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UBC Theses and Dissertations

Novelties Associated with a biodynamical interpretation of nuclear spin relaxation Werbelow, Lawrence Glen 1974

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NOVELTIES ASSOCIATED WITH A BIODYNAMICAL INTERPRETATION OF NUCLEAR SPIN RELAXATION by LAWRENCE GLEN WERBELOW B. S., Humboldt S t a t e U n i v e r s i t y , 1970 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n t h e department o f CHEMISTRY We a c c e p t tfhis t j i e s i s O a s confxjrming-tO/the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA A u g u s t , 1974 In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements tot an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. Department of C h e m i s t r y The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada Date j l j n e m, 1974 - i i -ABSTRACT Employing t h e s e m i c l a s s i c a l form o f t h e d e n s i t y o p e r a t o r t h e o r y o f dynamic p r o c e s s e s , t h e t r a n s i e n t n u c l e a r s p i n b e h a v i o r i s a n a l y z e d f o r a range o f m o t i o n a l parameters o f s i g n i f i c a n c e f o r b i o l o g i c a l i n t e r p r e -t a t i o n s o f n u c l e a r m a g n e t i c r e l a x a t i o n d a t a . Only r e l a x a t i o n w h i c h r e s u l t s s o l e l y from t h e r e o r i e n t a t i o n a l m o d u l a t i o n o f t h e v a r i o u s s p i n c o u p l i n g s i s c o n s i d e r e d . The ba t h c o r r e l a t i o n f u n c t i o n s w h i c h e n t e r i n t o t h e t h e o r y a r e assumed t o be c o m p l e t e l y c h a r a c t e r i z e d by two unique m o t i o n a l c o n s t a n t s (a dynamic symmetric t o p a p p r o x i m a t i o n ) . The e f f e c t s o f s l o w , a n i s o t r o p i c m o d u l a t i o n o f s p i n - s p i n and s p i n -m o l e c u l e i n t e r a c t i o n s on T^, T,,, T^ r a t i o s , 1^ r a t i o s , and Ov e r h a u s e r enhancements a r e d i s c u s s e d . I t i s r a t i o n a l i z e d t h a t i n g e n e r a l , any parameter dependent upon t h e s p e c t r a l d e n s i t y a t z e r o f r e q u e n c y i s i n d e -pendent o f t h e magnitude o f t h e asymmetry i n t h e m o t i o n . L i k e w i s e , p a r a -meters i n d e p e n d e n t o f t h e n e a r - z e r o f r e q u e n c y s p e c t r a l component o f t e n a r e s e n s i t i v e s o l e l y t o t h e magnitude o f t h e m o t i o n a l asymmetry. These c o n s i d e r a t i o n s a r e ex t e n d e d t o m u l t i s p i n systems and s p i n systems where n u c l e a r m a g n e t i c r e l a x a t i o n p roceeds by competing i n t e r -a c t i o n s c h a r a c t e r i z e d by n o n v a n i s h i n g i n t e r f e r e n c e o r c r o s s - c o r r e l a t i o n f u n c t i o n s . In g e n e r a l , one cann o t d e f i n e u n a m b i g i o u s l y a unique T-j o r 1^ i n such a case as t h e p r e d i c t e d decay i s m u l t i - ( n o n ) e x p o n e n t i a l . The s p i n b e h a v i o r i n such a s i t u a t i o n i s t h o r o u g h l y a n a l y z e d and many i n t r i -g u i n g p r e d i c t i o n s a r e p r e s e n t e d . I t i s seen t h a t t h e f a i l u r e o f a w h i t 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 o r a s i n g l e e x p o n e n t i a l decay o f t h e m o l e c u l a r c o r r e l a t i o n f u n c t i o n o f t e n l e a d s t o p r e d i c t i o n s o f extreme - i i i -n o n e x p o n e n t i a l i t y o f t h e m a g n e t i z a t i o n decay. W h i l e t h e r e s u l t s o f t h e s e c a l c u l a t i o n s a r e o f g e n e r a l i n t e r e s t , t h e y a r e e s p e c i a l l y p e r t i n -e n t f o r t h o s e c o n c e r n e d w i t h b i o l o g i c a l a p p l i c a t i o n s o f NMR. F i n a l l y , t h e i n i t i a l c o n c e p t s a re e x t e n d e d t o a d i s c u s s i o n o f t h e p o t e n t i a l l y p o w e r f u l P e r t u r b e d A n g u l a r C o r r e l a t i o n e x p e r i m e n t where i t i s shown t h a t t h e c o i n c i d e n c e c o u n t i n g r a t e w i l l i n g e n e r a l be i n f l u e n c e d by a n i s o t r o p i c m o d u l a t i o n o f t h e q u a d r u p o l a r p e r t u r b a t i o n . I t i s emphasized t h a t a c o n v e n t i o n a l i n t e r p r e t a t i o n o f r e l a x a t i o n d a t a i s im p r e g n a t e d w i t h h i d d e n , "extreme-narrowed", r e a s o n i n g . E x t e n s i v e f i g u r e s a r e p r o v i d e d w h i c h n o t o n l y f a c i l i t a t e e x p e r i m e n t a l a p p l i c a t i o n o f t h e c a l c u l a t i o n s , b u t a l s o p r o v i d e s t r i k i n g e v i d e n c e f o r t h e c a u t i o n which must be e x e r c i s e d i n any b i o d y n a m i c a l i n t e r p r e t a t i o n o f n u c l e a r s p i n r e l a x a t i o n . - i v-TABLE OF CONTENTS Page ABSTRACT - i i TABLE OF CONTENTS i v LIST OF TABLES v i i LIST OF FIGURES v i i i COMMENTS ON UNITS AND NOTATION x ACKNOWLEDGEMENTS x i i CHAPTER I . GENERAL INTRODUCTION 1 REFERENCES: CHAPTER I 7 CHAPTER I I . NUCLEAR MAGNETIC RELAXATION IN LIQUIDS: A COMPUTATIONAL GROUNDWORK 8 2.1. INTRODUCTION 8 2.2. THE SEMICLASSICAL FORM OF THE DENSITY OPERATOR THEORY OF RELAXATION 12 2.3. THE DYNAMICAL PROBLEM: THE CORRELATION FUNCTIONS 22 2.4. SUMMARY 35 REFERENCES: CHAPTER I I 38 CHAPTER I I I . ANISOTROPIC MOLECULAR MOTIONS AND THE NMR RELAXATION OBSERVABLES 42 3.1. INTRODUCTION 42 3.2. EFFECT OF ANISOTROPIC MOTIONS ON T-, AND T 2 VALUES 45 3.3. EFFECT OF ANISOTROPIC MOTIONS ON T ] AND T £ RATIOS .... 56 3.4. EFFECT OF ANISOTROPIC MOTIONS ON HOMONUCLEAR OVERHAUSER ENHANCEMENTS 64 3.5. SUMMARY 77 REFERENCES: CHAPTER I I I 81 -v-CHAPTER IV. DIPOLAR RELAXATION OF THREE-SPIN SYSTEMS 83 4.1. INTRODUCTION 83 4.2. RESUME OF PREVIOUS STUDIES 86 4.3. FORMULATION OF THE CALCULATION 92 4.4. RESULTS AND DISCUSSION 99 4.5. SUMMARY 119 REFERENCES: CHAPTER IV 121 CHAPTER V. INFLUENCE OF FINITE CROSS-CORRELATION TERMS BETWEEN PHYSICALLY DISTINCT RELAXATION MECHANISMS 124 5.1. INTRODUCTION 124 5.2. RESUME OF PREVIOUS STUDIES 130 5.3. FORMULATION OF THE CALCULATION 136 5.4. SOLUTION OF THE RELAXATION MATRIX 141 5.5. RESULTS AND DISCUSSION 149 5.6. SUMMARY 166 REFERENCES: CHAPTER V 169 CHAPTER V I . EFFECT OF MOLECULAR SHAPE AND FLEXIBILITY ON GAMMA-RAY DIRECTIONAL CORRELATIONS 172 6.1. INTRODUCTION 172 6.2. PERTURBED ANGULAR CORRELATIONS 174 6.3. ANISOTROPIC MOTION IN THE ABRAGAM-POUND LIMIT 181 6.4. ANISOTROPIC MOTION IN THE ADIABATIC LIMIT 183 6.5. RESULTS AND DISCUSSION 186 6.6. COMPARISON OF NMR AND PAC 202 REFERENCES: CHAPTER VI 206 CHAPTER V I I . CONCLUDING REMARKS 208 -vi-APPENDIX A. THE RELAXATION MATRIX 210 APPENDIX B. INFLUENCE OF SECOND ORDER FREQUENCY SHIFT TERMS •••• 215 APPENDIX C. QUADRUPOLAR RELAXATION OF SPIN 3/2 NUCLEI 224 APPENDIX D. NUCLEAR MAGNETIC RELAXATION FOR INDIVIDUAL TRAN-SITIONS OF AN AMX SPECTRUM: USE OF INTERFERENCE TERMS TO DETERMINE SIGNS OF SCALAR COUPLING CONSTANTS 230 - v i i -LIST OF TABLES T a b l e T i t l e Page 3.1. R e l a x a t i o n parameters i n v a r i o u s m o t i o n a l l i m i t s : P a r t I . 79 3.2. R e l a x a t i o n parameters i n v a r i o u s m o t i o n a l l i m i t s : P a r t I I . 80 5.1. C o n t r i b u t i o n s t o t h e r e l a x a t i o n m a t r i x f o r two i s o -c hronous s p i n s r e l a x e d by d i p o l a r , s h i f t a n i s o t r o p y , and s p i n - r o t a t i o n i n t e r a c t i o n s . 148 5.2. Summary o f i n t e r f e r e n c e term c h a r a c t e r i s t i c s . 168 D . l . L i n e w i d t h c o n t r i b u t i o n s t o i n d i v i d u a l t r a n s i t i o n s f o r an AMX t h r e e - s p i n s y s tem: D i p o l a r c o n t r i b u t i o n s . 235 D.2 L i n e w i d t h c o n t r i b u t i o n s t o i n d i v i d u a l t r a n s i t i o n s f o r an AMX t h r e e - s p i n system: I n t e r f e r e n c e c o n t r i b u t i o n s . 236 •vi n -LIST OF FIGURES F i g u r e T i t l e Page 2.1. A s s i g n i n g a m o l e c u l a r i n t e r p r e t a t i o n t o NMR r e l a x -a t i o n o b s e r v a b l e s . 37 53 3.1. E f f e c t o f a n i s o t r o p i c m o t i o n on T^ and T^ v a l u e s . 3.2. S p e c t r a l d e n s i t i e s f o r a t w o - s p i n system r o t a t i n g about an a x i s p e r p e n d i c u l a r t o t h e i n t e r n u c l e a r v e c t o r . 55 3.3. E f f e c t o f a n i s o t r o p i c m o t i o n on T^ r a t i o s . 59 3.4. E f f e c t o f a n i s o t r o p i c m o t i o n on r a t i o s . 61 3.5. E f f e c t o f a n i s o t r o p i c m o t i o n on t h e r a t i o , T - j / ^ - 63 3.6. Homonuclear O v e r h a u s e r enhancement as a f u n c t i o n o f i s o t r o p i c m o b i l i t y . 72 3.7. E f f e c t o f a n i s o t r o p i c m o t i o n on homonuclear O v e r h a u s e r enhancements. 74 3.8. Independence o f homonuclear O v e r h a u s e r enhancements on t h e magnitude o f m o t i o n a l a n i s o t r o p y . 76 4.1. N o n e x p o n e n t i a l d i p o l a r r e l a x a t i o n f o r t h e methyl group. 112 4.2. D i s s e c t i o n o f t h e m e t h y l group n o n e x p o n e n t i a l r e l a x a t i o n . 114 4.3. C o n t o u r p l o t s o f methyl group decay parameters f o r t h e l o n g i t u d i n a l m a g n e t i z a t i o n decay. 116 4.4. C o n t o u r p l o t s o f methyl group decay parameters f o r t h e t r a n s v e r s e m a g n e t i z a t i o n decay. 118 5.1. Decay pa r a m e t e r s f o r t h e l o n g i t u d i n a l r e l a x a t i o n as a f u n c t i o n o f magnitude o f s h i f t a n i s o t r o p y - d i p o l a r i n t e r a c t i o n c o n s t a n t s : E x t r e m e - n a r r o w i n g a p p r o x i m a t i o n . 159 5.2. Decay parameters f o r t h e l o n g i t u d i n a l r e l a x a t i o n as a f u n c t i o n o f magnitude o f s h i f t a n i s o t r o p y - d i p o l a r i n t e r a c t i o n c o n s t a n t s . 161 5.3. Decay pa r a m e t e r s f o r t h e l o n g i t u d i n a l as a f u n c t i o n o f s p i n m o b i l i t y (assuming d i p o l a r - s h i f t a n i s o t r o p y i n t e r f e r e n c e t e r m s ) . 163 - i x -F i g u r e T i t l e . Page 5.4. D i s s e c t i o n o f t h e n o n e x p o n e n t i a l (as a r e s u l t o f d i p o l a r - s h i f t a n i s o t r o p y c r o s s - t e r m s ) l o n g i t u d i n a l r e l a x a t i o n . 165 6.1. A t y p i c a l n u c l e a r d e - e x c i t a t i o n c a s c a d e . 178 6.2. A n g l e s d e f i n e d i n t h e P e r t u r b e d A n g u l a r C o r r e l a t i o n e x p e r i m e n t . 180 111m 6.3. O b s e r v a b l e a n i s o t r o p y ( f o r Cd) as a f u n c t i o n o f o v e r a l l m o b i l i t y . 193 111m 6.4. O b s e r v a b l e a n i s o t r o p y ( f o r Cd) as a f u n c t i o n o f i n t e r n a l m o b i l i t y : F a s t m o t i o n l i m i t . 195 111m 6.5. O b s e r v a b l e a n i s o t r o p y ( f o r Cd) as a f u n c t i o n o f i n t e r n a l m o b i l i t y : Slow m o t i o n l i m i t . 197 6.6. Decay scheme f o r 1 1 1 In and 1 1 1 m C d . 199 6.7. The i n t e g r a l a t t e h u a t i p n i f a c t o r f o r v a r i o u s i n t e r -m e d i a t e s t a t e l i f e t i m e s as a f u n t i o n o f i s o t r o p i c m o b i l i t y . 201 B . l . E f f e c t o f s e c o n d - o r d e r s h i f t c o r r e c t i v e terms. 221 B.2. G r a p h i c comparison o f r e l a x a t i o n , l i n e w i d t h , and s e c o n d - o r d e r c o r r e c t i v e t erms. 223 D . l . Energy l e v e l diagram f o r an AMX s p i n system. 238 -X-COMMENTS ON UNITS AND NOTATION The I n t e r n a t i o n a l System o f u n i t s and c o n v e n t i o n a l NMR n o t a t i o n i s used t h r o u g h o u t t h i s t h e s i s . D e v i a n t s a r e f u l l y e x p l a i n e d . However, i t i s u s e f u l t o summarize h e r e , t h e p a r t i c u l a r n o t a t i o n a l i d i o s y n c r a c i e s w h i c h a r e employed t o c l a s s i f y t h e r o t a t i o n a l d i f f u s i o n c o n s t a n t s ( u n i t s : s e c o n d s " ^ ) , r o t a t i o n a l c o r r e l a t i o n t i m e s ( u n i t s : s e c o n d s ) , s p e c t r a l d e n s i t i e s ( u n i t s : s e c o n d s " ) and a n g u l a r c o r r e l a t i o n f u n c t i o n s ( u n i t s : _ o seconds ). Symbol D e f i n i t i o n Page n o t a t i o n f i r s t appears D I s o t r o p i c r o t a t i o n a l d i f f u s i o n c o n s t a n t ( s c a l a r d i f f u s i o n t e n s o r ) . 26 D ± Symmetric t o p r o t a t i o n a l d i f f u s i o n c o n s t a n t ( d i f f u -s i o n p e r p e n d i c u l a r t o t h e p r i n c i p a l d i f f u s i o n a x i s ) . 29 D)( Symmetric top r o t a t i o n a l d i f f u s i o n c o n s t a n t ( d i f f u -* i n t s i o n p a r a l l e l t o t h e p r i n c i p a l d i f f u s i o n a x i s ) . 29 D. . D i f f u s i o n c o n s t a n t ( i n one d i m e n s i o n ) f o r an i n t e r n a l r o t o r a t t a c h e d t o an i s o t r o p i c framework. 30 D. D i f f u s i o n c o n s t a n t ( i n one d i m e n s i o n ) f o r an i n t e r n a l 1 r o t o r a t t a c h e d t o a symmetric to p framework. 33 T g , x . j A g e n e r a l , u n s p e c i f i e d r o t a t i o n a l c o r r e l a t i o n t i m e . 27,50 i 2 I s o t r o p i c . r o t a t i o n a l c o r r e l a t i o n t i m e o f a second rank s p h e r i c a l harmonic: = 1/6D. T C Symmetric t o p r o t a t i o n a l c o r r e l a t i o n t i m e : = 1/6D^. •• 47 x R Symmetric t o p r o t a t i o n a l c o r r e l a t i o n t i m e : = l / ( 4 D - 4 D j . 47 T R I Symmetric to p r o t a t i o n a l c o r r e l a t i o n t i m e : = 1 / ( D h - D x ) . 68 - x i -Symbol D e f i n i t i o n Page n o t a t i o n f i r s t a ppears k& C ? r ) ( t ) The t i m e c o r r e l a t i o n f u n c t i o n o f two v a r i a b l e s uht) and U £ ( t ) : H < U k ( t ) U , l ( 0 ) > . 16 g A n o r m a l i z e d c o r r e l a t i o n f u n c t i o n : = (-1) x <C K J t(t)>/<C K X'(0)>. 27 ?n ?n jj^ -U) The s p e c t r a l r e p r e s e n t a t i o n o f C k £ ( t ) : E ^ m I/O ^N ( 1 / 2 ) / C - ( t ) e x p ( i u j t ) d t . 16 j j ^ ( c j ) The s p e c t r a l r e p r e s e n t a t i o n o f g ^ ( t ) 27 J^(co) The s p e c t r a l r e p r e s e n t a t i o n o f 6 C nc£j-(t). 93 J * (u) The s p e c t r a l r e p r e s e n t a t i o n o f (1 -6£ n)C**(t). 93 Q ^ ( C J ) The H i l b e r t T r a n s f o r m o f J ^ w ) . - 15 q|^>) The H i l b e r t T r a n s f o r m o f j^-(u>). 218 One l a s t comment on n o t a t i o n i s i n o r d e r a t t h i s t i m e . Many p l o t s t h r o u g h o u t t h i s t h e s i s p l o t some o b s e r v a b l e v e r s u s D, D|(, D^, o r D..^ ^ T 2 ' T c ' TR' o r TR'^ o v e r m a n ^ decades °^ t n 1 s v a r i a b l e . In such a c a s e , t h e s c a l e reads Log(D) o r some f a c s i m i l e . As a l l t i m e s c a l e s used i n t h i s t h e s i s assume t h e second as t h e fundamental u n i t o f t i m e , o n l y mag-n i t u d e s o f t h e s e q u a n t i t i e s a r e r e l e v a n t and hence t h e j u s t i f i c a t i o n o f the n o t a t i o n . * I n t e r n a l r o t a t i o n on a s p h e r i c a l framework i s f o r m a l l y i d e n t i c a l t o symmetric t o p a n i s o t r o p i c m o t i o n i f t h e i d e n t i f i c a t i o n D n - D±= D- n t and D x= D i s made. A l s o , a t t i m e s i n t h i s t h e s i s , t h e q u a n t i t y D^ - D^ i s s i m p l y r e f e r r e d t o as t h e a n i s o t r o p y o f t h e m o t i o n . - x i i -ACKNOWLEDGEMENTS I would l i k e t o e x t e n d my a p p r e c i a t i o n t o Dr. A. G. M a r s h a l l f o r s u g g e s t i n g much o f t h e m a t e r i a l c o n s i d e r e d i n t h i s t h e s i s , and f o r c o n t i n u i n g g u i d a n c e and encouragement t h r o u g h o u t my t e n u r e a t t h e U n i v e r s i t y o f B r i t i s h C o l u mbia. I a l s o w i s h t o thank P r o f e s s o r s M. Bloom, L. H a l l , and B. D u n e l l f o r examples t h e y have s e t and t h e know-le d g e w h i c h t h e y have i m p a r t e d . The s u p p o r t o f a U n i v e r s i t y o f B r i t i s h C olumbia G r a d u a t e S c h o l a r -s h i p (1972-1974) i s a p p r e c i a t e d . Academic acknowledgements i n a r r e a r s , I w i s h t o e x p r e s s my warmest w i s h e s ( w i t h o u t p u b l i c e x p l a n a t i o n ) t o t h e f o l l o w i n g i n f l u e n c e s : To A l a n and M a r i l y n M a r s h a l l . To L a u r i e H a l l , P e t e r L e g z d i n s , Ben Malcomb, and Ian A r m i t a g e . To Merv Hanson. To Diane. To S t e p h a n i e . To my p a r e n t s . F i n a l l y , a measure o f acknowledgement must be ext e n d e d t o t h e nearby B r i t i s h C o l umbia c o u n t r y s i d e : "Heaven i s under o u r f e e t as w e l l as o v e r o u r heads", and t o Henry Thoreau whose words and t h o u g h t s have n u r t u r e d me many a t i m e . -1-CHAPTER I GENERAL INTRODUCTION In a p e r i o d s p a n n i n g t h e l a s t t w e n t y - f i v e y e a r s , N u c l e a r M a g n e t i c Resonance (NMR) has l e n t i t s e l f t o t h e t a s k o f p r o b i n g a v a s t range o f m o l e c u l a r i n f o r m a t i o n . S i m p l y s t a t e d , NMR i s t h e e x p e r i m e n t a l t e c h n i q u e by w h i c h one probes t h e m o l e c u l a r d e t a i l s embodied w i t h i n t h e n u c l e a r s p i n H a m i l t o n i a n . T h i s H a m i l t o n i a n i s i n s t r u c t i v e l y w r i t t e n as a sum o f a s m a l l number o f p h y s i c a l l y d i s t i n c t b i l i n e a r c o u p l i n g s ( i g n o r i n g m u l t i p l y i n g c o e f f i c i e n t s ) , £• = i- (I- S>)-B 0 + i - ( b + j )-r + I-C 0 + .... [1.1] where t h e r e s p e c t i v e r e p r e s e n t a t i v e c o u p l i n g s a r e t h e s p i n - e x t e r n a l f i e l d c o u p l i n g , t h e s p i n - s p i n c o u p l i n g , t h e s p i n - a n g u l a r momentum c o u p l i n g , e t c . A l t h o u g h t o f i r s t o r d e r , £ w i l l be i ndependent o f t i m e , t h e r e do e x i s t s m a l l , time-dependent c o n t r i b u t i o n s o f paramount i m p o r t a n c e . Hence, E q u a t i o n [1.1] i s o f t e n w r i t t e n as € = 6*0 + £ ( t ) ; .. . .. [1.2] where €{t) i s a s m a l l p e r t u r b a t i o n assumed ( f o r c o n v e n i e n c e ) t o have a z e r o t i m e a v e r a g e , <6?(t)> = 0. C o n v e n t i o n a l h i g h r e s o l u t i o n NMR s t u d i e s have h i s t o r i c a l l y r e l i e d on t h e i n t e r p r e t a t i o n o f l i n e p o s i t i o n s ( c h e m i c a l s h i f t s and c o u p l i n g con-s t a n t s ) and i n t e n s i t i e s . A n o t h e r way o f s t a t i n g t h i s f a c t i s t o say -2-t h a t o n l y t h e s p e c t r a l p r o p e r t i e s o f £ Q a r e u t i l i z e d ; t h e o n l y i n f o r m a t i o n o b t a i n e d c o n c e r n s t h e t r a c e o f t h e i n t e r a c t i o n t e n s o r s i n E q u a t i o n [1.1] and t h e " t r a c e l e s s " a s p e c t o f t h e s e t e n s o r s i s i n a d v e r t e n t l y i g n o r e d . However, th e a n a l y s i s o f l i n e s h a p e , o r more g e n e r a l l y , m a g n e t i c r e l a x a t i o n , i n p r i n c i p l e c o n t a i n s i n f o r m a t i o n on t h e c o m p l e t e t e n s o r i a l n a t u r e o f t h e s p i n - s p i n and s p i n - m o l e c u l e c o u p l i n g s . I n d eed, t h i s p r e s e n t s a l a r g e r e s e r v o i r o f i n f o r m a t i o n , a p o t e n t i a l w h i c h makes p a s t e x p l o i t a t i o n o f NMR seem meager. T h i s avenue o f approach has j u s t r e c e n t l y been t a p p e d i n an e x p l o r a t o r y f a s h i o n by t h e c h e m i s t . I t m i ght be mentioned t h a t t h e t r e n d towards u t i l i z a t i o n o f s p e c t r a l l i n e s h a p e i s i n vogue not o n l y i n NMR s t u d i e s , but i n o t h e r s p e c t r o s c o p i c s t u d i e s such as e l a s t i c and i n e l a s t i c photon and p a r t i c l e s c a t t e r i n g , IR a b s o r p t i o n e x p e r i m e n t s , and d i e l e c t r i c s u s c e p t i b i l i t y measurements. A t an e n t i r e l y d i f f e r e n t l e v e l , a n o t h e r t r e n d w h i c h has d e v e l o p e d i n r e c e n t y e a r s r e l a t e s t o t h e r e s e a r c h a c t i v i t i e s u n d e r t a k e n by a l a r g e number o f p h y s i o - c h e m i c a l a c a d e m i c i a n s , s t u d i e s o f " l i f e " on a m o l e c u l a r , s u b m o l e c u l a r , o r p h i l o s o p h i c a l p l a t e a u . T h i s t r e n d , stemming from r e c e n t s o c i a l - e c o n o m i c - m o r a l c o n f r o n t a t i o n s ( o r w h a t e v e r ) c e r t a i n l y does not d e t r a c t f r o m t h e i n h e r e n t i n t e r e s t i n t h i s f i e l d . J u s t i f i c a t i o n f o r t h e c o n t e n t o f t h i s t h e s i s i s t h e r e f o r e t w o - f o l d : (1) t o p r o v i d e i n s i g h t i n t o t h e i n t e r p r e t a t i o n o f t r a n s i e n t n u c l e a r s p i n b e h a v i o r , (2) as a means t o i l l u m i n a t e contemporary NMR r e s e a r c h e s i n t o complex b i o c h e m i c a l systems. R e l e v a n t background m a t e r i a l can be l o c a t e d i n any one o f s e v e r a l e x c e l l e n t m o n o g r a p h s 1 - 5 and r e v i e w s 6 - 1 0 w h i c h t r e a t b o t h t h e g e n e r a l -3-t h e o r y o f NMR r e l a x a t i o n and i t s a p p l i c a t i o n s t o b i o c h e m i c a l p r o b l e m s ! A d d i t i o n a l r e f e r e n c e s and s p e c i f i c a p p l i c a t i o n s can be found i n a l a r g e number o f r e c e n t a r t i c l e s stemming from t h e r e s e a r c h e s o f t h e groups headed by A. A l l e r h a n d ( I n d i a n a ) , S. Chan ( C a l T e c h ) , F. Noack ( S t u t t g a r t ) , B. Sykes ( H a r v a r d ) , and t h e b i o p h y s i c a l group a t Cambridge. T h i s i s o n l y a s a m p l i n g o f some o f t h e more a c t i v e p a r t i c i p a n t s i n t h e f i e l d , t h e num-ber making s i g n i f i c a n t c o n t r i b u t i o n s would e a s i l y demand a n o t h e r page o r two t o l i s t . F u r t h e r m o r e , a number o f c o n f e r e n c e p r o c e e d i n g s c o n c e r n i n g m a g n e t i c r e s o n a n c e a p p l i e d t o b i o l o g i c a l systems a r e i n p r i n t . ( F o r ex-ample, see r e f e r e n c e s 17,18.) I n t e r p r e t a t i o n o f NMR r e l a x a t i o n d a t a y i e l d s both s t a t i c i n f o r m a t i o n (bond d i s t a n c e s and a n g l e s , f i n e d e t a i l s o f t h e ground s t a t e n u c l e a r en-v i r o n m e n t , e l e c t r i c f i e l d g r a d i e n t s , e t c . ) and dynamic i n f o r m a t i o n ( c h e m i c a l and p h y s i c a l k i n e t i c s ) . In p a r t i c u l a r , t h i s t h e s i s i s o n l y c o n c e r n e d w i t h t h e r e l a t i o n s h i p between b i o m o l e c u l a r dynamics ( p h y s i c a l k i n e t i c s ) and t h e r e l a x a t i o n b e h a v i o r o f an ensemble o f n u c l e a r s p i n s . The s i t u a t i o n s c o v e r e d by such an a p proach a r e t h o s e where t h e l i n e b r e a d t h and r e l a t e d r e l a x a t i o n p a r a m e t e r s a r e d e t e r m i n e d s o l e l y by t h e a n i s o t r o p y i n t h e m a g n e t i c i n t e r a c t i o n s ( E q u a t i o n [ 1 . 1 ] ) c o u p l e d t o t h e m o l e c u l a r r o t a t i o n a l d i f f u s i o n . (We a l s o d i s m i s s t r a n s l a t i o n a l c o n t r i -b u t i o n s . ) The c o n t e n t o f t h i s t h e s i s i s n e i t h e r s t r i c t l y e x p e r i m e n t a l nor t h e o r e t i c a l i n n a t u r e but s t r i k e s a medium w h i c h c o u l d b e s t be c a t a l o g u e d under t h e heading " c o m p u t a t i o n a l " . The need f o r such an approach i s w a r r a n t e d by t h e l a c k o f communication between m u t u a l l y s k e p t i c a l seg--4-ments o f t h e s c i e n t i f i c community. On one hand, t h e r e e x i s t s e l e g a n t t r e a t m e n t s o f s p i n r e l a x a t i o n p r o b l e m s , y e t on the o t h e r hand, a p e r u s a l o f contempory b i o l o g i c a l l y o r i e n t e d NMR r e l a x a t i o n l i t e r a t u r e f i n d s much o f t h e d a t a i s i n t e r p r e t e d i n terms o f advances i n t h e t h e o r y t h r o u g h t h e y e a r 1955! F o r t h e most p a r t , t h i s u n s e t t l i n g s i t u a t i o n can be t r a c e d t o t h e f a c t t h a t one o f two a s s u m p t i o n s i s i n v a r i a b l y i n t r o d u c e d i n t o any c a l c u l a t i o n e x p l i c i t l y s o l v e d t o p r o v i d e a c o m p e l l i n g example o f t h e use-f u l n e s s o f t h e o r e t i c a l advances. U n f o r t u n a t e l y , one o f t h e s e two a p r i o r i a s s u m p t i o n s may be i n v a l i d i n a t y p i c a l b i o d y n a m i c a l s t u d y ; t h e s e two a s s u m p t i o n s b e i n g , (1) t h a t a u n i q u e t i m e c o n s t a n t c o m p l e t e l y c h a r -a c t e r i z e s t h e m i c r o s c o p i c decay o f m o l e c u l a r o r i e n t a t i o n , and (2) t h a t r e l a t i v e m o l e c u l a r o r i e n t a t i o n s e x i s t f o r a t i m e s h o r t compared w i t h t h e n a t u r a l (Larmor) t i m e s c a l e o f t h e p r e c e s s i n g s p i n . In t h i s t h e s i s , we e x p l o r e the p r e d i c t e d s p i n b e h a v i o r when one, o r b o t h o f t h e s e a s s u m p t i o n s f a i l . We s h a l l a l s o d i s m i s s t h e n o t i o n o f a t h i r d commonly advanced a s s u m p t i o n t h a t a u n i q u e t i m e c o n s t a n t c o m p l e t e l y c h a r a c t e r i z e s t h e m a c r o s c o p i c decay o f a n o n e q u i l i b r i u m m a g n e t i z a t i o n . Hence, t h e c o n t e n t o f t h i s t h e s i s w i l l p r o v i d e m e a n i n g f u l (and o f t e n v e r y g e n e r a l ) p r e d i c t i o n s o f s p i n b e h a v i o r i n s i t u a t i o n s where l i t t l e r e l e v a n t d i s c u s s i o n o r o t h e r i n t e r p r e t a t i o n a l a i d now e x i s t . The e x p e r i m e n t a l l i m i t a t i o n s o f NMR a r e w e l l known, l a c k o f s e n s i -t i v i t y and c o s t l y , complex i n s t r u m e n t a t i o n a r e two o f t h e t y p i c a l c r i t -i c i s m s . L e s s w e l l known i s t h e f a c t t h a t t h e r e i s an a n a l o g o u s t e c h n i q u e o f g r e a t p o t e n t i a l w h i c h , f o r t h e most p a r t , c i r c u m v e n t s t h e s e h a n d i c a p s . The p e r t u r b a t i o n o f t h e a n g u l a r c o r r e l a t i o n (PAC), w h i c h e x i s t s between a p a i r o f s u c c e s s i v e photons i n a n u c l e a r d e - e x c i t a t i o n c a s c a d e , d i --5-r e c t l y r e f l e c t s t h e r e l a x a t i o n o f a p o l a r i z e d s p i n e n s e m b l e . " 7 Whereas a s t r o n g e x t e r n a l Zeeman f i e l d i s e s s e n t i a l t o g e n e r a t e a p o l a r i z e d ensemble i n t h e NMR e x p e r i m e n t , i n t h e PAC e x p e r i m e n t , t h e s p i n s a r e " p o l a r i z e d by o b s e r v a t i o n " . O b s e r v a t i o n o f t h e f i r s t y-ray s e l e c t s an ensemble o f n u c l e i w h i c h i s c h a r a c t e r i z e d by a n o n - u n i f o r m p o p u l a t i o n o f m a g n e t i c s u b s t a t e s . N u c l e a r r e l a x a t i o n w i l l p e r t u r b t h i s t r a n s i e n t p o l a r i z a t i o n . The p r o b a b i l i t y o f t h e o b s e r v a t i o n o f t h e second y-ray f o r any g i v e n a n g l e w i t h r e s p e c t t o t h e f i r s t y-^ay can t h e n be quan-t i t a t i v e l y i n t e r p r e t e d i n terms o f i n i t i a l p o l a r i z a t i o n f a c t o r s and n u c l e a r r e l a x a t i o n f a c t o r s . O f t e n t h e s e r e l a x a t i o n f a c t o r s may reduce t o m u l t i p l e s o f c o n v e n t i o n a l NMR p a r a m e t e r s . A l t h o u g h t h e b u l k o f t h i s t h e s i s i s c o n c e r n e d w i t h NMR r e l a x a t i o n , t h e p o t e n t i a l o f PAC's ( u s e f u l -12 e x p e r i m e n t a l c o n c e n t r a t i o n s range down t o 10 M) i n b i o c h e m i c a l s t u d i e s j u s t i f i e s t h e a t t e n t i o n ( u n f o r t u n a t e l y c u r s o r y ) g i v e n t h i s t o p i c i n C h a p t e r V I . In g e n e r a l , t h e o u t l i n e o f t h i s t h e s i s i s q u i t e s t r a i g h t f o r w a r d . C h a p t e r I I summarizes t h e f o r m a l i s m n e c e s s a r y t o a s s i g n a m o l e c u l a r i n t e r p r e t a t i o n t o t h e b e h a v i o r o f t h e o b s e r v a b l e n u c l e a r paramagnetism. C h a p t e r I I I d e v e l o p s v e r y g e n e r a l c o n c e p t s w h i c h g r e a t l y a i d i n t h e r e d u c t i o n o f r e l a x a t i o n parameters f o r m o t i o n a l regimes where n a i v e e x t e n s i o n s o f c o n v e n t i o n a l i n t e r p r e t a t i o n s l e a d t o h o p e l e s s c o n t r a d i c t i o n s . The nex t two c h a p t e r s e x t e n d t h e s e i d e a s t o a r a t h e r s p e c i f i c r e l a x a t i o n phenomenon, t h e c r o s s - c o r r e l a t i o n o r i n t e r f e r e n c e problem. A l t h o u g h t h e i n f l u e n c e o f such terms has been, r e g a r d e d f o r t h e l a s t dozen y e a r s as a t h e o r e t i c a l c u r i o s i t y , t h e extended c a l c u l a t i o n s d e v e l o p e d i n -6-t h e s e two c h a p t e r s c l e a r l y show t h a t b i o c h e m i c a l s t u d i e s p r e s e n t a v e r y p r a c t i c a l p o t e n t i a l f o r p o s i n g e i t h e r a d i s c o u r a g i n g i n t e r p r e t a t i o n a l problem o r an avenue f o r a d d i t i o n a l i n s i g h t depending on t h e (un) f a m i l i a r i t y t h e e x p e r i m e n t a l i s t has w i t h t h e t h e o r y o f i n t e r f e r e n c e terms. F i n a l l y , t h e l a s t c h a p t e r e x t e n d s t h e i d e a s d e v e l o p e d i n C h a p t e r I I I t o t h e PAC e x p e r i m e n t . A s i d e from C h a p t e r I I , each c h a p t e r t r e a t s a s e l f - c o n t a i n e d t o p i c . Thus i t i s f e l t t o be more s a t i s f a c t o r y t o i n c l u d e a b i b l i o g r a p h y a t t h e end o f each i n d i v i d u a l c h a p t e r r a t h e r t h a n a c u m u l a t i v e b i b l i o g r a p h y . -7-REFERENCES: CHAPTER I 1. A. Abragam, The P r i n c i p l e s o f N u c l e a r Magnetism, C l a r e n d o n P r e s s , O x f o r d , 1961. 2. A. C a r r i n g t o n and A. D. M c L a c h l a n , I n t r o d u c t i o n t o M a g n e t i c Resonance, Harper-Row, New Y o r k , 1967. 3. C. P. S l i c h t e r , P r i n c i p l e s o f M a g n e t i c Resonance, Harper-Row, New Y o r k , 1963. 4. N. Bloembergen, N u c l e a r M a g n e t i c R e l a x a t i o n , B e n j a m i n , New Y o r k , 1961. 5. C. P. P o o l e and H. A. F a r a c h , R e l a x a t i o n i n M a g n e t i c Resonance, Academic P r e s s , New Y o r k , 1971. 6. H. G. H e r t z , P r o g r . NMR S p e c t . 3, 159 ( 1 9 6 7 ) . 7. M. D. Z e i d l e r , B er. Bunsenges. Phys. Chem. 75, 229 (1971). 8. G. C. L e v y , A c c o u n t s Chem. Res. 6, 161 (1973). 9. E. D. B e c k e r , R. R. Shoup, and T. C. F a r r a r , Pure A p p l . Chem. 32, 51 ( 1 9 7 2 ) . — 10. J . R. L y e r l a and D. M. G r a n t , MTP I n t e r n a t i o n a l Rev. S c i . 4, 155 (1972). 11. 0. J a r d e t z k y , Advan. Chem. Phys. 7., 499 ( 1 9 6 4 ) . 12. A. S. M i l d v a n and M. Cohn, Advan. Enzymology 33_, 1 (1 9 7 0 ) . 13. R. A. Dwek, Advan. M o l . R e l a x . P r o c e s s e s 4, 1 (1 9 7 2 ) . 14. F. Gurd and P. Keim, Methods Enzymology 27, 836 (1 9 7 3 ) . 15. G. A. G r a y , CRC C r i t . Rev. Biochem. 3, 247 (1 9 7 3 ) . 16. T. R. Krugh, t o be p u b l i s h e d i n S p i n L a b e l i n g : Theory and A p p l i c a t i o n s , e d i t e d by L. J . B e r l i n g e r , Academic P r e s s , New Y o r k , 1974. 17. C. F r a n c o n i , e d i t o r , M a g n e t i c Resonance i n B i o l o g i c a l R e s e a r c h , Gordon and B r e n c h , New Y o r k , 1971. 18. Volume 222 o f t h e P r o c e e d i n g s o f t h e New York Academy o f S c i e n c e s , 1973. 19. D. A. S h i r l e y and H. Haas, Ann. Rev. Phys. Chem. 23, 385 (1 9 7 2 ) . -8-CHAPTER I I NUCLEAR MAGNETIC RELAXATION IN LIQUIDS: A COMPUTATIONAL GROUNDWORK 2.1. INTRODUCTION A c o l l e c t i o n o f N l i k e s p i n s p l a c e d i n a c o n s t a n t m a g n e t i c f i e l d BQ(0,0,BQ) i s c h a r a c t e r i z e d by t h e m a c r o s c o p i c C u r i e e q u i l i b r i u m mag-n e t i z a t i o n , M e q = M < I Z > T = N Y f i ( Y 1 i I ( I + l ) B 0 / 3 k T ) [ 2 . 1 . 1 ] where y i s t h e g y r o m a g n e t i c r a t i o o f t h e n u c l e i and I z i s t h e e x p e c t a t i o n v a l u e o f t h e z component o f t h e n u c l e a r s p i n . The s u p e r s c r i p t "T", a n o t a t i o n used t h r o u g h o u t t h i s t h e s i s , i s w r i t t e n t o emphasize t h a t t h i s e x p r e s s i o n i s v a l i d f o r a s p i n system i n t h e r m a l harmony w i t h i t s s u r -r o u n d i n g s . The p r o c e s s by w h i c h a c o l l e c t i o n o f n u c l e a r m a g n e t i c d i p o l e s a p p r o a c h thermodynamic harmony a f t e r some i n i t i a l d i s t u r b a n c e , i s s i m p l y r e f e r r e d t o as n u c l e a r m a g n e t i c r e l a x a t i o n . The r e l a x a t i o n p r o c e s s i s c u s t o m a r i l y c h a r a c t e r i z e d by t h e p h y s i c a l l y d i s t i n c t l o n g i t u d i n a l and t r a n s v e r s e r e l a x a t i o n w h i c h measure t h e t i m e e v o l u t i o n o f t h e components o f t h e m a c r o s c o p i c m a g n e t i z a t i o n p a r a l l e l ( l o n g i t u d i n a l r e l a x a t i o n ) and p e r p e n d i c u l a r ( t r a n s v e r s e r e l a x a t i o n ) t o t h e a p p l i e d Zeeman f i e l d , BQ. S u f f i c e t o say t h a t t h e d i s t u r b a n c e o f M from e q u i l i b r i u m and t h e subsequent decay o f i t s o r t h o g o n a l components can be f o l l o w e d by a g r e a t -9-v a r i e t y o f e x p e r i m e n t a l t e c h n i q u e s . In t h e s i m p l e s t o f t e r m s , we may t h i n k o f n u c l e a r s p i n r e l a x a t i o n as n u c l e a r s p i n communication. Mechanisms whereby t h e s p i n s communicate d i r e c t l y w i t h t h e dynamic m o l e c u l a r s u r r o u n d i n g s * a r e m a n i f e s t e d i n l o n g i t u d i n a l r e l a x a t i o n whereas mechanisms wh i c h r e d i s t r i b u t e i n f o r m a t i o n w i t h i n t h e system o f s p i n s a r e m a n i f e s t e d i n t r a n s v e r s e r e l a x a t i o n . A l -though a s i n g l e t i m e c o n s t a n t w i l l r a r e l y d e s c r i b e p r e c i s e l y t h e s e r e -l a x a t i o n (communication) p r o c e s s e s , such a t i m e c o n s t a n t i s o f t e n i n t r o -duced as a v e r y good e m p i r i c a l , i f not t h e o r e t i c a l , p a r a m eter: ( d / d t ) B Q - M ( t ) = - X ( B Q - M ( t ) - M e q } B Q | ) [2.1.2] ( d / d t ) | B n x M ( t ) | = -X'|B X M ( t ) | [2.1.3] where x and A' a r e t h e f a m i l i a r NMR r e l a x a t i o n r a t e s 1/T^ and l / T ^ r e -s p e c t i v e l y . A l t h o u g h t h e c o n c e p t u a l f o u n d a t i o n s (and much o f t h e t e r m i n o l o g y ) o f s p i n r e l a x a t i o n d a t e back t o t h e e a r l y 1930's, i t was t h e famous paper by Bloembergen, Pound, and P u r c e l l 1 i n t h e l a t e 1940's w h i c h f i r s t e x p r e s s e d t h e e v o l u t i o n o f t h e m a c r o s c o p i c l o n g i t u d i n a l and t r a n s v e r s e n u c l e a r m a g n e t i z a t i o n s ( i . e . T-| and T^) as a f u n c t i o n o f m i c r o s c o p i c m o l e c u l a r p a r a m e t e r s . In t h e i r a p p r o a c h , w h i c h t r e a t e d two i d e n t i c a l * These f l u c t u a t i n g m o l e c u l a r d egrees o f freedom a r e o f t e n l o o s e l y r e -f e r r e d t o as t h e " l a t t i c e " o r t h e " b a t h " . We w i l l c a l l e v e r y t h i n g o t h e r t h a n t h e s p i n s , t h e l a t t i c e . Thus m o l e c u l a r p o s i t i o n s and o r i e n t a t i o n s as w e l l as i n t e r n u c l e a r c o o r d i n a t e s a r e l a t t i c e c o o r d i n a t e s . s p i n 1/2 n u c l e i , i t was assumed t h a t t h e r e l a x a t i o n mechanism i n d u c i n g t r a n s i t i o n s between th e Zeeman energy l e v e l s was t h e d i p o l e - d i p o l e i n t e r -a c t i o n . Time dependent p e r t u r b a t i o n t h e o r y was used t o c a l c u l a t e t h e t r a n s i t i o n p r o b a b i l i t y per u n i t t i m e between th e Zeeman l e v e l s . The s t o c h a s t i c n a t u r e o f t h e m o l e c u l a r m o t i o n was i n c o r p o r a t e d i n t h e t r a n s -i t i o n p r o b a b i l i t i e s t h r o u g h c o r r e l a t i o n f u n c t i o n s p a r a m e t e r i z i n g t h e l a t t i c e . The p h y s i c a l p i c t u r e p r e s e n t e d by such an approach was ex-t r e m e l y s a t i s f y i n g ; each s p i n " s e e s " a f l u c t u a t i n g l o c a l m a g n e t i c f i e l d p roduced by a n e i g h b o r w h i c h i n d u c e s t r a n s i t i o n s amongst i t s Zeeman l e v e l s and d r i v e s t h e n u c l e a r s p i n system towards a s t a t e o f e q u i l i b r i u m . T h i s a p proach t o q u a n t i t a t i v e r e l a x a t i o n t h e o r y , o f t e n r e f e r r e d t o i n t h e l i t e r a t u r e as t h e BPP t h e o r y , has formed t h e b a s i s f o r many f u n d a -mental c a l c u l a t i o n s ; two n o t e w o r t h y papers w h i c h e x e m p l i f y t h i s p r o c e d u r e 2 3 a r e t h o s e by Abragam and Pound and Solomon. A l t h o u g h t h e BPP t h e o r y l i e s a t t h e h e a r t o f t h e c o n c e p t u a l f o u n d a t i o n o f m a g n e t i c r e l a x a t i o n , i t s u f f e r s from many " q u a n t i t a t i v e d e f e c t s " . In t h e e n s u i n g y e a r s , many more e l a b o r a t e t h e o r i e s on n u c l e a r m a g n e t i c r e l a x a t i o n have g r a c e d t h e pages o f t h e l i t e r a t u r e . BPP c o n c e n t r a t e d t h e i r a t t e n t i o n on l i q u i d s and c a s t t h e i r a n a l y s i s i n a form p a r t i c u l a r l y w e l l s u i t e d t o t h e s t r u c t u r a l and dynamic d i s o r d e r w h i c h l i q u i d s p o s s e s s . S i n c e t h i s t i m e , q u i t e d i f f e r e n t , g e n e r a l approach 4 5 have p r o v e n f r u i t f u l f o r s i m p l e m o l e c u l a r s o l i d s and d i l u t e g a s e s . Yet o t h e r s p e c i a l i z e d approaches p r e d i c t r e l a x a t i o n b e h a v i o r i n e x o t i c forms o f m a t t e r such as t h e l i q u i d c r y s t a l l i n e s t a t e , ( a n t i ) f e r r o m a g n e t i c m a t e r i a l s o r t h e s u p e r - f l u i d s t a t e . As f o r t h e " s i m p l e " l i q u i d s t a t e , w h i c h i s our o n l y p r e s e n t c o n c e r n , numerous m a t h e m a t i c a l and p h y s i c a l 6-11 t e c h n i q u e s such as t h e d e n s i t y m a t r i x f o r m a l i s m , ~ t h e l i n e a r - r e s p o n s e 12 13 14 method o f Kubo, ' and t h e r e l a t e d approach o f A n d e r s o n , t h e F o k k e r -15 16 1 P l a n c k e q u a t i o n , p r o j e c t i o n o p e r a t o r s , t h e L i o u v i l l e r e p r e s e n t a t i o n , 20 21 and c o n c e p t s o f i r r e v e r s i b l e thermodynamics, ' t o m e n t i o n a few, have been a p p l i e d t o t h e problem. D i s c u s s i o n s o f t h e l i m i t a t i o n s o f v a r i o u s 22 approaches can a l s o be f o u n d i n the a r t i c l e s o f A r g y r e s and K e l l y , 23 24 R o b e r t s o n , and F u l t o n . E x t r e m e l y e n l i g h t e n i n g d i s c u s s i o n s o f many o f t h e p r e s e n t approaches t o r e l a x a t i o n can be found i n a p u b l i s h e d 25 s e r i e s o f l e c t u r e s . A s i d e from p r e s e n t i n g r e l e v a n t r e f e r e n c e s i n t h e p r e c e e d i n g p a r a -g r a p h , t h e i m p r e s s i v e f a c t i s t h e o b v i o u s , a g e n e r a l a c c o u n t o f t h e r e t u r n o f a d y n a m i c a l system t o e q u i l i b r i u m i s f a r from s i m p l e . A l t h o u g h no u n i v e r s a l t h e o r y o f r e l a x a t i o n e x i s t s , we s h a l l adopt t h e w e l l t e s t e d d e n s i t y m a t r i x a p p r o a c h , an approach a l m o s t u n i v e r s a l l y employed f o r q u a n t i t a t i v e c a l c u l a t i o n s o f m o b i l e n u c l e a r s p i n systems. -12-2.2 THE SEMICLASSICAL FORM OF THE DENSITY OPERATOR THEORY OF RELAXATION In t h e r e m a i n d e r o f t h i s s e c t i o n , we w i l l summarize t h e r e s u l t s o f t h i s t h e o r y o n l y so f a r as n e c e s s a r y i n o r d e r t o l a y t h e f o u n d a t i o n f o r l a t e r c a l c u l a t i o n s . The o r i g i n s o f t h i s method can be t r a c e d back t o B l o c h ' s a t t e m p t s a t f i n d i n g a f i r m t h e o r e t i c a l b a s i s f o r t h e p h e n o m e n o l o g i c a l r e l a x a t i o n e q u a t i o n s w h i c h b e a r h i s name. M e a n w h i l e , R e d f i e l d ' 7 i n d e p e n d e n t l y de-r i v e d a s i m i l a r t h e o r y . O f t e n however, R e d f i e l d ' s t r e a t m e n t i s c i t e d f o r c r e d i t as t h e n o t a t i o n a l changes he i n t r o d u c e d p r o v i d e a much s i m -p l i f i e d f o r m u l a t i o n f o r p r a c t i c a l a p p l i c a t i o n s . S u b s e q u e n t l y , v a r i o u s r e v i e w s o f t h i s o r i g i n a l work w i t h g e n e r a l i z a t i o n s and s i m p l i f i c a t i o n s a p p e a r e d i n t h e l i t e r a t u r e , t h e most n o t a b l e b e i n g t h e d i s c u s s i o n s by 26 8 9 Abragam, Hubbard, and Freed and F r a e n k e l . V e r y good i n t r o d u c t i o n s can a l s o be found i n S l i c h t e r ' s 1 0 book and H o f f m a n ' s 1 ^ r e v i e w a r t i c l e . A l t h o u g h t h e most r e a d a b l e a c c o u n t o f t h i s t o p i c i s R e d f i e l d ' s r e v i e w 7 8 a r t i c l e , p r o b a b l y t h e most g e n e r a l t r e a t m e n t i s t h a t o f Hubbard, and i t i s h i s f o r m a l i s m w h i c h w i l l g e n e r a l l y be adopted. We s h a l l a l s o assume a s e m i c l a s s i c a l approach (quantum t r e a t m e n t o f s p i n s - c l a s s i c a l t r e a t m e n t o f t h e l a t t i c e ) . The c l a s s i c a l t r e a t m e n t o f t h e l a t t i c e can be j u s t i f i e d r i g o r o u s l y by showing t h a t i t y i e l d s e s s e n t i a l l y t h e same r e s u l t s as a f u l l quantum m e c h a n i c a l t r e a t m e n t . I t i s shown i n quantum s t a t i s t i c a l mechanics t h a t an ensemble o f i d e n t i c a l n o n - i n t e r a c t i n g systems can be d e s c r i b e d by a reduced d e n s i t y o p e r a t o r , a ( t ) , o b t a i n e d from t h e system's c o m p l e t e d e n s i t y m a t r i x by -13-a v e r a g i n g o v e r t h e l a t t i c e degrees o f freedom.* The e x p e c t a t i o n v a l u e o f any s p i n o p e r a t o r A, averaged o v e r t h e ensemble i s g i v e n by <A> = T r [ a ( t ) A ] . [2.2.1] The t i m e e v o l u t i o n o f t h e d e n s i t y o p e r a t o r i s a s o l u t i o n o f t h e e q u a t i o n ( d / d t ) a ( t ) = -i[/£, a ( t ) ] [ 2 . 2 . 2 ] s u b j e c t t o t h e c o n d i t i o n t h a t f o r a l l v a l u e s o f t , T r [ a ( t ) ] = 1 . [2.2.3] The H a m i l t o n i a n p e r t i n e n t t o s p i n problems i s most c o n v e n i e n t l y w r i t t e n as fi£ = f i ( £ 0 + £ ( t n [ 2 . 2 . 4 ] where t h e s t a t i c H a m i l t o n i a n , £g (composed o f t h e Zeeman term and f i r s t o r d e r c o r r e c t i o n s ) , d e t e r m i n e s l i n e p o s i t i o n s and i n t e n s i t i e s . The f l u c t u a t i n g , time-dependent p e r t u r b a t i o n , c f ( t ) , d e t e r m i n e s l i n e w i d t h s . A l t h o u g h t h e H a m i l t o n i a n i t s e l f has t h e d i m e n s i o n s o f e n e r g y , E q u a t i o n [ 2 . 2 . 4 ] i m p l i e s t h a t cfQ and £{t) w i l l be t r e a t e d as though t h e y have u n i t s o f a n g u l a r f r e q u e n c y . F u r t h e r m o r e £{t) i s d e f i n e d so t h a t «£(t)>=0. The absence o f a s e c o n d a r y o s c i l l a t i n g f i e l d i s a l s o assumed. 6 ? ( t ) , t h e c o u p l i n g between t h e s p i n s and t h e m o l e c u l a r b a t h , may i t s e l f be composed o f many s e p a r a b l e c o u p l i n g s , a f a c t we s h a l l e l a b o r a t e on a t a l a t e r t i m e . F o r r e l a x a t i o n mechanism w h i c h i n v o l v e p a i r w i s e s p i n m a g n e t i c i n t e r a c t i o n s , * For e x c e l l e n t d i s c u s s i o n s o f t h e d e n s i t y m a t r i x , see t h e a c c o u n t s o f 27 28 29 Fano, T e r Haar, o r Tolman. -14-both r e o r i e n t a t i o n a l ( i n t r a m o l e c u l a r ) and r e l a t i v e t r a n s l a t i o n a l ( i n t e r -m o l e c u l a r ) c o n t r i b u t i o n s e f f e c t r e l a x a t i o n . A t t h i s j u n c t u r e , t h e sim -p l i f y i n g a s s u m p t i o n t h a t £ ( t) i s due s o l e l y t o t h e i n t r a m o l e c u l a r con-t r i b u t i o n s i s i n t r o d u c e d . I t i s a l s o c o n v e n i e n t t o i n t r o d u c e a " d e v i a t i o n " d e n s i t y o p e r a t o r , x ( t ) = a ( t ) - a T [ 2 . 2 . 5 ] where a1" i s t h e reduced d e n s i t y o p e r a t o r f o r a s p i n system i n t h e r m a l e q u i l i b r i u m , a T = e x p ( - ^ 0/kT)/Tr [exp(- £ Q / k T ] . [2. 2 . 6 ] In t h e h i g h t e m p e r a t u r e a p p r o x i m a t i o n ( £Q << k T ) , a T - (1 - c ? Q/kT)/Tr [3 ( ] [ 2 . 2 . 7 ] where 31, t h e u n i t m a t r i x , has d i m e n s i o n a l i t y e q u al t o t h e number o f de-g r e e s o f freedom i n t h e s p i n system. U s i n g t i m e dependent p e r t u r b a t i o n t h e o r y c a r r i e d t o second o r d e r , t h e r e s u l t i n g a p p r o x i m a t i o n f o r t h e e q u a t i o n o f m o t i o n i s s ( d / d t ) x ( t ) = i [ x ( t ) , £ 0 ] - \ / [ £ ( t ) , [ e x p ( i £ 0 T ) £ ( t - T ) e x p ( - i £ Q T ) ,x(t)]]dx. ^ — CO [2.2.8] A l l r e l a x i n g p e r t u r b a t i o n s o f i n t e r e s t can be w r i t t e n as a sum o v e r k o f a few t e r m s , each o f w h i c h i s t h e p r o d u c t o f pure s p i n o p e r a t o r s and pure l a t t i c e ( o r b a t h ) f u n c t i o n s , £ ( t ) = S S U k ( b a t h , t ) V k ( s p i n s ) . [2.2 . 9 ] 5 k ? ? -15-The summation o v e r r, i n c l u d e s a l l v a r i o u s r e l a x a t i o n pathways. The term pathway i s used t o denote e i t h e r d i s i m i l i a r p h y s i c a l r e l a x a t i o n mechanisms o r s i m i l i a r r e l a x a t i o n mechanisms o p e r a t i v e on d i f f e r e n t r e l a x a t i o n cen-t e r s . In w r i t i n g E q u a t i o n [ 2 . 2 . 9 ] , i t s h o u l d be kept i n mind that£(t) i s H e r m i t i a n . We assume t h a t t h e b a t h o p e r a t o r s a r e m o l e c u l e - f i x e d , i n t e r -im a c t i o n - d e p e n d e n t p a r a m e t e r s . The time-dependence i n U ? ( b a t h , t ) a r i s e s from t h e c l a s s i c a l m o l e c u l a r m o t i o n s w h i c h modulate t h e E u l e r i a n a n g l e s r e l a t i n g m o l e c u l e - f i x e d t o l a b - f i x e d frames o f r e f e r e n c e . U s i n g t h i s ex-p a n s i o n o f &{t), Hubbard has shown t h a t t h e e q u a t i o n o f m o t i o n f o r x ( t ) can c o m p a c t l y be r e w r i t t e n as ( d / d t ) x ( t ) = i [ x ( t ) , E+N] + R x ( t ) [2.2.10] where E E -.IOJQI . N and R a r e most e a s i l y d e f i n e d i f t h e m a t r i x expan-s i o n o f E q u a t i o n [2.2.10] i s s t u d i e d : ( d / d t ) < a | x ( t ) | a ' > = iw .<a|x(t)|a'> + f r " ( a ' ' a ' " ) a a a " a ' " X [ 6 a , a . , . < a | x ( t ) | a " > - 6^, , <a 1 " | x ( t ) | a 1 >] aa m u [2.2.11] — — C'a"a""<a"lx(t)h'"> a a z »n where N « ( . " . ' " ) =E1v E Q ^ a , , c , i v - u , c i i V [ i , , , ) ) < a " | V , k | « 1 ' V > < a 1 V | V « | „ ' " > Ot K j J6 [2.2.12] -16-cM(u) = j k j ^ z j d + expefiz/kDrVdz [2.2.12a] and a::.'....,,,, - E t J ^ . . ) +J^ (-a.a...)}<.|V*|."»<.-|vJ|."» E ,k£/ . \ ,,, i,,k| i v ivi,,£| i v Jr nK'"a l v ) < a lVJa ><a lVJ aa ' ' Z - ^ I V ?n a a ' 1 z;1 1 n 1 6 i I I I A" I I IV <a V J a ><a V a " > a a a a 1 C1 ' n [2.2.13] kp The f u n c t i o n , J ^ t a ) i s g i v e n by JJJJU) = \ ^ c j a ) ( t ) e x p ( i a . t ) d t [2.2.14] where C ^ ( t ) = <lAt + tn)U*(tn)> = <U*(t)U*(0)>. [2.2.15] The method o f d e r i v a t i o n i s based on t h e as s u m p t i o n t h a t t h e bath p r o v i d e s * a s u f f i c i e n t l y r a p i d f l u c t u a t i n g e n v i r o n m e n t f o r t h e s p i n . The prime i n * T h i s approach r e l a t e s ( d / d t ) x a t t i m e s t » x n t o x a t t = 0. A t t h i s k£/ p o i n t , an o p e r a t i o n a l d e f i n i t i o n o f T Q i s ; f o r t >>T Q , C ( t ) -* 0. The e x p r e s s i o n i s t h e f i r s t term i n a power s e r i e s e x p a n s i o n and i n o r d e r f o r t h e convergence t o be good, x a a . ( t ) s h o u l d not be v a s t l y d i f f e r e n t from x i ( 0 ) . T h i s i m p l i e s t h a t one can d e f i n e a range o f t i m e s such OtOt -j t h a t t >> i n f o r w h i c h x ( 0 ) = x ( t ) and y e t f o r w h i c h t << R~ U aa a a The p h y s i c a l s i g n i f i c a n c e o f both c o n d i t i o n s i s t h a t we never ask f o r i n f o r m a t i o n o v e r a t i m e i n t e r v a l comparable t o x n. -17-t h e summation denotes t h a t o n l y s e c u l a r terms ( i . e . oo , -to , , , . , << J a a a a R ) a r e r e t a i n e d . The a n g u l a r f r e q u e n c y i s d e f i n e d as u , = a a a a ' • -\ J A A -to , = ( E ' i - E ) / f i where E r e p r e s e n t s t h e energy o f t h e a e i g e n k e t : a a a a a r £Q|C*> = E |a>. The d i v e r g e n c e under t h e i n t e g r a l . i s c i r c u m v e n t e d by t a k i n g t h e Cauchy p r i n c i p a l v a l u e {J^) a t s i n g u l a r i t i e s o f t h e i n t e g r a n d . The C k J l ( t ) l s a r e d e f i n e d as t h e l a t t i c e c o r r e l a t i o n f u n c t i o n s and have t h e f o l l o w i n g c l a s s i c a l i n t e r p r e t a t i o n : S e l e c t from an e q u i l i b r i u m ensemble any one parameter and measure i t s v a l u e a t some t i m e w h i c h can be d e f i n e d as t g = 0. Next measure t h e same parameter ( i . e . k = i and ? = n i n E q u a t i o n [ 2 . 2 . 1 5 ] ; t h i s d e f i n e s t h e so c a l l e d a U t o - c o r r e l a t i o n f u n c t i o n ) a t some l a t e r t i m e , t . Then c a l c u l a t e t h e a p p r o p r i a t e p r o d u c t and p e r f o r m t h e denoted ensemble a v e r a g e . ( I f k f i and/or ? f n, t h i s p r o c e d u r e y i e l d s t h e c r o s s - c o r r e l a t i o n f u n c t i o n . ) The f u n c t i o n J (to) i s d e f i n e d as t h e s e m i c l a s s i c a l s p e c t r a l d e n s i t y o r s p e c t r a l r e p r e s e n -t a t i o n o f t h e ensemble au t o ( c r o s s ) - c o r r e l a t i o n f u n c t i o n . The m a t r i x R i s r e f e r r e d t o as t h e r e l a x a t i o n ( s u p e r ) m a t r i x and i s s t a n d a r d n o t a t i o n r e g a r d l e s s o f wh i c h t r e a t m e n t i s c o n s u l t e d . I t s h o u l d be emphasized t h a t t h i s i s n o t a t r a n s i t i o n p r o b a b i l i t y m a t r i x . The N m a t r i x n o t a t i o n i s unique t o Hubbard's f o r m a l i s m , a l t h o u g h a s i m i l a r n o t a t i o n i s i n t r o d u c e d i n R e d f i e l d ' s p u b l i c a t i o n s ( a l s o compare w i t h t h e V ( s u p e r ) m a t r i x used 30 i n v a r i o u s German c a l c u l a t i o n s ). However, f o r f u t u r e c a l c u l a t i o n s p r e s e n t e d i n t h i s t h e s i s , t h e i n f l u e n c e o f N w i l l be n e g l e c t e d , as w i l l now be d i s c u s s e d . F i r s t , i t i s easy t o j u s t i f y r e p l a c e m e n t o f t h e term l+exp(1iz/kT) by 2 i n E q u a t i o n [2.2.12a] and t o c o m p l e t e l y i g n o r e t h e z dependence i n -18-t h i s term f o r a l l v a l U e s o f z. F u r t h e r m o r e , i f we assume t h a t t h e a u t o ( c r o s s ) - c o r r e l a t i o n f u n c t i o n s a r e t h e sum o f d e c a y i n g e x p o n e n t i a l s w i t h d i f f e r e n t t i m e c o n s t a n t s (see next s e c t i o n ) , t h e n t h e s p e c t r a l d e n s i t y f u n c t i o n has t h e form,. J^(u)) ^ a . L l a ? + . [2 .2.16] R e c o g n i z i n g t h a t Qkri(w) = — * J k £ ( c o ) where * s t a n d s f o r t h e c o n v o l u t i o n TTU) . ?n o p e r a t i o n , i t i s seen t h a t Q(to) and J(co) form a H i l b e r t t r a n s f o r m p a i r , and hence, Q k £ ( co ) = = ] C ( " / a i ) a i K i . [2.2.17] ? n i 2 . 2 i 2 . 2 a . + co a . + co I t i s w e l l known t h a t t h e p h y s i c a l consequence o f i n c l u s i o n o f t h e s e 30 32 33 terms i s t o i n t r o d u c e a second o r d e r f r e q u e n c y s h i f t . ' ' These dy-namic l i n e s h i f t s a r e q u i t e s m a l l , a t t h e l a r g e s t , on t h e o r d e r o f a l i n e -w i d t h . T h i s f a c t i s e a s i l y seen f r o m E q u a t i o n [2.2.16] and [2 .2 .17 ] . F i r s t l y , n o t e t h a t J ( 0 ) i s on t h e o r d e r o f t h e o b s e r v a b l e n a t u r a l l i n e -w i d t h . Now i f a-j, o ^ , a n >>co ( e x t r e m e - n a r r o w i n g ) , then J ( 0 ) >> Q(to) L i k e w i s e , i f a-j, a n << co , a g a i n J ( 0 ) >> Q(co). O n l y i f a-j, o ^ , . . . , a * co , do t h e s e terms become comparable. Hence i f one c a l c u l a t e s t h e t r a n s v e r s e r e l a x a t i o n when t h e e x t r e m e - n a r r o w i n g a p p r o x i m a t i o n f a i l s , t h e m a g n e t i z a t i o n decay w i l l i n g e n e r a l be modulated by t h e i n t e r f e r e n c e o f s l i g h t l y d i f f e r e n t r e s o n a t i n g f r e q u e n c i e s , a l t h o u g h t h e decay e n v e l o p e w i l l be u n a f f e c t e d (see Appendix B f o r a co m p l e t e example). A p a r t i c u -l a r l y d e f i n i t i v e example o f a c a l c u l a t i o n where t h e N's a r e i n c l u d e d , and o f t h e c o m p l e x i t y i n v o l v e d i n b o t h t h e c a l c u l a t i o n s and t h e r e s u l t , -19-34 i s Hubbard's t r e a t m e n t o f f o u r i d e n t i c a l , s p i n 1/2 n u c l e i . T h e r e f o r e , i n t h e sake o f t r a c t a b i l i t y , t h e N dependence s h a l l be i g n o r e d t h r o u g h o u t t h i s work. However, t h i s s u b t l e p o i n t does j u s t i f y comment on t h e approx-i m a t i o n i n t r o d u c e d . Here t h e n , i s t h e p o i n t o f e m b a r k a t i o n f o r f u t u r e c o m p u t a t i o n a l s t u d i e s : (d/dt)<a| X(t)|a'>= ia) .<a|X(t)|a'> + S R„„ .„..„... <«| X ( t ) | a>"> . OtCX I I i l l CttX UC (Jt a a [2.2.18] Once h a v i n g chosen a c o n v e n i e n t s e t o f a's and o b t a i n e d t h e s o l u t i o n f o r x ( t ) , t h e o b s e r v a b l e v a l u e s o f < I z ( t ) > and < I + ( t ) > ( i . e . t h e l o n g i t u d i n a l and t r a n s v e r s e r e l a x a t i o n ; I + a r e t h e c o n v e n t i o n a l r a i s i n g and l o w e r i n g o p e r a t o r s ) , a r e c a l c u l a t e d from t h e a p p r o p r i a t e t r a c e . B e f o r e c o n t i n u i n g any f u r t h e r , i t m i g h t pay t o note why we a r e not c o n t e n t w i t h an a l l i a n c e w i t h BPP t h e o r y and i n s i s t on f o l l o w i n g a more cumbersome ap p r o a c h . O r i g -i n a l l y , t h e BPP t h e o r y a p p l i e d t o two l i k e s p i n 1/2 n u c l e i , hence o n l y m o l e c u l e s i n t h e non-degenerate I = ±1 s t a t e s have nonzero components o f m a g n e t i c moment and t o c a l c u l a t e < I z ( t ) > , i t was n e c e s s a r y o n l y t o c a l -c u l a t e t h e p r o b a b l e number o f m o l e c u l e s i n t h e s e s t a t e s as a f u n c t i o n o f t i m e . However, i f t h e m o l e c u l e c o n t a i n s t h r e e o r more l i k e s p i n s , t h e n t h e r e a r e d e g e n e r a t e Zeeman energy l e v e l s c o r r e s p o n d i n g t o a nonzero I z -U n f o r t u n a t e l y , i n t h i s c a s e , t h e c a l c u l a t i o n o f < I z ( t ) > i s dependent upon one's c h o i c e o f b a s i s e i g e n k e t s . A l t h o u g h t h e form o f both m a t r i c e s x ( t ) and I depend on what s e t o f b a s i s f u n c t i o n s a r e chosen t o span t h e s p i n s p a c e , t h e t r a c e o f x ( t ) I z i s independent o f t h i s c h o i c e o f b a s i s -20-* s t a t e s . . T h e r e f o r e , a d e n s i t y m a t r i x f o r m a l i s m y i e l d s t h e same r e s u l t f o r any o r t h o g o n a l c h o i c e o f e i g e n k e t s . A l s o , w i t h i n t h e framework o f t h e BPP t h e o r y , t h e r e i s no p r o v i s i o n t o i n c l u d e t h e e f f e c t o f s p a t i a l c o r r e l a t i o n s i n the f l u c t u a t i n g f i e l d s r e s p o n s i b l e f o r r e l a x a t i o n . L a s t l y , t h e BPP approach d e s c r i b e s o n l y t h e e v o l u t i o n o f p o p u l a t i o n s o f s t a t e s . In t h e language o f t h e d e n s i t y m a t r i x , BPP t h e o r y c o n s i d e r s o n l y t h e d i a g o n a l e lements o f a, and assumes t h a t a l l o f f d i a g o n a l elements a r e z e r o . However, a n o n v a n i s h i n g t r a n s v e r s e m a g n e t i z a t i o n i m p l i e s a c e r t a i n phase co h e r e n c e o f t h e s t a t e s (a phase c o h e r e n c e i s e x h i b i t e d by nonzero o f f d i a g o n a l elements o f a). Hence BPP t h e o r y i s u n a b l e t o t r e a t c o r r e c t l y , t h e t r a n s v e r s e r e l a x a t i o n . Many o t h e r p r a c t i c a l a d v a n t a g e s o f t h e d e n s i t y o p e r a t o r a p proach can be f o u n d i n papers d e a l i n g w i t h Double Resonance and o t h e r s a t u r a t i o n e f f e c t s where o f f d i a g o n a l e lements p l a y a dominant r o l e . However, a l t h o u g h s u p e r f i c i a l l y q u i t e d i f f e r e n t , t h e two approaches do s h a r e s i m i l a r i t i e s ; t h e BPP b e i n g a s p e c i a l c a s e o f t h e R e d f i e l d -B l o c h t h e o r y , and as shown i n Appendix A, many (but not a l l ! ) o f t h e 35 * T h i s f a i l i n g i s - t y p i f i e d i n a r e c e n t c a l c u l a t i o n by Nowak and M i l d v a n w h i c h a t t e m p t e d t o e x t e n d t h e BPP approach t o t h e c a s e o f t h r e e e q u i -v a l e n t s p i n o n e - h a l f n u c l e i by i n c l u d i n g a l l p o s s i b l e t r a n s i t i o n s be-tween t h e e i g h t l e v e l s o f t h e t h r e e - s p i n system ( o m i t t i n g o n l y t h e t r i p ! quantum t r a n s i t i o n ) . W h i l e t h e a l g e b r a l e a d i n g t o t h e i r e q u a t i o n s i s c o r r e c t , t h e s e t - u p i s n o t ; t h i s i n n o c e n t approach a p p l i e d t o a t h r e e s p i n system i s by no means o b v i o u s . As a m a t t e r o f f a c t , i t can e a s i l y be shown t h a t t h i s approach a p p l i e d t o N i d e n t i c a l s p i n s l e a d s t o t h e p r e d i c t i o n t h a t T" 1 = 2[W1 + (N-1)W 2 L so t h a t f o r l a r g e N, t h e r e l a x -a t i o n would depend s o l e l y upon t h e d o u b l e quantum t r a n s i t i o n s , c o n t r a r y t o i n t u i t i o n (W-j and W2 a r e t h e t r a n s i t i o n p r o b a b i l i t y p er u n i t t i m e f o r t h e s i n g l e and d o u b l e quantum t r a n s i t i o n s r e s p e c t i v e l y ) . -21-elements o f R have s i m p l e i n t e r p r e t a t i o n s i n more f a m i l i a r terms o f t r a n s i t i o n p r o b a b i l i t i e s o r l i n e w i d t h s . -22-2.3 THE DYNAMICAL PROBLEM - THE CORRELATION FUNCTION As seen i n E q u a t i o n [ 2 . 2 . 1 3 ] , t h e r e l a x a t i o n m a t r i x , R, i s com-posed o f p r o d u c t s o f s p i n m a t r i x e lements and c l a s s i c a l a n g u l a r c o r r e -l a t i o n f u n c t i o n s . In g e n e r a l , t h e s p i n and l a t t i c e o p e r a t o r s d e f i n e d i n E q u a t i o n [2.2.9] a r e components o f t h e s c a l a r p r o d u c t o f two i r r e -d u c i b l e s p h e r i c a l t e n s o r s o f rank L ( i . e . k = - L , - L + l , L - l , L ) . 38 T h i s has l e d A t k i n s , i n h i s superb r e v i e w a r t i c l e , t o r e f e r t o t h e s p i n r e l a x a t i o n problem as one w h i c h f a c t o r i z e s i n t o two d i s t i n c t and v e r y g e n e r a l c a l c u l a t i o n s - t h e g e o m e t r i c a l and t h e d y n a m i c a l components T h i s d i s t i n c t i o n i s most a p p r o p r i a t e . The e v a l u a t i o n o f t h e geomet-r i c a l a s p e c t i n v o l v e s t h e a p p l i c a t i o n o f t h e quantum t h e o r y o f a n g u l a r momentum t o e v a l u a t e t h e s p i n m a t r i x e l e m e n t s , a t e d i o u s but s t r a i g h t -f o r w a r d t a s k . The c o r r e l a t i o n f u n c t i o n s r e f l e c t t h e d y n a m i c a l a s p e c t s o f t h e problem. I t i s towards t h i s p r o s p e c t we now d i r e c t our a t t e n -t i o n . The r e l a x a t i o n mechanisms c o n s i d e r e d i n d e t a i l i n l a t e r s e c t i o n s o f t h i s t h e s i s a r e c h a r a c t e r i z e d by l a t t i c e p a rameters w h i c h i n g e n e r a l * a r e p r o p o r t i o n a l t o t h e s p h e r i c a l harmonics o f rank two. Hence, i t * The s p h e r i c a l harmonics o f rank two a r e d e f i n e d a s , Y°(n) = ( 5 / 1 6 7 r ) 1 / 2 ( 3 c o s 2 e - 1) Y ^ t f t ) = +(15/8Tr)^ 2sinecoseexp(±i(j)) Y ; 2 ( o J = (15/32ir) 1 / 2sin 2eexp(±2i<j>). k k k* The phase c o n v e n t i o n adopted t h r o u g h o u t t h i s t h e s i s i s Y^ {a) = (-1) Y L where t h e s u b s c r i p t L i s t h e rank and t h e s u p e r s c r i p t k i s t h e compon-e n t . F o r f u r t h e r p r o p e r t i e s o f t h e s p h e r i c a l h a r m o n i c s , t h e book by Rose ( r e f e r e n c e 68) i s an e x c e l l e n t s o u r c e . -23-w i l l p r ove n e c e s s a r y t o c a l c u l a t e t h e F o u r i e r T r a n s f o r m o f a n g u l a r c o r -r e l a t i o n f u n c t i o n s o f t h e f o r m , cj"(t) = £ ? < Y ^ ( n r ( t ) ) Y * ( n (0))> [2.3.1] where t h e U l < ( t ) i n E q u a t i o n [2.2.15] a r e now w r i t t e n as <% Y«(fi ( t ) ) . The p r o p o r t i o n a l i t y c o n s t a n t ( d i m e n s i o n s o f s e c " ) , 5 , i s d e f i n e d as t h e i n t e r a c t i o n c o n s t a n t o f t h e p a r t i c u l a r r e l a x a t i o n mechanism o f c o n c e r n ( n o t e t h a t t h i s d e f i n i t i o n o f t h e i n t e r a c t i o n c o n s t a n t w i l l i n g e n e r a l d i f f e r from most o t h e r s by a s m a l l n u m e r i c a l f a c t o r ) . To c a r r y o u t t h e ensemble a v e r a g e i n v o l v i n g t h e a n g u l a r c o r r e l a t i o n f u n c t i o n , we need t o i n v o k e t h e n o r m a l i z e d j o i n t p r o b a b i l i t y f u n c t i o n , P ( f i (t),^£^(0)) * x dQ dQ w h i c h d e s c r i b e s t h e p r o b a b i l i t y t h a t t h e r e l a x a t i o n v e c t o r K n c h a r a c t e r i z i n g pathway c, has an o r i e n t a t i o n r e l a t i v e t o t h e l a b system i n t h e range dft a t ti m e t and t h e r e l a x a t i o n v e c t o r f o r pathway n has an o r i e n t a t i o n i n dQ a t ti m e t = 0; * The r e l a x a t i o n v e c t o r i s a term commonly employed i n NMR r e l a x a t i o n t e r m i n o l o g y . Depending upon which r e l a x a t i o n mechanism i s c o n s i d e r e d , t h i s r e l a x a t i o n v e c t o r may d e f i n e t h e i n t e r n u c l e a r v e c t o r , t h e p r i n -c i p a l a x i s o f t h e e l e c t r i c f i e l d g r a d i e n t , t h e p r i n c i p a l a x i s o f t h e c h e m i c a l s h i f t t e n s o r , o r t h e p r i n c i p a l a x i s o f t h e s p i n - r o t a t i o n t e n s o r . The i n t r o d u c t i o n o f t h i s c o n c e p t i s u s e f u l because i t i s t h e r e o r i e n t a t i o n o f t h i s v e c t o r ( i f you w i s h , t h e p r i n c i p a l a x i s o f t h e i n t e r a c t i o n f i x e d c o o r d i n a t e system) w h i c h i n d u c e s r e l a x a t i o n . A l l r o t a t i o n s o f t h e m o l e c u l e w h i c h l e a v e t h e r e l a x a t i o n v e c t o r u n a f f e c t e d do not c o n t r i b u t e t o t h e r e l a x a t i o n r a t e , i . e . do not g i v e r i s e t o a l o c a l , f l u c t u a t i n g m a g n e t i c f i e l d . -24-C?n(t) = 5?S // Y2(^ ( t ) ) Y2 (% ( 0 ) ) P ( Q ? ( t )' t ; f in ( 0 ) ) d^ d f in ' [ 2 . 3 . 2 ] We summarize t h e c o n n e c t i o n between t h e m o l e c u l a r m o t i o n and t h e i n t r a -m o l e c u l a r r e l a x a t i o n r a t e i n the f o l l o w i n g way: The r o t a t i o n a l m o t i o n o f t h e r e l a x a t i o n v e c t o r s a r e d e s c r i b e d by t h e a p p r o p r i a t e p r o b a b i l i t y f u n c t i o n , P(ft ( t J . t ^ ^ O ) ) , from w h i c h t h e a n g u l a r c o r r e l a t i o n f u n c t i o n o f t h e second rank s p h e r i c a l harmonic i s c o n s t r u c t e d . P ( f i ? ( t ) ,t;tt^{0)) i s f o und from a proposed model f o r t h e r o t o r y m o t i o n o f t h e v e c t o r . T h i s model w i l l be c h a r a c t e r i z e d by a s e t o f parameters w h i c h can t h e n be deduced by comparison w i t h e x p e r i m e n t a l r e l a x a t i o n s t u d i e s . A l t h o u g h NMR l a t t i c e c o r r e l a t i o n f u n c t i o n s have been e v a l u a t e d hundreds o f t i m e s s t a r t i n g from v a r i o u s f i r s t p r i n c i p l e s i n s c a t t e r e d l i t e r a t u r e a p p e n d i c e s , p r o b a b l y t h e most p o w e r f u l and c o n s i s t e n t scheme o f e v a l u a t i o n r e l i e s on t h e v e r y g e n e r a l t r a n s f o r m a t i o n p r o p e r t i e s o f s p h e r i c a l t e n s o r s . T h i s approach has been f u l l y e x p l o i t e d i n a r e c e n t 39-42 s e r i e s o f papers by Paul Hubbard. However, t o m e r e l y summarize h i s r e s u l t s w o u l d do a g r e a t i n j u s t i c e t o t h e p e r s o n a l i t y o f t h e c e n t r a l problem - t h e problem o f a s s i g n i n g a m i c r o d y n a m i c a l i n t e r p r e t a t i o n t o a number - t h e m o l e c u l a r i n t e r p r e t a t i o n o f t h e c o r r e l a t i o n f u n c t i o n . F u r t h e r m o r e , t h e r e a r e many r e l a t e d p o i n t s c o n n e c t e d w i t h t h i s t o p i c t h a t need t o be d i s c u s s e d and q u a l i f i e d b e f o r e p r e s e n t i n g t h e f i n a l a s p e c t o f t h e groundwork chosen t o probe a s s o r t e d n u c l e a r m a g n e t i c r e l a x a t i o n p r o b l e m s . The t o p i c o f c o r r e l a t i o n f u n c t i o n s and s p e c t r o s c o p i c l i n e s h a p e s has e x p l o d e d i n t h e l a s t 10 y e a r s and p r o v i d e s t h e l i n k t o a much needed -25-u n i f i c a t i o n between s e e m i n g l y d i s t a n t forms o f dynamic s p e c t r o s c o p y . Good r e v i e w s o f g e n e r a l p r o p e r t i e s and u n i v e r s a l i t y o f c o r r e l a t i o n 43 f u n c t i o n s can be f o u n d i n the comprehensive r e v i e w a r t i c l e s by Gordon 44 and M o u n t a i n . However, t h i s f o r m a l i s m a l s o p o i n t s out a fundamental d i f f e r e n c e between NMR l i n e s h a p e t h e o r y and l i n e s h a p e t h e o r y f o r o t h e r r a d i a t i o n a b s o r p t i o n and s c a t t e r i n g s t u d i e s . Whereas t h e F o u r i e r T r a n s -form o f t h e bandshape c h a r a c t e r i s t i c o f t h e s e o t h e r forms o f s p e c t r o s c o p y can be i n t e r p r e t e d d i r e c t l y i n terms o f a m i c r o s c o p i c m o l e c u l a r p o s i t i o n / o r i e n t a t i o n c o r r e l a t i o n f u n c t i o n , t h e NMR bandshape y i e l d s o n l y - > • - » • <(B 0xM(t)-(B 0xM'(0)> [2.3.3] w h i c h ^ t o a f i r s t a p p r o x i m a t i o n , decays e x p o n e n t i a l l y w i t h t h e t i m e c o n s t a n t T ?. I t i s i m m e d i a t e l y r e a l i z e d t h a t E q u a t i o n [2.3.3] d e s c r i b e s t h e a u t o - c o r r e l a t i o n f u n c t i o n o f t h e m a c r o s c o p i c v a r i a b l e , JBQ><M(t) [. However, as we have a l r e a d y n o t e d , < I + ( t ) > and hence |BgxM(t)| can i t s e l f be e x p r e s s e d i n terms o f m i c r o s c o p i c c o r r e l a t i o n f u n c t i o n s . T h e r e f o r e , i t i s p o s s i b l e t o i n d i r e c t l y r e l a t e measured r e l a x a t i o n t i m e s t o m o l e c u l a r dynamics. Due t o t h i s f a c t , i t i s o f t e n s t a t e d t h a t NMR p r o v i d e s second hand, n o t second r a t e , d y n a m i c a l i n f o r m a t i o n . S t a t e d i n a s l i g h t l y d i f f e r e n t manner, NMR a n a l y s i s ( b o t h i n terms o f ( d / d t ) | B Q x M ( t ) | and ( d / d t ) B ^ M ( t ) ) d o e s not d i r e c t l y r e f l e c t t h e f r e -quency power spec t r u m o f t h e l o c a l f i e l d but o n l y t h e r e l a t i v e i n t e n s i t y o f a d i s c r e t e s p e c t r a l component o f t h i s f i e l d ( s e e E q u a t i o n [ 2 . 2 . 1 4 ] ) . Thus a t f i r s t g l a n c e , o t h e r s p e c t r o s c o p i c t e c h n i q u e s , i n p r i n c i p l e , c o n t a i n much more i n f o r m a t i o n about m o l e c u l a r dynamics th a n do NMR e x p e r i m e n t s . However, p r i n c i p l e i s f a r from p r a c t i c e . F o r i n s t a n c e , -26-i t i s o f t e n d i f f i c u l t t o s e p a r a t e t r a n s l a t i o n a l and r o t a t i o n a l m o b i l i t y i n some o t h e r s p e c t r o s c o p i c t e c h n i q u e s ; i n NMR t h i s i s o f t e n t r i v i a l as i m p l i e d by an e a r l i e r a s s u m p t i o n . A l s o , NMR t e c h n i q u e s y i e l d s i n g l e m o l e c u l e c o r r e l a t i o n f u n c t i o n s whereas i n t e r p r e t a t i o n o f o t h e r t e c h -n i q u e s i s o f t e n c l o u d e d by p a i r and h i g h e r o r d e r c o r r e l a t i o n f u n c t i o n s . F u r t h e r m o r e , NMR r e l a x a t i o n d a t a does have t h e advantage o f b e i n g ap-p l i c a b l e f o r a f a r w i d e r c l a s s o f e x p e r i m e n t a l c o n d i t i o n s , and much l i k e x - r a y d i f f r a c t i o n s t u d i e s , i s a b l e t o y i e l d i n f o r m a t i o n on i n d i v i d u a l atoms i n complex m o l e c u l e s . R e t u r n i n g t o t h e e v a l u a t i o n o f E q u a t i o n [ 2 . 3 . 2 ] , s i m p l e models o f 45 m o l e c u l a r r e o r i e n t a t i o n s u c h as t h e r o t a t i o n a l random w a l k p r o b l e m , 46 t h e r o t a t i o n a l L a n g e v i n e q u a t i o n , o r t h e r o t a t i o n a l d i f f u s i o n e q u a t i o n p r e d i c t e x p o n e n t i a l a n g u l a r c o r r e l a t i o n f u n c t i o n s and hence, a s i m p l e L o r e n t z i a n s p e c t r a l d e n s i t y . W h i l e t h e d i f f u s i o n e q u a t i o n and i t s r e l -a t i v e s may not be an e x a c t m o t i o n a l m o d e l , i t i s p r o b a b l y a v e r y good a p p r o x i m a t i o n f o r t h e d e s c r i p t i o n o f t h e r e o r i e n t a t i o n i n most l i q u i d s . In t h e f i r s t a p p l i c a t i o n s o f t h e r o t a t i o n a l d i f f u s i o n model f o r t h e c a l c u l a t i o n o f n u c l e a r m a g n e t i c r e l a x a t i o n b e h a v i o r , i s o t r o p i c r o t a t i o n a l d i f f u s i o n was assumed, r . i l n t h i s l i m i t , a g e n e r a l i z a t i o n o f P e t e r Debye's c l a s s i c t r e a t m e n t g i v e s the f o l l o w i n g e x p r e s s i o n f o r t h e a n g u l a r c o r -r e l a t i o n f u n c t i o n , where T 9 , t h e i s o t r o p i c r o t a t i o n a l c o r r e l a t i o n t i m e , i s d e f i n e d as C"n(t) - « k ^ - l ) k C ^ ( 0 ) e x p ( . t / x 2 ) [ 2 . 3 . 4 ] x 2 = (6D)" 1. [2.3.5] - 2 7 -The s c a l a r D, i s t h e i s o t r o p i c r o t a t i o n a l d i f f u s i o n c o n s t a n t ( d i m e n s i o n s o f t i m e - 1 ) . Note t h a t t h e form o f E q u a t i o n [ 2 . 3 . 4 ] a l s o a l l o w s t h e f o l l o w i n g i d e n t i f i c a t i o n t o be made, where g ^ t ) i s a reduced c o r r e l a t i o n f u n c t i o n , g K X , ( t ) = ( - 1 ) K S , ? i ?n K j - x . x ^ ( t J / C ^ 0 ) - O f t e n i n t h e l i t e r a t u r e , t h i s e x p r e s s i o n i s g e n e r a l i z e d t o a model i n d e p e n d e n t d e f i n i t i o n : The e f f e c t i v e r o t a t i o n a l c o r r e l a t i o n t i m e i s d e f i n e d as o n e - h a l f o f t h e a r e a under t h e c o r r e l a t i o n f u n c t i o n . A more t a n g i b l e q u a n t i t a t i v e i n t e r p r e t a t i o n can be a s s i g n e d i f t h e d e f i n i t i o n ( E q u a t i o n [ 2 , 3 . 5 ] ) i s s u b s t i t u t e d i n t o t h e famous E i n s t e i n r e l a t i o n s h i p , * T h i s d e f i n i t i o n has t h e advantage t h a t the shape o f th e c o r r e l a t i o n f u n c t i o n does not e n t e r i n t o t h e d e f i n i t i o n . E q u a t i o n [ 2 . 3 . 4 ] r e d u c e s t o a s p e c i a l l i m i t i n g c a s e . However, t h e term " e f f e c t i v e c o r r e l a t i o n 49 t i m e " so o f t e n r e f e r r e d t o i n t h e l i t e r a t u r e i s m i s l e a d i n g as t h e k -k E f o l l o w i n g s i m p l e example d e m o n s t r a t e s : Assuming g ' ( t ) = . exp(-t/x.), t h e n J-Q g ^ " K ( t ) e x p ( i w t ) d t = J T . / ( 1 + u> xf). However, E q u a t i o n [ 2 . 3 . 6 ] y i e l d s t h e e f f e c t i v e c o r r e l a t i o n t i m e , x ^  = ^  x... The d e f i n i t i o n i m p l i e s t h a t g k ' " k ( t ) = e x p ( - t / x g f f ) f o r w h i c h J g gj^~ k(t)exp(ia>t)dt = T g ^ / O + o^x 2^). To be c o m p a t i b l e w i t h o u r f i r s t e x p r e s s i o n , t h i s demands t h a t l. x./O + w2x2) = x e f f / ( l + " > 2 t 2 f f ) . As x e f f / ( l + <A 2 f f) = ? T . / ( 1 + a) T ? + co 2C(j T j ) 2 - T 2 ] ) , x g f f c a n n o t l i t e r a l l y mean " e f f e c t i v e " c o r r e l a t i o n t i m e u n l e s s CUT. << 1 f o r a l l i ( z e r o f i e l d o r ex t r e m e - n a r r o w i n g a p p r o x i m a t i o n ) o r t h e t r i v i a l case where i i s summed o v e r a s i n g l e e x p o n e n t i a l . To a v o i d t h i s misnomer, we s h a l l use t h e ka correct a n a l o g o u s term " r e d u c e d s p e c t r a l d e n s i t y " , j (co), d e f i n e d as ki the o n e - s i d e d F o u r i e r T r a n s f o r m o f g^{t), w h i c h , by d e f i n i t i o n , r e d u c e s t o t h e " e f f e c t i v e c o r r e l a t i o n t i m e " i n t h e l i m i t o f e x t r e m e - n a r r o w i n g . co 0 [ 2 . 3 . 6 ] -28-<e2> = 4Dt • [2.3.7] I t i s seen t h a t x ^ , t h e r o t a t i o n a l c o r r e l a t i o n t i m e o b t a i n e d from NMR r e l a x a t i o n s t u d i e s , c o r r e s p o n d s t o t h e t i m e t a k e n f o r a m o l e c u l a r -f i x e d v e c t o r t o r e o r i e n t a root-mean-square a n g l e o f /2/3 r a d i a n s . How-e v e r , t h i s d e f i n i t i o n i s o f l i m i t e d v a l i d i t y as i t p e r t a i n s t o t h e p a r -t i c u l a r model o f i s o t r o p i c d i f f u s i o n a l r e o r i e n t a t i o n . A more r e a l i s t i c degree o f a p p r o x i m a t i o n f o r m o l e c u l a r r e o r i e n t a t i o n assumes t h a t i n g e n e r a l , t h e d i f f u s i o n p r o c e s s i s no l o n g e r c o r r e c t l y d e s c r i b e d by a u n i q u e s c a l a r q u a n t i t y , but must be d e s c r i b e d by t h e more complete d i f f u s i o n t e n s o r . F o r t h i s g e n e r a l m o t i o n a l m o d e l , i t i s most c o n v e n i e n t t o c o n s i d e r t h e t i m e dependent t r a n s f o r m a t i o n from t h e i n t e r -a c t i o n f i x e d frame o f r e f e r e n c e t o t h e l a b frame v i a t h e i n t e r m e d i a t e t i m e i n d e p e n d e n t t r a n s f o r m a t i o n from t h e i n t e r a c t i o n f i x e d frame t o t h e frame w h i c h d i a g o n a l i z e s t h e m o l e c u l a r d i f f u s i o n t e n s o r . T h i s was n o t o f c o n c e r n f o r i s o t r o p i c m o t i o n as the d i f f u s i o n and i n t e r a c t i o n co-50 o r d i n a t e systems can always be chosen t o be c o l l i n e a r . P e r r i n was t h e f i r s t t o e x t e n d Debye's i d e a s i n t o t h i s r e a l m and c o n s i d e r t h e e f f e c t o f a n i s o t r o p i c r o t a t i o n a l Brownian m o t i o n on ( d i e l e c t r i c ) r e l a x a t i o n . S i n c e t h e i n i t i a l t r e a t m e n t o f P e r r i n , o t h e r e x t e n s i o n s o f t h e t h e o r y have been p r e s e n t e d i n t h e l i t e r a t u r e . P r o b a b l y t h e most o f t quoted 51 a r e t h e g e n e r a l i z a t i o n s p r o v i d e d by F a v r o . These i d e a s were f i r s t a p p l i e d t o t h e i n t e r p r e t a t i o n o f NMR r e -52 l a x a t i o n t i m e s i n a p a i r o f papers by Woessner. S h a r i n g t h e v i r t u e s o f p r o v i d i n g a more r e a l i s t i c s p i n m o t i o n a l model y e t y i e l d i n g t r a c t a b l e e x p e r i m e n t a l i n t e r p r e t a t i o n , t h e a n i s o t r o p i c r o t a t i o n a l d i f f u s i o n a l -29-model has grown i n p o p u l a r i t y and has s u b s e q u e n t l y been e x t e n s i v e l y d i s -53 54 55 56 c u s s e d by i n v e s t i g a t o r s such as Bopp, H u n t r e s s , M a r g a l i t , S h i m i z u , S t e e l e , 5 ' 7 and V a l i e v 5 ^ t o mention a few such c o n t r i b u t e r s . In g e n e r a l , t h e m o l e c u l a r m o t i o n a l d e s c r i p t i o n c o n t a i n s t h r e e em-p i r i c a l c o n s t a n t s , the t h r e e components o f t h e d i a g o n a l i z e d d i f f u s i o n t e n s o r . The c o r r e l a t i o n f u n c t i o n r e l e v a n t t o NMR r e l a x a t i o n i s no l o n g e r d e s c r i b e d by a s i n g l e e x p o n e n t i a l , but as a s u p e r p o s i t i o n o f f i v e d i s t i n c t e x p o n e n t i a l s , and hence, t h e s p e c t r a l d e n s i t y i s a sum o f f i v e d i s t i n c t t e r m s . More o f t e n , however, i t i s assumed t h a t t h e m o t i o n i s a p p r o x i m a t e d by a d i f f u s i o n t e n s o r c h a r a c t e r i z e d by o n l y two d i s t i n c t e i g e n v a l u e s , Dj_ and D)( ( i . e . an a x i a l l y symmetric e l l i p s o i d ) . In t h i s c a s e , t h e c o r r e l a t i o n f u n c t i o n i s a sum o f t h r e e e x p o n e n t i a l s . R e c e n t l y , t h e g e n e r a l i t y o f t h e s e r e s u l t s has been shown t o r e s u l t not from assuming a d i f f u s i o n a l p r o c e s s , 59 but o n l y from t h e symmetry o f t h e r e o r i e n t i n g body. B e s i d e s the asymmetry o f t h e d i f f u s i o n p r o c e s s o f t h e m o l e c u l a r frame-work, the d i r e c t i o n o f t h e r e l a x a t i o n v e c t o r r e l a t i v e t o t h e p r i n c i p a l d i f f u s i o n a x i s i s o f g r e a t i m p o r t a n c e . Each e x p o n e n t i a l i n t h e symmetric top c o r r e l a t i o n f u n c t i o n i s w e i g h t e d by an a n g u l a r f u n c t i o n whose argument i s d e f i n e d as the e n c l o s e d a n g l e between t h e s e two v e c t o r s . F u r t h e r m o r e , each e x p o n e n t i a l f a c t o r i s g i v e n i n terms o f m o l e c u l a r d i f f u s i o n c o n s t a n t s , hence, one o b t a i n s an i n t e r p r e t a t i o n o f t h e e f f e c t i v e s p e c t r a l d e n s i t y i n terms o f t r u e m o l e c u l a r r e o r i e n t a t i o n t i m e s as was p o s s i b l e f o r t h e i s o -t r o p i c c a s e , [ 2.3.8] There i s a n o t h e r c l o s e l y r e l a t e d s i t u a t i o n w h i c h i s o f i n t e r e s t f o r -30-t h e m o l e c u l a r i n t e r p r e t a t i o n o f t h e s p e c t r a l d e n s i t y . T h i s i s t h e c a s e where t h e r e l a x a t i o n v e c t o r undergoes r o t a t i o n a l m o t i o n about an a x i s w h i c h i s r i g i d l y f i x e d t o an o t h e r w i s e i s o t r o p i c a l l y r e o r i e n t i n g m o l e c u l a r backbone. I f one makes t h e i d e n t i f i c a t i o n , D u ->• D. t + D and D ± D, i t i s seen t h a t t h i s p roblem reduces t o t h e s ymmetric t o p c a s e . The r educed s p e c t r a l d e n s i t i e s have been decomposed i n t o m o l e c u l a r r e o r i e n t a t i o n t i m e s f o r more c o m p l i c a t e d m o t i o n a l m o d e l s 6 0 ' 6 1 and f o r CO f."i m u l t i p l e i n t e r n a l r o t a t i o n s . ' U n f o r t u n a t e l y , any but t h e s i m p l e s t o f m o t i o n a l models a r e c h a r a c t e r i z e d by so many p a r a m e t e r s , t h a t i n t e r -p r e t a t i o n o f e x p e r i m e n t a l d a t a r a p i d l y becomes h i g h l y u n d e r d e t e r m i n e d . U n t i l t h i s p o i n t , i t has been assumed t h a t t h e r o t a t i o n o f t h e m o l e c u l a r framework c a r r y i n g t h e r e l a x a t i o n v e c t o r r e s u l t s from a g r e a t number o f u n c o r r e l a t e d i n f i n i t e s i m a l s t e p s . T h i s l e a d s d i r e c t l y t o a m o l e c u l a r i n t e r p r e t a t i o n o f t h e c o r r e l a t i o n f u n c t i o n . The e n t i r e problem i s c l o u d e d whenever a non-Markovian p r o c e s s i s assumed f o r t h e r e o r i e n t a -t i o n a l m o del, t h a t i s , whenever i n e r t i a l e f f e c t s a r e c o n s i d e r e d and t h e a s s u m p t i o n o f d i f f u s i o n a l m o t i o n i s abandoned. I n e r t i a l e f f e c t s ( c h a r -a c t e r i z e d by s t o c h a s t i c , f i n i t e - s t e p r e o r i e n t a t i o n and n o n e x p o n e n t i a l c o r r e l a t i o n f u n c t i o n s ) a r e u n d o u b t e d l y i m p o r t a n t f o r s m a l l m o l e c u l e s i n n o n v i s c o u s l i q u i d s . T h i s a s p e c t o f NMR r e l a x a t i o n s t u d i e s , e s p e c i a l l y when i n t e r p r e t e d i n c o n j u n c t i o n w i t h o t h e r s p e c t r o s c o p i c t e c h n i q u e s , has commanded much r e c e n t i n t e r e s t from c h e m i c a l p h y s i c i s t s . However, as t h e p r e s e n t work f o c u s e s on t h e m o t i o n a l b e h a v i o r o f r e l a t i v e l y l a r g e m o l e c u l e s , t h e problem o f i n e r t i a l e f f e c t s w i l l prove t o be o f no m ajor c o n c e r n . The same argument cannot be used i f i n t e r n a l r o t a t i o n s a r e -31-c o n s i d e r e d . In t h i s c a s e , a second m odel, commonly used w i t h m e t hyl group r e o r i e n t a t i o n , c o n s i d e r s i n s t a n t a n e o u s random jumps among t h r e e o r i e n t a t i o n s 120° a p a r t . The r e s u l t o b t a i n e d from such a t r e a t m e n t i s e x p e r i m e n t a l l y a l m o s t i n d i s t i n g u i s h a b l e ( a l t h o u g h s h a r i n g no p h y s i c a l s i m i l a r i t y ) from t h e d i f f u s i o n a l a p p r o x i m a t i o n . However, t h i s l a t t e r a p p r o x i m a t i o n p r o v i d e s a c o n v e n i e n t m a t h e m a t i c a l form f o r d a t a a n a l y s i s . Good d i s c u s s i o n s c o n t r a s t i n g t h e s e two models o f methyl r e o r i e n t a t i o n 64 65 a r e c o n t a i n e d i n t h e l i t e r a t u r e . ' One f i n a l , r e l a t e d a s p e c t c o n c e r n i n g t h e c o r r e l a t i o n f u n c t i o n s h o u l d be t o u c h e d upon h e r e . A l l s i m p l e models o f d i f f u s i o n a l m o t i o n , no m a t t e r how c o m p l i c a t e d , r e d u c e t o t h e form 9 ^ ( t ) = ]C A . . e x p ( - t / x . . ) , where t h e summation ex t e n d s o v e r a s m a l l number o f d i s c r e t e t i m e c o n s t a n t s , x . . For i s o t r o p i c m o t i o n , a d e l t a f u n c t i o n d i s t r i b u t i o n o f t i m e c o n s t a n t s i s assumed; f o r s i m p l e a n i s o t r o p i c m o t i o n , a comb o f two o r t h r e e d e l t a f u n c t i o n s . More c o m p l i c a t e d models demand more " t e e t h " i n t h e comb. However, i n many NMR s t u d i e s , e s p e c i a l l y p o l y m e r i c o r s u r f a c e s t u d i e s , a continuum o f c o r r e l a t i o n t i m e c o n s t a n t s i s assumed }^6,67 A system i n w h i c h a continuum o f such t i m e c o n s t a n t s e x i s t would have t o be c o n s i d e r e d as a s p e c i a l case o f a m u l t i - p h a s e system. In many i n s t a n c e s , i t i s not a t a l l c l e a r when such an a s s u m p t i o n i s j u s t i f i e d and t h i s approach o f t e n appears t o be no more th a n a d a t a f i t t i n g d e v i c e which does l i t t l e more th a n add c o n f u s i o n . In t h e b e s t a p p r o a c h , g i v e n no a p r i o r i r e a s o n f o r assuming such a d i s t r i b u t i o n , one s h o u l d always seek t o f i t t h e r e l a x a t i o n d a t a w i t h c o r r e l a t i o n f u n c t i o n s ( a g r e e a b l y 0 [2.3.9] -32-m u l t i - e x p o n e n t i a l ) based on more r e a l i s t i c models o f m o l e c u l a r m o t i o n . Note however, t h a t n o n e x p o n e n t i a l o r m u l t i - e x p o n e n t i a l c o r r e l a t i o n f u n c t i o n s do not i m p l y n o n e x p o n e n t i a l m a g n e t i c r e l a x a t i o n (and v i c e -v e r s a ) ; t h e two l e v e l s o f i n f o r m a t i o n must always remain s e g r e g a t e d i n t h o u g h t . In c o n c l u s i o n o f t h i s s e c t i o n , t h e c l a s s i c a l e v a l u a t i o n o f t h e c o r r e l a t i o n f u n c t i o n s d e f i n e d by E q u a t i o n [ 2 . 3 . 1 ] i s i n t r o d u c e d . These f o r m u l a s f o l l o w d i r e c t l y ( a f t e r some m a n i p u l a t i o n ) from v a r i o u s e q u a t i o n s 39-42 c o n t a i n e d i n t h e s e r i e s o f papers by Hubbard mentioned i n t h e i n t r o -d u c t i o n o f t h i s s e c t i o n . I t can be shown by t h e use o f t h e t h e o r y o f r o t a t i o n a l Brownian m o t i o n t h a t t h e c o r r e l a t i o n f u n c t i o n o f two s p h e r i c a l harmonics dependent on fi and n^, i s g i v e n by, 2 < Y k ( ^ ( t ) ) Y ^ ( ^ ( 0 ) ) > = ( - 1 ) k 5 k j _ J l 5 " 1 ^ ( - D m Y 2 ( ^ ) Y - m ( ^ ) m=-2 x exp {-(6Dj_ + m 2 ( D M - D ± ) ) t | . [2.3.10] T h i s e v a l u a t i o n assumes t h e d i f f u s i o n i s c o m p l e t e l y c h a r a c t e r i z e d by t h e r o t a t i o n a l d i f f u s i o n c o n s t a n t s f o r r o t a t i o n s about a symmetry a x i s , D u, and f o r r o t a t i o n s about an a x i s p e r p e n d i c u l a r t o t h e symmetry a x i s , Dj_. I f DL = D n = D, t h e r i g h t hand s i d e o f E q u a t i o n [2.3.10] r e d u c e s t o t h e a d d i t i o n theorem f o r s p h e r i c a l h a r m o n i c s , t h a t i s , t h e r i g h t hand s i d e r e d u c e s t o (-1)^6. P. (cose )exp(-6Dt)/4Tr, as e x p e c t e d . The primed a n g l e s ( s t a t i c v a r i a b l e s i n t i m e ) r e f e r t o t h e p o l a r and a z i m u t h a l a n g l e s o f t h e v a r i o u s i n t e r a c t i o n c o o r d i n a t e systems w i t h r e s p e c t t o t h e m o l e c u l e f i x e d d i f f u s i o n r e f e r e n c e frame. The t i m e dependent, unprimed a n g l e s r e f e r t o t h e o b s e r v a b l e l a b frame. T h i s e q u a t i o n assumes t h a t the r e l a x -i n g s p i n s a r e r i g i d l y a t t a c h e d t o t h i s s ymmetric body. However, -33-t h e r e i s no p h y s i c a l d i s t i n c t i o n between t h i s c a s e and t h e s i t u a t i o n where t h e s p i n ( s ) a r e a t t a c h e d d i r e c t l y t o a f l e x i b l e f r a gment l o c a t e d on an o t h e r w i s e i s o t r o p i c a l l y r e o r i e n t i n g backbone. I n t h i s i n s t a n c e , E q u a t i o n [2.3.10] i s s t i l l v a l i d i f i s r e p l a c e d by D and D u i s r e -p l a c e d by D + D^ n t where D..^ i s t h e d i f f u s i o n c o n s t a n t c h a r a c t e r i z i n g t h e i n t e r n a l r o t o r . I f one assumes t h a t t h e a p p r o p r i a t e model f o r t h e i n t e r n a l m o t i o n i s 120° jumps, then i t can be shown r a t h e r s i m p l y t h a t t h e o n l y d i f f e r e n c e i s t h a t t h e e x p o n e n t i a l term i n E q u a t i o n [2.3.10] i s r e p l a c e d by exp|-(6D - 3 ( 3 - |m|) | m | v . j n t / 4 ) t j where v..^ i s t h e p r o b -a b i l i t y p e r u n i t time t h a t t h e system jumps from one o r i e n t a t i o n t o a n o t h e r . A second model we w i l l r e s o r t t o i n f u t u r e c a l c u l a t i o n s i s a s i m p l e e x t e n s i o n o f E q u a t i o n [ 2 . 3 . 1 0 ] . T h i s i s based on a symm e t r i c t o p e l l i p -s o i d w i t h an i n t e r n a l r o t o r a t t a c h e d a t an a r b i t r a r y a n g l e w i t h r e s p e c t t o t h e p r i n c i p a l a x i s o f t h e e l l i p s o i d . In t h i s c a s e , 2 2 < Y ^ ( n c ( t ) ) Y * ( n n ( 0 ) ) > = ( - D ^ ^ - 1 X) d n ' n " ( E 5 , ) n'=-2 n"=-2 x ( - l ) n " Y ^ " ( ^ ' ) Y - n " ( ^ ' ) e x p ( - n " 2 D . t ) x exp|-(6Dj_ + n , 2 ( D ( 1 - D ^ t } [2.3.11] where D. i s t h e d i f f u s i o n c o n s t a n t c h a r a c t e r i z i n g t h e i n t e r n a l r o t o r . The d o u b l e primed a n g l e s a r e f i x e d r e l a t i v e t o t h e i n t e r n a l r o t o r and B' i s t h e a n g l e d e f i n e d between t h e r o t o r a x i s and t h e p r i n c i p a l d i f -f u s i o n a x i s . The q u a n t i t i e s d , n , , ( B ' ) a r e el e m e n t s o f t h e r e d u c e d r o t a t i o n m a t r i x and a r e f u l l y d i s c u s s e d i n any e x p o s i t i o n on t h e t h e o r y o f a n g u l a r momentum.^ Assuming 120° s t e p s , t h e term exp(-n''D^.t) i s -34-r e p l a c e d by e x p ( - 3 ( 3 - | n " |) |n'' h i t t / 4 ) . One o f t h e v e r y i m p o r t a n t p r o p e r t i e s o f t h e two c o r r e l a t i o n f u n c t i o n s g i v e n , i s t h a t t h e r i g h t hand s i d e o f each e q u a t i o n i s dependent o n l y ka k upon t h e rank o f t h e i n i t i a l s p h e r i c a l harmonic. Hence, C ( t ) = (-1) Cn x 6, „C°°(t). T h i s f a c t i s v a l i d no m a t t e r what model o f m o t i o n a l k,-£ cn b e h a v i o r i s assumed and f o l l o w s d i r e c t l y from t h e g e n e r a l p r o p e r t i e s o f quantum m e c h a n i c a l c o r r e l a t i o n f u n c t i o n s o f i r r e d u c i b l e t e n s o r o p e r -a t o r s . -35-2.4 SUMMARY A f t e r a b r i e f i n t r o d u c t i o n t o t h e p r o b l e m , t h e t o o l s needed f o r the t w o - s t e p t r a n s l a t i o n from t h e m a c r o s c o p i c NMR o b s e r v a b l e s t o the m o l e c u l a r r e a l m o f i n t e r e s t have been p r e s e n t e d . A l t h o u g h t h e r e i s a l a r g e r e p e r t o i r e o f such t o o l s , we have i n t r o d u c e d an approach w h i c h i s f e l t t o be q u i t e adequate f o r o u r p u r p o s e s , t h e method e m p l o y i n g t h e s e m i c l a s s i c a l c o n s t r u c t i o n o f t h e e q u a t i o n s g o v e r n i n g t h e t i m e e v o l u t i o n o f t h e reduced d e n s i t y o p e r a t o r . Once t h e s p i n r e l a x a t i o n i s r e l a t e d t o t h e o t h e r n o n s p i n degrees o f freedom, t h e s p i n s no l o n g e r e n t e r i n t o t h e problem and one i s c o n f r o n t e d w i t h t h e d i f f i c u l t t a s k o f e v a l u a t i n g t h e l a t t i c e c o r r e l a t i o n f u n c t i o n s i n terms o f a p a r t i c u l a r m o t i o n a l model. O n l y i f t h e model i s chosen c o r r e c t l y , i s i t p o s s i b l e t o o b t a i n r e l i a b l e m o l e c u l a r dynamics. The e n t i r e i n t e r -p r e t a t i o n a l p r o b l e m , w h i c h has been d i s c u s s e d i n d e t a i l t h r o u g h o u t t h i s c h a p t e r , i s p i c t o r i a l l y o u t l i n e d i n F i g u r e 2.1. In a d d i t i o n t o t h e c r u x o f t h e i n t e r p r e t a t i o n a l p r o b l e m , many o f t h e a s s u m p t i o n s u n a v o i d a b l y e n c o u n t e r e d i n any t h o r o u g h d i s c u s s i o n o f l i q u i d s t a t e NMR r e l a x a t i o n were a l s o i n t r o d u c e d and d i s c u s s e d . The r a t h e r e x t e n s i v e b i b l i o g r a p h y w i l l p r o v i d e t h e needed s u p p l e m e n t a r y m a t e r i a l . In t h e f o l l o w i n g c h a p t e r s , E q u a t i o n s [ 2 . 2 . 1 8 ] , [ 2 . 3 . 1 0 ] , and [2.3.11] w i l l be a p p l i e d t o v a r i o u s problems o f c o n c e r n i n contemporary NMR r e l a x a t i o n s t u d i e s . The approach chosen w i l l be t o adopt t h e most r e a s o n a b l e m o t i o n a l a p p r o x i m a t i o n y e t one w h i c h y i e l d s a t r a c t a b l e i n t e r p r e t a t i o n (a t r a d e - o f f between r i g o r and the f a t e o f "over param-e t e r i z a t i o n " ) . - 3 6 -I t s h o u l d be kept i n mind t h a t t h e m o l e c u l a r i n t e r p r e t a t i o n o f t h e s p e c t r a l d e n s i t y i s an e x t r e m e l y r i s k y b u s i n e s s . H o p e f u l l y , t h i s t h e s i s can p r o v i d e some measure o f a s s i s t a n c e f o r t h e e x p e r i m e n t a l i s t . FIGURE 2.1 A s s i g n i n g a m o l e c u l a r i n t e r p r e t a t i o n t o NMR r e l a x a t i o n parameters MACROSCOPIC DOMAIN R e l a x a t i o n O b s e r v a b l e s ( < M + , z ( 0 ) M + , z ( t ) > o r Over h a u s e r e f f e c t s ) A A A A A A A A A A A A A A A A A A A A A MICROSCOPIC DOMAIN M o l e c u l a r I n t e r p r e t a t i o n S t a t i c s Dynamics-; Chemical P h y s i c a l T r a n s l a t i o n a l R e o r i e n t a t i o n a l I n e r t i a l D i f f u s i o n a l co —j i Realm o f R e l a x a t i o n Theory p r o p e r N e c e s s a r y a p r i o r i model f o r dynamics -38-REFERENCES: CHAPTER I I 1. N. Bloembergen, E. M. P u r c e l l , and R. V. Pound, Phys. Rev. 73, 679 (1948). 2. A. Abragam and R. V. Pound, Phys. Rev. 92, 943 ( 1 9 5 3 ) . 3. I . Solomon, Phys. Rev. 99, 559 ( 1 9 5 5 ) . 4. J . J e e n e r , Advan. Magn. Res. 3^ , 207 (1968). 5. M. Bloom, MTP I n t e r n a t i o n a l Rev. S c i . 4, 1 ( 1 9 7 2 ) . 6. R. K. Wangness and F. 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Tolman, P r i n c i p l e s o f S t a t i s t i c a l M e c h a n i c s , O x f o r d U n i v e r s i t y P r e s s , New Y o r k , 1938; Ch a p t e r IX. 30. H. P f e i f e r , Ann. P h y s i k 13, 174 (1 9 6 4 ) . 31. R. B r a c e w e l l , The F o u r i e r T r a n s f o r m and I t s A p p l i c a t i o n s , McGraw-H i l l , New Y o r k , 1968; C h a p t e r 12. 32. R e f e r e n c e 26; page 446. 33. G. K. F r a e n k e l , J . Chem. Phys. 4 2, 4275 ( 1 9 6 5 ) . 34. P. S. Hubbard, Phys. Rev. 128, 650 (1962). 35. T. Nowak and A. S. M i l d v a n , Biochem. 1J_, 2813 (1 9 7 2 ) . 36..".A.S. M i l d v a n , p r i v a t e communication. 37. L. G. Werbelow and A. G. M a r s h a l l , J . Amer. Chem. Soc. 95, 5132 (1 9 7 3 ) . 38. P. W. A t k i n s , Advan. M o l . R e l a x . P r o c e s s e s 2, 121 (1 9 7 2 ) . 39. P. S. Hubbard, Phys. Rev. 180, 319 ( 1 9 6 9 ) . 40. P.^S. Hubbard, J . Chem. Phys. 52, 563 (1 9 7 0 ) . 41. P. S. Hubbard, Phys. Rev. A6, 2421 ( 1 9 7 2 ) ; i b i d . , A8, 1429 (1973). 42. P. S. Hubbard, Phys. Rev. A9, 481 (1974). 43. R. G. Gordon, Advan. Magn. Res. 3_, 1 (1968). 44. R. D. M o u n t a i n CRC C r i t . Rev. S o l . S t a t e 1_, 5 ( 1 9 7 2 ) . 45. E. N. I v a n o v , Sov. Phys. JETP 18, 1041 (1964). -40-46. P. S. Hubbard, Phys. Rev. 1_3J[, 1155 (1963). 47. w. H. F u r r y , Phys. Rev. 107, 7 (1 9 5 7 ) . 48. P. Debye, P o l a r M o l e c u l e s , Dover, New Y o r k , 1929; C h a p t e r V. 49. A. G. M a r s h a l l , P. G. S c h m i d t , and B. D. S y k e s , Biochem. 1J_, 3875 (1 9 7 2 ) . 50. F. P e r r i n , J . Phys. Radium _5, 497 ( 1 9 3 4 ) ; i b i d . , 7, 1 (1 9 3 6 ) . 51. L. D. F a v r o , Phys. Rev. Vj_9, 53 ( 1 9 6 0 ) ; i b i d . , i n F l u c t u a t i o n  Phenomena i n S o l i d s , e d i t e d by R. B u r g e s s , Academic P r e s s , New Y o r k , 1965. 52. D. E. Woessner, J . Chem. Phys. 36, 1 ( 1 9 6 2 ) ; i b i d . , 37, 647 (1 9 6 2 ) . 53. T. T. Bopp, J . Chem. 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Res. 11_, 421 (1973). 64. D. E. Woessner and B. S. Snowden, Advan. M o l . R e l a x . P r o c e s s e s 3_, 181 (197 2 ) . 65. H. V e r s m o l d , Z. N a t u r f o r s c h . A25, 367 ( 1 9 7 0 ) ; i b i d . , J . Chem. Phys. 58, 5649 (1973) . -41-66. F. Noack, NMR B a s i c P r i n c i p l e s and P r o g r . 3, 83 ( 1 9 7 1 ) . 67. B. B l i c h a r s k i and J . S. B l i c h a r s k i , A c t a Phys. P o l o n . A41, 347 ( 1 9 7 2 ) . 68. M. E. Rose, E l e m e n t a r y Theory o f A n g u l a r Momentum, W i l e y , New Y o r k , 1957. -42-CHAPTER I I I ANISOTROPIC MOLECULAR MOTIONS AND THE NMR RELAXATION OBSERVABLES 3.1 INTRODUCTION-The model o f m o l e c u l a r r e o r i e n t a t i o n a l m o tions r e s p o n s i b l e f o r t h e m o d u l a t i o n o f t h e v a r i o u s s p i n i n t e r a c t i o n s i n n u c l e a r m a g n e t i c r e s o n -ance i s g e n e r a l l y assumed t o be i s o t r o p i c (a one-pa r a m e t e r , e x p o n e n t i a l c o r r e l a t i o n f u n c t i o n ; s p h e r i c a l t o p a p p r o x i m a t i o n ) and r a p i d compared to a t i m e s c a l e c h a r a c t e r i z e d by a Larmor p r e c e s s i o n ( t h e w h i t e s p e c t r a l d e n s i t y o r z e r o f i e l d a p p r o x i m a t i o n ) . A l t h o u g h o f t e n v a l i d f o r s i m p l e o r g a n i c m o l e c u l e s i n s o l u t i o n , both a s s u m p t i o n s a r e g e n e r a l l y i n v a l i d i n t y p i c a l b i o c h e m i c a l s t u d i e s . In t h i s c h a p t e r , i t w i l l be shown how the e f f e c t o f s l o w e r , a n i s o t r o p i c m o tions i n f l u e n c e NMR r e l a x a t i o n o b s e r v a b l e s f o r some s i m p l e c a s e s . As s u c c i n c t l y d e p i c t e d i n F i g u r e 2.1, the NMR e x p e r i m e n t i s h i g h l y u n d e r d e t e r m i n e d . F o r t r a c t a b i l i t y , t h e s i m p l e s t n o n t r i v i a l m o t i o n a l m odel, a hydrodynamic symmetric t o p , i s compared w i t h c o n c l u s i o n s i n f e r r e d from t h e as s u m p t i o n o f i s o t r o p i c m o b i 1 i t y . In t h e absence o f m a g n e t i c f i e l d g r a d i e n t s and c h e m i c a l exchange, t h e p r i n c i p a l r e l a x a t i o n mechanism f o r p r o t o n s and many o t h e r s p i n 1/2 n u c l e i i n l i q u i d s a r e i n t e r - and i n t r a m o l e c u l a r d i p o l e - d i p o l e i n t e r a c t i o n s . -43-I n t e r m o l e c u l a r r e l a x a t i o n can r e a d i l y be i d e n t i f i e d by e x p e r i m e n t s con-d u c t e d a t v a r y i n g degrees o f d i l u t i o n i n a nonmagnetic s o l v e n t , and f o r p r a c t i c a l p u r p o s e s , can be e x p e r i m e n t a l l y e l i m i n a t e d . In t h e f o l l o w i n g s e c t i o n s , we t r e a t o n l y t h e i n t r a m o l e c u l a r , d i r e c t d i p o l a r c o u p l i n g between a p a i r o f s p i n s , i and j . T h i s s i m p l e a p p r o x i m a t i o n t o s p i n b e h a v i o r has formed t h e b a s i s o f many d i s c u s s i o n s and t h e s e s h o u l d be c o n s u l t e d f o r an e n l i g h t e n i n g background i n t o t h i s s u b j e c t . 1 - ^ The d i r e c t , i n t r a m o l e c u l a r d i p o l a r H a m i l t o n i a n between a p a i r o f s p i n s i and j , can s i m p l y be e x p r e s s e d as However, f o r c o m p u t a t i o n a l p u r p o s e s , i t i s more c o n v e n i e n t l y e x p r e s s e d Q as a s c a l a r c o n t r a c t i o n o f two second rank s p h e r i c a l t e n s o r s , 2 [3.1 . 1 ] where b ( t ) i s both symmetric and t r a c e l e s s ( i .e.<c?/ n\ n-.-(t)> = 0 ) . [3.1. 2 ] k=-2 where t h e b a t h o r l a t t i c e f u n c t i o n s a r e d e f i n e d as 5 l J - ( f c / B ) 1 ' ^ . - ^ [3.1.3] and t h e s p i n o p e r a t o r s a r e d e f i n e d as V° = - ( 8 / 3 ) 1 / 2 [ I ^ I J z - l ( l { l J + f l j ) ] [3.1.4a] i j • v ± z [3.1.4b] [ 3 . 1 . 4 c ] -44-The l e n g t h o f t h e v e c t o r from t h e i t o t h e j n u c l e u s i s d e n oted by r . . and t h e p o l a r a n g l e s s p e c i f y i n g t h e d i r e c t i o n o f r . . i n t h e l a b o r a t o r y c o o r d i n a t e system ( i n w h i c h t h e components o f I . and I . a r e d e f i n e d ) a r e denoted by ti... F o r a t o t a l system o f N s p i n s ( l i k e o r u n l i k e ) , t h e r e a r e N ' d i p o l a r c o u p l i n g s ( r e l a x a t i o n pathways) and t h e t o t a l d i p o l a r H a m i l -2 torn'an i s w r i t t e n as N Wt)= L * ( D ) 1 j W SZ<?C(*)- ^3.1.5] i < j c The a p p r o p r i a t e s p e c t r a l d e n s i t i e s t o be employed i n t h i s s e c t i o n a r e e a s i l y o b t a i n e d from t h e s p e c t r a l d e c o m p o s i t i o n ( F o u r i e r T r a n s f o r m a -t i o n ) o f E q u a t i o n [ 2 . 3 . 1 0 ] . O n l y t h e a u t o - c o r r e l a t i o n f u n c t i o n s a r e r e t a i n e d i n c a l c u l a t i o n s p r e s e n t e d i n t h i s c h a p t e r . , That i s , t h e f o l l o w -i n g c o n d i t i o n i s assumed to;be i m p l i c i t l y v a l i d , In g e n e r a l , b oth a u t o - c o r r e l a t i o n and c r o s s - c o r r e l a t i o n s p e c t r a l den-s i t i e s must be c o n s i d e r e d as t h e d i p o l a r f i e l d s o f d i f f e r e n t p a i r w i s e i n t e r a c t i o n s a r e o f t e n c o r r e l a t e d . For t h e p r e s e n t however, E q u a t i o n [3.1.6] i s assumed. The d i p o l a r c r o s s - c o r r e l a t i o n problem i s f u l l y d i s c u s s e d i n C h a p t e r IV. -45-3.2 EFFECT OF ANISOTROPIC MOTION ON T ] AND T^ VALUES 9 C o n s i d e r a system o f N i d e n t i c a l n u c l e i i n e q u i v a l e n t p o s i t i o n s i n N 1 i d e n t i c a l m o l e c u l e s . I g n o r i n g c r o s s - t e r m s , both t h e l o n g i t u d i n a l and t r a n s v e r s e m a g n e t i z a t i o n s decay e x p o n e n t i a l l y . By t h e use o f E q u a t i o n s [2.2.18] and [ 2 . 3 . 1 0 ] , i t i s e a s i l y shown t h a t f o l l o w i n g a e-degree p u l s e ( o f d u r a t i o n t = e/o)-|), < I z ( t ) > - < I Z > T = (cose - l ) < I z > T e x p ( - t / T 1 ) [3.2.1] < I + ( t ) > = s i n G < I z > T e x p ( - t / T 2 ) . [3.2.2] As we a r e i n t e r e s t e d i n t h e i r r e v e r s i b l e decay o f I + ( o r e q u i v a l e n t l y I , I , o r I ) , t h e r a p i d o s c i l l a t o r y t e r m , e x p ( - i u ) n ( t + t 0 ) ) , has been x y u a i g n o r e d on t h e r i g h t hand s i d e o f E q u a t i o n [ 3 . 2 . 2 ] . The d e f i n e d time c o n s t a n t s a r e T" 1 = ( N - I M - J 1 ' - 1 ^ ) + 4J 2'- 2(2w 0)) [ 3 . 2 . 3 ] T"1 = ( N - 1 ) ( 3 J 0 0 ( 0 ) - S J 1 ' " 1 ^ ) + 2 J 2 ' " 2 ( 2 o ) 0 ) ) / 2 . [3.2.4] The s p e c t r a l d e n s i t i e s a p p e a r i n g i n t h e s e e q u a t i o n s a r e d e f i n e d as Ok'"k(co) = (-l) k£ 2(16^) - 1 j ( 3 c o s 2 3 - 1 ) 2 F Q ( D A , D^,U)) + 1 2 s i n 2 3 c o s 2 g F 1 ( D ± , D^w) 3 s i n 4 B F 2 ( D 1 5 D^ .w)} [3.2.5] + where ?n(D£ D u, u ) = [60^+ n 2 ( D u - D x)]/[(6D ±+ n 2 ( D R - D j ) 2 + u , 2 ] . [3.2.6] The s t a t e m e n t , "N i d e n t i c a l n u c l e i i n e q u i v a l e n t p o s i t i o n s " , i m p l i e s t h a t 5.: • = =5 and B .. = e., ., = 8. Hence, i n t h e above e q u a t i o n s , t h e -46-s u p e r f l u o u s s u b s c r i p t s i j , i j U,c) a r e dropped i n a l l r e l e v a n t terms. B e t a i s t h e e n c l o s e d a n g l e d e f i n e d by t h e p r i n c i p a l d i f f u s i o n a x i s and t h e i n t e r n u c l e a r v e c t o r . F o r a methyl group u n d e r g o i n g h i n d e r e d r o t a t i o n s about i t s t r i a d a x i s , N = 3 , g = TT / 2 , and D u and D ± c h a r a c t e r -i z e t h e s p i n n i n g o f t h e methyl group and t h e r o t a t i o n a l r e o r i e n t a t i o n o f t h e a x i s about which i t s p i n s r e s p e c t i v e l y ( h e n c e f o r t h , t h e q u a n t i t y D n " W l 1 1 D e 1oosely r e f e r r e d t o as t h e a n i s o t r o p y o f t h e m o t i o n ) . In t h e f o l l o w i n g d i s c u s s i o n , methyl group r e l a x a t i o n w i l l be s p e c i f i c a l l y r e f e r r e d t o because i t i s one o f t h e s i m p l e s t c a s e s where a n i s o t r o p i c m o t i o n a l e f f e c t s a r e u n d o u b t e d l y i m p o r t a n t . However, o t h e r i n t e r p r e t a t i o n s o f t h e p r e s e n t t r e a t m e n t w i l l be o b v i o u s and t h e approach adopted s h o u l d n o t be c o n s t r u e d as a l i m i t a t i o n o f t h e g e n e r a l v a l i d i t y o f t h e c o n c l u -s i o n s . F i g u r e 3.1 shows t h e r e l a x a t i o n b e h a v i o r o f t h e two m o t i o n a l models we w i s h t o compare. The s i m p l e s t model i s t o t r y t o f i t t h e e x p e r i m e n t a l T-| o r T 2 d a t a w i t h a s i n g l e " e f f e c t i v e " r o t a t i o n a l c o r r e l a t i o n t i m e , T 2 , u s i n g t h e a s s u m p t i o n t h a t J k ' " k ( w ) = (-1 ) k C 2 ( 4 i r ) " 1 ( 6 D ) ( 3 6 D 2 + c o 2 ) ' 1 [ 3 . 2 . 7 ] where 6D E ( t h e s e r e s u l t s a r e shown as t h e d o t t e d l i n e s i n t h e two g r a p h s ) . The s o l i d l i n e s i n each graph c o r r e s p o n d t o t h e more d e t a i l e d model ( E q u a t i o n [ 3 . 2 . 5 ] ) i n w h i c h t h e i n t e r n a l r o t a t i o n r a t e and a n g l e appear e x p l i c i t l y ( d i f f e r e n t s o l i d l i n e s c o r r e s p o n d t o m o l e c u l e s o f d i f f e r e n t s i z e as i n d i c a t e d ) . The most i n t e r e s t i n g f e a t u r e o f t h e T 1 c u r v e s i s t h a t f o r ma c r o m o l e c u l e s o f m o l e c u l a r w e i g h t g r e a t e r than c. a. 3 0 , 0 0 0 ( c u r v e s r and s) t h e ( s i n g l e , " e f f e c t i v e " ) c o r r e l a t i o n time •47-o b t a i n e d from E q u a t i o n [3.2.7] i s a good a p p r o x i m a t i o n t o t h e a c t u a l c o r r e l a t i o n t i m e f o r t h e r o t a t i o n a l a n i s o t r o p y , x R , where x R = (4D ] ( - 4D.J7 1 I f D„ » D^, then x R - (4D,,)"1'. T h i s f a c t i s e a s i l y deduced from E q u a t i o n [ 3 . 2 . 5 ] . D e f i n i n g x c = ( 6 0 ^ ) " ^ , i t i s seen t h a t i f x £ » ^ , x R < < W Q 1 » and w " 1 < < ( T R T C ) ^ 2 ' then J k'~ k(u>) - T r . [3.2.8] F u r t h e r m o r e , i f T , x R >>"^ and x c > > x R , Jk'~k(a>) - T~V2. [ 3 . 2 . 9 ] In e i t h e r c a s e , t h e s p e c t r a l d e n s i t i e s a r e a f u n c t i o n o f t h e r a t e o f i n t e r n a l m o b i l i t y a l o n e ; a t f i r s t s i g h t , a r a t h e r s t a r t l i n g r e v e l a t i o n . -1 1/2 A l s o note t h a t i f T ->• 0 o r u n » ( x D x „ ) , t h e s p e c t r a l d e n s i t i e s a r e K U K C i n d e p e n d e n t o f x R and approach a s y m p t o t i c v a l u e s d i f f e r i n g by a f a c t o r 2 2 o f f o u r ( i . e . (3cos 3 - 1) ) depending on w h i c h o f t h e two i n e q u a l i t i e s i s v i o l a t e d . For v a l u e s o f x £ < < ^ , t h e p l o t s f o r a g i v e n Tc would s i m p l y d e c r e a s e m o n o t o n i c a l l y between t h e s e l i m i t s as t h e a n i s o t r o p y o f t h e m o b i l i t y i n c r e a s e d . I t has o f t e n been s t a t e d t h a t one o b v i o u s drawback t o t h e i n t e r -p r e t a t i o n o f a g i v e n e x p e r i m e n t a l T-j v a l u e i s t h a t t h e s p e c t r a l d e n s i t y a t a nonzero f r e q u e n c y i s a do u b l e v a l u e d f u n c t i o n o f t h e c o r r e l a t i o n t i m e ( i . e . a uniqu e T^ c o u l d be i n t e r p r e t e d i n terms o f two d i f f e r e n t c o r r e l a t i o n t i m e s ) . Add t o t h i s a second degree o f m o t i o n a l freedom, t h e s p e c t r a l d e n s i t y (T^) now becomes a q u a d r u p l e - v a l u e d f u n c t i o n i n x £ and x R . But an i n c r e a s e i n m a g n e t i c f i e l d ( i n c r e a s e o f WQ) w i l l e f f e c -- 4 8 -t i v e l y s h i f t t h e l e f t - h a n d p o r t i o n o f t h e T, v e r s u s T d c u r v e s t o t h e I C 3 K r i g h t w h i l e l e a v i n g t h e r i g h t - h a n d p o r t i o n f i x e d ; t h u s by o b s e r v i n g whether t h e e x p e r i m e n t a l T-j d e c r e a s e s o r remains t h e same upon i n c r e a s i n g t h e m a g n e t i c f i e l d , one c o u l d d e c i d e whether t h e l o n g e r (x , x R > co"1) o r s h o r t e r ( x , x R < COQ1 ) c o r r e l a t i o n t i m e s a r e c o r r e c t . (However, i f T c >>u*0 > > T R ' n o dependence w i l l be not e d and t h u s , t h e l a t t e r case may be amb i g i o u s u n l e s s one has a good i d e a o f t h e magnitude o f e i t h e r T c o r T R ^ r o m ''"dependent d a t a . ) F i n a l l y , i t may be noted from t h e f i g u r e t h a t t he u s u a l i d e a t h a t r a p i d i n t e r n a l r o t a t i o n makes r e l a x a t i o n l e s s e f f i c i e n t i s v a l i d o n l y when x c <<UQ1 ; i n t e r n a l r o t a t i o n c l e a r l y makes r e l a x a t i o n more e f f i -c i e n t i n r e g i o n s where (co~x ) - 1 < x D < x ^ ' 1 0 U C K C An immediate i m p o r t a n t f e a t u r e o f t h e l ^ c u r v e s i s t h a t t h e s i m p l i -f i e d model ( E q u a t i o n [3 .2.7] ) g i v e s a v e r y m i s l e a d i n g p i c t u r e . F o r a g i v e n m a c r o m o l e c u l a r s i z e , t h e f i g u r e c l e a r l y shows t h a t t h e " e f f e c t i v e " c o r r e l a t i o n t i m e deduced from t h e broken l i n e w i l l n e v e r d e v i a t e v e r y much from t h e r o t a t i o n a l c o r r e l a t i o n t i m e o f t h e macr o m o l e c u l e i t s e l f , i n c o n t r a s t t o t h e r e s u l t above. T h i s a g a i n can be e a s i l y deduced from E q u a t i o n [ 3 .2 .5 ] . I f x c »u~Ql , t h e n J 0 0 ( 0 ) » - J 1 ' - 1 ^ ) , J 2 ' " 2 ( 2 co Q ) and T" 1 - 3 J 0 0 ( 0 ) . However, i f x £ « c o ~ ] , the n J 0 0 ( 0 ) - - J 1 ' " 1 ^ ) 2 -2 -1 00 - J ' (2<J0Q) and - 10J (0). The v a r i a t i o n between t h e s e extremes i s monotonic and hence, f o r a l l p r a c t i c a l p u r p o s e s , T^ i s dependent o n l y on t h e a r e a under t h e c o r r e l a t i o n f u n c t i o n f o r a l l v a l u e s o f t h e m o t i o n a l p a r a m e t e r s . Examining t h e s t r u c t u r e o f J 0 0 ( 0 ) , i t i s o b v i o u s t h a t i f J 0 0 ( 0 ) = C 2 ( 2 4 T T D X ) - 1 = 5 2 x (4*)" 1, [3.2.10] -49-and i f D u >> DL, J 0 0 ( 0 ) = 5 2 ( 3 c o s 2 B - l ) 2 ( 9 67rD1)"1. [3.2.11] I f i t i s assumed t h a t 6 = TT/2„ i t i s e a s i l y seen t h a t T c / 4 <_ J 0 0 ( 0 ) 4 T T £ ~ 2 <_ T The l o w e r l i m i t i s approached when x R << x c (D ( l >> D ) and t h e upper l i m i t i s approached when x R •> 0 (D„ -> Dj_). Note t h a t b o t h e x p r e s s i o n s [3.2.10] and [3.2.11] a r e independent o f t h e magnitude o f x R , t h e o r i g i n a l c o n t e n -* t i o n . A l s o n o t e t h a t t h i s argument i s i n d e p e n d e n t o f any i n e q u a l i t i e s c o n c e r n i n g WQ. ( T h i s i s not s t r i c t l y t r u e ; i f O J Q X c - 1, i t i s n o t pos-s i b l e t o w r i t e T^1 <= J ^ ( 0 ) . T h i s f a c t e x p l a i n s why c u r v e s p, q, r , and s i n t h e T 2 p l o t o f F i g u r e 3.1 cannot be superimposed. However, t h i s i s o f m i n o r c o n c e r n as c l e a r l y shown i n t h i s p l o t . ) Thus i t i s seen t h a t T-| f o r a methyl group on a macromolecule,•Equa-t i o n [ 3.2.7] may be used t o o b t a i n a good e s t i m a t e o f t h e r a t e o f i n t e r n a l r o t a t i o n o f t h a t m e t hyl group. L i k e w i s e , a T 2 measurement may be used t o o b t a i n an e s t i m a t e o f t h e o v e r a l l m o t i o n . A l s o , as can be seen from * An i n t e r e s t i n g f e a t u r e o f t h i s r e s u l t i s t h a t a l t h o u g h T^1 ( J u u ( 0 ) ) i s i n d e p e n d e n t o f t h e magnitude o f t h e m o t i o n a l asymmetry, i t does depend on t h e a n g l e between t h e r e l a x a t i o n v e c t o r and t h e p r i n c i p a l d i f f u s i o n a x i s . T h e r e f o r e , i n t h i s c a s e , i t c o u l d be argued t h a t i n d e e d , i n some i n s t a n c e s , t h e r e l e v a n t s p e c t r a l d e n s i t i e s depend on m o l e c u l a r s t a t i c s and o n l y i n d i r e c t l y on t h e dynamics! E x t e n s i o n o f t h i s argument i m p l i e s t h a t t h i s s t a t i c g e o m e t r i c a l f a c t o r m i ght more c o r r e c t l y be i d e n t i f i e d w i t h t h e s t a t i c i n t e r a c t i o n c o n s t a n t and n o t w i t h any d y n a m i c a l p a r a -meter. Hence, i n t h i s c a s e , a n i s o t r o p i c m o t i o n may be s a i d t o s i m p l y d e c r e a s e t h e " e f f e c t i v e " i n t e r a c t i o n c o n s t a n t . 1 1 However t h i s i s l i t t l e more t h a n a mnemonic d e v i c e and one s h o u l d a l w a y s speak i n terms o f t h e " e f f e c t i v e " s p e c t r a l d e n s i t y . -50-E q u a t i o n [ 3 .2 .11 ] , measurements p r o v i d e unique i n f o r m a t i o n i f t h e o v e r a l l m o b i l i t y (and t h e i n t e r a c t i o n c o n s t a n t s ) can be d e t e r m i n e d by a s e p a r a t e means. I n t h i s c a s e , a 1^ measurement c o u l d be used t o o b t a i n t h e average a n g l e a t wh i c h t h e i n t e r n a l r o t a t i o n o c c u r s . I t s h o u l d a l s o be i n s t r u c t i v e l y n oted t h a t a l l o f what has been s a i d can more d e m o n s t r a t i v e l y be deduced from a g r a p h i c a l c o n s t r u c t i o n o f the two c o n t r i b u t i n g L o r e n t z i a n s p e c t r a l e n v e l o p e s . As an example, see F i g u r e 3.2 f o r one such c o n s t r u c t i o n and i n t e r -p r e t a t i o n . A r e d u c e d , f r e q u e n c y w e i g h t e d , e x p o n e n t i a l s p e c t r a l d e n s i t y , 2 2-1 coJ(to.Tg) = toxgO + igco ) , i s p l o t t e d as a f u n c t i o n o f w i g . F o r the case c o n s i d e r e d i n t h i s s e c t i o n , t h e e f f e c t i v e s p e c t r a l d e n s i t y a t any g i v e n f r e q u e n c y i s g i v e n as a sum o f two c o n t r i b u t i o n s , t o j ( x D , x .co) = t o ( j 1 (co,T ) K C C + 3 j " (CO,T r ) ) /4 ; J ' ( C O , T C ) = T c ( 1 + T 2 C O 2 ) - 1 and j " ( t o , x R ) = T r ( 1 + x R c o 2 ) - 1 . These two c o n s t i t u e n t s p e c t r a l d e n s i t i e s a r e shown by t h e s o l i d c u r v e s where t h e x - a x i s i s d e f i n e d by t h e i d e n t i f i c a t i o n s , T Q E x c and i g E T R r e s p e c t i v e l y . I t i s assumed t h a t T d << x . The d o t t e d l i n e i s t h e l i m i t -K C i n g c a s e where a uniq u e c h a r a c t e r i z e s t h e c o r r e l a t i o n f u n c t i o n ( xg E T ^ ) . I t i s e a s i l y seen t h a t i f cogXc < 1, any i n t e r n a l m o b i l i t y ( f o r a g i v e n cog), w i l l c o n t r i b u t e l i t t l e , i f any, s p e c t r a l a m p l i t u d e . Hence, t h e c l i c h e , " i n t e r n a l m o b i l i t y always d e c r e a s e s t h e r e l a x a t i o n r a t e . " F o r COQT c > 1, t h i s i s g e n e r a l l y no l o n g e r v a l i d . F or example, i f cogX c = 10 and 1/10 < COQT r < 10, t h e n i n t e r n a l m o b i l i t y w i l l dominate t h e c o m p o s i t e s p e c t r a l d e n s i t y . A l t h o u g h we have j u s t p r e s e n t e d an i n t e r p r e t a t i o n o f j ( c o g , t g ) , i n t e r p r e t a t i o n s o f j ( 0 , x Q ) and j(2cog,Xg) can be s i m p l y deduced by l i k e r e a s o n i n g . I t cannot be s t r e s s e d t o o s t r o n g l y t h a t s p e c t r a l d e n s i t y g r a p h i c a l c o n s t r u c t i o n s such as F i g u r e 3.2 c o n t r i b u t e g r e a t l y -51-t o a p h y s i c a l u n d e r s t a n d i n g o f t h e " p s e u d o - c o n t r a d i c t o r y " r e s u l t s w h i c h a r i s e f o r s l o w , a n i s o t r o p i c m o d u l a t i o n s o f s p i n - i n t e r a c t i o n s . F o r u n l i k e n u c l e a r s p i n s I and S, t h e same p l o t s can be g e n e r a t e d and show much o f t h e same b e h a v i o r . 1 0 ' 1 2 There a r e some c o m p l i c a t i o n s i n t r o d u c e d i n t h i s c a s e however. F i r s t l y , c r o s s - r e l a x a t i o n may make t h e p r e c i s e d e f i n i t i o n o f a unique T-j o r T 2 i m p o s s i b l e . S e c o n d l y , i f t h e s p i n s a r e o f t h e same n u c l e a r s p e c i e s , the l o n g i t u d i n a l r e l a x a t i o n i s s e n s i t i v e t o t h e s p e c t r a l d e n s i t y a m p l i t u d e near z e r o f r e q u e n c y w h i c h causes t h e b e h a v i o r o f T, t o mimic t h a t o f T 0. -52-FIGURE 3.1: P l o t s o f p r o t o n ( t o p g r aph) and (bottom g r a p h ) v e r s u s t h e r e c i p r o c a l r o t a t i o n a l c o r r e l a t i o n t i m e f o r a methyl group on a m a c r o m o l e c u l e . Broken c u r v e s a r e o b t a i n e d from E q u a t i o n [ 3 . 2 . 7 ] ; t h e x a x i s t i m e argument i s i n t e r p r e t e d as i n t h e s e e q u a t i o n s . F o r t h e s o l i d c u r v e s , E q u a t i o n [3.2.5] y i e l d s s e p a r a t e l i n e s ( p , q , r , and s) f o r m a c r o m o l e c u l e s o f r e s p e c t i v e x c = I O " 8 , 5 , 1 0 " 8 , 1 0 " 7 , 5 , and 1 0 " 7 s e c ; t h e x a x i s f o r t h e s o l i d c u r v e s i s t o be i n t e r p r e t e d as i g E T r where x R c h a r a c t e r i z e s m o t i o n about t h e t r i a d a x i s and x c r o t a t i o n 8 -1 p e r p e n d i c u l a r t o t h i s a x i s . In a l l p l o t s , = 2-n- x 10 s e c " o and r„ u = 1.78 A . -54-FIGURE 3.2: P l o t o f a r e d u c e d , f r e q u e n c y w e i g h t e d , s p e c t r a l d e n s i t y c u r v e v e r s u s t h e u n i t l e s s q u a n t i t y , cox^. The upper d o t t e d c u r v e 2 2-1 c o r r e s p o n d s t o o>j = 10x2(1+ u> T^) w i t h t h e i d e n t i f i c a t i o n XQ E X 2 - The two s o l i d c u r v e s c o r r e s p o n d t o coj = ( 3 / 4 ) C J X R x (1 + w 2 x 2 ) - 1 and coj = (1/4)O>T (1 + ( A 2 ) - 1 w i t h t h e K C C a n a l o g o u s i d e n t i f i c a t i o n o f t h e g e n e r a l i z e d c o r r e l a t i o n t i m e X Q , w i t h x R and x c r e s p e c t i v e l y . - 5 5 --56-3.3 EFFECT OF ANISOTROPIC MOTIONS ON AND_T 2 RATIOS I t w i l l be n o t i c e d t h a t t o o b t a i n n u m e r i c a l v a l u e s f o r t h e c o r r e -l a t i o n t i m e s , t h e d i p o l a r i n t e r a c t i o n c o n s t a n t , £ , must be known. However, a r a t i o o f r e l a x a t i o n measurements a t two f r e q u e n c i e s m i g h t be e x p e c t e d , i n p r i n c i p l e , t o a l l o w t h e i n t e r a c t i o n c o n s t a n t and c o r r e l a t i o n t i m e t o 1 1 ft be found i n d e p e n d e n t l y . The e f f e c t s o f a n i s o t r o p i c m o t i o n on T ^ ( B Q ) / T ^ ( B Q ) , T ^ C B Q J / T ^ C B Q ) , and T-j (BQ)/T2(BQ) a r e shown i n F i g u r e s 3.3, 3.4, and 3.5 r e s p e c t i v e l y . 8 -1 For t h e sake o f d e f i n i t e n e s s , BQ = 2.35T (COQ = 2ir x 10 s e c " ) and B~ = 5.17T (U)Q = 2 . 2 U Q), t h e s e b e i n g t y p i c a l o p e r a t i n g f r e q u e n c i e s o f p r o t o n NMR s t u d i e s . The mode o f p l o t t i n g h o p e f u l l y f a c i l i t a t e s c o m p a r i s o n w i t h F i g u r e 3.1. In each o f t h e s e p l o t s , t h e a p p r o p r i a t e r a t i o i s p l o t t e d as a f u n c t i o n o f x R = T 0 - ( = ( 4 D U - 4 D J l ) " 1 ) f o r T = (SD±)~'[ = 10 r? , 1 0 " 8 * 5 , -8 -7.5 l.CJ "andilO - S e c . ( c u r v e s p, q, r , and s r e s p e c t i v e l y ) . The d o t t e d l i n e i s t h a t o b t a i n e d u s i n g E q u a t i o n [3 . 2.7] w i t h X Q now d e f i n e d as T^' C o n s i d e r f i r s t , t h e T^ r a t i o p l o t , F i g u r e 3.3. I t can be e a s i l y p r a t i o n a l i z e d t h a t t h e maximum v a l u e o f t h i s r a t i o i s (B Q / B Q ) and i s a s y m p t o t i c a l l y approached when T , T d >> t o l 1 o r x > c o l 1 and co" 1 < / x x D . C K U C U U C K The minimum v a l u e o f u n i t y i s approached when JQ « c o " 1. However, i t i s noted t h a t even f o r x £ > c o ^ 1, i t i s p o s s i b l e t o approach t h e v a l u e o f u n i t y f o r c e r t a i n r a t e s o f i n t e r n a l m o b i l i t y . A n a l o gous p l o t s o f T2(B Q ) /T2(B Q) and T^ ( B g ) / ^ ^ ) a r e shown i n F i g u r e s 3.4 and 3.5. I t i s i n t e r e s t i n g t o comment t h a t whereas T-. r a t i o s a r e n o t d o u b l e v a l u e d i n x 2 , t h e T 2 r a t i o s a r e . Thus we see t h e c u r i o u s d i s t i n c t i o n : T j s a r e d o u b l e v a l u e d , t h e i r r a t i o s a r e n o t ; T2 r a t i o s a r e d o u b l e v a l u e d , T i s a r e not (compare d o t t e d l i n e s i n F i g u r e s 3.1, 3.3, and -57-3.4). O t h e r c h a r a c t e r i s t i c s o f t h e s e p l o t s a r e s e l f - e v i d e n t and w i l l not be e l a b o r a t e d upon a t f u r t h e r l e n g t h . The i m p o r t a n t c o n c l u s i o n a r r i v e d a t from t h e s e p l o t s i s t h e f a c t t h a t t h e p r e s e n c e o f i n t e r n a l f l e x i b i l i t y ( o r more g e n e r a l l y , m o t i o n a l asymmetry) makes a d i r e c t i n t e r p r e t a t i o n o f s p e c t r a l d e n s i t y r a t i o s o f d u b i o u s w o r t h . A l s o , t h e f a c t t h a t t h e r e may be a 10% e r r o r i n t h e ex-p e r i m e n t a l r a t i o s (as w e l l as b e i n g d o u b l e v a l u e d ) adds t o t h e problem o f a m e a n i n g f u l i n t e r p r e t a t i o n . F u r t h e r m o r e , f o r t h i s t e c h n i q u e t o be o f any v a l u e , a t l e a s t one o f t h e two c o m p o s i t e s p e c t r a l d e n s i t i e s must be n e i t h e r b l a c k nor w h i t e , t h a t i s , C J Q T C - 1 o r U Q T R - 1. I t m i g h t be n o t e d t h a t u s i n g a p a i r o f f r e q u e n c y dependent r e l a x a t i o n t i m e s t o de-t e r m i n e t h e r e l a x a t i o n p a rameters may be viewed as t h e s i m p l e s t o f a l l p o s s i b l e NMR r e l a x a t i o n s p e c t r o s c o p y e x p e r i m e n t s - a v e r y p o w e r f u l t e c h -n i q u e i n i t s own r i g h t . 1 7 -58-FIGURE 3.3: P l o t s o f (220MHz)/T ](100MHz) as a f u n c t i o n o f m o t i o n a l a n i s o t r o p y (TQ =T^ = [4D I ( -4D X ] _ 1 ) f o r f o u r d i f f e r e n t v a l u e s o f T C ; T c = I O - 9 , I O " 8 , 5 , I O " 8 , and 1 0 " 7 , 5 s e c . ( c u r v e s p , q , r , and s r e s p e c t i v e l y ) . The d o t t e d c u r v e c o r r e s p o n d s t o i s o t r o p i c m o t i o n ( x n =T 9). -60-FIGURE 3.4: P l o t s o f T 2(220MHz)/T 2(100MHz) as a f u n c t i o n o f m o t i o n a l a n i s o t r o p y ( x Q = T R = [4D„ -4D^ ] ~ 1 ) f o r f o u r d i f f e r e n t v a l u e s o f x ; x = 1 0 " 9 , I O - 8 " 5 , 1 0 ~ 8 , and I O " 7 ' 5 s e c . c c ( c u r v e s p , q , r , and s r e s p e c t i v e l y ) . The d o t t e d c u r v e c o r r e s p o n d s t o i s o t r o p i c m o t i o n ( x n =x 9 ) . iq 2 $2 eg ^ Q (°CO)l5L(°CO>I -62-FIGURE 3.5: P l o t s o f T ](100MHz)/T 2(100MHz) as a f u n c t i o n o f m o t i o n a l a n i s o t r o p y ( Xg = x R E [ 4D ( j -4D^ ] _ 1 ) f o r f o u r d i f f e r e n t v a l u e s o f x : x = 1 0 " 9 , I O " 8 - 5 , 1 0 " 8 , and I O " 7 ' 5 s e c . c c ( c u r v e s p , q , r , and s r e s p e c t i v e l y ) . The d o t t e d c u r v e c o r r e s p o n d s t o i s o t r o p i c m o t i o n ( x n E X 9 ) . -64-3.4 EFFECT OF ANISOTROPIC MOTIONS ON NUCLEAR OVERHAUSER ENHANCEMENTS 1 0 In r e c e n t y e a r s , t h e N u c l e a r O v e r h a u s e r E f f e c t (NOE) has proven 19 t o be a v e r y p o w e r f u l t o o l f o r v a r i e d c h e m i c a l i n v e s t i g a t i o n s . Thus f a r , t h e p r i m a r y usage has been r e s t r i c t e d to m o l e c u l a r s t r u c t u r a l and i d e n t i f i c a t i o n s t u d i e s and f o r t h e c h a r a c t e r i z a t i o n o f v a r i o u s r e l a x -a t i o n pathways and r e l a t e d problems i n n u c l e a r m a g n e t i c r e l a x a t i o n . As an i n d i r e c t consequence o f t h e r e c e n t t r e n d toward b i o l o g i c a l NMR s t u d i e s , a n o t h e r usage o f t h e NOE emerges, as i t p r o v i d e s a no v e l method t o o b t a i n t h e q u a n t i t i e s p a r a m e t e r i z n n g t h e dynamics o f m o l e c u l a r p r o c e s s e s . I n t e r e s t i n t h i s f a c e t o f t h e NOE was a r o u s e d by r e c e n t homonuclear 20 Ove r h a u s e r s t u d i e s performed by A. A. Bothner-By and c o - w o r k e r s . R e l e -v a n t d i s c u s s i o n s o f t h e m o t i o n a l usage o f t h e NOE and e x p e r i m e n t a l ex-amples can be found i n t h e s e and r e l a t e d p a p e r s . C o n s i d e r an ensemble o f s p i n systems composed o f two n e c e s s a r i l y n o n e q u i v a l e n t s p i n o n e - h a l f n u c l e i s u b j e c t e d t o a l a r g e Zeeman f i e l d ; t h e ensemble b e i n g i n n o n e q u i l i b r i u m w i t h t h e s u r r o u n d i n g s . I t can e a s i l y be d e r i v e d from a Solomon-type t r e a t m e n t o r a more s o p h i s t i c a t e d d e n s i t y o p e r a t o r t r e a t m e n t , t h a t t h e time dependence o f t h e d e v i a t i o n m a g n e t i z a -t i o n s o f t h e two s p i n s , x [3.4.1a] [3.4.1b] obey the e q u a t i o n o f m o t i o n , ( d / d t ) Y = - © Y . [3.4.2] -65-/ i \ / I / T ! I / T , I S The v e c t o r Y i s d e f i n e d as I <-x l a n d i9 = I C T Q I. The expres-s i o n s f o r t h e elements o f © a r e , 1/T* E J 0 0 (u) I - a) s)/3 - J 1 ' " 1 ^ ) + 2 J 2 ' " 2 ( ( o I + u>s) [3.4.3a] 1 /T | S E - J 0 0 ( W i - oi s)/3 + 2 J 2 ' " 2 ( U l + ais) [3.4.3b] SI s w i t h T-| and T-j o b t a i n e d upon t h e exchange o f t h e a p p r o p r i a t e i n d i c e s . The v a r i o u s s p e c t r a l d e n s i t i e s a p p e a r i n g i n E q u a t i o n s [3.4.3] a r e assumed t o be a d e q u a t e l y d e s c r i b e d by E q u a t i o n [ 3 . 2 . 5 ] . The unique s o l u t i o n t o E q u a t i o n [3.4.2] i s dependent upon t h e i n i t i a l p r e p a r a t i o n o f t h e s p i n s y s t e m , a f a c e t o f t h e NMR e x p e r i m e n t l e f t t o th e d i s c r e t i o n o f t h e e x p e r i m e n t a l i s t . However, i n t h i s s i m p l e example, we w i s h t o c o n s i d e r o n l y t h e c l a s s i c s t e a d y s t a t e N u c l e a r O v e r h a u s e r E f f e c t . Assuming s p i n S i s s a t u r a t e d and t h e i n t e n s i t y o f s p i n I i s o b s e r v e d , t h e s t e a d y s t a t e boundary c o n d i t i o n s a r e , ( d / d t ) Y = 0 [3.4.4a] Y =( . [3.4.4b] The f r a c t i o n a l enhancement f a c t o r , n (n = l ^ t e a d y s t a t e / < I z > T ) , o f t h e ob s e r v e d s i g n a l can now be d e r i v e d from t h e s o l u t i o n o f E q u a t i o n [3.4.2] s u b j e c t t o t h e s t i p u l a t e d c o n d i t i o n s o f E q u a t i o n [ 3 . 4 . 4 ] : n = Y s Y j 1 ^ 0 0 ^ ! - w s ) - 6 J 2 ' " 2 ( a i I + w s ) ) ( - J 0 0 ( w r w ) + S J 1 ' " 1 ^ ) - 6 J 2 ' " 2 ( W l + u> s)) _ 1. [ 3 . 4 . 5 ] Obvious from E q u a t i o n [3.4.5] i s t h e f a c t t h a t n i s dependent upon t h e r a t i o o f u n l i k e c o m b i n a t i o n s o f s p e c t r a l d e n s i t i e s , a f a c t most p r o m i s i n g -66-f o r t h o s e s e e k i n g m i c o r o s c o p i c m o t i o n a l i n f o r m a t i o n from a m a c r o s c o p i c NMR o b s e r v a b l e . F u r t h e r i n s p e c t i o n shows t h a t i f t h e s p e c t r a l d e n s i t y i s n e i t h e r b l a c k nor w h i t e (which i s o f e n the c a s e f o r b i o l o g i c a l l y i n t e r e s t i n g m o l e c u l e s a t p r e s e n t e x p e r i m e n t a l f i e l d s t r e n g t h s ) , t h e n i n d e e d , t h e measurement o f n w i l l i n p r i n c i p l e y i e l d a t o o l t o q u a n t i t a t e m o l e c u l a r m o b i l i t y . As m ight be e x p e c t e d , t h e usage o f such i n f o r m a t i o n i s i n many ways analogous t o the i n f o r m a t i o n g a t h e r e d from t h e more f a m i l i a r s p e c t r a l d e n s i t y r a t i o s such as T^/T^ r a t i o s , ( B Q ) / T ^ ( B Q ) r a t i o s , o r T ^ B i J / l ^ B g ) r a t i o s b r i e f l y m entioned i n t h e l a s t s e c t i o n . U n f o r t u n a t e l y , as r e p e a t e d l y e m p h a s i z e d , a d e c i p h e r i n g o f t h e NMR m a c r o s c o p i c o b s e r v a b l e s i n t o a d e t a i l e d ( o r even m e a n i n g f u l ) d e s c r i p t i o n o f m o l e c u l a r m o b i l i t y , i s o f t e n c l o u d e d and ambiguous and must be a p p r o -ached w i t h c a r e . D i s c u s s i o n o f t h i s problem has appeared i n the l i t e r a t u r e i n c o n n e c t i o n w i t h i n t e r p r e t a t i o n o f r e l a x a t i o n t i m e s and r a t i o o f r e l a x -9 a t i o n t i m e s . We s h a l l now e x t e n d t h e s e c o n s i d e r a t i o n s t o t h e i n t e r p r e -t a t i o n o f n u c l e a r O v erhauser enhancements. D o d d r e l l e t . a l . 1 0 have p r e s e n t e d a s i m i l a r t r e a t m e n t t o t h e problem o u t l i n e d i n t h i s s e c t i o n , and e x c e p t f o r changes i n f o r m a l i s m , i s com-p l e t e l y e q u i v a l e n t . However, t h e i r r e s u l t s were s p e c i a l i z e d t o a compu-t a t i o n a l h e t e j ^ o n u c l e a r NOE s t u d y where s p i n S i s a p r o t o n and s p i n I , a 13 C n u c l e u s . In t h e s p e c i a l i z e d case c o n s i d e r e d h e r e , i t i s assumed t h a t t h e two n u c l e i a r e o f t h e same s p e c i e s (YJ = Y$) J but t h a t t h e i r r e s o n a n t f r e q u e n c i e s a r e s e p a r a t e d by a t l e a s t a few l i n e w i d t h s . The v a l i d i t y o f t h e a p p r o x i m a t i o n s , J 0 0 ^ - u>s) - J 0 0 ( 0 ) [3.4.6a] -67-1,-1 [3.4.6b] (coj + t o s ) - J 2 ' 2 ( 2 u 0 ) [ 3 . 4 . 6 c ] s u f f i c e s t o c o m p u t a t i o n a l l y d e f i n e t h e homonuclear a p p r o x i m a t i o n as i m p l i e d i n t h i s s e c t i o n . In F i g u r e 3.6, t h e o b s e r v a b l e n o r m a l i z e d f r a c t i o n a l l i n e - i n t e n s i t y , 1 + n, i s p l o t t e d f o r ? a - p a i r ' o f s p i n s r e o r i e n t i n g i s o t r o p i c a l l y ( i . e . t h e mot i o n i s c o m p l e t e l y c h a r a c t e r i z e d by a s i n g l e d i f f u s i o n c o n s t a n t ) . A g a i n , we s h a l l i d e n t i f y = (6D) " 1 . To p r e s e n t t h e r e s u l t s i n a f a s h i o n w h i c h i s i n d e p e n d e n t o f a p a r t i c u l a r e x p e r i m e n t a l Zeeman f i e l d , t h e d i m e n s i o n -l e s s q u a n t i t y , WQT2, i s chosen as t h e u n i t o f m o b i l i t y . The a s y m p t o t i c v a l u e s i n both t h e f a s t m o t i o n regime UQT2 << 1; 00 1-1 2 -2 J (0) - -J ' (COQ) - J ' (2COQ)) and slow m o tion regime (WQT2 >> 1; fin i i ? ? J u u ( 0 ) » -J'' (OJ 0) - J ' (2u 0 ) ) a r e e a s i l y p r e d i c t e d from E q u a t i o n [3 .4 .5] t o be 1.5 and 0.0 r e s p e c t i v e l y . The i n t r i g u i n g p r e d i c t i o n i n the homo-n u c l e a r c a s e i s t h e c o m p lete d i s a p p e a r a n c e o f the o b s e r v e d t r a n s i t i o n i n t h e s l o w m o t i o n l i m i t where th e m o t i o n a l l y a s s i s t e d mutual but oppo-s i t e f l i p s o f the two s p i n s c o m p l e t e l y dominate the r e l a x a t i o n t r a n s i t i o n o f Am = ±1 mutual f l i p s . O b s e r v a t i o n o f e i t h e r extreme v a l u e w i l l o f c o u r s e , o n l y s i g n i f y whether t h e m o t i o n i s " s l o w " o r " f a s t " . O b s e r v a t i o n o f an i n t e r m e d i a t e v a l u e can be d i r e c t l y t r a n s l a t e d i n t o a q u a n t i t a t i v e measure o f m o b i l i t y . However, i t i s o f paramount i m p o r t a n c e t o note t h a t t h i s s t r a i g h t f o r w a r d approach presumes i s o t r o p i c m o t i o n (and o f c o u r s e , o n l y i n t r a m o l e c u l a r d i p o l a r r e l a x a t i o n ) . As mentioned p r e v i o u s l y , t h e s i m p l e s t e x t e n s i o n o f t h i s p i c t u r e assumes motions c h a r a c t e r i z e d by two d i s t i n c t d i f f u s i o n a l c o n s t a n t s . -68-F i g u r e 3.7 i s a t o p o g r a p h i c a l ( c o n t o u r ) p l o t o f the n o r m a l i z e d f r a c t i o n a l l i n e - i n t e n s i t y as a f u n c t i o n o f o v e r a l l r o t a t i o n ( w g T c ) and t h e a n i s o -t r o p i c f l e x i b i l i t y (CUQT r). In t h i s s e c t i o n , we s h a l l a r b i t r a r i l y i d e n t i f y T r as (D„ - D J " 1 . As seen i n E q u a t i o n [ 3 . 2 . 5 ] , n w i l l depend e x p l i c i t l y upon t h e a n g l e 3. Three i l l u s t r a t i v e v a l u e s o f t h i s a n g l e a r e p r e s e n t e d i n F i g u r e 3.7. The a n g l e 3 = 54° was chosen t o i l l u s t r a t e t h e p a r t i c u l a r b e h a v i o r r e s u l t -i n g as 3 approaches t h e NMR magic a n g l e (3 m ag.j c = c o s ~ V 3 _ 1 54.7°). The p l o t f o r 3 = 36° t y p l i f i e s any p l o t f o r wh i c h 20° < 3 < 45° and t h e p l o t f o r 3 = 90° t y p l i f i e s t h e range 60° < 3 < 90°. N o t i c e t h a t o n l y s u b t l e d i f f e r e n c e s e x i s t between t h e s e two a n g u l a r r a n g e s . F u r t h e r m o r e , f o r 0° < 3 < 20°, t h e r e i s v e r y l i t t l e d e v i a t i o n from F i g u r e 3.6 f o r any r a t e o f i n t e r n a l r o t a t i o n ( f o r 3 = 0 ° t h e r e i s no a n i s o t r o p y dependence). F o r a l l r o t a t i o n a n g l e s (save t h o s e a p p r o a c h i n g t h e magic a n g l e ) , e x t r e m e l y r a p i d i n t e r n a l r o t a t i o n ( a s y m m e t r i c a l r e o r i e n t a t i o n ) r e s u l t s i n t h e c o n t o u r s o f c o n s t a n t 1 + n a p p r o a c h t h e l i m i t i n g v a l u e s o b t a i n e d i n t h e absence o f i n t e r n a l ( o r a s y m m e t r i c ) m o b i l i t y (compare a s y m p t o t i c v a l u e s w i t h t h o s e seen i n F i g u r e 3.6). T h i s f a c t i s t r u e r e g a r d l e s s o f t h e mag-n i t u d e o f ..the o v e r a l l m o t i o n . T h i s r a t h e r s u r p r i s i n g r e s u l t i s t o be c o n t r a s t e d w i t h t h e r e s u l t s o b t a i n e d i n r e f e r e n c e 10. A t i n t e r m e d i a t e r a t e s , t h e d e s c r i p t i o n becomes much more c o m p l i c a t e d due t o t h e i d e n t i t y s t r u g g l e between t h e v a r i a b l e s UQ, D_^(T ) , and (Tr), as r e v e a l e d by a r a t h e r l e n g t h y p e r u s a l o f E q u a t i o n [ 3 . 2 . 2 ] . F i g u r e 3.8 f a c i l i t a t e s t h i s a n a l y s i s . T h i s f i g u r e p l o t s c o n t o u r s o f t h e d i f f e r e n c e v a l u e s (Arotn) o f N0E enhancements. These d i f f e r e n c e v a l u e s a r e o b t a i n e d by -69-f i r s t c a l c u l a t i n g n assuming i n t e r n a l r o t a t i o n ( o f t h e magnitude i n d i c a t e d by t h e v e r t i c a l a x i s ) and the n s u b t r a c t i n g t h e v a l u e c a l c u l a t e d by assum-i n g no i n t e r n a l r o t a t i o n . T h i s q u a n t i t y i s t h e computed as a f u n c t i o n o f th e o v e r a l l i s o t r o p i c m o b i l i t y w h i c h i s p l o t t e d as t h e h o r i z o n t a l i n d e p e n -dent v a r i a b l e . T h e r e f o r e , t h i s n o n - n e g a t i v e q u a n t i t y i s o f t h e s i m p l e f o r m , \ o t n E \ > D j . ~ • [ 3 ' 4 - 7 ] F i g u r e 3.8 assumes t h a t 3 = 90°, but as mentioned e a r l i e r , t h i s w i l l g i v e the' q u a l i t a t i v e b e h a v i o r o v e r a l a r g e range o f i n t e r n a l g e o m e t r i e s . The f u l l meaning o f F i g u r e s 3.6-8 i s now d e v e l o p e d . I f , o n e n a i v e l y i n t e r p r e t s an Ov e r h a u s e r enhancement i n terms o f F i g u r e 3.6, how w i l l i t d i f f e r from a more e x a c t i n t e r p r e t a t i o n u s i n g F i g u r e 3.7? F i g u r e 3.6 shows t h a t between the l i m i t s 0.2 < n+1 < 1.3 ( o u t s i d e o f t h e s e l i m i t s , t h e NOE e x p e r i m e n t l o s e s i t s u s e f u l n e s s i n the p r e s e n t c o n t e x t ) , A n / A l o g U ^ ^ ) - -2. Hence t h e maximum e r r o r i n t r o -duced i n t o t h e i n t e r p r e t a t i o n o f T c by means o f p l o t 3.6 would o c c u r when U > 0 T r = I O " 0 - 3 ( D u - Dj_ - 2uQ) and UQTC = 1 0 " 0 , 5 {U± * u Q / 2 ) . As seen i n F i g u r e 3.8, A r Q t n would be on t h e o r d e r o f 0.6 r e s u l t i n g i n a 0 3 m i s i n t e r p r e t a t i o n o f x c by a f a c t o r o f 2 (=00 ' ). Any o t h e r p a i r o f d i f f u s i o n a l c o n s t a n t s would r e s u l t i n a s m a l l e r v a l u e o f A r o j . n » hence, a s m a l l e r m i s i n t e r p r e t a t i o n o f x . T h e r e f o r e , e x c e p t f o r a r a t h e r l i m i t e d range o f u n i q u e r o t a t i o n a l g e o m e t r i e s , i n t e r p r e t a t i o n o f NOE enhance-ments i s q u i t e s t r a i g h t f o r w a r d , y i e l d i n g t h e o v e r a l l r o t a t i o n a l d i f f u -s i o n c o n s t a n t w i t h i n a f a c t o r o f two o r l e s s . F i n a l l y , t h e s e p l o t s s u g g e s t a p r o m i s i n g , a l b e i t t e d i o u s , range o f e x p e r i m e n t s i f one i s b l e s s e d w i t h t h e f l e x i b i l i t y o f p e r f o r m i n g NOE -70-e x p e r i m e n t s a t v a r i o u s f i e l d s t r e n g t h s . T h i s c o u l d , i n f a v o r a b l e c i r c u m -s t a n c e s , y i e l d n ot o n l y t h e o v e r a l l r o t a t i o n a l r a t e s , but a l s o t h e r a t e o f i n t e r n a l r o t a t i o n . I n d e e d , t h e m o t i o n a l i n f o r m a t i o n o b t a i n a b l e from t h e NOE e x p e r i m e n t , e s p e c i a l l y when used i n c o n j u n c t i o n w i t h o t h e r s t u d i e s , appears t o be a p r o m i s i n g avenue o f s t u d y . The r e s u l t s p r e s e n t e d i n t h i s s e c t i o n bear a s t r o n g r e s e m b l a n c e t o t h o s e deduced from comparable s t u d i e s o f T 2 ( B Q ) / T 2 ( B 0 ) r a t i o s . A c l o s e r e x a m i n a t i o n o f t h i s b e h a v i o r can be t r a c e d t o t h e f a c t t h a t t h e o b s e r v a b l e parameter i n e i t h e r c a s e i s g i v e n by a r a t i o o f l i n e a r c o m b i n a t i o n s o f s p e c t r a l d e n s i t i e s , each c o m b i n a t i o n c o n t a i n i n g a term s e n s i t i v e s o l e l y t o t h e a r e a under t h e c o r r e l a t i o n f u n c t i o n , J°°(0). S i m i l a r l y , t h e r e s u l t s f o r t h e h e t e r o n u c l e a r O v e r h a u s e r e n h a n c e m e n t s 1 0 do not d i f f e r much from t h e c a l c u l a t i o n s p e r t i n e n t f o r T-| (Bg)/T-j ( B Q) r a t i o s . T h i s i s d i r e c t l y a t t r i b u t e d t o t h e f a c t t h a t n e i t h e r c o m b i n a t i o n c o n t a i n s a term w h i c h can be i n t e r p r e t e d as s i m p l y as "the a r e a under t h e c o r r e -l a t i o n f u n c t i o n " . ( u n l e s s t h e t r i v i a l c a s e o f e x t r e m e - n a r r o w i n g i s assumed). Viewed i n t h i s l i g h t , t h e r e s u l t s d e r i v e d i n t h i s s e c t i o n a r e s i m p l e e x t e n s i o n s o f some v e r y g e n e r a l c o n c e p t s . -71-FIGURE 3.6; P l o t o f t h e n o r m a l i z e d NOE f r a c t i o n a l l i n e i n t e n s i t y , 1 + n» as a f u n c t i o n o f i s o t r o p i c m o b i l i t y o f a p a i r o f homonuclear s p i n s r e l a x e d by d i p o l a r i n t e r a c t i o n s a l o n e . - 7 2 --73-FIQURE 3.7: Con t o u r ( t o p o g r a p h i c a l ) p l o t s o f t h e n o r m a l i z e d NOE f r a c t i o n a l l i n e i n t e n s i t y , 1 + n, as a f u n c t i o n o f b o t h o v e r a l l i s o t r o p i c m o b i l i t y and a n i s o t r o p y i n m o t i o n . T c i s d e f i n e d as ( 6 0 ^ ) " ^ and T r i = (D | ( - D ^ ) - 1 . The dependence o f t h e d i r e c t i o n o f t h e i n t e r n u c l e a r v e c t o r w i t h r e s p e c t t o t h e i n t e r n a l r o t o r a x i s i s d e p i c t e d f o r t h r e e a n g l e s and e x p l a i n e d i n t h e d i s c u s s i o n . C o n t o u r s a r e shown f o r (1+n) = 0.05, 0.25, 0.75, 1.25, and 1.45 -75-FIGURE 3.8: C o n t o u r ( t o p o g r a p h i c a l ) p l o t s o f t h e d i f f e r e n c e v a l u e s ( A r o t n ) o f t h e NOE enhancements ( c a l c u l a t e d assuming a n i s o t r o p i c m o b i l i t y minus t h a t c a l c u l a t e d assuming i s o t r o p i c m o b i l i t y ) as a f u n c t i o n o f " s i z e " and "shape" f o r t h e g i v e n a n g l e o f B = 90°. See E q u a t i o n [3.4.7] and r e l a t e d d i s c u s s i o n f o r f u r t h e r d e t a i l s . C o n t o u r s a r e shown f o r A .n = 0.01, r o t 0.05, 0.2, 0.4, and 0.6. -77-3.5 SUMMARY In t h i s c h a p t e r , we have seen w h i c h c o m b i n a t i o n s o f s p e c t r a l den-s i t i e s a r e probed i n t h e v a r i o u s s i m p l e r e l a x a t i o n e x p e r i m e n t s w h i c h a r e commonly performed. F o r s i m p l i c i t y , o n l y i n t r a m o l e c u l a r d i p o l a r r e l a x -a t i o n has been assumed. However, t h e e f f e c t o f a s ymmetric r e o r i e n t a t i o n on o t h e r i n t r a m o l e c u l a r r e l a x a t i o n mechanisms i s v e r y s i m i l a r t o t h e 21 p r e s e n t t r e a t m e n t . A l t h o u g h we have a n a l y z e d t h e e f f e c t o f one a d d i -t i o n a l degree o f freedom, i n p r i n c i p l e t h i s c o u l d be e x t e n d e d t o t r e a t m u l t i p l e i n t e r n a l r o t a t i o n s o r w h o l l y a n i s o t r o p i c m o t i o n s . F u r t h e r m o r e , we have o n l y c o n s i d e r e d e x p l i c i t l y t w o - s p i n r e l a x a t i o n w i t h t h e i m p l i e d a s s u m p t i o n t h a t more c o m p l i c a t e d s p i n systems w i l l be l i n e a r l y a d d i t i v e i n t h e v a r i o u s p a i r w i s e c o u p l i n g s . I t was shown t h a t J°°(0) ( s i m p l y r e l a t e d t o t h e a r e a under t h e c o r -r e l a t i o n f u n c t i o n ) i s i n d e p e n d e n t o f t h e magnitude o f asymmetry i n t h e m o t i o n a l p arameters and i s s e n s i t i v e t o t h e asymmetry o n l y t h r o u g h a k -k s t a t i c g e o m e t r i c a l f a c t o r . In g e n e r a l , J ' (u >> 0) i s a much more c o m p l i c a t e d f u n c t i o n o f t h e Larmor f r e q u e n c y and t h e r e l a t i v e magnitudes o f t h e c h a r a c t e r i s t i c m o t i o n a l p a r a m e t e r s . I f t h e m o t i o n i s c h a r a c t e r -i z e d by two m o t i o n a l c o n s t a n t s d i f f e r i n g by o r d e r s o f magnitude w i t h t h e s m a l l e r o f t h e s e two b e i n g on a t i m e s c a l e s l o w e r t h a n , but s i m i l a r t o k -k t h a t o f a Larmor p r e c e s s i o n , t h e n o f t e n J ' (cu » 0) i s dependent s o l e l y on t h e degree o f asymmetry i n t h e m o t i o n - i n d i r e c t c o n t r a s t t o J 0 0 ( 0 ) . In g e n e r a l , i t i s shown t h a t t h e i n c l u s i o n o f a n i s o t r o p i c m o t i o n w i l l i n f l u e n c e t o a l a r g e e x t e n t , t h e m o l e c u l a r i n t e r p r e t a t i o n o f c e r t a i n s p e c t r a l d e n s i t y c o m b i n a t i o n s , a l t h o u g h b o t h T0 and homonuclear O v e r h a u s e r -78-enhancements t e n d t o be r a t h e r i n s e n s i t i v e t o a s y m m e t r i c a l s p i n m o b i l i t y due t o t h e i r dependence on J ^ ( - O ) . F u r t h e r m o r e , t h e i n f l u e n c e o f t h e e f f e c t o f m o t i o n s s l o w on ti m e s c a l e o f t h e Larmor f r e q u e n c y g i v e r i s e t o a p p a r e n t c o n t r a d i c t i o n s i f i n t e r p r e t e d i n terms o f c o n v e n t i o n a l "extreme-narrowed" t h i n k i n g . T h i s i n t r o d u c t i o n was i n t e n d e d t o expose v e r y s i m p l e , y e t v e r y p r a c t i c a l a s p e c t s o f s l o w , a n i s o t r o p i c s p i n r e o r i e n t a t i o n . A l l t h e major p o i n t s o f emphasis a r e summarized i n T a b l e s 3.1 and 3.2. T a b l e 3.1 i s a g e n e r a l summary and T a b l e 3.2 s t r e s s e s t h e f a c t t h a t both T^ and T^ r e t a i n d i s t i n c t i v e forms i n v a r i o u s m o t i o n a l l i m i t s i f r e i d e n t i -f i c a t i o n o f i n t e r a c t i o n c o n s t a n t s o r dynamic pa r a m e t e r s a r e c o r r e c t l y made. In t h e next c h a p t e r , a much more s p e c i f i c c a l c u l a t i o n based on the same g e n e r a l i d e a s w i l l be p r e s e n t e d . TABLE 3.1 1 1 R e l a x a t i o n P a r a m e t e r s i n V a r i o u s M o t i o n a l L i m i t s : P a r t 1. ] < D ± co 0 > D A <o( < DM D A = D „ E D D 1 « D „ D ^ E D D 1 < D . . < U 0 D | | D L « W2 D N D A » a ) 2 XJ°°(0) f ( D ) f(DL,e) f ( D ) f ( D A , B ) f ( D i 5 3 ) f(DA,e) 7 0 0 / n\ f ( D ) f ( D x , B ) f ( D ) f ( D „ , 3 ) f ( D „ , 3 ) f ( D A , 3 ) 1/6D (3cos 3- 1 ) 2 / 2 4 D 1 12D/5co 2 2 2 6 s i n 3Dn/5coQ 2 3 s i n 3/ l c 1 C . 2ns 16D H n e - l S s m 3) ( 3 c o s 2 3 - l ) 2 / 8 0 D L 3 ( 3 c o s 2 3 - l ) \ / 5 w 2 x T 2 X T - 1/20D ( 3 c o s 2 3 - l ) 2 / 8 0 D x ( 3 c o s 2 3 - l ) 2/80Dj_ homonuclear Overhauser enhancement § 1/2 1/2 -1 -1 -1 -1 h e t e r o n u c l e a r O v e r h a u s e r YC72YJ enhancement Y s / 2 Y i g ( Y P Y S ) 9 ( Y I 5 Y S ) Y S / 2 Y I 9 ( Y P Y S ) n A s s u m p t i o n s : Two i d e n t i c a l s p i n s r e l a x e d s o l e l y by d i p o l a r i n t e r a c t i o n s ; J 0 0 ( t o ) i s c o r r e c t l y d e s c r i b e d ^o by E q u a t i o n [ 3 . 2 . 6 ] ; x _ 1 = ( Y 2 t i r " 3 ) 2 3 / 2 . I f 0 , a l l v a l u e s reduce t o the r e s p e c t i v e i s o t r o p i c v a l u e s . I f 3+cos" 1 ( 3 " 1 / 2 ) , a l j _ terms f o r T j 1 w i t h (3cos 2 3 -1) f a c t o r s a r e t o be r e p l a c e d by 3 s i n 2 3 ( l 6 - 1 5 s i n 2 3 ; ) / 1 6 D „ F o r T" 1, t h e r e p l a c e m e n t s a r e 9 s i n 3 ( 1 6 - 1 5 s i n 2 3 ) / l 60D,, and 3sin 2 3(16-15sin 2 3)/16D„ f o r D n < t o 0 and Dn>uQ r e s p e c t i v e l y . § I t i s assumed t h a t 3^  c o s - 1 ( 3 ~ 1 / / 2 ) . §§ I f s p i n S i s i r r a d i a t e d and s p i n I i s o b s e r v e d , g ( Y j , Y S ) = X [ - ( 1 + A ) 2 + 6 ( 1 - X ) 2 ] / [ ( 1 + A ) 2 + 6 ( 1 - X ) 2 + 3 ( 1 - X 2 ) 2 ] , where x= Y$ / Y J -V T 2 ( d T l / d D l ) D l l (dT 2 /dD A ) D n ( d T 1 / d D „ ) D l . ( d y d D , , ) TABLE 3.2 1 1 R e l a x a t i o n P a r a m e t e r s i n V a r i o u s M o t i o n a l L i m i t s : P a r t 2. u 0 > Dx w 0 > D ' i Dn=Dj.=D 72Da/5w 2 3a/10D 48 VD J0 D i < D i i < h i 0 72D | | C 4' 7 5 U Q 3a'/10D 1 2 a'"/D M 3 a 7 1 0 D A 1 w 0 ftcosVl^2 1 /P||\(3cos 2B-l) 2 96 D„D -0 + / 3 c o s 2 B - l \ ) ±\ s i n g y Dii Dx > > a )o T 2 0 \ D j s i n 2 B ( 1 _ ^ i r i 2 B ) 72Dj_a 1 / 5COQ 3 a 7 1 0 D ± 4 8 V D X ; -0 + + + -0 -0 n .a = ( Y 2 f i r " 3 ) 2 / 4 a' = a ( 3 C 0 S 2 B - l ) / 4 ? a 1 1 = a s i n B/2 a 1 1 '= 9 a s i n 2 3 ( 1 6 - 1 5 s i n 2 B ) / 1 6 O t h e r a s s u m p t i o n s a r e as d e s c r i b e d i n TABLE 3.1 -81-REFERENCES: CHAPTER I I I 1. I . Solomon, Phys. Rev. 99, 559 ( 1 9 5 5 ) . 2. H. S h i m i z u and S. F u j i w a r a , J . Chem. Phys. 34, 1501 ( 1 9 6 1 ) . 3. D. Geschke, Z. P h y s i k 212, 169 ( 1 9 6 8 ) . 4. R. Freeman, S. W i t t e k o e k , and R. R. E r n s t , J . Chem. Phys. 52, 1529 (1970) . — 5. D. M i c h e l , Ann. P h y s i k 27, 389 ( 1 9 7 1 ) . 6. H. S c h n e i d e r and H. S c h m i e d e l , Ann. P h y s i k 28, 346 ( 1 9 7 3 ) ; i b i d . , 357 ( 1 9 7 3 ) . ~~~ 7. R. K. H a r r i s and K. M. W o r v i l l , J . Magn. Res. 9, 383 ( 1 9 7 3 ) ; i b i d . , 394 ( 1 9 7 3 ) . 8. P. S. Hubbard, Rev. Mod. Phys. 33, 249 ( 1 9 6 1 ) . 9. L. G. Werbelow and A. G. M a r s h a l l , J . Amer. Chem. Soc. 95_, 5132 ( 1 9 7 3 ) . 10. D. D o d d r e l l , V. G l u s h k o , and A. A l l e r h a n d , J . Chem. Phys. 56, 3683 (1972). ~ 11. A. G. M a r s h a l l , P. G. S c h m i d t , and B. D. S y k e s , Biochem. 11, 3875 (1 9 7 2 ) . ~ 12. A. A l l e r h a n d , D. D o d d r e l l , and R. K o m o r o s k i , J . Chem. Phys. 55, 189 (1971) . — 13. G. Navon and A. L a n i r , J . Magn. Res. 8, 144 ( 1 9 7 2 ) . 14. H. B. C o a t e s , K. A. M c L a u c h l a n , I . D. C a m p b e l l , and C. E. M c C o l l , Biochem. B i o p h y s . A c t a 310, 1 ( 1 9 7 3 ) . 15. C. H. Fung, A. S. M i l d v a n , A. A l l e r h a n d , R. K o m o r o s k i , and M. C. S c r u t t o n , Biochem. 1_2, 620 ( 1 9 7 3 ) . 16. T. R. Krugh, t o be p u b l i s h e d i n S p i n L a b e l i n g : Theory and A p p l i c a t i o n s , e d i t e d by L. 0. B e r l i n e r , Academic P r e s s , New Y o r k , 1974. 17. F. Noack, NMR B a s i c P r i n c i p l e s and P r o g r e s s 3, 83 ( 1 9 7 1 ) . 18. L.G. Werbelow, J . Amer. Chem. S o c , t o be p u b l i s h e d . 19. J . H. Noggle and R. E. Schurmer, The N u c l e a r Overhauser E f f e c t , Academic P r e s s , New Y o r k , 1971. -82-20. P. B a l a r a m , A. A. B o t h n e r - B y , and E. B r e s l o w , Biochem. ]_2, 4695 ( 1 9 7 3 ) ; i b i d . , J . Amer. Chem. Soc..94, 4015 ( 1 9 7 2 ) . 21. W. T. H u n t r e s s , J . Chem. Phys. 48, 3524 (1968) . -83-CHAPTER IV DIPOLAR RELAXATION OF THREE-SPIN SYSTEMS 4.1 INTRODUCTION As was h i n t e d a t i n t h e p r e v i o u s c h a p t e r , t h e i n t e r p r e t a t i o n o f methyl group r e l a x a t i o n p l a y s an i m p o r t a n t r o l e i n a p p l i c a t i o n s . T h i s i s e s p e c i a l l y t r u e f o r b i o c h e m i c a l s t u d i e s , and i t w i l l be b e n e f i c i a l t o t a k e a c l o s e r l o o k a t some o f t h e p e c u l i a r i t i e s ' o f t h e ~ p r o b l e m . Many a t t e m p t s t o s t u d y t h e f l e x i b i l i t y a t s p e c i f i c s i t e s on macro-m o l e c u l e s from measurement o f n u c l e a r r e l a x a t i o n t i m e s have been based on t h e use o f -C0CH 3 o r -COCF^ g r o u p s , on a c c o u n t o f t h e i r s u p e r i o r s i g n a l - t o - n o i s e (compared w i t h -CH o r -CH^ g r o u p s ) and l a c k o f s c a l a r c o u p l i n g ( r e s u l t i n g i n a s i n g l e r e s o n a n c e ) . The e x p e r i m e n t t y p i c a l l y 2 i n v o l v e s e i t h e r c o v a l e n t " l a b e l i n g " a t t h e d e s i r e d s i t e o r use o f a methyl c o n t a i n i n g s m a l l m o l e c u l e w h i c h b i n d s and exchanges r a p i d l y and 3 r e v e r s i b l y t o t h e m a c r o m o l e c u l e , i n e i t h e r c a s e , t h e m a g n e t i c r e l a x -a t i o n b e h a v i o r f o r t h e methyl group a t t a c h e d t o t h e macromolecule a r e 4 r e a d i l y e x t r a c t e d from t h e d a t a . U n f o r t u n a t e l y , w h i l e a methyl group o f f e r s p r a c t i c a l advantages i n measurement o f t h e r e l a x a t i o n p a r a m e t e r s , the i n t e r p r e t a t i o n i n terms o f g r o s s and i n t e r n a l m o l e c u l a r m o t i o n i s c o m p l i c a t e d by two f a c t o r s p e c u l i a r t o methyl g r o u p s : F i r s t , t h e m o t i o n -84-o f any one methyl p r o t o n i s c l e a r l y c o r r e l a t e d t o t h a t o f t h e o t h e r two, l e a d i n g t o r a t h e r c o m p l i c a t e d e x p r e s s i o n s f o r t h e ( n o n - e x p o n e n t i a l ) de-c a y ; s e c o n d , i f t h e i n t e r n a l r o t a t i o n o f t h e methyl group i s s u f f i c i e n t l y f a s t , then s p i n - i n t e r n a l r o t a t i o n e f f e c t s may dominate t h e r e l a x a t i o n . D e s p i t e an e x t e n s i v e l i t e r a t u r e on t h e s u b j e c t , t h e r e i s a t p r e s e n t some c o n f u s i o n as t o t h e e x t e n t , o c c u r r e n c e , and i m p o r t a n c e o f nonexpon-e n t i a l n u c l e a r m a g n e tic r e l a x a t i o n f o r a methyl ( o r t r i f l u o r o m e t h y l ) group i n l i q u i d media b e i n g r e l a x e d by t h e i n t r a m o l e c u l a r d i p o l a r r e -5 T a x a t i o n mechanism. W h i l e n o n e x p o n e n t i a l r e l a x a t i o n has been p r e d i c t e d 6-15 and u n q u e s t i o n a b l y o b s e r v e d i n s o l i d s , n o n e x p o n e n t i a l e f f e c t s due t o d i p o l a r i n t e r f e r e n c e terms have not been c o n v i n c i n g l y o b s e r v e d f o r s m a l l m o l e c u l e s i n s o l u t i o n . * M a c r o m o l e c u l e s i n s o l u t i o n e x h i b i t m o t i o n a l p r o p e r t i e s i n t e r m e d i a t e between s o l i d s and l i q u i d s , and n o n e x p o n e n t i a l r e l a x a t i o n m i g h t t h u s be a n t i c i p a t e d f o r such s p e c i e s . However, t h e de-t a i l e d r e l a x a t i o n b e h a v i o r o f a -CH^ o r - C F 3 group a t t a c h e d t o a macro-m o l e c u l e i n s o l u t i o n has never been worked o u t , i n s p i t e o f t h e many a t t e m p t s t o s t u d y n u c l e a r r e l a x a t i o n f o r b i g m o l e c u l e s . I n t h i s c h a p t e r , p r e v i o u s work d e a l i n g w i t h d i p o l a r r e l a x a t i o n i n m u l t i - s p i n systems i s b r i e f l y r e v i e w e d , and H u b b a r d ' s 1 ^ ' 1 7 c r o s s - c o r r e l a t i o n t r e a t m e n t o f i n t r a -m o l e c u l a r d i p o l e - d i p o l e r e l a x a t i o n f o r a group o f t h r e e e q u i v a l e n t s p i n o n e - h a l f n u c l e i i s extended t o t h e c a s e o f l a r g e m o l e c u l e s i n s o l u t i o n (a) QT c > 1 ) . F i n a l l y , t h e qu e n c h i n g e f f e c t o f i n t e r n a l s p i n - r o t a t i o n i s c o n s i d e r e d q u a l i t a t i v e l y , and t h e i n t e r p r e t a t i o n o f methyl r e l a x a t i o n * See F i g u r e 7 and t h e r e l a t e d d i s c u s s i o n i n R e f e r e n c e 10 f o r a p o s s i b l e o b s e r v a t i o n i n a l i q u i d sample. -85-r a t e s i n terms o f m o l e c u l a r r o t a t i o n a l m o t i o n i s d i s c u s s e d . For problems i n v o l v i n g t h r e e o r more s p i n s , t h e s i m p l e s t c a s e i s t h a t o f N e q u i v a l e n t ( i . e . h a v i n g t h e same Larmor f r e q u e n c y ) n u c l e i s i t u a t e d a t e q u i v a l e n t p o s i t i o n s i n t h e m o l e c u l e . P r o v i d e d t h a t t h e mo t i o n o f any two n u c l e i i s u n c o r r e c t e d w i t h t h a t o f a t h i r d ( t h e " c r o s s -c o r r e l a t i o n f u n c t i o n s " a r e z e r o , E q u a t i o n [3.1.6] i s v a l i d ) , i t i s r e a d i l y shown t h a t t h e t o t a l r e l a x a t i o n r a t e i s j u s t t h e sum o f a l l t h e i n d i v i d -u a l p a i r w i s e d i p o l e - d i p o l e c o n t r i b u t i o n s as assumed i n t h e l a s t c h a p t e r . However, f o r a methyl group ( o r f o r any r i g i d frame c o n t a i n i n g a t l e a s t t h r e e n u c l e i t h e m o t i o n o f t h e t h i r d p r o t o n i s c l e a r l y d e t e r m i n e d by t h e m o t i o n o f t h e o t h e r two, and c r o s s c o r r e l a t i o n s may not i n p r i n c i p l e be n e g l e c t e d . V a r i o u s a t t e m p t s t o a c c o u n t f o r such c r o s s - c o r r e l a t i o n s w i l l now be l i s t e d b r i e f l y . -86-4.2 RESUME OF PREVIOUS STUDIES Working w i t h i n t h e framework o f t h e s e m i c l a s s i c a l d e n s i t y m a t r i x 18 t h e o r y o f r e l a x a t i o n , Hubbard i n h i s c l a s s i c a l paper c a l c u l a t e d t h e r e l a x a t i o n b e h a v i o r o f e i t h e r t h r e e o r f o u r e q u i v a l e n t s p i n o n e - h a l f n u c l e i p l a c e d a t t h e c o r n e r s o f an e q u i l a t e r a l t r i a n g l e (as f o r a methyl group) o r a t t h e c o r n e r s o f a r e g u l a r t e t r a h e d r o n . Assuming s p h e r i c a l l y symmetric r o t a t i o n a l d i f f u s i o n , and an " e x t r e m e - n a r r o w i n g " c o n d i t i o n , co^x^ « 1, he showed t h a t t h e l o n g i t u d i n a l r e l a x a t i o n was r e p r e s e n t e d by t h e sum o f two e x p o n e n t i a l s , but t h e r e s u l t a n t d i f f e r e d o n l y v e r y s l i g h t l y from t h e s i m p l e r c a l c u l a t i o n i n w h i c h c r o s s - c o r r e l a -20 t i o n s were c o m p l e t e l y n e g l e c t e d . * In a l a t e r p a p e r , Hubbard extended h i s f o u r - s p i n c a l c u l a t i o n t o c o r r e l a t i o n t i m e s l o n g e r t h a n t h e Larmor p e r i o d , and computed both t h e l o n g i t u d i n a l and t r a n s v e r s e r e l a x a t i o n . In t h i s c a s e , both decays a r e c o r r e c t l y r e p r e s e n t e d as a s u p e r p o s i t i o n o f t h r e e e x p o n e n t i a l s , r e d u c i n g t o two e x p o n e n t i a l s i n t h e l i m i t U ) 0 T 2 K < ^' a n c ' ^ o r l° n9it udinal r e l a x a t i o n ) i n t h e l i m i t >> 1. However, a g a i n t h e t h e o r y p r e d i c t e d o n l y v e r y s l i g h t d e v i a t i o n s from 23 t h e u n c o r r e l a t e d r e s u l t s . I t was t h i s work w h i c h l e d Abragam t o remark i n h i s famous monograph t h a t i n c l u s i o n o f c r o s s - c o r r e l a t i o n terms was o f t h e o r e t i c a l i n t e r e s t but o f no p r a c t i c a l i m p o r t a n c e , an o f t q uoted s t a t e m e n t . * The r e l a x a t i o n o f t h r e e and f o u r - s p i n systems was s i m u l t a n e o u s l y 19 a n a l y z e d (1957) by I . A l e k s a n d r o v . A l t h o u g h t h e r e a r e s u p e r f i c i a l s i m i l a r i t i e s between t h e two s e t s o f r e s u l t s , A l e k s a n d r o v ' s approach has t h e f a i l i n g i n t h a t BPP t h e o r y (which was t h e assumed s t a r t i n g p o i n t ) cannot be d i r e c t l y e x tended i n any s a t i s f y i n g f a s h i o n t o m u l t i s p i n systems (see C h a p t e r I I ) . -87-However, t h i s s t a t e m e n t was about t h r e e y e a r s premature as unques-5 t i o n a b l y d e m o n s t r a t e d i n t h e c a l c u l a t i o n where H i l t and Hubbard con-s i d e r e d an e q u i l a t e r a l t r i a n g l e o f i d e n t i c a l s p i n s w h i c h r o t a t e about a c r y s t a l - f i x e d a x i s p e r p e n d i c u l a r t o t h e p l a n e o f t h e t r i a n g l e . They c o n s i d e r e d v a r i o u s o r i e n t a t i o n s o f t h e t r i a n g l e w i t h r e s p e c t t o t h e e x t e r n a l l y a p p l i e d m a g n e t i c f i e l d d i r e c t i o n , and a l s o t r e a t e d a p o l y -c r y s t a l l i n e model. I t was shown t h a t t h e l o n g i t u d i n a l decay i s g i v e n by t h e sum o f f o u r e x p o n e n t i a l s , but more i m p o r t a n t l y , t h a t t h e p r e d i c t e d decay was m a r k e d l y n o n e x p o n e n t i a l and hence, s i g n i f i c a n t l y d i f f e r e n t from t h e u n c o r r e l a t e d r e s u l t . * Hubbard's most r e c e n t c o n t r i b u t i o n s assume an approach i n t e r m e d i a t e between t h e above l i m i t s o f a methyl group r i g i d l y bound t o a r o t a t i n g s p h e r e and a methyl a t t h e end o f an i n f i n i t e l y l o n g r o d . F i r s t , Hubbard c o n s i d e r e d a methyl group r i g i d l y a t t a c h e d a l o n g t h e symmetry a x i s o f a symmetric t o p m o l e c u l e u n d e r g o i n g a n i s o t r o p i c r o t a t i o n a l d i f f u s i o n ( f o r m a l l y e q u i v a l e n t t o a s p h e r i c a l t o p m o l e c u l e w i t h an i n t e r n a l l y r o t a t i o n methyl g r o u p ) . F i n a l l y , t h e t r e a t m e n t was e x t e n d e d t o an a s ymmetric t o p w i t h a r o t a t i n g methyl group a t t a c h e d a t an a r b i t r a r y a n g l e w i t h r e s p e c t t o t h e p r i n c i p a l m o l e c u l a r a x i s . 1 7 However, a l l * I t s h o u l d be n o t e d t h a t H i l t - H u b b a r d t h e o r y has l o n g been r e j e c t e d by many s o l i d s t a t e NMR p e o p l e d e s p i t e e x p e r i m e n t a l c o n f i r m a t i o n . The c r u x o f t h i s dilemma i s t h e j u s t i f i c a t i o n o f m u l t i p l e t i m e c o n s t a n t s needed t o c h a r a c t e r i z e t h e l o n g i t u d i n a l r e l a x a t i o n when s t r o n g i n t e r -m o l e c u l a r d i p o l a r c o u p l i n g s s h o u l d i n p r i n c i p l e e s t a b l i s h a unique s p i n t e m p e r a t u r e i n a t i m e s h o r t compared w i t h t h e t i m e s c a l e o f t h e s e m u l t i p l e t i m e c o n s t a n t s . Arguments pro and con e x i s t and t h e s i t u a t i o n 21 22 appears muddled a t b e s t . ' -88-e x p l i c i t c a l c u l a t i o n s were r e s t r i c t e d t o t h e e x t r e m e - n a r r o w i n g l i m i t , so t h a t o n l y t h e l o n g i t u d i n a l decay ( i d e n t i c a l t o t h e t r a n s v e r s e decay) was computed; i n t h e n e x t s e c t i o n , we e x t e n d t h e s e c a l c u l a t i o n s beyond t h e extreme n a r r o w i n g l i m i t , so t h a t t h e l o n g i t u d i n a l and t r a n s v e r s e decays must be t r e a t e d s e p a r a t e l y . A p a r t from t h e c a l c u l a t i o n s o f Hubbard, o t h e r a u t h o r s have a l s o c o n s i d e r e d t h e p roblem o f c r o s s - t e r m s between v a r i o u s p a i r w i s e d i p o l a r c o u p l i n g s i n t h e r e l a x a t i o n o f m u l t i - s p i n s y s t e m s . K a t t a w a r and E i s n e r 2 ' have t r e a t e d t h e s p e c i a l c a s e o f t h r e e i d e n t i c a l s p i n o n e - h a l f n u c l e i a t t h e c o r n e r s o f a 30°-120° i s o s c e l e s t r i a n g l e . The a c t u a l n u m e r i c a l r e -s u l t s i n t h i s paper a r e i n e r r o r , as p o i n t e d o u t i n R e f s . 25 and 29, d o u b t l e s s due t o t h e d i f f i c u l t y i n s o l u t i o n o f 14 s i m u l t a n e o u s d i f -f e r e n t i a l e q u a t i o n s , many o f w h i c h can be seen t o be l i n e a r l y dependent. 25 Z e i d l e r has c a r r i e d out a c a l c u l a t i o n on a s i m i l a r c a s e where two l i k e s p i n s and a n o n - i d e n t i c a l t h i r d s p i n form a t r i a n g u l a r arrangement. 26 R i c h a r d s has d e r i v e d t h e i n t e r e s t i n g r e s u l t t h a t i n t h e e x t r e m e - n a r -r o w i n g l i m i t , a system w i t h any number o f i d e n t i c a l s p i n s must have i d e n t i c a l (though not n e c e s s a r i l y a s i n g l e e x p o n e n t i a l ) t i m e - b e h a v i o r f o r b o th t r a n s v e r s e and l o n g i t u d i n a l d e c a y , even when c r o s s - c o r r e l a t i o n s a r e c o n s i d e r e d . I n t h e absence o f c r o s s - c o r r e l a t i o n s , t h i s s t a t e m e n t r e d u c e s t o t h e w e l l - k n o w n f a c t t h a t T-| = T 2 under e x t r e m e - n a r r o w i n g . 27 R u n n e l s , i n a r a t h e r g e n e r a l c a l c u l a t i o n , has proven f o r a system o f t h r e e e q u i v a l e n t s p i n s o n e - h a l f whose i n i t i a l p r e p a r a t i o n i s d e s c r i b -a b l e by a s p i n - t e m p e r a t u r e , t h e i n c l u s i o n o f c r o s s - c o r r e l a t i o n a lways 28 a c t s t o make t h e r e l a x a t i o n l e s s e f f i c i e n t ( s l o w e r d e c a y ) . F e n z k e , i n a c a l c u l a t i o n g e n e r a l i z e d t o any number o f e q u i v a l e n t s p i n s , i n --89-d e p e n d e n t l y a r r i v e s a t many o f t h e same c o n c l u s i o n s as Runnels demon-29 30 s t r a t e d . R e c e n t l y Pyper ' has a p p l i e d L i o u v i l l e r e p r e s e n t a t i o n f o r m a l i s m t o t h e c r o s s - c o r r e l a t i o n problem and has r e - d e r i v e d p r e v i o u s r e s u l t s t o s e r v e as c o m p e l l i n g examples o f t h e u s e f u l n e s s o f t h i s 31 32 app r o a c h . Buchner ' has approached t h e problem from t h e p o i n t o f view o f group t h e o r y . A n o t h e r prime s o u r c e o f q u a n t i t a t i v e knowledge about t h e i n f l u e n c e o f c r o s s - c o r r e l a t i o n s between d i p o l e - d i p o l e i n t e r a c t i o n s i n m a g n e t i c 33-37 r e s o n a n c e comes from a s e r i e s o f f i v e papers by S c h n e i d e r and 36 37 B l i c h a r s k i . ' S c h n e i d e r has d e r i v e d t h e l o n g i t u d i n a l and t r a n s v e r s e r e l a x a t i o n b e h a v i o r f o r a l m o s t e v e r y c o n c e i v a b l e arrangement o f t h r e e o r f o u r s p i n o n e - h a l f p a r t i c l e s , where t h e p a r t i c l e s a r e not neces-s a r i l y e q u i v a l e n t i n t h e m o l e c u l e and need not even be " l i k e " n u c l e i . ( I t s h o u l d be noted t h a t f o r c a l c u l a t i o n s t r e a t i n g " u n l i k e " n u c l e i , i t would i n g e n e r a l be e x p e c t e d t h a t c r o s s - r e l a x a t i o n e f f e c t s would l e a d t o n o n e x p o n e n t i a l r e l a x a t i o n even f o r a t w o - s p i n system. T h e r e -f o r e , i n c l u s i o n o f c r o s s - c o r r e l a t i o n e f f e c t s f o r a m u l t i s p i n s y s t e m , composed o f d i s s i m i l a r n u c l e i , l e a d s t o a v e r y c o m p l i c a t e d r e l a x a t i o n e x p r e s s i o n , and one t h a t s h o u l d be i n t e r p r e t e d w i t h c a r e . ) F o r c e r t a i n c o n f i g u r a t i o n s o f " l i k e " s p i n s , S c h n e i d e r t r e a t s a l l ranges o f t h e q u a n t i t y , F u r t h e r m o r e , h i s t r e a t m e n t can be adapted t o m o t i o n a l models i n c l u d i n g i s o t r o p i c o r a n i s o t r o p i c r o t a t i o n a l d i f f u s i o n o r i n -t e r n a l r o t a t i o n . In c a s e s where t h e two t r e a t m e n t s d e a l w i t h i d e n t i c a l m o l e c u l a r r o t a t i o n a l d y n a m i c s , S c h n e i d e r ' s r e s u l t s may be seen t o r e -duce t o Hubbard's. F o r v a r i o u s c o n f i g u r a t i o n s o f t h e t h r e e o r f o u r s p i n s , b oth t h e l o n g i t u d i n a l and t r a n s v e r s e r e l a x a t i o n a r e , i n g e n e r a l , -90-t h e r e s u l t a n t o f between t h r e e and seven e x p o n e n t i a l s . O t h e r examples o f t h e e f f e c t o f c r o s s - c o r r e l a t i o n i n c l u d e a paper 38 13 by N o g g l e , w h i c h shows t h a t f o r C i n a m e t h yl g r o u p , c r o s s - c o r r e -13 13 l a t i o n s between t h e C-H1 and H'-H" o r C-H" - r e l a x a t i o n v e c t o r s f o r t u i t o u s l y c a n c e l o u t ( t o f i r s t o r d e r ) , even though t h e o r d e r o f magnitude o f t h e s p e c t r a l d e n s i t y f o r c r o s s - c o r r e l a t i o n i s t h e same 39 as f o r a u t o c o r r e l a t i o n . Kuhlman e t . a l . have come t o a s i m i l a r 13 c o n c l u s i o n from t h e i r c o n s i d e r a t i o n o f O v e r h a u s e r e f f e c t s f o r CHg g r o u p s . I t s h o u l d be m e ntioned however, t h a t t h e e x a c t b e h a v i o r o f 13 t h e r e l a x i n g f o u r - s p i n s y s t e m , CH^ has not been c a r r i e d o u t i n de-t a i l , a l t h o u g h , as t h e s e l a s t two papers might i n d i c a t e , t h i s would p r o v i d e a m e a n i n g f u l , a l b e i t t e d i o u s , c a l c u l a t i o n . The e f f e c t o f c r o s s - c o r r e l a t i o n terms i n methyl r e l a x a t i o n and i t s e f f e c t s on t h e 4f) 41 13 13 42 i n t e r p r e t a t i o n o f CINDP, DNP, and C r e l a x a t i o n i n C H 3 groups has r e c e n t l y been r e a l i z e d . Among a l l t h e above l i s t e d e x a m p l e s , t h e one w h i c h most c l o s e l y a p p r o x i m a t e s t h e c r i t e r i a o f b e i n g p h y s i c a l l y r e a s o n a b l e w h i l e a t t h e same t i m e p a r a m e t r i c a l l y t r a c t a b l e i s t h a t o f a s ymmetric t o p m o l e c u l e t o w h i c h i s a t t a c h e d ( a t a r b i t r a r y a n g l e ) a methyl group w h i c h may 16 17 r o t a t e i n d e p e n d e n t l y o f t h e m o l e c u l e as a whole. ' U n f o r t u n a t e l y , t h e e x i s t i n g l i t e r a t u r e i s l i m i t e d t o t h e extreme n a r r o w i n g a p p r o x i -m a t i o n , and i s thus r e s t r i c t e d t o s m a l l m o l e c u l e s i n l i q u i d s . F o r p r e s e n t l y t y p i c a l m a g n e t i c f i e l d s o f 15-75 kg i n NMR, t h e extreme n a r -r o w i n g l i m i t w i l l be v i o l a t e d by m o l e c u l e s w i t h m o l e c u l a r w e i g h t g r e a t e r t h a n about 5000 i n aqueous s o l u t i o n ( p r o t e i n s , enzymes, membranes, n u c l e i c a c i d s , s y n t h e t i c p o l y m e r s ) , as w e l l as by s m a l l m o l e c u l e s i n -91-v i s c o u s media o r by s o l i d s near t h e m e l t i n g p o i n t s ; t h e b e h a v i o r o f a m e t h y l group on such m o l e c u l e s i s t h u s unknown. In s e c t i o n 4.3, we t r e a t t h e d i p o l e - d i p o l e i n t r a m o l e c u l a r r e l a x a t i o n f o r a methyl group on a symmetric t o p m o l e c u l e o f any s i z e , i n c l u d i n g t h e p o s s i b i l i t y o f i n t e r n a l r o t a t i o n . -92-4.3 FORMULATION OF THE CALCULATION In t h i s s e c t i o n , we d e r i v e t h e e x p e c t a t i o n v a l u e s o f I and I , u s i n g as b a s i s f u n c t i o n s t h e e i g h t s p i n s t a t e s o f a group o f t h r e e s p i n o n e - h a l f p a r t i c l e s , e x p r e s s e d i n terms o f t h e r e s p e c t i v e e i g e n v a l u e s o f I? I 2 2 , and I z , where I ] 2 = (I} + I 2 ) , I = ( I 1 2 + I 3 ) , and I 2 = I + I + I . The c h o i c e o f t h e e i g h t b a s i s s t a t e s i s based on t h e con-z 2 z 3 v e n t i o n , |T> = |1 3/2 3/2> |2> = | l 3/2 l/2> |3> = |1 3/2 ; i / 2 > |4> = |1 3/2 -3/2> |5> = |1 1/2 l/2> |6> = |1 1/2 - l / 2 > |7> E |0 1/2 l/2> |8> = 1° 1/2 - l / 2 > [4.3.1] where |a> E | 1 -|21 I z > * D e f i n i n 9 y i ( t ) ~ l ( x n _ x 4 4 ) + i ( x 2 2 _ x 3 3 } + I ^ s s + ^ ' X e e ' W ' [ 4 - 3 - 2 ] and i n t r o d u c i n g two o t h e r c o m b i n a t i o n s o f m a t r i x e l e m e n t s , 3 1 y 2 ( t ) = y ^ t ) - • 2-(x 1 1-X4 4) - -2"(X22"X33) [4.3.3] y 3 ( t ) = y , ( t ) - | ( x i r x 4 4 ) [4.3.4] where X„„. = <a| X(t)|a'> [4.3.5] -93-i t i s r e a d i l y shown t h a t 16 ( d / d t ) y , ( t ) + 12[ J ^ . ' - 1 (co0) + 2 J 2 ' - 2 ( 2 a 3 0 ) ] y 2 ( t ) + 16[-jJ ' _ 1(co 0) - J 2 ' - 2 ( 2 a ) 0 ) ] y 3 ( t ) [4 .3.6] ( d / d t ) y 2 ( t ) = - 4 J 2 » " 2 ( 2 c o 0 ) + J J ' _ 1U 0 ) + 4 J 2 ' - 2 ( 2 o ) ( ) ) ] y 1 ( t ) + [ - 3 j j > ) - J j ^ t c - Q ) - 2 J 2 ' - 2 ( 2 . 0 ) + 30 0 0 ( 0 ) + j j * " 1 («-<,) + 2 J 2 ' " 2 ( 2 U n ) ] y 2 ( t ) + 1 [ 6 J ° ° ( 0 ) + l O J 1 ' - 1 ^ ) + 4 J 2 ' _ 2 ( 2u ) 0 ) - 6 J ° ° ( 0 ) - lOjJ'-Vo) - 4 J 2 ' - 2 ( 2 a ) 0 ) ] y 3 ( t ) [4 .3 .7] ( d / d t ) y 3 ( t ) = [ - 0 a ' _ 1 U o ) - 2 J 2 ' " 2 (2co 0 ) - j j » " 1 ( a , 0 ) ] y 1 ( t ) + 7 [-9j ]| , " 1 («o 0 ) " 3 j J» _ 1 ( u . 0 ) + 1 2 J 2 ' - 2 ( 2 c o Q ) ] y 2 ( t ) + [9 J a ' - ] ( co 0 ) - j j * " 1 («-<,) - 4 J 2 ' - 2 ( 2 < , 0 ) ] y 3 ( t ) . [4 .3 .8 ] The s u b s c r i p t " a " l a b e l s t h e a u t o c o r r e l a t i o n s p e c t r a l d e n s i t i e s (c=n), and t h e s u b s c r i p t " c " l a b e l s t h e c r o s s - c o r r e l a t i o n s p e c t r a l d e n s i t i e s (c^n ; see E q u a t i o n s [2 .3.1] and [2 .3 .11 ] ) . A u n i q u e s o l u t i o n t o t h e s e t h r e e c o u p l e d d i f f e r e n t i a l e q u a t i o n s may be found from t h e i n i t i a l c o n d i t i o n s r e s u l t i n g from a p p l i c a t i o n o f an r f - p u l s e d i r e c t e d a l o n g t h e y - a x i s i n t h e r o t a t i n g f r a m e , c a u s i n g t h e * I t m i g h t be mentioned t h a t even i n t h i s s i m p l e s t o f systems o f t h r e e i d e n t i c a l s p i n o n e - h a l f p a r t i c l e s , t h e s t r a i g h t f o r w a r d c a l c u l a t i o n i s by no means t r i v i a l . The number o f elements i n t h e r e l a x a t i o n 3 4 m a t r i x i s (2 ) and each element i s a sum o f 4050 t e r m s . T r u e , many o f t h e elements o f R a r e z e r o o r can be deduced from symmetry ( s ee Appe n d i x A ) . However, t h e t a s k i s s t i l l t e d i o u s . -94-e q u i l i b r i u m m a g n e t i z a t i o n t o r o t a t e by e degrees ( r e l a x a t i o n e f f e c t s d u r i n g t h e p u l s e a r e i g n o r e d ) . T h e r e f o r e , a(t=0) = e x p ( - i e l y ) a T e x p ( + i e l y ) . [4. 3 . 9 ] Now, f o r t h e h i g h - t e m p e r a t u r e a p p r o x i m a t i o n , TIWQ « kT, a 1 = e x p ( - r i E / k T ) / T r [ e x p ( - - f i E / k T ) ] - ( % - (1iE/kT) )/Tr[3J] [4.3.10] where 3( d e n o t e s t h e u n i t m a t r i x . S u b s t i t u t i n g E q u a t i o n [4.3.10] i n t o [4.3 .9] and making note o f t h e f a c t t h a t E = - W g l z , a ( t = 0 ) = 1 + e x p ( - i e l ) I _ e x p ( + i e l v ) [4.3.11] 0 8kT y z y but e x p ( - i e l ^ ) I z e x p ( + i e l y ) = I x s i n e + I z c o s e . [4.3.12] T h e r e f o r e , a ( t = 0 ) , e v a l u a t e d i n t h e b a s i s o f [ 4 . 3 . 1 ] , ° \ 0 0 If n) 1 .L t o n t i s l n e l u u " J - J C 0 t H u ' u - u 0  a K Z ~ v ' " 8 "^TOT"! 0 0 0 0 , co t e 1 0 0 0 - c o t e [4.3.13] I t then f o l l o w s from E q u a t i o n s [4.3.2-4] and [4.3.13] t h a t y ^ O ) = ( c o s e - 1 ) < I Z > T [4.3.14a] y 2 ( 0 ) = ( 1 / 6 ^ ( 0 ) [4.3.14b] 3cote /3 0 0 0 : o •:o /3 c o t e 2 0 0 : o '.-0 0 2 - c o t e /3 0 '.. 0 -0 0 0 /3 - 3 c o t e 0 i 0 . 0 0 0 0 0 .J c o t e 1 0 0 0 0 0 1 - c o t e 0 0 0 0 0 0 0 c o t e 0 0 0 0 0 0 1 -95-y 3 ( 0 ) = ( V 4 ) y 1 ( 0 ) . [4. 3 . 1 4 c ] In o r d e r t o s o l v e E q u a t i o n s [4.3.6-8] s u b j e c t t o c o n d i t i o n s [ 4 . 3 . 1 4 ] , i t i s n e c e s s a r y t o choose a m o l e c u l a r system and d e f i n e a model f o r i t s r o t a t i o n a l dynamics. F o r t h e g e n e r a l ( b u t s t i l l m o d e r a t e l y t r a c t a b l e ) system o f a symmetric t o p m o l e c u l e w i t h t h e symmetry a x i s o f a r o t a t i n g methyl group a t t a c h e d a t an a n g l e 6 w i t h r e s p e c t t o t h e symmetry a x i s o f t h e t o p , t h e a p p r o p r i a t e s p e c t r a l d e n s i t i e s a r e o b t a i n e d by F o u r i e r T r a n s f o r m a t i o n o f E q u a t i o n [2.3 .11] , J ak'- k(S) = (-Dk|o(4-f 1 ( 1 / 4 ) ( 3 c 2 o A - f ^ 9 ° 4 0 V r 3 J \ ( 6 D X ) 2 + ( k ^ ) 2 + 3sin 26cos 2 B (5D, + p t |) + (3/4)sin 4B ( 2 D , + 4D | () (5D^ + D„ ) 2 + (ko) 0) 2 (2D L + 4D 4 | ) 2 + (ko>Q)2 + (9/8)sin 4B (6D j_ +-4D.) + ( 3 / 2 ) s i n 2 B ( l + c o s 2 B ) (5D1 +Du +4Dn.) (6D L + 4 D . ) 2 + (kco 0) 2 (5Dj_ + D„ + 4 D i ) 2 + (ku>Q)2 + ( 3 / 8 ) [ ( l + c o s 2 g ) 2 + 4 c o s 2 3 ] ( 2 D 1 + 4(D U+ D.)) ) (2D X + 4(D„ + D ^ ) 2 + ( k ^ ) 2 J [4.3 . 15 ] J c k - k ( S ) - J a k'~ kH> - \ ^ ^ % - ± ^ ^ 2 c 0 a 0 8 0 V r 3 / ( (6D J_+ 4D n.) 2 + ( k U Q ) 2 + ( 3 / 2 ) s i n 2 B ( l + cos 2B ) (5D 1 + D,, + 4D.) (5DX + D u + 4 D . ) 2 + (ko) 0) 2 + 0 / 8 ) [ ( l + c o s 2 B ) 2 + 4cos 2B ] ( 2D 1 + 4(D„ + D.)) \ (2D± + 4(D„ + D ^ ) 2 + ( k ^ ) 2 J . [4.3.16] -96-The magnitude o f t h e c r o s s - c o r r e l a t i o n s p e c t r a l d e n s i t i e s , w h i c h i n con-t r a s t t o t h e a u t o c o r r e l a t i o n f u n c t i o n s , may be p o s i t i v e o r n e g a t i v e , depend on t h e r e l a t i v e o r i e n t a t i o n s o f t h e i n t e r n u c l e a r v e c t o r s . Equa-t i o n [4.3.16] was d e r i v e d from [2.3.11] assuming \$ - 'f'J^  = ^ 3 and 6 = 6^ = TT/2. In t h e s e e q u a t i o n s , Dj_ i s t h e d i f f u s i o n c o n s t a n t f o r r o t a t i o n a l d i f f u s i o n about an a x i s p e r p i n d i c u l a r t o t h e symmetry a x i s o f t h e symmetric t o p , D M i s t h e p a r a l l e l d i f f u s i o n c o n s t a n t , and i s t h e d i f f u s i o n c o n s t a n t f o r t h e i n t e r n a l r o t a t i o n o f t h e methyl group w i t h r e s p e c t t o t h e m o l e c u l a r frame. By t h e u s u a l methods, t h e t h r e e s i m u l t a n e o u s , f i r s t o r d e r , l i n e a r homogeneous d i f f e r e n t i a l e q u a t i o n s ( [ 4 . 3 . 6 - 8 ] ) w i t h c o n s t a n t c o e f f i c i e n t s may be s o l v e d d e t e r m i n a n c y o r by L a p l a c e T r a n s f o r m methods. The form o f t h e s o l u t i o n f o r t h e e x p e c t a t i o n v a l u e o f I i s g i v e n by, T T 3 <I > - <I > = <I >'(cose - 1) E A .exp(-x.t) [4.3.17] z z z i = 1 i 1 f o r an i n i t i a l p r e p a r a t i o n o f t h e system by a e - p u l s e . I n i t i a l c o n d i -t i o n s d i c t a t e t h a t E = 1. The p r e - e x p o n e n t i a l and e x p o n e n t i a l f a c t -o r s would be e x c e e d i n g l y b u l k y t o l i s t i n c l o s e d f o r m , but t h e l o n g i -t u d i n a l m a g n e t i z a t i o n i s r e a d i l y d i s p l a y e d i n g r a p h i c a l form (see F i g u r e s 4.1-4). S i n c e <I >^ = 0 ( t h e r e i s no e q u i l i b r i u m m a g n e t i z a t i o n i n t h e x-y A p l a n e ) , < I x ( t ) > = T r [ x ( t ) I x ] = T r [ a ( t ) I x ] , [4.3.18] and i t s u f f i c e s t o d e t e r m i n e t h e t i m e e v o l u t i o n o f a ( t ) . On s e t t i n g up t h e d e n s i t y m a t r i x e q u a t i o n s u s i n g t h e same b a s i s f u n c t i o n s as f o r t h e -97-l o n g i t u d i n a l c a l c u l a t i o n , i t becomes e v i d e n t t h a t t h e a p p r o p r i a t e p a r a -m e t r i z a t i o n i s i n terms o f t h r e e l i n e a r l y i n d e p e n d e n t c o m b i n a t i o n s o f m a t r i x elements o f a, c o n v e n i e n t l y chosen a s , q ^ t ) = T r [ a ( t ) I x ] = /3 R e [ a ] 2 + o^] + 2 R e [ a 2 3 ] + R e [ a 5 g + a y 8 ] [4.3.19a] q 2 ( t ) = R e [ a 5 6 + c ? 8 ] [4.3.19b] q 3 ( t ) = 2 R e [ a 2 3 ] + q 2 ( t ) = q ^ t ) - /3 R e [ a ] 2 + a ^ ] [4.3. 1 9 c ] where Re i s s h o r t h a n d f o r " r e a l p a r t o f " . W i t h t h e s e d e f i n i t i o n s , and t h e h e l p o f E q u a t i o n s [ 2 . 2 . 1 1 - 1 5 ] , t h e e q u a t i o n s f o r t h e t i m e r a t e o f change o f q ^ , q 2 , and q 3 may be o b t a i n e d , ( d / d t ) q i ( t ) = [-3(J°°(0) + J°°(0)) + B J ^ U Q ) + jj' _ 1(u) 0) - 2 J 2 ' - 2 ( 2 u n ) + 2 J 2 ' - 2 ( 2 o J o ) ] q 1 ( t ) + [ 6 ( - j J -- 1 ( U o ) + J 2 ' - 2 ( 2 a ) 0 ) ) ] q 2 ( t ) + [ 6 ( J ° ° ( 0 ) - J 2 ' " 2 ( 2 a > n ) ) ] q 3 ( t ) [4.3.20] ( d / d t ) q 2 ( t ) = C J a ' ' 1 ^ ) - J ^ ' ^ ^ ^ d ) + [ 2 ( J a ' - 1 ( c o Q ) - j J'~V n)) - 2 ( J 2 ' - 2 ( 2 u ) 0 ) - J 2 ' " 2 ( 2 c o 0 ) ) ] q 2 ( t ) + [-(J°°(0) - 0 ° ° ( 0 ) ) + (-J a'~V n) + j J » " 1 ( o . 0 ) ) ] q 3 ( t ) [4.3.21] ( d / d t ) q 3 ( t ) = [J a' _ 1(u> n) - J c ' ~ 1 ( w 0 ) 3 q l ( t ) + [ 3 ( - J a ' " 1 ( a , 0 ) " ^ ' ^ S ^ + 6 J 2 ' - 2 ( 2 c o n ) ] q 2 ( t ) + [ - 3 ( J ° ° ( 0 ) - J ° ° ( 0 ) ) + 2(2J a'- 1(a ) ( )) + 0 j ' _ 1 U 0 ) ) - 2( J 2 ' - 2 ( 2 a ) Q ) + 2 J 2 ' - 2 ( 2 u ) 0 ) ) ] q 3 ( t ) . [4.3.22] The i n i t i a l boundary c o n d i t i o n s can e a s i l y be found from E q u a t i o n [ 4 . 3 . 1 3 ] , -98-q-|(0) = s i n e < I z > [4.3.23a] q 2 ( 0 ) = ( l / 6 ) q n ( 0 ) q 3 ( 0 ) = ( l / 2 ) q - , (0) [4.3.23b] [4 . 3 . 2 3 c ] The system o f d i f f e r e n t i a l e q u a t i o n s ( [ 4 . 3 . 2 0 - 2 2 ] ) s u b j e c t t o t h e i n i t i a l c o n d i t i o n s ( [ 4 . 3 . 2 3 ] ) , y i e l d s a s o l u t i o n o f t h e f o r m , 3 A g a i n t h e B's a r e c o n s t r a i n e d , t h e sum b e i n g e q u a l t o u n i t y . T h i s s o l u t i o n i s a d a p t e d t o t h e r o t a t i n g frame i n w h i c h t h e r a p i d o s c i l l a t o r y t i m e dependence has been s u p p r e s s e d ( i . e . t h e t r u e s o l u t i o n t o R e d f i e l d ' s e q u a t i o n s y i e l d s a r e s u l t where t h e r i g h t hand s i d e o f E q u a t i o n [4.3.24] i s m u l t i p l i e d by t h e t e r m , exp(-io>Q(t + t ' ) ) , where t ' i s an i n i t i a l phase f a c t o r ) . As w i t h t h e l o n g i t u d i n a l r e l a x a t i o n , i t i s n o t w o r t h l i s t i n g t h e p r e - e x p o n e n t i a l and e x p o n e n t i a l f a c t o r s a n a l y t i c a l l y ; t h e g r a p h i c a l r e s u l t s a r e a n a l y z e d i n t h e D i s c u s s i o n . <I ( t ) > = s i n e < I > A L. T E B . e x p ( - Y , t ) . [4.3.24] i = l -99-4.4 RESULTS AND DISCUSSION B e f o r e p r o c e e d i n g t o a g e n e r a l d i s c u s s i o n o f t h e n u m e r i c a l r e s u l t s , i t w i l l be b e n e f i c i a l t o examine c e r t a i n a s y m p t o t i c l i m i t s o f t h e r e l a x -a t i o n b e h a v i o r . D i s r e g a r d i n g c r o s s - c o r r e l a t i o n terms ( i . e . , s e t t i n g a l l J c ( w ) = 0 ) , i t can be seen from E q u a t i o n s [4.3.6-8] and [4.3.20-22] r e s p e c t i v e l y t h a t ( < I z ( t ) > - < I z > T ) [ ( c o s e - l ) < I z > T ] - 1 = e x p [ 2 ( J a ' ~ 1 ( a ) 0 ) - 4 J ? r 2 ( 2 u ) 0 ) ) t ] [4 .4.1] and < I x ( t ) > ( s i n e < I z > T r 1 = e x p [ ( - 3 J ° ° ( 0 ) + S j J ' " 1 ^ ) - 2 J 2 ' - 2 ( 2 c o Q ) ) t ] . [4 .4.2] I t i s noted t h a t t h e s e e x p r e s s i o n s a r e i d e n t i c a l w i t h t h o s e i n t r o d u c e d from i n t u i t i o n i n t h e l a s t c h a p t e r . For t h e f a s t m o t i o n l i m i t , 6 D . » 2u>n ( J k ' " k ( k u ) n ) = (-1 ) kJ?° (0 ) ) , -*• u a , c u a,c 1 g c o n s i d e r e d p r e v i o u s l y , y-|(t) and y 2 ( t ) a r e no l o n g e r c o u p l e d t o Y 3 ( t ) . In t h i s c a s e , E q u a t i o n s [4 .3.6] and [4 .3.7] reduce t o ( d / d t ) y i ( t ) = 2 [ - 5 J ° ° ( 0 ) - J ° ° ( 0 ) ] y i ( t ) + [ 1 2 J ° ° ( 0 ) ] y 2 ( t ) [4 .4.3] ( d / d t ) y 2 ( t ) = [ - j f ( O ) + J ° ° ( 0 ) ] y i ( t ) + 4[-J°°(0) + J ° ° ( 0 ) ] y 2 ( t ) . [4 .4.4] F u r t h e r m o r e , i n t h i s l i m i t , q-,(t) c y ^ t ) [4 .4.5] -100-and q 2 ( t ) - y 2 ( t ) . [4. 4 . 6 ] That i s , t h e l o n g i t u d i n a l and t r a n s v e r s e m a g n e t i z a t i o n s decay i n an i d e n t i c a l manner. However, i n t h i s l i m i t , i t pro v e s more i n s t r u c t i v e t o d e f i n e t h e f o l l o w i n g c o m b i n a t i o n s o f m a t r i x e l e m e n t s , P ^ t ) -cc y ^ t ) - y 2 ( t ) -cc q ] ( t ) - q 2 ( t ) [4.4.7a] P 2 ( t ) s y 2 ( t ) « q 2 ( t ) . [4.4.7b] E q u a t i o n s [4.3.6] and [4.3.7] can be used t o d e t e r m i n e t h e e q u a t i o n s o f mo t i o n o f t h e p ' s , ( d / d t ) P l ( t ) = 3[-3J°°(0) - J ° ° ( 0 ) ] P l ( t ) + 5[-J°°(0) + J°°(0)]p 2(t) [4.4.8] ( d / d t ) p 2 ( t ) = [-J°°(0) + J°°(0)] P l(t) + 5[-J°°(0) + J°°(0)]p 2(t) [4.4.9] where ( < I 7 ( t ) > - <I > T ) [ ( c o s e -1)<I > T ] _ 1 = < I v ( t ) > [ s i n e < I > T ] - 1 E p , ( t ) + p ? ( t ) . T h i s c o m b i n a t i o n i s much more i n f o r m a t i v e than t h o s e chosen by Hubbard. A s i d e from b e i n g a e s t h e t i c a l l y p l e a s i n g ( o r t h o g o n a l c o m b i n a t i o n s ) , t h i s c o m b i n a t i o n emphasizes t h e p h y s i c a l b a s i s o f t h e r e s u l t i n g b i e x p o n e n t i a l decay o f m a g n e t i z a t i o n : The q u a r t e t s t a t e s (p-j's) r e l a x i n a p h y s i c a l l y d i s t i n c t f a s h i o n from t h e d o u b l e t s t a t e s ( p 2 ' s ) and hence, t h e r e s u l t a n t o b s e r v a b l e m a g n e t i z a t i o n , b e i n g t h e sum o f t h e s e two c o n t r i b u t i o n s , can-not be c h a r a c t e r i z e d by a uniq u e t i m e c o n s t a n t . I f D. > D ±> t o Q , t h e n j£'~ k(! k a ) n) ~ Ja'~k(k(V a s s e e n f r o m E c 1 u a t i o n -101-[ 4 . 3 . 1 6 ] . In t h i s l i m i t , f u r t h e r s i m p l i f i c a t i o n s a r e p o s s i b l e and E q u a t i o n s [4.4.8] and [4.4.9] reduce t o ( d / d t j p ^ t ) = -~\2J00{0)p^{t) ; p ^ O ) = 5/6 [4.4.10a] ( d / d t ) p 2 ( t ) = 0 ; p 2 ( 0 ) = 1/6 [4.4.10b] w i t h t h e s o l u t i o n p ^ t ) + p 2 ( t ) = (1 + 5 e x p ( - 1 2 J 0 0 ( 0 ) t ) ) / 6 . [4.4.11] Thus, t h e q u a r t e t system and d o u b l e t systems a r e c o m p l e t e l y i s o l a t e d f rom mutual c o m m u n i c a t i o n , and f u r t h e r m o r e , t h e d o u b l e t systems never l o s e t h e i r r e s i d u a l p o l a r i z a t i o n as a r e s u l t o f i n t r a m o l e c u l a r d i p o l a r i n t e r a c t i o n s . Note t h a t t h i s c o n c l u s i o n i s i n d e p e n d e n t o f t h e a n g l e o f a t t a c h m e n t , B ( a s i d e from t h e e x t r e m e l y u n l i k e l y s i t u a t i o n where B e q u a l s t h e NMR magic a n g l e ) , and i s dependent o n l y on whether t h e c r o s s - c o r r e -l a t i o n and a u t o c o r r e l a t i o n s p e c t r a l d e n s i t i e s a r e i d e n t i c a l i n magnitude. A l t h o u g h l i m i t i n g e x p r e s s i o n s can be a n a l y z e d q u i t e r e a d i l y , we have chosen t o e x p r e s s a l l g e n e r a l r e s u l t s g r a p h i c a l l y i n F i g u r e s 4.1-4. F o r t h e s e p l o t s , i t was assumed t h a t t h e methyl group a x i s was c o i n c i -d ent w i t h t h e symmetry a x i s o f t h e symmetric t o p t o w h i c h t h e m e t h y l was a t t a c h e d ( i . e . B= 0° i n E q u a t i o n s [4.3.15] and [ 4 . 3 . 1 6 ] ) , as i t was found i n g e n e r a l t h a t t h e n o n - e x p o n e n t i a l c h a r a c t e r o f t h e r e l a x a t i o n was most e x a g g e r a t e d f o r t h i s c a s e . In any i n s t a n c e , e x p l i c i t d i s p l a y o f t h i s angle-dependence would be u n p r o f i t a b l y l e n g t h y . For each p l o t , t h e m a g n e t i c f i e l d was chosen as t y p i c a l f o r h i g h r e s o l u t i o n NMR e x p e r i m e n t s : 8 -1 B Q = 2.35T (cog = 2TT x 10 s e c " ). F i n a l l y , f o r each p l o t , an i n t e r p r o t o n d i s t a n c e o f 1.8 A was assumed s i n c e X-ray and microwave v a l u e s average -102-c l o s e t o t h i s v a l u e . F i g u r e 4.1 shows p l o t s o f l o g ( d e v i a t i o n m a g n e t i z a t i o n ) v e r s u s t i m e f o r t h r e e v a l u e s o f D x, t h e d i f f u s i o n c o n s t a n t f o r r o t a t i o n a l d i f f u s i o n about an a x i s p e r p e n d i c u l a r t o t h e symmetry a x i s o f t h e symmetric t o p . The upper p l o t c o r r e s p o n d s t o an extreme-narrowed c a s e , 6Dj_>> lo^, and i n t h i s l i m i t t h e t r a n s v e r s e and l o n g i t u d i n a l m a g n e t i z a t i o n s have t h e same time-dependence so o n l y one p l o t i s r e q u i r e d ; t h e m i d d l e p l o t s show t h e l o n g i t u d i n a l ( l e f t - h a n d p l o t ) and t r a n s v e r s e ( r i g h t - h a n d p l o t ) m a g n e t i z a t i o n f o r a c a s e where 6 D ± = LOQ; t h e bottom p l o t s g i v e t h e l o n g i t u d i n a l ( l e f t - h a n d p l o t ) and t r a n s v e r s e ( r i g h t - h a n d p l o t ) magnet-i z a t i o n s f o r a m e t hyl group on a v e r y l a r g e m o l e c u l e i n s o l u t i o n ( o r s m a l l m o l e c u l e i n v e r y v i s c o u s s o l u t i o n ) , 6 D X < ion. The e x t r e m e - n a r -rowed ca s e ( F i g . 4.1a) i s i d e n t i c a l t o Hubbard's F i g . 1 e x c e p t f o r a change i n n o t a t i o n o f t h e a b s c i s s a s c a l e . From c o m p a r i s o n o f c u r v e s P-V f o r each g r a p h , f o r a symmetric t o p o f any s i z e , r e l a x a t i o n (whether l o n g i t u d i n a l o r t r a n s v e r s e ) i s most n o n e x p o n e n t i a l when t h e t o p i s most p r o l a t e and/or e x h i b i t s f a s t e s t i n t e r n a l m o t i o n . F o r l a r g e m o l e c u l e s (bottom two p l o t s ) , t h e l o n g i t u d i n a l m a g n e t i z a t i o n decays v e r y n e a r l y a c c o r d i n g t o a s i n g l e e x p o n e n t i a l , w h i l e t h e t r a n s v e r s e decay i s marked-l y n o n e x p o n e n t i a l . 27 Two g e n e r a l p r o p e r t i e s commented upon by Runnels a r e m a n i f e s t i n F i g u r e 4.1. F i r s t , t h e i n c l u s i o n o f c r o s s - c o r r e l a t i o n terms always r e -t a r d s t h e r e l a x a t i o n , and s e c o n d , t h e i n i t i a l s l o p e o f each c u r v e ap-p r o a c h e s t h e s l o p e o f t h e s i n g l e e x p o n e n t i a l decay o b t a i n e d when c r o s s -c o r r e l a t i o n i s n e g l e c t e d (see F i g u r e 4.2). -103-T h i s l a t t e r f a c t can be e a s i l y be d e r i v e d from E q u a t i o n s [ 4 . 3 . 6 ] , [ 4 . 3 . 1 4 ] , [ 4 . 3 . 2 0 ] , and [ 4 . 3 . 2 3 ] . I t f o l l o w s d i r e c t l y t h a t f o r s u f -f i c i e n t l y s m a l l T, ( d / d t j y ^ x ) - y ] ( 0 ) [ 2 J a ' - 1 (COQ) - 8 J 2 ' " 2 ( 2 t o Q ) ] and ( d / d t ) q i ( x ) - [-3J°°(-0) + 5 J a ' ~ V 0 ) - 2 J 2 ' " 2 ( 2 a ) ( ) ) ] q 1 ( 0 ) . I t i s seen t h a t t h e i n i t i a l decay i s ind e p e n d e n t o f any c r o s s - c o r r e l a t i o n s p e c t r a l d e n s i t i e s and i s i d e n t i c a l t o t h e r e s u l t o b t a i n e d assuming a d d i t i v e p a i r w i s e i n t e r a c t i o n s . T h i s o b s e r v a t i o n w h i c h i s a n a l y t i c a l l y j u s t i f i e d , has g r e a t p r a c t i c a l i m p l i c a t i o n s . F i n a l l y , i t s h o u l d be noted t h a t i n F i g u r e 4.1 and 4.2, t h e a b s c i s s a i s i n s p e c i f i c u n i t s o f seconds r a t h e r t h a n t h e more g e n e r a l "reduced t i m e " v a r i a b l e as seems t o be t h e common c o n v e n t i o n . 1 ^ ' 2 4 T h i s i s be c a u s e , u n l e s s w o r k i n g w i t h i n t h e c o n f i n e s o f t h e l i m i t 6Dj_ >>WQ, t h e e x p l i c i t f i e l d dependence d e t r a c t s from t h e u s e f u l n e s s o f t h i s form o f p r e s e n t a t i o n . A l t h o u g h t h e s c a l e as shown i n F i g u r e 4.1 i s s p e c i f i c f o r t h e c a s e o f an i s o l a t e d -CH^ group i n a 2.35T f i e l d , t h e graphs a r e s t i l l r a t h e r g e n e r a l i n t h e i r p r e s e n t a t i o n . 19 F o r example, i f t h e g y r o m a g n e t i c r a t i o were s m a l l e r (as f o r F ) , t h e shapes o f a l l c u r v e s would remain t h e same, b u t t h e t i m e s c a l e would be c o n t r a c t e d ( s l o w e r r e l a x a t i o n ) . I f r were s m a l l e r , t h e shapes o f t h e c u r v e s would a g a i n remain t h e same, and t h e t i m e s c a l e would be expanded ( f a s t e r r e l a x a t i o n ) . F o r d i f f e r e n t v a l u e s o f WQ, t h e c u r v e s w i l l a g a i n r e t a i n t h e same s h a p e s , p r o v i d e d t h a t t h e r o t a t i o n a l d i f f u s i o n c o n s t a n t s a r e changed by t h e a p p r o p r i a t e f a c t o r . In F i g u r e 4.2, c u r v e V o f F i g u r e 4.1b i s r e p r o d u c e d ( c u r v e d l i n e ) , and r e s o l v e d i n t o i t s t h r e e component e x p o n e n t i a l s ( t h e t h r e e s o l i d l i n e s ) , and compared t o t h e r e s u l t o b t a i n e d when c r o s s - c o r r e l a t i o n i s -104-n e g l e c t e d ( d o t t e d l i n e ) . The a c t u a l e q u a t i o n o f t h e c u r v e is, ( < I z ( t ) > - < I z > T ) / ( c o s e - l ) < I z > T = 0.125exp(-0.678t) + 0.171exp(-12.5t) + 0 . 7 0 4 e x p ( - 5 . 3 4 t ) . [4.4.12] Note t h a t t h e y - i n t e r c e p t s o f t h e t h r e e s o l i d l i n e s a r e : l o g ( 0 . 1 2 5 ) , 1og(0.125 + 0.704) and l o g ( l ) ; t h i s seems t o be t h e most u s e f u l way t o show t h e l i m i t i n g b e h a v i o r a t l o n g t i m e s . From t h i s t y p i c a l p l o t , i t i s c l e a r t h a t t h e i n i t i a l s l o p e o f t h e c u r v e approaches t h a t o f t h e d o t t e d l i n e . The most g e n e r a l d i s p l a y o f t h e p r e s e n t c a l c u l a t i o n s i s p r o v i d e d by F i g u r e s 4.3 and 4.4, wh i c h show t h e r e l a t i v e c o n t r i b u t i o n s and t i m e c o n s t a n t s o f t h e t h r e e e x p o n e n t i a l s f o r t h e l o n g i t u d i n a l ( F i g u r e 4.3) and t r a n s v e r s e ( F i g u r e 4.4) m a g n e t i z a t i o n s . The p r e - e x p o n e n t i a l f a c t o r s a r e shown i n t h e upper t h r e e p l o t s o f each f i g u r e , and t h e r e l a t i v e t i m e -c o n s t a n t s i n t h e l o w e r two p l o t s . T h i s manner o f p r e s e n t a t i o n s e r v e s t h e same purpose as an e x t e n s i v e d i g i t a l t a b l e o r a t h r e e - d i m e n s i o n a l p l o t d e p i c t i n g t h e mutual e f f e c t s o f s i z e , shape, and i n t e r n a l f l e x i -b i l i t y on t h e r e l a x a t i o n b e h a v i o r o f t h e s p i n s . These two f i g u r e s a r e s i m p l y c o n t o u r p l o t s and a r e i n t e r p r e t e d as one would use a t o p o g r a p h -i c a l map. F o r example, say one w i s h e s t o d e t e r m i n e t h e r e l a x a t i o n be-h a v i o r o f an i s o l a t e d , f l e x i b l e methyl group a t t a c h e d t o t h e backbone o f a l a r g e m o l e c u l e i n s o l u t i o n . F o r t h e purpose o f s e r v i n g as an ex-ample, assume t h a t t h e l a r g e m o l e c u l e i s r o u g h l y s p h e r i c a l i n shape and t h e system can be c h a r a c t e r i z e d by t h e t h r e e d i f f u s i o n c o n s t a n t s , d± = D(l = 1 0 9 sec'J and = 1 0 1 0 secT1 R e f e r r i n g t o F i g u r e 4.3, i t i s -105-f o u n d t h a t A 2 - * 3 - 4A^ and t h e p r e - e x p o n e n t i a l s f o r t h e two d i s t i n c t i v e terms a r e ( A 2 + A^) - 0.92 and A^ - 0.08. Of c o u r s e , t h e same r e l a x a t i o n would be e x p e c t e d i f l O D ^ = D„ = 1 0 1 0 s e c - 1 and D i = 1 0 9 s e c " 1 ( t h i s i s due t o t h e f a c t t h a t f o r t h e s e p l o t s , 8 = 0°, t h e r e f o r e , (D ( | + D.) i s e f f e c t i v e l y a s i n g l e v a r i a b l e ) . F i g u r e s 4.3 and 4.4 a r e most i n f o r m a t i v e when used i n c o n j u n c t i o n w i t h F i g u r e 4.1. By r e f e r r i n g t o F i g u r e 4.1, i t i s now p o s s i b l e t o c o n c l u d e how s m a l l t h e r a t i o o f A -|/x 2 ( a n d / o r A 3 / A 2 ) and a l s o how comparable A^ and A 2 (and/or A^) must be i n magnitude i n o r d e r t h a t r e l a x a t i o n be m a r k e d l y n o n e x p o n e n t i a l . A l s o , o n l y t h e r a t i o s o f t h e e x p o n e n t i a l f a c t o r s a r e shown i n F i g u r e s 4.3 and 4.4. T h e r e f o r e , r e l a t i v e , n o t a b s o l u t e b e h a v i o r i s d e p i c t e d i n t h e s e ' f i g u r e s . A g a i n , F i g u r e 4.1 shows some a b s o l u t e p l o t s f o r f i x e d c h o i c e s o f D., D n, and Dj_. W i t h t h e a i d o f F i g u r e s 4.3 and 4.4, i t i s p o s s i b l e t o make q u i t e p r e -c i s e q u a l i t a t i v e i n t e r p o l a t i o n s o r e x t r a p o l a t i o n s . In o r d e r f o r non-e x p o n e n t i a l b e h a v i o r t o be m a n i f e s t , i t i s n e c e s s a r y both t h a t a t l e a s t two o f t h e p r e - e x p o n e n t i a l f a c t o r s (A's o r B's) be o f t h e same o r d e r o f magnitude and t h a t t h e r a t i o o f t h e c o r r e s p o n d i n g e x p o n e n t i a l s d i f f e r a p p r e c i a b l y from u n i t y . From F i g u r e s 4.3 and 4.4, i t i s seen t h a t non-e x p o n e n t i a l r e l a x a t i o n i s most e v i d e n t f o r v e r y p r o l a t e t o p s and/or when i n t e r n a l r o t a t i o n i s f a s t compared t o D .^ In t h e e x t r e m e - n a r r o w i n g l i m i t (upper r i g h t hand r e g i o n o f each p l o t ) , A 3 and B 3 approach z e r o and/or x 3 A 2 - 1, i n e i t h e r e v e n t , t h e r e s u l t r e d u c e s t o t h e two e x p o n e n t i a l s 16 o b t a i n e d by Hubbard. I t s h o u l d a l s o be mentioned t h a t t h e r e g i o n s o f t h e p l o t s where > (D. + D ( )) have l i t t l e p h y s i c a l meaning s i n c e (even f o r an e x t r e m e l y f l a t t e n e d o b l a t e t o p ) d o e s n ' t d i f f e r a p p r e c i a b l y -106-from t h e v a l u e f o r D , r A f i n a l i n t r i g u i n g f e a t u r e o f t h e t r a n s v e r s e m a g n e t i z a t i o n i s t h a t f o r a [(D„ + D i)/D J_] r a t i o o f 7 /4 , t h e r e l a x a t i o n approaches t h a t o f a s i n g l e e x p o n e n t i a l , t h a t i s , B 2 1 .00 as seen i n F i g u r e 4 . 4 . T h i s b e h a v i o r has been noted f o r t h e extreme-narrowed c a l c u l a t i o n . Examina-t i o n o f i t s s o u r c e f o r t h e nonextreme-narrowed c a s e p r o v i d e s a few more novel p r e d i c t i o n s c o n c e r n i n g t h e m a g n e t i z a t i o n d e c a y s . I f D ( < to n (J?°(0) » -jl'ZVn) * J ? ' ^ 2 ( 2 w n ) ) , t h e s i m p l i f i c a t i o n o f t h e l o n g i -a , C a , C U a , C U t u d i n a l r e l a x a t i o n b e h a v i o r ( E q u a t i o n s [ 4 . 3 . 6 - 8 ] ) i s not r e a d i l y a c h i e v e d . 1/2 However, u n l e s s ( 24D^D i ) 1 > COQ, t h e r e l a x a t i o n i s o b v i o u s l y a p p r o x i -mated v e r y w e l l by a uniq u e e x p o n e n t i a l . In s t a r k c o n t r a s t , f o r t h e t r a n s v e r s e r e l a x a t i o n i n t h i s s l ow m o t i o n l i m i t , E q u a t i o n s [ 4 . 3 . 2 0 - 2 2 ] reduce t o , ( d / d t ) q i ( t ) = - 3 [ J ° ° ( 0 ) + 3J°°(0)] q i(t) + 6 J ° ° ( 0 ) q 3 ( t ) [ 4 . 4 . 1 2 ] ( d / d t ) q 3 ( t ) = 3 [ - J ° ° ( 0 ) + J°°(0)]q 3(t) [ 4 . 4 . 1 3 ] from w h i c h i t f o l l o w s t h a t < I x ( t ) > = <V°)> { e x p ( - 3 ( J a 0 0 ( 0 ) + J ° ° ( 0 ) ) t ) + exp ( -3 ( J a 0 0C0) - J ° ° ( 0 ) ) t ) } . [ 4 . 4 . 1 4 ] I f J ^ ° ( 0 ) = 0 ( c r o s s - c o r r e l a t i o n s p e c t r a l d e n s i t i e s v a n i s h ) , t h i s ex-p r e s s i o n r e d u c e s t o a s i n g l e e x p o n e n t i a l . F o r 6 = 0 ° , e x a m i n a t i o n o f E q u a t i o n [ 4 . 3 . 1 6 ] shows t h a t i f [(D\ + D ^ / D j = 7/4 , J ° ° ( 0 ) does i n d e e d e q u a l z e r o . I t s h o u l d be n o t e d t h a t i r r e g a r d l e s s o f t h e c h o i c e o f g, t h e r e w i l l i n g e n e r a l e x i s t a unique r a t i o o f r o t a t i o n a l c o n s t a n t s f o r -107-w h i c h a l l c r o s s - t e r m s i d e n t i c a l l y v a n i s h . As o b v i o u s l y shown by F i g u r e 4.3, t h i s i s not t r u e f o r l o n g i t u d i n a l r e l a x a t i o n . From E q u a t i o n [4.4.12-13] n o t e t h e n o v e l t y p r e d i c t e d f o r t h e t r a n s -v e r s e decay i f D x < OJQ w i t h t h e f u r t h e r r e q u i r e m e n t , D.. >> Dj_ ( n o t e we make no s t i p u l a t i o n t h a t >> COQ o r t h a t B = 0 ° ! ) . In t h i s c a s e , q ] ( t ) - \ e x p ( - 6 J 0 0 ( 0 ) t ) [4.4.15] q 3 ( t ) - q 3 ( 0 ) = 1/2 . [4.4.16] Indeed, t h e imposed c o n d i t i o n s appear v e r y m i l d w i t h t h e p r e d i c t i o n o f v e r y marked n o n e x p o n e n t i a l decay. T h i s b e h a v i o r i s q u i t e a p p a r e n t i n t h e t h e o r e t i c a l p l o t i n F i g u r e 4.1e. I t i s i n t e r e s t i n g t o comment on t h e r e l a t i o n s h i p between t h e c r o s s - c o r r e l a t i o n s p e c t r a l a m p l i t u d e and t h e i n f l u e n c e o f t h e s e terms. When J , = J , a maximal e f f e c t i s s e e n ; a c f o r J c = 0, no e f f e c t i s seen. In g e n e r a l , t h e c r o s s - c o r r e l a t i o n s p e c -t r a l d e n s i t i e s f a l l between t h e s e l i m i t s , and hence, t h e o b s e r v a b l e e f f e c t w i l l l i e between t h e c o r r e s p o n d i n g e x t r e m i t i e s o f b e h a v i o r . E x t e n s i o n o f t h e t r e a t m e n t t o -CF^ groups f o l l o w s d i r e c t l y . The i n t e r - f l u o r i n e d i s t a n c e i s t y p i c a l l y 2.21+0.04.^, and t h e F m a g n e t o g y r i c r a t i o i s (25179/26753) t h a t f o r p r o t o n s ; t h u s f o r t h e same v a l u e o f 19 and i d e n t i c a l m o l e c u l a r d y n a m i c s , t h e F d i p o l e - d i p o l e r e l a x a t i o n r a t e w i l l be down by a f a c t o r o f ( 2 . 5 2 / 2 . 6 7 ) 4 ( 1 . 7 7 / 2 . 2 1 ) 6 - (1/5) t h a t f o r a p r o t o n methyl group. From F i g u r e s 4.3 and 4.4, i t i s e v i d e n t t h a t t h e b e s t system i n whic h t o o b s e r v e n o n e x p o n e n t i a l decay i n m a g n e t i z a t i o n s h o u l d be a mole-c u l e p o s s e s s i n g a methyl group whose i n t e r n a l r o t a t i o n r a t e i s f a s t -108-compared t o r e o r i e n t a t i o n o f t h e m o l e c u l e as a whole. F o r s m a l l mole-c u l e s i n l i q u i d s t a t e , t h i s means t h a t f o r t h e e f f e c t o f c r o s s - c o r r e l a -t i o n s t o be n o t i c e a b l e , D. w i l l be r e s t r i c t e d t o c a s e s where t h e m e t h y l group r o t a t i o n approaches t h e " f r e e d i f f u s i o n " l i m i t when i s on t h e 1/2 13 -1 o r d e r o f (kT/1 + u , n ) = 1 0 s e c , ( I n i s t h e moment o f i n e r t i a metnyI methyl f o r t h e m ethyl g r o u p ) . J u s t such an example i s p r o v i d e d by l i q u i d a c e t o n i t r i l e ( t h e m o l e c u l e f o r w h i c h c r o s s - c o r r e l a t i o n was f i r s t mani-f e s t e d i n t h e s o l i d s t a t e ); by comparing p r o t o n and n i t r o g e n r e l a x a t i o n t i m e s i n CH^CN, i t may be shown t h a t t h e r o t a t i o n about t h e C-N a x i s i s 43 about t e n t i m e s as f a s t as r e o r i e n t a t i o n o f t h a t a x i s i t s e l f . However, t h e p r o t o n r e l a x a t i o n was o b s e r v e d t o be e x p o n e n t i a l . T h i s i n c o n g r u i t y i s r e s o l v e d by a s t u d y o f t h e t e m p e r a t u r e dependence o f T-j, w h i c h shows t h a t a s p i n - r o t a t i o n i n t e r a c t i o n i s p r e s e n t and p r o v i d e s t h e dominant i n t r a m o l e c u l a r r e l a x a t i o n mechanism above 25°C. M o r e o v e r , i t has r e -44 45 46 c e n t l y been argued ' and d e m o n s t r a t e d t h a t t h e m a g n e t i c f i e l d due t o i n t e r n a l r o t a t i o n can a l s o produce a s p i n - r o t a t i o n i n t e r a c t i o n ; t h u s even f o r methyl groups on l a r g e m o l e c u l e s , t h e p r e s e n c e o f f r e e o r n e a r -l y f r e e i n t e r n a l r o t a t i o n s h o u l d i n t r o d u c e a l a r g e ( e x p o n e n t i a l ) s p i n -r o t a t i o n r e l a x a t i o n c o n t r i b u t i o n w h i c h s h o u l d mask any n o n e x p o n e n t i a l e f f e c t s due t o d i p o l e - d i p o l e i n t e r a c t i o n s . 47 The s p i n - i n t e r n a l r o t a t i o n i n t e r a c t i o n g e n e r a l l y has t h e f o r m ( ^ s p i n - r o t a t i o n = ! kT *~ Vo?SR > £4.4.17] where I i s t h e moment o f i n e r t i a o f t h e i n t e r n a l r o t o r , C i s t h e c o r -a a r e s p o n d i n g i n t e r n a l r o t a t i o n s p i n - r o t a t i o n c o n s t a n t , and x C D i s t h e -109-c o r r e l a t i o n t i m e f o r changes i n magnitude o f t h e s p i n - r o t a t i o n i n t e r -a c t i o n . T<.p i s d i s t i n g u i s h e d i n two ways from t h e ave r a g e c o r r e l a t i o n t i m e , f o r changes i n magnitude o f t h e d i p o l e - d i p o l e i n t e r a c t i o n . F i r s t , may be t h o u g h t o f as t h e ave r a g e t i m e between " c o l l i s i o n s " w h i c h change e i t h e r the p r i n c i p a l a x i s d i r e c t i o n o r r a t e o f t h e i n -t e r n a l r o t a t i o n , so t h a t becomes s h o r t e r as t h e r o t a t i o n a l m o t i o n becomes more h i n d e r e d ( t h i s may be c o n t r a s t e d w i t h X2» w h i c h may be r e -garded as t h e average t i m e i t t a k e s f o r t h e d i p o l e - d i p o l e a x i s t o r e -o r i e n t t h e o r d e r o f a r a d i a n ) . T h e r e f o r e , T2 becomes l o n g e r as the r o t a t i o n becomes more h i n d e r e d , and t h u s , t h e d i p o l e - d i p o l e and s p i n -r o t a t i o n i n t e r a c t i o n s g e n e r a t e o p p o s i t e t e m p e r a t u r e dependence f o r r e -l a x a t i o n , w h i c h p r o v i d e s a means o f r e s o l v i n g t h e i r r e l a t i v e c o n t r i b u t i o n s . Second, x<.R i s i n g e n e r a l s h o r t e r t h a n T^. W h i l e t h e i m p o r t a n c e o f 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 i s 47 n o t o r i o u s l y d i f f i c u l t t o q u a n t i t a t i v e l y a s s e s s i n advance, t h e form o f 48 49 t h e r e l a x a t i o n i s w e l l e s t a b l i s h e d f o r r o t a t i o n a l d i f f u s i o n o f s p h e r i c a l , ' 50 51 52 and symmetric ' ' t o p s , and r e c e n t l y f o r a more g e n e r a l m o t i o n a l model 53 f o r symmetric t o p s . In any c a s e , t h e r e i s w e l l documented e v i d e n c e t h a t s p i n - r o t a t i o n i s an i m p o r t a n t c o n t r i b u t i o n t o r e l a x a t i o n f o r b o t h -CH^ 48 54 46 58 groups ' and -CF^ groups ' on s m a l l m o l e c u l e s i n l i q u i d s . There i s , however, one s i t u a t i o n i n w h i c h n o n e x p o n e n t i a l decay s h o u l d be o b s e r v a b l e i n l i q u i d s . In o r d e r t o emphasize t h e d i p o l e - d i p o l e i n t e r a c t i o n compared t o t h e s p i n i n t e r n a l r o t a t i o n , systems w i t h r e l a t i v e l y h i n d e r e d i n t e r n a l m o t i o n ( y e t r a p i d compared t o t h e r e o r i e n t a t i o n o f t h e m o l e c u l e as a whole) p r o v i d e s p r o m i s i n g s u b j e c t s . F u r t h e r m o r e , i f t h e o v e r a l l m o t i o n i s on t h e -no-o r d e r o f t h e Larmor f r e q u e n c y , i t has been shown t h a t t h e t r a n s v e r s e decay i s v e r y s e n s i t i v e t o t h e e f f e c t s o f f i n i t e c r o s s - c o r r e l a t i o n terms ( i n t h e same l i m i t , t h e l o n g i t u d i n a l r e l a x a t i o n i s a l m o s t i n d e p e n d e n t o f t h e s e t e r m s) and m i g h t p r o v i d e o b s e r v a b l e e f f e c t s i n many s t u d i e s . -111-FIGURE 4.1: P l o t s o f m a g n e t i z a t i o n ( d e v i a t i o n from t h e r m a l e q u i l i b r i u m ) v e r s u s t i m e , f o l l o w i n g a e-degree p u l s e d i r e c t e d a l o n g t h e y - a x i s o f a r o t a t i n g f r a m e , f o r a methyl group a t t a c h e d a l o n g t h e symmetry a x i s o f a symmetric t o p m o l e c u l e . F o r p l o t s (b) and ( d ) , C ( t ) = f o r p l o t s ( c ) and ( e ) . < I z ( t ) > - <I Z> T (cose - 1) <I z> T 5 ( t ) = < I x ( t ) > s i n e <I Z> ^ f o r p l o t ( a ) , l o n g i t u d i n a l = t r a n s v e r s e m a g n e t i z a t i o n = ? ( t ) . 8 - 1 0 F o r a l l p l o t s , ion = 2TT X 10 sec , r ^ _ ^ = 1.8A . F o r p l o t ( a ) , D± = 1 0 1 0 s e c " 1 ; f o r p l o t s (b) and ( c ) , Dj_ = 1 0 8 s e c " 1 ; f o r p l o t s (d) and ( e ) , = 1 0 7 s e c " 1 . F o r p l o t ( a ) , c u r v e s P,Q,...V c o r r e s p o n d t o r e s p e c t i v e v a l u e s o f [D. + D ( J] o f 1, ( 7 / 4 ) , 4, 6, 10, 20, and 100 x 1 0 1 0 s e c " 1 ; f o r p l o t s (b) and (c) ( o r (d) and ( e ) ) , c u r v e s P,Q,...V c o r r e s p o n d t o t h e same [9D.j + D(( ) / D x ] - r a t i o s as f o r p l o t ( a ) . See t e x t f o r f u r t h e r e x p l a n a t i o n ' ' and d i s c u s s i o n . -1 1 2 --113-FIGURE 4.2: R e s o l u t i o n o f a p l o t o f t h e l o n g i t u d i n a l m a g n e t i z a t i o n v e r s u s t i m e i n t o i t s ( t h r e e ) component e x p o n e n t i a l s . Curve denotes p l o t V o f F i g u r e 4 . 1 ( b ) . D o t t e d l i n e d e n o t e s l i n e a r r e s u l t o b t a i n e d by n e g l e c t i n g c r o s s - c o r r e l a t i o n e f f e c t s . The t h r e e s o l i d l i n e s ( i n o r d e r o f i n c r e a s i n g s l o p e ) a r e p l o t s o f l o g ( 0 . 1 2 5 e x p [ - 0 . 6 7 8 t ] ) 5 l o g ( ( 0 . 1 2 5 + 0 . 7 0 4 ) e x p [ - 5 . 3 4 t ] ) , and l o g ( e x p [ - 1 2 . 5 t ] ) v e r s u s t . See E q u a t i o n [4.4.12] and accompanying d i s c u s s i o n . -115-FIGURE 4.3: N o n - e x p o n e n t i a l i t y o f t h e t i m e - d e c a y f o r t h e l o n g i t u d i n a l m a g n e t i z a t i o n , d i s p l a y e d as c o n t o u r p l o t s , where each c o n t o u r i s a l i n e o f c o n s t a n t magnitude f o r e i t h e r p r e -e x p o n e n t i a l f a c t o r s (A.) o r e x p o n e n t i a l r a t i o s A . / X . I J K from E q u a t i o n [ 4 . 3 . 1 7 ] , as a f u n c t i o n o f D,, and (D ( | + D..). -117-FIGURE 4.4: N o n - e x p o n e n t i a l i t y o f t h e t i m e - d e c a y f o r t h e t r a n s v e r s e m a g n e t i z a t i o n , d i s p l a y e d as c o n t o u r p l o t s , where each c o n t o u r i s a l i n e o f c o n s t a n t magnitude f o r e i t h e r p r e -e x p o n e n t i a l f a c t o r s (B.) o r e x p o n e n t i a l r a t i o s Y-/Y|, I J K from E q u a t i o n [ 4 . 3 . 2 4 ] , as a f u n c t i o n o f and (D(( + D.. ). -119-4.5 SUMMARY I t has been shown t h a t i n t r a m o l e c u l a r d i p o l e - d i p o l e i n d u c e d r e l a x -a t i o n f o r an i s o l a t e d methyl group can be m a r k e d l y n o n e x p o n e n t i a l , e s -p e c i a l l y f o r i n t e r n a l l y r o t a t i n g methyl ( o r t r i f l u o r o m e t h y l ) groups on medium s i z e d m o l e c u l e s i n s o l u t i o n . F o r l a r g e m o l e c u l e s i n s o l u t i o n , i t has been shown t h a t t h e t r a n s v e r s e r e l a x a t i o n may be e x t r e m e l y non-e x p o n e n t i a l , whereas a u n i q u e may be used t o c h a r a c t e r i z e t h e l o n g -i t u d i n a l r e l a x a t i o n . However, u n l e s s t h e i n t e r n a l m o t i o n i t s e l f i s a p p r e c i a b l y h i n d e r e d , t h e s p i n - r o t a t i o n r e l a x a t i o n w i l l c o n t r i b u t e and p r o b a b l y mask any non-e x p o n e n t i a l e f f e c t e x p e r i m e n t a l l y . Thus, t o d e t e c t n o n e x p o n e n t i a l decay i n m a g n e t i z a t i o n , t h e methyl group i n t e r n a l r o t a t i o n s h o u l d be f a s t com-pared t o r o t a t i o n o f t h e m o l e c u l e as a w h o l e , but s l o w compared t o a " f r e e d i f f u s i o n " model. F o r -CF^ g r o u p s , t h e e f f e c t w i l l p r o b a b l y n e v e r be o b s e r v a b l e i n l i q u i d s , because t h e d i p o l e - d i p o l e r e l a x a t i o n r a t e i s s l o w e r by a f a c t o r o f f i v e , and t h e s p i n - r o t a t i o n c o u p l i n g i s much l a r g e r t h a n f o r p r o t o n s . F o r -CF^, n o n e x p o n e n t i a l r e l a x a t i o n would be e x p e c t e d o n l y when t h e i n t e r n a l r o t a t i o n o f t h e -CF^ group i s v e r y " h i n d e r e d " y e t t h e r o t a t i o n a l a n i s o t r o p y d i f f e r s a p p r e c i a b l y from u n i t y . There i s one o t h e r way i n wh i c h f l u o r i n e m a g n e t i c r e l a x a t i o n m i g h t be n o n e x p o n e n t i a l , and t h a t i s due t o p o s s i b l e c r o s s - c o r r e l a t i o n e f f e c t s between competing r e l a x a t i o n p r o c e s s e s . T h i s problem i s f o r m a l l y s i m i l a r t o t h e p r e s e n t c a l c u l a t i o n and w i l l be t r e a t e d i n t h e n e x t c h a p t e r . F i n a l l y , i t has been shown t h a t a p l o t o f e i t h e r t h e t r a n s v e r s e o r l o n g i t u d i n a l m a g n e t i z a t i o n v e r s u s t i m e g i v e s an i n i t i a l s l o p e w h i c h i s -120-t h e same as would be o b t a i n e d by n e g l e c t i n g a l l c r o s s - c o r r e l a t i o n e f f e c t s , f u r t h e r compounding t h e e x p e r i m e n t a l d i f f i c u l t y o f d e t e c t i n g nonexpon-e n t i a l i t y ( a l t h o u g h making a s i m p l e i n t e r p r e t a t i o n e a s i e r ) . N e v e r t h e l e s s , an i n t e r n a l l y r o t a t i n g methyl group on a l a r g e m o l e c u l e i n s o l u t i o n s h o u l d show n o n e x p o n e n t i a l r e l a x a t i o n . The f a c t t h a t n o n - e x p o n e n t i a l r e l a x a t i o n has n o t been r e p o r t e d f o r such systems i s v e r y u n d e r s t a n d a b l e . The e f f e c t s p r e d i c t e d , a l t h o u g h q u i t e d r a m a t i c , a r e v e r y tenuous and a r e e a s i l y quenched by s p i n - i n t e r n a l r o t a t i o n o r i n t e r m o l e c u l a r d i p o l a r i n t e r -a c t i o n s . T h i s b e i n g s o , one immediate s o u r c e o f knowledge t h e p r e s e n c e , o r absence o f t h e s e e f f e c t s p r o v i d e s can be l i k e n e d t o t h e i n f o r m a t i o n p r o v i d e d by Ov e r h a u s e r enhancements; a means t o d e t e r m i n e t h e magnitude o f t h e i n t r a m o l e c u l a r d i p o l a r r e l a x a t i o n pathways i n comparison w i t h t h e o v e r a l l r e l a x a t i o n r a t e . -121-REFERENCES: CHAPTER IV 1. L. G. Werbelow and A. G. M a r s h a l l , J . Mag. Res. 1J_, 299 (1 9 7 3 ) . 2. E. W. B i t t n e r and J . T. G e r i g , J . Amer. Chem. Soc. 92_, 5001 (1 9 7 2 ) . 3. B. D. Sykes and M. S c o t t , Ann. Rev. B i o p h y s . B i o e n g . 1_, 251 (1 9 7 2 ) . 4. B. D. S y k e s , P. G. S c h m i d t , and G. R. S t a r k , J . B i o l . Chem. 245, 1180, (1 9 7 0 ) . 5. R. L. H i l t and P. S. Hubbard, Phys. Rev. 134, A392 ( 1 9 6 4 ) . 6. M. F. Baud and P. S. Hubbard, Phys. Rev. 170, 384 ( 1 9 6 8 ) . 7. S. A l b e r t and J . A. R i p m e e s t e r , J . Chem. Phys. 57, 2641 (1 9 7 2 ) . 8. J . L. C a r o l a n and T. A. S c o t t , J . Magn.Res. 2, 243 (1 9 7 0 ) . 9. K. van P u t t e and G. J . N. Egmond, J . Magn. Res. 4, 236 (1971). 10. K. van P u t t e , J . Magn. Res. 5, 367 (1 9 7 1 ) . 11. L. J . B u r n e t t and B. H. M u l l e r , Chem. Phys. L e t t . 18, 553 (1 9 7 3 ) . 12. G. A. DeWitt and M. Bloom, Can. J . Phys. 47, 1195 (1 9 6 9 ) . 13. M. Meh r i n g and H. Raber, J . Chem. Phys. 59, 1116 (1 9 7 3 ) . 14. J . C u t n e l l and L. V e r d u i n , J . Chem. Phys. 59, 259 ( 1 9 7 3 ) . 15. P. S. A l l e n , A. W. K. Khazada, and C. A. M c D o w e l l , M o l . Phys. 25, 1273 (19 7 3 ) . 16. P. S. Hubbard, J . Chem. Phys. 51_, 1647 (1969). 17. P. S. Hubbard, J . Chem. Phys. 52, 563 (1 9 7 0 ) . 18. P. S. Hubbard, Phys. Rev. 109, 1153 ( 1 9 5 8 ) ; i b i d . , JJJ_, 1746 ( 1 9 5 8 ) ; i b i d . , Ph. D. T h e s i s , H a r v a r d , 1958. 19. I . V. A l e k s a n d r o v , S o v i e t Phys. Doklady 3_, 110 (1 9 5 8 ) . 20. P. S. Hubbard, Phys. Rev. 128, 650 (1 9 6 2 ) . 21. P r o f . S. Emid, p r i v a t e c o m munication. 22. P r o f . M. Bloom, p r i v a t e communication -122-23. A. Abragam, The P r i n c i p l e s o f N u c l e a r Magnetism, C l a r e n d o n P r e s s , O x f o r d , 1961; page 293. 24. G. W. K a t t a w a r and M. E i s n e r , Phys. Rev. 126, 1054 (1962). 25. M. D. Z e i d l e r , Ber. Bunsenges. physik.Chem. 72. 481 (1968). 26. P. M. R i c h a r d s , Phys. Rev. ]3Z, 27 ( 1 9 6 3 ) . 27. L. K. R u n n e l s , Phys. Rev. 134, A28 (1964). 28. D. F e n z k e , Ann. P h y s i k ' 1 6 281 ( 1 9 6 5 ) ; D. Fenzke and H. S c h n e i d e r , Ann. P h y s i k 19., 321 (1967). 29. N. C. P y p e r , M o l . Phys. 21_, 1 (1971). 30. N. C. P y p e r , M o l . Phys. 22, 433 (1971). 31. W. Buchner, J . Magn. Res. 11_, 46 (1973). 32. W. Buchner, J . Magn. Res. J_2, 82 (1973). 33. H. S c h n e i d e r , Ann. P h y s i k 1 3 , 313 (1963). 34. H. S c h n e i d e r , Ann. P h y s i k 1 6 , 135 (1965). 35. H. S c h n e i d e r , Z. N a t u r f o r s c h . 19a, 510 (1964). 36. J . S. B l i c h a r s k i and H. S c h n e i d e r , Ann. P h y s i k 22, 306 ( 1 9 6 9 ) . 37. H. S c h n e i d e r and J . S. B l i c h a r s k i , Ann. P h y s i k 23, 139 (1969). 38. J . H. N o g g l e , J . Bhys. Chem. 72. 1324 (1968). 39. K. F. Kuhlmann, D. M. G r a n t , and R. K. H a r r i s , J . Chem. Phys. 52, 3439 (1970). 40. R. G. L a w l e r , p r i v a t e c o m munication. 41. W. Buchner, J . Magn. Res. 4, 90 (1971). 42. E. W e l l s , p r i v a t e communication. 43. D. E. Woessner, B. S. Snowden, and E. T. Strom, M o l . Phys. 14, 265 (1968). 44. A. S. D u b l i n and S. I . Chan, J . Chem. Phys. 46, 4533 (1968). 45. C. T. Schmidt and S. I . Chan, J . Magn. Res. 5_, 151 (1971). -123-46. T. E. Burke and S. I . Chan, J . Magn. Res. 2, 120 (1 9 7 0 ) . 47. C. D e v e r e l l , M o l . Phys. 18, 319 (1970). 48. D. K. Green and J . G. P o w l e s , P r o c . Phys. Soc. 85, 87 (1965). 49. P. S. Hubbard, Phys. Rev. 131_, 1155 (1963). 50. H. J . Bender and M. D. Z e i d l e r , B er. Bunsenges. p h y s i k Chem. 75.* 236 (1971 ). 51. C. H. Wang, D. M. G r a n t , and J . R. L y e r l a , J . Chem. Phys 55, 4674 (1971 ). 52. C. H. Wang, J . Magn. Res. 9, 75 (1973). 53. R. E. D. McClung, J . Chem. Phys. 57, 5478 ( 1 9 7 2 ) . 54. R. G. P a r k e r and J . J o n a s , J . Magn. Res. 6_, 106 (1 9 7 2 ) . -124-CHAPTER V INFLUENCE OF FINITE CROSS-CORRELATION TERMS BETWEEN PHYSICALLY DISTINCT RELAXATION MECHANISMS 1 5.1 INTRODUCTION Fo r n u c l e a r m a g n e t i c r e l a x a t i o n s t u d i e s i n t h e 1 i q u i d s t a t e , i t i s i n v a r i a b l y o b s e r v e d t h a t t h e l o n g i t u d i n a l and t r a n s v e r s e components o f m a g n e t i z a t i o n o f a p e r t u r b e d s p i n system decay e x p o n e n t i a l l y t o t h e i r r e s p e c t i v e t h e r m a l e q u i l i b r i u m v a l u e s . These decays can thus be c h a r -a c t e r i z e d by t h e r e s p e c t i v e r e l a x a t i o n c o n s t a n t s and x^ ( o r by t h e commonly quoted t i m e c o n s t a n t s T^ ^ = l.A-| 2)' * n c o n t r a s t t o e x p e r i m e n t , t h e o r y p r e d i c t s t r u l y e x p o n e n t i a l r e l a x a t i o n o f e i t h e r p r o c e s s t o be t h e e x c e p t i o n , not t h e r u l e . A p r i o r i , i t i s p o s s i b l e t o p r e d i c t true_ ex-p o n e n t i a l decay o n l y i n t h e c a s e where t h e s t a t i c Zeeman H a m i l t o n i a n c o n s i s t s o f a s i n g l e p a i r o f energy l e v e l s * ( i . e . t h e s p i n - s p a c e i s f u l l y spanned by two e i g e n k e t s ) , a d i s c o n c e r t i n g f a c t f o r t h e p e r f e c t i o n -i s t . F o r example, c o n s i d e r t h e q u a d r u p o l a r r e l a x a t i o n o f a s i n g l e n u c l e u s * A n o v e l c a l c u l a t i o n j e o p a r d i z e s even t h i s s i m p l e s t a t e m e n t . I t has been s u g g e s t e d t h a t t h e p r e s e n c e o f h i d d e n v a r i a b l e s i n - quantum mech-a n i c s would cause a r e l a x i n g s p i n 1/2 n u c l e u s t o approach e q u i l i b r i u m i n a n o n e x p o n e n t i a l f a s h i o n (however s l i g h t ) . W i s h i n g t o a v o i d t h e p h i l o s o p h i c a l and e p i s t e m o l o g i c a l i m p l i c a t i o n s o f t h i s t o p i c , we o n l y make note o f t h i s u n i q u e paper: R.K. Wangsness, Phys. Rev. 160, 1190 (19 6 7 ) . -125-w i t h s p i n I > 1. I f t h e extreme-narrowed l i m i t i s v i o l a t e d , i t i s known t h a t a u n i q u e ^ does not e x i s t and t h e r e l a x a t i o n i s g i v e n by a sum o f two o r more e x p o n e n t i a l s (see A ppendix C f o r f u l l d e t a i l s ) . As a n o t h e r c a s e i n p o i n t , c o n s i d e r t h e s i t u a t i o n where both q u a d r u p o l a r and c h e m i c a l s h i f t a n i s o t r o p y i n t e r a c t i o n s r e l a x a s i n g l e n u c l e a r s p i n ( f o r n u c l e i c o n t a i n i n g a l a r g e number o f n u c l e o n s , where th e c h e m i c a l s h i f t a n i s o t r o p y may be on t h e o r d e r o f t e n s o f thousands o f ppm, t h i s example i s by no means c o n t r i v e d ) . A g a i n , m u l t i e x p o n e n t i a l r e l a x a t i o n would r e s u l t from t h e i n t e r f e r e n c e between t h e s e competing r e l a x a t i o n mechanisms. I s o l a t e d one s p i n systems a r e r a r e ; t h e y a r e i n c l u d e d as examples i n t h i s i n t r o d u c t i o n o n l y t o p o i n t out even i n t h e s i m p l e s t o f s p i n s y s t e m s , n o n e x p o n e n t i a l r e l a x a t i o n i s a p o s s i b i l i t y . F o r a t w o - s p i n s y s t e m , t h e d i s c u s s i o n i s not c o n f i n e d t o such l i m i t e d examples. I f t h e two s p i n s a r e i s o c h r o n o u s s p i n 1/2 n u c l e i , p a i r w i s e ( o r m u l t i - ) i n t e r f e r e n c e terms between th e i n d i r e c t d i p o l a r o r d i r e c t d i p o l a r , and t h e c h e m i c a l s h i f t a n i s o t r o p y i n t e r a c t i o n s g i v e r i s e t o n o n e x p o n e n t i a l b e h a v i o r . I f t h e two n u c l e i a r e " u n l i k e " and t h e r e i s e i t h e r a ( d i r e c t ) d i p o l a r o r s c a l a r ( i n d i r e c t d i p o l a r ) c o u p l i n g so t h a t one n u c l e u s may "sample" t h e s t a t e o f t h e o t h e r n u c l e u s , t h e n c r o s s - r e l a x a t i o n c a u s e s t h e m a g n e t i z a t i o n decay o f each s p i n t o be b i -4 e x p o n e n t i a l even i n t h e absence o f a l l i n t e r f e r e n c e e f f e c t s . F o r a t h r e e - s p i n s y s t e m , i n c l u s i v e c a l c u l a t i o n s b e g i n t o become unmanageable and t h e p o s s i b i l i t y o f t r u e e x p o n e n t i a l r e l a x a t i o n o f any g r o u p i n g o f i n d i s t i n g u i s h a b l e s p i n s becomes remote, a l t h o u g h i n most i n s t a n c e s , t h e d e v i a t i o n i s e x p e r i m e n t a l l y u n o b s e r v a b l e . As t h e s i m -p l e s t example o f a t h r e e - s p i n s y s t e m , c o n s i d e r t h r e e i d e n t i c a l s p i n 1/2 -126-n u c l e i . Even i f o n l y t h e d i p o l a r i n t e r a c t i o n i s o p e r a t i v e , i n t e r f e r e n c e terms a r i s e between t h e d i f f e r e n t p a i r w i s e i n t e r a c t i o n s , as f o r t h e two d i p o l e - d i p o l e i n t e r a c t i o n s f o r any one p r o t o n i n a methyl group. Of c o u r s e , i f o t h e r r e l a x a t i o n mechanisms c o n t r i b u t e t o t h e decay o f mag-n e t i z a t i o n and/or t h e t h r e e s p i n s a r e not " a l i k e " and/or one o r more o f t h e s p i n s have 1 ^ 1 , t h e s i t u a t i o n i s t h a t much more c o m p l i c a t e d and c o n s i d e r a t i o n must be g i v e n t o such o b s c u r e p o s s i b i l i t i e s as i n t e r f e r e n c e terms a r i s i n g from r e l a x a t i o n due t o s c a l a r c o u p l i n g o f t h e 2nd k i n d f o r two n u c l e i c o u p l e d t o a common n u c l e u s w i t h a n o n v a n i s h i n g q u a d r u p o l e moment.^ To summarize, n o n e x p o n e n t i a l ( i . e . m u l t i e x p o n e n t i a l ) r e l a x a t i o n f o r nonexchanging systems w i l l i n g e n e r a l , be e x p e c t e d when t h e r e i s , (1) q u a d r u p o l a r r e l a x a t i o n i n t h e nonextreme narrowed l i m i t , (2) c r o s s -r e l a x a t i o n , * o r (3) i n t e r f e r e n c e terms o f t h e f i r s t o r second k i n d . I n t e r f e r e n c e terms o f t h e f i r s t k i n d , w h i c h might a p p r o p r i a t e l y be c a l l e d "Hubbard t e r m s " o r "homo-terms", a r i s e from a s i n g l e r e l a x a t i o n mechanism o p e r a t i v e on d i f f e r e n t n u c l e a r s p i n s . I n t e r f e r e n c e terms o f t h e second k i n d , w h i c h m i g h t be c a l l e d " B l i c h a r s k i t e r m s " o r " h e t e r o -t e r m s " , a r i s e from two o r more d i f f e r e n t r e l a x a t i o n mechanisms o p e r a t i v e a t t h e same n u c l e u s . I n t e r f e r e n c e terms o f t h e " t h i r d k i n d " , d i f f e r e n t r e l a x a t i o n mechanisms on d i f f e r e n t c e n t e r s ( e . g . i n d i r e c t d i p o l a r i n t e r -f e r e n c e w i t h d i r e c t d i p o l a r ) , a r e not g i v e n f u r t h e r c o n s i d e r a t i o n as * Sometimes i t i s found i n l i t e r a t u r e t h a t t h e term c r o s s - r e l a x a t i o n i s used t o denote changes i n r e l a x a t i o n t i m e s r e s u l t i n g from t h e i n c o r -p o r a t i o n o f c r o s s - c o r r e l a t i o n f u n c t i o n s i n t o t h e r e l a x a t i o n e q u a t i o n s . Ft T h i s i s u n f o r t u n a t e as the two terms s h a r e no r e l a t i o n t o each o t h e r . -127-t h e y appear t o be o f so l i t t l e i m p o r t a n c e a t p r e s e n t . Hubbard terms c o u l d a l s o be d e f i n e d as i n t e r f e r e n c e terms between r e l a x a t i o n p a t h -ways c h a r a c t e r i z e d by i n t e r a c t i o n c o n s t a n t s o f t h e same f u n c t i o n a l form whereas B l i c h a r s k i terms a r e o f d i f f e r e n t f u n c t i o n a l form. A c t u a l l y , t h e d i v i s i o n o f i n t e r f e r e n c e terms i n t o two s u b c l a s s e s i s c o m p l e t e l y a r b i t r a r y s i n c e both s h a r e a common h e r i t a g e , t h e y both r e s u l t from non-z e r o c r o s s - c o r r e l a t i o n terms o f one m a g n e t i c i n t e r a c t i o n w i t h a n o t h e r m a g n e t i c i n t e r a c t i o n . The c l a s s i f i c a t i o n i s h i s t o r i c a l , due t o two d i r e c t i o n s o f c o n c e n t r a t i o n i n t h e l i t e r a t u r e . A l t h o u g h c r o s s - c o r r e l a t i o n terms w i l l be p r e s e n t whenever t h e s p i n system has t h r e e o r more r i g i d l y f i x e d s p i n s o r has a n u c l e u s b e i n g r e -l a x e d by two d i f f e r e n t mechanisms " o f t h e same c o r r e l a t i o n t i m e " , t h e e x p e r i m e n t a l r a r i t y o f n o n e x p o n e n t i a l r e l a x a t i o n i m p l i e s t h a t t h e ap-p r o p r i a t e c r o s s - t e r m s a r e v a n i s h i n g l y s m a l l o r e n t e r i n t o t h e c a l c u l a -t i o n s i n such a c l a n d e s t i n e f a s h i o n as t o m i n i m i z e t h e i r i m p o r t a n c e . T h e r e f o r e , t h e u s u a l approach t o r e l a x a t i o n c a l c u l a t i o n s i s t o s i m p l y n e g l e c t t h e s e terms w h i c h r e s u l t i n m u l t i e x p o n e n t i a l b e h a v i o r . I t i s t r u l y u n f o r t u n a t e t h a t n a t u r e has been so b e n e v o l e n t t o t h e exp e r i m e n -t a l i s t because t h e e x p e r i m e n t a l o b s e r v a t i o n o f i n t e r f e r e n c e e f f e c t s would g r e a t l y enhance our u n d e r s t a n d i n g o f t h e r e l a x a t i o n p r o c e s s and a l s o p r o v i d e an a d d i t i o n a l avenue t o o b t a i n c h e m i c a l and p h y s i c a l i n -f o r m a t i o n from r e l a x a t i o n measurements. However, i n f u t u r e s t u d i e s , such a s s u m p t i o n s may v e r y l i k e l y p r ove unwarranted and. t h e more c o r r e c t d e s c r i p t i o n w i l l need be a p p l i e d . The purpose o f t h i s c h a p t e r i s t o examine t h e e f f e c t and i m p o r t a n c e -128-o f t h e s e i n t e r f e r e n c e o r c r o s s - c o r r e l a t i o n t e r m s , (a t a s k begun i n C h a p t e r IV) and t o j u s t i f y such a c a l c u l a t i o n a f t e r a d m i t t i n g t o t h e p r e s e n t e x p e r i m e n t a l o d d i t y o f such e f f e c t s . An immediate j u s t i f i c a t i o n f o r c o n c e r n stems from t h e p r e s e n t t r e n d toward usage o f l a r g e s t a t i c f i e l d s w i t h d a t a a q u i s i t i o n by way o f a ti m e domain s i g n a l ( F o u r i e r T r a n s f o r m t e c h n i q u e s ) . The immediate consequence i s i n c r e a s e d s e n s i -t i v i t y w h i c h means l a r g e r m o l e c u l e s ( b i o m o l e c u l e s ) can be s t u d i e d on a r o u t i n e b a s i s . I n c r e a s e d s e n s i t i v i t y a l s o means a l a r g e v a r i e t y o f n u c l e i can be f e a s i b l y s t u d i e d by t h e NMR t e c h n i q u e , and p r e v i o u s r e -l a x a t i o n s t u d i e s , t h e m a j o r i t y o f wh i c h were done on p r o t o n s , s h o u l d by no means p r e j u d i c e f u t u r e i n t e r p r e t a t i o n s o r d i c t a t e a "norm" f o r r e -l a x a t i o n b e h a v i o r . S e c o n d l y , an o f t e n d i s c u s s e d consequence o f l a r g e f i e l d s i s t h a t t h e c h e m i c a l s h i f t a n i s o t r o p y i n t e r a c t i o n i n c r e a s e s as t h e second power o f t h e a p p l i e d f i e l d so t h a t p e c u l i a r i t i e s r e l a t i n g t o t h i s pathway o f r e l a x a t i o n may become m a n i f e s t . E x i s t i n g c a l c u l a t i o n s d e a l i n g w i t h t h e e f f e c t o f c r o s s - t e r m s a r e l i m i t e d i n v a l i d i t y on two grounds: They t r e a t o n l y i s o t r o p i c m o l e c u l a r r e o r i e n t a t i o n and a r e o f t e n based on an a s s u m p t i o n o f r a p i d m o t i o n . How-e v e r , t h e m o t i o n o f a s p i n o r a s e t o f i s o l a t e d s p i n s a t t a c h e d t o a b u l k y framework (such as a s m a l l b i o p o l y m e r ) i s l i k e l y t o be h i g h l y a n i s o t r o p i c (due t o i n t e r n a l m o t i o n s ) and/or f a i l t o s a t i s f y t h e extreme-narrowed a p p r o x i m a t i o n . In p a r t i c u l a r , t h e r e m a i n d e r o f t h i s c h a p t e r w i l l be c o n c e r n e d w i t h e x a m i n i n g t h e e f f e c t o f a n i s o t r o p i c m o t i o n o f t h e s p i n system and t h e f a i l u r e o f t h e extreme-narrowed a p p r o x i m a t i o n when c r o s s - c o r r e l a t i o n -129-terms a r e i n c l u d e d i n t h e r e l a x a t i o n c a l c u l a t i o n s . O n l y " B l i c h a r s k i t e rms" w i l l be d i s c u s s e d , s i n c e t h e p r e v i o u s c h a p t e r d i s c u s s e d "Hubbard te r m s " t r e a t e d i n t h e same m o t i o n a l l i m i t s . 7 I t i s w e l l known t h a t h i g h l y a n i s o t r o p i c m o t i o n has pronounced e f f e c t s on t h e i m p o r t a n c e o f Hubbard terms and hence t h e p r e s e n t e x t e n s i o n s h o u l d prove i n t e r e s t i n g . To h e l p o r i e n t t h e r e a d e r , t h e second s e c t i o n i s d e v o t e d t o a u s e f u l resume o f p r e v i o u s papers c o n c e r n i n g c r o s s - c o r r e l a t i o n c a l c u -l a t i o n s . * In s e c t i o n t h r e e and f o u r , t h e t h e o r y i s f o r m u l a t e d w i t h i n t h e framework o f t h e s e m i - c l a s s i c a l r e l a x a t i o n m a t r i x and t h e r e s u l t a n t c a l c u l a t i o n s a r e p r e s e n t e d , f o l l o w e d by a d i s c u s s i o n o f major c o n c l u s i o n s i n S e c t i o n 5. * U n f o r t u n a t e l y t h i s approach i s n e c e s s i t a t e d as l i t e r a t u r e a c c o u n t s o f p r e v i o u s work on a l l f a c e t s o f c r o s s - c o r r e l a t i o n terms compose a g l a r i n g o m i s s i o n from p u b l i s h e d work a l t h o u g h two r e v i e w s a r e i n t h e p l a n n i n g ; J.S. B l i c h a r s k i i n Advan. Magn. Res. and L.G. Werbelow i n i n Advan. M o l . R e l a x . P r o c e s s e s . -130-5.2 RESUME OF PREVIOUS STUDIES In t h i s s e c t i o n , p r e v i o u s e x p o s i t i o n s o f t h e problem o f c r o s s -c o r r e l a t i o n terms a r i s i n g i n t h e d e n s i t y m a t r i x t h e o r y o f n u c l e a r mag-n e t i c r e l a x a t i o n a r e r e v i e w e d . T h i s b r i e f summary i s n o t i n t e n d e d t o be e x h a u s t i v e , but r a t h e r t o s e r v e as a g u i d e t o r e l e v a n t background and p e r i p h e r a l m a t e r i a l . As an i n t r o d u c t i o n , i t may be s a i d t h a t i f t h e n u c l e a r r e l a x a t i o n p r o c e s s i s dominated by v a r i o u s i n t e r a c t i o n s w i t h t he "same c o r r e l a t i o n t i m e " , c r o s s - c o r r e l a t i o n ( i n t e r f e r e n c e ) terms w i l l a p pear i n t h e r e l a x -a t i o n e q u a t i o n s . E x a c t l y what i s meant by t h e "same c o r r e l a t i o n t i m e " w i l l be c l a r i f i e d l a t e r i n t h e t e x t . The p h y s i c a l e f f e c t o f t h e s e c r o s s - t e r m s i s t o cause a p e r t u r b e d n u c l e a r s p i n system t o r e l a x i n a n o n e x p o n e n t i a l f a s h i o n . The f i r s t i n t e r e s t i n c r o s s - c o r r e l a t i o n e f f e c t s a r o s e i n a n s w e r i n g c e r t a i n e x p e r i m e n t a l q u e s t i o n s c o n n e c t e d w i t h e l e c t r o n i c r e l a x a t i o n . In ESR, t h e rrij dependence o f l i n e w i d t h s was e x p l a i n e d by n o t i n g t h a t c r o s s -terms between t h e g - t e n s o r a n i s o t r o p y and t h e e l e c t r o n - n u c l e a r d i p o l a r o i n t e r a c t i o n s c o u l d l e a d t o such b e h a v i o r . V e r y good summaries on t h e s u b j e c t o f c r o s s - c o r r e l a t i o n t e r m s , and o f m a g n e t i c r e l a x a t i o n i n g e n e r a l 9 9 10 a r e t h e a r t i c l e s by F r e e d and F r a e n k e l . ' R e f e r e n c e 10 i n p a r t i c u l a r d i s c u s s e s a t g r e a t l e n g t h t h e a p p l i c a t i o n s o f c r o s s - t e r m s i n ESR r e l a x -a t i o n s t u d i e s . The i m p o r t a n c e o f . c r o s s - t e r m s i n n u c l e a r r e l a x a t i o n was f i r s t examined i n depth when i t was r e a l i z e d t h a t i t m i g h t be p o s s i b l e t o u t i l i z e t h e o c c u r r e n c e o f such terms as a d i a g o n i s t i c t o o l . J u s t as -131-t h e c r o s s - c o r r e l a t i o n terms between the g - t e n s o r a n i s o t r o p y and t h e e l e c t r o n - n u c l e a r d i p o l a r i n t e r a c t i o n s p e r m i t e x p e r i m e n t a l d e t e r m i n a t i o n o f t h e s i g n o f t h e h y p e r f i n e s p l i t t i n g , 1 0 a c r o s s - t e r m between t h e a n i s o t r o p y i n t h e c h e m i c a l s h i f t t e n s o r and t h e n u c l e a r d i p o l a r i n t e r -a c t i o n s s h o u l d a l l o w one t o d e t e r m i n e t h e a b s o l u t e s i g n o f J , t h e s c a l a r c o u p l i n g c o n s t a n t . A number o f papers have t r e a t e d an AX s p e c t r u m , namely t h e s c a l a r - c o u p l e d system composed o f two n o n - i d e n t i c a l , s p i n 1/2 11-13 n u c l e i . In t h e s e t r e a t m e n t s , i t i s assumed t h a t t h e "A" n u c l e u s i s r e l a x e d s o l e l y by t h e d i p o l a r i n t e r a c t i o n and t h a t t h e "X" n u c l e u s i s a d d i t i o n a l l y r e l a x e d by a second mechanism w i t h t h e " s a m e - c o r r e l a t i o n t i m e " (e. g . c h e m i c a l s h i f t a n i s o t r o p y ) . The i n t e r f e r e n c e terms a r e m a n i f e s t e d i n t h e appearance o f unequal l i n e w i d t h s w i t h i n t h e X - m u l t i p l e t ( i . e . d i f f e r e n t T^'s f o r d i f f e r e n t t r a n s i t i o n s ) . * Mackor and MacLean have c o i n e d t h e p h rase "anomalous r e l a x a t i o n " t o d e s c r i b e t h i s r e s u l t o f i n c l u s i o n o f c r o s s - c o r r e l a t i o n terms i n t h e r e l a x a t i o n e q u a t i o n s . A l t h o u g h o n l y a c e r t a i n c r o s s - t e r m was c o n s i d e r e d i n t h e s e t r e a t m e n t s (deemed t h e most l i k e l y o b s e r v a b l e i n t e r f e r e n c e ) , p a i r w i s e i n t e r f e r e n c e * These l i n e w i d t h changes a r e not t o be c o n f u s e d w i t h o t h e r f r e q u e n t l y d i s c u s s e d l i n e w i d t h changes due t o t h e c o r r e l a t i o n o f a c l a s s i c a l r a n -dom m a g n e t i c f i e l d ( s t o c h a s t i c f i e l d model) produced a t two d i f f e r e n t 14-19 n u c l e a r s i t e s . T h i s phenomenon a l s o c auses v a r i a t i o n s o f l i n e -w i d t h w i t h i n each m u l t i p l e t o f a two s p i n system ( e . g . i n an AB system) a l t h o u g h f o r an AX system ( J ^ « ^^y)> t h i s e f f e c t v a n i s h e s . The a n a l y s i s o f c r o s s - c o r r e l a t i o n terms u s i n g t h e random l o c a l f i e l d ap-p r o x i m a t i o n , e s p e c i a l l y i n c o n j u n c t i o n w i t h d o u b l e r e s o n a n c e e x p e r i -ments, can be f r u i t f u l (see above r e f e r e n c e s ) , but does not bear d i r e c t l y on t h e p r e s e n t d i s c u s s i o n . -132-e f f e c t s between o t h e r r e l a x a t i o n mechanisms i n s c a l a r c o u p l e d systems 4 20 c o u l d a l s o y i e l d t h e same i n f o r m a t i o n . ' However, i t can be shown t h a t t h i s i s not t r u e f o r t h e i n t e r f e r e n c e terms c o n s i d e r e d i n t h e l a s t c h a p t e r (see Appendix D). A paper w h i c h g i v e s a v e r y good i n s i g h t i n t o 20 t h e e f f e c t o f c r o s s - c o r r e l a t i o n terms i n J - c o u p l e d s p e c t r a i s Anderson's a n a l y s i s o f t h e symmetry p r e s e n t i n t h e R e d f i e l d r e l a x a t i o n m a t r i x . O t h e r r e l e v a n t papers d e a l i n g w i t h t h e g e n e r a l t o p i c a r e : V o i d and 21 Gutowsky's a n a l y s i s o f a s p i n 1/2 - s p i n 3/2 s c a l a r c o u p l e d s y s t e m , 22 S y k o r a ' s i n t e r e s t i n g d i s c u s s i o n o f t h e o c c u r r e n c e and i m p o r t a n c e o f 23 c r o s s - t e r m s , and Hoffman's t e r s e d i s c u s s i o n o f c r o s s - t e r m s i n h i s ex-c e l l e n t e x p o s i t i o n o f m a g n e t i c r e l a x a t i o n . Up t o t h i s p o i n t , t h e papers c i t e d have d e a l t w i t h c r o s s - c o r r e l a t i o n terms i n an e n l i g h t e n i n g and f u n d a m e n t a l , y e t v e r y l i m i t e d c o n t e x t . They have e i t h e r c o n s i d e r e d w h i c h r e l a x a t i o n mechanisms can m u t u a l l y i n t e r f e r e , o r d i s c u s s e d t h e e x t r a c t i o n o f p r a c t i c a l i n f o r m a t i o n from t h e s e t e r m s . In a s e r i e s o f papers " I n t e r f e r e n c e E f f e c t s i n N u c l e a r M a g n e t i c R e l a x a t i o n I - V I I I " , J . S. B l i c h a r s k i , * 0 1 t h o r o u g h l y e x p l o r e s t h e e f f e c t o f c r o s s - c o r r e l a t i o n s i n a q u a n t i t a t i v e manner. H i s t r e a t m e n t s c o n s i d e r systems o f l i k e o r u n l i k e s p i n s b e i n g r e l a x e d by d i p o l a r , c h e m i c a l s h i f t a n i s o t r o p y , q u a d r u p o l a r (where a p p l i c a b l e ) , and s p i n - r o t a t i o n i n t e r a c t i o n s . The f i r s t t h r e e terms can m u t u a l l y i n t e r f e r e , t h e s p i n - r o t a t i o n mechanism b e i n g i n c l u d e d i n the c a l c u l a t i o n , s i n c e , i n most c a s e s , i t w i l l be the d o m i n a t i n g masking e f f e c t o f the i n t e r f e r e n c e t erms. Both l o n g i t u d i n a l and t r a n s v e r s e r e l a x a t i o n b e h a v i o r i s examined f o r 1,2,3,4, o r 6 - s p i n systems i n t h e l i m i t o f r a p i d i s o t r o p i c r e o r i e n t a t i o n . C e n t r a l t o * An i n c l u s i v e summary o f t h i s work ( i n P o l i s h ) a p p e ars i n " I n s t y t u t F i z y k i J a d r o w e j U K r a k o w i e " R e p o r t INP No. 792/PL (1972) . -133-B l i c h a r s k i ' s t r e a t m e n t s i s t h e a s s u m p t i o n o f d e g e n e r a t e t r a n s i t i o n s , and i t i s w e l l known t h a t when t h e spectrum c o n t a i n s m u l t i p l y d e g e n e r a t e t r a n s i t i o n s , t h e l i n e s h a p e i s i n g e n e r a l a sum o f L o r e n t z i a n s w i t h v a r i o u s w i d t h s , o r , e q u i v a l e n t l y t h a t t h e decay o f m a g n e t i z a t i o n i s m u l t i -e x p o n e n t i a l ( n o n e x p o n e n t i a l ) , < I z ( t ) > - < I Z > T = E a i e x p ( - \ i t ) [5.2.1] < I x ( t ) > = £ 3 i e x p ( - y i t ) . [5.2 . 2 ] P l o t s o f t h e a . j , B. , A . . , and t h e y^ as a f u n c t i o n o f t h e r a t i o o f c o u p l -i n g c o n s t a n t s o f t h e i n t e r f e r i n g mechanisms can be found i n v a r i o u s papers i n t h i s s e r i e s by B l i c h a r s k i . These papers p r o v i d e t h e most compre-h e n s i v e a c c o u n t o f c r o s s - c o r r e l a t i o n terms and t h e i r i n f l u e n c e on n u c l e a r m a g n e t i c r e l a x a t i o n . A l t h o u g h t h e b a s i c t h e o r y b e h i n d t h e e f f e c t o f c r o s s - c o r r e l a t i o n terms i s w e l l d e f i n e d , e x p e r i m e n t a l o b s e r v a t i o n s o f t h e r e s u l t a n t n o n e x p o n e n t i a l n u c l e a r r e l a x a t i o n a r e r a t h e r s c a n t ( e x c e p t i n g o f c o u r s e t h e w e a l t h o f e x p e r i m e n t a l s t u d i e s i n ESR w o r k ) . T h i s s h o r t a g e can be a t t r i b u t e d t o masking and quenching e f f e c t s , and b e c a u s e , u n t i l v e r y r e c e n t l y , o n l y 19 p r o t o n (and t o a l e s s e r e x t e n t , F) r e l a x a t i o n t i m e s o f s m a l l m o l e c u l e s were r o u t i n e l y and r e l i a b l y measured; and i n such c a s e s , one would n o t e x p e c t n o n e x p o n e n t i a l e f f e c t s t o be o b s e r v a b l e . R e l a t i v e l i n e - b r o a d e n i n g o f v a r i o u s components o f the s c a l a r c o u p l e d 12 t w o - s p i n system C F H C ^ has been o b s e r v e d , and as mentioned b e f o r e , t h i s f a c t i s d i r e c t l y a t t r i b u t e d t o t h e i n f l u e n c e o f c r o s s - c o r r e l a t i o n terms. I n t e r f e r e n c e e f f e c t s a r e d e f i n i t e l y r e s p o n s i b l e f o r n o n e x p o n e n t i a l n u c l e a r -134-r e l a x a t i o n f o r t h e f l u o r i n e n u c l e i , i n C F g C ^ ^ ' ( f r e o n 12) and ^ B F ^ 3 0 . Each o f t h o s e o b s e r v a t i o n s was made a t e x t r e m e l y low t e m p e r a t u r e s j u s t above t h e m e l t i n g p o i n t o f t h e r e s p e c t i v e compounds. A t h i g h e r temper-a t u r e s , t h e e f f e c t i s o b s c u r e d by t h e dominance o f t h e s p i n - r o t a t i o n i n t e r a c t i o n . In c o n t r a s t t o t h e e x p e r i m e n t a l l y o b s e r v e d e f f e c t o f Hubbard terms i n s o l i d s , t h e e f f e c t o f B l i c h a r s k i terms has t o o u r knowledge n e v e r been c o n c l u s i v e l y v e r i f i e d i n s o l i d s t a t e s t u d i e s . T h i s i s e a s i l y r a t i o n a l i z e d by assuming t h a t i n t e r f e r e n c e e f f e c t s a r e quenched by t h e s t r o n g s t a t i c d i p o l a r i n t e r a c t i o n s w h i c h m a i n t a i n a common s p i n 32 33 t e m p e r a t u r e . * ' A l s o t h e r e i s one example o f p o s s i b l e c r o s s - c o r r e -34 l a t i o n e f f e c t s i n a gaseous sample o f HD. Of c o u r s e , t h e s t a n d a r d d e n s i t y m a t r i x t h e o r y i s not t h e o n l y mathe-m a t i c a l f o r m u l a t i o n o f r e l a x a t i o n t h e o r y . In some o t h e r a p p r o a c h e s , c r o s s -c o r r e l a t i o n terms can be i n c o r p o r a t e d i n t o t h e f o r m a l i s m f o r the same p h y s i c a l r e a s o n s w i t h many o f t h e same p h y s i c a l consequences. I n con-t r a s t , Kubo's l i n e a r r e s p o n s e t h e o r y i s i n a d e q u a t e t o t r e a t g e n e r a l r e l a x a t i o n problems o f a spectrum c o n t a i n i n g d e g e n e r a t e t r a n s i t i o n s as i t p r e d i c t s L o r e n t z i a n l i n e s h a p e s even i n t h e p r e s e n c e o f c r o s s - t e r m s . Thus l i n e a r r e s p o n s e t h e o r y i s i l l - s u i t e d f o r t h e problem a t hand as i t f a i l s t o p r e d i c t t h e known s p i n b e h a v i o r , a l t h o u g h a c l o s e r l o o k a t 35 t h i s problem shows some o f t h e c r i t i c i s m i s u n j u s t i f i e d . F i n a l l y , we c i t e t h e f o r m u l a t i o n o f R e d f i e l d ' s t h e o r y i n L i o u v i l l e s pace. In a * Indeed, t h i s e x p l a n a t i o n s u g g e s t s t h a t w h i l e Hubbard and B l i c h a r s k i terms appear s u p e r f i c i a l l y a l i k e , t h e r e a r e r a d i c a l d i f f e r e n c e s , s i n c e Hubbard terms a p p a r e n t l y a r e not c o m p l e t e l y quenched i n s o l i d s . -135-p a i r o f p a p e r s , P y p e r J D t h o r o u g h l y d i s c u s s e s r e l a x a t i o n i n t h i s " s u p e r " m a n i f o l d and, a l t h o u g h d i s c u s s i n g Hubbard terms i n g r e a t d e t a i l , makes o n l y f l e e t i n g comments on B l i c h a r s k i terms ( b u t r e c a l l , f o r t h e most p a r t , t h e o n l y d i f f e r e n c e between t h e s e was s e m a n t i c s ) . These papers by Pyper a r e recommended, as t h i s a p p roach t o r e l a x a t i o n i n g e n e r a l , and t o t h e i n f l u e n c e o f c r o s s - c o r r e l a t i o n terms i n p a r t i c u l a r , appears t o be v e r y p r o m i s i n g . From t h i s b r i e f i n t r o d u c t i o n t o t h e problem a t hand, i t i s a p p a r e n t t h a t t h e term " c r o s s - c o r r e l a t i o n problem" i s d i s c u s s e d i n a r a t h e r r e -s t r i c t e d sense. A l s o , t h e c o n n o t a t i o n a s s o c i a t e d w i t h t h e term c r o s s -c o r r e l a t i o n t o u c h es on o t h e r r e l a t e d (and u n r e l a t e d ) t o p i c s f a m i l i a r t o t h e p r a c t i c i n g NMR s p e c t r o s c o p i s t . -136-5.3 FORMULATION OF THE CALCULATION As mentioned i n s e c t i o n 2.2, t h e t o t a l s p i n H a m i l t o n i a n can conven-i e n t l y be d i v i d e d i n t o two p a r t s , €=&Q + £ ( t ) , where t h e s t a t i c H a m i l -t o n i a n (composed o f t h e Zeeman term and f i r s t o r d e r c o r r e c t i o n s ) , £Q , d e t e r m i n e s l i n e p o s i t i o n s and i n t e n s i t i e s , and t h e f l u c t u a t i n g t i m e -dependent p e r t u r b a t i o n , £ ( t ) , d e t e r m i n e s t h e l i n e w i d t h s ! d f ( t ) , t h e s p i n -l a t t i c e c o u p l i n g , may i n i t s e l f be composed o f many s e p a r a b l e c o u p l i n g s , w h i c h f o r emphasis, may be w r i t t e n as b i l i n e a r c o u p l i n g s . F o r example; I 1 - II 1 ( t ) - I i + I n - - ( ^ n . ( t ) - a.i )-B Q + I - t ^ t J - J ^ t ) + I . j - B . f t ) [5.3.1] where t h e s u b s c r i p t s r e f e r t o n u c l e i , J E T r [ j ] / 3 , a = T r [ # ] / 3 , and I E as t h e second r a n k u n i t t e n s o r . The c o u p l i n g s a r e i n a n g u l a r f r e -quency u n i t s . The r e s p e c t i v e c o u p l i n g s a r e t h e d i r e c t d i p o l e ( D ) , i n -d i r e c t d i p o l e o r a n i s o t r o p i c s p i n - s p i n ( I D ) , s c a l a r ( S C ) , q u a d r u p o l a r (Q), c h e m i c a l s h i f t a n i s o t r o p y ( C S A ) , s p i n r o t a t i o n ( S R ) , and t h e random f i e l d (RF) c o u p l i n g s . I t i s assumed t h a t t h e e f f e c t s o f i n t e r m o l e c u l a r r e l a x a t i o n , c h e m i c a l exchange, and f i e l d i n h o m o g e n i t i e s c o m p r i s e Of c o u r s e , u s u a l l y one o r two mechanisms dominate t h e r e l a x a t i o n o f a g i v e n n u c l e u s (and o t h e r mechanisms may v a n i s h c o m p l e t e l y f o r n u c l e i i n c e r t a i n e n v i r o n m e n t s ) . I n t h i s c h a p t e r , o n l y s p i n 1/2 n u c l e i w i l l be c o n s i d e r e d , and t h e r e f o r e 6?g must v a n i s h . F u r t h e r m o r e , i t s h a l l be -137-assumed t h a t t h e s p i n system i s s u f f i c i e n t l y d i l u t e i n an i n e r t s o l v e n t , f r e e from p a r a m a g n e t i c i m p u r i t i e s , e x t r a n e o u s f i e l d g r a d i e n t s and chem-i c a l exchange so t h a t can be i g n o r e d . The term c ? ^ a r i s e s because o f a t i m e dependence o f e i t h e r t h e i s o t r o p i c s c a l a r c o u p l i n g c o n s t a n t ( s c a l a r c o u p l i n g o f t h e 1 s t k i n d ) o r t h e p r o j e c t i o n quantum number o f one o f t h e c o u p l e d s p i n s ( s c a l a r c o u p l i n g o f t h e 2nd k i n d ) . In t h e absence o f c h e m i c a l exchange and i f n e i t h e r o f t h e n u c l e a r s p i n s i s r e l a x e d by t h e q u a d r u p o l a r i n t e r a c t i o n t h i s r e l a x a t i o n mechanism may be i g n o r e d . M o d u l a t i o n o f t h e a n i s o t r o p i c p o r t i o n o f t h e s p i n - s p i n i n d i r e c t d i p o l a r c o u p l i n g t e n s o r , J , by m o l e c u l a r m o t i o n can cause r e l a x a t i o n i n a f a s h i o n a n a l o g o u s t o r e l a x a t i o n i n d u c e d by t h e d i r e c t d i p o l a r c o u p l i n g . 37 T h i s seldom d i s c u s s e d form o f r e l a x a t i o n may be o f i m p o r t a n c e f o r n u c l e i w i t h a l a r g e number o f n u c l e o n s , but t h i s c o u p l i n g w i l l be i g -nored i n any f u r t h e r d i s c u s s i o n s i n t h i s paper. Thus, w i t h t h e s e a s s u m p t i o n s , i t i s o n l y n e c e s s a r y t o i n c l u d e t h e t h r e e r e m a i n i n g i n t e r -a c t i o n s i n t h e r e l a x a t i o n c a l c u l a t i o n s , 8{t) = c ? D ( t ) + c ^ ^ U ) + < ^ S R ^ ^ ' The t h e o r y of S^{t) a n d c ? < - R ( t ) , as w e l l as t h e e x p e r i m e n t a l con-s e q u e n c e s , i s w e l l documented and have been d i s c u s s e d i n p r e v i o u s chap-t e r s . In c o n t r a s t , ^ ^ ( t ) , a l t h o u g h l o n g d i s c u s s e d , has u n t i l v e r y r e c e n t l y escaped t h e e x p e r i m e n t a l i s t ' s e f f o r t s o f d e t e c t i o n ( f o r a l l p r a c t i c a l p u r p o s e s ) . But w i t h l a r g e r m a g n e t i c f i e l d s now a v a i l a b l e f o r r o u t i n e work, and t h e r e c e n t emphasis on r e l a x a t i o n s t u d i e s o f n u c l e i o t h e r t h a n p r o t o n s , i t i s becoming a p p a r e n t t h a t . d t ^ ( t ) may p l a y a l a r g e r r o l e i n t h e i n t e r p r e t a t i o n o f f u t u r e r e l a x a t i o n d a t a , even though i t may not dominate t h e m a g n e t i z a t i o n decay. Indeed, i n a d d i t i o n t o -138-38 39 r e c e n t e x p e r i m e n t a l s t u d i e s c o n f i r m i n g t h i s i n t e r a c t i o n , ' even t h e re m a r k a b l e "7/6" e f f e c t u n i q u e l y c h a r a c t e r i s t i c o f t h i s c o u p l i n g has 40 a p p a r e n t l y been o b s e r v e d . Each o f t h e s e i n t e r a c t i o n s i n t u r n , can be w r i t t e n as t h e sum o f p r o d u c t s o f s p i n and l a t t i c e o p e r a t o r s . The f o l l o w i n g e x p a n s i o n s a r e most u s e f u l : n 2 f D(t) +£ C S A(t) +£ S R(t)=§ Z E u U . j v U . j f i - S i j ) _ n 1 A=SR,CSA i = l k=-l ^ A ; i l M J i l*-*-'! where t h e l a t t i c e p arameters f o r t h e two, o n e - s p i n i n t e r a c t i o n s a r e g i v e n by, U ( C S A ) i = ? C S A / - 1 ) k ^ i ( t ) ) ^ 5 - 3 - 3 ] u ( s R ) i = H ) k M C ] o i k + ( ^ ) 1 / 2 A C k E J k„ S j k ' k l ) Y 2 k ' ( f i i ^ ' [ 5.3.4] where £ C S A - = (2TT/1 5 ) 1 / / 2 Y - J B Q ( A O ) and J Q = J . J . S c r i p t i and j r e f e r t o n u c l e i . The s p i n o p e r a t o r s f o r t h e s e i n t e r a c t i o n s a r e g i v e n by, V(CSA) i = <8/3) 1 / 2lj ; v f ^ , . = +1* [5.3.5] V ( S R ) i " ! z • V(SR),. " K IJ1 •  [5-3'6] The e x p a n s i o n f o r t h e d i p o l a r i n t e r a c t i o n i s g i v e n by E q u a t i o n s [ 3 . 1 . 3 - 4 ] , The time dependent a n g l e s , ( t ) , d e f i n e t h e o r i e n t a t i o n o f t h e p r i n c i p a l a x i s o f t h e i c h e m i c a l s h i f t ( o r s p i n - r o t a t i o n ) c o u p l i n g t e n s o r w i t h -139-r e s p e c t t o t h e l a b frame. The o t h e r n o t a t i o n i s r e g a r d e d as s t a n d a r d . I t i s u s u a l l y s t a t e d t h a t m a g n e t i c p e r t u r b a t i o n s s h o u l d b e s t be w r i t t e n as a p r o d u c t o f s p h e r i c a l t e n s o r s w h i c h can be r educed t o a . s c a l a r c o n t r a c t i o n o f i n d i v i d u a l components. T h i s approach i s e x t r e m e l y f r u i t f u l , as t h e t r a n s f o r m a t i o n p r o p e r t i e s o f s p h e r i c a l t e n s o r s under ( m o l e c u l a r ) r o t a t i o n s i s v e r y w e l l known (by d e f i n i t i o n ! ) and s i m p l i f i e s t h e a n a l y s i s o f t h e r e l a x a t i o n problem. A s i d e from n o r m a l i z i n g c o e f f i -c i e n t s ( t h e n o t a t i o n used i n E q u a t i o n s [ 3 . 1 . 3 - 4 ] and [ 5 . 3 . 3 - 6 ] i s chosen f o r s i m p l i c i t y - n o t r i g o r ) , i t i s seen t h a t t h e d i p o l a r c o u p l i n g i s w r i t t e n as a s c a l a r c o n t r a c t i o n o f second rank t e n s o r s (as a r e c 2 g , @JQ)> whereas t h e s p i n - r o t a t i o n c o u p l i n g i s seen t o be a c o n t r a c t i o n o f f i r s t s p h e r i c a l t e n s o r s (as a r e <£ R [-). F i n a l l y , we n o t e t h a t t h e CSA c o u p l -i n g i s u n i q u e i n t h a t i t cannot be w r i t t e n as a s c a l a r c o n t r a c t i o n o f two s p h e r i c a l t e n s o r s ( a n o t h e r way o f s a y i n g t h i s i s t o say t h a t t h e i n t e r -a c t i o n i s not r o t a t i o n a l l y i n v a r i a n t ) , t h u s g i v i n g r i s e t o t h e i n t r i g u -i n g i n e q u a l i t y o f T-| and T 2 even f o r a s i n g l e n u c l e u s i n t h e extreme-narrowed a p p r o x i m a t i o n . In w r i t i n g t h e s e e x p a n s i o n s , c e r t a i n a s s u m p t i o n s have n e c e s s a r i l y been made. The s p i n - r o t a t i o n i n t e r a c t i o n t e n s o r i s assumed t o be d i a g o n -i z a b l e w i t h two i n d e p e n d e n t components, C„ and C J A C = C , , - ^ ) . A l s o , i n a d d i t i o n t o t h e a n g u l a r momentum o p e r a t o r s ( i . e . t h e f i r s t term i n Equa-t i o n [ 5 . 3 . 4 ] ) , t h e l a t t i c e f u n c t i o n s may a l s o depend on t h e o r i e n t a t i o n 41 o f t h e m o l e c u l e t h r o u g h t h e second rank s p h e r i c a l h a r m o n i c s . Of c o u r s e i f t can be w r i t t e n as a m u l t i p l e o f t h e u n i t m a t r i x , t h i s o r i e n t a t i o n dependence v a n i s h e s . The c h e m i c a l s h i f t a n i s o t r o p y i n t e r a c t i o n c o n s t a n t 1/2 1/2 has been w r i t t e n as = (2 i r/15) ' Y 1-' Bg (o, l -aL) = (2TT/15) u^bo.. -140-I m p l i c i t i n t h i s a s s u m p t i o n i s the f a c t t h a t t h e asymmetic second r a n k s c r e e n i n g t e n s o r , £ , i s i s o t r o p i c i n one p l a n e and can be f u l l y des-c r i b e d by o n l y two inde p e n d e n t components and a ( l; t h e e l e c t r o n i c s c r e e n i n g p a r a l l e l t o t h e a x i s o f t h e s h i f t t e n s o r and t h e s h i e l d i n g p e r p e n d i c u l a r t o t h i s a x i s . A l t h o u g h c a l c u l a t i o n s can be c a r r i e d o ut «v 42 u s i n g t h e g e n e r a l form o f 3, t h e s i m p l i f i c a t i o n o f two in d e p e n d e n t components w i l l be made, e q u i v a l e n t t o t h e as s u m p t i o n t h a t t h e s p i n l i e s 43 a t a s i t e C 3 v o r h i g h e r symmetry. (However, f o r c u b i c s y m m e t r y , t h i s i n t e r a c t i o n v a n i s h e s . ) I t i s a l s o assumed t h a t t h e t r a c e o f t h e s h i e l d -i n g t e n s o r , (2a x+a^)/3, i s i n c l u d e d i n £ Q . F u r t h e r m o r e , t o a v o i d c o m p l i c a t i o n s i n v o l v i n g Hubbard t e r m s , a system o f two i s o c h r o n o u s s p i n s ( w i t h n e c e s s a r i l y d e g e n e r a t e t r a n s i t i o n s ) i s c o n s i d e r e d . With t h e s e p r e l i m i n a r i e s b e h i n d , l e t us now t u r n o u r a t t e n t i o n t o t h e c a l c u l a t i o n o f t h e s p i n b e h a v i o r . -141-5.4 SOLUTION OF THE RELAXATION MATRIX As e i g e n s t a t e s o f t h e Zeeman H a m i l t o n i a n , we s h a l l choose t h e s i m p l e u n c o u p l e d b a s i s . a-j > - |1> = | + +> a^> = |2> = |+-> a 3 > = |3> = |-+> a 4 > = |4> , l " > [ 5 . 4 . 2 ] With t h e s e s t a t e s so c h o s e n , t h e m a t r i x e l ements o f E q u a t i o n [2.2.13] can be s i m p l y e v a l u a t e d . As s t a t e d so o f t e n , o n l y terms w h i c h c o n n e c t degen-e r a t e t r a n s i t i o n s ( t h e s e c u l a r c o n t r i b u t i o n s ) need t o be c o n s i d e r e d . F o r t h e c a l c u l a t i o n o f t h e l o n g i t u d i n a l r e l a x a t i o n , t h i s e n a b l e s one t o n e g l e c t a l l o f f - d i a g o n a l e l ements i n the d e n s i t y m a t r i x e x c e p t x 2 3 a n d x 32- ^ i s c o n v e n i e n t t o t h i n k o f t h e s e v a l u e s o f x ( t ) as d e f i n i n g a s i x - d i m e n s i o n a l v e c t o r ( L x ^ U ) , x 2 2 ( t ) , x 2 3 ( t ) , x 3 2 U ) , x 3 3 ( t ) , x 4 4 ( t ) ] ; X a g ( t ) = < a | x ( t ) j 3 > ) . U s i n g E q u a t i o n s [ 5 . 3 . 5 ] , [ 5 . 3 . 6 ] , and [ 3 . 1 . 4 ] , i t i s now q u i t e s t r a i g h t f o r w a r d t o c o n s t r u c t a t a b l e such as T a b l e 5.1 whi c h l i s t s t h e v a r i o u s c o n t r i b u t i o n s t o t h e r e l a x a t i o n m a t r i x . A l t h o u g h o n l y 10 e l ements a r e l i s t e d , t h e o t h e r 26 terms can be deduced from the sym-m e t r y i n c o r p o r a t e d i n t o t h e c o n s t r u c t i o n ( i . e . d e f i n i t i o n ) o f R; R = R = R - R and R - p a a ' 3 3 ' gg ' a a ' V a 3 ' 3 " 3 ' 3 a ' a 5 - a , 5 - a ' , 5 - 3 , 5 - 3 ' " ijwjja >34<B1 = K a a'gg. e x c e p t f o r t h e s h i f t a n i s o t r o p y - d i p o l a r c r o s s term c o n t r i b u -t i o n s i n which c a s e R, . = -R , O D , ( s e e Appendix A ) . In t h i s n o t a t i o n , \p i s t h e s p i n i n v e r s i o n o p e r a t o r ( e . g . ^ |1> = |4>, \|J|2> = |3>). -142-F u r t h e r m o r e , t h e terms R-j-|-|-| and R 2 2 2 2 i n t h i s T a b l e a r e r e d u n d a n t s i n c e t h e s e terms g i v e t h e t o t a l r a t e a t w h i c h s t a t e s 1 and 2 [ r e s p e c t i v e l y ) a r e d e p l e t e d by t r a n s i t i o n s t o o t h e r l e v e l s . Note t h a t t h e two symmetry c o n d i t i o n s imposed upon R i m p l y t h a t R - j , ^2233' ^2323' a n c ' ^2332 c a n have no d i p o l a r - s h i f t a n i s o t r o p y c r o s s - t e r m s . Of t h e 25 p o s s i b l e c o n t r i b u t i o n s t o each o f t h e 36 elements o f R ( i . e . R , , = E r ? T 1 ' R R ' )» o n l y 13 o f t h e s e c o n t r i b u t i o n s may be non-c >n z e r o . F i v e a r e t h e c o n v e n t i o n a l " a u t o " c o n t r i b u t i o n s , 6 R ,„„,, f o u r Cn a a 66 terms a r i s e from t h e c r o s s - t e r m s between CSA-j 2 a n c ' t h e d i p o l a r mechanism, CSA -CSA and t h e r e m a i n i n g f o u r terms a r e pseudo c r o s s - t e r m s o f t h e form R i j C D C D o r R i j . C r o s s - t e r m s between CSA^ 2 o r t h e d i p o l a r mechanism and SR-j 2 n e c e s s a r i l y v a n i s h s i n c e t h e t i m e dependent l a t t i c e p a r a m e t e r s a r e c h a r -a c t e r i z e d by d i f f e r e n t t r a n s f o r m a t i o n p r o p e r t i e s under r o t a t i o n s . The c o r r e l a t i o n f u n c t i o n s between t h e s e terms a r e o f t h e form <d. ( t ) Y 2 ( t ) Y 2 ( 0 ) > o r < J j ( t ) Y 2 ( 0 ) > . I f i t can be assumed t h a t t h e m o l e c u l a r a n g u l a r momen-tum and o r i e n t a t i o n a r e i n d e p e n d e n t , and f u r t h e r m o r e , t h a t a l l i n i t i a l o r i e n t a t i o n s o f a n g u l a r momentum a r e e q u a l l y p r o b a b l e , t h e n t h e s e c o r r e l a -t i o n f u n c t i o n s r e d u c e t o < J j . ( t ) Y 2 ( t ) Y 2 ( 0 ) > = < J j ( t ) > < Y 2 ( t ) Y 2 ( 0 ) > = 0 [5.4.3] < 0 j ( t ) Y k ( 0 ) > = < J j ( t ) > < Y k ( 0 ) > = 0 f o r a l l j , k, and i s i n c e < Y 2 ( t ) > = 0 and < J ^ ( t ) > = 0. T h e r e f o r e , i n E q u a t i o n [ 2 . 2 . 1 3 ] , C ^ ( a l l t ) = 0 -* J kJ-(u) = 0 + R C n = 0 f o r a l l k and a. By i n s p e c t i o n , i t can be seen t h a t t h e t i m e dependency o f t h e s i x e l e m e n t s o f x a r e not i n d e p e n d e n t . I t i s now c o n v e n i e n t t o i n t r o d u c e -143-t h e f o l l o w i n g f o u r o r t h o g o n a l l i n e a r c o m b i n a t i o n s o f m a t r i x e l e m e n t s , y-,(t) = X l l ( t ' ) - x 4 4 ( t ) [ 5 . 4 . 4 ] y 2 ( t ) = x 2 2 ( t ) - x 3 3 ( t ) [ 5 . 4 . 5 ] y 3 ( t ) = - X ] 1 ( t ) - x 4 4 ( t ) + x 2 2 ( t ) + x 3 3 ( t ) [ 5 . 4 . 6 ] y 4 ( t ) - x 3 2 ( t ) + x 2 3 ( t ) [ 5 . 4 . 7 ] Note t h a t y-j ( t ) has been d e f i n e d such t h a t y ^ t ) = x n ( t ) - x 4 4 ( t ) = T r [ x ( t ) I z ] = < I z ( t ) > - < I Z > T [ 5 . 4 . 8 ] I t t h e n f o l l o w s from T a b l e 5.1 t h a t ( d / d t ) y i ( t ) = [ A - , - 4 A 2 + 2 ( A ] ] + A 2 2 ) + $ j ] + s f j y ^ t ) + [ 2 ( A ] ] - A 2 2 ) + s ] 1 - * 2 2 ] y 2 ( t ) + [ 2 ( r ] + r 2 ' ) ] y 3 ( t ) + [ 2 ( r ] + r 2 ) ] y 4 ( t ) [ 5.4.9] ( d / d t ) y 2 ( t ) = [ 2 ( A ] ] - A 2 ? ) + O ] 1 - $ 2 2 ] y i ( t ) + [ - § A Q + A ] + 2 ( A ] ] + A 2 2 ) + + $ 2 2 ] y 2 ( t ) + [ 2 ( r ] ' - r 2 : : ) ] y 3 ( t ) + [ j ( r j - r 2 ' ) ] y 4 ( t ) [5.4.10] ( d / d t ) y 3 ( t ) = [ 4 ( r ] + r 2 ) ] Y l ( t ) + [ 4 ( r ] - r 2 ) ] y 2 ( t ) + [2A-, + 4 ( A ] ] + A 2 2 ) + 2 ( $ ] ] + * f ) ] y 3 ( t ) + [2A-, + 8 A ] 2 + 4 $ ] 2 ] y 4 ( t ) [5.4.11] ( d / d t ) y 4 ( t ) = [ 2 ( r ] + r 2 ) ] y i ( t ) + [ | ( r j - r 2 ) ] y 2 ( t ) + [>, + 4 A ] 2 + 2 $ ] 2 ] y 3 ( t ) + " f C A f j 1 + A 2 2 - 2 A j 2 ) - (.Jl + $ 2 2 - * Q 2) + 2 ( A ^ + A 2 2 ) -144-+ o]1 + $22]y4Ct) £5.4.12] k -k where t h e f o l l o w i n g s h o r t h a n d n o t a t i o n has been i n t r o d u c e d : = J ^ ' (kcjg); A k J ~ J C S A k - C S A . ( S ) ; r k ~ ° C S A k - D ( k u 0 ) ; E J S R . - S R . ( k c o0 }' The g e n e r a l s o l u t i o n f o r a system o f n f i r s t o r d e r homogeneous l i n e a r d i f f e r e n t i a l e q u a t i o n s w i t h c o n s t a n t c o e f f i c i e n t s i s g i v e n by, n y,-(t) = S a , , e x p ( x . t ) [5.4.13] 1 j = l 1 J J where t h e a., a r e d e t e r m i n e d by t h e a p p r o p r i a t e boundary c o n d i t i o n s . The s o l u t i o n can be o b t a i n e d by m a t r i x methods, a l g e b r a i c methods, o r by L a p l a c e T r a n s f o r m methods, but i n any c a s e , w i l l i n v o l v e d e t e r m i n a t i o n o f t h e r o o t s o f a n degree e q u a t i o n , a t a s k w h i c h i s t e d i o u s f o r any but a system composed o f o n l y two c o u p l e d e q u a t i o n s . The m a t r i x a p proach i s used t o s o l v e t h e p r e s e n t problem and y i e l d s t h e f o l l o w i n g form o f 44 s o l u t i o n , y ( t ) = toexp(b)to_1y(0) ; b ^ a t t l [5.4.14] where 111 i s a n o n s i n g u l a r t r a n s f o r m a t i o n m a t r i x , fat)1 i s i t s i n v e r s e , h i s t h e d i a g o n a l s i m i l a r i t y m a t r i x o f 91 , y ( t ) i s the v e c t o r [ y - | ( t ) , ^2^' y 3 ^ ) ' y ^ t J L and 91 i s t h e a p p r o p r i a t e c o e f f i c i e n t m a t r i x ( i . e . ( d / d t ) y ( t ) = £1y(t)). The d i a g o n a l e lements o f b a r e t h e e i g e n v a l u e s o f Sl and t h e columns o f t D a r e t h e c o r r e s p o n d i n g e i g e n v e c t o r s . By p e r u s a l o f E q u a t i o n s [5.4.9-12], i t i s a p p a r e n t t h a t i n g e n e r a l , t h e decay o f m a g n e t i z a t i o n i s n o n e x p o n e n t i a l ( t h e sum o f f o u r e x p o n e n t i a l s ) a l t h o u g h i n t h e absence ( o r n e g l e c t ) o f c r o s s - c o r r e l a t i o n t e r m s , one would e x p e c t t h e decay t o be d e s c r i b e d as t h e sum o f o n l y two e x p o n e n t i a l s . -145-I f t h e s p e c t r a l d e n s i t y a t u>Q i s i d e n t i c a l f o r both c h e m i c a l s h i f t and s p i n - r o t a t i o n i n t e r a c t i o n s , t h e n a uni q u e e x p o n e n t i a l w i l l s u f f i c e , y ^ t ) = y 1(0 ' )exp[(A 1 - 4 A 2 + 4A ] + 2^)t = y^OexpC t / T Q ) . [ 5 . 4 . 1 5 ] O t h e r l i m i t i n g c a s e s a r e i r ^ , A^ J -> 0 where t h e decay i s g i v e n by y-] ( t ) = y-j ( 0 ) e x p ( ( - A - | - 4 A 2 ) t ) and , A K ->• 0 i n w h i c h cas e t h e decay I ' l l 22 i s g i v e n by ( t ) = ^ { e x p ^ A - j t ) + exp (4A^ t ) ] . These l i m i t i n g v a l u e s a r e v e r y f a m i l i a r from s t a n d a r d t r e a t m e n t s . More g e n e r a l l y , we w i l l f i n d i t c o n v e n i e n t t o d e f i n e t h e c o e f f i c i e n t o f y-j ( t ) i n E q u a t i o n [ 5 . 4 . 9 ] as T" 1. To s o l v e t h e g e n e r a l p r o b l e m , seven c o o r d i n a t e systems must be s p e c i f i e d as w e l l as t h e ti m e ( i n ) d e p e n d e n t n a t u r e o f t h e r o t a t i o n mat-r i c e s r e l a t i n g t h e s e v a r i o u s c o o r d i n a t e s . These a r e ; (1 ) t h e l a b f r a m e , (2) t h e m o l e c u l a r frame i n wh i c h t h e d i f f u s i o n t e n s o r i s d i a g o n a l , (3-6) c o o r d i n a t e frames w h i c h d i a g o n a l i z e each o f t h e two s p i n - r o t a t i o n c o u p l i n g t e n s o r s and t h e frames w h i c h d i a g o n a l i z e each o f t h e two c h e m i c a l s h i f t t e n s o r s , and (7 ) t h e d i p o l a r r e l a x a t i o n frame w i t h t h e z - a x i s o r i e n t e d a l o n g the i n t e r n u c l e a r v e c t o r . As mentioned e a r l i e r , t h e r e o r i e n t a t i o n o f t h e d i p o l a r and c h e m i c a l s h i f t frames w i t h r e s p e c t t o t h e l a b frame has a l r e a d y been e x p l i c i t l y i n c l u d e d i n t h e l a t t i c e p a rameters ( E q u a t i o n s [ 3 . 1 . 3 ] and [ 5 . 3 . 3 ] ) . However, t h e s t a t i c o r i e n t a t i o n o f t h e s e v a r i o u s c o o r d i n a t e systems r e l a t i v e t o t h e m o l e c u l a r frame i n f l u e n c e s the magnitudes o f t h e v a r i o u s s p e c t r a l d e n s i t i e s a p p e a r i n g i n E q u a t i o n [ 2 . 2 . 1 3 ] and must now be ex-amined i n d e t a i l . C o n s i d e r f i r s t t h e c o r r e l a t i o n f u n c t i o n s A^, A ^ J , and -146-r ^ . We w i l l assume the c o r r e l a t i o n f unc t i ons are a c c u r a t e l y desc r ibed by Equation [2 .3 .10 ] ( i . e . the mo lecu la r frame dynamics are f u l l y des -c r i bed by on ly two c l a s s i c a l d i f f u s i o n cons t an t s , D ( 1 f o r motions about a symmetry ax i s and f o r motions about an a x i s pe rpend i cu l a r to t h i s major a x i s ) . In these equa t i ons , i t i s assumed tha t the sp ins are r i g i d l y a t tached to t h i s symmetric body. S ince the p r i n c i p a l concern of t h i s paper i s w i th l a rge mo l e cu l e s , and o f t en l a rge molecules are g l o b u l a r i n na tu re , t h i s equat ion and a l l f o l l o w i n g equat ions w i l l a l s o hold t rue i f we assume D E D± and D..^ = D() - where D. ^ i s the c l a s -s i c a l d i f f u s i o n constant c h a r a c t e r i s t i c o f an i n t e r n a l r o t o r a t tached to t h i s bu lky framework. The a r b i t r a r y attachment of an i n t e r n a l r o t o r to an asymmetric top cannot be t r ea ted as s imply as shown here (see Equat ion [4 .3 .15-16 ] ) . I t then f o l l o w s from Equations [ 2 . 3 . 1 0 ] , [ 3 . 1 . 3 ] , and [ 5 . 3 . 3 . ] , 0 r k £ ( k a ) n ) = ( - l ) k £ ? 6. /t) T7r (3cos 2 e ' - D (3cos 2e' - 1 ) — ^ — . ?n 0 c n k,-£\5/ 16rr c n ( 6 D j 2 + ^ 2 30 2 2 2 D J . + 4 D n + s i n e ' s i n e'cos(2(d)' - *')) * ^ 3 2 7 1 ? C n (2D, + 4 D . / + ( k . n ) 2 30 5 D i + D u + n - c o s e ' c o s e ' s i n e ' s i n e ' c o s U ' - <$>') 8T C n C n Z n (5DJ_ + D J 2 + ( k c o Q ) 2 • [ 5 .4 .16 ] (Reca l l tha t the primed angles r e f e r to mo lecu l e- f i xed a n g l e s . ) I t i s apparent tha t the magnitude of the spec t r a l d e n s i t i e s are dependent on ly 45 upon the rank of the sphe r i c a l harmonic and not on i t s component. A l s o . J = J . Cn nc -147-For t h e c a l c u l a t i o n o f , i t i s n e c e s s a r y t o e v a l u a t e t h e v a r i o u s a n g u l a r momentum and a n g u l a r momentum-angular p o s i t i o n , c o r r e l a t i o n f u n c -t i o n s . T h i s i s by no means a t r i v i a l problem and a t t e n d s much a t t e n t i o n i n contempory s t u d i e s . An e x p r e s s i o n f o r ( T ^ ) S R assuming a n i s o t r o p i c 46 m o l e c u l a r r e o r i e n t a t i o n i s i t s e l f q u i t e complex, and t h e i n c o r p o r a t i o n o f t h i s e x p r e s s i o n i n t o t h e p r e s e n t c a l c u l a t i o n s w o u l d n e c e s s i t a t e t h e e x a m i n a t i o n o f (1) t h e e f f e c t o f asymmetry i n t h e s p i n - r o t a t i o n c o u p l i n g t e n s o r , (2) an a n a l y s i s o f r e o r i e n t a t i o n a l models ( i f i t i s n o t p o s s i b l e t o s e p a r a t e t h e a n g u l a r p o s i t i o n a l r e o r i e n t a t i o n a l and a n g u l a r momentum r e o r i e n t a t i o n a l t i m e s c a l e s ) , and (3) t h e e f f e c t o f asymmetry i n m o l e c u l a r r e o r i e n t a t i o n . T h e r e f o r e , we s h a l l make t h e t a c i t a s s u m p t i o n t h a t t h e s p i n - r o t a t i o n mechanism i s i n o p e r a t i v e , = 0. The s p i n - r o t a t i o n i n t e r -a c t i o n c o u l d be r e t a i n e d as a p h e n o m e n o l o g i c a l masking p a r a m e t e r , but t h i s a p p r oach would l e n d l i t t l e t o t h e p r e s e n t t r e a t m e n t . We s h a l l r e t u r n t o t h i s p o i n t l a t e r . S u b s t i t u t i o n o f E q u a t i o n [5.4.16] i n t o E q u a t i o n s [5.4.9-12] and t h e subsequent s o l u t i o n (by means o f E q u a t i o n s [5.4. 14]) o f t h e s e c o u p l e d e q u a t i o n s s u b j e c t t o t h e boundary c o n d i t i o n s , y ^ O ) = (cose - 1 ) < I Z > T y2(o) = 0 y3(o) = o y 4(0) = 0 [5.4.17] f o r v a r i o u s r a n g e s o f g e o m e t r i e s , r e l a t i v e magnitudes o f t h e i n t e r a c t i o n c o n s t a n t s , and shapes and s i z e s o f m o l e c u l a r frameworks has been examined. -148-TABLE 5.1 C o n t r i b u t i o n s t o t h e R e l a x a t i o n M a t r i x f o r two i d e n t i c a l s p i n s r e l a x e d by d i p o l a r , s h i f t a n i s o t r o p y and s p i n - r o t a t i o n i n t e r a c t i o n s element o f r e l a x a t i o n R m a t r i x D DCSA,-CSA. K l J RCSA.-D R S R i - S R j ' l l l l 11 77 A ] - 2 A 2 2A-] ' + 2 A " -2r] - 2 r 2 >1 + $ 1 1 1 2 2 '1123 Ml 33 -A-,/2 -A-,/2 - y 2 -2A -2A -2A 22 1 12 1 11 1 2 r ! r 1 + r 2  11 r l 2r. 22 1 12 1 .11 ?1 k1144 2A, 0 ' 2 2 2 2 11 77 •A Q /3+A 1 2A-J 1 + 2 A " 2 r - 2 r ' * 2 2 l2223 A-,/2 2A 12 2 r Q / 3 - 2 r 2 / 3 o ] 2 7 2 3 3 V 3 '2323 v2332 A Q/3+A 1 - 8 / 3 ( A N + A 2 2 - 2 A 0 2 ) 0 A Q / 3 + 2 A ] ] + 2 A 2 2 0 11 22 ,o 12 >0 "*0 +2$o + J 1 + * 2 2 u N o t a t i o n : s u p e r s c r i p t s r e f e r t o n u c l e i ; A K E j k ' ~ k ( k U ( ) ) , AJ J" E j£sJJ k_ C S A (ko» 0), r k s J C S A k - D ( S ) ' a n d *k S JS^R.<V" -149-5.5 RESULTS AND DISCUSSION The c o m p l i c a t e d form o f both t h e s p e c t r a l d e n s i t i e s i n E q u a t i o n [5.4.16] and t h e c o u p l e d e q u a t i o n s o f e v o l u t i o n ( [ 5 . 4 . 9 - 1 2 ] ) p r e c l u d e s a s i m u l t a n e o u s a n a l y s i s o f v a r i o u s e f f e c t s on t h e l o n g i t u d i n a l m a g n e t i -z a t i o n . S t i l l , a few exemplory c o n c l u s i o n s can be deduced from E q u a t i o n [ 5 . 4 . 1 6 ] . I f 6p = c o s _ 1 ( l / / 3 ) , t h e n n e i t h e r t h e c r o s s - t e r m s nor t h e d i p o l a r 2 2 s p e c t r a l d e n s i t i e s depend on t h e 6D 1/((6D i_) + (kcoQ) ) term. Thus, when D u > > D x ' 6 C ? L ~ ^0 ^ f o r k = 0 t e r m s t n i s l a t t e r s t i p u l a t i o n i s s u p e r -f l u o u s ) , i s a p p r o x i m a t e l y (but n o t i d e n t i c a l l y ) e q u al t o c o s - 1 ( l / / 3 ) , and e^ S A a r e much d i f f e r e n t from c o s _ 1 ( l / / 3 ) , t h e n t h e terms w i l l 1»2 assume l a r g e r magnitudes t h a n t h e A^ terms even i f t h e s t r e n g t h o f t h e CSA i n t e r a c t i o n i s much l e s s t h a n t h a t o f t h e d i p o l a r i n t e r a c t i o n ( ? D » E,^^ ). T h e r e f o r e , t h e r e l a t i v e o r i e n t a t i o n s o f t h e r e l a x a t i o n 1 s 2 c o o r d i n a t e systems may have an immense i n f l u e n c e oh t h e i m p o r t a n c e o f c r o s s - t e r m s . F o r a n o t h e r example, f o r t h e geometry where and -1 1 ' 2 - cos ( 1 / / 3 ) , a l l c r o s s - t e r m s v a n i s h even i f t h e two r e l a x a t i o n mechan-isms a r e o f comparable magnitude ~ A l s o a p p a r e n t from 1 »2 t h e s e e q u a t i o n s i s a seldom emphasized r e s u l t o f d i a g n o s t i c v a l u e : i n t h e s l o w m o t i o n r e g i m e , i n t e r n a l r o t a t i o n l e a d s t o s p e c t r a l d e n s i t i e s ( s a y a t t h e Larmor f r e q u e n c y ) w h i c h a r e l a r g e r i n magnitude t h a n i n t h e 47 48 absence o f i n t e r n a l f l e x i b i l i t y . ' S i n c e each m o l e c u l e would p r e s e n t a v i r t u a l l y u nique c a s e , i t i s i m p r a c t i c a l t o e x h a u s i v e l y a n a l y z e t h e g e n e r a l r e s u l t s o u t l i n e d t h u s f a r . I n s t e a d , two r e p r e s e n t a t i v e g e o m e t r i e s a r e chosen f o r d i s c u s s i o n . The -150-f i r s t c o r r e s p o n d s t o t h a t i m p l i e d i n t h e p i o n e e r i n g work o f B l i c h a r s k i , geometry J_, and t h e second c o r r e s p o n d s t o t h e a p p r o x i m a t e geometry f o r two s p i n 1/2 n u c l e i a t t h e c o r n e r s o f a t e t r a h e d r o n , geometry I_I_. In J_, a l l r e l a x a t i o n v e c t o r s a r e c o l l i n e a r ; i n Jl, e' = T T / 2 , ercn c o s " (1/3), (cj>'D - <f)^SA ) = T T / 6 , and (<j>p - <j> S^A ) = 5TT/6. F o r geometry I I , t h e p r i n c i p a l a x i s o f each c h e m i c a l s h i f t t e n s o r i s t a k e n as c o l -l i n e a r w i t h t h e c o v a l e n t bond a x i s and t h a t t h e p r i n c i p a l d i f f u s i o n a x i s i s d i r e c t e d a l o n g t h e bond w h i c h c o n n e c t s t h e c e n t r a l c a r b o n ( s a y ) t o t h e r e s t o f t h e m o l e c u l a r frame. By no means do we w i s h t o i m p l y any g e n e r a l i z a t i o n s o f t h i s a s s u m p t i o n ; t h i s c h o i c e i s made f o r t h e s o l e purpose o f p r o v i d i n g a grounds f o r a n u m e r i c a l example. I t mi g h t pay to n ote t h a t e x p e r i m e n t a l l y , i t pr o v e s q u i t e d i f f i c u l t t o r e l a t e s h i f t a n i s o t r o p i c s t o t h e m o l e c u l a r frame. F i n a l l y , i n t h e n u m e r i c a l c a l c u -8 -1 l a t i o n s , i t i s assumed t h a t yB n = 6 x 10 s e c " . F i g u r e s 5.1-3 g i v e t h e r e l a x a t i o n p a r a m e t e r s w h i c h c h a r a c t e r i z e t h e decay o f t h e l o n g i t u d i n a l m a g n e t i z a t i o n f o l l o w i n g a e-pulse. The decay, i n g e n e r a l , i s t h e sum o f f o u r e x p o n e n t i a l s , but when t h e two 11 22 12 s h i f t t e n s o r : p r i n c i p a l axes a r e c o l l i n e a r , A^ = A^ = A^ = A^ and 1 2 = = r^. In t h i s c a s e , t h e decay reduces t o t h e sum o f two expo-n e n t i a l s because o n l y E q u a t i o n [5.4.9] and t h e sum o f [5.4 .11 ] and [5.4 . 12 ] remain c o u p l e d , ( d / d t ) y ] ( t ) = [ A 1 - 4 A 2 + 4 A 1 ] y 1 ( t ) + [ 4 r i ] ( y 3 ( t ) + y 4 ( t ) ) [5.5 . 1 ] ( d / d t i ) ( y 3 ( t ) + y 4 ( t ) ) = [ 1 2 1 ^ ( t ) + [3A 1 + 1 2 A ] ] ( y 3 ( t ) + y 4 ( t ) ) . [5.5 . 2 ] -151-I t i s c o n v e n i e n t t o w r i t e <I ( t ) > - < I z > T ^ ^ f = 2~i *.exp(-\.t) = JL, a.exp ( -e.t') (cose - 1)<I > i = l 1 1 i = l 1 1 [5.5.3] where £ a. = 1, c. = X . [ A 1 - 4 A 2 + 2A^ + 2 A " ] " , and t ' = t [ A 1 - 4 A 2 11 22 + 2A-J + 2A-| ] . F o r geometry J_ and t h e summations e x t e n d o v e r two and f o u r terms r e s p e c t i v e l y . F i g u r e s 5.1-3 g i v e the n o r m a l i z e d magnitudes o f X-|, A 2 , and a - j ' ( a l s o X^ , X^ , a 2 , and i f a p p l i c a b l e ) under v a r i o u s c o n d i t i o n s . F i g u r e 5.1 g i v e s t h e r e l a x a t i o n p a rameters assuming the s i m p l e s t o f a l l c o n c e i v a b l e s i t u a t i o n s : Geometry J_; i s o t r o p i c m o b i l i t y ; ionx2 << 1. The r e l a x a t i o n p a rameters a r e p l o t t e d as a f u n c t i o n o f t h e i n t e r a c t i o n 24 c o n s t a n t s m a g n i t u d e s . T h i s p l o t c o r r e s p o n d s t o B l i c h a r s k i ' s c a l c u l a t i o n and p r o v i d e s a b a s i s f o r c o m p a r i s o n . F i g u r e 5.2 shows th e r e l a t i o n s h i p between th e degree o f nonexponen-t i a l r e l a x a t i o n and t h e r e l a t i v e magnitudes o f t h e two c o r r e l a t e d r e l a x -a t i o n mechanisms ( i n t h i s c a s e , d i p o l e - d i p o l e and c h e m i c a l s h i f t a n i s o -t r o p y ) . F i g u r e s 5.2A and B c o r r e s p o n d t o geometry J_, 5.2C and D t o geometry P l o t s 5.2A and C r e p r e s e n t s o l u t i o n s f o r D x = D u = 1 0 7 s e c " ^ 8 -1 and p l o t s 5.2B and D f o r = = 10 s e c " . E x p o n e n t i a l and pre-exponen-t i a l f a c t o r s a r e p l o t t e d v e r s u s t h e d i m e n s i o n l e s s r a t i o o f i n t e r a c t i o n -1 3 2 * c o n s t a n t s , ?Q5A?[) = (<°oAar )/(3^ >Y )• As e x p e c t e d , t h e decay i s d e s c r i b e d by a unique T-j when 5 ^ >> o r when « The g r e a t e s t d e v i -4 -1 * Assuming ? D = 2 x 10 Hz, a r a t i o £ C S A ? D - 1, i m p l i e s t h a t Aa - 300 ppm. -152-a t i o n from e x p o n e n t i a l decay o c c u r s when 4 > S Q ^ Q J^ > 2- However, i t i s e v i d e n t t h a t d e v i a t i o n s from e x p o n e n t i a l decay a r e r a t h e r s i m i l a r f o r b oth slow m o t i o n and f a s t m o t i o n p l o t s ( s l i g h t l y more pronounced i n t h e s l ow m o t i o n l i m i t ) , w i t h o n l y s u b t l e d i f f e r e n c e s a r i s i n g from t h e d i f f e r e n t g e o m e t r i e s i n v e s t i g a t e d ; however, geometry can be a d o m i n a t i n g f a c t o r as d i c u s s e d p r e v i o u s l y . F i g u r e 5.3 p r e s e n t s t h e e f f e c t o f "shape" and " s i z e " on t h e impor-t a n c e o f t h e c r o s s - t e r m s . In each o f t h e s e p l o t s , a s i n g l e r a t i o o f i n t e r a c t i o n c o n s t a n t s has been assumed, ?rj?cSA = 3 " ^ o t s 5-3A and B 7 8 -1 denote a c o n s t a n t " s i z e " c o r r e s p o n d i n g t o D x = 10 o r 10 s e c " r e s p e c -t i v e l y . The r e l a x a t i o n p a rameters a r e t h e n d e t e r m i n e d as a f u n c t i o n o f t h e "shape" v a r i a b l e , D„. P l o t s 5.3C and D on t h e o t h e r hand, a r e based on a c o n s t a n t "shape" (Dj_ = D^; s p h e r i c a l t o p a p p r o x i m a t i o n ) , but d i f f e r -e n t i n t e r n a l g e o m e t r i e s (J_ and J_I_ r e s p e c t i v e l y ) . Note t h a t when = 6D^, t h e r e l a x a t i o n b e h a v i o r becomes q u i t e complex as m i g h t be e x p e c t e d , and may change r a d i c a l l y o v e r q u i t e s m a l l v a r i a t i o n s i n " s i z e " . I t must be kept i n mind t h a t t h e i n t e r p l a y between e^, e£<-A , a)^, fy^^ , D^, _1 1 > 2 i » 2 D,( and Z^^Q » forms a most i n t r i c a t e a rrangement, and t h e g e n e r a l i t y o f t h e c o n c l u s i o n s drawn from any one f i g u r e n e c e s s a r i l y p r o v i d e s o n l y a p a r t i a l p i c t u r e . From t h e q u a n t i t a t i v e s p i n b e h a v i o r shown i n F i g u r e s 5.1-3, i t appears t h a t f o r a l l p r a c t i c a l p u r p o s e s , r a t h e r minor d e v i a t i o n s from e x p o n e n t i a l decay a r e g e n e r a l l y t o be e x p e c t e d . Even when t h e p r e -e x p o n e n t i a l f a c t o r s a r e a p p r o x i m a t e l y e q u a l , t h e e x p o n e n t i a l f a c t o r s u s u a l l y d i f f e r by a f a c t o r o f l e s s t h a n two; when t h e e x p o n e n t i a l f a c t o r s a r e s i g n i f i c a n t l y d i f f e r e n t , one p r e - e x p o n e n t i a l f a c t o r u s u a l l y dominates - 1 5 3 -t h e o t h e r s . One f e a t u r e i s c l e a r - i n t h e i n s t a n c e s examined h e r e , even a marked a n i s o t r o p y i n r e o r i e n t a t i o n has an a l m o s t n e g l i g i b l e i n f l u e n c e i n m a g n i f y i n g t h e n o n e x p o n e n t i a l i t y o f t h e decay. An i n t e r e s t i n g a s p e c t o f t h e e f f e c t o f c r o s s - t e r m s on r e l a x a t i o n i s e x h i b i t e d i n F i g u r e 5 . 4 . The s o l i d l i n e shows t h e r e l a x a t i o n b e h a v i o r 8 1 1 f o r geometry J_ when = D u = 1 0 s e c " and ? d 5 Q S A = 3 . The v a r i o u s p a r a m e t e r s may be d e t e r m i n e d from F i g u r e 5 . 3 . The two t h i n s o l i d l i n e s r e p r e s e n t t h e c o n s t i t u e n t e x p o n e n t i a l s o f t h e c o m p o s i t e decay. F i n a l l y , t h e broken l i n e shows t h e l o n g i t u d i n a l m a g n e t i z a t i o n decay e x p e c t e d under t h e same c i r c u m s t a n c e s when a l l c r o s s - c o r r e l a t i o n f u n c t i o n s a r e i g n o r e d . The i m p o r t a n t c o n c l u s i o n t o be drawn i s t h a t t h e i n c l u s i o n o f c r o s s - c o r r e l a t i o n terms always r e s u l t s i n r e l a x a t i o n w h i c h i s l e s s e f f i c i e n t ( c o n t r a r y t o p o s s i b l e i n t u i t i o n ) and t h a t t h e i n i t i a l decay i s i n d e p e n d e n t o f whether c r o s s - t e r m s a r e i n c l u d e d o r n o t . These con-c l u s i o n s can r e a d i l y be d e r i v e d from t h e boundary c o n d i t i o n s ( E q u a t i o n s [ 5 . 4 . 1 7 ] ) and E q u a t i o n s [ 5 . 4 . 9 - 1 2 ] , [ 5 . 5 . 3 ] and [ 5 . 4 . 1 5 ] . S i n c e y 2 ( 0 ) = y 3 ( 0 ) = y 4 ( 0 ) = 0 , i t d i r e c t l y f o l l o w s t h a t ( d / d t J y ^ O ) = T~^, and t h a t i a.\. = T ^ , where T ^ i s d e f i n e d as th e c o e f f i c i e n t o f y - i ( t ) i n E q u a t i o n [ 5 . 4 . 9 ] . F u r t h e r m o r e , a t a s h o r t t i m e l a t e r , T (T \. « 1 f o r a l l i ) , e l [ 5 . 5 . 4 ] but i g n o r i n g c r o s s - t e r m s one h a s , ( d / d t ) y ] ( t ) - T"1 + T " 2 x [ 5 . 5 . 5 ] I t t h e n f o l l o w s t h a t -154-^ a^ 2 ! >_ ( Xlcia. ) 2 ! = T ~ 2 T [ 5 . 5 . 6 ] and hence th e s l o p e i s always more p o s i t i v e . T h i s argument i l l u s t r a t e s t h a t t h e p r e s e n c e o f c r o s s - c o r r e l a t i o n terms always r e t a r d s r e l a x a t i o n , w h i l e t h e decay a t s h o r t t i m e s approaches t h e decay o b t a i n e d i n t h e absence o f c r o s s - t e r m s . I n no i n s t a n c e have d i f f u s i o n c o n s t a n t s l a r g e r t h a n U± ^ = 1 0 ^ s e c " ^ been c o n s i d e r e d . Owing t o t h e c l o s e r e l a t i o n s h i p between th e o r i g i n o f 49-51 t h e c h e m i c a l s h i f t t e n s o r and t h e s p i n - r o t a t i o n t e n s o r , i t i s e x p e c t e d t h a t t h e s p i n - r o t a t i o n mechanism w i l l dominate t h e r e l a x a t i o n f o r m o b i l e s p i n systems w h i c h have s i z a b l e C Q S A ? ^ r a t i o s . F o r r a p i d r o t a t i o n a l m o t i o n c h a r a c t e r i z e d r a t h e r a r b i t r a r i l y by D u ^> 1 0 ^ s e c ~ \ t h e ( e x p o n e n t i a l ) c o n t r i b u t i o n o f s p i n - r o t a t i o n s h o u l d be enough t o mask any e x p e r i m e n t a l d e t e c t i o n o f t h e ( n o n e x p o n e n t i a l ) r e l a x a t i o n r e s u l t i n g from t h e c r o s s - t e r m s d i s c u s s e d i n t h i s c h a p t e r . From E q u a t i o n s [ 5 . 4 . 9 - 1 2 ] , i t may be seen t h a t q u a n t i t a t i v e i n c o r p o r a t i o n o f t h e s p e c t r a l d e n s i t i e s , , i n t o t h e r e l a x a t i o n c a l c u l a t i o n s would amount t o f i n d i n g " s p e c i a l g e o m e t r i e s " i n ' t h e p r e s e n t f o r m u l a t i o n w h i c h would m i n i m i z e t h e i n f l u e n c e o f t h e c r o s s - t e r m s ( d e c o u p l i n g y-|(t) and y 2 ( t ) from y 3 ( t ) and y ^ ( t ) ) . I f , $ ^ = then i t i s o b v i o u s t h a t t h e decay i s e x p o n e n t i a l . Q u a l i t a t i v e l y , t h e s p i n - r o t a t i o n mechanism r e l a x e s a s p i n p o p u l a t i o n i n d e p e n d e n t l y o f o t h e r mechanism and thus t e n d s t o s t r a i g h t e n o u t any c u r v a t u r e i n t h e o b s e r v e d time-domain r e l a x a t i o n . S p i n - r o t a t i o n may be t h o u g h t o f as an e n f o r c e r o f t h e e x p o n e n t i a l law obeyed by s m a l l m o l e c u l e s ( c o n t a i n i n g n u c l e i w i t h i n h e r e n t l y l a r g e s h i f t a n i s o t r o p i e s ) w h i c h m i g h t o t h e r w i s e e x h i b i t a more c o l o r f u l r e l a x a t i o n b e h a v i o r . -155-Al though o n l y t h e l o n g i t u d i n a l r e l a x a t i o n has been d e s c r i b e d i n any d e t a i l , c a l c u l a t i o n s have been e x t e n d e d and examined f o r t h e t r a n s -v e r s e r e l a x a t i o n a l s o . In g e n e r a l , t h e same c o n c l u s i o n s a p p l y . To c a l c u l a t e t h e t r a n s v e r s e r e l a x a t i o n , we s t a r t by d e f i n i n g t h e f o l l o w i n g o r t h o g o n a l c o m b i n a t i o n s o f m a t r i x e l e m e n t s , = R e ( x 1 2 + x 1 3 + x 2 4 + W = T r ^ ( t ) l x ] [ 5 . 5 .7 ] q2(t) = R e ( - x 1 2 • • x 3 4 + x 1 3 + x 2 4 } [ 5 .5 .8 ] q 3 ( t ) = Re(x-, 2 - x34> [ 5 . 5 . 9 ] q4(t) £ R e ( x 1 3 - x 2 4 ) f [ 5 .5 .10 ] from w h i c h , by analogous p r o c e e d u r e s as i n t r o d u c e d i n t h e l a s t s e c t i o n , i t f o l l o w s t h a t (assuming o n l y c? D and c 3 C S A a r e o p e r a t i v e ) , (d/dt)qi(t) = [-3A q/2 + 5A 1 /2 - A 2 + A ] 1 + A 2 2 - | ( A J ] + A 2 2 ) ] q ] ( t ) + [ A ] 1 - A 2 2 + f ( - A N + A 2 2 ) ] q 2 ( t ) + [ r ] - 3 r 2 + | ( r j + 5 r 2 ) ] q 3 ( t ) + [ r 2 - 3r] + f ( r 2 + 5 r J ) ] q 4 ( t ) [ 5 .5 .11 ] ( d / d t ) q 2 ( t ) = [ A ] 1 - A 2 2 + ^(-AUQ + A 2 2 ) ] q ] ( t ) + [ A Q / 6 - A.,/2 + A 2 + A ] 1 + A 2 2 - fuj1 + A 2 2 ) ] q 2 ( t ) + [ r ] + r 2 + | r j - 2 r 2 ] q 3 ( t ) + [-r] - r 2 - | r 2 + 2 r Q ] q 4 ( t ) [ 5 .5 .12 ] ( d / d t ) q 3 ( t ) = [ | ( r j + 5 r 2 ) + r ] - 3 r 2 ] q i ( t ) / 2 + [ | ( r j - 3 r 2 ) + r] + r 2 ] q 2 ( t ) / 2 -156-+ [-5AQ/6 + - A 0 + 4AJ 1 + 2A 2 2 - | A 2 2 ] q 3 ( t ) + [-|AQ + 4 A J 2 ] q 4 ( t ) [ 5 .5 .13 ] ( d / d t ) q 4 ( t ) = [ | ( r 2 + 5r J ) + v\ - 3 r ] ] q i ( t ) / 2 + [- | ( r 2 - 3 r Q ) - r ] - r 2 ] x q 2 ( t ) / 2 + [-|A Q + 4 A J 2 ] q 3 ( t ) + [-5A Q/6 + L}/Z - A £ + 4 A 2 2 + 2A] 1 - § A n ] q 4 ( t ) . [ 5 .5 .14 ] The s i m p l e s t r e d u c t i o n o f t h e s e e q u a t i o n s i s r e a l i z e d i f A ^ J = A^ and * 24 25 = r ^ . In t h i s c a s e ( c o n s i d e r e d p r e v i o u s l y by B l i c h a r s k i ' ) , t h e e q u a t i o n s reduce t o ( d / d t ) q i ( t ) = [-3A Q/2 + 5A 1 /2 - A 2 + 2A 1 - 8 A 0 / 3 ] q i ( t ) + [-2r ] + 4 r Q ] x ( q 3 ( t ) + q 4 ( t ) ) [ 5 .5 .15 ] ( d / d t ) ( q 3 ( t ) + q 4 ( t ) ) = [ - 2 r 1 + 4 r - ] q j ( t ) i i : [ r 3 A 0 / 2 + A ^ 2 - A 2 + 10A-, - 8A Q /3 x ( q 3 ( t ) + q 4 ( t ) ) . [ 5 .5 .16 ] (The d i s c r e p a n c y between t h e s e e q u a t i o n s and t h o s e a p p e a r i n g i n B l i c h a r s k i ' s 25 paper can be t r a c e d t o s i g n e r r o r s i n h i s e q u a t i o n s . ) B e f o r e l e a v i n g t h i s p o i n t b e h i n d , one comment uniq u e t o the t r a n s -v e r s e decay i s i n o r d e r . I f 6D L < GUQ, t h e n t h e a p p r o x i m a t e e q u a t i o n s o f m o t i o n f o r q-,(t) and q 3 ( t ) + q 4 ( t ) a r e ( d / d t ) q ] ( t ) - [-3A Q/2 - 8 A 0 / 3 ] q i ( t ) + 4 r Q ( q 3 ( t ) + q 4 ( t ) ) [5 .5 .17 ] ( d / d t ) ( q 3 ( t ) + q 4 ( t ) ) = 4 r Q q i ( t ) + [-3AQ/2 - 8 A Q / 3 ] ( q 3 ( t ) + q 4 ( t ) ) . [ 5 .5 .18 ] -157-S i n c e q ^ O ) = < I X ( 0 ) > and q 3 ( 0 ) + q 4 ( 0 ) = 0, < I x ( t ) > / < I x ( 0 ) > = l { e x p ( A 1 t ) + exp.(x2t)| [5.5.19] where ^ ^ z ~ 3 a(/ 2 " 8 Ao / 3 ± ^ro* ^ A 0 ~ A 0 ~ r 0 ' n o t l c e t n a t t n e decay w i l l be e x t r e m e l y n o n e x p o n e n t i a l w i t h one t i m e c o n s t a n t a l m o s t two o r d e r s o f magnitude l a r g e r t h a n t h e o t h e r ( w i t h e q u a l p r e - e x p o n e n t i a l f a c t o r s ! ) . T h i s one example p o i n t s o u t q u i t e c l e a r l y t h a t t h e f a i l u r e o f an e x t r e m e - n a r r o w i n g a p p r o x i m a t i o n may have d r a m a t i c e f f e c t s on t h e t r a n s v e r s e decay. - 1 5 8 -FIGURE 5.1: P l o t s o f t h e decay c o n s t a n t s ( u n i t l e s s ) v e r s u s t h e r e l a t i v e m agnitudes o f t h e two i n t e r f e r i n g r e l a x a t i o n mechanisms ( d i p o l a r - s h i f t a n i s o t r o p y ) assuming t h a t t h e p r i n c i p a l axes o f t h e s h i f t t e n s o r s a r e c o l l i n e a r . The decay i s g i v e n by [ < I z ( t ) > - < I z > T ] [ < I z > T ( c o s e - l ) ] _ 1 = a 1 e x p ( - e 1 t ' ) + ( l - a 1 ) e x p ( - e 2 t ' ) where e. =*.JTQ and t ' = t / T Q . T ^ i s t h e c o e f f i c i e n t o f y - j ( t ) '•>• • f i n , E q u a t i o n [ 5 . 4 . 9 ] . -160-FIGURE 5.2: P l o t s o f t h e n o r m a l i z e d decay c o n s t a n t s ( u n i t l e s s ) v e r s u s t h e r e l a t i v e magnitudes o f t h e two i n t e r f e r i n g r e l a x a t i o n mechan-isms ( d i p o l a r - s h i f t a n i s o t r o p y ) . The c a l c u l a t e d v a l u e s c h a r -a c t e r i z e t h e l o n g i t u d i n a l decay o f m a g n e t i z a t i o n , [ < I z ( t ) > - < I z > T ] [ ( c o s e - 1 ) < I Z > T ] _ 1 = z a . e x p f - e . t ' ) , 1 _ i where J a. = 1 , I a.,^. = 1, e.. = X ^ T Q , and t ' = t / T Q . T Q i s th e c o e f f i c i e n t o f y-,(t) i n E q u a t i o n [ 5 . 4 . 9 ] . The a b s c i s s a i s 1 3 p l o t t e d as t h e l o g a r i t h m o f t h e r a t i o , S ^ S Q (=BQAar / 3 y f i ) . F i g u r e s (A) and (B) a r e f o r geometry J_, (C) and (D) a r e f o r geometry J_J_ (see t e x t ) . F i g u r e s (A) and (C) a r e f o r D(| = DL = 1 0 7 s e c " 1 , and (B) and (D) a r e f o r D„ = D = 1 0 8 s e c " 1 . -162-FIGURE 5.3: P l o t s o f t h e decay c o n s t a n t s ( u n i t l e s s ) v e r s u s d i f f e r e n t hydrodynamic " e f f e c t i v e " shapes and s i z e s o f m o l e c u l a r frameworks. The c a l c u l a t e d v a l u e s c h a r a c t e r i z e t h e l o n g i -t u d i n a l decay o f m a g n e t i z a t i o n , [ < I z ( t ) > - < I z > T ] [ ( c o s e - l ) < I z > T ] " 1 = E a.expf-e.f) , where z a. = 1, E c^.e.. = 1, = A ^ T Q, and t 1 = t / T n I n F i g u r e s (A) and ( B ) , t h e s e p a r a m e t e r s a r e p l o t t e d as a 7 - 1 8 - 1 f u n c t i o n o f D,, assuming = 10 s e c " and = 10 s e c " r e s p e c t i v e l y . In F i g u r e s (C) and ( D ) , t h e s e p arameters a r e p l o t t e d as a f u n c t i o n o f D|( assuming D^/D^ = 1. F i g u r e (D) i s t h e c a l c u l a t i o n performed w i t h geometry J_, and F i g u r e s ( A ) , ( B ) , and (C) a r e done assuming geometry II_ (see t e x t ) . A l l p l o t s assume ?n/?^Qfl = 3. -1 6 3 --164-FIGURE 5.4: The l o n g i t u d i n a l decay o f m a g n e t i z a t i o n f o r a p a i r o f s p i n s 8 -1 i n t h e l i m i t : D,, = D L=10 s e c " , geometry I_, and Sp / ^ C S A = 3. The t h i c k s o l i d upper c u r v e i s t h e t h e o r e t i c a l p r e d i c t i o n , t h e broken l i n e i s t h e same c a l c u l a t i o n e x c e p t a l l c r o s s -c o r r e l a t i o n terms have been s e t equal t o z e r o . The s o l i d c u r v e i s b i e x p o n e n t i a l ; t h e c o n s t i t u e n t decays a r e shown i n t h e l i g h t e r l i n e s . The a b s c i s s a i s i n u n i t s o f T Q ( t h e i n v e r s e o f t h e c o e f f i c i e n t o f y-,(t) i n E q u a t i o n [ 5 . 4 . 9 ] ) . -166-5.6 SUMMARY T h i s c h a p t e r and t h e p r e c e e d i n g one have a n a l y z e d t h e s i g n i f i c a n c e o f c r o s s - c o r r e l a t i o n s i n t h e a n a l y s i s o f n u c l e a r m a g n e t i c r e l a x a t i o n d a t a , p a r t i c u l a r l y f o r l a r g e m o l e c u l e s i n s o l u t i o n . The major p o i n t s may now be summarized. (1) Whereas a n i s o t r o p i c r e o r i e n t a t i o n m a g n i f i e s t h e e f f e c t o f Hubbard t e r m s , t h i s paper has shown t h a t n o n s p h e r i c a l d i f f u s i o n gen-e r a l l y has a much s m a l l e r e f f e c t on B l i c h a r s k i t erms. I t s h o u l d a l s o be n oted t h a t when r e o r i e n t a t i o n i s i s o t r o p i c , t h e e x p e r i m e n t a l e f f e c t o f Hubbard terms i s n e g l i g i b l e , but B l i c h a r s k i terms may s t i l l be s i g -n i f i c a n t . (2) F o r c r o s s - c o r r e l a t i o n terms o f t h e second ( B l i c h a r s k i ) k i n d , we have n o t e d t h a t c e r t a i n r e l a t i o n s h i p s between t h e i n t e r n a l r e l a x -a t i o n c o o r d i n a t e systems may s e r v e t o m a g n i f y t h e s e terms. In f a c t , i t i s p o s s i b l e t h a t n o n e x p o n e n t i a l decay o r decay by means o f t h e c h e m i c a l s h i f t a n i s o t r o p y (CSA) mechanism i t s e l f may dominate t h e ob-s e r v e d r e l a x a t i o n , even when t h e CSA i n t e r a c t i o n c o n s t a n t i s s m a l l com-par e d t o t h a t o f t h e d i p o l a r (D) i n t e r a c t i o n . F o r m o l e c u l e s s u f f i c i e n t -l y l a r g e t h a t t h e e x t r e m e - n a r r o w i n g a p p r o x i m a t i o n no l o n g e r h o l d s , f u r t h e r p e c u l i a r i t i e s a r i s e from p a r t i c u l a r r e l a t i o n s h i p s between t h e magnitudes o f t h e p r i n c i p a l d i f f u s i o n c o n s t a n t s and t h e Larmor f r e -quency. A l t h o u g h s i m i l a r g e o m e t r i c a l c o n s i d e r a t i o n s a p p l y t o Hubbard t e r m s , t h e r e i s i n t h a t case l e s s v a r i a b i l i t y i n r e l a x a t i o n c o r r e s p o n d -i n g t o d i f f e r e n t i n t e r n a l o r i e n t a t i o n s . F u r t h e r m o r e , i n t h e slow m o tion l i m i t , e x t e n s i o n o f B l i c h a r s k i ' s o r i g i n a l c a l c u l a t i o n shows t h a t t h e -167-t r a n s v e r s e r e l a x a t i o n may be e x t r e m e l y n o n e x p o n e n t i a l . (3) The i n c l u s i o n o f e i t h e r t y p e o f c r o s s - c o r r e l a t i o n f u n c t i o n r e s u l t s i n l e s s e f f e c t i v e r e l a x a t i o n than would be f ound from a c a l -c u l a t i o n i n v o l v i n g j u s t t h e a p p r o p r i a t e a u t o - c o r r e l a t i o n f u n c t i o n s , t h a t i s , t h e decay c u r v e a t any g i v e n t i m e shows a s m a l l e r ( i n magni-t u d e ) s l o p e when c r o s s - c o r r e l a t i o n s a r e i n c l u d e d . The i n i t i a l decay (as t i m e goes t o z e r o ) i s t h e same, whether c r o s s - c o r r e l a t i o n s a r e i n -c l u d e d o r n o t . (4) There a r e s e v e r a l c i r c u m s t a n c e s w h i c h w i l l p r e c l u d e o b s e r v a t i o n o f n o n e x p o n e n t i a l r e l a x a t i o n o f t h e k i n d d i s c u s s e d i n t h e s e c h a p t e r s . P a r a m a g n e t i c i m p u r i t i e s o r nearby n u c l e a r s p i n s w i l l c o n t r i b u t e i n t e r -m o l e c u l a r r e l a x a t i o n ( e x p o n e n t i a l ) , masking any n o n e x p o n e n t i a l e f f e c t . S p i n - r o t a t i o n i n t e r a c t i o n s may mask B l i c h a r s k i t e r m s , s i n c e a l a r g e CSA i n t e r a c t i o n c o n s t a n t i m p l i e s a l a r g e SR i n t e r a c t i o n c o n s t a n t . F i n a l l y , o t h e r i n t r a m o l e c u l a r mechanisms may a l s o mask ( o r c o m p l i c a t e ) nonexponen-t i a l r e l a x a t i o n and a r e not d i s c u s s e d f u r t h e r h e r e . (5) A l t h o u g h the p r e s e n t paper t r e a t s a t w o - s p i n s y s t e m , w h i l e C h a p t e r IV n e c e s s a r i l y t r e a t s a t h r e e - s p i n s y s t e m , t h e c o n c l u s i o n s i n e i t h e r c a s e c o u l d e a s i l y be e x t e n d e d ( a t l e a s t q u a l i t a t i v e l y ) t o l a r g e r s p i n systems and t o systems c o n t a i n i n g c h e m i c a l l y d i f f e r e n t n u c l e i ( J ' h e t e r o " - n u c l e i ) . In c o n c l u s i o n , a l t h o u g h n o n e x p o n e n t i a l n u c l e a r m a g n e t i c r e l a x a t i o n i s a t p r e s e n t an e x p e r i m e n t a l c u r i o s i t y , t h e p r e s e n t t h e o r y s u g g e s t s t h a t f o r f u t u r e NMR s t u d i e s o f l a r g e m o l e c u l e s i n s o l u t i o n a t h i g h ex-t e r n a l m a g n e t i c f i e l d s , n o n e x p o n e n t i a l r e l a x a t i o n may w e l l become e v i -d ent e x p e r i m e n t a l l y , and can p r o v i d e a p p r e c i a b l e a d d i t i o n a l i n f o r m a t i o n , and c o n f u s i o n , about the s p i n e n v i r o n m e n t and m o t i o n a l dynamics. •168-TABLE 5.2 1 1 SUMMARY: EFFECT OF INTERFERENCE TERMS HUBBARD (FIRST KIND) BLICHARSKI(SECOND KIND) C r i t e r i a o f d e f i n i t i o n : example: s i g n r e v e r s a l o f r e l a x a t i o n terms under s p i n i n v e r s i o n ? o b s e r v e d i n s o l i d s ? o b s e r v e d i n l i q u i d s ? lis w^.. \To (d/dt)M A r a p i d , i s o t r o p i c c • r - ts •r— £ S- •I— 1 +-> re o on CD • 1 — o + J o (O B + J s- to o Q. o E •r— var r a p i d , a n i s o t r o p i c s l o w , i s o t r o p i c s l o w , a n i s o t r o p i c d i p o l e - d i p o l e no yes CD 3C0CD 2CH 3 •1/Ti -1/T, n i l pronounced n i l f(c5) / f(cn) d i p o l e - s h i f t a n i s o t r o p y y e s no B F 3 , C F 2 C 1 2 , CHFC1 2 -1 / T , -1/T 2 m a r g i n a l m a r g i n a l m a r g i n a l ( l o n g i t u d i n a l ) p r o n o u n c e d ( t r a n s v e r s e ) n i l ( l o n g i t u d i n a l ) m a r g i n a l ( l o n g i t u d i n a l ) p r o n o u n c e d ( t r a n s v e r s e ) p r o n o u n c e d ( t r a n s v e r s e ) n The l a s t f o u r i t e m s i n t h i s TABLE a r e g e n e r a l i z a t i o n s and a r e not v a l i d i n a l l c i r c u m s t a n c e s . See t e x t f o r f u r t h e r e x p l a n a t i o n . * A l l s t a t e m e n t s a r e based on t h e a s s u m p t i o n o f t h r e e , e q u a l l y spaced , i d e n t i c a l n u c l e i . -169-REFERENCES: CHAPTER V 1. L. G. Werbelow and A. G. M a r s h a l l , M o l . Phys. , ( 1 9 7 4 ) . 2. A. Abragam, P r i n c i p l e s d f N u c l e a r Magnetism, C l a r e n d o n P r e s s , O x f o r d , 1961; page 441T 3. P. S. Hubbard, J . Chem. Phys. 53, 985 ( 1 9 7 0 ) . 4. R e f e r e n c e 2; page 295. 5. A. Kumar, N. R. K r i s h u a , and B. D. Nageswara Rao, M o l . Phys. 18, 11 (1970 ) . 6. A. A. B r o o k s , J . D. C u t n e l l , E. 0. S t j s k a l , and V. W. W e i s s , J . Chem. Phys. 49, 1571 (1969). 7. L. G. Werbelow and A. G. M a r s h a l l , J . Magn. Res. V\_, 299 (1973). 8. H. M. M c C o n n e l l , J . Chem. Phys.-25, 709 (1956). 9. J . M. F r e e d and G. K. F r a e n k e l , J . Chem. Phys. 39, 326 (1963). 10. G. K. F r a e n k e l , J . Bhys. Chem. 71_, 139 ( 1 9 6 7 ) . 11. H. S h i m i z u , J . Chem. Phys. 40, 3357 (1964). 12. E. L. Mackor and C. MacLean, J . Chem. Phys. 44, 64 (1966). 13. R. M. L y n d e n - B e l l , P r o g . NMR S p e c t . 2, 163 (1 9 6 7 ) . 14. R e f e r e n c e 2; page 508. 15. P. W. A t k i n s , M o l . Phys. J_2, 125 (1967). 16. R. Freeman, S. W i t t e k o e k , and R. R. E r n s t , J . Chem. Phys. 52, 1529 (1970). 17. T. N. K h a z a n o v i c h and V. Yu. Z i t s e r m a n , M o l . Phys. 21, 65 (1 9 7 1 ) . 18. B. D. Nageswara Rao, Adv. Magn. Res. 4, 271 (1968). 19. A. Kumar and B. D. Nageswara Rao, J . Magn. Res. 8, 1 (1 9 7 2 ) . 20. J . M. A n d e r s o n , M o l . Phys. 8, 505 (1 9 6 4 ) . 21. R. L. V o i d and H. S. Gutowsky, J . Chem. Phys. 47, 4782 ( 1 9 6 7 ) . -170-22. S. S y k o r a , J . Chem. Phys. 54, 2469 (1971). 23. R. Hoffman, Adv. Magn. Res. 4, 87 ( 1 9 6 8 ) . 24. J . S. B l i c h a r s k i , Phys. L e t t . 24A, 608 ( 1 9 6 7 ) . 25. J . S. B l i c h a r s k i , A c t a Phys. P o l o n . 36, 211 ( 1 9 6 9 ) . 26. J . S. B l i c h a r s k i , A c t a Phys. P o l o n . A38, 19 ( 1 9 7 0 ) . 27. J . S. B l i c h a r s k i and W. N o s e l , A c t a Phys. P o l o n . A38, 25 ( 1 9 7 0 ) . 28. J . S. B l i c h a r s k i , W. N o s e l , and H. S c h n e i d e r , Ann. P h y s i k 27 17 ( 1 9 7 1 ) . 29. J . S. B l i c h a r s k i , P r o c . X V I t h Congress AMPERE, page 680, 1971. 30. J . S. B l i c h a r s k i and W. N o s e l , A c t a Phys. P o l o n . A42, 223 ( 1 9 7 2 ) . 31. J . S. B l i c h a r s k i , P r o c . X V I I t h Congress AMPERE, 1971; u n p u b l i s h e d . 32. P. R i g n e y and J . V i r l e t , J . Chem. Phys. 51_, 3807 ( 1 9 6 9 ) . 33. N. Boden and R. F o l l a n d , M o l . Phys. 21_, 1123 (1971 ). 34. B. D. Nageswara Rao and L. R. A n d e r s , Phys. Rev. 140, A112 ( 1 9 6 5 ) . 35. P. W. A t k i n s , M o l . Phys. 21_, 97 ( 1 9 7 1 ) . 36. N. C. P y p e r , M o l . Phys. 21_, 1 ( 1 9 7 1 ) ; i b i d . , 22, 433 (1971 ). 37. J . S. B l i c h a r s k i , Z. N a t u r f o r s c h . 27a, 1355 ( 1 9 7 2 ) . 38. G. C. L e v y , D. W. W h i t e , and F. A. L. A n e t , J . Magn. Res. 6_, 453 ( 1 9 7 2 ) . 39. H. W. S p i e s s , D. S c h w e i t z e r , and U. H a e b e r l e n , J . Magn. Res. £, 444 ( 1 9 7 3 ) . 40. H. W. S p i e s s and H. Mahnke, Ber. Bunsenges. physik.Chem. 7J5, 990 ( 1 9 7 2 ) . 41. P. S. Hubbard, Phys. Rev. 13J_, 1155 ( 1 9 6 3 ) . 42. J . S. B l i c h a r s k i , Z. N a t u r f o r s c h . 27a, 1456 ( 1 9 7 2 ) . 43. A. D. Buckingham and S. M. Malm, M o l . Phys. 22, 1127 ( 1 9 7 1 ) . 44. J . H. W i l k i n s o n , The A l g e b r a i c E i g e n v a l u e P r o b l e m , C l a r e n d o n P r e s s , O x f o r d , 1965; page 30. 45. P. S. Hubbard, Phys. Rev. A6, 2412 ( 1 9 7 2 ) . -171-46. C. S. Wang, J . Magn. Res. 9, 75 (1973). 47. D. D o d d r e l l , V. G l u s h k o , and A. A l l e r h a n d , J . Chem. Phys. 56, 3683 (1972) . ~~~ 48. L. G. Werbelow and A. G. M a r s h a l l , J . Amer. Chem. Soc. 95, 5132 (1 9 7 3 ) . 49. J . M. Deutch and J . S. Waugh, J . Chem. Phys. 43, 1914 (1965). 50. H. W. S p i e s s , D. S c h w e i t z e r , U. H a e b e r l e n , and K. H. H a u s s e r , J . Magn. Res. 5_, 101 (1971). 51. R. V. R e i d and A. Chu, Phys. Rev. A9, 609 (1 9 7 4 ) . -172-CHAPTER VI EFFECT OF MOLECULAR SHAPE AND FLE X I B I L I T Y ON GAMMA-RAY DIRECTIONAL CORRELATIONS 1 6.1. INTRODUCTION N u c l e a r M a g n e t i c Resonance i s not t h e o n l y t e c h n i q u e a v a i l a b l e t o t h e p h y s i c a l b i o c h e m i s t whereby n u c l e a r r e l a x a t i o n can be used t o e l u c i -d a t e m a c r o m o l e c u l a r dynamics. The p o t e n t i a l o f P e r t u r b e d A n g u l a r C o r r e -l a t i o n (PAC) e x p e r i m e n t s as a means towards m o t i o n a l and s t r u c t u r a l i n f o r m a t i o n c o n c e r n i n g b i o l o g i c a l l y i n t e r e s t i n g m o l e c u l e s has been d i s -2-9 • • c u s s e d i n s e v e r a l r e c e n t p u b l i c a t i o n s . I n complete, a n a l o g y w i t h i t s n e a r > e l a t ; i v e , NMR;* a thor o u g h u n d e r s t a n d i n g o f t h e e f f e c t s o f m o l e c u l a r m o t i o n on a n g u l a r c o r r e l a t i o n s i s n e c e s s a r y i n o r d e r t o a p p l y t h i s t e c h -n i q u e e f f e c t i v e l y t o t h e s t u d y o f b i o l o g i c a l s y s t e m s . The a n g u l a r c o r r e l a t i o n e x p e c t e d f o r a s o l i d sample o r f o r s p e c i e s o f low m o l e c u l a r w e i g h t d i s s o l v e d i n i s o t r o p i c l i q u i d s i s w e l l known. R e c e n t l y , s e v e r a l c a l c u l a t i o n s have p o i n t e d o u t t h e e f f e c t s on a n g u l a r c o r r e l a t i o n s o f s l o w ( c o r r e l a t i o n t i m e s g r e a t e r t h a n a few nanoseconds) 13-17 i s o t r o p i c r o t a t i o n a l d i f f u s i o n and s l o w r o t a t i o n a l d i f f u s i o n i n one 1 g d i m e n s i o n . However, t h e e f f e c t o f a n i s o t r o p i c m o t i o n s has not been commented upon i n t h e l i t e r a t u r e . The i m p o r t a n c e o f a n i s o t r o p i c r o t a t i o n a l d i f f u s i o n has been demon-s t r a t e d f o r e x p e r i m e n t a l l y o b s e r v e d q u a n t i t i e s i n such t e c h n i q u e s as N u c l e a r M a g n e t i c Resonance (see C h a p t e r s I I - V ) and f l u o r e s c e n c e d e p o l a r --173-i z a t i o n . ~ F u r t h e r m o r e , i t has been shown t h a t i n t e r n a l r o t a t i o n a l m o t i o n s can m a r k e d l y a f f e c t t h e r e s u l t s o f N u c l e a r M a g n e t i c Resonance (see C h a p t e r s I I - V ) , f l u o r e s c e n c e d e p o l a r i z a t i o n , and p a r a m a g n e t i c " s p i n 22 l a b e l " e x p e r i m e n t s . Knowledge o f t h e e f f e c t o f a n i s o t r o p i c r o t a t i o n a l d i f f u s i o n and a l s o o f i n t e r n a l r o t a t i o n on gamma-ray a n g u l a r c o r r e l a t i o n s i s t h u s o f g r e a t i n t e r e s t . The c a l c u l a t i o n s p r e s e n t e d i n t h i s s e c t i o n a r e adapted t o both an e x t r e m e - n a r r o w i n g l i m i t ( r a p i d m o t i o n ) and an a d i a b a t i c l i m i t ( s l o w m o t i o n ) ; t h e two l i m i t i n g c a ses span n e a r l y t h e whole range o f r a t e s o f r o t a t i o n a l r e o r i e n t a t i o n . -174-6.2. PERTURBED ANGULAR CORRELATIONS In t h e r e a l m o f c o n v e n t i o n a l n u c l e a r s t u d i e s , t h e t h e o r y o f a n g u l a r c o r r e l a t i o n s i s p r o b a b l y one o f t h e b e s t and most comprehensive t h e o r i e s o f n u c l e a r phenomena due t o t h e f a c t t h a t i t i s based on v e r y g e n e r a l symmetry p r i n c i p l e s . I t i s t h e i s o t r o p y o f t h r e e - d i m e n s i o n a l space w i t h i t s i m p l i c a t i o n : o f c o n s e r v a t i o n o f a n g u l a r momentum t h a t i s t h e fundamental r e a s o n f o r t h e e x i s t e n c e o f a n i s o t r o p i c a n g u l a r c o r r e l a t i o n s and d i s t r i b u t i o n s . When two gamma-rays from a r a d i o a c t i v e n u c l e u s a r e e m i t t e d i n a de-e x c i t a t i o n c a s c a d e and d e t e c t e d by a c o i n c i d e n c e s p e c t r o m e t e r , t h e co-i n c i d e n c e c o u n t i n g r a t e may depend s t r o n g l y on t h e a n g l e between t h e p r o p a g a t i o n v e c t o r s , and o f t h e two gamma-rays. T h i s a n g u l a r c o r -r e l a t i o n may be p e r t u r b e d ( a t t e n u a t e d ) by t h e i n t e r a c t i o n o f e x t r a n u c l e a r f i e l d s w i t h t h e n u c l e a r moments i n t h e i n t e r m e d i a t e s t a t e o f t h e c a s c a d e . F i g u r e 6.1 i l l u s t r a t e s an e n e r g y - l e v e l scheme f o r a y-y n u c l e a r decay p r o c e s s . The t r a n s i t i o n f r o m t h e i n i t i a l n u c l e a r s t a t e i t o t h e i n t e r -m e d i a t e s t a t e i s accompanied by e m i s s i o n o f t h e f i r s t p h o t o n , y-j. I f t h e l i f e t i m e o f t h e i n t e r m e d i a t e s t a t e and t h e c o u p l i n g o f t h e n u c l e u s t o i t s e n v i r o n m e n t a r e o f the p r o p e r m a g n i t u d e , t h e n m o l e c u l a r r o t a t i o n can cause t h e n u c l e a r s p i n v e c t o r t o change i t s d i r e c t i o n w i t h r e s p e c t t o a l a b o r a t o r y - f i x e d system. A t some ti m e t a f t e r e m i s s i o n o f y-|, t h e r e i s a t r a n s i t i o n from t h e i n t e r m e d i a t e s t a t e t o t h e f i n a l s t a t e , f , accompanied * The term a n g u l a r c o r r e l a t i o n g e n e r a l l y r e f e r s t o both d i r e c t i o n a l and p o l a r i z a t i o n c o r r e l a t i o n s . The e x p e r i m e n t we w i s h t o d i s c u s s o n l y ob-s e r v e s t h e d i r e c t i o n a l c o r r e l a t i o n and t h i s t i t l e would be more a p p r o p r i a t e . However, bound by c o n v e n t i o n , t h e term a n g u l a r c o r r e l a t i o n w i l l be used. -175-by t h e e m i s s i o n o f a second p h o t o n , Y 2 -The phenomena o f e n t r y i n t o , e x i t f r o m , and p e r t u r b a t i o n w h i l e i n t h e i n t e r m e d i a t e s t a t e a r e s e p a r a b l e . I f Y ] i s e m i t t e d i n d i r e c t i o n k-|, and Y 2 i s e m i t t e d i n d i r e c t i o n k 2 , the n t h e a n g u l a r c o r r e l a t i o n function, 3 8^(kp k 2 , t ) , f o r a p a r t i c u l a r d e l a y t i m e between o b s e r v a t i o n o f Y-| and Y 2 i s g i v e n by & ( k , , k ? , t ) ^ e x p t - t / x j S S [ ( 2 k , + 1 ) " 1 / 2 Y ^ K f i J A . ( 1 ) ] 1 L IN H k 1 k 2 N ] N 2 1 K l 1 K l x G J j l j j 2(t) [ ( 2 k 2 + 1 ) ~ 1 / 2 Y J j 2 ( n 2 ) A k ( 2 ) ] . [6.2.1] The s u b s c r i p t s 1 and 2 r e f e r t o r a d i a t i o n s 1 and 2, and t h e arguments o f each s p h e r i c a l harmonic r e f e r t o the a n g l e s between an a r b i t r a r y c o o r d i n a t e frame and t h e d i r e c t i o n o f e m i s s i o n o f t h e i n d i c a t e d gamma-r a y , as shown i n F i g u r e 6.2. The r a d i a t i o n p a r a m e t e r s , A. ( i ) depend o n l y on t h e s p i n s and m u l t i p o l a r i t i e s a s s o c i a t e d w i t h t h e i t r a n s i t i o n , and t h e c o u p l i n g o f t h e i n t e r m e d i a t e - s t a t e n u c l e u s w i t h i t s e n v i r o n m e n t N N 11 i s c o m p l e t e l y d e s c r i b e d by t h e p e r t u r b a t i o n f a c t o r G . 1 . 2 ( t ) , K 1 K 2 G ^ 2 ( t ) - g [ ( 2 k , • D (2k 2 • 1 ) ] V 2 ^ ^kjJ a b x (i-™bX)<n'biA(t>iv<rai'iA(t)ira;>* • [6'2'2] Im > and |m'>:are i n t e r m e d i a t e n u c l e a r s u b s t a t e s i m m e d i a t e l y a f t e r t h e 1 a 1 a e m i s s i o n o f y-j > \\> a n d hb> a r e s u b s t a t e s i m m e d i a t e l y b e f o r e e m i s s i o n o f Y 2 , A N D A ( t ) i s t h e e v o l u t i o n o p e r a t o r f o r t h e i n t e r m e d i a t e s t a t e . -176-\ m, m 9 m~/ The I / a r e t h e f a m i l i a r v e c t o r c o u p l i n g c o e f f i c i e n t s ( 3 J sym- m m ' b o l s ) 2 4 In o r d e r t h a t t h e r e be an a n g u l a r c o r r e l a t i o n , t h e i n t e r m e d i a t e s t a t e must have n u c l e a r s p i n >_ 1. Such a n u c l e u s may have a q u a d r u p o l e moment, and i n most s t u d i e s o f m a t e r i a l s i n a condensed phase, t h e quad-r u p o l a r i n t e r a c t i o n r e p r e s e n t s t h e major p e r t u r b i n g i n f l u e n c e on t h e a n g u l a r c o r r e l a t i o n . I t i s o f t e n a good a p p r o x i m a t i o n - and c e r t a i n l y a c o m p u t a t i o n a l c o n v e n i e n c e - t o assume t h a t t h e e l e c t r i c f i e l d g r a d i e n t r e s p o n s i b l e f o r t h e q u a d r u p o l a r i n t e r a c t i o n has a x i a l symmetry. When t h i s i s t h e c a s e , t h e p e r t u r b a t i o n f a c t o r f o r a s i n g l e n u c l e u s , e v a l u a t e d i n a c o o r d i n a t e system whose z - a x i s i s p a r a l l e l t o t h e p r i n c i p a l a x i s o f t h e i n t e r a c t i o n , i s g i v e n by G k Nk (t> =X) [ ( 2 k l + i ) ( 2 k 2 + D : 1 ^ 1 1 kA(l 1 k2) 1 K2 m 1 c \m' -m N / \m' -m N / x e x p ( - i ( E m - E m . ) t ) . [6.2.3] 2 2 The f o l l o w i n g d e f i n i t i o n s - a r e " e m p l o y e d : E m = (e qQ/fi)[3m - 1(1+1)] x ( 4 1 ( 2 1 + 1 ) ) _ 1 , (eQ) i s t h e n u c l e a r q u a d r u p o l e moment, (eq) i s t h e p r i n c i p a l v a l u e o f t h e e l e c t r i c f i e l d g r a d i e n t t e n s o r , and m i s t h e quantum number f o r p r o j e c t i o n o f n u c l e a r s p i n on t h e z - a x i s . I n p r a c -t i c e , o n l y a few terms c o n t r i b u t e t o t h e e x p a n s i o n s e r i e s f o r l ^ (Equa-t i o n [ 6 . 2 . 1 ] ) , and o f t h e s e , o f t e n o n l y one, o r two a t t h e most, a r e s i g n i f i c a n t ( o t h e r t h a n t h e z e r o o r d e r t e r m ) . -177-FIGURE 6.1: Sche m a t i c e n e r g y - l e v e l diagram f o r a gamma-gamma cascad e from an e x c i t e d n u c l e a r s t a t e o f s p i n I . t o a ground s t a t e o f s p i n I f t h r o u g h an i n t e r m e d i a t e s t a t e o f s p i n I . Y 2 -179-FIGURE 6.2: A n g u l a r c o o r d i n a t e s o f t h e p r o p a g a t i o n d i r e c t i o n s , k-j and 1<2> o f s u c c e s s i v e gamma r a y s from a c a s c a d e , w i t h r e s p e c t t o an a r b i t r a r y frame o f r e f e r e n c e (see E q u a t i o n [ 6 . 2 . 1 ] ) . -181 -6.3 ANISOTROPIC MOTION IN THE ABRAGAM-POUND REGIME In t h e i r d e r i v a t i o n o f t h e p e r t u r b a t i o n f a c t o r f o r a r a p i d l y r e -o r i e n t i n g n u c l e a r s p i n , Abragam and P o u n d ^ ' ^ employed.methods a n a l o g o u s t o t h o s e used t o o b t a i n N u c l e a r M a g n e t i c Resonance (NMR) r e l a x a t i o n t i m e s . When th e a n g u l a r c o r r e l a t i o n e x p e r i m e n t i s c o n d u c t e d i n t h e absence o f any a p p l i e d s t a t i c ( e x t e r n a l ) f i e l d , t h e " e x t r e m e - n a r r o w i n g " approxima-t i o n (WQTQ << 1) w i l l p e r t a i n . In a d d i t i o n , i f TQ << x^ (where x^ i s t h e mean l i f e t i m e o f t h e i n t e r m e d i a t e s t a t e ; t h i s i s on t h e o r d e r o f 10-100 nanoseconds f o r a l l cascades o f p r a c t i c a l importance,), and x Q « p 1/CJQXQ (where i s t h e fundamental q u a d r u p o l e f r e q u e n c y ) , Abragam and Pound showed t h a t t h e p e r t u r b a t i o n f a c t o r becomes a s i m p l e e x p o n e n t i a l , a n d - t h e - a n g u l a r c o r r e l a t i o n f u n c t i o n r e duces t o t h e f o l l o w i n g f o r m , Wtfv l<2> *) = x j j 1 e x p ( - t / T N ) S A k ( l ) A k ( 2 ) e x p ( - X k t ) P k ( c o s n ) . . [ 6 . 3 . 1 ] k The e n c l o s e d a n g l e between 1<-| and "k^ 1 S denoted by Q. I n p a r t i c u l a r , f o r an a x i a l l y s ymmetric q u a d r u p o l e i n t e r a c t i o n , t h e e x p o n e n t i a l f a c t o r i n E q u a t i o n [6.3.1] i s g i v e n by X | ( . f j ( k < k t 1 >WW) 2 - " ^ - ^ ( e W v [ 6 . 3 . 2 ] In E q u a t i o n s [ 6 . 3 . 1 - 2 ] , i t i s assumed t h a t t h e r o t a t i o n a l d i f f u s i o n i s i s o t r o p i c . In t h i s c a s e , i t can be r a t i o n a l i z e d t h a t t h e r o t a t i o n a l c o r r e l a t i o n t i m e r e d u c e s t o t h e f a m i l i a r NMR t i m e c o n s t a n t f i r s t i n t r o -duced i n C h a p t e r I I . A c o m p a r i s o n o f t h e d e r i v a t i o n o f E q u a t i o n [ 6 . 3 . 2 ] w i t h t h e e x p r e s -s i o n s o b t a i n e d f o r t h e NMR r e l a x a t i o n t i m e , T ] , shows t h a t b o t h ( 1 / T ^ and A r a r e s i m p l y p r o p o r t i o n a l t o a s p e c t r a l d e n s i t y a t z e r o f r e q u e n c y -182-o f a c o r r e l a t i o n f u n c t i o n o f a second r a n k s p h e r i c a l harmonic. Thus E q u a t i o n [ 6 . 3 . 2 ] may be m o d i f i e d t o a c c o u n t f o r a n i s o t r o p i c r o t a t i o n a l d i f f u s i o n s i m p l y by r e p l a c i n g by f(D | ( , D ^ e ) where f(D ) ( , UL,B) may be r e a d i l y o b t a i n e d from a m o d i f i e d form o f E q u a t i o n [ 3 . 2 . 5 ] ; f ( D n , D x , e ) = (3cos2e - l ) 2 ( 2 4 D i ) " 1 + 3sin 20cos 2e(5D x + D j " 1 + (3/4)sin 4e(2Dj_ + 4D„ ) _ 1 . [6.3.3] In the p r e s e n t c a s e (no f r e q u e n c y dependence i n t h e p e r t i n e n t s p e c t r a l d e n s i t i e s ) , t h i s can i n f o r m a t i v e l y be w r i t t e n as f ( D t , D, ,6) = (eoj-^i +M^()We + • 9 i D 1 _ z _ q | ) 2 s i n 4 9 - - L 5 D L + D n 2 ( 5 D L + D , , ) ^ + 2D | () [6.3.4] Theta i s t h e p o l a r a n g l e w h i c h r e l a t e s t h e e l e c t r i c f i e l d g r a d i e n t p r i n -c i p a l a x i s t o t h e p r i n c i p a l ( m o l e c u l e - f i x e d ) a x i s system o f t h e r o t a t i o n a l d i f f u s i o n t e n s o r . I t i s assumed t h a t t h e m o l e c u l e i s d y n a m i c a l l y q u i t e w e l l a p p r o x i m a t e d by a symmetric t o p . F i n a l l y , f o r a s p h e r i c a l r o t o r where D() = DL= D, one r e c o v e r s t h e Abragam-Pound r e s u l t where f ( D ) = (6D)~^ = x2• Thus, whenever , D"1 << T ^ , ^ and t h e r e a r e no a p p l i e d s t a t i c f i e l d s , t h e dependence o f t h e a n g u l a r c o r r e l a t i o n on m o l e c u l a r symmetry can be found from E q u a t i o n s [ 6 . 3 . 1 - 4 ] . -183-6.4. ANISOTROPIC MOTION IN THE ADIABATIC LIMIT C o n s i d e r t h e case i n w h i c h t h e two d i f f u s i o n c o n s t a n t s o f a symmetric r o t o r a r e each s m a l l compared t o t h e fundamental f r e q u e n c y , cog, o f t h e q u a d r u p o l e i n t e r a c t i o n For a s i n g l e n u c l e u s f i x e d t o a symmetric r o t o r m o l e c u l e , t h e " a t o m i c frame" can be d e f i n e d as a c o o r d i n a t e system whose z - a x i s i s p a r a l l e l t o t h e p r i n c i p a l a x i s o f t h e e l e c t r i c f i e l d g r a d i e n t t e n s o r and a " l a b frame" can be chosen such t h a t i t s z - a x i s i s p a r a l l e l t o k-,. The r o t a t i o n s between t h e a t o m i c frame and t h e l a b frame a r e c o n v e n i e n t l y d i v i d e d i n t o a r o t a t i o n from t h e a t o m i c frame t o t h e d i f f u s i o n t e n s o r p r i n c i p a l a x i s system f o l l o w e d by a r o t a t i o n from t h e d i f f u s i o n t e n s o r frame t o t h e l a b frame. Now i f t h e d i f f u s i o n t e n s o r frame changes i t s o r i e n t a t i o n s l o w l y compared t o a p e r i o d o f t h e q u a d r u p o l a r H a m i l t o n i a n f o r t h e i n t e r m e d i a t e s t a t e , t h e r e w i l l be no t r a n s i t i o n s i n d u c e d between t h e v a r i o u s s u b s t a t e s * o f t h e i n t e r m e d i a t e s t a t e . In t h i s a d i a b a t i c l i m i t , t h e forms o f E q u a t i o n s [6.2.1] and [6.2.3] a r e p r e s e r v e d , e x c e p t t h a t t h e argument o f t h e second s p h e r i c a l harmonic i s now time-dependent. Thus, t h e o n l y cause o f decay o f t h e p e r t u r b a t i o n f u n c t i o n , G ^1^2(t) when viewed from t h e l a b o r a t o r y * T h i s a d i a b a t i c a p p r o x i m a t i o n y i e l d s r e s u l t s d i f f e r e n t from o t h e r ap-p r o a c h e s . A d i s c u s s i o n o f t h i s problem has been p r e s e n t e d by Lynden-23 B e l l . However, t h e d i f f e r e n c e s f o r PAC s t u d i e s a r e , on t h e most p a r t , beyond e x p e r i m e n t a l v e r i f i c a t i o n and need not c o n c e r n us h e r e . [ 6 . 4 . 1 ] -184-axes i s t h e r a n d o m i z a t i o n o f m o l e c u l a r o r i e n t a t i o n . 24 T a k i n g advantage o f t h e p r o p e r t i e s o f t h e D m a t r i c e s , one i s l e d a f t e r a few s t e p s t o t h e e x p r e s s i o n ( i r r e l e v a n t f a c t o r s a r e o m i t t e d ) < £ g ( k r k 2 , t ) > = S A k ( l ) A k ( 2 ) G k k ( t ) P k ( c o s n ) X ] 4 ^ * ( e ) Y P ( e ) x e x p ( - E k p t ) [ 6 . 4 . 2 ] 2 where E, = k ( k + l ) D x + p ( D H - D J , e d e f i n e s t h e a n g l e between t h e p r i n -"kp ixe E / i I k\2 \ n ' -n N/ c i p a l ax s o f t h e d i f f u s i o n and f i e l d g r a d i e n t t e n s o r s , and G k k ( t ) = «i i e x p [ - i ( E -E , ) t ] . Two f e a t u r e s o f t h i s e x p r e s s i o n may \ n 1 -n N/ n n be noted a t t h i s s t a g e : (1) The a n g l e between t h e d e t e c t o r s appears s e p a r a t e l y as the argument o f a Legendre p o l y n o m i a l , as i s t h e c a s e f o r i s o t r o p i c d i f f u s i o n , and (2) when t h e d i f f u s i o n becomes v e r y s low ( D h , << T ^ 1 ) , t h e r i g h t - h a n d s i d e o f E q u a t i o n [6.4.2] r e d u c e s t o S A k ( l ) A k ( 2 ) G k k ( t ) P k ( c o s n ) [ 6 . 4 . 3 ] w h i c h i s j u s t t h e f a m i l i a r p o l y c r y s t a l l i n e r e s u l t e x p e c t e d i n t h e absence o f m o t i o n . 1 1 A s i g n i f i c a n t f e a t u r e o f E q u a t i o n [ 6 . 4 . 2 ] i s t h a t when the " a n g l e o f a t t a c h m e n t " , t h e t a , i s z e r o d e g r e e s , t h e o b s e r v e d a n g u l a r c o r -r e l a t i o n i s u n a f f e c t e d by r o t a t i o n a l d i f f u s i o n a t any r a t e about t h e 25 * More g e n e r a l l y i t has been r a t i o n a l i z e d t h a t i f t h e p e r t u r b a t i o n i s such t h a t i t does not i n t r o d u c e a p r i v i l e g e d d i r e c t i o n i n space ( l i q u i d and p o l y c r y s t a l l i n e s o u r c e s ) , t h e n t h e p e r t u r b a t i o n t e r m , G k l k 2 ( t ) must be i n v a r i a n t under r o t a t i o n s . The r e s u l t o f such a demand i s £he f a c t t h a t t h e a n g u l a r c o r r e l a t i o n f u n c t i o n w i l l always c o n t a i n a s e p a r a b l e P i,(cosft) f a c t o r . Hence, t h e a n g u l a r d i s t r i b u t i o n i s n o t a l t e r e d , but M M o n l y d i m i n i s h e d and t h e term G . 1 . 2 ( t ) i s r e f e r r e d t o as an a t t e n u a t i o n 1 2 c o e f f i c i e n t . -185-symmetry a x i s . F i n a l l y , f o r t h e s p e r i c a l r o t o r , f o r w h i c h D|( = D^, t h e s p h e r i c a l harmonic a d d i t i o n theorem reduces E q u a t i o n [ 6 . 4 . 2 ] t o t h e p r e v i o u s l y 13 o b t a i n e d r e s u l t , < & ( k r k 2 , t ) > = ^ A k ( l ) A k ( 2 ) G k k ( t ) P k ( c o s n ) e x p ( - k ( k + l ) D t ) . [ 6 . 4 . 4 ] E q u a t i o n [ 6 . 4 . 2 ] t h u s g i v e s t h e p r e d i c t e d P e r t u r b e d A n g u l a r C o r r e l a t i o n f o r m o l e c u l e s o f a symmetric t o p a p p r o x i m a t i o n , and w i l l be v a l i d when-e v e r t h e d i f f u s i o n a l r o t a t i o n a l r e o r i e n t a t i o n o f t h e m o l e c u l e i s s u f f i -c i e n t l y s l o w t h a t D^, D^<< Wq. E x t e n d i n g t h e s e r e s u l t s t o t h e pro b l e m o f i n t e r n a l m o t i o n s i s s t r a i g h t f o r w a r d . When t h e m o t i o n o f t h e m o l e c u l e as a whole i s r a p i d (Abragam-Pound l i m i t ) one may make the f o l l o w i n g i d e n t i f i c a t i o n i n E q u a t i o n [ 6 . 3 . 3 ] o r [ 6 . 3 . 4 ] ; D^-D^E D. where Dj_ now c h a r a c t e r i z e s t h e o v e r a l l i s o t r o p i c m o b i l i t y (Dj_ ->• D) and D-n^ c h a r a c t e r i z e s t h e s i n g l e degree o f i n t e r n a l m o b i l i t y . When t h e m o t i o n o f t h e m o l e c u l e as a whole i s s l o w and t h e i n t e r n a l m o b i l i t y i s r a p i d compared t o the o v e r a l l m o t i o n , but s l o w compared t o t h e q u a d r u p o l e f r e q u e n c y (D^, D 1- n t < < "Q)» t h e n t h e same i n t e r p r e t a t i o n can be a p p l i e d t o E q u a t i o n [ 6 . 3 . 6 ] . The i n t e r m e d i a t e c a s e , f a s t i n t e r n a l m o b i l i t y and slow o v e r a l l m o t i o n poses a number o f problems and has not been s a t i s f a c t o r i l y s o l v e d . -186-6.5. DISCUSSION I t i s i n s t r u c t i v e t o i l l u s t r a t e t h e c a l c u l a t e d b e h a v i o r o f t h e a n g u l a r c o r r e l a t i o n i n terms o f m e a s u r e a b l e q u a n t i t i e s . One c o n v e n i e n t q u a n t i t y i s t h e t i m e - i n t e g r a t e d ( o r i n t e g r a l ) a n i s o t r o p y , <&> = (W(^,~)/W(^/2,-)) -1 [ 6 . 5 . 1 ] where th e t i m e - i n t e g r a t e d a n g u l a r c o r r e l a t i o n f u n c t i o n i s d e f i n e d as ' S^n»") = T N ] f°%& ( kr k2> t ) e x p ( - t / x N ) . [ 6 . 5 . 2 ] Tjyi i s t h e mean l i f e t i m e o f t h e i n t e r m e d i a t e s t a t e ( x-, , ^ = x^£n(2)). n i s t h e e n c l o s e d a n g l e d e f i n e d by k-j and k^. F i g u r e s 6.3-5 show th e dependence o f <^ l> on t h e a n g l e o f a t t a c h -ment o f t h e l a b e l and e i t h e r m o l e c u l a r symmetry o r r a t e o f i n t e r n a l r o t a t i o n . A l t h o u g h s e v e r a l i s o t o p e s such as ^ C o , ^ 2 Z n , ^ 9 m S n , ^ 3 3 B a , 1 8 1 T a , 1 9 9 m H g , o r 2 0 7 m P b * c o u l d p o s s i b l y be used i n PAC s t u d i e s , most * The p r i m a r y l i m i t a t i o n s on s u i t a b l e n u c l e i i s t h a t I >_ 1 and t h a t t h e p a r e n t n u c l e u s has a h a l f - l i f e o f hours o r days. F u r t h e r m o r e , the i n t e r m e d i a t e s t a t e l i f e t i m e must be l o n g enough so t h a t t h e a t t e n u a t i o n w i l l be m a n i f e s t , y e t s h o r t enough so t h a t t h e a n i s o t r o p y i s not com-p l e t e l y a t t e n u a t e d . Of c o u r s e t h e magnitude o f t h e i n t e g r a l a t t e n u a t i o n , Gj <| <( 0 0) depends upon both t h e s t r e n g t h o f t h e i n t e r a c t i o n and d u r a t i o n o f i n t e r a c t i o n . No e x t r a n u c l e a r f i e l d s a r e s t r o n g enough t o m e a s u r a b l y p e r t u r b an a n g u l a r c o r r e l a t i o n i n w h i c h t h e n u c l e a r l i f e t i m e i s l e s s than a few t e n s o f p i c o s e c o n d s . In a d d i t i o n , t h e f a c t t h a t a n g u l a r c o r r e l a t i o n s can be measured o n l y i f one e s t a b l i s h e s a g e n e r i c r e l a t i o n -s h i p between t h e two r a d i a t i o n s s e t s a p r a c t i c a l upper l i m i t on t h e i n t e r m e d i a t e s t a t e l i f e t i m e . F o r l i f e t i m e s >_ 10 s e c o n d s , a c c i d e n t a l c o i n c i d e n c e s o u t w e i g h t r u e c o i n c i d e n c e s and t h e PAC t e c h n i q u e l o s e s i t s -187-l a b e l i n g e x p e r i m e n t s have employed t h e 247-kev s t a t e o f Cd. The decay scheme f o r t h i s n u c l e i i s shown i n F i g u r e 6.6. The v a r i o u s p l o t s ( F i g u r e s 6.3-5) have been g e n e r a t e d from t h e parameters a s s o c i a t e d w i t h VE3) Y 2 ( E 2 ) 111m t h e 11/2 >>>>>>>>> 5/2 >>>>>>>> 1/2 decay o f Cd. Fo r c o n v e n i e n c e , t h e s e v a l u e s a r e r e p r o d u c e d h e r e : A Q ( 1 ) = A Q ( 2 ) = 1, A 2 ( l ) = -0.535, A 2 ( 2 ) = -0.334, A 4 ( l ) = 0.617, and A 4 ( 2 ) = 0.007. A l l o t h e r c a s c a d e c o e f f i c i e n t s a r e i d e n t i c a l l y equal t o z e r o . I n g e n e r a l , due t o symmetry c o n s i d e r a t i o n s , i t can be shown t h a t f o r d i r e c t i o n a l y-Y PAC e x p e r i m e n t s , o n l y even terms c o n t r i b u t e t o t h e summation i n E q u a t i o n [ 6 . 2 . 1 ] . F u r t h e r -more, s i n c e A 2 ( 1 ) A 2 ( 2 ) >> A 4 ( 1 ) A 4 ( 2 ) , f o r a l l p r a c t i c a l p u r p o s e s , o n l y one n o n t r i v i a l t erm c o n t r i b u t e s t o t h i s summation. F i n a l l y , t h e f a c t o r FOOTNOTE CONTINUED u s e f u l n e s s . T h e r e f o r e , f o r o u r p u r p o s e s , we a r e r e s t r i c t e d t o n u c l e i -9 -6 where 10 sec <_ T ^ <_ 10 sec w h i c h u s u a l l y i m p l i e s Ml o r E2 decay o f t h e i n t e r m e d i a t e s t a t e . ( I t i s i n t e r e s t i n g t o note t h a t t h i s i s t h e t y p i c a l range f o r u s a b l e Mossbauer i s o t o p e s . ) In F i g u r e 6.7, t h e i n t e -g r a l a t t e n u a t i o n f a c t o r (which a p p r o x i m a t e l y , equals. . $3 2 / ( 3 A 2 ( l ) A 2 ( 2 ) ) ) i s p l o t t e d v e r s u s an i s o t r o p i c c o r r e l a t i o n t i m e f o r v a r i o u s v a l u e s o f T ^ . F o r T 2 < CJQ^ , t h e Abragam-Pound r e s u l t i s a p p l i e d ; f o r x 2 > co^ , t h e a d i a b a t i c l i m i t i s a p p l i e d . F o r i n t e r m e d i a t e t i m e s , t h e c u r v e i s i n t e r p o l a t e d ( t h i s i s j u s t i f i e d as i t has been shown t h a t t h e c u r v e behaves q u i t e s m o o t h l y as one passes from one extreme t o t h e o t h e r : see r e f e r e n c e 1 4 ) . Note t h a t a c o n s t a n t v a l u e f o r Wq i s assumed. T h i s p l o t h e l p s t o i n d i c a t e why, i n o r d e r t o o b t a i n c h e m i c a l l y i n t e r e s t i n g i n f o r -m a t i o n , i t i s d e s i r a b l e ( n e c e s s a r y ) f o r 10 nanosec < x^ < 100 nanosec. I f t h e n u c l e a r l i f e t i m e f a l l s o u t s i d e o f t h i s r a n g e , i t i s d i f f i c u l t t o o b t a i n m o t i o n a l i n f o r m a t i o n . I t i s i n t e r e s t i n g t o note t h a t i n some ways, t h i s i s q u i t e a n a l o g o u s t o the s i t u a t i o n when one employs T-j o r T 9 r a t i o s f o r m o t i o n a l i n f o r m a t i o n (see s e c t i o n 3.3). -188-G k k ( t ) f o r a p o l y c r y s t a l l i n e sample (see E q u a t i o n [ 6 . 4 . 3 ] ) i s g i v e n by I I k \ 2 n' -n N J e x P ( - i ( E n - E n , ) t ) = (7 + 13coswpt + 10cos2ojpt + 5cos3u)gt)/35 [ 6 . 5 . 3 ] 2 where cog=3e qQ/20fi. A t y p i c a l ( y e t a r b i t r a r y ) v a l u e f o r t h e q u a d r u p o l e 2 9 c o u p l i n g c o n s t a n t i s c h o s e n , |e qQ/fi]2TT = 10 Hz. F o r a l l p r a c t i c a l p u r p o s e s , t h e o r d i n a t e a x i s c o u l d a l s o be i n t e r -p r e t e d i n terms o f t h e i n t e g r a l a t t e n u a t i o n c o e f f i c i e n t , G ^ 0 0 ) , where G k k ( » ) = x j J 1 G k k ( t ) e x p ( - t / T N ) d t and G k k ^ = G k k ^ t ^ 4 7 T YP ( e ) Y P ( e ) e x p ( - E k p t ) . [ 6 . 5 . 4 ] T h i s r e s u l t s from t h e f a c t t h a t <iH> as d e f i n e d i n E q u a t i o n [6.4.1] can be w r i t t e n as = [1 + A 2(1)A 2(2)G 2 2(»)][1/(1 - \ A 2 ( 1 ) A 2 ( 2 ) G 2 2 ( ~ ) ] - 1. [ 6 . 5 . 5 ] S i n c e A 2 ( 1 )A 2(2)G 2 2(°°) < 1, a s t a n d a r d e x p a n s i o n o f t h e f r a c t i o n a l t erm y i e l d s <9> = ! A 2(1)A 2(2)G 2 2(») , [6.5.6] and hence, <!H> <= G 2 2(°°). F i g u r e 6.3 shows t h e e f f e c t o f a t t a c h i n g a l a b e l a t v a r i o u s a n g l e s t o a p r o l a t e symmetric t o p m o l e c u l e . The shape o f t h e m o l e c u l e has been f i x e d by s e t t i n g t h e r a t i o , D„/Dj_ = 8, w h i l e t h e " s i z e " o f t h e m o l e c u l e i s v a r i e d c o n t i n u o u s l y by v a r y i n g D x. I t i s e v i d e n t t h a t t h e dependence o f <iU> on t h e a n g l e o f a t t a c h m e n t o f t h e l a b e l t o t h i s q u i t e asymmetric N.n \ -189-m o l e c u l e i s weak, but m e a s u r a b l e . I t s h o u l d a l s o be noted t h a t i n t h e f a s t m o t i o n l i m i t , r a p i d m o t i o n makes <^ i> l a r g e r because t h e r e l a x -a t i o n c o n s t a n t i s i n v e r s e l y p r o p o r t i o n a l t o t h e d i f f u s i o n c o n s t a n t ; w h i l e i n t h e a d i a b a t i c l i m i t , r a p i d m o t i o n makes <!H> s m a l l e r because t h e r e l e v a n t r e l a x a t i o n c o n s t a n t s a r e p r o p o r t i o n a l t o t h e d i f f u s i o n c o e f f i c i e n t s . The a n i s o t r o p y f o r 6 = 0° i n t h i s f i g u r e i s t h e same as would be o b s e r v e d f o r a s p h e r i c a l m o l e c u l e w i t h D(1 = Dj_. F i g u r e s 6.4 and 6.5 show t h e e f f e c t o f i n t e r n a l r o t a t i o n on > f o r a l a b e l bound t o a s p h e r i c a l m o l e c u l e i n such a way t h a t an i n t e r n a l r o t a t i o n about j u s t one bond can o c c u r . The f i g u r e s show how < !H > v a r i e s a c c o r d i n g t o t h e s i z e o f t h e m o l e c u l e , f o r a number o f f i x e d c h o i c e s f o r i n t e r n a l r o t a t i o n r a t e and a n g l e o f a t t a c h m e n t ; t h e a n g l e o f a t t a c h m e n t i n t h i s c a s e i s t h e a n g l e between t h e p r i n c i p a l a x i s o f t h e e l e c t r i c f i e l d g r a d i e n t t e n s o r and t h e i n t e r n a l r o t a t i o n a x i s . S e v e r a l f e a t u r e s o f t h e s e p l o t s d e s e r v e comment. F i r s t , from F i g u r e 6.4, i t appears t h a t even v e r y f a s t i n t e r n a l r o t a t i o n w i l l a f f e c t t h e a t t e n u a t i o n f a c t o r o n l y i f t h e a n g l e o f a t t a c h m e n t i s s i g n i f i c a n t l y d i f f e r e n t from z e r o degrees - t h e most s u b s t a n t i a l e f f e c t i s f o r t h e NMR magic a n g l e , e - c o s ~ ^ ( l / / 3 ) . P h y s i c a l l y , we have noted t h a t v e r y r a p i d i n t e r n a l r o t a t i o n " e f f e c t i v e l y " reduces t h e magnitude o f t h e quad-r u p o l a r i n t e r a c t i o n and t h u s i n c r e a s e s t h e a t t e n u a t i o n f a c t o r ( r e d u c e s t h e p e r t u r b a t i o n ) . In c o n t r a s t , i n t e r n a l r o t a t i o n a c t s t o d e c r e a s e t h e o b s e r v e d a n i s o t r o p y when both t h e i n t e r n a l r o t a t i o n and r o t a t i o n o f t h e m o l e c u l e as a whole a r e s l o w , as seen i n F i g u r e 6.5. P r a c t i c a l l y s p e a k i n g , F i g u r e 6.5 shows t h a t even when r e o r i e n t a t i o n o f t h e m o l e c u l e as a whole i s s l o w , i n t e r n a l r o t a t i o n r e s u l t s i n l i t t l e change i n t h e -190-o b s e r v e d a n i s o t r o p y u n l e s s t h e r a t e o f t h e i n t e r n a l r o t a t i o n i s a p p r e c i -a b l y f a s t e r than t h e r a t e o f r e o r i e n t a t i o n o f t h e m o l e c u l e as a whole. For t h e most p a r t , t h e b e h a v i o r i s r e a d i l y a p p a r e n t from t h e a s y m p t o t i c v a l u e s o f t h e t i m e - i n t e g r a l a t t e n u a t i o n f a c t o r . Making t h e a s s u m p t i o n s , cog » T^,1, D. n t; D -> 0, and e f 0°, [ 6 . 3 . 4 ] , [ 6 . 4 . 2 ] , and [6.5. 1 ] t h a t JQ > : > T~\\ ' ^ i n t ' ^ * ®' a n c ' 9 ^ ^ f o l l o w s d i r e c t l y from E q u a t i o n s <&> * "3 j ( 3 c o s 2 e - T ) 2 + 12cos 2 6 s i n 2 9 + 3 s l n % ) * 1 + T N D i n t 1 + V i n t ' x A 2 ( 1 ) A 2 ( 2 ) . [6.5 . 7 ] F u r t h e r m o r e , i f T^D^^. << 1, t h i s r e duces t o t h e r e s u l t o b t a i n e d i n th e absence o f i n t e r n a l m o t i o n s . I f T ^ D . ^ » 1, < H > = |Q A 2 ( l ) A 2 ( 2 ) ( 3 c o s 2 6 - l ) 2 . [ 6 . 5 . 8 ] I t i s i n t e r e s t i n g t o note t h a t i n e i t h e r l i m i t , t h e a n i s o t r o p y i s i n d e -pendent o f t h e r a t e o f i n t e r n a l m o t i o n and depends s o l e l y on s t a t i c geo-m e t r i c a l f a c t o r s . F i n a l l y , i n t e r n a l r o t a t i o n , whether f a s t o r s l o w , has no e f f e c t on t h e a n g u l a r c o r r e l a t i o n when t h e a n g l e o f att a c h m e n t i s z e r o d e g r e e s . In c o n c l u s i o n , t h e c a l c u l a t i o n s and p l o t t e d r e s u l t s i n t h i s s h o r t e x p o s i t i o n have shown t h a t gamma-gamma a n g u l a r c o r r e l a t i o n s a f f o r d a p a r t i c u l a r l y a t t r a c t i v e means f o r s t u d y o f s p e c i f i c s i t e s on l a r g e m o l e c u l e s i n d i l u t e s o l u t i o n , s i n c e t h e o b s e r v e d parameter ( t i m e - i n t e -g r a t e d a n i s o t r o p y ) i s r e l a t i v e l y i n s e n s i t i v e t o t h e shape o f t h e l a r g e m o l e c u l e (as shown i n F i g u r e 6.3) but can be changed q u i t e m a r k e d l y i n the p r e s e n c e o f l o c a l f l e x i b i l i t y ( i n t e r n a l r o t a t i o n ) a t t h e s i t e o f -191-a t t a c h m e n t o f t h e r a d i o a c t i v e t r a c e r t o t h e l a r g e m o l e c u l e ; t h e s e changes c o n t a i n i n f o r m a t i o n on b o t h t h e r a t e o f i n t e r n a l m o t i o n as w e l l as on t h e geometry ( a n g l e o f a t t achment o f t h e t r a c e r ) f o r t h e complex. In c o m b i n a t i o n w i t h t h e p r e v i o u s l y e s t a b l i s h e d advantages t h a t t h e concen--12 t r a t i o n s e n s i t i v i t y approaches 10 M, t h e e x p e r i m e n t a l measurement a r e s i m p l e , h i g h l y p e n e t r a t i n g gamma-rays a r e i d e a l l y s u i t e d f o r i n v i v o s t u d i e s , and t h e a p p a r a t u s i s a l l c o m m e r c i a l l y a v a i l a b l e , t h e c a l c u l a t i o n s i n t h i s c h a p t e r c o n s i d e r a b l y enhance t h e a p p e a l o f s u c h s t u d i e s . F i n a l l y , t h e a t t e n t i v e eyes may note t h e extreme s i m i l a r i t y o f t h i s d i s c u s s i o n and t h a t o f C h a p t e r I I I . The o b v i o u s r e l a t i o n s h i p between PAC and NMR i s not by c o i n c i d e n c e , and some t h o u g h t p r o v o k i n g comments con-c e r n i n g t h i s r a p p o r t a r e now d i s c u s s e d . -192-FIGURE 6.3: P l o t o f t h e t i m e - i n t e g r a t e d a n i s o t r o p y v e r s u s l o g ( 6 D x ) f o r a p r o l a t e symmetric t o p h a v i n g t h e r a t i o D^/D^ = 8. In t h e f a s t m o t i o n l i m i t ( l e f t - h a n d s e t o f c u r v e s ) , t h e i n t e g r a l a n i s o t r o p y v a r i e s w i t h a n g l e o f a t t a c h m e n t , 6, i n t h e o r d e r , 60°:> 90 o.>30°,>0°. In t h e a d i a b a t i c l i m i t ( r i g h t - h a n d s e t o f c u r v e s ) , t h e i n t e g r a l a n i s o t r o p y v a r i e s w i t h 9 i n t h e o r d e r , 0°> 30°> 60°> 90°. In both l i m i t s , t h e v a r i o u s p arameters c h a r a c t e r i s t i c o f t h e decay a r e g i v e n i n s e c t i o n 6.5. -194-FIGURE 6.4: P l o t o f t h e i n t e g r a l a n i s o t r o p y i n t h e f a s t m o t i o n l i m i t v e r s u s l o g ( 6 D ) , where D i s t h e r o t a t i o n a l d i f f u s i o n c o n s t a n t f o r a s p h e r i c a l m o l e c u l e . Each f a m i l y o f c u r v e s c o r r e s p o n d s t o a f i x e d c h o i c e o f "attachment a n g l e " as l i s t e d i n t h e f i g u r e . For i n d i v i d u a l c u r v e s (A) t h r o u g h ( D ) , t h e i n t e r n a l 12 11 r o t a t i o n a l d i f f u s i o n c o n s t a n t , D- ^, i s equal t o 10 , 1 0 , 1 0 ^ , and 1 0 9 sec"^ r e s p e c t i v e l y . The l o w e s t c u r v e i n each s e t (dashed l i n e ) i s t h e i n t e g r a l a n i s o t r o p y i n t h e absence o f i n t e r n a l r o t a t i o n , o r e q u i v a l e n t l y (see t e x t ) , f o r z e r o " attachment a n g l e " . For a l l c u r v e s , t h e v a r i o u s p a r a m e t e r s needed t o g e n e r a t e t h e s e p l o t s a r e g i v e n i n s e c t i o n 6.5. A N I S O T R O P Y A N I S O T R O P Y ANISOTROPY -196-FIGURE 6.5: P l o t s o f t h e i n t e g r a l a n i s o t r o p y v e r s u s l o g ( D), where D i s t h e r o t a t i o n a l d i f f u s i o n c o n s t a n t f o r a s p e r i c a l m o l e c u l e ( i n t h e a d i a b a t i c l i m i t ) . The two s e t s o f c u r v e s c o r r e s p o n d t o t h e two c h o i c e s f o r t h e i n t e r n a l r o t a t i o n a l d i f f u s i o n c o n s t a n t , D ^ n t 5 shown on t h e f i g u r e . The i n t e g r a l a n i s o t r o p y v a r i e s w i t h d i f f e r e n t a n g l e s o f a t t a c h m e n t ; t h e v a l u e s 0°, 30°, 60°, and 90° c o r r e s p o n d t o c u r v e s a-d r e s p e c t i v e l y . The r e s u l t f o r e= 0° i s t h e same as f o r no i n t e r n a l r o t a t i o n a t a l l and p r o v i d e s a c o m p a r i s o n . The v a r i o u s parameters needed t o g e n e r a t e t h e s e p l o t s a r e g i v e n i n s e c t i o n 6.5. - 1 9 7-A A A A A i I D ^ CO CM i - Q OOLx]AdOH10SINV -198-FIGURE 6.6: Decay scheme o f ^ I n and 1 1 1 m C d showing t h e photon c a s c a d e s r e l e v a n t f o r a n g u l a r c o r r e l a t i o n s t u d i e s . - 1 9 9 -111 In t,2=2.8 days ti,2 =0.12 ns ti/, = 49min t,/2=85 ns ENERGY (KeV) 420 397 SPIN 7/2 11/2 247 5/2 stable 4 o 1/2 -200-FIGURE 6.7: P l o t o f t h e i n t e g r a l a t t e n u a t i o n c o e f f i c i e n t v e r s u s an i s o t r o p i c c o r r e l a t i o n t i m e f o r v a r i o u s i n t e r m e d i a t e s t a t e 8 -1 l i f e t i m e s (assuming I = 5/2, u>g = 1.5 x 10 sec ). The l e f t and r i g h t - h a n d p o r t i o n s o f each c u r v e a re g i v e n by t h e Abragam-Pound and a d i a b a t i c a p p r o x i m a t i o n s r e s p e c t i v e l y . The i n t e r m e d i a t e r e g i o n i s t h e i n t e r p o l a t e d b e h a v i o r . - 2 0 1 --202-6.6. CONNECTION BETWEEN NMR AND PAC B e f o r e c o n c l u d i n g t h i s c h a p t e r , we w i s h t o emphasize t h e c h a p t e r ' s r e l a t i o n s h i p w i t h t h e p r e v i o u s m a t e r i a l p r e s e n t e d i n t h i s t h e s i s w h i c h has e x c l u s i v e l y d e a l t w i t h NMR r e l a x a t i o n s t u d i e s . N u c l e a r M a g n e t i c Resonance i s o n l y one i s o l a t e d usage o f c h e m i c a l * i n f o r m a t i o n g a t h e r e d from t r a n s i e n t n u c l e a r p o l a r i z a t i o n e x p e r i m e n t s . In NMR, a b r u t e f o r c e method i s adopted t o a c h i e v e t h i s g o a l . A s t r o n g m a g n e tic f i e l d l i f t s t h e degeneracy o f t h e s p i n s t a t e s and v a r i o u s ex-p e r i m e n t s a r e p e r f o r m e d i n w h i c h t h e m a g n e t i c d i p o l a r p o l a r i z a t i o n i s m o n i t o r e d . A n o t h e r a l t e r n a t i v e t o t h i s a p p r o a c h , w h i c h does not r e l y d i r e c t l y on t h e p r e s e n c e o f t h e Zeeman f i e l d , i s t o s i m p l y l e t " n a t u r e " p o l a r i z e t h e s p i n ensemble. Two e x p e r i m e n t a l t e c h n i q u e s a r e now b e i n g performed by c h e m i s t s w h i c h do j u s t t h i s . One i s t h e PAC e x p e r i m e n t , 26 27 the o t h e r i s t h e muon d e p o l a r i z a t i o n e x p e r i m e n t . ' A l t h o u g h t h e muon probe has g r e a t p o t e n t i a l , a t p r e s e n t t h e a p p l i c a t i o n o f t h i s t e c h n i q u e i s s e v e r e l y hampered by many c o n s i d e r a t i o n s , t h e f o r e m o s t b e i n g a g e n e r a l l a c k o f s e l e c t i v i t y and s p e c i f i c i t y . The PAC e x p e r i m e n t i s one s t e p b e t t e r t h a n t h e muon e x p e r i m e n t s i n c e i t i s much e a s i e r t o c h e m i c a l l y m a n i p u l a t e r a d i o a c t i v e n u c l i d e s t h a n p o s i t i v e muons. In any c a s e , as f a r as t h e n o n e x p e r i m e n t a l i s t g o e s , the t h r e e t e c h n i q u e s s h a r e much i n common, t h e main d i f f e r e n c e b e i n g t h e l a c k o f s e n s i t i v i t y o f NMR due t o t h e f a c t t h a t t h e e n e r g i e s i n v o l v e d a r e p r e c i s e l y t h o s e o f the t r a n s i -t i o n s i n d u c e d . In t h e s e a l t e r n a t i v e t e c h n i q u e s , e n e r g e t i c photons o r * Here we use t h e g e n e r a l term p o l a r i z a t i o n t o s i g n i f y any s p i n system t h a t cannot be c h a r a c t e r i z e d by a d e n s i t y m a t r i x w h i c h i s a m u l t i p l e o f t h e u n i t m a t r i x . -203-6 10 e l e c t r o n s a r e t h e e x p e r i m e n t a l o b s e r v a b l e w i t h a r e s u l t a n t 10 - 10 i n c r e a s e i n s e n s i t i v i t y . L e t us now c o n s i d e r t h e s i m i l a r i t i e s between NMR and PAC i n g r e a t e r d e t a i l . A t f i r s t t h o u g h t , i t m i g h t seem t h a t t h e PAC t e c h n i q u e i s r e a l l y n o t h i n g more than a NMR e x p e r i m e n t where t h e z e r o f i e l d a p p r o x i m a t i o n i s always v a l i d . The c o n n e c t i o n between NMR and PAC t e c h n i q u e s i s some-what more s u b t l e and can p r o b a b l y be seen most r e a d i l y i f a d e n s i t y 17 28-o p e r a t o r f o r m a l i s m i s used t o e x p r e s s t h e a n g u l a r c o r r e l a t i o n f u n c t i o n , ' ( k r k 2 , t ) = T r [ a ( k 2 ) a ( k r t ) ] . [ 6 . 6 . 1 ] ->-The d e n s i t y m a t r i x a(k-|,0) d e s c r i b e s t h e n u c l e a r system i m m e d i a t e l y me a f t e r t h e e m i s s i o n o f the f i r s t r a d i a t i o n i n t h e d i r e c t i o n k^ a t t i t = 0. The d e n s i t y m a t r i x a ( k 2 ) c o r r e s p o n d s t o t h e second t r a n s i t i o n a t a l a t e r t i m e t . Due t o t h e i n t e r a c t i o n w i t h e x t r a n u c l e a r p e r t u r b a t i o n s , a(k-|,t) i s d i f f e r e n t from i t s v a l u e a t t = 0 and can be d e t e r m i n e d from t h e von Neumann E q u a t i o n ( E q u a t i o n [ 2 . 2 . 2 ] ) . The s t a n d a r d approach t h e n r e s o l v e s t h e d e n s i t y o p e r a t o r i n t o a m u l t i p o l e e x p a n s i o n (Fano's s t a t i s t i c a l t e n s o r s o r s t a t e m u l t i p o l e s ) . I f t h e m a c r o s c o p i c e n v i r o n -ment o f t h e s p i n s i s i s o t r o p i c , t h e n each term i n t h i s m u l t i p o l e expan-s i o n e v o l v e s i n d e p e n d e n t l y i n t i m e , ° k ( t ) = G k k ( t ) a k ( 0 > - C6.6.2] The i n d e x k i s t h e p o l e o r d e r (a 2 - p o l e ) and N l a b e l s each o f t h e 2k+l components o f ^ ( t ) . These G's may be c a l l e d p e r t u r b a t i o n , a t t e n u a t i o n , o r r e l a x a t i o n c o e f f i c i e n t s , and as t h e n o t a t i o n r e a d i l y i m p l i e s , a r e i d e n t i c a l l y e q u a l t o t h o s e a t t e n u a t i o n f a c t o r s i n t r o d u c e d e a r l i e r i n t h e -204-c h a p t e r . In p a r t i c u l a r , t h e PAC e x p e r i m e n t probes o n l y t h o s e p o l e s w i t h even k. In NMR, t h i s same f o r m a l i s m can be a p p l i e d . However, o n l y t h e e v o l u t i o n o f t h e k = 1 component i s o b s e r v e d i n NMR r e l a x a t i o n e x p e r i m e n t s . Thus, i n t h i s a p p r o a c h , we see t h a t NMR and PAC a r e o n l y two i s o l a t e d c a s e s o f a h i e r a r c h i c a l s t r u c t u r e o f n u c l e a r r e l a x a t i o n . 1 ^ R e l a x a t i o n t i m e s measured by c o n v e n t i o n a l NMR t e c h n i q u e s c h a r a c t e r i z e t h e i r r e v e r s i b l e b e h a v i o r o f o n l y t h e ( m a g n e t i c ) d i p o l e p o l a r i z a t i o n . In a c o n v e n t i o n a l PAC e x p e r i m e n t , one o b s e r v e s under s u i t a b l e c o n d i t i o n s , damping c o n s t a n t s b e l o n g i n g t o h i g h e r rank m u l t i p o l a r i z a t i o n s ( e . g . ( e l e t r i c ) q u a d r u p o l e and h e x a d e c a p o l e p o l a r i z a t i o n s ) . I t i s o b v i o u s t h a t e x c i t i n g g e n e r a l -i z a t i o n s and u n i f y i n g t h o u g h t s m i g h t be d e r i v e d from a more t h o r o u g h approach t o n u c l e a r s p i n r e l a x a t i o n a l o n g t h e s e l i n e s . I t i s v e r y i n t e r e s t i n g t o note t h e a p p a r e n t s i m i l a r i t y o f t h i s approach w i t h a n o t h e r broad i n t e r p r e t a t i o n o f t h e NMR r e l a x a t i o n e x p e r -iment. Most common forms o f s p e c t r o s c o p y w h i c h probe t h e r e o r i e n t a t i o n a l dynamics o f f l u i d s o l u t i o n s can q u i t e g e n e r a l l y be c a t e g o r i z e d i n terms o f w h i c h rank s p h e r i c a l harmonic t h e l i n e s h a p e r e f l e c t s ( t h i s may be th o u g h t t o a r i s e from assuming a m u l t i p o l e e x p a n s i o n o f the s o l u t i o n t o th e r o t a t i o n a l d i f f u s i o n e q u a t i o n ) . F o r example, d i e l e c t r i c r e l a x a t i o n and IR a b s o p t i o n probe Y ^ ( f i ) , NMR, f l u o r e s c e n c e d e p o l a r i z a t i o n , and Raman and R a y l e i g h s c a t t e r i n g probe Y,,(ft), second harmonic l i g h t s c a t t e r i n g probes Y - ^ f t ) , and so on. The u s e f u l n e s s o f v i e w i n g NMR as a s p e c i a l case o f more g e n e r a l p r i n c i p l e s has been e x p l o i t e d r e c e n t l y and has g r e a t l y a i d e d o u r u n d e r s t a n d i n g o f s e e m i n g l y u n r e l a t e d s p e c t r o s c o p i c + u • 32,33 t e c h n i q u e s . ' In a comment, we w i l l b r i e f l y view a somewhat more p r a c t i c a l compar--205-i s o n o f t h e two t e c h n i q u e s . I t i s p o s s i b l e t o r e w r i t e t h e Abragam-Pound r e s u l t ( E q u a t i o n [6.3.2]) a s , i - k ( k + 1) j , k ( k + 1) -2 I r f i , o - i A k 2Tp- j 1 ~ 41(1 + 1) - 3 " ( * [ 6 - 6 - 3 ] T-| i s n o t h i n g more t h a n t h e f a m i l i a r s p i n - l a t t i c e r e l a x a t i o n time so common t o NMR d i s c u s s i o n s . Hence, f o r t h e s p i n 5/2 Cd, x 2 = 2^/ 8 T-]» a n d i n t h i s i n s t a n c e , t h e q u a n t i t a t i v e c o n n e c t i o n i s c o m p l e t e . However, as the r e l a x a t i o n c o n s t a n t s measure d i f f e r e n t p h y s i c a l p a r a m e t e r s , one must not i n g e n e r a l e x p e c t t o be a b l e t o e x p r e s s t h e damping c o n s t a n t s appear-i n g i n t h e PAC p e r t u r b a t i o n f a c t o r s i n terms o f c o n v e n t i o n a l NMR r e l a x a -t i o n v a r i a b l e s . F u r t h e r d i s c u s s i o n s p e r t a i n i n g t o n u m e r i c a l r e l a t i o n -17 OR s h i p s can be f o u n d i n G a b r i e l ' s a r t i c l e s . ' T h i s b r i e f i n t r o d u c t i o n t o t h e PAC e x p e r i m e n t i s o f c o u r s e l a c k i n g i n d e t a i l and t h e n e c e s s a r y t h e o r e t i c a l background. However, f u r t h e r i n f o r m a t i o n on t h i s n ovel t e c h n i q u e can be r e a d i l y o b t a i n e d from t h e numerous r e f e r e n c e s c i t e d i n t h i s c h a p t e r . -206-REFERENCES: CHAPTER VI 1. A. G. M a r s h a l l , L. G. Werbelow, and C. F. Meares, J . Chem. Phys. 57, 364 (197 2 ) . 2. T. K. L e i p e r t , J . D. B a l d e s c h w i e l e r , and D. A. S h i r l e y , N a t u r e 220, 907 (196 8 ) . 3. C. F. Meares, R. G. B r y a n t , J . D. B a l d e s w i e l e r , and D. A. S h i r l e y , P r o c . Nat. Acad. S c i . 64, 1155 (1969). 4. D. A. S h i r l e y , J . Chem. Phys. 53, 465 (1 9 7 0 ) . 5. C. F. Meares and D. G. Westmoreland, P r o c . C o l d S p r i n g Harbor Symp. Quant. B i o l . 36, 511 (1 9 7 1 ) . 6. C. F. Meares. M. W. Sunberg, and J . D. B a l d e s c h w i e l e r , P r o c . Nat. Acad. S c i . 69, 3718 (1972). 7. D. A. Goodwin, C. F. Meares, and C. H. Song, R a d i o l o g y 105, 699 (1972). 8. R. Bauer, P. L i m k i l d e , and 0. G l o m s e t , Phys. Rev. L e t t . ' 3 2 , 340 (1 9 7 4 ) . 9. J . C. G l a s s and G. G r a f , N a t u r e 226, 635 (1970). 10. A. Abragam and R. V. Pound, Phys. Rev. 92_, 943 (1 9 5 3 ) . 11. H. F r a u e n f e l d e r and R. M. S t e f f e n , i n A l p h a - , B e t a - , and Gamma-Ray  S p e c t r o s c o p y , e d i t e d by K. S i e g b a h n , N o r t h - H o l l a n d , Amsterdam, 1965; pages 9 9 7 f f . 12. A. J . F e r g u s o n , A n g u l a r C o r r e l a t i o n Methods i n Gamma-Ray S p e c t r o s c o p y , N o r t h - H o l l a n d , Amsterdam, 1965. 13. A. G. M a r s h a l l and C. F. Meares, J . Chem. Phys. 56, 1226 (1 9 7 2 ) . 14. R. M. L y n d e n - B e l l , M o l . Phys. 21_, 891 (1971 ). 15. M. Blume, N u c l . Phys. A!67, 81 (1971). 16. D. S p a n j a a r d and F. Hartmann-Boutron, J . P h y s i q u e 30, 975 (1969). 17. H. G a b r i e l , Phys. Rev. 181_, 506 (1969). 18. D. A. S h i r l e y , J . Chem. Phys. 53, 465 (1 9 7 1 ) . 19. J . R. Lombardi and G. A. D a f f o r n , J . Chem. Phys. 44, 3882 (1966). 20. T. Tao, i n M o l e c u l a r L uminescence, e d i t e d by E. C. L i m , B e n j a m i n , New Y o r k , 1969; pages 8 6 1 f f . -207-21. G. Weber, J . Chem. Phys. 55, 2399 (1971). 22. D. W a l l a c h , J . Chem. Phys. 47, 5258 (1967). 23. R. M. L y n d e n - B e l l , M o l . Phys. 26, 979 (1973). 24. M. E. Rose, E l e m e n t a r y Theory o f A n g u l a r Momentum, W i l e y , New Y o r k , 1957. 25. R. M. S t e f f e n , i n A n g u l a r C o r r e l a t i o n s i n N u c l e a r D i s i n t e g r a t i o n s , e d i t e d by H. van Kr u g t e n and B. van N o o i j e n , Rotterdam U n i v e r s i t y P r e s s , G r o n i n g e n , 1971; page 1. 26. A. Schenck, D. L. W i l l i a m s , J . H. Brewer, K. M. Crowe, and R. F. Jo h n s o n , Chem. Phys. L e t t . U, 544 (1972). 27. J . H. Brewer, Ph. D. T h e s i s , B e r k e l e y , 1972. 28. H. G a b r i e l 3 a n d J . B o s s e , i n A n g u l a r C o r r e l a t i o n s i n N u c l e a r D i s i n t e - g r a t i o n s , Rotterdam U n i v e r s i t y P r e s s , G r o n i n g e n , 1971; page 394. 29. E. M a t t h a i a s , B. O l s e n , D. A. S h i r l e y , R. M. S t e f f e n , and J . E. Tem p i e t o n , Phys. Rev. A4, 1626 (1971). 30. P. Da R. Andrade, J . D. R o g e r s , and A. Vasquez, Phys. Rev. 188, 571 ( 1 9 6 9 ) ; i b i d . , B3, 1052 (1971 ). 31. A. L o p e z - G a r c i a , Z. P h y s i k 253, 187 (1 9 7 2 ) . 32. R. D. M o u n t a i n , CRC C r i t . Rev. S o l . S t a t e 1, 5 (1972). 33. R. Gordon, Adv. Magn. Res. 3_, 1 (1968). -208-CHAPTER V I I CONCLUDING REMARKS H o p e f u l l y , t h e m a t e r i a l d e v e l o p e d i n C h a p t e r s I I I - V I w i l l a i d t h e b i o c h e m i s t who r e l i e s on n u c l e a r s p i n r e l a x a t i o n as an a i d i n h i s s e a r c h f o r c l u e s t o u n r a v e l t h e c o m p l e x i t y and f u n c t i o n o f b i o l o g i c a l s t r u c t u r e s . The p r i m a r y theme d e v e l o p e d , emphasized t h a t t h e b i o c h e m i s t must abandon many c o n v e n t i o n a l i d e a s one a t t r i b u t e s t o t r a n s i e n t s p i n b e h a v i o r . T r e a d i n g on v i r g i n s o i l can be as e m b a r r a s i n g as i t i s r e w a r d i n g . I t i s a l l t o o easy t o s e a r c h f o r g e n e r a l a n a l o g i e s from p r o t o n r e l a x a t i o n e x p e r i m e n t s o r from r e s u l t s o b t a i n e d from r e l a t i v e l y s m a l l , m o t i o n a l l y i s o t r o p i c s p i n systems. A d o p t i n g such an approach s h o u l d be u n d e r t a k e n o n l y w i t h extreme c a u t i o n ; t h i s t h e s i s c l e a r l y has shown many i n s t a n c e s where s i m p l e i n d u c t i o n may l e a d t o g r a v e c o n c e p t u a l d i f f i c u l t i e s . A l s o , i t was commented t h a t NMR p r e s e n t s a h i g h l y u n d e r d e t e r m i n e d problem f o r t h e e x p e r i m e n t a l i s t . However, on a more o p t i m i s t i c n o t e , i t was seen t h a t i n many i n s t a n c e s , t h e c o n v e n t i o n a l r e l a x a t i o n t i m e s were f u n c t i o n s o f o n l y one t y p e o f m o t i o n ( i n a d d i t i o n t o s t a t i c m o l e c u l a r g e o m e t r i c a l c o n s t a n t s ) w h i c h g r e a t l y reduced t h e p o s s i b i l i t y o f m i s i n t e r -p r e t a t i o n . I n d e e d , i t i s f e l t t h a t NMR w i l l c o n t i n u e t o e x t e n d i t s u s e f u l n e s s as a p h y s i c a l t e c h n i q u e f o r b i o l o g i c a l s t u d i e s , e s p e c i a l l y when used i n c o n j u n c t i o n w i t h o t h e r m o t i o n a l t r a c e r t e c h n i q u e s . As f o r e x t e n s i o n s o f t h i s work, t h e r e a r e numerous pat h s one c o u l d e x p l o r e . The o b v i o u s e x t e n s i o n i s a c o m p l i m e n t a r y e x p e r i m e n t a l s t u d y -209-on model systems w h i c h would be of, g r e a t i n t e r e s t t o t h o s e p e r s o n s i n t e r -e s t e d s o l e l y i n t h e p l e a s u r e o f u n d e r s t a n d i n g s p i n r e l a x a t i o n . As f o r p r a c t i c a l e x t e n s i o n s o f a p p l i e d r e l a x a t i o n t h e o r y t o problems o f b i o -c h e m i c a l i n t e r e s t , t h e most f r u i t f u l c a l c u l a t i o n s would a t t e m p t t o t h r o u g h l y examine r e l a x a t i o n i n o r d e r e d systems. These systems a r e one c a s e o f extreme a n i s o t r o p i c m o t i o n w h i c h cannot be c o r r e c t l y t r e a t e d by t h e methods used i n t h i s t h e s i s . F i n a l l y , t h e t h e o r e t i c a l minded would be most i n t e r e s t e d i n e x t e n d i n g t h e g e n e r a l t h e o r y o f n u c l e a r m u l t i p o l e r e l a x a t i o n and r e l a x a t i o n t h e o r i e s w h i c h a r e a p p l i c a b l e when m o l e c u l a r m o t i o n s a r e on t h e o r d e r o f t h e r i g i d l a t t i c e l i n e w i d t h . -210-APPENDIX A THE RELAXATION MATRIX A l l c a l c u l a t i o n s p r e s e n t e d i n t h i s t h e s i s r e s o r t t o d e t e r m i n g t h e s o l u -t i o n t o t h e e q u a t i o n o f m o t i o n o f t h e d e n s i t y o p e r a t o r i n w h i c h t h e elements o f a ( o r x = cr - a T ) a r e d e s c r i b e d by c o u p l e d f i r s t o r d e r d i f f e r -e n t i a l e q u a t i o n s w i t h c o n s t a n t c o e f f i c i e n t s , ( d / d t ) x , = iu l X . + R i ,, m x . • [ A . l ] aa aa aa • i • • • aa a a aa a a T h i s e q u a t i o n was f u l l y d i s c u s s e d i n C h a p t e r I I . In t h i s A p p e n d i x , a d i s c u s s i o n o f t h e p h y s i c a l and c o m p u t a t i o n a l i m p l i c a t i o n s o f v a r i o u s terms i n t h i s e q u a t i o n i s p r e s e n t e d . F i r s t , we may ask i f any o f t h e elements o f x have a " p h y s i c a l " meaning. Presuming t h e e i g e n b a s i s |a> i s n o n d e g e n e r a t e , t h e d i a g o n a l elements o f x can be s i m p l y i n t e r p r e t e d as s t a t e p s e u d o p o p u l a t i o n s , o r more c o r r e c t l y , as t h e r m a l d e v i a t i o n p o p u l a t i o n s . As t h e t o t a l number o f s p i n s i s c o n s t a n t , S x = 0 [A.2] t * "aa a ( d / d t ) x _ = 0 . [A.3] aa I f d e g e n e r a t e energy l e v e l s a r e p r e s e n t , t h e r e i s no o p e r a t i o n a l s i g n i -f i c a n c e i n t h e c o n c e p t o f t h e number o f s p i n s as b e i n g i n one o f t h e d e g e n e r a t e e i g e n k e t s ; o n l y s p i n s h a v i n g d i f f e r e n t e n e r g i e s a r e d i s t i n -g u i s h a b l e i n t h i s problem. As f o r t h e o f f - d i a g o n a l e l e m e n t s , t h e r e i s no u n i v e r s a l i n t e r p r e t a t i o n , a l t h o u g h f o r s p i n p r o b l e m s , i n a b a s i s w h i c h -211-d i a g o n a l i z e s C^Q, t h e s e can be i n t e r p r e t e d as a measure o f t h e phase coh e r e n c e o f t h e s p i n s ( t h e net t r a n s v e r s e m a g n i t i z a t i o n ) . In t h e absence o f r e l a x a t i o n , t h e e lements o f t h e t e t r a d i c R, a r e z e r o . I t i s seen t h a t i n t h i s i n s t a n c e , t h e d e n s i t y m a t r i x e l e m e n t s undergo a s i m p l e o s c i l l a t o r y m o t i o n *aa" 4 e ^ a a ' V £A.4] a t t h e n a t u r a l f r e q u e n c y , co ,. I t i s r a t i o n a l i z e d t h a t e lements o f R aa c o n n e c t i n g e l e m e n t s o f x w h i c h have d i f f e r e n t n a t u r a l f r e q u e n c i e s , t e n d t o a v e r a g e o u t o v e r a p e r i o d o f t i m e e q u a l t o (co , - co ,, ,,, . T h i s aa a a i s t r u e o n l y i f t h e e lements o f x undergo undamped m o t i o n d u r i n g t h i s i n t e r v a l . In o t h e r w ords, R , .I •(• < < I to i - co , , 1 1 1 I . [A. 5] aa a a 1 aa a a 1 L J Terms whose n a t u r a l f r e q u e n c i e s d i f f e r by more than a few l i n e w i d t h s ( t h e so c a l l e d n o n s e c u l a r t e r m s ) can s a f e l y be i g n o r e d , and f o r most p r a c t i c a l p u r p o s e s , R , i , i , i = 0 i f c o ^ c o , , , , , . T h i s f a c t a l l o w s r a a a a a a a a a g r e a t s i m p l i f i c a t i o n i n t h e number o f elements o f x t h a t need t o be c a l c u l a t e d i n a p p l i c a t i o n s . F u r t h e r m o r e , f o r l o n g i t u d i n a l r e l a x a t i o n , o n l y terms where co , - 0 need be c o n s i d e r e d . The e lements i n v o l v e d J aa f o r t r a n s v e r s e r e l a x a t i o n have d i f f e r e n t n a t u r a l f r e q u e n c i e s and hence cannot c o u p l e w i t h the l o n g i t u d i n a l e l e m e n t s , t h e two s e t s o f e l e m e n t s r e l a x i n d e p e n d e n t l y . The r e l a x a t i o n H a m i l t o n i a n cannot d i r e c t l y c o u p l e e l e m e n t s o f t h e d e n s i t y m a t r i x h a v i n g d i f f e r e n t n a t u r a l f r e q u e n c i e s . I f t h e r e a r e no d e g e n e r a t e t r a n s i t i o n s , t h e s e s t a t e m e n t s may be s i m p l i f i e d : A l l d i a g o n a l e lements ( l o n g i t u d i n a l r e l a x a t i o n ) r e l a x i n d e --212-p e n d e n t l y o f a l l o f f - d i a g o n a l elements and v i c e - v e r s a . In the absence o f o v e r l a p p i n g r e s o n a n c e s , t h i s a l l o w s a s i m p l e i n t e r p r e t a t i o n o f r e l a x -a t i o n e l ements o f t h e form R , ',. E q u a t i o n [ A . l ] becomes ( d / d t ) x . = ( i w , + R , , ) x . [A.6] aa aa aa aa 'Aaa where t h e t i m e dependence o f x a a i i s o f the form o f a damped o s c i l l a t i o n . A l i n e c e n t e r e d a t to , has a L o r e n t z i a n shape o f h a l f - w i d t h R , , (_"|) aa aa aa v ' and t h e i d e n t i f i c a t i o n - R . • • , = ( T : 1 ) , [A.7] aa aa d. a-Hx i s r e a d i l y made. I t i s noted t h a t t h e t r a n s v e r s e r e l a x a t i o n i s not a f f e c t e d by o t h e r l e v e l s w h i c h a r e not d i r e c t l y c o n c e r n e d w i t h t h e t r a n -s i t i o n under c o n s i d e r a t i o n . The l o n g i t u d i n a l r e l a x a t i o n o f i n d i v i d u a l t r a n s i t i o n s have no such s i m p l e c o r r e s p o n d e n c e . The l o n g i t u d i n a l m a g n e t i z a t i o n c o r r e s p o n d i n g t o any g i v e n t r a n s i t i o n i s a f u n c t i o n o f a l l o t h e r Zeeman l e v e l s . However, a r e l a t e d q u a n t i t y , ^ a i a > a c L > does have a s i m p l e i n t e r p r e t a t i o n as t h e p r o b a b i l i t y p e r u n i t t i m e o f a t r a n s i t i o n from s t a t e a1 t o s t a t e a t h r o u g h i n t e r a c t i o n s w i t h t h e m o l e c u l a r s u r r o u n d i n g s . U s i n g t h e s t a n d a r d n o t a t i o n o f c l a s s i c a l t r a n s i t i o n p r o b a b i l i t i e s , R i i = W , . [A.8] a a aa a a - R i s t h e t o t a l r a t e a t which s t a t e a i s d e p l e t e d by t r a n s i t i o n s t o aaaa o t h e r l e v e l s , and t h u s , R = Z—• R i i [A.9] aaaa i , a a aa a fa -213-w h i c h can be w r i t t e n as £ _ R . , = 0 . [A.10] i a a aa a In the BPP t h e o r y , one i m p l i c i t l y t r e a t s o n l y s t a t e p o p u l a t i o n s ( d i a g o n a l e l e ments o f x)> and hence t h i s approach can be seen t o be a s p e c i a l i z e d c a s e ( o f t e n i n a p p l i c a b l e ) o f t h e more g e n e r a l d e n s i t y o p e r a t o r a p p r o a c h . To summarize, terms o f t h e r e l a x a t i o n m a t r i x o f t h e form R , , can be a aa a d e s c r i b e d as l i n e w i d t h t e r m s , t h e R , , as t r a n s i t i o n p r o b a b i l i t y t e r m s , a a aa r J and a l l o t h e r s s i m p l y as " i n t e r a c t i o n " t erms. I t s h o u l d be emphasized t h a t t h e r e l a x a t i o n m a t r i x i s n o t a t r a n s i t i o n m a t r i x . I t i s seen t h a t terms o f t h e form R , , must be r e a l n e g a t i v e q u a n t i t i e s and t h a t the aa aa n R ,', must be r e a l n o n - n e g a t i v e q u a n t i t i e s . aaa a The s e c u l a r a p p r o x i m a t i o n s m e n t i o n e d , s i m p l i f y c a l c u l a t i o n s immensely. An e x a m i n a t i o n o f t h e c o n s t r u c t i o n o f R and t h e H e r m i t i a n c h a r a c t e r o f the r e l a x i n g i n t e r a c t i o n s shows many more s i m p l i f i c a t i o n s a r e r e a d i l y a c h e i v e d . I t i s o b v i o u s t h a t R , , , , , , = R ,,,,, , • [A.11] aa a a a a a a L i k e w i s e , due t o t h e f a c t t h a t t h e s p i n o p e r a t o r s o c c u r i n a d j o i n t p a i r s ( V k = ( - l ) k V ~ k ) , R i II 111 = R i i [A. 12] aa a a a aa a T h e r e f o r e , R = R ii ••• I = R I it = R ••• •! • a a a a a a a a a a a a a a a a . A d d i t i o n a l r e l a t i o n s a r e o b t a i n e d i f we note t h e b e h a v i o r o f c e r t a i n terms under s p i n i n v e r s i o n . F o r i n d i r e c t o r d i r e c t d i p o l a r and quadru-p o l a r c o u p l i n g s , t h e s p i n o p e r a t o r s a r e components o f a second rank - 2 1 4 -s p h e r i c a l t e n s o r , and f o r s h i f t a n i s o t r o p y and s p i n - r o t a t i o n , t h e s p i n o p e r a t o r s a r e components o f a f i r s t rank s p h e r i c a l t e n s o r . A d i r e c t c a l c u l a t i o n shows t h a t <^a|V k L|^a'> = ( - l ) L < a | V ^ | a ' > [ A . 1 3 ] where i s t h e s p i n i n v e r s i o n o p e r a t o r ( r o t a t e s a l l s p i n s by TT) and L i s t h e t e n s o r i a l rank o f t h e s p i n o p e r a t o r s . Hence any terms a p p e a r i n g w i t h t h e a p p r o p r i a t e a u t o - c o r r e l a t i o n f u n c t i o n s (c=n i n E q u a t i o n [ 2 . 2 . 1 3 ] ) , R i I I I I I ~ R i - i , i i , I I I v i s „ [ A . 1 4 J F o r c r o s s - t e r m s , i f L = L', r e l a t i o n s h i p [ A . 1 4 ] s t i l l h o l d s , whereas i f L f V , t h e r e l a t i o n R, , I , I I , I I I = - R i i i i i i L A . 1 5 J ijjaipa i|;a y a aa a a i s v a l i d . To d e m o n s t r a t e how d r a s t i c a l l y t h e s e v a r i o u s symmetries and approx-i m a t i o n s reduce the magnitude o f t h e c a l c u l a t i o n , c o n s i d e r t h e f o l l o w i n g 2 2 example. F o r two d d e n t i c a l s p i n 1/2 p a r t i c l e s , x has ( 2 ) = 1 6 elements 2 2 2 and R has ( ( 2 ) ) o r 2 5 6 e l e m e n t s . For t h e l o n g i t u d i n a l r e l a x a t i o n , o n l y s i x e lements o f x a r e r e l e v a n t ( f o u r f o r t r a n s v e r s e r e l a x a t i o n ) . Of t h e 3 6 elements o f R t h a t need be e v a l u a t e d , o n l y f i v e need t o be c a l c u l a t e d , t h e o t h e r s b e i n g f ound from E q u a t i o n s [ A . 1 0 ] , [ A . 1 1 ] , [ A . 1 2 ] , [ A . 1 4 ] , o r [ A . 1 5 ] . -215-APPENDIX B INFLUENCE OF THE SECOND ORDER FREQUENCY SHIFT TERMS As was mentioned i n C h a p t e r I I , i f t h e e x t r e m e - n a r r o w i n g approx-i m a t i o n f a i l s , c e r t a i n i m a g i n a r y c o r r e c t i v e terms must be i n c o r p o r a t e d 1-4 i n t o t h e e q u a t i o n s g o v e r n i n g t h e t i m e e v o l u t i o n o f t h e d e n s i t y o p e r a t o r . In p a r t i c u l a r , t h e m o t i o n a l e q u a t i o n o f any element o f x can be w r i t t e n as ( d / d t ) x i = N a t u r a l f r e q u e n c y term + R e l a x a t i o n terms aa + Second o r d e r c o r r e c t i v e terms [ B . l ] In a l l c a l c u l a t i o n s p r e s e n t e d i n t h i s t h e s i s , t h e i n f l u e n c e o f t h e s e l a t t e r terms has been i g n o r e d . In t h i s A p p e n d i x , t h e i m p o r t a n c e o f t h e s e terms i s examined by means o f a s i m p l e example. The s i m p l e s t o f a l l ( n o n t r i v i a l ) r e l a x a t i o n c a l c u l a t i o n s t r e a t s two i d e n t i c a l s p i n 1/2 n u c l e i r e l a x e d by d i p o l a r communication (a case t r e a t e d e x t e n s i v e l y i n C h a p t e r I I I ) . The i n f l u e n c e o f t h e s e second o r d e r c o r r e c -t i o n s on any e l e m e n t , Y ,, can be w r i t t e n as (see E q u a t i o n [ 2 . 2 . 1 0 - 1 1 ] ) , aa ( d / d t ) x . = i ' * N ( a " a " ' ) [ 6 , ,,,x . . - S . .X >>• J x ' ' Aaa • , I I , V a a Aaa aa A a a a a «k'"k<K"„1v + 'lvV"> a a a ,k X <a 1 V|V" k|a'''>[6 , , , l X , , - 6 l l X . . . . ] [B.2] 1 1 a a aa aa a a k -k k -k where Q ' (CD) and J ' (w) form a H i l b e r t T r a n s f o r m p a i r , Qk'"k(u>) = - ^ * J k'~ k(w) . [B.3] -216-C h o o s i n g as b a s i s s t a t e s , |1> = |++> |2> = (|+-> + |-+>)//2 |3> = (|+-> - |-+>)//2 |4> = |-> , [B . 4 ] i t i s s i m p l y shown t h a t t h e l o n g i t u d i n a l r e l a x a t i o n i s g i v e n by < I z ( t ) > - < I 2 > T = (<I Z (0 )> - < I 2 > T ) e x p { ( - R 1 1 4 4 - R 1 1 2 2 ) t } t B - 5 ^ where R ] ] 4 4 = 2 J 2 ' ~ 2 ( 2 O J 0) and R 1 1 2 2 = - J 1 ' - 1 ^ ) + 2 J 2 ' " 2 ( 2 O J 0 ) . T h i s r e s u l t i s o b t a i n e d from t h e complete r e l a x a t i o n d e s c r i p t i o n o f E q u a t i o n [ B . l ] , but i s i d e n t i c a l t o t h e c a l c u l a t i o n where t h e second o r d e r s h i f t terms a r e i g n o r e d . T h i s can be seen by e x a m i n a t i o n o f E q u a t i o n [ B . 2 ] . The o n l y nonzero terms o f t h e m a t r i x N a r e , N( l l ) = Q ° ° ( 0 ) / 6 - Q 1 ' " 1 ( co 0 ) /2 + Q 2 ' " 2 (2o ) 0 ) [B . 6 ] N(22) = 2 Q ° ° ( 0 ) / 3 - Q 1 > _ 1 (u)J) [B . 7 ] N(44) = N ( l l ) . [B . 8 ] Hence, f o r t h e l o n g i t u d i n a l r e l a x a t i o n where o n l y t h e c o u p l i n g o f X-J-JJ x 2 2 and X44 need t o be c o n s i d e r e d ; a ' ' = a ' 1 1 = 1, 2, o r 4, and a ' = a = 1, 2, o r 4. T h e r e f o r e t h e o n l y terms t h a t a r i s e i n E q u a t i o n [B . 2 ] a r e o f t h e form [6 ,,x 11 - 5 11X ,,] = 0! Indeed, t h i s r e s u l t can aa aa aa aa be g e n e r a l i z e d and i t f o l l o w s t h a t i t i s c o m p l e t e l y j u s t i f i e d t o i g n o r e t h e s e c o r r e c t i v e terms i n any l o n g i t u d i n a l r e l a x a t i o n c a l c u l a t i o n . For t h e t r a n s v e r s e r e l a x a t i o n c a l c u l a t i o n , o n l y two ele m e n t s o f x -217-need t o be t r e a t e d , x- ] 2 a n d x 2 4 - ^ 1 S s i m p l y shown t h a t ( d / d t ) x 1 2 = i u 1 2 x 1 2 + R 1 2 1 2 X 1 2 + R 1 2 2 4 x 2 4 + i ^ 2 2 ) " N ( l l ) ) x 1 2 [B.9] ( d / d t ) x 2 4 = i u 24 x24 + R 2 4 1 2 x 1 2 + R 2 4 2 4 x 2 4 + i ( N ( 4 4 ) " N ( 2 2 ) )x 2 4 -[B.10] S i n c e N ( l l ) = N ( 4 4 ) , co-|2 = O O 2 4 E -WQ, and R-|2-|2 = R 2 4 2 4 ' t n i s c . p a i r ° f e q u a t i o n s can be reg r o u p e d and r e w r i t t e n a s , (d / d t ) < I + ( t ) > = ( d / d t ) ( x 1 2 + x 2 4 ) = ( d / d t ) y i ( t ) = ^ ( t ) + i v y 2 ( t ) [ B . l l ] ( d / d t ) y 2 ( t ) = ( d / d t ) ( x 1 2 - x 2 4 ) = i y y ^ t ) + i p y 2 ( t ) [B.12] where $ = -iwg + R-|2-|2 + R ] 2 2 4 a n d v = (N(22') - N ( l l ) ) . S u p p r e s s i n g t h e r a p i d Larmor p r e c e s s i o n , t h e t r a n s v e r s e decay can be w r i t t e n a s , < I + ( t ) > / < I + ( 0 ) > = ^ e x p [ ( R ] 2 1 2 + R 1 2 2 4 ) t ] | e x p ( i y t ) + e x p ( - i y t ) } [B.13] where R ] 2 ] 2 + R ] 2 2 4 = -|o 0 0(0) + |o] ' _ 1 (u>0) - 2J 2 ' ~ 2 ( 2 u > 0 ) . I t i s c l e a r l y seen t h a t t h e p h y s i c a l e f f e c t o f t h e i m a g i n a r y terms g i v e r i s e t o a m o d u l a t i o n o f t h e decay e n v e l o p e . F o r s i m p l i c i t y , i f we f u r t h e r assume t h a t J k ' " k ( u>) = ( - l ) k 5 2 T 2 ( l + L O 2 T 2 )~\ t h e n Q k ," k (u>) = u>T 2J k'" k(u)). T h i s a l l o w s one t o w r i t e E q u a t i o n [B.13] a s , < I + ( t ) > / < I + ( 0 ) > = ex p | ( - 3 -5(1 + x 2 ) " 1 - 2(1 + 4 x 2 ) - 1 ) t c 2 x 2 / 2 } x c o s | ( x ( l + x 2 ) " 1 + 4 x ( l + 4 x 2 ) " 1 ) U 2 x 2 / 2 J ; x = W Q T 2 . [B.14] -218-Th i s c o u l d a l s o be i n s t r u c t i v e l y r e w r i t t e n as < I + ( t ) > / < I + ( 0 ) > = R e [ ( ( - 1 / T 2 - 1-0)^2/2^ ) t ) ] [B.15] where T-j and T 2 a r e c o n v e n t i o n a l l y d e f i n e d . In F i g u r e B . l , l o g ( < I + ( t ) > / < I + ( 0 ) > ) i s p l o t t e d as a f u n c t i o n o f t h e t h e u n i t l e s s v a r i a b l e , t g x 2 , f o r f i v e d i f f e r e n t v a l u e s o f togx,, (tooT2 = 1 0 ± 2 ^ , l O 1 ^ , and 10^; c u r v e s p - t c o r r e s p o n d t o i n c r e a s i n g v a l u e s o f COQT 2 r e s p e c t i v e l y ) . The upper and l o w e r t r i a n g u l a r l a b e l e d c u r v e s c o r r e s p o n d t o t h e l i m i t s o f togx,, >> 1 and COQX 2 << 1 r e s p e c t i v e l y . The open squares d e p i c t t h e p r e d i c t e d decay when t h e i m a g i n a r y terms a r e i g n o r e d ( f o r t h e c a s e , U Q T 2 = 1 ) . I t i s o b v i o u s t h a t even i n t h i s optimum i n s t a n c e , t h e i n c l u s i o n o f t h e i m a g i n a r y c o r r e c t i v e terms b o r d e r s on b e i n g n e g l i b l e . A diagram w h i c h summarizes t h e s e f a c t s i s F i g u r e B.2 ( n o t e t h e s i m i l a r i t y t o F i g u r e 3.2). The r e d u c e d , f r e q u e n c y w e i g h t e d s p e c t r a l 2 2 - 1 2 2 2 2 - 1 d e n s i t i e s , u)j(to) = -r 2co(l + to x 2 ) ~ and toq(to) = to x 2 ( l + to x 2 ) as w e l l as wj(0) = U JT 2 a r e p l o t t e d as a f u n c t i o n o f t ox 2 . D e a l i n g w i t h o r d e r s o f m a g n i t u d e , j ( 0 ) i s a measure o f t h e l i n e w i d t h , j ( t o Q ) i s a measure o f (1/ T . j ) , and q(tog) i s a measure o f t h e magnitude o f t h e second o r d e r s h i f t terms. T h i s p l o t shows v e r y e x p l i c i t l y t h a t q(cog) w i l l be o f s i g n i f i c a n c e o n l y when J ( U ) Q ) - q(tog) - j ( 0 ) , t h a t i s , when COQT 2 - 1. I n t r o d u c t i o n o f i n t e r n a l m o t i o n s can be d e s c r i b e d i n t h e same f a s h i o n as i n C h a p t e r I I I . I t can be r a t i o n a l i z e d t h a t t h e e f f e c t o f a n i s o t r o p i c m o b i l i t y o n l y m i n i m i z e s t h e im p o r t a n c e o f t h e s e second o r d e r s t a t i c c o r r e c t i o n s . A l t h o u g h o n l y a s i m p l i f i e d c a s e has been e x p l i c i t l y t r e a t e d i n t h i s -219-s e c t i o n , t e d i o u s g e n e r a l i z a t i o n s f o l l o w d i r e c t l y . However, w e i g h i n g t h e c o m p l i c a t i o n s i n t r o d u c e d a g a i n s t t h e i n s i g h t and r i g o r g a i n e d , such an approach has not been adopted. 1. H. G a b r i e l , Phys. S t a t . S o l . 23, 195 ( 1 9 6 7 ) . 2. R. A. Hoffman, Advan. Magn. Res. 4_, 87 (1970) . 3. H. P f e i f e r , Ann. P h y s i k V3, 174 (1964) . 4. G. K. F r a e n k e l , J . Chem. Phys. 42, 4275 (1965) . -220-FIGURE B . l : The t r a n s v e r s e decay i s p l o t t e d as a f u n c t i o n o f t h e u n i t l e s s v a r i a b l e , t£ x^ f o r f i v e d i f f e r e n t v a l u e s o f CJQT2 (WQT£ = +2/3 +1/3 0 10— ' , 10— ' , and 10 ; c u r v e s p - t c o r r e s p o n d t o i n c r e a s i n g v a l u e s o f COQT^ ) . The upper and l o w e r t r i a n g u l a r l a b e l e d c u r v e s c o r r e s p o n d t o t h e l i m i t s o f > > - l a n c l w Q t 2 < k 1 r e s p e c t i v e l y . The open sq u a r e s d e p i c t t h e p r e d i c t e d ( l i n e a r ) b e h a v i o r when t h e i m a g i n a r y ( second o r d e r s h i f t t e r m s ) c o r r e c t i o n s a r e i g n o r e d ( u ) n T 9 = 1 i s assumed). -2 2 1--222-FIGURE B . 2 : P l o t s o f r e d u c e d , f r e q u e n c y w e i g h t e d , s p e c t r a l d e n s i t i e s , 2 2-1 2 coj(w) = 12^(1+ co T^) and toq(<o) = co TgjCco) as w e l l as <oj(0) Etox9 a r e p l o t t e d as a f u n c t i o n o f I D T 9 . -2 2 3--224-APPENDIX C QUADRUPOLE RELAXATION OF I = 3/2 NUCLEI In r e c e n t y e a r s , n u c l e a r r e l a x a t i o n s t u d i e s o f s p i n 3/2 n u c l e i i n -v o l v e d i n c h e m i c a l exchange have proven v e r y f r u i t f u l f o r t h e s t u d y o f b i o m o l e c u l e s i n s o l u t i o n . ^ I n t e r p r e t a t i o n assumes t h a t t h e r e l a x a t i o n a t each s i t e i s c h a r a c t e r i z e d by a uniqu e t i m e c o n s t a n t . In C h a p t e r V, i t was mentioned t h a t t h i s a s s u m p t i o n w i l l n ot be v a l i d f o r a q u a d r u p o l a r r e l a x e d n u c l e u s ( w i t h I >_ 3/2) when e x t r e m e - n a r r o w i n g i s v i o l a t e d ( i . e . f o r most b i o m o l e c u l e T C ' S ) . AS a l a r g e p o r t i o n o f t h i s t h e s i s has d e a l t w i t h t h e e f f e c t s o f a n i s o t r o p i c m o t i o n s and n o n - e x p o n e n t i a l r e l a x a t i o n , a b r i e f m e n t i o n o f t h e n o v e l i t y o f t h i s phenomenon i s i n o r d e r . A l t h o u g h t h i s f a c e t o f q u a d r u p o l a r r e l a x a t i o n has been known e v e r 2 s i n c e B l o c h f i r s t commented on t h i s p o i n t , o n l y r e c e n t l y have q u a n t i -3 4 t a t i v e n u m e r i c a l e x p r e s s i o n s been g i v e n i n t h e l i t e r a t u r e . ' However, t h e e f f e c t o f a n i s o t r o p i c m o t i o n s on t h e p e r t u r b a t i o n has not been commented upon. T h i s i s a v e r y r e a l problem as a t y p i c a l s t u d y may t r e a t t h e r e l a x a t i o n b e h a v i o r o f a s p e c i e s s u c h as C/^-Hg/ where t h e CI n u c l e u s r o t a t e s r a p i d l y about t h e macromolecule-Hg bond. The q u a d r u p o l a r H a m i l t o n i a n can be c o n v e n i e n t l y e x p r e s s e d i n t h e f o l l o w i n g f o r m , £ n ( t ) = i f U k ( t ) V k [ C . l ] . ^ k=-2 where U k ( t ) = ( - l ) K U / 1 8 0 ) 1 / 2 (^-f-j r2k(ti(t)) [ C 2 ] -225-V° = 3 I 2 - I 2 [C . 3 ] V ± ] = + [ I Z , I ± ] + / 2 / 6 [ C 4 ] V ± 2 = I 2 / 2 / 6 . [C . 5 ] F o r s i m p l i c i t y , i t i s assumed t h a t t h e q u a d r u p o l a r n u c l e u s has I = 3/2 ( e . g . 7 L i , 1 ] B , 2 3 N a , 3 3 S , 3 5 C 1 , 3 7 C I , 7 9 B r , 8 1 B r , e t c . ) and t h a t t h e n u c l e a r e n vironment i s c o m p l e t e l y c h a r a c t e r i z e d by an a x i a l l y symmetric e l e c t r i c f i e l d g r a d i e n t (n=0). Upon c h o o s i n g t h e b a s i s f u n c t i o n s |IM>, |1> = |3/2 3/2> |2> = 13/2 l/2> |3> = 13/2 -l/2> |4> = 13/2 -3/2> , [C . 6 ] a r a t h e r s t r a i g h t f o r w a r d c a l c u l a t i o n u s i n g t h e s e m i c l a s s i c a l form o f t h e d e n s i t y o p e r a t o r t h e o r y o f r e l a x a t i o n y i e l d s t h e f o l l o w i n g p a i r o f c o u p l e d e q u a t i o n s , ( d / d t ) y ] ( t ) = s e c o 1 ' - 1 ^ ) - J 2 ' " 2 ( 2 w 0 ) ] y i ( t ) + 36EJ 1 , _ 1 (a3 0 ) + J 2 ' - 2 ( 2 w 0 ) ] y 2 ( t ) [C . 7 ] ( d / d t ) y 2 ( t ) = S G L J 1'"^^) + J 2 , ~ 2 ( 2 a > 0 ) ] y 1 ( t ) + seEJ1'"'1^) - J 2 ' " 2 ( 2 W o ) ] y 2 ( t ) [C . 8 ] where y ^ t ) = T r [ x l z ] = < I z ( t ) > - < I z > T = | ( X l l - x 4 4 ) + j(x22 - x 3 3 ) [ C 9 ] -226-1 3 y 2 ( t ) - "2"(x44 " x-|) + 2"(x 3 3 - x 2 2 ) • [C.10] The s p e c t r a l d e n s i t i e s a r e g i v e n by t h e o n e - s i d e d F o u r i e r T r a n s f o r m o f the a p p r o p r i a t e a u t o c o r r e l a t i o n f u n c t i o n s (composed from t h e l a t t i c e f u n c t i o n s as d e f i n e d i n E q u a t i o n [ C . 2 ] ) . F u r t h e r m o r e , t h e Hubbard 5 k k -k 00 r e l a t i o n s , (-1) J ' (ku>g) = J (kt jg) , a r e v a l i d , and t h e second o r d e r f r e q u e n c y s h i f t s v a n i s h i d e n t i c a l l y . S i m i l i a r i l y , q ^ t ) = T r [ I + x ( t ) ] = < I + ( t ) > = / 3 ( x 1 2 + x 3 4 ) + 2 x 2 3 [ C . l l ] A s h o r t c a l c u l a t i o n y i e l d s , ( d / d t ) q ] ( t ) = 3 6 [ - J 0 0 ( 0 ) + J 1 » - 1 ( o ) ( J ) - 2 Q 1 ' - 1 ( a , 0 ) ] q 1 ( t ) + 36[ J°°(0) - J 2 ' ' 2 ( 2 u 0 ) + 2 ( 2 Q 1 ' - 1 ( ( , 0 ) + Q 2 ' _ 2 ( 2 a ) ( ) ) ) ] q 2 ( t ) [ C 1 2 ] ( d / d t ) q 2 ( t ) = 36[ J 1 ' - 1 ^ ) - J 2 ' " 2 ( 2 a ) 0 ) + 2 ( Q 1 ' - 1 ( a ) 0 ) + Q 2 7 2 U 0 ) ) ^ 2 ( t ) [C.13] where q 2 ( t ) = 2 x 2 3 . [C.14] A l t h o u g h p r e v i o u s c a l c u l a t i o n s d i d not m e n t i o n t h e second o r d e r c o r -r e c t i o n s ( i . e . f i n i t e Q ' s ) , i t can be seen t h a t t h e y a r e o f l i t t l e i m p o r t a n c e and a r e h e n c e f o r t h i g n o r e d . The s o l u t i o n s f o r y-|(t) and q-j(t) s u b j e c t t o t h e i n i t i a l p r e p a r a t i o n o f t h e s p i n system by a e p u l s e , y ^ O ) = (cose - 1 ) < I Z > T y 2 ( 0 ) = - ( 3 / 5 ^ ( 0 ) q-j (0) = s i n e < I z > T q 2 ( 0 ) = ( 2 / 5 ) q i ( 0 ) [ C 1 5 ] -227-a r e found t o be, y-,(t) q-,(t) [ C . 1 6 ] [ C 1 7 ] From t h e s e e q u a t i o n s , i t i s o b v i o u s t h a t i f J (0 ) = -J 1,-1 2,-2 (2co 0 t h e n both decays a r e d e s c r i b a b l e by a s i n g l e e x p o n e n t i a l . However, i n g e n e r a l both decays a r e b i e x p o n e n t i a l . I t i s i n t e r e s t i n g t o note t h e form o f t h e i n i t i a l decay o f e i t h e r t h e t r a n s v e r s e o r l o n g i t u d i n a l r e l a x a t i o n . Abragam's t r e a t m e n t ( r e c a s t i n t h e p r e s e n t n o t a t i o n ) g i v e s t h e f o l l o w i n g e x p r e s s i o n s f o r t h e r e l a x -a t i o n , U n f o r t u n a t e l y , t h e v a l i d i t y o f t h e s e e x p r e s s i o n s i s l i m i t e d t o s p i n 1=1 n u c l e i ( o r a w h i t 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 ) . However, as t-K), s u b s t i t u t i o n o f E q u a t i o n [ C . 1 5 ] i n t o [ C . 7 ] and [ C . 1 2 ] shows v e r y c l e a r l y t h a t E q u a t i o n s [ C . 1 8 ] and [ C . 1 9 ] c o m p l e t e l y d e s c r i b e t h e i n i t i a l decay o f t h e m a g n e t i z a t i o n s i r r e g a r d l e s s o f e i t h e r o f t h e above a s s u m p t i o n s ! The p r e s e n c e o f a n i s o t r o p i c m o t i o n s has l i t t l e e f f e c t on t h e r e l a x -a t i o n b e h a v i o r . F o r example, f o r s l o w , i s o t r o p i c m o t i o n s , J 0 0 ( t o 0 ) / J 0 0 ( 2 u > 0 = 4 . I f an a d d i t i o n a l degree o f m o t i o n a l freedom i s assumed, o f t e n t h i s - J 1 ' " 1 ^ ) + 4 J 2 ' " 2 ( 2 ( o 0 ) } [ C . 1 8 ] [ C . 1 9 ] -228-r a t i o w i l l a pproach u n i t y even though t h e mot i o n s p e r p e n d i c u l a r t o t h e symmetry a x i s o f t h e d i f f u s i o n t e n s o r a r e slow on t h e ti m e s c a l e o f t h e i n v e r s e Larmor f r e q u e n c y (see t h e main d i s c u s s i o n s i n C h a p t e r I I I ) . T h e r e f o r e , i f a n i s o t r o p i c m o t i o n i s assumed, t h e l o n g i t u d i n a l r e l a x a t i o n may s t i l l be a p p r o x i m a t e d q u i t e w e l l by a s i n g l e e x p o n e n t i a l . F u r t h e r -more, i t w i l l be r e c a l l e d t h a t J ^ ( 0 ) , t o a good a p p r o x i m a t i o n , i s i n -s e n s i t i v e t o m o t i o n s o t h e r t h a n t h o s e p e r p e n d i c u l a r t o t h e symmetry a x i s o f t h e d i f f u s i o n t e n s o r . T h e r e f o r e , a d i s c u s s i o n o f a n i s o t r o p i c m o t i o n s i n t h i s problem h o l d s l i t t l e i n t e r e s t . B e f o r e d i s m i s s i n g t h i s t o p i c however, i t i s w o r t h w h i l e t o note t h a t t h e most unique p r e d i c t i o n t h a t t h e s e c a l c u l a t i o n s have t o o f f e r seems t o have ne v e r been n o t e d i n p r e -3 7 v i o u s c o n s i d e r a t i o n s o f t h i s p r o b l e m . ' In l i g h t o f t h e p r e c e e d i n g d i s c u s s i o n , and f o r s i m p l i c i t y , we assume t h a t t h e r e o r i e n t a t i o n p r o c e s s i s c h a r a c t e r i z e d by a uniq u e x,,. In t h i s nn 9 9 9 9 i c a s e , J (co) = (e qQ/fi) x 2/720 (1 + w T 2 ) . F o r CO QT 2 > 1, E q u a t i o n s [C.16] and [C.17] redu c e t o t h e a s y m p t o t i c form where a E ^ q Q / f O S ^ O ^ g - r , , ) ^ ) - 1 . T h i s shows v e r y e x p l i c i t l y t h a t under t h e s e c i r c u m s t a n c e s , t h a t e x c e p t f o r an i n i t i a l p a r t i a l l o s s o f x-y p o l a r -i z a t i o n , < I + ( t ) > decays on t h e same ti m e s c a l e as t h e l o n g i t u d i n a l m a g n e t i -z a t i o n . In o t h e r words, a d e c r e a s i n g m o b i l i t y l e a d s t o a d e c r e a s e i n t h e o b s e r v a b l e l i n e w i d t h ( t h i s s t a t e m e n t assumes t h a t t h e r a p i d l y d e c a y i n g component o f m a g n e t i z a t i o n w i l l be broadened t o t h e p o i n t o f t o t a l [C.20] q , ( t ) = q ^ O ) |3exp ( -2a (co 0 x 2 ) 2 t ) + 2 e x p ( - 5 a t / 2 ) \ /5 [ C 2 1 ] -229-o b s c u r i t y ) . . No c l a i m s a r e made f o r t h e p r a c t i c a l consequences o f t h i s p r e d i c t i o n . F u r t h e r m o r e , i t s h o u l d be borne i n mind t h a t t h e range o f v a l i d i t y o f t h i s p r e d i c t i o n i s l i m i t e d by t h e f a c t t h a t << ( r e l a x -a t i o n m a t r i x e l e m e n t ) " ^ and t h i s p u t s a s t r i n g e n t l i m i t on maximal 2 2 - 1 - 1 v a l u e s o f e qQ/fi (e qQ/fi « > T 2 < uo^ " However, ^ i s i s an i n t r i -g u i n g t h o u g h t and s h o u l d l e n d i t s e l f r e a d i l y t o e x p e r i m e n t a l v e r i f i c a t i o n f o r n u c l e i such as \ i o r w h i c h have s m a l l Q and l a r g e y. 1. B.D. Sykes and M.D. S c o t t , Ann. Rev. B i o p h y s . B i o e n g . 1_, 27 ( 1 9 7 2 ) . 2. R.K. Wangsness and F. B l o c h , Phys. Rev. 89, 728 ( 1 9 5 3 ) . 3. P.S. Hubbard, J . Chem. Phys. 53, 985 ( 1 9 7 0 ) . 4. N.C. P y p e r , M o l . Phys. 2]_, 1 ( 1 9 7 1 ) . 5. P.S. Hubbard, Phys. Rev. 180, 319 ( 1 9 6 9 ) . 6. A. Abragam, P r i n c i p l e s o f N u c l e a r Magnetism, C l a r e n d o n P r e s s , O x f o r d , 1961; pg. 311. 7. T.E. B u l l , J . Magn. Res. 8, 344 ( 1 9 7 2 ) . * An a n a l o g o u s phenomenon i s p r e d i c t e d when t h e i n f l u e n c e o f c o r r e l a t e d m o t i o n s a r e c o n s i d e r e d i n t h r e e - s p i n systems where i t i s f o u n d t h a t t h e t r a n s v e r s e decay i s w e l l a p p r o x i m a t e d as a sum o f two e x p o n e n t i a l s i r r e g a r d l e s s o f t h e magnitude o f o v e r a l l m o b i l i t y ( s e e C h a p t e r I V ) . I f t h e o v e r a l l m o b i l i t y o f t h e t r i a d i s reduced w h i l e t h e a n i s o t r o p y i n t h e m o t i o n i s i n c r e a s e d , one o f t h e t i m e c o n s t a n t s i n c r e a s e s , t h e o t h e r d e c r e a s e s . I f t h e r e l a t i v e magnitude d i f f e r by o r d e r s o f mag-n i t u d e , t h e n i t i s v e r y c o n c e i v a b l e t h a t one c o u l d c o n c l u d e a T^, i n e x a c t a n a l o g y w i t h t h e p r e s e n t c a s e . -230-APPENDIX D NUCLEAR MAGNETIC RELAXATION FOR INDIVIDUAL TRANSITIONS OF AN AMX SPECTRUM: USE OF INTERFERENCE TERMS TO DETERMINE SIGNS OF SCALAR COUPLING CONSTANTS I t has been shown t h a t t h e p r e s e n c e o f f i n i t e i n t e r f e r e n c e terms between two p h y s i c a l 1 y d i f f e r e n t m a g n e t i c r e l a x a t i o n mechanisms, such as between d i p o l a r and s h i f t a n i s o t r o p y . . i n t e r a c t i o n s , may p r o v i d e f o r d e t e r -m i n a t i o n o f t h e a b s o l u t e s i g n o f t h e s c a l a r c o u p l i n g c o n s t a n t ( t r a c e o f the i n d i r e c t d i p o l a r c o u p l i n g t e n s o r ) from c o m p a r i s o n o f t h e r e l a t i v e 2-4 l i n e w i d t h s i n an AX spectrum. F o r s p i n systems c o n t a i n i n g t h r e e o r more c o u p l e d s p i n o n e - h a l f n u c l e i , a d d i t i o n a l i n t e r f e r e n c e terms a r i s e from c r o s s - c o r r e l a t i o n e f f e c t s between p h y s i c a l l y 1 i k e r e l a x a t i o n mech-5 a n i s m s , as between two d i p o l e - d i p o l e i n t e r a c t i o n s , so i t i s n a t u r a l t o i n q u i r e whether t h e s e e f f e c t s can be used t o f i n d t h e s i g n s o f c o u p l i n g c o n s t a n t s . F i n a l l y , t h e answer t o t h i s q u e s t i o n s h o u l d p r o v i d e i n s i g h t i n t o t h e r e l a t i o n between t h e s e two d i s t i n c t c l a s s e s o f i n t e r f e r e n c e 5 6 e f f e c t s , ' t h i s a s p e c t p r o v i d i n g t h e p r i m a r y i n t e r e s t o f t h i s d i s c u s s i o n . The s i m p l e s t s u i t a b l e system o f t h r e e s p i n o n e - h a l f n u c l e i i s t h e "AMX" c a s e , f o r wh i c h J A M « (v f t - v M ) ; J A X « ( v A - v x ) , and J M X « ( v M - v ^ ) . A s c h e m a t i c energy l e v e l d i a g r a m i s shown i n F i g u r e D . l . F o r d e f i n i t e n e s s i n t h e d i a g r a m , i t i s supposed t h a t | J A M | = 2 | J A X [ = 4 | J M X | ; th e s i g n c o n v e n t i o n i s t h a t a s t a t i c m a g n e t i c f i e l d a p p l i e d i n t h e z-d i r e c t i o n and a p o s i t i v e s c a l a r c o u p l i n g c o n s t a n t l e a d t o a l o w e r energy f o r a n t i p a r a l l e l than f o r p a r a l l e l s p i n s . Because t h e s c a l a r c o u p l i n g i s weak, an u n c o u p l e d b a s i s s e t d i a g o n a l i z e s t h e s t a t i c H a m i l t o n i a n t o f i r s t o r d e r : -231-|1> = |+++> |2> = |++-> |3> H |+-+> |4> = |-++> |5> E |+—> |6> = |-+-> |7> = I—+> |8> = | — > ; | i > = | I ^ l V . [ D . l ] The energy s c a l e o f F i g u r e D.l i s c o m p l e t e l y a r b i t r a r y , and t h e r e l a t i v e m agnitudes o f t h e s c a l a r c o u p l i n g s have been e x a g g e r a t e d f o r v i s i b i l i t y . A t w e l v e - l i n e spectrum ( t h r e e q u a r t e t s ) i s o b s e r v e d , f o r w h i c h each l i n e c o r r e s p o n d s t o one o f t h e t w e l v e t r a n s i t i o n s shown i n t h e r i g h t - h a n d diagram f o r F i g u r e D . l . The problem w i t h t h e AMX spectrum i s t h a t f o r g i v e n magnitudes o f t h e t h r e e c o u p l i n g c o n s t a n t s , t h e r e a r e e i g h t , . p o s s i b l e energy diagrams w h i c h would y i e l d i d e n t i c a l s pectrum ( i d e n t i c a l l i n e p o s i t i o n s ) ; t h e diagrams d i f f e r a c c o r d i n g t o t h e c h o i c e o f a b s o l u t e s i g n o f each o f t h e t h r e e c o u p l i n g c o n s t a n t s (see l e f t - h a n d p o r t i o n o f F i g u r e D . l ) . F o r example, i f J ^ , J ^ , and a r e a l l p o s i t i v e , t h e l o w - f r e q u e n c y l i n e s i n each q u a r t e t c o r r e s p o n d t o r e s p e c t i v e A-,, M-,, and X-, t r a n s i t i o n s ; i f a l l t h e J ' s a r e n e g a t i v e , t h e same t r a n s i t i o n s now c o r r e s p o n d t o t h e h i g h -f r e q u e n c y l i n e s i n each q u a r t e t , and so f o r t h t h r o u g h a l l e i g h t combina-t i o n s . The p r i n c i p a l means f o r o b t a i n i n g r e l a t i v e and a b s o l u t e s i g n s o f c o u p l i n g c o n s t a n t s a r e from t h e a n a l y s i s o f 1 i n e p o s i t i o n s and i n t e n s i t i e s f o l l o w i n g e i t h e r d o u b l e i r r a d i a t i o n o r o r i e n t a t i o n i n an e l e c t r i c f i e l d o r i n a l i q u i d c r y s t a l ; we now show t h a t t h e same i n f o r m a t i o n can be o b t a i n e d ( i n p r i n c i p l e ) from s c r u t i n y o f t h e r e l a t i v e l i n e w i d t h s o f i n d i v i d u a l t r a n s i t i o n s o f an AMX spectrum. T h i s p r e s e n t e x p o s i t i o n i s -232-c l o s e l y a k i n t o a s i m i l a r problem o f s i g n d e t e r m i n a t i o n o f h y p e r f i n e c o u p l i n g c o n s t a n t s i n E l e c t r o n S p i n Resonance s p e c t r a . 7 As shown i n Appendix A, one can make t h e i d e n t i f i c a t i o n -RiJu • ( T 2 > ^ • CD-2] T h e r e f o r e , t o d e t e r m i n e t h e complete s p e c t r a l l i n e s h a p e f u n c t i o n , a t o t a l o f 12 r e l a x a t i o n m a t r i x e lements ( o f t h e p o s s i b l e 4096) must be c a l c u l a t e d , c o r r e s p o n d i n g t o t h e 12 t r a n s i t i o n s shown i n F i g u r e D . l . For t h e p r e s e n t d i s c u s s i o n , we assume o n l y d i p o l a r i n t e r a c t i o n s a r e o p e r a t i v e . D i p o l a r c o n t i b u t i o n s t o t h e l i n e w i d t h f o r each t r a n s i t i o n o f t h e AMX spec t r u m a r e l i s t e d i n T a b l e D.l and the f i r s t column o f T a b l e D.2 w i t h t h e n o t a t i o n l i s t e d b eneath t h e T a b l e s ( t h e s p e c t r a l d e n s i t i e s a r e d e f i n e d by E q u a t i o n s [2.2.14] and [ 3 . 1 . 3 ] ) . O n l y s i x t r a n s i t i o n s a r e g i v e n due t o t h e symmetry o f t h e r e l a x a t i o n m a t r i x e l e m e n t s : R i j k * = % i > j > k ^ J i l-D'3-1 where \\i i s t h e s p i n i n v e r s i o n o p e r a t o r ; ij>|l> = |8>, ip|2> = |7>, ii\3> = |6>, and ip|4> = |5>. R e l a t i o n [D.3] s t i l l h o l d s when c r o s s - c o r r e l a t i o n between  d i f f e r e n t d i p o l e - d i p o l e i n t e r a c t i o n s a r e i n c l u d e d . With t h e a i d o f F i g u r e D.l and T a b l e D . l , i t can be e a s i l y j u s t i f i e d t h a t even when c r o s s - c o r r e l a -t i o n s a r e n e g l e c t e d , t h e l i n e w i d t h s w i t h i n a g i v e n m u l t i p l e t a r e d i f f e r e n t ; f o r a g i v e n s e t o f r e l a t i v e s c a l a r c o u p l i n g s i g n s , t h e i n n e r two l i n e s o f a g i v e n q u a r t e t w i l l be b r o a d e r ( o r n a r r o w e r ) t h a n the o u t e r two l i n e s . I n c l u s i o n o f c r o s s - c o r r e l a t i o n terms m a g n i f i e s t h i s d i f f e r e n c e . P r a c t i -c a l l y s p e a k i n g , i f a w h i t e s p e c t r a l d e n s i t y i s assumed ( v a l i d f o r a l l but th e l a r g e s t m o l e c u l e s i n o r d i n a r y s o l v e n t s ) , and a l l d i p o l a r i n t e r a c t i o n c o n s t a n t s have s i m i l a r m a g n i t u d e s , the e x p e c t e d v a r i a t i o n i n l i n e w i d t h -233-between members o f a m u l t i p l e t s h o u l d be o f t h e o r d e r o f 20%. F u r t h e r , i f t he s p e c t r a l d e n s i t i e s f o r c r o s s - t e r m s a r e o n l y (1/4) as l a r g e as f o r a u t o - c o r r e l a t i o n terms (a " t y p i c a l " v a l u e ) , t h e l i n e w i d t h v a r i a t i o n i n -c r e a s e s t o about 30%. When d i p o l a r r e l a x a t i o n i s t h e o n l y r e l a x a t i o n mechanism p r e s e n t , o n l y r e l a t i v e s i g n s o f t h e J's a r e a v a i l a b l e . F o r example, by ex a m i n i n g t h e "A" q u a r t e t , t h e r e l a t i v e s i g n s o f and may be e s t a b l i s h e d ; by e x a m i n a t i o n o f an a d d i t i o n a l q u a r t e t , t h e r e l a t i v e s i g n s o f a l l t h r e e J's a r e d e t e r m i n e d (no a d d i t i o n a l i n f o r m a t i o n d e r i v e s from e x a m i n a t i o n o f a l l t h r e e q u a r t e t s r a t h e r than j u s t t w o ) . There a r e s p e c i a l c i r c u m -s t a n c e s f o r wh i c h t h e l i n e w i d t h v a r i a t i o n may be much g r e a t e r than i n d i -c a t e d above, as when t h e extreme-narrowed a p p r o x i m a t i o n f a i l s ( l a r g e m o l e c u l e s and/or v i s c o u s m e d i a ) , o r when t h e p a i r w i s e i n t e r a c t i o n c o n s t a n t s -3 -3 ar e v e r y d i f f e r e n t ( i . e . y y n r „ >> o r << y .y r . ) , o r when t h e r e i s an i S O -t r o p i c r e o r i e n t a t i o n and t h e r e l a t i v e o r i e n t a t i o n s o f t h e i n t e r n u c l e a r v e c t o r s s a t i s f y u n i q u e r e l a t i o n s h i p s w i t h r e s p e c t t o t h e p r i n c i p a l d i f f u -s i o n a x i s . We s h a l l not d i s c u s s f u r t h e r t h e s p e c i f i c e v a l u a t i o n o f t h e a p p r o p r i a t e c o r r e l a t i o n f u n c t i o n s o r elements o f R i n v a r i o u s l i m i t s , s i n c e t h e o b j e c t i v e o f t h i s Appendix i s m e r e l y t o c o n t r a s t t h e q u a l i t a t i v e i n f l u e n c e o f t h e two a f o r e m e n t i o n e d c a t e g o r i e s o f i n t e r f e r e n c e terms. A l t h o u g h o n l y t r a n s v e r s e r e l a x a t i o n i s d i s c u s s e d h e r e , a s i m i l a r a n a l y s i s c o u l d be a p p l i e d t o l o n g i t u d i n a l r e l a x a t i o n f o r each t r a n s i t i o n . The d i f f i c u l t y i s t h a t each t r a n s i t i o n w i l l no l o n g e r be c h a r a c t e r i z e d by a u n i q u e "T^', and no p r a c t i c a l i n f o r m a t i o n p e r t a i n i n g t o l o n g i t u d i n a l r e l a x a t i o n can be o b t a i n e d from knowledge o f any i n d i v i d u a l element o f R. I f o n l y one o f t h e t h r e e c o u p l e d n u c l e i i s r e l a x e d by an a d d i t i o n a l -234-i n t e r f e r e n c e pathway (as from c r o s s - c o r r e l a t i o n between d i p o l e - d i p o l e and c h e m i c a l s h i f t a n i s o t r o p y f o r a n u c l e u s o t h e r than a p r o t o n ) , t h e n i t i s q u i t e s t r a i g h t f o r w a r d t o show t h a t a b s o l u t e s i g n s o f a l l t h r e e s c a l a r c o u p l i n g s can be d e t e r m i n e d from t h e l i n e w i d t h s o f i n d i v i d u a l l i n e s i n the v a r i o u s q u a r t e t s . T h i s i s e v i d e n t w i t h t h e use o f t h e l a s t two columns i n T a b l e D.2 c o u p l e d w i t h t h e f a c t t h a t R.., „ = -R,., .,, ,„ f o r the s h i f t a n i s o t r o p y - d i p o l a r c r o s s - t e r m s ( i n t h i s T a b l e , i t i s assumed t h a t o n l y n u c l e u s "M" i s r e l a x e d by s h i f t a n i s o t r o p y i n t e r a c t i o n s : t h e s p e c t r a l d e n s i t i e s a r e d e f i n e d by E q u a t i o n s [ 2 . 2 . 1 4 ] , [ 3 . 1 . 3 ] , and [ 5 . 3 . 3 ] ) . The a b s o l u t e s i g n d e t e r m i n a t i o n o f c o u r s e depends on knowing t h e a b s o l u t e s i g n o f t h e c h e m i c a l s h i f t a n i s o t r o p y . In c o n c l u s i o n , whereas i n t e r f e r e n c e terms between d i f f e r e n t r e l a x -a t i o n mechanisms a l l o w f o r t h e d e t e r m i n a t i o n o f t h e a b s o l u t e s i g n s o f s c a l a r c o u p l i n g c o n s t a n t s , i n t e r f e r e n c e terms between l i k e r e l a x a t i o n mechanisms can a t b e s t d e t e r m i n e r e l a t i v e s i g n s o f J ' s . I n o r d e r t h a t t h e e x p e r i m e n t a l d e t e c t i o n o f d i f f e r e n c e s i n l i n e w i d t h s between members o f a p a r t i c u l a r AMX q u a r t e t n o t be masked by t h e p r e s e n c e o f f i e l d inhomo-g e n i t y , i n t e r m o l e c u l a r r e l a x a t i o n , o r s p i n - r o t a t i o n e f f e c t s , t h e l i n e w i d t h measurement would b e s t be c o n d u c t e d i n a d i l u t e s o l u t i o n o f t h e m o l e c u l e o f i n t e r e s t , i r v - a c d e u t e r a t e d s o l v e n t (such as g l y c e r o l - d g ) . 1. L. G. Werbelow and A. G. M a r s h a l l , Chem. Phys. L e t t . 22, 568 (1973). 2. H. S h i m i z u , J . Chem. Phys. 40, 3357 (1964). 3. J . M. A n d e r s o n , M o l . Phys. 8, 505 (1964). 4. E. L. Mackor and C. MacLean, J . Chem. Phys. 44, 64 (1966). 5. L. G. Werbelow and A. G. M a r s h a l l , J . Magn. Res. 11, 299 (1973). 6. L. G. Werbelow and A. G. M a r s h a l 1 , M o l . Phys. , ( 1 9 7 4 ) . 7. J . H. F r e e d and G. K. F r a e n k e l , J . Chem. Phys. 40, 1815 (1964). TABLE D . l 1 1 L i n e w i d t h c o n t r i b u t i o n s t o i n d i v i d u a l t r a n s i t i o n s f o r an AMX t h r e e s p i n system: P a r t 1. L i n e w i d t h element D i p o l a r c o n t r i b u t i o n s - J 2 E ( ^ ? ) • 1/2 [ J ^ U n ) + J ^ U { ) • j l ? ( „ n ) + • • j j ^ ) ] 0 2 3535 R 1 2 1 2 " ( ' / 3 > J r, c (V"t' + ^nc'VV R ' 3 1 3 - " 6 C 4 J > > + 4 J > > + J > n - 5 > + J « < V » S > ] " 4 W " J„VvV - J 2 ? U 5 + » { ) + l / 2 [ j l e ( B . ) + jj t(„ {) + 0 J e ( „ n ) + j j ^ ) + j l ? ( U { , + j j ^ , ] 0 2 R 2 5 2 5 R 1 3 1 3 " t l ' ' 3 ) J n ? ( V u E ) + ^ns'W - J n « ( < V \ > + VZC J j t ( » n ) - J 1 ^ ) + J j ? ( . n ) • o;{(.5) • 0 ^ ) • ] R 2 6 2 6 R 1 4 1 4 - 0 / 3 ) J ° 5 ( v » 5 ) + 2J 2 5(„ (.+ l„ {) ^ N o t a t i o n : Greek s u b s c r i p t s r e f e r t o n u c l e u s A ( n ) , n u c l e u s M (c), and n u c l e u s X (c). A r a b i c s u b s c r i p t s s u b s c r i p t s r e f e r t o s t a t i o n a r y s t a t e s (see Eq. [ D . l ] ) f o r an uncoupled r e p r e s e n t a t i o n . J k g ( w ) 1 S t h e s p e c t r a l d e n s i t y a t co f o r t h e d i p o l e - d i p o l e i n t e r a c t i o n between n u c l e i a and g . TABLE D.2 L i n e w i d t h c o n t r i b u t i o n s t o i n d i v i d u a l t r a n s i t i o n s f o r an AMX t h r e e s p i n system: P a r t 2. S h i f t a n i s o t r o p y c o n t r i b u t i o n s D i p o l e - s h i f t a n i s o t r o p y c r o s s - t e r m c o n t r i b u t i o n s 2 J 1 ( U ) 2 ^ ) - ( 8 / 3 ) J ° ( 0 ) + 2 j J ( ^ ) ( 8 / 3 ) J ° ( 0 ) + 2 J ^ ) ( 8 / 3 ) ^ ( 0 ) - ( 8 / 3 ) ^ ( 0 ) - 2 J 1 (a) ) + 2 J 1 U ) ,1 - 2 J 1 (a) ) ( 8 / 3 ) J ° n C ( 0 ) + ( 8 / 3 ) ^ ( 0 ) - 2 J 1 (to ) - 2 J 1 r ( a j ) 0 ,0 2 J ' U ) -2J 1 J t o ) 2 J J ( W ? ) L i n e w i d t h e l ement '1212 D i p o l e - d i p o l e c r o s s - t e r m c o n t r i b u t i o n s • (4/3 ) J ° (0) + J ^ U J ^3535 ( 4/3 ) JnW(0 ) - J n ^ ( a > ? ) '1313 '2525 '1414 - ( 4 / 3 ) J ° c ( 0 ) + J1 U ) '2626 ( 4 / 3 ) J ° AO) + J1 c(a> ) ^ N o t a t i o n : Same as f o r p r e c e e d i n g T a b l e w i t h t he f o l l o w i n g a d d i t i o n s ; J (to) i s the s p e c t r a l d e n s i t y a t a k to f o r t h e a u t o c o r r e l a t i o n f u n c t i o n o f t h e s h i f t a n i s o t r o p y i n t e r a c t i o n , Jv (to) i s t h e s p e c t r a l d e n s i t y ct py a t to f o r t h e s h i f t a n i s o t r o p y ( o p e r a t i v e a t n u c l e u s a)and the d i p o l a r ( o p e r a t i v e a t n u c l e i e and y) c r o s s t e r m s , and J a g y ( 5 ( u ) i s t h e s p e c t r a l d e n s i t y a t to f o r t h e d i p o l a r - d i p o l a r c r o s s c o r r e l a t i o n terms between t h e p a i r w i s e i n t e r a c t i o n s a-g and y-s . -237-FIGURE D J : The energy l e v e l d i a g r a m f o r an AMX t h r e e s p i n system. The r i g h t - h a n d p a r a l l e l o g r a m shows t h e v a r i o u s t r a n s i t i o n s w h i c h form t h e A, M, and X q u a r t e t s . S c a l a r c o u p l i n g removes t h e o t h e r w i s e d e g e n e r a t e t r a n s i t i o n s A-j, h^, A^, A^; M-j, M^, M^, M^; and X-|, X 2 , X^, X^. The e i g h t p o s s i b l e e nergy l e v e l diagrams ( s c a l a r c o u p l i n g i n c l u d e d ) a re shown i n t h e l e f t hand p o r t i o n o f t h i s f i g u r e . PUBLICATIONS A. Refereed Journals : 1. L.G. Werbelow, "Homonuclear Overhauser Enhancements as Probes of Molecular M o b i l i t y " , J . Amer. Chem. S o c , submitted. 2. L.G. Werbelow and A.G. Marsha l l , "Anisotropic Reorientat ion and Non-exponential Nuclear Magnetic Re laxat ion" , Mol. Phys., accepted and in press. 3. L.G. Werbelow and A.G. Marsha l l , "Nuclear Magnetic Relaxation for Individual Transit ions of an AMX Spectrum; Use of Interference Terms to Determine Signs of Scalar Coupling Constants", Chem. Phys. Let t . 22, 568-571 (1973). 4. L.G. Werbelow and A.G. Marsha l l , " Internal Rotation and Non-exponential Methyl Nuclear Relaxation for Macromolecules", J . Mag. Res. ]_]_, 299-313 (1973). 5. I.M. Armitage, L.D. H a l l , A.G. Marsha l l , and L.G. Werbelow, "Determination of Molecular Configurat ion from Lanthanide Induced Proton NMR Chem-i ca l S h i f t s " , in Nuclear Magnetic Resonance Sh i f t Reagents, Academic Press Inc . , 1973, pg. 313-339. 6. L.G. Werbelow and A.G. Marsha l l , " Internal Rotation and Methyl Proton Magnetic Relaxation for Macromolecules", J . Amer. Chem. Soc. 95, 5132-5134 (1973). 7. I.M. Armitage, L.D. H a l l , A.G. Marsha l l , and L.G. Werbelow, "Use of Lanthanide Nuclear Magnetic Sh i f t Reagents in Determination of Molecular Conf igurat ions" , J . Amer. Chem. Soc. 95, 1437-1443 (1973). 8. M.P. Hanson and L.G. Werbelow, "Co l l inear Co l l i s i ons of an Atom and Str ing O s c i l l a t o r " , J . Chem. Phys. 58, 3669-3675 (1973). 9. A.G. Marsha l l , L.G. Werbelow, and C F . Meares, "E f fect of Molecular Shape and F l e x i b i l i t y on Gamma-Ray Direct iona l Cor re l a t i ons " , J . Chem. Phys. 57_, 364-370 (1972); J . Chem. Phys. 57, 4508 (1972). 10. L.G. Werbelow, " Interference Effects in Nuclear Magnetic Re laxat ion" , Adv. Mol. Relax. Processes, in progress. 

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