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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 U n i v e r s i t y of Sussex, M.Sc. The U n i v e r s i t y  1974  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 t h i s t h e s i s as conforming to t h e ^ e q u ^ r e d standard  THE UNIVERSITY OF BRITISH COLUMBIA JUNE 1985 ©  P.S.PHILLIPS, 1985  In p r e s e n t i n g  this  thesis  in  partial  fulfilment  requirements for an advanced degree at the The B r i t i s h Columbia, I agree that freely  available  for  the  reference  Library and  this  scholarly  by  the  Department  or  understood  that  f i n a n c i a l gain  by  may his  copying  shall  not  be or  granted her  be  allowed  DEPT. OF CHEMISTRY  Date: JUNE 20th  1985  representatives.  it  agree  thesis Head  of  of It  for my is  or p u b l i c a t i o n of t h i s t h e s i s f o r  permission.  The U n i v e r s i t y of B r i t i s h 2036 Main M a l l , Vancouver, Canada V6T 1Y6  make  study. I f u r t h e r of  the  University  shall  that permission f o r e x t e n s i v e copying purposes  of  Columbia  without  my  written  Abstract A new  numerical  vs.  a n a l y s i s method, d i s p e r s i o n  absorption  p l o t s ( D I S P A ) , has  method may  be u s e d 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  and  been d e v e l o p e d  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  addition  i t provides  f o r ESR.  analytical  a method of a u t o m a t i c  This studies  tool.  phasing  In  for  m a g n e t i c r e s o n a n c e s p e c t r a . Numerous e x a m p l e s o f i t s a p p l i c a t i o n s , b o t h s i m u l a t e d and w i t h e m p h a s i s on and  processing  briefly  spin-probe  experimental  s t u d i e s . The  methods used f o r t h e s e  are  digital  presented, acquisition  s t u d i e s are  also  discussed.  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 h a v e been p e r f o r m e d . The  T,'s  of  c o m p l e x were m e a s u r e d by NMR.  The  c o m p l e x were m e a s u r e d by ESR t h e o r y . The principal  two  and  1 3  C  and  enriched 6 3  l i n e - w i d t h s of analysed  nickel  Cu  by R e d f i e l d  rotational diffusion  the p y r o l l i d i n e d e r i v a t i v e i n toluene.  a diffusion  H  s e t s o f r e s u l t s were c o m b i n e d t o g i v e  e l e m e n t s of the  t i m e t h a t ESR  2  spin-probes  and  NMR  This  tensor  i s the  the for  first  s t u d i e s have been combined t o measure  tensor. A general  strategy f o r t h i s approach i s  presented. ESR light  data  from p r e v i o u s  o f t h e new  work has  r e s u l t s . The  been r e - a n a l y s e d  a n a l y s i s shows t h a t  commonly u s e d 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 misleading.  ii  is  in  the  the extremely  Table of Contents  Abstract L i s t of T a b l e s L i s t of F i g u r e s Acknowledgements  i i x i i xiv xix PART  1.  DISPERSION V S . ABSORPTION PLOTS:  DISPA 1  1.  2.  INTRODUCTION  2  1.1  3  THE B A S I C THEORY OF DISPA  5  2.1  The DISPA C i r c l e  5  2.2  Lorentzian  Lines Distributed  2.3  The E f f e c t  o f S a t u r a t i o n on L o r e n t z i a n  2.4  D i s t r i b u t i o n i n Linewidths of L o r e n t z i a n Lines D i s t r i b u t i o n i n Resonant Frequency of Lorentzian Lines  11  Distribution Amplitude  13  2.5 2.6  3.  A Brief History  i n Amplitude Lines  8 ..8  10  i n Resonant F r e q u e n c y and  2.7  Modulation  Broadening  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 D y s o n i a n L i n e  16  THE HILBERT TRANSFORM AND DATA PRESENTATION  18  3.1  Generating  18  3.2  Pre-Processing  3.3  The D i f f e r e n c e P l o t  23  3.3.1  The I n d e x 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  29  the Dispersion  Spectrum  o f t h e S p e c t r u m f o r t h e FFT  iii  Plot  ...21  3.3.3 3.4 4.  5.  6.  Difference Plot  30  The Gaussian D i f f e r e n c e P l o t  33  INSTRUMENTAL DIAGNOSTICS AND APPLICATIONS  34  4.1  Time Constant  34  4.2  Noise  35  4.3  Baseline Artefacts  36  4.4  A m p l i f i e r Phasing  37  4.5  Microwave-Bridge Phasing  38  4.6  Saturation  38  4.7  Modulation  39  4.8  Line T r u n c a t i o n and Padding  40  THE AUTOMATIC PHASING OF SPECTRA  43  5.1  Basic Theory of Phase C o r r e c t i o n  43  5.2  Use of DISPA p l o t s f o r Phase C o r r e c t i o n  45  APPLICATIONS TO LINE SHAPE ANALYSIS IN LIQUIDS  52  6.1  Classification  54  6.2  Notes on the S i m u l a t i o n s  6.3  D e t e c t i n g Two Superimposed L o r e n t z i a n L i n e s  6.4  D e t e c t i n g Two Superimposed G a u s s i a n L i n e s  6.5  D e t e c t i n g Two Overlapping  Lorentzian Lines  6.6  D e t e c t i n g Two Overlapping  Gaussian L i n e s  65  6.7  D e t e c t i n g Combinations of L o r e n t z i a n and Gaussian L i n e s  67  D e t e c t i n g and Measuring Unresolved Couplings  69  6.8 6.9 7.  The Absorption  of Lobes and P l o t s  56  ....59 ...62  Hyperfine  A p p l i c a t i o n s t o Line-Shape A n a l y s i s of Solids  EXPERIMENTAL EXAMPLES  ..57  71 73  iv  7.1  8.  Temperature Dependence of Unresolved Hyperfine  73  7.2  Mixtures  75  7.3  Unresolved  7.4  Using DISPA p l o t s t o Detect S a t e l l i t e s  7.5  The D e t e c t i o n of Chemical  of Spin Probes Hyperfine Coupling Constants  76 77  Exchange.  Solvation Effects  78  7.6  The Spectrum of Grey P i t c h  79  7.7  G r a p h i t e Spectra  80  7.8  Coal Spectra  81  7.9 7.10  Wood Spectra N i t r o x i d e s i n the Slow-Motional Powder Spectra  84 Regime and 86  CONCLUSIONS  88  8.1  Summary of R e s u l t s  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 S u b s t i t u e n t s  101  9.4  Choice  of C e n t r a l Metal  102  v  10.  11 .  9.4.1  C e n t r a l M e t a l f o r ESR E x p 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  ....103  Experiments  GENERAL THEORY  104  10.1  Introduction  t o R e d f i e l d Theory  10.2  On S p e c t r a l D e n s i t i e s  111  10.3  C h o i c e of t h e A x i s System  113  10.4  H y d r o d y n a m i c M o d e l s f o r R o t a t i o n a l D i f f u s i o n 116  GENERAL EXPERIMENTAL 11.1  105  119  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 Carbodithioates  119  11.2  Transition Metal Dithiocarbamates  121  11.3  Preparation  122  of S o l u t i o n s  PART  3.  ELECTRON S P I N RESONANCE STUDIES 124 12.  ESR THEORY  1 25  12.1  The I s o t r o p i c ESR S p e c t r u m  125  12.2  The ESR P r o b l e m : Equation  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 M o d e l f o r an A s y m m e t r i c Rotor  140  Spin R o t a t i o n a l Relaxation  141  12.5 13.  Development of t h e R e d f i e l d  ESR EXPERIMENTAL  144 vi  Preparation of  13.2  Preparation of Copper(II) Dithiocarbamate Complexes  13.3  14.  6 3  13.1  Copper(II) Chloride  Preparation of Copper-free Nickel  144 144  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 o f t h e s o l u t i o n s f o r ESR  146  13.6  ESR Sample T u b e s  146  13.7  R e c o r d i n g ESR S p e c t r a  147  13.8  T e m p e r a t u r e M e a s u r e m e n t i n ESR e x p e r i m e n t s  13.9  F i e l d C a l i b r a t i o n o f ESR S p e c t r a  150  13.10 C o l l e c t i o n a n d A n a l y s i s o f ESR S p e c t r a  151  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  for the  Spin Hamiltonian 14.2  ..150  152  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 T ) « \  Approximation  155  14.2.2 The {(ji T ) «\  Approximation  156  Z  0  C  2  a  c  14.2.3 The w « c j a  0  Approximation  156  14.3  C o n t r i b u t i o n s From t h e N u c l e a r Zeeman Term  14.4  The F i r s t  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 P a r a m a g n e t i c  159  and Second Order  Contribution  Broadening  ..157 158  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 M a g n e t i c  161  Field  Inhomogeneity  14.5.7 S p e c t r o m e t e r P h a s i n g vii  161  14.5.8 Time C o n s t a n t  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 A s s u m p t i o n  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  Using the Approximations  169  15.4  I n v e r s i o n of Data w i t h t h e A x i a l Approximation  170  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  15.5 15.6  I n t e r p r e t i n g Data  15.7  Conclusions  from I s o t r o p i c  I n v e r s i o n s .174 178  PART 4. NMR  STUDIES 180  NMR THEORY 16.1  16.2  Chemical  181 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 t h e CSA Term  183  Quadrupolar  184  Relaxation vi i i  16.3  Spin R o t a t i o n a l Relaxation  186  16.4  C h o i c e o f T, E x p e r i m e n t  187  16.4.1 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 16.4.2 The S a t u r a t i o n R e c o v e r y E x p e r i m e n t 16.4.3 I n v e r s i o n R e c o v e r y v s . S a t u r a t i o n Recovery 17.  18.  NMR  EXPERIMENTAL  ....187 ...188 189 190  17.1  Preparation  17.2  NMR  17.3  Powder S p e c t r a  190  17.4  T, M e a s u r e m e n t s  191  17.5  A n a l y s i s o f NMR  NMR  o f t h e s o l u t i o n s f o r NMR  sample t u b e s  190 190  Data  193  ERROR DISCUSSION  195  18.1  On A p p r o x i m a t i n g t h e S p e c t r a l D e n s i t i e s  195  18.2  Residual  195  Contributions  to Relaxation  18.2.1 I n t e r m o l e c u l a r  Dipolar Relaxation  ....196  18.2.2 I n t r a m o l e c u l a r  Dipolar Relaxation  ....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  19.  ..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, M e a s u r e m e n t s  199  NMR 19.1  RESULTS AND DISCUSSION 1 3  C  Results  201 201  19.2  D e u t e r i u m T, R e s u l t s  202  19.3  Discussion  203  ix  PART 5. COMMENTS ON THE COMBINED NMR-ESR STUDIES 204 20.  COMBINED ESR AND NMR RESULTS AND DISCUSSION  205  20.1  Introduction  205  20.2  Comments on D a t a I n v e r s i o n  205  20.3  The D i f f u s i o n T e n s o r  206  20.4  The H y d r o d y n a m i c M o d e l  208  20.5  Summary o f t h e R e s u l t s  211  20.6  A Strategy Tensors  212  20.7  f o r Measurement o f D i f f u s i o n  F i n a l Remarks  217  PART 6. NOTES ON THE DIGITAL ACQUISITION OF ESR SPECTRA 219 21.  THE D I G I T A L ACQUISITION  OF ESR SPECTRA  220  21.1  Introduction  220  21.2  The H a r d w a r e  222  21.3  The B a s i c P r o b l e m s 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. o f P o i n t s C o l l e c t e d . The N y q u i s t Criterion  227  21.6  F i l t e r i n g Methods  228  21.7  Interpolation  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  233  21.10  Baseline  21.11  I n t e g r a t i o n of Spectra  21.12  A d d i t i o n and S u b t r a c t i o n  21.13  Shifting  and F i t t i n g  Fitting  and F l a t t e n i n g  234 237  of Spectra  240  Spectra  243  APPENDICES 247 22.  APPENDICES  248  22.1  Nomenclature  248  22.2  The H NMR  22.3  NMR  S p e c t r a l Parameters  250  22.4  ESR S p e c t r a l P a r a m e t e r s  252  22.5  Comparison o f R e d f i e l d and Other T h e o r i e s  22.6  Hamiltonian  22.7  N o t e s 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  Ring Pucker  257  22.9  The F a s t - m o t i o n a l  22.10  ESR L i n e - w i d t h  Data  260  22.11  NMR  Relaxation  Data. Deuterium  261  22.12  NMR  Relaxation  Data.  2  Spectrum of P y r o l l i d i n e  249  i n a Spherical Basis  254  Limit  References  1 3  C  ...253  258  -  264 267  xi  L I S T OF TABLES 7.1 T e m p e r a t u r e d e p e n d e n c e o f t h e u n r e s o l v e d hyperfine 7.2 I d e n t i f i c a t i o n  o f a n d n o t e s on t h e c o a l  samples  84  8.1 Summary o f r e s u l t s 11.1  f o r s i m p l e DISPA p l o t s  Microanalyses for Dithiocarbamates  11.2 S o l u b i l i t i e s o f m e t a l 12.1  12.2 The M a t r i x E l e m e n t s  dtc's  123  Spectrum Recording C o n d i t i o n s  14.1  Effect  139 148  o f h y p e r f i n e on c o r r e l a t i o n  times  o f P h a s e on O b s e r v e d L i n e - w i d t h s  Spectral densities  15.2 ESR d a t a 15.3  133  f o r R e d f i e l d Theory  13.1  15.1  88 122  Spectral density Frequencies  14.2 E f f e c t  74  161 162 166  inverted with approximations  .....170  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  toluene  171  15.4 A x i a l a p p r o x i m a t i o n u s e d 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 u s e d w i t h CuMeOddtc  172  15.6  Comparison of T ' S  174  C  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 19.1 C T,'s  178 201  1 3  19.2 D e u t e r i u m  T/s  202  19.3  The d i f f u s i o n t e n s o r f r o m  20.1  The d i f f u s i o n t e n s o r  20.2 F r i c t i o n c o e f f i c i e n t s 20.3  1 3  C  2  and H d a t a  203 207  f o r the probe  Predicted diffusion coefficients xi i  210 ..210  22.1  2  H r e l a x a t i o n times f o r neat d  9  pyrollidine  249  22.2  L i n e - w i d t h data f o r CuPydtc i n c h l o r o f o r m  ...260  22.3  T=3 1 OK. 61 . 4MHz  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  261  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  22.15  T=333K. 100.7MHz  '  265 266  xi i i  L I S T OF FIGURES 2.1 T y p i c a l  Cole-Cole format DISPA p l o t  8  2.2 DISPA f o r superimposed L o r e n t z i a n L i n e s  10  2.3 DISPA c i r c l e  12  in polar  coordinates  2.4 DISPA 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  13  2.5 DISPA p l o t f o r a p o o r l y phased l i n e . 0=10°  16  2.6 The Dysonian Line-shape  17  3.1 Flow-chart f o r g e n e r a t i n g the d i s p e r s i o n 3.2 Diagrammatic d e f i n i t i o n of a d i f f e r e n c e  data plot  23 25  3.3 Diagrammatic d e f i n i t i o n of the lobe parameters  26  3.4 L i n e a r and square root plots  27  indexed  difference  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 difference plots  28  3.6 Logarithmic indexed p l o t as a line-width/sweep-width.  29  3.7 P o l a r d i f f e r e n c e  plot  3.8 The a b s o r p t i o n d i f f e r e n c e 3.9 T y p i c a l  function  30 plot  Cole-Cole and d i f f e r e n c e  4.1 E f f e c t of a l a r g e  PSD f i l t e r  31 DISPA p l o t  on a DISPA p l o t  4.2 E f f e c t of noise on a DISPA p l o t 4.3 DISPA p l o t showing the e f f e c t of  35 36  baseline  artefacts 4.4 DISPA p l o t  32  37 f o r a mis-phased microwave b r i d g e  38  4.5 DISPA p l o t f o r an overmodulated l i n 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  40 40  4.7 DISPA p l o t f o r a t r u n c a t e d l i n e with unresolved hyperfine  41  xiv  4.8 E f f e c t of v a r i o u s padding schemes on the DISPA plot  42  5.1 R a d i a l d i f f e r e n c e angles  46  5.2 D i f f e r e n c e p l o t of phase angle  plots  for various  lobe asymmetry as a  function 46  5.3 Flow c h a r t f o r the automatic phase of s p e c t r a 5.4 Phase e r r o r  phase  as a f u n c t i o n  correction 48  of l i n e - p o s i t i o n  49  5.5 The c e n t e r l i n e f o r Fremies s a l t 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 ) automatic phase correction  50  5.6 The r a d i a l d i f f e r e n c e the diagram above  50  p l o t corresponding to  5.7 An u n i d e n t i f i e d r a d i c a l before ( l i g h t and a f t e r (heavy l i n e ) automatic phase correction 6.1 The i n f l u e n c e  of i n t e g r a t i o n  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  line) 51  on r e s o l u t i o n  54  plots  55  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  difference  plots  56  6.4 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  58  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  59  6.6 DISPA p l o t  f o r superimposed Gaussian l i n e s  60  6.7 DISPA p l o t  f o r superimposed Gaussian l i n e s  60  6.8 DISPA p l o t  f o r superimposed Gaussian l i n e s  61  6.9 DISPA p l o t  f o r superimposed Gaussian l i n e s  61  6.10 DISPA p l o t  f o r superimposed Gaussian l i n e s  62  6.11 DISPA p l o t  f o r superimposed Gaussian l i n e s  62  6.12 DISPA p l o t  for overlapping Lorentzian l i n e s  63  xv  6.13 DISPA p l o t  f o r overlapping  Lorentzian lines  64  6.14 DISPA p l o t  for overlapping  Lorentzian  lines  64  6.15 DISPA p l o t  for overlapping Lorentzian l i n e s  65  6.16 DISPA p l o t  f o roverlapping Gaussian l i n e s  66  6.17 DISPA p l o t  f o roverlapping  66  Gaussian l i n e s  6.18 DISPA p l o t f o r a m i x t u r e Gaussian l i n e s  of a L o r e n t z i a n and  6.19 DISPA p l o t f o r a m i x t u r e Gaussian l i n e s  of a L o r e n t z i a n and  6.20 DISPA p l o t f o r a m i x t u r e Gaussian l i n e s  of a L o r e n t z i a n and  67  68  68  \  6.21 DISPA p l o t f o r a m i x t u r e Gaussian l i n e s 6.22 DISPA p l o t  of a L o r e n t z i a n and 69  f o r unresolved  6.23 C a l i b r a t i o n c h a r t 7.1 D i f f e r e n c e p l o t s 7.2 The DISPA p l o t  hyperfine  for unresolved  70  hyperfine  f o r TEMPO a n d TEMPONE  f o r two s u p e r i m p o s e d  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  7.4 The DISPA p l o t  f o r CuMe dtc  6 5  2  7.5 A S q u a r e - r o o t d i f f e r e n c e  plot  i n toluene  77  78 showing Chemical Exchange  7.7 The DISPA p l o t  7.9 DISPA p l o t s  76  showing  satellites  7.8 DISPA p l o t  74  spin  labels  7.6 DISPA p l o t  71  for grey-pitch  80  f o r SP1 g r a p h i t e for various coal  81 samples  7.10 DISPA p l o t s f o r v a r i o u s c o a l s a m p l e s 7.11 DISPA p l o t f o r n a t u r a l d e c a y e d wood xvi  79  82 83 85  7.12 DISPA p l o t irradiation  f o r d e c a y e d wood a f t e r  7.13 DISPA p l o t  f o r d e c a y e d wood a f t e r  and  85 irradiation  relaxation  86  7.14 DISPA p l o t  f o r a powder s p e c t r u m  87  7.15 DISPA p l o t  for a nitroxide  87  i n a membrane  9.1 The g e n e r a l s t r a t e g y 9.2 T y p i c a l  metal  98  dithiocarbamate  101  10.1 A x i s s y s t e m f o r t e n s o r s 12.1 T y p i c a l  metal  12.2 T r a n s i t i o n  115  dithiocarbamate  spectrum  diagram  128 132  13.1 The S p e c t r o m e t e r  149  14.1 S p e c t r a l d e n s i t i e s vs. frequency 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 Approximations  156  15.1 P y r o l l i d i n e dt c l i n e - w i d t h  Density  data  1 57 165  15.2 The e f f e c t o f a n i s o t r o p y on T J / T p l o t s  176  15.3 A 77/T p l o t  177  f r o m p r e v i o u s work  17.1 S c h e m a t i c o f t h e p u l s e s e q u e n c e  191  17.2 T y p i c a l  IR d a t a  set  192  17.3 T y p i c a l  SR d a t a  set  193  20.1 The p r o b e a s an e l l i p s o i d  209  21.1 B l o c k d i a g r a m o f t h e a c q u i s i t i o n s y s t e m  223  21.2 F l o w - c h a r t f o r t h e s o f t w a r e a c q u i s i t i o n system  224  of the  21.3 F l o w 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 flattening  xvi i  237  21.4 The e f f e c t integration  o f low d a t a d e n s i t y  on 238  21.5 I n t e g r a t i o n  errors  vs.  data density  239  21.6 I n t e g r a t i o n  errors  vs.  data density  240  21.7 I n t e g r a t i o n  errors  vs.  data density  240  21.8 S p e c t r u m o f t h e f r e e  spin-probe  242  21.9 S p e c t r u m o f f r e e a n d bound s p i n - p r o b e  243  21.10 S p e c t r u m o f a bound s p i n - p r o b e  243  22.1 C h e m i c a l  s h i f t value f o r n i c k e l  dtc's  22.2 C o u p l i n g p a r a m e t e r s  (Hz) f o r n i c k e l  22.3 The powder  for NiEt dtc  spectrum  2  xvi i i  251 dtc's  252 252  Acknowledgements I w i s h t o s i n c e r e l y t h a n k D r . F.G. H e r r i n g f o r h i s help, guidance I would  also like  Dr.R.F.Snider for  a n d p a t i e n c e t h r o u g h o u t my r e s e a r c h . t o thank D r . K . A . M i t c h e l l and  f o r u s e f u l comments on t h e t h e s i s ,  Dr.A.Storr  t h e e i g h t y e a r l o a n o f h i s m o l e c u l a r m o d e l s a n d Kam  S u k u l , Mike for t h e i r  H a t t o n a n d o t h e r members o f t h e e l e c t r o n i c  never  ending maintainence  shop  of the equipment.  I a l s o wish t o acknowledge the Chemistry Department of the U n i v e r s i t y  of B r i t i s h Columbia  for providing  financial  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 e c o n o m i c 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.  xix  "Onwards always  Onwards,  In Si I e nee and in Gl oom. . . "  1  1  " F u n g u s t h e Bogeyman", Raymond B r i g g s , H a m i s h London (1977) xx  Hamilton,  PART 1.  DISPERSION VS. ABSORPTION  1  PLOTS: DISPA  1. Any  line  amplitude,  INTRODUCTION  i n a s p e c t r u m may  p o s i t i o n and  s h a p e . The  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 the  line.  The  be c h a r a c t e r i s e d by i t s  position gives  amplitude i s u s u a l l y a  of the  species associated  structural  information.  shape i s u s u a l l y assumed t o be L o r e n t z i a n the  line  width  and  with  The  the w i d t h  of  i s t h e n u s e d t o c o m p l e t e t h e c h a r a c t e r i s a t i o n . The  of t h e  l i n e can  give  information  about the  species'  dynamics. In r e c e n t  years there  extracting motional very is  important,  has  information  the  and  from the L o r e n t z i a n s u c h s t u d i e s . To the  on C o l e - C o l e  interest in line-widths. It is sure  that the  line-width will  r e l a x a t i o n t i m e and  species. A quick  analysing  f r o m ESR  f o r t h e s e s t u d i e s , t o be  i n f a c t L o r e n t z i a n , or the  reflect  been g r e a t  not  line-shape  t h i s end  line-shape p l o t s (1)  we  but  w o u l d be  not  deviations  of g r e a t  value  s p e c t r a . The only detects  also identifies  deviation.  I t i s thus u s e f u l as b o t h a d i a g n o s t i c  analytical  tool.  dispersion  signal will  the cause of  its  circle.  I f the  circular If  the  t h e n t h e DISPA p l o t i s a  s i g n a l i s d i s t o r t e d by,  2  the  corresponding  p r o d u c e an a p p r o x i m a t e l y  signal i s Lorentzian  from  and  g r a p h . S u c h a p l o t i s known ( h e r e ) a s a DISPA p l o t . absorption  of  deviations  behaviour,  s i g n a l vs.  to  method i s b a s e d  Lorentzian  A p l o t o f an a b s o r p t i o n  the  h a v e d e v e l o p e d a method  o f ESR  and  accurately  hence the motion of  e a s y method f o r a s s e s s i n g  line  for instance,  a  3 second l i n e ,  then a d i s t o r t e d e l l i p s e  graph's d e v i a t i o n from a c i r c l e uniquely provide  thus  a method o f q u a n t i t a t i n g t h e shape o f a l i n e a n d f o r f e a t u r e s . H o w e v e r , i t s h o u l d be  emphasised that the u t i l i t y information  obtained,  obtained.  only  i s c h a r a c t e r i s t i c (often  s o ) o f t h e d i s t o r t i n g m e c h a n i s m . DISPA p l o t s  detecting unresolved  is  i s p r o d u c e d . The  takes  o f DISPA l i e s n o t o n l y w i t h t h e  but a l s o w i t h the ease w i t h which i t  A DISPA p l o t r e q u i r e s  no s p e c i a l e x p e r i m e n t a n d  a few m i n u t e s a t m o s t . S i m u l a t i o n s ,  can  take  1.1  A BRIEF HISTORY  f o r example,  days.  Dispersion  vs. a b s o r p t i o n  dielectric  studies  p l o t s h a v e been l o n g  (as Cole-Cole  plots) to  used i n  analyse  r e l a x a t i o n m e c h a n i s m s (1). T h e r e have been b r i e f a l l u s i o n s (2)(3)(4),  t o DISPA p l o t s i n m a g n e t i c r e s o n a n c e l i t e r a t u r e but  i t s f u l l potential asa tool  not  realised until  (5)(6)(7)(8). by  d e v e l o p e d f o r NMR by M a r s h a l l  o f p r o d u c i n g ESR d i s p e r s i o n  p r o b l e m was c i r c u m v e n t e d b y u s i n g  relations  more e f f i c i e n t  algorithm  utilising  ( F F T ) was i n t r o d u c e d  apparent that data considerably  processing  spectra.  from  absorption  impractical until a  the f a s t  (10).  hampered  the Kramers-Kronig  (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  s p e c t r a . However, t h i s m e t h o d p r o v e d  transform  a n a l y s i s was  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  the d i f f i c u l t i e s  This  i n line-shape  Fourier  I t r a p i d l y became  f o r ESR s p e c t r a was  d i f f e r e n t ( a n d 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 a n d t h e d e v e l o p m e n t o f DISPA f o r t h e s e  techniques NMR  diverged. Marshall continued  (11)(12)(13)  to refine  a n d H e r r i n g and P h i l l i p s  (10)(14)  t h e method f o r u s e w i t h ESR. O n l y t h e l a t t e r be d i s c u s s e d  2  here.  two DISPA f o r developed  research  will  2  Much o f t h i s work i s a p p l i c a b l e t o NMR a n d some 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 .  overlap  2. THE BASIC THEORY OF DISPA A line  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 ,  p o s i t i o n and shape. A t h e o r e t i c a l treatment should  take a l l of these  variables into  o f DISPA p l o t s  acccount.  Unfortunately, general a n a l y t i c a l expressions t h e c a s e s do n o t e x i s t simple  area,  so t h e b a s i c theory  f o r most o f  i s restricted to  d i s t r i b u t i o n s o f L o r e n t z i a n l i n e s . More c o m p l e x c a s e s  h a v e 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 C I R C L E  (A(CJ)) a n d d i s p e r s i o n  In magnetic resonance t h e a b s o r p t 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 7BiT  \  0  =  * \ C 0 >  D ( a , )  =  1  +  T  2  ( - ) l0  2  Z +  cjQ  +  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 , are  2  B  2  T  i  T  2  2  1+TlU-o> H 7 BiT T 0  dispersion,  7  1  ( 2 2  '  1 )  v for absorption, u f o r  3  3  S t r i c t l y s p e a k i n g , f o r ESR, t h e s i g n a l s a r e t h e d i f f e r e n t i a l s o f t h e s 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  v  Ab ~ 1 + (bp7 +S 2  „ -  Apb  2  " " 1 + (bpF+S  ( 2  where A i s the amplitude of v a t resonance (~M yBy),  2 )  b = T ,  0  2  *  2  2  p = (CJ-O) ) and S i s the s a t u r a t i o n term ( 7 B T T ) . I f we 0  note that  1  v=bpu we can e l i m i n a t e  Eqn.2.2 and a f t e r r e a r r a n g i n g s a t u r a t i o n , i.e.,  the parameter p from  we get ( i f there  i s no  S<<1)  2  u or  if  2  u  + v 2  2  = bAv  + (v-ibA)  2  = (ibA)  2  (2.3)  we l e t £bA=R then we get  2  u  This  + (v-R)  2  = R  2  i s a c i r c l e of r a d i u s R d i s p l a c e d along  (2.4)  the v a x i s by R  ( F i g . 2 . 1 ) . R i s p r o p o r t i o n a l t o the spectrum l i n e - w i d t h and amplitude. A p l o t of u vs. circle and  v superimposed on a r e f e r e n c e  c i r c l e (a  of r a d i u s equal t o the maximum of v, R, i n t h i s case  d i s p l a c e d along  ( i n the Cole-Cole  the v a x i s by R) i s c a l l e d a DISPA p l o t  format).  Such a p l o t i s only a c i r c l e f o r  7 a pure L o r e n t z i a n  line. Generally  a distorted ellipse i s  ( s e e F i g . 2 . 1 ) . The d i s t o r t i o n  observed  the mechanism c a u s i n g  i s characteristic for  d e v i a t i o n s from L o r e n t z i a n  behaviour.  N o t e f r o m Eqn.2.4 t h a t t h e f r e q u e n c y d e p e n d e n c e ( t h e p t e r m ) h a s been e l i m i n a t e d independent  so t h a t DISPA p l o t s a r e  of l i n e p o s i t i o n . A l s o ,  i f the spectrum i s  scaled to a fixed amplitude before 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 d e p e n d e n c e o f DISPA p l o t s a r e r e m o v e d . In magnetic considered distributed  resonance  as c o m p o s i t e s  the s p e c t r a l l i n e s can o f t e n  of,many L o r e n t z i a n  lines  i n a m p l i t u d e , frequency 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  cases are  discussed  below. (AR)  F i g u r e 2.1. T y p i c a l C o l e - C o l e f o r m a t 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 t h e r e f e r e n c e c i r c l e . The d i a m o n d s a r e t h e data.  be  8 2.2 LORENTZIAN  LINES DISTRIBUTED  The a m p l i t u d e ,  A, i n Eqn.2.2  I N AMPLITUDE  i s simply  a height  scaling  f a c t o r . So p r o v i d i n g t h a t t h e l i n e s have t h e same w i d t h p o s i t i o n adding Lorentzian DISPA c i r c l e  height  f o r DISPA s t u d i e s t h i s e f f e c t i s  o b s e r v e d . The c o r o l l a r y t o t h i s  changes o n l y  i s that anything  the spectrum amplitude  modulation changes, a m p l i f i e r gain DISPA p l o t f o r t h a t l i n e . non-Lorentzian. Lorentzian  j u s t produces a  o f l a r g e r r a d i u s . As t h e s p e c t r a a r e a l w a y s  scaled to a constant not  l i n e s together  and  This  Another p o i n t  line,  that  ( s m a l l power and etc.)  does not a f f e c t the  i s t r u e even i f the l i n e i s i s that, for a single  a change i n l i n e - w i d t h o n l y a f f e c t s t h e  a m p l i t u d e a n d so i s n o t o b s e r v e d i n DISPA p l o t s . T h e s e rather t r i v i a l that,  r e s u l t s lead t o the important  for a single Lorentzian  line,  conclusion  homogeneous  broadening  m e c h a n i s m s do n o t a f f e c t t h e 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 m e c h a n i s m s a r e ; l i f e t i m e b r o a d e n i n g due t o e x c h a n g e o r m o t i o n , d i p o l a r b r o a d e n i n g a n d microwave s a t u r a t i o n .  2.3 THE EFFECT OF SATURATION ON LORENTZIAN If the  we e l i m i n a t e p f r o m o u r e q u a t i o n s  LINES  as b e f o r e ,  but r e t a i n  s a t u r a t i o n t e r m S we g e t  2  v (1+S) + u  2  = Abv  (2.5)  9 if  we  l e t q=v/(1+S) t h e n  2  u  As p o i n t e d o u t  +  we  obtain  (vq-bA/q)  by Abragam (4)  Marshall  (11),  parallel  t o t h e u a x i s The  this  2  = (ibA/q)  and  shown n u m e r i c a l l y  p o s i t i o n and  The still  vide  absorption line  Lorentzian, only  Similarly  i t s w i d t h and  transform*  c o m p u t e r p r o g r a m j u s t has and  appear e x p l i c i t l y  behaviour  from t h a t of  a  However, t h e d i s p e r s i o n l i n e of t h e a b s o r p t i o n  a Lorentzian data  generates  s e t of  the c o r r e s p o n d i n g  h e i g h t ) d i s p e r s i o n l i n e . The  f r o m t h e DISPA  9  h e i g h t and  elliptical  is different  i s generated  by a H i l b e r t  change.  d e p e n d e n c e on m i c r o w a v e power  (16).  given width  scaled to a  amplitude  f o r t h e d i s p e r s i o n l i n e . The  saturated absorption line  not  effect  i n t h e p r e s e n c e of s a t u r a t i o n i s  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  w i d t h and  this  are  supra.  a r i s e s because the amplitude  The  the  w i d t h of the e l l i p s e  because the a b s o r p t i o n s p e c t r a are  amplitude,  by  l e n g t h of w h i c h d e p e n d s on  a l s o a f u n c t i o n o f m i c r o w a v e p o w e r , but  constant  (2.6)  i s an e l l i p s e whose m a j o r a x i s i s  m i c r o w a v e power. The  disappears  2  a (in  s a t u r a t i o n term does  so t h e e l l i p t i c a l  behaviour  disappears  plot.  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 unsaturated, systems. (17)  line.  for linear,  i.e.,  10 2.4 DISTRIBUTION  I_N LINEWIDTHS OF LORENTZIAN L I N E S  I f we assume t h e l i n e  i s composed o f s e v e r a l  Lorentzian  l i n e s o f t h e same r e s o n a n t f r e q u e n c y , b u t d i f f e r e n t relaxation  t i m e s t h e n we h a v e t o s u b s t i t u t e Ab=ZA.b. i n t o i  Eqn.2.3,  5  If  i  i  h e n c e we g e t / c i r c l e s  2  u) t  + [f  (v -iA b )3 /  /  2  = [f  /  2  iA.b,] ]  (2.7)  where u.  =  1  -/ -/ 1+(bp ) +S  (2.8)  etc.  2  /  Eqn.2.7 r e p r e s e n t s and  a s e t of c i r c l e s each of r a d i u s  d i s p l a c e d by b ^ / 2 , w h i c h , when summed, w i l l  d i s t o r t e d oblate e l l i p s e D  b./2  form a  (Fig.2.2). ID  F i g u r e 2.2. DISPA f o r s u p e r i m p o s e d Lorentzian Lines.  5  A a n d b a r e l i n k e d by t h e a r e a o f t h e 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 t h e 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 parameters as independent v a r i a b l e s .  The r a d i u s o f t h e i ' t h c i r c l e d e p e n d s on b o t h t h e l i n e - w i d t h and c o n c e n t r a t i o n  of species  i s o i t i s n o t 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 statements about the d i s t o r t i o n maybe f o r t h e c a s e o f i = 2 . The d i s t r i b u t i o n  except  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 h a s been discussed  by C o l e & D a v i d s o n  2.5 DISTRIBUTION  (18).  I N RESONANT FREQUENCY OF LORENTZIAN L I N E S  I n t h i s c a s e we s u b s t i t u t e p= I p.  s o t h a t A^. r e p r e s e n t s center  t h e s h i f t of t h e i ' t h l i n e  of the spectrum, w . 0  form f o r t h i s  i n t o Eqn.2.3,  Eqn.2.3  i s i n an  s u b s t i t u t i o n , b u t c a n be r e c a s t  where  from t h e  inconvenient into 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 coordinates definition  i s g i v e n by r = 2 R c o s 0 = A b c o s 0 . See F i g . 2 . 3 f o r a of terms.  12  F i g u r e 2.3. D I S P A c i r c l e coordinates.  The a n g l e ,  8,  i s given  in polar  by  6 = arctan{D(cj)/A(cj)} = arctan{bp}  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  Ir. /  i  get  (2.10)  The c o m p o s i t e  d i f f i c u l t to v i s u a l i s e , but (Fig.2.4).  1  r e p r e s e n t s one c i r c l e . T h e 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 ,  ellipse  we  = Abcos{arctanZ[b(p+A.)]}  '  Note that t h i s equation  d e p e n d s on A..  frequency  (2.9)  for a given frequency,  f i g u r e i n t h i s case  p,  is  i t w i l l be a d i s t o r t e d p r o l a t e  1 3  ID  F i g u r e 2.4. DISPA f o r o v e r l a p p i n g lines.  Lorentzian  2.6 DISTRIBUTION IN RESONANT FREQUENCY AND AMPLITUDE T h i s i s an important unresolved  d i s t r i b u t i o n as i t i n c l u d e s the case of  h y p e r f i n e s t r u c t u r e . The DISPA p l o t  i s simply  given by  Zr. = bIA.cos{arctanZ[(b.p+A.)]} /  The exact  (2.11)  i  shape of the DISPA p l o t w i l l depend on A.  the amplitudes are d i s t r i b u t e d , distorted prolate e l l i p s e . If  and how  but i t w i l l be, as above, a i s constant  and A . f o l l o w f  the b i n o m i a l d i s t r i b u t i o n , then v r e p r e s e n t s a l i n e with unresolved  hyperfine coupling  (approximately  the V o i g t  l i n e - s h a p e ) . In the l i m i t of A —> 0 Eqn.2.11 g i v e s the result  f o r a Gaussian l i n e (A. d i s t r i b u t e d normally) or a  14  Voigt  (19)  line  (A^ d i s t r i b u t e d a s 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  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  absorption  t oallow f o r  (10)(20)  modulation are  v  _ AbF(k) k=0 1+b (p+kw m) y  2  2  2  "  1 -  m  K m  i s a Bessel  modulation is  _ A b F ( k ) (p+kcj ) £ o HFTp+kwF"  F ( k ) = J ? {7B /CJ  where  J  =  m  (2.12)  70  function of the f i r s t  m  k i n d , co  f r e q u e n c y , B^ i s t h e m o d u l a t i o n a m p l i t u d e a n d 7  t h e e l e c t r o n gyromagnetic  ratio.  I f we c o m b i n e A a n d F ( k )  ( t o g i v e A.) a n d n o t e t h e s i m i l a r i t y o f kw ^  i s the  k  a n d A. m  i  in  Eqn.2.11 we g e t  Ir. /'  which  = l A ^ c o s U r c t a n t l b ^ p + k c j p ]} k  (2.13)  k  i s t h e same a s E q n . 2 . 1 1 , a s u p e r p o s i t i o n o f L o r e n t z i a n  lines distributed amplitude  i n frequency  ( t h e harmonics  o f u^) a n d  (a f u n c t i o n o f t h e m o d u l a t i o n a m p l i t u d e and  frequency).  The shape o f t h e DISPA p l o t i s d i f f e r e n t  from  15 that caused by unresolved CJm «A. i  hyperfine coupling as generally  . ( a t 1 0 0 k H z , u> =30mG) ma n d Ai. i s d i s t r i b u t e d a c c o r d i n g  the Bessel f u n c t i o n s a s opposed t o a Pascal t r i a n g l e . Although  in p r a c t i c e the d i f f e r e n c e s are o f t e n obscured by  noise.  2.8  T H E E F F E C T OF DISPERSION  For a poorly phased spectrometer and  i t s quadrature  the observed  s i g n a l O(w)  Q(CJ) a r e g i v e n b y (21)  0(u>) = A ( c j ) c o s ( e ) + D ( c j ) s i n ( 0 ) Qioj)  w h e r e A(w)  and D(u)  = D(a))cos(0)-A(w)sin(0)  are the a b s o r p t i o n  signals r e s p e c t i v e l y . The angle angle'  (2.14)  and  dispersion  6 i s c a l l e d the  'phase  o r 'phase'. From C a r t e s i a n geometry f o r the r o t a t i o n  of c o o r d i n a t e s ,  the l o c u s o f (0(w),Q(w)) (the DISPA p l o t ) i s  j u s t the l o c u s o f  (A(W),D(CJ))  r o t a t e d by  L o r e n t z i a n then d i s p e r s i o n leakage  simply  6.  I f A(w) i s  gives a rotated  circle  (Fig.2.5). For other  f u n c t i o n a l forms o f A(w), t h e  effect  i s t h e same, t h e D I S P A p l o t s a r e j u s t r o t a t e d b y 6.  T h i s i s the o n l y mechanism t h a t c a u s e s asymmetric DISPA p l o t s f o r spectra with symmetric absorption b a s e l i n e and  l i n e s ( f r e e from  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  of the e f f e c t o f spectrometer  phase can be found  i n (22) a n d  16 Sect.5.2.  DM  F i g u r e 2.5. DISPA p l o t f o r a p o o r l y phased l i n e . 8=10°.Corresponding spectrum on the left.  2.9 THE DYSONIAN LINE The Dysonian l i n e - s h a p e a r i s e s from paramagnetic c e n t e r s i n a conductor (23).  The exact form of the l i n e - s h a p e depends  on the system; conductor t h i c k n e s s , d i f f u s i o n r a t e of the center etc. (and  , but the spectrum i s , i n a l l c a s e s , s i m i l a r t o  i n some cases a c t u a l l y  i s ) an admixture of 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 . A simulated spectrum i s shown i n F i g . 2 . 6 . The DISPA p l o t the  circle  i s s i m i l a r t o F i g . 2 . 5 , except that  i s r o t a t e d =*45°.  F i g u r e 2.6.  The  Dysonian  Line-shape.  3. THE HILBERT TRANSFORM AND  DATA PRESENTATION  3.1  GENERATING THE DISPERSION SPECTRUM  ESR  d i s p e r s i o n s p e c t r a are very d i f f i c u l t  t o produce  d i r e c t l y . A l s o i t i s not c u r r e n t l y p o s s i b l e to produce the a b s o r p t i o n and d i s p e r s i o n s p e c t r a s i m u l t a n e o u s l y , which p r e s e n t s very s e r i o u s s c a l i n g problems. The d i s p e r s i o n spectrum  can, however, be obtained via  relations  Kramers-Kronig  (9).  A(u'  D(w')  )  =  +7T-  'Pf  = -*-'P£  D(OJ)—  ,  A(u)^,  (3.1)  The P denotes use of the Cauchy p r i n c i p l e v a l u e . T h i s i s more u s u a l l y known as a H i l b e r t transform and  i s completely  g e n e r a l f o r a l l spectroscopy except n o n - l i n e a r systems.  6  Eqn.3.1 can be s o l v e d n u m e r i c a l l y as i t stands. However, the computation  i s very slow and the pole (at o) =u)  problems. The  0  can cause  transform can be done very e f f i c i e n t l y once i t  i s r e c o g n i s e d as a c o n v o l u t i o n i n t e g r a l  6  (24)  so that  A H i l b e r t transform can be done with any s p e c t r o s c o p i c data s e t . However, the quadrature spectrum so obtained may not be the true quadrature s i g n a l (see Sect.2.3 on s a t u r a t i o n ) . T h i s g e n e r a l l y does not a f f e c t DISPA's use as a d i a g n o s t i c t o o l , but care should be taken when making comparisons with NMR data, where the t r u e quadrature s i g n a l i s always a v a i l a b l e . 18  19  F-Hll  where F "  1  D ( w )  ZPo>'  }  = F- {D( )}F-MCJ-M 1  i s the reverse Fourier transform.  1  but  F"'!"" }  so  F" {D(CJ)} 1  (3.2)  W  7  8  i . rr. s g n (CJ)  =  = i .TTF-  1  {A(CJ) }  .sgn(cj)  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 b y u s i n g t h e fast-Fourier-transform (FFT) algorithm  (25)(26).  A  few words  of c a u t i o n a r e needed a s t h e FFT a l g o r i t h m i s n o t e x a c t l y equivalent t o the analytic transform. spectrum  F i r s t , t h e ESR  i s c o l l e c t e d a s a f u n c t i o n o f time  in units  p r o p o r t i o n a l t oGauss. I t i s nevertheless a spectrum frequency  i nthe  d o m a i n . T h i s means t h a t , i f o n e w i s h e s t o m a i n t a i n  an a n a l o g y to generate  w i t h FT-NMR, o n e h a s t o u s e t h e i n v e r s e  transform  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 decay  (FID); m u l t i p l y by t h e sgn f u n c t i o n ; then do a forward transform  t o recover the d i s p e r s i o n spectrum. Secondly the  d i s c o n t i n u i t y o f the sgn function should l i e a t t h e o r i g i n of t h e data s e t . F o r a data s e t generated  by an F F T t h e  ordinate i s not defined ( i t s an array index), the o r i g i n , a s T h e 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 simply a s i g n change centered a t t h e o r i g i n . The t r a n s f o r m from t h e time domain t o t h e frequency domain is c a l l e d t h e forward transform and i s defined a s S(w) = F { G ( t ) } = J G ( t ) e - ' 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 - { S ( c o ) } = $*- J S ( t ) e ' d t 7  8  / w  1  ,  ,  + / w  20 such  lies  at the N/2th data p o i n t ( f o r N data p t s . ) .  even, then  strictly  speaking  t h e N/2  a f t e r m u l t i p l i c a t i o n by t h e sgn discontinuity. convenient.  In p r a c t i c e  a b s o r p t i o n spectrum so s h o u l d be  array be  i t can  is usually  i t -lies  on  be m u l t i p l i e d by ±1  imaginary. This w i l l  f u n c t i o n and  be  the product  as  of  spectrum  which  regarded  as t h e  a r r a y . The  c h o i c e of w h i c h i s a matter  transformed  be  the  also  sgn  to give  the  i n the r e a l p a r t of  p a r t of the spectrum  of c o n v e n t i o n  l i b e r t i e s as t o the d i r e c t i o n  data  transform w i l l  t h e n m u l t i p l i e d by  so one  The  imaginary  i m a g i n a r y p a r t of t h e  resultant  will  the  imaginary parts.  forward  dispersion  a  r e a l and  loaded to the  f o r t h e t r a n s f o r m . The  imaginary  undefined  of t h e a d j a c e n t p o i n t s ) . F i n a l l y  a l g o r i t h m uses independent  part  f u n c t i o n as  be  (This i s e q u i v a l e n t t o s e t t i n g the pole  E q n . 3 . 4 e q u a l t o one FFT  point w i l l  If N i s  the  i s real  can  and  take  o f t h e 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 important of  t o k e e p t r a c k o f where t h e d a t a a r e a t e a c h  the transform. This i s e s p e c i a l l y true i f the absorption  s p e c t r u m ' s maximum d o e s n o t time domain spectrum imaginary by t h e sgn  l i e a t the o r i g i n  (FID) w i l l  c o n t a i n both  (N/2) real  c o m p o n e n t s (and b o t h p a r t s have t o be  as  the  and  multiplied  f u n c t i o n ) . T h i s can c r e a t e problems w i t h  minicomputers for  stage  as t h e y o f t e n o n l y p e r m i t  the forward transform. Furthermore,  c o n s i s t of 2  n  r e a l d a t a t o be t h e d a t a must  (where n i s i n t e g r a l ) e q u a l l y s p a c e d  p o i n t s . A l s o the a b s o r p t i o n spectrum  used  data  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 are d i s c u s s e d i n the next  3.2 PRE-PROCESSING  i n ESR. The l a t t e r section.  OF THE SPECTRUM FOR THE  The d e r i v a t i v e s p e c t r u m  c a n be r e a d i l y  Simpsons Rule or t h e t r a p e z o i d a l  present problems.  discussed  i n Sect.21.10.  b u t c a r e must be  (27).  B a s e l i n e o f f s e t and  B a s e l i n e c o r r e c t i o n methods a r e The e f f e c t s o f p o o r 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 U n l i k e NMR,  FFT  integrated using  rule.,  t a k e n t o a v o i d low d a t a d e n s i t i e s drift  two p r o b l e m s  i n Sect.4.3.  ESR d a t a a r e n o t 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 a r e a v a i l a b l e , b u t a r e much s l o w e r more c o n v e n i e n t spaced  (28).  Itis  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  data s e t . This i s d i s c u s s e d i n Sect.21.7.  A data s e t of 2 adding  t o r u n , e.g.  data  n  data p o i n t s i s e a s i l y produced  e x t r a data p o i n t s ; padding.  The p a d d i n g  by  s h o u l d be  done s y m m e t r i c a l l y ( s o t h a t a n y a r t e f a c t s a r e s y m m e t r i c ) i n s u c h 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 spectrum  or a t the ends of t h e spectrum.  i n the  (The e n d s o f t h e  d a t a s e t s h o u l d be a t z e r o ) . The e f f e c t s o f p a d d i n g discussed  are  i n Sect.4.8.  The optimum m e t h o d seems t o be t o p a d t h e d a t a by interpolation  t o 1024 p t s ; r e d u c e  the data  s e t b a c k 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 p a d 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 t h e H i l b e r t  transform,  then r e c o v e r only t h e p a r t of the d i s p e r s i o n spectrum  22 corresponding chart  t o t h e 512 p o i n t a b s o r p t i o n s p e c t r u m . A f l o w  f o r the basic procedure  i s given  i n Fig.3.1.  (This i s  f o r t h e L S I - 1 1 c o m p u t e r , f o r a l a r g e r c o m p u t e r i t w o u l d be b e t t e r t o double  t h e number o f p o i n t s i.e.  , do a 2048 p o i n t  transform). FLATTEN DERIVATIVE DATA  INTEGRATE DERIVATIVE DATA  ABSORPTION DATA  PAD & CENTER ABSORPTION DATA  REVERSE FFT  MULTIPLY BY S G N AT CENTER OF FID  FORWARD FFT  SCALE 4 EXTRACT DISPERSION DATA  SAVE DISPERSION DATA  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 data from e q u a l l y spaced derivative data.  23  3.3  THE  DIFFERENCE PLOT  The  Cole-Cole  format  ( F i g . 2 . 1 ) , but  f o r DISPA p l o t s  is quite  striking  i s u n s a t i s f a c t o r y f o r many p u r p o s e s a s per-cent  observed  distortion  i s o n l y a few  of the  circle's  r a d i u s . A p l o t of t h e d i f f e r e n c e b e t w e e n  reference c i r c l e  and  the e x p e r i m e n t a l  9  The  r a d i a l d i f f e r e n c e , AR,  be more u s e f u l . between the  reference c i r c l e ' s  experimental  u s i n g the v e r t i c a l angle  corresponding  over  p l o t as o r d i n a t e would  r a d i u s and  (difference  the d i s t a n c e of  d i s p e r s i o n p o i n t s and  t o i m p l e m e n t on a c o m p u t e r and  are  offer  i s inconvenient  i n ESR  as the  frequency  of but  frequency.  i s both  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  t h e a r r a y i n d e x of t h e a b s o r p t i o n s i g n a l ,  polar angle,  8,  d i s c u s s i n g these  9  advantages  s a m p l e d e p e n d e n t . T h r e e o t h e r schemes h a v e  been d e v e l o p e d , A(w),  no  very  i n v e s t i g a t e d a number o f t h e s e ,  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 s p e c t r u m  and  as  the  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  instrument  an  circle)  o r d i n a t e use. Other schemes, such  p o i n t s on t h e r e f e r e n c e c i r c l e ,  a b s c i s s a . M a r s h a l l has  This  the  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  for experimental  difficult  reference  p o i n t f r o m t h e c e n t e r of t h e r e f e r e n c e  i s t h e most c o n v e n i e n t  the  of the e x p e r i m e n t a l i t i s necessary  I , and  d a t a . However,  t o d e f i n e some  signal, the  before  conventions.  A t low d a t a d e n s i t i e s t h i s i s n o t a v e r y u s e f u l f o r m a t a s t h e d a t a p o i n t s become t o o s p r e a d o u t . T h i s i s n o t a p r o b l e m w i t h ESR, b u t s o m e t i m e s o c c u r s i n NMR.  24  The r e f e r e n c e radius i s equal absorption the  circle  i s d e f i n e d as a c i r c l e  t o t h a t o f t h e maximum h e i g h t  and d i s p l a c e d a l o n g  reference  be n o t e d t h a t t h e  absorption  amplitude  absorption  s i g n a l was a l w a y s s c a l e d t o 25000,  to  of the  t h e a b s c i s s a by t h e r a d i u s o f  ( R ) . I t should  circle  whose  i s f i x e d by s c a l i n g . F o r t h i s work t h e  a DISPA r e f e r e n c e  circle  outside of the reference  of r a d i u s  10cm. Any  i s considered  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  corresponding displacement  t o be a p o s i t i v e  within the c i r c l 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  negative.  a p e r - c e n t a g e of t h e r e f e r e n c e  circle's  radius, R , 0  as  i.e.,  AR= ( R ^ ^ - R Q  ) 1 0 0 / R . The s p e c t r u m h a s t o be p h a s e d s u c h t h a t  the double  integral  O  i s positive.  be swept f r o m l e f t i s taken  t o correspond  the c e n t e r data  to right  (i.e.  The s p e c t r u m i s assumed t o , the l e f t  t o low f i e l d ) .  I t i s convenient i f  of the spectrum corresponds t o the center  s e t . T h i s c a n be r e a d i l y a c h i e v e d  comparisons of v a r i o u s data definition  of t h e spectrum  by p a d d i n g and makes  sets e a s i e r .  1 0  of the d i f f e r e n c e p l o t i s given  1 0  of the  A diagrammatic i n Fig.3.2  I f DISPA i s used f o r a s y m m e t r i c s p e c t r a a n o t h e r d e f i n i t i o n o f t h e c e n t e r s h o u l d be f o u n d . The p o i n t corresponding to g a s f o u n d by Hydes (29) algorithm i s p r o b a b l y a good c h o i c e . /  ?  0  It  i s convenient  lobe height  (=AR  ma;c  t o d e f i n e two more parameters, the  ), and the lobe s e p a r a t i o n ,  AS=(S^-S )/R , where 0  S  r  i s the p o s i t i o n of the l e f t  i s the p o s i t i o n of the r i g h t l o b e .  1 1  lobe and  Another u s e f u l  parameter i s the lobe asymmetry A^, d e f i n e d as; A^ = A^-A^.; where A, i s ARmax„ f o r the l e f t /  lobe and Ar  r i g h t l o b e . A diagrammatic d e f i n i t i o n can be found  1 1  i s that f o r the  f o r these parameters  i n Fig.3.3.  The lobe s p l i t t i n g was found t o have a weak dependence on the spectrum parameters and was not very u s e f u l as a d i a g n o s t i c parameter. A normal s p l i t t i n g was taken t o 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  F i g u r e 3.3. D i a g r a m m a t i c d e f i n i t i o n o f t h e lobe parameters f o r a t y p i c a l d i f f e r e n c e • plot.  Subsequent r e f e r e n c e s  t o t h e t e r m DISPA p l o t w i l l  be  taken  t o mean t h e d i f f e r e n c e t y p e DISPA p l o t r a t h e r t h a n t h e Cole-Cole are  type  p l o t . Unless  otherwise  f o r a l i n e with unresolved  protons  s t a t e d a l l examples  hyperfine coupling;  12  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 o f  1 .OG.  3.3.1  THE INDEX DIFFERENCE The a b s c i s s a  the a b s o r p t i o n necessary). follows.  i s simply  data  PLOT a f u n c t i o n of the index of  array scaled to a s u i t a b l e value ( i f  The a b s c i s s a , F ( x ) , i s c a l c u l a t e d a s  27  F(x) = f ( I  max  F(x) = f ( I - I  where I  max  - I ) .k  I < I  ).k  I > I  max  i s t h e i n d e x a t w h i c h maximum  max  max  (3.3)  absorption  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 the index of the d a t a a r r a y . P l o t s f o r F ( l ) where ' f i s one ( l i n e a r ) ,  ' f i s 1/ ( s q u a r e - r o o t ) ,  (ln) and ' f i s L o r e n t z i a n  ' f i s logarithmic  ( L z ) , a r e shown i n  Fig.3.4-Fig.3.5  3.4. L i n e a r and square difference plots.  Figure  root  indexed  28  LZ(I)  F i g u r e 3.5. LZ, indexed  L o g a r i t h m i c , LN, a n d L o r e n t z i a n , difference plots.  The l i n e a r p l o t h a s t h e u n f o r t u n a t e baseline  r e g i o n and c o m p r e s s i n g  difficult  of e x p a n d i n g the  t h e r e g i o n of i n t e r e s t .  i s o f no c o n s e q u e n c e f o r n u m e r i c a l it  effect  computations,  This  b u t makes  t o u s e f o r m a n u a l 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 w o r k . 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, b u t t h e c h o i c e of t h e w i d t h o f 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 s q u a r e r o o t  plot  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 t h e s h o u l d e r s o f t h e m a i n p e a k . The m a i n a d v a n t a g e o f plots  index  i s that the o r d i n a t e i s independent of the  experimental  d a t a so t h a t the p l o t t i n g  with p a t h o l o g i c a l data  s e t s . The m a i n d i s a d v a n t a g e  the lobe s e p a r a t i o n of index p l o t s (Fig.3.6)  routine i s stable i s that  i s l i n e - w i d t h dependent.  29  0.16  F i g u r e 3.6. L o g a r i t h m i c i n d e x e d p l o t function line-width/sweep-width.  The g l i t c h  i n the t a i l s  of the p l o t s  as a  i s a truncation  artefact.  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 t h e most n a t u r a l way o f e x p r e s s i n g them. The p o l a r angle, in  6,  i s given  Fig.3.7.  i n Fig.3.2. A t y p i c a l plot  i s shown  30  T h i s p l o t d i s t r i b u t e s the data ( b a s e l i n e data has  has.  compressed w i t h r e s p e c t  the advantage t h a t the  symmetry  i n a s a t i s f a c t o r y manner t o the main peak)  difference plot retains  (about the v e r t i c a l  p o s s i b l e to unambiguously a s s i g n (Fig.3.2)  lies  i n . The  r e s u l t a n t p l o t may  THE  plot  i t i s not  the quadrant t h a t  always  6  jump a r o u n d .  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  3.3.3  any  a x i s ) t h a t the Cole-Cole  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 ,  and  This  spectra.  ABSORPTION DIFFERENCE PLOT  Using  a sigmoid  type f u n c t i o n as  a l s o d i s t r i b u t e the data w o u l d be  the h y p e r b o l i c  sigmoid,  i t i s the  as  the a b s c i s s a  required. A natural  choice  tangent because, besides  being  i n t e g r a l of a L o r e n t z i a n , i . e . ,  d o u b l e i n t e g r a l of the  o r i g i n a l data  s e t may  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  be  will  a  cause  the useful  31  considerable i n s t a b i l i t y  i n such a p l o t . A more s u i t a b l e  (approximately sigmoidal) f u n c t i o n can be c o n s t r u c t e d from the a b s o r p t i o n data using Eqn.3.3, with A(o>), used i n s t e a d of ' f . T h i s g i v e s more s t a b l e p l o t s than the p o l a r d i f f e r e n c e p l o t , but the symmetry i s l o s t . The example f o r unresolved h y p e r f i n e c o u p l i n g i s given i n F i g . 3 . 8 . U n f o r t u n a t e l y the a b s o r p t i o n p l o t s loop-the-loop i f p a r t i a l l y  r e s o l v e d f e a t u r e s are  p r e s e n t . T h i s i s very e n t e r t a i n i n g , but can s e r i o u s l y hamper numerical a n a l y s i s of the p l o t s . The p r i n c i p a l advantage of these p l o t s i s that they are e a s i l y compared with the C o l e - C o l e type p l o t s as both are p l o t t e d as a f u n c t i o n of A(a>).  M(u)  Figure 3.8.  The a b s o r p t i o n d i f f e r e n c e p l o t .  An a b s o r p t i o n d i f f e r e n c e p l o t along with i t s corresponding Cole-Cole p l o t  i s shown i n F i g . 3 . 9 . (The s c a l e  of the a b s c i s s a f o r the d i f f e r e n c e p l o t  i s halved with  r e s p e c t t o the Cole-Cole p l o t ) . The o r d i n a t e i s the  32 dispersion per-cent  s i g n a l f o r t h e C o l e - C o l e type p l o t and t h e  radial difference  f o r m a t o f DISPA p l o t will  n o t be u s e d  i s preferred  diagnostic plots linear  This  by some w o r k e r s , b u t i t  here.  F i g u r e 3.9. Typical DISPA p l o t .  Generally  f o r the difference p l o t .  C o l e - C o l e and  the absorption  work ( u n l e s s  difference  t y p e o f p l o t was u s e d f o r  otherwise stated a l l difference  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 t h e  i n d e x e d p l o t was u s e d f o r n u m e r i c a l w o r k . P o l a r  (when s t a b l e ) required.  were u s e f u l where a e s t h e t i c  p l o t s were  plots  33 3.4 THE GAUSSIAN DIFFERENCE PLOT The  Gaussian line-shape  line-shape,  are very  and i t s c l o s e r e l a t i v e ,  common i n ESR s p e c t r o s c o p y .  of t h e 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 its This  l i n e - w i d t h (except l e dMarshall  the Voigt The s h a p e  i s independent of  for truncation effects,  t o propose a r e n o r m a l i s a t i o n  whereby a G a u s s i a n l i n e w o u l d g i v e a c i r c u l a r  Sect.4.8). procedure Cole-Cole  plot  ( 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 t h e DISPA p l o t w o u l d 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 behaviour. very  u s e f u l , b u t an e f f i c i e n t ,  algorithm  T h i s w o u l d be  r e l i a b l e and c o n s i s t e n t  r e m a i n s t o be d e v e l o p e d .  1 2  The p r o b l e m s t h a t a r i s e  a r e a s 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 s p e c t r u m i s n o t an a n a l y t i c f u n c t i o n a n d i s n o t a m e n a b l e t o r a p i d e v a l u a t i o n . T h i s makes t h e g e n e r a t i o n difficult,  of a reference  circle  b) A l o o k - u p t a b l e f o r t h e d i s p e r s i o n l i n e ,  i n t e r p o l a t i o n , was u s e d by M a r s h a l l unstable  numerical  (12),  but t h i s  with  i s very  i n t h e p r e s e n c e o f n o i s e , c ) The s h a p e o f t h e  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 o f t h e p o i n t s on t h e l o c u s o f t h e 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  d i f f e r e n c e p l o t i s t h u s n o t p o s s i b l e , d) U s i n g table f o r a reference a large table of our c u r r e n t  (=10  S  a look-up  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  points)  i s r e q u i r e d , beyond t h e c a p a c i t y  c o m p u t e r . T h e s e p r o b l e m s 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  1 2  reference  n o t be p u r s u e d  here.  M a r s h a l l has demonstrated t h e u t i l i t y of t h e technique (12), but h i s a l g o r i t h m i n v o l v e s a Gaussian reference l i n e of a r b i t r a r y w i d t h .  4. INSTRUMENTAL DIAGNOSTICS Instrumental  AND  a r t e f a c t s should  APPLICATIONS  be removed b e f o r e  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 . especially  t r u e f o r DISPA, w h i c h i s v e r y  d i s t o r t i o n s . However, t h i s i d e n t i f y and thus particular  any  This i s sensitive to line  s e n s i t i v i t y may  be u s e d t o  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  i t c a n be u s e d f o r t h e a u t o m a t i c  phasing  of  spectra. I n 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  of s o u r c e s ;  s a t u r a t i o n , overmodulation,  misphasing,  over-filtering,  frequency  noise  are discussed  bridge  n o i s e and l o w  and o f f s e t ) . These p r o b l e m s  i n turn 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 unresolved  a m p l i f i e r and  high frequency  (baseline d r i f t  f r o m a number  a l l examples a r e f o r a s i n g l e l i n e  hyperfine coupling unless otherwise  with  stated.  4.1 TIME CONSTANT The i n f l u e n c e o f 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 (PSD) a m p l i f i e r s time (30).  on l i n e - s h a p e  i s w e l l documented  However, t h e s i z e o f t h i s d i s t o r t i o n  under-estimated. effect  constant  1 3  An e x a m p l e i s shown i n F i g . 4 . 1 .  i s e a s i l y avoided  1/100th t h e t i m e sharpest  line  i s often  i t takes  by s e t t i n g t o scan  the time-constant  to  (peak-to-peak) the  i n the spectrum. Nevertheless  1 3  This  i t i s worth  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 h a s an a s y m m e t r i c t i m e 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 t h e 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 t i m e c o n s t a n t a n d t h e 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  filter  i s as marked. IttJX  F i g u r e 4.1. E f f e c t of a l a r g e PSD f i l t e r on a DISPA p l o t . Spectrum on the l e f t . 3.2G l i n e - w i d t h , 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 filter.  4.2  NOISE  Reducing 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 i n c r e a s e s the noise l e v e l . The i s thus an  important  i n t e g r a t i o n of  (apparent)  problems have been encountered very d i f f i c u l t  s e n s i t i v i t y of DISPA to n o i s e  f a c t o r . The  d e r i v a t i v e improves the  filter  SNR  the  substantially.  with n o i s e , but  i t can  No be  to f l a t t e n the b a s e l i n e f o r a n o i s y spectrum.  An example (unresolved h y p e r f i n e c o u p l i n g as b e f o r e ) i s shown i n Fig.4.2.  36  F i g u r e 4.2. E f f e c t o f n o i s e on a DISPA p l o t . The SNR ( p e a k - p e a k 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 be removed b e f o r e d o i n g any DISPA a n a l y s e s  (Sect.21.10).  e x a m p l e o f i t s e f f e c t s a r e shown i n F i g . 4 . 3 . N o t e t h a t example i s f o r a s m a l l d r i f t / o f f s e t the o r i g i n a l d e r i v a t i v e spectrum.  must An  this  t h a t i s not v i s i b l e i n  37  F i g u r e 4.3. DISPA p l o t s h o w i n g t h e e f f e c t o f 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 o f +0.1% o f t h e p e a k - p e a k h e i g h t . A d r i f t o f +0.2% a t t h e r i g h t s i d e c a u s e s t h e same d i s t o r t i o n .  4.4 An  A M P L I F I E R PHASING ESR  s p e c t r o m e t e r has  a number o f a m p l i f i e r s  be p h a s e d c o r r e c t l y . O n l y amplifier affects plots  front  (Sect.2.2)  correctly  14  modulation  are r e a d i l y  though.  This control  does not a f f e c t  automatic  frequency  phased or b a s e l i n e  amplifier  phased or asymmetric  DISPA a n a l y s e s a r e 1 f ,  The  s h o u l d be c o r r e c t l y  a r i s e . The  problems  panel c o n t r o l  (10).  to  phase-sensitive-detection  the l i n e a m p l i t u d e s and  amplifier will  has  the  t h a t have  output  lines will  c u r e d , b u t may  only  t h e DISPA control artefacts  should also  be  r e s u l t . These  n o t be o b s e r v e d  until  attempted.  F o r p h a s e c h a n g e s o f ^90°, o r 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 , b u t g e n e r a l l y one h a s t o d e l i b e r a t e l y m i s - s e t 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 phase t o see such e f f e c t s  38  4.5 MICROWAVE-BRIDGE PHASING R e f l e c t i o n mode c a v i t i e s are arranged such that  the  d i s p e r s i o n component i s e l i m i n a t e d by the automatic frequency c o n t r o l a m p l i f i e r . However, s p u r i o u s r e f l e c t i o n s i n the wave guides can give r i s e  to d i s p e r s i o n  leakage. A l s o  some b r i d g e designs do not have an automatic frequency c o n t r o l or have to be phased  f o r each experiment  The e f f e c t of d i s p e r s i o n leakage Fig.4.4 The lobe asymmetry  e.g.(31).  (poor phasing) i s shown i n  (Sect.3.3) i s a l i n e a r  function  of the phase angle and can be used f o r a u t o m a t i c a l l y phasing the spectrum.  {(22),  Sect.5.2)  F i g u r e 4 . 4 . DISPA p l o t f o r a mis-phased microwave b r i d g e . A pure L o r e n t z i a n l i n e with 6=2°.  4.6  SATURATION  S a t u r a t i o n does not a f f e c t a pure L o r e n t z i a n ( S e c t . 2 . 3 ) . However, i t does a f f e c t composite  line  the DISPA p l o t s of  l i n e s . The d i s t o r t i o n of the DISPA p l o t f o r  39 composite  l i n e s i s a f u n c t i o n of the r a t i o of  the  l i n e - w i d t h s to each other or t h e i r s e p a r a t i o n , Sect.6.5).  (see  S a t u r a t i o n b r o a d e n s l i n e s so t h i s w i l l c h a n g e  the  D I S P A 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 the i n d i v i d u a l l i n e s . G e n e r a l l y the DISPA d i s t o r t i o n w i l l decrease.  4.7  MODULATION  Modulation  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).  difference p l o t for overmodulation  The  i s shown i n F i g . 4 . 5 .  t h a t i t i s s i m i l a r t o , but d i s t i n g u i s h a b l e from d i s t o r t i o n caused  by u n r e s o l v e d  the  h y p e r f i n e c o u p l i n g as i t  changes with modulation  amplitude.  c a l i b r a t e the modulation  amplitude  i s u s e f u l for quick checks  Note  One  can  in fact crudely  u s i n g D I S P A (32)  of the modulation  and  amplitude,  which changes whenever the c a v i t y i s changed.  F i g u r e 4.5. D I S P A 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 of CuPydtc 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 respectively.  this  40  4.8 L I N E TRUNCATION AND PADDING L o r e n t z i a n and Dysonian sweep w i d t h  l i n e s have v e r y l o n g t a i l s .  i s t o o narrow these 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 a p p e a r i n t h e DISPA p l o t s .  (The b o u n d s o f t h e  i n t e g r a l , E q n . 3 . 1 , s h o u l d be i n f i n i t e ) . of  I f the  at least  t h i s problem.  U s i n g a sweep w i d t h  10X t h e p e a k - p e a k w i d t h o f t h e l i n e w i l l In practice  avoid  5x i s u s u a l l y a d e q u a t e a s most tails  (i.e.  , decay r a p i d l y  real  l i n e s have G a u s s i a n  five  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  after  i s unavoidable  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 u n m a n a g e a b l y large data sets or too small data d e n s i t i e s . spectrum  h e l p s reduce  the problem,  Padding t h e  but a r t e f a c t s  still  o c c u r . The e f f e c t s o f t r u n c a t i o n (on a 'ramp' padded s p e c t r u m ) 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 Lorentzian l i n e f o r v a r i o u s values of line-width/sweep-width.  The  v-lobes are additive  s o t h a t t h e most o b v i o u s e f f e c t i s  41  t h a t the raised.  tails  and  (Fig.4.7)  qualitative  end  p o i n t s of  T h i s does not  i n t e r p r e t a t i o n of  the d i f f e r e n c e p l o t interfere with  are  the  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 c a u s e d by t r u n c a t i o n .  zero  If  the  b a s e l i n e goes t o z e r o  by  interactive baseline  ( i t ' s usually forced  f l a t t e n i n g ) then a l l  reasonable  p a d d i n g schemes a r e  equivalent  does not  then e x t r a a r t e f a c t s , which depend  go  to zero  t h e p a d d i n g scheme u s e d , a r e i n t e r p o l a t i v e padding, zero tail  padding  (padding the  spectrum w i t h the and  ramp p a d d i n g  linearly  to zero  padding  left  f i r s t and  to padding with zeroes.  introduced.  and  last  The  data  s i d e s of  points  s p e c t r u m w i t h a s m a l l DC  o f f s e t are  zeroes), the  decays  s p e c t r u m ) , on  shown i n  on  respectively)  (padding with a f u n c t i o n that from the ends of the  If i t  e f f e c t s of  (padding w i t h right  to  a  Fig.4.8.  42  Figure 4.8. E f f e c t of various padding schemes on t h e D I S P A 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 o f 0.1% o f t h e d e r i v a t i v e amplitude. I i s the i n t e r p o l a t i v e padding. 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 padding.  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 a d d i n g and  ramp p a d d i n g , e x c e p t t h a t t h e v - l o b e s f r o m more e a s i l y r e c o g n i s e d a s an a r t e f a c t occur with  interpolative  padding.  ramp p a d d i n g a r e  than t h e w-lobes  that  5. THE AUTOMATIC PHASING OF SPECTRA Dispersion  leakage i s c h a r a c t e r i s e d  C o l e - C o l e DISPA p l o t  by r o t a t i o n o f t h e  (Sect.2.1) which g i v e s  asymmetric d i f f e r e n c e p l o t . This  rise  t o an  suggests 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 m a g n e t i c spectra. Discussion  here, w i l l  c a s e . A more g e n e r a l in  (22).  5.1  discussion  Applications  m e t h o d i s now  be r e s t r i c t e d  t o NMR  resonance  t o t h e ESR  (by t h e a u t h o r ) c a n - b e f o u n d  are dealt  with  i n (33)  and  this  i n commercial use.  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 , / . e. ,  Q(CJ) ,  1 5  H{S(CJ)} = Q(w) where  S(a>) = A (to) c o s ( 6) +D (co) s i n ( 6) Q(CJ)  and  A(co) a n d D(CJ)  = D(u)cos(6)-A(u)sin(0)  are the absorption  (5.1)  and d i s p e r s i o n  r e s p e c t i v e l y . They a l s o f o r m a H i l b e r t t r a n s f o r m  signal  pair.  F o r f u r t h e r d i s c u s s i o n s e e (34)(35). Also note that H{H{S(a>)}} •= -S(co). H , h e r e , d e n o t e s t h e H i l b e r t t r a n s f o r m ; n o t t o be c o n f u s e d w i t h t h e H a m i l t o n i a n , w h i c h i s n o t u s e d i n t h i s part of the t h e s i s . 1 5  43  44  H{A(u)}  Now c o n s i d e r a phase  = Q(w)  correction  <j> t o S(w) t o g i v e  (5.2)  S'(o>).  A l s o d r o p t h e (w) f o r c o n v e n i e n c e . Hence  S' = Scos(<*>)-Qsin(tf>)  Q' = Q c o s ( 0 ) + S s i n ( 0 )  substituting  (5.3)  back we g e t  S' = Acos0costf> + Dsin0sintf> - Dcos0sintf> + Asin0sintf>  (5.4)  which reduces t o  S' = A c o s ( 6-<p)+Dsin(6-<J>) and  similarly  Hilbert  (5.5)  f o r 0' ( w h i c h may a l s o be o b t a i n e d f r o m t h e  t r a n s f o r m o f S ' ) . T h i s may be r e p e a t e d s o t h a t  t h e i ' t h c o r r e c t i o n we g e t f o r S'  after  45  S(CJ) = kcos(6-L<t>. ) - D s i n ( 0 - p . )  (5.6)  T h i s forms the b a s i s of phase c o r r e c t i o n . I f we c o u l d characterise  0 d i r e c t l y then only one phase c o r r e c t i o n i s  <t> i s set t o 0 i n Eqn. 1.5.  required,  t h a t , we can i n c r e m e n t a l l y correction  change 4> u n t i l the d e s i r e d  i s achieved. In t h i s case we need some c r i t e r i o n  to e s t a b l i s h when 10^=0. either  However, i f we cannot do  16  DISPA p l o t s are a candidate i n  case.  5.2 USE OF DISPA PLOTS FOR PHASE CORRECTION To c h a r a c t e r i s e  0 explicitly,  requires  that a known r e l a t i o n  e x i s t s between the phase angle, 0, and some parameter of the d i f f e r e n c e p l o t . For the i t e r a t i v e method, the f u n c t i o n a l form of the r e l a t i o n need not be known, but i t should be monotonic with a unique zero f o r 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 d i f f e r e n c e p l o t s of s i n g l e L o r e n t z i a n superimposed L o r e n t z i a n coupling  l i n e s and unresolved  (Fig.5.1) i t can be d e s c r i e d  asymmetry, A^=d, i s l i n e a r l y  lines,  hyperfine  that the lobe  r e l a t e d to 0 (Fig.5.2) f o r  0<3O°.  1 6  T h i s i s the b a s i s of manual phase c o r r e c t i o n . The c r i t e r i o n used i s that of b a s e l i n e 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 p h a s e a n g l e s ; 0, 1 , 2 , 3, 5, 7 a n d 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 a s y m m e t r y as a f u n c t i o n o f phase a n g l e f o r v a r i o u s spectra.  This  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 t h e  47  increments,  <j>, c o n t r o l l e d by t h e  i s the p r o p o r t i o n a l i t y The  basic flow chart  details  see  procedures The  (22).  For  b e t w e e n 6 and  constant  i s shown i n F i g . 5 . 3 .  to note that the  (see F i g . 5 . 7 ) .  as there  phase a n g l e  For f u r t h e r  p r e d i c a t e s on t h e s p e c t r u m  needs a c e n t e r o f s y m m e t r y , t h e t e c h n i q u e  centered  0<3O°.  being  w h i c h i t u s u a l l y i s f o r ESR s o l u t i o n  I t i s important  lines  d for  where k  f u r t h e r examples o f phase c o r r e c t i o n  whole procedure  unresolved  0.^ = k l d . ,  (36)(37)(35)(33)(38).  see  centro-symmetric, spectra.  relation  on the  1 7  spectrum works f o r  The s p e c t r u m s h o u l d be  i s a s m a l l dependence o f the line-position  the spectrum i s l o c a t e d from the  only  (Fig.5.4).  The  observed center of  s m o o t h e d power s p e c t r u m  and  s h i f t e d as necessary.  I f t h e c e n t e r o f 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 t h a n -ve c u r v a t u r e (e.g. a t r i p l e t ) , t h e n t h e a l g o r i t h m has t o be m o d i f i e d . The s p e c t r u m has t o be smoothed t o remove t h e v a l l e y s , t h e n t h e v a l u e o f 6 e s t a b l i s h e d w i t h t h a t s p e c t r u m . The v a l u e s o 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. 1 7  START PHASER  SELECT PEAKS  PHASE TO GIVE + VE INTEGRAL  CENTER DATA AND PAD  HILBERT TRANSFORM  YES 0>45°  SET 0 = 90°  ESTIMATE 0  F i g u r e 5.3. Flow c h a r t f o r t h e automatic 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 line-position.  Examples f o r the phase c o r r e c t i o n of a s i n g l e l i n e and 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 ) and a f t e r (heavy 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 to the diagram above.  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 ( 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 . <j>=*20°. I n t h i s c a s e t h e d i s p e r s i o n l e a k a g e was n o t a p p a r e n t 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  L I N E SHAPE ANALYSIS IN  DISPA i s a p p l i c a b l e t o any f r o m any is  k i n d of  thus endless.  spectra  s a m p l e . The H e r e we  composed of  line-shapes  are  two  spectroscopic  line-shape  r a n g e of p o s s i b l e  will  restrict  line-shapes  the d i s c u s s i o n  G a u s s i a n or L o r e n t z i a n  commonly e n c o u n t e r e d  i n magnetic  s o l u t i o n s . F u r t h e r m o r e , the  restricted  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 )  be  analysed  p l o t s are The  by  or  fully  resolved  inspection. Also  u s u a l l y c o m p l e x and  e q u a l l y a p p l i c a b l e t o NMR  o r any  However, i t s h o u l d not  noise  with  be  can  to  be  is  usually DISPA  analyse. utility  of  r e s u l t s are  technique  though.  resolved derivative  resolved absorption  spectra  spectra  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 s p r e s e n t ) so  respect  other  resonance  line  corresponding  s p e c t r a . The  noted that  necessarily give  (Fig.6.l)(or if  spectra  main o b j e c t i v e i s t o d e m o n s t r a t e the ESR  Such  discussion w i l l  difficult  DISPA p l o t s f o r a n a l y s i n g  do  the  to  lines.  s p e c t r a of  observed. P a r t i a l l y ,  LIQUIDS  t o NMR  the  r e s u l t s are  spectra, which gives  spectrum.  52  slightly the  restricted  absorption  F i g u r e 6.1. The i n f l u e n c e o f i n t e g r a t i o n on r e s o l u t i o n , a) P u r e G a u s s i a n l i n e , b) P u r e 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 o f d i f f e r e n t w i d t h s a n d 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  C L A S S I F I C A T I O N OF  The  amplitudes  are  r a r e l y adequate c r i t e r i a  plot. the  and  LOBES  s e p a r a t i o n of t h e  l o b e s of a DISPA p l o t  to completely  I t i s thus d e s i r a b l e to c l a s s i f y  lobes  the o v e r a l l  somehow. DISPA p l o t s were s i m u l a t e d  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 separation, widths  and  line-shape.  DISPA p l o t s g e n e r a l l y r e t a i n fall  characterise  i n t o one  of the  two  I t was  the  s h a p e s of  f o r p a i r s of  amplitude, found t h a t  the  t h e i r mammiform c h a r a c t e r  s e q u e n c e s shown i n F i g . 6 . 2 .  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  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 .  and  The  G  L-W  If  the  s p e c t r a l l i n e s a r e a s y m m e t r i c t h e n t h e DISPA p l o t s w i l l a s y m m e t r i c , one and  the other  lobe w i l l  G ( 2 ) - G ( 3 ) - G has  been o b s e r v e d  p o i n t s of the  be m o d i f i e d  by  of the  asymmetric  s e q u e n c e . The  plots.  sequences  sequence  of c o u r s e  but be  not  at  sequence. Furthermore, the  plots  instrumental artefacts (Fig.6.3); for poor phasing  l o b e s t o g i v e a s y m m e t r i c l o b e s . The  l o b e s can  sequence  (Fig.6.1O-Fig.6.11),  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 ; antisymmetric  p a r t of t h e  i n s t a n c e , the  s e q u e n c e G ( 2 ) - G ( 3 ) - G ( 4 ) . P l o t s may  intermediate may  i n one  i n a n o t h e r p a r t of t h e  shown a r e a p p r o x i m a t e . F o r  the  fall  be  vary  s u b s t a n t i a l l y , but  adds i n curvature  usually only  with  55  lobe separation  L(l)  weak  L(2)  strong  L(2)  weak  W(l)  G(2)  strong  w  G(2) we ok  weak  w  G(4)  F i g u r e 6.2. plots.  Classification  of difference  56  Figure 6.3. Miscellaneous 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 s y m m e t r i c l o b e s , b) v - l o b e s , c ) s h a r p G - l o b e s , d) a n t i s y m m e t r i c lobes.  6.2 NOTES ON THE SIMULATIONS AND The  PLOTS  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  their  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 c o r r e s p o n d e n c e one-to-one; the diagrams i l l u s t r a t e  i s not  t h e 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 f r o m many s i m u l a t i o n s a s 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 combination All  d i f f e r e n c e p l o t s pass through  asymmetric p l o t s the l e f t separately.  and r i g h t  I t i s important  t o note  of l i n e s .  the o r i g i n  so f o r  lobes are c l a s s i f i e d that 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 t h e o r d i n a t e i f t h e direction  of f i e l d  p a i r s of the l i n e s  scan  i s reversed or the order of the  i s reversed.  57  All  numeric  i n f o r m a t i o n i s given as a r a t i o  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 considered The  t o have a c o n s t a n t  amplitude,  second l i n e , added t o t h e r e f e r e n c e  with  line i s  w i d t h and p o s i t i o n . line,  i s variable,  i t s w i d t h a n d h e i g h t a r e d i v i d e d by t h e c o r r e s p o n d i n g for  the reference l i n e  line-height variable  ratios  line  t o o b t a i n t h e l i n e - w i d t h and  r e s p e c t i v e l y . The p o s i t i o n  the l e f t  the -1.5  line  s h i f t means t h e v a r i a b l e l i n e i s  of the reference l i n e . A p o s i t i v e  to the right  i s 4.5G t o t h e l e f t  All observed  shifts line  places  l i n e means t h a t t h e v a r i a b l e l i n e  o f t h e r e f e r e n c e . As m e n t i o n e d  changing t h e s i g n of t h e s h i f t through  shift  o f t h e r e f e r e n c e . Hence a s h i f t o f  f r o m a 3.0G r e f e r e n c e  reflect  of the  i s d e f i n e d i n terms of t h e l i n e - w i d t h of t h e  reference l i n e . A negative to  earlier,  j u s t causes the lobes t o  the ordinate. and r a t i o s a r e r e s t r i c t e d  i s unresolved  such that the  a n d 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 n o t i n c l u d e p h a s i n g  or baseline  artefacts,  w h i c h i n f l u e n c e t h e symmetry o f t h e l o b e s . A l s o n o t e although  values  that  t h e a b s o r p t i o n l i n e s a s p l o t t e d may a p p e a r t o be  readily 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 are often obscured  by n o i s e .  6.3 DETECTING TWO  SUPERIMPOSED LORENTZIAN L I N E S  T h i s c a s e m i g h t be e x p e c t e d a spin label  i n two s i t e s ,  g i v i n g broad l i n e s ,  t o occur  when a s y s t e m  contains  one s i t e , bound a n d t h e r e f o r e  the other  s i t e unbound a n d t h u s g i v i n g  58 sharp l i n e s . I f we think i n terms of c i r c l e p l o t s , Eqn.1.3, it  i s immediately obvious that v a r y i n g the r a t i o of the  heights  of the l i n e  i s i n d i s t i n g u i s h a b l e from v a r y i n g the  r a t i o s of the widths of the l i n e s ; the radius of the c i r c l e i s p r o p o r t i o n a l to The  amplitude/line-width.  DISPA p l o t s are of the W-type, the amplitude of  which v a r i e s with the l i n e - w i d t h and amplitude (Fig.6.4-Fig.6.6).  ratios.  As expected l i n e - h e i g h t v a r i a t i o n s are  i n d i s t i n g u i s h a b l e from l i n e - w i d t h v a r i a t i o n s . IQOl  F i g u r e 6.4. 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 corresponding derivative spectra, for various line-width r a t i o s , but the same amplitude. R a t i o s f o r the s p e c t r a a r e 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 s u p e r i m p o s e d 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 derivative spectra, f o r various line-height 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 t h e 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  T h i s c a s e may o c c u r different  organic  for liquids  radicals  GAUSSIAN L I N E S (or s o l i d s )  with unresolved  c o u p l i n g . Superimposing a second Gaussian the amplitude  t e r m , A^. , i n Eqn. 1.11.  thus  t o be ( a n d i s ) m a i n l y  expected  are given  i n Fig.6.6-Fig.6.11.  c o n t a i n i n g two hyperfine  just  redistributes  The l o b e b e h a v i o u r G-type. S e v e r a l  is  examples  F i g u r e 6.6. DISPA p l o t f o r superimposed Gaussian l i n e s f o r v a r i o u s l i n e - w i d t h r a t i o s . L i n e s are the same h e i g h t . R a t i o s for the s p e c t r a a r e ; 0.0, 1.5, 2.0, 2.5, 3.0.  F i g u r e 6.7. DISPA p l o t f o r superimposed Gaussian 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 2.7. 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, 0.7, 1.0.  61  F i g u r e 6.8. DISPA p l o t f o r superimposed Gaussian 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 ; 0.0, 0.03, 0.07.  F i g u r e 6.9. DISPA p l o t f o r superimposed Gaussian 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 ; 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 Gaussian l i n e s f o r various ratios. Line-width r a t i o i s s p e c t r a a r e ; 1.0, 2.0, 3.0,  superimposed line-height 3.3. R a t i o s f o r ».  F i g u r e 6.11. DISPA p l o t f o r s u p e r i m p o s e d Gaussian l i n e s f o r various line-height 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 L I N E S  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 different  g - s h i f t s . A case of s p e c i a l  c h e m i c a l exchange, ligand  sites  {vide  supra)  interest  have  i s that of  where, f o r i n s t a n c e , t h e exchange  causes a g - s h i f t .  may  of a  63  The  simplest case i s f o r two  and height being (Fig.6.12). The amplitudes and  split  l i n e s of equal  a p a r t . L-type lobes are  combinations f o r d i f f e r e n t  amplitude  obtained  linewidths,  s p l i t t i n g s are e n d l e s s , but a p a t t e r n does  emerge. If the l i n e s are of the same width, but  different  amplitude an asymmetric L-type p l o t  (Fig.6.13  and F i g . 6 . 1 4 ) .  As the s e p a r a t i o n of the l i n e i n c r e a s e s  asymmetry i n c r e a s e s . The  s m a l l e r of the two  forms a G(2)-type lobe. The If  larger  the l i n e s are of d i f f e r e n t  types dominate. The ( F i g . 6 . 1 5 ) . The lines split L(2)-type  i s obtained  plots  the  lobes e v e n t u a l l y  lobe remains G-type. widths as w e l l ,  W-L  are an asymmetric W-type  asymmetry of the lobes i n c r e a s e s as  the  apart, with the s m a l l e r lobe becomeing an  for large s p l i t t i n g s .  F i g u r e 6.12. DISPA p l o t f o r o v e r l a p p i n g Lorentzian l i n e s for various s p l i t t i n g s . L i n e s are the same height and width. S p l i t t i n g s f o r the spectrum are; 0.0, 1.0, 1.5, 2.0, 2.5, 3.0.  lobe  1001  F i g u r e 6.13. D I S P A p l o t f o r o v e r l a p p i n g Lorentzian lines for various s p l i t t i n g s . L i n e s a r e t h e 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 t h e s p e c t r u m a r e ; 0.0 1.0, 1.5, 2.0, 2.5.  F i g u r e 6.14. D I S P A p l o t f o r o v e r l a p p i n g Lorentzian lines for various s p l i t t i n g s . W i d t h r a t i o i s 2.0. The h e i g h t s a r e t h e same. S p l i t t i n g s f o r t h e s p e c t r u m a r e ; 0.0 0.2, 0.5, 0.7, 1.0. N o t e t h e s i m i l a r i t y t o next f i g u r e .  65 2001  F i g u r e 6.15. DISPA p l o t f o r o v e r l a p p i n g 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 f o r the spectrum are; 0.0, 0.5, 2.0, 3.0. The s h i f t s are negative.  The r a i s e d contribution  6.6  tails  i n each case are due to a v-type  from t r u n c a t i o n . (Sect.4.8).  DETECTING TWO  OVERLAPPING GAUSSIAN LINES  T h i s case i s of i n t e r e s t because  organo-sulfur r a d i c a l s ,  which occur i n c o a l , are g - s h i f t e d w i t h - r e s p e c t - t o ' o r d i n a r y ' r a d i c a l s , which a l s o occur i s c o a l . The  ESR  s p e c t r a of c o a l are thus expected to be two overlapped Gaussian l i n e s . Two  Gaussian l i n e s of the same width and h e i g h t j u s t  redistribute  the amplitudes of the c i r c l e s i n Eqn.1.11, with  more i n t e n s i t y appearing i n the wings of the DISPA p l o t the l i n e s s p l i t (Fig.6.16).  as  a p a r t . T h i s g i v e s r i s e to G-type behaviour  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 Gaussian l i n e s f o r various s p l i t t i n g s . Lines a r e t h e same h e i g h t a n d w i d t h . S p l i t t i n g s f o r t h e s p e c t a r a r e ; 0.0, 1.5, 2.0, 2.5, 3.0.  If  t h e l i n e s a r e n o t t h e same a m p l i t u d e  asymmetric G lobes occur.  or height  then  (Fig.6.17). 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 Gaussian l i n e s f o r various s p l i t t i n g s . Lines a r e t h e 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 t h e s p e c t r a a r e ; 0.0, 1.5, 2.0, 2.5.  It  should  be n o t e d t h a t t h e a m p l i t u d e  here a r e not a r b i t r a r y .  Smaller  similar  ratios  effects.  Larger  ratios  chosen  r a t i o s give s m a l l e r , but give derivative spectra  that  are p a r t i a l l y  r e s o l v e d and thus were not used.  6.7 DETECTING COMBINATIONS OF LORENTZIAN AND GAUSSIAN LINES This case may seem rather c o n t r i v e d , but can be used t o account f o r the shape of DISPA p l o t s from c o a l . The lobe behaviour i s s u p e r f i c i a l l y overlapping  s i m i l a r t o the cases f o r  Gaussian l i n e s , but on the whole the p l o t s are  unique t o mixed L o r e n t z i a n and Gaussian  lines.  (Fig.6.18-Fig.6.21) The  f o l l o w i n g f i g u r e s are r e p r e s e n t a t i v e of a range of  many p o s s i b l e combinations of l i n e s .  loot  F i g u r e 6.18. DISPA p l o t f o r a mixture of a L o r e n t z i a n and Gaussian l i n e s f o r v a r i o u s width r a t i o s . Amplitudes are the same. S p l i t t i n g s f o r the s p e c t r a a r e ; 0.2, 0.5, 1.0. (The L o r e n t z i a n i s the r e f e r e n c e l i n e ) .  F i g u r e 6.19. DISPA p l o t f o r a mixture of a L o r e n t z i a n and Gaussian l i n e s f o r v a r i o u s width r a t i o s . Amplitudes are the same. R a t i o s f o r the s p e c t r a a r e ; 0.2, 0.5, 1.0. (The L o r e n t z i a n i s the r e f e r e n c e l i n e ) . IQOl  F i g u r e 6.20. DISPA p l o t f o r a mixture of a L o r e n t z i a n and Gaussian l i n e s f o r v a r i o u s width r a t i o s . Amplitudes are the same. R a t i o s f o r the s p e c t r a a r e ; 2.0, 2.5, 3.0, 3.5, 4.0. (The L o r e n t z i a n i s the r e f e r e n c e line).  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 o f 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. W i d t h 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 h e r e ) f o r t h e s p e c t r a a r e ; 0.0 1.0, 2.0, 3.0. (The Gaussian i s the reference l i n e ) .  6.8 DETECTING AND MEASURING UNRESOLVED HYPERFINE Unresolved  hyperfine coupling  i n the n i t r o x i d e type t h i s type  of system  i s very  spin labels.  common i n ESR,  unresolved  easily  ( E q n . 1 . 1 1 ) seems q u i t e u n m a n a g e a b l e .  hyperfine coupling  guessed so t h a t  amplitudes,  i s o f t e n known o r c a n be  ' i ' i n Eqn.6.11 i s d e f i n e d .  unknown i s A , t h e h y p e r f i n e  constructed  responsible f o r  A^. , f o l l o w t h e b i n o m i a l d i s t r i b u t i o n  o n l y one t y p e  notably  The DISPA e q u a t i o n f o r  H o w e v e r , t h e number a n d s p i n o f t h e n u c l e i the  COUPLINGS  splitting  of spin a c a l i b r a t i o n  from the d i f f e r e n c e  constant.  chart  plots.  The  so t h e o n l y I f there i s  i s easily  F i g u r e 6.22. D I S P A p l o t f o r u n r e s o l v e d hyperfine c o u p l i n g f o r v a r i o u s reduced coupling constants (coupling constant/natural line-width). Hyperfine 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 protons.  The h e i g h t  of the lobes  calibration Fig.6.23. coupling  plot  are related  f o r 2 —>  to A (Fig.6.22).  12 s p i n h a l f  nuclei  i s shown i n  The a p p a r e n t r e d u c e d c o u p l i n g c o n s t a n t constant  line-width.  A  i s the  expressed as a f r a c t i o n of the observed  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 t h e no. o f s p i n 1/2 n u c l e i c o u p l e d to the e l e c t r o n .  6.9 APPLICATIONS TO LINE-SHAPE ANALYSIS OF I n 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  SOLIDS  l i n e s occur the  analysis discussed  i n t h e s e c t i o n s above i s a p p l i c a b l e .  However, symmetric  ( o r near  p a t t e r n s can a r i s e ,  symmetric), s i n g l e l i n e ,  that are not Gaussian or L o r e n t z i a n i n  nature. For instance, a r a d i c a l with a small g - t e n s o r may g i v e r i s e shoulders.  powder  t o a symmetric  T h i s c o u l d be m i s t a k e n  l i n e s . A l s o , asymmetric  orthorhombic  line with  small  f o r superimposed  d i f f e r e n c e p l o t s may a r i s e  s l i g h t l y a n i s o t r o p i c g-tensors  Gaussian from  (or g-tensors averaged  by  72 motion  (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 .  p a t t e r n s i m u l a t i o n s a r e needed difference plot  Powder  t o c l a r i f y t h i s . The G a u s s i a n  ( 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  this case. The e x a m p l e s  g i v e n i n t h e next s e c t i o n been  interpreted  i n terms 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 such 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 of the technique as a p p l i e d t o s o l i d s and not as 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 of the spectrum. A d e t a i l e d a n a l y s i s o f D I S P A 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 commercial  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  7.1  EXAMPLES  TEMPERATURE DEPENDENCE OF UNRESOLVED HYPERFINE  If 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 temperature dependent then the r e s i d u a l l i n e - w i d t h (Sect. 14.5) i s a l s o temperature  dependent and t h i s hampers spin  relaxation  s t u d i e s . 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 to n i t r o x i d e type spin-probes  i s quite large (Fig.7.1,  (40)).  Proving t h a t 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 temperature  independent  i s very d i f f i c u l t ,  the e f f e c t s are  masked by motional l i n e - b r o a d e n i n g . One method of s o l v i n g t h i s problem i s t o f i x the c o r r e l a t i o n time of the probe and then study the DISPA p l o t s as a f u n c t i o n of temperature. p r a c t i c e , t h i s i s achieved by examining s e r i e s of non-polar  In  the probe i n a  s o l v e n t s at temperatures  chosen  such  that the v i s c o s i t y / t e m p e r a t u r e (r)/T) remains c o n s t a n t . T h i s of course assumes t h a t the r o t a t i o n a l c o r r e l a t i o n  time  depends o n l y on 77/T and that the e f f e c t of the s o l v e n t on r o t a t i o n a l a n i s o t r o p y can be n e g l e c t e d . DISPA p l o t s of TEMPONE  18  were obtained i n pentane,  heptane, decane and dodecane f o r 7j/T values of 0.001 and 1  0.002 cP K " , for the  19  over a temperature  (0) l i n e  range of -20°C t o +70°C  ( f o r which motional e f f e c t s are the  s m a l l e s t ) . W i t h i n experimental e r r o r no dependence of the DISPA p l o t  temperature  (Table 7.1) and hence of the  18  TEMPONE = 4-oxo-2,2,6,6,-tetramethylpiperidine-1-oxyl TEMPO = 2 , 2 , 6 , 6 , t e t r a m e t h y l p i p e r i d i n e - 1 - o x y l . cP=centi-Poise. 1 9  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 a n d TEMPONE, s h o w i n g t h e l a r g e u n r e s o l v e d hyperfine coupling contribution to the l i n e - w i d t h s . TEMPO i s on t h e l e f t .  Tj/T 0.001  Solvent Hexane  Line-width  %R  24.4  1.46  11.2  46.7  1.44  10.8  -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  Heptane  0.002  Temp °C  Hexane  ,  Table 7.1. T e m p e r a t u r e d e p e n d e n c e o f t h 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 f o r TEMPONE  75 7.2 MIXTURES OF S P I N PROBES E s t a b l i s h i n g the degree host molecule are  of b i n d i n g of a s p i n - l a b e l  i s important i f meaningful motional  much i n g - s h i f t  t h i s c a n be v e r y d i f f i c u l t t o  m e a s u r e . DISPA i s i d e a l l y  suited  f o r a n a l y s i n g such a case.  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 m i x e d to  studies  t o be done. A s t h e bound a n d unbound l a b e l s w i l l n o t  differ  To  to the  w i t h TEMPONE ( l i n e - w i d t h =*1.5G  r e p r e s e n t t h e unbound p r o b e )  ratio.  2 0  The DISPA p l o t  i n d i v i d u a l probes  i n a 1:3 c o n c e n t r a t i o n  i s quite different  (Fig.7.2) as might  from t h a t of t h e  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 c a n be o b t a i n e d i f a calibration of  c h a r t f o r the system  c a n be p r e p a r e d  t h e unbound a n d t h e p u r e b o u n d p r o b e  F i g u r e 7.2. The DISPA p l o t superimposed spin l a b e l s .  are required).  f o r two  An a p p l i c a t i o n o f t h i s t o a r e a l c a s e  2 0  (the widths  2 1  i s shown i n F i g . 7 . 3 .  T h i s was one o f a s e r i e s o f s a m p l e s p r e p a r e d by L . F . Y i p i n an a t t e m p t t o q u a n t i f y t h e m e t h o d . T a k e n f r o m work p e r f o r m e d by E.Lam i n t h i s l a b o r a t o r y 2 1  (41).  76 Here a spin label i s partitioned between a c e l l membrane ('bound probe') and the surrounding f l u i d spectrum  (free probe). The  shows some d i s t o r t i o n , but i t s clear from the DISPA  exactly what i s causing i t ; a mixture of free and bound probes accompanied by a s l i g h t g-shift with-respect-to each other.  J  r  Figure 7.3. The DISPA plot for an amphipathic spin-probe incorporated into red blood c e l l s in the presence of c r y s t a l s of mono sodium urate mono-hydrate. The difference plot i s for the center l i n e .  7.3 UNRESOLVED HYPERFINE COUPLING CONSTANTS It i s useful to determine what proportion of the residual line-width of the copper dithiocarbamate class of spin-probes i s due to unresolved hyperfine coupling. The dimethyl derivative i s a simple case to study as i t has twelve equivalent protons. The difference plots for t h i s probe are shown in Fig.7.4.  The unresolved hyperfine  coupling contribution can be measured from these p l o t s . From Fig.6.23  and Fig.7.4, we get a reduced coupling constant of  77 0.13, which corresponds  t o an unresolved h y p e r f i n e c o u p l i n g  constant of 0.6G f o r an observed  l i n e - w i d t h of 4.6G with  c o u p l i n g t o 12 p r o t o n s . T h i s agrees  reasonably w e l l with  v a l u e s from s i m u l a t i o n s (0.3G (42)) and by comparison with the p e r - d e u t e r a t e d compound  (0.4G via Bales formulas  (43)).  The  s l i g h t l y high value can be a t t r i b u t e d t o the s a t e l l i t e s  and  line  t r u n c a t i o n . I t n e v e r t h e l e s s p r o v i d e s a good  s t a r t i n g point for simulations. s.ox  65  F i g u r e 7.4. The DISPA p l o t f o r C u M e d t c i n t o l u e n e . The h i g h - f i e l d l i n e . 2  7.4 USING DISPA PLOTS TO DETECT SATELLITES The  a b s c i s s a of a d i f f e r e n c e p l o t  i s u s u a l l y chosen t o  compress the b a s e l i n e and expand the resonance  r e g i o n of a  spectrum. However, the converse case can be u s e f u l i f we wish t o d e t e c t s a t e l l i t e s square-root p l o t The  i n the wings. Fig.7.5  shows the  f o r p e r - d e u t e r a t e d CuPydtc i n d-chloroform.  c e n t r a l peaks are due t o 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  78 from the deuterons, r e s i d u a l hydrogen The two outer peaks are due to the lines),  3 3  S  and  1 3  C  satellites.  satellites  (four (44)(45).  which are i n the wings of the spectrum  2+sx  F i g u r e 7.5. A Square-root d i f f e r e n c e showing s a t e l l i t e s .  plot  7.5 THE DETECTION OF CHEMICAL EXCHANGE. SOLVATION EFFECTS P y r i d i n e exchanges with d i t h i o c a r b a m a t e s (46) t i m e - s c a l e and consequently broadens  on the ESR  the l i n e s .  mixture of p y r i d i n e and toluene the broadening, case, i s about  in this  2 Gauss (from 5G), but otherwise the l i n e i s  unchanged. The DISPA p l o t very d i f f e r e n t  For a 50:50  i s shown i n F i g . 7 . 6 . The p l o t i s  from that obtained i n the absence  of p y r i d i n e  (Fig.7.4 and corresponds to the DISPA of two overlapped L o r e n t z i a n l i n e s of s i m i l a r width and i n t e n s i t y showing that the broadening changing the magnetic  (Fig.6.13)),  i s due i n part t o the p y r i d i n e  parameters  of the dt c and not j u s t by  79 changing  i t s motion.  F i g u r e 7.6. DISPA p l o t showing Chemical Exchange. The h i g h - f i e l d l i n e of C u P y d t c i n 50:50 t o l u e n e / p y r i d i n e . 65  7.6 THE SPECTRUM OF GREY PITCH Grey p i t c h any  i s a standard sample f o r ESR (47). I t i s not of  great i n t e r e s t  have a s i n g l e  other than  f o r that purpose,  l i n e ESR spectrum. The DISPA p l o t  but i t does (Fig.7.7)  shows t h a t the l i n e may be c o n s i d e r e d as a s u p e r p o s i t i o n of two (or more) Gaussian  or 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  widths and h e i g h t s (see Fig.6.5, Fig.6.9 and Fig.6.18) and thus the sample c o n t a i n s two (or more) r a d i c a l s  or r a d i c a l  s i t e s . However, t h i s e f f e c t may be due t o a s i n g l e with a small orthorhombic  g-tensor.  radical  80  F i g u r e 7.7.  7.7  The  DISPA p l o t  for grey-pitch.  GRAPHITE SPECTRA  Graphite  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  Dysonian  l i n e - s h a p e ( F i g . 2 . 6 ) . The  Fig.7.8  i s not t y p i c a l  unknown, b u t of and  (48).  c a s t s no l i g h t for coal,  l i k e domains An  shoulder observed  of t h e D y s o n i a n  line,  22  The  DISPA p l o t  on t h e p r o b l e m , vide  infra,  but  i s not very  i t may  w h i c h may  in  i t s origin  i t s a m p l i t u d e v a r i e s w i t h t h e d e g r e e and  intercalation.  a control  should give a  type useful  be r e g a r d e d  as  contain graphitic  (49).  interesting possibility  t y p e shown i n F i g . 2 . 6 i s t o u s e  f o r Dysonian  l i n e s of  the auto-phase  the  algorithm  ( S e c t . 5 ) t o remove t h e d i s p e r s i o n component o f t h e l i n e t h e n t o c h a r a c t e r i s e t h e r e m a i n i n g l i n e by p o s i t i o n and phase  2 2  is  T h i s s a m p l e was pure.  and  i t s width,  correction.  o b t a i n e d from Dr.F.Aubke and  i s extremely  81  7.8 COAL SPECTRA Coal  2 3  gives a single line  spectrum. Various  examples a r e  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 a s 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 d e g r e e s o f o v e r l a p . One  probably  line  c o r r e s p o n d s t o a c a r b o n based r a d i c a l and t h e o t h e r  a sulphur  b a s e d r a d i c a l . The p l o t s were b l i n d  terms of i n c r e a s i n g l i n e  separation  ranked i n  ( l o b e asymmetry). Note  t h a t t h e s p e c t r a a r e a l m o s t 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 t h e sulphur  content,  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 s t u d y  i s obviously  warranted.  2 3  The s a m p l e s were o b t a i n e d f r o m D r . T a o o f 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 b y D r . N . R . J a g a n n a t h a n .  Figure 7.9.  DISPA p l o t s f o r v a r i o u s c o a l samples, (cont. o v e r l e a f ) .  F i g u r e 7. 10. DISPA p l o t s f o r v a r i o u s c o a l samples. ( c o n t . from p r e v i o u s page).  84 SAMPLE NO.  COMMENTS  1  Coronach. Saskatchewan l i g n i t e . contain organic sulphur.  2  Forestburg. pyrite.  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 More o r g a n i c s u l p h u r t h a n a b o v e s a m p l e s .  4  F o r d i n g A d i t 23. S.E. B.C. b i t u m e o u s c o a l . M o r e p y r i t e than above samples, but l e s s o r g a n i c sulphur.  5  S u k u n k a . A l b e r t a b i t u m e o u s c o a l . More p y r i t e , l e s s s u l p h u r than above samples.  6  D e v c o . Nova S c o t i a c o a l . H i g h s u l p h u r c o a l . 2-3% sulphur.  7  Minto.  SW  WOOD SPECTRA  The  s p e c t r a and  various  stages  High p y r i t e c o a l .  of p h o t o - i r r a d i a t i o n a r e 2  width,  (Fig.7.11)  but  slightly  c o n s i s t e n t w i t h two d i f f e r e n t amplitudes  2 4  sulphur  notes  on  shown i n  radical concentration  ( F i g . 7 . 1 3 ) . One  These samples were o b t a i n e d F o r e s t r y department.  may  r a d i c a l s of and  similar  g-values.  then s l o w l y  After  increases decays  original  p o s t u l a t e t h a t one  f r o m L a i Hong o f t h e  but  bitumeous  sulphur.  of the r a d i c a l s  o v e r a number o f d a y s , b a c k t o n e a r t h e concentration  <0.5%  N a t u r a l d e c a y e d wood g i v e s a DISPA  i r r a d i a t i o n t h e p r o p o r t i o n of one ( F i g . 7 . 1 2 ) . The  o f and  7%  may  DISPA p l o t s f o r d e c a y e d A s p e n wood a t  Fig.7.11-Fig.7.13. " spectrum  pyrite  A l b e r t a . Sub-bitumeous c o a l  T a b l e 7.2. I d e n t i f i c a t i o n the c o a l samples.  7.9  <0.5%  radical  UBC  85 i s a c h e m i c a l d e c a y p r o d u c t and t h e o t h e r o f p h o t o l y t i c o r i g i n . As w i t h reveal  the c o a l  very l i t t l e ,  Figure wood.  samples, the o r i g i n a l  spectra  b u t t h e DISPA p l o t s a r e q u i t e d i f f e r e n t .  7.11. DISPA p l o t  for natural  F i g u r e 7.12. DISPA p l o t f o r d e c a y e d after irradiation.  decayed  wood  86  7.10 NITROXIDES I N THE SLOW-MOTIONAL REGIME AND POWDER SPECTRA Both t h e spectra  and t h e 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 ) . I n t h i s c a s e an i n d e x e d p l o t useful  as the absorption  plot  i s unstable.  i s more  However,  neither  type of d i f f e r e n c e  plot  only  f o r c o m p l e t e n e s s a n d t o d e m o n s t r a t e one  of  been i n c l u d e d  i s r e a d i l y i n t e r p r e t a b l e and have  t h e l i m i t a t i o n s t h e DISPA t e c h n i q u e . V a r i a b l e  experments not been  may y i e l d  useful  investigated.  temperature  r e s u l t s , but t h i s approach has  87 BO.PZ  ir  rr  F i g u r e 7.14. DISPA p l o t f o r a powder s p e c t r u m . The s p e c t r u m o f C u P y d t c d o p e d i n t o the corresponding n i c k e l s a l t . Log-index plot.  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 membrane. L o g - i n d e x p l o t .  in a  8.  8.1  CONCLUSIONS  SUMMARY OF RESULTS  LOBE DESCRIPTION  DIAGNOSIS  Lobes of d i f f e r e n t amplitude, but otherwise symmetric  Mis-phased spectrometer or c l o s e o v e r l a p of similar lines  Similar drops  Phase-sensitive-detection a m p l i f i e r time constant too l a r g e  t o above, but r i g h t  T a i l s of lobes  tail  Line truncation  raised going  lobes  Unresolved h y p e r f i n e c o u p l i n g , overmodulation or any d i s t r i b u t i o n i n line-position  Twin negative going  lobes  Superimposed l i n e s , e. g. a d i s t r i b u t i o n of c o r r e l a t i o n times f o r a one s p e c i e s  Twin p o s i t i v e  Asymmetric  See  lobes  Table 8.1. Summary of r e s u l t s DISPA p l o t s .  rules  of thumb  f o r simple  8.2 RULES-OF-THUMB These r u l e s  of thumb e s s e n t i a l l y  summarise the r e s u l t s of  Sect.6. a) W-lobe behaviour  i s characteristic  s u p e r p o s i t i o n of l i n e s of equal resonant different  widths.  88  of a  frequency, but  89  b) G - l o b e b e h a v i o u r  is characteristic  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 l i n e amplitude center  decaying  o f symmetry  of a symmetric  frequencies, with the  as t h e d i s t a n c e of t h e l i n e  from t h e  increases.  c ) C o m b i n i n g a ) and b) a n d r e t a i n i n g a c e n t e r o f symmetry r e s u l t s  i n a mixture  o f G - l o b e and W-lobe  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 d) A s y m m e t r i c l o b e s  result.  from l i q u i d  spectra are ( i nthe  abscence of instrumental a r t e f a c t s ) c h a r a c t e r i s t i c more l i n e s o f d i f f e r e n t w i d t h resonant  or amplitude  o f two o r  at different  frequencies.  e) S o l i d care  behavior.  should  s t a t e s p e c t r a c a n g i v e s i m p l e DISPA p l o t s , b u t  be u s e d when u s i n g t h e g e n e r a l i s a t i o n s a b o v e .  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  simple  a n a l y s i s method t h a t e n a b l e s u s , f o r t h e f i r s t a g e n e r a l , but concrete  time,  their  t h e most o u t s t a n d i n g  sensitivity  Spectra  t o get  grasp 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 Probably  numerical  information.  f e a t u r e o f DISPA p l o t s a r e  t o d e v i a t i o n s from L o r e n t z i a n  that are almost i n d i s t i n g u i s h a b l e , give  behaviour. very  d i f f e r e n t DISPA p l o t s and b e c a u s e we have d e v e l o p e d a s o u n d b a s i s f o r DISPA, t h e s e quantitatively qualitatively  (e.g. (e.g.  p l o t s a r e , i n most  unresolved  cases,  hyperfine coupling)  or  c o a l s a m p l e s ) i n t e r p r e t a b l e . Of more  i m m e d i a t e r e l e v a n c e , DISPA c a n be e x p l o i t e d i n  spin-probe  90 s t u d i e s to assess spectrometer performance i n v e s t i g a t e t h e ESR determine  their  s p e c t r a of paramagnetic  suitablity  species  to  f o r such s t u d i e s .  A l t h o u g h much o f t h e i n f o r m a t i o n obtained  and a l s o t o  by o t h e r m e t h o d s (e.g.  f r o m DISPA c a n  be  s i m u l a t i o n s ) t h e s e methods  a r e 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  often give  s o l u t i o n s . DISPA n e e d s no s p e c i a l e q u i p m e n t  non-unique  (other than a  c o m p u t e r ) o r e x p e r i m e n t s and c a n g i v e a d e f i n i t i v e  solution.  PART  RELAXATION  STUDIES  BY  91  2.  MAGNETIC  RESONANCE  9. INTRODUCTION TO THE MOTIONAL STUDIES  9.1 In  INTRODUCTION r e c e n t y e a r s i t i s h a s become a p p a r e n t , e s p e c i a l l y i n  biochemistry, that  molecular structure  intimately  to function,  as  related  i s not only  but i s p r o b a b l y as important  the 'chemistry'. Unfortunately, there i s a paucity  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  approach, developed i n the l a s t  to r e l a t e molecular The  reorientation  problems, however, a r e q u i t e  to molecular  function  structure.  formidable. F i r s t l y  to r e l a t e the experimentally a c c e s s i b l e correlation  few y e a r s , i s  one h a s  parameters t o the  f o r t h e m o t i o n . T h i s t h e n h a s t o be  related  t o the molecular motion, which f i n a l l y  related  t o m o l e c u l a r g e o m e t r y . E a c h one o f t h e s e s t e p s  constitute  major and c h a l l e n g i n g  Molecular rotation)  reorientation  i n solution  (often  loosly referred studied  response of the s o l u t i o n t o a r a d i a t i o n  the  must be  i.e.,  For such  of t h e molecule i n  s o l u t i o n , t h e i n t e r a c t i o n between t h e f i e l d  field,  t o as  by m e a s u r i n g t h e  field.  the reorientation  m o l e c u l e must d e p e n d on t h e o r i e n t a t i o n that  alone  a r e a s of r e s e a r c h .  i s generally  measurements t o r e f l e c t  h a s t o be  with the  of t h e molecule i n  t h e i n t e r a c t i o n and t h e r a d i a t i o n  field  anisotropic.  G e n e r a l l y t h e response of a system a t a frequency C J , J(CJ), to a perturbing  field,  T i s g i v e n by  92  (50)  93  (9.1)  J(u) = F{G(t) }  where F { x }  d e n o t e s t h e F o u r i e r t r a n s f o r m and  correlation  f u n c t i o n , i s given  G(t)  The  <>  by  tensor property  that interacts  frame. This  t h e two  tensors  i n t o the  with  same  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 e x a m p l e t r a n s f o r m i n g frame we  (9.2)  d e n o t e s an e n s e m b l e a v e r a g e . I t i s  convenient to transform reference  the  = <[T(0).F(0)].[T(t).F(t)]>  where F i s a m o l e c u l a r the f i e l d .  G(t),  to the  molecular  get  (9.3)  where D i s t h e W i g n e r r o t a t i o n m a t r i x and Euler angles r e l a t i n g molecular and  j i s the  v e c t o r and simple  2 5  frame,  'm'  (a/37) a r e  the l a b o r a t o r y ( f i e l d ) and  'k' a r e  rank of the t e n s o r  3 for a tensor).  2 5  the  the  frame t o  the  tensor element i n d i c e s  (1 f o r a s c a l a r ,  2 for a  Tensor products are  quite  i n a s p h e r i c a l b a s i s , hence  S c a l a r s are here.  rotationally  i n v a r i a n t and  a r e o f no  interest  94  »-V  "*?!.<-" V!j'ri,'»E/ [.#7(t)]  (9.4)  t  Note that the time dependence i s c a r r i e d e n t i r e l y by the 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)  <D  -L  {0)  B  {t)  a' L  (9  t>  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 design 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 . The s u b s c r i p t s  ' a ' a n d 'b' i n  Eqn.9.5 denotes 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 only depends on t h e o r i e n t a t i o n o f a given molecule, but also on the o r i e n t a t i o n o f i t s neighbours. 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 correlation function  (intermolecular  i n t e r a c t i o n s do not  influence the rotation o r the molecules interaction with the f i e l d ) . When a * b , g e n e r a l l y ,  then i t i s c a l l e d a  multi-particle correlation function  (neighbouring  molecules  interact). Multi-particle correlation functions are extremely d i f f i c u l t  t o i n t e r p r e t and there  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 t h e m t o s i n g l e p a r t i c l e correlation functions, instance,  w h i c h a r e more m a n a g e a b l e . F o r  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 , w h e r e 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 correlation functions  because o f the c o n t r i b u t i o n s  i n d u c e d d i p o l e s . On t h e o t h e r studies,  where  multi-particle from  h a n d i f we d o i n f r a - r e d ( I R )  95 F  i s t h e t i m e d e r i v a t i v e o f t h e d i p o l e moment, we  obtain  single p a r t i c l e c o r r e l a t i o n functions. Although the dipoles are  coupledfluctuations  i n them a r e n o t ,  or only  weakly  so. The  d i p o l e moment i s a v e c t o r 1  0  spectral d e n s i t i e s , J'  1  and J ^  1  property,  h e n c e o n l y two  are observable.  Light  s c a t t e r i n g o r Raman s t u d i e s , w h i c h i n v o l v e t h e polarisibility, J  1 , 0  ,  7  J '  ±  7  ,  2  a tensor, 0  J ' ,  J  2  ,  ±  i  ,  gives 2  J '  ± 2  five .  spectral densities,  Obviously  t h e more d a t a t h e  better. Magnetic resonance, the subject thesis, it  i s an o b v i o u s c h o i c e  differs  f o rmotional  of the  s t u d i e s . However,  i n a number o f ways f r o m t h e m e t h o d s m e n t i o n e d •  above. There a r e s e v e r a l c h o i c e s (quadrupole, chemical s h i f t discussed  of t h i s p a r t  in detail  f o r the i n t e r a c t i o n tensor  anisotropy  etc.;  these are  i n the next s e c t i o n ) . Furthermore, 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 , t h e o p t i c a l methods a r e restricted and or  t o pure solvents  or concentrated  ESR may be u s e d w i t h d i l u t e  s o l u t i o n s . NMR  s o l u t i o n s (i.e.  ,  10%  solute  l e s s ) and t h e t h e dynamics of s o l u t e s not j u s t neat  solvents  c a n be s t u d i e d . U n f o r t u n a t e l y  a s i n g l e frequency technique.  Unlike  where t h e w h o l e c o r r e l a t i o n f u n c t i o n Fourier  transform  resonance can only frequencies, Also  of t h e J ( w ) , give  the other  techniques  i s a v a i l a b l e ( i t sthe  the line-shape),  magnetic  s p e c t r a l d e n s i t i e s f o r two  the spectrometer  the s e n s i t i v i t y  magnetic resonance i s  frequency and zero  frequency.  of magnetic resonance a r i s e s from i t s  96 selectively can  (ESR  only detects paramagnetic s p e c i e s ) . This  make i t d i f f i c u l t  t o o b t a i n enough  i n f o r m a t i o n t o measure t h e d i f f u s i o n has  been n e c e s s a r y  e.g.  (51),  to  t o c o m b i n e NMR  independent  tensor.  with optical  o r 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 w i t h ESR  studies,  s t u d i e s , e.g.  o b t a i n enough i n f o r m a t i o n . H e r e , f o r t h e  c o m b i n e NMR  In the p a s t i t  first  s t u d i e s t o measure a  time,  (52), we  diffusion  tensor. A f l o w c h a r t of the multi-technique be and  restricted  general  strategy for a  a p p r o a c h i s shown i n F i g . 9 . 1 . T h i s work  t o the  fast motional  t h e Debye d i f f u s i o n m o d e l . The  using simple For  an  hydrodynamic  case  (Redfield  results will  be  will  theory) examined  models.  b v e r v e i w of s t r a t e g i e s i n m o l e c u l a r  (53)(54)(55)  ( o p t i c a l m e t h o d s ) , (56)(57),  s t u d i e s ) and  (58)(59)  (multi-technique  dynamics  (theoretical approach).  see  LINEWIDTH STUDIES  LINESHAPE STUDIES  SLOWMOTIONRL THEORY  Tl EXPERIMENTS  RELAX ATION TIM ES  FASTMOTIONAL THEORIES  REDUCED SPECTRAL DENSITIES  CORRELATION FUNCTIONS  DECONVOLUTION METHODS  BAND-SHAPE STUDIES  OPT I C f l L S T U D I E S  MOTIONAL MODELS  DIFFUSION TENSORS  HYDRODYNAMICS  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 geometric 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 m a g n e t i c r e s o n a n c e .  98  9.2  CHOICE OF  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  chemistry  of the d i t h i o c a r b a m a t e  c o m b i n e NMR for  and  but  exercise  motion of the  or the  spectra  of the  s m a l l and  the  occur.  l o n g a s one  t h e p r o b e and  fairly  purpose  the  the  of t o a.  latter  case  local  spin-probe i t s e l f ) ;  the  results interpreted carefully I n our  case p e r t u r b a t i o n  i t s interaction with  structure probe as  is irrelevant  i n terms of  the  nature  the  and  does  solvent  t o the o v e r a l l  of not  structure  solvent.  stable. Their  are  relatively  small s i z e minimises  spin-labelling  small  and  perturbation  t h e y h a v e been e x t e n s i v e l y u s e d f o r  s p i n - p r o b e and  of  spin-probe in  ( d i s t o r t i o n of t h e  n i t r o x i d e type spin-probes  p r o b l e m s and  that the  from the concept  i n t e r p r e t s the data  m o t i o n of t h e The  advantages  d i f f e r e n t from s t u d y i n g  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 and  the  probe.  s p i n - l a b e l . In the  i s important  a r t e f a c t s can  out  tensor  the m o t i o n of the m a c r o m o l e c u l e i s deduced  l o c a l m o t i o n by  must be  spin  to  of probe i s  whereby a s p i n - p r o b e i s a t t a c h e d  m a c r o m o l e c u l e and  perturbation  to discuss  t h e m o t i o n of t h e  s o l v e n t , or  diffusion  o r i g i n a l choice  be p o i n t e d  is entirely  spin-labelling,  from the  measure the  c l a s s e s of  i s to study  flexible  c l a s s of s p i n - p r o b e and  i t i s appropriate  i t should  s o l u t i o n . This  as  s t u d i e s and  t h i s probe over other Firstly,  the  ESR  t h e p r o b e s i n s o l u t i o n . The  historical, of  S P I N PROBE  biological  s t u d i e s . However, f o r  'pure'  99 motional  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  large unresolved line-width  hyperfine  coupling  contribution to the  ( w h i c h may b e o v e r c o m e b y d e u t e r a t i o n ,  is generally very tedious  t o do);  but this  they do n o t have  simple  geometries, by v i r t u e o f t h e p r o t e c t i n g groups; t h e line-widths  have a r e l a t i v e l y l a r g e s p i n  rotation  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 times u n r e l i a b l e ; the hyperfine  anisotropy  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 small so that  i s small  (see  Eqn.20.2),  this also decreases the r e l i a b i l i t y of the determined c o r r e l a t i o n times;  they have a three  l i n e spectrum so that  the 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 l i n e s a r e needed, three t o g e t t h e tensor  four  and one f o r t h e  s p i n - r o t a t i o n term); and f i n a l l y there a r e no r e a d i l y a v a i l a b l e diamagnetic analogs, used t o provide  more  t h u s NMR s t u d i e s c a n n o t b e  information.  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 spin-probes  c a n be overcome 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 copper acetyl-acetonate Unfortunately  by K i v e l s o n  complexes  et al who u s e d  (60)(61)(62)(63).  these complexes have very l a r g e g a n i s o t r o p i c s  so 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 term, a s  w i t h n i t r o x i d e s , and a l s o by l i n e o v e r l a p . A l s o t h e chemistry  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.  d i f f i c u l t t o change t h e i r geometry i n a systematic However, these  i t is manner.  c o m p o u n d s 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 , w h e r e L = 0 (64), N (65) (66), S (67) (68) (69) (70),  100 Se  (71) a n d 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  b a s e d l i g a n d s g i v e c o m p l e x e s 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 a n d Se b a s e d l i g a n d s a l l g i v e s p l i t t i n g s . MSi, t y p e  Of  axially  hyperfine  compounds have n e i t h e r o f t h e s e  p r o b l e m s and a l l have s i m i l a r (i.e.,  (undesirable)  spin Hamiltonian  parameters.  s y m m e t r i c t e n s o r s w i t h A =*80G a n d g - 2 . 0 4 ) . o  o  t h i s c l a s s o f compounds, t h e d i t h i o c a r b a m a t e s  are 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 , isotopic  and o t h e r w i s e .  (44) (69) (70)(75)(76)(77)(7  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 8)(7 9)(80) (81)  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  pioneered  et al. (82)(83)(84).  via  by H e r r i n g  s e c o n d a r y amines  26  isotopic  c o m p l e x e s was Preparation i s  s u b s t i t u t i o n s . The l i g a n d s  form complexes w i t h a wide range o f m e t a l s so t h a t  m u l t i n u c l e a r NMR o f t h e d i a m a g n e t i c with these study  very  (see Sect.11) which a r e r e a d i l y  available with various also  and g e n e r a l l y  complexes i s p o s s i b l e  compounds. D i t h i o c a r b a m a t e s  biological  have been u s e d t o  s y s t e m s (86). A l s o , a t t e m p t s h a v e been made  to 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 studies  2 6  (74).  G i b s o n (85) was t h e f i r s t t o do m o t i o n a l s t u d i e s o f a d i t h i o c a r b a m a t e , b u t he u s e d m i x e d c o p p e r 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 o f 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  C H O I C E OF PROBE S U B S T I T U E N T S  The  general  formula  R'',  R''',  t h e same. The  Typical metal  9.2.  Figure  R',  for a metal (II) dithiocarbamate  R'''' R',  R''  a n d R''',  molecular The  R'''  1  p a i r s may  i n a p p e n d i x 22.1). T h e s e  geometries. requirements f o r the c h o i c e  complex should eliminate  soluble  of R a r e ; the  ( f o r t h e NMR  studies),  relaxation contributions  be s t a b l e  m o i e t y has  compound the  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  2 7  and  f o r d e u t e r i u m NMR  plane  and  these  and  complex  s t u d i e s , the  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  bonds l i e out of the MS„  (to  from i n t e r n a l motion  t o s i m p l i f y the use of h y d r o d y n a m i c m o d e l s ) , the  27  and  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 u s t be r e a s o n a b l y  should  are,  constitute a  for the R s u b s t i t u e n t s  nomenclature are given  s u b s t i t u e n t s may  dithiocarbamate.  are not n e c e s s a r i l y , but u s u a l l y  c y c l i c group. (Abbreviations general  is  bonds should  alkyl C-D not  The p y r r o l e d e r i v a t i v e has 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). Generally though d i t h i o c a r b a m a t e s complexes a r e e x t r e m e l y s t a b l e and a r e e x t e n s i v e l y used i n a n a l y t i c a l chemistry (79).  be  1 02  r e l a t e d by  symmetry  (see  Sect.20.6).  In g e n e r a l , s m a l l or r i g i d d e r i v a t i v e s t h a t have low methyl  derivative  f o r NMR  has  the r e l a t i v e l y  s o l u t i o n , a s do 11.2),  s t u d i e s (82)  C  was  work r e q u i r e s a  a poorly defined structure (in  very  s m a l l due  i n Table  wish  to s t e r i c  the  hindrance.  s t u d i e s (88)(84)  p y r o l l i d i n e d e r i v a t i v e s behave  CHOICE OF  that  similarly.  to maintain  simple  geometries  to a s s i s t s h o u l d be  the chosen  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  t o d i v a l e n t m e t a l s . F o r ESR, paramagnetic,  f o r NMR,  a r e needed, but  t h e c o m p l e x must  diamagnetic.  Clearly  choice  be  two  different  t h e r e s u l t i n g c o m p l e x e s must  be  isostructural.  9.4.1  CENTRAL METAL FOR The  the  CENTRAL METAL  a n a l y s i s of the d a t a , the c e n t r a l metal  metals  was  but m o l e c u l a r models i n d i c a t e t h a t the m o t i o n of  e t h y l and  such  i s too  soluble ethyl derivative  A l s o i t i s known f r o m p r e v i o u s ESR  As we  1 3  The  (it is  t h e most o f t h e o t h e r d e r i v a t i v e s  e t h y l groups i s probably  9.4  11.2).  pyrollidine derivative  s t u d i e s , but  u s e d . T h i s d e r i v a t i v e has  (see T a b l e  a s i m p l e g e o m e t r y ) , but  s t u d i e s . The  somewhat b e t t e r f o r NMR c o m p r o m i s e and  solubilities  i s g o o d f o r ESR  r e a d i l y d e u t e r a t e d and insoluble  substituents give  c e n t r a l metal  ESR  EXPERIMENTS  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 spin  103 r o t a t i o n t e r m r e q u i r e s t h a t we must have a t l e a s t  four  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 c o p p e r and v a n a d i u m . U n f o r t u n a t e l y t h e v a n a d i u m complexes o x i d i s e  readily  t o V=0  type complexes  (74),  which are not very s o l u b l e . A l s o the geometric simplicity  i s d e s t r o y e d . The c o p p e r c o m p l e x e s a r e 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 . C o p p e r h a s two 6 3  C u and  9.4.2  6 5  Cu.  The  6 3  C u was  CENTRAL METAL FOR  NMR  isotopes,  used f o r h i s t o r i c a l r e a s o n s .  EXPERIMENTS  A metal with a zero nuclear spin  i s u s e f u l , but not  n e c e s s a r y . O b v i o u s c a n d i d a t e s a r e N i , Zn a n d P d .  The  z i n c c o m p l e x e s a r e d i s t o r t e d t e t r a h e d r a l and a r e 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 n o t u s e d . P a l l a d i u m i s i n t e r e s t i n g b e c a u s e i t h a s a n o n - z e r o s p i n and c a n be s t u d i e d d i r e c t l y Unfortunately  i t s gyromagnetic r a t i o  e x p l o i t with the a v a i l a b l e  by  NMR.  i s too small to  s p e c t r o m e t e r s (CJ -10MHZ a t o  4 . 7 T ) . The n i c k e l c o m p l e x e s a r e i s o m o r p h o u s w i t h t h e c o p p e r c o m p l e x e s , have z e r o s p i n and t h u s were u s e d . H o w e v e r , t h e y a r e much l e s s s o l u b l e t h a n t h e c o r r e s p o n d i n g copper complexes c o m p l e x e s ) and t h i s  (as a r e t h e p a l l a d i u m  somewhat l i m i t s  t h e s e c o m p l e x e s f o r NMR  studies.  the u s e f u l n e s s of  10.  GENERAL THEORY  M o t i o n a l r e l a x a t i o n theory f o r magnetic resonance i s d o m i n a t e d 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 Kubo-Tomita t h e o r y as developed stochastic Liouville  by K i v e l s o n (93), a n d t h e  t h e o r y a s e x p l o i t e d by F r e e d  Kubo-Tomita t h e o r y has a r e l a t i v e l y interpretation, unwarranted  transitions  converging  physical  its ability  t o handle  (94). S t o c h a s t i c L i o u v i l l e  probably the d e f i n i t i v e slow-motion  easy  (39).  b u t i t s p o p u l a r i t y h a s waned due t o t h e  controversy surrounding  degenerate  (90)(91)(92),  relaxation  theory i s  theory and can handle t h e  c a s e . However, i t i s b a s e d on a s l o w l y series,  f o r which  i t i sdifficult  to assign  p h y s i c a l meaning. A l s o t h e computer time r e q u i r e d f o r these calculations  i s not j u s t i f i e d  cases. R e d f i e l d series  relaxation  t h e o r y i s b a s e d on a r a p i d l y  converging  (for short c o r r e l a t i o n  interpretation relaxation. first  f o r the simple  2 8  and i s w e l l s u i t e d  p r i n c i p a l problem  regime.  with a l l relaxation  the e v a l u a t i o n of t h e c o r r e l a t i o n motion.  f o r ESR s t u d i e s i f  a r e u s e d where n e c e s s a r y , b u t  be u s e d i n t h e s l o w - m o t i o n a l  The  physical  f o r d e s c r i b i n g NMR  I t c a n be r e a d i l y a d a p t e d  order wavefunctions  cannot  times) with a simple  theories .is  function for molecular  S e v e r a l a p p r o a c h e s h a v e been made t o t h i s  problem  (50)(96),  b u t a l l t h e s e t h e o r i e s c o n v e r g e t o t h e Debye  diffusion  case  i n the fast motional l i m i t . Also the theory  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  2 8  f o r t h e Debye  An e x p a n s i o n t o , a n d 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 t e r m s i s g i v e n by S i l l e s c u a n d K i v e l s o n (95) 104  1 05 d i f f u s i o n . For  t h a t reason  extensions w i l l  o n l y Debye d i f f u s i o n and i t s  be d i s c u s s e d i n t h i s w o r k . R e v i e w s o f  o t h e r a p p r o a c h e s a r e g i v e n by S t e e l e (96)  10.1  INTRODUCTION TO  Redfield  and M c C l u n g  f o r nuclear 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). ESR  can  and  Freed  be  found  R e d f i e l d theory  i n p a p e r s by F r e e d a n d  Fraenkel  i n h o m o g e n e i t y makes v e r y  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 s o f ESR  spectra  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 's 2  spectral field  l i n e w i d t h s . T h i s however, i s not  inhomogeneity i s the p r i n c i p a l  broadening.  I n t h i s c a s e T,'s  of R e d f i e l d t h e o r y outline  for T/s  in detail  (Sect.12  Sect.16).  |a>,  and  T 's 2  and  source  (91)  of  and  the  NMR  where  line development  i s t h e same and  a  brief  differ  to merit separate d i s c u s s i o n s .  element of the r e l a x a t i o n m a t r i x  i s g i v e n by  from  and  so f o r NMR,  a r e m e a s u r e d . The  i s g i v e n b e l o w . However ESR  sufficiently  An  (90)  (98).  G e n e r a l l y magnetic f i e l d little  (50).  REDFIELD THEORY  r e l a x a t i o n theory  for  the  (R)  for a  state,  29  F r e e d (98) d r o p s t h e f a c t o r o f 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 o f a h a l f i s o f t e n i n t r o d u c e d . 2 9  106  (r) aa' 00' ~  2J  aaP' 0  (w  a' a00 • 0/3'  P' P  )  "  J  a' aP' p *P' 0 ((  (r)  - J  aa'  0/3'  (10.1)  aa00  0/3  where t h e p r i m e s d e n o t e t h e c o r r e s p o n d i n g u p p e r s p i n 0 r e p r e s e n t s any o t h e r will  spin  state  d e p e n d on t h e r e l a x a t i o n  represents a l l other spin hyperfine  splittings)  i n the system  rate  states  i n the system  (e.g. f r o m  and takes i n t o account the f a c t  r e l a x a t i o n pathways a r e not n e c e s s a r i l y  the  e x c i t a t i o n pathway  follows.  ( i t s choice  required) and 7  the  densities  (the leading  f o r each t r a n s i t i o n  states,  that  the reverse of  t e r m ) . The s p e c t r a l  f r e q u e n c y , to, a r e g i v e n a 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 o f 1/2 h e r e a s i t  s i m p l i f i e s the spectral densities  i n the case of i s o t r o p i c  diffusion):  aa' PP'  = F{G  where F { X } d e n o t e s t h e F o u r i e r  F{G(t)} = £  G(t)e"  aa* 00  (10.2)  transform.  / w r  dt  (10.3)  107  The  correlation  function G(t) i s  = <H,(t)  .H,(t+r)  (10.4)  where H, i s t h e 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 e n s e m b l e a v e r a g e . D r o p p i n g  convenience,  a a ' a n d 0/3' f o r  t h e time dependent p e r t u r b a t i o n  H, ( t )  i s g i v e n by,  (10.5)  + e(t) X  The  X i s used t o l a b e l t h e d i f f e r e n t  Hamiltonian; hyperfine coupling, c o u p l i n g etc. The p a r a m e t e r  type of i n t e r a c t i o n  spin-rotation,  quadrupole  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 . I n 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 . I n ESR  t h i s t e r m may be d r o p p e d  s c a l e much f a s t e r  than T  to the r e l a x a t i o n  (82).  The  2  a s i t o s c i l l a t e s on a t i m e  and t h e r e f o r e does n o t c o n t r i b u t e  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) a n d m a g n e t i c  H(Mt)  It  =.Z.A  rX;  interaction  t e n s o r s (F) so  X;  Ff (t)  i s more c o n v e n i e n t t o d e f i n e t h e t e n s o r s i n a  spherical  basis  (90)(99)(100)(101)  so t h a t  (10.6)  108  (-nV  m) (l  -Z .A. .F: . ( t ) = I i,j  ij  H e r e we laboratory frame.  3 0  molecular  rank  o f t h e t e n s o r and m=-/  latter  coincident. This  where a,0,7  —>  +/ i n the molecular  thus c a r r i e s the time dependence f o r magnetic  interaction  tensor i s  i n t o t h e l a b o r a t o r y frame w i t h t h e  that the m o l e c u l a r  matrices  (10.7)  the magnetic t e n s o r s i n the  r o t a t i o n . The  transformed  ^  have d e f i n e d t h e s p i n o p e r a t o r s  f r a m e and  The  , -m)  m  ij  where / i s t h e  F  and  magnetic  interaction  assumption  tensors  i s r e a d i l y done w i t h W i g n e r  are  rotation  (99)(103)(104).  are the E u l e r angles  r e l a t i n g the  molecular  f r a m e component q w i t h t h e l a b o r a t o r y f r a m e component m. Euler angles  c a r r y the time dependence of t h e r o t a t i o n .  m a g n e t i c r e s o n a n c e we  o n l y have s c a l a r  respectively) interactions.  As  scalar  and  tensor  interactions  3 0  (/=0  The For  and  2  are  The c h o i c e o f t r a n s f o r m i n g t h e 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 t h e e i g e n v a l u e s o f q u a n t u m 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 leave the o p e r a t o r s i n the 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 o f Eqn.10.7 (102), the main consequence of t h i s i s whether t h e ( - 1 ) term i n Eqn.10.7 i s a c o e f f i c i e n t o f A o r F . m  109  rotationally relaxation. we g e t f r o m  r  ^  where  invariant  t h e y do n o t c o n t r i b u t e  So, d r o p p i n g t h e / ( i m p l i c i t l y  ao'  (m)  i  t  type of i n t e r a c t i o n s  (m)  = <a\h \a'>.  t e n s o r s obey t h e f o l l o w i n g  (l,m)*  =  K  <a|A|a'>  <W* aa'  similarly  (  (10.10)  symmetry  relations  _ m r/.-m; 1 )  A  (10.11)  = <a'|A Ia> a s A i s H e r m i t i a n h e n c e  A  and  i t to 2),  d e f i n e d a s X) a n d  K  also  p/3  v and u denote t h e d i f f e r e n t  Spherical  setting  (Eqn.10.4-Eqn.10.8)  mm'qq'  (previously  t o the  m  {  .^m (-m) a' a K  f o r F a n d t h e 00'  t e r m s s o Eqn.10.9  (10.12)  becomes  110  G .  (  ,<t> = Z  M f l f l aa pp  mcj  t h e m' a n d  J»  A Q  ,  K-  <D* ( t ) D ( t + r ) > F ^ V " ^ ( 1 0 . 1 3 )  m)  m  M^,Q  terms d i s a p p e a r  m  q  q  m  v  u  because of theo r t h o g o n a l i t y  r e l a t i o n s of t h e Wigner r o t a t i o n m a t r i c e s  (99)(104).  Note  t h a t t h e 00' s u b s c r i p t r e v e r s e s b e c a u s e o f E q n . 10.12 It  i s convenient  t o d e f i n e a reduced  correlation  f u n c t i o n , g^ mq  qmq = < Dmq * ( t ) Dmq( t + r ) >  (10.14)  mn  and a l s o a reduced time dependent  s p e c t r a l d e n s i t y (as t h i s  i s the only  term),  j mq (CJ) = F { gmq ( t ) }  (10.15)  where F { x } i s t h e F o u r i e r t r a n s f o r m a s d e f i n e d b e f o r e (Eqn.1.8). F i n a l l y we g e t  J  ,««.(«) = Z  aa'00'  v,n  (m)  Z K  m,q  and hence from Eqn.10.1, R  v  a  (  a f l  a  ,  m)  (  q)  (q)  h ~ i(a>)F - F n^.^rhq  « ^ .  v  u  (10.16)  111  Explicit tensors  interaction  ( F ) and t h e s p i n o p e r a t o r s (A) a r e g i v e n i n  (90)(105) ESR  e x p r e s s i o n s f o r the magnetic  a n d a p p e n d i x 22.6.  E x p a n s i o n of Eqn.10.16 f o r the  case i s d i s c u s s e d f u r t h e r The  e v a l u a t i o n of the c o r r e l a t i o n  r e m a i n s one  function  (Eqn.10.14)  o f t h e most c h a l l e n g i n g a s p e c t s o f m o l e c u l a r  d y n a m i c s . The magnetic  i n Sect.12.3.  b a s i c goal of m o l e c u l a r dynamics  resonance  i s t o d e d u c e g(co) f r o m  studies  J(CJ),  by  which, i n  p r i n c i p l e c a n 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 m e a s u r e m e n t s .  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 to  m e a s u r e a s t h e y a r e i n d e p e n d e n t o f t h e m o t i o n a l model  used. T h i s g r e a t l y from d i f f e r e n t  facilitates  comparisons between  Also i t i s usual to assign r e s u l t s and  Reviews  (106).  some p h y s i c a l m e a n i n g t o t h e  i n s t e a d . T h i s g e n e r a l l y hampers  correlation comparisons  s t u d i e s and t e c h n i q u e s . 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 f o u n d i n (50)(96). in  variables.  so a d i f f u s i o n m o d e l i s assumed a n d  times c a l c u l a t e d different  results  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 t h a n i n d e p e n d e n t  of  quantities  Three  U s e f u l d i s c u s s i o n s c a n a l s o be  found  f a c t s emerge f r o m t h e s e r e v i e w s ; t h e Debye  d i f f u s i o n m o d e l and  i t s e x t e n s i o n s a r e t h e most  model;  of f a s t m o t i o n or i s o t r o p i c m o t i o n a l l  i n the l i m i t  m o d e l s r e d u c e t o t h e Debye m o d e l ;  and c u r r e n t l y  successful  only  Debye m o d e l i s d e v e l o p e d f o r t h e c a s e o f a n i s o t r o p i c  the motion.  1 12 For  t h e s e r e a s o n s t h e Debye m o d e l i s t h e most w i d e l y  rotational  diffusion  theory.  The Debye m o d e l 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 u n d e r g o a random s m a l l s t e p rotation  used  ( r o t a t i o n a l Brownian motion) about  independent axes ( u s u a l l y ,  angular three  but not n e c e s s a r i l y  a x e s ) . The r a t e s o f r o t a t i o n  are characterised  the molecular 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^ a n d R^, t h e p r i n c i p a l e l e m e n t s o f the  rotational  diffusion  t e n s o r , R. T h i s d i f f u s i o n  c o n s i d e r e d t o be i n d e p e n d e n t translational diffusion  {i.e.,  tensor i s  not coupled t o ) the  t e n s o r . The p r o b l e m i s t o r e l a t e  these 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 j(o>). F o r i s o t r o p i c Debye d i f f u s i o n  densities,  ( t h e most commonly  used  m o d e l f o r NMR), j ( u ) i s g i v e n by  j(a>) =  -c  (10.17)  c  1 + (CJT )  where T  c  2  i s known a s t h e c o r r e l a t i o n  t i m e a n d T =1/6R, where 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 . I n t h i s c a s e j(a>) i s referred  t o a s t h e Debye s p e c t r a l  confused with the spectral For  anisotropic  d e n s i t y , n o t t o be  d e n s i t y , J(a>).  m o t i o n we g e t f o r j ( w )  (98)(107)  113'  j  /  f  ( u )  = .L,  = X..(w)  -Ik-Ik 1  +  (  c  J  T  //t  )  (10.18)  2  where  are a combination of t h er o t a t i o n a l  constants  and the T ' S are the eigenvalues  diffusion  f o r the  rotor which are combinations of t h er o t a t i o n a l constants. densities  Explicit  3 1  will  diffusion  expansion of t h e reduced s p e c t r a l  be g i v e n  i n the a p p r o p r i a t e  section.  G e n e r a l l y we c a l c u l a t e t h e X t e r m s d i r e c t l y least  squares f i t of the r e l a x a t i o n times.  methods t o g i v e t h e d i f f u s i o n  Newton-Raphson  tensor.  CHOICE OF THE AXIS SYSTEM  An e s s e n t i a l p a r t o f R e d f i e l d t h e o r y theory  a n d Debye  diffusion  i s t e n s o r s . As t h e t e n s o r s a r e n o t n e c e s s a r i l y  diagonal reference  i n t h e same frame  i t i s convenient t o introduce a  f r a m e o r a x i s s y s t e m ; a frame  one o f t h e t e n s o r s Assignment  i n which a t least  i s diagonal.  of the a x i s system  of a number o f r e f e r e n c e 3 1  o r by a  The r e d u c e d  s p e c t r a l d e n s i t i e s c a n be t h e n i n v e r t e d u s i n g  10.3  asymmetric  i s c o n f u s e d by the  frames. I n our case t h e r e are  choice five  T h e r e i s some c o n f u s i o n o f n o m e n c l a t u r e 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) a s d e f i n e d by Eqn. 1 0 . 2 , j ( w ) i s t h e ' r e d u c e d s p e c t r a l d e n s i t y ' a s d e f i n e d by E q n . 1 0 . 1 5 . The t e r m '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 c a s e o f 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 t h e 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 d e n o t e d b y X ( f o l l o w i n g Freeds 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 the  magnetic  f r a m e . We inertial  frame, the d i f f u s i o n  a r e not i n t e r e s t e d frame  i s not u s e d .  g i v e a time dependent We not  frame, the m o l e c u l a r frame,  are t r y i n g  3 2  frame and t h e  in inertial The  diffusion  laboratory  a good c h o i c e f o r a r e f e r e n c e frame  would  i s not  used.  tensor, thus t h i s i s  f r a m e . However, we  do  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 and c h e m i c a l s h i f t  so t h e  frame  t e n s o r so t h i s  t o measure t h e d i f f u s i o n  assume t h e d i f f u s i o n  models  inertial  frames  ( g , A,  quadrupole  t e n s o r s ) . I n o u r s y s t e m t h e 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 e a c h o t h e r and w i t h t h e molecular  f r a m e . As t h e c h o i c e o f m o l e c u l a r frame i s  somewhat a r b i t r a r y it  (because of t h e h i g h degree of symmetry)  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 t h e g and A t e n s o r s  ( w h i c h by c o n v e n t i o n a r e d e f i n e d  such t h a t  I i n a right-handed coordinate  |A 1  s y s t e m ) . The  |>|A ZZ  1  1  |>|A XX  1  1  3  ^ '  a x i s system i s thus  3 2  The i n e r t i a l frame i s o f r e l e v a n c e t o t h 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 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 = I * I . So t h a t c h o i c e o f a x e s i n t h i s f r a m e w o u l d be e s s e n t i a l l y a r b i t r a r y . y  Z  x  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 s y s t e m 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.  , the x a x i s i s p a r a l l e l  i n t h e MS axis  4  plane  and  i s perpendicular  t o the O N  perpendicular t o t h e MS„  used t o d e f i n e the E u l e r a n g l e s notably the chemical A simplistic  t o t h e C-N  the quadrupole  R, <-w  33  i s the  of  frame, but w i t h R  >R  =*R yy  . R zz  xx  tensor  r o t a t i o n a l d i f f u s i o n constant  for  XX  r o t a t i o n a b o u t t h e x a x i s etc. 3 3  tensor.  tensor  are the p r i n c i p a l elements of the d i f f u s i o n where R  be  of r o t a t i o n a l d i f f u s i o n  would g i v e a d i f f u s i o n  z  tensors,  xx  etc.  the  p l a n e . T h i s frame w i l l  t e n s o r and  reference  bond and  for a l l other  interpretation  the d i t h i o c a r b a m a t e s c o i n c i d e n t t o the  shift  bond, the y a x i s i s  This  i s important  t o note  as  R, r a t h e r t h a n D, i s u s e d h e r e . D b e i n g r e s e r v e d f o r t h e 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 d e n o t e 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 . F u t h e r m o r e t h i s R s h o u l d n o t 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 molecular R  zz  >R  xx  >R  and d i f f u s i o n  that the  frames a r e c o i n c i d e n t  (i.e.,  ) a n d do n o t a l l o w s u c h p r o m i s c u o u s a s s i g n m e n t o f  yy  c  the r e f e r e n c e  3  frame.  10.4 HYDRODYNAMIC MODELS FOR ROTATIONAL DIFFUSION According constant  t o t h e hydrodynamic model  the rotational  diffusion  a b o u t a g i v e n a x i s ' i ' i s g i v e n by (106)  R:  where L^. i s t h e t o r q u e  1  = \i*  (10.19)  about t h e ' i  1  t h a x i s a n d i s g i v e n by  L. = \.t(V)  (10.20)  where X^. i s t h e f r i c t i o n c o e f f i c i e n t a n d f ( V ) i s a f u n c t i o n of m o l e c u l a r i t s volume,  geometry. F o r a sphere, f ( V ) i s p r o p o r t i o n a l t o f o r an e l l i p s o i d  (or other  shapes) t h e  r e l a t i o n s h i p i s more c o m p l e x . The i m p o r t a n t to note that the d i f f u s i o n molecular  i sdirectly  a r e c a l c u l a t e d by s o l v i n g t h e S t o k e s - N a v i e r  f o r t h e 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 d a t e o n l y e l l i p s o i d s h a v e been but  related t o the  geometry.  The t o r q u e s equation  tensor  p o i n t though i s  s o l u t i o n s f o r more c o m p l e x  considered,  shapes a r e , i n p r i n c i p l e ,  11 7 possible. Solutions  f o r two  t y p e s of b o u n d a r y  h a v e a l s o been i n v e s t i g a t e d ; t h e and  the  'stick*  the  by  r o t a t i o n ) . This  P e r r i n (108)  'slip'  solvent  and  to the  p r o b l e m was  later  t o the volume of  the  solvent  done by Hu  Acrivos The and  and  Z w a n z i g (110)  s t i c k model i s f a i r l y  coefficients the  for e l l i p s o i d s are tables d i f f e r  m o l e c u l a r v o l u m e has to note that  The  solvent  damping i s  due  r o t a t i o n . This  s o l u t i o n and  a l s o Youngren  successful with T a b l e s of  given  slightly  been d e f i n e d  s t i c k models g i v e  rough spheres w i t h  like  results  in  this and  identical  ionic  species  friction (110)(111)(106).  b e c a u s e o f t h e way  (106).  f o r extreme g e o m e t r i e s  Also  3  paper  (111).  3  s l i p and  scale  in a classic  during  and  there  time  F a v r o (109).  a numerical  for t r a n s l a t i o n a l d i f f u s i o n . *  Note t h a t  the  frictional  swept a s i d e  boundary c o n d i t i o n r e q u i r e s was  i.e.,  j u s t assumes t h a t the  s o l u t e , the  the  the  zero,  (on  solved  r e p e a t e d by  boundary c o n d i t i o n  doesn't penetrate  solute  condition  l a t t e r case  molecules at  i n t e r f a c e i s assumed t o be  i s a s o l v a t i o n cage s t u c k of the  boundary  boundary c o n d i t i o n . In the  t a n g e n t i a l v e l o c i t y of solvent-solute  'slip'  conditions  the  It is interesting  ( d i s k s and friction  needles)  the  coefficients.  s l i p boundary c o n d i t i o n s  give  stick  (114).  * 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 t h e s t i c k b o u n d a r y 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 o f two (113); swept v o l u m e i s t h e o n l y i m p o r t a n t parameter i n t h i s case.  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).  T h e s e o t h e r m o d e l s a r e more e l e g a n t  make r e a l i s t i c  the solvent i s continuous  l e v e l ) . However, t h e y a r e f o r s p h e r e s  c a n n o t be e x t e n d e d t o o t h e r g e o m e t r i e s . little  use.  they  assumptions about the the s o l v e n t  ( h y d r o d y n a m i c m o d e l s assume molecular  i n that  As s u c h  at the  o n l y and they a r e of  11. GENERAL The  experimental  EXPERIMENTAL  m e t h o d s c a n 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 a n d NMR  differ  c o n s i d e r a b l y and  are discussed 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 t h e s p i n - p r o b e s dithiocarbamate identical  (transition  c o m p l e x e s ) and t h e i r  f o r ESR a n d NMR  s o l u t i o n s i s almost  and i t i s c o n v e n i e n t  i n one ( t h i s ) c h a p t e r . E a c h s p i n - p r o b e  For  the  1 3  C  studies 6 3  F o r ESR s t u d i e s  11.1  1 3  C  to discuss i t  i s isotopically  s u b s t i t u t e d as a p p r o p r i a t e . F o r t h e d e u t e r i u m studies perdeuterated  metal  relaxation  a m i n e s were u s e d i n t h e p r e p a r a t i o n s . enriched carbon  d i s u l p h i d e was u s e d .  C u was u s e d a s t h e c e n t r a l  metal.  PREPARATION OF SODIUM DITHIOCARBAMATES  AND  CARBODITHIOATES All  compounds were p r e p a r e d  mixing  of  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 (69)  by t h e s t a n d a r d p r o c e d u r e  and t h e a p p r o p r i a t e secondary  amine  (115).  CS  The  2  + KOH + R N H  dithiocarbamate  2  ->  R NCSi + K  derivative  to the xanthate. D e t a i l s are given  119  +  2  i s formed below.  +  H 0 2  preferentially  1 20 The  secondary  alcoholic  sodium  disulphide  amine (O.lmmol) was d i s s o l v e d i n  hydroxide  (100ml  o f 0.1M). C a r b o n  (0.1M) i n e t h a n o l (^SOml) was t h e n a d d e d  dropwise  o v e r a p e r i o d o f 30 m i n . t o t h e s t i r r e d m i x t u r e , (n.b. t h e order of a d d i t i o n react with CS double  salt  The  2  i s c r i t i c a l a s b o t h KOH a n d t h e amine  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 t h e amino  (115) r e s p e c t i v e l y ) .  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  f r o m h o t e t h a n o l o r by p r e c i p i t a t i o n  by r e c r y s t a l l i s 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 w e r e more than t h e c o r r e s p o n d i n g sodium difficult high  to purify  soluble  s a l t s a n d h e n c e were more  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  (=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  stored  u n d e r n i t r o g e n i n t h e d a r k a t =-20°C t o m i n i m i s e decomposition. All hydrogen checked  p e r - d e u t e r a t e d s a l t s were c h e c k e d by NMR  end  (none was d e t e c t e d ) . P u r i t y  via a n a l y s i s o f t h e c o p p e r  below T a b l e product.  for residual  or n i c k e l  o f t h e s a l t s was s a l t s (see  11.1 b e l o w ) a s t h e s e compounds a r e t h e d e s i r e d  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. ( O t h 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)).  Listing  i s by  parent amine. * dimethylamine 1 5  N  dimethylamine  d -dimethylamine * diethylamine * 3  pyrollidine d -pyrollidine 9  All  s u b s t i t u t i o n at the CS  2  All  G  g r o u p . 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 , prepared  1 3  compounds were p r e p a r e d w i t h and w i t h o u t  b u t were  f o r use f o r m i c r o a n a l y s i s . s t a r t i n g m a t e r i a l s are commercially  ( A l d r i c h and Merck, Sharpe and  available  Dohme).  11.2 TRANSITION METAL DITHIOCARBAMATES T h e s e a r e s i m p l y p r e p a r e d by m i x i n g a q u e o u s s o l u t i o n s o f t h e appropriate t r a n s i t i o n metal  s a l t s and the p o t a s s i u m  dithiocarbamate. Further d e t a i l s are given i n Sect.13. c o p p e r s a l t s were p r e p a r e d  The  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 microanalysis,  3 5  (see Table  11.1)  1 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 . N and a r e n o t 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 u s e f o r a n a l y s i s i s an e x p e n s i v e w a s t e .  1 3  C  122  COMPOUND  C  H  N  CuPydtc  33.98(33.78)  4.44(4.45)  7.87(7.88)  CuEt dtc  33. 17(33.36)  5.60(5.60)  7.80(7.79)  d -NiPydtc 9  32.36(32.70)  NiPydtc  34.20(34.20)  2  7.47(7.63) 4.59(4.59)  8.03(7.98)  Table 11.1. Microanalyses for D i t h i o c a r b a m a t e s . ( ) denotes c a l c u l a t e d v a l u e . Other numbers are the a n a l y s i s . The analyses were performed by P.Borda of the UBC Chem. Dept.  11.3  PREPARATION OF SOLUTIONS  Toluene  was  used  i n p r e v i o u s s t u d i e s (116),  but the n i c k e l  d e r i v a t i v e s are not s o l u b l e enough i n t h i s s o l v e n t to permit NMR  r e l a x a t i o n s t u d i e s . Other  (see Table  11.2)  s o l v e n t s were i n v e s t i g a t e d  and c h l o r o f o r m was  found to be the best  compromise. Accurate v i s c o s i t y data are a v a i l a b l e f o r c h l o r o f o r m (117).  P r e v i o u s s t u d i e s (43)(42)(88)  that the r e l a t i v e l y low  indicate  b o i l i n g p o i n t (70 °C) i s not  too  restrictive. U n f o r t u n a t e l y c h l o r o f o r m r e a c t s with dithiocarbamates  i n the presence  metal  of water (118)  and  i t is  e s s e n t i a l t h a t the s o l v e n t i s dry. All  samples were s e a l e d under vacuum a f t e r  freeze-pump-thaw c y c l e s t o remove d i s s o l v e d oxygen. Thermal degradation products decomposition  (from s e a l i n g the tubes)  initiate  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 h a s t o be t a k e n Chloroform  explaining  p o s s i b l y weakly coordinates  the greater  Alternatively coordination  when s e a l i n g t h e s a m p l e s .  solubility  t o the copper  of t h e copper  complexes.  t h e r e maybe an i n t e r m o l e c u l a r N i - S i n the s o l i d  energy and lower  leading to increased  solubility,  suggests otherwise the c r y s t a l packing  (119).  lattice  but c r y s t a l l o g r a p h i c evidence  Bulky  alkyl  groups i n t e r f e r e  (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.) t h u s i n c r e a s i n g t h e s o l u b i l i t y .  SOLVENT  COMPLEX  Dichloromethane Dichloroethane Trichioroethane Carbon Tetrachloride Tr i c h l o r o e t h y l e n e Nitrobenzene  NiPydtc  Nitromethane Acetone Toluene Benzene Tetrahydrofuran Ethanol Methanol Cyclohexane Pentane Chloroform n it it it it it it it  S O L U B I L I T Y mg/ml 0 .9 0 .4  it  n n  1. 1 . 1  <0  it  <0 . 1 high , but complexes <0 . 1 =0 . 1  tt  it it  =0 =0 =0 =0 ~0 =0  tt  tt tt  n n  NiEt dtc NiHxmdtc NiMpdtc NiMe d t c CuEt d t c CuMe d t c CuPydtc CuOc d t c PdEt d t c 2  2 2  2  Solubilities  .1 .1  oo  30  2  T a b l e 11.2.  .2 .2 .2  .2 0 1 .5 as 37 30 <0 . 1 2 50 20 40  tt  2  it  with  of metal  dfc's.  PART 3  ELECTRON SPIN RESONANCE STUDIES  /  124  12. ESR  THEORY  T h e r e a r e a number of a p p r o a c h e s t o ESR t h e o r y ; Kubo-Tomita theory (39);  Redfield  theory  (93);  stochastic Liouville 36  theory  relaxation  (90)(91)(92).  Stochastic Liouville  i s t h e most r i g o r o u s a p p r o a c h and  can  be  i n t o the slow motional  regime  (see appendix  K u b o - T o m i t a t h e o r y has  the advantage of b e i n g  s t r a i g h t - f o r w a r d w i t h an e a s y most of t h e t e r m s . physical  interpretation,  t h e c o n t e x t o f NMR H e r e we  Redfield  develop  (i.e.,  The  extended  22.9). fairly  interpretation  t h e o r y a l s o has  a  with zero order  for  simple  been d e v e l o p e d  mainly  in  wave-functions).  R e d f i e l d t h e o r y f o r t h e ESR  order wavefunctions, discussed  physical  b u t has  theory  case  w h i c h s u r p r i s i n g l y has  not  with  first  been  before. B r o w n i a n r o t a t i o n a l d i f f u s i o n m o d e l and  spin-rotation contribution  the  to r e l a x a t i o n are a l s o d i s c u s s e d  briefly.  12.1 The  THE  ISOTROPIC ESR  SPECTRUM  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 with a single nucleus  H = 0B.g.S - 0 .B.g  .1  is  + S.A.I  where B i s a s t a t i c m a g n e t i c f i e l d ,  3 6  F o r a u s e f u l , but d a t e d  reveiw 125  (121)  (12.1)  g i s t h e g t e n s o r , g„  see  (120).  is  126  the chemical  tensor, A i s the hyperfine  I and S a r e t h e e l e c t r o n and n u c l e a r  operators  it  spin  magneton  vector  respectively.  I f we d e f i n e B t o be a l o n g then  coupling  /3 i s t h e B o h r m a g n e t o n , 0^ i s t h e n u c l e a r  tensor, and  shift  the z a x i s ,  f o r i s o t r o p i c motion tensors  i.e.,  i n Eqn.12.1  B=B^k,  average out so  r e d u c e s t o (121)  H = g 0B i o  z  2  - q P Bi n  where g , g^ a n d A 0  z z  n  +  other  The n u c l e a r terms The  z  0  2  +  +  (12.2)  Zeeman t e r m i s s m a l l c o m p a r e d w i t h t h e  ( S e c t . 1 4 . 3 ) a n d c a n be n e g l e c t e d .  final  i.e.,  +  terms have t h e i r u s u a l meanings  t e r m o f 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 theory,  |°[s I. + S.I ]  a r e the t r a c e s of the corresponding  0  t e n s o r s . The r e m a i n i n g (122).  A I i  H = H  0  t r e a t e d by p e r t u r b a t i o n  + H'  J  where  H' = | ° | ^ S I _ + S_I +  and  H  The s e c o n d o r d e r  +  0  = g 0B S o  energies  Z  Z  + AQI^S^  a r e given  by (123)  (12.3)  1 27  E  = E° + <n|H'|n> + § <nlB'lm><m|g'[n> m m n^m E - E_ 1  n  T?  E  m  „s  z  0  i . r I (1 + 1 )~m.. (m. ±1 )1 + . A m.m4 +.—_ o _ — r ' T ' ~r— 4 g / ? B Lgo0B m + Aom^.m^J J  R  m  m  0  s i g n of m  |m , 1 ^ . >. m^  system;  projected  0  |m>  o  4  )  112.5) K  l  5  are a r b i t r a r y basis  i s the e l e c t r o n  o n t o t h e z a x i s and nu  projection  <  i n t h e s e c o n d o r d e r t e r m i s t h e same a s t h e  a n d |n> a n d  $  2  , , ~  -  P  5  0  where t h e s i g n  the  m  0  1  m  A  « f l n q pB m  =  (  1  elements of  s p i n quantum  number  i s the corresponding  f o r t h e 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 , co , b e t w e e n |m> 0  and  |n> ( t o  s e c o n d o r d e r ) i s r e a d i l y o b t a i n e d f r o m E q n . 1 2 . 5 . (The selection  r u l e s a r e Am =1 and Am.=0).  AE -mn  comn  = g p-B 0  0  +  A m 0  ;  +  2  fJp  | l ( I 1> " »  where I i s t h e n u c l e a r s p i n q u a n t u m For  6 3  Cu  (and  6 5  C u ) 1=3/2  (12.6)  +  B o  and A  number. 0  l i n e s w i t h t h e nu =-3/2  i s n e g a t i v e t h u s we  will  see f o u r  a t low f i e l d .  Each  line  will  be s h i f t e d f r o m i t s z e r o o r d e r p o s i t i o n by ^ I G a t 9GHz.  A t y p i c a l s p e c t r u m i s shown i n F i g . 1 2 . 1 . (The s p e c t r u m p a r a m e t e r s a r e g i v e n i n t h e a p p e n d i x 2 2 . 4 ) . The v a r i a t i o n o f line-height  ( w i d t h ) i s due t o m o l e c u l a r m o t i o n a n d i s  128 discussed  i n the next  section.  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  first  order wavefunctions are a l s o r e a d i l y  by p e r t u r b a t i o n  theory.  i  ^  i  ,  v  <nIH'Im>i n  If  we u s e  |a'>  t o show t h a t  (12.7)  n>  m  | a> t o d e n o t e t h e low e n e r g y  w a v e f u n c t i o n and i s easy  obtained  the h i g h energy  first  order  wavefunction, then i t  1  29  |a> = \-i,m.> + p j + i , m . - 1 >  |o'>  For denoted  = |+i,m.> + q j - i ,  convenience, thebasis  1  /  + 1>  (12.8)  s t a t e s , E q n . 1 2 . 8 , w i l l be  |-m> a n d |+m> r e s p e c t i v e l y .  a> =  m  Hence  -m> + p |+m-1> c  1  m  |a'> = |+m> + q |-m+1> m  (12.9)  where  p . • ifteo [  12.2  1 ( 1 + 1 }  THE ESR PROBLEM: DEVELOPMENT OF THE REDFIELD EQUATION Development o f t h e R e d f i e l d  that  "  f o r NMR i n f o u r  b) f i r s t  major  t h e o r y f o r ESR d i f f e r s  from  ways; a ) T ' s a r e r e q u i r e d , 2  o r d e r w a v e f u n c t i o n s (vide supra) h a v e t o be u s e d ,  c ) t h e s p e c t r a l d e n s i t i e s c a n n o t be s i m p l i f i e d ( S e c t . 1 4 . 2 ) ,  130 d)the f i r s t  order c o r r e c t i o n s  so  0  to the l i n e - p o s i t i o n are large  t h a t B ^ B . T h e s e f a c t o r s make t h e e x p a n s i o n o f t h e  Redfield equation extremely tedious straight-forward).  (albeit  Only the s a l i e n t features  p r e s e n t e d h e r e . The d e v e l o p m e n t f o l l o w s Park  (42).  that  w i l l be s u g g e s t e d by  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  symbolic algebra  reduction  t h e a i d of a  (124).  program  We make t h e 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 relaxation within a spin state  -(1/T ) 22  i sT  2  (125)  that the  i.e.,  ( r )  m  i  = (R , , ) aa' aa' m.  (12.11)  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 ) 7 aa aa aa 7 aa aa aa 7 7 7 7 7 7 m.  (12.12)  J  where 7 l a b e l s t h e sum o v e r t h e h y p e r f i n e the  l i n e b e i n g measured and t h e s u p e r s c r i p t  real part the  of the r e l a x a t i o n r a t e .  (small  3 7  )  frequency s h i f t s  ' r ' denotes the  (The i m a g i n a r y p a r t  (97)(126)).  d e n s i t i e s , J(co), a r e expanded a s g i v e n  3 7  s t a t e s , m^. l a b e l s  gives  The s p e c t r a l  by N o r d i o  (99)  i.e.,  T h e s e a r e of t h e order of t h e s t a t i c second order c o r r e c t i o n t o t h e l i n e - p o s i t i o n s a n d were n o t i n c l u d e d i n the a n a l y s i s .  131  J  (-q)^(q) aa'  ao'  where A, F a r e g i v e n spectral  (12.13)  v, u m, q v  i n a p p e n d i x 22.6 a n d t h e r e d u c e d  d e n s i t y , j(o>), i s g i v e n by E q n . 1 0 . 1 5 . The  transition  frequencies are discussed  12.2.1 THE TRANSITION The t r a n s i t i o n  FREQUENCIES  frequencies  densities are assigned 12.1 The t r a n s i t i o n o p e r a t o r s , A, c o u p l e wavefunctions.  those  f o r the  frequency  d e p e n d s on how t h e  the f i r s t  order  modify the t r a n s i t i o n  are those  associated with the S  zero frequency. attached +  and S I z  +  , result  z  The p s e u d o - s e c u l a r  t o the nuclear  (90).  and I  z  Secular  operators,  CJ , r e s u l t i n g  o p e r a t o r s . The n o n - s e c u l a r +  +  terms i.e.,  terms a r e those  transitions,  from t h e S I  equation  to the transition  a s s o c i a t e d w i t h the term  0  probabilities at  f r e q u e n c i e s . The t e r m s o f t h e r e l a x a t i o n  frequency  u=u ±u  part of the  part of the  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  the I  spectral  by F i g . 1 2 . 2 a n d shown i n T a b l e  The s e c o n d o r d e r  wavefunctions  below.  operators.  terms,  from  1 32  ZERO FIELD  B„  HYPERFINE  F i g u r e 12.2. T r a n s i t i o n d i a g r a m . N o t e t h a t t h i s diagram corresponds t o a frequency-swept experiment. In the (usual) f i e l d - s w e p t experiment a low energy t r a n s i t i o n corresponds t o a higher resonance field.  N o t e t h a t t h e s i g n s o f t h e m. assigned dtc's  t o the spectrum  i t i s negative  c o m p o n e n t s c a n o n l y be  i f we know t h e s i g n o f A . F o r  (127).  0  133  STATE  a a ' a n d a'a transitions  a n d a'a' transitions  aa  OPERATOR <*f>  t  2  CO  CO  (*:->•  + 0 0  res  a  CO  a  res  co  0  -co  res  t  0  <*°>>  +1 +1  t  CO  t  CO  res res  co  0  a g  a  res  -co  a  Table 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 , f d e n o t e s t h a t t h e o p e r a t o r s do n o t c o n n e c t 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 /2; fcV e s > o 2m/CJ a  =a  0  +  A  12.3  THE F I N A L EQUATION The i n t e r a c t i o n s o f i n t e r e s t h e r e a r e t h e h y p e r f i n e  c o u p l i n g tensor (denoted F > a  (The  and t h e g - t e n s o r  (denoted F )  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 determined  e m p i r i c a l l y , S e c t . 1 2 . 5 ) . As b o t h t h e s e t e n s o r s a r e a x i a l l y symmetric  ( S e c t . 1 4 . 1 ) q=0  E q n . 1 2 . 1 3 become  (for T ) 2  a l w a y s , h e n c e Eqn.12.12 a n d  134  aa' aa'  m  J6  % L  a  a  a  a  a a  g  - a'a + F  9  9 aa> V a  o o r m ; (- ) F  a  K  ( A  93  a  m  +  3»  aa a a »  9  =<a|A^|a'>  where  r«; r-m^i 3  aa  aa  9 , a .  J  (12.15)  etc.  aa The wavefunctions  a r e s e l e c t e d a c c o r d i n g t o the  s u b s c r i p t s of the s p e c t r a l d e n s i t i e s i n Eqn.12.12. Care must be taken t o keep track of the running v a r i a b l e X. I t should a l s o be noted  that the a's a r e the f i r s t  (Eqn.12.9) i n t h i s The  order  wavefunctions  case.  o p e r a t o r s and magnetic i n t e r a c t i o n tensor elements  are given by (90) convenience.  and are reproduced  Note that B  i n appendix 22.6 f o r  i s one element of a v e c t o r  z  operator B, 0  (B = B = 0) and i s not equal t o the s t a t i c y  because of the f i r s t  order frequency  field,  shift.  The c o n t r i b u t i o n of the i n d i v i d u a l matrix elements t o each J i s given i n Table for  12.2. The elements i n d i v i d u a l (ffi)  Eqn.12.14 a r e given by the row denoted A ' o  a p p r o p r i a t e , and the column 3 > aa  >  aa  etc.,  A' y  , or as  again as r e q u i r e d .  (The s u b s c r i p t r e v e r s a l a r i s e s from the Hermitian Eqn.10.12). The a b b r e v i a t i o n s a r e  terms  (~m)  property,  135  2  v  = Cx  z  = -Cy  m  = Nuclear  B  2  y  =  2  i n magnetic  field  units  [K-m(m+1)]  = [K-m(m-1)]  K  =1(1+1) 2g 0B o  3  s p i n quantum n o . f o r l i n e a t  = Line-position  z  x  2  { 0  a  )  =  ^  0  j  z  { ( J  (w±a) = j(a) J  •> oo  ) a  res  ±CJ ) a  For example, c o n s i d e r the e v a l u a t i o n of t h e 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 t h e 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 J , , term a aa a  A ^ V - ^ J O . F ^ F W aa'  t h a t co=w  r  alone i s  es  (12.16)  a' a  The s p i n H a m i l t o n i a n p a r a m e t e r , experiment.  i.e.,  i n Eqn.12.14  F^, i s known  The o p e r a t o r e l e m e n t s  from  are non-secular, i S I , +  +  +u> . H e n c e t h e m a t r i x e l e m e n t f o r t h e o p e r a t o r a  so  136  <o'|S I |a ><a |S_I_|a'>  \Z  +  +  7  (12.17)  7  The wave f u n c t i o n i s reversed f o r the second  p a r t because  i t ' s H e r m i t i a n . Note e s p e c i a l l y t h a t the running v a r i a b l e , 7 must a l s o be r e v e r s e d . The wave f u n c t i o n s , a,  a r e given by  Eqn.12.9. The e l e c t r o n o p e r a t o r s are evaluated f i r s t  because  they e l i m i n a t e the most terms so we get  Z(<+m|+q <-m+1 | ) 11 |m^x-m^ | +p (<+m-1 11_ | -m>)  I  wi  +  (12.18)  m  where m i s the m. value f o r the l i n e of i n t e r e s t and m i s 7 1  the running v a r i a b l e . E v a l u a t i n g the nuclear o p e r a t o r s Eqn.12.18 reduces to  •k(S '  T h i s equation all  m  x 7  +/  )U 7  .y)  (12.19)  7  i s only non-zero when m^m-1, thus  (as i s with  the terms) only one of the running quantum numbers i s  r e t a i n e d . I f we note that x  ,=y  then the term  reduces to  y^, i . e . , Eqn.12.17 i s  {i->„/ . 8  l , p  !' 4 l !  <12  -  20>  1 37 T h e 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 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  TEJX , 0 0  where X o i s 0  rotor (see Sect.12.4.).  This  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 term i n Eqn.12.14 f o r each of t h e s p e c t r a l d e n s i t i e s , J , i n Eqn.12.12. F o r s i m p l i c i t y o n l y the e v a l u a t e d o p e r a t o r terms a r e g i v e n i n the t a b l e o v e r l e a f .  138  -HO)  +  CO  N  CO  E  -7.  E  XlCM  > -ho  > -HO)  4.  X|CM  N  - f c  + >  en > -to  -to  -to  -to  -^0  K|CN  + N  I  > — loo  a  4*  + >  CO  I  E  ca -|vo  4*  4.  « N CO  co >  5  4*  -loo  B -Hvo  CO N  -to  -Hto  i co <  o co  o  o ra  O  <  <  I  <  Ol  <  Cn  + o>  O  <  tji  O  O CO  O  <  *  <  Ol  Ol  <  CO CO  +  I CO  <  T a b l e 12.2. The M a t r i x E l e m e n t s f o r R e d f i e l d T h e o r y . The e l e m e n t s i n c l u d e t h e ( - 1 ) t e r m a n d a l s o n e g a t e d so t h a t T i s +ve (Eqn.12.11). m  2  I  <  Ol  1 39  T h e 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  1  2  Ti =l/l2^j(0){8(Am +2A m+G) g  2  +  8 C ( K - m ) ( A [ m + 1 ] + 2 A )} 8 2  2  + j(a){3A(K-m +2Cm)} + j(w-a){A(K-m +6Cm)} 2 2  ++ jj ((CCJJ)) {{ 66 (( AA mm ++ 2A 2A mm+G) + 1 2 C ( K - m ) (Am+A ) 88 _. + X 8 + j (cj+a) { 6 A ( K - m )} 2  (12.21)  2  A p p r o x i m a t i n g o> ±a> =>a> re5  fl  1  a n d o> =>0 ( s e e S e c t . 1 4 . 2 . ) we g e t  0  fl  Ti =1/2 j(0){A(3K+5m )+8(2A 2  m+G) 8  2  + 8C(K-m )(A[m+1]+2A  )+6CAm} 8  2  + j (CJ) { A ( 7 K - m ) + 6 ( 2 A  m+G) 8  2  + 12C(K-m )(Am+A^)+4CAm}j  + X  (12.22)  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 . )  0  A = (F ) a  2  2  G = B (F°) z g  2  A  = B F°F° z a g  g  and F °a  = i ( xx A - Ayy ) 2  Fg°  = i x( xg - ^yy g 2  y  )  140 Eqn. 12.22 c a n be r e a d i l y to  r e c o v e r t h e r e s u l t s g i v e n by o t h e r  (128)(129)(130)(131)(99) coefficient first  See  appendix  order equation  spectral  etc.  The  The on  contributions. this to zero.  from  order  t o o b t a i n t h e two  linewidth.  The l a t t e r c a n  i n part t o experimental artefacts  (dipolar  the solvent, unresolved hyperfine coupling  contribution spectral  principle)  order  inverted  and a r e s i d u a l  see Sect.14.5.1),  rotation  (The  0  theories).  densities  be a t t r i b u t e d  = B .  2  i s d e r i v e d by s e t t i n g  d a t a c a n be r e a d i l y  broadening  B  22.5 f o r comments on c o m p a r i n g t h e s e c o n d  terms w i t h other The  by l e t t i n g  workers  'C i d e n t i f i e s t h e second  The  i n t o a c u b i c i n m.  rearranged  b u t i s m a i n l y due t o t h e s p i n (see Sect.12.5).  densities  c a n be f u r t h e r  t o o b t a i n t h e elements  inverted ( i n  o f t h e d i f f u s i o n t e n s o r R.  r e l a t i o n s h i p between t h e s p e c t r a l t h e d i f f u s i o n model u s e d . We w i l l  densities  and R depend  c o n c e n t r a t e on t h e  Debye d i f f u s i o n m o d e l , b e c a u s e o f i t s s i m p l e r e l a t i o n s h i p t o molecular  geometry a s w e l l  as the other reasons  outlined i n  Sect.10.4.  12.4 The  THE DEBYE DIFFUSION MODEL FOR AN ASYMMETRIC ROTOR 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 o f an asymmetric  rotor  a r e g i v e n by F r e e d  o n l y one e i g e n v a l u e in  Freeds  (98) a n d F a v r o  (one r e d u c e d  (109).  spectral  I n . our  case  d e n s i t y , =1/5X O* 0  n o t a t i o n ) i s measurable because o f t h e near  symmetry o f t h e 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 . U s i n g  axial  Freeds  141 equations  and  o b t a i n the  an a l g e b r a i c m a n i p u l a t i o n p r o g r a m  following  .  A-o o  where  =  relation  12R  a  The  x,y,z  +3R z  )  +  3CJ (R 2  CJ"  + 8(2R -3R  R  = R  + R  x  -R  )  s—z— )u + 144R a 2  (12.23)  + R  y  z  a = R xR y + R xR z +  reduced  can  2  zR R y  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 .  o n l y h a v e two it  (sR  we  for anisotropic diffusion  2  s R  —  (124)  spectral densities,  i s not p o s s i b l e t o i n v e r t the data  j ( 0 ) and  We  J(CJ), so  for a l l three  d i f f u s i o n c o n s t a n t s . M o r e o v e r , Eqn.12.23 i s s y m m e t r i c w i t h - r e s p e c t - t o the the case  interchange  o f R^  o f a x i a l d i f f u s i o n , we  i n f o r m a t i o n to i n v e r t the data  need two (this  Sect.15.4).  T h i s d a t a may  essentially  t h e o b j e c t i v e of t h e  12.5  and  be o b t a i n e d  R^  so, except  extra pieces  for  of  i s discussed further in f r o m NMR  and  that i s  thesis.  S P I N ROTATIONAL RELAXATION  As a m o l e c u l e moment. The  r o t a t e s the e l e c t r o n s generate  motion of the e l e c t r o n s i s not  t o t h e m o t i o n of t h e m o l e c u l a r  frame, they  a magnetic  rigidly  coupled  lag slighty,  t h i s moment c a n c o u p l e w i t h t h e n u c l e a r s p i n s ( t h e NMR  so case)  142  or,  the unpaired electron  (132)(133)(134). The  ESR c a s e )  i s related  (and i s c h a r a c t e r i s e d  coupling  by t h e s p i n - r o t a t i o n  t h e l a g and hence a l a r g e r  i s modulated then r e l a x a t i o n  c a n be c h a r a c t e r i s e d  diffusion)  tensor)  (thelarger the coupling). I f  can occur.  M o d u l a t i o n o f t h e a n g u l a r momentum o c c u r s d u r i n g and  coupling.  t o the electronic  t h e a n g u l a r momentum o f t h e m o l e c u l e  momentum t h e l a r g e r the  (the  i s known a s s p i n - r o t a t i o n a l  magnitude of t h e c o u p l i n g  structure and  This  spin  collision  (for the case of i s o t r o p i c time T . . Modulation o f t h e  by a c o r r e l a t i o n  1  spin-rotation  tensor  itself,  c o n s i d e r e d t o be n e g l i g i b l e  by c o l l i s i o n , (135)  (i.e.,  i s usually  T » T .). T h e c  a n g u l a r momentum a n d h e n c e i t s c o r r e l a t i o n clearly  related  to the strength  of t h e  t o r q u e s a n d t h e moments o f i n e r t i a information  i s thus a v a i l a b l e  spin-rotation calculated  r^.,is  intermolecular  of the molecule.  Valuable  f r o m r.. U n f o r t u n a t e l y t h e  i s usually  unknown a n d h a s t o b e  i s open t o q u e s t i o n . A l s o t h e i n e r t i a  anisotropic  characterised  importantly,  f r o m one r e l a x a t i o n  i n the case of anisotropic contribution  t i m e . These  times  t i m e . More and T. i s u n c l e a r ,  t h e r e l a t i o n s h i p between  spin-rotation  tensor i s  s o t h e a n g u l a r momentum i s  b y more t h a n one c o r r e l a t i o n  c a n n o t be o b t a i n e d  especially  time,  f r o m t h e g - t e n s o r . The a c c u r a c y o f s u c h  calculations generally  tensor  l  motion so although the  to relaxation  measured  (Sect.14.5) i t i s not p o s s i b l e  motional  information  c a n be r e a d i l y t o e x t r a c t any  f r o m i t . One i m p o r t a n t p o i n t  though i s  143 that  T. i s p r o p o r t i o n a l  to temperature(T)  1  i s p r o p o r t i o n a l t o 1/T.  This accounts  b e h a v i o u r of the spectrum temperature.  (136),  f o r the  whereas T •*  quadratic  l i n e - w i d t h s as a f u n c t i o n  of  13. ESR E X P E R I M E N T A L P r e p a r a t i o n and p u r i f i c a t i o n of the copper dithiocarbamate  spin-probes  i s straight-forward.  P r e c i s e and accurate a c q u i s i t i o n and a n a l y s i s of s p e c t r a i s l e s s easy,  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  use o f d i g i t a l t e c h n i q u e s . e a r l i e r w o r k (88)  and a l s o i n P a r t . 6 ) . DISPA ( d i s p e r s i o n  absorption p l o t s , Part.1) in o p t i m i s i n g spectrum  13.1  PREPARATION. OF  6 3  Isotopically enriched  was a l s o f o u n d  t o be a g r e a t a i d  C Q P P E R ( I I ) CHLORIDE (99.99% AERE H a r w e l l )  6 3  C u metal  was  n i t r i c a c i d and the s o l u t i o n  T h e 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  concentrated  vs.  a c q u i s i t i o n and a n a l y s i s .  dissolved i n concentrated evaporated.  ( T h i s has been d i s c u s s e d i n  h y d r o c h l o r i c a c i d t o form  6 3  from  copper(II)  chlor ide. The  c h l o r i d e was u s e d  i n preference  t o the n i t r a t e as  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 thermally s t a b l e so that excess  a c i d c a n b e r e a d i l y r e m o v e d by h e a t i n g .  13.2 P R E P A R A T I O N OF C Q P P E R ( I I ) D I T H I O C A R B A M A T E A s l i g h t excess  of t h e a p p r o p r i a t e sodium  COMPLEXES  dithiocarbamate  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 copper brown copper  c h l o r i d e (=*20mg, 0.15mMol) i n s o l u t i o n . T h e  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 . was n e c e s s a r y  I n some c a s e s i t  t o a d d e t h a n o l t o b r e a k up t h e 144  1 45 water/chloroform emulsions that  form  i n the presence of the  complex. After  s e p a r a t i o n , t h e c h l o r o f o r m s o l u t i o n was  s e v e r a l t i m e s w i t h w a t e r and  then f i l t e r e d .  The  washed  solution  a l l o w e d t o e v a p o r a t e and t h e r e s i d u e d r i e d a t 80°C. c o m p l e x 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  was  The  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 b a t h 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 e x p e r i m e n t s the n i c k e l analog at a l e v e l  6 3  copper  o f 0.1%  complex i s doped i n t o i t s w/w.  Consequently  the host  n i c k e l c o m p l e x must c o n t a i n <0.005% (50ppm) o f m i x e d  isotope  copper  i m p u r i t y . However, ( f o r unknown r e a s o n s ) a l l t h e  nickel  s a l t s t r i e d gave n i c k e l c o m p l e x e s  c o p p e r . To c i r c u m v e n t t h i s p r o b l e m  containing  the n i c k e l  =0.1%  s a l t s had  be t r e a t e d a s f o l l o w s : An e x c e s s o f t h e a q u e o u s n i c k e l was  salt  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  dithiocarbamate s a l t , d i s c a r d e d . The The  to  the s o l u t i o n  filtrate  i s now  p u r e n i c k e l c o m p l e x was  sodium  f i l t e r e d and t h e r e s i d u e  copper  free  These n i c k e l studies.  ESR).  t h e n p r e p a r e d by a d d i n g a q u e o u s  dithiocarbamate to the f i l t r a t e ,  recrystallising  (<0.001% by  filtering  the r e s i d u e t o g i v e g r e e n - b l a c k  s a l t s were a l s o u s e d  f o r t h e NMR  and  crystals.  relaxation  146 13.4  POLYCRYSTALLINE ESR SPECTRA  The  polycrystalline  s p e c t r a of the copper  c o m p l e x e s 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 p a r a m e t e r s  were  (137).  o b t a i n e d b y 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  Studies  i n c h l o r o f o r m g l a s s e s were n o t s u c c e s s f u l d u e t o t h e formation of t r i p l e t  13.5  (138).  s t a t e s , probably dimers  PREPARATION OF THE SOLUTIONS FOR ESR  S p e c t r o g r a d e o r d , - c h l o r o f o r m was d r i e d o v e r Type 4A molecular  s i e v e s and then f u r t h e r p u r i f i e d  by p a s s a g e  just prior  t h r o u g h an a l u m i n a c o l u m n . A 10ml  t o use  s o l u t i o n o f 0.2  t o 0.7mM o f t h e c o m p l e x was p r e p a r e d . A p p r o x 0.5ml o f t h e solution  transferred  by s e v e r a l <10" It  5  t o a ' f l a m e d o u t ' ESR t u b e a n d d e g a s s e d  freeze-pump-thaw c y c l e s t o a p r e s s u r e o f  T o r r on a g r e a s e l e s s vacuum l i n e a n d t h e n s e a l e d o f f .  s h o u l d be n o t e d t h a t m o l e c u l a r s i e v e s remove t h e  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 . T h e s e i m p u r i t i e s a r e removed b y t h e All  alumina.  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 b a g  u n d e r a n a t m o s p h e r e o f d r y n i t r o g e n . The s a m p l e s were a t =-20°C i n t h e d a r k t o p r e v e n t d e c o m p o s i t i o n  13.6 The  stored  (118).  ESR SAMPLE TUBES s a m p l e t u b e s were c o n s t r u c t e d f r o m s p e c i a l  thin walled  p y r e x t u b i n g (5mm OD,4mm ID) t o m a x i m i s e t h e s a m p l e v o l u m e . E a c h t u b e was f i l l e d  t o a d e p t h o f 3-4 cm ( t h i s  minimises  147 temperature  g r a d i e n t s due t o c o n v e c t i o n ; s e e S e c t . 1 4 . 6 ) . and  s e a l e d o f f a t 4-5 cm distilling  out of t h e c a v i t y  a gap o f =1cm allow  ( t h i s prevents the s o l v e n t  from  a r e a ) . C a r e was t a k e n t o l e a v e  b e t w e e n t h e t o p o f t h e s o l v e n t and t h e s e a l t o  for liquid  expansion.  13.7 RECORDING ESR SPECTRA All  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  ESR s p e c t r o m e t e r magnet  and M k l l  (139)  X-band homodyne  (Fig.13.1) which c o n s i s t e d o f : a  Fieldial  c o n t r o l ; an HP716B k l y s t r o n  s u p p l y a n d sweep u n i t ; a h o m e - b u i l t u n i t ; an I t h a c o D y n a t r a c k  AFC;  100 kHz  391A p h a s e - l o c k  I2in  power  modulation  amplifier.  The  m i c r o w a v e 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 ,  three port c i r c u l a t o r , Schottky detector  d i o d e a n d a m i c r o w a v e b u c k i n g arm. M i c r o w a v e power  was  m e a s u r e d w i t h an HP431C power m e t e r and t h e f r e q u e n c y m e a s u r e d w i t h an HP5246L  frequency counter  5256A p l u g - i n m o d u l e . The c a v i t y system control  and t h e t e m p e r a t u r e u n i t . The m a g n e t i c  fitted  acquisition  controlled  by a V a r i a n E257  f i e l d was c a l i b r a t e d w i t h a  to a microprocesser controlled system  (140).  with a  was f i t t e d w i t h a dewar  V a r i a n E500 p r o t o n m a g n e t o m e t e r . The s p e c t r o m e t e r interfaced  was  digital  ( s e e 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 unit  via  t h e 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 summarised i n Table  13.1.  was  f o r a l l samples.  These a r e  tape  148 The w h o l e 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 s t a n d a r d samples t o check  forsensitivity,  tested  with  amplifier  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 a n d 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 u s e o f t h e DISPA t e c h n i q u e was made f o r t h i s purpose  (10)(14)  & Part.1.  One c o m p l e t e s p e c t r u m was a l w a y s r e c o r d e d a t room t e m p e r a t u r e t o check f o r paramagnetic  CONTROL Microwave  Freq.  Microwave  Power  Modulation  impurities.  SETTINGS/COMMENTS ^9.04GHz 1  1-5mW, f o r c h l o r o f o r m s o l v e n t . s a t u r a t e a t =200mW (42) <0.8G. N a r r o w e s t o b s e r v e d l i n e  Cudtc s i s 3.0G.  S c a n Time & Sweep W i d t h  5 m i n s f o r 25G, 50G o r 100G s c a n s a s a p p r o p r i a t e . E a c h l i n e was r e c o r d e d separately.  Time C o n s t a n t  125mS o r 400mS (=1/100 o f t h e t i m e t o sweep t h e l i n e .  Temperature Sampling  rate  T a b l e 13.1.  15SCFH N  2  flow rate with C0 /(CH ) CO coolant (88). 2  3  2.5Hz o r =750 p t s . p e r l i n e . Spectrum R e c o r d i n g C o n d i t i o n s .  2  UUJ D£  _i  DflTfl TEM  t£C£ CCD XO  -©  CC DZ  U J —•  <n  VID TERM  CrC UJ t - O  H0> 0_ U J X :  ace o I —iUJ  D  at >-  I to u.  KENf (EDT TAPE U N I T  149  •  o a ct . xio  ace  —ui in TL u  -ID  zee: D D Ml-  N  —I—I  g  OUT DI/1 Z D  Zh CXUJ  (  UJ  m D rx cu  H C  eet/i am > 3 a u  I  t i— a a _iee 3UI a* DD  x > a  0>  D  OS  acccc UJUI o»i-  aui tna az  a>  C C D U I U L C  uce  V  ce  UJI-  i- > — az Ul=3 3Z  X  t-utx  UJ >  1 1-  >-  UJtE_J  zuia. u»a_ C L I O  ua  =  cn J  CCI-  Z U J U J I -  0 - -~  atnz a aui  3 Z 0 3  Z=3  tea. • UIDCv X  (m)  UJCC  D U J C J 4  SCAN DRIVE  I  U I O  ecu  I  URSE NUflTOR  -1L  D  >-0£  aui »_/ • a. uia ZD  UJ  utter  Figure  O U  13.1.  The  Spectrometer.  OL I10  Cd >c-CC-J tOUJCL. >-»0J O D  iCO-CO  150 13.8 The  TEMPERATURE  MEASUREMENT JJN ESR EXPERIMENTS  t e m p e r a t u r e was m e a s u r e d w i t h a c o p p e r / c o n s t a n t a n  thermocouple  inserted  i n t o t h e t o p of t h e c a v i t y  near t o the  s a m p l e . C a r e was t a k e n n o t t o d e c o u p l e t h e c a v i t y . The t h e r m o c o u p l e EMF was a m p l i f i e d  lOOOx by a l o w d r i f t  a m p l i f i e r w h i c h was c o n n e c t e d t o t h e d i g i t a l  D.C.  acquisition  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 t h e t e m p e r a t u r e f o r lOOmS ( w h i c h was a v e r a g e d l a t e r t o e l i m i n a t e n o i s e ) a t t h e s t a r t and e n d o f e a c h r e c o r d e d s p e c t r u m .  13.9 F I E L D CALIBRATION OF ESR SPECTRA Each  s p e c t r u m 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 t h e 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 a s  f o l l o w s . The m a g n e t o m e t e r was t a p p e d t o p r o v i d e r . f . proton precession  (the  f r e q u e n c y = 14MHz) f o r t h e 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 s a m p l e d  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 corresponding Fieldial  v o l t a g e a n d s t o r e d by t h e 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 t h e n c o p i e d a s a t a b l e frequency the  vs.  Fieldial  (magnetometer  v o l t a g e ) t o the magnetic  tape u n i t a t  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 t h e 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 . p a r a m e t e r s were o b t a i n e d f r o m t h e f i t , t h e s l o p e ( G a u s s / v o l t ) and t h e i n t e r c e p t left  edge o f t h e s p e c t r u m ) .  (the absolute f i e l d a t the  Two  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 a s f o l l o w s . E a c h l i n e o f t h e spectrum  was r e c o r d e d a n d c a l i b r a t e d  individually.  The g a i n ,  s c a n w i d t h a n d t i m e c o n s t a n t were a d j u s t e d t o s u i t e a c h peak using 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 temperature  table,  a n d o t h e r d a t a were r e c o r d e d on m a g n e t i c t a p e  as  16 b i t X-Y d a t a p o i n t p a i r s . The m a g n e t i c t a p e 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 processed  file  and  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  t h e DEC L S I - 1 1  minicomputer  l o c a t e d i n t h e 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 a s f o l l o w s (88).  The p e a k  t o p s were l o c a t e d a u t o m a t i c a l l y ( A m d a h l p r o g r a m ) o r interactively  (DEC L S I - 1 1  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 t h e 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 locally  fitting  l o c a t e d and then  to a straight  found e x a c t l y  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 t h e b a s e l i n e . The r e s u l t s c a n t h e n c o n v e r t e d t o G a u s s by means o f t h e c a l i b r a t i o n written to a f i l e  The r e d u c e d  temperature  spectral densities,  be  t a b l e and  as l i n e - w i d t h and l i n e - p o s i t i o n s ,  with the corresponding  by  and microwave  along  frequency.  j ( 0 ) , j (co) and t h e 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 least-square-fit for to  a brief  (14J)  i n the l i n e - w i d t h s (see appendix  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 )  get the d i f f u s i o n tensor i s d i s c u s s e d i n Sect.15  Sect.20.  22.7  and  14. ESR ERROR DISCUSSION The  errors arise  approximations  from f o u r p r i n c i p a l  inherent i n the theory;  a r t e f a c t s ; experimental  instrumental  a r t e f a c t s ; computational  Many o f t h e e x p e r i m e n t a l assessed  sources;  artefacts.  a n d i n s t r u m e n t a l a r t e f a c t s c a n be  w i t h t h e a i d o f DISPA a n d s u b s e q u e n t l y  e l i m i n a t e d . Computational  minimised or  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 t h e program d e s i g n , the accuracy  of t h e 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 data  analysis.  14.1  THE A X I A L SYMMETRY APPROXIMATION FOR THE SPIN HAMILTONIAN  This approximation  e l i m i n a t e s a l l 0 a n d ±1 t e n s o r  c o n s i d e r a b l y s i m p l i f y i n g E q n . 1 2 . 1 3 . However, t h i s  elements i s atthe  e x p e n s e o f 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 t h e '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  r o t a t i o n s that interchange The very  t h e 'zz'  and 'yy' components.  magnetic p a r a m e t e r s f o r CuPyDtc  little  f o r the a l k y l  (these values  s u b s t i t u t e d copper  complexes) a r e  152  from  vary  dithiocarbamate  1 53  A  = -119MHz  q  A yy  A zz  =2.022  3  XX  XX  = -106MHz  g  =2.018  = -474MHz  g zz = 2.088  ^yy 3  That i s , approximately, but not exactly  axially  symmetric.  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 a r e g i v e n b y ((90)(101)  = 4. ( A  ± 2 F  F°  a  2  a  = / | [A 3  [ zz  xx  A  yy  xx  )  )1  +A  -i(A 2  -  and i n appendix 22.6).  yy  J  F  F°  g  ± 2  g  = v/§[g ['zz 3  =  2  i(g  'xx  +g  -i(g 2  -  'xx  ( 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 symmetric).  g  yy  )  )1 j  basis)  1 54  In t h e r e l a x a t i o n e q u a t i o n m u l t i p l i e d o r squared. a b o v e we f i n d  2  (FI" )  (Eqn.12.22) t h e t e r m s a r e c r o s s  Substituting  2  = 40MHz  2  (F" a  +2+2 = -0.01MHz a g  The  2  * 4X10-  )  0 0 F a a  FI^F„  2  given  that  a  (Fg )  the parameters  0,  6  8  =  a x i a l symmetry a p p r o x i m a t i o n  = 86000MHz  2  = -70.0MHz  U  F"  2  6x10"  2  amounts t o d r o p p i n g  t h e F~  t e r m a n d r e t a i n i n g t h e F^ t e r m s i n E q n . 1 2 . 1 3 . U s i n g t h e d a t a above, t h i s w i l l term  so the t o t a l e r r o r  =0.2%, w i t h i n  14.2 The  i n t r o d u c e an e r r o r  o f =0.05% f o r e a c h F  for t h i s approximation  experimental  w i l l be  error.  ON APPROXIMATING SPECTRAL DENSITIES s p e c t r a l d e n s i t i e s a r e o f t h e form  j(w)  =  -c 1+  where u> i s : spectral  <^  line,  r e s  i  (OJT  (14.1) c  )  2  the t r a n s i t i o n frequency  for a given  or w , the hyperfine coupling fl  ( i A o ) , o r a m i x e d t e r m , u>  frequency  ±u , o r z e r o . I n o u r c a s e  155 oi  res  -56.9  Grad s"  1  and r  c  i s 10-200pS. I n NMR  2  (ur ) « 1 c  (see  S e c t . 1 8 . 1 ) . a n d t h i s t e r m c a n be d r o p p e d . H o w e v e r , f o r ESR this  i s not the case  this  k i n d n e e d more c a r e f u l  investigated  ( s e e F i g . 1 4 . 1 ) and a p p r o x i m a t i o n s of examination. This i s r e a d i l y  by s u b s t i t u t i n g  the appropriate approximations  i n t o E q n . 1 2 . 2 2 . I t s h o u l d be n o t e d t h a t following inverted  approximations are necessary via  the spectral  14.2.1 THE The frequency  (g> T U Q 0  2  )«1  frequency &  APPROXIMATION  r  i s a p p r o x i m a t e l y t h e Larmor  f o r the electron,  u . From F i g . 1 4 . 1  setting  CJO7"  t e r m s must be e x p l i c i t l y  calculations.  i f t h e d a t a a r e t o be  densities.  that c  some o f t h e  0  i t s clear  —> 0 i s n o t a good a p p r o x i m a t i o n . The included  i n a l l the  156  110.0  LOG FREQUENCY  S p e c t r a l d e n s i t i e s vs.  F i g u r e 14.1. frequency.  2  14.2.2 THE —(w — r ) « 1 a  (Hz)  APPROXIMATION  c  ' F o r r <120pS t h i s approximation c  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%, s e e F i g . 1 4 . 2 i . e . , and  «cjn  *• APPROXIMATION  T h i s w o u l d a l l o w u s t o s e t oo  ± a> => C J a n d a>  Both of these  error the  approximations  (<0.5%, s e e F i g . 1 4 . 2 ) .  =>  0  res 0  f l  j ( 0 ) are experimentally i n d i s t i n g u i s h a b l e ) .  14.2.3 THE w —a  u.  j(w )  res  a  introduce  negligible  i s very  u s e f u l as the  This  r e l a x a t i o n t i m e d e p e n d s 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 a t B (=B )=B_. z res °  t h i s does not imply  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 t h e m/=-3/2 l i n e . E r r o r s f o r t h e o t h e r l i n e s a r e l e s s than h a l f of t h i s .  14.3 CONTRIBUTIONS FROM THE NUCLEAR ZEEMAN The n u c l e a r  Zeeman t e r m  TERM  ( E q n . 1 2 . 2 . ) makes s m a l l  contributions to the t r a n s i t i o n  frequencies  and hence t o t h e  s p e c t r a l d e n s i t i e s . F o r an X-band ESR s p e c t r o m e t e r a field  o f 0.32T) t h e L a r m o r f r e q u e n c y  a b o u t 4MHz. T h i s frequency  of  6 3  C u and  i s s m a l l compared t o t h e e l e c t r o n  (=9GHz). I t i s n o t q u i t e n e g l i g i b l e w i t h  to the hyperfine c o u p l i n g frequency hyperfine coupling  frequency  (i.e. 6 5  , at  Cu i s resonant respect  (220MHz). However, t h e  c o n t r i b u t e s <1%  (Sect.14.5.5)  158  t o t h e s p e c t r a l d e n s i t i e s . Hence t h e n u c l e a r contribution  14.4  Zeeman  c a n be n e g l e c t e d .  THE F I R S T AND SECOND ORDER CONTRIBUTION  The  second order c o n t r i b u t i o n  'C  t e r m s i n Eqn.12.22) i s e a s i l y c a l c u l a t e d  equation. For t y p i c a l j(0)=lOpS first  reduced  t o the r e l a x a t i o n  care has t o taken  from  this  s p e c t r a l d e n s i t i e s of  and j(co)=5pS, t h i s c o n t r i b u t i o n  order correction  times (the  i s <0.5%. H o w e v e r ,  s h i f t s t h e l i n e w i d t h s by 5% a n d one  t o u s e t h e l i n e - p o s i t i o n s , B^, n o t B , i n 0  Eqn.12.22.  14.5  THE RESIDUAL LINEWIDTH  A number o f f a c t o r s than  those  subtracting  influence  the observed  discussed i n Sect.12.  The l i n e w i d t h  the t h e o r e t i c a l l i n e w i d t h  known a s t h e r e s i d u a l  linewidth  after  and i s determined i s the s p i n - r o t a t i o n  ( S e c t . 1 2 . 5 . ) . However, t h e r e a r e s e v e r a l  contributions.  other  (from Eqn.12.22) i s  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 term  linewidth,  other  (small)  These a r e d i s c u s s e d below.  14.5.1 DIPOLAR BROADENING This arises  from  electron-nuclear spin  between the s o l v e n t p r o t o n s can  be m i n i m i s e d  interactions  and the copper complex. I t  by t h e u s e o f 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 shows t h a t t h i s c o n t r i b u t i o n  CuMedtc i n C D C 1  3  and CHC1  3  i s 0.04G i n a l i n e w i d t h o f  1 59 3.6G.  {i.e.,  1% a t m a x i m u m ) .  38  14.5.2 PARAMAGNETIC BROADENING T h i s a r i s e s from  (unpaired) 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 c o p p e r c o m p l e x e s t h u s at high concentrations  show  broadening  3  (>10" M), 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 at the usual working (=10""M). A n o t h e r s o u r c e  concentration  of u n p a i r e d  electrons i s  d i s s o l v e d o x y g e n . T h i s must be removed by f r e e z e - t h a w - p u m p c y c l e s o f t h e s o l u t i o n on a vacuum line.  14.5.3 SOLVENT COORDINATION Solvent e.g.(46). unknown  coordination w i l l  modulate the l i n e w i d t h s  The i n f l u e n c e o f w a t e r (except  s e e (118)),  (wet s o l v e n t s ) i s  although  i t i s unlikely t o  c o o r d i n a t e w i t h t h e c o m p l e x e s . The s o l v e n t s were nevertheless  thoroughly  14.5.4 INTERNAL  d r i e d before use.  MOTION  The C-N bond h a s s i g n i f i c a n t d o u b l e bond c h a r a c t e r (142) a n d t h e r e  i s no e v i d e n c e  bond o n t h e NMR t i m e pyrollidine the  of r o t a t i o n about the  s c a l e (125).  The r i n g o f t h 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  r e l a x a t i o n via t h e u n r e s o l v e d  hyperfine  c o n t r i b u t i o n . However, t h e u n r e s o l v e d 3 8  coupling  hyperfine  The e f f e c t i s e v e n s m a l l e 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 , <0.1%, s e e S e c t . 1 4 . 5 . 5 .  coupling  160 contribution  {vide infra)  fluctuations  i n the conformation  b o t h been shown t o be  and t h e c o n t r i b u t i o n  from  ( a p p e n d i x 22.8) h a v e  negligible.  14.5.5 UNRESOLVED HYPERFINE The p r o t o n s c o u p l e t o t h e u n p a i r e d e l e c t r o n on t h e copper  g i v i n g r i s e t o broadening  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 and  from t h e u n r e s o l v e d (14),  simulations  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  (42)  CuMedtc  show t h e 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 t h e narrowest  l i n e by 10% (0.3G on 3 G ) .  3 9  T h i s e f f e c t c a n be  m i n i m i s e d by t h e u s e o f p e r d e u t e r a t e d compounds o r by use o f c o r r e c t i o n p r o c e d u r e s on  T  c  (143).  However, t h e e f f e c t s  a r e s m a l l ( T a b l e 14.1) a n d do n o t a f f e c t t h e  observed trends.  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 , a b o u t t h e same a 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 .  161  TEMP°C  T (D)pS  T (H)pS  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  c  -50 -30 -20 0 20 30 45  T (corr)pS C  c  1 08±4 55±2 41 ± 1 25±0.5 17+1.0 14±0.5 11±0.5  T a b l e 14.1. E f f e c t o f 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 . D a t a i s f r o m (88). (D) denotes the c o r r e l a t i o n time f o r t h e p e r - d e u t e r o d e r i v a t i v e . (H) d e n o t e s t h e 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 m e t h o d were used.  14.5.6 MAGNETIC F I E L D INHOMOGENEITY Typically  <5mG a c r o s s t h e ESR s a m p l e .  (manufacturers spec.) This i s small line-width  o f dtc's  with  (>3G) a n d c a n be  respect t o the  ignored.  14.5.7 SPECTROMETER PHASING Line-heights (dispersion  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  l e a k a g e ) of t h e 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 badly with in  the past  (131).  line-height  The l i n e - w i d t h s  a n a l y s i s methods  used  (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 misadjustments (Table  14.2) a n d t h i s  uses the l i n e - w i d t h s  i s not a problem  i f the analysis  d i r e c t l y a s i s done  here.  1 62  Phase  Angle 0  Observed* Line-width  Line-width from H e i g h t s 1 .000 1 .005 1.010 1 .029 1 .060 1.101 1.151  1 .000 1 .006 1 .006 1 .006 1.013 1.014 1 .022  2 5 10 15 20  T a b l e 14.2. E f f e c t o f P h a s e on O b s e r v e d L i n e - w i d t h s . * - T h e r e i s an e r r o r o f =±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  effect  of t h e 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 d o c u m e n t e d (30). broadening of the narrowest negligible <1%  line  (<0.1%). Broadening  Line  by t h e t i m e c o n s t a n t i s  from t h e m o d u l a t i o n i s  (= 30mG w i t h 0.8G m o d u l a t i o n )  i n t h e worst case and  i s t y p i c a l l y much l e s s .  14.6  TEMPERATURE INHOMOGENEITY  A temperature  g r a d i e n t a c r o s s t h e sample w i l l  observed l i n e w i d t h and cause  inaccurate  a f f e c t the  thermocouple  r e a d i n g s . T h i s h a s b e e n p r e v i o u s l y i n v e s t i g a t e d (88).  The  t e m p e r a t u r e 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 o f 0.05°C. A 1°C s h i f t will  i n temperature  c h a n g e t h e l i n e - w i d t h by 3% o f t h e l i n e - w i d t h i n t h e  worst case  (m.=3/2 l i n e a t =*-50°C), b u t w o u l d  be t y p i c a l l y  163 =0.5%  14.7 The  of  the  line-width.  F I T T I N G ARTEFACTS AND NOISE influence  extensively  of n o i s e  i n the  on p e a k f i t t i n g  literature  experimental conditions points  is  e.g.(144).  used here  discussed Under  (an SNR o f  f o r a f i t ) we e x p e c t a n o i s e  the  50:1  w i t h 50  r e l a t e d e r r o r of  <1.0%  (145). Fitting  a r t e f a c t s a r i s e because the  window a b o u t t h e window i s t o o l a r g e the  peak) a r e  small  cubic  fitting  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  noise  a p r o b l e m f o r i n t e r a c t i v e peak level  i s very high) a s the  c o n t r o l s the  window w i d t h and  routines  line-width  the c o r r e c t  I f the  too  (unless  of  to a cubic.  becomes a p r o b l e m , i f i t i s  i s not  the  (within in a  noise  shape. T h i s the  fitted  data  operator  p o s i t i o n . With the  i s unknown t o t h e  window i s d i f f i c u l t ,  fitting  automated  p r o g r a m and  e s p e c i a l l y i f the  line-widths  vary w i t h i n a spectrum o r set o f spectra.  p r o b l e m was  investigated  errors  i n the  s w e e p - w i d t h ) and  14.8 The  ±0.5%  i n the  ±20mG (<1%  This  Typically  of  the  l i n e - w i d t h f o r both methods.  F I E L D CALIBRATION AND CAVITY SHIFT Varian  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  free proton precession in  i n p r e v i o u s work (88).  l i n e p o s i t i o n s are  choice  s o l u t i o n , (i.e.,  diamagnetic  shift).  frequency instead  t h e m e t e r i s not  of  that  corrected  the  of a p r o t o n  for  T y p i c a l l y t h e m e t e r r e a d s =0.1G  the high.  1 64 The c o r r e c t value can be recovered by m u l t i p l y i n g the meter reading by 234.868/234.874. However, we tap the probe to o b t a i n the proton p r e c e s s i o n frequency d i r e c t l y  so t h i s  problem i s not r e l e v a n t . The gate t i m e s *  0  f o r the t r a c k i n g Gaussmeter and the  frequency counter are =*0.1s, t h i s l e a d s to an e r r o r f i e l d c a l i b r a t i o n of <(sweep-rate)*(gate-time),  i n the  or <l5mG f o r  a 5min., 25G scan. T h i s w i l l average out to a systematic c a l i b r a t i o n e r r o r of = -5mG  f o r a low-to-high f i e l d  sweep.  However, t h i s only a f f e c t s the a b s o l u t e l i n e - p o s i t i o n s , not l i n e - w i d t h s or s p l i t t i n g s hence the e r r o r  i s negligible.  The Gaussmeter probe i s o u t s i d e the ESR c a v i t y so a field  s h i f t with respect to the c e n t e r of the c a v i t y i s  expected. T h i s was checked with Fremys s a l t . The center of  a 1 X 1 0 ~ M s o l u t i o n of Fremys s a l t 4  i n a 0.05M s o l u t i o n of  potassium carbonate was measured 7X. The mean f i e l d c o r r e c t e d f o r microwave frequency s h i f t s was (the  line  position  3238.355±0.003G  g a t i n g e r r o r i n t h i s case should be <1mG). Using.  g =2.00545 and A =13.0910 (130), 0  the l i n e - p o s i t i o n , to  0  second o r d e r , should be 3238.450,  i.e.,  a c a v i t y s h i f t of  0.095G should be added to the Gaussmeter r e a d i n g s . The cavity shift  f o r t u i t o u s l y c a n c e l s the e r r o r of the  Gaussmeter to w i t h i n 0.01G.  9 0  $  To o b t a i n an accuracy of 1:10 (=3mG) one must count the frequency f o r at l e a s t 10 c y c l e s , i.e.,* 0.1s f o r 13MHz, the Larmor frequency of a proton i n a 3000G f i e l d s  15. ESR RESULTS AND DISCUSSION  15.1  ESR RESULTS 6 3  Line-width data f o r C u d -Pydtc 9  o b t a i n e d over a temperature  i n d - c h l o r o f o r m were  r a n g e o f -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 d a t a a r e i n a p p e n d i x 22. 10.  0.0 -50  J  L  J  -2510  L  J  OlO  I  L  TEMPERATURE  J  L  2510  5o:o  J  l_  [°C1  F i g u r e 1 5 . 1 . P y r o l l i d i n e dt c data.  line-width  L i n e - w i d t h s f o r 3 1 0 , 323 a n d 333K were o b t a i n e d b y 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 spectral densities,  i n t o E q n . 1 2 . 2 2 . The r e d u c e d  j ( 0 ) , j(a>) a n d t h e r e s i d u a l  f a c t o r w e r e o b t a i n e d via  line-width  a l e a s t s q u a r e s f i t (141).  r e s u l t s a r e shown i n T a b l e 1 5 . 1 .  165  The  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 1 5 . 1 . S p e c t r a l d e n s i t i e s f o r CuPydtc in c h l o r o f o r m . F i t t i n g e r r o r s on the parameters are a l l <1%.  1 3  By combining  2  these r e s u l t s with the C or H data, we can,  in p r i n c i p l e ,  f o r a Debye d i f f u s i o n model, o b t a i n the  d i f f u s i o n tensor. This i s discussed i n Part.5.  15.2 APPROXIMATE METHODS FOR DATA ANALYSIS In view of the l a r g e body of ESR data a v a i l a b l e  from  p r e v i o u s s t u d i e s i t i s u s e f u l t o examine Eqn.12.23 ( X d e t a i l t o g a i n some i n s i g h t  0 0  )  in  i n t o the d i f f u s i o n tensor,  without r e s o r t i n g t o NMR s t u d i e s t o do so. There are two approaches  t o t h i s , s i m u l a t i o n s or approximations. Three  approximations are of i n t e r e s t here: The i s o t r o p i c assumption,  R^=R^*=R=R, which i s u s e f u l f o r o b t a i n i n g order z  of magnitude v a l u e s . The a x i a l approximation with R^=R^=R^ and the f a s t a n i s o t r o p i c approximation, with R^>>R^,R « The 2  l a t t e r two approximations a r e expected t o be v a l i d on geometric grounds and both g i v e unique v a l u e s f o r R^.  167  15.2.1  SIMULATIONS  One  can  densities but  simulate data  t o p r o d u c e t a b l e s of  f o r v a r i o u s v a l u e s of t h e d i f f u s i o n  t h i s approach i s not  u s e f u l u n l e s s the  v a l u e s a r e a l r e a d y known. H o w e v e r , we  can  A l s o we  know t h a t t h e d i f f u s i o n  p o s i t i v e . F u r t h e r m o r e we s u p p o s e t h a t R^>R^,R  2  tensors,  approximate get  magnitude f i g u r e s f o r the tensor elements  spectral  order  (vide  infra).  tensor elements are a l l  a l s o h a v e good r e a s o n s  so one  of  can  limit  the  to  s i z e of  the  tables. By  a u t o m a t i c a l l y matching experimental  within  5%)  values  f o r the p o s s i b l e d i f f u s i o n  solution  w i t h s i m u l a t e d v a l u e s one  i s not p o s s i b l e , but  can  data  o b t a i n a set  elements.  the data  A  15.2.2 THE  (R x =R z ) and  of  unique  obtained  system i s c o n s i s t e n t w i t h motion approximately s •* ymmetric  ( t o say  for  our  axially  R x >R y ,R z .  ISOTROPIC ASSUMPTION  Eqn.12.23 r e d u c e s i n t h i s c a s e  X o(") = 0  The  interesting  two  e s t i m a t e s of the  o b t a i n e d ; one  W  2  |  3  6  R  (15.1)  2  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 isotropic  correlation  f r o m j ( 0 ) , R ( 0 ) , and  R(to). T h e s e two  to  time  the other  values are only equal  i s that are  f r o m j(a>),  i f the motion i s  168  isotropic.  The r a t i o , R(0)/R(CJ), t h u s g i v e s a m e a s u r e o f  the a n i s o t r o p y of t h e motion. criterion will  T h i s may be a u s e f u l  t o apply t o n i t r o x i d e type spin-probes, but i t  n o t be p u r s u e d  here.  15.2.3 THE FAST MOTIONAL APPROXIMATION On g e o m e t r i c  g r o u n d s we e x p e c t R^ t o be l a r g e r  R^, o r R^. The a p p r o x i m a t i o n R >>R^,R x  useful.  z  than  t h u s may be  I n t h i s c a s e Eqn.12.23 r e d u c e s t o  Xoo(0) = 12(R  + R ) z  x  oo(u  > =  CJ" +  ™ SJ* i* ±* L±»ll x  x  y  (4R co)  2  x  z  ( 1 5  + [ 12R (R +R ) ] x  x  .2)  2  y  S i m u l a t i o n s show t h a t R^ a n d (R^+R^) a r e u n d e r e s t i m a t e d if The  t h e ESR d a t a a r e i n v e r t e d u s i n g t h i s errors i nR  approximation.  a r e (R +R )/R % a n d t h o s e f o r (R +R ) X  a b o u t t w i c e a s much.*  y 1  z  X  x  T h e s e e r r o r s a r e =20% f o r o u r  c a s e , b u t 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 starting  values f o r thed i f f u s i o n  non-linear  inversion  useful  tensor f o r the  methods.  15.2.4 THE A X I A L APPROXIMATION Eqn.12.23 r e d u c e s t o  1  y  * T h e s e r e s u l t s a r e f o r 9GHz, t h e e r r o r s c h a n g e w i t h f r e q u e n c y , b u t n o t i n a s y s t e m a t i c manner.  169  ,  *  A  0  n  s  —p—  = 12R  .  /  x  X (w) 0 0  12R  = +  for  p  8(5R  (2R 2  p  +R  p  ) (5R  x—p  +2R  +R  5R  o(0)  (2R  +R  p—x  2  R  p x  x  2  +2Rx)CJ  +R ) p  +3CJ  +  2  (R  +R  p—x-  [ 12R  p  (2R  )  x  , ,  C  -  S  (15.3)  +R  p  ) ]  2  t h i s a p p r o x i m a t i o n . A g a i n R^ a n d (R^+R^)=R^ a r e  underestimated, v a l u e s than  but i t works w e l l  f o r a wider  range of  t h e 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  typically  <5% e r r o r  and  for R  i f |(R^-R^J/R^|<0.3  x  the % error  f o r R = 1 0 ( R -R ) / R 2  P  Providing  y  z  P  t h e m o t i o n i s a p p r o x i m a t e l y a x i a l we c a n g e t 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 Sect.15.4.  N o t e 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  for obtaining starting  values.  1 5 . 3 USING THE APPROXIMATIONS The  r e s u l t s o f a p p l y i n g t h e above a p p r o x i m a t i o n s  in Table  1 5 . 1 a r e shown b e l o w .  t o the data  1 70  Temp  R(0)  R(«)  R  4.8  X 48.9  37. 1  323  16.3  43.7  90.1  5.1  67. 1  333  20.2  38.9  82.5  4.3  54.4  temperature z  shows good a g r e e m e n t  I f we c a n r e m a i n c o n f i d e n t t h a t  f o r a wide temperature  range,  the a x i a l  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  reasonable isotropic  15.4  starting  approximation infra.  The  i s u s e f u l because i t p r o v i d e s  values  f o r the a x i a l approximation.  values give order-of-magnitude  The  values as expected.  INVERSION OF DATA WITH THE A X I A L APPROXIMATION  Data o b t a i n e d and  10.1  f o r the complete tensor a t the high  (Sect.20.3).  f a s t R^ a p p r o x i m a t i o n  8.2  inverted with  f o r the a x i a l approximation  w i t h t h e known v a l u e s  R  R +R y 7.3  14.5  data  R^,-  P  310  T a b l e 1 5 . 2 . ESR d a t a approximations.  The  R  R  X 63.8  from t h i s a p p r o x i m a t i o n  i s only u s e f u l f o r examining The  toluene  i s rather scattered  trends.  r e s u l t s of t h e v a r i o u s approximations i s shown i n T a b l e 15.3  f o r Pydtc i n  171  Temp  R(0)  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  R(u) 35. 1 66.7 41.2 46.0 51 .7 51 .4 46.0 47.0 47.2 42.7  38.7 96.7 55.5 68. 1 87.8 97.2 89.7 102.0 110.0 96.0  2.1 2.5 3.5 4.1 4.8 5.8 6.7 8.0 8.9 10.0  37.6 93.3 51 .2 61 .8 77.4 81.8  71 .0 77.2 79.7 65.8  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 u s e d CuPydtc i n t o l u e n e .  B o t h R^  and  R^  increase w i t h temperature  have r e s p e c t i v e a c t i v a t i o n  i m p l i e s the motion (i.e.  =9.5kJ/mol i f i t w e r e ) , f o r t h e two  Table CuMeOddtc  15.4  interesting  but a l s o  and T a b l e  up  15.5  2  The  i n two  2  solvents,  d a t a are r a t h e r poor, but  R^=200, f o u r t i m e s f a s t e r ,  i n the cyclohexane,  the l o c a l  be  show t h e r e s u l t s f o r  but not  s o l v e n t s h a v e t h e same d i e l e c t r i c reflect  would  t h e damping m e c h a n i s m i s  T h i s i m p l i e s t h a t the long hydrocarbon coiled  as i t  dependent  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 and  and  motions.  cyclohexane.*  in cyclohexane  and  result  the a c t i v a t i o n energy  (a 17 c a r b o n - c h a i n d e r i v a t i v e )  t o l u e n e and  as e x p e c t e d  i s not o n l y not v i s c o s i t y  , i s not hydrodynamic,  different  with  e n e r g i e s o f 7±2kJ/mol  13±0.5kJ/mol. T h i s i s an e x t r e m e l y  2.3 2.7 4.0 4.6 5.4 6.8 8.4 9.6 10.8 12.8  tail  f o r the  R^  probe  i n toluene.  of t h e probe  i n t o l u e n e . As  is  these  constant t h i s effect  must  s t r u c t u r e o f t h e s o l v e n t , a l t h o u g h what  * T h e l a t t e r d a t a were t a k e n this laboratory.(146)  f r o m a s t u d y done by M.Yu.  in  172  that  i s , i s unclear. Temp  r?/T  24.3 31 .2 32.8 37. 1 52. 1 62.6 73.3  3.0.5 2.66 2.57 2.37 1.81 1 .52 1 .29  c  R*  97.3 86.1 83.6 77.5 60.3 52. 1 47.2  23.6 61 .6 53.6 31 .2 27.1 126.0 53.9  T  R* 1 .7 1.9 2.1 2.2 2.9 3.0 3.6  24.8 62.6 55.2 32.8 30.0 132.0 58.6  1.0 1 .0 1.2 1.3 1.9 1 .6 2.1  T a b 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 i s t a k e n f o r m p r e v i o u s w o r k . n/T i s C P / K X 1 0 . c  3  Temp  T?/T  10.0 46.9 51 .3 56.5 78.0 88.9 98.3  2.32 1 .33 1 .25 1.17 0.91 0.81 0.74  c  R*  107.3 52.2 48.2 43.6 29.8 27.9 23. 1  59 232 205 153 198 131 94  T  R +R y  R*  z  1 .6 3.1 3.3 3.6 4.9 5.6 6.4  60 244 218 1 64 224 1 53 11 5  Table 15.5. A x i a l a p p r o x i m a t i o n used CuMeOddtc i n t o l u e n e .  For comparable  v a l u e s o f TJ/T, T  £  0.8 1.6 1.7 1.9 2.6 3.1 3.7  with  i s similar  i n both  cases so t h a t t h e 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 t h e b e h a v i o u r o f R . H o w e v e r , Some c a u t i o n h a s 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 b e c a u s e j(o)«j(0) and as such be b e t t e r t o i n v e r t  i s subject to large errors.  the data d i r e c t l y f o r R  and R x  than  via t h e r e d u c e d s p e c t r a l  densities.  a p p r o x i m a t i o n p r e d i c a t e s on t h e m o t i o n axial  over t h e e n t i r e  temperatures. Perhaps  temperature  range  I t might rather p  Also this  being approximately not just at high  more i m p o r t a n t l y , t h e r e s u l t s  i n Table  173 15.4 a n d T a b l e 15.5 show t h a t R  i s =1GHz; t h e s l o w - m o t i o n a l P  regime  (see appendix  2 2 . 9 ) . I t s n o t c l e a r what t h e e f f e c t o f  this i s .  15.5 DIRECT INVERSION USING THE ISOTROPIC ASSUMPTION If  we assume t h a t t h e m o t i o n  i s isotropic  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 f o r m ; E q n . 1 5 . 1 . I n s u c h a c a s e we can i n v e r t t h e d a t a d i r e c t l y  r a t h e r t h a n via  work by P a r k et  d e n s i t i e s . T h i s was done i n e a r l i e r and  i n f a c t most m o t i o n a l d a t a (ESR, NMR  s c a t t e r i n g ) a r e o b t a i n e d via  the s p e c t r a l  and  al  light  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 r  as c a l c u l a t e d  c  calculation first  (82)  by P a r k i f t h e m o t i o n  0 0  i s anisotropic.  and  The  i s c o m p l i c a t e d by s e c o n d o r d e r t e r m s , b u t t o  o r d e r i t i s e a s y t o s e e f r o m Eqn.15.1  j(u)/j(0) =  (1+O T 2  2  ) -  that  1  c  hence  r  1  = CJ- [ j ( 0 ) / j ( o ) - 1 ]  Comparison of the v a l u e s of T is  fairly  2  calculated  from P a r k s method  s t r a i g h t - f o r w a r d , but g i v e n t h e c o m p l e x i t y of t h e  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), relating  (15.4)  T  £  t o the d i f f u s i o n  obtain a functional  (Eqn. 12.23),  tensor i s d i f f i c u l t .  form f o r r  from s i m u l a t i o n s  We c a n though.  174 Temp  r Park  r (Eqn.15.4)  c  -30 -19 -9 0 9 20 29 39 49 56  c  83 61 39 38 37 33 27 25 23 19  83 61 48 40 33 27 22 19 17 15  T a b l e 15.6. C o m p a r i s o n of T ' S . P a r k s method vs. t h i s work. D a t a i s f r o m Table.15.3 and p r e v i o u s work. C  It  i s i n t e r e s t i n g t o note that the T  calculated  from  Eqn.15.4 do n o t d e v i a t e f r o m 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 a s much a s P a r k s r e s u l t s  15.6  do.  INTERPRETING DATA FROM ISOTROPIC INVERSIONS  The most p o p u l a r m o d e l f o r i n t e r p r e t i n g g i v e n by  where V i s a h y d r o d y n a m i c m o l e c u l a r volume,  (148).  plots i s  (147)  r  Boltzmann  T^VS.TJ/T  c  = p  term  +  r  (15.5)  0  (Sect.20.4.), u s u a l l y  the  T i s the t e m p e r a t u r e and k i s the  c o n s t a n t and  T  0  i s a 'free-rotor'  correlation  time  T h i s i s a s o u r c e o f some c o n t r o v e r s y a s t h e m o d e l i s  unsatisfactory  f o r a number o f r e a s o n s ( o t h e r t h a n t h o s e  t h a t c a n be l e v e l l e d a g a i n s t t h e h y d r o d y n a m i c  model i n  175  general). zero,  For  a purely  h y d r o d y n a m i c model t h e  i n a number o f s t u d i e s  o b s e r v e d . I t has  been p r o p o s e d t h a t  f o r by a d d i n g a ' f r e e - r o t o r ' correlation rates.  The  where  intercept  t h i s can  correct  expression  is  7  = T T 0  0  S  n  corresponding  o f  r  1  (88)(149)(150).  i n a number o f c a s e s to non-linearity  reason to b e l i e v e is  is  0  the  o f the  This  that  is  is  plots. Finally  such p l o t s are  0  the  there  i s no  meaningful  if  non-linearity  are  the  anisotropic.  Given that then the  +  observed i n t e r c e p t  m o d e l i m p l i c i t l y a s s u m e s i s o t r o p i c m o t i o n and  motion  ~Po ) Po7'o J /  o f 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  c o r r e l a t i o n t i m e . Secondly the  p r o b a b l y due  been  However,  hydrodynamic c o r r e l a t i o n t i m e , p  f r a c t i o n a l population  negative  the  has  is  be a c c o u n t e d  c o r r e l a t i o n time.  t i m e s c a n n o t be a d d e d , o n l y  i s the  their  a finite  intercept  the  question  intercepts  i s , "can  including anisotropic Assuming t h a t  and  we a c c o u n t f o r t h e s e e f f e c t s by  motion i n the  the  real,  model ?"  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 d e p e n d e n c e , t h e n we h a v e  R  . = R A obs x  For  the  and  E  3  3  will  e  ~  E  l  /  R  T  + R A e" y *  E 2 / R T  be r e l a t e d  The  t o the  E  3  /  R  T  (15.6)  ' a c t i v a t i o n e n e r g i e s ' , E,,  s h o u l d be a l l e q u a l t o the E^(88).  + R A e" z J 3  2  h y d r o d y n a m i c model the  viscous flow, A  1i  a c t i v a t i o n energy  collision friction  frequencies,  A  1 f  coefficients for  E  for A  2  the  and  2  176  molecule  ( S e c t . 2 0 . 4 ) . F o r c o n v e n i e n c e we w i l l  s e m i - e m p i r i c a l model. collision r  use a  The a c t i v a t i o n e n e r g i e s a n d t h e  f r e q u e n c i e s were c h o s e n  ^  s u c h t h a t R =10R =20R ,  x  y  z  = 20pS a t 310K a n d t h e s i m u l a t e d a c t i v a t i o n e n e r g y f o r  obs  T ^  was 1 3 . 6 k J / m o l .  observed c o r r e l a t i o n  ( O b t a i n e d f r o m p r e v i o u s w o r k ) . The t i m e was c a l c u l a t e d u s i n g Eqn.15.4 a n d  E q n . 1 5 . 6 . The r e s u l t s a r e shown i n F i g . 1 5 . 2 . Some  results  f r o m 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 m o t i o n were a l s o c a l c u l a t e d  f o r comparison.  120.0  100.0 to  80.0 u ZD CC  o  60.0 -  40.0 to  20.0  0.0 0 0  10.0  20.0  30.0  40.0  VISC/T  50.0  60.0  70.0  80.0  CuP/K)  F i g u r e 1 5 . 2 . The e f f e c t o f a n i s o t r o p y on T J / T p l o t s . The d a s h e d 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 t h e f u n c t i o n a l form of the p l o t i n experimentally inaccessible regions.  The g r a p h s show a b r e a k  i n t h e s l o p e a t about  30MP/K.  Above t h i s p o i n t t h e graphs a r e a l m o s t i d e n t i c a l , below p o i n t t h e m a t c h i s p o o r , b u t b o t h show a f i n i t e  this  intercept.  1 77  T h i s i m p l i e s t h a t T may b e a m e a s u r e o f m o t i o n a l 0  a n i s o t r o p y , not an a l t e r n a t i v e r e l a x a t i o n mechanism.  i  0. 01  0.0  i  i  I  i  1  1  1  10.0 20.0 30.0 40.0 50.0 60.0 70.0 VISC/T (uP/K) F i g u r e 15.3. A T J / T p l o t from p r e v i o u s work. This plot i s typical ofa l l previous 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 u p i s o f t e n s m a l l e r . T h e d o t t e d l i n e i s t h e same a s i n the p r e v i o u s f i g u r e .  The d e v i a t i o n f o r  7}/T<30MP/K  80.0  however does 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 mechanism maybe i m p o r t a n t a t h i g h temperatures.  T a b l e 15.7 s h o w s t h a t i n t h i s  regime the s p i n - r o t a t i o n term accounts line-width o f the narrowest  temperature  f o r 80% o f t h e  l i n e and 50% o f t h e b r o a d e s t  l i n e and 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 b e 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 and produce  a systematic artefact that i s  responsible f o r the observed curvature, (c./. Table  15.6).  178  It  s h o u l d be n o t e d t h a t Eqn.15.6 i s n o t a p r o p o s e d  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  that,  i f the motion  i s anisotropic  t h e n we  a c c o u n t f o r t h e n o n - l i n e a r i t y o f t h e T • vs.  partially  p l o t s and w h o l l y a c c o u n t f o r t h e p r e s e n c e of t h e is,T  i n t e r c e p t . That  0  i s an a r t e f a c t o f t h e  assuming  an  77/T  finite  isotropic  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 of  can  dangers  i s o t r o p i c m o d e l when d o i n g m o t i o n a l s t u d i e s .  Temp(°C)  Larmor F r e q .  Zero F r e q . 2.2 13.6  -30  Residual  0.5 1.5  0.7 0.7  + 10  1 .0 6.0  0.3 1.0  1.9 1 .9  +60  0.5 2.9  0.4 1.2  3.4 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 at various temperatures, i n G a u s s , f o r C u P y d t c i n t o l u e n e . The f i r s t row of e a c h e n t r y i s f o r t h e n a r r o w e s t l i n e , t h e s e c o n d row i s f o r t h e b r o a d e s t l i n e  15.7 We  CONCLUSIONS  have demonstrated  not o n l y  i s information l o s t ,  i n t r o d u c e d . The probably  3  t h a t by a s s u m i n g  0 3  free-rotor  arises  i s o t r o p i c motion  that  but a l s o that a r t e f a c t s are  t e r m p o s t u l a t e d by P e c o r a  from the n o n - l i n e a r i t y  introduced  (147) into  * T h e NMR e q u a t i o n s a r e s i m i l a r , b u t more c o m p l e x t h a n f o r t h e ESR c a s e and i t i s d i f f i c u l t t o d e m o n s t r a t e f o r t h e g e n e r a l c a s e t h a t a r t e f a c t s a r i s e . E a c h c a s e h a s t o be considered separately.  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 fact anisotropic. fact that  The  isotropic assumption also obscures  the a c t i v a t i o n  c o n s t a n t s may  coefficients  f o r the  be d i f f e r e n t , w h i c h c a s t s  diffusion  serious  doubts  the v a l i d i t y of the 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 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 and  t h e e n t r y o f some o f t h e  slow-motional regime are The  The  inversion  t o be  via  the  p r o c e s s i s l i n e a r and  a p p r o x i m a t i o n s and  'noisy'.  inherently precision  I t s not  unreliable  the  The  independent  spectral densities  because j ( w ) « j ( 0 ) , or whether  w i l l be \/3 m o r e p r e c i s e  / J 0  'noisy'  than the  tend  the an  anisotropic  data probably  gives the  though. a s s u m p t i o n of i s o t r o p i c d i f f u s i o n does not r e f l e c t the  t r e n d s for the  individual  elements, in fact i t produces very misleading D e s p i t e i t s s i m p l i c i t y the abandoned and  are  constants  r e f l e c t i o n of the a c c u r a c y ( r e l i a b i l i t y ) of  trends that  the use  spectral  c l e a r whether the method i s  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  results  the  o f t h e m o r e d i r e c t m e t h o d s u s e d by P a r k i s  illusion; R  a better  derivative  obscured.  of the m o t i o n a l model u s e d . However, d i f f u s i o n o b t a i n e d via  on  effects,  tensor elements in to  data are c o n v e n i e n t l y inverted  densities.  the  i" 's c  of s p e c t r a l  replaced  densities  l e s s m i s l e a d i n g and  densities.  be Although  i s not w i t h o u t p r o b l e m s they  p r o b a b l y r e f l e c t our  u n d e r s t a n d i n g of the p r o b l e m s  tensor  results.  i s o t r o p i c model should by s p e c t r a l  produce  better.  lack  of  PART 4.  NMR STUDIES  180  16. NMR r e l a x a t i o n discussed  theory  NMR  i s w e l l d e v e l o p e d and i s  i n a number o f t e x t s  mechanisms t h a t  contribute  THEORY  (91)(92).  to the relaxation  n u c l e u s . The most e f f i c i e n t m e c h a n i s m relaxation  T h e r e a r e many  and f o r n u c l e i w i t h  times of a  i s quadrupolar  I>1/2 t h i s  i s t h e dominant  mechanism and i n d e e d t h e o n l y mechanism t h a t  n e e d be  c o n s i d e r e d . F o r t h i s r e a s o n t h e r e h a v e been  numerous  relaxation e. g. (102) several  (151)  using quadrupolar  (149)  (152)  (153)  nuclei  . F o r 1 = 1/2 n u c l e i  c o m p e t i n g r e l a x a t i o n mechanisms; p r o t o n  coupling 'thio'  studies  i s usually  the p r i n c i p a l  there are dipolar f o r dtc's  o n e . However  the  c a r b o n i s remote from t h e p r o t o n s and t h e o t h e r  m e c h a n i s m s a r e more i m p o r t a n t . The one o f most i n t e r e s t i s relaxation  due t o t h e c h e m i c a l s h i f t  mechanism and q u a d r u p o l a r r e l a x a t i o n The o t h e r m e c h a n i s m s make s m a l l discussed b r i e f l y  i n the error  anisotropy  (CSA). T h i s  a r e d i s c u s s e d below.  contributions section  and a r e  (Sect.18).  181  4  182 16.1 CHEMICAL SHIFT ANISOTROPY (CSA) The CSA c o n t r i b u t i o n rotor  to the  1 3  C  r e l a x a t i o n time f o r a  planar  (102)  i s g i v e n by  1  Ti  = (3/lO)cog6jf.(G,D)  (16.1)  where  f(0,D) = (3/4R ) " |acos 0 + b s i n 0 - c s i n f l c o s 6 | 1  2  2  2  2  r  and  a =  (1/3) [ 4 R + ( T J - 1 ) R  b=  ( 1 / 3 ) [ 4 R ^ + ( T 7 - 1 ) R +(T +1 ) R ]  +(77+1 ) R l  2  2  v  2  2  ; c  c =  (16.2)  ?  Z  d/9)(77-3) (|x^)  (16.3)  2  z  5  where n = ( 6 - 6 ) / 6 , 8 i s t h e a n g l e o f 6 , t h e t r a c e l e s s x  y  z  3  x  component o f o f t h e CSA t e n s o r , axis, and  x  f r o m an a r b i t r a r i l y  ( b u t z must be p e r p e n d i c u l a r  i s defined  by F i g . 1 0 . 1 ) ,  CSA t e n s o r  i s co-linear with  thus reduces t o  t o the plane of the r o t o r  R^, R^ a n d R  elements of the d i f f u s i o n tensor.  chosen x  £  F o r a die,  are the p r i n c i p a l 0=0 as the  t h e m o l e c u l a r frame.  1 3  C  Eqn.16.1  183  J(w)  1  r  Note t h e s i m i l a r i t y 7}&CJ=0.  data  2  2  = ( 4 R ) " [4R +(r?-1 ) R^+(r?+1 ) R x  ]  z  (16.4)  o f t h i s e q u a t i o n w i t h Eqn.12.23 f o r  I n p r i n c i p l e we c a n c o m b i n e t h e  ( i f T?*0), o r t h e d e u t e r i u m d a t a  1 3  C  d a t a w i t h t h e ESR  (or both) t o o b t a i n t h e  diffusion tensor.  16.1.1 ISOLATING THE CSA TERM For  1 3  i s o t r o p i c d i f f u s i o n t h e CSA  C  relaxation  time  i s g i v e n by  Ti  1  2  2  = (9/10)(J§6 ,(1+TJ /3)T  (16.5)  c  I n o u r c a s e TJ=2.12 a n d 6^=76.9ppm. U s i n g a v a l u e o f 20pS for  T we g e t a r e l a x a t i o n  relaxation the  time  CSA t e r m  multifield  the  this  d e p e n d e n t s o we c a n s e p a r a t e  f r o m a l l t h e o t h e r r e l a x a t i o n m e c h a n i s m s by  NMR e x p e r i m e n t s . M o s t n o t a b l y we c a n s e p a r a t e  out t h e s p i n similar  i s field  time of 38s. Note t h a t  rotation  (SR) c o n t r i b u t i o n  which  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  CSA t e r m . To s e p a r a t e t h e t e r m s we d e f i n e  i s usually to  (154)  184  1  = au  1  = b  T" cs a T" res.  2  obs.  so f o r experiments with the CXP200 and WH400 we get  ^obs.^OO-^obs.UoO  2  2  " a(200 -400 )  1  Thus we can determine  'a' and T" . U n f o r t u n a t e l y cs a T"' ^ T ' and the e r r o r i n 'a' i s thus l a r g e , which csa res . ' J  1  3  compounds the experimental e r r o r s  16.2  .  QUADRUPOLAR RELAXATION  The r e l a x a t i o n rotor  in  time of an 1=1 nucleus i n a p l a n a r  i s g i v e n by ( f o r 77= 0)  (102)  2  1/T, = 3/8x J(0)  where x, the quadrupolar s p l i t t i n g  (16.6)  2  constant i s e qQ/ft (Q i s  the quadrupole moment and q the e l e c t r i c J(0)=f(O,D) which  asymmetric  i s d e f i n e d by Eqn.16.2,  f i e l d g r a d i e n t ) and  185  but where  a = R +R s  z  2  2  b = (Rz +Rs )cos c/> + (Rv+Rs ) s i n < 6 - ^x-£v-cos c/>sin cj> R +R 2  2  7  c  2  = ^|z^-cos <6  y  +  z  2  2  x  s  z  w h e r e 8 i s now t h e p o l a r a n g l e  2  - -^|x^|^-cos 0sin <6  -^z^^-sin *  s  s  (16.7)  s  (C-D) bond a n g l e w i t h  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  (angle with  respect respect  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 the a x i s system  p r e v i o u s l y d e s c r i b e d . Note that  this  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 s y m m e t r y r e l a t e d , (e.g. f o r t w o n u c l e i ' a and  ' b ' , <t>a±<t>,+nn/2, b  where n i s i n t e g r a l )  For t h e p y r o l l i d i n e d e r i v a t i v e there a r e two magnetically d i s t i n c t types o f deuterium.  We t h u s n e e d a  t h i r d piece o f information t o invert the data t oget the 1  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 . The "N r e l a x a t i o n time obvious,  but impractical choice  times from ESR o r  1 3  C spectra a r e other  There a r e two q u a d r u p o l a r compounds, D a n d '*N The V  o f p y r i d i n e (151). we g e t t h a t  (vide  1 B  infra).  i s an  Relaxation  possibilities.  nuclei of interest i n our  N case  i s very s i m i l a r t o that  I f we a s s u m e i s o t r o p i c m o t i o n a s b e f o r e  1  186  Ti  for  1fl  N,  1  =  24X T  x-5MHz and ( f r o m ESR)  <0.1mS. H e n c e , T  2  will  (16.8)  2  c  r = 2 0 p S . Hence T, w i l l c  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 o f =3kHz. T h i s f a c t c o u p l e d frequency  w i t h t h e low  (11MHz a t 4.7T) a n d l o w s e n s i t i v i t y  means t h a t  1  be  *N T, f o r o u r s y s t e m measurement  f e a s i b l e with the a v a i l a b l e  resonant  of the n u c l e u s a r e not  spectrometers.  D e u t e r i u m measurements however, a r e p o s s i b l e . T h i s nucleus  i s 10x more s e n s i t i v e  t h e T,'s a r e a p p r o x i m a t e l y  1  than  °N a n d x-lOOkHz so t h a t  1 sec.  16.3 S P I N ROTATIONAL RELAXATION The T, f o r s p i n r o t a t i o n a l  relaxation  i s g i v e n by S p e i s s  as  (107)  T  i  1  where 6 i s t h e i n e r t i a tensor  -  **?  i h  e  t e n s o r , c.  j  c  h  T  (  j  i s the spin  6  -  9  )  rotation  i n the i n e r t i a l frame, T . i s the a n g u l a r  correlation  1  momentum  time.  As w i t h ESR t h e s p i n - r o t a t i o n t e n s o r may from the c h e m i c a l  shift  tensor data."  1  The  be  calculated  spin-rotation  ** T h i s h a s 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 s y m m e t r i c m e t a l 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 o f t h i s method t o C i n a s y m m e t r i c t r a n s i t i o n m e t a l c o m p l e x e s i s open t o q u e s t i o n . 1 3  187  contribution  can  often  by m e a s u r i n g T, o v e r spin-rotation  be e x t r a c t e d  a wide temperature  T, w h e r e a s a l l o t h e r  m e c h a n i s m s v a r y a s 1/T. of r e a s o n a b l e  16.4  accuracy case  as i t  relaxation  However, e v e n i f s p i n - r o t a t i o n i s obtained  data  r a n g e . The  term dominates at h i g h temperature  v a r i e s w i t h temperature,  w i t h t h e ESR  from e x p e r i m e n t a l  data  i t s interpretation i s , as  ( S e c t . 1 2 . 5 ) , not c u r r e n t l y  possible.  CHOICE OF T, EXPERIMENT  A number o f methods f o r m e a s u r i n g T ^ S h a v e been p r o p o s e d (157) (158)(159)(160)(161)(162)(163) r e s p e c t t o p r e c i s i o n and  and  analysed  with  (164)(165)(157)(166)  efficiency  (158) (167) (168) (169) (170) (171) (172) (173) (17 4) . A good introduction two  t o t h e v a r i o u s methods i s g i v e n  most r e l i a b l e methods a r e i n v e r s i o n  saturation regarded  recovery  ( S R ) . The  a s the f a s t e r  methods, a l t h o u g h  former  recovery  method i s  (for a given precision)  t h e r e i s some c o n f u s i o n  l i t e r a t u r e about t h i s .  i n (17 5). The  The  i n s e n s i t i v e to instrument  (IR)  and  generally o f the two  i n the  early  l a t t e r method i s r e l a t i v e l y s e t t i n g s and  t h u s p r o b a l y more  accurate.  16.4.1 THE INVERSION RECOVERY EXPERIMENT T h e r e has  been a l o t o f d i s c u s s i o n  e f f i c a c y o f I R vs. SR t e c h n i q u e s (176)(174) efficient  of the  relative  (157)(158)(167)  B a s i c a l l y t h e I R m e t h o d i s c o n s i d e r e d more b e c a u s e a)  i t has  t w i c e the dynamic range o f  188 t h e SR t e c h n i q u e  f r o m -m  (the data a r e spread  as o p p o s e d t o 0 t o m^ f o r S R ) . b ) The m principle,  recoverable a t time  +  a  value  B  to m is,  a5 a>  in  z e r o . One d o e s n ' t h a v e t o  wait f o r S T / s t o get the value. In p r a c t i c e these 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 c o u l d n o t be o b t a i n e d a t t i m e  z e r o . The 90° a n d 180°  p u l s e s h a v e t o be s e t i n d e p e n d e n t l y  (because t h e p u l s e  shape i s n o t p e r f e c t ) t o w i t h i n 1° (17 2). maintained  (i.e.,  there  a  li6  and  t h e t r a n s m i t t e r must be s t a b l e  f o r t h e d u r a t i o n on t h e e x p e r i m e n t ) .  A l s o each  pulse  s e q u e n c e must c o n t a i n a 5T, (minimum) d e l a y (168)(169)(170) T  t  s o one h a s t o e s t a b l i s h a n a p p r o x i m a t e  t o s e t t h i s delay c o r r e c t l y . A l l these  to long set-up  times  which can abrogate  (=5hrs i n t h e case  factors lead  o f t h e CXP200),  t h e dynamic range advantage t h a t t h e  IR e x p e r i m e n t h a s o v e r  t h e SR e x p 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 t h e n e e d f o r t h e 5T, w a i t between p u l s e s , b u t one h a s t o c o l l e c t  four times as  much d a t a a s w i t h t h e I R e x p e r i m e n t t o a c h i e v e precision. Also the m  B  t h e same  v a l u e c a n n o t be o b t a i n e d a t t i m e  z e r o , g e t t i n g t h i s v a l u e c a n a c c o u n t one t h i r d o f t h e data c o l l e c t i o n 5  t i m e . H o w e v e r , t h i s method i s q u i t e  * 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 . * There a r e m u l t i p l e p u l s e sequences t h a t minimise t h i s p r o b l e m (177)(178)(179), but they i n c r e a s e set-up time further 6  189 i n s e n s i t i v e t o t h e p u l s e l e n g t h s e t t i n g and hence t o t h e r.f.  i n h o m o g e n e i t y and t h e o f f s e t  l e n g t h and o f f s e t  (180).  The  s h o u l d be s e t r e a s o n a b l y  c o r r e c t value t o maximise s e n s i t i v i t y ,  pulse  c l o s e to the  b u t do n o t h a v e  t o be e x a c t . T h i s c o n s i d e r a b l y r e d u c e s t h e s e t - u p B e c a u s e t h e SR method i s i n s e n s i t i v e t o artefacts  i t s h o u l d be more a c c u r a t e  for a given precision.  time.  instrument  than  t h e IR method  U n f o r t u n a t e l y , t h i s method  only  * gives correct results  f o r s a m p l e s where T < < T . 2  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 , efficient and  then  IR i s t h e more  method t o u s e . I n p r a c t i c e  insensitivity  to artefacts  t h e ease of  o f t h e SR method  set-up overides  t h e t h e o r e t i c a l e f f i c i e n c y a d v a n t a g e o f t h e I R method for  s a m p l e s w i t h l o n g T ^ s . Hence t h e IR method was  t o measure t h e d e u t e r i u m for the  1 3  C  relaxation  T/s  and t h e SR method was  measurements.  used used  17. NMR EXPERIMENTAL  17.1 The  PREPARATION OF THE SOLUTIONS FOR NMR s a m p l e s were p r e p a r e d  except that the saturated isotopically  s o l u t i o n of the appropriate  from copper f r e e n i c k e l c h l o r i d e  but otherwise  manner a s t h e c o p p e r  17.2  (Sect.13),  s u b s t i t u t e d n i c k e l c o m p l e x was u s e d . The n i c k e l  s a l t s were p r e p a r e d (Sect.3.3),  a s w i t h t h e ESR s a m p l e s  t h e y were p r e p a r e d  i n t h e same  complexes.  NMR SAMPLE TUBES  S a m p l e s f o r u s e on t h e WH400 were s e a l e d using  standard  in short a small  i n 5mm NMR t u b e s  m e t h o d s . S a m p l e s f o r t h e CXP200 were  lengths  (3.5-4.Ocm) o f 7mm o.d.  ' b u b b l e ' was l e f t ,  sealed  glass tubing.  Only  t o minimise r . f . inhomogeneity.  ( H o w e v e r , t h e ' b u b b l e ' must be l a r g e e n o u g h t o a c c o m o d a t e thermal  expansion of the s o l v e n t ) .  S o l u t i o n s o f t h e p y r o l l i d i n e d e r i v a t i v e were as needed r a t h e r than s t o r e d because o f problems decomposition  17.3 The  of t h e samples  prepared with  (Sect.3.5).  POWDER SPECTRA 1 3  C chemical  shift  a n i s o t r o p y was m e a s u r e d f r o m  powder s p e c t r a u s i n g s t a n d a r d (181).  The p u l s e  cross polarisation  s e q u e n c e u s e d i s shown b e l o w .  190  1 3  C  techniques  191 90° phase  -rrn—"—rn 6pS 2mS  25mS  4S  F i g u r e 17.1. S c h e m a t i c o f t h e p u l s e s e q u e n c e u s e d t o o b t a i n t h e powder s p e c t r u m .  17.4 Tj_ MEASUREMENTS Deuterium  s p e c t r a were i n i t i a l l y  phase a l t e r n a t i n g  inversion  [( X)  i  8  o  - -(  - ( - X )  i  8  0  - T - ( + X )  +  Typically  r  +  X  )  g  recovery (IR) experiment.  -5T ,( X)  o  +  1  9  0  - 5 T  1  r e c o r d e d a t 30.7MHz u s i n g a  l  , ( - X )  8  i  0  8  0  -r-(-X)  9  - T - ( - X )  9  0  0  -5  T  - 5 T  l  1  ]  2000 s c a n s w e r e u s e d . L a t e r e x p e r i m e n t s were  p e r f o r m e d on t h e WH400 u s i n g t h e B r u k e r I R e x p e r i m e n t . A typical  s e tof r e s u l t s  i s shown i n F i g . 1 7 . 2 .  1 92  Tl  JIX. jil^ J l \  J-u jX.  J^P  A  F i g u r e 17.2. T y p i c a l I R d a t a s e t . D e l a y t i m e s a r e 3 0 , 5 0 , 7 0 , 100, 130, 1 5 0 , 1 8 0 , 250, 500 a n d 1000mS. 61.4MHz, T=310K.  1 3  C  spectra  were r e c o r d e d a t 100MHz u s i n g t h e WH400's  software f o rthe saturation 50MHz  recovery experiment  and a l s o a t  (CXP200) u s i n g t h e f o l l o w i n g p h a s e a l t e r n a t i n g  sequence.  [20{( X) +  -20{(+X)  9 0  9 0  -r ( X) r  -T -(+X) 5  * where T >T =50mS. T y p i c a l l y 2  S  are  shown below  +  9 0  g o  }-r-( X) +  }-r-(-X)  g o  g o  ]  200 scans were used. Some data  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 a n d 1 0 0 s . 100.6MHz, T=310K.  17.5 A N A L Y S I S OF NMR T, d a t a w e r e a n a l y s e d  DATA by t h e s t a n d a r d method  (175)  of  m e a s u r i n g 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 semi-log p l o t equations.  of m a g n e t i s a t i o n  vs.  to a  delay using the following  1 94  in  = m (1 - k e ~  l n ( m -m ) = l n ( k ) - T / T , oo  where m  )  00  f  hence  T / T l  oo  ^  •  i s the equilibrium magnetisation  (== m^ f o r r > 5 T , )  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 an i n v e r s i o n r e c o v e r y (IR)  experiment.  o r k=2 f o r  18. As  with  ESR  ESR  are  a number of  s o u r c e s of  d i f f e r e n t f r o m ESR,  a much l a r g e r  role  i n NMR  error.  The  instrumental  experiments than  in  experiments.  18.1 The  play  ERROR DISCUSSION  t h e r e are  experimental errors artefacts  NMR  ON  APPROXIMATING THE  spectral  densities  SPECTRAL D E N S I T I E S  are  of  the  form  L  =  c  (18.1)  1 + (o>r )  2  c  where u and  T  i s the  i s a l i n e a r c o m b i n a t i o n of  C  f o r an  asymmetric r o t o r .  of  100MHz ( f o r  of  <0.02, i . e . ,  within  proton  13  C  at  an  error  T  £  the the  nucleus being rotational  i s lO-200pS and  9.4T), g i v i n g  (cor )  2  u  eigenvalues  i s a maximum  a maximum v a l u e  i n a s s u m i n g j ( 0 ) = j(co) of <2%,  t h i s approximation  i s inappropriate  well to  f o r 400MHz  spectra.  RESIDUAL CONTRIBUTIONS TO  There are  RELAXATION  a number of c o n t r i b u t i o n s  magnitude v a r i e s  greatly  to r e l a x a t i o n ,  from system to system. i s the  significant  1/2  s o u r c e of  many s o u r c e s of  r e l a x a t i o n . For  r e l a x a t i o n . The  195  spin  m a j o r one  their  For  quadrupolar n u c l e i , quadrupolar relaxation  are  studied  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  note that  18.2  L a r m o r f r e q u e n c y of  only  nuclei,  there  f o r dt c' s  have  196  been d i s c u s s e d i n S e c t . 1 6 . , discussed  other minor c o n t r i b u t i o n s are  below.  18.2.1 INTERMOLECULAR DIPOLAR RELAXATION This arises  from  the t r a n s l a t i o n a l  s o l v e n t p a s t t h e s o l u t e . The  relaxation  ( i n t h e s o l u t e ) by a n u c l e u s  S  Ti  1  where D = h'^y.y  ' i 's  the  the  time of a s p i n I  ( i n the s o l v e n t ) i s given  2  * D S ( S + 1 )N7j/kT  For p r o t o n s  3  1  limit  and  t h a t the r e l a x a t i o n  of t h a t of I , which i s r e a s o n a b l e 4 7  i n chloroform d i f f u s i n g past a  h a v e S = i , -q t h e v i s c o s i t y  o f 3 0 0 0 0 s , an e r r o r  for the  1 3  C  i n our  system.  in  of S i s y.»J •  if 1 3  C  S  nucleus  a t T=300K i s 0.57cP a n d  p r o t o n number d e n s i t y i s 7 . 5 X 1 0 a T,  (18.2)  r : . T h i s a s s u m e s Debye d i f f u s i o n i s  fast motional  independent  2 1  o f 0.1%  3  spins/cm . f o r a 30s T,  This  be  the  gives  typical  As d e u t e r o - c h l o r o f o r m  used as s o l v e n t the e r r o r w i l l  4 7  of  (182)  by  we  diffusion  was  negligible.  I n t h e o r y t h e c h l o r i n e a t o m s make a lOx l a r g e r c o n t r i b u t i o n t h a n t h e 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 e y a r e q u a d r u p o l a r n u c l e i ) and t h e y 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 t h e r e l a x a t i o n o f t h e solute.  1 97  18.2.2 INTRAMOLECULAR The  1  DIPOLAR RELAXATION  "N nucleus bound t o the  1 3  C may cause d i p o l a r  r e l a x a t i o n . The T, f o r t h i s i s given by  Ti  1  = ^ 1 ^ ( 1 ^ + 1 )D J(CJ) 2  In the f a s t motional l i m i t , J (CJ) = 1 Or  c  (182)  (18.3)  f o r i s o t r o p i c motion  hence  2  1/T, = 3 / 8 D T  For  (18.4)  c  our case r ==20pS at 300K and r=1.33A (183),  giving a  c  T, of » 500s. However, r  i s estimated from ESR data c  assuming i s o t r o p i c motion. The c o r r e l a t i o n time f o r the motion a f f e c t i n g the d i p o l a r r e l a x a t i o n with respect  t o the C-N bond) i s probably longer and T,  may be s u f f i c i e n t l y contribution  (end-over-end  short  t o make a s i g n i f i c a n t  t o the r e l a x a t i o n .  18.2.3 FLUCTUATIONS IN THE SCALAR COUPLINGS Relaxation 1 3  C  relaxation  1  of the *N nucleus c o n t r i b u t e s via  the c o u p l i n g ,  mechanism i s given by  The T  t  t o the for this  198  T^  oc (2/3)J S(S+1 ) T '  1  2  r'  where  assuming  fast  L  =  isotropic  N  (18.5)  motion. For our case we have  and T^=0.1mS (see Sect. 16.2). J ^ ^ I S H z a negligible  S=1  hence T^lSOOOs,  contribution.  18.2.4 INTERNAL MOTION C o n t r i b u t i o n s from p y r o l l i d i n e negligible  (see appendix  is a similar  size  r i n g pucker are  22.8). The d i e t h y l  to the p y r o l l i d i n e  derivative  f o r ESR  data (88).  stereochemistry i s d i f f e r e n t ,  the e t h y l  g i v e the same r e s u l t s  derivative and both  However, i t s groups are  probably v e r t i c a l w i t h - r e s p e c t - t o the plane of the molecule. There i s a l a r g e s t e r i c hindrance between them so one w i l l geometry  s t i c k up and one down. F l u c t u a t i o n s i n t h i s  are u n l i k e l y  to a f f e c t  the  times, but t h i s conformation w i l l coefficients may  for rotation  C  relaxation  reduce the d i f f u s i o n  about the x and z axes. T h i s  account f o r the i n c o m p a t i b i l i t y  and the D-NMR and ESR  1 3  results.  of the  1 3  C  results  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 m  B  was  v a l u e u s e d . To m i n i m i s e interactively  Multi-exponential  fitted  this effect  sensitive t o the  the experimental  t o an 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 n o t n e e d a n D I v a l u e . T h e s e m e t h o d s  give  b  reasonable  (i.e.  r e s u l t s w i t h p o o r m^ v a l u e s  but do n o t g i v e r e l i a b l e included  i n the data  s e t (43).  A l s o t h e s e m e t h o d s c a n be  p u l s e l e n g t h s and r . f .  However, t h e i r  (as i s t h e case  , a t low SNR),  r e s u l t s u n l e s s a n m^ v a l u e i s  used t o c o r r e c t f o r m i s - s e t inhomogeneity.  data  use w i t h s m a l l n o i s y d a t a  here) i s q u e s t i o n a b l e  sets  (192).  18.4 ERRORS I N Tj_ MEASUREMENTS T, m e a s u r e m e n t s o f l o w s e n s i t i v i t y relaxation  times a r e extremely  spectrometer. days,  I nour case  nuclei  with  long  d e m a n d i n g on t h e  t h e experiment  may l a s t  three  d u r i n g w h i c h t h e t r a n s m i t t e r must r e m a i n s t a b l e , t h e  s p i n n i n g r a t e must be c o n s t a n t , t h e m a g n e t i c f i e l d not d r i f t sources  and t h e temperature  oferror arise  impurities,  temperature  out o f t h e c o i l  from r . f . inhomogeneity,  Other  paramagnetic  g r a d i e n t s and d i f f u s i o n o f t h e probe  (157).  There i s l i t t l e  t h a t one c a n do a b o u t v a r i a b l e  t r a n s m i t t e r o r temperature suspect  should be.constant.  should  instability,  other than  spinner, t o reject  d a t a . T e m p e r a t u r e i n s t a b i l i t y was n o t a p r o b l e m ,  200  a l t h o u g h t e m p e r a t u r e g r a d i e n t s o f >1°C a r e p r e s e n t a c r o s s the  coil  (193).  The f i e l d  o f t h e CXP 200 a p p e a r s t o be  s t a b l e . E x p e r i m e n t s w i t h t h e WH400 were d o n e w i t h a d e u t e r i u m l o c k . SR e x p e r i m e n t s a r e i n s e n s i t i v e inhomogeneity, but i t can cause problems  i n IR experiments.  T h i s e f f e c t c a n be m i n i m i s e d by u s i n g s h o r t The e f f e c t s o f t h e p r o b e d i f f u s i n g  to r . f .  sample t u b e s .  out o f 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 s a m p l e s . I t i s e s s e n t i a l t o remove p a r a m a g n e t i c they g r e a t l y  i m p u r i t i e s when m e a s u r i n g  reduce the r e l a x a t i o n  (e. g. (175)(182)(194). copper-free nickel  l o n g T,'s a s  time  A l l s a m p l e s were p r e p a r e d f r o m s a l t s and deoxygenated  before use.  19. NMR RESULTS AND  DISCUSSION  c  19.1 _]_f__ RESULTS The  1 3  C  relaxation  r e s u l t s a r e shown i n T a b l e  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 relative are  reliability  averages  appendix  19.1. The  serve as a guide t o the  o f t h e r e l a x a t i o n t i m e s . The r e s u l t s  o f , o r 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  22.12 f o r t h e c o m p l e t e  Temp K  50.3MHz  data).  100.6MHz  T,(CSA)  310  22±1  10±1  60±10  323  16±1  11.5±1  125±20  333  11.5±2  T a b l e 19.1.  1 3  C  8±1  165±30  T/s forEt dtc. 2  The CSA t e n s o r was d e t e r m i n e d  (sees).  f r o m t h e powder  b u t t h e r e i s some a m b i g u i t y i n t h e a s s i g n m e n t  spectrum,  of t h e y and z  c o m p o n e n t s o f t h a t t e n s o r . (The x component c a n be u n i q u e l y determined  from t h e d i p o l a r  see a p p e n d i x  splitting  o f t h e powder  2 2 . 3 ) . As we h a v e an o v e r - d e t e r m i n e d  p i e c e s o f r e l a x a t i o n d a t a a n d 3 unknowns to  pattern,  system  (5  ) we s h o u l d be a b l e  r e s o l v e t h e 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  rotation axis  i s a b o u t t h e C-N bond (i.e.,  i n t e r c h a n g e o f t h e y a n d z c o m p o n e n t s i s t h e most c o n t r i b u t i o n t o the relaxation  the  important  t i m e ) . I f we assume R » R x  i n E q n . 16.4 t h e n  i s independent 201  o f r). Hence  within  ,R y  z  202 e x p e r i m e n t a l e r r o r we c a n n o t  distinguish  between t h e y and z  c o m p o n e n t s o f t h e CSA t e n s o r . (The c o r o l l a r y , o f c o u r s e , i s t h a t we d o n ' t need t o a n y w a y ! )  19.2 The  DEUTERIUM Tj_ RESULTS deuterium  (see appendix  relaxation  r e s u l t s a r e shown i n T a b l e  22.11 f o r t h e c o m p l e t e  Temp K  data).  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  0.187  0.242  333  19.2  NA  T a b l e 19.2. Deuterium Times a r e sees.  T / s f o r d-9  Py dtc. 2  The  61.4MHz a n d 30.7MHz r e s u l t s a r e i d e n t i c a l a s e x p e c t e d .  The  s p e c t r a f o r t h e 333K r e s u l t s show e x t r a p e a k s  indicating  t h a t t h e s a m p l e i s d e c o m p o s i n g . T h e s e T, v a l u e s a r e s u s p e c t . The  quadrupolar  splitting  constant  solution  i s not a v a i l a b l e ,  constant  f o r a l k y l compounds (195).  used.  f o r o u r compound i n  however t h e s e v a l u e s a r e n e a r l y A v a l u e o f 175kHz was  203 19.3  DISCUSSION  The two d e u t e r i u m and,  T,'s c a n be c o m b i n e d w i t h t h e  1 3  C  data  u s i n g Eqn.16.4 a n d Eqn.16.7 s o l v e d f o r t h e d i f f u s i o n  tensor  using non-linear l e a s t - s q u a r e s - f i t procedures  (196).  The r e s u l t s a r e shown i n b e l o w .  Temp K  R  310  R  x  R  y  z  non-convergent  323  60.2  8.9  -0.9  333  46.9  10.5  -1.7  T a b l e 19.3. and H d a t a  The d i f f u s i o n  tensor  from  1 3  C  2  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 low q u a l i t y o f t h e values time  1 3  C  data.*  8  Also simulations using  f o r the d i f f u s i o n tensor  i s essentially  show t h a t t h e  with the large e r r o r s i n the  The d e u t e r i u m and  8  1 3  i n d e p e n d e n t o f R^, t h i s may  i n c o r r e c t convergence of the f i t t i n g coupled  a t t r i b u t a b l e to the known  C relaxation lead to  routine, especially 1 3  C  T/s.  d a t a c a n be c o m b i n e d w i t h t h e ESR  data  sucessfully inverted (Part.5).  * The  1 3  C  d a 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 t h e ESR  data.  PART 5.  COMMENTS ON THE COMBINED NMR-ESR STUDIES  204  20. COMBINED ESR  20.1  AND  NMR  RESULTS AND  DISCUSSION  INTRODUCTION  The NMR  and ESR  r e s u l t s have been combined to o b t a i n the  r o t a t i o n a l d i f f u s i o n tensor f o r MPy dtc i n c h l o r o f o r m . 2  Although  the r e s u l t s span a l i m i t e d temperature range they  demonstrate the u t i l i t y  of combined ESR-NMR s t u d i e s to  obtain d i f f u s i o n t e n s o r s . A l s o they provide a s t a r t i n g p o i n t to  explore some approximation  d i f f u s i o n t e n s o r s from ESR  methods f o r e x t r a c t i n g  data alone. Furthermore the  r e s u l t s can be used to check the v a l i d i t y of the hydrodynamic model f o r r o t a t i o n a l  20.2  diffusion.  COMMENTS ON DATA INVERSION  I n v e r s i o n of n o n - l i n e a r equations especially  can be  problematic,  i f good s t a r t i n g values are not a v a i l a b l e , or  there are l a r g e e r r o r s i n the d a t a . Some workers studying d i f f u s i o n t e n s o r s with NMR  (151)(180)  have r e s o r t e d to  i n c r e m e n t a l l y searching a range of p o s s i b l e v a l u e s f o r R^ and R . z  of  R^,  T h i s i s a reasonable approach because the range  values f o r the R's  are q u i t e c o n s t r a i n e d . (They are a l l  p o s i t i v e and one can use hydrodynamic models to get order of magnitude v a l u e s for them). For our system good s t a r t i n g values f o r R  x  approximation to  and R +R  y  z  are a v a i l a b l e by u s i n g the J  •  methods o u t l i n e d i n Sect.15. However, one  i n v e r t the data i n i t i a l l y  to ensure these  approximations  are v a l i d so the g e n e r a l i t y of t h i s approach i s 205  has  206  questionable. One s e r i o u s p r o b l e m a r i s e s when i n v e r t i n g b e c a u s e Eqn.12.23 interchange completely  i s symmetric w i t h respect t o the  o f R^ a n d R^  f  i.e.  , X(o) and X(0) a r e n o t  i n d e p e n d e n t 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 on g e o m e t r i c  well  f o r t h i s f u n c t i o n a s we know,  g r o u n d s , t h a t R^>R^,R  z  (and a l l a r e p o s i t i v e ) .  T h i s a p p r o a c h a l s o g i v e s a good p h y s i c a l f e e l The d a t a were f i n a l l y  to get the f i n a l  Incremental  searching  0  d a t a t o g e t good  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 (196)  f o r X o»  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 o f t h e ESR a n d d e u t e r i u m  procedure  t h e ESR d a t a  Newton-Raphson  values.  i s a very  one d o e s n ' t have a r e a s o n a b l e  starting  i n e f f i c i e n t approach i f  i d e a o f t h e d o m a i n o f R,  especially  i f t h e 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 c a n  be g r e a t l y  i m p r o v e d by s e a r c h i n g f o r r a t i o s o f t h e  r e l a x a t i o n times 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 . reduces  t h e s y s t e m a t i c e r r o r s i n t h e data and h a l v e s t h e  amount o f d a t a t o be s e a r c h e d . for  A failure  t o o b t a i n a match  the 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  p r o g r a m , 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  i s very useful i n the i n i t i a l  20.3 THE DIFFUSION Five  approach  stages of data r e d u c t i o n .  TENSOR  i n d e p e n d e n t p i e c e s o f i n f o r m a t i o n were o b t a i n e d .  ESR s p e c t r a l d e n s i t i e s , one  This  1 3  C  two d e u t e r i u m  Two  r e l a x a t i o n times and  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 h a v e a n overdetermined  system and 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  equations are non-linear i t i s best to t r y t o invert the data i n groups Grouping useful  of three and then c r o s s - c h e c k the r e s u l t s .  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 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 done) 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 converge.  The  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  deuterium  r e l a x a t i o n t i m e s a n d j ( 0 ) ( t h e m o r e 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 h e r e s u l t s a r e shown i n T a b l e 2 0 . 1 . T h e 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 . T h e 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 from  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 . T h e e r r o r s c o r r e s p o n d to the range of 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 that 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  Temp(K)  R  system.  R X  R z  y  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.  The d i f f u s i o n  tensor.  208  20.4 One  THE HYDRODYNAMIC MODEL goal of measuring d i f f u s i o n  geometry  o f 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  f o r our probe i s w e l l d e f i n e d test  tensors i s to e s t a b l i s h the  the v a l i d i t y  s o we c a n u s e o u r r e s u l t s t o  of the hydrodynamic  used t o r e l a t e t h e d i f f u s i o n  geometry  model, t h e u s u a l  model  t e n s o r t o t h e geometry.  The d i m e n s i o n s o f t h e p r o b e c a n 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 h a s t o a c c o u n t f o r d e a d - v o l u m e  (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 penetrate t h e  gaps b e t w e e n  t h e s u l p h u r s . The v o l u m e  of t h e probe thus  a p p e a r s l a r g e r t o t h e s o l v e n t t h a n t h e t r u e v o l u m e . The probe thus approximates a rounded p a r a l l e l e p i p e d d i m e n s i o n s x , y , z , o f 1.6x0.5x0.25 nm. H y d r o d y n a m i c 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 ,  theories  but f o r t u n a t e l y the  p r o b e i s w e l l a p p r o x i m a t e d by a n e l l i p s o i d 1.75x0.65x0.3 nm. ( s e e F i g . 2 0 . 1 )  with  with dimensions  209  F i g u r e 2 0 . 1 . The p r o b e a s 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 V a n - d e r - W a a l s shape. D o t t e d l i n e i s e l l i p s o i d a l approximation.  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 f r o m t h e  hydrodynamic  m o d e l s g i v e n by P e r r i n  Acrivos  (111)  (YA)  Rj  (108)  and Youngren  and  are  t  /  C  llp  K*  *  3  = 3kT/47ra rj(4C ) /  3  = 3kT/47ra r (X p p- ) ?  /  1  (20.1)  2  w h e r e ' a ' (=x/2) i s t h e l e n g t h o f t h e m a j o r s e m i - a x i s ; are calculated  from P e r r i n ' s  X. a r e t h e c o e f f i c i e n t s f  YA's  e q u a t i o n s as o u t l i n e d  g i v e n by YA and p  1 f  p  2  in  imply t h a t ,  although the  c o e f f i c i e n t s are d i f f e r e n t ,  (88);  ( a and b i n  n o t a t i o n ) a r e t h e s e m i - a x i s r a t i o s , z/2a a n d y / 2 a .  t h a t Eqn.20.1  C.  Note  diffusion  t h e t e m p e r a t u r e dependence  is  210 i d e n t i c a l , both a r e v i s c o s i t y The  appropriate  slip stick  friction  dependent. coefficients  x  y  z  0.22  1.46  0.635  0.098  0.362  0.328  Table 20.2. F r i c t i o n c o e f f i c i e n t s probe.  The  f o r our probe a r e  f o r 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 are thus R  R  x  R  y  230 8.1  34 2.2  79 2.4  T=323 slip stick  267 9.4  40 2.6  92 2.8  T=333 slip stick  300 10.6  45 2.9  103 3.2  Table 20.3. P r e d i c t e d d i f f u s i o n 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 magnitude i n e r r o r . to predict  z  T=31 0 slip stick  This  coefficients  t h a t a r e an o r d e r o f  i s not unexpected, but the f a i l u r e  the observed order of the d i f f u s i o n  is disturbing.  A l s o t h e r e l a t i v e m a g n i t u d e s do n o t a g r e e  w e l l . The o b 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 x p m o d e l g i v e s =4 a n d t h e s t i c k m o d e l =3. T h i s scaling  coefficients  the r e s u l t s  the s l i p  implies  t o l i e b e t w e e n t h e two b o u n d a r y  that  21 1 c o n d i t i o n s , as has satisfactory;  the  (197),  been s u g g e s t e d tensor  more s l i p c h a r a c t e r  i s not  e l e m e n t s behave d i f f e r e n t l y .  than the other  two  c a v i t y model f o r m o t i o n : A  significant population  of the m o l e c u l e s o c c u p i e s  liquid  freely  that are  l a r g e enough t o p e r m i t  r o t a t e , or a t  the x - a x i s  (which  a s m a l l number o f t h e  e f f e c t of  other  energy f o r the  observations  discussed  only  sweep o u t the  decreasing to that  the  expected  i s also consistent with  i n ( S e c t . 1 5 ) and  other  much  the  workers  . The  p e c u l i a r o r d e r i n g o f R^  e x p l a i n , b u t may p o s s i b l y due  20.5  reflect  SUMMARY OF  R^  is difficult  local anisotropy  THE  be  fails  i n t h e medium,  and  NMR  s t u d i e s can  r o t a t i o n a l d i f f u s i o n tensor  c ) The  chloroform.  drawn f r o m t h i s w o r k , a) be  is  behaviour  has  combined  of our  The  extremely  hydrodynamic model, w h i l e not  to account f o r the  It  o f t h e p r o b e , b)  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 misleading,  to  RESULTS  been d e m o n s t r a t e d t h a t ESR give the  and  t o a weak c o o r d i n a t i o n w i t h t h e  T h r e e c o n c l u s i o n s may  to  r a t e and  to  allow  f r e e r o t o r s has  rotation relative  from hydrodynamic m o d e l s . T h i s  (198)  of  (which  in  jumps, about  l a r g e enough t o axes  i n c r e a s i n g the d i f f u s i o n  activation  ESR  two  population  cavities  s m a l l e s t volume), but  c a v i t i e s are  f r e e r o t a t i o n about the l a r g e r v o l u m e s ) . The  the  This  the m o l e c u l e  l e a s t undergo l a r g e angle  sweeps out  has  modes o f m o t i o n .  i s c o n s i s t e n t with the  the  R^  misleading,  probe. There i s  212 still  scope f o r d e v e l o p i n g  incorporating  t h i s model, but a  l a r g e a n g l e jumps ( d r s o l v e n t  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  20.6  theory cavities),  m e d i a may be n e e d e d .  A STRATEGY FOR MEASUREMENT OF DIFFUSION TENSORS  Firstly  i t should  be commented t h a t o b t a i n i n g  measurements from d i l u t e  NMR r e l a x a t i o n  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 very  e r r o r prone. Recent  d e v e l o p m e n t s i n NMR t e c h n o l o g y h a v e i m p r o v e d t h i s but  instrument time i s s t i l l  any  strategy  that  situation,  a t a premium. F o r t h i s  reason  r e d u c e s u s e o f NMR s p e c t r o m e t e r s i s o f  interest. Secondly, the o b j e c t i v e of the exercise  i s t o obtain  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 times for a solute fast-motional not  i n the  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 r e  a useful objective. Studies  i n t e r e s t i n g , but rather  i n neat l i q u i d s a r e  restrictive.  Slow-motional  studies  i n NMR h a v e 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 , b u t i t r e m a i n s t o be d e m o n s t r a t e d  that d i f f u s i o n tensors  c a n be r e l i a b l y e x t r a c t e d  from  such  spectra. G i v e n t h e a b o v e comments, how d o e s one d e s i g n to obtain  the best  information?  As d i s c u s s e d  a probe  previously the  probe should  be s t a b l e w i t h a w e l l d e f i n e d  sufficiently  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 t h e  p r o b e must be a b l e pieces  of data.  t o provide  g e o m e t r y a n d be  a t l e a s t three  independent  213 Consider the requirements There to  f o r a n NMR o n l y s t u d y  first.  s h o u l d be o n l y o n e 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 h e m e a s u r e d T,, i.e.,  only quadrupole  n u c l e i s h o u l d be  used, 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 couplings looks promising  (199)(200).  There  s h o u l d be a t  least three magnetically d i s t i n c t n u c l e i . Furthermore o r i e n t a t i o n of the major a x i s of the quadrupole  the  tensor f o r  t h e t h e s e n u c l e i s h o u l d be such 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 and one p o l a r a n g l e *90° a r e needed 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 inequivalent deuterons nucleus  t h e r e a r e two m a g n e t i c a l l y  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  i s i n t h e p l a n e . We u s e d  1 3  1  C , b u t "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 . O u r t h i r d n u c l e u s was a n 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  problem  w i t h NMR. Two s u i t a b l e n u c l e i c a n b e 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 . Also the r e l a x a t i o n times 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 the azimuthal angle o f the major a x i s of the quadrupole  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 separated, azimuthal angles of 20° and 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 . For these reasons to another  i t i s common t o r e s o r t  spectroscopic technique t o obtain the extra  i n f o r m a t i o n . I R , L S 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 E S R m e a s u r e m e n t s t o g e t  214 t h e e x t r a d a t a . ESR  a l l o w s t h e use o f d i l u t e  r e q u i r e s a paramagnetic  solutions,  s p e c i e s . In our c a s e t h i s  but  is easily  a c h i e v e d by c h a n g i n g t h e c e n t r a l m e t a l o f o u r p r o b e . T h i s however, i s the major  restriction  to t h i s approach; t h e r e  must be d i a m a g n e t i c and p a r a m a g n e t i c a n a l o g s o f t h e p r o b e . The  d e s i g n of paramagnetic probes  the l i g h t  i s best considered i n  of Eqn.12.22. I n our c a s e 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. H o w e v e r , i n g e n e r a l  most o f t h e r e d u c e d s p e c t r a l d e n s i t i e s h a v e t o be but the b a s i c  retained,  f o r m o f Eqn.12.22 i s u n c h a n g e d by t h e s e  extra  t e r m s so f o r t h e p u r p o s e o f t h i s d i s c u s s i o n we c a n u s e following expression for  Ti  1  =  where t h e sum  T . 2  I J ( 0 ) . [ A ( K + m ) + A m + G] 2  k  K  over k i s used t o denote  non-secular terms, j ( w ) , a r e  (20.2)  rotor  t h a t more t h a n  i s retained.  not  f o r convenience.  we n o t e t h a t t h e s p e c t r u m w i l l  (=21+1) l i n e s so t h e l a r g e r  one  The  n o t n e g l i g i b l e , b u t do  t h e a r g u m e n t s so t h e y a r e d r o p p e d  Firstly  + X  8  e i g e n v a l u e f o r the asymmetric  affect  the  contain  K  the n u c l e a r s p i n coupled t o the  e l e c t r o n t h e b e t t e r . F o r i n s t a n c e , v a n a d i u m , 1=7/2, w i t h eight  lines gives better s t a t i s t i c s  which g i v e s three l i n e s . o r t h o r h o m b i c and I>1 o b t a i n e d f r o m t h e ESR  than n i t r o g e n ,  I f the magnetic  1=1,  tensors are  t h e c o m p l e t e d i f f u s i o n t e n s o r may  be  s p e c t r u m . H o w e v e r , i f one w i s h e s t o do  215 via  this least for  the  2k+1  the  s p e c t r a l d e n s i t i e s the  lines,  residual  2k  f o r the  s p e c t r u m must c o n t a i n a t  k j ( 0 ) and  l i n e - w i d t h , X.  j(o>) t e r m s and  A l s o the e i g e n v a l u e s  linearly  independent. Obtaining  does not  guarantee t h a t the complete d i f f u s i o n  found.  f r o m two  suffice  i f direct  It t h e ESR  nuclei  then  are  not  three spectral d e n s i t i e s t e n s o r can  H o w e v e r , i f t h e d a t a a r e t o be c o m b i n e d w i t h  results  one  a two  l i n e ESR  i n v e r s i o n methods a r e  NMR  spectrum  will  used.  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 s p e c t r u m t o o b t a i n r e l i a b l e d a t a . As  m.  the  d e p e n d e n c e of l i n e - w i d t h i s c a r r i e d a l m o s t e n t i r e l y  by  hyperfine anisotropy  quite  l a r g e , but fails  not  approximations  so l a r g e t h a t t h e  discussed  e x a m i n e d . The  minimised  by  be a c h i e v e d c o u p l i n g and responsible  keeping  keeping f o r the  the  T h i s can  the w i d t h  be  the be  should also also  be  be  residual l i n e - w i d t h small. This  the g-anisotropy  can  hyperfine  s m a l l . The  s p i n - r o t a t i o n term, the  latter  is  major  residual line-width. t o note t h a t f o r n i t r o x i d e s the  (iru=0) c a r r i e s v e r y  f a c t most o f t h i s  line.  itself  o v e r l a p . L i n e o v e r l a p can  is interesting line  have t o  by u s i n g p r o b e s w i t h no u n r e s o l v e d  c o n t r i b u t i o n t o the  center  i n Sect.14 w i l l  the  criterion  interest. Also  0  line  s h o u l d be  fast-motional  hyperfine coupling, A ,  large to prevent  In  (A i n E q n . 2 0 . 2 ) t h i s  f o r the t e m p e r a t u r e range of  It  be  little  motional  i n f o r m a t i o n i s c a r r i e d by  information. the  m.=1  t u r n e d t o a d v a n t a g e t h o u g h , by s u b t r a c t i n g  of the c e n t e r  line  from the o u t e r  two  lines  and  216  then subtracting the widths of these two l i n e s , the -relaxation  Eqn.12.22  i s thus considerably  simplified.  To summarise, the ideal probe for ESR/NMR studies should have the following requirements.  Paramagnetic and diamagnetic structural Magnetic tensors that are not At least two non-coincident  analogs.  isotropic.  a x i a l 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 s p l i t t i n g should be such that I i n e - w i d t h / l i n e - s p l i t t i n g < 0 . 2 . The nuclei for NMR quadrupolar.  studies should be  The probe should possess enough symmetry that the orientation of the d i f f u s i o n tensor i s known.  The dtc class of probes c e r t a i n l y  f i t most of the above  requirements. Their p r i n c i p a l f a i l i n g s are the  low  s o l u b i l i t y of the nickel complexes and the a x i a l symmetry of the hyperfine coupling and g-terisors. However, as demonstrated here this i s just an inconvenience,  they can  s t i l l be used to obtain the d i f f u s i o n tensor. Also their geometry can, and has been, t a i l o r e d for motional It may  be possible to t a i l o r their chemistry  studies.  for work in  aqueous solvents and also improve the s o l u b i l i t y of the nickel, complexes.  217  20.7  F I N A L REMARKS The  p e r s i s t e n c e o f t h e 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 c a n be a s c r i b e d t o t h r e e f a c t o r s . For molecules t h i s theory  t h a t a r e much l a r g e r  than the s o l v e n t  i s q u i t e a c c u r a t e . However, how l a r g e t h e p r o b e  molecule/solvent molecule h y d r o d y n a m i c model f a i l s  s i z e r a t i o h a s t o be b e f o r e t h e isstill  unknown, a l t h o u g h  there  (201)(202).  h a v e been some r e c e n t a d v a n c e s i n t h i s a r e a  The  i s o t r o p i c model h a s been e x t e n s i v e l y u s e d t o i n t e r p r e t motional  r e s u l t s . T h i s approach  has p r o b a b l y  obscured the  i n a d e q u a c i e s o f t h e hydrodynamic model and i f i t s use persists  i tw i l l  s e r i o u s l y hamper p r o g r e s s  dynamics s t u d i e s . A l s o the f r i c t i o n 3  a ,  i n molecular  coefficients  t h e 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  p r o d u c e s a 30% v a r i a t i o n This w i l l  greatly  scale with i n 'a'  i n the c a l c u l a t e d c o r r e l a t i o n  i n f l u e n c e t h e agreement o r o t h e r w i s e  time. with  the hydrodynamic t h e o r y . F o r i n s t a n c e , i s t h e Van-der-Waals r a d i u s t h e a p p r o p r i a t e measure of t h e d i m e n s i o n s o f molecules  rotating  i n s o l u t i o n ? The a p p a r e n t  agreement o f t h e ' s l i p '  better  r a t h e r t h a n t h e ' s t i c k ' m o d e l may b e  a r e s u l t ofa systematic underestimation of the molecular dimensions. The  c o n c e p t u a l s i m p l i c i t y of t h e hydrodynamic model  makes i t v e r y a t t r a c t i v e . The model r e m a i n s t o b e d e v e l o p e d for  non-spheroidal molecules.  approach  Youngren's and A c r i v o s ' s  a l l o w s t h i s , b u t t h e gap s t i l l  remains.  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 , both of which  have been t h o r o u g h l y  s t u d i e d a n d 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  t h e s e two c a s e s , t h e n m o d e l s 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 in  t h e s o l v e n t and ' f r e e - r o t a t i o n '  (203)(204)(136)(205)(206)  will  i n e r t i a l m o d e l s (50) e x t e n d e d  i n solvent cavities  n e e d t o be d e v e l o p e d , t o a n i s o t r o p i c motion.  orthe Further  s t u d i e s o f t h e 'probe i n a s o l v e n t ' t y p e , a s i s d i s c u s s e d here, w i l l  be n e e d e d t o p o i n t t h e way f o r t h e o r e t i c a l  developments o f t h a t  kind.  PART  6.  NOTES ON THE D I G I T A L ACQUISITION OF ESR  219  SPECTRA  21. THE D I G I T A L ACQUISITION OF ESR SPECTRA  21.1  INTRODUCTION  This section w i l l spectra  address  the d i g i t a l  f r o m two p o i n t s o f v i e w ,  a c q u i s t i o n o f ESR  namely, t h e development o f  the s o f t w a r e and hardware o f t h e d i g i t a l  acquisition  system;  the development o f t h e a l g o r i t h m s f o r t h e 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 . Digital acquisition see  (116)(207)  (211)(212)(213)),  systems a r e not n o v e l  {e.g.  (140)(208)(209)(198)(210)  f o r examples see  but a t t h e time o f c o n s t r u c t i o n of our  system, the micro-processer technology  (for reviews  was a t t h e l e a d i n g - e d g e o f  memory was e x p e n s i v e )  and t h e p e r s o n a l  c o m p u t e r h a d n o t been i n v e n t e d . Some o f t h e a s p e c t s o f d e s i g n and methodology r e f l e c t  t h i s . Notably,  t h e use o f  o f f - l i n e data p r o c e s s i n g and t h e l a c k o f r e a l - t i m e a v e r a g i n g facilities. E a r l y s p e c t r a were c o l l e c t e d o n p a p e r - t a p e processed (140).  on t h e Amdahl 470 a t t h e UBC c o m p u t i n g  Later, this link  center  t o t h e Amdahl was u p g r a d e d by t h e  purchase of a magnetic tape computer p l a c e s v i r t u a l l y  u n i t . The u s e o f a main  frame  n o r e s t r i c t i o n s o n memory o r  speed, so the p r i m o r d i a l nature  o f our microprocesser (a  F a i r c h i l d F 8 ) was n o t a g r e a t d i s a d v a n t a g e . further  and  improved by t h e a c q u i s i t i o n  The s y s t e m was  o f a DEC L S I - 1 1  micro-computer. T h i s computer p e r m i t t e d t h e use o f interactive graphics  (which were n o t a v a i l a b l e  220  f o r the  221  m a i n - f r a m e a t t h a t t i m e ) , but and  s p e e d now had  restrictions  t o be c o n s i d e r e d .  A n a l y s i s o f d i g i t a l ESR s p e c t r a t h a n f o r most d i g i t a l the a b s c i s s a  methods f o r m a n i p u l a t i n g  overview w i l l algorithms information personal  data.  not  ESR s p e c t r a  i s sadly  be b r i e f l y  ?',  the  i s r e d u c e d by 2-3 o r d e r s  q u a l i t y and  basic this  9  software).  The  the  time  reasons  s p e c t r u m measurement i s the a n a l y s i s time f o r  o f magnitude;  be m a n i p u l a t e d i n a manner t h a t using analog  bother t o  (especially considering  i m p r o v e d by a t l e a s t a f a c t o r o f f i v e ;  impossible  techniques  l a c k i n g . Such a n  t h a t m i g h t be a s k e d i s , 'why  t h r e e - f o l d ; the p r e c i s i o n of  s p e c t r a can  scattered  r e v i e w e d and  n e e d e d t o d e v e l o p t h e h a r d w a r e and  spectra  numerical  be o f i n t e r e s t t o a n y b o d y a t t a c h i n g a  d i g i t i s e ESR s p e c t r a  are  and  an overview o f the  c o m p u t e r t o a n ESR s p e c t r o m e t e r . "  A question  complicated  spectra are  be a t t e m p t e d h e r e , b u t  used w i l l should  Algorithms  digital  l i t e r a t u r e , but  needed f o r p r o c e s s i n g  i s more  s p e c t r a because of the v a r i a b i l i t y o f  (field-sweep)  throughout the  i n memory s i z e  methods (e.g.DISPA).  digital  is difficult or Thus b o t h  the  q u a n t i t y o f 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 increased.  9  * T h e 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 o f a few b r o a d l i n e s t h a t a r e m e a s u r e d 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 t h e a b s e n c e o f s o 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 ( s e e (214)(215) for 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 block diagram simplified  of t h e s y s t e m  i s given in Fig.21.1.  f l o w - c h a r t f o r the s o f t w a r e  Briefly  the system  o p e r a t e s a s f o l l o w s . The  e n t e r s t h e sample i d e n t i f i c a t i o n starts  etc.  via  information written then taken  i s m e a s u r e d and  operator  the  scan i s  sample  to the tape. F i e l d / a m p l i t u d e data  s i m u l t a n e o u s l y and  t o the tape  Fig.21.2  t h e t e r m i n a l and  t h e a c g u i s t i o n p r o g r a m . When t h e f i e l d  s t a r t e d the temperature  written  i s shown i n  A  are  16 b i t d a t a p o i n t p a i r s  i n b l o c k s of 50 p a i r s . B e t w e e n e a c h  b l o c k t h e G a u s s m e t e r r e a d i n g and  the c o r r e s p o n d i n g  voltage are c o l l e c t e d . This i s continued u n t i l t h e s c a n when a c q u i s i t i o n  i s stopped.  The  measured a g a i n , the microwave frequency c a l i b r a t i o n data w r i t t e n be t r a n s f e r r e d t o t h e DEC  t h e end  temperature  i s t a k e n and  t o t h e t a p e . The LSI-11  Fieldial  data  for analysis.  file  of is  the can  then  READY  FREQUENCY COUNTER (F/C) 1(3  CHANNELS)  0) *1  O M-  iQ iQ C C  cn  i->-  rt  <X>-  SAMPLE & HOLD  16 BIT ADC  ANALOGUE MULTIPLEXOR  DIGITAL MULTIPLEXOR  LSI-11 COMPUTER  n> 1  HOLD  to  o • 1  3 t— • Cd in •< cn  n- o (D o 3 *"  ADC ON  SAMPLE & HOLD  ADC  READY  S/H  COMMAND DECODER  HOLD•  9719 TAPE FORMATTER  cr u. cc  11 DATA LATCH  START/STOP SCAN-LIMITS  9800 TAPE DRIVE  9718 TAPE FORMATTER  RUN PORT 0  0)  3  CC • CJ -x U J  SELECT X.T.Y  a  Q)  oiict * iu TL  SELECT CHANNEL  PORT 1 VIDEO TERMINAL  CPU & CLOCK  PROGRAM  BUFFER RESET  —  _  -.i  FBUG MONITOR  :TAPE DATA  BUS  4K MEMORY  F-8  TAPE ; FLAGS: 9717A TAPE  DflTfl  PROCESSING SYSTEM  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  TEMP. RECORD  PAD DATA RECORD  COLLECT X-T DATA  DELAT LOOP  GET CALIB DATA  VIDEO PLOT NO TES  NO  STOP 7  SO PTS. ?  TES  END DATA RECORD  F i g u r e 21.2. Flow-chart f o r the software of the a c q u i s i t i o n system.  225 21.3  THE  BASIC PROBLEMS IN ACQUIRING ESR  A number o f n o i s e s o u r c e s  i n ESR,  SPECTRA  t h a t a r e masked by  the  i n h e r e n t t i m e c o n s t a n t o f t h e c h a r t r e c o r d e r , become apparent  when one  first  records a d i g i t a l  n o i s e s o u r c e s depend on t h e s p e c t r o m e t e r ,  spectrum. but  These  typical  e x a m p l e s 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 latter in  i s modulation  of the spectrum  the Gaussmeter probe.  essential.  ( A l s o see  most s e v e r e  of  by t h e m o d u l a t i o n  Sect.14.8)  the c e n t e r of f i e l d  some m e t h o d of c a l i b r a t i n g the r e a l problem m e c h a n i s m . The non-linear equally  t h e sweep i s e s s e n t i a l .  i s incremented  fashion resulting  spaced  d a t a and  requirements. for  reliable  spaced  5 0  data  so  However,  inconsistent  and  although spaced  T h i s has a g r e a t i n f l u e n c e of  d a t a ; d o u b l i n g computer-memory  A l s o enough p o i n t s must be c o l l e c t e d  d i s c u s s e d here  5 0  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  spectrum  interpolation  Both  sweep  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  on t h e s o f t w a r e d e s i g n a s field  i n an  i n a data set that,  w i t h - r e s p e c t - t o magnetic f i e l d .  both  sweep a r e v a r i a b l e  i s the n o i s e from the F i e l d i a l  field  coils  C a r e f u l placement of t h i s probe i s  Magnet n o i s e p o s e s a number of s e r i o u s p r o b l e m s . t h e w i d t h and  the  of the d a t a . Most of t h e  r e l a t e s t o t h i s problem.  to allow software  Once an e q u a l l y  set i s c r e a t e d the software development i s  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 a n d 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 c a n a l s o be n o n - l i n e a r , a l t h o u g h we h a v e n o t d e t e c t e d t h i s on o u r s y s t e m . A l g o r i t h m s t o s h i f t and e x p a n d / c o m p r e s s t h e d a t a a r e t h u s an e s s e n t i a l p a r t o f t h e s o f t w a r e .  226  relatively  21.4  straight-forward.  ADC RESOLUTION  There  are four b a s i c  the F i e l d i a l  types of data c o l l e c t e d  v o l t a g e (X-data, the f i e l d ) ,  by o u r  system,  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 , t h e a m p l i t u d e ) , thermocouple Gaussmeter  data  ( t h e t e m p e r a t u r e ) and Gaussmeter  data are already a v a i l a b l e  need not concern us h e r e . For d i s p l a y Y-data  form and  5 1  p u r p o s e s a 10 b i t  5 2  resolution  of t h e  a r e q u i t e a d e q u a t e . However i f e x t e n s i v e n u m e r i c a l  manipulation  i s t o be done 12 o r 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 for  in digital  d a t a . The  with d i g i t i s a t i o n  the X-data,  10 b i t s  is fine,  n o i s e (116)(219). buti f ,  Similarly  f o r e x a m p l e , one  w i s 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 o v e r a 1000G sweep t o avoid amplitude d i s t o r t i o n r e q u i r e d . Moreover, opposed  t o -10V  (220),  the F i e l d i a l  then  13-14  bits  are  v o l t a g e r u n s f r o m 0-5V, a s  t o +10V f o r t h e Y d a t a s o i f one u s e s  the  same ADC f o r b o t h d a t a s e t s a n d no a m p l i f i e r s a n o t h e r 2 b i t s are required.  In practice,  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 w i s h t o m e a s u r e t o t e m p e 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 1  5 3  over a wide 6  t e m p e r a t u r e r a n g e . The  A r e s o l u t i o n o f a t l e a s t 1:10 i s r e q u i r e d f o r Gaussmeter measurements, i . e . , 2 0 b i t s . 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) h a v e a r e s o l u t i o n o f 2 w h e r e n i s t h e number o f ' b i t s ' o f 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) 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 o f 0.1° o v e r a r a n g e o f -70°C t o +120°C i s r e q u i r e d f o r t h i s work. 5 2  n  5 3  227  thermocouple  voltage i s amplified  15 b i t ADC, a t l e a s t ,  21.5 The  1000X t o ±3V (max.) s o a  i s desirable.  NCK OF POINTS COLLECTED. THE NYQUIST CRITERION maximum number o f u s e f u l d a t a p o i n t s t h a t c a n be  collected  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 The  a n d c o m p u t e r memory a n d  X signal-to-noise  ratio  averaging  i s done t h e r e i s l i t t l e  more t h a n  2K p o i n t s . " A v e r a g i n g  5  speed.  (SNR) i s =2000:1 s o u n l e s s t o be g a i n e d by c o l l e c t i n g i s not meaningful  a r e c o l l e c t e d a t a r a t e above t h e N y q u i s t band l i m i t e d a m p l i f i e r =1/time-constant, One c a n a r r a n g e the scan  the Nyquist  rate  5 5  i f data  frequency. For a is  typically  <5Hz, o r <2000 p o i n t s f o r u s .  t o decrease  t h e time c o n s t a n t and i n c r e a s e  t i m e s o t h a t more p o i n t s a r e g a t h e r e d ,  p o i n t s o c c u p i e s 64 K b y t e s ,  1/4 o f t h e a v a i l a b l e  b u t 16K computer  memory. T h i s means t h a t t h e d a t a p r o c e s s i n g h a s t o be done ' o f f - t h e - d i s c ' , which  i s slow,  r a t h e r than  ' i n core'. Also,  ' i n c o r e ' p r o c e s s i n g o f >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 s o t h e d a t a h a s  t o be b o x - c a r r e d b a c k t o <4000 p o i n t s b e f o r e u s e . T h i s o f course  5 4  improves  the apparent  1 0  SNR, b u t i t i s j u s t a s  A K i s 1024 o r 2 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 c o m p u t e r s , 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 u s e d w i t h Fourier transforms. N o t e t h a t h e r e t h e N y q u i s t r a t e i s d e t e r m i n e d by t h e a m p l i f i e r b a n d - w i d t h , n o t t h e s p e c t r u m ( a s 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 o f N y q u i s t r a t e s a n d ESR s p e c t r a s e e (219) (221) (27) . 5 5  228  efficient  to c o l l e c t  c o n s t a n t . The  l e s s data with a higher  PSD t i m e  c o n c l u s i o n t o be drawn i s t h a t , f o r  s y s t e m , 2000 p o i n t s i s t h e maximum s i z e  our  f o r a spectrum.  Though i n g e n e r a l more w o u l d be d e s i r a b l e .  21.6  F I L T E R I N G METHODS  Filtering  (222)(223)(224)(225)(226)(227).  studied will  (smoothing) methods have been f a i r l y e x t e n s i v e l y  be d i s c u s s e d h e r e ,  analog  (time-constant)  An  important  acquired, data  by eye  (Sect.21.8) and  filtering.  does not  Filtering  set i s  improve the p r e c i s i o n o f the  i t i s purely cosmetic  t h e a p p a r e n t SNR). a r e not v e r y  filtering  methods  p o i n t t o note i s , t h a t once a d a t a  filtering  (145),  box-car  O n l y two  (i.e.,  i t only  i s important  improves  a s many  algorithms  s t a b l e i n the presence o f n o i s e . A l s o  searching  is difficult.  The  Nyquist  pass a m p l i f i e r  rate,  (N^), for data  a c q u i r e d w i t h a low  is  N  where T i s t h e t i m e  f  = (2irr)"  constant  1  (21.1)  ( i n v e r s e band-width) o f the  a m p l i f i e r . T h i s means t h a t we h a v e t h e c h o i c e o f a c q u i r i n g l a r g e amount o f d a t a a t a low noisy-data)  and  time  constant  numerically f i l t e r i n g  (i.e.  ,  i t (with a box-car  in  229  t h i s c a s e ) , or of c o l l e c t i n g data  {i.e.,  at a higher  Box-car f i l t e r i n g adjacent  points  s m a l l e r amounts o f  time c o n s t a n t ; (228)  j u s t c o n s i s t s of  ( S e c t . 2 1 . 8 ) so c o n s i d e r  collecting  'n' p o i n t s w i t h a SNR o f S  *m'  (n/m w i l l be i n t e g r a l ) ,  by  points  analog  0  filtered  filtering) averaging  the case of and b o x - c a r r i n g  t h e SNR, S, i s t h e n  given  (229)  S = So/fn/m)  with a corresponding  (21.2)  d e c r e a s e i n r e s o l u t i o n o f n/m.  maximum number o f m e a n i n g f u l p o i n t s a c q u i r e d constant  r  0  i n a scan time,  n = N,T  If  to  f o r a time  T, i s f r o m Eqn.21.1  A  (21.3)  T/T  Q  we i n c r e a s e t h e t i m e c o n s t a n t  meaningful  Now t h e  t o T t h e n we g e t m'  points  m' <* T / T  (21.4)  We c o u l d o f 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 t h a n t h e N y q u i s t frequency,  b u t no f u r t h e r  information  i s obtained;  the data  230 s h o u l d be  a c q u i r e d at a r a t e c o n s i s t e n t w i t h the  b a n d - w i d t h . Hence we  get  amplifier  f r o m Eqn.21.2-Eqn.21.4  S = ScVCrAc) = S i/(n/m' )  (21.5)  0  i.e.,  b o x - c a r r i n g c o n f e r s no a d v a n t a g e o v e r  f o r a g i v e n scan  of the n o i s e . Analog  filters  an a s y m m e t r i c manner (30). thus  filtering  s h i f t and  and  the  called  1/f  noise  several  The  offset),  box-car  w i t h a l a r g e time  average,  so  which  than  t o do  averaging  i s the  r e a l - t i m e a v e r a g i n g . Memory and of t h e L S I - 1 1  approach f o r us, but f a s t e r c o m p u t e r was  one  acquisition  and m a n i p u l a t i o n o f a l a r g e number o f p o i n t s . T h i s can  restrictions  scan  constant.  p e n a l t y f o r box-car  o v e r c o m e by  The  In  t i m e . Thus, i t i s b e t t e r t o r a p i d l y  ( n o i s y ) s p e c t r a and  long scan  and  the  methods.  p r o d u c e a l a r g e amount o f  (baseline d r i f t  i n c r e a s e s w i t h scan  nature  B o x - c a r r i n g j u s t broadens  i s t h e more d e s i r a b l e o f t h e two  spectrometers  two  broaden the peaks i n  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 . p r a c t i c e ESR  methods  t i m e . However, t h i s a n a l y s i s i g n o r e s  f a c t o r s , t h e d i s t o r t i o n c a u s e d by  p e a k s and  analog  limit  speed  t h e u s e f u l n e s s of  i t w o u l d be w o r t h w h i l e  be  this  pursuing  if a  available.  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 t h e maximum t i m e c o n s t a n t c o n s i s t e n t  231  with a distortion  (i.e.,  f r e e spectrum  sweep t h e n a r r o w e s t  feature  <l/50 of t h e t i m e t o  o f i n t e r e s t (30)).  The d a t a a r e  t h e n b o x - c a r r e d on t h e L S I - 1 1 down t o 2 0 4 8 , 1024 o r 512 p o i n t s d e p e n d i n g o n t h e end u s e .  21.7  INTERPOLATION  As m e n t i o n e d p r e v i o u s l y m a i n l y due t o n o i s e .  t h e sweep i s n o t q u i t e  The d a t a  terms of t h e magnetic  are thus not e q u a l l y  a n a l y s i s . H o w e v e r , most a l g o r i t h m s , execute f a s t e r i f the data e.g.  (227)  smoothing  spaced i n  f i e l d . F o r most p u r p o s e s t h i s  consequence, p r o v i d i n g b o t h t h e X and Y d a t a  algorithms,  linear,  notably  are equally  t h e FFT (25)(26) r e q u i r e the data  and  i s o f no  a r e used f o r integration,  s p a c e d . Some Savitsky-Golay  t o be e q u a l l y  spaced. For  these 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 data t o o b t a i n an e q u a l l y  spaced data  s e t . Here, data  i n t e r p o l a t e d and/or box-carred  (vide  supra)  were  t o 5 1 2 , 1024 o r  2048 p o i n t s , d e p e n d i n g on t h e e n d u s e . P o w e r s - o f - t w o a r e used t o m a i n t a i n graphics  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 a n d t h e  terminal.  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 , Lagrangian piecewise  5 6  i n t e r p o l a t i o n , cubic polynomial.  5 6  s p l i n e s , or a s l i d i n g  Lagrange i n t e r p o l a t i o n i s extremely  A n'th order polynomial i s f i t t e d t o p o i n t s ' i ' t o 'i+n' and t h e d a t a i n t e r p o l a t e d f o r t h i s r a n g e o f p o i n t s . T h e p o l y n o m i a l i s t h e n f i t t e d t o t h e 'i+n' t o 'i+2n' p o i n t s etc.. A cubic s p l i n e i s a piecewise cubic polynomial f i t with the extra c o n s t r a i n t that the d e r i v a t i v e s at the e n d - p o i n t s must b e c o n t i n u o u s b e t w e e n a d j a c e n t 'pieces' (230).  232 unstable  (231) e s p e c i a l l y  i n the presence  s p l i n e s a l s o show i n s t a b i l i t y Sliding low-data avoided and  p o l y n o m i a l s can  21.8  the spectrum  and  time).  BOX-CAR INTERPOLATION AND F I L T E R I N G  sets.  method o f r e d u c i n g t h e s i z e o f  data  I t c a n a l s o be u s e d t o p r o d u c e a n e q u a l l y s p a c e d  from an u n e q u a l l y spaced  f o l l o w s , each  (X,Y)  determined  data  set. Briefly,  data point i s loaded  an a r r a y ( t h e b o x - c a r ) . The  i n t o an e l e m e n t o f  index of the a r r a y element i s  by t h e r a n g e o f v a l u e s t h a t X l a y s i n . F o r  10<X<19 t h e y a r e a s s i g n e d t o e l e m e n t 2 etc. summed and  data  i t works a s  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 e l e m e n t  loaded  The  p i e c e s o f d a t a summed i n a p a r t i c u l a r another  a r r a y ; the  Y-data.  When t h i s p r o c e s s  empty b o x - c a r s  extrapolation.  5 7  running average,  1. F o r  Y values  i n t o the a r r a y element determined  c o r r e s p o n d i n g X v a l u e s a s d e s c r i b e d a b o v e . The  and  at  ( C u b i c s p r o v i d e d no improvement  interpolation  This i s a convenient  set  of n o i s e .  We u s e d a q u a d r a t i c p o l y n o m i a l  low d a t a d e n s i t i e s .  i n c r e a s e the  i n the presence  seriously distort  (232).  densities  of n o i s e . Cubic  box-car  filled  by l i n e a r  (Note t h a t t h i s which  by t h e  number o f are stored i n  i n d e x numbers b e i n g d e t e r m i n e d i s complete  are  the data are  as f o r t h e averaged  interpolation or  i s n o t t h e same a s a  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 n o t c o l l e c t e d . T h e s e p o r t i o n s of t h e 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 s u b s e q u e n t c a l c u l a t i o n s .  233 G e n e r a l l y the data were i n t e r p o l a t e d to 2 box-carred  back to 2  apparent SNR p o i n t s and l o s s of  21.9  p o i n t s . Box-carring  'm'  the f i n a l number. The  (n/m).  then  number of  penalty i s a r e s o l u t i o n  58  PEAK SEARCHING AND  b a s e l i n e was  points  improves the  by v/(n/m) where 'n' i s the i n i t i a l  FITTING  Automatic peak f i t t i n g was l o c a t e d (vide  achieved as f o l l o w s . F i r s t infra)  and  the n o i s e  (standard d e v i a t i o n ; 6) determined. The for peaks only using data  f u r t h e r than  b a s e l i n e to a v o i d spurious f i t s . The d i s t i n g u i s h i n g high-frequency and  n  level  data were 56's  searched  from the  d i f f i c u l t i e s of  n o i s e mixed with the  the s i g n a l i t s e l f were avoided  the  signal  by stepping through the  data at i n t e r v a l s of 5% of the sweep-width. Extrema were l o c a t e d by changes i n the s i g n of the slope of the j o i n i n g adjacent  step p o i n t s . T h i s of course  produces  problems i n l o c a t i n g peaks with widths l e s s than sweep-width. I f the SNR but then d i f f i c u l t i e s  i s low  line  5% of  the step s i z e can be  reduced,  i n i d e n t i f i n g broad peaks occur  slope change must be g r e a t e r than  the  (the  the noise l e v e l ) . T h i s  method cannot be g e n e r a l i s e d to a l l types of s p e c t r a , but i t i s 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  a p p l i c a t i o n s , notably s p e c t r a from n i t r o x i d e s i n membranes.  5 8  N o t e t h a t f o r a spectrum where the box-car i s much s m a l l e r than the l i n e - w i d t h , as i n our case, the e f f e c t on r e s o l u t i o n can be ignored. T h i s i s not t r u e f o r the time c o n s t a n t , where the time-constant approaches the time taken to scan the l i n e .  234  Once a peak  i s f o u n d i t c a n be f i t t e d  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  a r e more r e l i a b l e  p e a k s and c u b i c s f o r n a r r o w p e a k s . set  to half  of the s t e p s i z e .  the  initial  f i t i s ambiguous  automatically  (locally) to a  5 9  The  fitting  I f the noise  level  f o r broad window i s i s h i g h or  t h e s i z e o f t h e window i s  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 p e r f o r m a n c e c a n be f o u n d i n  (88).  The p r o b l e m s o u t l i n e d a b o v e a r e a l l e l i m i n a t e d by interactive  fitting.  window s i z e  via  The  operator  a graphics  s e l e c t s t h e peaks and  t e r m i n a l . However, t h i s a p p r o a c h  i s more t i m e c o n s u m i n g a n d i n t r o d u c e s a d e g r e e o f subjectivity  21.10 For  when v e r y n o i s y s p e c t r a a r e  BASELINE FITTING AND  fitted.  FLATTENING  l i n e a r b a s e l i n e a r t e f a c t s a d d e d 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  where b i s t h e b a s e l i n e d r i f t  (21.6)  and c i s t h e DC o f f s e t .  use t h e s u b s c r i p t s / and r t o denote t h e l e f t  and  I f we  right  s i d e s of the spectrum then  5 9  T h a t i s b r o a d and n a r r o w w i t h - r e s p e c t - t o t h e f i t t i n g window. The f i t t i n g window b e i n g t h e number o f p o i n t s f i t t e d to the peak, i n the v i c i n i t y of the peak.  235  f  =  If  the  can  f  b(x )  d e n s i t i e s , as of  c from the d a t a .  the b r o a d e s t l i n e e.g.  i f one  this will If  large data  s e t s , or  g e n e r a l l y not  the  s i g n a l has  Ay =  can  +  y; y;  = f'U,)  choose our  be  spectrum.  (234),  r e l y on  6 0  having  and  large  areas  set with high data  densities  case. of  b a s e l i n e . We  left  baseline  baseline,  a center  r  Most  the  the  the  + r(x )  low  a t l e a s t 5x  i n the  uses a s m a l l data  t h i s to e s t a b l i s h the  symmetry t h e n we  can  use  have  + b ( x , ) + b ( x ) + 2c r  r i g h t data  a d e r i v a t i v e spectrum, f ' ( x ^ ) = - f ' ( x  6 0  However, t o  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  but  we  1  f (x^ ) = f' (x^_ )=0  s w e e p - w i d t h must be  flattening algorithms, of  i.e.,  baseline  requires very the  (21.7)  + c  r  d e t e r m i n e b and  meet t h i s c r i t e r i o n  width  b(x, ) + c  l  spectrum goes t o the  t h e n we  data  u)  points  (21.8)  such t h a t , f o r  ), hence  I d e a l l y one c o u l d a v e r a g e t h e d a t a i n p r o p o r t i o n t o t h e s l o p e o f t h e l i n e , b u t 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 =  we c a n now  b ( x ) + b ( x ) + 2c z  (21 .9)  r  use a n o t h e r 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  center  i s a f u n c t i o n o f t h e b a s e l i n e . However, f o r a s i n g l e the c e n t e r position For rapidly,  of the spectrum i s s t e e p  i s not very  so t h a t t h e  cross-over  s e n s i t i v e to small baseline  dt c s t u d i e s , where a s i n g l e l i n e ,  line,  changes.  scanned  fairly  i s c o l l e c t e d , we c a n assume t h a t t h e d r i f t i s  n e g l i g i b l e , h e n c e we c a n d e t e r m i n e t h e DC o f f s e t  easily.  The a b o v e m e t h o d s l e n d t h e m s e l v e s w e l l t o i n t e r a c t i v e graphics  methods o f b a s e l i n e c o r r e c t i o n . The  c o r r e c t s the data  f o r b and c via  However, i t i s i m p o s s i b l e  the terminal cursor  to uniquely  t h e s e m e t h o d s and s p e c t r a w i t h  operator keys.  d e t e r m i n e b and c by  large d r i f t s  should  be  r e j e c t e d . The s e n s i t i v i t y  o f t h i s a p p r o a c h c a n be  considerably  the i n t e g r a l while c o r r e c t i n g the  by o b s e r v i n g  d e r i v a t i v e data 'b'  f o r b a n d c . The s e n s i t i v i t y  and 'c' i s i n c r e a s e d  by n a n d n  i s the r e s o l u t i o n of the g r a p h i c s for  6 1  interactive baseline  2  t o changes i n  r e s p e c t i v e l y , where n  terminal.  flattening  improved  i s shown  0 n e c o u l d expand t h e b a s e l i n e , but t h i s noisy spectra.  6 1  The  algorithm  i n Fig.21.3  i s not u s e f u l  with  237  DISPLAY DERIVATIVE SPECTRUM  INCREMENTALLY CORRECT SPECTRUM  INTEGRATE SPECTRUM & DISPLAY  INCREMENTALLY CORRECT SPECTRUM YES DISPLAY INTEGRATED SPECTRUM XlO  INCREMENTALLY 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 baseline flattening.  21.11 INTEGRATION OF SPECTRA The i n t e g r a t i o n o f ESR s p e c t r a a n d t h e a s s o c i a t e d e r r o r s h a s been d i s c u s s e d (238)(239)  i n a number o f p a p e r s  and w i l l  n o t be r e v i e w e d  (235)(236)(237)  here.  A l l integrations  were p e r f o r m e d u s i n g Simpsons r u l e w i t h a n i n t e r p o l a t e d d a t a s e t . C a r e h a s t o be t a k e n  t o remove b a s e l i n e a r t e f a c t s a s  238  the  i n t e g r a l i s very s e n s i t i v e  serious  t o them {vide supra).  No  p r o b l e m s were e n c o u n t e r e d , e x c e p t a t l o w d a t a  densities  where random s t e p s a r e i n t r o d u c e d  spectrum  (Fig.21.4).  i n t o the  J  Y  1—  F i g u r e 21.4. The e f f e c t on i n t e g r a t i o n .  This effect  i s discussed  X  j  o f low d a t a  in detail  i  1  density  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  error  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 .  of data d e n s i t y  on t h e a v e r a g e  integration  (The  ' a v e r a g e d S i m p s o n s ' i s j u s t t h e a v e r a g e o f two S i m p s o n s  rule  integrations  (27)).  s t a r t i n g at d i f f e r e n t data points;  The d a t a d e n s i t y  peak-to-peak  line-width.  see  i s s i m p l y t h e number o f p o i n t s  per  1.5  •SIMPSONS RULE A f l V E R R G E D SIMPSONS x T R A P E Z O I D A L RULE  1.0 0.5 o  0.0  ct: ii i  -0.5 CD  o  -1.0 -1.5 -2.0 -2.5 0.0  1.0  2.0  3.0  4.0  5.0  6.0  7.0  8.0  DATA D E N S I T Y 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. data 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  •SIMPSONS RULE A A V E R A G E D SIMPSONS X T R A P E Z 0 I D A L RULE  1.0 0.5 D  Ql  0.0  ce: LU  -0.5h  CD  o -1.0 -1.5 -2.0 -2.5 0.0  _L  1.0  2.0  3.0  4.0  5.0  6.0  7.0  DATA D E N S I T Y 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. data d e n s i t y f o r t h e a r e a of an a b s o r p t i o n signal.  8.0  240  1.5  •SIMPSONS RULE A f l V E R f l G E D SIMPSONS XTRRPEZOIDRL RULE  1.0 0.5 0.0  D DC  LU  -0.5  CD  o -1.0 -1.5 -2.0 -2.5 0.0  _i_  1.0  2.0  3.0  4.0  5.0  6.0  7.0  8.0  DflTfl D E N S I T Y F i g u r e 2 1 . 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 to the area.  Note t h a t  this error  i s entirely  due t o l o w d a t a d e n s i t y a n d  not  noise  or t r u n c a t i o n . Also  are  n o t u n u s u a l i n NMR a n d f o r some c o m m e r c i a l ESR d a t a  note that  such low d e n s i t i e s  stations.  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  o f ESR s p e c t r a  i s non-trivial.  The  p o s i t i o n o f t h e s p e c t r u m d e p e n d s on t h e m i c r o w a v e  frequency, the magnetic f i e l d and the r a d i c a l s g - s h i f t . the  data are not equally  number o f a c q u i r e d 6 2  can  points  Also  spaced and t h e sweep-width and i s not c o n s t a n t .  6 2  (Shifting  The s w e e p - w i d t h i s c o n s t a n t t o w i t h i n 1%, h o w e v e r c a u s e 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 .  this  241  spectra  i s described  The easily  i n more d e t a i l b e l o w ) .  unequal data  taken care  described.  spacing  o f by  I f Gauss are  and  number of d a t a  i n t e r p o l a t i o n , as u s e d as  previously  the X-data then changes i n  s w e e p - w i d t h and  f i e l d are  changes the  c a l i b r a t i o n parameters before  data  two  t o Gauss. G e n e r a l l y  integers before  use  and  points i s  r e a d i l y d e a l t w i t h , one  the care  data has  of the d e s i r e d p o r t i o n of the  i s converted t o be  simply  converting back  to  t a k e n t o keep t r a c k  spectrum.  C o m p e n s a t i n g f o r m i c r o w a v e s h i f t s i s more d i f f i c u l t the  g - v a l u e s must be  the  same and  via  interactive graphics.  known. One  and  matching the  assume t h e  t h e n be  a t o n e ' s l e i s u r e . H o w e v e r , one s p e c t r a have g - s h i f t An  spectra  the  best  (after  g-factors  subtracted has  t o be  d i f f e r e n c e s , the  are  interpolation matched  (interactively)  careful  r e s u l t s can  e x a m p l e o f t h i s method i s shown i n  Fig.21.8-Fig.21.10  as  approach i s  s w e e p - w i d t h s ) a r e d i s p l a y e d and  i n t e r a c t i v e l y . They c a n  misleading.  can  c o r r e c t a c c o r d i n g l y , but The  the  i f the be  242  1  r  ir  F i g u r e 21.8. S p e c t r u m o f t h e spin-probe.  free  »>B  F i g u r e 21.9. S p e c t r u m o f f r e e a n d bound spin-probe i n red blood c e l l ghosts.  243  F i g u r e 21.10. S p e c t r u m o f a bound s p i n - p r o b e . Found by s u b t r a c t i n g t h e f r e e probe spectrum from the combination spectrum above.  21.13 This  SHIFTING SPECTRA i s again  misleading. underlying  0  theory. order  the t r a n s i t i o n  i n the appropriate  hu> = g 0 B m o  + Am  o  frequency  i s g i v e n by  units)  0  S  or  o u t a b o v e c a n be  I t i s t h u s p e r t i n e n t t o d i s c u s s some o f t h e  To f i r s t (with A  n o n - t r i v i a l and as p o i n t e d  i n terms of the l i n e - p o s i t i o n ,  m  (21.10)  IS  B ,  244  B  =  Z  hco/q p  swept e x p e r i m e n t s , a> i s t h e  w h e r e , b e c a u s e we do f i e l d microwave If  (21.11)  0  frequency.  we w i s h t o c o m p e n s a t e a l i n e - p o s i t i o n frequency to'  microwave  B'  z  If  =  new  then  =  (g co'/goCj)B 0  (21.12)  2  I f we want t o s h i f t t h e w h o l e s p e c t r u m ,  AB  for a  ^[co'/go-co/go  A=B  Z  ~B  Z  then  ,  3  (21.13)  t h e g - v a l u e s a r e n o t known t h e y must be f o u n d by o t h e r  means e.g. generally g-shifts  Hydes a l g o r i t h m (29).  However,  g-shifts  are  s m a l l a n d t h e r a t i o g / g o c a n be s e t t o o n e . 0  may  be c o m p e n s a t e d  f o r by t h e same f o r m u l a s  (with  «'=&>), b u t t h i s c a n 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 , If  for instance.  b o t h t h e microwave and g - v a l u e s a r e s h i f t e d  correction  i s most e a s i l y  done i n t e r a c t i v e l y ,  obscures the u n d e r l y i n g assumptions  one h a s t o  but as  then the this  especially  245 c a r e f u l when u s i n g t h i s The  approach.  c e n t e r of t h e r e s o n a n c e ,  correspond  other  a r e c o r r e c t e d s i m p l y as  does not n e c e s s a r i l y  spectrum. In t h i s case  where  the  AB  A = I ' - I  and  I a r e t h e new  calibration.  See  and  (21.14)  old values  Sect.13.9.  c o m p e n s a t e d f o r by  f o r the  i n t e r c e p t of  Changes i n sweep-width  trivial,  sweep i s t h e new  however, they  at  interpolated  sweep-width.  above c o r r e c t i o n p r o c e d u r e s  continuous,  are  interpolated  i n t e r v a l s o f sweep/N, where N i s t h e number of  The  the  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 data are  p o i n t s and  data  follows.  B' = B + z z  1  0  t o t h e c e n t e r of t h e sweep o r , more i m p o r t a n t l y ,  w i t h t h e c e n t e r of any  1  B ,  are  superficially  i m p l i c i t l y assume t h a t t h e d a t a  whereas the d a t a a r e d i s c r e t e . T h i s  are  causes  problems because the d e s i r e d s h i f t s are r a r e l y  integral  m u l t i p l e s of the data  sweep-width  r e s o l u t i o n . Changing the  i s e v e n more p r o b l e m a t i c a s t h a t i s e q u i v a l e n t t o a t h a t v a r i e s a c r o s s t h e s p e c t r u m . The c o n v e r t i n g the spectrum x-data  problem i s solved  t o G a u s s and  i n t e r p o l a t i n g the spectrum t o get the p o i n t s to the d e s i r e d shifted/expanded  shift  s p e c t r u m . The  by  then corresponding original  246 s p e c t r u m 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 several or, large,  A l s o c a r e must be t a k e n  s h i f t s a r e done.  t o adopt a c o n s i s t e n t s i g n  convention  f o r the d i r e c t i o n  of the s h i f t s .  convention  i s A=X'-X, where X'  Here the  i s the c o r r e c t e d v a l u e and X  i s t h e o r i g i n a l v a l u e and A i s t h e d e s i r e d  shift.  APPENDICES  247  22.  22.1 For  APPENDICES  NOMENCLATURE the purposes of t h i s  t h e s i s metal  (dithiocarbamate) complexes  carbodithioate  a r e denoted as f o l l o w s ;  where M i s t h e c e n t r a l m e t a l a n d R i s a n a l k y l substituent  on one e n d o f t h e c o m p l e x ,  Me  methyl  Et  ethyl  Py  pyrollidinyl  Oc  n-octyl  Mp  morpholinyl  Hxm  hexamethyleniminyl  Ocm  octamethyleniminyl  Od  octadecyl  6 3  6 3  or ring  i s abbreviated  C u E t 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  2  is abbreviated generic  2  which a r e as f o l l o w s .  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 ) C u ( I I ) to  MR dtc,  title  to  6 3  C u P y d t c . The M R d t c 2  of dithiocarbamates,  Cu(II)  are given the  d e n o t e d 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 ) o r c o p p e r ( I l ) . The 'imine' nomenclature i s a r c h a i c  a n d now o n l y  d o u b l e b o n d e d t o a c a r b o n . The c o r r e c t 'azacycloalkane',  but the older  248  r e f e r s t o an NH  nomenclature i s  n o m e n c l a t u r e w i l l be  249 retained  here.  22.2 THE f H NMR SPECTRUM OF PYROLLIDINE The d e u t e r i u m at  spectrum  of neat d  30.7MHz a n d 61.4MHz.  9  A spectrum  p y r o l l i d i n e was r e c o r d e d c o n s i s t i n g of three  lines  was o b t a i n e d ; one l i n e due t o t h e a m i n e d e u t e r o n a n d two peaks,  s e p a r a t e d by 1.16ppm, due t o t h e m e t h y l e n e  The l o w f i e l d groups  peak was a s s i g n e d t o t h e p a i r o f  methylene  a t t a c h e d t o t h e n i t r o g e n and t h e h i g h f i e l d  assigned t o the other resolution  limited  (ring) methylenes.  peak  L i n e - w i d t h s were  (1Hz f o r t h e WH400 a n d 5Hz f o r t h e  C X P 2 0 0 ) . From t h e K a r p l u s e q u a t i o n (240) deuteron c o u p l i n g around  we w o u l d  expect the  t h e r i n g t o be <0.2Hz a n d w i l l 1  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 "N r e l a x e s make any c o n t r i b u t i o n t o t h e s p e c t r u m . peaks a r e s u f f i c i e n t l y w e l l nickel  deuterons.  make  too fast to  The two  methylene  s e p a r a t e d (l.76ppm  f o r the (241).  s a l t ) t h a t o v e r l a p p r e s e n t s no p r o b l e m  R e l a x a t i o n time f o r t h e neat p y r o l l i d i n e were r e c o r d e d at  310K a n d 323K on t h e CXP200 a t 30.7MHz a n d a r e shown i n  T a b l e 22.1 Temp 310 323  Ring methylene  N methylene  2.43 1.89  2.40 1.75  Amine  deuteron  1.36 0.67  Table 22.1. H r e l a x a t i o n times f o r neat d p y r o l l i d i n e . T i m e s a r e s e e s . The N m e t h y l e n e is adjacent to nitrogen. 2  9  250 The  relaxation  times w i l l  n o t be  t o comment t h a t t h e r e l a x a t i o n deuterons  times f o r the  are c o n s i s t e n t with a x i a l d i f f u s i o n  pyrollidine, values  i n t e r p r e t e d here other  as might  be e x p e c t e d . The  i s t o set the c y c l e time  (5T,)  methylene of  the  p r i m a r y use o f f o r the  t h e T,  The  NMR  have a s h o r t e r  T,.  SPECTRAL PARAMETERS  assignments  and F i g . 2 2 . 2 The  f o r t h e NMR  parameters  a r e shown i n F i g . 2 2 . 1  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  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 f r o m t h i s and  the c r o s s o v e r s of the d i s p e r s i o n  The  x component was  which  limit  f o r the d i t h i o c a r b a m a t e complex, which being a  larger molecule w i l l  22.3  these  inversion  r e c o v e r y e x p e r i m e n t s . T h e s e v a l u e s p r o v i d e an u p p e r for  than  spectrum  spectrum  (not shown).  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  i s l a r g e s t a l o n g t h e C-N  bond.  from  251  (9.4) (48.4)  I  I  (204) Q  CH-CH  ^  2  / Cr%CH  N206.2 o 208.8*  2  \  V  ^  ^  ^PH  C—N  3  \ CH-  12.4 43.8 12.8* 48.9* F i g u r e 22.1. Chemical 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 C D C 1 . E x t . TMS r e f . * d e n o t e s v a l u e s f o r a 1M C e n r i c h e d K E t d t c i n D 0. ( ) denote s o l i d s t a t e v a l u e s , t h e CSA t e n s o r i s t r a c e l e s s . 3 1 3  +  2  2  7.0  126.8*138.8* n  7.2  n  CHrCH  13.0  2  13.  15  /  N—G  s  CHgCH,  t  C><>  15,  N  s  /  I 1 CHrCH.  \ CHiCH, t_l_J 3.3t 3  5.3  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 C D C 1 . f d e n o t e s v a l u e s f o r 1M s o l n . o f p o t a s s i u m d e r i v a t i v e . N c o u p l i n g s a r e f r o m 10" M s o l n . o f C , N enriched methyl d e r i v a t i v e . ) 3  1 5  1 3  3  1 5  F i g u r e 2 2 . 3 . The powder s p e c t r u m f o r N i E t d t c a t 50.3MHz. The a r r o w d e n o t e s t h e e x t e r n a l benzene r e f e r e n c e . 2  22.4 ESR SPECTRAL PARAMETERS The m a g n e t i c t e n s o r  p a r a m e t e r s f o r CuPyDtc a r e  -119MHz  A A  0  •-- 2.022  g ^yy  -• 2.018  -474MHz  *zz  = 2.088 '•  g  = 2.043  zz  A  XX  -106MHz yy  A  g 3  XX  = -233MHz  0  (44)  253 22.5  COMPARISON OF REDFIELD AND OTHER THEORIES  An a l t e r n a t i v e a p p r o a c h wave-functions  to dealing with f i r s t  i s t o do a v a n - V l e c k  transform  order (242)  so t h a t  t h e p e r t u r b a t i o n t e r m s a r e c a r r i e d by t h e o p e r a t o r s r a t h e r than t h e w a v e - f u n c t i o n s . simplified  (  w  —>  m o d i f i e d , i.e.  f r e q u e n c i e s a r e then  )• However, t h e s p e c t r a l d e n s i t i e s a r e  ,  j(co  where w  The t r a n s i t i o n  r e 5  ) ->  i s thef i r s t  j(co )(1+f)  (22.1)  0  f r e q u e n c y , co i s  order t r a n s i t i o n  the zero order t r a n s i t i o n  frequency  0  and f i s t h e van-Vleck  c o r r e c t i o n term. Using a T a y l o r expansion co res  i t i s easy  o f Eqn.22.1 w i t h  t o show t h a t f=co co j ( 0 ) j (co) where co i s t h e a a 0  hyperfine coupling frequency,  i.e.,  thef i r s t  c o r r e c t i o n s are products of t h e reduced  order  spectral  Comparison of our r e s u l t s w i t h other workers order  i s c o m p l i c a t e d by t h e d i f f e r i n g  approximation set  densities. to first  degrees of  u s e d by t h e v a r i o u s w o r k e r s .  H o w e v e r , i f we  co —> co a n d co —> 0 i n a l l c a s e s a n d d r o p a l l t e r m s res a ^ u 0  of  o r d e r h i g h e r t h a n oi /m  in  our work) then a comparison  a  second  order  (Redfield  vs.  (which are e q u i v a l e n t t o C  (C) t e r m o c c u r s  a corresponding more-or-less  0  f term  2  terms  i s f e a s i b l e . Wherever a i n o u r e q u a t i o n t h e r e s h o u l d be  i n theother theories. This i s  the case, b u t g i v e n the d i f f e r e n c e s i n approach l i n e a r - r e s p o n s e vs.  stochastic L i o u v i l l e ) the  254  a g r e e m e n t c a n n o t be e x p e c t e d not c l e a r  t o be e x a c t . F o r i n s t a n c e i t i s  which terms correspond  to the c o r r e c t i o n  for B  0  to  i n our theory.  22.6  HAMILTONIAN I_N A SPHERICAL BASIS The  spherical  r e l e v a n t magnetic  interaction  b a s i s a r e g i v e n by  tensors i n a  (90)(99)  i(g  - g  yy  * ^xx A  zz  -4(A  xx  +A  )]  yy j  v/f g  -i(g  )  +g  )  The  ±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 .  The  r e l e v a n t second rank  tensor operators are  255  22.7  NOTES ON UNITS FOR  Relaxation spectra  ESR  theories are usually derived  a r e measured i n Gauss.  (what a r e t h e u n i t s o f T, For  this  reason a b r i e f  work i s g i v e n If line  6 3  This  ? (243))  i n energy u n i t s ;  can l e a d t o  below.  the peak-to-peak  2  _ 1  we  then  T i2  For  T,  problems.  d i s c u s s i o n of u n i t s used i n t h i s  H i s the h a l f - w i d t h at h a l f height  H=(7rT )  confusion  and p r o g r a m m i n g  for Lorentzian  w i d t h of the d e r i v a t i v e i s thus  H = ^§AH 2 pp  as  (22.2)  get  1  =  t o be i n t h e u s u a l  TT^|AH  2  (22.3)  pp  units  (sec r a d ~ ) AH 1  must be i n PP  Hz.  To c o n v e r t  f r o m G a u s s we  6  use *  D I S P A shows t h a t t h e dt c l i n e s a r e v e r y c l o s e t o Lorentzian. " O u r e x p e r i m e n t i s f i e l d swept s o t h i s c o n v e r s i o n i s n o t s t r i c t l y v a l i d . H o w e v e r , t h i s c a u s e s no p r o b l e m s , e v e n i n the s l o w - m o t i o n a l regime (39)(244). 6 3  6  ESR  256  hv  = g/JB  ? ( H z ) = h~ g/3B ( G a u s s ) 1  hence  w i t h t h e B o h r m a g n e t o n , /3 and P l a n c k s in their  usual  units.  AH  Hence we  For  (Hz) = A-'go/JAH  h,  defined  (Gauss) PP  T i = ir^q ph' 1  1  0  the l i n e - p o s i t i o n s  c o r r e c t i o n ) we  constant,  get  PP  and t h u s  (22.4)  AH  (Gauss)  (22.5)  ( i g n o r i n g the small  first  order  have  ? ( H z ) = h~ g ^ B ^ ( G a u s s ) 1  (22.6)  z  and f o r t h e g a n i s o t r o p y  Af  n o t e t h a t B^  (Ag)  1  S  (Hz) = /r g/3B  i s the p o s i t i o n  i s n o t t h e same a s B  0  s o Av  z  (Gauss)  (22.7)  of the l i n e  of i n t e r e s t ,  d e p e n d s on  line-position.  which  257  Finally  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 (MHz) = 2.8047(g/g )a (Gauss) 0  where q  (22.8)  o  i s t h e 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  g  splitting allow  g  c o n s t a n t may  f o r any  the second  be o b t a i n e d f r o m t h e s p e c t r u m  temperature  o r d e r terms  dependence, but  i t only appears  t y p i c a l v a l u e s we  _ 1  find  modifications.  1  that  j i s =1nS  1  and R ^ I G r a d s " .  In f a c t  i t i s essential  t o use t h e s e u n i t s  co* t e r m i n  troublesome.  state NiPydtc i s puckered  the plane of the molecule  (183),  bond i n t h e r i n g m e t h y l e n e s  by  some 0.03nm o u t  t h e p o l a r a n g l e of t h e  i s t h u s ±11°  different  'ripple'  on t h e t i m e s c a l e o f t h e NMR  ' r i p p l e s ' much f a s t e r  p o l a r angle averages (There  relaxation.  than the r e l a x a t i o n  out and  of C-D  from  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 t h e r i n g  valid.  to  PYROLLIDINE RING PUCKER  In the s o l i d  ring  may  i n t h e a b o v e e q u a t i o n s w i t h no  Eqn.12.23 i s e s p e c i a l l y  ON  It is  as our base u n i t s . These  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 t h e p r o g r a m . The  22.8  the  and R i s sec r a d " . S u b s t i t u t i n g i n  t h u s s e n s i b l e t o use nS a n d GHz be u s e d d i r e c t l y  in  so t h e l i t e r a t u r e v a l u e i s a d e q u a t e .  D i m e n s i o n a l a n a l y s i s o f Eqn.12.22 g i v e s us t h a t u n i t s of j ( c o ) i s r a d s  to  the  may  I f the  time then  the p l a n a r a p p r o x i m a t i o n i s  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 fast  on  the  258  the  ESR time s c a l e  scale).  I f the the r i n g  activation scale  (245)  and hence a l s o on the NMR time is essentially rigid  energy f o r the r i n g f l i p  the r e l a x a t i o n  {i.e.,  the  i s high) on the NMR time  times are a s u p e r p o s i t i o n of the the  times f o r each r i n g conformation. However, as sine and c o s i n e f u n c t i o n s are f a i r l y (the out  polar  angles r e q u i r e d ) the r e s u l t s once again average  (within  'ripple'  l i n e a r i n between 20° and 40°  experimental e r r o r ) t o the p l a n a r c a s e . I f the  i s on the same time s c a l e as the r e l a x a t i o n (=1s)  then the problem  i s complex and the f l u c t u a t i o n  c o n t r i b u t e t o the o v e r a l l r e l a x a t i o n However, we can r e a d i l y t e s t  will  of the molecule.  f o r t h i s e f f e c t as we have an  over-determined system. We j u s t have to check f o r c o n s i s t e n c y between the two deuterium r e l a x a t i o n the  two corresponding ESR reduced s p e c t r a l  r e s u l t s and  d e n s i t i e s . The  r e s u l t s a t both 310 and 323K are e n t i r e l y c o n s i s t e n t  with  each o t h e r , the o r i g i n a l (planar) approximation i s thus valid.  22.9  THE FAST-MOTIONAL LIMIT  Redfield i.e.  theory i s only, v a l i d  i n the f a s t motional l i m i t ,  , t h e molecular motion must be on a time s c a l e much  faster  than the r e l a x a t i o n  s m a l l e s t element motion  PP  2  of the d i f f u s i o n t e n s o r . I f R~**T  Z  then the  i s s a i d t o be i n the slow-motional regime and  Redfield R»AH  1  time, R' >>T , where R i s the  theory breaks down. The above equation reduces t o  /200,  where R i s i n Grad s  _ 1  and AH  PP  i s the  259 peak-to-peak l i n e - w i d t h ,  i n Gauss, of the broadest l i n e i n  the spectrum. In t h i s and p r e v i o u s work (88) breaks down (i.e.  the theory  , j(co) or the SR term become negative) when  AH^>20G. For an a r b i t r a r y  l i m i t of 10% t h i s equation  corresponds t o an R of 1 Grad s ~  1  as found by the  approximation methods i n Sect.15. f o r the cases where  j (CJ)=0.  260 22.10 ESR L I N E - W I D T H Peak-to-peak  DATA 6 3  line-width data for CuPydtc  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 G H z . Mean l i n e - p o s i t i o n s t o l o w f i e l d ) w e r e 3270G, 3 1 9 2 G , 3115G a n d 3 0 3 9 G . Temp ° C -44.3 -39.0 -35.4 -34.7 -30.0 -25.0 -20.0 -15.1 -10.5 -5.1 0.0 4.0 9.8 19.9 30.4 35.9 40.9 46.5 50. 1 55.6 61 .0  Width in Gauss 3.62 3.35 3.28 3.30 3.12 3.09 2.95 2.99 3.00 3.04 2.97 3.06 3.00 3.14 3.23 3.32 3.40 3.23 3.50 3.71 3.83  5.45 4.99 4.85 4.76 4.58 4.35 4.18 4.10 4.12 4.01 3.90 3.96 3.91 3.84 3.96 4.15 4.15 4.16 4.23 4.30 4.45  10.74 10.06 9.54 9.01 8.44 7.76 7.32 6.89 6.95 6.07 6.11 5.99 5.90 5.69 4.04 6.25 5.37 5.29 5.49 5.56 5.41  19.99 19.12 16.57 16.11 15.46 13.58 12.43 1 1 .53 1 1 .46 10.47 10.21 9.43 9.18 8.48 7.70 7.59 7.57 7.49 7.33 7.13 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 of field- position. High-field first.  (high  261 22.11  NMR R E L A X A T I O N DATA. D E U T E R I U M  Deuterium  T, r e l a x a t i o n  data data f o r d NiPydtc i n 9  chloroform. Delays a r e i n seconds Amplitudes  unless otherwise  noted.  are i n a r b i t r a r y u n i t s . N denotes the  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 the n i t r o g e n . R denotes o t h e r two m e t h y l e n e s . Delay 0.02 0.03 0.05 0.07 0.10 0.13 0.15 0.18 0.25 0.50 1 .00 1 .50  the magnetisation amplitude  T i s the  temperature.  N -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  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.  Delay 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  N -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  T a b l e 2 2 . 4 . T=323K. 61.4MHz.  R -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  f o r the  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 2 0 . 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 2 2 . 5 . T=333K. 61.4MHz  Delay  40.00 60.00 80.00 100.00 120.00 140.00 160.00 180.00 200.00 220.00 240.00 260.00 280.00 300.00  N -4.00 -2.80 -1.10 0.0 0.70 1 .50 2.10 3.00 3.90 4.20 4.80 5.20 5.80 5.95  R -5.20 -3.20 -2.70 -1 .55 -0.90 0.0 0.60 1 .30 1 .50 2.50 3.10 3.30 4.00 4.30  T a b l e 2 2 . 6 . T = 3 1 0 K . 30.7MHz. T i m e s i n mS,  Delay 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 2 2 . 7 . T= 310K. 30.7MHz. T i m e s i n mS.  Delay 40.00 60.00 80.00 100.00 120.00 140.00 160.00 180.00 200.00 220.00 240.00 260.00 280.00 300.00  N -6.40 -4.75 -2.75 -1 .80 0.50 1.10 1 .60 2.50 3.50 4.10 4.80 5.50 6.10 6.60  R -7.70 -6.35 -4.50 -4.00 -2.40 -1 .60 -0.70 0.40 0.60 1 .30 1 .90 2.90 3.35 4.20  Table 2 2 . 8 . T=323K. 30.7MHz. T i m e s i n  22.12 NMR R E L A X A T I O N DATA. 1 3  C T, r e l a x a t i o n  substitution Amplitudes  1 3  C  data data f o r N i E t d t c 2  in chloroform,  on t h e C S moiety. Delays a r e i n seconds. 2  a r e i n a r b i t r a r y u n i t s . T i s the temperature, Time 1 .0 2.0 3.0 4.0 5.0 7.0 9.0 15.0 20.0 80.0  Amp. 1 .1 1 .7 2.1 2.8 3.7 4.6 5.6 8.7 10.5 17.4  i . T=310K. 5 0 . 3 M H z .  Time 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  Time 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 50.0 150.0  Amp. 2.40 4.15 5.40 6.40 7.00 7.50 7.90 8.20 8.50 9.00  Table 22.11. T=323K. 50.3MHz.  Time  Amp.  2.0 5.0 10.0 15.0 20.0 30.0 100.0  4.55 8, 65 13, 95 17, 40 19, 30 21 ,40 24, 72  T a b l e 22.12. T=323K.  Time 1.0 2.0 3.0 4.0 5.0 7.0 9, .0 15, ,0 20, 0 30, 0 100, ,0 T a b l e 22.13. T=333K.  Time 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 50.0 150.0  100.7MHz,  Amp. 1 .55 2.70 3.90 5.15 5.90 7.50 9.20 12.60 14.30 16.00 17.10 50.3MHz.  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