T E R O F F I D F O R W A R D F F T S C A L E 4 E X T R A C T D I S P E R S I O N D A T A S A V E D I S P E R S I O N D A T A F i g u r e 3.1. F l o w - c h a r t f o r g e n e r a t i n g t h e d i s p e r s i o n d a t a from e q u a l l y spaced d e r i v a t i v e d a t a . 23 3.3 THE DIFFERENCE PLOT The C o l e - C o l e format f o r DISPA p l o t s i s q u i t e s t r i k i n g ( F i g . 2 . 1 ) , but i s u n s a t i s f a c t o r y f o r many purposes as the o b s e r v e d d i s t o r t i o n i s o n l y a few per-cent of the r e f e r e n c e c i r c l e ' s r a d i u s . A p l o t of the d i f f e r e n c e between the r e f e r e n c e c i r c l e and the e x p e r i m e n t a l p l o t as o r d i n a t e would be more u s e f u l . 9 The r a d i a l d i f f e r e n c e , AR, ( d i f f e r e n c e between the r e f e r e n c e c i r c l e ' s r a d i u s and the d i s t a n c e of an e x p e r i m e n t a l p o i n t from the c e n t e r of the r e f e r e n c e c i r c l e ) i s the most c o n v e n i e n t o r d i n a t e use. Other schemes, such as u s i n g the v e r t i c a l d i f f e r e n c e s or the d i f f e r e n c e i n p o l a r a n g l e f o r e x p e r i m e n t a l d i s p e r s i o n p o i n t s and the c o r r e s p o n d i n g p o i n t s on the r e f e r e n c e c i r c l e , a r e v e r y d i f f i c u l t t o implement on a computer and o f f e r no advantages over the r a d i a l d i f f e r e n c e p l o t . There i s a wide c h o i c e of a b s c i s s a . M a r s h a l l has i n v e s t i g a t e d a number of t h e s e , but r e s t r i c t e d h i s c h o i c e t o f u n c t i o n s of spectrum f r e q u e n c y . T h i s i s i n c o n v e n i e n t i n ESR as the f r e q u e n c y i s both i n s t r u m e n t and sample dependent. Three o t h e r schemes have been d e v e l o p e d , u s i n g as a b s c i s s a ; the a b s o r p t i o n s i g n a l , A(w), the a r r a y i n d e x of the a b s o r p t i o n s i g n a l , I , and the p o l a r a n g l e , 8, of the e x p e r i m e n t a l d a t a . However, b e f o r e d i s c u s s i n g t h e s e i t i s n e c e s s a r y t o d e f i n e some c o n v e n t i o n s . 9 A t low d a t a d e n s i t i e s t h i s i s not a v e r y u s e f u l format as t h e d a t a p o i n t s become too s p r e a d o u t . T h i s i s not a problem w i t h ESR, but sometimes o c c u r s i n NMR. 24 The r e f e r e n c e c i r c l e i s d e f i n e d as a c i r c l e whose r a d i u s i s e q u a l t o t h a t of the maximum h e i g h t of the a b s o r p t i o n and d i s p l a c e d a l o n g t h e a b s c i s s a by the r a d i u s of the r e f e r e n c e c i r c l e ( R ) . I t s h o u l d be noted t h a t the a b s o r p t i o n a m p l i t u d e i s f i x e d by s c a l i n g . For t h i s work the a b s o r p t i o n s i g n a l was always s c a l e d t o 25000, c o r r e s p o n d i n g t o a DISPA r e f e r e n c e c i r c l e of r a d i u s 10cm. Any d i s p l a c e m e n t o u t s i d e of the r e f e r e n c e i s c o n s i d e r e d t o be a p o s i t i v e r a d i a l d i f f e r e n c e and d i s p l a c e m e n t s w i t h i n the c i r c l e n e g a t i v e . The r a d i a l d i f f e r e n c e , A R , i s u s u a l l y e x p r e s s e d as a p e r - c e n t a g e of the r e f e r e n c e c i r c l e ' s r a d i u s , R 0 , i.e., A R = ( R ^ ^ - R Q ) 1 0 0 / R O . The spectrum has t o be phased such t h a t the d o u b l e i n t e g r a l i s p o s i t i v e . The spectrum i s assumed t o be swept from l e f t t o r i g h t (i.e. , the l e f t of the spectrum i s t aken t o c o r r e s p o n d t o low f i e l d ) . I t i s c o n v e n i e n t i f the c e n t e r of the spectrum c o r r e s p o n d s t o the c e n t e r of the da t a s e t . T h i s can be r e a d i l y a c h i e v e d by padding and makes comparisons of v a r i o u s data s e t s e a s i e r . 1 0 A diagrammatic d e f i n i t i o n of the d i f f e r e n c e p l o t i s g i v e n i n F i g . 3 . 2 1 0 I f DISPA i s used f o r asymmetric s p e c t r a a n o t h e r d e f i n i t i o n of the c e n t e r s h o u l d be found. The p o i n t c o r r e s p o n d i n g t o g / ? 0 as found by Hydes (29) a l g o r i t h m i s p r o b a b l y a good c h o i c e . It i s convenient to define two more parameters, the lobe height (=AR m a ; c ), and the lobe separation, AS=(S^-S )/R0, where i s the position of the l e f t lobe and Sr i s the position of the right l o b e . 1 1 Another useful parameter i s the lobe asymmetry A^, defined as; A^ = A^ -A^ .; where A, i s AR „ for the l e f t lobe and A i s that for the / max r right lobe. A diagrammatic d e f i n i t i o n for these parameters can be found in Fig.3.3. 1 1 The lobe s p l i t t i n g was found to have a weak dependence on the spectrum parameters and was not very useful as a diagnostic parameter. A normal s p l i t t i n g was taken to mean AS^0.5; a wide s p l i t t i n g >0.7; a narrow s p l i t t i n g <0.3. 26 Figure 3 .3 . Diagrammatic d e f i n i t i o n of the lobe p a r a m e t e r s f o r a t y p i c a l d i f f e r e n c e • p l o t . Subsequent r e f e r e n c e s t o the term DISPA p l o t w i l l be ta k e n t o mean the d i f f e r e n c e type DISPA p l o t r a t h e r than the C o l e - C o l e t y p e p l o t . U n l e s s o t h e r w i s e s t a t e d a l l examples a r e f o r a l i n e w i t h u n r e s o l v e d h y p e r f i n e c o u p l i n g ; 12 p r o t o n s c o u p l i n g t o 3G l i n e w i t h a c o u p l i n g c o n s t a n t of 1 .OG. 3.3.1 THE INDEX DIFFERENCE PLOT The a b s c i s s a i s s i m p l y a f u n c t i o n of the index of the a b s o r p t i o n d a t a a r r a y s c a l e d t o a s u i t a b l e v a l u e ( i f n e c e s s a r y ) . The a b s c i s s a , F ( x ) , i s c a l c u l a t e d as f o l l o w s . 27 F ( x ) = f ( I - I ) .k I < I max max F ( x ) = f ( I - I max ).k I > I max ( 3 . 3 ) where I max i s t h e i n d e x a t w h i c h maximum a b s o r p t i o n o c c u r s ; k i s some s c a l i n g f a c t o r ; f ( I ) i s a f u n c t i o n o f t h e i n d e x o f t h e d a t a a r r a y . P l o t s f o r F ( l ) w h e r e ' f i s one ( l i n e a r ) , ' f i s 1/ ( s q u a r e - r o o t ) , ' f i s l o g a r i t h m i c ( l n ) a n d ' f i s L o r e n t z i a n ( L z ) , a r e shown i n F i g . 3 . 4 - F i g . 3 . 5 F i g u r e 3 . 4 . L i n e a r a n d s q u a r e r o o t i n d e x e d d i f f e r e n c e p l o t s . 28 LZ(I) F i g u r e 3 .5. L o g a r i t h m i c , LN, and L o r e n t z i a n , LZ, indexed d i f f e r e n c e p l o t s . The l i n e a r p l o t has the u n f o r t u n a t e e f f e c t of expanding the b a s e l i n e r e g i o n and co m p r e s s i n g the r e g i o n of i n t e r e s t . T h i s i s of no consequence f o r n u m e r i c a l c o m p u t a t i o n s , but makes i t d i f f i c u l t t o use f o r manual i n t e r p r e t a t i o n s . The l o g p l o t i s u s e f u l f o r g e n e r a l d i a g n o s t i c work. The L o r e n t z i a n p l o t i s v e r y good f o r d i a g n o s t i c work, but the c h o i c e of the w i d t h of t h e L o r e n t z i a n i s a r b i t r a r y . The square r o o t p l o t i s u s e f u l f o r p i c k i n g out s a t e l l i t e s and o t h e r peaks i n the s h o u l d e r s of the main peak. The main advantage of index p l o t s i s t h a t the o r d i n a t e i s independent of the e x p e r i m e n t a l d a t a so t h a t the p l o t t i n g r o u t i n e i s s t a b l e w i t h p a t h o l o g i c a l d a t a s e t s . The main d i s a d v a n t a g e i s t h a t the l o b e s e p a r a t i o n of inde x p l o t s i s l i n e - w i d t h dependent. ( F i g . 3 . 6 ) 29 0.16 F i g u r e 3 .6 . L o g a r i t h m i c indexed p l o t as a f u n c t i o n l i n e - w i d t h / s w e e p - w i d t h . The g l i t c h i n the t a i l s of the p l o t s i s a t r u n c a t i o n a r t e f a c t . 3.3.2 THE POLAR DIFFERENCE PLOT As DISPA p l o t s a r e c i r c u l a r , p o l a r c o o r d i n a t e s a r e the most n a t u r a l way of e x p r e s s i n g them. The p o l a r a n g l e , 6, i s g i v e n i n F i g . 3 . 2 . A t y p i c a l p l o t i s shown i n F i g . 3 . 7 . 30 T h i s p l o t d i s t r i b u t e s the d a t a i n a s a t i s f a c t o r y manner ( b a s e l i n e d a t a compressed w i t h r e s p e c t t o the main peak) and has the advantage t h a t the d i f f e r e n c e p l o t r e t a i n s any symmetry (about the v e r t i c a l a x i s ) t h a t t h e C o l e - C o l e p l o t has. I t s main d i s a d v a n t a g e i s i n s t a b i l i t y , i t i s not always p o s s i b l e t o unambiguously a s s i g n t h e quadrant t h a t 6 ( F i g . 3 . 2 ) l i e s i n . The r e s u l t a n t p l o t may jump around. T h i s i s e s p e c i a l l y n o t i c a b l e w i t h p o o r l y phased s p e c t r a . 3.3.3 THE ABSORPTION DIFFERENCE PLOT U s i n g a s i g m o i d type f u n c t i o n as t h e a b s c i s s a w i l l a l s o d i s t r i b u t e the d a t a as r e q u i r e d . A n a t u r a l c h o i c e would be the h y p e r b o l i c t a n g e n t because, b e s i d e s b e i n g s i g m o i d , i t i s the i n t e g r a l of a L o r e n t z i a n , i . e . , the d o u b l e i n t e g r a l of the o r i g i n a l d a t a s e t may be a u s e f u l a b s c i s s a . However, b a s e l i n e a r t e f a c t s c o u l d cause 31 considerable i n s t a b i l i t y in such a p l o t . A more suitable (approximately sigmoidal) function can be constructed from the absorption data using Eqn.3.3, with A(o>), used instead of ' f . This gives more stable plots than the polar difference p l o t , but the symmetry i s l o s t . The example for unresolved hyperfine coupling i s given in Fig.3.8. Unfortunately the absorption plots loop-the-loop i f p a r t i a l l y resolved features are present. This is very entertaining, but can seriously hamper numerical analysis of the pl o t s . The p r i n c i p a l advantage of these plots i s that they are e a s i l y compared with the Cole-Cole type plots as both are plotted as a function of A(a>). M(u) F i g u r e 3 . 8 . The absorption difference p l o t . An absorption difference plot along with i t s corresponding Cole-Cole plot i s shown in Fig.3.9. (The scale of the abscissa for the difference plot i s halved with respect to the Cole-Cole p l o t ) . The ordinate i s the 32 d i s p e r s i o n s i g n a l f o r the C o l e - C o l e type p l o t and the p e r - c e n t r a d i a l d i f f e r e n c e f o r the d i f f e r e n c e p l o t . T h i s format of DISPA p l o t i s p r e f e r r e d by some w o r k e r s , but i t w i l l not be used h e r e . F i g u r e 3.9. T y p i c a l C o l e - C o l e and d i f f e r e n c e DISPA p l o t . G e n e r a l l y the a b s o r p t i o n t ype of p l o t was used f o r d i a g n o s t i c work ( u n l e s s o t h e r w i s e s t a t e d a l l d i f f e r e n c e p l o t s i n t h i s work a r e a b s o r p t i o n d i f f e r e n c e p l o t s ) and the l i n e a r i n d e x e d p l o t was used f o r n u m e r i c a l work. P o l a r p l o t s (when s t a b l e ) were u s e f u l where a e s t h e t i c p l o t s were r e q u i r e d . 33 3.4 THE GAUSSIAN DIFFERENCE PLOT The G a u s s i a n l i n e - s h a p e and i t s c l o s e r e l a t i v e , the V o i g t l i n e - s h a p e , a r e v e r y common i n ESR s p e c t r o s c o p y . The shape of the d i f f e r e n c e p l o t f o r a G a u s s i a n l i n e i s independent of i t s l i n e - w i d t h (except f o r t r u n c a t i o n e f f e c t s , Sect.4.8). T h i s l e d M a r s h a l l t o propose a r e n o r m a l i s a t i o n p r o c e d u r e whereby a G a u s s i a n l i n e would g i v e a c i r c u l a r C o l e - C o l e p l o t ( l i n e a r d i f f e r e n c e p l o t ) so t h a t the DISPA p l o t would now r e f l e c t d e v i a t i o n s from G a u s s i a n b e h a v i o u r . T h i s would be ve r y u s e f u l , but an e f f i c i e n t , r e l i a b l e and c o n s i s t e n t a l g o r i t h m remains t o be d e v e l o p e d . 1 2 The problems t h a t a r i s e are as f o l l o w s : a) The G a u s s i a n d i s p e r s i o n spectrum i s not an a n a l y t i c f u n c t i o n and i s not amenable t o r a p i d n u m e r i c a l e v a l u a t i o n . T h i s makes the g e n e r a t i o n of a r e f e r e n c e c i r c l e d i f f i c u l t , b) A l o o k - u p t a b l e f o r the d i s p e r s i o n l i n e , w i t h i n t e r p o l a t i o n , was used by M a r s h a l l (12), but t h i s i s v e r y u n s t a b l e i n the presence of n o i s e , c) The shape of the d i f f e r e n c e p l o t i s independent of l i n e - w i d t h , but the p o s i t i o n s of the p o i n t s on t h e l o c u s of the d i f f e r e n c e p l o t a r e n o t , c.f. Eqn.1.10. S i m p l y s u b t r a c t i n g a r e f e r e n c e d i f f e r e n c e p l o t i s thus not p o s s i b l e , d) U s i n g a l o o k - u p t a b l e f o r a r e f e r e n c e d i f f e r e n c e p l o t i s a l s o p o s s i b l e , but a l a r g e t a b l e (=10S p o i n t s ) i s r e q u i r e d , beyond the c a p a c i t y of our c u r r e n t computer. These problems need f u r t h e r i n v e s t i g a t i o n , but w i l l not be pursued h e r e . 1 2 M a r s h a l l has demonstrated t h e u t i l i t y of t h e t e c h n i q u e (12), but h i s a l g o r i t h m i n v o l v e s a G a u s s i a n r e f e r e n c e l i n e of a r b i t r a r y w i d t h . 4. INSTRUMENTAL DIAGNOSTICS AND APPLICATIONS I n s t r u m e n t a l a r t e f a c t s s h o u l d be removed b e f o r e any k i n d of l i n e - s h a p e a n a l y s i s i s u n d e r t a k e n . T h i s i s e s p e c i a l l y t r u e f o r DISPA, which i s v e r y s e n s i t i v e t o l i n e d i s t o r t i o n s . However, t h i s s e n s i t i v i t y may be used t o i d e n t i f y and thus h e l p remove i n s t r u m e n t a l d i s t o r t i o n . I n p a r t i c u l a r i t can be used f o r the a u t o m a t i c p h a s i n g of s p e c t r a . In ESR s p e c t r o s c o p y d i s t o r t i o n may a r i s e from a number of s o u r c e s ; s a t u r a t i o n , o v e r m o d u l a t i o n , a m p l i f i e r and b r i d g e m i s p h a s i n g , o v e r - f i l t e r i n g , h i g h f r e q u e n c y n o i s e and low frequ e n c y n o i s e ( b a s e l i n e d r i f t and o f f s e t ) . These problems a r e d i s c u s s e d i n t u r n i n the f o l l o w i n g s e c t i o n s . As b e f o r e a l l examples a r e f o r a s i n g l e l i n e w i t h u n r e s o l v e d h y p e r f i n e c o u p l i n g u n l e s s o t h e r w i s e s t a t e d . 4.1 TIME CONSTANT The i n f l u e n c e of the p h a s e - s e n s i t i v e - d e t e c t i o n (PSD) a m p l i f i e r s time c o n s t a n t on l i n e - s h a p e i s w e l l documented (30). However, the s i z e of t h i s d i s t o r t i o n i s o f t e n u n d e r - e s t i m a t e d . 1 3 An example i s shown i n F i g . 4 . 1 . T h i s e f f e c t i s e a s i l y a v o i d e d by s e t t i n g the t i m e - c o n s t a n t t o 1/100th the time i t t a k e s t o scan (peak-to-peak) the s h a r p e s t l i n e i n the spectrum. N e v e r t h e l e s s i t i s worth 1 3 A v e r t i c a l l y mounted c h a r t r e c o r d e r has an asymmetric time c o n s t a n t due t o g r a v i t y . T h i s may o p e r a t e i n the o p p o s i t e sense of the p h a s e - s e n s i t i v e - d e t e c t i o n a m p l i f i e r s time c o n s t a n t and the two f a c t o r s p a r t i a l l y c a n c e l . T h i s i s not apparent u n t i l d i g i t a l d a t a a r e used. 34 35 checking a standard spectrum with DISPA to ensure that the phase-sensitive-detection amplifier f i l t e r i s as marked. IttJX F i g u r e 4 .1 . E f f e c t of a large PSD f i l t e r on a DISPA plo t . Spectrum on the l e f t . 3.2G line-width, sweep-rate 5G/min. The low amplitude symmetric lobes are for a 0.125s f i l t e r . The other lobes are for a 4.0s f i l t e r . 4.2 NOISE Reducing the phase-sensitive-detection amplifier f i l t e r increases the noise l e v e l . The s e n s i t i v i t y of DISPA to noise i s thus an important factor. The integration of the derivative improves the (apparent) SNR s u b s t a n t i a l l y . No problems have been encountered with noise, but i t can be very d i f f i c u l t to f l a t t e n the baseline for a noisy spectrum. An example (unresolved hyperfine coupling as before) i s shown in Fig.4.2. 36 F i g u r e 4.2. E f f e c t of n o i s e on a DISPA p l o t . The SNR (peak-peak s i g n a l / p e a k - p e a k b a s e l i n e n o i s e ) i s 10:1. 4.3 BASELINE ARTEFACTS DISPA i s v e r y s e n s i t i v e t o b a s e l i n e d i s t o r t i o n and t h i s must be removed b e f o r e d o i n g any DISPA a n a l y s e s ( S e c t . 2 1 . 1 0 ) . An example of i t s e f f e c t s a re shown i n F i g . 4 . 3 . Note t h a t t h i s example i s f o r a s m a l l d r i f t / o f f s e t t h a t i s not v i s i b l e i n the o r i g i n a l d e r i v a t i v e spectrum. 37 F i g u r e 4 .3 . DISPA p l o t showing the e f f e c t of b a s e l i n e a r t e f a c t s . The d i s t o r t i o n i s f o r a DC o f f s e t of +0.1% of the peak-peak h e i g h t . A d r i f t of +0.2% a t the r i g h t s i d e causes the same d i s t o r t i o n . 4.4 AMPLIFIER PHASING An ESR s p e c t r o m e t e r has a number of a m p l i f i e r s t h a t have t o be phased c o r r e c t l y . Only t h e p h a s e - s e n s i t i v e - d e t e c t i o n a m p l i f i e r has f r o n t p a n e l c o n t r o l though. T h i s c o n t r o l o n l y a f f e c t s the l i n e a m p l i t u d e s and does not a f f e c t the DISPA p l o t s ( S e c t . 2 . 2 ) (10).14 The a u t o m a t i c f r e q u e n c y c o n t r o l a m p l i f i e r s h o u l d be c o r r e c t l y phased or b a s e l i n e a r t e f a c t s w i l l a r i s e . The m o d u l a t i o n a m p l i f i e r o u t p u t s h o u l d a l s o be c o r r e c t l y phased or asymmetric l i n e s w i l l r e s u l t . These problems a r e r e a d i l y c u r e d , but may not be o b s e r v e d u n t i l DISPA a n a l y s e s a r e a t t e m p t e d . 1 f , F o r phase changes of ^90°, or m u l t i p l e s t h e r e o f , some e f f e c t s a r e o b s e r v e d , but g e n e r a l l y one has t o d e l i b e r a t e l y m i s - s e t the p h a s e - s e n s i t i v e - d e t e c t i o n a m p l i f i e r phase t o see such e f f e c t s 38 4.5 MICROWAVE-BRIDGE PHASING Reflection mode c a v i t i e s are arranged such that the dispersion component i s eliminated by the automatic frequency control amplifier. However, spurious r e f l e c t i o n s in the wave guides can give r i s e to dispersion leakage. Also some bridge designs do not have an automatic frequency control or have to be phased for each experiment e.g.(31). The ef f e c t of dispersion leakage (poor phasing) i s shown in Fig.4.4 The lobe asymmetry (Sect.3.3) is a l i n e a r function of the phase angle and can be used for automatically phasing the spectrum. {(22), Sect.5.2) F i g u r e 4 . 4 . DISPA plot for a mis-phased microwave bridge. A pure Lorentzian l i n e with 6=2°. 4.6 SATURATION Saturation does not aff e c t a pure Lorentzian l i n e (Sect.2.3). However, i t does a f f e c t the DISPA plots of composite l i n e s . The d i s t o r t i o n of the DISPA plot for 39 c o m p o s i t e l i n e s i s a f u n c t i o n o f t h e r a t i o o f t h e l i n e - w i d t h s t o e a c h o t h e r o r t h e i r s e p a r a t i o n , ( s e e S e c t . 6 . 5 ) . S a t u r a t i o n b r o a d e n s l i n e s s o t h i s w i l l c h a n g e t h e DISPA p l o t d e p e n d i n g on t h e s a t u r a t i o n c h a r a c t e r i s t i c s o f t h e i n d i v i d u a l l i n e s . G e n e r a l l y t h e DISPA d i s t o r t i o n w i l l d e c r e a s e . 4.7 MODULATION M o d u l a t i o n h a s b e e n d i s c u s s e d p r e v i o u s l y (10). The d i f f e r e n c e p l o t f o r o v e r m o d u l a t i o n i s shown i n F i g . 4 . 5 . N o t e t h a t i t i s s i m i l a r t o , b u t d i s t i n g u i s h a b l e f r o m t h e d i s t o r t i o n c a u s e d by u n r e s o l v e d h y p e r f i n e c o u p l i n g a s i t c h a n g e s w i t h m o d u l a t i o n a m p l i t u d e . One c a n i n f a c t c r u d e l y c a l i b r a t e t h e m o d u l a t i o n a m p l i t u d e u s i n g DISPA (32) a n d t h i s i s u s e f u l f o r q u i c k c h e c k s o f t h e m o d u l a t i o n a m p l i t u d e , w h i c h c h a n g e s w h e n e v e r t h e c a v i t y i s c h a n g e d . F i g u r e 4.5. DISPA p l o t f o r an o v e r m o d u l a t e d l i n e . H i g h - f i e l d l i n e o f C u P y d t c i n c h l o r o f o r m . N a t u r a l l i n e - w i d t h i s 3.21G. M o d u l a t i o n l e v e l s a r e 0.50G, 1.67G a n d 2.63G r e s p e c t i v e l y . 40 4.8 LINE TRUNCATION AND PADDING L o r e n t z i a n and Dysonian l i n e s have v e r y l o n g t a i l s . I f the sweep w i d t h i s t o o narrow t h e s e l i n e s a r e t r u n c a t e d and a r t e f a c t s appear i n the DISPA p l o t s . (The bounds of t h e i n t e g r a l , Eqn.3.1, s h o u l d be i n f i n i t e ) . U s i n g a sweep w i d t h of a t l e a s t 10X the peak-peak w i d t h of the l i n e w i l l a v o i d t h i s problem. In p r a c t i c e 5x i s u s u a l l y adequate as most r e a l l i n e s have G a u s s i a n t a i l s (i.e. , decay r a p i d l y a f t e r f i v e l i n e - w i d t h s ) . U n f o r t u n a t e l y t h i s problem i s u n a v o i d a b l e w i t h s i m u l a t e d s p e c t r a , l a r g e sweep w i d t h s g i v e unmanageably l a r g e d a t a s e t s or too s m a l l d a t a d e n s i t i e s . Padding the spectrum h e l p s reduce the problem, but a r t e f a c t s s t i l l o c c u r . The e f f e c t s of t r u n c a t i o n (on a 'ramp' padded spectrum) a r e shown i n F i g . 4 . 6 F i g u r e 4 . 6 . DISPA p l o t f o r a t r u n c a t e d L o r e n t z i a n l i n e f o r v a r i o u s v a l u e s of l i n e - w i d t h / s w e e p - w i d t h . The v - l o b e s a r e a d d i t i v e so t h a t the most o b v i o u s e f f e c t i s 41 t h a t the t a i l s and end p o i n t s of the d i f f e r e n c e p l o t a r e r a i s e d . ( F i g . 4 . 7 ) T h i s does not i n t e r f e r e w i t h the q u a l i t a t i v e i n t e r p r e t a t i o n of the p l o t s . , F i g u r e 4.7. DISPA p l o t f o r a t r u n c a t e d l i n e w i t h u n r e s o l v e d h y p e r f i n e c o u p l i n g . The d o t t e d l i n e i n d i c a t e s the superimposed v - l o b e s caused by t r u n c a t i o n . I f the b a s e l i n e goes t o z e r o ( i t ' s u s u a l l y f o r c e d t o z e r o by i n t e r a c t i v e b a s e l i n e f l a t t e n i n g ) then a l l r e a s o n a b l e p a dding schemes a r e e q u i v a l e n t t o p a d d i n g w i t h z e r o e s . I f i t does not go t o z e r o then e x t r a a r t e f a c t s , which depend on the p a d d i n g scheme used, a r e i n t r o d u c e d . The e f f e c t s of i n t e r p o l a t i v e p a d d i n g , z e r o p a d d i n g (padding w i t h z e r o e s ) , t a i l p a d d i n g (padding the l e f t and r i g h t s i d e s of t h e spectrum w i t h the f i r s t and l a s t d a t a p o i n t s r e s p e c t i v e l y ) and ramp padding ( p a d d i n g w i t h a f u n c t i o n t h a t d ecays l i n e a r l y t o z e r o from th e ends of t h e s p e c t r u m ) , on a spectrum w i t h a s m a l l DC o f f s e t a r e shown i n F i g . 4 . 8 . 42 F i g u r e 4 . 8 . E f f e c t of v a r i o u s p a d d i n g schemes on the DISPA p l o t . L o r e n t z i a n l i n e w i t h a DC o f f s e t of 0.1% of the d e r i v a t i v e a m p l i t u d e . I i s t h e i n t e r p o l a t i v e p a d d i n g . R i s ramp p a d d i n g . Z i s z e r o p a d d i n g . T i s t a i l p a d d i n g . There i s l i t t l e t o choose between i n t e r p o l a t i v e p adding and ramp p a d d i n g , except t h a t t h e v - l o b e s from ramp pa d d i n g a r e more e a s i l y r e c o g n i s e d as an a r t e f a c t than t h e w-lobes t h a t o c c u r w i t h i n t e r p o l a t i v e p a d d i n g . 5. THE AUTOMATIC PHASING OF SPECTRA D i s p e r s i o n l e a k a g e i s c h a r a c t e r i s e d by r o t a t i o n of the C o l e - C o l e DISPA p l o t (Sect.2.1) which g i v e s r i s e t o an asymmetric d i f f e r e n c e p l o t . T h i s s u g g e s t s the p o s s i b i l i t y of a method of a u t o m a t i c phase c o r r e c t i o n of magnetic resonance s p e c t r a . D i s c u s s i o n h e r e , w i l l be r e s t r i c t e d t o the ESR c a s e . A more g e n e r a l d i s c u s s i o n (by the a u t h o r ) can-be found i n (22). A p p l i c a t i o n s t o NMR a r e d e a l t w i t h i n (33) and t h i s method i s now i n commercial use. 5.1 BASIC THEORY OF PHASE CORRECTION C o n s i d e r a s i g n a l , S(a>), and i t s H i l b e r t t r a n s f o r m , Q(CJ) , / . e. , 1 5 H{S(CJ)} = Q(w) where S(a>) = A (to) cos ( 6) +D (co) s i n ( 6) Q(CJ) = D ( u ) c o s ( 6 ) - A ( u ) s i n ( 0 ) (5.1) and A(co) and D(CJ) a r e the a b s o r p t i o n and d i s p e r s i o n s i g n a l r e s p e c t i v e l y . They a l s o form a H i l b e r t t r a n s f o r m p a i r . 1 5 F o r f u r t h e r d i s c u s s i o n see (34)(35). A l s o note t h a t H{H{S(a>)}} •= -S(co). H , h e r e , denotes the H i l b e r t t r a n s f o r m ; not t o be c o n f u s e d w i t h the H a m i l t o n i a n , which i s not used i n t h i s p a r t of the t h e s i s . 43 44 H{A(u)} = Q(w) (5.2) Now c o n s i d e r a phase c o r r e c t i o n <j> t o S(w) t o g i v e S'(o>). A l s o d r o p the (w) f o r c o n v e n i e n c e . Hence S' = Scos(<*>)-Qsin(tf>) Q' = Qcos ( 0)+Ssin ( 0 ) (5.3) s u b s t i t u t i n g back we get S' = Acos0costf> + Dsin0sintf> - Dcos0sintf> + Asin0sintf> (5.4) which r e d u c e s t o S' = Acos( 6-<p)+Dsin(6-<J>) (5.5) and s i m i l a r l y f o r 0' (which may a l s o be o b t a i n e d from the H i l b e r t t r a n s f o r m of S ' ) . T h i s may be r e p e a t e d so t h a t a f t e r the i ' t h c o r r e c t i o n we get f o r S' 45 S(CJ) = kcos(6-L<t>. )-Dsin(0 - p . ) (5.6) This forms the basis of phase correction. If we could characterise 0 d i r e c t l y then only one phase correction i s required, <t> i s set to 0 in Eqn. 1.5. However, i f we cannot do that, we can incrementally change 4> u n t i l the desired correction i s achieved. In thi s case we need some c r i t e r i o n to establish when 10^=0.16 DISPA plots are a candidate in either case. 5.2 USE OF DISPA PLOTS FOR PHASE CORRECTION To characterise 0 e x p l i c i t l y , requires that a known re l a t i o n exists between the phase angle, 0, and some parameter of the difference p l o t . For the i t e r a t i v e method, the functional form of the re l a t i o n need not be known, but i t should be monotonic with a unique zero for i t , or i t s f i r s t d i f f e r e n t i a l , at 0=0. From difference plots of single Lorentzian l i n e s , superimposed Lorentzian l i n e s and unresolved hyperfine coupling (Fig.5.1) i t can be descried that the lobe asymmetry, A^=d, is l i n e a r l y related to 0 (Fig.5.2) for 0<3O°. 1 6 This i s the basis of manual phase correction. The c r i t e r i o n used i s that of baseline symmetry about the peak being phased. F i g u r e 5.1. R a d i a l d i f f e r e n c e p l o t s f o r v a r i o u s phase a n g l e s ; 0, 1 , 2 , 3, 5, 7 and 10 d e g r e e s . 20.0 -CO *- 18.0 -PHASE BNGLE F i g u r e 5.2. D i f f e r e n c e p l o t l o b e asymmetry as a f u n c t i o n of phase a n g l e f o r v a r i o u s s p e c t r a . T h i s s u g g e s t s u s i n g an i t e r a t i v e c o r r e c t i o n w i t h the 47 i n c r e m e n t s , <j>, c o n t r o l l e d by the r e l a t i o n 0.^ = k l d . , where k i s the p r o p o r t i o n a l i t y c o n s t a n t between 6 and d f o r 0 < 3 O ° . The b a s i c f l o w c h a r t i s shown i n F i g . 5 . 3 . For f u r t h e r d e t a i l s see (22). For f u r t h e r examples of phase c o r r e c t i o n p r o c e d u r e s see (36)(37)(35)(33)(38). The whole procedure p r e d i c a t e s on the spectrum b e i n g c e n t r o - s y m m e t r i c , which i t u s u a l l y i s f o r ESR s o l u t i o n s p e c t r a . I t i s i m p o r t a n t t o note t h a t the spectrum o n l y needs a c e n t e r of symmetry, the t e c h n i q u e works f o r u n r e s o l v e d l i n e s (see F i g . 5 . 7 ) . 1 7 The spectrum s h o u l d be ce n t e r e d as t h e r e i s a s m a l l dependence of the observed phase a n g l e on the l i n e - p o s i t i o n ( F i g . 5 . 4 ) . The c e n t e r of the spectrum i s l o c a t e d from the smoothed power spectrum and s h i f t e d as n e c e s s a r y . 1 7 I f the c e n t e r of symmetry has +ve c u r v a t u r e (e.g. a d o u b l e t ) , r a t h e r than -ve c u r v a t u r e (e.g. a t r i p l e t ) , then the a l g o r i t h m has t o be m o d i f i e d . The spectrum has t o be smoothed t o remove the v a l l e y s , then the v a l u e of 6 e s t a b l i s h e d w i t h t h a t spectrum. The v a l u e so o b t a i n e d i s then used t o c o r r e c t the o r i g i n a l spectrum. START PHASER SELECT PEAKS PHASE TO GIVE + VE INTEGRAL CENTER DATA AND PAD HILBERT T R A N S F O R M 0>45° YES SET 0 = 90° ESTIMATE 0 F i g u r e 5 . 3 . Flow c h a r t f o r the a u t o m a t i c phase c o r r e c t i o n of s p e c t r a . 49 2.0 -2.0 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 SPECTRUM SHIFT (FRACTIONAL SWEEP WIDTH) F i g u r e 5.4. P h a s e e r r o r a s a f u n c t i o n o f l i n e - p o s i t i o n . E x a m p l e s f o r t h e p h a s e c o r r e c t i o n o f a s i n g l e l i n e a n d a m u l t i p l e t a r e shown i n F i g . 5 . 5 - F i g . 5 . 7 F i g u r e 5.5. The c e n t e r l i n e f o r F r e m i e s s a l t b e f o r e ( l i g h t l i n e ) a n d a f t e r ( h e a v y l i n e ) a u t o m a t i c p h a s e c o r r e c t i o n . (#=*-8°). F i g u r e 5.6. The r a d i a l d i f f e r e n c e p l o t c o r r e s p o n d i n g t o t h e d i a g r a m a b o v e . 51 F i g u r e 5 . 7 . An u n i d e n t i f i e d r a d i c a l b e f o r e ( l i g h t l i n e ) and a f t e r (heavy l i n e ) a u t o m a t i c phase c o r r e c t i o n . <j>=*20°. In t h i s case the d i s p e r s i o n l e a k a g e was not apparent u n t i l a s i m u l a t i o n was a t t e m p t e d . 6 . APPLICATIONS TO LINE SHAPE ANALYSIS IN LIQUIDS DISPA i s a p p l i c a b l e t o any s p e c t r o s c o p i c l i n e - s h a p e from any k i n d of sample. The range of p o s s i b l e l i n e - s h a p e s i s t h u s e n d l e s s . Here we w i l l r e s t r i c t the d i s c u s s i o n t o s p e c t r a composed of two G a u s s i a n or L o r e n t z i a n l i n e s . Such l i n e - s h a p e s a r e commonly e n c o u n t e r e d i n magnetic resonance s p e c t r a of s o l u t i o n s . F u r t h e r m o r e , the d i s c u s s i o n w i l l be r e s t r i c t e d t o c a s e s where a s i n g l e ( u n r e s o l v e d ) l i n e i s o b s e r v e d . P a r t i a l l y , or f u l l y r e s o l v e d s p e c t r a can u s u a l l y be a n a l y s e d by i n s p e c t i o n . A l s o the c o r r e s p o n d i n g DISPA p l o t s a r e u s u a l l y complex and d i f f i c u l t t o a n a l y s e . The main o b j e c t i v e i s t o demonstrate the u t i l i t y of DISPA p l o t s f o r a n a l y s i n g ESR s p e c t r a . The r e s u l t s a r e e q u a l l y a p p l i c a b l e t o NMR or any o t h e r t e c h n i q u e though. However, i t s h o u l d be n o t e d t h a t r e s o l v e d d e r i v a t i v e s p e c t r a do not n e c e s s a r i l y g i v e r e s o l v e d a b s o r p t i o n s p e c t r a ( F i g . 6 . l ) ( o r even n o t i c e a b l y d i s t o r t e d s p e c t r a , e s p e c i a l l y i f n o i s e i s p r e s e n t ) so the r e s u l t s a r e s l i g h t l y r e s t r i c t e d w i t h r e s p e c t t o NMR s p e c t r a , which g i v e s the a b s o r p t i o n s pectrum. 52 F i g u r e 6.1. The i n f l u e n c e of i n t e g r a t i o n on r e s o l u t i o n , a) Pure G a u s s i a n l i n e , b) Pure L o r e n t z i a n l i n e , c) Two L o r e n t z i a n l i n e s of d i f f e r e n t w i d t h s and p o s i t i o n s , d) Two L o r e n t z i a n l i n e s of d i f f e r e n t w i d t h s . 54 6.1 CLASSIFICATION OF LOBES The a m p l i t u d e s and s e p a r a t i o n of the l o b e s of a DISPA p l o t are r a r e l y adequate c r i t e r i a t o c o m p l e t e l y c h a r a c t e r i s e the p l o t . I t i s t h u s d e s i r a b l e t o c l a s s i f y the o v e r a l l shapes of the l o b e s somehow. DISPA p l o t s were s i m u l a t e d f o r p a i r s of l i n e s f o r a wide range of c o m b i n a t i o n s of a m p l i t u d e , s e p a r a t i o n , w i d t h s and l i n e - s h a p e . I t was found t h a t the DISPA p l o t s g e n e r a l l y r e t a i n t h e i r mammiform c h a r a c t e r and f a l l i n t o one of the two sequences shown i n F i g . 6 . 2 . The G sequence i s c h a r a c t e r i s t i c of G a u s s i a n l i n e s , the L-W sequence i s c h a r a c t e r i s t i c of L o r e n t z i a n l i n e s . I f the s p e c t r a l l i n e s are asymmetric then the DISPA p l o t s w i l l be a symmetric, one l o b e w i l l f a l l i n one p a r t of the sequence and the o t h e r i n a n other p a r t of the sequence. The sequences shown a r e a p p r o x i m a t e . For i n s t a n c e , the sequence G(2)-G(3)-G has been obse r v e d ( F i g . 6 . 1 O - F i g . 6 . 1 1 ) , but not the sequence G ( 2 ) - G ( 3 ) - G ( 4 ) . P l o t s may of c o u r s e be a t i n t e r m e d i a t e p o i n t s of the sequence. F u r t h e r m o r e , the p l o t s may be m o d i f i e d by i n s t r u m e n t a l a r t e f a c t s ( F i g . 6 . 3 ) ; f o r i n s t a n c e , t r u n c a t i o n adds i n v - l o b e s ; poor p h a s i n g adds i n a n t i s y m m e t r i c l o b e s t o g i v e asymmetric l o b e s . The c u r v a t u r e of the l o b e s can v a r y s u b s t a n t i a l l y , but u s u a l l y o n l y w i t h asymmetric p l o t s . 55 weak G(2) strong G(2) we ok lobe separation G(4) L(l) L(2) strong L(2) weak W(l) w weak w F i g u r e 6.2. C l a s s i f i c a t i o n of d i f f e r e n c e p l o t s . 56 F i g u r e 6 . 3 . M i s c e l l a n e o u s c l a s s i f i c a t i o n of d i f f e r e n c e p l o t s . a ) a symmetric l o b e s , b) v - l o b e s , c) sharp G - l o b e s , d) a n t i s y m m e t r i c l o b e s . 6.2 NOTES ON THE SIMULATIONS AND PLOTS The d i f f e r e n c e p l o t s a r e shown a l o n g w i t h t h e i r c o r r e s p o n d i n g d e r i v a t i v e s p e c t r a . The co r r e s p o n d e n c e i s not one-to-one; the diagrams i l l u s t r a t e t he t r e n d s o n l y . The p l o t s were s e l e c t e d from many s i m u l a t i o n s as b e i n g r e p r e s e n t a t i v e of a p a r t i c u l a r c o m b i n a t i o n of l i n e s . A l l d i f f e r e n c e p l o t s pass t h r o u g h t h e o r i g i n so f o r asymmetric p l o t s the l e f t and r i g h t l o b e s a r e c l a s s i f i e d s e p a r a t e l y . I t i s im p o r t a n t t o note t h a t the l e f t and r i g h t l o b e s undergo m i r r o r r e f l e c t i o n about the o r d i n a t e i f the d i r e c t i o n of f i e l d scan i s r e v e r s e d or the o r d e r of t h e p a i r s of the l i n e s i s r e v e r s e d . 57 A l l numeric i n f o r m a t i o n i s g i v e n as a r a t i o w i t h r e s p e c t t o a r e f e r e n c e l i n e . The r e f e r e n c e l i n e i s c o n s i d e r e d t o have a c o n s t a n t a m p l i t u d e , w i d t h and p o s i t i o n . The second l i n e , added t o the r e f e r e n c e l i n e , i s v a r i a b l e , i t s w i d t h and h e i g h t a r e d i v i d e d by the c o r r e s p o n d i n g v a l u e s f o r the r e f e r e n c e l i n e t o o b t a i n the l i n e - w i d t h and l i n e - h e i g h t r a t i o s r e s p e c t i v e l y . The p o s i t i o n of the v a r i a b l e l i n e i s d e f i n e d i n terms of the l i n e - w i d t h of the r e f e r e n c e l i n e . A n e g a t i v e s h i f t means the v a r i a b l e l i n e i s t o the l e f t of the r e f e r e n c e l i n e . A p o s i t i v e s h i f t p l a c e s the l i n e t o t h e r i g h t of the r e f e r e n c e . Hence a s h i f t of -1.5 from a 3.0G r e f e r e n c e l i n e means t h a t the v a r i a b l e l i n e i s 4.5G t o the l e f t of the r e f e r e n c e . As mentioned e a r l i e r , c h a n g i n g t h e s i g n of the s h i f t j u s t causes the l o b e s t o r e f l e c t t h r o u g h the o r d i n a t e . A l l s h i f t s and r a t i o s a r e r e s t r i c t e d such t h a t the obs e r v e d l i n e i s u n r e s o l v e d and o n l y s l i g h t l y d i s t o r t e d . The s i m u l a t i o n s do not i n c l u d e p h a s i n g or b a s e l i n e a r t e f a c t s , which i n f l u e n c e the symmetry of the l o b e s . A l s o note t h a t a l t h o u g h t h e a b s o r p t i o n l i n e s as p l o t t e d may appear t o be r e a d i l y d i s t i n g u i s h a b l e , i n p r a c t i c e the d i f f e r e n c e s a r e o f t e n o b s c u r e d by n o i s e . 6.3 DETECTING TWO SUPERIMPOSED LORENTZIAN LINES T h i s case might be e x p e c t e d t o occur when a system c o n t a i n s a s p i n l a b e l i n two s i t e s , one s i t e , bound and t h e r e f o r e g i v i n g b r oad l i n e s , t he o t h e r s i t e unbound and thus g i v i n g 58 sharp l i n e s . If we think in terms of c i r c l e plots, Eqn.1.3, i t i s immediately obvious that varying the r a t i o of the heights of the li n e i s indistinguishable from varying the ratios of the widths of the l i n e s ; the radius of the c i r c l e i s proportional to amplitude/line-width. The DISPA plots are of the W-type, the amplitude of which varies with the line-width and amplitude r a t i o s . (Fig.6.4-Fig.6.6). As expected line-height variations are indistinguishable from line-width variations. IQOl F i g u r e 6.4. DISPA plots for superimposed Lorentzian l i n e s and the i r corresponding derivative spectra, for various line-width r a t i o s , but the same amplitude. Ratios for the spectra are 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0. 59 F i g u r e 6.5. DISPA p l o t s f o r superimposed L o r e n t z i a n l i n e s and t h e i r c o r r e s p o n d i n g d e r i v a t i v e s p e c t r a , f o r v a r i o u s l i n e - h e i g h t r a t i o s . L i n e - w i d t h r a t i o i s 3.33. The sequences j u s t r e v e r s e s ( s l o w l y ) f o r r a t i o s > 0.2. R a t i o s f o r the s p e c t r a a r e 0.0, 0.1, 0.2, 0.3, 0.5, 0.7, 1.0, 1.2, 1.5, 2.0, 3.0, 6.4 DETECTING TWO SUPERIMPOSED GAUSSIAN LINES T h i s c a s e may occur f o r l i q u i d s (or s o l i d s ) c o n t a i n i n g two d i f f e r e n t o r g a n i c r a d i c a l s w i t h u n r e s o l v e d h y p e r f i n e c o u p l i n g . S u p e r i m p o s i n g a second G a u s s i a n j u s t r e d i s t r i b u t e s the a m p l i t u d e term, A^ . , i n Eqn. 1.11. The l o b e b e h a v i o u r i s thus e x p e c t e d t o be (and i s ) m a i n l y G-type. S e v e r a l examples ar e g i v e n i n F i g . 6 . 6 - F i g . 6 . 1 1 . F i g u r e 6.6. DISPA plot for superimposed Gaussian l i n e s for various line-width r a t i o s . Lines are the same height. Ratios for the spectra are; 0.0, 1.5, 2.0, 2.5, 3.0. F i g u r e 6.7. DISPA plot for superimposed Gaussian l i n e s for various line-height r a t i o s . Line-width r a t i o i s 2.7. Ratios for spectra are; 0.0, 0.2, 0.5, 0.7, 1.0. 61 F i g u r e 6.8. DISPA plot for superimposed Gaussian l i n e s for various line-height r a t i o s . Line-width r a t i o i s 3.3. Ratios for spectra are; 0.0, 0.03, 0.07. F i g u r e 6.9. DISPA plot for superimposed Gaussian l i n e s for various line-height r a t i o s . Line-width r a t i o i s 3.3. Ratios for spectra are; 0.1, 0.3, 0.5,0.8. 62 F i g u r e 6.10. DISPA p l o t f o r superimposed G a u s s i a n l i n e s f o r v a r i o u s l i n e - h e i g h t r a t i o s . L i n e - w i d t h r a t i o i s 3.3. R a t i o s f o r s p e c t r a a r e ; 1.0, 2.0, 3.0, ». F i g u r e 6.11. DISPA p l o t f o r superimposed G a u s s i a n l i n e s f o r v a r i o u s l i n e - h e i g h t r a t i o s . L i n e - w i d t h r a t i o i s 2.0. R a t i o s f o r s p e c t r a a r e ; 0.0, 0.2, 0.5, 1.0, 2.0, =>. 6.5 DETECTING TWO OVERLAPPING LORENTZIAN LINES R a d i c a l s o c c u p y i n g two d i f f e r e n t s i t e s {vide supra) may have d i f f e r e n t g - s h i f t s . A case of s p e c i a l i n t e r e s t i s t h a t of c h e m i c a l exchange, where, f o r i n s t a n c e , t h e exchange of a l i g a n d c auses a g - s h i f t . 63 The simplest case is for two l i n e s of equal amplitude and height being s p l i t apart. L-type lobes are obtained (Fig.6.12). The combinations for d i f f e r e n t linewidths, amplitudes and s p l i t t i n g s are endless, but a pattern does emerge. If the lines are of the same width, but d i f f e r e n t amplitude an asymmetric L-type plot i s obtained (Fig.6.13 and Fig.6.14). As the separation of the l i n e increases the asymmetry increases. The smaller of the two lobes eventually forms a G(2)-type lobe. The larger lobe remains G-type. If the l i n e s are of d i f f e r e n t widths as well, W-L lobe types dominate. The plots are an asymmetric W-type (Fig.6.15). The asymmetry of the lobes increases as the l i n e s s p l i t apart, with the smaller lobe becomeing an L(2)-type for large s p l i t t i n g s . F i g u r e 6.12. DISPA plot for overlapping Lorentzian l i n e s for various s p l i t t i n g s . Lines are the same height and width. S p l i t t i n g s for the spectrum are; 0.0, 1.0, 1.5, 2.0, 2.5, 3.0. 1001 F i g u r e 6.13. DISPA p l o t f o r o v e r l a p p i n g L o r e n t z i a n l i n e s f o r v a r i o u s s p l i t t i n g s . L i n e s a re the same w i d t h . H e i g h t r a t i o i s 0.33. S p l i t t i n g s f o r the spectrum a r e ; 0.0 1.0, 1.5, 2.0, 2.5. F i g u r e 6.14. DISPA p l o t f o r o v e r l a p p i n g L o r e n t z i a n l i n e s f o r v a r i o u s s p l i t t i n g s . W idth r a t i o i s 2.0. The h e i g h t s a r e the same. S p l i t t i n g s f o r the spectrum a r e ; 0.0 0.2, 0.5, 0.7, 1.0. Note the s i m i l a r i t y t o next f i g u r e . 65 2001 Figure 6 .15. DISPA plot for overlapping Lorentzian lines for various s p l i t t i n g s . Width r a t i o i s 3.33. Height r a t i o i s 0.33. S p l i t t i n g s for the spectrum are; 0.0, 0 .5 , 2.0, 3.0. The s h i f t s are negative. The raised t a i l s in each case are due to a v-type contribution from truncation. (Sect.4.8). 6.6 DETECTING TWO OVERLAPPING GAUSSIAN LINES This case i s of interest because organo-sulfur r a d i c a l s , which occur in coal, are g-shifted with-respect-to 'ordinary' r a d i c a l s , which also occur i s coal . The ESR spectra of coal are thus expected to be two overlapped Gaussian l i n e s . Two Gaussian lines of the same width and height just r e d i s t r i b u t e the amplitudes of the c i r c l e s in Eqn.1.11, with more intensity appearing in the wings of the DISPA plot as the l i n e s s p l i t apart. This gives r i s e to G-type behaviour (Fig.6.16). 66 F i g u r e 6.16. DISPA p l o t f o r o v e r l a p p i n g G a u s s i a n l i n e s f o r v a r i o u s s p l i t t i n g s . L i n e s a r e the same h e i g h t and w i d t h . S p l i t t i n g s f o r the s p e c t a r a r e ; 0.0, 1.5, 2.0, 2.5, 3.0. I f the l i n e s a r e not the same a m p l i t u d e or h e i g h t then asymmetric G l o b e s o c c u r . ( F i g . 6 . 1 7 ) . 40Q1 F i g u r e 6.17. DISPA p l o t f o r o v e r l a p p i n g G a u s s i a n l i n e s f o r v a r i o u s s p l i t t i n g s . L i n e s a r e the same w i d t h . H e i g h t r a t i o i s 0.333. S p l i t t i n g s f o r the s p e c t r a a r e ; 0.0, 1.5, 2.0, 2.5. I t s h o u l d be noted t h a t t h e a m p l i t u d e r a t i o s chosen here a r e not a r b i t r a r y . S m a l l e r r a t i o s g i v e s m a l l e r , but s i m i l a r e f f e c t s . L a r g e r r a t i o s g i v e d e r i v a t i v e s p e c t r a t h a t are p a r t i a l l y resolved and thus were not used. 6.7 DETECTING COMBINATIONS OF LORENTZIAN AND GAUSSIAN LINES This case may seem rather contrived, but can be used to account for the shape of DISPA plots from c o a l . The lobe behaviour i s s u p e r f i c i a l l y similar to the cases for overlapping Gaussian l i n e s , but on the whole the plots are unique to mixed Lorentzian and Gaussian l i n e s . (Fig.6.18-Fig.6.21) The following figures are representative of a range of many possible combinations of l i n e s . loot Figure 6.18. DISPA plot for a mixture of a Lorentzian and Gaussian l i n e s for various width r a t i o s . Amplitudes are the same. S p l i t t i n g s for the spectra are; 0.2, 0.5, 1.0. (The Lorentzian i s the reference l i n e ) . Figure 6.19. DISPA plot for a mixture of a Lorentzian and Gaussian l i n e s for various width r a t i o s . Amplitudes are the same. Ratios for the spectra are; 0.2, 0.5, 1.0. (The Lorentzian i s the reference l i n e ) . IQOl Figure 6.20. DISPA plot for a mixture of a Lorentzian and Gaussian l i n e s for various width r a t i o s . Amplitudes are the same. Ratios for the spectra are; 2.0, 2.5, 3.0, 3.5, 4.0. (The Lorentzian i s the reference l i n e ) . 69 4021 F i g u r e 6.21. DISPA p l o t f o r a m i x t u r e of a L o r e n t z i a n and G a u s s i a n l i n e s f o r v a r i o u s s p l i t t i n g s . A m p l i t u d e s r a t i o i s 0.2. Width r a t i o i s 4.0. S p l i t t i n g s ( n e g a t i v e here) f o r the s p e c t r a a r e ; 0.0 1.0, 2.0, 3.0. (The G a u s s i a n i s the r e f e r e n c e l i n e ) . 6.8 DETECTING AND MEASURING UNRESOLVED HYPERFINE COUPLINGS U n r e s o l v e d h y p e r f i n e c o u p l i n g i s v e r y common i n ESR, n o t a b l y i n t h e n i t r o x i d e t y p e s p i n l a b e l s . The DISPA e q u a t i o n f o r t h i s t y p e of system (Eqn.1.11) seems q u i t e unmanageable. However, t h e number and s p i n of the n u c l e i r e s p o n s i b l e f o r the u n r e s o l v e d h y p e r f i n e c o u p l i n g i s o f t e n known or can be e a s i l y guessed so t h a t ' i ' i n Eqn.6.11 i s d e f i n e d . The a m p l i t u d e s , A^ . , f o l l o w the b i n o m i a l d i s t r i b u t i o n so the o n l y unknown i s A , the h y p e r f i n e s p l i t t i n g c o n s t a n t . I f t h e r e i s o n l y one t y p e of s p i n a c a l i b r a t i o n c h a r t i s e a s i l y c o n s t r u c t e d from the d i f f e r e n c e p l o t s . Figure 6.22. DISPA p l o t f o r u n r e s o l v e d h y p e r f i n e c o u p l i n g f o r v a r i o u s reduced c o u p l i n g c o n s t a n t s ( c o u p l i n g c o n s t a n t / n a t u r a l l i n e - w i d t h ) . H y p e r f i n e s p l i t t i n g i s f o r 12 e q u i v a l e n t p r o t o n s . The h e i g h t of the l o b e s a r e r e l a t e d t o A ( F i g . 6 . 2 2 ) . A c a l i b r a t i o n p l o t f o r 2 —> 12 s p i n h a l f n u c l e i i s shown i n F i g . 6 . 2 3 . The apparent r e d u c e d c o u p l i n g c o n s t a n t i s the c o u p l i n g c o n s t a n t e x p r e s s e d as a f r a c t i o n of the o b s e r v e d l i n e - w i d t h . 71 F i g u r e 6.23. C a l i b r a t i o n c h a r t f o r u n r e s o l v e d h y p e r f i n e c o u p l i n g . The numbers r e f e r t o the no. of s p i n 1/2 n u c l e i c o u p l e d t o the e l e c t r o n . 6.9 APPLICATIONS TO LINE-SHAPE ANALYSIS OF SOLIDS In s o l i d s where G a u s s i a n or L o r e n t z i a n l i n e s o c c u r t h e a n a l y s i s d i s c u s s e d i n the s e c t i o n s above i s a p p l i c a b l e . However, symmetric (or near s y m m e t r i c ) , s i n g l e l i n e , powder p a t t e r n s can a r i s e , t h a t a r e not G a u s s i a n or L o r e n t z i a n i n n a t u r e . For i n s t a n c e , a r a d i c a l w i t h a s m a l l o r t h o r h o m b i c g - t e n s o r may g i v e r i s e t o a symmetric l i n e w i t h s m a l l s h o u l d e r s . T h i s c o u l d be m i s t a k e n f o r superimposed G a u s s i a n l i n e s . A l s o , asymmetric d i f f e r e n c e p l o t s may a r i s e f r om s l i g h t l y a n i s o t r o p i c g - t e n s o r s ( o r g - t e n s o r s averaged by 72 m o t i o n (39)) r a t h e r t h a n f r o m two o v e r l a p p i n g l i n e s . Powder p a t t e r n s i m u l a t i o n s a r e n e e d e d t o c l a r i f y t h i s . The G a u s s i a n d i f f e r e n c e p l o t ( S e c t . 3 . 4 ) w o u l d p r o b a b l y be v e r y u s e f u l i n t h i s c a s e . The e x a m p l e s g i v e n i n t h e n e x t s e c t i o n b e e n i n t e r p r e t e d i n t e r m s o f t h e a n a l y s i s f o r s o l u t i o n s p e c t r a . As s u c h t h e s e r e s u l t s s h o u l d be r e g a r d e d a s a d e m o n s t r a t i o n o f t h e p o t e n t i a l u t i l i t y o f t h e t e c h n i q u e a s a p p l i e d t o s o l i d s a n d n o t a s a d e f i n i t i v e i n t e r p r e t a t i o n o f t h e s p e c t r u m . A d e t a i l e d a n a l y s i s o f DISPA p l o t s f o r s o l i d s w i l l be w o r t h p u r s u i n g , b e c a u s e c o a l a n d wood, two s o l i d s o f g r e a t c o m m e r c i a l v a l u e , b o t h g i v e s i n g l e l i n e ESR s p e c t r a . 7. EXPERIMENTAL EXAMPLES 7.1 TEMPERATURE DEPENDENCE OF UNRESOLVED HYPERFINE If the unresolved hyperfine coupling i s temperature dependent then the residual line-width (Sect. 14.5) is also temperature dependent and t h i s hampers spin relaxation studies. The unresolved hyperfine coupling contribution to nitroxide type spin-probes i s quite large (Fig.7.1, (40)). Proving that the unresolved hyperfine coupling i s temperature independent i s very d i f f i c u l t , the ef f e c t s are masked by motional line-broadening. One method of solving t h i s problem i s to f i x the cor r e l a t i o n time of the probe and then study the DISPA plots as a function of temperature. In practice, t h i s i s achieved by examining the probe in a series of non-polar solvents at temperatures chosen such that the viscosity/temperature (r)/T) remains constant. This of course assumes that the rot a t i o n a l c o r r e l a t i o n time depends only on 77/T and that the eff e c t of the solvent on rotational anisotropy can be neglected. DISPA plots of TEMPONE18 were obtained in pentane, heptane, decane and dodecane for 7j/T values of 0.001 and 0.002 cP K" 1, 1 9 over a temperature range of -20°C to +70°C for the (0) l i n e (for which motional e f f e c t s are the smallest). Within experimental error no temperature dependence of the DISPA plot (Table 7.1) and hence of the 18TEMPONE = 4-oxo-2,2,6,6,-tetramethylpiperidine-1-oxyl TEMPO = 2,2,6,6,tetramethylpiperidine-1-oxyl. 1 9 cP=centi-Poise. 73 74 u n r e s o l v e d h y p e r f i n e c o u p l i n g , was o b s e r v e d . F i g u r e 7.1. D i f f e r e n c e p l o t s f o r TEMPO and TEMPONE, showing the l a r g e u n r e s o l v e d h y p e r f i n e c o u p l i n g c o n t r i b u t i o n t o the l i n e - w i d t h s . TEMPO i s on the l e f t . Tj/T S o l v e n t Temp °C L i n e - w i d t h %R 0.001 Hexane 24.4 1.46 11.2 Heptane , 46.7 1.44 10.8 0.002 Hexane -23.3 1.38 14.5 Heptane -3.0 1.37 15.3 Decane 47.8 1.38 15.6 Dodecane 72.7 1.40 15.6 Table 7.1. Temperature dependence of the u n r e s o l v e d h y p e r f i n e c o u p l i n g f o r TEMPONE 75 7.2 MIXTURES OF SPIN PROBES E s t a b l i s h i n g the degree of b i n d i n g of a s p i n - l a b e l t o the host m o l e c u l e i s i m p o r t a n t i f m e a n i n g f u l m o t i o n a l s t u d i e s a r e t o be done. As the bound and unbound l a b e l s w i l l not d i f f e r much i n g - s h i f t t h i s can be v e r y d i f f i c u l t t o measure. DISPA i s i d e a l l y s u i t e d f o r a n a l y s i n g such a c a s e . To s i m u l a t e t h i s , TEMPO ( l i n e - w i d t h = 3.0G t o r e p r e s e n t a bound s p i n - p r o b e ) was mixed w i t h TEMPONE ( l i n e - w i d t h =*1.5G t o r e p r e s e n t the unbound probe) i n a 1:3 c o n c e n t r a t i o n r a t i o . 2 0 The DISPA p l o t i s q u i t e d i f f e r e n t from t h a t of the i n d i v i d u a l probes ( F i g . 7 . 2 ) as might be e x p e c t e d from S e c t . 6 . 4 . S e m i - q u a n t i t a t i v e r e s u l t s can be o b t a i n e d i f a c a l i b r a t i o n c h a r t f o r the system can be p r e p a r e d (the w i d t h s of t he unbound and the pure bound probe a r e r e q u i r e d ) . Figure 7.2. The DISPA p l o t f o r two superimposed s p i n l a b e l s . An a p p l i c a t i o n of t h i s t o a r e a l c a s e 2 1 i s shown i n F i g . 7 . 3 . 2 0 T h i s was one of a s e r i e s of samples p r e p a r e d by L . F . Y i p i n an attempt t o q u a n t i f y the method. 2 1 T a k e n from work performed by E.Lam i n t h i s l a b o r a t o r y (41). 76 Here a spin label is partitioned between a c e l l membrane ('bound probe') and the surrounding fluid (free probe). The spectrum shows some distortion, but its clear from the DISPA exactly what is causing i t ; a mixture of free and bound probes accompanied by a slight g-shift with-respect-to each other. Figure 7.3. The DISPA plot for an amphipathic spin-probe incorporated into red blood cells in the presence of crystals of mono sodium urate mono-hydrate. The difference plot is for the center line. 7.3 UNRESOLVED HYPERFINE COUPLING CONSTANTS It is useful to determine what proportion of the residual line-width of the copper dithiocarbamate class of spin-probes is due to unresolved hyperfine coupling. The dimethyl derivative is a simple case to study as i t has twelve equivalent protons. The difference plots for this probe are shown in Fig.7.4. The unresolved hyperfine coupling contribution can be measured from these plots. From Fig.6.23 and Fig.7.4, we get a reduced coupling constant of J r 77 0.13, which corresponds to an unresolved hyperfine coupling constant of 0.6G for an observed line-width of 4.6G with coupling to 12 protons. This agrees reasonably well with values from simulations (0.3G (42)) and by comparison with the per-deuterated compound (0.4G via Bales formulas (43)). The s l i g h t l y high value can be attributed to the s a t e l l i t e s and l i n e truncation. It nevertheless provides a good s t a r t i n g point for simulations. 7.4 USING DISPA PLOTS TO DETECT SATELLITES The abscissa of a difference plot i s usually chosen to compress the baseline and expand the resonance region of a spectrum. However, the converse case can be useful i f we wish to detect s a t e l l i t e s in the wings. Fig.7.5 shows the square-root plot for per-deuterated CuPydtc in d-chloroform. The central peaks are due to unresolved hyperfine coupling s.ox F i g u r e 7.4. The DISPA plot for 6 5CuMe 2dtc in toluene. The h i g h - f i e l d l i n e . 78 from the deuterons, residual hydrogen and 1 3C s a t e l l i t e s . The two outer peaks are due to the 3 3S s a t e l l i t e s (four l i n e s ) , which are in the wings of the spectrum (44)(45). 2+sx Figure 7.5. A Square-root difference plot showing s a t e l l i t e s . 7.5 THE DETECTION OF CHEMICAL EXCHANGE. SOLVATION EFFECTS Pyridine exchanges with dithiocarbamates (46) on the ESR time-scale and consequently broadens the l i n e s . For a 50:50 mixture of pyridine and toluene the broadening, in th i s case, i s about 2 Gauss (from 5G), but otherwise the li n e i s unchanged. The DISPA plot i s shown in Fig.7.6. The plot i s very d i f f e r e n t from that obtained in the absence of pyridine (Fig.7.4 and corresponds to the DISPA of two overlapped Lorentzian l i n e s of similar width and intensity (Fig.6.13)), showing that the broadening i s due in part to the pyridine changing the magnetic parameters of the dt c and not just by 79 changing i t s motion. Figure 7.6. DISPA plot showing Chemical Exchange. The h i g h - f i e l d l i n e of 6 5CuPydtc in 50:50 toluene/pyridine. 7.6 THE SPECTRUM OF GREY PITCH Grey p i t c h i s a standard sample for ESR (47). It i s not of any great interest other than for that purpose, but i t does have a single l i n e ESR spectrum. The DISPA plot (Fig.7.7) shows that the l i n e may be considered as a superposition of two (or more) Gaussian or Lorentzian l i n e s of d i f f e r e n t widths and heights (see Fig.6.5, Fig.6.9 and Fig.6.18) and thus the sample contains two (or more) ra d i c a l s or r a d i c a l s i t e s . However, this e f f e c t may be due to a single r a d i c a l with a small orthorhombic g-tensor. 80 F i g u r e 7.7. The DISPA p l o t f o r g r e y - p i t c h . 7.7 GRAPHITE SPECTRA G r a p h i t e i s a t w o - d i m e n s i o n a l c o n d u c t o r and s h o u l d g i v e a Dysonian l i n e - s h a p e ( F i g . 2 . 6 ) . The s h o u l d e r o b s e r v e d i n F i g . 7 . 8 i s not t y p i c a l of the Dysonian l i n e , i t s o r i g i n i s unknown, but i t s a m p l i t u d e v a r i e s w i t h the degree and type of i n t e r c a l a t i o n . (48).22 The DISPA p l o t i s not v e r y u s e f u l and c a s t s no l i g h t on the problem, but i t may be r e g a r d e d as a c o n t r o l f o r c o a l , vide i n f r a , which may c o n t a i n g r a p h i t i c l i k e domains (49). An i n t e r e s t i n g p o s s i b i l i t y f o r Dysonian l i n e s of t h e t y p e shown i n F i g . 2 . 6 i s t o use the auto-phase a l g o r i t h m ( S e c t . 5 ) t o remove the d i s p e r s i o n component of the l i n e and then t o c h a r a c t e r i s e the r e m a i n i n g l i n e by i t s w i d t h , p o s i t i o n and phase c o r r e c t i o n . 2 2 T h i s sample was o b t a i n e d from Dr.F.Aubke and i s e x t r e m e l y p u r e . 81 7.8 COAL SPECTRA C o a l 2 3 g i v e s a s i n g l e l i n e spectrum. V a r i o u s examples are shown o v e r l e a f . The p l o t s may be i n t e r p r e t e d as a L o r e n t z i a n and G a u s s i a n l i n e w i t h v a r i o u s degrees of o v e r l a p . One l i n e p r o b a b l y c o r r e s p o n d s t o a carbon based r a d i c a l and the o t h e r a s u l p h u r based r a d i c a l . The p l o t s were b l i n d ranked i n terms of i n c r e a s i n g l i n e s e p a r a t i o n ( l o b e asymmetry). Note t h a t the s p e c t r a a r e almost i n d i s t i n g u i s h a b l e . From T a b l e 7.2 i t s c l e a r t h a t t h e r a n k i n g c o r r e l a t e s w e l l w i t h the s u l p h u r c o n t e n t , an i m p o r t a n t measure g i v e n Canada's c u r r e n t c o n c e r n w i t h a c i d r a i n . A more d e t a i l e d study i s o b v i o u s l y w a r r a n t e d . 2 3 The samples were o b t a i n e d from Dr.Tao of t h e C o a l I n s t i t u t e UBC. The s p e c t r a t a k e n by Dr.N.R.Jagannathan. Figure 7 . 9 . DISPA plots for various coal samples, (cont. o v e r l e a f ) . F i g u r e 7. samples. 10. DISPA p l o t s f o r v a r i o u s ( c o n t . from p r e v i o u s p age). c o a l 84 SAMPLE NO. COMMENTS 1 Coronach. Saskatchewan l i g n i t e . <0.5% p y r i t e may c o n t a i n o r g a n i c s u l p h u r . 2 F o r e s t b u r g . SW A l b e r t a . Sub-bitumeous c o a l <0.5% p y r i t e . 3 Onakawana. N . O n t a r i o . May c o n t a i n o r g a n i c s u l p h u r More o r g a n i c s u l p h u r than above samples. 4 F o r d i n g A d i t 23. S.E. B.C. bitumeous c o a l . More p y r i t e than above samples, but l e s s o r g a n i c s u l p h u r . 5 Sukunka. A l b e r t a bitumeous c o a l . More p y r i t e , but l e s s s u l p h u r than above samples. 6 Devco. Nova S c o t i a c o a l . High s u l p h u r bitumeous c o a l . 2-3% s u l p h u r . 7 M i n t o . High p y r i t e c o a l . 7% s u l p h u r . T a b l e 7.2. I d e n t i f i c a t i o n of and notes on the c o a l samples. 7.9 WOOD SPECTRA The s p e c t r a and DISPA p l o t s f o r decayed Aspen wood a t v a r i o u s s t a g e s of p h o t o - i r r a d i a t i o n a r e shown i n F i g . 7 . 1 1 - F i g . 7 . 1 3 . 2 " N a t u r a l decayed wood g i v e s a DISPA spectrum ( F i g . 7 . 1 1 ) c o n s i s t e n t w i t h two r a d i c a l s of s i m i l a r w i d t h , but s l i g h t l y d i f f e r e n t a m p l i t u d e s and g - v a l u e s . A f t e r i r r a d i a t i o n the p r o p o r t i o n of one of the r a d i c a l s i n c r e a s e s ( F i g . 7 . 1 2 ) . The r a d i c a l c o n c e n t r a t i o n then s l o w l y decays over a number of d a y s , back t o near the o r i g i n a l c o n c e n t r a t i o n ( F i g . 7 . 1 3 ) . One may p o s t u l a t e t h a t one r a d i c a l 2 4 These samples were o b t a i n e d from L a i Hong of t h e UBC F o r e s t r y department. 85 i s a c h e m i c a l decay p r o d u c t and the o t h e r of p h o t o l y t i c o r i g i n . As w i t h the c o a l samples, the o r i g i n a l s p e c t r a r e v e a l v e r y l i t t l e , but the DISPA p l o t s a r e q u i t e d i f f e r e n t . F i g u r e 7.11. DISPA p l o t f o r n a t u r a l decayed wood. F i g u r e 7.12. DISPA p l o t f o r decayed wood a f t e r i r r a d i a t i o n . 86 7.10 NITROXIDES IN THE SLOW-MOTIONAL REGIME AND POWDER  SPECTRA Both the s p e c t r a and the d i f f e r e n c e p l o t s a r e complex ( F i g . 7 . 1 4 - F i g . 7 . 1 5 ) . In t h i s c a s e an indexed p l o t i s more u s e f u l as the a b s o r p t i o n p l o t i s u n s t a b l e . However, n e i t h e r t y p e of d i f f e r e n c e p l o t i s r e a d i l y i n t e r p r e t a b l e and have o n l y been i n c l u d e d f o r compl e t e n e s s and t o demonstrate one of the l i m i t a t i o n s the DISPA t e c h n i q u e . V a r i a b l e t e m p e r a t u r e experments may y i e l d u s e f u l r e s u l t s , but t h i s a p p r o a c h has not been i n v e s t i g a t e d . 87 BO.PZ i r rr F i g u r e 7.14. DISPA p l o t f o r a powder spectrum. The spectrum of CuPydtc doped i n t o the c o r r e s p o n d i n g n i c k e l s a l t . L o g - i n d e x p l o t . 80, OZ F i g u r e 7.15. DISPA p l o t f o r a n i t r o x i d e i n a membrane. Log - i n d e x p l o t . 8. CONCLUSIONS 8.1 SUMMARY OF RESULTS LOBE DESCRIPTION DIAGNOSIS Lobes of d i f f e r e n t amplitude, but otherwise symmetric Similar to above, but right t a i l drops T a i l s of lobes raised Twin positive going lobes Twin negative going lobes Mis-phased spectrometer or close overlap of similar lines Phase-sensitive-detection amplifier time constant too large Line truncation Unresolved hyperfine coupling, overmodulation or any d i s t r i b u t i o n in li n e - p o s i t i o n Superimposed l i n e s , e. g. a d i s t r i b u t i o n of corre l a t i o n times for a one species Asymmetric lobes See rules of thumb Table 8.1. Summary of results for simple DISPA p l o t s . 8.2 RULES-OF-THUMB These rules of thumb e s s e n t i a l l y summarise the results of Sect.6. a) W-lobe behaviour i s c h a r a c t e r i s t i c of a superposition of l i n e s of equal resonant frequency, but di f f e r e n t widths. 88 89 b) G-lobe b e h a v i o u r i s c h a r a c t e r i s t i c of a symmetric s u p e r p o s i t i o n of l i n e s of d i f f e r e n t f r e q u e n c i e s , w i t h the l i n e a m p l i t u d e d e c a y i n g as the d i s t a n c e of the l i n e from the c e n t e r of symmetry i n c r e a s e s . c) Combining a) and b) and r e t a i n i n g a c e n t e r of symmetry r e s u l t s i n a m i x t u r e of G-lobe and W-lobe b e h a v i o r . M u l t i l o b e d i f f e r e n c e p l o t s may r e s u l t . d) Asymmetric l o b e s from l i q u i d s p e c t r a a r e ( i n the abscence of i n s t r u m e n t a l a r t e f a c t s ) c h a r a c t e r i s t i c of two or more l i n e s of d i f f e r e n t w i d t h or a m p l i t u d e a t d i f f e r e n t r e s o nant f r e q u e n c i e s . e) S o l i d s t a t e s p e c t r a can g i v e s i m p l e DISPA p l o t s , but c a r e s h o u l d be used when u s i n g t h e g e n e r a l i s a t i o n s above. 8.3 CONCLUSIONS DISPA i s an e x p e r i m e n t a l l y and c o n c e p t u a l l y s i m p l e n u m e r i c a l a n a l y s i s method t h a t e n a b l e s us, f o r the f i r s t t i m e , t o get a g e n e r a l , but c o n c r e t e g r a s p of s p e c t r a l l i n e - s h a p e s ; a u s e f u l a d d i t i o n t o l i n e - w i d t h and l i n e - p o s i t i o n i n f o r m a t i o n . P r o b a b l y the most o u t s t a n d i n g f e a t u r e of DISPA p l o t s a r e t h e i r s e n s i t i v i t y t o d e v i a t i o n s from L o r e n t z i a n b e h a v i o u r . S p e c t r a t h a t a r e almost i n d i s t i n g u i s h a b l e , g i v e v e r y d i f f e r e n t DISPA p l o t s and because we have d e v e l o p e d a sound b a s i s f o r DISPA, t h e s e p l o t s a r e , i n most c a s e s , q u a n t i t a t i v e l y (e.g. u n r e s o l v e d h y p e r f i n e c o u p l i n g ) or q u a l i t a t i v e l y (e.g. c o a l samples) i n t e r p r e t a b l e . Of more immediate r e l e v a n c e , DISPA can be e x p l o i t e d i n s p i n - p r o b e 90 s t u d i e s t o a s s e s s s p e c t r o m e t e r performance and a l s o t o i n v e s t i g a t e the ESR s p e c t r a of paramagnetic s p e c i e s t o determine t h e i r s u i t a b l i t y f o r such s t u d i e s . A l t h o u g h much of the i n f o r m a t i o n from DISPA can be o b t a i n e d by o t h e r methods (e.g. s i m u l a t i o n s ) t h e s e methods are u s u a l l y e x t r e m e l y l a b o r i o u s and o f t e n g i v e non-unique s o l u t i o n s . DISPA needs no s p e c i a l equipment ( o t h e r than a computer) or e x p e r i m e n t s and can g i v e a d e f i n i t i v e s o l u t i o n . P A R T 2. R E L A X A T I O N S T U D I E S B Y M A G N E T I C R E S O N A N C E 91 9. INTRODUCTION TO THE MOTIONAL STUDIES 9.1 INTRODUCTION In r e c e n t y e a r s i t i s has become a p p a r e n t , e s p e c i a l l y i n b i o c h e m i s t r y , t h a t m o l e c u l a r s t r u c t u r e i s not o n l y i n t i m a t e l y r e l a t e d t o f u n c t i o n , but i s p r o b a b l y as i m p o r t a n t as the ' c h e m i s t r y ' . U n f o r t u n a t e l y , t h e r e i s a p a u c i t y of t e c h n i q u e s f o r d e t e r m i n i n g m o l e c u l a r geometry i n s o l u t i o n . One p o s s i b l e a p p r o a c h , developed i n the l a s t few y e a r s , i s to r e l a t e m o l e c u l a r r e o r i e n t a t i o n t o m o l e c u l a r s t r u c t u r e . The p r o b l e m s , however, are q u i t e f o r m i d a b l e . F i r s t l y one has t o r e l a t e the e x p e r i m e n t a l l y a c c e s s i b l e parameters t o the c o r r e l a t i o n f u n c t i o n f o r the mo t i o n . T h i s then has t o be r e l a t e d t o the m o l e c u l a r m o t i o n , which f i n a l l y has t o be r e l a t e d t o m o l e c u l a r geometry. Each one of t h e s e s t e p s a l o n e c o n s t i t u t e major and c h a l l e n g i n g a r e a s of r e s e a r c h . M o l e c u l a r r e o r i e n t a t i o n ( o f t e n l o o s l y r e f e r r e d t o as r o t a t i o n ) i n s o l u t i o n i s g e n e r a l l y s t u d i e d by measuring the response of the s o l u t i o n t o a r a d i a t i o n f i e l d . For such measurements t o r e f l e c t the r e o r i e n t a t i o n of the m o l e c u l e i n the s o l u t i o n , the i n t e r a c t i o n between the f i e l d w i t h the m o l e c u l e must depend on the o r i e n t a t i o n of t h e m o l e c u l e i n t h a t f i e l d , i . e . , t h e i n t e r a c t i o n and the r a d i a t i o n f i e l d must be a n i s o t r o p i c . G e n e r a l l y t h e response of a system a t a fr e q u e n c y C J , J ( C J ) , t o a p e r t u r b i n g f i e l d , T i s g i v e n by (50) 92 93 J ( u ) = F { G ( t ) } (9.1) where F{x} d e n o t e s the F o u r i e r t r a n s f o r m and G ( t ) , the c o r r e l a t i o n f u n c t i o n , i s g i v e n by where F i s a m o l e c u l a r t e n s o r p r o p e r t y t h a t i n t e r a c t s w i t h the f i e l d . The <> denotes an ensemble average. I t i s c o n v e n i e n t t o t r a n s f o r m the two t e n s o r s i n t o the same r e f e r e n c e frame. T h i s i s most c o n v e n i e n t l y done i n a s p h e r i c a l b a s i s so f o r example t r a n s f o r m i n g t o the m o l e c u l a r frame we get where D i s t h e Wigner r o t a t i o n m a t r i x and (a/37) a r e the E u l e r a n g l e s r e l a t i n g the l a b o r a t o r y ( f i e l d ) frame t o the m o l e c u l a r frame, 'm' and 'k' a r e the t e n s o r element i n d i c e s and j i s t h e rank of the t e n s o r (1 f o r a s c a l a r , 2 f o r a v e c t o r and 3 f o r a t e n s o r ) . 2 5 Tensor p r o d u c t s a r e q u i t e s i m p l e i n a s p h e r i c a l b a s i s , hence 2 5 S c a l a r s a r e r o t a t i o n a l l y i n v a r i a n t and are of no i n t e r e s t h e r e . G ( t ) = < [ T ( 0 ) . F ( 0 ) ] . [ T ( t ) . F ( t ) ] > (9.2) (9.3) 94 »-V "*?!.<-" V!j'ri,'»E/ t[.#7(t)] ( 9 . 4 ) N o t e t h a t t h e t i m e d e p e n d e n c e i s c a r r i e d e n t i r e l y by t h e W i g n e r r o t a t i o n m a t r i x . H e n c e we f i n d f o r G ( t ) G(t) - <DL{0) a'BL{t) t> (9'5) Eqn.9.4 & Eqn.9.5 h o l d t h e key t o e x p e r i m e n t a l d e s i g n f o r m o l e c u l a r r o t a t i o n s t u d i e s . T h e s u b s c r i p t s ' a' a n d 'b' i n Eqn . 9 . 5 d e n o t e s d i f f e r e n t m o l e c u l e s ; t h e e v a l u a t i o n o f Eqn . 9 . 5 n o t o n l y d e p e n d s on t h e o r i e n t a t i o n o f a g i v e n m o l e c u l e , b u t a l s o on t h e o r i e n t a t i o n o f i t s n e i g h b o u r s . When a=b, a l w a y s , G ( t ) i s c a l l e d a s i n g l e p a r t i c l e c o r r e l a t i o n f u n c t i o n ( i n t e r m o l e c u l a r i n t e r a c t i o n s do n o t i n f l u e n c e t h e r o t a t i o n o r t h e m o l e c u l e s i n t e r a c t i o n w i t h t h e f i e l d ) . When a*b, g e n e r a l l y , t h e n i t i s c a l l e d a m u l t i - p a r t i c l e c o r r e l a t i o n f u n c t i o n ( n e i g h b o u r i n g m o l e c u l e s i n t e r a c t ) . M u l t i - p a r t i c l e c o r r e l a t i o n f u n c t i o n s a r e e x t r e m e l y d i f f i c u l t t o i n t e r p r e t a n d t h e r e i s no s a t i s f a c t o r y way o f r e l a t i n g them t o s i n g l e p a r t i c l e c o r r e l a t i o n f u n c t i o n s , w h i c h a r e more m a n a g e a b l e . F o r i n s t a n c e , i f we p e r f o r m d i e l e c t r i c s t u d i e s , where F i s t h e d i p o l e moment o f t h e m o l e c u l e , we o b t a i n m u l t i - p a r t i c l e c o r r e l a t i o n f u n c t i o n s b e c a u s e o f t h e c o n t r i b u t i o n s f r o m i n d u c e d d i p o l e s . On t h e o t h e r h a n d i f we do i n f r a - r e d ( I R ) s t u d i e s , where 95 F i s the time d e r i v a t i v e of t h e d i p o l e moment, we o b t a i n s i n g l e p a r t i c l e c o r r e l a t i o n f u n c t i o n s . A l t h o u g h the d i p o l e s a r e c o u p l e d f l u c t u a t i o n s i n them a r e no t , or o n l y weakly so. The d i p o l e moment i s a v e c t o r p r o p e r t y , hence o n l y two s p e c t r a l d e n s i t i e s , J1' 0 and J 1 ^ 1 a r e o b s e r v a b l e . L i g h t s c a t t e r i n g or Raman s t u d i e s , which i n v o l v e the p o l a r i s i b i l i t y , a t e n s o r , g i v e s f i v e s p e c t r a l d e n s i t i e s , J 1 , 0 , J 7 ' ± 7 , J 2 ' 0 , J 2 , ± i , J 2 ' ± 2 . O b v i o u s l y the more data the b e t t e r . M a gnetic resonance, the s u b j e c t of t h i s p a r t of the t h e s i s , i s an o b v i o u s c h o i c e f o r m o t i o n a l s t u d i e s . However, i t d i f f e r s i n a number of ways from the methods mentioned • above. There a r e s e v e r a l c h o i c e s f o r the i n t e r a c t i o n t e n s o r ( q u a d r u p o l e , c h e m i c a l s h i f t a n i s o t r o p y etc.; t h e s e a r e d i s c u s s e d i n d e t a i l i n the next s e c t i o n ) . F u r t h e r m o r e , the t e c h n i q u e i s q u i t e s e n s i t i v e , the o p t i c a l methods are r e s t r i c t e d t o pure s o l v e n t s or c o n c e n t r a t e d s o l u t i o n s . NMR and ESR may be used w i t h d i l u t e s o l u t i o n s (i.e. , 10% s o l u t e or l e s s ) and the the dynamics of s o l u t e s not j u s t neat s o l v e n t s can be s t u d i e d . U n f o r t u n a t e l y magnetic resonance i s a s i n g l e f r e q u e n c y t e c h n i q u e . U n l i k e the o t h e r t e c h n i q u e s where the whole c o r r e l a t i o n f u n c t i o n i s a v a i l a b l e ( i t s the F o u r i e r t r a n s f o r m of the J ( w ) , the l i n e - s h a p e ) , magnetic resonance can o n l y g i v e s p e c t r a l d e n s i t i e s f o r two f r e q u e n c i e s , t h e s p e c t r o m e t e r f r e q u e n c y and z e r o f r e q u e n c y . A l s o the s e n s i t i v i t y of magnetic resonance a r i s e s from i t s 96 s e l e c t i v e l y (ESR o n l y d e t e c t s paramagnetic s p e c i e s ) . T h i s can make i t d i f f i c u l t t o o b t a i n enough independent i n f o r m a t i o n t o measure the d i f f u s i o n t e n s o r . In the p a s t i t has been n e c e s s a r y t o combine NMR w i t h o p t i c a l s t u d i e s , e.g. (51), or t o p e r f o r m m u l t i - n u c l e a r NMR s t u d i e s , e.g. (52), t o o b t a i n enough i n f o r m a t i o n . Here, f o r the f i r s t t i m e , we combine NMR s t u d i e s w i t h ESR s t u d i e s t o measure a d i f f u s i o n t e n s o r . A f l o w c h a r t of the g e n e r a l s t r a t e g y f o r a m u l t i - t e c h n i q u e approach i s shown i n F i g . 9 . 1 . T h i s work w i l l be r e s t r i c t e d t o the f a s t m o t i o n a l case ( R e d f i e l d t h e o r y ) and the Debye d i f f u s i o n model. The r e s u l t s w i l l be examined u s i n g s i m p l e hydrodynamic models. For an bverveiw of s t r a t e g i e s i n m o l e c u l a r dynamics see (53)(54)(55) ( o p t i c a l methods), (56)(57), ( t h e o r e t i c a l s t u d i e s ) and (58)(59) ( m u l t i - t e c h n i q u e a p p r o a c h ) . LINE-SHAPE STUDIES SLOW-MOTIONRL THEORY LINE-WIDTH STUDIES RELAX TIM ATION ES T l EXPERI-MENTS FAST-MOTIONAL THEORIES REDUCED SPECTRAL DENSITIES DECON-VOLUTION METHODS CORRELATION FUNCTIONS MOTIONAL MODELS BAND-SHAPE STUDIES OPT I C f l L S T U D I E S DIFFUSION TENSORS HYDRO-DYNAMICS MOLECULAR GEOMETRY F i g u r e 9.1. The g e n e r a l s t r a t e g y f o r o b t a i n i n g g e o m e t r i c i n f o r m a t i o n from m o t i o n a l s t u d i e s by magnetic r e s o n a n c e . 98 9.2 CHOICE OF SPIN PROBE The o b j e c t i v e of t h i s t h e s i s i s t o e x p l o i t the f l e x i b l e c h e m i s t r y of the d i t h i o c a r b a m a t e c l a s s of s p i n - p r o b e and t o combine NMR and ESR s t u d i e s and measure the d i f f u s i o n t e n s o r f o r the probes i n s o l u t i o n . The o r i g i n a l c h o i c e of probe i s h i s t o r i c a l , but i t i s a p p r o p r i a t e t o d i s c u s s the advantages of t h i s probe over o t h e r c l a s s e s of s p i n probe. F i r s t l y , i t s h o u l d be p o i n t e d out t h a t the purpose of the e x e r c i s e i s t o s t u d y the motion of the s p i n - p r o b e i n s o l u t i o n . T h i s i s e n t i r e l y d i f f e r e n t from s t u d y i n g the motion of the s o l v e n t , or from the concept of s p i n - l a b e l l i n g , whereby a s p i n - p r o b e i s a t t a c h e d t o a. macromolecule and the motion of the macromolecule i s deduced from the s p e c t r a of the s p i n - l a b e l . In the l a t t e r case p e r t u r b a t i o n i s i m p o r t a n t ( d i s t o r t i o n of the l o c a l s t r u c t u r e or t h e l o c a l motion by the s p i n - p r o b e i t s e l f ) ; t h e probe must be s m a l l and the r e s u l t s i n t e r p r e t e d c a r e f u l l y as a r t e f a c t s can o c c u r . In our c a s e p e r t u r b a t i o n i s i r r e l e v a n t as l o n g as one i n t e r p r e t s the d a t a i n terms of t h e n a t u r e of the probe and i t s i n t e r a c t i o n w i t h the s o l v e n t and does not t r y t o g e n e r a l i s e the c o n c l u s i o n s t o the o v e r a l l s t r u c t u r e and motion of t h e s o l v e n t . The n i t r o x i d e t ype s p i n - p r o b e s are r e l a t i v e l y s m a l l and f a i r l y s t a b l e . T h e i r s m a l l s i z e m i n i m i s e s p e r t u r b a t i o n problems and they have been e x t e n s i v e l y used f o r b i o l o g i c a l s p i n - p r o b e and s p i n - l a b e l l i n g s t u d i e s . However, f o r 'pure' 99 m o t i o n a l s t u d i e s t h e y h a v e a number o f p r o b l e m s : T h e y h a v e a l a r g e u n r e s o l v e d h y p e r f i n e c o u p l i n g c o n t r i b u t i o n t o t h e l i n e - w i d t h ( w h i c h may be o v e r c o m e by d e u t e r a t i o n , b u t t h i s i s g e n e r a l l y v e r y t e d i o u s t o d o ) ; t h e y do n o t h a v e s i m p l e g e o m e t r i e s , by v i r t u e o f t h e p r o t e c t i n g g r o u p s ; t h e l i n e - w i d t h s h a v e a r e l a t i v e l y l a r g e s p i n r o t a t i o n c o n t r i b u t i o n , t h i s c a n make e x t r a c t i o n o f t h e c o r r e l a t i o n t i m e s u n r e l i a b l e ; t h e h y p e r f i n e a n i s o t r o p y i s s m a l l s o t h a t t h e nu d e p e n d e n c e o f t h e l i n e - w i d t h i s s m a l l ( s e e E q n . 2 0 . 2 ) , t h i s a l s o d e c r e a s e s t h e r e l i a b i l i t y o f t h e d e t e r m i n e d c o r r e l a t i o n t i m e s ; t h e y h a v e a t h r e e l i n e s p e c t r u m s o t h a t t h e d i f f u s i o n t e n s o r c a n n o t be d e t e r m i n e d ( a t l e a s t f o u r l i n e s a r e n e e d e d , t h r e e t o g e t t h e t e n s o r a n d one f o r t h e s p i n - r o t a t i o n t e r m ) ; a n d f i n a l l y t h e r e a r e no r e a d i l y a v a i l a b l e d i a m a g n e t i c a n a l o g s , t h u s NMR s t u d i e s c a n n o t be u s e d t o p r o v i d e more i n f o r m a t i o n . Many o f t h e p r o b l e m s a s s o c i a t e d w i t h t h e n i t r o x i d e s p i n - p r o b e s c a n be o v e r c o m e by t h e u s e o f m e t a l c o m p l e x e s . T h i s a p p r o a c h was p i o n e e r e d by K i v e l s o n et al who u s e d c o p p e r a c e t y l - a c e t o n a t e c o m p l e x e s (60)(61)(62)(63). U n f o r t u n a t e l y t h e s e c o m p l e x e s h a v e v e r y l a r g e g a n i s o t r o p i c s s o a n a l y s i s i s c o m p l i c a t e d by t h e s p i n - r o t a t i o n t e r m , a s w i t h n i t r o x i d e s , a n d a l s o by l i n e o v e r l a p . A l s o t h e c h e m i s t r y o f t h e s e c o m p l e x e s i s i n f l e x i b l e , e. g. i t i s d i f f i c u l t t o c h a n g e t h e i r g e o m e t r y i n a s y s t e m a t i c m anner. H o w e v e r , t h e s e compounds a r e p a r t o f a l a r g e c l a s s o f ML, c o m p l e x e s , where L = 0 (64), N (65) (66), S (67) (68) (69) (70), 100 Se (71) and v a r i o u s c o m b i n a t i o n s t h e r e o f (72)(73). Oxygen based l i g a n d s g i v e complexes w i t h l a r g e g - a n i s o t r o p i e s (74), P, N and Se based l i g a n d s a l l g i v e ( u n d e s i r a b l e ) h y p e r f i n e s p l i t t i n g s . MSi, type compounds have n e i t h e r of these problems and a l l have s i m i l a r s p i n H a m i l t o n i a n p a r a m e t e r s . ( i . e . , a x i a l l y symmetric t e n s o r s w i t h Ao=*80G and g o - 2 . 0 4 ) . Of t h i s c l a s s of compounds, the d i t h i o c a r b a m a t e s a r e the e a s i e s t t o prepare i n a wide range of s u b s t i t u t i o n s , i s o t o p i c and o t h e r w i s e . They a r e a l s o w e l l c h a r a c t e r i s e d (44) (69) (70)(75)(76)(77)(7 8)(7 9)(80) (81) and g e n e r a l l y v e r y s t a b l e . ESR m o t i o n a l s t u d i e s w i t h these complexes was p i o n e e r e d by H e r r i n g et al. (82)(83)(84).26 P r e p a r a t i o n i s via secondary amines (see Sect.11) which a r e r e a d i l y a v a i l a b l e w i t h v a r i o u s i s o t o p i c s u b s t i t u t i o n s . The l i g a n d s a l s o form complexes w i t h a wide range of m e t a l s so t h a t m u l t i n u c l e a r NMR of the d i a m a g n e t i c complexes i s p o s s i b l e w i t h t h e s e compounds. D i t h i o c a r b a m a t e s have been used t o study b i o l o g i c a l systems (86). A l s o , a t t e m p t s have been made t o s y n t h e s i s e water s o l u b l e d e r i v a t i v e s f o r b i o l o g i c a l s t u d i e s (74). 2 6 G i b s o n (85) was the f i r s t t o do m o t i o n a l s t u d i e s of a d i t h i o c a r b a m a t e , but he used mixed copper i s o t o p e s f o r h i s i n v e s t i g a t i o n . H i s work i s of h i s t o r i c a l i n t e r e s t o n l y . 101 9.3 CHOICE OF PROBE SUBSTITUENTS The g e n e r a l f o r m u l a f o r a m e t a l ( I I ) d i t h i o c a r b a m a t e i s F i g u r e 9 .2 . T y p i c a l m e t a l d i t h i o c a r b a m a t e . R', R'', R''', R'''' a r e n o t n e c e s s a r i l y , b u t u s u a l l y a r e , t h e same. The R', R'' a n d R''', R''' 1 p a i r s may c o n s t i t u t e a c y c l i c g r o u p . ( A b b r e v i a t i o n s f o r t h e R s u b s t i t u e n t s a n d g e n e r a l n o m e n c l a t u r e a r e g i v e n i n a p p e n d i x 22.1). T h e s e s u b s t i t u e n t s may be r e a d i l y c h a n g e d t o g i v e a w i d e r a n g e o f m o l e c u l a r g e o m e t r i e s . The r e q u i r e m e n t s f o r t h e c h o i c e o f R a r e ; t h e compound must be r e a s o n a b l y s o l u b l e ( f o r t h e NMR s t u d i e s ) , t h e c o m p l e x s h o u l d be r i g i d w i t h a w e l l d e f i n e d g e o m e t r y ( t o e l i m i n a t e r e l a x a t i o n c o n t r i b u t i o n s f r o m i n t e r n a l m o t i o n a n d t o s i m p l i f y t h e u s e o f h y d r o d y n a m i c m o d e l s ) , t h e c o m p l e x s h o u l d be s t a b l e 2 7 a n d f o r d e u t e r i u m NMR s t u d i e s , t h e a l k y l m o i e t y h a s t o be s u b s t i t u t e d s u c h t h a t a t l e a s t two C-D b o n d s l i e o u t o f t h e M S „ p l a n e a n d t h e s e b o n d s s h o u l d n o t be 2 7 T h e p y r r o l e d e r i v a t i v e h a s a v e r y d e s i r a b l e g e o m e t r y , h o w e v e r i t i s u n s t a b l e (87). G e n e r a l l y t h o u g h d i t h i o c a r b a m a t e s c o m p l e x e s a r e e x t r e m e l y s t a b l e a n d a r e e x t e n s i v e l y u s e d i n a n a l y t i c a l c h e m i s t r y (79). 1 02 r e l a t e d by symmetry (see S e c t . 2 0 . 6 ) . In g e n e r a l , s m a l l or r i g i d s u b s t i t u e n t s g i v e d e r i v a t i v e s t h a t have low s o l u b i l i t i e s (see T a b l e 11.2). The methyl d e r i v a t i v e i s good f o r ESR s t u d i e s (82) ( i t i s r e a d i l y d e u t e r a t e d and has a s i m p l e geometry), but i s t o o i n s o l u b l e f o r NMR s t u d i e s . The p y r o l l i d i n e d e r i v a t i v e was somewhat b e t t e r f o r NMR s t u d i e s , but 1 3 C work r e q u i r e s a compromise and the r e l a t i v e l y s o l u b l e e t h y l d e r i v a t i v e was used. T h i s d e r i v a t i v e has a p o o r l y d e f i n e d s t r u c t u r e ( i n s o l u t i o n , as do the most of t h e o t h e r d e r i v a t i v e s i n Table 11.2), but m o l e c u l a r models i n d i c a t e t h a t the motion of the e t h y l groups i s p r o b a b l y v e r y s m a l l due t o s t e r i c h i n d r a n c e . A l s o i t i s known from p r e v i o u s ESR s t u d i e s (88)(84) t h a t the e t h y l and p y r o l l i d i n e d e r i v a t i v e s behave s i m i l a r l y . 9.4 CHOICE OF CENTRAL METAL As we w i s h t o m a i n t a i n s i m p l e g e o m e t r i e s t o a s s i s t the a n a l y s i s of t h e d a t a , the c e n t r a l m e t a l s h o u l d be chosen such the complex i s square p l a n a r . T h i s r e s t r i c t s our c h o i c e t o d i v a l e n t m e t a l s . For ESR, t h e complex must be p a r a m a g n e t i c , f o r NMR, d i a m a g n e t i c . C l e a r l y two d i f f e r e n t m e t a l s a r e needed, but the r e s u l t i n g complexes must be i s o s t r u c t u r a l . 9.4.1 CENTRAL METAL FOR ESR EXPERIMENTS The c e n t r a l m e t a l s h o u l d have a n u c l e a r s p i n > 1 ( t o s o l v e f o r t h r e e d i f f u s i o n c o n s t a n t s and a s p i n 103 r o t a t i o n term r e q u i r e s t h a t we must have a t l e a s t f o u r o b s e r v a b l e l i n e s ) . The two most s u i t a b l e c a n d i d a t e s a r e copper and vanadium. U n f o r t u n a t e l y the vanadium complexes o x i d i s e r e a d i l y t o V=0 type complexes (74), which are not v e r y s o l u b l e . A l s o the g e o m e t r i c s i m p l i c i t y i s d e s t r o y e d . The copper complexes are v e r y s t a b l e and w e l l c h a r a c t e r i s e d . Copper has two i s o t o p e s , 6 3 C u and 6 5 C u . The 6 3 C u was used f o r h i s t o r i c a l r e a s o n s . 9.4.2 CENTRAL METAL FOR NMR EXPERIMENTS A m e t a l w i t h a z e r o n u c l e a r s p i n i s u s e f u l , but not n e c e s s a r y . Obvious c a n d i d a t e s a r e N i , Zn and Pd. The z i n c complexes a r e d i s t o r t e d t e t r a h e d r a l and are t h u s not i s o s t r u c t u r a l w i t h the copper complexes (89) and were not used. P a l l a d i u m i s i n t e r e s t i n g because i t has a non-zero s p i n and can be s t u d i e d d i r e c t l y by NMR. U n f o r t u n a t e l y i t s gyromagnetic r a t i o i s t o o s m a l l t o e x p l o i t w i t h the a v a i l a b l e s p e c t r o m e t e r s (CJo-10MHZ a t 4.7T). The n i c k e l complexes a r e isomorphous w i t h t h e copper complexes, have z e r o s p i n and t h u s were used. However, they a r e much l e s s s o l u b l e than the c o r r e s p o n d i n g copper complexes (as a r e the p a l l a d i u m complexes) and t h i s somewhat l i m i t s the u s e f u l n e s s of t h e s e complexes f o r NMR s t u d i e s . 10. GENERAL THEORY M o t i o n a l r e l a x a t i o n t h e o r y f o r magnetic resonance i s dominated by t h r e e t h e o r i e s ; R e d f i e l d t h e o r y (90)(91)(92), Kubo-Tomita t h e o r y as de v e l o p e d by K i v e l s o n (93), and the s t o c h a s t i c L i o u v i l l e t h e o r y as e x p l o i t e d by F r e e d (39). Kubo-Tomita t h e o r y has a r e l a t i v e l y easy p h y s i c a l i n t e r p r e t a t i o n , but i t s p o p u l a r i t y has waned due t o the unwarranted c o n t r o v e r s y s u r r o u n d i n g i t s a b i l i t y t o handle degenerate t r a n s i t i o n s (94). S t o c h a s t i c L i o u v i l l e t h e o r y i s p r o b a b l y the d e f i n i t i v e r e l a x a t i o n t h e o r y and can handle the slow-motion c a s e . However, i t i s based on a s l o w l y c o n v e r g i n g s e r i e s , f o r which i t i s d i f f i c u l t t o a s s i g n p h y s i c a l meaning. A l s o the computer time r e q u i r e d f o r thes e c a l c u l a t i o n s i s not j u s t i f i e d f o r the s i m p l e r e l a x a t i o n c a s e s . R e d f i e l d t h e o r y i s based on a r a p i d l y c o n v e r g i n g s e r i e s ( f o r s h o r t c o r r e l a t i o n t i m e s ) w i t h a s i m p l e p h y s i c a l i n t e r p r e t a t i o n 2 8 and i s w e l l s u i t e d f o r d e s c r i b i n g NMR r e l a x a t i o n . I t can be r e a d i l y adapted f o r ESR s t u d i e s i f f i r s t o r d e r w a v e f u n c t i o n s a r e used where n e c e s s a r y , but cannot be used i n the s l o w - m o t i o n a l regime. The p r i n c i p a l problem w i t h a l l r e l a x a t i o n t h e o r i e s . i s the e v a l u a t i o n of the c o r r e l a t i o n f u n c t i o n f o r m o l e c u l a r m o t i o n . S e v e r a l approaches have been made t o t h i s problem (50)(96), but a l l these t h e o r i e s converge t o t h e Debye d i f f u s i o n case i n the f a s t m o t i o n a l l i m i t . A l s o the t h e o r y of a n i s o t r o p i c motion has o n l y been d e v e l o p e d f o r the Debye 2 8 An e x p a n s i o n t o , and p h y s i c a l i n t e r p r e t a t i o n o f , h i g h e r o r d e r terms i s g i v e n by S i l l e s c u and K i v e l s o n (95) 1 0 4 1 05 d i f f u s i o n . For t h a t reason o n l y Debye d i f f u s i o n and i t s e x t e n s i o n s w i l l be d i s c u s s e d i n t h i s work. Reviews of the o t h e r approaches are g i v e n by S t e e l e (96) and McClung (50). 10.1 INTRODUCTION TO REDFIELD THEORY R e d f i e l d r e l a x a t i o n t h e o r y f o r n u c l e a r magnetic resonance i s d e a l t w i t h i n a number of t e x t s (91)(97). R e d f i e l d t h e o r y f o r ESR can be found i n papers by F r e e d and F r a e n k e l (90) and F r e e d (98). G e n e r a l l y magnetic f i e l d inhomogeneity makes v e r y l i t t l e c o n t r i b u t i o n t o the l i n e w i d t h s of ESR s p e c t r a and hence i t i s p o s s i b l e t o a c c u r a t e l y measure T 2's from the s p e c t r a l l i n e w i d t h s . T h i s however, i s not so f o r NMR, where f i e l d inhomogeneity i s the p r i n c i p a l s o u r c e of l i n e b r o a d e n i n g . In t h i s case T,'s are measured. The development of R e d f i e l d t h e o r y f o r T / s and T 2's i s t h e same and a b r i e f o u t l i n e i s g i v e n below. However ESR and NMR d i f f e r s u f f i c i e n t l y i n d e t a i l t o m e r i t s e p a r a t e d i s c u s s i o n s . (Sect.12 and S e c t . 1 6 ) . An element of the r e l a x a t i o n m a t r i x (R) f o r a s t a t e , |a>, i s g i v e n by (91)29 2 9 F r e e d (98) drops the f a c t o r of 2 i n t h i s e q u a t i o n . I t a r i s e s from the d e f i n i t i o n used f o r the s p e c t r a l d e n s i t i e s , where a f a c t o r of a h a l f i s o f t e n i n t r o d u c e d . 106 (r) aa' 00' ~ 2J aaP' 0 P' P a' a00 • (w0/3' ) " J a' aP' p((*P' 0 (r) - J aa' 0/3' 0/3 aa00 (10.1) where the primes denote the c o r r e s p o n d i n g upper s p i n s t a t e s , 0 r e p r e s e n t s any o t h e r s p i n s t a t e i n the system ( i t s c h o i c e w i l l depend on the r e l a x a t i o n r a t e r e q u i r e d ) and 7 r e p r e s e n t s a l l o t h e r s p i n s t a t e s i n the system (e.g. from h y p e r f i n e s p l i t t i n g s ) and t a k e s i n t o account the f a c t t h a t the r e l a x a t i o n pathways are not n e c e s s a r i l y the r e v e r s e of the e x c i t a t i o n pathway (the l e a d i n g t e r m ) . The s p e c t r a l d e n s i t i e s f o r each t r a n s i t i o n f r e q u e n c y , to, a r e g i v e n as f o l l o w s . (Some a u t h o r s i n t r o d u c e a f a c t o r of 1/2 here as i t s i m p l i f i e s t he s p e c t r a l d e n s i t i e s i n the case of i s o t r o p i c d i f f u s i o n ) : aa' PP' = F{G aa* 00 (10.2) where F{X} denotes t h e F o u r i e r t r a n s f o r m . F { G ( t ) } = £ G ( t ) e " / w r d t (10.3) 107 The c o r r e l a t i o n f u n c t i o n G ( t ) i s = <H,(t) .H,(t+r) (10.4) where H, i s the p e r t u r b i n g H a m i l t o n i a n . The a n g l e b r a c k e t s denote an ensemble average. D r o p p i n g aa' and 0/3' f o r c o n v e n i e n c e , the time dependent p e r t u r b a t i o n i s g i v e n by, The X i s used t o l a b e l the d i f f e r e n t type of i n t e r a c t i o n H a m i l t o n i a n ; h y p e r f i n e c o u p l i n g , s p i n - r o t a t i o n , quadrupole c o u p l i n g etc. The parameter e ( t ) i s the r . f . f i e l d used f o r d e t e c t i o n . In p u l s e d NMR t h i s i s z e r o d u r i n g o b s e r v a t i o n . In ESR t h i s term may be dropped as i t o s c i l l a t e s on a time s c a l e much f a s t e r than T 2 and t h e r e f o r e does not c o n t r i b u t e t o the r e l a x a t i o n (82). The H a m i l t o n i a n terms a r e themselves c o m b i n a t i o n s of s p i n o p e r a t o r s (A) and magnetic i n t e r a c t i o n t e n s o r s (F) so H, ( t ) + e ( t ) (10.5) X H(Mt) =.Z.A r X ;Ff X ; ( t ) (10.6) I t i s more c o n v e n i e n t t o d e f i n e the t e n s o r s i n a s p h e r i c a l b a s i s (90)(99)(100)(101) so t h a t 1 0 8 -Z .A. .F : . ( t ) = I i , j ij ij m ( - n V m)F(l , -m) ^ ( 1 0 . 7 ) where / i s the rank of the t e n s o r and m=-/ —> +/ Here we have d e f i n e d t h e s p i n o p e r a t o r s i n the l a b o r a t o r y frame and the magnetic t e n s o r s i n the m o l e c u l a r f r a m e . 3 0 The l a t t e r t h u s c a r r i e s the time dependence f o r m o l e c u l a r r o t a t i o n . The magnetic i n t e r a c t i o n t e n s o r i s t r a n s f o r m e d i n t o the l a b o r a t o r y frame w i t h the assumption t h a t the m o l e c u l a r and magnetic i n t e r a c t i o n t e n s o r s a r e c o i n c i d e n t . T h i s i s r e a d i l y done w i t h Wigner r o t a t i o n m a t r i c e s (99)(103)(104). where a,0,7 a r e the E u l e r a n g l e s r e l a t i n g t h e m o l e c u l a r frame component q w i t h the l a b o r a t o r y frame component m. The E u l e r a n g l e s c a r r y the time dependence of t h e r o t a t i o n . F or magnetic resonance we o n l y have s c a l a r and t e n s o r (/=0 and 2 r e s p e c t i v e l y ) i n t e r a c t i o n s . As s c a l a r i n t e r a c t i o n s a r e 3 0 The c h o i c e of t r a n s f o r m i n g the o p e r a t o r s i n t o t h e m o l e c u l a r frame or t r a n s f o r m i n g the magnetic i n t e r a c t i o n t e n s o r s i n t o the l a b o r a t o r y ( o b s e r v e r s ) frame i s a r b i t r a r y . As the e i g e n v a l u e s of quantum m e c h a n i c a l o p e r a t o r s a r e by d e f i n i t i o n o b s e r v a b l e s , i t i s p h i l o s o p h i c a l l y more s a t i s f a c t o r y t o l e a v e the o p e r a t o r s i n t h e l a b o r a t o r y frame. A l s o t h e r e i s a c h o i c e i n t h e d e f i n i t i o n of Eqn.10.7 (102), the main consequence of t h i s i s whether th e ( - 1 ) m term i n Eqn.10.7 i s a c o e f f i c i e n t of A or F . 109 r o t a t i o n a l l y i n v a r i a n t they do not c o n t r i b u t e t o the r e l a x a t i o n . So, d r o p p i n g the / ( i m p l i c i t l y s e t t i n g i t t o 2 ) , we get from (Eqn.10.4-Eqn.10.8) ^r mm'qq' ao' p/3 i t where v and u denote the d i f f e r e n t type of i n t e r a c t i o n s ( p r e v i o u s l y d e f i n e d as X) and K(m) = <a\h(m)\a'>. (10.10) S p h e r i c a l t e n s o r s obey the f o l l o w i n g symmetry r e l a t i o n s K(l,m)* = ( _ 1 ) m A r / . - m ; (10.11) a l s o <a|A|a'> = <a'|A Ia> as A i s H e r m i t i a n hence A<W* m {.^mK(-m) (10.12) aa' a' a and s i m i l a r l y f o r F and the 00' terms so Eqn.10.9 becomes 110 G M. f l f l,<t> = Z K(-m) <D* ( t ) D m ( t + r ) > F ^ V " ^ ( 1 0 . 1 3 ) aa pp mcj J » A Q , M ^ , Q mq mq v u the m' and terms d i s a p p e a r because of the o r t h o g o n a l i t y r e l a t i o n s of the Wigner r o t a t i o n m a t r i c e s (99)(104). Note t h a t the 00' s u b s c r i p t r e v e r s e s because of Eqn. 10.12 I t i s c o n v e n i e n t t o d e f i n e a reduced c o r r e l a t i o n f u n c t i o n , g^ mq qmn = <D*(t)D ( t + r ) > (10.14) mq mq mq and a l s o a reduced s p e c t r a l d e n s i t y (as t h i s i s the o n l y time dependent t e r m ) , j (CJ) = F{g ( t ) } (10.15) mq mq where F{x} i s the F o u r i e r t r a n s f o r m as d e f i n e d b e f o r e (Eqn.1.8). F i n a l l y we get J ,««.(«) = Z Z K(m) h(~m)i(a>)F(-q)F(q) (10.16) aa' 0 0 ' v,n m,q v a f l , n^.^rhq v u and hence from Eqn.10.1, R a a « ^ . 111 E x p l i c i t e x p r e s s i o n s f o r the magnetic i n t e r a c t i o n t e n s o r s (F) and the s p i n o p e r a t o r s (A) are g i v e n i n (90)(105) and appendix 22.6. E x p a n s i o n of Eqn.10.16 f o r the ESR case i s d i s c u s s e d f u r t h e r i n S e c t . 1 2 . 3 . The e v a l u a t i o n of the c o r r e l a t i o n f u n c t i o n (Eqn.10.14) remains one of the most c h a l l e n g i n g a s p e c t s of m o l e c u l a r dynamics. The b a s i c g o a l of m o l e c u l a r dynamics s t u d i e s by magnetic resonance i s t o deduce g(co) from J ( C J ) , which, i n p r i n c i p l e can be o b t a i n e d f o r r e l a x a t i o n t i m e measurements. 10.2 ON SPECTRAL DENSITIES Reduced s p e c t r a l d e n s i t i e s a r e e x t r e m e l y u s e f u l q u a n t i t i e s t o measure as they are independent of the m o t i o n a l model used. T h i s g r e a t l y f a c i l i t a t e s comparisons between r e s u l t s from d i f f e r e n t e x p e r i m e n t a l methods. U n f o r t u n a t e l y t h e r e a r e o f t e n more s p e c t r a l d e n s i t i e s than independent v a r i a b l e s . A l s o i t i s u s u a l t o a s s i g n some p h y s i c a l meaning t o the r e s u l t s and so a d i f f u s i o n model i s assumed and c o r r e l a t i o n t i m e s c a l c u l a t e d i n s t e a d . T h i s g e n e r a l l y hampers comparisons of d i f f e r e n t s t u d i e s and t e c h n i q u e s . Reviews of the v a r i o u s r o t a t i o n a l d i f f u s i o n models can be found i n (50)(96). U s e f u l d i s c u s s i o n s can a l s o be found i n (106). Three f a c t s emerge from t h e s e r e v i e w s ; the Debye d i f f u s i o n model and i t s e x t e n s i o n s are the most s u c c e s s f u l model; i n the l i m i t of f a s t motion or i s o t r o p i c motion a l l models reduce t o the Debye model; and c u r r e n t l y o n l y the Debye model i s developed f o r the case of a n i s o t r o p i c motion. 1 12 For t h e s e reasons the Debye model i s the most w i d e l y used r o t a t i o n a l d i f f u s i o n t h e o r y . The Debye model i s , b r i e f l y , as f o l l o w s . The m o l e c u l e i s c o n s i d e r e d t o undergo a random s m a l l s t e p a n g u l a r r o t a t i o n ( r o t a t i o n a l Brownian motion) about t h r e e independent axes ( u s u a l l y , but not n e c e s s a r i l y the m o l e c u l a r a x e s ) . The r a t e s of r o t a t i o n a r e c h a r a c t e r i s e d by t h r e e d i f f u s i o n c o n s t a n t s R^, R^ and R^, the p r i n c i p a l elements of the r o t a t i o n a l d i f f u s i o n t e n s o r , R. T h i s d i f f u s i o n t e n s o r i s c o n s i d e r e d t o be independent {i.e., not c o u p l e d t o ) the t r a n s l a t i o n a l d i f f u s i o n t e n s o r . The problem i s t o r e l a t e t hese d i f f u s i o n c o n s t a n t s t o t h e reduced s p e c t r a l d e n s i t i e s , j(o>). For i s o t r o p i c Debye d i f f u s i o n (the most commonly used model f o r NMR), j(u) i s g i v e n by j(a>) = - c 1 + (CJT ) 2 c (10.17) where T i s known as the c o r r e l a t i o n time and T =1/6R, where c c R i s t h e i s o t r o p i c d i f f u s i o n c o n s t a n t . In t h i s case j(a>) i s r e f e r r e d t o as the Debye s p e c t r a l d e n s i t y , not t o be c o n f u s e d w i t h the s p e c t r a l d e n s i t y , J(a>). For a n i s o t r o p i c motion we get f o r j(w) (98)(107) 1 1 3 ' j / f ( u ) = . L , - I k - I k = X . . ( w ) ( 1 0 . 1 8 ) 1 + ( c J T / / t ) 2 where a r e a c o m b i n a t i o n of the r o t a t i o n a l d i f f u s i o n c o n s t a n t s and the T ' S are the e i g e n v a l u e s f o r t h e asymmetric r o t o r w h i c h a r e c o m b i n a t i o n s of t h e r o t a t i o n a l d i f f u s i o n c o n s t a n t s . E x p l i c i t e x p a n s i o n of the reduced s p e c t r a l d e n s i t i e s 3 1 w i l l be g i v e n i n the a p p r o p r i a t e s e c t i o n . G e n e r a l l y we c a l c u l a t e the X terms d i r e c t l y o r by a l e a s t s q u a r e s f i t of the r e l a x a t i o n t i m e s . The reduced s p e c t r a l d e n s i t i e s can be then i n v e r t e d u s i n g Newton-Raphson methods t o g i v e the d i f f u s i o n t e n s o r . 1 0 . 3 CHOICE OF THE AXIS SYSTEM An e s s e n t i a l p a r t of R e d f i e l d t h e o r y and Debye d i f f u s i o n t h e o r y i s t e n s o r s . As the t e n s o r s a r e not n e c e s s a r i l y d i a g o n a l i n the same frame i t i s c o n v e n i e n t t o i n t r o d u c e a r e f e r e n c e frame o r a x i s system; a frame i n which a t l e a s t one of t h e t e n s o r s i s d i a g o n a l . Assignment of the a x i s system i s c o n f u s e d by the c h o i c e of a number of r e f e r e n c e frames. I n our case t h e r e a r e f i v e 3 1 There i s some c o n f u s i o n of nomenclature i n t h e l i t e r a t u r e as ' s p e c t r a l d e n s i t y ' i s o f t e n synonymous w i t h 'Debye s p e c t r a l d e n s i t y ' . Here ' s p e c t r a l d e n s i t y ' w i l l r e f e r t o J(co) as d e f i n e d by Eqn. 1 0 . 2 , j(w) i s the 'reduced s p e c t r a l d e n s i t y ' as d e f i n e d by E q n . 1 0 . 1 5 . The term 'Debye s p e c t r a l d e n s i t y ' w i l l r e f e r t o the reduced s p e c t r a l d e n s i t y f o r the case of i s o t r o p i c d i f f u s i o n . F o r a n i s o t r o p i c Debye d i f f u s i o n the r e d u c e d s p e c t r a l d e n s i t y w i l l be denoted by X ( f o l l o w i n g F r e e d s n o t a t i o n ) . 1 14 p o s s i b l e c h o i c e s ; the l a b o r a t o r y frame, the m o l e c u l a r frame, the magnetic frame, the d i f f u s i o n frame and the i n e r t i a l frame. We a r e not i n t e r e s t e d i n i n e r t i a l models so the i n e r t i a l frame i s not u s e d . 3 2 The l a b o r a t o r y frame would g i v e a time dependent d i f f u s i o n t e n s o r so t h i s i s not used. We a r e t r y i n g t o measure the d i f f u s i o n t e n s o r , thus t h i s i s not a good c h o i c e f o r a r e f e r e n c e frame. However, we do assume the d i f f u s i o n frame i s c o i n c i d e n t w i t h the m o l e c u l a r frame. There a r e s e v e r a l magnetic frames (g, A, q u a d r u p o l e and c h e m i c a l s h i f t t e n s o r s ) . In our system the g and A t e n s o r s a r e c o i n c i d e n t w i t h each o t h e r and w i t h the m o l e c u l a r frame. As the c h o i c e of m o l e c u l a r frame i s somewhat a r b i t r a r y (because of the h i g h degree of symmetry) i t i s c o n v e n i e n t t o d e f i n e the axes of the m o l e c u l a r frame u s i n g the g and A t e n s o r s (which by c o n v e n t i o n are d e f i n e d such t h a t |A |>|A |>|A I i n a r i g h t - h a n d e d c o o r d i n a t e 1 Z Z 1 1 X X 1 1 ^ ' 3 s y s t e m ) . The a x i s system i s t h u s 3 2 The i n e r t i a l frame i s of r e l e v a n c e t o the s p i n - r o t a t i o n c o n t r i b u t i o n t o r e l a x a t i o n . However t h i s c o n t r i b u t i o n i s d e t e r m i n e d e m p i r i c a l l y and i t i s d i f f i c u l t t o e x t r a c t i n e r t i a l i n f o r m a t i o n from i t . A l s o the i n e r t i a t e n s o r i s a x i a l l y symmetric ( I y = I Z * I x . So t h a t c h o i c e of axes i n t h i s frame would be e s s e n t i a l l y a r b i t r a r y . 1 1 5 F i g u r e 10.1 . A x i s system f o r t e n s o r s . Note t h a t t h i s f i g u r e i s drawn i n the l e f t - h a n d e d c o o r d i n a t e system f o r c o n v e n i e n c e . Y i s -Y i n the r i g h t - h a n d e d c o o r d i n a t e system. i.e. , t h e x a x i s i s p a r a l l e l t o the O N bond, th e y a x i s i s i n the MS 4 p l a n e and p e r p e n d i c u l a r t o the C-N bond and the z a x i s i s p e r p e n d i c u l a r t o the MS„ p l a n e . T h i s frame w i l l be used t o d e f i n e the E u l e r a n g l e s f o r a l l o t h e r t e n s o r s , n o t a b l y t h e c h e m i c a l s h i f t t e n s o r and the q u a d r u p o l e t e n s o r . A s i m p l i s t i c i n t e r p r e t a t i o n of r o t a t i o n a l d i f f u s i o n of the d i t h i o c a r b a m a t e s would g i v e a d i f f u s i o n t e n s o r c o i n c i d e n t t o the r e f e r e n c e frame, but w i t h R >R =*R . R xx yy zz xx etc. a r e t h e p r i n c i p a l elements of the d i f f u s i o n t e n s o r R, 3 3 where R i s the r o t a t i o n a l d i f f u s i o n c o n s t a n t f o r <-w XX r o t a t i o n about the x a x i s etc. T h i s i s i m p o r t a n t t o note as 3 3 R, r a t h e r than D, i s used h e r e . D b e i n g r e s e r v e d f o r the t r a n s l a t i o n a l d i f f u s i o n t e n s o r . Some t e x t s use D t o denote b o t h t r a n s l a t i o n and r o t a t i o n a l d i f f u s i o n t e n s o r s . Futhermore t h i s R s h o u l d not be c o n f u s e d w i t h t h e r e l a x a t i o n m a t r i x , which i s u s u a l l y s u b s c r i p t e d . 1 16 some t h e o r e t i c a l d e r i v a t i o n s assume a p r i o r i t h a t the m o l e c u l a r and d i f f u s i o n frames a r e c o i n c i d e n t ( i . e . , R >R >R ) and do not a l l o w such promiscuous assignment of zz xx yy c 3 the r e f e r e n c e frame. 10.4 HYDRODYNAMIC MODELS FOR ROTATIONAL DIFFUSION A c c o r d i n g t o the hydrodynamic model the r o t a t i o n a l d i f f u s i o n c o n s t a n t about a g i v e n a x i s ' i ' i s g i v e n by (106) R:1 = \i* (10.19) where L^ . i s the to r q u e about the ' i 1 t h a x i s and i s g i v e n by L. = \.t(V) (10.20) where X^. i s the f r i c t i o n c o e f f i c i e n t and f ( V ) i s a f u n c t i o n of m o l e c u l a r geometry. For a sp h e r e , f ( V ) i s p r o p o r t i o n a l t o i t s volume, f o r an e l l i p s o i d (or o t h e r shapes) t h e r e l a t i o n s h i p i s more complex. The im p o r t a n t p o i n t though i s t o note t h a t the d i f f u s i o n t e n s o r i s d i r e c t l y r e l a t e d t o the m o l e c u l a r geometry. The t o r q u e s a r e c a l c u l a t e d by s o l v i n g the S t o k e s - N a v i e r e q u a t i o n f o r the a p p r o p r i a t e geometry and boundary c o n d i t i o n s . To date o n l y e l l i p s o i d s have been c o n s i d e r e d , but s o l u t i o n s f o r more complex shapes a r e , i n p r i n c i p l e , 1 1 7 p o s s i b l e . S o l u t i o n s f o r two t y p e s of boundary c o n d i t i o n s have a l s o been i n v e s t i g a t e d ; the ' s l i p ' boundary c o n d i t i o n and the ' s t i c k * boundary c o n d i t i o n . In the l a t t e r case the t a n g e n t i a l v e l o c i t y of the s o l v e n t m o l e c u l e s a t the s o l v e n t - s o l u t e i n t e r f a c e i s assumed t o be z e r o , i.e., t h e r e i s a s o l v a t i o n cage s t u c k t o the s o l u t e (on the time s c a l e of the r o t a t i o n ) . T h i s problem was s o l v e d i n a c l a s s i c paper by P e r r i n (108) and l a t e r r e p e a t e d by F a v r o (109). The ' s l i p ' boundary c o n d i t i o n j u s t assumes t h a t the s o l v e n t doesn't p e n e t r a t e the s o l u t e , t h e f r i c t i o n a l damping i s due t o the volume of s o l v e n t swept a s i d e d u r i n g r o t a t i o n . T h i s boundary c o n d i t i o n r e q u i r e s a n u m e r i c a l s o l u t i o n and t h i s was done by Hu and Zwanzig (110) and a l s o Youngren and A c r i v o s (111). The s t i c k model i s f a i r l y s u c c e s s f u l w i t h i o n i c s p e c i e s and f o r t r a n s l a t i o n a l d i f f u s i o n . 3 * T a b l e s of f r i c t i o n c o e f f i c i e n t s f o r e l l i p s o i d s a r e g i v e n i n (110)(111)(106). Note t h a t the t a b l e s d i f f e r s l i g h t l y because of the way the m o l e c u l a r volume has been d e f i n e d (106). I t i s i n t e r e s t i n g t o note t h a t f o r extreme g e o m e t r i e s ( d i s k s and n e e d l e s ) the s l i p and s t i c k models g i v e i d e n t i c a l f r i c t i o n c o e f f i c i e n t s . A l s o rough spheres w i t h s l i p boundary c o n d i t i o n s g i v e s t i c k l i k e r e s u l t s (114). 3* I o n i c s p e c i e s have w e l l d e f i n e d s o l v a t i o n s h e l l s so the s t i c k boundary c o n d i t i o n s a r e r e a s o n a b l e ( 1 1 2 ) . The s l i p and s t i c k s o l u t i o n s f o r t r a n s l a t i o n a l d i f f u s i o n d i f f e r o n l y by a f a c t o r of two (113); swept volume i s the o n l y i m p o r t a n t parameter i n t h i s c a s e . 1 18 A d i s c u s s i o n of o t h e r m o t i o n a l models i s g i v e n i n (50)(96). These o t h e r models a r e more e l e g a n t i n t h a t they make r e a l i s t i c assumptions about the the s o l v e n t (hydrodynamic models assume the s o l v e n t i s c o n t i n u o u s a t the m o l e c u l a r l e v e l ) . However, they a r e f o r spheres o n l y and cannot be extended t o o t h e r g e o m e t r i e s . As such they are of l i t t l e use. 11. GENERAL EXPERIMENTAL The e x p e r i m e n t a l methods can be d i v i d e d i n t o two t y p e s , a c q u i s i t i v e and p r e p a r a t i v e . Data a c q u i s i t i o n i n ESR and NMR d i f f e r c o n s i d e r a b l y and ar e d i s c u s s e d s e p a r a t e l y i n the a p p r o p r i a t e s e c t i o n s ( S e c t . 1 2 and S e c t . 1 6 ) . P r e p a r a t i o n of the s p i n - p r o b e s ( t r a n s i t i o n m e t a l d i t h i o c a r b a m a t e complexes) and t h e i r s o l u t i o n s i s almost i d e n t i c a l f o r ESR and NMR and i t i s c o n v e n i e n t t o d i s c u s s i t i n one ( t h i s ) c h a p t e r . Each s p i n - p r o b e i s i s o t o p i c a l l y s u b s t i t u t e d as a p p r o p r i a t e . For the d e u t e r i u m r e l a x a t i o n s t u d i e s p e r d e u t e r a t e d amines were used i n the p r e p a r a t i o n s . For the 1 3 C s t u d i e s 1 3 C e n r i c h e d c a r b o n d i s u l p h i d e was used. For ESR s t u d i e s 6 3 C u was used as t h e c e n t r a l m e t a l . 11.1 PREPARATION OF SODIUM DITHIOCARBAMATES AND CARBODITHIOATES A l l compounds were p r e p a r e d by the s t a n d a r d p r o c e d u r e of m i x i n g s t o i c h i o m e t r i c q u a n t i t i e s of carbon d i s u l p h i d e , sodium h y d r o x i d e and the a p p r o p r i a t e secondary amine (69) (115). CS 2 + KOH + R 2NH -> R 2NCSi + K + + H 20 The d i t h i o c a r b a m a t e d e r i v a t i v e i s formed p r e f e r e n t i a l l y t o the x a n t h a t e . D e t a i l s a r e g i v e n below. 119 1 20 The secondary amine (O.lmmol) was d i s s o l v e d i n a l c o h o l i c sodium h y d r o x i d e (100ml of 0.1M). Carbon d i s u l p h i d e (0.1M) i n e t h a n o l (^SOml) was then added d r o p w i s e over a p e r i o d of 30 min. t o the s t i r r e d m i x t u r e , (n.b. t h e o r d e r of a d d i t i o n i s c r i t i c a l as b o t h KOH and the amine r e a c t w i t h CS 2 i r r e v e r s i b l y t o form a x a n t h a t e and the amino doub l e s a l t (115) r e s p e c t i v e l y ) . The r e s u l t a n t m i x t u r e was p u r i f i e d by r e c r y s t a l l i s a t i o n from hot e t h a n o l or by p r e c i p i t a t i o n from a c o l d s a t u r a t e d s o l u t i o n w i t h e t h e r . (The p o t a s s i u m s a l t s were more s o l u b l e than the c o r r e s p o n d i n g sodium s a l t s and hence were more d i f f i c u l t t o p u r i f y i n h i g h y i e l d s ) . Y i e l d s were r e a s o n a b l y h i g h (=70%) i n a l l c a s e s . The f i n a l p r o d u c t s were s t o r e d under n i t r o g e n i n the dark a t =-20°C t o m i n i m i s e d e c o m p o s i t i o n . A l l p e r - d e u t e r a t e d s a l t s were checked f o r r e s i d u a l hydrogen by NMR (none was d e t e c t e d ) . P u r i t y of the s a l t s was checked via a n a l y s i s of the copper or n i c k e l s a l t s (see below T a b l e 11.1 below) as the s e compounds a r e the d e s i r e d end p r o d u c t . 121 The f o l l o w i n g p o t a s s i u m d i t h i o c a r b a m a t e s were made. (Oth e r s a r e d e s c r i b e d i n p r e v i o u s work (88)). L i s t i n g i s by p a r e n t amine. * d i m e t h y l a m i n e 1 5 N di m e t h y l a m i n e d 3 - d i m e t h y l a m i n e * d i e t h y l a m i n e * p y r o l l i d i n e d 9 - p y r o l l i d i n e A l l compounds were p r e p a r e d w i t h and w i t h o u t 1 3G s u b s t i t u t i o n a t the CS 2 group. The compounds a r e a n n o t a t e d w i t h an a s t e r i s k a r e c o m m e r c i a l l y a v a i l a b l e , but were p r e p a r e d f o r use f o r m i c r o a n a l y s i s . A l l s t a r t i n g m a t e r i a l s a r e c o m m e r c i a l l y a v a i l a b l e ( A l d r i c h and Merck, Sharpe and Dohme). 11.2 TRANSITION METAL DITHIOCARBAMATES These a r e s i m p l y p r e p a r e d by m i x i n g aqueous s o l u t i o n s of the a p p r o p r i a t e t r a n s i t i o n m e t a l s a l t s and the p o t a s s i u m d i t h i o c a r b a m a t e . F u r t h e r d e t a i l s a r e g i v e n i n S e c t . 1 3 . The copper s a l t s were p r e p a r e d f o r a l l the n o n - i s o t o p i c a l l y 3 5 s u b s t i t u t e d d i t h i o c a r b a m a t e l i g a n d s and s u b m i t t e d f o r m i c r o a n a l y s i s , (see T a b l e 11.1) 3 5 D e u t e r i u m a n a l y s i s i s not r o u t i n e l y a v a i l a b l e . 1 5 N and 1 3 C a r e not r e s o l v a b l e u s i n g m i c r o a n a l y s i s so t h e i r use f o r a n a l y s i s i s an e x p e n s i v e waste. 122 COMPOUND C H N CuPydtc 33.98(33.78) 4.44(4.45) 7.87(7.88) CuEt 2dtc 33. 17(33.36) 5.60(5.60) 7.80(7.79) d 9-NiPydtc 32.36(32.70) 7.47(7.63) NiPydtc 34.20(34.20) 4.59(4.59) 8.03(7.98) Table 11.1. Microanalyses for Dithiocarbamates. ( ) denotes calculated value. Other numbers are the analysis. The analyses were performed by P.Borda of the UBC Chem. Dept. 11.3 PREPARATION OF SOLUTIONS Toluene was used in previous studies (116), but the nickel derivatives are not soluble enough in t h i s solvent to permit NMR relaxation studies. Other solvents were investigated (see Table 11.2) and chloroform was found to be the best compromise. Accurate v i s c o s i t y data are available for chloroform (117). Previous studies (43)(42)(88) indicate that the r e l a t i v e l y low b o i l i n g point (70 °C) i s not too r e s t r i c t i v e . Unfortunately chloroform reacts with metal dithiocarbamates in the presence of water (118) and i t i s ess e n t i a l that the solvent i s dry. A l l samples were sealed under vacuum after freeze-pump-thaw cycles to remove dissolved oxygen. Thermal degradation products (from sealing the tubes) i n i t i a t e decomposition of the p y r o l l i d i n e dithiocarbamate solutions 123 and g r e a t c a r e has t o be tak e n when s e a l i n g the samples. C h l o r o f o r m p o s s i b l y weakly c o o r d i n a t e s t o the copper e x p l a i n i n g the g r e a t e r s o l u b i l i t y of the copper complexes. A l t e r n a t i v e l y t h e r e maybe an i n t e r m o l e c u l a r Ni-S c o o r d i n a t i o n i n the s o l i d l e a d i n g t o i n c r e a s e d l a t t i c e energy and lower s o l u b i l i t y , but c r y s t a l l o g r a p h i c e v i d e n c e s u g g e s t s o t h e r w i s e (119). B u l k y a l k y l groups i n t e r f e r e w i t h the c r y s t a l p a c k i n g (the d i o c t y l d e r i v a t i v e i s a t h i c k o i l a t room temp.) thus i n c r e a s i n g the s o l u b i l i t y . SOLVENT COMPLEX SOLUBILITY mg/ml Dichloromethane N i P y d t c 0 .9 D i c h l o r o e t h a n e it 0 .4 T r i c h i o r o e t h a n e n 1 . 1 Carbon n <0 . 1 T e t r a c h l o r i d e Tr i c h l o r o e t h y l e n e it <0 . 1 N i t r o b e n z e n e tt h i g h , but complexes Nitromethane <0 . 1 Acetone it =0 . 1 Toluene it =0 .2 Benzene tt =0 .2 T e t r a h y d r o f u r a n =0 .2 E t h a n o l tt =0 . 1 Methanol tt ~0 . 1 Cyclohexane n =0 .2 Pentane tt 0 C h l o r o f o r m n 1 .5 n N i E t 2 d t c as 37 it NiHxmdtc 30 it NiMpdtc <0 . 1 it NiMe 2 d t c 2 it CuEt 2 d t c 50 it CuMe 2 d t c 20 it CuPydtc 40 it CuOc 2 d t c oo it PdEt 2 d t c 30 T a b l e 11.2. S o l u b i l i t i e s of m e t a l d f c ' s . PART 3 ELECTRON SPIN RESONANCE STUDIES / 1 2 4 12. ESR THEORY There a r e a number of approaches t o ESR r e l a x a t i o n t h e o r y ; Kubo-Tomita t h e o r y (93); s t o c h a s t i c L i o u v i l l e t h e o r y (39); R e d f i e l d t h e o r y (90)(91)(92).36 S t o c h a s t i c L i o u v i l l e t h e o r y i s the most r i g o r o u s approach and can be extended i n t o the slow m o t i o n a l regime (see appendix 22.9). Kubo-Tomita t h e o r y has the advantage of b e i n g f a i r l y s t r a i g h t - f o r w a r d w i t h an easy p h y s i c a l i n t e r p r e t a t i o n f o r most of t h e terms. R e d f i e l d t h e o r y a l s o has a s i m p l e p h y s i c a l i n t e r p r e t a t i o n , but has been developed m a i n l y i n the c o n t e x t of NMR (i.e., w i t h z e r o o r d e r w a v e - f u n c t i o n s ) . Here we d e v e l o p R e d f i e l d t h e o r y f o r the ESR case w i t h f i r s t o r d e r w a v e f u n c t i o n s , which s u r p r i s i n g l y has not been d i s c u s s e d b e f o r e . The Brownian r o t a t i o n a l d i f f u s i o n model and the s p i n - r o t a t i o n c o n t r i b u t i o n t o r e l a x a t i o n are a l s o d i s c u s s e d b r i e f l y . 12.1 THE ISOTROPIC ESR SPECTRUM The s p i n H a m i l t o n i a n , H, f o r an u n p a i r e d e l e c t r o n i n t e r a c t i n g w i t h a s i n g l e n u c l e u s i s (121) H = 0B.g.S - 0 .B.g .1 + S.A.I (12.1) where B i s a s t a t i c magnetic f i e l d , g i s the g t e n s o r , g„ i s 3 6 For a u s e f u l , but d a t e d r e v e i w see (120). 125 126 the c h e m i c a l s h i f t t e n s o r , A i s the h y p e r f i n e c o u p l i n g t e n s o r , /3 i s the Bohr magneton, 0^ i s the n u c l e a r magneton and I and S are the e l e c t r o n and n u c l e a r s p i n v e c t o r o p e r a t o r s r e s p e c t i v e l y . I f we d e f i n e B t o be a l o n g the z a x i s , i.e., B=B^k, then f o r i s o t r o p i c motion t e n s o r s i n Eqn.12.1 average out so i t r educes t o (121) H = g o 0 B z i 2 - qnPnBziz + A 0 I z i 2 + | ° [ s + I . + S . I + ] ( 1 2 . 2 ) where g 0 , g^ and A 0 a r e the t r a c e s of the c o r r e s p o n d i n g t e n s o r s . The r e m a i n i n g terms have t h e i r u s u a l meanings (122). The n u c l e a r Zeeman term i s s m a l l compared w i t h the o t h e r terms (Sect.14.3) and can be n e g l e c t e d . The f i n a l term of Eqn.12.2 i s a l s o s m a l l , but n o n - n e g l i g i b l e and i s c o n v e n i e n t l y t r e a t e d by p e r t u r b a t i o n t h e o r y , i.e., H = H 0 + H' where H' = | ° | ^ S + I _ + S_I +J and H 0 = g o 0 B Z S Z + AQI^S^ (12.3) The second o r d e r e n e r g i e s a r e g i v e n by (123) 1 27 E = E° + <n|H'|n> + § <nlB'lm><m|g'[n> ( 1 2 < 4 ) m m 1 1 n^m E - E_ n m T? « f l n m J . R m m 4 - A i . r I (1 + 1 )~m.. (m. ±1 )1 , , ~ K l E „ = q 0 p B 0 m + A0m.m +.—_ o P _ — r ' T - ' ~ r — 112.5) m z s 5 4 g 0 / ? B 0 Lgo0Bom5 + Aom^.m J^ where the s i g n i n the second o r d e r term i s the same as the s i g n of m$ and |n> and |m> a r e a r b i t r a r y b a s i s elements of the system; |m ,1^ . >. m^  i s the e l e c t r o n s p i n quantum number p r o j e c t e d onto the z a x i s and nu i s the c o r r e s p o n d i n g p r o j e c t i o n f o r the n u c l e a r s p i n quantum number. The t r a n s i t i o n f r e q u e n c y , co0, between |m> and |n> ( t o second o r d e r ) i s r e a d i l y o b t a i n e d from Eqn.12.5. (The s e l e c t i o n r u l e s a r e Am =1 and Am.=0). co AE mn -mn = g 0 p-B 0 + A 0m ; + 2 f J p B o | l ( I + 1> " » (12.6) where I i s the n u c l e a r s p i n quantum number. For 6 3 C u (and 6 5 C u ) 1=3/2 and A 0 i s n e g a t i v e t h u s we w i l l see f o u r l i n e s w i t h the nu =-3/2 a t low f i e l d . Each l i n e w i l l be s h i f t e d from i t s z e r o o r d e r p o s i t i o n by ^IG a t 9GHz. A t y p i c a l spectrum i s shown i n F i g . 1 2 . 1 . (The spectrum parameters a r e g i v e n i n the appendix 22.4). The v a r i a t i o n of l i n e - h e i g h t ( w i d t h ) i s due t o m o l e c u l a r motion and i s 128 d i s c u s s e d i n the next s e c t i o n . J . J B F i g u r e 12.1. T y p i c a l m e t a l d i t h i o c a r b a m a t e s p e c t r u m . C a l i b r a t i o n i n t e r v a l i s 50G. The f i r s t o r d e r w a v e f u n c t i o n s a r e a l s o r e a d i l y o b t a i n e d by p e r t u r b a t i o n t h e o r y . i ^ i , v <nIH'Im>i n m n> (12.7) I f we use | a> t o denote the low energy f i r s t o r d e r w a v e f u n c t i o n and |a'> t h e h i g h energy w a v e f u n c t i o n , then i t i s easy t o show t h a t 1 29 |a> = \-i,m.> + pj+i,m.-1> |o'> = |+i,m.> + q j - i , m / + 1> (12.8) For c o n v e n i e n c e , the b a s i s s t a t e s , Eqn.12.8, w i l l be denoted |-m> and |+m> r e s p e c t i v e l y . Hence a> = -m> + p |+m-1> 1 cm1 |a'> = |+m> + qm|-m+1> (12.9) where p . • ifteo [ 1 ( 1 + 1 } " 12.2 THE ESR PROBLEM: DEVELOPMENT OF THE REDFIELD EQUATION Development of the R e d f i e l d t h e o r y f o r ESR d i f f e r s from t h a t f o r NMR i n f o u r major ways; a ) T 2 ' s a r e r e q u i r e d , b) f i r s t o r d e r w a v e f u n c t i o n s (vide supra) have t o be used, c) the s p e c t r a l d e n s i t i e s cannot be s i m p l i f i e d ( S e c t . 1 4 . 2 ) , 130 d ) t h e f i r s t o r d e r c o r r e c t i o n s t o the l i n e - p o s i t i o n a r e l a r g e so t h a t B ^ B 0 . These f a c t o r s make the e x p a n s i o n of the R e d f i e l d e q u a t i o n e x t r e m e l y t e d i o u s ( a l b e i t s t r a i g h t - f o r w a r d ) . Only the s a l i e n t f e a t u r e s w i l l be p r e s e n t e d h e r e . The development f o l l o w s t h a t s u g g ested by Park (42). The r e s u l t s were c r o s s - c h e c k e d w i t h the a i d of a sym b o l i c a l g e b r a r e d u c t i o n program (124). We make the p h e n o m e n o l o g i c a l i d e n t i f i c a t i o n t h a t the r e l a x a t i o n w i t h i n a s p i n s t a t e i s T 2 (125) i . e . , - ( 1 / T 2 ) = (R , , ) ( r ) (12.11) 2 mi aa' aa' m. 20 , , (u , ) - Z J , , (a , ) - Z J , , (a , ) aaa a a a y a aaa aa y a aa a a a 7 7 7 7 7 7 •y(r) - Z J , , (u , ) - Z J (CJ ) (12.12) 7 aa aa aa 7 aa aa aa 7 7 7 7 7 7 J m. where 7 l a b e l s the sum over the h y p e r f i n e s t a t e s , m^. l a b e l s the l i n e b e i n g measured and the s u p e r s c r i p t ' r ' denotes the r e a l p a r t of the r e l a x a t i o n r a t e . (The i m a g i n a r y p a r t g i v e s the ( s m a l l 3 7 ) f r e q u e n c y s h i f t s (97)(126)). The s p e c t r a l d e n s i t i e s , J(co), a r e expanded as g i v e n by N o r d i o (99) i . e . , 3 7 T h e s e a r e of the o r d e r of the s t a t i c second o r d e r c o r r e c t i o n t o the l i n e - p o s i t i o n s and were not i n c l u d e d i n the a n a l y s i s . 131 J aa' ao' (-q)^(q) (12.13) v, u m, q v where A, F a r e g i v e n i n appendix 22.6 and t h e reduced s p e c t r a l d e n s i t y , j(o>), i s g i v e n by Eqn.10.15. The t r a n s i t i o n f r e q u e n c i e s a r e d i s c u s s e d below. 12.2.1 THE TRANSITION FREQUENCIES The t r a n s i t i o n f r e q u e n c i e s f o r t h e s p e c t r a l d e n s i t i e s a r e a s s i g n e d by Fi g . 1 2 . 2 and shown i n Ta b l e 12.1 The t r a n s i t i o n f r e q u e n c y depends on how the o p e r a t o r s , A, c o u p l e t h e f i r s t o r d e r p a r t of the w a v e f u n c t i o n s . The second o r d e r p a r t of the w a v e f u n c t i o n s modify t h e t r a n s i t i o n p r o b a b i l i t i e s a t those f r e q u e n c i e s . The terms of the r e l a x a t i o n e q u a t i o n are o f t e n c a t e g o r i s e d a c c o r d i n g t o t h e t r a n s i t i o n f r e q u e n c y a s s o c i a t e d w i t h the term (90). S e c u l a r terms are those a s s o c i a t e d w i t h the S z and I z o p e r a t o r s , i.e., z e r o f r e q u e n c y . The p s e u d o - s e c u l a r terms a r e those a t t a c h e d t o the n u c l e a r t r a n s i t i o n s , CJ , r e s u l t i n g from the I + and S z I + o p e r a t o r s . The n o n - s e c u l a r terms, u=u0±u , r e s u l t from the S + I + o p e r a t o r s . 1 32 ZERO FIELD B„ HYPERFINE F i g u r e 12.2. T r a n s i t i o n diagram. Note t h a t t h i s diagram c o r r e s p o n d s t o a frequency-swept e x p e r i m e n t . In the ( u s u a l ) f i e l d - s w e p t experiment a low energy t r a n s i t i o n c o r r e s p o n d s t o a h i g h e r resonance f i e l d . Note t h a t the s i g n s of the m. components can o n l y be a s s i g n e d t o the spectrum i f we know the s i g n of A 0. For dtc's i t i s n e g a t i v e (127). 133 STATE aa and a'a' aa' and a'a t r a n s i t i o n s t r a n s i t i o n s OPERATOR < * f > 2 t CO + 0 0 res a (*:->• CO a CO res 0 co -co res a <*°>> 0 t t CO res +1 +1 t CO res a g 0 co -co res a T a b l e 1 2 . 1 . S p e c t r a l d e n s i t y F r e q u e n c i e s , f denotes t h a t t h e o p e r a t o r s do not connect t h e s e s t a t e s t o f i r s t o r d e r . co a=A 0/2; fcV es = a>o + 2m/CJA 12.3 THE FINAL EQUATION The i n t e r a c t i o n s of i n t e r e s t here a r e the h y p e r f i n e c o u p l i n g t e n s o r (denoted F a> and the g - t e n s o r (denoted F ) (The s p i n - r o t a t i o n term a l s o c o n t r i b u t e s , but i s d e t e r m i n e d e m p i r i c a l l y , S e c t . 1 2 . 5 ) . As b o t h these t e n s o r s a r e a x i a l l y symmetric (Sect.14.1) q=0 a l w a y s , hence Eqn.12.12 and Eqn.12.13 become ( f o r T 2 ) 134 aa' aa' m J6% L a a a a a - aa'a g 9 9aa> V a + Fo Fo ( Arm; K(-m) + r«; r-m^i a 9 a » 9 » 9 , a . 3 aa 3a a 3aa a a J where =<a|A^|a'> etc. (12.15) aa The wavefunctions are selected according to the subscripts of the spectral densities in Eqn.12.12. Care must be taken to keep track of the running variable X. It should also be noted that the a's are the f i r s t order wavefunctions (Eqn.12.9) in thi s case. The operators and magnetic interaction tensor elements are given by (90) and are reproduced in appendix 22.6 for convenience. Note that B i s one element of a vector z operator (B = B = 0) and i s not equal to the s t a t i c f i e l d , y B 0, because of the f i r s t order frequency s h i f t . The contribution of the individual matrix elements to each J i s given in Table 12.2. The elements individual terms (ffi) (~m) for Eqn.12.14 are given by the row denoted A ' A' , or as o y appropriate, and the column 3aa> aa> etc., again as required. (The subscript reversal arises from the Hermitian property, Eqn.10.12). The abbreviations are 135 v = Cx 2 z = -Cy 2 m = N u c l e a r s p i n quantum no. f o r l i n e a t Bz = L i n e - p o s i t i o n i n magnetic f i e l d u n i t s x 2 = [K-m(m+1)] y 2 = [K-m(m-1)] K =1(1+1) 2 g o 0 B z 3 0 0 { a ) = ^ { ( J a ) j (w±a) = j(a) ±CJ ) •> oo J res a For example, c o n s i d e r the e v a l u a t i o n of the c o n t r i b u t i o n of o p e r a t o r elements ±2 f o r the h y p e r f i n e c o u p l i n g c o n t r i b u t i o n t o the J , , term i n Eqn.12.14 i . e . , a aa a A ^ V - ^ J O . F ^ F W ( 1 2 . 1 6 ) aa' a' a The s p i n H a m i l t o n i a n parameter, F^, i s known from e x p e r i m e n t . The o p e r a t o r elements a r e n o n - s e c u l a r , i S + I + , so t h a t co=w +u> . Hence the m a t r i x element f o r t h e o p e r a t o r r es a a l o n e i s 136 \Z <o'|S +I +|a 7><a 7|S_I_|a'> (12.17) The wave function is reversed for the second part because i t ' s Hermitian. Note especially that the running variable, 7 must also be reversed. The wave functions, a, are given by Eqn.12.9. The electron operators are evaluated f i r s t because they eliminate the most terms so we get IZ(<+m|+qwi<-m+1 | ) 11 + |m^x-m^ | +pm(<+m-1 11_ | -m>) (12.18) where m i s the m. value for the l i n e of interest and m is 1 7 the running variable. Evaluating the nuclear operators Eqn.12.18 reduces to •k(S m + /x ) U . y ) (12.19) ' 7 7 7 This equation i s only non-zero when m^m-1, thus (as i s with a l l the terms) only one of the running quantum numbers i s retained. If we note that x ,=y then the term reduces to y^, i . e . , Eqn.12.17 i s { i - > „ / 8 . l , p ! ' l ! 4 < 1 2 - 2 0 > 1 37 The r e d u c e d s p e c t r a l d e n s i t y i s g i v e n by T E J X 0 0 , where X 0 o i s an e i g e n v a l u e o f t h e a s y m m e t r i c r o t o r ( s e e S e c t . 1 2 . 4 . ) . T h i s p r o c e s s h a s t o be r e p e a t e d w i t h e a c h o p e r a t o r e l e m e n t f o r e v e r y t e r m i n Eqn.12.14 f o r e a c h o f t h e s p e c t r a l d e n s i t i e s , J , i n E q n . 1 2 . 1 2 . F o r s i m p l i c i t y o n l y t h e e v a l u a t e d o p e r a t o r t e r m s a r e g i v e n i n t h e t a b l e o v e r l e a f . 138 -HO) + N CO -7. CO E XlCM E > -ho 4 . > -HO) + > 4* X | C M N - f c ca -|vo E I a - ^ 0 CO > — loo en > - to -to + > I K|CN + N 4* 4 . 4* « N CO 5 B -Hvo co > - l o o CO N -to -to -to -Hto i co < o co < o ra < o < O Ol I Cn < + o> < O t j i < O CO O CO < O O l + CO < * O l < I CO I O l < T a b l e 12.2. The M a t r i x Elements f o r R e d f i e l d Theory. The elements i n c l u d e t h e ( - 1 ) m term and a l s o negated so t h a t T 2 i s +ve (Eqn.12.11). 1 39 The f i n a l e x p r e s s i o n , o b t a i n e d f r o m T a b l e 12.1 a n d T a b l e 12.2 i s T i 1 = l / l 2 ^ j ( 0 ) { 8 ( A m 2 + 2 A g m + G ) + 8 C ( K - m 2 ) ( A [ m + 1 ] + 2 A )} 8 + j ( a ) { 3 A ( K - m 2 + 2 C m ) } + j ( w - a ) { A ( K - m 2 + 6 C m ) } + j (CJ) {6(Am 2 + 2A m-8 . + j (cj+a) {6A(K-m 2 )} + j (CJ) {6(Am 2 + 2A m+G) + 1 2C(K-m 2) (Am+A ) 8 _ 8 + X (12.21) A p p r o x i m a t i n g o>re5±a>fl=>a>0 a n d o>fl=>0 ( s e e S e c t . 14.2.) we g e t T i 1 = 1 / 2 j ( 0 ) { A ( 3 K + 5 m 2 ) + 8 ( 2 A m+G) 8 + 8 C ( K - m 2 ) ( A [ m + 1 ] + 2 A )+6CAm} 8 + j (CJ) {A(7K-m 2 )+6(2A m+G) 8 + 12C(K-m 2)(Am+A^)+4CAm } j + X ( 1 2 . 2 2 ) where X i s t h e r e s i d u a l l i n e - w i d t h ( s e e S e c t . 1 4 . 5 . ) a n d A = ( F 0 ) 2 G = B 2 ( F ° ) 2 A = B F ° F ° a z g g z a g F ° = i ( A - A ) F ° = i ( g - g ) a 2 xx yy g 2 y x x ^yy 140 Eqn. 12.22 can be r e a d i l y r e a r r a n g e d i n t o a c u b i c i n m. t o r e c o v e r the r e s u l t s g i v e n by ot h e r workers (128)(129)(130)(131)(99) by l e t t i n g B 2 = B 0 . (The c o e f f i c i e n t 'C i d e n t i f i e s t he second o r d e r c o n t r i b u t i o n s . The f i r s t o r d e r e q u a t i o n i s d e r i v e d by s e t t i n g t h i s t o z e r o . See appendix 22.5 f o r comments on comparing the second o r d e r terms w i t h o t h e r t h e o r i e s ) . The d a t a can be r e a d i l y i n v e r t e d t o o b t a i n the two s p e c t r a l d e n s i t i e s and a r e s i d u a l l i n e w i d t h . The l a t t e r can be a t t r i b u t e d i n p a r t t o e x p e r i m e n t a l a r t e f a c t s ( d i p o l a r b r o a d e n i n g from the s o l v e n t , u n r e s o l v e d h y p e r f i n e c o u p l i n g etc. see S e c t . 1 4 . 5 . 1 ) , but i s m a i n l y due t o the s p i n r o t a t i o n c o n t r i b u t i o n (see S e c t . 1 2 . 5 ) . The s p e c t r a l d e n s i t i e s can be f u r t h e r i n v e r t e d ( i n p r i n c i p l e ) t o o b t a i n the elements of the d i f f u s i o n t e n s o r R. The r e l a t i o n s h i p between the s p e c t r a l d e n s i t i e s and R depend on the d i f f u s i o n model used. We w i l l c o n c e n t r a t e on the Debye d i f f u s i o n model, because of i t s s i m p l e r e l a t i o n s h i p t o m o l e c u l a r geometry as w e l l as the o t h e r r e a s o n s o u t l i n e d i n Sect. 1 0 . 4 . 12.4 THE DEBYE DIFFUSION MODEL FOR AN ASYMMETRIC ROTOR The e i g e n v a l u e s f o r r o t a t i o n a l d i f f u s i o n of an asymmetric r o t o r a r e g i v e n by Fr e e d (98) and Fa v r o (109). I n . our c a s e o n l y one e i g e n v a l u e (one reduced s p e c t r a l d e n s i t y , =1/5X 0 O * i n F r e e d s n o t a t i o n ) i s measurable because o f the near a x i a l symmetry of the magnetic i n t e r a c t i o n t e n s o r . U s i n g F r e e d s 141 e q u a t i o n s and an a l g e b r a i c m a n i p u l a t i o n program (124) we can o b t a i n the f o l l o w i n g r e l a t i o n f o r a n i s o t r o p i c d i f f u s i o n . 1 2 R ( R + 3 R ) + 3 C J 2 (R - R ) A-o o = a — s z s—z— (12.23) CJ" + 8(2R 2-3R )u 2 + 144R 2 a where R = R + R + R s x y z R = R R + R R + R R a x y x z z y The x,y,z axes a r e a s s i g n e d as d i s c u s s e d i n S e c t . 1 0 . 3 . We o n l y have two reduced s p e c t r a l d e n s i t i e s , j ( 0 ) and J(CJ), so i t i s not p o s s i b l e t o i n v e r t the data f o r a l l t h r e e d i f f u s i o n c o n s t a n t s . Moreover, Eqn.12.23 i s symmetric w i t h - r e s p e c t - t o the i n t e r c h a n g e of R^ and R^ so, except f o r the case of a x i a l d i f f u s i o n , we need two e x t r a p i e c e s of i n f o r m a t i o n t o i n v e r t the d a t a ( t h i s i s d i s c u s s e d f u r t h e r i n S e c t . 1 5 . 4 ) . T h i s d a t a may be o b t a i n e d from NMR and t h a t i s e s s e n t i a l l y the o b j e c t i v e of the t h e s i s . 12.5 SPIN ROTATIONAL RELAXATION As a m o l e c u l e r o t a t e s the e l e c t r o n s g e n e r a t e a magnetic moment. The motion of the e l e c t r o n s i s not r i g i d l y c o u p l e d t o t h e motion of the m o l e c u l a r frame, they l a g s l i g h t y , so t h i s moment can c o u p l e w i t h the n u c l e a r s p i n s (the NMR case) 142 o r , the u n p a i r e d e l e c t r o n s p i n (the ESR case) (132)(133)(134). T h i s i s known as s p i n - r o t a t i o n a l c o u p l i n g . The magnitude of t h e c o u p l i n g i s r e l a t e d t o the e l e c t r o n i c s t r u c t u r e (and i s c h a r a c t e r i s e d by the s p i n - r o t a t i o n t e n s o r ) and the a n g u l a r momentum of the m o l e c u l e ( the l a r g e r the momentum the l a r g e r t h e l a g and hence a l a r g e r c o u p l i n g ) . I f the c o u p l i n g i s modulated then r e l a x a t i o n can o c c u r . M o d u l a t i o n of the a n g u l a r momentum o c c u r s d u r i n g c o l l i s i o n and can be c h a r a c t e r i s e d ( f o r the case of i s o t r o p i c d i f f u s i o n ) by a c o r r e l a t i o n time T . . M o d u l a t i o n of the 1 s p i n - r o t a t i o n t e n s o r i t s e l f , by c o l l i s i o n , i s u s u a l l y c o n s i d e r e d t o be n e g l i g i b l e (135) (i.e., T » T .). The c l a n g u l a r momentum and hence i t s c o r r e l a t i o n t i m e , r ^ . , i s c l e a r l y r e l a t e d t o the s t r e n g t h of the i n t e r m o l e c u l a r t o r q u e s and the moments of i n e r t i a of the m o l e c u l e . V a l u a b l e i n f o r m a t i o n i s t h u s a v a i l a b l e from r.. U n f o r t u n a t e l y the s p i n - r o t a t i o n t e n s o r i s u s u a l l y unknown and has t o be c a l c u l a t e d from t h e g - t e n s o r . The a c c u r a c y of such c a l c u l a t i o n s i s open t o q u e s t i o n . A l s o the i n e r t i a t e n s o r i s g e n e r a l l y a n i s o t r o p i c so the a n g u l a r momentum i s c h a r a c t e r i s e d by more than one c o r r e l a t i o n t i m e . These t i m e s cannot be o b t a i n e d from one r e l a x a t i o n t i m e . More i m p o r t a n t l y , the r e l a t i o n s h i p between and T. i s u n c l e a r , e s p e c i a l l y i n the c a s e of a n i s o t r o p i c motion so a l t h o u g h the s p i n - r o t a t i o n c o n t r i b u t i o n t o r e l a x a t i o n can be r e a d i l y measured (Sect.14.5) i t i s not p o s s i b l e t o e x t r a c t any m o t i o n a l i n f o r m a t i o n from i t . One i m p o r t a n t p o i n t though i s 143 t h a t T. i s p r o p o r t i o n a l t o t e m p e r a t u r e ( T ) (136), whereas T 1 •* i s p r o p o r t i o n a l t o 1/T. T h i s a c c o u n t s f o r the q u a d r a t i c b e h a v i o u r of the spectrum l i n e - w i d t h s as a f u n c t i o n of t e m p e r a t u r e . 13. ESR EXPERIMENTAL P r e p a r a t i o n a n d p u r i f i c a t i o n o f t h e c o p p e r d i t h i o c a r b a m a t e s p i n - p r o b e s i s s t r a i g h t - f o r w a r d . P r e c i s e a n d a c c u r a t e a c q u i s i t i o n a n d a n a l y s i s o f s p e c t r a i s l e s s e a s y , b u t i s c o n s i d e r a b l y e n h a n c e d by t h e u s e o f d i g i t a l t e c h n i q u e s . ( T h i s h a s b e e n d i s c u s s e d i n e a r l i e r work (88) a n d a l s o i n P a r t . 6 ) . DISPA ( d i s p e r s i o n vs. a b s o r p t i o n p l o t s , P a r t . 1 ) was a l s o f o u n d t o be a g r e a t a i d i n o p t i m i s i n g s p e c t r u m a c q u i s i t i o n a n d a n a l y s i s . 13.1 PREPARATION. OF 6 3 C Q P P E R ( I I ) CHLORIDE I s o t o p i c a l l y e n r i c h e d ( 9 9 . 9 9 % AERE H a r w e l l ) 6 3 C u m e t a l was d i s s o l v e d i n c o n c e n t r a t e d n i t r i c a c i d a n d t h e s o l u t i o n e v a p o r a t e d . The r e s i d u e was r e p e a t e d l y c r y s t a l l i s e d f r o m c o n c e n t r a t e d h y d r o c h l o r i c a c i d t o f o r m 6 3 c o p p e r ( I I ) c h l o r i d e . The c h l o r i d e was u s e d i n p r e f e r e n c e t o t h e n i t r a t e a s i t i s l e s s d e l i q u e s c e n t . I t i s a l s o t h e r m a l l y s t a b l e s o t h a t e x c e s s a c i d c a n be r e a d i l y r e m o v e d by h e a t i n g . 13.2 PREPARATION OF C Q P P E R ( I I ) DITHIOCARBAMATE COMPLEXES A s l i g h t e x c e s s o f t h e a p p r o p r i a t e s o d i u m d i t h i o c a r b a m a t e s a l t i n a q u e o u s s o l u t i o n was a d d e d t o t h e i s o t o p i c a l l y e n r i c h e d c o p p e r c h l o r i d e (=*20mg, 0.15mMol) i n s o l u t i o n . The brown c o p p e r c o m p l e x p r e c i p i t a t e was e x t r a c t e d f r o m t h e a q u e o u s p h a s e by s h a k i n g w i t h c h l o r o f o r m . I n some c a s e s i t was n e c e s s a r y t o a d d e t h a n o l t o b r e a k up t h e 1 4 4 1 45 w a t e r / c h l o r o f o r m emulsions t h a t form i n the presence of the complex. A f t e r s e p a r a t i o n , the c h l o r o f o r m s o l u t i o n was washed s e v e r a l t i m e s w i t h water and then f i l t e r e d . The s o l u t i o n was a l l o w e d t o evaporate and the r e s i d u e d r i e d a t 80°C. The complex was r e c r y s t a l l i s e d by d i s s o l v i n g i n a minimum amount of b o i l i n g c h l o r o f o r m and r a p i d l y c o o l e d i n an i c e bath t o form b l u e - b l a c k c r y s t a l s . 13.3 PREPARATION OF COPPER-FREE NICKEL COMPLEXES FOR ESR MATRIX EXPERIMENTS In m a t r i x experiments the 6 3 c o p p e r complex i s doped i n t o i t s n i c k e l a n a l o g at a l e v e l of 0.1% w/w. Consequently the h o s t n i c k e l complex must c o n t a i n <0.005% (50ppm) of mixed i s o t o p e copper i m p u r i t y . However, ( f o r unknown reasons) a l l the n i c k e l s a l t s t r i e d gave n i c k e l complexes c o n t a i n i n g =0.1% copper. To c i r c u m v e n t t h i s problem the n i c k e l s a l t s had t o be t r e a t e d as f o l l o w s : An e x c e s s of the aqueous n i c k e l s a l t was added t o a s o l u t i o n of t h e a p p r o p r i a t e sodium d i t h i o c a r b a m a t e s a l t , the s o l u t i o n f i l t e r e d and the r e s i d u e d i s c a r d e d . The f i l t r a t e i s now copper f r e e (<0.001% by ESR). The pure n i c k e l complex was then p r e p a r e d by a d d i n g aqueous sodium d i t h i o c a r b a m a t e t o t h e f i l t r a t e , f i l t e r i n g and r e c r y s t a l l i s i n g the r e s i d u e t o g i v e g r e e n - b l a c k c r y s t a l s . These n i c k e l s a l t s were a l s o used f o r the NMR r e l a x a t i o n s t u d i e s . 146 13.4 POLYCRYSTALLINE ESR SPECTRA The p o l y c r y s t a l l i n e s p e c t r a of the copper complexes were r e c o r d e d i n t o l u e n e g l a s s e s . The g and A parameters were o b t a i n e d by s i m u l a t i o n p r o c e d u r e s due t o T a i t (137). S t u d i e s i n c h l o r o f o r m g l a s s e s were not s u c c e s s f u l due t o the f o r m a t i o n of t r i p l e t s t a t e s , p r o b a b l y dimers (138). 13.5 PREPARATION OF THE SOLUTIONS FOR ESR Sp e c t r o g r a d e or d , - c h l o r o f o r m was d r i e d over Type 4A m o l e c u l a r s i e v e s and then f u r t h e r p u r i f i e d j u s t p r i o r t o use by passage t h r o u g h an alu m i n a column. A 10ml s o l u t i o n of 0.2 to 0.7mM of the complex was p r e p a r e d . Approx 0.5ml of the s o l u t i o n t r a n s f e r r e d t o a 'flamed out' ESR tube and degassed by s e v e r a l freeze-pump-thaw c y c l e s t o a p r e s s u r e of <10" 5 T o r r on a g r e a s e l e s s vacuum l i n e and then s e a l e d o f f . I t s h o u l d be noted t h a t m o l e c u l a r s i e v e s remove the s t a b i l i s i n g agent ( e t h a n o l ) from c h l o r o f o r m which then r a p i d l y o x i d i s e s t o phosgene, c h l o r i n e and hydrogen c h l o r i d e . These i m p u r i t i e s a r e removed by the a l u m i n a . A l l s o l u t i o n m a n i p u l a t i o n s were done i n a g l o v e bag under an atmosphere of dr y n i t r o g e n . The samples were s t o r e d a t =-20°C i n the dark t o pr e v e n t d e c o m p o s i t i o n (118). 13.6 ESR SAMPLE TUBES The sample tubes were c o n s t r u c t e d from s p e c i a l t h i n w a l l e d p y r e x t u b i n g (5mm OD,4mm ID) t o maximise t h e sample volume. Each tube was f i l l e d t o a dep t h of 3-4 cm ( t h i s m i n i m i s e s 147 temperature g r a d i e n t s due t o c o n v e c t i o n ; see S e c t . 1 4 . 6 ) . and s e a l e d o f f a t 4-5 cm ( t h i s p r e v e n t s the s o l v e n t from d i s t i l l i n g out of the c a v i t y a r e a ) . Care was t a k e n t o l e a v e a gap of =1cm between the t o p of the s o l v e n t and the s e a l t o a l l o w f o r l i q u i d e x p a n s i o n . 13.7 RECORDING ESR SPECTRA A l l s p e c t r a were r e c o r d e d on a h o m e - b u i l t X-band homodyne ESR s p e c t r o m e t e r (139) ( F i g . 1 3 . 1 ) which c o n s i s t e d o f : a I 2 i n magnet and M k l l F i e l d i a l c o n t r o l ; an HP716B k l y s t r o n power s u p p l y and sweep u n i t ; a h o m e - b u i l t AFC; 100 kHz m o d u l a t i o n u n i t ; an I t h a c o D y n a t r a c k 391A p h a s e - l o c k a m p l i f i e r . The microwave b r i d g e was a r e f l e c t i v e homodyne d e s i g n u s i n g a TE-102 c a v i t y , t h r e e p o r t c i r c u l a t o r , S c h o t t k y d e t e c t o r d i o d e and a microwave b u c k i n g arm. Microwave power was measured w i t h an HP431C power meter and the f r e q u e n c y was measured w i t h an HP5246L f r e q u e n c y c o u n t e r f i t t e d w i t h a 5256A p l u g - i n module. The c a v i t y was f i t t e d w i t h a dewar system and the temperature c o n t r o l l e d by a V a r i a n E257 c o n t r o l u n i t . The magnetic f i e l d was c a l i b r a t e d w i t h a V a r i a n E500 p r o t o n magnetometer. The s p e c t r o m e t e r was i n t e r f a c e d t o a m i c r o p r o c e s s e r c o n t r o l l e d d i g i t a l a c q u i s i t i o n system (140). (see S e c t . 6 ) . The s p e c t r a , a l o n g w i t h c a l i b r a t i o n d a t a , were r e c o r d e d on a Kennedy 9800 tape u n i t via the F8 m i c r o p r o c e s s e r . The same r e c o r d i n g c o n d i t i o n s were m a i n t a i n e d f o r a l l samples. These a r e summarised i n T a b l e 13.1. 148 The whole s p e c t r o m e t e r was p e r i o d i c a l l y t e s t e d w i t h s t a n d a r d samples t o check f o r s e n s i t i v i t y , a m p l i f i e r phase, magnet s t a b i l i t y , m o d u l a t i o n a m p l i t u d e and d i s p e r s i o n l e a k a g e . E x t e n s i v e use of the DISPA t e c h n i q u e was made f o r t h i s purpose (10)(14) & P a r t . 1 . One complete spectrum was always r e c o r d e d a t room temp e r a t u r e t o check f o r paramagnetic i m p u r i t i e s . CONTROL Microwave F r e q . Microwave Power M o d u l a t i o n Scan Time & Sweep Width Time C o n s t a n t Temperature Sampling r a t e SETTINGS/COMMENTS ^9.04GHz 1-5mW, f o r c h l o r o f o r m s o l v e n t . C u d t c 1 s s a t u r a t e a t =200mW (42) <0.8G. Narrowest o b s e r v e d l i n e i s 3.0G. 5 mins f o r 25G, 50G or 100G scans as a p p r o p r i a t e . Each l i n e was r e c o r d e d s e p a r a t e l y . 125mS or 400mS (=1/100 of the time t o sweep the l i n e . 15SCFH N 2 f l o w r a t e w i t h C 0 2 / ( C H 3 ) 2 C O c o o l a n t (88). 2.5Hz or =750 p t s . per l i n e . T a b l e 13.1. Spectrum R e c o r d i n g C o n d i t i o n s . 149 CrC U J t - O t£C£ -© H0> _ i CC DZ U J —• DflTfl TEM (EDT UNIT CCD XO U U J D£ VID TERM <n at >-I t o u . KENf TAPE 0_ U J X : ace o I —iUJ ace D • o a ct . xio —ui in TL -ID u z e e : D D M l - N — I — I g O U T D I / 1 ( Z D Z h CXUJ U J H C m eet/i D am rx > 3 cu a u I t (m) 0 > U J > acccc a> U J U I o » i -C C D U I U L C x > I a D U J C J 4 t e a . • U I D C v X D X t-utx U J C C Z= 3 OS aui tna a z uce Z U J U J I -3 Z 0 3 U I O ecu -1L URSE NUflTOR URSE NUflTOR D U J ut-t-er I V 1 i— a a _iee 3 U I a * DD c e UJI-i - >— a z U l = 3 3Z SCAN DRIVE 1- >-UJtE_J zuia. SCAN DRIVE u»a_ C L I O CCI-a a tnz aui u a = cn J 0 - -~ >-0£ aui »_/ • a. uia ZD O U OL I-10 Cd >-c-CC-J tOUJCL. >-»0-J O D iCO-CO Figure 13.1. The S p e c t r o m e t e r . 150 13.8 TEMPERATURE MEASUREMENT JJN ESR EXPERIMENTS The temperature was measured w i t h a c o p p e r / c o n s t a n t a n thermocouple i n s e r t e d i n t o the t o p of the c a v i t y near t o the sample. Care was taken not t o decouple the c a v i t y . The thermocouple EMF was a m p l i f i e d lOOOx by a low d r i f t D.C. a m p l i f i e r which was connected t o the d i g i t a l a c q u i s i t i o n u n i t . The m i c r o p r o c e s s e r r e c o r d e d the temperature f o r lOOmS (which was averaged l a t e r t o e l i m i n a t e n o i s e ) a t the s t a r t and end of each r e c o r d e d spectrum. 13.9 FIELD CALIBRATION OF ESR SPECTRA Each spectrum was c a l i b r a t e d a b s o l u t e l y u s i n g the V a r i a n t r a c k i n g Gaussmeter. The c a l i b r a t i o n p r o c e d u r e was as f o l l o w s . The magnetometer was tapped t o p r o v i d e r . f . (the p r o t o n p r e c e s s i o n f r e q u e n c y = 14MHz) f o r the HP f r e q u e n c y c o u n t e r . The c o u n t e r was sampled a u t o m a t i c a l l y e v e r y 50 d a t a p o i n t s of the spectrum, c o n c u r r e n t l y w i t h the c o r r e s p o n d i n g F i e l d i a l v o l t a g e and s t o r e d by the m i c r o p r o c e s s e r . T h i s i n f o r m a t i o n was then c o p i e d as a t a b l e (magnetometer f r e q u e n c y vs. F i e l d i a l v o l t a g e ) t o the magnetic tape u n i t a t the end of each spectrum. A l l r e q u i r e d c a l i b r a t i o n d a t a was then r e c o v e r e d from the t a b l e u s i n g a l e a s t - s q u a r e s - f i t . Two parameters were o b t a i n e d from the f i t , t he s l o p e ( G a u s s / v o l t ) and the i n t e r c e p t ( t h e a b s o l u t e f i e l d a t the l e f t edge of the s p e c t r u m ) . 151 13.10 COLLECTION AND ANALYSIS OF ESR SPECTRA The s p e c t r a were a n a l y s e d as f o l l o w s . Each l i n e of the spectrum was r e c o r d e d and c a l i b r a t e d i n d i v i d u a l l y . The g a i n , scan w i d t h and time c o n s t a n t were a d j u s t e d t o s u i t each peak u s i n g the c r i t e r i a g i v e n i n p r e v i o u s work (88). Each l i n e w i t h i t s a s s o c i a t e d c a l i b r a t i o n t a b l e , temperature and o t h e r d a t a were r e c o r d e d on magnetic tape as 16 b i t X-Y d a t a p o i n t p a i r s . The magnetic tape was r e a d a t the computing c e n t e r and t r a n s f e r r e d t o a l i n e f i l e and p r o c e s s e d t h e r e . A l t e r n a t i v e l y t h e d a t a were t r a n s f e r r e d t o the DEC LSI-11 minicomputer l o c a t e d i n the ESR l a b o r a t o r y . The s p e c t r a were a n a l y s e d as f o l l o w s (88). The peak tops were l o c a t e d a u t o m a t i c a l l y (Amdahl program) or i n t e r a c t i v e l y (DEC LSI-11 programs) and f i t t e d t o a c u b i c . T h i s was s o l v e d t o l o c a t e the e x a c t extremum. The c r o s s o v e r s were a l s o a p p r o x i m a t e l y l o c a t e d and then found e x a c t l y by l o c a l l y f i t t i n g t o a s t r a i g h t l i n e and f i n d i n g i t s i n t e r s e c t i o n w i t h the b a s e l i n e . The r e s u l t s can then be c o n v e r t e d t o Gauss by means of the c a l i b r a t i o n t a b l e and w r i t t e n t o a f i l e as l i n e - w i d t h and l i n e - p o s i t i o n s , a l o n g w i t h the c o r r e s p o n d i n g t e m p e r a t u r e and microwave f r e q u e n c y . The reduced s p e c t r a l d e n s i t i e s , j ( 0 ) , j (co) and the r e s i d u a l l i n e - w i d t h were e x t r a c t e d from Eqn.12.22 u s i n g a l e a s t - s q u a r e - f i t (14J) i n the l i n e - w i d t h s (see appendix 22.7 f o r a b r i e f d i s c u s s i o n of u n i t s ) . I n v e r s i o n of j ( 0 ) and j ( o ) t o get t h e d i f f u s i o n t e n s o r i s d i s c u s s e d i n Sect.15 and Se c t . 2 0 . 14. ESR ERROR DISCUSSION The e r r o r s a r i s e from f o u r p r i n c i p a l s o u r c e s ; a p p r o x i m a t i o n s i n h e r e n t i n the t h e o r y ; i n s t r u m e n t a l a r t e f a c t s ; e x p e r i m e n t a l a r t e f a c t s ; c o m p u t a t i o n a l a r t e f a c t s . Many of the e x p e r i m e n t a l and i n s t r u m e n t a l a r t e f a c t s can be a s s e s s e d w i t h the a i d of DISPA and s u b s e q u e n t l y m i n i m i s e d or e l i m i n a t e d . C o m p u t a t i o n a l a r t e f a c t s . c a n be e l i m i n a t e d by c a r e f u l program d e s i g n . T h e o r e t i c a l a p p r o x i m a t i o n s were i n v e s t i g a t e d c a r e f u l l y as they a f f e c t the program d e s i g n , the a c c u r a c y of the r e s u l t s and g e n e r a l l y s i m p l i f y the d a t a a n a l y s i s . 14.1 THE AXIAL SYMMETRY APPROXIMATION FOR THE SPIN  HAMILTONIAN T h i s a p p r o x i m a t i o n e l i m i n a t e s a l l 0 and ±1 t e n s o r elements c o n s i d e r a b l y s i m p l i f y i n g Eqn.12.13. However, t h i s i s a t the expense of some i n f o r m a t i o n . R o t a t i o n s t h a t i n t e r c h a n g e the ' zz ' and 'xx' t e n s o r components a r e i n d i s t i n g u i s h a b l e from r o t a t i o n s t h a t i n t e r c h a n g e the ' zz ' and 'yy' components. The magnetic parameters f o r CuPyDtc ( t h e s e v a l u e s v a r y v e r y l i t t l e f o r the a l k y l s u b s t i t u t e d copper d i t h i o c a r b a m a t e complexes) a r e 152 1 53 A = -119MHz q = 2 . 0 2 2 X X 3 X X A = -106MHz g = 2 . 0 1 8 yy ^yy A = -474MHz g = 2.088 zz 3zz T h a t i s , a p p r o x i m a t e l y , b u t n o t e x a c t l y a x i a l l y s y m m e t r i c . The m a g n e t i c i n t e r a c t i o n t e n s o r s ( i n a s p h e r i c a l b a s i s ) a r e g i v e n by ((90)(101) a n d i n a p p e n d i x 2 2 . 6 ) . F± 2 = 4. ( A - A ) F ± 2 = i ( g - g ) a 2 xx yy g 2 'xx yy F ° = / | [A - i ( A +A )1 F ° = v/§[g - i ( g +g )1 a 3 [ zz 2 xx yy J g 3 [ ' z z 2 'xx j ( t h e ±1 e l e m e n t s a r e z e r o b e c a u s e t h e t e n s o r s a r e s y m m e t r i c ) . 1 54 In the r e l a x a t i o n e q u a t i o n (Eqn.12.22) the terms a r e c r o s s m u l t i p l i e d or squared. S u b s t i t u t i n g the parameters g i v e n above we f i n d t h a t (FI" 2) 2 = 40MHz 2 a (F" ) 2 = 86000MHz 2 a +2+2 0 0 F I ^ F „ = -0.01MHz F " F U a g a a = -70.0MHz ( F g 2 ) 2 * 4X10- 6 0 , 8 = 6x10" 2 The a x i a l symmetry a p p r o x i m a t i o n amounts t o d r o p p i n g the F~ term and r e t a i n i n g the F^ terms i n Eqn.12.13. U s i n g the d a t a above, t h i s w i l l i n t r o d u c e an e r r o r of =0.05% f o r each F term so the t o t a l e r r o r f o r t h i s a p p r o x i m a t i o n w i l l be =0.2%, w i t h i n e x p e r i m e n t a l e r r o r . 14.2 ON APPROXIMATING SPECTRAL DENSITIES The s p e c t r a l d e n s i t i e s a r e of the form j(w) = -c (14.1) 1 + (OJT ) 2 c where u> i s : < ^ r e s i the t r a n s i t i o n frequency f o r a g i v e n s p e c t r a l l i n e , or wfl, the h y p e r f i n e c o u p l i n g f r e q u e n c y ( i A o ) , or a mixed term, u> ±u , or z e r o . I n our case 155 oi -56.9 Grad s" 1 and r i s 10-200pS. In NMR (ur ) 2 « 1 (see res c c S e c t . 1 8 . 1 ) . and t h i s term can be dropped. However, f o r ESR t h i s i s not the case (see Fi g . 1 4 . 1 ) and a p p r o x i m a t i o n s of t h i s k i n d need more c a r e f u l e x a m i n a t i o n . T h i s i s r e a d i l y i n v e s t i g a t e d by s u b s t i t u t i n g the a p p r o p r i a t e a p p r o x i m a t i o n s i n t o Eqn.12.22. I t s h o u l d be noted t h a t some of the f o l l o w i n g a p p r o x i m a t i o n s a r e n e c e s s a r y i f the d a t a a r e t o be i n v e r t e d via the s p e c t r a l d e n s i t i e s . 14.2.1 THE ( g > 0 T ) 2 « 1 APPROXIMATION U Q The f r e q u e n c y &r i s a p p r o x i m a t e l y the Larmor f r e q u e n c y f o r the e l e c t r o n , u 0 . From Fig.14.1 i t s c l e a r t h a t s e t t i n g —> 0 i s not a good a p p r o x i m a t i o n . The CJO7"c terms must be e x p l i c i t l y i n c l u d e d i n a l l the c a l c u l a t i o n s . 156 110.0 LOG FREQUENCY (Hz) F i g u r e 14.1. S p e c t r a l d e n s i t i e s v s . f r e q u e n c y . 14.2.2 THE (w r ) 2 « 1 APPROXIMATION — a — c  ' For r c<120pS t h i s a p p r o x i m a t i o n does not i n t r o d u c e any s i g n i f i c a n t e r r o r (<0.5%, see F i g . 1 4 . 2 i . e . , j(w f l) and j ( 0 ) are e x p e r i m e n t a l l y i n d i s t i n g u i s h a b l e ) . 14.2.3 THE w « c j n APPROXIMATION —a *• T h i s would a l l o w us t o s e t oo ± a> => C J 0 and a> => res a res u0. Both of these a p p r o x i m a t i o n s i n t r o d u c e n e g l i g i b l e e r r o r (<0.5%, see F i g . 1 4 . 2 ) . T h i s i s v e r y u s e f u l as the the r e l a x a t i o n time depends on o n l y two s p e c t r a l d e n s i t i e s , j ( 0 ) and j ( w ) . However, t h i s does not imply t h a t B (=B )=B_. z res ° 157 TflU(pS) F i g u r e 14.2. L i n e - w i d t h E r r o r s f o r t h e S p e c t r a l D e n s i t y A p p r o x i m a t i o n s . The p l o t i s f o r the m/=-3/2 l i n e . E r r o r s f o r the o t h e r l i n e s a r e l e s s t h a n h a l f of t h i s . 14.3 CONTRIBUTIONS FROM THE NUCLEAR ZEEMAN TERM The n u c l e a r Zeeman term (Eqn.12.2.) makes s m a l l c o n t r i b u t i o n s t o the t r a n s i t i o n f r e q u e n c i e s and hence t o the s p e c t r a l d e n s i t i e s . For an X-band ESR s p e c t r o m e t e r (i.e. , a t a f i e l d of 0.32T) t h e Larmor f r e q u e n c y of 6 3 C u and 6 5 C u i s about 4MHz. T h i s i s s m a l l compared t o the e l e c t r o n r e s o n a n t f r e q u e n c y (=9GHz). I t i s not q u i t e n e g l i g i b l e w i t h r e s p e c t t o the h y p e r f i n e c o u p l i n g f r e q u e n c y (220MHz). However, t h e h y p e r f i n e c o u p l i n g f r e q u e n c y c o n t r i b u t e s <1% (Sect.14.5.5) 158 t o the s p e c t r a l d e n s i t i e s . Hence the n u c l e a r Zeeman c o n t r i b u t i o n can be n e g l e c t e d . 14.4 THE FIRST AND SECOND ORDER CONTRIBUTION The second ord e r c o n t r i b u t i o n t o the r e l a x a t i o n t i m e s (the 'C terms i n Eqn.12.22) i s e a s i l y c a l c u l a t e d from t h i s e q u a t i o n . For t y p i c a l reduced s p e c t r a l d e n s i t i e s of j ( 0 ) = l O p S and j(co)=5pS, t h i s c o n t r i b u t i o n i s <0.5%. However, f i r s t o r d e r c o r r e c t i o n s h i f t s t h e l i n e w i d t h s by 5% and one c a r e has t o taken t o use the l i n e - p o s i t i o n s , B^, not B 0, i n Eqn.12.22. 14.5 THE RESIDUAL LINEWIDTH A number of f a c t o r s i n f l u e n c e the ob s e r v e d l i n e w i d t h , o t h e r than t h o s e d i s c u s s e d i n S e c t . 1 2 . The l i n e w i d t h a f t e r s u b t r a c t i n g the t h e o r e t i c a l l i n e w i d t h (from Eqn.12.22) i s known as the r e s i d u a l l i n e w i d t h and i s de t e r m i n e d e m p i r i c a l l y . The p r i n c i p a l c o n t r i b u t i o n i s the s p i n - r o t a t i o n term ( S e c t . 1 2 . 5 . ) . However, t h e r e a re s e v e r a l o t h e r ( s m a l l ) c o n t r i b u t i o n s . These a r e d i s c u s s e d below. 14.5.1 DIPOLAR BROADENING T h i s a r i s e s from e l e c t r o n - n u c l e a r s p i n i n t e r a c t i o n s between the s o l v e n t p r o t o n s and the copper complex. I t can be m i n i m i s e d by the use of d e u t e r a t e d s o l v e n t s . Comparison of p e r d e u t e r a t e d CuMedtc i n CDC1 3 and CHC1 3 shows t h a t t h i s c o n t r i b u t i o n i s 0.04G i n a l i n e w i d t h of 1 59 3.6G. {i.e., 1% a t maximum). 3 8 14.5.2 PARAMAGNETIC BROADENING T h i s a r i s e s from ( u n p a i r e d ) e l e c t r o n - e l e c t r o n i n t e r a c t i o n s . The copper complexes thus show b r o a d e n i n g a t h i g h c o n c e n t r a t i o n s (>10" 3M), but t h i s c o n t r i b u t i o n i s n e g l i g i b l e a t the u s u a l w o r k i n g c o n c e n t r a t i o n (=10""M). Another so u r c e of u n p a i r e d e l e c t r o n s i s d i s s o l v e d oxygen. T h i s must be removed by freeze-thaw-pump c y c l e s of the s o l u t i o n on a vacuum l i n e . 14.5.3 SOLVENT COORDINATION S o l v e n t c o o r d i n a t i o n w i l l modulate the l i n e w i d t h s e.g.(46). The i n f l u e n c e of water (wet s o l v e n t s ) i s unknown (except see (118)), a l t h o u g h i t i s u n l i k e l y t o c o o r d i n a t e w i t h the complexes. The s o l v e n t s were n e v e r t h e l e s s t h o r o u g h l y d r i e d b e f o r e use. 14.5.4 INTERNAL MOTION The C-N bond has s i g n i f i c a n t double bond c h a r a c t e r (142) and t h e r e i s no e v i d e n c e of r o t a t i o n about the bond on the NMR time s c a l e (125). The r i n g of t h e p y r o l l i d i n e i s s l i g h t l y p u c k e r e d and may c o n t r i b u t e t o the r e l a x a t i o n via the u n r e s o l v e d h y p e r f i n e c o u p l i n g c o n t r i b u t i o n . However, the u n r e s o l v e d h y p e r f i n e c o u p l i n g 3 8 The e f f e c t i s even s m a l l e r f o r the p y r o l l i d i n e d e r i v a t i v e , <0.1%, see S e c t . 1 4 . 5 . 5 . 160 c o n t r i b u t i o n {vide infra) and the c o n t r i b u t i o n from f l u c t u a t i o n s i n the c o n f o r m a t i o n (appendix 22.8) have both been shown t o be n e g l i g i b l e . 14.5.5 UNRESOLVED HYPERFINE The p r o t o n s c o u p l e t o the u n p a i r e d e l e c t r o n on the copper g i v i n g r i s e t o b r o a d e n i n g from the u n r e s o l v e d h y p e r f i n e c o u p l i n g . DISPA s t u d i e s (14), s i m u l a t i o n s (42) and s t u d i e s w i t h o r d i n a r y and p e r - d e u t e r a t e d 6 3 C u M e d t c show the c o u p l i n g c o n s t a n t t o be =0.5G, b r o a d e n i n g the narrowest l i n e by 10% (0.3G on 3 G ) . 3 9 T h i s e f f e c t can be m i n i m i s e d by the use of p e r d e u t e r a t e d compounds or by use of c o r r e c t i o n p r o c e d u r e s (143). However, t h e e f f e c t s on T a r e s m a l l (Table 14.1) and do not a f f e c t the c observ e d t r e n d s . 3 9 For the p y r o l l i d i n e d e r i v a t i v e s the e f f e c t i s even l e s s , 0.04G i n a 3.6G l i n e , about the same as d i p o l a r b r o a d e n i n g from the s o l v e n t . 161 TEMP°C T c ( D ) p S T c ( H ) p S TC(corr)pS -50 -30 -20 0 20 30 45 105±2 55±1 41 ±2 26±0.5 17±1 15±0.5 13±0.5 1 04±2 52±2 40±1 24±0.5 16±1 14±0.5 11±0.5 1 08±4 55±2 41 ± 1 25±0.5 17+1.0 14±0.5 11±0.5 Ta b l e 14.1. E f f e c t of h y p e r f i n e on c o r r e l a t i o n t i m e s . Data i s from (88). (D) denotes the c o r r e l a t i o n time f o r the p e r - d e u t e r o d e r i v a t i v e . (H) denotes the normal d e r i v a t i v e , ( c o r r ) denotes t h a t l i n e - w i d t h s c o r r e c t e d by B a l e s method were used. 14.5.6 MAGNETIC FIELD INHOMOGENEITY T y p i c a l l y <5mG a c r o s s the ESR sample. ( m a n u f a c t u r e r s spec.) T h i s i s s m a l l w i t h r e s p e c t t o the l i n e - w i d t h of dtc's (>3G) and can be i g n o r e d . 14.5.7 SPECTROMETER PHASING L i n e - h e i g h t s a r e v e r y s e n s i t i v e t o any m i s p h a s i n g ( d i s p e r s i o n l e a k a g e ) of the s p e c t r o m e t e r and t h i s i n t e r f e r e s b a d l y w i t h l i n e - h e i g h t a n a l y s i s methods used i n t h e p a s t (131). The l i n e - w i d t h s (but not the l i n e - s h a p e s ) a r e not s e n s i t i v e t o phase m i s a d j u s t m e n t s ( T a b l e 14.2) and t h i s i s not a problem i f the a n a l y s i s uses the l i n e - w i d t h s d i r e c t l y as i s done h e r e . 1 62 Phase A n g l e Observed* L i n e - w i d t h L i n e - w i d t h from H e i g h t s 0 1 .000 1 .006 1 .006 1 .006 1.013 1.014 1 .022 1 .000 1 .005 1.010 1 .029 1 .060 1.101 1.151 2 5 10 15 20 T a b l e 14.2. E f f e c t of Phase on Observed L i n e - w i d t h s . *-There i s an e r r o r of =±5mG a s s o c i a t e d w i t h t h i s measurement. 14.5.8 TIME CONSTANT AND MODULATION The e f f e c t of the time c o n s t a n t and m o d u l a t i o n a m p l i t u d e on ESR s p e c t r a i s w e l l documented (30). L i n e b r o a d e n i n g of the narrowest l i n e by the time c o n s t a n t i s n e g l i g i b l e (<0.1%). Broadening from the m o d u l a t i o n i s <1% (= 30mG w i t h 0.8G m o d u l a t i o n ) i n the worst case and i s t y p i c a l l y much l e s s . 14.6 TEMPERATURE INHOMOGENEITY A te m p e r a t u r e g r a d i e n t a c r o s s t h e sample w i l l a f f e c t the observ e d l i n e w i d t h and cause i n a c c u r a t e thermocouple r e a d i n g s . T h i s has been p r e v i o u s l y i n v e s t i g a t e d (88). The temperature g r a d i e n t was <0.5°C w i t h an o v e r a l l s t a b i l i t y / a c c u r a c y of 0.05°C. A 1°C s h i f t i n temp e r a t u r e w i l l change t h e l i n e - w i d t h by 3% of the l i n e - w i d t h i n the worst case (m.=3/2 l i n e a t =*-50°C), but would be t y p i c a l l y 163 =0.5% of the l i n e - w i d t h . 14.7 FITTING ARTEFACTS AND NOISE The i n f l u e n c e of n o i s e on peak f i t t i n g i s d i s c u s s e d e x t e n s i v e l y i n the l i t e r a t u r e e.g.(144). Under the e x p e r i m e n t a l c o n d i t i o n s used here (an SNR of 50:1 w i t h 50 p o i n t s f o r a f i t ) we expect a n o i s e r e l a t e d e r r o r of <1.0% (145). F i t t i n g a r t e f a c t s a r i s e because the d a t a ( w i t h i n i n a window about the peak) a r e f i t t e d t o a c u b i c . I f the f i t t i n g window i s t o o s m a l l n o i s e becomes a problem, i f i t i s t o o l a r g e the c u b i c no l o n g e r a d e q u a t e l y a p p r o x i m a t e s the peak shape. T h i s i s not a problem f o r i n t e r a c t i v e peak f i t t i n g ( u n l e s s the n o i s e l e v e l i s v e r y h i g h ) as the o p e r a t o r c o n t r o l s t h e window w i d t h and p o s i t i o n . W i t h the automated r o u t i n e s t h e l i n e - w i d t h i s unknown t o the program and c h o i c e of the c o r r e c t window i s d i f f i c u l t , e s p e c i a l l y i f the l i n e - w i d t h s v a r y w i t h i n a spectrum o r s e t of s p e c t r a . T h i s problem was i n v e s t i g a t e d i n p r e v i o u s work (88). T y p i c a l l y e r r o r s i n the l i n e p o s i t i o n s a r e ±20mG (<1% of the sweep-width) and ± 0 . 5 % i n the l i n e - w i d t h f o r both methods. 14.8 FIELD CALIBRATION AND CAVITY SHIFT The V a r i a n Gaussmeter i s ( i n c o r r e c t l y ) c a l i b r a t e d w i t h the f r e e p r o t o n p r e c e s s i o n f r e q u e n c y i n s t e a d of t h a t of a p r o t o n i n s o l u t i o n , (i.e., the meter i s not c o r r e c t e d f o r the d i a m a g n e t i c s h i f t ) . T y p i c a l l y the meter reads =0.1G h i g h . 1 64 The correct value can be recovered by multiplying the meter reading by 234.868/234.874. However, we tap the probe to obtain the proton precession frequency d i r e c t l y so th i s problem i s not relevant. The gate times* 0 for the tracking Gaussmeter and the frequency counter are =*0.1s, thi s leads to an error in the f i e l d c a l i b r a t i o n of <(sweep-rate)*(gate-time), or <l5mG for a 5min., 25G scan. This w i l l average out to a systematic c a l i b r a t i o n error of = -5mG for a low-to-high f i e l d sweep. However, t h i s only a f f e c t s the absolute l i n e - p o s i t i o n s , not line-widths or s p l i t t i n g s hence the error i s n e g l i g i b l e . The Gaussmeter probe i s outside the ESR cavity so a f i e l d s h i f t with respect to the center of the cavity is expected. This was checked with Fremys s a l t . The center l i n e of a 1X10~ 4M solution of Fremys s a l t in a 0.05M solution of potassium carbonate was measured 7X. The mean f i e l d position corrected for microwave frequency s h i f t s was 3238.355±0.003G (the gating error in t h i s case should be <1mG). Using. g0=2.00545 and A0=13.0910 (130), the l i n e - p o s i t i o n , to second order, should be 3238.450, i.e., a cavity s h i f t of 0.095G should be added to the Gaussmeter readings. The cavity s h i f t f o r t u i t o u s l y cancels the error of the Gaussmeter to within 0.01G. 9 0 To obtain an accuracy of 1:10$ (=3mG) one must count the frequency for at least 10s cycles, i.e.,* 0.1s for 13MHz, the Larmor frequency of a proton in a 3000G f i e l d 15. ESR RESULTS AND DISCUSSION 15.1 ESR RESULTS L i n e - w i d t h data f o r 6 3 C u d 9 - P y d t c i n d - c h l o r o f o r m were o b t a i n e d over a temp e r a t u r e range of -50°C t o +70°C. The r e s u l t s a r e shown i n F i g . 1 5 . 1 . The da t a a r e i n appendix 22. 10. 0.0 -50 J L J L J I L J L J l_ -2510 OlO 2510 TEMPERATURE [°C1 5o:o F i g u r e 15.1. P y r o l l i d i n e dt c l i n e - w i d t h d a t a . L i n e - w i d t h s f o r 310, 323 and 333K were o b t a i n e d by i n t e r p o l a t i o n and s u b s t i t u t e d i n t o Eqn.12.22. The reduced s p e c t r a l d e n s i t i e s , j ( 0 ) , j(a>) and the r e s i d u a l l i n e - w i d t h f a c t o r were o b t a i n e d via a l e a s t squares f i t (141). The r e s u l t s a r e shown i n T a b l e 15.1. 165 1 66 Temp(K) j(0) j(a>) Residual 310 0.0115 0.00448 0.0176 323 0.0102 0.00381 0.0205 333 0.00825 0.00428 0.0373 Table 15.1. Spectral densities for CuPydtc in chloroform. F i t t i n g errors on the parameters are a l l <1%. By combining these results with the 1 3C or 2H data, we can, in p r i n c i p l e , for a Debye d i f f u s i o n model, obtain the di f f u s i o n tensor. This i s discussed in Part.5. 15.2 APPROXIMATE METHODS FOR DATA ANALYSIS In view of the large body of ESR data available from previous studies i t i s useful to examine Eqn.12.23 ( X 0 0 ) in d e t a i l to gain some insight into the d i f f u s i o n tensor, without resorting to NMR studies to do so. There are two approaches to t h i s , simulations or approximations. Three approximations are of interest here: The isotropic assumption, R^ =R^ *=Rz=R, which i s useful for obtaining order of magnitude values. The a x i a l approximation with R^=R^=R^ and the fast anisotropic approximation, with R^>>R^,R 2« The l a t t e r two approximations are expected to be v a l i d on geometric grounds and both give unique values for R^. 167 15.2.1 SIMULATIONS One can s i m u l a t e d a t a t o produce t a b l e s of s p e c t r a l d e n s i t i e s f o r v a r i o u s v a l u e s of the d i f f u s i o n t e n s o r s , but t h i s approach i s not u s e f u l u n l e s s the approximate v a l u e s a r e a l r e a d y known. However, we can get o r d e r of magnitude f i g u r e s f o r the t e n s o r elements ( v i d e i n f r a ) . A l s o we know t h a t the d i f f u s i o n t e n s o r e lements are a l l p o s i t i v e . Furthermore we a l s o have good r e a s o n s t o suppose t h a t R^>R^,R2 so one can l i m i t the s i z e of the t a b l e s . By a u t o m a t i c a l l y matching e x p e r i m e n t a l d a t a ( t o say w i t h i n 5%) w i t h s i m u l a t e d v a l u e s one can o b t a i n a s e t of v a l u e s f o r the p o s s i b l e d i f f u s i o n e l ements. A unique s o l u t i o n i s not p o s s i b l e , but the d a t a o b t a i n e d f o r our system i s c o n s i s t e n t w i t h motion a p p r o x i m a t e l y a x i a l l y symmetric (R =R ) and R >R ,R . •* x z x y z 15.2.2 THE ISOTROPIC ASSUMPTION Eqn.12.23 reduces i n t h i s case t o X 0 o ( " ) = W 2 | 3 6 R 2 (15.1) The i n t e r e s t i n g f e a t u r e of t h i s a p p r o x i m a t i o n i s t h a t two e s t i m a t e s of the i s o t r o p i c c o r r e l a t i o n t i m e a r e o b t a i n e d ; one from j ( 0 ) , R ( 0 ) , and the o t h e r from j(a>), R(to). These two v a l u e s a r e o n l y e q u a l i f t h e motion i s 168 i s o t r o p i c . The r a t i o , R(0)/R(CJ), thus g i v e s a measure of the a n i s o t r o p y of the motio n . T h i s may be a u s e f u l c r i t e r i o n t o a p p l y t o n i t r o x i d e type s p i n - p r o b e s , but i t w i l l not be pursued h e r e . 15.2.3 THE FAST MOTIONAL APPROXIMATION On geometric grounds we expect R^ t o be l a r g e r than R^, or R^. The a p p r o x i m a t i o n Rx>>R^,Rz thus may be u s e f u l . In t h i s case Eqn.12.23 reduces t o Xoo(0) = 12(R + R ) z x o o ( u > = ™xSJ*xi*y±*zL±»ll ( 1 5 . 2 ) CJ" + (4R co) 2 + [ 12R (R +R ) ] 2 x x x y S i m u l a t i o n s show t h a t R^ and (R^+R^) a r e u n d e r e s t i m a t e d i f t he ESR da t a a r e i n v e r t e d u s i n g t h i s a p p r o x i m a t i o n . The e r r o r s i n R a r e (R +R )/R % and t h o s e f o r (R +R ) X y z X x y about t w i c e as much.* 1 These e r r o r s a r e =20% f o r our ca s e , but t h i s a p p r o x i m a t i o n does p r o v i d e u s e f u l s t a r t i n g v a l u e s f o r the d i f f u s i o n t e n s o r f o r the n o n - l i n e a r i n v e r s i o n methods. 15.2.4 THE AXIAL APPROXIMATION Eqn.12.23 reduces t o * 1These r e s u l t s a r e f o r 9GHz, the e r r o r s change w i t h f r e q u e n c y , but not i n a s y s t e m a t i c manner. 1 6 9 * , n s 5 R +R A 0 o ( 0 ) = —p— 1 2 R ( 2 R +R ) p x p . / x 1 2 R ( 2 R + R ) ( 5 R + R + 3 C J 2 ( R + R ) , , C - S X 0 0 ( w ) = p x—p p—x p—x- ( 1 5 . 3 ) + 8 ( 5 R 2 + 2 R R +2R2)CJ2 + [ 1 2 R ( 2 R + R ) ] 2 p p x x p x p f o r t h i s a p p r o x i m a t i o n . A g a i n R^ and (R^+R^)=R^ a r e u n d e r e s t i m a t e d , but i t works w e l l f o r a wider range of v a l u e s than the p r e v i o u s a p p r o x i m a t i o n . The e r r o r s a r e t y p i c a l l y <5% e r r o r f o r R x i f |(R^-R^J/R^|< 0 . 3 and the % e r r o r f o r R = 1 0 ( R -R ) 2/R P y z P P r o v i d i n g the motion i s a p p r o x i m a t e l y a x i a l we can get a r e a s o n a b l e e s t i m a t e f o r R^. T h i s i s d i s c u s s e d f u r t h e r i n S e c t . 1 5 . 4 . Note t h a t t h i s method r e q u i r e s a n o n - l i n e a r i n v e r s i o n and so i s not u s e f u l f o r o b t a i n i n g s t a r t i n g v a l u e s . 1 5 . 3 USING THE APPROXIMATIONS The r e s u l t s of a p p l y i n g the above a p p r o x i m a t i o n s t o the d a t a i n T a b l e 1 5 . 1 a r e shown below. 1 70 Temp R(0) R(«) R X R P R X R +R y 310 14.5 37. 1 63.8 4.8 48.9 7.3 323 16.3 43.7 90.1 5.1 67. 1 8.2 333 20.2 38.9 82.5 4.3 54.4 10.1 T a b l e 15.2. ESR d a t a i n v e r t e d w i t h a p p r o x i m a t i o n s . The d a t a f o r the a x i a l a p p r o x i m a t i o n shows good agreement w i t h the known v a l u e s f o r the complete t e n s o r a t the h i g h t e m p e r a t u r e ( S e c t . 2 0 . 3 ) . I f we can remain c o n f i d e n t t h a t R^,-Rz f o r a wide temperature range, the a x i a l a p p r o x i m a t i o n i s a u s e f u l method f o r i n v e r t i n g ESR d a t a , vide infra. The f a s t R^ a p p r o x i m a t i o n i s u s e f u l because i t p r o v i d e s r e a s o n a b l e s t a r t i n g v a l u e s f o r the a x i a l a p p r o x i m a t i o n . The i s o t r o p i c v a l u e s g i v e o r d e r - o f - m a g n i t u d e v a l u e s as e x p e c t e d . 15.4 INVERSION OF DATA WITH THE AXIAL APPROXIMATION Data o b t a i n e d from t h i s a p p r o x i m a t i o n i s r a t h e r s c a t t e r e d and i s o n l y u s e f u l f o r ex a m i n i n g t r e n d s . The r e s u l t s of t h e v a r i o u s a p p r o x i m a t i o n s f o r Pydtc i n t o l u e n e i s shown i n T a b l e 15.3 171 Temp R(0) R(u) 29.7 19.3 -8.8 -0.2 9.0 19.5 28.8 39.4 49. 1 51 .4 4.2 5.1 7.0 8.1 9.6 11.6 13.9 16.0 17.7 20.0 35. 1 66.7 41.2 46.0 51 .7 51 .4 46.0 47.0 47.2 42.7 37.6 93.3 51 .2 61 .8 77.4 81.8 71 .0 77.2 79.7 65.8 2.1 2.5 3.5 4.1 4.8 5.8 6.7 8.0 8.9 10.0 38.7 96.7 55.5 68. 1 87.8 97.2 89.7 102.0 110.0 96.0 2.3 2.7 4.0 4.6 5.4 6.8 8.4 9.6 10.8 12.8 T a b l e 15.3. A x i a l a p p r o x i m a t i o n used w i t h CuPydtc i n t o l u e n e . Both R^ and R^ i n c r e a s e w i t h t e m p e r a t u r e as e x p e c t e d and have r e s p e c t i v e a c t i v a t i o n e n e r g i e s of 7±2kJ/mol and 13±0.5kJ/mol. T h i s i s an e x t r e m e l y i n t e r e s t i n g r e s u l t as i t i m p l i e s the motion i s not o n l y not v i s c o s i t y dependent (i.e. , i s not hydrodynamic, t h e a c t i v a t i o n energy would be =9.5kJ/mol i f i t were), but a l s o the damping mechanism i s d i f f e r e n t f o r the two m o t i o n s . T a b l e 15.4 and T a b l e 15.5 show the r e s u l t s f o r CuMeOddtc (a 17 c a r b o n - c h a i n d e r i v a t i v e ) i n two s o l v e n t s , t o l u e n e and c y c l o h e x a n e . * 2 The d a t a a r e r a t h e r poor, but R^ i s a p p r o x i m a t e l y c o n s t a n t i n b o t h c a s e s . R^-50 f o r the probe i n c y c l o h e x a n e and R^=200, f o u r t i m e s f a s t e r , i n t o l u e n e . T h i s i m p l i e s t h a t the l o n g h y d r o c a r b o n t a i l of the probe i s c o i l e d up i n the c y c l o h e x a n e , but not i n t o l u e n e . As t h e s e s o l v e n t s have the same d i e l e c t r i c c o n s t a n t t h i s e f f e c t must r e f l e c t the l o c a l s t r u c t u r e of t h e s o l v e n t , a l t h o u g h what * 2The l a t t e r d a t a were t a k e n from a study done by M.Yu. i n t h i s l a b o r a t o r y . ( 1 4 6 ) 172 t h a t i s , i s u n c l e a r . Temp r?/T Tc R* R* 24.3 3.0.5 97.3 23.6 1 .7 24.8 1.0 31 .2 2.66 86.1 61 .6 1.9 62.6 1 .0 32.8 2.57 83.6 53.6 2.1 55.2 1.2 37. 1 2.37 77.5 31 .2 2.2 32.8 1.3 52. 1 1.81 60.3 27.1 2.9 30.0 1.9 62.6 1 .52 52. 1 126.0 3.0 132.0 1 .6 73.3 1 .29 47.2 53.9 3.6 58.6 2.1 Tab l e 15.4. A x i a l a p p r o x i m a t i o n used w i t h CuMeOddtc i n c y c l o h e x a n e . r c i s taken form p r e v i o u s work. n/T i s C P / K X 1 0 3 . Temp T?/T Tc R* Ry+Rz R* 10.0 2.32 107.3 59 1 .6 60 0.8 46.9 1 .33 52.2 232 3.1 244 1.6 51 .3 1 .25 48.2 205 3.3 218 1.7 56.5 1.17 43.6 153 3.6 1 64 1.9 78.0 0.91 29.8 198 4.9 224 2.6 88.9 0.81 27.9 131 5.6 1 53 3.1 98.3 0.74 23. 1 94 6.4 1 1 5 3.7 Table 15.5. A x i a l a p p r o x i m a t i o n used w i t h CuMeOddtc i n t o l u e n e . For comparable v a l u e s of TJ/T, T £ i s s i m i l a r i n bo t h c a s e s so t h a t the i s o t r o p i c a p p r o x i m a t i o n c o m p l e t e l y o b s c u r e s the b e h a v i o u r of R . However, Some c a u t i o n has t o x be e x e r c i s e d when u s i n g t h i s a p p r o x i m a t i o n because j(o)«j(0) and as such i s s u b j e c t t o l a r g e e r r o r s . I t might be b e t t e r t o i n v e r t the da t a d i r e c t l y f o r R and R r a t h e r x p than via the reduced s p e c t r a l d e n s i t i e s . A l s o t h i s a p p r o x i m a t i o n p r e d i c a t e s on the motion b e i n g a p p r o x i m a t e l y a x i a l over the e n t i r e t e mperature range not j u s t a t h i g h t e m p e r a t u r e s . Perhaps more i m p o r t a n t l y , the r e s u l t s i n T a b l e 173 15.4 and T a b l e 15.5 show t h a t R i s =1GHz; the s l o w - m o t i o n a l P regime (see appendix 22.9). I t s not c l e a r what the e f f e c t of t h i s i s . 15.5 DIRECT INVERSION USING THE ISOTROPIC ASSUMPTION I f we assume t h a t the motion i s i s o t r o p i c then the s p e c t r a l d e n s i t i e s assume a s i m p l e form; Eqn.15.1. In such a case we can i n v e r t the d a t a d i r e c t l y r a t h e r than via the s p e c t r a l d e n s i t i e s . T h i s was done i n e a r l i e r work by Park et al (82) and i n f a c t most m o t i o n a l d a t a (ESR, NMR and l i g h t s c a t t e r i n g ) a r e o b t a i n e d via t h i s a p p r o x i m a t i o n . The q u e s t i o n t h u s a r i s e s , as t o the r e l a t i o n s h i p between X 0 0 and rc as c a l c u l a t e d by Park i f the motion i s a n i s o t r o p i c . The c a l c u l a t i o n i s c o m p l i c a t e d by second o r d e r terms, but t o f i r s t o r d e r i t i s easy t o see from Eqn.15.1 t h a t j ( u ) / j ( 0 ) = ( 1 + O 2 T 2 ) - 1 c hence r = CJ- 1 [ j ( 0 ) / j (o)-1 ] 2 (15.4) Comparison of the v a l u e s of T c a l c u l a t e d from P a r k s method i s f a i r l y s t r a i g h t - f o r w a r d , but g i v e n the c o m p l e x i t y of the t h e o r e t i c a l e x p r e s s i o n s f o r j ( 0 ) and j(cu), (Eqn. 12.23), r e l a t i n g T£ t o the d i f f u s i o n t e n s o r i s d i f f i c u l t . We can o b t a i n a f u n c t i o n a l form f o r r from s i m u l a t i o n s though. 174 Temp r c P a r k r c(Eqn.15.4) -30 83 83 -19 61 61 -9 39 48 0 38 40 9 37 33 20 33 27 29 27 22 39 25 19 49 23 17 56 19 15 T a b l e 15.6. Comparison of T C ' S . P a r k s method vs. t h i s work. Data i s from Table.15.3 and p r e v i o u s work. I t i s i n t e r e s t i n g t o note t h a t the T c a l c u l a t e d from Eqn.15.4 do not d e v i a t e from l i n e a r i t y a t h i g h t e m p e r a t u r e s as much as P a r k s r e s u l t s do. 15.6 INTERPRETING DATA FROM ISOTROPIC INVERSIONS The most p o p u l a r model f o r i n t e r p r e t i n g T ^ V S . T J / T p l o t s i s g i v e n by (147) r c = p + r 0 (15.5) where V i s a hydrodynamic term (Sect.20.4.), u s u a l l y the m o l e c u l a r volume, T i s the temperature and k i s the Boltzmann c o n s t a n t and T 0 i s a ' f r e e - r o t o r ' c o r r e l a t i o n time (148). T h i s i s a s o u r c e of some c o n t r o v e r s y as the model i s u n s a t i s f a c t o r y f o r a number of reasons ( o t h e r than t h o s e t h a t can be l e v e l l e d a g a i n s t the hydrodynamic model i n 175 g e n e r a l ) . For a p u r e l y hydrodynamic model the i n t e r c e p t i s z e r o , i n a number of s t u d i e s a f i n i t e i n t e r c e p t has been o b s e r v e d . I t has been proposed t h a t t h i s can be acc o u n t e d f o r by adding a ' f r e e - r o t o r ' c o r r e l a t i o n t i m e . However, c o r r e l a t i o n t i m e s cannot be added, o n l y the c o r r e s p o n d i n g r a t e s . The c o r r e c t e x p r e s s i o n i s 7 0 0 S = TnTo f r 1 ~Po )+Po7'o J / where i s the hydrodynamic c o r r e l a t i o n t i m e , p 0 i s the f r a c t i o n a l p o p u l a t i o n of f r e e l y r o t a t i n g m o l e c u l e s and r 0 t h e i r c o r r e l a t i o n t i m e . S e c o n d l y the o b s e r v e d i n t e r c e p t i s n e g a t i v e i n a number of cas e s (88)(149)(150). T h i s i s p r o b a b l y due t o n o n - l i n e a r i t y of the p l o t s . F i n a l l y the model i m p l i c i t l y assumes i s o t r o p i c motion and t h e r e i s no reason t o b e l i e v e t h a t such p l o t s are m e a n i n g f u l i f the m otion i s a n i s o t r o p i c . G i v e n t h a t the i n t e r c e p t s and n o n - l i n e a r i t y a r e r e a l , then the q u e s t i o n i s , "can we account f o r t h e s e e f f e c t s by i n c l u d i n g a n i s o t r o p i c motion i n the model ?" Assuming t h a t the d i f f u s i o n p r o c e s s has an A r r h e n i u s t e m p e r a t u r e dependence, then we have R . = R A i e ~ E l / R T + R A 2 e " E 2 / R T + R A 3 e " E 3 / R T (15.6) obs x 1 y * z J For the hydrodynamic model the ' a c t i v a t i o n e n e r g i e s ' , E,, E 2 and E 3 s h o u l d be a l l e q u a l t o t h e a c t i v a t i o n energy f o r v i s c o u s f l o w , E^(88). The c o l l i s i o n f r e q u e n c i e s , A 1 f A 2 and A 3 w i l l be r e l a t e d t o the f r i c t i o n c o e f f i c i e n t s f o r the 176 m o l e c u l e ( S e c t . 2 0 . 4 ) . For conv e n i e n c e we w i l l use a s e m i - e m p i r i c a l model. The a c t i v a t i o n e n e r g i e s and the c o l l i s i o n f r e q u e n c i e s were chosen such t h a t R =10R =20R , ^ x y z robs = 20pS a t 310K and the s i m u l a t e d a c t i v a t i o n energy f o r T ^ was 13.6kJ/mol. ( O b t a i n e d from p r e v i o u s work). The obs e r v e d c o r r e l a t i o n time was c a l c u l a t e d u s i n g Eqn.15.4 and Eqn.15.6. The r e s u l t s a r e shown i n F i g . 1 5 . 2 . Some r e s u l t s from e a r l i e r work a r e a l s o shown i n F i g . 1 5 . 3 . R e s u l t s f o r i s o t r o p i c motion were a l s o c a l c u l a t e d f o r comparison. 1 2 0 . 0 1 0 0 . 0 -to 8 0 . 0 -u ZD CC 6 0 . 0 -o 4 0 . 0 -to 2 0 . 0 0 . 0 0 0 1 0 . 0 2 0 . 0 3 0 . 0 4 0 . 0 5 0 . 0 6 0 . 0 7 0 . 0 8 0 . 0 VISC/T CuP/K) F i g u r e 1 5 . 2 . The e f f e c t of a n i s o t r o p y on TJ /T p l o t s . The dashed l i n e i s f o r i s o t r o p i c m o t i o n . The d o t t e d e x t e n s i o n s show the f u n c t i o n a l form of the p l o t i n e x p e r i m e n t a l l y i n a c c e s s i b l e r e g i o n s . The graphs show a break i n t h e s l o p e a t about 3 0 M P / K . Above t h i s p o i n t the graphs a r e alm o s t i d e n t i c a l , below t h i s p o i n t t h e match i s p o o r , but both show a f i n i t e i n t e r c e p t . 1 77 T h i s i m p l i e s t h a t T 0 may be a m e a s u r e o f m o t i o n a l a n i s o t r o p y , n o t an a l t e r n a t i v e r e l a x a t i o n m e c h a n i s m . 0. 01 i i i I i 1 1 1 0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 VI S C / T (uP/K) F i g u r e 1 5 . 3 . A T J / T p l o t f r o m p r e v i o u s work. T h i s p l o t i s t y p i c a l o f a l l p r e v i o u s r e s u l t s . A l l show t h e same f o r m e x c e p t a t low T J / T v a l u e s where t h e c u r v e up i s o f t e n s m a l l e r . T he d o t t e d l i n e i s t h e same a s i n t h e p r e v i o u s f i g u r e . T h e d e v i a t i o n f o r 7 } / T < 3 0 M P / K h o w e v e r d o e s i n d i c a t e t h a t a n a d d i t i o n a l d i f f u s i o n m e c h a n i s m maybe i m p o r t a n t a t h i g h t e m p e r a t u r e s . T a b l e 15.7 shows t h a t i n t h i s t e m p e r a t u r e r e g i m e t h e s p i n - r o t a t i o n t e r m a c c o u n t s f o r 8 0 % o f t h e l i n e - w i d t h o f t h e n a r r o w e s t l i n e a n d 50% o f t h e b r o a d e s t l i n e a n d o v e r a l l 50% o f t h e r e l a x a t i o n . T h e l e a s t - s q u a r e s - f i t may n o t be v e r y r e l i a b l e u n d e r t h e s e c o n d i t i o n s a n d p r o d u c e a s y s t e m a t i c a r t e f a c t t h a t i s r e s p o n s i b l e f o r t h e o b s e r v e d c u r v a t u r e , ( c . / . T a b l e 1 5 . 6 ) . 178 I t s h o u l d be noted t h a t Eqn.15.6 i s not a proposed model f o r f i t t i n g the e x p e r i m e n t a l d a t a , i t i s merely t o demonstrate t h a t , i f the motion i s a n i s o t r o p i c then we can p a r t i a l l y account f o r the n o n - l i n e a r i t y of the T • vs. 77/T p l o t s and w h o l l y account f o r the presence of the f i n i t e i n t e r c e p t . That i s , T 0 i s an a r t e f a c t of the i s o t r o p i c a p p r o x i m a t i o n . T h i s example c l e a r l y demonstrates the dangers of assuming an i s o t r o p i c model when d o i n g m o t i o n a l s t u d i e s . Temp(°C) Zero F r e q . Larmor F r e q . R e s i d u a l -30 2.2 0.5 0.7 13.6 1.5 0.7 + 10 1 .0 0.3 1.9 6.0 1.0 1 .9 +60 0.5 0.4 3.4 2.9 1.2 3.4 T a b l e 15 .7. R e l a t i v e r e l a x a t i o n c o n t r i b u t i o n s a t v a r i o u s t e m p e r a t u r e s , i n Gauss, f o r CuPydtc i n t o l u e n e . The f i r s t row of each e n t r y i s f o r the narrowest l i n e , the second row i s f o r the b r o a d e s t l i n e 15.7 CONCLUSIONS We have demonstrated t h a t by assuming i s o t r o p i c motion t h a t not o n l y i s i n f o r m a t i o n l o s t , but a l s o t h a t a r t e f a c t s are i n t r o d u c e d . The f r e e - r o t o r term p o s t u l a t e d by P e c o r a (147) p r o b a b l y 0 3 a r i s e s from the n o n - l i n e a r i t y i n t r o d u c e d i n t o * 3The NMR e q u a t i o n s a r e s i m i l a r , but more complex than f o r the ESR case and i t i s d i f f i c u l t t o demonstrate f o r the g e n e r a l case t h a t a r t e f a c t s a r i s e . Each case has t o be c o n s i d e r e d s e p a r a t e l y . 179 p l o t s by a s s u m i n g i s o t r o p i c m o t i o n , when i t i s i n f a c t a n i s o t r o p i c . The i s o t r o p i c a s s u m p t i o n a l s o o b s c u r e s t h e f a c t t h a t t h e a c t i v a t i o n c o e f f i c i e n t s f o r t h e d i f f u s i o n c o n s t a n t s may be d i f f e r e n t , w h i c h c a s t s s e r i o u s d o u b t s on t h e v a l i d i t y o f t h e h y d r o d y n a m i c m o d e l . A l s o o t h e r e f f e c t s , s u c h a s t h e p r o p o s e d c h a i n c o i l i n g f o r t h e MeOd d e r i v a t i v e a n d t h e e n t r y o f some o f t h e t e n s o r e l e m e n t s i n t o t h e s l o w - m o t i o n a l r e g i m e a r e o b s c u r e d . The d a t a a r e c o n v e n i e n t l y i n v e r t e d via t h e s p e c t r a l d e n s i t i e s . The i n v e r s i o n p r o c e s s i s l i n e a r a n d i n d e p e n d e n t o f t h e m o t i o n a l m o d e l u s e d . However, d i f f u s i o n c o n s t a n t s o b t a i n e d via a p p r o x i m a t i o n s a n d t h e s p e c t r a l d e n s i t i e s t e n d t o be ' n o i s y ' . I t s n o t c l e a r w h e t h e r t h e m e t h o d i s i n h e r e n t l y u n r e l i a b l e b e c a u s e j ( w ) « j ( 0 ) , o r w h e t h e r t h e p r e c i s i o n o f t h e more d i r e c t m e t h o d s u s e d by P a r k i s an i l l u s i o n ; R / J 0 w i l l be \/3 more p r e c i s e t h a n t h e a n i s o t r o p i c v a l u e s b e c a u s e o f a v e r a g i n g . The ' n o i s y ' d a t a p r o b a b l y g i v e s a b e t t e r r e f l e c t i o n o f t h e a c c u r a c y ( r e l i a b i l i t y ) o f t h e r e s u l t s t h o u g h . The a s s u m p t i o n o f i s o t r o p i c d i f f u s i o n d o e s n o t p r o d u c e t r e n d s t h a t r e f l e c t t h e t r e n d s f o r t h e i n d i v i d u a l t e n s o r e l e m e n t s , i n f a c t i t p r o d u c e s v e r y m i s l e a d i n g r e s u l t s . D e s p i t e i t s s i m p l i c i t y t h e i s o t r o p i c m o d e l s h o u l d be a b a n d o n e d a n d i"c's r e p l a c e d by s p e c t r a l d e n s i t i e s . A l t h o u g h t h e u s e o f s p e c t r a l d e n s i t i e s i s n o t w i t h o u t p r o b l e m s t h e y a r e l e s s m i s l e a d i n g a n d p r o b a b l y r e f l e c t o u r l a c k o f u n d e r s t a n d i n g o f t h e p r o b l e m s b e t t e r . PART 4. NMR STUDIES 180 16. NMR THEORY NMR r e l a x a t i o n t h e o r y i s w e l l d e v e l o p e d and i s d i s c u s s e d i n a number of t e x t s (91)(92). There a r e many mechanisms t h a t c o n t r i b u t e t o the r e l a x a t i o n t i m e s of a n u c l e u s . The most e f f i c i e n t mechanism i s q u a d r u p o l a r r e l a x a t i o n and f o r n u c l e i w i t h I>1/2 t h i s i s the dominant mechanism and indeed the o n l y mechanism t h a t need be c o n s i d e r e d . For t h i s reason t h e r e have been numerous r e l a x a t i o n s t u d i e s u s i n g q u a d r u p o l a r n u c l e i e. g. (102) (151) (149) (152) (153) . For 1 = 1/2 n u c l e i t h e r e a re s e v e r a l competing r e l a x a t i o n mechanisms; p r o t o n d i p o l a r c o u p l i n g i s u s u a l l y the p r i n c i p a l one. However f o r dtc's the ' t h i o ' c arbon i s remote from the p r o t o n s and the o t h e r mechanisms a r e more i m p o r t a n t . The one of most i n t e r e s t i s r e l a x a t i o n due t o the c h e m i c a l s h i f t a n i s o t r o p y (CSA). T h i s mechanism and q u a d r u p o l a r r e l a x a t i o n a r e d i s c u s s e d below. The o t h e r mechanisms make s m a l l c o n t r i b u t i o n s and a r e d i s c u s s e d b r i e f l y i n the e r r o r s e c t i o n ( S e c t . 1 8 ) . 181 4 182 16.1 CHEMICAL SHIFT ANISOTROPY (CSA) The CSA c o n t r i b u t i o n t o the 1 3 C r e l a x a t i o n time f o r a p l a n a r r o t o r i s g i v e n by (102) T i 1 = ( 3 / l O ) c o g 6 j f . ( G , D ) (16.1) where f(0,D) = (3/4R r ) " 1 |acos 20 + b s i n 2 0 - c s i n 2 f l c o s 2 6 | (16.2) and a = (1/3) [4R v +(TJ -1 ) 2R +(77+1 ) 2 R l b = ( 1 / 3 ) [ 4 R ^ + ( T 7 - 1 ) 2 R ; c+(T ? + 1 ) 2 R Z ] c = d / 9 ) ( 7 7 - 3 ) 2 ( | x ^ ) ( 1 6 . 3 ) z 5 where n=(6 -6 )/6 , 8 i s the a n g l e of 6 , the t r a c e l e s s x y z x 3 x component of of the CSA t e n s o r , from an a r b i t r a r i l y chosen x a x i s , (but z must be p e r p e n d i c u l a r t o the p l a n e of the r o t o r and i s d e f i n e d by F i g . 1 0 . 1 ) , R^, R^ and R £ a r e t h e p r i n c i p a l elements of the d i f f u s i o n t e n s o r . For a die, 0 = 0 as the 1 3 C CSA t e n s o r i s c o - l i n e a r w i t h t h e m o l e c u l a r frame. Eqn. 1 6 . 1 t h u s r e d u c e s t o 183 J(w) = (4R r ) " 1 [4R x+(r?-1 ) 2R^+(r?+1 ) 2 R z ] (16.4) Note the s i m i l a r i t y of t h i s e q u a t i o n w i t h Eqn.12.23 f o r 7}&CJ=0. In p r i n c i p l e we can combine the 1 3 C da t a w i t h the ESR da t a ( i f T?*0), or the d e u t e r i u m d a t a (or both) t o o b t a i n the d i f f u s i o n t e n s o r . 16.1.1 ISOLATING THE CSA TERM For i s o t r o p i c d i f f u s i o n the CSA 1 3 C r e l a x a t i o n time i s g i v e n by T i 1 = (9/10)(J§6 2,(1+TJ 2/3)T c (16.5) In our case TJ=2.12 and 6^=76.9ppm. U s i n g a v a l u e of 20pS f o r T we get a r e l a x a t i o n time of 38s. Note t h a t t h i s r e l a x a t i o n time i s f i e l d dependent so we can s e p a r a t e the CSA term from a l l the o t h e r r e l a x a t i o n mechanisms by m u l t i f i e l d NMR e x p e r i m e n t s . Most n o t a b l y we can s e p a r a t e out the s p i n r o t a t i o n (SR) c o n t r i b u t i o n which i s u s u a l l y s i m i l a r i n s i z e and i n f a c t d i r e c t l y r e l a t e d t o (154) the CSA term. To s e p a r a t e the terms we d e f i n e 184 T"1 = au 2 cs a T"1 = b res. obs. so for experiments with the CXP200 and WH400 we get ^obs.^OO-^obs.UoO " a(200 2-400 2 ) Thus we can determine 'a' and T"1 . Unfortunately cs a J T"' ^T' 1 and the error in 'a' i s thus large, which c s a r e s . 3 ' compounds the experimental errors in . 16.2 QUADRUPOLAR RELAXATION The relaxation time of an 1=1 nucleus in a planar asymmetric rotor i s given by (for 77= 0) (102) 1/T, = 3/8x 2J(0) (16.6) where x, the quadrupolar s p l i t t i n g constant i s e2qQ/ft (Q i s the quadrupole moment and q the e l e c t r i c f i e l d gradient) and J(0)=f(O,D) which i s defined by Eqn.16.2, 185 b u t where a = R +R z s b = (R +R )cos 2c/> + (R +R ) s i n 2 < 6 - ^x-£v-cos 2 c/>sin 2 cj> z s v s R +R 7 z s c = ^ | z ^ - c o s 2 < 6 + - ^ z ^ ^ - s i n 2 * - - ^ | x ^ | ^ - c o s 2 0 s i n 2 < 6 ( 1 6 . 7 ) y s x s z s where 8 i s now t h e p o l a r a n g l e (C-D) bond a n g l e w i t h r e s p e c t t o t h e z - a x i s a n d <j> i s t h e p l a n a r a n g l e ( a n g l e w i t h r e s p e c t t o t h e x a x i s ) . T h e c h o i c e o f a x i s i s a r b i t r a r y , we w i l l u s e t h e a x i s s y s t e m p r e v i o u s l y d e s c r i b e d . N o t e t h a t t h i s f u n c t i o n i s s y m m e t r i c w i t h r e s p e c t t o R^ a n d R^ i f t h e p l a n a r a n g l e s a r e symmetry r e l a t e d , (e.g. f o r two n u c l e i ' a 1 a n d 'b', <t> ±<t>,+nn/2, where n i s i n t e g r a l ) a b F o r t h e p y r o l l i d i n e d e r i v a t i v e t h e r e a r e two m a g n e t i c a l l y d i s t i n c t t y p e s o f d e u t e r i u m . We t h u s n e e d a t h i r d p i e c e o f i n f o r m a t i o n t o i n v e r t t h e d a t a t o g e t t h e r o t a t i o n a l d i f f u s i o n t e n s o r . T h e 1"N r e l a x a t i o n t i m e i s an o b v i o u s , b u t i m p r a c t i c a l c h o i c e (vide infra). R e l a x a t i o n t i m e s f r o m ESR o r 1 3 C s p e c t r a a r e o t h e r p o s s i b i l i t i e s . T h e r e a r e two q u a d r u p o l a r n u c l e i o f i n t e r e s t i n o u r compounds, D a n d '*N V The 1 B N c a s e i s v e r y s i m i l a r t o t h a t o f p y r i d i n e (151). I f we assume i s o t r o p i c m o t i o n a s b e f o r e we g e t t h a t 186 T i 1 = 2 4 X 2 T c (16.8) f o r 1 f lN, x-5MHz and (from ESR) r c=20pS. Hence T, w i l l be <0.1mS. Hence, T 2 w i l l be <*0.1mS, i m p l y i n g a h a l f - h e i g h t l i n e - w i d t h of =3kHz. T h i s f a c t c o u p l e d w i t h the low resonant f r e q u e n c y (11MHz a t 4.7T) and low s e n s i t i v i t y of the n u c l e u s means t h a t 1 *N T, f o r our system measurement are not f e a s i b l e w i t h the a v a i l a b l e s p e c t r o m e t e r s . Deuterium measurements however, a r e p o s s i b l e . T h i s n u c l e u s i s 10x more s e n s i t i v e than 1 °N and x-lOOkHz so t h a t the T,'s a r e a p p r o x i m a t e l y 1 s e c . 16.3 SPIN ROTATIONAL RELAXATION The T, f o r s p i n r o t a t i o n a l r e l a x a t i o n i s g i v e n by S p e i s s (107) as T i 1 - * * ? i h e j c h T j ( 1 6 - 9 ) where 6 i s the i n e r t i a t e n s o r , c. i s the s p i n r o t a t i o n t e n s o r i n the i n e r t i a l frame, T . i s the a n g u l a r momentum c o r r e l a t i o n t i m e . As w i t h ESR the s p i n - r o t a t i o n t e n s o r may be c a l c u l a t e d from the c h e m i c a l s h i f t t e n s o r d a t a . " 1 The s p i n - r o t a t i o n ** T h i s has been done s u c c e s s f u l l y f o r f l u o r i n e i n a number h i g h l y symmetric metal f l u o r i d e s (155)(156). The a p p l i c a b i l i t y of t h i s method t o 1 3 C i n asymmetric t r a n s i t i o n m e t a l complexes i s open t o q u e s t i o n . 187 c o n t r i b u t i o n can o f t e n be e x t r a c t e d from e x p e r i m e n t a l d a t a by measuring T, over a wide temperature range. The s p i n - r o t a t i o n term dominates at h i g h temperature as i t v a r i e s w i t h t e m p e r a t u r e , T, whereas a l l o t h e r r e l a x a t i o n mechanisms v a r y as 1/T. However, even i f s p i n - r o t a t i o n d a t a of r e a s o n a b l e a c c u r a c y i s o b t a i n e d i t s i n t e r p r e t a t i o n i s , as w i t h the ESR case ( S e c t . 1 2 . 5 ) , not c u r r e n t l y p o s s i b l e . 16.4 CHOICE OF T, EXPERIMENT A number of methods f o r measuring T ^ S have been proposed (157) (158)(159)(160)(161)(162)(163) and a n a l y s e d w i t h r e s p e c t t o p r e c i s i o n and e f f i c i e n c y (164)(165)(157)(166) (158) (167) (168) (169) (170) (171) (172) (173) (17 4) . A good i n t r o d u c t i o n t o the v a r i o u s methods i s g i v e n i n (17 5). The two most r e l i a b l e methods a r e i n v e r s i o n r e c o v e r y (IR) and s a t u r a t i o n r e c o v e r y (SR). The former method i s g e n e r a l l y r e g a r d e d as the f a s t e r ( f o r a g i v e n p r e c i s i o n ) of the two methods, a l t h o u g h t h e r e i s some c o n f u s i o n i n the e a r l y l i t e r a t u r e about t h i s . The l a t t e r method i s r e l a t i v e l y i n s e n s i t i v e t o i n s t r u m e n t s e t t i n g s and t h u s p r o b a l y more a c c u r a t e . 16.4.1 THE INVERSION RECOVERY EXPERIMENT There has been a l o t of d i s c u s s i o n of the r e l a t i v e e f f i c a c y of IR vs. SR t e c h n i q u e s (157)(158)(167) (176)(174) B a s i c a l l y the IR method i s c o n s i d e r e d more e f f i c i e n t because a) i t has t w i c e the dynamic range of 188 the SR t e c h n i q u e ( t h e d a t a a r e spread from -ma t o +m a > a 5 as opposed t o 0 t o m^ f o r SR). b) The m B v a l u e i s , i n p r i n c i p l e , r e c o v e r a b l e a t time z e r o . One doesn't have t o wai t f o r S T / s t o get the v a l u e . In p r a c t i c e t h e s e advantages a r e r a t h e r s m a l l . R e l i a b l e r e s u l t s f o r n i a c o u l d not be o b t a i n e d a t time z e r o . The 90° and 180° p u l s e s have t o be s e t i n d e p e n d e n t l y (because the p u l s e shape i s not p e r f e c t ) t o w i t h i n 1° (17 2).li6 and ma i n t a i n e d t h e r e (i.e., the t r a n s m i t t e r must be s t a b l e f o r the d u r a t i o n on the e x p e r i m e n t ) . A l s o each p u l s e sequence must c o n t a i n a 5T, (minimum) d e l a y (168)(169)(170) so one has t o e s t a b l i s h an approximate T t t o s e t t h i s d e l a y c o r r e c t l y . A l l t h e s e f a c t o r s l e a d t o l o n g s e t - u p times (=5hrs i n the case of the CXP200), which can ab r o g a t e the dynamic range advantage t h a t the IR experiment has over the SR exp e r i m e n t , which i s e a s i e r t o s e t up. 16.4.2 THE SATURATION RECOVERY EXPERIMENT T h i s method e l i m i n a t e s the need f o r the 5T, w a i t between p u l s e s , but one has t o c o l l e c t f o u r t i m e s as much d a t a as w i t h the IR experiment t o a c h i e v e the same p r e c i s i o n . A l s o the m B v a l u e cannot be o b t a i n e d a t time z e r o , g e t t i n g t h i s v a l u e can account one t h i r d of the da t a c o l l e c t i o n t i m e . However, t h i s method i s q u i t e * 5 The e q u i l i b r i u m m a g n e t i s a t i o n . * 6 There a r e m u l t i p l e p u l s e sequences t h a t m i n i m i s e t h i s p r oblem (177)(178)(179), but th e y i n c r e a s e s e t - u p time f u r t h e r 189 i n s e n s i t i v e t o the p u l s e l e n g t h s e t t i n g and hence t o the r . f . inhomogeneity and the o f f s e t (180). The p u l s e l e n g t h and o f f s e t s h o u l d be set r e a s o n a b l y c l o s e t o the c o r r e c t v a l u e t o maximise s e n s i t i v i t y , but do not have t o be e x a c t . T h i s c o n s i d e r a b l y reduces the s e t - u p t i m e . Because the SR method i s i n s e n s i t i v e t o i n s t r u m e n t a r t e f a c t s i t s h o u l d be more a c c u r a t e than the IR method f o r a g i v e n p r e c i s i o n . U n f o r t u n a t e l y , t h i s method o n l y * g i v e s c o r r e c t r e s u l t s f o r samples where T 2<<T 1. 16.4.3 INVERSION RECOVERY VS. SATURATION RECOVERY The f i n a l c o n c l u s i o n i s t h a t if_ t h e i n s t r u m e n t i s p e r f e c t and p r o p e r l y s e t - u p , then IR i s the more e f f i c i e n t method t o use. In p r a c t i c e the ease of s e t - u p and i n s e n s i t i v i t y t o a r t e f a c t s of the SR method o v e r i d e s the t h e o r e t i c a l e f f i c i e n c y advantage of the IR method f o r samples w i t h l o n g T ^ s . Hence the IR method was used t o measure the d e u t e r i u m T / s and the SR method was used f o r the 1 3 C r e l a x a t i o n measurements. 17. NMR EXPERIMENTAL 17.1 PREPARATION OF THE SOLUTIONS FOR NMR The samples were p r e p a r e d as w i t h the ESR samples ( S e c t . 1 3 ) , e x cept t h a t the s a t u r a t e d s o l u t i o n of the a p p r o p r i a t e i s o t o p i c a l l y s u b s t i t u t e d n i c k e l complex was used. The n i c k e l s a l t s were pr e p a r e d from copper f r e e n i c k e l c h l o r i d e ( S e c t . 3 . 3 ) , but o t h e r w i s e they were p r e p a r e d i n the same manner as the copper complexes. 17.2 NMR SAMPLE TUBES Samples f o r use on the WH400 were s e a l e d i n 5mm NMR tubes u s i n g s t a n d a r d methods. Samples f o r the CXP200 were s e a l e d i n s h o r t l e n g t h s (3.5-4.Ocm) of 7mm o.d. g l a s s t u b i n g . Only a s m a l l 'bubble' was l e f t , t o m i n i m i s e r . f . inhomogeneity. (However, the 'bubble' must be l a r g e enough t o accomodate t h e r m a l e x p a n s i o n of the s o l v e n t ) . S o l u t i o n s of the p y r o l l i d i n e d e r i v a t i v e were p r e p a r e d as needed r a t h e r than s t o r e d because of problems w i t h d e c o m p o s i t i o n of the samples ( S e c t . 3 . 5 ) . 17.3 POWDER SPECTRA The 1 3 C c h e m i c a l s h i f t a n i s o t r o p y was measured from 1 3 C powder s p e c t r a u s i n g s t a n d a r d c r o s s p o l a r i s a t i o n t e c h n i q u e s (181). The p u l s e sequence used i s shown below. 190 191 90° phase - r r n — " — r n 6pS 2mS 25mS 4S F i g u r e 17.1. Schematic of the p u l s e sequence used t o o b t a i n t h e powder spectrum. 17.4 Tj_ MEASUREMENTS Deuterium s p e c t r a were i n i t i a l l y r e c o r d e d a t 30.7MHz u s i n g a phase a l t e r n a t i n g i n v e r s i o n r e c o v e r y (IR) e x p e r i m e n t . [ ( + X ) i 8 o - r - ( + X ) g o - 5 T 1 , ( + X ) l 8 0 - r - ( - X ) 9 0 - 5 T l - ( - X ) i 8 0 - T - ( + X ) 9 0 - 5 T 1 , ( - X ) i 8 0 - T - ( - X ) 9 0 - 5 T 1 ] T y p i c a l l y 2000 scans were used. L a t e r e x p e r i m e n t s were performed on the WH400 u s i n g t h e Bruker IR e x p e r i m e n t . A t y p i c a l s e t of r e s u l t s i s shown i n F i g . 1 7 . 2 . 1 92 Tl A J ^ P J-u jX. JIX. jil^ J l \ Figure 17.2. T y p i c a l IR d a t a s e t . D e l a y t i m e s a r e 30, 50, 70, 100, 130, 150, 180, 250, 500 and 1000mS. 61.4MHz, T=310K. 1 3 C s p e c t r a were r e c o r d e d a t 100MHz u s i n g the WH400's s o f t w a r e f o r the s a t u r a t i o n r e c o v e r y experiment and a l s o a t 50MHz (CXP200) u s i n g the f o l l o w i n g phase a l t e r n a t i n g sequence. [ 2 0 { ( + X ) 9 0 - r r ( + X ) 9 0 } - r - ( + X ) g o - 2 0 { ( + X ) 9 0 - T 5 - ( + X ) g o } - r - ( - X ) g o ] * where T >T2=50mS. Ty p i c a l l y 200 scans were used. Some data S are shown below 193 J F i g u r e 17.3. T y p i c a l SR d a t a s e t . The d e l a y t i m e s a r e 2, 5, 10, 15, 20, 30 and 100s. 100.6MHz, T=310K. 17.5 ANALYSIS OF NMR DATA T, d a t a were a n a l y s e d by t h e s t a n d a r d method (175) of measuring peak h e i g h t s and d o i n g a l e a s t - s q u a r e s - f i t t o a s e m i - l o g p l o t of m a g n e t i s a t i o n vs. d e l a y u s i n g the f o l l o w i n g e q u a t i o n s . 1 94 in = m (1 - k e ~ T / T l ) f 0 0 h e n c e l n ( m -m ) = l n ( k ) - T / T , oo ^ • where moo i s t h e e q u i l i b r i u m m a g n e t i s a t i o n (== m^ f o r r>5T,) a n d k=1 f o r a s a t u r a t i o n r e c o v e r y (SR) e x p e r i m e n t o r k=2 f o r an i n v e r s i o n r e c o v e r y ( I R ) e x p e r i m e n t . 18. NMR ERROR DISCUSSION As w i t h ESR t h e r e are a number of s o u r c e s of e r r o r . The e x p e r i m e n t a l e r r o r s a r e d i f f e r e n t from ESR, i n s t r u m e n t a l a r t e f a c t s p l a y a much l a r g e r r o l e i n NMR e x p e r i m e n t s than i n ESR e x p e r i m e n t s . 18.1 ON APPROXIMATING THE SPECTRAL DENSITIES The s p e c t r a l d e n s i t i e s are of the form = Lc (18.1) 1 + ( o > r c ) 2 where u i s the Larmor f r e q u e n c y of the n u c l e u s b e i n g s t u d i e d and TC i s a l i n e a r c o m b i n a t i o n of the r o t a t i o n a l e i g e n v a l u e s f o r an asymmetric r o t o r . T£ i s lO-200pS and u i s a maximum of 100MHz ( f o r 1 3 C a t 9.4T), g i v i n g (cor ) 2 a maximum v a l u e of <0.02, i . e . , an e r r o r i n assuming j ( 0 ) = j(co) of <2%, w e l l w i t h i n e x p e r i m e n t a l e r r o r . However, i t i s i n t e r e s t i n g t o note t h a t t h i s a p p r o x i m a t i o n i s i n a p p r o p r i a t e f o r 400MHz p r o t o n s p e c t r a . 18.2 RESIDUAL CONTRIBUTIONS TO RELAXATION There a r e a number of c o n t r i b u t i o n s t o r e l a x a t i o n , t h e i r magnitude v a r i e s g r e a t l y from system t o system. For q u a d r u p o l a r n u c l e i , q u a d r u p o l a r r e l a x a t i o n i s t h e o n l y s i g n i f i c a n t s o u r c e of r e l a x a t i o n . For s p i n 1/2 n u c l e i , t h e r e a r e many s o u r c e s of r e l a x a t i o n . The major one f o r dt c' s have 195 196 been d i s c u s s e d i n S e c t . 1 6 . , o t h e r minor c o n t r i b u t i o n s a r e d i s c u s s e d below. 18.2.1 INTERMOLECULAR DIPOLAR RELAXATION T h i s a r i s e s from the t r a n s l a t i o n a l d i f f u s i o n of the s o l v e n t p a s t the s o l u t e . The r e l a x a t i o n time of a s p i n I ( i n the s o l u t e ) by a n u c l e u s S ( i n the s o l v e n t ) i s g i v e n by (182) T i 1 * D 2S(S+1 )N7j/kT (18.2) where D = h'^y.y r : 3 . T h i s assumes Debye d i f f u s i o n i n ' i ' s i s 1 the f a s t m o t i o n a l l i m i t and t h a t the r e l a x a t i o n of S i s independent of t h a t of I , which i s r e a s o n a b l e i f y.»JS• For p r o t o n s 4 7 i n c h l o r o f o r m d i f f u s i n g p a s t a 1 3 C n u c l e u s we have S = i , -q the v i s c o s i t y a t T=300K i s 0.57cP and the p r o t o n number d e n s i t y i s 7.5X10 2 1 s p i n s / c m 3 . T h i s g i v e s a T, of 30000s, an e r r o r of 0.1% f o r a 30s T, t y p i c a l f o r the 1 3 C i n our system. As d e u t e r o - c h l o r o f o r m was used as s o l v e n t the e r r o r w i l l be n e g l i g i b l e . 4 7 In t h e o r y the c h l o r i n e atoms make a lOx l a r g e r c o n t r i b u t i o n than the p r o t o n . However t h e i r r e l a x a t i o n t i m e s a r e v e r y f a s t ( t h ey a r e q u a d r u p o l a r n u c l e i ) and they a r e u n l i k e l y t o make any c o n t r i b u t i o n t o the r e l a x a t i o n of the s o l u t e . 1 97 18.2.2 INTRAMOLECULAR DIPOLAR RELAXATION The 1"N nucleus bound to the 1 3C may cause dipolar relaxation. The T, for t h i s i s given by (182) T i 1 = ^ 1 ^ ( 1 ^ + 1 ) D 2 J ( C J ) ( 1 8 . 3 ) In the fast motional l i m i t , for isotropic motion J (CJ) = 1 Or hence c 1/T, = 3/8D 2 T c (18.4) For our case rc==20pS at 300K and r=1.33A (183), giving a T, of » 500s. However, r i s estimated from ESR data c assuming is o t r o p i c motion. The c o r r e l a t i o n time for the motion a f f e c t i n g the dipolar relaxation (end-over-end with respect to the C-N bond) i s probably longer and T, may be s u f f i c i e n t l y short to make a s i g n i f i c a n t contribution to the relaxation. 18.2.3 FLUCTUATIONS IN THE SCALAR COUPLINGS Relaxation of the 1*N nucleus contributes to the 1 3C relaxation via the coupling, The T t for t h i s mechanism i s given by 198 T^1 oc (2/3)J 2S(S+1 ) T ' where r ' = LN (18.5) assuming fast isotropic motion. For our case we have S=1 and T^=0.1mS (see Sect. 16.2). J^^ISHz hence T^lSOOOs, a n e g l i g i b l e contribution. 18.2.4 INTERNAL MOTION Contributions from p y r o l l i d i n e ring pucker are ne g l i g i b l e (see appendix 22.8). The di e t h y l derivative i s a similar size to the p y r o l l i d i n e derivative and both give the same results for ESR data (88). However, i t s stereochemistry i s d i f f e r e n t , the ethyl groups are probably v e r t i c a l with-respect-to the plane of the molecule. There i s a large s t e r i c hindrance between them so one w i l l stick up and one down. Fluctuations in t h i s geometry are unlikely to a f f e c t the 1 3C relaxation times, but t h i s conformation w i l l reduce the d i f f u s i o n c o e f f i c i e n t s for rotation about the x and z axes. This may account for the incompatibility of the 1 3C results and the D-NMR and ESR r e s u l t s . 199 18.3 ERRORS FROM DATA ANALYSIS The s e m i - l o g d a t a a n a l y s i s method i s v e r y s e n s i t i v e t o the m B v a l u e used. To m i n i m i s e t h i s e f f e c t t he e x p e r i m e n t a l d a t a was i n t e r a c t i v e l y f i t t e d t o an e x p o n e n t i a l . M u l t i - e x p o n e n t i a l f i t t i n g methods a r e a v a i l a b l e (166)(168)(184)(185)(186)(187)(188)(189)(190)(191) which i n p r i n c i p l e do not need an D I b v a l u e . These methods g i v e r e a s o n a b l e r e s u l t s w i t h poor m^ v a l u e s (i.e. , a t low SNR), but do not g i v e r e l i a b l e r e s u l t s u n l e s s an m^ v a l u e i s i n c l u d e d i n the d a t a s e t (43). A l s o t h e s e methods can be used t o c o r r e c t f o r m i s - s e t p u l s e l e n g t h s and r . f . inhomogeneity. However, t h e i r use w i t h s m a l l n o i s y d a t a s e t s (as i s the case here) i s q u e s t i o n a b l e (192). 18.4 ERRORS IN Tj_ MEASUREMENTS T, measurements of low s e n s i t i v i t y n u c l e i w i t h l o n g r e l a x a t i o n t i m e s a r e e x t r e m e l y demanding on the s p e c t r o m e t e r . In our ca s e the experiment may l a s t t h r e e days, d u r i n g which the t r a n s m i t t e r must remain s t a b l e , the s p i n n i n g r a t e must be c o n s t a n t , the magnetic f i e l d s h o u l d not d r i f t and the t e m p e r a t u r e s h o u l d b e . c o n s t a n t . Other s o u r c e s of e r r o r a r i s e from r . f . inhomogeneity, paramagnetic i m p u r i t i e s , t emperature g r a d i e n t s and d i f f u s i o n of the probe out of the c o i l (157). There i s l i t t l e t h a t one can do about v a r i a b l e s p i n n e r , t r a n s m i t t e r o r tempe r a t u r e i n s t a b i l i t y , o t h e r than t o r e j e c t s u s p e c t d a t a . Temperature i n s t a b i l i t y was not a problem, 200 a l t h o u g h temperature g r a d i e n t s of >1°C a r e p r e s e n t a c r o s s the c o i l (193). The f i e l d of the CXP 200 appe a r s t o be s t a b l e . Experiments w i t h the WH400 were done w i t h a deu t e r i u m l o c k . SR ex p e r i m e n t s a r e i n s e n s i t i v e t o r . f . inhomogeneity, but i t can cause problems i n IR ex p e r i m e n t s . T h i s e f f e c t can be m i n i m i s e d by u s i n g s h o r t sample t u b e s . The e f f e c t s of the probe d i f f u s i n g out of t h e c o i l can a l s o be m i n i m i s e d by u s i n g s h o r t samples. I t i s e s s e n t i a l t o remove paramagnetic i m p u r i t i e s when mea s u r i n g l o n g T,'s as they g r e a t l y reduce the r e l a x a t i o n time (e. g. (175)(182)(194). A l l samples were p r e p a r e d from c o p p e r - f r e e n i c k e l s a l t s and deoxygenated b e f o r e use. 19. NMR RESULTS AND DISCUSSION 19.1 _]_f_c_ RESULTS The 1 3 C r e l a x a t i o n r e s u l t s a r e shown i n T a b l e 19.1. The e r r o r s a r e e s t i m a t e s o n l y and j u s t s e r v e as a guide t o the r e l a t i v e r e l i a b i l i t y of the r e l a x a t i o n t i m e s . The r e s u l t s a r e a v e r a g e s o f , or s e l e c t e d v a l u e s from, s e v e r a l r u n s . (See appendix 22.12 f o r the complete d a t a ) . Temp K 50.3MHz 100.6MHz T,(CSA) 310 22±1 10±1 60±10 323 16±1 11.5±1 125±20 333 11.5±2 8±1 165±30 Ta b l e 19.1. 1 3 C T / s f o r E t 2 d t c . ( s e e s ) . The CSA t e n s o r was d e t e r m i n e d from the powder spectrum, but t h e r e i s some a m b i g u i t y i n the assignment of the y and z components of t h a t t e n s o r . (The x component can be u n i q u e l y d e t e r m i n e d from the d i p o l a r s p l i t t i n g of t h e powder p a t t e r n , see appendix 22.3). As we have an o v e r - d e t e r m i n e d system (5 p i e c e s of r e l a x a t i o n d a t a and 3 unknowns ) we s h o u l d be a b l e t o r e s o l v e the a m b i g u i t y from t h e r e l a x a t i o n d a t a . However, the f a s t e s t r o t a t i o n a x i s i s about the C-N bond (i.e., the i n t e r c h a n g e of the y and z components i s the most i m p o r t a n t c o n t r i b u t i o n t o the r e l a x a t i o n t i m e ) . I f we assume R » R ,R x y z i n Eqn. 16.4 then i s independent of r). Hence w i t h i n 201 202 e x p e r i m e n t a l e r r o r we cannot d i s t i n g u i s h between the y and z components of the CSA t e n s o r . (The c o r o l l a r y , of c o u r s e , i s t h a t we don't need t o anyway!) 19.2 DEUTERIUM Tj_ RESULTS The d e u t e r i u m r e l a x a t i o n r e s u l t s a re shown i n Ta b l e 19.2 (see a p p e n d i x 22.11 f o r the complete d a t a ) . Temp K 30.7MHz 61.4MHz N R N R 310 0.167 0.207 0.162 0.200 323 0.185 0.265 0.185 0.270 333 NA 0.187 0.242 T a b l e 19.2. D e u t e r i u m T / s f o r d-9 P y 2 d t c . Times are se e s . The 61.4MHz and 30.7MHz r e s u l t s a re i d e n t i c a l as e x p e c t e d . The s p e c t r a f o r the 333K r e s u l t s show e x t r a peaks i n d i c a t i n g t h a t t h e sample i s decomposing. These T, v a l u e s a r e s u s p e c t . The q u a d r u p o l a r s p l i t t i n g c o n s t a n t f o r our compound i n s o l u t i o n i s not a v a i l a b l e , however t h e s e v a l u e s a r e n e a r l y c o n s t a n t f o r a l k y l compounds (195). A v a l u e of 175kHz was used. 203 19.3 DISCUSSION The two d e u t e r i u m T,'s can be combined w i t h the 1 3 C d a t a and, u s i n g Eqn.16.4 and Eqn.16.7 s o l v e d f o r the d i f f u s i o n t e n s o r u s i n g n o n - l i n e a r l e a s t - s q u a r e s - f i t p r o c e d u r e s (196). The r e s u l t s a r e shown i n below. Temp K R R R x y z 310 non-convergent 323 60.2 8.9 -0.9 333 46.9 10.5 -1.7 Tab l e 19.3. The d i f f u s i o n t e n s o r from 1 3 C and 2H d a t a The u n s a t i s f a c t o r y r e s u l t s a r e p r o b a b l y a t t r i b u t a b l e t o the low q u a l i t y of the 1 3 C d a t a . * 8 A l s o s i m u l a t i o n s u s i n g known v a l u e s f o r the d i f f u s i o n t e n s o r show t h a t the 1 3 C r e l a x a t i o n time i s e s s e n t i a l l y independent of R^, t h i s may l e a d t o i n c o r r e c t convergence of the f i t t i n g r o u t i n e , e s p e c i a l l y c o u p l e d w i t h the l a r g e e r r o r s i n the 1 3 C T / s . The d e u t e r i u m d a t a can be combined w i t h the ESR d a t a and s u c e s s f u l l y i n v e r t e d ( P a r t . 5 ) . * 8The 1 3 C da t a a r e a l s o i n c o m p a t i b l e w i t h the ESR d a t a . PART 5. COMMENTS ON THE COMBINED NMR-ESR STUDIES 204 20. COMBINED ESR AND NMR RESULTS AND DISCUSSION 20.1 INTRODUCTION The NMR and ESR results have been combined to obtain the rotational d i f f u s i o n tensor for MPy 2dtc in chloroform. Although the results span a limited temperature range they demonstrate the u t i l i t y of combined ESR-NMR studies to obtain d i f f u s i o n tensors. Also they provide a s t a r t i n g point to explore some approximation methods for extracting di f f u s i o n tensors from ESR data alone. Furthermore the results can be used to check the v a l i d i t y of the hydrodynamic model for rotational d i f f u s i o n . 20.2 COMMENTS ON DATA INVERSION Inversion of non-linear equations can be problematic, especially i f good starting values are not ava i l a b l e , or there are large errors in the data. Some workers (151)(180) studying d i f f u s i o n tensors with NMR have resorted to incrementally searching a range of possible values for R^, R^ and Rz. This i s a reasonable approach because the range of values for the R's are quite constrained. (They are a l l positive and one can use hydrodynamic models to get order of magnitude values for them). For our system good st a r t i n g values for R and R +R are available by using the x y z J • approximation methods outlined in Sect.15. However, one has to invert the data i n i t i a l l y to ensure these approximations are v a l i d so the generality of t h i s approach i s 205 206 q u e s t i o n a b l e . One s e r i o u s problem a r i s e s when i n v e r t i n g the ESR d a t a because Eqn.12.23 i s symmetric w i t h r e s p e c t t o the i n t e r c h a n g e of R^ and R^f i.e. , X(o) and X(0) a r e not c o m p l e t e l y independent f u n c t i o n s . However, an i n c r e m e n t a l s e a r c h works r e a s o n a b l y w e l l f o r t h i s f u n c t i o n as we know, on g e o m e t r i c grounds, t h a t R^>R^,Rz (and a l l a r e p o s i t i v e ) . T h i s approach a l s o g i v e s a good p h y s i c a l f e e l f o r X 0o» The d a t a were f i n a l l y i n v e r t e d w i t h an i n c r e m e n t a l s e a r c h of the ESR and d e u t e r i u m d a t a t o get good s t a r t i n g v a l u e s , f o l l o w e d by i n v e r s i o n u s i n g a Newton-Raphson pr o c e d u r e (196) t o get t h e f i n a l v a l u e s . I n c r e m e n t a l s e a r c h i n g i s a v e r y i n e f f i c i e n t approach i f one doesn't have a r e a s o n a b l e i d e a of the domain of R, e s p e c i a l l y i f the d a t a c o n t a i n s e r r o r s . The e f f i c i e n c y can be g r e a t l y improved by s e a r c h i n g f o r r a t i o s of the r e l a x a t i o n t i m e s and r a t i o s of s p e c t r a l d e n s i t i e s . T h i s reduces the s y s t e m a t i c e r r o r s i n the da t a and h a l v e s the amount of da t a t o be s e a r c h e d . A f a i l u r e t o o b t a i n a match f o r t he r a t i o s u s u a l l y i n d i c a t e s a fundamental e r r o r i n the program, e.g. a s s i g n i n g t h e axes i n c o r r e c t l y . T h i s approach i s v e r y u s e f u l i n the i n i t i a l s t a g e s of d a t a r e d u c t i o n . 20.3 THE DIFFUSION TENSOR F i v e independent p i e c e s of i n f o r m a t i o n were o b t a i n e d . Two ESR s p e c t r a l d e n s i t i e s , two d e u t e r i u m r e l a x a t i o n t i m e s and one 1 3 C r e l a x a t i o n t i m e . As we o n l y need t h r e e independent 207 p i e c e s o f i n f o r m a t i o n t o g e t t h e d i f f u s i o n t e n s o r we have an o v e r d e t e r m i n e d s y s t e m a n d t h u s a c h o i c e o f s o l u t i o n s . As t h e e q u a t i o n s a r e n o n - l i n e a r i t i s b e s t t o t r y t o i n v e r t t h e d a t a i n g r o u p s o f t h r e e a n d t h e n c r o s s - c h e c k t h e r e s u l t s . G r o u p i n g t h e two p i e c e s o f ESR d a t a w i t h a t h i r d i s n o t u s e f u l ( u n l e s s an i n c r e m e n t a l s e a r c h i s d o n e ) b e c a u s e o f t h e symmetry a l l u d e d t o i n S e c t 20.2; t h e d a t a w i l l n o t c o n v e r g e . T h e 1 3 C d a t a p r o v e d t o be u n r e l i a b l e ( s e e S e c t . 1 9 . 3 ) s o t h e o n l y r o u t e was t o i n v e r t t h e two d e u t e r i u m r e l a x a t i o n t i m e s a n d j ( 0 ) ( t h e more r e l i a b l e o f t h e two ESR s p e c t r a l d e n s i t i e s ) a n d c r o s s - c h e c k t o s e e i f j(a>) i s c o r r e c t . T he r e s u l t s a r e shown i n T a b l e 2 0 . 1 . The 333K p y r o l l i d i n e NMR d a t a a r e n o t r e l i a b l e b e c a u s e t h e s a m p l e d e c o m p o s e s . The v a l u e s shown f o r t h i s t e m p e r a t u r e a r e f r o m an i n c r e m e n t a l s e a r c h o f t h e ESR d a t a . The e r r o r s c o r r e s p o n d t o t h e r a n g e o f R v a l u e s t h a t g i v e s p e c t r a l d e n s i t i e s t h a t a r e w i t h i n 5% o f t h e o b s e r v e d v a l u e s . T h e s e e r r o r s a r e q u i t e r e s p e c t a b l e f o r an u n d e r d e t e r m i n e d s y s t e m . Temp(K) R X R y R z 310 57.6 6.5 2.6 323 82.5 7.3 2.3 333 90110 4± 3 1 0 ± 5 T a b l e 20.1. T h e d i f f u s i o n t e n s o r . 208 20.4 THE HYDRODYNAMIC MODEL One g o a l of measuring d i f f u s i o n t e n s o r s i s t o e s t a b l i s h the geometry of a m o l e c u l e i n s o l u t i o n . The s o l u t i o n geometry f o r our probe i s w e l l d e f i n e d so we can use our r e s u l t s t o t e s t the v a l i d i t y of the hydrodynamic model, the u s u a l model used t o r e l a t e the d i f f u s i o n t e n s o r t o the geometry. The d i m e n s i o n s of the probe can be e s t a b l i s h e d by a c o m b i n a t i o n of c r y s t a l l o g r a p h i c d a t a and m o l e c u l a r models. A l s o one has t o account f o r dead-volume (113). The s o l v e n t i s of f i n i t e s i z e and, f o r i n s t a n c e , cannot p e n e t r a t e the gaps between the s u l p h u r s . The volume of the probe t h u s appears l a r g e r t o the s o l v e n t than the t r u e volume. The probe thus a p p r o x i m a t e s a rounded p a r a l l e l e p i p e d w i t h d i mensions x , y , z , of 1.6x0.5x0.25 nm. Hydrodynamic t h e o r i e s have o n l y been d e v e l o p e d f o r e l l i p s o i d s , but f o r t u n a t e l y the probe i s w e l l a pproximated by an e l l i p s o i d w i t h d i m e n s i o n s 1.75x0.65x0.3 nm. (see F i g . 2 0 . 1 ) 209 F i g u r e 20 . 1 . The probe as an e l l i p s o i d . S c a l e i s 1cm=0.5nm. S o l i d l i n e i s Van-der-Waals shape. D o t t e d l i n e i s e l l i p s o i d a l a p p r o x i m a t i o n . The d i f f u s i o n c o e f f i c i e n t s f o r an e l l i p s o i d from t h e hydrodynamic models g i v e n by P e r r i n (108) and Youngren and A c r i v o s (YA) (111) are R j t / C * = 3kT/47ra 3rj(4C / ) K*llp = 3kT/47ra 3r ?(X /p 1p- 2) (20.1) where 'a' (=x/2) i s the l e n g t h of the major s e m i - a x i s ; C. a r e c a l c u l a t e d from P e r r i n ' s e q u a t i o n s as o u t l i n e d i n (88); Xf. a r e t h e c o e f f i c i e n t s g i v e n by YA and p 1 f p 2 (a and b i n YA's n o t a t i o n ) a r e the s e m i - a x i s r a t i o s , z/2a and y/2a. Note t h a t Eqn.20.1 imply t h a t , a l t h o u g h the d i f f u s i o n c o e f f i c i e n t s a r e d i f f e r e n t , the temperature dependence i s 210 i d e n t i c a l , both a r e v i s c o s i t y dependent. The a p p r o p r i a t e f r i c t i o n c o e f f i c i e n t s f o r our probe a r e x s l i p 0.22 s t i c k 0.098 Tab l e 20.2. F r i c t i o n probe. y z 1.46 0.635 0.362 0.328 c o e f f i c i e n t s f o r t h e The p r e d i c t e d d i f f u s i o n c o e f f i c i e n t s a r e thus Rx Ry Rz T=31 0 s l i p 230 34 79 s t i c k 8.1 2.2 2.4 T=323 s l i p 267 40 92 s t i c k 9.4 2.6 2.8 T=333 s l i p 300 45 103 s t i c k 10.6 2.9 3.2 Ta b l e 20.3. P r e d i c t e d d i f f u s i o n c o e f f i c i e n t s f o r MPydtc. Both boundary c o n d i t i o n s g i v e r e s u l t s t h a t a r e an orde r of magnitude i n e r r o r . T h i s i s not unexpected, but the f a i l u r e t o p r e d i c t the ob s e r v e d o r d e r of the d i f f u s i o n c o e f f i c i e n t s i s d i s t u r b i n g . A l s o t h e r e l a t i v e magnitudes do not agree w e l l . The ob s e r v e d v a l u e f o r R /R i s =7 w h i l s t the s l i p x p model g i v e s =4 and the s t i c k model =3. T h i s i m p l i e s t h a t s c a l i n g the r e s u l t s t o l i e between the two boundary 21 1 c o n d i t i o n s , as has been suggested (197), i s not s a t i s f a c t o r y ; the t e n s o r elements behave d i f f e r e n t l y . R^ has more s l i p c h a r a c t e r than the o t h e r two modes of m o t i o n . T h i s i s c o n s i s t e n t w i t h t h e c a v i t y model f o r m o t i o n : A s i g n i f i c a n t p o p u l a t i o n of the m o l e c u l e s o c c u p i e s c a v i t i e s i n the l i q u i d t h a t a r e l a r g e enough t o p e r m i t the m o l e c u l e t o f r e e l y r o t a t e , or a t l e a s t undergo l a r g e a n g l e jumps, about the x - a x i s (which sweeps out the s m a l l e s t volume), but o n l y a s m a l l number of t h e c a v i t i e s a r e l a r g e enough t o a l l o w f r e e r o t a t i o n about th e o t h e r two axes (which sweep out much l a r g e r v o l u m e s ) . The p o p u l a t i o n of f r e e r o t o r s has the e f f e c t of i n c r e a s i n g t h e d i f f u s i o n r a t e and d e c r e a s i n g the a c t i v a t i o n energy f o r the r o t a t i o n r e l a t i v e t o t h a t e x p e c t e d from hydrodynamic models. T h i s i s a l s o c o n s i s t e n t w i t h the ESR o b s e r v a t i o n s d i s c u s s e d i n (Sect.15) and o t h e r workers (198) . The p e c u l i a r o r d e r i n g of R^ and R^ i s d i f f i c u l t t o e x p l a i n , but may r e f l e c t l o c a l a n i s o t r o p y i n the medium, p o s s i b l y due t o a weak c o o r d i n a t i o n w i t h the c h l o r o f o r m . 20.5 SUMMARY OF THE RESULTS Three c o n c l u s i o n s may be drawn from t h i s work, a) I t has been demonstrated t h a t ESR and NMR s t u d i e s can be combined t o g i v e t h e r o t a t i o n a l d i f f u s i o n t e n s o r of the probe, b) The i s o t r o p i c model f o r r o t a t i o n a l d i f f u s i o n i s e x t r e m e l y m i s l e a d i n g , c) The hydrodynamic model, w h i l e not m i s l e a d i n g , f a i l s t o account f o r t h e b e h a v i o u r of our probe. There i s 212 s t i l l scope f o r d e v e l o p i n g t h i s model, but a t h e o r y i n c o r p o r a t i n g l a r g e a n g l e jumps (dr s o l v e n t c a v i t i e s ) , a n i s o t r o p i c motion and d i s c o n t i n u o u s media may be needed. 20.6 A STRATEGY FOR MEASUREMENT OF DIFFUSION TENSORS F i r s t l y i t s h o u l d be commented t h a t o b t a i n i n g NMR r e l a x a t i o n measurements from d i l u t e s o l u t i o n s of i n s e n s i t i v e n u c l e i i s e x t r e m e l y time consuming and v e r y e r r o r prone. Recent developments i n NMR t e c h n o l o g y have improved t h i s s i t u a t i o n , but i n s t r u m e n t time i s s t i l l a t a premium. For t h i s reason any s t r a t e g y t h a t reduces use of NMR s p e c t r o m e t e r s i s of i n t e r e s t . S e c o n d l y , the o b j e c t i v e of the e x e r c i s e i s t o o b t a i n s i n g l e p a r t i c l e c o r r e l a t i o n t i m e s f o r a s o l u t e i n the f a s t - m o t i o n a l regime. M u l t i - p a r t i c l e c o r r e l a t i o n times a re not a u s e f u l o b j e c t i v e . S t u d i e s i n neat l i q u i d s a r e i n t e r e s t i n g , but r a t h e r r e s t r i c t i v e . S l o w - m o t i o n a l s t u d i e s i n NMR have y e t t o be d e v e l o p e d . S l o w - m o t i o n a l s t u d i e s i n ESR a r e b e t t e r d e v e l o p e d , but i t remains t o be demonstrated t h a t d i f f u s i o n t e n s o r s can be r e l i a b l y e x t r a c t e d from such s p e c t r a . G i v e n the above comments, how does one d e s i g n a probe t o o b t a i n the be s t i n f o r m a t i o n ? As d i s c u s s e d p r e v i o u s l y the probe s h o u l d be s t a b l e w i t h a w e l l d e f i n e d geometry and be s u f f i c i e n t l y s o l u b l e t o p e r m i t NMR s t u d i e s . I n a d d i t i o n the probe must be a b l e t o p r o v i d e a t l e a s t t h r e e independent p i e c e s of d a t a . 213 C o n s i d e r t h e r e q u i r e m e n t s f o r a n NMR o n l y s t u d y f i r s t . T h e r e s h o u l d be o n l y o ne r e l a x a t i o n m e c h a n i s m c o n t r i b u t i n g t o t h e m e a s u r e d T,, i.e., o n l y q u a d r u p o l e n u c l e i s h o u l d be u s e d , a l t h o u g h r e c e n t work u s i n g p r o t o n d i p o l e - d i p o l e c o u p l i n g s l o o k s p r o m i s i n g (199)(200). T h e r e s h o u l d be a t l e a s t t h r e e m a g n e t i c a l l y d i s t i n c t n u c l e i . F u r t h e r m o r e t h e o r i e n t a t i o n o f t h e m a j o r a x i s o f t h e q u a d r u p o l e t e n s o r f o r t h e t h e s e n u c l e i s h o u l d be s u c h t h a t a t l e a s t two d i f f e r e n t a z i m u t h a l a n g l e s a n d one p o l a r a n g l e * 9 0 ° a r e n e e d e d t o c h a r a c t e r i s e t h e i r d i s p o s i t i o n s . A l s o t h e m a j o r a x e s must n o t be o r t h o g o n a l . I n o u r p r o b e t h e r e a r e two m a g n e t i c a l l y i n e q u i v a l e n t d e u t e r o n s on t h e p y r o l l i d i n e r i n g . T h e t h i r d n u c l e u s i s i n t h e p l a n e . We u s e d 1 3 C , b u t 1"N o r Pd c o u l d , i n p r i n c i p l e , h a v e b e e n u s e d . Our t h i r d n u c l e u s was an u n s u c c e s s f u l c h o i c e , b u t t h i s seems t o be a g e n e r a l p r o b l e m w i t h NMR. Two s u i t a b l e n u c l e i c a n be f o u n d , b u t f i n d i n g a t h i r d i s d i f f i c u l t . A l s o t h e r e l a x a t i o n t i m e s a r e r e l a t i v e l y i n s e n s i t i v e t o t h e a z i m u t h a l a n g l e o f t h e m a j o r a x i s o f t h e q u a d r u p o l e t e n s o r . I d e a l l y t h e a z i m u t h a l a n g l e s s h o u l d be w e l l s e p a r a t e d , a z i m u t h a l a n g l e s o f 2 0 ° a n d 2 5 ° , f o r e x a m p l e , may n o t p r o v i d e e n o u g h d i s c r i m i n a t i o n t o o b t a i n r e l i a b l e r e s u l t s . F o r t h e s e r e a s o n s i t i s common t o r e s o r t t o a n o t h e r s p e c t r o s c o p i c t e c h n i q u e t o o b t a i n t h e e x t r a i n f o r m a t i o n . IR, LS a n d Raman s t u d i e s h a v e b e e n c o m b i n e d w i t h NMR t o t h i s e n d , b u t t h e s e t e c h n i q u e s r e s t r i c t one t o n e a t l i q u i d s o r s t r o n g (>10%) s o l u t i o n s . H e r e we h a v e c o m b i n e d ( f o r a f i r s t t i m e ) NMR a n d ESR m e a s u r e m e n t s t o g e t 214 the e x t r a d a t a . ESR a l l o w s the use of d i l u t e s o l u t i o n s , but r e q u i r e s a paramagnetic s p e c i e s . In our case t h i s i s e a s i l y a c h i e v e d by changing the c e n t r a l m e t a l of our probe. T h i s however, i s the major r e s t r i c t i o n t o t h i s approach; t h e r e must be d i a m a g n e t i c and paramagnetic a n a l o g s of the probe. The d e s i g n of paramagnetic probes i s b e s t c o n s i d e r e d i n the l i g h t of Eqn.12.22. In our case t h i s e q u a t i o n s i m p l i f i e s c o n s i d e r a b l y because of a x i a l symmetry. However, i n g e n e r a l most of the reduced s p e c t r a l d e n s i t i e s have t o be r e t a i n e d , but the b a s i c form of Eqn.12.22 i s unchanged by t h e s e e x t r a terms so f o r the purpose of t h i s d i s c u s s i o n we can use the f o l l o w i n g e x p r e s s i o n f o r T 2. T i 1 = I J(0).[A(K+m 2) + A m + G] + X (20.2) k K 8 where the sum over k i s used t o denote t h a t more than one e i g e n v a l u e f o r the asymmetric r o t o r i s r e t a i n e d . The n o n - s e c u l a r terms, j ( w ) , a r e not n e g l i g i b l e , but do not a f f e c t the arguments so they a r e dropped f o r c o n v e n i e n c e . F i r s t l y we note t h a t the spectrum w i l l c o n t a i n K (=21+1) l i n e s so the l a r g e r the n u c l e a r s p i n c o u p l e d t o the e l e c t r o n the b e t t e r . For i n s t a n c e , vanadium, 1=7/2, w i t h e i g h t l i n e s g i v e s b e t t e r s t a t i s t i c s than n i t r o g e n , 1=1, which g i v e s t h r e e l i n e s . I f the magnetic t e n s o r s a r e o r t h o r h o m b i c and I>1 the complete d i f f u s i o n t e n s o r may be o b t a i n e d from the ESR spectrum. However, i f one wishes t o do 215 t h i s via the s p e c t r a l d e n s i t i e s the spectrum must c o n t a i n a t l e a s t 2k+1 l i n e s , 2k f o r the k j ( 0 ) and j(o>) terms and one f o r the r e s i d u a l l i n e - w i d t h , X. A l s o the e i g e n v a l u e s a r e not l i n e a r l y independent. O b t a i n i n g t h r e e s p e c t r a l d e n s i t i e s does not guarantee t h a t the complete d i f f u s i o n t e n s o r can be found. However, i f the d a t a a r e t o be combined w i t h NMR r e s u l t s from two n u c l e i then a two l i n e ESR spectrum w i l l s u f f i c e i f d i r e c t i n v e r s i o n methods a r e used. I t i s d e s i r a b l e t o have a l a r g e range of l i n e - w i d t h s i n the ESR spectrum t o o b t a i n r e l i a b l e d a t a . As the m. dependence of l i n e - w i d t h i s c a r r i e d a lmost e n t i r e l y by the h y p e r f i n e a n i s o t r o p y (A i n Eqn.20.2) t h i s s h o u l d be q u i t e l a r g e , but not so l a r g e t h a t the f a s t - m o t i o n a l c r i t e r i o n f a i l s f o r the t emperature range of i n t e r e s t . A l s o the a p p r o x i m a t i o n s d i s c u s s e d i n Sect.14 w i l l have t o be examined. The h y p e r f i n e c o u p l i n g , A 0, i t s e l f s h o u l d a l s o be l a r g e t o p r e v e n t l i n e o v e r l a p . L i n e o v e r l a p can a l s o be m i n i m i s e d by k e e p i n g the r e s i d u a l l i n e - w i d t h s m a l l . T h i s can be a c h i e v e d by u s i n g probes w i t h no u n r e s o l v e d h y p e r f i n e c o u p l i n g and k e e p i n g the g - a n i s o t r o p y s m a l l . The l a t t e r i s r e s p o n s i b l e f o r the s p i n - r o t a t i o n term, t h e major c o n t r i b u t i o n t o the r e s i d u a l l i n e - w i d t h . I t i s i n t e r e s t i n g t o note t h a t f o r n i t r o x i d e s the c e n t e r l i n e (iru=0) c a r r i e s v e r y l i t t l e m o t i o n a l i n f o r m a t i o n . In f a c t most of t h i s i n f o r m a t i o n i s c a r r i e d by the m.=1 l i n e . T h i s can be t u r n e d t o advantage though, by s u b t r a c t i n g the w i d t h of the c e n t e r l i n e from the o u t e r two l i n e s and 216 then subtracting the widths of these two lines, the -relaxation E q n . 1 2 . 2 2 is thus considerably simplified. To summarise, the ideal probe for ESR/NMR studies should have the following requirements. Paramagnetic and diamagnetic structural analogs. Magnetic tensors that are not isotropic. At least two non-coincident axial tensors. The geometry should be well defined,, but easily tailored. The g-tensor anisotropy should be small and the hyperfine tensor anisotropy large. The isotropic hyperfine splitting should be such that Iine-width/line-splitting< 0 . 2 . The nuclei for NMR studies should be quadrupolar. The probe should possess enough symmetry that the orientation of the diffusion tensor is known. The dtc class of probes certainly f i t most of the above requirements. Their principal failings are the low solubility of the nickel complexes and the axial symmetry of the hyperfine coupling and g-terisors. However, as demonstrated here this is just an inconvenience, they can s t i l l be used to obtain the diffusion tensor. Also their geometry can, and has been, tailored for motional studies. It may be possible to tailor their chemistry for work in aqueous solvents and also improve the solubility of the nickel, complexes. 217 20.7 FINAL REMARKS The p e r s i s t e n c e of the hydrodynamic model f o r i n t e r p r e t i n g m o t i o n a l • s t u d i e s can be a s c r i b e d t o t h r e e f a c t o r s . For m o l e c u l e s t h a t a r e much l a r g e r than the s o l v e n t t h i s t h e o r y i s q u i t e a c c u r a t e . However, how l a r g e the probe m o l e c u l e / s o l v e n t m o l e c u l e s i z e r a t i o has t o be b e f o r e the hydrodynamic model f a i l s i s s t i l l unknown, a l t h o u g h t h e r e have been some r e c e n t advances i n t h i s a r e a (201)(202). The i s o t r o p i c model has been e x t e n s i v e l y used t o i n t e r p r e t m o t i o n a l r e s u l t s . T h i s approach has p r o b a b l y o b s c u r e d the i n a d e q u a c i e s of the hydrodynamic model and i f i t s use p e r s i s t s i t w i l l s e r i o u s l y hamper p r o g r e s s i n m o l e c u l a r dynamics s t u d i e s . A l s o the f r i c t i o n c o e f f i c i e n t s s c a l e w i t h a 3 , the l a r g e s t m o l e c u l a r a x i s . A 10% v a r i a t i o n i n 'a' produces a 30% v a r i a t i o n i n the c a l c u l a t e d c o r r e l a t i o n t i m e . T h i s w i l l g r e a t l y i n f l u e n c e the agreement or o t h e r w i s e w i t h t h e hydrodynamic t h e o r y . For i n s t a n c e , i s the Van-der-Waals r a d i u s the a p p r o p r i a t e measure of the di m e n s i o n s of mo l e c u l e s r o t a t i n g i n s o l u t i o n ? The apparent b e t t e r agreement of the ' s l i p ' r a t h e r than the ' s t i c k ' model may be a r e s u l t of a s y s t e m a t i c u n d e r e s t i m a t i o n of the m o l e c u l a r d i m e n s i o n s . The c o n c e p t u a l s i m p l i c i t y of the hydrodynamic model makes i t v e r y a t t r a c t i v e . The model remains t o be de v e l o p e d f o r n o n - s p h e r o i d a l m o l e c u l e s . Youngren's and A c r i v o s ' s a p p r o a c h a l l o w s t h i s , but the gap s t i l l r e m a i n s . Obvious c a n d i d a t e s f o r development a r e c a r b o n - t e t r a c h l o r i d e and 218 carbon d i s u l p h i d e , b o t h of which have been t h o r o u g h l y s t u d i e d and have s i m p l e g e o m e t r i e s . I f the model f a i l s f o r these two c a s e s , then models a c c o u n t i n g f o r d i s c o n t i n u i t i e s i n the s o l v e n t and ' f r e e - r o t a t i o n ' i n s o l v e n t c a v i t i e s (203)(204)(136)(205)(206) w i l l need t o be dev e l o p e d , or the i n e r t i a l models (50) extended t o a n i s o t r o p i c m o t i o n . F u r t h e r s t u d i e s of the 'probe i n a s o l v e n t ' t y p e , as i s d i s c u s s e d here, w i l l be needed t o p o i n t the way f o r t h e o r e t i c a l developments of t h a t k i n d . PART 6. NOTES ON THE DIGITAL ACQUISITION OF ESR SPECTRA 219 21. THE DIGITAL ACQUISITION OF ESR SPECTRA 21.1 INTRODUCTION T h i s s e c t i o n w i l l a d d r ess the d i g i t a l a c q u i s t i o n of ESR s p e c t r a from two p o i n t s of vie w , namely, t h e development of the s o f t w a r e and hardware of the d i g i t a l a c q u i s i t i o n system; the development of the a l g o r i t h m s f o r the p r o c e s s i n g and a n a l y s i s of the s p e c t r a . D i g i t a l a c q u i s i t i o n systems a r e not n o v e l ( f o r r e v i e w s see (116)(207) f o r examples see (140)(208)(209)(198)(210) (211)(212)(213)), but a t the time of c o n s t r u c t i o n of our system, the m i c r o - p r o c e s s e r was a t t h e l e a d i n g - e d g e of te c h n o l o g y {e.g. memory was e x p e n s i v e ) and the p e r s o n a l computer had not been i n v e n t e d . Some of t h e a s p e c t s of d e s i g n and methodology r e f l e c t t h i s . N o t a b l y , the use of o f f - l i n e d a t a p r o c e s s i n g and the l a c k of r e a l - t i m e a v e r a g i n g f a c i l i t i e s . E a r l y s p e c t r a were c o l l e c t e d on pa p e r - t a p e and p r o c e s s e d on the Amdahl 470 a t the UBC computing c e n t e r (140). L a t e r , t h i s l i n k t o the Amdahl was upgraded by the purchase of a magnetic tape u n i t . The use of a main frame computer p l a c e s v i r t u a l l y no r e s t r i c t i o n s on memory or speed, so the p r i m o r d i a l n a t u r e of our m i c r o p r o c e s s e r (a F a i r c h i l d F8) was not a g r e a t d i s a d v a n t a g e . The system was f u r t h e r improved by the a c q u i s i t i o n of a DEC LSI-11 micro-computer. T h i s computer p e r m i t t e d t h e use of i n t e r a c t i v e g r a p h i c s (which were not a v a i l a b l e f o r t h e 220 221 main-frame a t t h a t t i m e ) , but r e s t r i c t i o n s i n memory s i z e and speed now had t o be c o n s i d e r e d . A n a l y s i s of d i g i t a l ESR s p e c t r a i s more c o m p l i c a t e d than f o r most d i g i t a l s p e c t r a because of the v a r i a b i l i t y of the a b s c i s s a ( f i e l d - s w e e p ) d a t a . A l g o r i t h m s and n u m e r i c a l methods f o r m a n i p u l a t i n g d i g i t a l s p e c t r a a r e s c a t t e r e d throughout the l i t e r a t u r e , but an overview of the t e c h n i q u e s needed f o r p r o c e s s i n g ESR s p e c t r a i s s a d l y l a c k i n g . Such an overv i e w w i l l not be attempted h e r e , but the b a s i c a l g o r i t h m s used w i l l be b r i e f l y r eviewed and t h i s i n f o r m a t i o n s h o u l d be of i n t e r e s t t o anybody a t t a c h i n g a p e r s o n a l computer t o an ESR s p e c t r o m e t e r . " 9 A q u e s t i o n t h a t might be asked i s , 'why b o t h e r t o d i g i t i s e ESR s p e c t r a ?', ( e s p e c i a l l y c o n s i d e r i n g the time needed t o d e v e l o p the hardware and s o f t w a r e ) . The reasons are t h r e e - f o l d ; the p r e c i s i o n of spectrum measurement i s improved by a t l e a s t a f a c t o r of f i v e ; the a n a l y s i s time f o r s p e c t r a i s reduced by 2-3 o r d e r s of magnitude; d i g i t a l s p e c t r a can be m a n i p u l a t e d i n a manner t h a t i s d i f f i c u l t or i m p o s s i b l e u s i n g a n a l o g methods (e.g.DISPA). Thus both the q u a l i t y and q u a n t i t y of i n f o r m a t i o n a v a i l a b l e from a spectrum i s s u b s t a n t i a l l y i n c r e a s e d . * 9The s o f t w a r e was d e v e l o p e d f o r m o t i o n a l s t u d i e s o n l y , i.e. , s p e c t r a c o n s i s t i n g of a few broad l i n e s t h a t a r e measured a c c u r a t e l y . T h i s i s r e f l e c t e d by the absence of so f t w a r e r e l a t i n g t o s p e c t r a l a n a l y s i s (see (214)(215) f o r examples) o r d e c o n v o l u t i o n programs (216)(217)(218). 222 21.2 THE HARDWARE A b l o c k diagram of the system i s g i v e n i n F i g . 2 1 . 1 . A s i m p l i f i e d f l o w - c h a r t f o r the s o f t w a r e i s shown i n F i g . 2 1 . 2 B r i e f l y the system o p e r a t e s as f o l l o w s . The o p e r a t o r e n t e r s the sample i d e n t i f i c a t i o n etc. via the t e r m i n a l and s t a r t s the a c g u i s t i o n program. When the f i e l d scan i s s t a r t e d the temperature i s measured and the sample i n f o r m a t i o n w r i t t e n t o the t a p e . F i e l d / a m p l i t u d e d a t a a r e then t a k e n s i m u l t a n e o u s l y and 16 b i t d a t a p o i n t p a i r s w r i t t e n t o the tape i n b l o c k s of 50 p a i r s . Between each b l o c k t h e Gaussmeter r e a d i n g and the c o r r e s p o n d i n g F i e l d i a l v o l t a g e a r e c o l l e c t e d . T h i s i s c o n t i n u e d u n t i l the end of the scan when a c q u i s i t i o n i s stopped. The temperature i s measured a g a i n , the microwave f r e q u e n c y i s t a k e n and the c a l i b r a t i o n d a t a w r i t t e n t o the t a p e . The d a t a f i l e can then be t r a n s f e r r e d t o the DEC LSI-11 f o r a n a l y s i s . 0) *1 O M-iQ iQ C C cn n> i->-rt to o • 3 t—1 • in •< cn n- o (D o 3 *" a 0) Q) 3 Cd <X>- SAMPLE & HOLD HOLD 1SAMPLE & HOLD READY ANALOGUE MULTI-PLEXOR 16 BIT ADC ADC ON SELECT X.T.Y S/H HOLD• COMMAND DECODER VIDEO TERMINAL — _ -.i FBUG MONITOR FREQUENCY COUNTER (F/C) 1(3 CHANNELS) DIGITAL MULTI-PLEXOR SELECT CHANNEL ADC READY oi i -ct * iu TL C C • C J cr - x U J u. cc 1 1 DATA LATCH PORT 0 PORT 1 BUS START/STOP SCAN-LIMITS RUN CPU & CLOCK PROGRAM RESET TAPE ; FLAGS: :TAPE DATA 4K MEMORY LSI-11 COMPUTER 9719 TAPE FORMATTER 9718 TAPE FORMATTER 9800 TAPE DRIVE 9717A TAPE BUFFER DflTfl PROCESSING SYSTEM F - 8 MICRO-COMPUTER ro co • RESET \ . / 224 SELECT OPTION UPDATE FILE NOS. SEARCH FOR FILE END OF SPECTRUM CHECK FLAGS MV. FREO. RECORD ENTER SPECTRUM SETTINGS TEMP. RECORD WRITE TRAILER TABLES PAD DATA RECORD TEMP. RECORD COLLECT X-T DATA VIDEO PLOT TES DELAT LOOP GET CALIB DATA STOP 7 NO NO SO PTS. ? TES END DATA RECORD F i g u r e 21.2. F l o w - c h a r t f o r the s o f t w a r e of the a c q u i s i t i o n system. 225 21.3 THE BASIC PROBLEMS IN ACQUIRING ESR SPECTRA A number of n o i s e sources i n ESR, t h a t are masked by the i n h e r e n t time c o n s t a n t of the c h a r t r e c o r d e r , become apparent when one f i r s t r e c o r d s a d i g i t a l spectrum. These n o i s e s o u r c e s depend on the s p e c t r o m e t e r , but t y p i c a l examples a r e ; d i s p e r s i o n l e a k a g e , a c o u s t i c n o i s e , magnet n o i s e and m i s c e l l a n e o u s c r o s s - t a l k . The most severe of the l a t t e r i s m o d u l a t i o n of the spectrum by the m o d u l a t i o n c o i l s i n the Gaussmeter probe. C a r e f u l placement of t h i s probe i s e s s e n t i a l . ( A l s o see Sect.14.8) Magnet n o i s e poses a number of s e r i o u s problems. Both the w i d t h and the c e n t e r of f i e l d sweep a r e v a r i a b l e 5 0 so some method of c a l i b r a t i n g the sweep i s e s s e n t i a l . However, the r e a l problem i s the n o i s e from the F i e l d i a l sweep mechanism. The f i e l d i s incremented i n an i n c o n s i s t e n t and n o n - l i n e a r f a s h i o n r e s u l t i n g i n a d a t a s e t t h a t , a l t h o u g h e q u a l l y spaced w i t h - r e s p e c t - t o t i m e , i s not e q u a l l y spaced w i t h - r e s p e c t - t o magnetic f i e l d . T h i s has a g r e a t i n f l u e n c e on the s o f t w a r e d e s i g n as i t n e c e s s i t a t e s the c o l l e c t i o n of both f i e l d d a t a and spectrum d a t a ; d o u b l i n g computer-memory r e q u i r e m e n t s . A l s o enough p o i n t s must be c o l l e c t e d t o a l l o w f o r r e l i a b l e i n t e r p o l a t i o n of the d a t a . Most of the s o f t w a r e d i s c u s s e d here r e l a t e s t o t h i s problem. Once an e q u a l l y spaced d a t a s e t i s c r e a t e d the s o f t w a r e development i s 5 0 T h e f r o n t p a n e l s e t t i n g s of the F i e l d i a l a r e not v e r y a c c u r a t e and v a r y s l o w l y w i t h t i m e . The f i e l d sweep can a l s o be n o n - l i n e a r , a l t h o u g h we have not d e t e c t e d t h i s on our system. A l g o r i t h m s t o s h i f t and expand/compress the d a t a a r e thus an e s s e n t i a l p a r t of the s o f t w a r e . 226 r e l a t i v e l y s t r a i g h t - f o r w a r d . 21.4 ADC RESOLUTION There are f o u r b a s i c t y p e s of d a t a c o l l e c t e d by our system, t h e F i e l d i a l v o l t a g e ( X - d a t a , the f i e l d ) , the s i g n a l from t h e p h a s e - s e n s i t i v e - d e t e c t o r , PSD, (Y - d a t a , the a m p l i t u d e ) , thermocouple d a t a ( t h e t e m p e r a t u r e ) and Gaussmeter d a t a . The Gaussmeter d a t a a r e a l r e a d y a v a i l a b l e i n d i g i t a l form and need not concern us h e r e . 5 1 For d i s p l a y purposes a 10 b i t 5 2 r e s o l u t i o n of the Y-data are q u i t e adequate. However i f e x t e n s i v e n u m e r i c a l m a n i p u l a t i o n i s t o be done 12 or more b i t s a r e d e s i r a b l e t o a v o i d problems w i t h d i g i t i s a t i o n n o i s e (116)(219). S i m i l a r l y f o r the X-data, 10 b i t s i s f i n e , but i f , f o r example, one wis h e s t o m a i n t a i n a 0.1G r e s o l u t i o n over a 1000G sweep t o a v o i d a m p l i t u d e d i s t o r t i o n (220), then 13-14 b i t s a r e r e q u i r e d . Moreover, the F i e l d i a l v o l t a g e runs from 0-5V, as opposed t o -10V t o +10V f o r t h e Y data so i f one uses the same ADC f o r b o t h d a t a s e t s and no a m p l i f i e r s a n o ther 2 b i t s a r e r e q u i r e d . In p r a c t i c e , the X r e s o l u t i o n i s l i m i t e d by n o i s e t o =12 b i t s . We wi s h t o measure t o tempe r a t u r e t o =0.01° ( i . e . , <1% a t room t e m p e r a t u r e ) 5 3 over a wide temperature range. The 5 1 A r e s o l u t i o n of a t l e a s t 1:10 6 i s r e q u i r e d f o r Gaussmeter measurements, i . e . ,20 b i t s . 5 2 A n a l o g t o d i g i t a l c o n v e r t e r s (ADC's) have a r e s o l u t i o n of 2n where n i s the number of ' b i t s ' of r e s o l u t i o n . Hence a 10 b i t r e s o l u t i o n i s =1:1000) 5 3 T h i s i s a r e q u i r e m e n t f o r b i o l o g i c a l s t u d i e s . An a c c u r a c y of 0.1° over a range of -70°C t o +120°C i s r e q u i r e d f o r t h i s work. 227 thermocouple v o l t a g e i s a m p l i f i e d 1000X t o ±3V (max.) so a 15 b i t ADC, a t l e a s t , i s d e s i r a b l e . 21.5 NCK OF POINTS COLLECTED. THE NYQUIST CRITERION The maximum number of u s e f u l d a t a p o i n t s t h a t can be c o l l e c t e d i s l i m i t e d by t h r e e f a c t o r s ; X n o i s e , a m p l i f i e r band-width and computer memory and speed. The X s i g n a l - t o - n o i s e r a t i o (SNR) i s =2000:1 so u n l e s s a v e r a g i n g i s done t h e r e i s l i t t l e t o be g a i n e d by c o l l e c t i n g more than 2K p o i n t s . 5 " A v e r a g i n g i s not m e a n i n g f u l i f data a r e c o l l e c t e d a t a r a t e above the N y q u i s t f r e q u e n c y . For a band l i m i t e d a m p l i f i e r the N y q u i s t r a t e 5 5 i s =1 / t i m e - c o n s t a n t , t y p i c a l l y <5Hz, or <2000 p o i n t s f o r us. One can a r r a n g e t o de c r e a s e the time c o n s t a n t and i n c r e a s e the scan time so t h a t more p o i n t s a r e g a t h e r e d , but 16K p o i n t s o c c u p i e s 64 K b y t e s , 1/4 of the a v a i l a b l e computer memory. T h i s means t h a t the d a t a p r o c e s s i n g has t o be done ' o f f - t h e - d i s c ' , which i s slo w , r a t h e r than ' i n c o r e ' . A l s o , ' i n c o r e ' p r o c e s s i n g of >16K d a t a p o i n t s i s s u f f i c i e n t l y slow t o make i n t e r a c t i v e p r o c e s s i n g u n u s e f u l so the da t a has t o be b o x - c a r r e d back t o <4000 p o i n t s b e f o r e use. T h i s of co u r s e improves the apparent SNR, but i t i s j u s t as 5 4 A K i s 1024 or 2 1 0 and i s a c o n v e n i e n t s i z e u n i t t o use w i t h computers, e s p e c i a l l y i f t h e d a t a a r e t o be used w i t h F o u r i e r t r a n s f o r m s . 5 5 N o t e t h a t here the N y q u i s t r a t e i s d e t e r m i n e d by the a m p l i f i e r band-width, not the spectrum (as i n NMR). The N y q u i s t r a t e f o r a L o r e n t z i a n i s i n f i n i t e so t h e system band-width i s always the l i m i t i n g f a c t o r . F o r f u r t h e r d i s c u s s i o n of N y q u i s t r a t e s and ESR s p e c t r a see (219) (221) (27) . 228 e f f i c i e n t t o c o l l e c t l e s s d a t a w i t h a h i g h e r PSD time c o n s t a n t . The c o n c l u s i o n t o be drawn i s t h a t , f o r our system, 2000 p o i n t s i s the maximum s i z e f o r a spectrum. Though i n g e n e r a l more would be d e s i r a b l e . 21.6 FILTERING METHODS F i l t e r i n g (smoothing) methods have been f a i r l y e x t e n s i v e l y s t u d i e d (222)(223)(224)(225)(226)(227). Only two methods w i l l be d i s c u s s e d h e r e , box-car f i l t e r i n g (Sect.21.8) and ana l o g ( t i m e - c o n s t a n t ) f i l t e r i n g . An i m p o r t a n t p o i n t t o note i s , t h a t once a d a t a s e t i s a c q u i r e d , f i l t e r i n g does not improve the p r e c i s i o n of the d a t a (145), i t i s p u r e l y c o s m e t i c (i.e., i t o n l y improves the apparent SNR). F i l t e r i n g i s i m p o r t a n t as many a l g o r i t h m s a r e not v e r y s t a b l e i n the p r e s ence of n o i s e . A l s o s e a r c h i n g by eye i s d i f f i c u l t . The N y q u i s t r a t e , ( N ^ ) , f o r d a t a a c q u i r e d w i t h a low pass a m p l i f i e r i s N f = ( 2 i r r ) " 1 (21.1) where T i s the time c o n s t a n t ( i n v e r s e band-width) of the a m p l i f i e r . T h i s means t h a t we have the c h o i c e of a c q u i r i n g l a r g e amount of d a t a a t a low time c o n s t a n t (i.e. , n o i s y - d a t a ) and n u m e r i c a l l y f i l t e r i n g i t ( w i t h a box-car i n 229 t h i s c a s e ) , or of c o l l e c t i n g s m a l l e r amounts of f i l t e r e d d a t a {i.e., a t a h i g h e r time c o n s t a n t ; a n a l o g f i l t e r i n g ) Box-car f i l t e r i n g (228) j u s t c o n s i s t s of a v e r a g i n g a d j a c e n t p o i n t s (Sect.21.8) so c o n s i d e r the case of c o l l e c t i n g 'n' p o i n t s w i t h a SNR of S 0 and b o x - c a r r i n g t o *m' p o i n t s (n/m w i l l be i n t e g r a l ) , the SNR, S, i s then g i v e n by (229) w i t h a c o r r e s p o n d i n g decrease i n r e s o l u t i o n of n/m. Now the maximum number of m e a n i n g f u l p o i n t s a c q u i r e d f o r a t i m e c o n s t a n t r 0 i n a scan t i m e , T, i s from Eqn.21.1 S = So/fn/m) (21.2) n = N,T A T/TQ (21.3) I f we i n c r e a s e the time c o n s t a n t t o T then we get m' m e a n i n g f u l p o i n t s m' <* T/T (21.4) We c o u l d of c o u r s e a c q u i r e a t a r a t e f a s t e r than t h e N y q u i s t f r e q u e n c y , but no f u r t h e r i n f o r m a t i o n i s o b t a i n e d ; t h e d a t a 230 s h o u l d be a c q u i r e d a t a r a t e c o n s i s t e n t w i t h the a m p l i f i e r b a nd-width. Hence we get from Eqn.21.2-Eqn.21.4 S = ScVCrAc) = S0i/(n/m' ) (21.5) i . e . , b o x - c a r r i n g c o n f e r s no advantage over a n a l o g methods f o r a g i v e n scan t i m e . However, t h i s a n a l y s i s i g n o r e s two f a c t o r s , t h e d i s t o r t i o n caused by f i l t e r i n g and the n a t u r e of the n o i s e . Analog f i l t e r s s h i f t and broaden the peaks i n an asymmetric manner (30). B o x - c a r r i n g j u s t broadens the peaks and t h u s i s the more d e s i r a b l e of the two methods. The a n a l y s i s a l s o assumes t h a t the n o i s e i s G a u s s i a n . In p r a c t i c e ESR s p e c t r o m e t e r s produce a l a r g e amount of so c a l l e d 1/f n o i s e ( b a s e l i n e d r i f t and o f f s e t ) , which i n c r e a s e s w i t h scan t i m e . Thus, i t i s b e t t e r t o r a p i d l y scan s e v e r a l ( n o i s y ) s p e c t r a and box-car average, than t o do one l o n g scan w i t h a l a r g e time c o n s t a n t . The p e n a l t y f o r box-car a v e r a g i n g i s t h e a c q u i s i t i o n and m a n i p u l a t i o n of a l a r g e number of p o i n t s . T h i s can be overcome by r e a l - t i m e a v e r a g i n g . Memory and speed r e s t r i c t i o n s of the LSI-11 l i m i t the u s e f u l n e s s of t h i s a pproach f o r us, but i t would be w o r t h w h i l e p u r s u i n g i f a f a s t e r computer was a v a i l a b l e . In p r a c t i c e a compromise i s used, d a t a a r e a c q u i r e d a t the N y q u i s t r a t e u s i n g the maximum time c o n s t a n t c o n s i s t e n t 231 w i t h a d i s t o r t i o n f r e e spectrum (i.e., <l/50 of the time t o sweep the n a r r o w e s t f e a t u r e of i n t e r e s t (30)). The d a t a a re then b o x - c a r r e d on the LSI-11 down t o 2048, 1024 or 512 p o i n t s depending on the end use. 21.7 INTERPOLATION As mentioned p r e v i o u s l y the sweep i s not q u i t e l i n e a r , m a i n l y due t o n o i s e . The d a t a a r e t h u s not e q u a l l y spaced i n terms of the magnetic f i e l d . For most purposes t h i s i s of no consequence, p r o v i d i n g b oth the X and Y da t a a re used f o r a n a l y s i s . However, most a l g o r i t h m s , n o t a b l y i n t e g r a t i o n , e x e c u t e f a s t e r i f the da t a a re e q u a l l y spaced. Some a l g o r i t h m s , e.g. t h e FFT (25)(26) and S a v i t s k y - G o l a y smoothing (227) r e q u i r e the da t a t o be e q u a l l y spaced. For t h e s e reasons i t i s d e s i r a b l e t o i n t e r p o l a t e the da t a t o o b t a i n an e q u a l l y spaced d a t a s e t . Here, d a t a were i n t e r p o l a t e d and/or b o x - c a r r e d (vide supra) t o 512, 1024 or 2048 p o i n t s , depending on the end use. Powers-of-two a r e used t o m a i n t a i n c o m p a t i b i l t y w i t h FFT r o u t i n e s and the g r a p h i c s t e r m i n a l . A number o f i n t e r p o l a t i o n schemes a r e p o s s i b l e , L a g r a n g i a n i n t e r p o l a t i o n , c u b i c s p l i n e s , or a s l i d i n g p i e c e w i s e p o l y n o m i a l . 5 6 Lagrange i n t e r p o l a t i o n i s e x t r e m e l y 5 6 A n ' t h o r d e r p o l y n o m i a l i s f i t t e d t o p o i n t s ' i ' t o 'i+n' and the d a t a i n t e r p o l a t e d f o r t h i s range of p o i n t s . The p o l y n o m i a l i s t h e n f i t t e d t o the 'i+n' t o 'i+2n' p o i n t s etc.. A c u b i c s p l i n e i s a p i e c e w i s e c u b i c p o l y n o m i a l f i t w i t h the e x t r a c o n s t r a i n t t h a t the d e r i v a t i v e s a t the e n d - p o i n t s must be c o n t i n u o u s between a d j a c e n t ' p i e c e s ' (230). 232 u n s t a b l e (231) e s p e c i a l l y i n the presence of n o i s e . Cubic s p l i n e s a l s o show i n s t a b i l i t y i n the presence of n o i s e . S l i d i n g p o l y n o m i a l s can s e r i o u s l y d i s t o r t the spectrum a t l o w - d a t a d e n s i t i e s (232). We used a q u a d r a t i c p o l y n o m i a l and a v o i d e d low d a t a d e n s i t i e s . ( C u b i c s p r o v i d e d no improvement and i n c r e a s e the i n t e r p o l a t i o n t i m e ) . 21.8 BOX-CAR INTERPOLATION AND FILTERING T h i s i s a c o n v e n i e n t method of r e d u c i n g the s i z e of d a t a s e t s . I t can a l s o be used t o produce an e q u a l l y spaced d a t a s e t from an u n e q u a l l y spaced d a t a s e t . B r i e f l y , i t works as f o l l o w s , each (X,Y) d a t a p o i n t i s l o a d e d i n t o an element of an a r r a y (the b o x - c a r ) . The index of the a r r a y element i s d e t e r m i n e d by the range of v a l u e s t h a t X l a y s i n . For i n s t a n c e , a l l X v a l u e s 0<X<9 a r e a s s i g n e d t o element 1. For 10<X<19 they a r e a s s i g n e d t o element 2 etc. The Y v a l u e s a r e summed and l o a d e d i n t o the a r r a y element determined by the c o r r e s p o n d i n g X v a l u e s as d e s c r i b e d above. The number of p i e c e s of d a t a summed i n a p a r t i c u l a r box-car a r e s t o r e d i n a n o t h e r a r r a y ; the index numbers b e i n g d e t e r m i n e d as f o r the Y-data. When t h i s p r o c e s s i s complete the d a t a a r e averaged and empty b o x - c a r s f i l l e d by l i n e a r i n t e r p o l a t i o n or e x t r a p o l a t i o n . 5 7 (Note t h a t t h i s i s not t h e same as a ru n n i n g average, which i s a c o n v o l u t i o n w i t h a box-car (233).) 5 7 The f i r s t and l a s t few p o i n t s i n a sweep were o f t e n not c o l l e c t e d . These p o r t i o n s of the d a t a were z e r o - f i l l e d , f l a g g e d and i g n o r e d i n subsequent c a l c u l a t i o n s . 233 Generally the data were interpolated to 2n points then box-carred back to 2 points. Box-carring improves the apparent SNR by v/(n/m) where 'n' i s the i n i t i a l number of points and 'm' the f i n a l number. The penalty i s a resolution loss of (n/m). 5 8 21.9 PEAK SEARCHING AND FITTING Automatic peak f i t t i n g was achieved as follows. F i r s t the baseline was located (vide infra) and the noise l e v e l (standard deviation; 6) determined. The data were searched for peaks only using data further than 56's from the baseline to avoid spurious f i t s . The d i f f i c u l t i e s of distinguishing high-frequency noise mixed with the signal and the signal i t s e l f were avoided by stepping through the data at intervals of 5% of the sweep-width. Extrema were located by changes in the sign of the slope of the l i n e joining adjacent step points. This of course produces problems in locating peaks with widths less than 5% of the sweep-width. If the SNR i s low the step size can be reduced, but then d i f f i c u l t i e s in i d e n t i f i n g broad peaks occur (the slope change must be greater than the noise l e v e l ) . This method cannot be generalised to a l l types of spectra, but i t is extremely r e l i a b l e when t a i l o r e d for s p e c i f i c applications, notably spectra from nitroxides in membranes. 5 8Note that for a spectrum where the box-car i s much smaller than the line-width, as in our case, the e f f e c t on resolution can be ignored. This i s not true for the time constant, where the time-constant approaches the time taken to scan the l i n e . 234 Once a peak i s found i t can be f i t t e d ( l o c a l l y ) t o a c u b i c or q u a d r a t i c . Q u a d r a t i c s are more r e l i a b l e f o r broad peaks and c u b i c s f o r narrow p e a k s . 5 9 The f i t t i n g window i s set t o h a l f of the s t e p s i z e . I f the n o i s e l e v e l i s h i g h or the i n i t i a l f i t i s ambiguous the s i z e of the window i s a u t o m a t i c a l l y i n c r e a s e d . F u r t h e r d e t a i l s of the a l g o r i t h m and i t s performance can be found i n (88). The problems o u t l i n e d above are a l l e l i m i n a t e d by i n t e r a c t i v e f i t t i n g . The o p e r a t o r s e l e c t s the peaks and window s i z e via a g r a p h i c s t e r m i n a l . However, t h i s approach i s more time consuming and i n t r o d u c e s a degree of s u b j e c t i v i t y when v e r y n o i s y s p e c t r a a r e f i t t e d . 21.10 BASELINE FITTING AND FLATTENING For l i n e a r b a s e l i n e a r t e f a c t s added t o a peak f ' ( x ) we have an o b s e r v e d s i g n a l y' = f ' ( x ) + b(x) + c (21.6) where b i s the b a s e l i n e d r i f t and c i s the DC o f f s e t . I f we use the s u b s c r i p t s / and r t o denote the l e f t and r i g h t s i d e s of the spectrum then 5 9 That i s broad and narrow w i t h - r e s p e c t - t o the f i t t i n g window. The f i t t i n g window b e i n g the number of p o i n t s f i t t e d t o the peak, i n the v i c i n i t y of the peak. 235 f ul) b(x, ) + c = f b ( x r ) + c (21.7) I f the spectrum goes t o the b a s e l i n e i . e . , f 1 (x^ ) = f' (x^_ )=0 then we can d etermine b and c from the d a t a . However, t o meet t h i s c r i t e r i o n r e q u i r e s v e r y l a r g e d a t a s e t s , or low d a t a d e n s i t i e s , as the sweep-width must be a t l e a s t 5x the w i d t h of the b r o a d e s t l i n e i n the s p e c t r u m . 6 0 Most b a s e l i n e f l a t t e n i n g a l g o r i t h m s , e.g. (234), r e l y on h a v i n g l a r g e a r e a s of s i g n a l - f r e e spectrum a v a i l a b l e t o d e f i n e the b a s e l i n e , but i f one uses a s m a l l data s e t w i t h h i g h d a t a d e n s i t i e s t h i s w i l l g e n e r a l l y not be the c a s e . I f the s i g n a l has a c e n t e r of symmetry then we can use t h i s t o e s t a b l i s h the b a s e l i n e . We have Ay = y ; + y ; = f'U,) + r ( x r ) + b ( x , ) + b ( x r ) + 2c (21.8) we can choose our l e f t and r i g h t d a t a p o i n t s such t h a t , f o r a d e r i v a t i v e spectrum, f ' ( x ^ ) = - f ' ( x ), hence 6 0 I d e a l l y one c o u l d average the d a t a i n p r o p o r t i o n t o the s l o p e of the l i n e , but t h i s i s d i f f i c u l t t o do i n r e a l - t i m e . 236 Ay = b ( x z ) + b ( x r ) + 2c (21 .9) we can now use another s e t of symmetric p o i n t s t o s o l v e f o r b and c. S t r i c t l y we s h o u l d i t e r a t e as the spectrum c e n t e r i s a f u n c t i o n of the b a s e l i n e . However, f o r a s i n g l e l i n e , the c e n t e r of the spectrum i s s t e e p so t h a t the c r o s s - o v e r p o s i t i o n i s not v e r y s e n s i t i v e t o s m a l l b a s e l i n e changes. For dt c s t u d i e s , where a s i n g l e l i n e , scanned f a i r l y r a p i d l y , i s c o l l e c t e d , we can assume t h a t the d r i f t i s n e g l i g i b l e , hence we can determine the DC o f f s e t e a s i l y . The above methods l e n d t hemselves w e l l t o i n t e r a c t i v e g r a p h i c s methods of b a s e l i n e c o r r e c t i o n . The o p e r a t o r c o r r e c t s the d a t a f o r b and c via the t e r m i n a l c u r s o r keys. However, i t i s i m p o s s i b l e t o u n i q u e l y determine b and c by these methods and s p e c t r a w i t h l a r g e d r i f t s s h o u l d be r e j e c t e d . The s e n s i t i v i t y of t h i s approach can be improved c o n s i d e r a b l y by o b s e r v i n g the i n t e g r a l w h i l e c o r r e c t i n g the d e r i v a t i v e d a t a f o r b and c. The s e n s i t i v i t y t o changes i n 'b' and 'c' i s i n c r e a s e d by n and n 2 r e s p e c t i v e l y , where n i s the r e s o l u t i o n of the g r a p h i c s t e r m i n a l . 6 1 The a l g o r i t h m f o r i n t e r a c t i v e b a s e l i n e f l a t t e n i n g i s shown i n F i g . 2 1 . 3 6 1 0 n e c o u l d expand the b a s e l i n e , but t h i s i s not u s e f u l w i t h n o i s y s p e c t r a . 237 DISPLAY DERIVATIVE SPECTRUM INCRE-MENTALLY CORRECT SPECTRUM INTEGRATE SPECTRUM & DISPLAY INCRE-MENTALLY CORRECT SPECTRUM YES DISPLAY INTEGRATED SPECTRUM XlO INCRE-MENTALLY CORRECT SPECTRUM YES SAVE ABSORPTION SPECTRUM F i g u r e 21.3. Flow c h a r t f o r i n t e r a c t i v e b a s e l i n e f l a t t e n i n g . 21.11 INTEGRATION OF SPECTRA The i n t e g r a t i o n of ESR s p e c t r a and the a s s o c i a t e d e r r o r s has been d i s c u s s e d i n a number of papers (235)(236)(237) (238)(239) and w i l l not be re v i e w e d h e r e . A l l i n t e g r a t i o n s were performed u s i n g Simpsons r u l e w i t h an i n t e r p o l a t e d d a t a s e t . Care has t o be take n t o remove b a s e l i n e a r t e f a c t s as 238 the i n t e g r a l i s v e r y s e n s i t i v e t o them {vide supra). No s e r i o u s problems were e n c o u n t e r e d , except a t low d a t a d e n s i t i e s where random s t e p s a r e i n t r o d u c e d i n t o the spectrum ( F i g . 2 1 . 4 ) . J 1— Y X j 1 i F i g u r e 21.4. The e f f e c t of low d a t a d e n s i t y on i n t e g r a t i o n . T h i s e f f e c t i s d i s c u s s e d i n d e t a i l by P h i l l i p s and H e r r i n g (27). The e f f e c t of d a t a d e n s i t y on the average i n t e g r a t i o n e r r o r f o r t h r e e a l g o r i t h m s i s shown i n F i g . 2 1 . 5 - F i g . 2 1 . 7 . (The 'averaged Simpsons' i s j u s t t h e average of two Simpsons r u l e i n t e g r a t i o n s s t a r t i n g a t d i f f e r e n t d a t a p o i n t s ; see (27)). The d a t a d e n s i t y i s s i m p l y the number of p o i n t s per peak-to-peak l i n e - w i d t h . 1.5 1.0 0.5 o 0.0 ct: ii i -0.5 CD o -1.0 -1.5 --2.0 -2.5 •SIMPSONS RULE AflVERRGED SIMPSONS x T R A P E Z O I D A L RULE 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 DATA DENSITY F i g u r e 21.5. I n t e g r a t i o n e r r o r s vs. d a t a d e n s i t y f o r the i n t e g r a t i o n of a d e r i v a t i v e t o an a b s o r p t i o n s i g n a l . 1.5 1.0 0.5 D 0.0 Ql ce: LU -0.5h CD o -1.0 --1.5 --2.0 --2.5 •SIMPSONS RULE AAVERAGED SIMPSONS X T R A P E Z 0 I D A L RULE _ L 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 DATA DENSITY F i g u r e 21.6. I n t e g r a t i o n e r r o r s vs. d a t a d e n s i t y f o r the a r e a of an a b s o r p t i o n s i g n a l . 8.0 240 1.5 1.0 0.5 D 0.0 DC LU -0.5 CD o -1.0 -1.5 -2.0 -2.5 •SIMPSONS RULE AflVERflGED SIMPSONS XTRRPEZOIDRL RULE 0.0 1.0 2.0 _ i_ 3.0 4.0 5.0 DflTfl DENSITY 6.0 7.0 8.0 F i g u r e 21.7. I n t e g r a t i o n e r r o r s vs. d a t a d e n s i t y f o r the i n t e g r a t i o n of a d e r i v a t i v e t o t h e a r e a . Note t h a t t h i s e r r o r i s e n t i r e l y due t o low da t a d e n s i t y and not n o i s e or t r u n c a t i o n . A l s o n o t e t h a t such low d e n s i t i e s a r e not unu s u a l i n NMR and f o r some commercial ESR d a t a s t a t i o n s . 21.12 ADDITION AND SUBTRACTION OF SPECTRA The a d d i t i o n and s u b t r a c t i o n of ESR s p e c t r a i s n o n - t r i v i a l . The p o s i t i o n of the spectrum depends on the microwave f r e q u e n c y , t h e magnetic f i e l d and the r a d i c a l s g - s h i f t . A l s o the d a t a a r e not e q u a l l y spaced and the sweep-width and number of a c q u i r e d p o i n t s i s not c o n s t a n t . 6 2 ( S h i f t i n g 6 2 The sweep-width i s c o n s t a n t t o w i t h i n 1%, however t h i s can cause l a r g e a r t e f a c t s when s u b t r a c t i n g s p e c t r a . 241 s p e c t r a i s d e s c r i b e d i n more d e t a i l b e l o w ) . The unequal d a t a s p a c i n g and number of d a t a p o i n t s i s e a s i l y t aken c a r e of by i n t e r p o l a t i o n , as p r e v i o u s l y d e s c r i b e d . I f Gauss a r e used as the X-data then changes i n sweep-width and f i e l d a r e r e a d i l y d e a l t w i t h , one s i m p l y changes the two c a l i b r a t i o n parameters b e f o r e c o n v e r t i n g the d a t a t o Gauss. G e n e r a l l y the d a t a i s c o n v e r t e d back t o i n t e g e r s b e f o r e use and c a r e has t o be taken t o keep t r a c k of the d e s i r e d p o r t i o n of the spectrum. Compensating f o r microwave s h i f t s i s more d i f f i c u l t as the g - v a l u e s must be known. One can assume the g - f a c t o r s a r e the same and c o r r e c t a c c o r d i n g l y , but the b e s t approach i s via i n t e r a c t i v e g r a p h i c s . The s p e c t r a ( a f t e r i n t e r p o l a t i o n and matching the sweep-widths) a r e d i s p l a y e d and matched i n t e r a c t i v e l y . They can then be s u b t r a c t e d ( i n t e r a c t i v e l y ) a t one's l e i s u r e . However, one has t o be c a r e f u l i f the s p e c t r a have g - s h i f t d i f f e r e n c e s , the r e s u l t s can be m i s l e a d i n g . An example of t h i s method i s shown i n F i g . 2 1 . 8 - F i g . 2 1 . 1 0 242 1 r ir F i g u r e 21.8. Spectrum of the f r e e s p i n - p r o b e . »>B F i g u r e 21.9. Spectrum of f r e e and bound s p i n - p r o b e i n r e d b l o o d c e l l g h o s t s . 243 F i g u r e 21.10. Spectrum of a bound s p i n - p r o b e . Found by s u b t r a c t i n g the f r e e probe spectrum from the c o m b i n a t i o n spectrum above. 21.13 SHIFTING SPECTRA T h i s i s a g a i n n o n - t r i v i a l and as p o i n t e d out above can be m i s l e a d i n g . I t i s thus p e r t i n e n t t o d i s c u s s some of the u n d e r l y i n g t h e o r y . To f i r s t o r d e r the t r a n s i t i o n f r e q u e n c y i s g i v e n by ( w i t h A 0 i n the a p p r o p r i a t e u n i t s ) hu> = g o 0 B o m + A 0m m (21.10) S IS or i n terms of the l i n e - p o s i t i o n , B , 2 4 4 B Z = hco/q0p ( 2 1 . 1 1 ) where, because we do f i e l d swept e x p e r i m e n t s , a> i s the microwave f r e q u e n c y . I f we w i s h t o compensate a l i n e - p o s i t i o n f o r a new microwave f r e q u e n c y t o ' then B'z = ( g 0 c o ' / g o C j ) B 2 ( 2 1 . 1 2 ) I f we want t o s h i f t the whole spectrum, A = B Z ~ B Z , then A B = ^ [ c o ' / g o - c o / g o 3 ( 2 1 . 1 3 ) I f the g - v a l u e s a r e not known t h e y must be found by o t h e r means e.g. Hydes a l g o r i t h m (29). However, g - s h i f t s a r e g e n e r a l l y s m a l l and the r a t i o g 0/go can be s e t t o one. g - s h i f t s may be compensated f o r by the same f o r m u l a s ( w i t h «'=&>), but t h i s can be v e r y m i s l e a d i n g , u n l e s s i t i s known t h a t t h e s h i f t i s due t o s o l v e n t e f f e c t s , f o r i n s t a n c e . I f b o t h the microwave and g - v a l u e s a r e s h i f t e d then the c o r r e c t i o n i s most e a s i l y done i n t e r a c t i v e l y , but a s t h i s o b s c u r e s the u n d e r l y i n g a s s u m p t i o n s one has t o e s p e c i a l l y 245 c a r e f u l when u s i n g t h i s approach. The c e n t e r of t h e resonance, B 0 , does not n e c e s s a r i l y c o r r e s p o n d t o the c e n t e r of the sweep o r , more i m p o r t a n t l y , w i t h the c e n t e r of any o t h e r spectrum. In t h i s case the d a t a a r e c o r r e c t e d s i m p l y as f o l l o w s . B' = B + AB z z where A = I ' - I (21.14) 1 1 and I a r e the new and o l d v a l u e s f o r the i n t e r c e p t of the c a l i b r a t i o n . See S e c t . 1 3 . 9 . Changes i n sweep-width a r e compensated f o r by i n t e r p o l a t i o n . Assuming the change i s l i n e a r a c r o s s the spectrum, the d a t a a r e i n t e r p o l a t e d a t i n t e r v a l s of sweep/N, where N i s t h e number of i n t e r p o l a t e d p o i n t s and sweep i s t h e new sweep-width. The above c o r r e c t i o n p r o c e d u r e s a r e s u p e r f i c i a l l y t r i v i a l , however, they i m p l i c i t l y assume t h a t the d a t a a r e c o n t i n u o u s , whereas the d a t a a r e d i s c r e t e . T h i s causes problems because the d e s i r e d s h i f t s a r e r a r e l y i n t e g r a l m u l t i p l e s of the d a t a r e s o l u t i o n . Changing th e sweep-width i s even more p r o b l e m a t i c as t h a t i s e q u i v a l e n t t o a s h i f t t h a t v a r i e s a c r o s s the spectrum. The problem i s s o l v e d by c o n v e r t i n g the spectrum x-data t o Gauss and then i n t e r p o l a t i n g the spectrum t o get t h e p o i n t s c o r r e s p o n d i n g t o t h e d e s i r e d s h i f t e d / e x p a n d e d spectrum. The o r i g i n a l 2 4 6 spectrum s h o u l d have a l a r g e d a t a d e n s i t y as e r r o r s can accumulate r a p i d l y i f s e v e r a l o r , l a r g e , s h i f t s a r e done. A l s o c a r e must be taken t o adopt a c o n s i s t e n t s i g n c o n v e n t i o n f o r the d i r e c t i o n of the s h i f t s . Here the c o n v e n t i o n i s A=X'-X, where X' i s the c o r r e c t e d v a l u e and X i s the o r i g i n a l v a l u e and A i s the d e s i r e d s h i f t . APPENDICES 247 2 2 . APPENDICES 2 2 . 1 NOMENCLATURE For the purposes of t h i s t h e s i s m e t a l c a r b o d i t h i o a t e ( d i t h i o c a r b a m a t e ) complexes a r e denoted as f o l l o w s ; M R 2dtc, where M i s the c e n t r a l m e t a l and R i s an a l k y l or r i n g s u b s t i t u e n t on one end of the complex, which a r e as f o l l o w s . Me me t h y l Et e t h y l Py p y r o l l i d i n y l Oc n - o c t y l Mp m o r p h o l i n y l Hxm h e x a m e t h y l e n i m i n y l Ocm o c t a m e t h y l e n i m i n y l Od o c t a d e c y l Hence b i s ( d i e t h y l - N - c a r b o d i t h i o a t e ) 6 3 C u ( I I ) i s a b b r e v i a t e d t o 6 3 C u E t 2 d t c and b i s ( p y r o l i d i n y l - N - c a r b o d i t h i o a t e ) 6 3 C u ( I I ) i s a b b r e v i a t e d t o 6 3 C u P y d t c . The MR 2dtc a r e g i v e n the g e n e r i c t i t l e of d i t h i o c a r b a m a t e s , denoted dtc's, where M i s a d i v a l e n t m e t a l , u s u a l l y n i c k e l ( I I ) or c o p p e r ( I l ) . The 'im i n e ' nomenclature i s a r c h a i c and now o n l y r e f e r s t o an NH double bonded t o a c a r b o n . The c o r r e c t n omenclature i s ' a z a c y c l o a l k a n e ' , but t h e o l d e r nomenclature w i l l be 248 249 r e t a i n e d h e r e . 22.2 THE f H NMR SPECTRUM OF PYROLLIDINE The d e u t e r i u m spectrum of neat d 9 p y r o l l i d i n e was r e c o r d e d at 30.7MHz and 61.4MHz. A spectrum c o n s i s t i n g of t h r e e l i n e s was o b t a i n e d ; one l i n e due t o the amine d e u t e r o n and two peaks, s e p a r a t e d by 1.16ppm, due t o the methylene d e u t e r o n s . The low f i e l d peak was a s s i g n e d t o the p a i r of methylene groups a t t a c h e d t o the n i t r o g e n and the h i g h f i e l d peak a s s i g n e d t o the o t h e r ( r i n g ) m e t h y l e n e s . L i n e - w i d t h s were r e s o l u t i o n l i m i t e d (1Hz f o r t h e WH400 and 5Hz f o r t h e CXP200). From the K a r p l u s e q u a t i o n (240) we would expect the deu t e r o n c o u p l i n g around t h e r i n g t o be <0.2Hz and w i l l make no c o n t r i b u t i o n t o r e l a x a t i o n . The 1"N r e l a x e s t o o f a s t t o make any c o n t r i b u t i o n t o the spectrum. The two methylene peaks a r e s u f f i c i e n t l y w e l l s e p a r a t e d (l.76ppm f o r the n i c k e l s a l t ) t h a t o v e r l a p p r e s e n t s no problem (241). R e l a x a t i o n time f o r the neat p y r o l l i d i n e were r e c o r d e d at 310K and 323K on the CXP200 a t 30.7MHz and a r e shown i n Ta b l e 22.1 Temp R i n g methylene N methylene Amine d e u t e r o n 310 2.43 2.40 1.36 323 1.89 1.75 0.67 Ta b l e 22 . 1 . 2H r e l a x a t i o n t i m e s f o r neat d 9 p y r o l l i d i n e . Times a r e s e e s . The N methylene i s a d j a c e n t t o n i t r o g e n . 250 The r e l a x a t i o n times w i l l not be i n t e r p r e t e d here o t h e r than t o comment t h a t the r e l a x a t i o n t i m e s f o r the methylene d e u t e r o n s a r e c o n s i s t e n t w i t h a x i a l d i f f u s i o n of the p y r o l l i d i n e , as might be e x p e c t e d . The p r i m a r y use of t h e s e v a l u e s i s t o s e t the c y c l e time (5T,) f o r the i n v e r s i o n r e c o v e r y e x p e r i m e n t s . These v a l u e s p r o v i d e an upper l i m i t f o r t h e T, f o r the d i t h i o c a r b a m a t e complex, which b e i n g a l a r g e r m o l e c u l e w i l l have a s h o r t e r T,. 22.3 NMR SPECTRAL PARAMETERS The a s s i g n m e n t s f o r the NMR parameters a r e shown i n Fig.22.1 and F i g . 2 2 . 2 The c h e m i c a l s h i f t a n i s o t r o p y was measured from F i g . 2 2 . 3 . L i n e - p o s i t i o n s were a s s i g n e d from t h i s spectrum and the c r o s s o v e r s of the d i s p e r s i o n spectrum (not shown). The x component was a s s i g n e d from the d i p o l a r s p l i t t i n g which i s l a r g e s t a l o n g the C-N bond. 251 (9.4) (48.4) I I C H - C H 2 N-/ C r % C H 2 12.4 43.8 12.8* 48.9* (204) Q 206.2 o 208.8* ^ \ ^ ^ ^ P H 3 V C — N \ C H -F i g u r e 22.1. C h e m i c a l s h i f t v a l u e f o r n i c k e l dtc's i n ppm. 0.1M s o l n i n CDC1 3. E x t . TMS r e f . * denotes v a l u e s f o r a 1M 1 3 C e n r i c h e d K + E t 2 d t c i n D 20. ( ) denote s o l i d s t a t e v a l u e s , the CSA t e n s o r i s t r a c e l e s s . 126.8*138.8* n n C H r C H 2 13.0 7.0 15 13. / C H g C H , t N — G C><> s s 5.3 15, N / 7.2 I 1 C H r C H . \ C H i C H , t _ l _ J 3 3.3t F i g u r e 22.2. C o u p l i n g parameters (Hz) f o r n i c k e l dtc's. 0.1M s o l n . i n CDC1 3. f denot e s v a l u e s f o r 1M s o l n . of p o t a s s i u m d e r i v a t i v e . 1 5 N c o u p l i n g s a r e from 10" 3M s o l n . of 1 3 C , 1 5 N e n r i c h e d m e t h y l d e r i v a t i v e . ) F i g u r e 22.3. The powder spectrum f o r N i E t 2 d t c a t 50.3MHz. The arrow denotes the e x t e r n a l benzene r e f e r e n c e . 22.4 ESR SPECTRAL PARAMETERS The magnetic t e n s o r parameters f o r CuPyDtc a r e (44) A -119MHz g • - 2.022 XX 3 XX A -106MHz g -• 2.018 yy ^yy A zz -474MHz *zz '• = 2.088 A 0 = -233MHz g 0 = 2.043 253 22.5 COMPARISON OF REDFIELD AND OTHER THEORIES An a l t e r n a t i v e approach t o d e a l i n g w i t h f i r s t o r d e r w a v e - f u n c t i o n s i s t o do a va n - V l e c k t r a n s f o r m (242) so t h a t the p e r t u r b a t i o n terms a r e c a r r i e d by the o p e r a t o r s r a t h e r than the w a v e - f u n c t i o n s . The t r a n s i t i o n f r e q u e n c i e s a r e then s i m p l i f i e d ( w —> )• However, the s p e c t r a l d e n s i t i e s a r e m o d i f i e d , i.e. , j ( c o r e 5 ) -> j(co 0 )(1+f) (22.1) where w i s the f i r s t o r d e r t r a n s i t i o n f r e q u e n c y , co0 i s the z e r o o r d e r t r a n s i t i o n f r e q u e n c y and f i s the van-Vl e c k c o r r e c t i o n term. U s i n g a T a y l o r e x p a n s i o n of Eqn.22.1 w i t h co i t i s easy t o show t h a t f=co0co j (0 ) j (co) where co i s the res a a h y p e r f i n e c o u p l i n g f r e q u e n c y , i . e . , the f i r s t o r d e r c o r r e c t i o n s a r e p r o d u c t s of the reduced s p e c t r a l d e n s i t i e s . Comparison of our r e s u l t s w i t h o t h e r workers t o f i r s t o r d e r i s c o m p l i c a t e d by the d i f f e r i n g degrees of a p p r o x i m a t i o n used by the v a r i o u s w o r k e r s . However, i f we set co —> co0 and co —> 0 i n a l l c a s e s and drop a l l terms r e s u a ^ of o r d e r h i g h e r than oia/m0 (which a r e e q u i v a l e n t t o C 2 terms i n our work) then a comparison i s f e a s i b l e . Wherever a second o r d e r (C) term o c c u r s i n our e q u a t i o n t h e r e s h o u l d be a c o r r e s p o n d i n g f term i n the o t h e r t h e o r i e s . T h i s i s m o r e - o r - l e s s the c a s e , but g i v e n the d i f f e r e n c e s i n approach ( R e d f i e l d vs. l i n e a r - r e s p o n s e vs. s t o c h a s t i c L i o u v i l l e ) the 254 agreement cannot be e x p e c t e d t o be e x a c t . For i n s t a n c e i t i s not c l e a r which terms c o r r e s p o n d t o the c o r r e c t i o n f o r B 0 t o i n our t h e o r y . 22.6 HAMILTONIAN I_N A SPHERICAL BASIS The r e l e v a n t magnetic i n t e r a c t i o n t e n s o r s i n a s p h e r i c a l b a s i s a r e g i v e n by (90)(99) The ±1 elements a r e z e r o because the t e n s o r s a re symmetric. The r e l e v a n t second rank t e n s o r o p e r a t o r s a r e i ( g - g ) * ^xx yy A -4(A +A )] zz xx yy j v/f g - i ( g +g ) 255 22.7 NOTES ON UNITS FOR ESR R e l a x a t i o n t h e o r i e s a r e u s u a l l y d e r i v e d i n energy u n i t s ; ESR s p e c t r a a r e measured i n Gauss. T h i s can l e a d t o c o n f u s i o n (what a r e the u n i t s of T, ? (243)) and programming problems. For t h i s reason a b r i e f d i s c u s s i o n of u n i t s used i n t h i s work i s g i v e n below. I f H i s the h a l f - w i d t h a t h a l f h e i g h t f o r L o r e n t z i a n l i n e 6 3 the peak-to-peak w i d t h of the d e r i v a t i v e i s thus H = ^ § A H (22.2) 2 pp as H = ( 7 r T 2 ) _ 1 we then get T i 1 = TT^|AH (22.3) 2 2 pp For T, t o be i n the u s u a l u n i t s (sec r a d ~ 1 ) A H must be i n PP Hz. To c o n v e r t from Gauss we u s e 6 * 6 3 D I S P A shows t h a t the dt c l i n e s a r e v e r y c l o s e t o L o r e n t z i a n . 6"Our experiment i s f i e l d swept so t h i s c o n v e r s i o n i s not s t r i c t l y v a l i d . However, t h i s c a u s e s no problems, even i n the s l o w - m o t i o n a l regime (39)(244). 256 hv = g/JB hence ?(Hz) = h~ 1 g/3B (Gauss) (22.4) w i t h the Bohr magneton, /3 and P l a n c k s c o n s t a n t , h, d e f i n e d i n t h e i r u s u a l u n i t s . Hence we get AH (Hz) = A-'go/JAH (Gauss) PP PP and thus T i 1 = ir^q0ph' 1 AH (Gauss) (22.5) For the l i n e - p o s i t i o n s ( i g n o r i n g the s m a l l f i r s t o r d e r c o r r e c t i o n ) we have ? z ( H z ) = h~ 1g^B^(Gauss) (22.6) and f o r the g a n i s o t r o p y (Ag) Af (Hz) = /r 1g/3B (Gauss) (22.7) S z note t h a t B^ i s the p o s i t i o n of the l i n e of i n t e r e s t , which i s not the same as B 0 so Av depends on l i n e - p o s i t i o n . 257 F i n a l l y the h y p e r f i n e s p l i t t i n g i s g i v e n by A 0(MHz) = 2 . 8 0 4 7 ( g / g g ) a o ( G a u s s ) (22.8) where qg i s the f r e e - e l e c t r o n g - v a l u e . The h y p e r f i n e s p l i t t i n g c o n s t a n t may be o b t a i n e d from the spectrum t o a l l o w f o r any temperature dependence, but i t o n l y appears i n the second o r d e r terms so t h e l i t e r a t u r e v a l u e i s adequate. D i m e n s i o n a l a n a l y s i s of Eqn.12.22 g i v e s us t h a t the u n i t s of j ( c o ) i s r a d s _ 1 and R i s sec r a d " 1 . S u b s t i t u t i n g i n t y p i c a l v a l u e s we f i n d t h a t j i s =1nS and R ^ I G r a d s " 1 . I t i s thus s e n s i b l e t o use nS and GHz as our base u n i t s . These may be used d i r e c t l y i n the above e q u a t i o n s w i t h no m o d i f i c a t i o n s . In f a c t i t i s e s s e n t i a l t o use t h e s e u n i t s t o a v o i d f l o a t i n g - p o i n t o v e r f l o w i n the program. The co* term i n Eqn.12.23 i s e s p e c i a l l y t r o u b l e s o m e . 22.8 ON PYROLLIDINE RING PUCKER In the s o l i d s t a t e N i P y d t c i s p u c k e r e d by some 0.03nm out of the p l a n e of the m o l e c u l e (183), t h e p o l a r a n g l e of t h e C-D bond i n the r i n g methylenes i s t h u s ±11° d i f f e r e n t from the v a l u e f o r t h e p l a n a r r i n g . T h i s means t h a t the r i n g may ' r i p p l e ' on the time s c a l e of the NMR r e l a x a t i o n . I f the r i n g ' r i p p l e s ' much f a s t e r than the r e l a x a t i o n t ime then the p o l a r a n g l e averages out and the p l a n a r a p p r o x i m a t i o n i s v a l i d . (There i s some e v i d e n c e t h a t t h e r i p p l e i s f a s t on 258 the ESR time scale (245) and hence also on the NMR time scal e ) . If the the ring is e s s e n t i a l l y r i g i d {i.e., the act i v a t i o n energy for the ring f l i p i s high) on the NMR time scale the relaxation times are a superposition of the the times for each ring conformation. However, as sine and cosine functions are f a i r l y linear in between 20° and 40° (the polar angles required) the results once again average out (within experimental error) to the planar case. If the 'ripp l e ' i s on the same time scale as the relaxation (=1s) then the problem i s complex and the fluctuation w i l l contribute to the ove r a l l relaxation of the molecule. However, we can readily test for t h i s effect as we have an over-determined system. We just have to check for consistency between the two deuterium relaxation results and the two corresponding ESR reduced spectral densities. The results at both 310 and 323K are entire l y consistent with each other, the o r i g i n a l (planar) approximation i s thus v a l i d . 22.9 THE FAST-MOTIONAL LIMIT Redfield theory i s only, v a l i d in the fast motional l i m i t , i.e. , the molecular motion must be on a time scale much faster than the relaxation time, R'1>>T2, where R i s the smallest element of the di f f u s i o n tensor. If R~**TZ then the motion i s said to be in the slow-motional regime and Redfield theory breaks down. The above equation reduces to R » A H /200, where R i s in Grad s _ 1 and AH i s the PP PP 259 peak-to-peak line-width, in Gauss, of the broadest l i n e in the spectrum. In th i s and previous work (88) the theory breaks down (i.e. , j(co) or the SR term become negative) when AH^>20G. For an arb i t r a r y l i m i t of 10% t h i s equation corresponds to an R of 1 Grad s~ 1 as found by the approximation methods in Sect.15. for the cases where j (CJ)=0. 260 22.10 ESR LINE-WIDTH DATA P e a k - t o - p e a k l i n e - w i d t h d a t a f o r 6 3 C u P y d t c i n t o l u e n e . Mean m i c r o w a v e f r e q u e n c y was 9.06 GHz. Mean l i n e - p o s i t i o n s ( h i g h t o low f i e l d ) were 3270G, 3192G, 3115G a n d 3039G. Temp ° C W i d t h i n G a u s s -44.3 3.62 5.45 10.74 19.99 -39.0 3.35 4.99 10.06 19.12 -35.4 3.28 4.85 9.54 16.57 -34.7 3.30 4.76 9.01 16.11 -30.0 3.12 4.58 8.44 15.46 -25.0 3.09 4.35 7.76 13.58 -20.0 2.95 4.18 7.32 12.43 -15.1 2.99 4.10 6.89 1 1 .53 -10.5 3.00 4.12 6.95 1 1 .46 -5.1 3.04 4.01 6.07 10.47 0.0 2.97 3.90 6.11 10.21 4.0 3.06 3.96 5.99 9.43 9.8 3.00 3.91 5.90 9.18 19.9 3.14 3.84 5.69 8.48 30.4 3.23 3.96 4.04 7.70 35.9 3.32 4.15 6.25 7.59 40.9 3.40 4.15 5.37 7.57 46.5 3.23 4.16 5.29 7.49 50. 1 3.50 4.23 5.49 7.33 55.6 3.71 4.30 5.56 7.13 61 .0 3.83 4.45 5.41 7.13 T a b l e 22.2. L i n e - w i d t h d a t a f o r C u P y d t c i n c h l o r o f o r m . W i d t h s i n o r d e r o f f i e l d - p o s i t i o n . H i g h - f i e l d f i r s t . 261 22.11 NMR RELAXATION DATA. DEUTERIUM D e u t e r i u m T, r e l a x a t i o n d a t a d a t a f o r d 9 N i P y d t c i n c h l o r o f o r m . D e l a y s a r e i n s e c o n d s u n l e s s o t h e r w i s e n o t e d . A m p l i t u d e s a r e i n a r b i t r a r y u n i t s . N d e n o t e s t h e m a g n e t i s a t i o n a m p l i t u d e o f t h e two m e t h y l e n e s a d j a c e n t t o t h e n i t r o g e n . R d e n o t e s t h e m a g n e t i s a t i o n a m p l i t u d e f o r t h e o t h e r two m e t h y l e n e s . T i s t h e t e m p e r a t u r e . D e l a y 0.02 0.03 0.05 0.07 0.10 0.13 0.15 0.18 0.25 0.50 1 .00 1 .50 -8.43 -7.49 -3.88 -1 .60 1 .60 4.10 5.80 7.70 1 0.52 14.61 15.91 15.60 N R -10.42 -9.88 -6.21 -4.20 -1 .35 1 .50 2.90 4.80 8.70 14.85 16.80 17.80 T a b l e 2 2 . 3 . T=310K. 61.4MHz. D e l a y 0.02 0.03 0.05 0.07 0.10 0.13 0.15 0.18 0.25 0.35 0.50 0.70 1 .20 1 .70 -7.39 -6.69 -3.70 -2.14 0.50 2.40 3.60 4.10 8.11 10.21 12.50 13.32 13.60 13.27 N -9.00 -8.70 -5.80 -4.51 -2.05 0.0 1.10 2.65 5.80 8.85 12.20 13.90 15.12 14.90 R T a b l e 22.4. T=323K. 61.4MHz. 262 Delay 0.03 0.05 0.10 0.15 0.20 0.25 0.30 0.40 0.50 0.70 N -11.60 -6.80 -1 .20 3.10 7.60 1 1 .00 13.20 16.40 18.50 20. 10 R -13.40 -8.80 -3.60 0.0 3.50 7.00 9.10 12.80 15.60 18.30 T a b l e 22.5. T=333K. 61.4MHz Delay N R 40.00 -4.00 -5.20 60.00 -2.80 -3.20 80.00 -1.10 -2.70 100.00 0.0 -1 .55 120.00 0.70 -0.90 140.00 1 .50 0.0 160.00 2.10 0.60 180.00 3.00 1 .30 200.00 3.90 1 .50 220.00 4.20 2.50 240.00 4.80 3.10 260.00 5.20 3.30 280.00 5.80 4.00 300.00 5.95 4.30 T a b l e 22.6. T=310K. 30.7MHz. T i m e s i n mS, D e l a y 20.00 30.00 70.00 90.00 1 10.00 120.00 150.00 200.00 300.00 N -8.80 -6.00 -2.30 -1 .70 0.50 0.70 2.50 4.90 7.80 R -9.20 -9.20 -5.40 -3.20 -2.00 -1 .60 0.0 2.50 5.80 T a b l e 22.7. T= 310K. 30.7MHz. T i m e s i n mS. D e l a y N R 40.00 -6.40 -7.70 60.00 -4.75 -6.35 80.00 -2.75 -4.50 100.00 -1 .80 -4.00 120.00 0.50 -2.40 140.00 1.10 -1 .60 160.00 1 .60 -0.70 180.00 2.50 0.40 200.00 3.50 0.60 220.00 4.10 1 .30 240.00 4.80 1 .90 260.00 5.50 2.90 280.00 6.10 3.35 300.00 6.60 4.20 Table 22.8. T=323K. 30.7MHz. T i m e s i n 22.12 NMR RELAXATION DATA. 1 3 C 1 3 C T, r e l a x a t i o n d a t a d a t a f o r N i E t 2 d t c i n c h l o r o f o r m , s u b s t i t u t i o n on t h e C S 2 m o i e t y . D e l a y s a r e i n s e c o n d s . A m p l i t u d e s a r e i n a r b i t r a r y u n i t s . T i s t h e t e m p e r a t u r e , T i m e Amp. 1 .0 1 .1 2.0 1 .7 3.0 2.1 4.0 2.8 5.0 3.7 7.0 4.6 9.0 5.6 15.0 8.7 20.0 10.5 80.0 17.4 i . T=310K. 50.3MHz. T i m e 2.0 5.0 10.0 15.0 100.0 Amp. 3.40 4.90 1 1 .75 15.15 20.20 Table 22.10. T=310K. 100.7MHz T i m e Amp. 5.0 2.40 10.0 4.15 15.0 5.40 20.0 6.40 25.0 7.00 30.0 7.50 35.0 7.90 40.0 8.20 50.0 8.50 150.0 9.00 Table 22.11. T=323K. 50.3MHz. T i m e 2.0 5.0 10.0 15.0 20.0 30.0 100.0 Amp. 4.55 8, 13, 17, 19, 21 , 24, 65 95 40 30 40 72 T a b l e 22.12. T=323K. 100.7MHz, T i m e 1.0 2.0 3.0 4.0 5.0 7.0 9, 15, 20, 30, 100, .0 ,0 0 0 ,0 Amp. 1 .55 2.70 3.90 5.15 5.90 7.50 9.20 12.60 14.30 16.00 17.10 T a b l e 22.13. T=333K. 50.3MHz. T i m e 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 50.0 150.0 Amp. 8.55 1 1 .90 13.90 15.00 16.00 16.40 16.85 16.90 17.05 17.40 T a b l e 22.14. T=333K. 50.3MHz. Time Amp. 2.0 3.70 5.0 7.45 10.0 1 1 .25 15.0 13.70 20.0 14.80 30.0 15.70 100.0 19.95 Table 22.15. T=333K. 100.7MHz. REFERENCES 1. K.S.Cole and R.H.Cole, J.Chem.Phys. 9, 341, (1941) 2. C.P.Poole, "Electron Spin Resonance 2nd ed.", Chap.12H, W i l e y , 1983 3. M.A.Hemminga, J.H.Reinders and P.A.De J a g e r , J.Magn. Reson. 58, 428, (1984) 4. A.Abragam, "Principles of Nuclear Magnetism", C h a p . I l l , O.U.P., 1983 5. A . G . M a r s h a l l and D.C.Roe, J. Magn.Reson. 33, 551 , ( 1978) 6. A . G . 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