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

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NOVELTIES A S S O C I A T E D WITH A BIODYNAMICAL INTERPRETATION OF NUCLEAR S P I N RELAXATION  by  LAWRENCE GLEN WERBELOW B. S., H u m b o l d t S t a t e U n i v e r s i t y , 1 9 7 0  A T H E S I S SUBMITTED IN PARTIAL F U L F I L M E N T OF THE REQUIREMENTS  FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY  i n t h e department of CHEMISTRY  We a c c e p t  tfhis  tjiesisOas confxjrming-tO/the r e q u i r e d  THE U N I V E R S I T Y OF B R I T I S H A u g u s t , 1974  COLUMBIA  standard  In p r e s e n t i n g  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 r e q u i r e m e n t s tot  an advanced degree a t the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t 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 r e f e r e n c e and  I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e f o r s c h o l a r l y purposes may by h i s r e p r e s e n t a t i v e s .  study.  copying of t h i s  be g r a n t e d by the Head of my  Department or  I t i s understood t h a t c o p y i n g 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 g a i n s h a l l not be a l l o w e d w i t h o u t written permission.  Department of  Chemistry  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  thesis  my  -i  i-  ABSTRACT  Employing t h e s e m i c l a s s i c a l dynamic p r o c e s s e s ,  form o f t h e d e n s i t y o p e r a t o r theory o f  the t r a n s i e n t nuclear spin behavior  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  t a t i o n s o f nuclear magnetic r e l a x a t i o n results  data.  Only r e l a x a t i o n  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 modulation  couplings  i s considered.  The b a t h  correlation  t h e t h e o r y a r e a s s u m e d t o be c o m p l e t e l y motional  constants  The  which  of the various spin into  c h a r a c t e r i z e d by two u n i q u e  (a dynamic symmetric t o p a p p r o x i m a t i o n ) . o f s p i n - s p i n and s p i n -  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 , a n d O v 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 rationalized  p a r a m e t e r d e p e n d e n t upon t h e s p e c t r a l  t h a t i n g e n e r a l , any  density a t zero frequency  pendent o f t h e magnitude o f t h e asymmetry i n t h e m o t i o n . meters  interpre-  f u n c t i o n s which enter  e f f e c t s o f slow, a n i s o t r o p i c modulation  molecule  i s analyzed f o r  independent o f the near-zero  are s e n s i t i v e  solely  frequency  spectral  t o t h e magnitude o f t h e motional  i s inde-  Likewise,  para-  component o f t e n asymmetry.  These c o n s i d e r a t i o n s a r e e x 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 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 functions. 1^ i n s u c h  a case  molecular  a u n i q u e T-j o r  as t h e p r e d i c t e d d e c a y i s m u l t i - ( n o n ) e x p o n e n t i a l .  i n such  a situation  p r e d i c t i o n s are presented.  spectral  inter-  interference or cross-correlation  I n g e n e r a l , one c a n n o t d e f i n e u n a m b i g i o u s l y  spin behavior guing  p r o c e e d s by c o m p e t i n g  density approximation correlation  i s thoroughly  analyzed  a n d many  intri-  I t i s seen t h a t t h e f a i l u r e o f a  or a single exponential  The  white  decay o f t h e  f u n c t i o n o f t e n leads t o p r e d i c t i o n s o f extreme  -i i i-  nonexponentiality these  calculations  ent f o r those Finally, potentially is  of the magnetization are of general  concerned  powerful  concepts  While  i n t e r e s t , they  with biological  the i n i t i a l  decay.  applications of  data  figures  of the quadrupolar  i s emphasized t h a t a conventional  i s impregnated  with hidden,  i n general  interpretation of  be i n f l u e n c e d  relaxation  "extreme-narrowed", reasoning.  are provided which not only f a c i l i t a t e  w h i c h m u s t be e x e r c i s e d i n a n y b i o d y n a m i c a l relaxation.  where i t  perturbation.  experimental  of the c a l c u l a t i o n s , but also provide s t r i k i n g evidence  spin  pertin-  NMR.  C o r r e l a t i o n experiment  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  It  are e s p e c i a l l y  are extended t o a d i s c u s s i o n o f the  Perturbed Angular  by a n i s o t r o p i c m o d u l a t i o n  the results of  Extensive  application  f o r the caution  interpretation  of nuclear  - i v-  TABLE OF  CONTENTS  Page  ABSTRACT  -  i i  TABLE OF CONTENTS  iv  L I S T OF T A B L E S  v i i  L I S T OF FIGURES  viii  COMMENTS ON UNITS AND NOTATION  x  ACKNOWLEDGEMENTS CHAPTER I .  CHAPTER I I .  x i i  GENERAL INTRODUCTION  1  REFERENCES: CHAPTER I  7  NUCLEAR MAGNETIC RELAXATION IN L I Q U I D S : A COMPUTATIONAL GROUNDWORK  8  2.1.  INTRODUCTION  8  2.2.  THE S E M I C L A S S I C A L 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 CHAPTER I I I .  38  ANISOTROPIC MOLECULAR MOTIONS AND THE  NMR  RELAXATION OBSERVABLES  42  3.1.  INTRODUCTION  3.2.  EFFECT OF ANISOTROPIC MOTIONS ON T-, AND T  2  VALUES  45  3.3.  EFFECT OF ANISOTROPIC MOTIONS ON T  £  RATIOS ....  56  3.4.  EFFECT OF ANISOTROPIC MOTIONS ON HOMONUCLEAR OVERHAUSER  3.5.  ENHANCEMENTS  SUMMARY REFERENCES: CHAPTER I I I  42  ]  AND T  64 77 81  -v-  CHAPTER I V .  DIPOLAR RELAXATION OF T H R E E - S P I N 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  CHAPTER V.  119  REFERENCES: CHAPTER IV  121  INFLUENCE OF F I N I T E CROSS-CORRELATION TERMS BETWEEN P H Y S I C A L L Y D I S T I N C T 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 CHAPTER V I .  EFFECT OF MOLECULAR SHAPE AND F L E X I B I L I T Y GAMMA-RAY DIRECTIONAL CORRELATIONS  169 ON 172  6.1.  INTRODUCTION  6.2.  PERTURBED ANGULAR  6.3.  ANISOTROPIC MOTION I N THE ABRAGAM-POUND L I M I T  181  6.4.  ANISOTROPIC MOTION I N THE A D I A B A T I C L I M I T  183  6.5.  RESULTS AND DISCUSSION  186  6.6.  COMPARISON OF NMR AND PAC  202  CHAPTER V I I .  172 CORRELATIONS  174  REFERENCES: CHAPTER V I  206  CONCLUDING REMARKS  208  -vi-  APPENDIX A.  THE RELAXATION MATRIX  210  APPENDIX B.  INFLUENCE OF SECOND ORDER FREQUENCY S H I F T TERMS ••••  215  APPENDIX C.  QUADRUPOLAR RELAXATION OF S P I N 3/2 NUCLEI  224  APPENDIX D.  NUCLEAR MAGNETIC RELAXATION FOR INDIVIDUAL TRANS I T I O N S OF AN AMX SPECTRUM: USE OF INTERFERENCE TERMS TO DETERMINE SIGNS OF SCALAR COUPLING CONSTANTS 230  -vii-  L I S T OF  Table  3.1.  3.2.  5.1.  TABLES  Title  Page  Relaxation Part I.  parameters i n various  motional  Relaxation Part I I .  parameters i n various motional  limits: 79 limits: 80  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 h r o n o u s 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 t e r m c h a r a c t e r i s t i c s .  168  D.l.  Linewidth contributions to individual 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 t e m : D i p o l a r c o n t r i b u t i o n s .  235  Linewidth contributions to individual 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 t e m : 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  D.2  •vi n L I S T OF  Figure  2.1.  FIGURES  Title  Page  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 ation observables.  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^  3.2.  Spectral  densities  a b o u t an a x i s  a n d T^  f o r a two-spin system  relax37 values.  53  rotating  p e r p e n d i c u l a r to the i n t e r n u c l e a r  vector.  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^  ratios.  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  ratios.  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 / ^ -  3.6.  H o m 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 as a f u n c t i o n isotropic mobility.  3.7.  3.8.  Effect of anisotropic enhancements. Independence  m o t i o n on h o m o n u c l e a r  of homonuclear  on t h e m a g n i t u d e  63 of 72  Overhauser 74  Overhauser  enhancements  of motional anisotropy.  4.1.  Nonexponential d i p o l a r r e l a x a t i o n  4.2.  Dissection  4.3.  Contour p l o t s o f methyl group decay parameters f o r 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 decay.  116  C o n t o u r p l o t s o f methyl group decay the 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  4.4.  5.1.  5.2.  5.3.  f o r the methyl group.  76  of the methyl 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 .  112 114  parameters f o r  D e c a y p a 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 anisotropy-dipolar i n t e r a c t i o n constants: Extreme-narrowing approximation.  159  Decay p a 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 anisotropy-dipolar interaction constants.  161  D e c a y p a 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 of 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 interference terms).  163  -ix-  Figure  5.4.  Title.  Page  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 anisotropy cross-terms) longitudinal relaxation.  165  6.1.  A typical nuclear de-excitation  178  6.2.  Angles defined experiment.  cascade.  i n the Perturbed Angular  Correlation 180  111m  6.3.  Observable anisotropy ( f o r overall mobility.  6.4.  Observable anisotropy ( f o r i n t e r n a l m o b i l i t y : Fast motion  Cd)  as a f u n c t i o n  of  Cd) as a f u n c t i o n limit.  of  Cd) as a f u n c t i o n limit.  of  193  111m  195  111m  6.5.  Observable anisotropy ( f o r i n t e r n a l m o b i l i t y : Slow motion  6.6.  D e c a y scheme f o r  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 mobility.  201  B.l.  E f f e c t of second-order s h i f t  221  B.2.  Graphic comparison of r e l a x a t i o n , l i n e w i d t h , second-order c o r r e c t i v e terms.  D.l.  1 1 1  Energy l e v e l diagram  I n and  1 1 1 m  Cd.  197 199  corrective  f o r an AMX  spin  terms.  system.  and 223 238  -X-  COMMENTS ON  The  International  used throughout it  i s useful  which  UNITS AND  System o f u n i t s  this thesis.  densities  and  c o n v e n t i o n a l NMR  are f u l l y  to summarize h e r e , the p a r t i c u l a r  are employed t o c l a s s i f y  seconds"^),  Deviants  NOTATION  rotational  (units:  the r o t a t i o n a l  correlation  seconds"  ) and  times  notation i s  explained.  notational diffusion  (units:  idiosyncracies constants  seconds),  angular correlation  However,  (units:  spectral  functions  (units:  _ o  seconds  ).  Symbol  D  Isotropic diffusion  D  D  Definition  ±  )(  D. . int  *  D. 1  rotational tensor).  diffusion  Page n o t a t i o n f i r s t appears  constant  (scalar 26  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 sion p e r p e n d i c u l a r to the p r i n c i p a l d i f f u s i o n a x i s ) .  29  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 sion p a r a l l e l to the p r i n c i p a l d i f f u s i o n a x i s ) .  29  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 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 f r a m e w o r k .  internal 30  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 r o t o r a t t a c h e d to a symmetric top framework.  internal 33  Tg,x.j  A general, unspecified  i2  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 time of a rank s p h e r i c a l harmonic: = 1/6D.  second  T  C  Symmetric top r o t a t i o n a l  correlation  time:  = 1/6D^. •••  x  R  Symmetric top r o t a t i o n a l  correlation  time: = l/(4D-4Dj.  47  T  R I  Symmetric top r o t a t i o n a l  correlation  time:  68  rotational  correlation  time.  27,50  = 1/(D -D ). h  x  47  -xi-  Symbol  C  k& ? r )  Definition  (t)  Page n o t a t i o n f i r s t appears  The t i m e c o r r e l a t i o n f u n c t i o n uht)  g  and U ( t ) : £  o f two v a r i a b l e s  H <U (t)U (0)>. k  16  , l  A normalized c o r r e l a t i o n function: x <C  jj^-U)  K J t  (t)>/<C '(0)>.  27  K X  ?n  ?n  The s p e c t r a l  ^  = (-1)  of C  I/O  m (1/2)/  representation  C  k £  (t):  E  ^  N  -(t)exp(iujt)dt.  16  jj^(cj)  The s p e c t r a l  representation  of g ^ ( t )  J^(co)  The s p e c t r a l  representation  of  J*  The s p e c t r a l  representation  o f (1 - 6 £ ) C * * ( t ) .  (u)  6 Cn  27  c£j-(t).  93  93  n  The H i l b e r t T r a n s f o r m o f J ^ w ) . -  Q^(CJ)  15  One l a s t comment o n 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 . q|^>) The H i l b e r t T r a n s f o r m o f j^-(u>). throughout t h i s t h e s i s ^ 2' T  the this  T  c ' R' T  scale  o  r  T  R'^  o  v  e  r  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 |(  m  a  n  ^  t h e s i s assume t h e s e c o n d  °^  t  n  1  s  variable.  are relevant  In such a c a s e ,  As a l l t i m e s c a l e s  as t h e f u n d a m e n t a l  D..^  used i n  unit of time, only  mag-  and hence t h e j u s t i f i c a t i o n o f  notation.  * Internal  r o t a t i o n on a s p h e r i c a l  symmetric and D = x  is  decades  r e a d s L o g ( D ) o r some f a c s i m i l e .  nitudes o f these q u a n t i t i e s the  Many p l o t s 218  top anisotropic  D i s made.  simply referred  Also,  motion  framework i s f o r m a l l y  identical to  i f the i d e n t i f i c a t i o n D  n  - D= ±  a t times i n t h i s t h e s i s , the quantity  t o as t h e a n i s o t r o p y  o f the motion.  D-  nt  D^ - D^  -xi i-  ACKNOWLEDGEMENTS  I would l i k e t o extend  my a p p r e c i a t i o n t o D r . 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 c o n t i n u i n g guidance  and encouragement t h r o u g h o u t  University of British B l o o m , L. H a l l ,  The ship  Columbia.  a n d B. D u n e l l  ledge which they support  i n this  have  I a l s o wish  t h e s i s , and f o r  my t e n u r e  at the  t o t h a n k P r o f e s s o r s M.  f o r examples they  h a v e s e t a n d t h e know-  imparted.  of a University  of British  Columbia Graduate  Scholar-  (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 wish  wishes  t o express  my w a r m e s t  (without public explanation) t o the following i n f l u e n c e s : To A l a n a n d M a r i l y n M a r s h a l l . To  Laurie Hall,  To  Merv  Diane.  To  Stephanie.  Finally, British  Hanson.  To  To my  P e t e r L e g z d i n s , Ben M a l c o m b , a n d I a n A r m i t a g e .  parents.  a m e a s u r e o f a c k n o w l e d g e m e n t m u s t be e x t e n d e d t o t h e n e a r b y  C o l u m b i a c o u n t r y s i d e : "Heaven i s u n d e r o u r f e e t as w e l l  o u r h e a d s " , and t o Henry T h o r e a u whose words and t h o u g h t s me many a t i m e .  have  as o v e r nurtured  -1CHAPTER I GENERAL INTRODUCTION  In  a p e r i o d spanning  R e s o n a n c e (NMR) molecular  has  lent  information.  by w h i c h one  itself Simply  twenty-five years, Nuclear  to the task of probing s t a t e d , NMR  probes the m o l e c u l a r  spin Hamiltonian. of a small  the l a s t  i s the experimental  b i l i n e a r couplings  0  +  i-( b + j  )-r  +  I-C  where the 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 c o u p l i n g , the  s p i n - s p i n c o u p l i n g , the  to f i r s t order, £  will  be  0 +  [1.1]  zero time  i s a small  +£(t)  [1.1]  are the s p i n - e x t e r n a l  i n d e p e n d e n t o f t i m e , t h e r e do  ;  exist  Hence,  ..  intensities.  .  [1.2]  ..  ( f o r c o n v e n i e n c e ) t o have a  0.  h i g h r e s o l u t i o n NMR  i n t e r p r e t a t i o n of l i n e  s t a n t s ) and  field  s p i n - a n g u l a r momentum c o u p l i n g , e t c .  p e r t u r b a t i o n assumed  a v e r a g e , <6?(t)> =  Conventional the  (ignoring  i s o f t e n w r i t t e n as  € = 6*0 w h e r e €{t)  sum  ....  s m a l l , time-dependent c o n t r i b u t i o n s o f paramount importance. Equation  nuclear  coefficients),  £• = i - ( I - S > ) - B  Although  of  technique  i s i n s t r u c t i v e l y w r i t t e n as a  number o f p h y s i c a l l y d i s t i n c t  multiplying  a v a s t range  d e t a i l s embodied w i t h i n the  This Hamiltonian  Magnetic  s t u d i e s have h i s t o r i c a l l y  p o s i t i o n s (chemical A n o t h e r way  s h i f t s and  of s t a t i n g  this fact  relied  coupling i s to  con-  say  on  -2-  t h a t only the obtained and  the  spectral properties of £  concerns the  t r a c e of the  " t r a c e l e s s " aspect of these tensors a n a l y s i s of lineshape,  in principle  contains  s p i n - s p i n and reservoir of  the  spin-molecule  This  exploratory  t r e n d towards u t i l i z a t i o n  i n NMR  s t u d i e s , but  inelastic dielectric At  an  on  the  i n other  p h o t o n and  Indeed, t h i s  susceptibility  chemist.  presents  a  d e t r a c t from the  s t u d i e s s u c h as  research  activities  Justification to provide  confrontations  This  f o r the  content  "life"  and and  developed  on  a a  of t h i s  molecular, recent not  field. thesis i s therefore  two-fold:  i n t e r p r e t a t i o n of t r a n s i e n t nuclear researches  spin into  systems.  Relevant background m a t e r i a l excellent monographs  1 - 5  and  can  reviews  6 - 1 0  be  in  large  t r e n d , stemming f r o m  a means t o i l l u m i n a t e c o n t e m p o r a r y NMR  complex b i o c h e m i c a l  only  experiments,  ( o r w h a t e v e r ) c e r t a i n l y does  interest in this  i n s i g h t i n t o the  ( 2 ) as  that  elastic  u n d e r t a k e n by  academicians, studies of  inherent  tapped  i s i n vogue not  s c a t t e r i n g , IR a b s o r p t i o n  submolecular, or philosophical plateau. social-economic-moral  the  large  j u s t r e c e n t l y been  of s p e c t r a l lineshape  of  e x p l o i t a t i o n of  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  number o f p h y s i o - c h e m i c a l  behavior,  nature  measurements.  recent years r e l a t e s to the  (1)  ignored.  I t m i g h t be m e n t i o n e d  spectroscopic  particle  [1.1]  i n Equation  complete t e n s o r i a l  a v e n u e o f a p p r o a c h has the  information  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 ,  couplings.  f a s h i o n by  only  is inadvertently  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  seem m e a g e r .  i n an  information  u t i l i z e d ; the  i n t e r a c t i o n tensors  However, the  NMR  are  Q  located  i n any  one  which t r e a t both the  of  several  general  -3-  t h e o r y o f NMR Additional  relaxation  and  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  number o f r e c e n t a r t i c l e s h e a d e d by A. B.  Sykes  Chan ( C a l T e c h ) ,  to l i s t .  in a large  of the F.  groups  Noack  (Stuttgart),  This  i s only  i n the f i e l d ,  t h e num-  c o n t r i b u t i o n s w o u l d e a s i l y demand a n o t h e r  Furthermore,  a number o f c o n f e r e n c e  magnetic resonance a p p l i e d to b i o l o g i c a l ample, see  problems!  found  the b i o p h y s i c a l group a t Cambridge.  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  ber making s i g n i f i c a n t two  be  stemming f r o m t h e r e s e a r c h e s  A l l e r h a n d ( I n d i a n a ) , S.  ( H a r v a r d ) , and  a sampling  i t s a p p l i c a t i o n s to biochemical  proceedings  systems are  page  concerning  in print.  (For  r e l a x a t i o n data y i e l d s  both  static  information  ( b o n d 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 the ground s t a t e n u c l e a r  vironment,  field  electric  g r a d i e n t s , e t c . ) and  physical kinetics).  and  concerned  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  k i n e t i c s ) and situations and  anisotropy molecular  In p a r t i c u l a r , t h i s  the r e l a x a t i o n behavior covered  related  by  such  relaxation  i n the magnetic rotational  o f an  thesis  i s only (physical  ensemble of n u c l e a r s p i n s .  an a p p r o a c h a r e t h o s e w h e r e t h e parameters are determined  interactions  diffusion.  (We  en-  dynamic i n f o r m a t i o n  (chemical  breadth  ex-  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  The  or  (Equation  line  s o l e l y by  [1.1]) coupled  also dismiss translational  the  to  the  contri-  butions. ) The  content  theoretical  of t h i s  i n nature  under the  heading  warranted  by  but  thesis  is neither s t r i c t l y  strikes  "computational".  experimental  a medium w h i c h c o u l d b e s t be The  need f o r s u c h  an  the l a c k of communication between m u t u a l l y  nor catalogued  approach i s skeptical  seg-  -4-  ments o f the s c i e n t i f i c treatments  community.  of spin r e l a x a t i o n  On  one  hand, t h e r e e x i s t s  p r o b l e m s , y e t on  o f c o n t e m p o r y b i o l o g i c a l l y o r i e n t e d NMR  elegant  the o t h e r hand, a p e r u s a l  relaxation  literature  f i n d s much  of the data  i s i n t e r p r e t e d i n terms o f advances i n the t h e o r y t h r o u g h  year  For the most p a r t , t h i s u n s e t t l i n g  1955!  the f a c t  t h a t one  calculation  o f two  explicitly  assumptions  situation  is invariably  U n f o r t u n a t e l y , one  priori  be  in a typical  a s s u m p t i o n s may  assumptions being,  natural  this  both o f these third  (1) t h a t a u n i q u e t i m e  assumptions f a i l .  We  shall  of t h i s  thesis will  d i s c u s s i o n or other i n t e r p r e t a t i o n a l  tivity icisms.  experimental  and  of great potential  these char-  (2)  that  s h o r t compared w i t h  the  spin.  a l s o dismiss the notion of  a i d now  known i s t h e f a c t  are well  magnetization. (and  often  very  relevant  two  known, l a c k o f of the  sensi-  typical  t h a t t h e r e i s an a n a l o g o u s  correlation  a  exist.  f o r the most p a r t , c i r c u m v e n t s  p e r t u r b a t i o n of the angular  or  completely  i n s i t u a t i o n s where l i t t l e  l i m i t a t i o n s o f NMR  which,  constant  provide meaningful  c o s t l y , complex i n s t r u m e n t a t i o n are  Less w e l l  use-  a  completely  decay of a n o n e q u i l i b r i u m  general) p r e d i c t i o n s of s p i n behavior  The  two  study;  o r i e n t a t i o n , and  commonly a d v a n c e d 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  Hence, the c o n t e n t  any  e x p l o r e t h e p r e d i c t e d s p i n b e h a v i o r when o n e ,  c h a r a c t e r i z e s the macroscopic  The  constant  s c a l e of the precessing  t h e s i s , we  of these  biodynamical  o r i e n t a t i o n s e x i s t f o r a time  (Larmor) time  In  invalid  the m i c r o s c o p i c decay of m o l e c u l a r  r e l a t i v e molecular  traced to  s o l v e d to p r o v i d e a c o m p e l l i n g example of the advances.  acterizes  be  introduced into  fulness of t h e o r e t i c a l  two  can  the  these  (PAC), which e x i s t s  crittechnique  handicaps. between  a p a i r of 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 cascade, d i -  -5-  rectly  reflects therelaxation of a polarized  a strong  external  Zeeman f i e l d  i sessential  spin  ensemble."  Whereas  7  t o generate a polarized  e n s e m b l e 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 "polarized  by o b s e r v a t i o n " .  Observation o f thef i r s t  ensemble o f n u c l e i which i s c h a r a c t e r i z e d of magnetic substates. polarization. for  population  perturb this  transient  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  interpreted  respect to thef i r s t  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  Often these r e l a x a t i o n  to m u l t i p l e s  NMR p a r a m e t e r s .  of conventional  i sconcerned with  y-ray  y-^ay c a n t h e n be q u a n -  nuclear relaxation factors.  thesis  s e l e c t s an  by a n o n - u n i f o r m  Nuclear relaxation w i l l  any g i v e n a n g l e w i t h  titatively  y-ray  f a c t o r s may r e d u c e  Although t h e bulk o f t h i s  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  (useful  -12 experimental justifies  c o n c e n t r a t i o n s r a n g e down t o 10  theattention  (unfortunately  M) i n b i o c h e m i c a l  studies  cursory) given t h i s topic i n  Chapter V I . In g e n e r a l ,  theoutline of this thesis  i squite  straightforward.  Chapter 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 molecular  interpretation t o the behavior of theobservable nuclear Chapter I I I develops very general reduction  of relaxation  concepts which g r e a t l y  parameters f o rmotional  extensions of conventional  interpretations  lead  the as  influence  a i d i n the  regimes where n a i v e t o hopeless  The n e x 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 phenomenon, t h e c r o s s - c o r r e l a t i o n  paramagnetism.  or interference  contradictions.  specific relaxation  problem.  Although  o f s u c h t e r m s h a s been, r e g a r d e d f o r t h e l a s t d o z e n y e a r s  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 developed i n  -6these  two c h a p t e r s  practical problem  potential  III  show t h a t b i o c h e m i c a l  studies present a very  f o r posing e i t h e r a discouraging  o r an avenue f o r a d d i t i o n a l  familiarity terms.  clearly  interpretational  i n s i g h t depending on t h e ( u n )  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  Finally,  the l a s t chapter extends  t o t h e PAC e x p e r i m e n t .  a self-contained topic.  A s i d e from  the ideas developed  Chapter  Thus i t i s f e l t  I I , each chapter  treats  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 e a c h i n d i v i d u a l than a cumulative b i b l i o g r a p h y .  i n Chapter  chapter rather  -7-  REFERENCES: CHAPTER I  1.  A. A b r a g a m , T h e P r i n c i p l e s o f N u c l e a r M a g n e t i s m , O x f o r d , 1961.  2.  A. C a r r i n g t o n a n d A. D. M c L a c h l a n , H a r p e r - R o w , New Y o r k , 1 9 6 7 .  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 R e s o n a n c e , H a r p e r - R o w , New Y o r k , 1963.  4.  N. B l o e m b e r g e n ,  5.  C. P. P o o l e a n d H. A. F a r a c h , R e l a x a t i o n A c a d e m i c P r e s s , New Y o r k , 1 9 7 1 .  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 e r . B u n s e n g e s . P h y s .  8.  G. C. L e v y , A c c o u n t s  9.  E. D. B e c k e r , R. R. S h o u p , a n d T. C. F a r r a r , P u r e A p p l . Chem. 3 2 , 51 ( 1 9 7 2 ) .  Nuclear Magnetic  Clarendon Press,  Introduction t o Magnetic  Resonance,  R e l a x a t i o n , B e n j a m i n , New Y o r k , 1 9 6 1 .  Chem. R e s . 6,  i n Magnetic  Resonance,  Chem. 7 5 , 229 ( 1 9 7 1 ) .  161 ( 1 9 7 3 ) .  —  10. J . R. L y e r l a  a n d D. M. G r a n t , MTP I n t e r n a t i o n a l  11. 0. J a r d e t z k y , A d v a n . Chem. P h y s .  R e v . S c i . 4, 155 ( 1 9 7 2 ) .  7., 4 9 9 ( 1 9 6 4 ) .  12. A. S. M i l d v a n a n d M. C o h n , A d v a n . E n z y m o l o g y 33_, 1 ( 1 9 7 0 ) . 13. R. A. Dwek, A d v a n . 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. G u r d  a n d P. K e i m , M e t h o d s E n z y m o l o g y 2 7 , 8 3 6 ( 1 9 7 3 ) .  15. G. A. G r a y , CRC C r i t .  R e v . B i o c h e m . 3 , 247 ( 1 9 7 3 ) .  16. T. R. K r u g h , 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 : T h e o r y a n d 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 , A c a d e m i c P r e s s , New Y o r k , 1 9 7 4 . 17. C. F r a n c o n i , e d i t o r , M a g n e t i c R e s o n a n c e i n B i o l o g i c a l and B r e n c h , New Y o r k , 1 9 7 1 .  R e s e a r c h , Gordon  18. V o l u m e 2 2 2 o f t h e P r o c e e d i n g s o f t h e New Y o r k Academy o f S c i e n c e s , 1 9 7 3 . 19. D. A. S h i r l e y  a n d H. H a a s , A n n . R e v . P h y s .  Chem. 2 3 , 3 8 5 ( 1 9 7 2 ) .  -8-  CHAPTER I I NUCLEAR MAGNETIC RELAXATION IN L I Q U I D S : A COMPUTATIONAL GROUNDWORK  2.1.  INTRODUCTION A collection  of N like  spins  placed  i n a constant  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  magnetic  field  e q u i l i b r i u m mag-  netization,  M  eq  =  M<I > Z  T  = N fi( 1iI(I+l)B /3kT) Y  Y  where y i s t h e gyromagnetic r a t i o value  o f t h e n u c l e i and I  o f t h e z component o f t h e n u c l e a r  n o t a t i o n used throughout t h i s expression roundings.  i s valid  [2.1.1]  0  spin.  t h e s i s , i s w r i t t e n t o emphasize t h a t  f o r a s p i n system i n thermal  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  t o as n u c l e a r magnetic r e l a x a t i o n .  customarily transverse  i s the expectation  The s u p e r s c r i p t "T", a  harmony w i t h  disturbance,  this  i t s sur-  o f nuclear magnetic  a p p r o a c h t h e r m o d y n a m i c harmony a f t e r some i n i t i a l referred  z  dipoles  i s simply  The r e l a x 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 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  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  of the macroscopic magnetization perpendicular  (transverse  parallel  ( l o n g i t u d i n a l r e l a x a t i o n ) and  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 ,  S u f f i c e to say that the disturbance subsequent decay o f i t s orthogonal  of M  BQ.  from e q u i l i b r i u m and t h e  c o m p o n e n t s c a n be f o l l o w e d  by a  great  -9-  v a r i e t y o f experimental In  techniques.  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  as n u c l e a r s p i n c o m m u n i c a t i o n .  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 longitudinal within  though a s i n g l e laxation  surroundings* are manifested i n  r e l a x a t i o n whereas mechanisms which  t h e system  of spins are manifested  time constant w i l l  (d/dt)B -M(t) Q  redistribute information  i n transverse relaxation. A l -  r a r e l y describe p r e c i s e l y these r e -  (communication) p r o c e s s e s , such  duced as a v e r y good e m p i r i c a l ,  relaxation  a time constant  i fnot t h e o r e t i c a l ,  = -X(B -M(t) - M Q  e q  i s often  parameter:  }B |)  [2.1.2]  Q  ( d / d t ) | B x M ( t ) | = -X'|B X M ( t ) |  [2.1.3]  n  w h e r e x a n d A' a r e t h e f a m i l i a r NMR  intro-  relaxation  r a t e s 1/T^ a n d l / T ^ r e -  spectively. Although of  spin  the conceptual  relaxation  foundations  date back t o t h e e a r l y  p a p e r by B l o e m b e r g e n , P o u n d , a n d P u r c e l l expressed  ( a n d much o f t h e t e r m i n o l o g y )  1  1 9 3 0 ' s , i t was t h e f a m o u s i n the late  the e v o l u t i o n of the macroscopic  1940's w h i c h  longitudinal  first  and t r a n s v e r s e  nuclear magnetizations  ( i . e . T-| a n d T^) a s a f u n c t i o n o f m i c r o s c o p i c  molecular  In t h e i r approach,  parameters.  which  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 degrees o f freedom a r e o f t e n l o o s e l y r e ferred  t o as t h e " l a t t i c e "  o t h e r than  or the "bath".  the spins, the l a t t i c e .  orientations  as w e l l  We w i l l  Thus m o l e c u l a r  call  everything  p o s i t i o n s and  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  coordinates.  spin  1/2  n u c l e i , i t was  transitions action.  assumed t h a t  between the  Time dependent p e r t u r b a t i o n p r o b a b i l i t y per  stochastic  nature of  the  unit  t h e o r y was  The  physical  picture  p r o d u c e d by l e v e l s and  by  drives  the  nuclear spin  the  BPP  c a l c u l a t i o n s ; two  it  t h o s e by the  suffers  A b r a g a m and h e a r t of  the  relaxation  t h e o r y , has  BPP in  incorporated  Pound  trans-  parameterizing  s u c h an  the  a p p r o a c h was  exfield  theory, often  formed the  basis  equilibrium.  referred  on  to  in  f o r many f u n d a procedure  3 and  Solomon.  conceptual  Although the  foundation of magnetic defects".  In the  BPP  theory  relaxation,  e n s u i n g y e a r s , many  nuclear magnetic r e l a x a t i o n  have g r a c e d  the  the l i t e r a t u r e . concentrated t h e i r attention  a form p a r t i c u l a r l y w e l l  which l i q u i d s possess. have p r o v e n f r u i t f u l  suited  o f m a t t e r s u c h as or  the  the  on  l i q u i d s and  to the  cast  s t r u c t u r a l and  their  analysis  dynamic  disorder  Since t h i s time, quite  d i f f e r e n t , general approach 4 5 for simple molecular s o l i d s and d i l u t e g a s e s . Yet  other s p e c i a l i z e d approaches p r e d i c t  materials  The  i n the  noteworthy papers which exemplify t h i s  f r o m many " q u a n t i t a t i v e  more e l a b o r a t e t h e o r i e s pages of  the  Zeeman l e v e l s .  system towards a s t a t e of  2  at  inter-  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 a m o n g s t i t s Zeeman  l i t e r a t u r e as  lies  dipole-dipole  "sees" a f l u c t u a t i n g l o c a l magnetic  This approach to q u a n t i t a t i v e  mental  presented  inducing  used t o c a l c u l a t e  time between the  m o l e c u l a r m o t i o n was  t r e m e l y s a t i s f y i n g ; each s p i n  are  the  p r o b a b i l i t i e s through c o r r e l a t i o n functions  lattice.  the  r e l a x a t i o n mechanism  Zeeman e n e r g y l e v e l s was  transition  ition  the  liquid  super-fluid  relaxation  behavior in exotic  crystalline state,  (anti)  state.  "simple"  As  f o r the  forms  ferromagnetic liquid  state,  which  i s o u r 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  6-11 t e c h n i q u e s s u c h 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 , ~ 12 13 method o f Kubo, ' and  the r e l a t e d  approach  and  the  and  concepts of i r r e v e r s i b l e  been a p p l i e d  t o the problem.  thermodynamics,  linear-response  of Anderson,  15 16 Planck equation, projection operators, the L i o u v i l l e 20  '  21  physical  14  the  Fokker-  1 representation,  to mention  Discussions of the l i m i t a t i o n s  a few,  have  of various 22  approaches  c a n a l s o be f o u n d  i n t h e a r t i c l e s o f A r g y r e s and  Kelly,  23 24 Robertson, 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 a p p r o a c h e s t o r e l a x a t i o n c a n be f o u n d i n a p u b l i s h e d series of  lectures.  25  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 the preceeding paragraph, the impressive f a c t r e t u r n of a dynamical no u n i v e r s a l  i s the obvious, a general account of  system  to equilibrium  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  d e n s i t y m a t r i x a p p r o a c h , an a p p r o a c h  the  i s f a r from s i m p l e . shall  adopt  the well  Although tested  almost 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 of mobile nuclear spin  systems.  -12-  2.2  THE  S E M I C L A S S I C A L FORM OF  In t h e r e m a i n d e r this  of this  s e c t i o n , we  will  RELAXATION  summarize the r e s u l t s  of  i n order to l a y the foundation f o r  calculations. The  origins  at finding  o f t h i s method can  a firm  equations which  theoretical  bear  appeared  original  Hubbard,  a l s o be f o u n d  Although  introduced provide  applications.  de-  is cited a much  Subsequently,  work w i t h g e n e r a l i z a t i o n s and  8  and  F r e e d and  in Slichter's  1 0  t h e most r e a d a b l e account 7  assume a s e m i c l a s s i c a l treatment  Fraenkel. book and of this  r i g o r o u s l y by  as a f u l l  will  sim-  various  simplifications  Very good  The  introductions  Hoffman's ^ review 1  topic  i s t h a t o f Hubbard, We  shall  (quantum t r e a t m e n t o f s p i n s classical  o p e r a t o r , a ( t ) , o b t a i n e d from  and  also classical can  t h e same  treatment.  i s shown i n q u a n t u m s t a t i s t i c a l n o n - i n t e r a c t i n g systems can  8  treatment of the l a t t i c e  showing t h a t i t y i e l d s e s s e n t i a l l y  quantum m e c h a n i c a l  article.  i s R e d f i e l d ' s review  g e n e r a l l y be a d o p t e d .  approach  of the l a t t i c e ) .  be j u s t i f i e d  identical  9  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 h i s formalism which  It  independently  7  i n t h e l i t e r a t u r e , t h e m o s t 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 Abragam,  results  c h a n g e s he  formulation for practical  reviews of t h i s  article,  Meanwhile, Redfield'  attempts relaxation  O f t e n however, R e d f i e l d ' s t r e a t m e n t  f o r c r e d i t as t h e n o t a t i o n a l plified  be t r a c e d b a c k t o B l o c h ' s  b a s i s f o r the phenomenological  h i s name.  rived a s i m i l a r theory.  it  DENSITY OPERATOR THEORY OF  t h e o r y o n l y so f a r a s n e c e s s a r y  later  can  THE  m e c h a n i c s t h a t an e n s e m b l e o f  be d e s c r i b e d by a r e d u c e d d e n s i t y  the system's complete  density matrix  by  -13-  averaging  over the l a t t i c e  of any s p i n operator  degrees o f freedom.*  time e v o l u t i o n o f t h edensity  (d/dt)a(t) subject  .  [2.2.1]  operator  i s a s o l u t i o n o f the equation  = -i[/£, a ( t ) ]  t o t h e c o n d i t i o n that f o r a l l values  [2.2.2] of t ,  Tr[a(t)] = 1 .  The  Hamiltonian  order  [2.2.3]  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  fi£  where t h e s t a t i c  =  fi(£  Hamiltonian,  0  [2.2.4]  £ g ( c o m p o s e d o f t h e Zeeman t e r m a n d f i r s t  c o r r e c t i o n s ) , determines l i n e  Although the Hamiltonian  itself  p o s i t i o n s and i n t e n s i t i e s .  Q  between t h e s p i n s  o f many s e p a r a b l e For  linewidths. Equation  field  so t h a t  i s a l s o assumed.  a n d t h e m o l e c u l a r b a t h , may i t s e l f  couplings,  The  be t r e a t e d a s t h o u g h t h e y  F u r t h e r m o r e £{t) i s d e f i n e d  absence o f a secondary o s c i l l a t i n g  coupling  determines  has t h e d i m e n s i o n s o f e n e r g y ,  [ 2 . 2 . 4 ] i m p l i e s t h a t cf a n d £{t) w i l l units of angular frequency.  w r i t t e n as  + £ ( t n  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 ) ,  The  value  A , a v e r a g e d o v e r t h e e n s e m b l e i s g i v e n by  <A> = T r [ a ( t ) A ]  The  The e x p e c t a t i o n  a f a c t we s h a l l  elaborate  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  have «£(t)>=0.  6?(t), the be c o m p o s e d  on a t a l a t e r  time.  spin magnetic i n t e r a c t i o n s ,  * For excellent discussions o f the density matrix, 27 28 29 Fano, T e r Haar, o r Tolman.  see t h e accounts o f  -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  molecular) contributions effect relaxation. plifying  assumption that  tributions It  translational  (inter-  A t t h i s j u n c t u r e , t h e sim-  £ ( 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-  i s introduced.  i s also convenient t o introduce a "deviation" density  (t)  x  =  a(t) - a  operator,  [2.2.5]  T  where a " 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 1  i n thermal  equilibrium,  a  e x p ( - ^ /kT)/Tr[exp(- £ / k T ] .  =  T  0  [2.2.6]  Q  I n 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  w h e r e 31,  T  - (1 - c / k T ) / T r [ 3 ( ]  [2.2.7]  ?  Q  t h e u n i t m a t r i x , h a s d i m e n s i o n a l i t y e q u a l t o t h e number o f d e -  grees o f freedom  i n the spin  system.  Using time 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 the r e s u l t i n g  approximation f o r t h e equation o f motion i s  (d/dt) (t) = i[ (t),£ ] x  x  0  \ ^  /  order,  s  [ £ ( t ) , [ e x p ( i £ T ) £ ( t - T ) e x p ( - i £ T ) ,x(t)]]dx. 0  Q  — CO  [2.2.8]  All  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 c a n be w r i t t e n a s 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 lattice  ( o rbath)  functions,  £(t)  = SS 5  U (bath,t)V (spins) . k  k  ?  k  ?  [2.2.9]  -15The  s u m m a t i o n o v e r r, i n c l u d e s a l l v a r i o u s  pathway  relaxation  i s used t o denote e i t h e r d i s i m i l i a r  or s i m i l i a r  physical  pathways.  The t e r m  relaxation  mechanisms  r e l a x a t i o n m e c h a n i s m s o p e r a t i v e on d i f f e r e n t  relaxation  cen-  ters. In w r i t i n g Hermitian.  E q u a t i o n [ 2 . 2 . 9 ] , i t s h o u l d be k e p t i n m i n d  that£(t) i s  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 ,  inter-  im action-dependent parameters. from t h e c l a s s i c a l  i n U (bath,t) arises  m o l e c u l a r motions which modulate  relating molecule-fixed p a n s i o n o f &{t),  The t i m e - d e p e n d e n c e  to lab-fixed  Hubbard  ?  the Eulerian angles  frames o f r e f e r e n c e .  Using t h i s ex-  h a s 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 )  c a n c o m p a c t l y be r e w r i t t e n a s (d/dt) (t) = i [ ( t ) , x  E+N] + R ( t )  x  w h e r e E E -.IOJQI .  N and R a r e most e a s i l y  s i o n o f Equation [2.2.10]  (d/dt)<a|x(t)|a'>  [2.2.10]  x  i fthe matrix  expan-  i s studied:  = iw a  defined  .<a|x(t)|a'> +  fr"(a' ' a ' " ) a " a ' "  a  X [6 , .,.<a|x(t)|a"> a  a  —a a  - 6^, , <a " | ( t ) | a >] 1  1  x  —z » "lx(t)h'"> aa nC'a"a"" <a  m  u  [2.2.11]  where  N«(.".'")  =E  E  1v  Ot  Kj  Q^ ,, ,iv-u, a  c  c i  iV  ,,,))<a"|V, |« ' ><a k  [ i  1  V  1 V  |V«|„'">  J6  [2.2.12]  -16-  cM( =  + expefiz/kDrVdz  jkj^zjd  u)  [2.2.12a]  and  a::.'....,,,, - E t J ^ . . ) +J^(- . ...)}<.|V*|."»<.-|vJ|."» a  E  aa ' ' Z - ^ I V iv  6 i  J  ,k£/ ?n  a  . \ al v ) <'a  a  r K'"a  ,,, i,,k| i v ivi,,£| Vz;a><a V n  l J lJ 1  n  A"  I I I  a a  IV <a  II  a  1  a  1  V J a  C  1  ><a  1  V  a">  ' n  1  [2.2.13]  kp  The f u n c t i o n , J ^ t a ) i s g i v e n b y JJJJU) = \  ^ cj (t)exp(ia.t)dt  [2.2.14]  a  )  where  C^(t) =  <lAt + t )U*(t )>  The m e t h o d o f d e r i v a t i o n a sufficiently  p o i n t , an o p e r a t i o n a l  from x  environment f o r t h e spin  (d/dt)x a t times t » definition of  i s the f i r s t  f o r the convergence  = <U*(t)U*(0)>.  T Q  is;  x  n  to x  for t  *  . The prime i n  a t t = 0. k£/  >>TQ,  C  At this  ( t ) -* 0. T h e  t e r m i n a power s e r i e s e x p a n s i o n a n d i n o r d e r  t o be g o o d , x  a a  . ( t ) s h o u l d n o t be v a s t l y  different  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 which x(0) = x ( t )  a n d y e t f o r w h i c h t << R~  U  The  [2.2.15]  i s based on t h e a s s u m p t i o n t h a t t h e bath p r o v i d e s  rapid fluctuating  * T h i s approach r e l a t e s  expression  n  n  physical  aa a 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  information over a time i n t e r v a l  comparable  a  i s t h a t we n e v e r a s k f o r to x . n  -17-  the  summation denotes t h a t o n l y  ( i . e . oo  s e c u l a r terms  R  ) are retained. aa  a  -to  a  ,  The a n g u l a r  ' a  a  £Q|C*> = E |a>.  The  C  ( t ) s are defined  the  following classical  l  i s defined  d e f i n e d a s t g = 0.  ? = n i n Equation  {J^)  as t h e l a t t i c e  correlation  and  defined  tation is  defines  t.  ( i . e . k = i and aUto-correlation  Then c a l c u l a t e t h e a p p r o p r i a t e (If k f  i and/or ? f  the c r o s s - c o r r e l a t i o n function.)  regardless  (cross)-correlation function.  o f which treatment  i s not a t r a n s i t i o n  i s consulted.  probability matrix.  unique t o Hubbard's f o r m a l i s m ,  although  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 30  in  v a r i o u s German c a l c u l a t i o n s i n this  n,  (to)  represenThe m a t r i x  and i s s t a n d a r d  I t should  this  notation  be e m p h a s i z e d  The N m a t r i x  that  notation i s  the V (super)matrix  used  However, f o r f u t u r e c a l c u l a t i o n s  thesis, the influence of 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 . First, by  i t i s easy t o j u s t i f y  2 i n Equation  R  a s i m i l a r notation i s introduced  in  ).  product  The f u n c t i o n J  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  o f t h e ensemble auto  presented  a t some t i m e w h i c h c a n  t h e so c a l l e d  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  this  of the integrand.  S e l e c t f r o m an e q u i l i b r i u m  perform t h e denoted ensemble average.  procedure y i e l d s  is  this  by  f u n c t i o n s and have  N e x t m e a s u r e t h e same p a r a m e t e r  time,  A  eigenket:  circumvented  at singularities  interpretation:  f u n c t i o n ) a t some l a t e r  =  A  under t h e i n t e g r a l . i s  value  [2.2.15];  <<  as u ,  the energy o f t h e a  ensemble any one p a r a m e t e r and measure i t s v a l u e be  ,., a  r  The d i v e r g e n c e  t h e Cauchy p r i n c i p a l  a  J  -\  represents a  taking k J l  frequency  •  = ( E ' i - E ) / f i where E  aa  , -to , ,  aa  J  replacement o f t h e term  [2.2.12a] and t o c o m p l e t e l y  ignore  l+exp(1iz/kT)  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. (cross)-correlation  Furthermore,  i f we assume t h a t t h e a u t o  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 time c o n s t a n t s (see next s e c t i o n ) , then t h e s p e c t r a l function  has t h e form,.  J^(u)) ^ a . L l a ?  Recognizing  t h a t Q ri(w) = —  o p e r a t i o n , i t i s seen  +  .  [2.2.16]  * J ( 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  k  k £  . ?n  TTU)  and  density  t h a t Q(to) a n d J(co) f o r m a H i l b e r t t r a n s f o r m  pair,  hence, Q (co) =  = ] C i  k£  ? n  It  i swell  i  2 . 2 a . + co  known t h a t t h e p h y s i c a l  ( "  .  i i i 2 . 2 a . + co /  a  )  a  K  [2.2.17]  consequence o f i n c l u s i o n  o f these  30 32 3 3 terms  i s t o i n t r o d u c e a second  namic l i n e s h i f t s width.  This fact  Firstly,  ...,  a  '  These dyline-  i s e a s i l y seen from E q u a t i o n [2.2.16] and [2.2.17].  note t h a t J ( 0 )  L i k e w i s e , i f a-j,  '  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  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  Now i f a - j , o ^ ,  width.  order frequency s h i f t .  a a  n  * co , do t h e s e t e r m s  >>co  n  line-  (extreme-narrowi n g ) , then J ( 0 )  << co , a g a i n J ( 0 )  >> Q(co).  become c o m p a r a b l e .  >> Q(to)  i f a-j, o ^ ,  Only  Hence i f o n e 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  approximation  the m a g n e t i z a t i o n decay w i l l  by t h e i n t e r f e r e n c e  of  i n g e n e r a l be m o d u l a t e d  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  will larly  be u n a f f e c t e d ( s e e A p p e n d i x  B f o r a complete  example).  A  fails,  envelope  particu-  d e f i n i t i v e e x a m p l e o f a c a l c u l a t i o n w h e r e 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 both 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-  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  nuclei.  34  Therefore,  i n the sake o f t r a c t a b i l i t y , the N dependence s h a l l  be  t h i s work.  comment on t h e  imation  However, t h i s  s u b t l e p o i n t does j u s t i f y  ignored  throughout approx-  introduced.  Here t h e n , i s the p o i n t of embarkation  for future  computational  studies:  ( d / d t ) < a | ( t ) | a ' > = ia)  S  .<a| (t)|a'> +  X  X  R„„ .„..„... <«| X ( t ) | a>">  I I  OtCX  a  i l l  CttX  a  UC  .  (Jt  [2.2.18] Once h a v i n g  chosen a convenient  s e t o f a's  x ( t ) , the observable values of < I ( t ) >  and  z  and  transverse relaxation;  o p e r a t o r s ) , are c a l c u l a t e d any  further,  w i t h BPP inally,  i t m i g h t pay  t h e o r y and t h e BPP  molecules  I  +  obtained the s o l u t i o n f o r  <I (t)>  the a p p r o p r i a t e t r a c e .  t o n o t e why  we  and  lowering  Before continuing alliance  f o l l o w i n g a more cumbersome a p p r o a c h .  t h e o r y a p p l i e d t o two  i n the non-degenerate I  like  = ±1  s p i n 1/2  c u l a t e the probable  number o f m o l e c u l e s  z  However, i f t h e m o l e c u l e  Orig-  n u c l e i , hence o n l y  s t a t e s have n o n z e r o c o m p o n e n t s  t o c a l c u l a t e < I ( t ) > , i t was  there are degenerate  longitudinal  a r e n o t c o n t e n t w i t h an  m a g n e t i c moment and  time.  ( i . e . the  +  are the conventional r a i s i n g  from  i n s i s t on  and  necessary  only to  of  cal-  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  c o n t a i n s t h r e e o r more l i k e  Zeeman e n e r g y l e v e l s c o r r e s p o n d i n g  s p i n s , then  to 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 ( t ) > i s d e p e n d e n t upon z  one's c h o i c e o f b a s i s e i g e n k e t s . and  I  space,  Although  the form  o f both m a t r i c e s x ( t )  d e p e n d on w h a t s e t o f b a s i s f u n c t i o n s a r e c h o s e n t o s p a n t h e s p i n the t r a c e of  x  (t)I  z  i s independent  of t h i s  choice of basis  -20-  * states..  Therefore,  f o r any orthogonal the  a density matrix  choice of eigenkets.  BPP t h e o r y , t h e r e  correlations  formalism  yields  t h e same  result  A l s o , w i t h i n t h e framework o f  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  i n the f l u c t u a t i n g  fields  responsible f o r relaxation.  L a s t l y , t h e BPP a p p r o a c h 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 states.  I n t h e l a n g u a g e 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  the diagonal are zero. certain  e l e m e n t s o f a, a n d a s s u m e s t h a t a l l o f f d i a g o n a l  However, a n o n v a n i s h i n g  correctly,  elements o f a).  p l a y a dominant  Many o t h e r  a p p r o a c h c a n be f o u n d  Double Resonance and o t h e r  elements  implies a  i s unable t o t r e a t  practical  advantages  i n papers d e a l i n g w i t h  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  elements  role.  However, a l t h o u g h share  Hence BPP t h e o r y  the transverse relaxation.  of t h e d e n s i t y operator  Bloch  transverse magnetization  only  p h a s e c o h e r e n c e o f t h e s t a t e s ( a p h a s e 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 diagonal  do  considers  similarities;  superficially  quite different,  t h e BPP b e i n g  a special  t h e o r y , a n d a s shown i n A p p e n d i x A, many  t h e two a p p r o a c h e s  case of the R e d f i e l d (but not a l l ! )  of the  35 * This  failing  i s -typified  i n a recent calculation  which attempted t o extend valent spin one-half  t h e BPP a p p r o a c h t o t h e c a s e o f t h r e e  nuclei  c o r r e c t , t h e set-up s p i n system be  prediction  While the algebra i s not; this  t h a t T"  = 2[W  the  (W-j a n d W  2  1  +  equations i s  approach a p p l i e d t o a  As a m a t t e r  of fact,  approach a p p l i e d t o N i d e n t i c a l 1  a t i o n w o u l d depend s o l e l y to i n t u i t i o n  (omitting only the t r i p !  leading to their  innocent  i s by no means o b v i o u s .  shown t h a t t h i s  equi-  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 b e -  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 quantum t r a n s i t i o n ) .  by Nowak a n d M i l d v a n  three  i tcan e a s i l y  spins leads  to the  (N-1)W L s o t h a t f o r l a r g e N, t h e r e l a x 2  upon t h e d o u b l e quantum t r a n s i t i o n s ,  are the t r a n s i t i o n  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  probability  contrary  per u n i t time f o r  respectively).  -21elements of transition  R have s i m p l e i n t e r p r e t a t i o n s probabilities  or  linewidths.  i n more f a m i l i a r  terms  of  -22-  2.3 THE  DYNAMICAL PROBLEM - THE  As s e e n i n E q u a t i o n  CORRELATION FUNCTION  [ 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 lation in  functions.  Equation  ducible  In general,  e l e m e n t s and c l a s s i c a l  the spin  angular  and l a t t i c e o p e r a t o r s  defined  [ 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  spherical  tensors o f rank L ( i . e . k = -L, - L + l ,  corre-  irre-  L - l ,L).  38 T h i s has spin  led Atkins,  relaxation  very general  i n h i s superb review a r t i c l e ,  p r o b l e m a s one w h i c h f a c t o r i z e s  calculations  - the geometrical  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 . rical  aspect involves  of  task.  i n t o two d i s t i n c t a n d  and t h e d y n a m i c a l  The e v a l u a t i o n  components  o f t h e geomet-  elements, a tedious but s t r a i g h t -  The c o r r e l a t i o n f u n c t i o n s  the problem.  to the  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 forward  to refer  r e f l e c t the dynamical  I t i s t o w a r d s t h i s p r o s p e c t we  now  d i r e c t our  aspects atten-  tion.  of  The r e l a x a t i o n m e c h a n i s m s 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 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 r a m e t e r s w h i c h i n g e n e r a l  * are  proportional  * The s p h e r i c a l  to the spherical  h a r m o n i c s o f r a n k two.  h a r m o n i c s o f r a n k two a r e d e f i n e d Y°(n) = ( 5 / 1 6 7 r )  1 / 2  (3cos e 2  -  Hence, i t  as,  1)  Y ^ t f t ) = +(15/8Tr)^ sinecoseexp(±i(j)) 2  Y; (oJ 2  =  (15/32ir) sin eexp(±2i<j>). 1/2  2  k The p h a s e c o n v e n t i o n a d o p t e d t h r o u g h o u t t h i s t h e s i s where t h e s u b s c r i p t ent.  For further  Rose ( r e f e r e n c e  L i s t h e r a n k and t h e s u p e r s c r i p t  properties  68)  i s Y^  of the spherical  i s an e x c e l l e n t  source.  k k* {a) = (-1)  k i s the  Y  compon-  h a r m o n i c s , t h e b o o k by  L  -23-  will  prove necessary t o c a l c u l a t e  relation  functions  the Fourier  Transform o f angular  o f the form,  cj"(t)  = £ ? <Y^(n (t))Y*(n r  where t h e U ( t ) i n E q u a t i o n l <  (0))>  [2.3.1]  [ 2 . 2 . 1 5 ] a r e now w r i t t e n  a s <% Y«(fi ( t ) ) .  (dimensions o f s e c " ), 5 , i s defined  The  proportionality  the  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  constant  concern  (note that  general  d i f f e r f r o m most o t h e r s by a s m a l l  out  numerical  factor).  we  P(fi (t),^£^(0))  which describes the p r o b a b i l i t y that  the relaxation  vector  n  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  in  t h e r a n g e dft a t t i m e t a n d t h e r e l a x a t i o n  an  orientation  i n dQ  * The r e l a x a t i o n terminology.  axis  a t t i m e t = 0;  v e c t o r may d e f i n e  chemical  shift  tensor.  The i n t r o d u c t i o n  reorientation  rotations  field  the internuclear gradient,  vector,  to the relaxation field.  rate,  of the  induces r e l a x a t i o n . A l l  o f the molecule which leave the r e l a x a t i o n  f l u c t u a t i n g magnetic  because i t i s t h e  ( i f you wish, the p r i n c i p a l axis  f i x e d c o o r d i n a t e system) which  of the  of the spin-rotation  o f t h i s concept i s useful  of this vector  the prin-  the principal axis  tensor, or the principal axis  not contribute  local,  relaxation  D e p e n d i n g upon w h i c h r e l a x a t i o n m e c h a n i s m i s c o n s i d e r e d ,  of the e l e c t r i c  interaction  v e c t o r f o r pathway n has  v e c t o r i s a t e r m commonly e m p l o y e d i n NMR  relaxation  cipal  do  carry  *  characterizing  this  To  the angular c o r r e l a t i o n function,  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 ,  K  as  t h i s d e f i n i t i o n of the interaction constant w i l l i n  t h e ensemble average i n v o l v i n g  x dQ dQ  cor-  vector  unaffected  i . e . do n o t g i v e r i s e  to a  -24-  ?n ?S  C  (t =)5  // 2(^ 2 % Y  (  (t))Y  (0))P(Q (t) ?  '  t;fi  n  (0))d  ^ n dfi  ' [2.3.2]  We s u m m a r i z e t h e c o n n e c t i o n molecular  between t h e 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 t h e f o l l o w i n g way:  of the relaxation vectors are described f u n c t i o n , P(ft ( t J . t ^ ^ O ) ) ,  m o t i o n and t h e i n t r a The r o t a t i o n a l  by t h e a p p r o p r i a t e  from which t h e angular  is  f o u n d f r o m a p r o p o s e d model  T h i s model w i l l  ?  ,t;tt^{0))  be c h a r a c t e r i z e d by a s e t o f p a r a m e t e r s w h i c h c a n t h e n  NMR  hundreds o f times  lattice correlation starting  l i t e r a t u r e appendices, of e v a l u a t i o n r e l i e s spherical  P(fi (t)  function  f o r the r o t o r y motion o f the vector.  be d e d u c e d by c o m p a r i s o n w i t h e x p e r i m e n t a l Although  probability  correlation  of 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 .  motion  tensors.  relaxation studies.  f u n c t i o n s have been  from v a r i o u s f i r s t  probably  principles  t h e most powerful  on t h e v e r y g e n e r a l  evaluated i n scattered  and c o n s i s t e n t scheme  transformation properties of  T h i s a p p r o a c h has been f u l l y  exploited i n a  recent  39-42  s e r i e s o f p a p e r s by P a u l  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  injustice to the personality of the central  problem - t h e problem o f a s s i g n i n g a microdynamical number - t h e m o l e c u l a r  points connected with  t h a t n e e d t o be d i s c u s s e d a n d q u a l i f i e d  before  presenting  o f t h e groundwork chosen t o probe a s s o r t e d  relaxation The exploded  to a  interpretation of the correlation function.  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  aspect  interpretation  nuclear  this  topic  the final magnetic  problems.  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 h a s i n the l a s t  10 y e a r s  and p r o v i d e s  the link  t o a much n e e d e d  -25-  unification  between s e e m i n g l y  Good r e v i e w s functions  of general  can  be  found  d i s t a n t forms of dynamic  p r o p e r t i e s and i n the  universality  comprehensive review  spectroscopy.  of  correlation  articles  by  Gordon  43  44 and  Mountain.  difference radiation form can  However, t h i s  b e t w e e n NMR  formalism  l i n e s h a p e t h e o r y and  a b s o r p t i o n and  scattering  interpreted  orientation  directly  correlation  a  fundamental  lineshape theory  studies.  o f the bandshape c h a r a c t e r i s t i c be  a l s o p o i n t s out  f o r other  Whereas t h e F o u r i e r  of these  o t h e r forms o f  spectroscopy  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  f u n c t i o n , t h e NMR  bandshape y i e l d s  Trans-  position/  only  ->•-»•  <(B xM(t)-(B xM'(0)> 0  which^to  a first  constant T .  approximation,  I t i s immediately  ?  [2.3.3]  0  decays e x p o n e n t i a l l y w i t h the realized  that Equation  the a u t o - c o r r e l a t i o n f u n c t i o n of the macroscopic H o w e v e r , a s we itself  be  Therefore, times NMR  have a l r e a d y n o t e d ,  expressed  +  i t i s p o s s i b l e to i n d i r e c t l y  to molecular  dynamics.  in a slightly  (d/dt)|B xM(t)| Q  and  variable, hence  Due  to t h i s  fact,  can  second r a t e , dynamical  (d/dt)B^M(t))does  spectral  local  component o f t h i s  information.  a n a l y s i s (both  not  field  directly  but  i s f a r from  i n terms  reflect  the  only the r e l a t i v e  field  c o n t a i n much m o r e i n f o r m a t i o n a b o u t m o l e c u l a r However, p r i n c i p l e  relaxation  i t i s often stated that  (see Equation  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 ,  experiments.  JBQ><M(t) [.  |BgxM(t)|  r e l a t e measured  d i f f e r e n t m a n n e r , NMR  quency power s p e c t r u m o f t h e of a d i s c r e t e  and  [2.3.3] d e s c r i b e s  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 .  p r o v i d e s second hand, not  Stated  <I (t)>  time  in  freintensity  [2.2.14]).  principle,  dynamics than practice.  of  For  do  NMR  instance,  -26-  it  is often d i f f i c u l t  in  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  implied  by an e a r l i e r a s s u m p t i o n .  molecule niques  to separate t r a n s l a t i o n a l  correlation  NMR  by p a i r and  relaxation  p l i c a b l e f o r a f a r wider x-ray d i f f r a c t i o n atoms i n c o m p l e x Returning  this  rotational  mobility  is often t r i v i a l  techniques y i e l d  f u n c t i o n s whereas i n t e r p r e t a t i o n  i s often clouded  Furthermore,  A l s o , NMR  and  as  single  of other tech-  higher order correlation functions.  d a t a does have t h e a d v a n t a g e o f b e i n g  class of experimental  studies, i s able to y i e l d  ap-  c o n d i t i o n s , and much  i n f o r m a t i o n on  like  individual  molecules.  to the e v a l u a t i o n o f Equation  m o l e c u l a r r e o r i e n t a t i o n such  [ 2 . 3 . 2 ] , s i m p l e models  as t h e r o t a t i o n a l  random w a l k  problem,  of 45  46 the r o t a t i o n a l  Langevin  equation,  or the r o t a t i o n a l  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 Lorentzian spectral a t i v e s may  not  approximation In  the f i r s t  calculation  be an e x a c t m o t i o n a l f o r the d e s c r i p t i o n  assumed, r.iln t h i s  "n  9  t h e d i f f u s i o n e q u a t i o n and model, i t i s probably  of the r e o r i e n t a t i o n  simple its rel-  a very  i n most  good  liquids.  d i f f u s i o n model f o r t h e  limit,  a g e n e r a l i z a t i o n o f P e t e r Debye's  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 the a n g u l a r c o r -  function,  C  where T ,  hence, a  equation  of nuclear magnetic 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  treatment  relation  While  a p p l i c a t i o n s of the r o t a t i o n a l  d i f f u s i o n was classic  density.  diffusion  the  (t)  [2.3.4]  -« ^-l) C^(0)exp(.t/x ) k  k  2  isotropic  rotational  x  2  =  (6D)" . 1  correlation  time, i s defined  as  [2.3.5]  -27-  The of  scalar time  - 1  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  ).  following  d i f f u s i o n constant (dimensions  t h a t t h e form o f E q u a t i o n [ 2 . 3 . 4 ]  Note  also  allows the  i d e n t i f i c a t i o n t o be made, co [2.3.6]  0 where g ^ t ) i s a r e d u c e d c o r r e l a t i o n  function,  g  ?i  K X ,  ( t ) = (-1) S, K  ?n  x ^ ( t J / C ^ t o a model  0  ) -  Kj-x.  Often i n the l i t e r a t u r e , t h i s expression i s generalized  independent d e f i n i t i o n : The e f f e c t i v e  rotational  correlation  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  function.  A more t a n g i b l e  i fthe  quantitative  interpretation  (Equation [ 2 , 3 . 5 ] )  definition  c a n be a s s i g n e d  i s substituted  i n t o t h e famous  Einstein  relationship, * T h i s d e f i n i t i o n has t h e a d v a n t a g e function  l i m i t i n g case.  time" sooften referred following J-Q  yields implies =  K  T./(1  the effective correlation  T./(1  k  o^x ^). 2  l  . x./O  + a) T ?  "effective"  + u> xf).  time, x ^  that g ' " ( t ) = exp(-t/x  Tg^/O +  ?  t oi n the literature i smisleading k -k  + wx) =x 2  + co C(j 2  2  -  = ^ x...  the  [2.3.6]  The d e f i n i t i o n  ) f o r w h i c h J g gj^~ (t)exp(ia>t)dt  /(l + "> t 2  e f f  T  2  ] ) ,  x  2  g f f  f f  ).  expression,  As x  e f f  /(l +  cannot l i t e r a l l y  this  <A  2 ff  )  mean  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  extreme-narrowing approximation) o r the t r i v i a l  correct  ( t ) = . exp(-t/x.),  k  g f f  2  T j )  as t h e  E  However, E q u a t i o n  To be c o m p a t i b l e w i t h o u r f i r s t  correlation  over a s i n g l e  g '  reduces  correlation 49  g^" (t)exp(iwt)dt = J  demands t h a t =  However, t h e term " e f f e c t i v e  s i m p l e example demonstrates: Assuming  k  o fthe correlation  Equation [ 2 . 3 . 4 ]  does n o t e n t e r i n t o t h e d e f i n i t i o n .  to a s p e c i a l  then  t h a t t h e shape  c a s e w h e r e i i s summed  exponential.  T o a v o i d t h i s m i s n o m e r , we s h a l l u s e t h e ka analogous term "reduced s p e c t r a l d e n s i t y " , j (co), d e f i n e d a s ki  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),  to t h e " e f f e c t i v e  correlation  w h i c h , by d e f i n i t i o n , r e d u c e s  time" i n the l i m i t o f extreme-narrowing.  -28-  <e > = 2  It  4Dt  i s seen t h a t x ^ , the r o t a t i o n a l  NMR  relaxation  fixed  [2.3.7]  correlation  s t u d i e s , corresponds  time  definition  i s of limited  t i c u l a r model o f i s o t r o p i c A more r e a l i s t i c  validity  diffusional  for a  o f /2/3  from molecular-  radians.  How-  i t p e r t a i n s to the  par-  reorientation. f o r molecular  process  i s no  reorientation  longer  correctly  by a u n i q u e s c a l a r q u a n t i t y , b u t m u s t be d e s c r i b e d by t h e  complete d i f f u s i o n convenient action  as  degree o f approximation  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 described  obtained  to the time taken  v e c t o r to r e o r i e n t a root-mean-square angle  ever, t h i s  time  •  tensor.  For t h i s  to c o n s i d e r the time  fixed  general  model, i t i s most  dependent t r a n s f o r m a t i o n from the  frame o f r e f e r e n c e to the  l a b frame v i a the  independent t r a n s f o r m a t i o n from the  frame which d i a g o n a l i z e s the m o l e c u l a r of concern  motional  f o r i s o t r o p i c m o t i o n as  interaction  and  inter-  intermediate  fixed  diffusion tensor.  the d i f f u s i o n  more  frame to T h i s was  interaction  the not  co-  50 o r d i n a t e systems can the f i r s t  to extend  a l w a y s be  Debye's i d e a s  of anisotropic rotational Since the  initial  into this  B r o w n i a n m o t i o n on  treatment  have been p r e s e n t e d  c h o s e n t o be  collinear. realm  and  Perrin  was  c o n s i d e r the  (dielectric)  effect  relaxation.  o f P e r r i n , other extensions of the  theory  i n the l i t e r a t u r e .  P r o b a b l y the 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 . T h e s e i d e a s w e r e 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 laxation providing  times  i n a p a i r o f p a p e r s by W o e s s n e r .  a more r e a l i s t i c  experimental  spin motional  Sharing  the v i r t u e s  model y e t y i e l d i n g  i n t e r p r e t a t i o n , the a n i s o t r o p i c r o t a t i o n a l  tractable  diffusional  of  -29-  model  has  cussed  g r o w n i n p o p u l a r i t y and  by  Steele, ' 5  i n v e s t i g a t o r s s u c h as B o p p , and  7  The  53  D  )(  description contains  f u n c t i o n r e l e v a n t t o NMR  hence, the  spectral  55  Shimizu,  three  56  em-  diffusion  r e l a x a t i o n i s no  as a s u p e r p o s i t i o n o f f i v e d e n s i t y i s a sum  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  ( i . e . an  function results  Margalit,  of f i v e  longer distinct  distinct  M o r e o f t e n , h o w e v e r , 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 and  54  t h r e e components o f the d i a g o n a l i z e d  correlation  e x p o n e n t i a l s , and  been e x t e n s i v e l y d i s -  such c o n t r i b u t e r s .  motional  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 , b u t  terms.  Huntress,  5  c o n s t a n t s , the  tensor.  subsequently  V a l i e v ^ t o m e n t i o n a few  In g e n e r a l , t h e m o l e c u l a r pirical  has  axially  i s a sum has  symmetric e l l i p s o i d ) .  of three exponentials.  b e e n shown t o r e s u l t  not  d i s t i n c t e i g e n v a l u e s , Dj_ In t h i s  Recently,  case,  the  correlation  the g e n e r a l i t y of  from assuming a d i f f u s i o n a l  these  process,  59 but o n l y from t h e symmetry o f the r e o r i e n t i n g body. Besides  the asymmetry o f the d i f f u s i o n  work, the d i r e c t i o n diffusion top  a x i s i s of great  correlation  i s d e f i n e d as  enclosed  angle  to the  Each e x p o n e n t i a l by an  angular  between t h e s e  two  as was  principal  i n the  vectors.  of the e f f e c t i v e  r e o r i e n t a t i o n times  frame-  symmetric  f u n c t i o n whose a r g u m e n t  i n terms o f m o l e c u l a r  interpretation  terms of t r u e m o l e c u l a r tropic  importance.  f a c t o r i s given  o b t a i n s an  of the m o l e c u l a r  relaxation vector relative  f u n c t i o n i s weighted  the  each e x p o n e n t i a l h e n c e , one  of the  process  Furthermore,  diffusion spectral  constants,  density in  p o s s i b l e f o r the  iso-  case, [2.3.8]  There i s another  closely  related  s i t u a t i o n which i s of i n t e r e s t  for  -30-  the  molecular  where the which  relaxation  is rigidly  backbone. it  i n t e r p r e t a t i o n of  i s seen t h a t The  makes t h e  identification,D  densities  internal  rotations.  o f m o t i o n a l models are pretation Until  ->• D.  u  of experimental this point,  uncorrelated  '  symmetric top  U n f o r t u n a t e l y , any by  model, that  i t has  been assumed t h a t the  relaxation  i n f i n i t e s i m a l steps.  correlation functions) nonviscous l i q u i d s . when i n t e r p r e t e d has as  This aspect of  NMR  in conjunction with  concern.  The  problem of  '  and  6 1  the  for  This  are  The  entire  a  problem  reorienta-  c o n s i d e r e d and  Inertial  relaxation  and  effects  (char-  molecules  in  especially  techniques,  physicists.  However,  r e l a t i v e l y large  p r o v e t o be  used i f i n t e r n a l  the  nonexponential  studies,  behavior of  effects will  great  leads d i r e c t l y to  other spectroscopic  same a r g u m e n t c a n n o t be  inter-  the  i s assumed f o r t h e  effects  the m o t i o n a l  inertial  simplest  r o t a t i o n of  undoubtedly important f o r small  p r e s e n t w o r k f o c u s e s on  m o l e c u l e s , the  6 0  molecular  v e c t o r r e s u l t s from a  commanded much r e c e n t i n t e r e s t f r o m c h e m i c a l the  the  finite-step reorientation  are  D,  ±  underdetermined.  correlation function.  i s , whenever i n e r t i a l  stochastic,  D  molecular  case.  but  d a t a r a p i d l y becomes h i g h l y  assumption o f d i f f u s i o n a l motion i s abandoned. by  axis  s o many p a r a m e t e r s , t h a t  i s c l o u d e d whenever a non-Markovian p r o c e s s  acterized  + D and  t  case  f."i  m o l e c u l a r i n t e r p r e t a t i o n of the  tional  i s the  have been decomposed i n t o  characterized  m o l e c u l a r framework c a r r y i n g number o f  This  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  CO multiple  density.  otherwise i s o t r o p i c a l l y reorienting  t h i s problem reduces to the  reduced s p e c t r a l  reorientation  spectral  v e c t o r u n d e r g o e s r o t a t i o n a l m o t i o n a b o u t an  f i x e d t o an  I f one  the  of  rotations  no  major are  -31-  considered.  In t h i s  group r e o r i e n t a t i o n , orientations  c a s e , a s e c o n d m o d e l , commonly considers  120° a p a r t .  used w i t h  methyl  i n s t a n t a n e o u s random j u m p s among  three  The r e s u l t o b t a i n e d f r o m s u c h a t r e a t m e n t i s  experimentally almost i n d i s t i n g u i s h a b l e 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  (although sharing  no  physical  approximation.  However, t h i s  latter  approximation provides a convenient mathematical  form f o r data  analysis.  Good  discussions  contrasting  t h e s e two m o d e l s o f m e t h y l  reorientation  64 65 are  contained i n the l i t e r a t u r e . One  should  final,  related  aspect concerning the c o r r e l a t i o n  be t o u c h e d upon h e r e .  no m a t t e r how  '  A l l s i m p l e models o f d i f f u s i o n a l  c o m p l i c a t e d , r e d u c e t o t h e f o r m 9 ^ ( t ) = ]C  where t h e summation e x t e n d s o v e r a s m a l l x.. is  function  For i s o t r o p i c motion, a d e l t a  motion,  A..exp(-t/x..),  number o f d i s c r e t e t i m e  function  distribution  constants,  o f time constants  a s s u m e d ; 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 t w o o r t h r e e d e l t a  functions.  M o r e c o m p l i c a t e d m o d e l s demand more " t e e t h "  H o w e v e r , i n many NMR a continuum  studies, especially  of correlation  i n t h e comb.  polymeric or surface  studies,  time c o n s t a n t s i s assumed ^6,67 }  [2.3.9] 0 A system to  i n which  a continuum  be c o n s i d e r e d as a s p e c i a l  o f such time c o n s t a n t s e x i s t would case o f a m u l t i - p h a s e system.  i n s t a n c e s , i t i s n o t a t a l l c l e a r when s u c h an a s s u m p t i o n and  this  approach  which does l i t t l e no a p r i o r i seek  o f t e n appears  is justified  data with  device  In the b e s t approach, g i v e n  reason f o r assuming such a d i s t r i b u t i o n ,  to f i t the relaxation  I n many  t o be no m o r e t h a n a d a t a f i t t i n g  more t h a n add c o n f u s i o n .  have  correlation  one s h o u l d a l w a y s  functions (agreeably  -32-  m u l t i - e x p o n e n t i a l ) b a s e d on m o r e r e a l i s t i c m o d e l s o f m o l e c u l a r Note however, t h a t n o n e x p o n e n t i a l f u n c t i o n s do  not  v e r s a ) ; t h e two  imply  motion.  or multi-exponential c o r r e l a t i o n  nonexponential  m a g n e t i c r e l a x a t i o n (and  vice-  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  in  thought. In c o n c l u s i o n o f t h i s correlation formulas  s e c t i o n , the  functions defined  follow directly  by  classical  Equation  evaluation of  the  [2.3.1] i s introduced.  ( a f t e r some m a n i p u l a t i o n )  These  from v a r i o u s  equations  39-42 contained  i n the  duction of t h i s rotational  s e r i e s o f p a p e r s by section.  I t can  be  Hubbard shown by  fi  <Y (^(t))Y^(^(0))>  (-1) 5 _ 5"  =  k  and  n^,  k  k j  i s given 2  and If  d i f f u s i o n constants  f o r r o t a t i o n s a b o u t an D  L  = D  addition  n  m  =  D,  the  (static  [2.3.10]  ±  i s completely  c h a r a c t e r i z e d by  diffusion  refer  to the observable  spins are  )exp(-6Dt)/4Tr,  interaction coordinate  fixed  reference  rigidly  frame.  lab frame.  attached  the  f o r r o t a t i o n s about a symmetry a x i s ,  axis perpendicular  P. ( c o s e  spherical  - D ))t|.  The  [2.3.10] reduces to  as e x p e c t e d . p o l a r and  systems w i t h  hand  The  u  respect  the  side  primed  azimuthal to the  time dependent, unprimed  This equation  to t h i s  D,  t o t h e s y m m e t r y a x i s , Dj_.  v a r i a b l e s i n time) r e f e r to the  of the v a r i o u s  ing  M  r i g h t hand s i d e o f E q u a t i o n  (-1)^6.  of  m  theorem f o r s p h e r i c a l harmonics, t h a t i s , the r i g h t  reduces to angles  of the theory  (-D Y2(^)Y- (^)  2  e v a l u a t i o n assumes t h e d i f f u s i o n  intro-  by,  ^ m=-2  1  J l  x exp {-(6Dj_ + m ( D  rotational  the use  i n the  B r o w n i a n 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  h a r m o n i c s d e p e n d e n t on  This  mentioned  angles molecule  angles  assumes t h a t the r e l a x -  s y m m e t r i c body.  However,  -33-  there  i s no p h y s i c a l d i s t i n c t i o n  where t h e s p i n ( s ) a r e a t t a c h e d on  an o t h e r w i s e  Equation placed the  isotropically  [2.3.10] i s s t i l l  by D + D ^  internal  internal  nt  motion  directly  to a f l e x i b l e  valid  i f  In t h i s  instance,  i s the diffusion  u  i s re-  constant c h a r a c t e r i z i n g  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 s 120° j u m p s , t h e n  i t c a n be shown r a t h e r s i m p l y term i n Equation  that  [2.3.10]  r e p l a c e d by e x p | - ( 6 D - 3 ( 3 - |m|) | m | v . j / 4 ) t j w h e r e v . . ^ i s t h e p r o b n t  ability  p e r u n i t time  t h a t t h e s y s t e m jumps f r o m one o r i e n t a t i o n t o a n o t h e r .  A s e c o n d model we w i l l  resort to i n future calculations i s a  extension o f Equation  [2.3.10].  s o i d w i t h an i n t e r n a l  rotor attached  to  fragment l o c a t e d  i s r e p l a c e d by D and D  the only d i f f e r e n c e i s t h a t t h e exponential is  c a s e and t h e s i t u a t i o n  r e o r i e n t i n g backbone.  w h e r e D..^  rotor.  between t h i s  the principal  This  i s based on a symmetric t o p e l l i p a t an a r b i t r a r y a n g l e w i t h  axis of the e l l i p s o i d .  In t h i s 2  <Y^(n (t))Y*(n (0))> c  n  = (-D^^x  case, 2  1  n'=-2  X)  d  n'n"  ( E 5 , )  n"=-2  n  n  + n  , 2  (D  ( 1  2  - D ^ t }  [2.3.11]  w h e r e D. i s t h e d i f f u s i o n c o n s t a n t  characterizing the internal  The  relative  B' i s t h e a n g l e fusion axis.  respect  (-l) "Y^"(^')Y- "(^')exp(-n" D.t)  x exp|-(6Dj_  double primed angles  simple  are fixed  to the internal  rotor.  r o t o r and  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 -  The q u a n t i t i e s d , , , ( B ' ) a r e e l e m e n t s o f t h e r e d u c e d  rotation matrix  n  and a r e f u l l y  o f a n g u l a r momentum.^  discussed  i n a n y e x p o s i t i o n on t h e t h e o r y  A s s u m i n g 120° s t e p s , t h e t e r m e x p ( - n ' ' D ^ . t ) i s  -34-  replaced  by e x p ( - 3 ( 3 - | n " |) |n'' h  One o f t h e v e r y given,  i  t  t/4).  important properties  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  i s t h a t t h e r i g h t hand s i d e o f e a c h e q u a t i o n  i s dependent o n l y ka  upon t h e r a n k o f t h e i n i t i a l x 6, „C°°(t). k,-£ cn  This  s p h e r i c a l harmonic.  fact i s valid  no m a t t e r w h a t m o d e l o f  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 o f quantum m e c h a n i c a l ators.  Hence, C  from the general  correlation functions  (-1)  (t) =  k  Cn motional properties  of i r r e d u c i b l e tensor  oper-  -35-  2.4  SUMMARY After a brief  the  introduction  two-step t r a n s l a t i o n  to the problem, the t o o l s  f r o m 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 to 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 . l a r g e r e p e r t o i r e o f s u c h t o o l s , we is felt the  t o be q u i t e a d e q u a t e  semiclassical  evolution is  f o r o u r p u r p o s e s , t h e method  task of evaluating  the l a t t i c e  p a r t i c u l a r m o t i o n a l model.  correlation  O n l y i f t h e model  this  p r o b l e m , w h i c h has  chapter, i s p i c t o r i a l l y In  addition  assumptions  liquid  to the  s t a t e NMR  The  r e l a x a t i o n were a l s o  rather extensive bibliography w i l l  entire  i n any  inter-  throughout  2.1.  crux of the i n t e r p r e t a t i o n a l  unavoidably encountered  difficult  correctly, is  in detail  i n Figure  no  i n terms o f a  i s chosen  been d i s c u s s e d  outlined  the spins  functions  time  relaxation  i s confronted with the  p o s s i b l e to o b t a i n r e l i a b l e m o l e c u l a r dynamics.  pretational  the  Once t h e s p i n  t o the o t h e r nonspin degrees o f freedom,  which  employing  c o n s t r u c t i o n of the equations governing the  l o n g e r e n t e r i n t o t h e p r o b l e m a n d one  it  Although there i s a  have i n t r o d u c e d an a p p r o a c h  of the reduced d e n s i t y o p e r a t o r .  related  needed f o r  p r o b l e m , many o f  thorough d i s c u s s i o n  i n t r o d u c e d and d i s c u s s e d .  p r o v i d e t h e needed  of  The  supplementary  material. In  the f o l l o w i n g  [2.3.11] w i l l NMR  relaxation  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  be a p p l i e d studies.  to various The  problems  approach  of concern i n  chosen w i l l  be t o a d o p t t h e m o s t  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 interpretation eterization").  contemporary  a  tractable  (a t r a d e - o f f between r i g o r and t h e f a t e o f " o v e r param-  -36-  It should the  spectral  thesis  be k e p t i n m i n d t h a t  density  can provide  the molecular i n t e r p r e t a t i o n  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 .  some m e a s u r e  of assistance  f o r the  Hopefully,  of this  experimentalist.  FIGURE 2.1 Assigning  A A MACROSCOPIC DOMAIN A A A A A A A A A A A A A Relaxation Observables A ( <M , (0)M , (t)> or A A Overhauser e f f e c t s ) A A A +  z  +  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  MICROSCOPIC DOMAIN  Molecular Interpretation  Statics  Chemical  Translational  Inertial  Dynamics-;  Physical  Reorientational  Diffusional  z  Realm o f R e l a x a t i o n Theory proper  Necessary a p r i o r i model f o r d y n a m i c s  co  —j i  -38-  REFERENCES: CHAPTER I I  1.  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York,  -42-  CHAPTER I I I ANISOTROPIC MOLECULAR MOTIONS AND THE NMR  3.1  RELAXATION OBSERVABLES  INTRODUCTIONThe  model o f m o l e c u l a r  modulation  reorientational  o f the various spin interactions  a n c e i s g e n e r a l l y a s s u m e d t o be i s o t r o p i c correlation to  a time  function; spherical  i n nuclear magnetic  top approximation)  organic molecules typical  approximation).  i n s o l u t i o n , both  biochemical  studies.  Although  assumptions  In t h i s  and r a p i d  often valid  compared  f o r some s i m p l e c a s e s .  t h e NMR e x p e r i m e n t  chapter, i t w i l l  As s u c c i n c t l y  i s h i g h l y underdetermined.  simplest nontrivial  motional  spectral  f o r simple  are generally invalid  t h e 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 t i o n s i n f l u e n c e NMR observables  reson-  (a one-parameter, exponential  s c a l e c h a r a c t e r i z e d by a L a r m o r p r e c e s s i o n ( t h e w h i t e  density or zero f i e l d  in  motions r e s p o n s i b l e f o r the  be shown how relaxation  d e p i c t e d i n F i g u r e 2.1,  For t r a c t a b i l i t y , the  model, 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 assumption  of isotropic  mobi1ity. In  t h e absence o f magnetic f i e l d  the p r i n c i p a l nuclei  g r a d i e n t s and chemical  r e l a x a t i o n mechanism f o r p r o t o n s  in liquids  are inter-  exchange,  a n d many o t h e r s p i n 1/2  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-  Intermolecular  r e l a x a t i o n c a n r e a d i l y be i d e n t i f i e d  ducted a t varying practical  degrees o f d i l u t i o n  behavior consulted  This  simple  .  In the f o l l o w i n g  f o r an e n l i g h t e n i n g b a c k g r o u n d  and t h e s e s h o u l d  into this  subject.  intramolecular d i p o l a r Hamiltonian  i and j , can s i m p l y  coupling  approximation to spin  h a s f o r m e d t h e b a s i s o f many d i s c u s s i o n s  The d i r e c t , spins  eliminated  the intramolecular, d i r e c t d i p o l a r  between a p a i r o f s p i n s , i and j .  con-  i n a nonmagnetic s o l v e n t , and f o r  p u r p o s e s , c a n be e x p e r i m e n t a l l y  s e c t i o n s , we t r e a t o n l y  by e x p e r i m e n t s  1 -  be  ^  between a p a i r o f  be e x p r e s s e d a s  [3.1.1]  where  b ( t ) i s b o t h s y m m e t r i c a n d t r a c e l e s s ( i .e.<c?/ \ -.-(t)> = 0 ) . n  However, f o r c o m p u t a t i o n a l as  n  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  a s c a l a r c o n t r a c t i o n o f two s e c o n d r a n k s p h e r i c a l  tensors,  expressed  Q  2 [3.1.2] k=-2 where t h e b a t h o r l a t t i c e  5  and  l  J  -  ( f c / B )  the spin operators  V°  functions are defined  ij •  ' ^ . - ^  1  are defined  = -(8/3)  v  1 / 2  [I^I  ± z  as  J z  [3.1.3]  as  - l(l{lJ + f l j ) ]  [3.1.4a]  [3.1.4b]  [3.1.4c]  -44-  The  length  r . . and  of the  the  polar  c o o r d i n a t e system ti...  d e n o t e d by N  2  vector  i  angles specifying ( i n which the For  a total  'dipolar couplings  torn'an i s w r i t t e n  from the  to the the  j  nucleus i s denoted  d i r e c t i o n of  components o f  I . and  system of N spins  L  N  appropriate spectral  e a s i l y o b t a i n e d from the  retained  In g e n e r a l ,  [2.3.10].  c o n s i d e r e d as often  [ 3 . 1 . 6 ] i s assumed. discussed  densities  t o be  dipolar  Hamil-  decomposition  Only the  auto-correlation  i n Chapter  the  and  (Fourier  IV.  Transforma-  functions  That i s , the  cross-correlation  For  section  are follow-  valid,  dipolar fields  correlated.  The  ^3.1.5]  employed i n t h i s  spectral  i s assumed t o ; b e i m p l i c i t l y  i n t e r a c t i o n s are  total  are  c  both a u t o - c o r r e l a t i o n  s i t i e s m u s t be  are  u n l i k e ) , there  C  i n c a l c u l a t i o n s presented i n t h i s chapter.,  ing condition  defined)  Z<?(*)-  S  *(D)1jW  i<j  tion) of Equation  the  laboratory  as  W are  I . are  ( l i k e or  ( r e l a x a t i o n p a t h w a y s ) and  t)=  The  r . . i n the  by  the  spectral  of d i f f e r e n t  den-  pairwise  p r e s e n t however, E q u a t i o n  dipolar cross-correlation  problem i s  fully  -45-  3.2 EFFECT OF ANISOTROPIC MOTION ON T Consider N  identical  1  and  a system  of N identical  molecules.  9  i n equivalent positions i n the longitudinal  decay e x p o n e n t i a l l y .  [2.2.18] and [ 2 . 3 . 1 0 ] , i t i s e a s i l y  By t h e u s e o f E q u a t i o n s  shown t h a t f o l l o w i n g  a e-degree  pulse  = e/o)-|),  <I (t)>  - <I >  z  = (cose - l ) < I > e x p ( - t / T )  T  z  [3.2.2]  T  z  2  i n the irreversible  , I , or I ), the rapid x y  i g n o r e d on t h e r i g h t  1  = sinG<I > exp(-t/T ).  +  As we a r e i n t e r e s t e d  [3.2.1]  T  Z  <I (t)>  I  nuclei  I g n o r i n g c r o s s - t e r m s , both  transverse magnetizations  (of d u r a t i o n t  AND T^ V A L U E S  ]  oscillatory  decay o f I  (or equivalently  exp(-iu) (t + t ) ) , u a  term,  hand s i d e o f E q u a t i o n  +  n  [3.2.2].  0  has been  The d e f i n e d  time  constants are  T"  1  T"  1  = (N-IM-J '- ^) + 4J '- (2w )) 1  1  2  [3.2.3]  2  0  = (N-1)(3J  The  spectral  O '" (co) k  k  0 0  (0)  - S J  densities  2  + 3sin BF (D  - 1  1  ^ )  + 2J '" (2o) ))/2 2  j  2  0  .  i n these equations  (3cos 3 2  - 1) F (D ,  a r e d e f i n e d as  D^,U)) + 1 2 s i n 3 c o s g F  2  Q  [3.2.4]  2  A  D^.w)}  4  2  ' "  appearing  = (-l) £ (16^) k  1  1 5  2  1  (D , ±  [3.2.5]  where  ? (D n  £  D , ) = [60^+ n ( D 2  u  u  - D )]/[(6D + n ( D 2  u  x  ±  - D j ) + u, ]. 2  R  2  [3.2.6] The  statement,  5.: • =  =5  "N i d e n t i c a l  nuclei  i n equivalent positions",  a n d B .. = e., ., = 8.  implies that  Hence, i n t h e above e q u a t i o n s , t h e  D^w)  -46-  U,c)  superfluous subscripts i j , i j Beta and  i s the enclosed  the i n t e r n u c l e a r vector.  rotations ize  n  In  F o r a methyl  the s p i n n i n g o f t h e methyl  "  W  l  1  1  D  d e f i n e d by t h e p r i n c i p a l  a x i s , N = 3,  about i t st r i a d  of t h e a x i s about which D  angle  the following  group undergoing  g = T T / 2 , and D  and D  u  i t spins respectively  terms.  diffusion  g r o u p and t h e r o t a t i o n a l  1oosely r e f e r r e d  e  a r e dropped i n a l l r e l e v a n t  axis  hindered character-  ±  reorientation  (henceforth, the quantity  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 ) .  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  specifically  r e f e r r e d t o b e c a u s e i t i s one o f t h e s i m p l e s t c a s e s w h e r e a n i s o t r o p i c motional  effects  of the present not  are undoubtedly  treatment  be c o n s t r u e d  will  important.  be o b v i o u s  as a l i m i t a t i o n  However, o t h e r  interpretations  and t h e a p p r o a c h a d o p t e d  of the general  validity  should  of the conclu-  sions. F i g u r e 3.1 shows t h e r e l a x a t i o n we w i s h T-| o r T using  t o compare. 2  the assumption  k  graphs).  "effective"  2  (these r e s u l t s  (Equation  lines  [3.2.5])  appear e x p l i c i t l y  models  rotational  correlation  time,  T , 2  that  k  The s o l i d  o f t h e two m o t i o n a l  i s to t r y to f i t the experimental  = (-1 ) C ( 4 i r ) " ( 6 D ) ( 3 6 D  k  w h e r e 6D E  model  The s i m p l e s t model  data w i t h a s i n g l e  J '" (w)  behavior  1  2  + co )' 2  a r e shown a s t h e d o t t e d  i n each graph correspond i n which the i n t e r n a l  (different solid  d i f f e r e n t s i z e as i n d i c a t e d ) .  [3.2.7]  1  lines  lines  t o t h e more  rotation  correspond  The m o s t i n t e r e s t i n g  i n t h e two detailed  r a t e and  to molecules  angle of  feature of the T  curves  i s t h a t f o r macromolecules o f molecular weight  g r e a t e r than  30,000  ( 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  1  c. a.  •47-  o b t a i n e d from Equation [3.2.7] correlation If  D„ »  time f o rt h e r o t a t i o n a l  D^, t h e n x  This fact =  (60^)"^,  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 anisotropy, x  R  , where x  R  = (4D  ](  - 4D.J7  - (4D,,)" '. 1  R  i s e a s i l y deduced from E q u a t i o n [ 3 . 2 . 5 ] .  i t i s seen  that  i fx  £  »  ^ , x  <  <  R  W  Q  1  » and w "  Defining x 1  <  <  (  T  R  T  C  ) ^  c  '  2  then J '~ (u>) k  Furthermore,  i fT , x  > R  >"^  .  k  and x  T  r  [3.2.8]  >>x ,  c  R  J '~ (a>) - T~V2. k  In  either case, the spectral  internal  k  densities are a function o f the rate of  mobility alone; a t f i r s t  A l s o note t h a t i f T independent  ofx  R  K  > • 0 oru  -1 U  n  and approach 2  [3.2.9]  sight, a rather startling  » ( x  D  x „ )  1/2  K C  , the spectral  revelation.  densities are  asymptotic values d i f f e r i n g  by a f a c t o r  2  of  f o u r ( i . e . ( 3 c o s 3 - 1) ) d e p e n d i n g  is  violated.  For values o f x  < £  <  on w h i c h  o f t h e two i n e q u a l i t i e s  ^ , the plots f o ra given T  c  would  simply  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 mobility increased. It  h a s o f t e n been s t a t e d t h a t o n e o b v i o u s d r a w b a c k 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 at  a nonzero  time  i s a double  valued function o f the c o r r e l a t i o n  ( i . e . a u n i q u e T^ c o u l d be i n t e r p r e t e d  correlation the spectral and  frequency  x  R  .  times).  Add t o t h i s a second  i n terms  degree  o f two d i f f e r e n t  o f motional  freedom,  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  B u t an i n c r e a s e i n m a g n e t i c  field  ( i n c r e a s e o f WQ)  will  in x effec-  £  1  -48-  tively  shift  the left-hand  p o r t i o n o f t h e T, v e r s u s T I  right while leaving  the right-hand portion  fixed;  w h e t h e r 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 r e m a i n s the magnetic or T  c  ( x , x  shorter *0  >>u  > > T  field,  R'  n  c  o  r  T  R ^  r  o  when x cient  <<UQ ; i n t e r n a l  (co~x ) U  fied  immediate  in  size,  - 1  less e f f i c i e n t  c l e a r l y makes <x  D  <x  K  1  ^ '  1  t o the  correlation  r e s u l t above. c  0 0  i svalid  Examining  J  0 0  only effi-  0  i s that the simpliFora  shows t h a t t h e " e f f e c t i v e " never d e v i a t e very  time o f t h e macromolecule T h i s a g a i n c a n be e a s i l y  itself, deduced  , then J ( 0 ) » - J ' ^ ) , J '" (2co ) «co~ , then J ( 0 ) - - J ' " ^ )  l  0 0  Q  ]  £  The v a r i a t i o n  2  X  1  =  - 1  2  1  between t h e s e  2  1  Q  extremes  p u r p o s e s , T^ i s d e p e n d e n t o n l y  function  = C (24TTD )-  1  0 0  the structure ofJ  (0)  that  r e l a x a t i o n more  line will  and hence, f o r a l l p r a c t i c a l  on t h e a r e a u n d e r t h e c o r r e l a t i o n parameters.  clearly  t h e broken  2 -2 -1 00 ' (2<J0Q) a n d - 10J (0).  i s monotonic  case  C  the figure  from Equation [3.2.5]. I f x »u~ a n d T" - 3 J ( 0 ) . However, i f x - J  t h e f i g u r e that the usual idea  C  time deduced from  the rotational  contrast  1  be n o t e d a n d t h u s , t h e l a t t e r  important feature o f the l ^ curves  given macromolecular correlation  > co" )  R  (However, i f  ( E q u a t i o n [3.2.7]) g i v e s a very m i s l e a d i n g p i c t u r e .  model  much f r o m  from  rotation  i n r e g i o n s where  An  times are c o r r e c t .  r o t a t i o n makes r e l a x a t i o n  1  c  increasing  data.)  i t may be n o t e d  internal  t h e same upon  u n l e s s one has a good i d e a o f t h e m a g n i t u d e o f e i t h e r  ''"dependent  m  Finally, rapid  1  dependence w i l l  may be a m b i g i o u s T  < COQ ) c o r r e l a t i o n  o  t h u s by o b s e r v i n g  c o u l d d e c i d e w h e t h e r t h e l o n g e r (x , x  one R  curves t o t h e  d  C3 K  5  0 0  2  f o r a l l values o f the motional (0),  i t i s obvious  x (4*)" , 1  that i f  [3.2.10]  -49-  and  i f D  D,  >>  u  L  J  If  0 0  ( 0 ) = 5 (3cos B 2  - l)  2  ( 9 67rD )" .  2  i t i s a s s u m e d t h a t 6 = TT/2„ i t i s e a s i l y  The  lower l i m i t  i s a p p r o a c h e d when x  i s a p p r o a c h e d when x and  R  << x  R  •> 0 (D„ -> Dj_).  [3.2.11]  1  1  <_ J  seen t h a t T / 4 c  (D  c  >>  (l  (0)4TT£~  0 0  D ) and t h e u p p e r  2  <_  limit  Note t h a t both e x p r e s s i o n s [3.2.10] of x ,  [3.2.11] are independent o f the magnitude  the o r i g i n a l  R  conten-  * tion.  A l s o n o t e t h a t t h i s a r g u m e n t i s i n d e p e n d e n t o f any  c o n c e r n i n g WQ.  (This i s not s t r i c t l y  s i b l e t o w r i t e T^ s i n the T  2  <= J ^ ( 0 ) .  1  - 1, i t i s n o t p o s -  c  T h i s f a c t e x p l a i n s why  p l o t o f F i g u r e 3.1  o f m i n o r c o n c e r n as c l e a r l y  t r u e ; i f OJQX  c u r v e s p, q, r , and  c a n n o t be s u p e r i m p o s e d .  shown i n t h i s  [ 3 . 2 . 7 ] may  be u s e d t o o b t a i n a g o o d  r o t a t i o n o f t h a t methyl group.  * An  interesting  feature of this  independent o f the magnitude on t h e a n g l e b e t w e e n axis.  that this  static  with the s t a t i c meter.  m e a s u r e m e n t may  be a r g u e d t h a t  d e n s i t i e s depend  on t h e d y n a m i c s !  used  1  (J  u u  from  (0)) is depend  diffusion  i n d e e d , i n some  on m o l e c u l a r s t a t i c s  Extension of this  f a c t o r m i g h t more c o r r e c t l y  argument be  implies  identified  i n t e r a c t i o n c o n s t a n t a n d n o t w i t h any d y n a m i c a l p a r a -  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  decrease the " e f f e c t i v e "  interaction  constant.  1 1  be s a i d  spectral  density.  to simply  However t h i s  m o r e t h a n a mnemonic d e v i c e a n d one s h o u l d a l w a y s s p e a k the " e f f e c t i v e "  be  A l s o , as c a n be s e e n  i s t h a t a l t h o u g h T^  case, i t could  geometrical  internal  o f t h e m o t i o n a l asymmetry, i t does  instances, the relevant spectral indirectly  2  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  Therefore, in this  and o n l y  motion.  result  macromolecule,•Equa-  estimate o f the rate of  Likewise, a T  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  However, t h i s i s  plot.)  Thus i t i s s e e n t h a t T-| f o r a m e t h y l g r o u p on a tion  inequalities  is  i n terms  little of  T  -50-  [3.2.11],  Equation overall  mobility  measurements p r o v i d e  (and t h e i n t e r a c t i o n c o n s t a n t s )  a s e p a r a t e means.  In this  t h e average angle a t which  should  also  be i n s t r u c t i v e l y  more d e m o n s t r a t i v e l y  As  spectral  considered  i s given  envelopes.  3.2  f o r one such c o n s t r u c t i o n  , i s p l o t t e d as a f u n c t i o n o f w i g .  C  T (1  =  c  + T CO ) 2  will  I t i s assumed t h a t T  cliche, COQT  and  c  "internal  > 1,  this  1/10 <  spectral  d  like  density  COQT  r  and j " ( t o , x )  - 1  = T (1  R  << x . C  characterizes i f cogX  little,  < 10, t h e n  r  c  The d o t t e d  c  line  graphical  C + x co ) R  mobility  i f any, spectral amplitude.  no l o n g e r internal  valid.  R  given  Hence, t h e For c  = 10  dominate t h e composite  A l t h o u g h we h a v e j u s t p r e s e n t e d a n i n t e r p r e t a t i o n o f  I t c a n n o t be s t r e s s e d constructions  too strongly that  such as F i g u r e  3.2 c o n t r i b u t e  .  (xg E T ^ ) .  (for a  Q  - 1  i s the l i m i t -  F o r e x a m p l e , i f cogX  mobility will  2  curves  the correlation function  < 1, a n y i n t e r n a l  )  1  and i g E T  i n t e r p r e t a t i o n s o f j ( 0 , x ) a n d j(2cog,Xg) c a n be s i m p l y  reasoning.  t o ( j (co,T  m o b i l i t y always decreases t h e r e l a x a t i o n r a t e . "  i s generally  density.  j(cog,tg), by  seen t h a t  contribute  2  by t h e i d e n t i f i c a t i o n s , T Q E x  i s defined  c a s e where a u n i q u e  cog),  a t any g i v e n  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  K  i seasily  F o r the case  D  T h e s e two c o n s t i t u e n t  It  density,  a s a sum o f two c o n t r i b u t i o n s , t o j ( x , x .co) = K C  r  respectively.  and i n t e r -  spectral  section, the effective spectral density  + 3 j " (CO,T ))/4; J ' ( C O , T )  where t h e x - a x i s  o f t h e two  2-1  + igco )  i n this  frequency  ing  construction  A reduced, frequency weighted, exponential  coJ(to.Tg) = t o x g O  It  n o t e d t h a t a l l o f what has been s a i d c a n  an e x a m p l e , s e e F i g u r e  2  be u s e d t o  r o t a t i o n occurs.  be d e d u c e d f r o m a g r a p h i c a l  Lorentzian  pretation.  the internal  i f the  c a n be d e t e r m i n e d b y  c a s e , a 1^ m e a s u r e m e n t c o u l d  obtain  contributing  unique i n f o r m a t i o n  deduced  spectral greatly  -51-  to a p h y s i c a l understanding arise  f o r slow,  spins  I a n d S,  show much o f t h e same b e h a v i o r .  introduced  i n t h i s case however.  the  precise definition  the  s p i n s a r e o f t h e same n u c l e a r  is  results  which  a n i s o t r o p i c modulations of s p i n - i n t e r a c t i o n s .  For u n l i k e n u c l e a r and  of the "pseudo-contradictory"  1 0  '  t h e same p l o t s c a n be g e n e r a t e d 1 2  T h e r e a r e some  Firstly,  o f a u n i q u e T-j o r T  c r o s s - r e l a x a t i o n may make 2  impossible.  o f T,  Secondly, i f  species, the l o n g i t u d i n a l r e l a x a t i o n  s e n s i t i v e to the s p e c t r a l density amplitude  causes t h e behavior  complications  near zero  t o mimic t h a t o f T . 0  frequency  which  -52-  FIGURE 3.1:  Plots of proton the r e c i p r o c a l  ( t o p g r a p h ) and  graph)  versus  r o t a t i o n a l c o r r e l a t i o n time f o r a methyl  group  on a m a c r o m o l e c u l e .  Broken  [3.2.7]; the x axis these equations. yields  for  curves are o b t a i n e d from  time argument i s i n t e r p r e t e d  For the s o l i d  curves, Equation  s e p a r a t e l i n e s ( p , q , r , and  respective x  c  the s o l i d  = IO"  (bottom  8 , 5  curves  c h a r a c t e r i z e s motion  ,  10" , 8  i s t o be  10"  Equation  as  in  [3.2.5]  s) f o r macromolecules 7 , 5  , and  interpreted  about the t r i a d  axis  10"  sec;  7  as i g E T and  x  c  r  of  the x where  u  axis.  In a l l p l o t s ,  x  R  rotation 8  perpendicular to this o and r„ = 1.78 A .  axis  = 2-n- x 10  -1 sec"  -54-  FIGURE 3.2:  Plot of a reduced, frequency weighted,  spectral  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^. 2 c o r r e s p o n d s t o o>j = XQ E X -  (1  + w x ) 2  2  - 1  u>  The u p p e r d o t t e d c u r v e  2-1 with the i d e n t i f i c a t i o n  T^)  c u r v e s c o r r e s p o n d t o coj =  The two s o l i d  2  x  10x2(1+  (1 + ( A )  a n d coj = (1/4)O>T  K  2  C  of the generalized  time XQ, with x  respectively.  and x  c  - 1  (3/4)CJX  with the  C  analogous i d e n t i f i c a t i o n R  density  correlation  R  -55-  -56-  3.3  EFFECT OF ANISOTROPIC MOTIONS ON It will  lation  AND_T  2  RATIOS  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  times, the dipolar  values f o r the corre-  i n t e r a c t i o n c o n s t a n t , £ , m u s t be known.  However,  a r a t i o o f r e l a x a t i o n m e a s u r e m e n t s a t two f r e q u e n c i e s m i g h t be  expected,  in  time t o  principle,  to allow the interaction  1 be f o u n d  constant  and c o r r e l a t i o n  1 ft  independently.  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^CBQJ/T^CBQ),  and  T-j ( B Q ) / T 2 ( B Q ) a r e shown i n F i g u r e s 3 . 3 , 3.4, a n d 3.5 8  For  t h e s a k e o f d e f i n i t e n e s s , BQ = 2.35T (COQ = 2ir (U)Q = 2 . 2 U Q ) , t h e s e  5.17T NMR  studies.  F i g u r e 3.1.  typical  The mode o f p l o t t i n g I n each o f t h e s e  a function of x -8  being  = T -(=(4D  R  0  s e c " ) a n d B~ =  operating frequencies of proton  hopefully facilitates  comparison with  p l o t s , the appropriate r a t i o = (SD )~'  - 4D )" ) f o r 1  U  x 10  respectively. -1  J l  r  ±  T  i s plotted  as  = 10 ? , 1 0 " * ,  [  8  5  -7.5  l.CJ  "andilO  - 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 ) .  line  i s that obtained  The d o t t e d  S  Consider  first,  [ 3 . 2 . 7 ] w i t h X Q now d e f i n e d a s T^'  using Equation t h e T^  r a t i o p l o t , F i g u r e 3.3.  I t c a n be  easily  p rationalized  t h a t t h e maximum v a l u e o f t h i s  a s y m p t o t i c a l l y a p p r o a c h e d when T , T C  d  >>  tol  K  1  or x  U  The minimum v a l u e o f u n i t y i s a p p r o a c h e d when J t h a t even f o r x  noted  unity f o r certain  £  i s (BQ/BQ)  ratio  Q  > col C  U  «  co" .  < /x  x . D  C K  However, i t i s  1  mobility.  F i g u r e s 3.4 a n d 3.5.  I t i s interesting  are not double  in x ,  double  1  U 1  A n a l o g o u s p l o t s o f T 2 ( B Q ) / T 2 ( B Q ) a n d T^ ( B g ) / ^ ^ )  distinction:  and co"  > co^ , i t i s p o s s i b l e t o approach the value o f  rates of internal  valued  1  and i s  2  T j s a r e double  the T  2  t o comment  ratios  valued, their  v a l u e d , T i s a r e n o t (compare d o t t e d  are.  a r e shown i n  t h a t w h e r e a s T-. r a t i o s  T h u s we s e e t h e c u r i o u s  r a t i o s a r e n o t ; T2 lines  ratios are  i n F i g u r e s 3.1, 3.3, and  -57-  3.4). be  Other c h a r a c t e r i s t i c s of  e l a b o r a t e d upon a t f u r t h e r  that  these plots  important conclusion  arrived  the  presence of  flexibility  at from these p l o t s  perimental of  be  ratios  a meaningful  o f any  value, at  neither  noted that termine the possible  NMR  Also,  the  (as w e l l  l e a s t one  of  the  white, that  using a p a i r of relaxation  t h e r e may  right.  spectral be  density  a 10%  error  Furthermore, f o r t h i s two is,  will  motional ratios  of  i n the  ex-  composite spectral C J Q T  C  -  1 or  U Q T  R  be  v i e w e d as  the  problem  -  densities 1.  be must  I t might times to  simplest  be  de-  of a l l  spectroscopy experiments - a very powerful 1 7  not  fact  technique to  frequency dependent r e l a x a t i o n  p a r a m e t e r s may  relaxation  i s the  b e i n g d o u b l e v a l u e d ) adds t o the  interpretation.  b l a c k nor  n i q u e i n i t s own  fact that as  and  ( o r more g e n e r a l l y ,  a s y m m e t r y ) makes a d i r e c t i n t e r p r e t a t i o n o f dubious worth.  self-evident  length.  The  internal  are  tech-  -58-  FIGURE 3.3:  Plots  of  anisotropy  (220MHz)/T (100MHz) as a f u n c t i o n ]  (TQ =T^  values of T ; C  T  c  = [4D  = IO  - 9  I(  -4D  X  , IO"  ] 8 , 5  _ 1  ,  ) f o r four different I O " , and 1 0 " 8  ( c u r v e s p , q , r , and s r e s p e c t i v e l y ) . corresponds  t o i s o t r o p i c motion  of motional  ( x  7 , 5  sec.  The d o t t e d c u r v e n  =T ). 9  -60-  FIGURE 3.4:  Plots  o f T ( 2 2 0 M H z ) / T ( 1 0 0 M H z ) as a f u n c t i o n 2  anisotropy  2  ( x  Q  = T  R  = [4D„ - 4 D ^ ] ~  values of x ; x = 10" , I O c c 9  - 8  5  t o i s o t r o p i c motion  8  ( x  7  The d o t t e d n  motional  ) f o r four different  1  " , 1 0 ~ , and I O " '  ( c u r v e s p , q , r , and s r e s p e c t i v e l y ) . corresponds  of  =x ). 9  5  sec.  curve  iq  2  $2  eg  (°CO) 5L(°CO>I l  ^  Q  -62-  FIGURE 3.5:  Plots  o f T (100MHz)/T (100MHz) as a f u n c t i o n ]  anisotropy  2  ( Xg = x  R  E [ 4D  (j  values of x : x = 10" , I O " c c 9  -4D^ ] 8 - 5  ,  _  to i s o t r o p i c motion  ) f o r four different  1 0 " , and I O " ' 8  ( c u r v e s p , q , r , and s r e s p e c t i v e l y ) . corresponds  1  of motional  ( x  7  The d o t t e d n  E X  9  ) .  5  sec.  curve  -64-  3.4  EFFECT OF ANISOTROPIC MOTIONS ON NUCLEAR OVERHAUSER 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  ENHANCEMENTS  10  (NOE) h a s p r o v e n 19  t o be a v e r y p o w e r f u l far,  the primary  identification ation  tool  usage has been r e s t r i c t e d  investigations.  problems i n n u c l e a r magnetic  biological  u s a g e o f t h e NOE e m e r g e s , as i t p r o v i d e s a n o 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 processes. Interest  relax-  relaxation.  an i n d i r e c t c o n s e q u e n c e o f t h e r e c e n t t r e n d t o w a r d  NMR s t u d i e s , a n o t h e r  Thus  to m o l e c u l a r s t r u c t u r a l and  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  pathways and r e l a t e d As  f o r v a r i e d chemical  i n this  parameteriznngthe  f a c e t o f t h e NOE was a r o u s e d  dynamics o f m o l e c u l a r  by r e c e n t  homonuclear 20  Overhauser s t u d i e s performed vant  discussions o f the motional  a m p l e s c a n be f o u n d Consider nonequivalent  Rele-  u s a g e o f t h e NOE a n d e x p e r i m e n t a l e x -  i n t h e s e and r e l a t e d  papers.  an e n s e m b l e o f s p i n s y s t e m s c o m p o s e d o f t w o n e c e s s a r i l y spin one-half nuclei  the ensemble being be  b y A. A. B o t h n e r - B y a n d c o - w o r k e r s .  d e r i v e d from  s u b j e c t e d t o a l a r g e Zeeman  i n nonequilibrium with the surroundings.  field;  I t can e a s i l y  a S o l o m o n - t y p e 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 treatment, 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 , [3.4.1a]  [3.4.1b]  x obey t h e e q u a t i o n o f m o t i o n ,  (d/dt)Y = - © Y  .  [3.4.2]  -65/i  The  vector Y  i s defined  \  / I / T !  a s I <- l a n d  i9  x  = I  C  I/T,  T  I S  I.  Q  The expres-  sions f o r t h e elements o f © a r e , E J (u)  1/T*  00  1/T| E - J S  SI w i t h T-| The  (  1  W i  a n d T-j o b t a i n e d  - oi )/3 + 2 J ' " ( 2  described  by E q u a t i o n  [3.4.3a]  s  + ai )  [3.4.3b]  s  U l  i nEquations  indices.  [ 3 . 4 . 3 ] a r e assumed  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 However, i n t h i s  steady  simple  state Nuclear  example,  Overhauser  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  observed, t h e steady  s t a t e boundary c o n d i t i o n s a r e , (d/dt)Y  = 0  Y =(  fractional  [3.4.4a]  .  [3.4.4b]  e n h a n c e m e n t f a c t o r , n (n = l ^  o b s e r v e d s i g n a l c a n now be d e r i v e d  t e a d y  s t a t e  1  -6J '" (  0  0  ^ ! - w ) - 6J '" (ai 2  s  2  2  I  / < I > ) , of the T  z  from t h e s o l u t i o n o f Equation  t o t h e s t i p u l a t e d conditions o f Equation  n =YsYj ^ 2  + u>)  [ 3 . 4 . 2 ] i s d e p e n d e n t upon t h e i n i t i a l  t o consider only the c l a s s i c  subject  I  [3.2.5].  the 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 .  The  2  upon t h e e x c h a n g e o f t h e a p p r o p r i a t e  unique s o l u t i o n t o Equation  preparation  Effect.  2  2  s  s p e c t r a l d e n s i t i e s appearing  t o be a d e q u a t e l y  we w i s h  + 2J '" ((o  1  s  s  various  The  0 0  - a) )/3 - J ' " ^ )  I  + w ))(-J s  0 0  (w  [3.4.4]:  r  w ) + SJ  1  '"  + u> )) . _1  W l  s  Obvious from Equation  [3.4.2]  1  ^) [3.4.5]  [ 3 . 4 . 5 ] i s t h e f a c t t h a t n i s d e p e n d e n t 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  promising  -66-  for  those  NMR  observable.  is  seeking micoroscopic motional  n e i t h e r b l a c k nor w h i t e  (which  i s ofen  molecular mobility.  macroscopic  such  ratios  biologically  a tool  to quantitate  the usage o f such  as T^/T^  from  information  t h e more  (BQ)/T^(BQ)  ratios,  b r i e f l y mentioned  density  s t r e n g t h s ) , then  to the i n f o r m a t i o n gathered  density ratios  r a t i o s , or T ^ B i J / l ^ B g )  field  in principle yield  As m i g h t be e x p e c t e d ,  i n many ways a n a l o g o u s spectral  the case f o r  at present experimental  i n d e e d , the measurement o f n w i l l  familiar  a  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  i n t e r e s t i n g molecules  is  i n f o r m a t i o n from  i n the l a s t  section.  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 macroscopic of  observables  molecular m o b i l i t y , i s often clouded  ached w i t h c a r e . in  into a detailed  Discussion of this  connection with interpretation  ( o r even m e a n i n g f u l ) and  problem  description  a m b i g u o u s and m u s t be has  of r e l a x a t i o n  appeared times  and  NMR  i n the  approliterature  r a t i o of  relax-  9 ation times.  We  shall  now  extend  t a t i o n of nuclear Overhauser Doddrell outlined  et. a l .  in this  have p r e s e n t e d  0  except  However, t h e i r  h e t e j ^ o n u c l e a r NOE  interpre-  enhancements.  s e c t i o n , and  pletely equivalent. tational  1  these c o n s i d e r a t i o n s to the  a similar  treatment  to the  problem  f o r c h a n g e s i n f o r m a l i s m , i s comr e s u l t s were s p e c i a l i z e d  s t u d y where s p i n S i s a p r o t o n  and  t o a compuspin I, a  13 C nucleus. t h a t t h e two  In the s p e c i a l i z e d nuclei  case  c o n s i d e r e d h e r e , i t i s assumed  a r e o f t h e same s p e c i e s (YJ  resonant  frequencies are separated  validity  of the  = Y$)J  by a t l e a s t a few  but  that  linewidths.  their The  approximations, J  0  0  ^  - u> ) s  - J  0 0  (0)  [3.4.6a]  -67-  1,-1  (coj suffices implied  + to ) - J ' (2u ) 2  s  t o computationally define i nt h i s  [3.4.6b]  the homonuclear approximation as  observable normalized 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  spins  motion i s completely characterized = (6D)" .  reorienting  shall identify  is  independent o f a p a r t i c u l a r experimental  00  1-1  (0) - - J ' fin i J ( 0 ) » -J'' u u  i n both the  site  o f the  f a s t motion regime  The i n t r i g u i n g p r e d i c t i o n  complete d i s a p p e a r a n c e o f the  two s p i n s  o f Am = ±1 m u t u a l f l i p s . course, only  dimension-  UQT2 << 1;  motionally  i n the  a s s i s t e d mutual  c o m p l e t e l y dominate the  this  straightforward  only  intramolecular  homo-  but  oppo-  relaxation transition  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  s i g n i f y whether the motion i s "slow" o r " f a s t " .  measure o f m o b i l i t y .  [3.4.5]  observed t r a n s i t i o n  o f a n i n t e r m e d i a t e v a l u e c a n be d i r e c t l y t r a n s l a t e d  As  the  0  0  s l o w m o t i o n l i m i t where the  flips  Zeeman f i e l d ,  which  2 -2  n u c l e a r case i s the the  results in a fashion  (COQ) - J ' (2COQ)) a n d s l o w m o t i o n r e g i m e (WQT2 >> 1; i ? ? (OJ ) - J ' (2u )) are e a s i l y p r e d i c t e d from E q u a t i o n  t o be 1.5 a n d 0.0 r e s p e c t i v e l y .  in  Again,  WQT2, i s c h o s e n a s t h e u n i t o f m o b i l i t y .  asymptotic values  J  To p r e s e n t t h e  1  isotropically (i.e. the  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 ) .  we  The  [3.4.6c]  0  section.  In F i g u r e 3.6, the  less quantity,  2  Observation  into a quantitative  However, i t i s o f paramount i m p o r t a n c e t o note a p p r o a c h presumes i s o t r o p i c m o t i o n dipolar  o f course,  relaxation).  mentioned p r e v i o u s l y ,  assumes m o t i o n s c h a r a c t e r i z e d  (and  the  simplest  extension o f this  by t w o d i s t i n c t d i f f u s i o n a l  that  picture  constants.  -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  (contour) p l o t o f the normalized  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 tropic f l e x i b i l i t y T  r  as (D„ As  (CUQT ).  Three i l l u s t r a t i v e  The  angle  ing  as  [3.2.5], n w i l l  a p p r o a c h e s t h e NMR m a g i c a n g l e  identify  r o t a t i o n angles  (save  rotation  of constant  absence o f i n t e r n a l  i n F i g u r e 3.7.  (3 g.j ma  =  c  cos~V3  _ 1  result-  54.7°).  The  Notice that only subtle  ranges.  Furthermore, f o r  d e v i a t i o n f r o m F i g u r e 3.6 f o r a n y r a t e  ( f o r 3 = 0 ° t h e r e i s no a n i s o t r o p y d e p e n d e n c e ) .  rotation  internal  upon t h e a n g l e  the p a r t i c u l a r behavior  two a n g u l a r  0° < 3 < 2 0 ° , t h e r e i s v e r y l i t t l e  with those  arbitrarily  a n y p l o t f o r w h i c h 20° < 3 < 45° a n d t h e p l o t  d i f f e r e n c e s e x i s t between t h e s e  contours  ) and t h e a n i s o -  are presented  3 = 90° t y p l i f i e s t h e r a n g e 60° < 3 < 90°.  rapid  c  depend e x p l i c i t l y  values of t h i s angle  p l o t f o r 3 = 36° t y p l i f i e s  all  T  s e c t i o n , we s h a l l  3 = 54° was c h o s e n t o i l l u s t r a t e  of internal  w  1  3.  for  ( g  DJ" .  seen i n Equation  3  In t h i s  r  rotation  fractional  those  approaching  (asymmetrical  t h e magic a n g l e ) ,  reorientation) results  1 + n approach the l i m i t i n g  For  extremely i n the  values obtained  i n the  ( 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  values  seen i n F i g u r e 3.6).  n i t u d e o f ..the o v e r a l l  motion.  This fact  i s t r u e r e g a r d l e s s o f t h e mag-  This rather surprising  contrasted with the results obtained  r e s u l t i s t o be  i n r e f e r e n c e 10.  At intermediate  r a t e s , t h e d e s c r i p t i o n becomes much m o r e 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 U Q , D_^(T ) , and  r  a rather lengthy perusal of Equation  [3.2.2].  (T ), F i g u r e 3.8  a s r e v e a l e d by facilitates  this analysis. This f i g u r e plots contours o f the d i f f e r e n c e values A o f N0E e n h a n c e m e n t s . 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 rot  ( n)  by  -69-  first by  calculating  n assuming i n t e r n a l  the v e r t i c a l axis)  i n g no i n t e r n a l the  overall  and t h e n s u b t r a c t i n g  rotation.  This quantity  isotropic mobility  dent v a r i a b l e .  rotation  ( o f the magnitude  indicated  the value calculated  by a s s u m -  i s t h e computed as a f u n c t i o n  which i s p l o t t e d  as t h e h o r i z o n t a l  Therefore, t h i s non-negative quantity  of  indepen-  i s o f the simple  form, \ o t  n  E  \> j. D  ~  •  3 = 9 0 ° , b u t as m e n t i o n e d e a r l i e r , t h i s  F i g u r e 3.8 a s s u m e s t h a t  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 The  full  interprets  F i g u r e 3.6, how w i l l F i g u r e 3.7? (outside  7 ]  geometries.  an O v e r h a u s e r e n h a n c e m e n t i n t e r m s o f  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  U> T  r  = IO"  A n / A l o g U ^ ^ ) - -2. of T  ( D - Dj_ - 2u )  0 - 3  Q  u  s e e n i n F i g u r e 3.8,  A  r  Q  t  n  using  b e t w e e n t h e l i m i t s 0.2 < n+1 < 1.3  F i g u r e 3.6 shows t h a t  present context),  0  4  will  i t d i f f e r f r o m a more e x a c t i n t e r p r e t a t i o n  duced i n t o t h e i n t e r p r e t a t i o n when  range o f i n t e r n a l  ' -  m e a n i n g o f F i g u r e s 3.6-8 i s now d e v e l o p e d .  If,one naively  the  [ 3  c  i t s usefulness i n  Hence t h e maximum e r r o r  by means o f p l o t 3.6 w o u l d and  UT Q  C  = 10"  occur  * u /2).  {U  0 , 5  ±  intro-  Q  As  w o u l d 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  misinterpretation diffusional smaller  of x  c  by a f a c t o r o f 2 (=00 ' ) .  constants would r e s u l t i n a smaller  misinterpretation  range o f unique r o t a t i o n a l ments i s q u i t e sion  value of  A r o  j.n»  hence, a  Therefore, except f o ra rather  geometries, interpretation  straightforward,  constant within Finally,  of x .  Any o t h e r p a i r o f  y i e l d i n g the overall  limited  o f NOE e n h a n c e rotational  diffu-  a f a c t o r o f two o r l e s s .  these plots  suggest a promising, a l b e i t tedious,  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  range o f  o f performing  NOE  -70-  experiments  at various f i e l d  stances, y i e l d of internal  not only the overall  rotation.  t h e NOE e x p e r i m e n t ,  those  results  examination  of this  parameter i n e i t h e r spectral  r a t e s , but a l s o the rate  information obtainable  i n this  s e c t i o n bear  a strong resemblance t o  2  behavior  2  is directly  of linear  w h i c h c a n be i n t e r p r e t e d  lation  function".(unless the t r i v i a l  Viewed  i n this  as s i m p l y as " t h e a r e a case  concepts.  solely  do n o t d i f f e r ratios.  This  contains  under t h e c o r r e -  o f extreme-narrowing  the results derived i n this  e x t e n s i o n s o f some v e r y g e n e r a l  1 0  that n e i t h e r combination  a term  light,  sensitive  of  S i m i l a r l y , the  p e r t i n e n t f o r T-| (Bg)/T-j ( B Q )  to the fact  combinations  c o n t a i n i n g a term  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).  attributed  closer  c a n 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  i s g i v e n by a r a t i o  the calculations  A  0  f o r the heteronuclear Overhauser enhancements  much f r o m  from  i n conjunction with other s t u d i e s ,  comparable s t u d i e s o f T ( B Q ) / T ( B ) r a t i o s .  case  circum-  avenue o f s t u d y .  d e n s i t i e s , each c o m b i n a t i o n  to the area results  rotational  e s p e c i a l l y when u s e d  presented  deduced from  This could, i n favorable  Indeed, the motional  a p p e a r s t o be a p r o m i s i n g The  strengths.  i s assumed).  section are simple  -71-  FIGURE 3.6;  P l o t of the normalized  NOE  fractional line  as a f u n c t i o n  of i s o t r o p i c mobility  spins relaxed  by d i p o l a r  intensity, 1 +  of a pair of  interactions  alone.  n»  homonuclear  -72-  -73-  FIQURE 3.7:  Contour line  (topographical) p l o t s of the normalized  i n t e n s i t y , 1 + n, as a f u n c t i o n o f both o v e r a l l  m o b i l i t y and a n i s o t r o p y and  NOE  T i r  = (D  | (  -D^)  - 1  .  i n motion.  T  c  i s defined  as  fractional isotropic (60^)"^  The d e p e n d e n c e o f t h e d i r e c t i o n  internuclear vector with  respect  to the internal  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  of the  rotor axis  i n the discussion.  C o n t o u r s a r e shown f o r (1+n) = 0 . 0 5 , 0 . 2 5 , 0.75, 1.25, and  1.45  -75-  FIGURE 3.8:  Contour  (topographical) plots of the difference  (  ) o f t h e NOE  A r  o  t  n  enhancements  a n i s o t r o p i c m o b i l i t y minus t h a t mobility)  as a f u n c t i o n  a n g l e o f B = 90°.  for further details. 0.05, 0.2, 0.4, a n d  calculated  of " s i z e "  See E q u a t i o n  (calculated  assuming  assuming  isotropic  and "shape" f o r t h e g i v e n  [ 3 . 4 . 7 ] and r e l a t e d  C o n t o u r s a r e shown 0.6.  values  for A  .n = rot  discussion 0.01,  -77-  3.5  SUMMARY In t h i s  sities  are  c h a p t e r , we  probed i n the  commonly p e r f o r m e d . ation on  has  other  For  been assumed.  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 denvarious  simple  r e l a x a t i o n experiments which  s i m p l i c i t y , only However, t h e  are  intramolecular dipolar relax-  e f f e c t of asymmetric r e o r i e n t a t i o n  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  similar to  the  21 present  treatment.  tional  we  internal  explicitly  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 the  various  I t was  this  r o t a t i o n s or wholly  have o n l y c o n s i d e r e d  in  have a n a l y z e d  degree of freedom, i n p r i n c i p l e  multiple we  Although  pairwise  t h e e f f e c t o f one  c o u l d be  extended to t r e a t  a n i s o t r o p i c motions.  two-spin  Furthermore,  r e l a x a t i o n w i t h the  s p i n systems w i l l  addi-  be  linearly  implied additive  couplings.  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  area  under the  cor-  relation  f u n c t i o n ) i s i n d e p e n d e n t o f the magnitude o f asymmetry i n  the  motional  p a r a m e t e r s and  a  static  geometrical  complicated  i s s e n s i t i v e t o the asymmetry o n l y t h r o u g h k -k  factor.  In g e n e r a l , J  f u n c t i o n o f the  '  (u >>  Larmor frequency  and  0)  the  i s a much more relative  magnitudes  of the c h a r a c t e r i s t i c motional  parameters.  I f the motion i s c h a r a c t e r -  i z e d by  differing  orders  two  motional  smaller of these  two  constants being  on  t h a t o f a Larmor p r e c e s s i o n , s o l e l y on J  0 0  the  by  a time scale slower k -k then o f t e n J  (cu  '  of magnitude w i t h than,  »  0)  but  similar  the to  i s dependent  d e g r e e o f asymmetry i n t h e m o t i o n - i n d i r e c t  contrast  to  (0). In g e n e r a l ,  will  i t i s shown t h a t t h e  inclusion  i n f l u e n c e to a l a r g e e x t e n t , the molecular  spectral  density combinations,  although  both T  0  of a n i s o t r o p i c motion i n t e r p r e t a t i o n of c e r t a i n and  homonuclear Overhauser  -78-  e n h a n c e m e n t s 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 due  to their  dependence on J ^ ( - O ) .  e f f e c t o f m o t i o n s s l o w on t i m e to apparent  practical  rise  thinking.  i n t r o d u c t i o n was i n t e n d e d t o e x p o s e v e r y s i m p l e , y e t v e r y aspects o f slow, a n i s o t r o p i c  3.1  i s a general  and  T^ r e t a i n  made.  give  i n terms o f c o n v e n t i o n a l  spin reorientation.  major p o i n t s o f emphasis a r e summarized i n T a b l e s  fication  the influence of the  s c a l e o f t h e Larmor frequency  contradictions i f interpreted  "extreme-narrowed" This  Furthermore,  spin mobility  summary a n d T a b l e  distinctive  of interaction  a n d 3.2.  3.2 s t r e s s e s t h e f a c t  forms i n v a r i o u s motional  ideas w i l l  be  presented.  the Table  t h a t both  limits  c o n s t a n t s o r dynamic parameters  I n t h e n e x t c h a p t e r , a much more s p e c i f i c  t h e same g e n e r a l  3.1  A l l  T^  i f reidenti-  are correctly  calculation  b a s e d on  TABLE 3.1 i n Various Motional 11  R e l a x a t i o n Parameters  Limits:  co  ] < ± D  0  D  >  P a r t 1.  A  <o  < M D  (  D «D„  D =D„ED  XJ°°(0) 7  xT  0 0  /  n\  D ^ E D  1  A  homonuclear Overhauser enhancement  < D  ..  D D « | |  < U  L  W  D D »a)  2  A  N  f ( D )  f(D ,B)  f(D 3)  f(D ,e)  f ( D )  f(D ,B)  f ( D )  f(D„,3)  f(D ,3)  1/6D  ( 3 c o s 3- 1 ) / 2 4 D  f(D„,3) 2 3 s i n 3/  L  x  2  1  A  12D/5co  6sin  2  2  2  0  f(D ,e)  i 5  2 3D /5coQ  16D  n  1/20D  (3cos 3-l) /80D  -1  -1  A  . 2s  lc  1 C  n  2  ( 3 c o s 3 - l ) /80Dj_  2  2  x  3(3cos 3-l ) \ / 5 w  n e - l S s m 3)  H  (3cos 3-l) /80D  2  2  A  2  2  L  §  1/2  1/2  heteronuclear Overhauser YC72YJ enhancement n Assumptions: by E q u a t i o n If  1  f(D)  XT-  2  D  Y /2 s  9(Y  g(Y Y )  Y i  P  S  -1  I 5  Y /2  Y )  S  S  -1  9(Y Y )  Y I  P  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 ( t o ) i s c o r r e c t l y d e s c r i b e d ^o If = (Y tir" ) 3/2. 0 , a l l values reduce t o t h e 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 .  Two i d e n t i c a l [3.2.6];  3+cos" ( 3 " 1  1 / 2  ),  _  1  x  S  0 0  2  a l j _ terms  3  f  o  r  2  Tj  1  For T" , t h e replacements a r e 9 s i n 1  with  ( 3 c o s 3 - 1 ) f a c t o r s a r e t o be r e p l a c e d by  3sin 3(l6-15sin 3 )/16D„  2  2  3 ( 1 6 - 1 5 s i n 3 ) / l 60D,, a n d 3 s i n 3 ( 1 6 - 1 5 s i n 3 ) / 1 6 D „ f o r 2  2  2  D <to n  2  ;  a n d Dn>uQ  0  respectively. § I t i s a s s u m e d t h a t 3^ c o s §§  I f spin S i s irradiated w h e r e x= Y $ / J Y  - 1  (3~  1 / / 2  ).  and s p i n I i s o b s e r v e d , g (  ,Y )=X[-(1+A) +6(1-X) ]/[(1+A) +6(1-X) +3(1-X ) ] 2  Y j  S  2  2  2  2  2  ,  2  TABLE 3.2 Relaxation Parameters i nVarious Motional 11  u  w  0  Dn Dj.=D =  72Da/5w  2  V  (  2  T  d  T  l  0  48 VD  /  d  D  l )  D  l  (dT /dD ) 2  A  D n  D l  .a a' a  1 1  a  1 1  =  x  i  < D  72D  ii  < h i  ||C4  D  0  ' 75UQ  3a'/10D  a'"/D  3a710D  A  2  2  ±  (Y fir" ) /4 2  3  2  a(3C0S B-l)/4 ? = a s i n B/2 =  2  '= 9 a s i n 3 ( 1 6 - 1 5 s i n B ) / 1 6 2  2  Other assumptions a r e as d e s c r i b e d  i n TABLE 3.1  >>a)  1  2 2 1 0 / ftcosVl^ 3 c o s B - l \ 1 /P||\(3cos B-l) 96 D„D) \ s i n g y T20\Dj 2 _^ w  D  72Dj_a / 5COQ  M  3a710D  1  ii x o  s  i  n  B  (  1  i  2  r  i  2  B  )  4 8 V D  -0  -0  +  +  +  +  -0 -0  (dydD,,)  n  >D  l  (dT /dD„) . 1  0  P a r t 2.  'i D  3a/10D J  > D  Limits:  X  ;  ±  -81-  REFERENCES: CHAPTER I I I  1.  I. Solomon, Phys.  Rev.  2.  H.  S h i m i z u and  S.  F u j i w a r a , J . Chem. P h y s .  3.  D.  G e s c h k e , Z.  P h y s i k 212,  4.  R. F r e e m a n , S. W i t t e k o e k , and (1970) .  5.  D.  M i c h e l , Ann.  6.  H. S c h n e i d e r and 357 ( 1 9 7 3 ) .  H.  7.  R. K. H a r r i s and 394 ( 1 9 7 3 ) .  K. M.  8.  P.  9.  L. G.  S.  N a v o n a n d A.  34,  G.  1501  (1961).  (1968). R.  R.  E r n s t , J . Chem. P h y s .  Phys.  Schmidt,  D o d d r e l l , and  1529  (1971). P h y s i k 28, ~~~  3 3 , 249  346  9,  (1973);  383  R.  B. D.  Sykes,  Komoroski,  L a n i r , J . Magn. R e s .  8,  ibid.,  (1961). 95_, 5132  A l l e r h a n d , J . Chem. P h y s .  and  ibid.,  (1973);  M a r s h a l l , J . Amer. Chem. S o c .  G l u s h k o , a n d A.  52, —  W o r v i l l , J . Magn. Res.  Mod.  1 1 . A. G. M a r s h a l l , P. G. (1972).  13. G.  169  S c h m i e d e l , Ann.  W e r b e l o w and A.  12. A. A l l e r h a n d , D. (1971) .  (1955).  P h y s i k 2 7 , 389  H u b b a r d , Rev.  10. D. D o d d r e l l , V. (1972).  9 9 , 559  144  56, ~  Biochem. 11, ~  J . Chem. P h y s .  (1973).  3683  3875  55, —  189  (1972).  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 , Biochem. B i o p h y s . A c t a 310, 1 (1973).  15.  C. H. F u n g , A. S. M i l d v a n , A. A l l e r h a n d , R. S c r u t t o n , B i o c h e m . 1_2, 620 ( 1 9 7 3 ) .  and  C.  Komoroski,  E. M c C o l l ,  and M.  C.  16. T. R. K r u g h , 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 : T h e o r y 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 , A c a d e m i c P r e s s , New Y o r k , 1974. 17.  F. N o a c k , NMR  B a s i c P r i n c i p l e s and  18.  L.G.  19.  J . H. N o g g l e a n d R. A c a d e m i c P r e s s , New  P r o g r e s s 3, 83  W e r b e l o w , J . Amer. Chem. S o c , E. S c h u r m e r , The Y o r k , 1971.  (1971).  t o be p u b l i s h e d . Nuclear Overhauser  Effect,  -82-  20. P. B a l a r a m , A. A. B o t h n e r - B y , a n d E. B r e s l o w , B i o c h e m . ]_2, 4 6 9 5 ( 1 9 7 3 ) ; i b i d . , J . Amer. Chem. S o c . . 9 4 , 4 0 1 5 ( 1 9 7 2 ) . 2 1 . W. T. H u n t r e s s , J . Chem. P h y s . 4 8 , 3524 ( 1 9 6 8 ) .  -83-  CHAPTER IV DIPOLAR RELAXATION OF  4.1  THREE-SPIN SYSTEMS  INTRODUCTION As was  methyl  hinted at i n the previous c h a p t e r , the i n t e r p r e t a t i o n  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  is  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  to  take a closer  on t h e u s e o f - C 0 C H  coupling  3  (compared  (resulting  containing  be  w i t h -CH  o r -CH^  The  and  s i t e s on  b i n d s and  macro-  lack of  2  based  superior  experiment  at the desired s i t e  small molecule which  beneficial  t i m e s have been  groups)  i n a single resonance).  This  the~problem.  o r -COCF^ g r o u p s , on a c c o u n t o f t h e i r  involves either covalent "labeling" methyl  i t will  at s p e c i f i c  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  signal-to-noise  in applications.  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  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  of  scalar  typically  o r use o f a  exchanges  rapidly  and  3 reversibly  to the macromolecule,  a t i o n behavior f o r the methyl  group 4  r e a d i l y e x t r a c t e d from the d a t a . offers the  practical  advantages  interpretation  c o m p l i c a t e d by two  i n terms factors  i n e i t h e r case, the magnetic  relax-  a t t a c h e d t o the macromolecule  U n f o r t u n a t e l y , w h i l e a methyl  i n measurement o f t h e r e l a x a t i o n o f g r o s s and  internal  p e c u l i a r t o methyl  First,  group  parameters,  molecular motion  groups:  are  the  is motion  -84-  o f a n y one m e t h y l  proton  leading  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-  cay;  to rather  i s clearly  second, i fthe i n t e r n a l  f a s t , then s p i n - i n t e r n a l  correlated  to that  o f t h e other two,  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  r o t a t i o n e f f e c t s may d o m i n a t e 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 o n t h e s u b j e c t ,  there  i s at present  some c o n f u s i o n a s t o t h e e x t e n t , o c c u r r e n c e , a n d i m p o r t a n c e o f n o n e x p o n ential  nuclear magnetic r e l a x a t i o n  f o r a methyl  group i n l i q u i d media being r e l a x e d  (or trifluoromethyl)  by t h e i n t r a m o l e c u l a r  dipolar re5  T a x a t i o n mechanism.  While nonexponential  relaxation  has been  predicted  6-15 and  unquestionably observed  dipolar  interference  molecules properties  in solids,  t e r m s h a v e n o t been c o n v i n c i n g l y  in solution.*  Macromolecules  relaxation  p r e v i o u s work d e a l i n g  with  1  one-half nuclei c  dipolar  r e v i e w e d , and H u b b a r d ' s ^ '  molecular dipole-dipole  Q  motional  > 1).  f o r such s p e c i e s . 3  However, t h e de-  group a t t a c h e d t o a macro-  has n e v e r b e e n w o r k e d o u t , i n s p i t e o f t h e many  attempts t o study nuclear r e l a x a t i o n  (a) T  observed f o r small  i n solution exhibit  b e h a v i o r o f a -CH^ o r - C F  molecule i n solution  briefly  e f f e c t s due t o  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 tailed  nonexponential  relaxation  f o r b i g molecules.  relaxation 1 7  In this  i n multi-spin  cross-correlation  treatment o f i n t r a -  molecules  F i n a l l y , the quenching e f f e c t o f i n t e r n a l  in a liquid  discussion  sample.  spin  in solution  spin-rotation 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  observation  systems i s  f o r a group o f t h r e e e q u i v a l e n t  i s extended t o t h e case o f large  * See F i g u r e 7 and t h e r e l a t e d  chapter,  relaxation  i n R e f e r e n c e 10 f o r a  possible  -85-  rates  i n terms o f m o l e c u l a r r o t a t i o n a l motion i s d i s c u s s e d . For  problems i n v o l v i n g three  that of N equivalent situated  positions i n the molecule.  m o t i o n o f a n y two n u c l e i  pairwise  group  that of a t h i r d  [3.1.6] i s v a l i d ) ,  ( o r f o r any r i g i d  n u c l e i the motion of the t h i r d  neglected.  will  Equation  that the (the "crossi ti s readily  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  the motion o f t h e other be  with  Provided  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 -  However, f o r a methyl three  i s uncorrected  functions" are zero,  shown t h a t t h e t o t a l ual  case i s  ( i . e . h a v i n g t h e same L a r m o r f r e q u e n c y ) n u c l e i  at equivalent  correlation  o r more s p i n s , t h e s i m p l e s t  Various  now be l i s t e d  proton  t w o , and c r o s s  frame c o n t a i n i n g  i s clearly  chapter. at least  d e t e r m i n e d by  c o r r e l a t i o n s may n o t i n p r i n c i p l e  attempts t o account f o r such c r o s s - c o r r e l a t i o n s  briefly.  -86-  4.2  RESUME OF  PREVIOUS STUDIES  Working w i t h i n the framework o f the s e m i c l a s s i c a l  density matrix  18 t h e o r y o f r e l a x a t i o n , Hubbard relaxation  behavior  in his classical  of e i t h e r three or four equivalent spin  nuclei  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  methyl  group) or at the corners  spherically  represented  «  1, he  by t h e sum  only very s l i g h t l y  d i f f u s i o n , and  showed t h a t t h e  o f two  triangle  an  neglected.*  In a l a t e r  Assuming  relaxation  i n which  cross-correla20  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 times  longer than the  p e r i o d , and  transverse  In t h i s  computed both  case,  both  the l o n g i t u d i n a l  decays are c o r r e c t l y  and  was  the r e s u l t a n t d i f f e r e d  from the s i m p l e r c a l c u l a t i o n  t i o n s were c o m p l e t e l y  one-half  "extreme-narrowing"  longitudinal  e x p o n e n t i a l s , but  the  (as f o r a  of a regular tetrahedron.  symmetric r o t a t i o n a l  c o n d i t i o n , co^x^  paper c a l c u l a t e d  represented  Larmor  relaxation.  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 0 2 ^' ' ^ l° 9it dinal 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 the 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  U )  T  K <  a n c  o  r  n  u  23 the uncorrelated r e s u l t s .  I t was  t h i s work w h i c h l e d Abragam  remark i n h i s famous monograph t h a t i n c l u s i o n o f t e r m s was  of theoretical  o f t quoted * The  o f t h r e e and  ( 1 9 5 7 ) by  similarities has  practical  i n t h a t BPP  p o i n t ) c a n n o t be d i r e c t l y systems  f o u r - s p i n s y s t e m s was 19  I. Aleksandrov.  b e t w e e n t h e two  the f a i l i n g  multispin  b u t o f no  cross-correlation importance,  an  statement.  relaxation  analyzed  interest  to  Although  simultaneously  there are  superficial  sets of r e s u l t s , Aleksandrov's theory  extended  (see Chapter I I ) .  ( w h i c h was i n any  t h e assumed  satisfying  approach starting  f a s h i o n to  -87-  However, t h i s  s t a t e m e n t was  about three years  p r e m a t u r e 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 w h e r e H i l t and s i d e r e d an  equilateral  a crystal-fixed  t r i a n g l e of i d e n t i c a l  axis perpendicular  to the  considered  various o r i e n t a t i o n s of the  externally  a p p l i e d magnetic f i e l d  c r y s t a l l i n e model. by  the  sum  d e c a y was from the  I t was  plane  of the  d i r e c t i o n , and  triangle.  respect  to  also treated a  uncorrelated  the poly-  hence, s i g n i f i c a n t l y  predicted  different  result.*  Hubbard's most r e c e n t between the  and  They  l o n g i t u d i n a l decay i s given  o f f o u r e x p o n e n t i a l s , b u t more i m p o r t a n t l y , t h a t t h e markedly nonexponential  con-  s p i n s which r o t a t e about  triangle with  shown t h a t t h e  Hubbard  above l i m i t s  c o n t r i b u t i o n s assume an  o f a methyl  s p h e r e and  a methyl  at the  end  considered  a methyl  group r i g i d l y  group r i g i d l y  o f an  infinitely  attached  approach  bound t o a r o t a t i n g  long  along  intermediate  the  rod.  First,  Hubbard  symmetry a x i s o f  symmetric top molecule undergoing a n i s o t r o p i c r o t a t i o n a l  diffusion  ( 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  internally  r o t a t i o n methyl  group).  Finally,  the treatment  asymmetric top w i t h a r o t a t i n g methyl angle *  with  respect  I t should many s o l i d crux  be  to the  principal  group attached  molecular  noted that Hilt-Hubbard  s t a t e NMR  theory  o f t h i s dilemma i s the j u s t i f i c a t i o n  molecular  axis. has  people despite experimental  needed t o c h a r a c t e r i z e t h e  appears muddled a t b e s t .  a t an 1 7  should  However, a l l  l o n g been r e j e c t e d confirmation.  and  con  the  The  inter-  unique  time s c a l e of  e x i s t and  by  constants  in principle establish a  Arguments pro 22 '  an  arbitrary  of m u l t i p l e time  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 21  extended to  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  d i p o l a r couplings  m u l t i p l e time constants.  was  a  the  these  situation  -88-  explicit that  c a l c u l a t i o n s were r e s t r i c t e d to the  o n l y the  longitudinal  decay ( i d e n t i c a l to the  was  computed; i n the  the  extreme narrowing l i m i t ,  d e c a y s m u s t be  A p a r t from the considered the couplings  the  the  corners of  sults  ferential  that  calculations  special  of  longitudinal  and  beyond  transverse  Hubbard, o t h e r a u t h o r s have  of m u l t i - s p i n  systems.  also  triangle.  i n e r r o r , as  difficulty  The  p o i n t e d out  i n s o l u t i o n of  e q u a t i o n s , many o f w h i c h c a n  be  2  numerical  i n R e f s . 25  and  simultaneous  s e e n t o be  Eisner '  one-half nuclei  actual  14  dipolar  K a t t a w a r and  case of three i d e n t i c a l spin  a 30°-120° i s o s c e l e s  to the  extend these c a l c u l a t i o n s the  so  decay)  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  relaxation  i n t h i s paper are  d o u b t l e s s due  so  transverse  limit,  separately.  problem of  i n the  have t r e a t e d  n e x t s e c t i o n , we  treated  extreme-narrowing  linearly  at  re-  29, dif-  dependent.  25 Zeidler like  has  spins  c a r r i e d out  a c a l c u l a t i o n on  and  a non-identical  has  derived  third  a s i m i l a r case where  spin  form a t r i a n g u l a r  two  arrangement.  26 Richards rowing  limit,  identical  the  a system w i t h  (though not  interesting result that any  number o f  necessarily  b o t h t r a n s v e r s e and  longitudinal  are  considered.  absence of  reduces to the  i d e n t i c a l spins  a single exponential)  for  In the  i n the  well-known f a c t that  d e c a y , e v e n when  cross-correlations, T-|  = T  extreme-narmust have  time-behavior cross-correlations this  statement  under extreme-narrowing.  2  27 Runnels,  in a rather  three equivalent spins a b l e by  acts in  general  o n e - h a l f whose i n i t i a l  a s p i n - t e m p e r a t u r e , the  t o make t h e  c a l c u l a t i o n , has  relaxation  a calculation generalized  i n c l u s i o n of  less e f f i c i e n t t o any  proven f o r a system  preparation  is  describ-  cross-correlation  (slower decay).  of  always 28  Fenzke,  number o f e q u i v a l e n t s p i n s ,  in-  -89-  dependently 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 a s R u n n e l s demon29  strated.  Recently Pyper  30 '  has a p p l i e d  formalism to the cross-correlation results  Buchner  view o f group  p r o b l e m and has r e - d e r i v e d  '  36  relaxation  and  the longitudinal  and t r a n s v e r s e  behavior f o r almost every c o n c e i v a b l e arrangement o f three 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-  equivalent  i n t h e m o l e c u l e a n d n e e d n o t e v e n be " l i k e "  be n o t e d t h a t  f o rcalculations  be e x p e c t e d t h a t  to nonexponential  fore,  i n magnetic 33-37  o f f i v e p a p e r s by S c h n e i d e r  S c h n e i d e r has d e r i v e d  i t would i n general lead  interactions  influence  37 '  spin  should  knowledge about t h e  between d i p o l e - d i p o l e  r e s o n a n c e comes f r o m a s e r i e s  Blicharski.  of  theory.  of c r o s s - c o r r e l a t i o n s  (It  previous  has a p p r o a c h e d t h e p r o b l e m from t h e p o i n t  Another prime source o f q u a n t i t a t i v e  sarily  representation  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  approach.  or four  Liouville  relaxation  treating  "unlike"  cross-relaxation  effects  effects  f o ra multispin  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 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  configurations  spins,  of "like"  nuclei,  even f o r a t w o - s p i n system.  inclusion of cross-correlation  with  Schneider treats  care.)  nuclei.  would Theresystem,  relaxation For c e r t a i n  a l l ranges o f t h e  quantity,  F u r t h e r m o r e , h i s t r e a t m e n t c a n be a d a p t e d t o m o t i o n a l  models i n c l u d i n g  isotropic or anisotropic  ternal  rotation.  rotational  diffusion or i n -  I n c a s e s w h e r e t h e two t r e a t m e n t s d e a l  with  identical  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 s e e n t o r e duce t o Hubbard's. spins,  For various configurations  both t h e l o n g i t u d i n a l  of the three or four  and t r a n s v e r s e r e l a x a t i o n  are, i n general,  -90-  the  resultant  of  b e t w e e n t h r e e and  Other examples of by  Noggle,  lations  38  13  between the  magnitude of the  C-H  out  groups.  i n a methyl 13  or  order),  density  C-H"  include  group,  Kuhlman e t .  al.  even though the  39  a  vectors order  i s the  h a v e come t o a  the  of same  similar 13  of Overhauser e f f e c t s f o r  be m e n t i o n e d h o w e v e r , t h a t  paper  cross-corre-  -relaxation  for cross-correlation  from t h e i r c o n s i d e r a t i o n I t should  C  H'-H"  (to f i r s t  for autocorrelation.  conclusion  13  for  and  1  spectral  exponentials.  e f f e c t of c r o s s - c o r r e l a t i o n  w h i c h shows t h a t  f o r t u i t o u s l y cancel  as  the  seven  CHg  exact behavior  of  13 the  relaxing  tail,  four-spin  a l t h o u g h , as  system,  CH^  t h e s e l a s t two  has  not  calculation.  cross-correlation  terms i n methyl  relaxation  i n t e r p r e t a t i o n of  CINDP,  and  has  recently  approximates the  above l i s t e d  criteria  same t i m e p a r a m e t r i c a l l y to which i s attached  the  DNP,  13  rowing  of  examples, the  being  physically  tractable  i s that  the  m o l e c u l e as  i s thus r e s t r i c t e d to  t y p i c a l magnetic f i e l d s  limit will  be  violated  small of  acids,  one  of  in  13  CH  the  3  groups  which most  closely  reasonable while at  of a symmetric top  a whole.  42  synthetic  molecules  15-75  kg  (proteins,  p o l y m e r s ) , as  '  well  as  may  Unfortunately,  in liquids.  i n NMR,  the  molecule  group which 17  extreme narrowing  by m o l e c u l e s w i t h  t h a n a b o u t 5000 i n aqueous s o l u t i o n nucleic  effect  i t s e f f e c t s on  C relaxation  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 to the  presently  and  The  (at a r b i t r a r y angle) a methyl 16  i n d e p e n d e n t l y of  m a t i o n , and  de-  been r e a l i z e d .  Among a l l t h e  rotate  41  in  papers might i n d i c a t e , t h i s would  provide a meaningful, a l b e i t tedious,  4f)  been c a r r i e d out  the  approxiFor  extreme  molecular weight  nargreater  enzymes, membranes, by  small  molecules  in  -91-  viscous  m e d i a o r by s o l i d s n e a r t h e m e l t i n g  a methyl group  on s u c h m o l e c u l e s i s t h u s unknown.  t r e a t the d i p o l e - d i p o l e on a s y m m e t r i c internal  p o i n t s ; the behavior of  intramolecular  t o p m o l e c u l e o f any  rotation.  I n s e c t i o n 4.3,  r e l a x a t i o n f o r a methyl  size,  we  group  i n c l u d i n g the p o s s i b i l i t y  of  -92-  4.3  FORMULATION OF THE CALCULATION In t h i s  using  as b a s i s f u n c t i o n s  one-half I? I  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  2 2  particles, z  I  + I . 2 3 vention, z  The c h o i c e  of I  and I ,  t h e e i g h t s p i n s t a t e s o f a group o f three  expressed  , and I , where I  values  ]  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  = (I  2  spin  + I ), I = ( I  }  2  of the eight basis  1  2  states  + I ) , and I 3  2  = I  of +  i s b a s e d on t h e c o n -  z  |T> = |1 3/2 3/2> |2> =  |l  |3> =  |1 3/2 ; i / 2 >  3/2 l / 2 >  |4> = |1 3/2 -3/2> |5> =  |1  1/2 l / 2 >  |6> =  |1  1/2 - l / 2 >  |7>  |0 1/2 l / 2 >  E  |8> = where  y  i  1  1  I  l n 44 ( x  ~  ( t )  and  |a> E | -|2 z *  _ x  >  )+  D e f i n i n  i  ( x  22  i n t r o d u c i n g two o t h e r  _ x  33  1/2 - l / 2 >  1° 9  }  I^ss+^'Xee'W  +  combinations o f matrix  3  y (t)  = y^t)  2  y (t) 3  [4.3.1]  '  [ 4  - 3  2 ]  elements,  1  - • -(x -X4 ) 2  11  4  = y,(t) - | ( x  i  r  x  4  4  -2"(X22"X33)  [4.3.3]  )  [4.3.4]  where X„„.  = <a| (t)|a'> X  [4.3.5]  -93-  it  i s r e a d i l y shown  that  16  (d/dt)y,(t) + 12[ J ^ . '  - 1  (co ) + 2 J ' - ( 2 a 3 ) ] y ( t ) 2  0  2  0  2  + 16[-jJ' (co ) - J ' - ( 2 a ) ) ] y ( t ) _1  (d/dt)y (t) 2  2  0  2  0  - 4J »" (2co ) +  =  2  + [-3jj>) + 2J '" (2 2  2  + 4J ' 2  0  JJ' U ) _1  2  U n  2  0  2  1 [6J°°(0)  )]y (t) + - 6J°°(0)  -  + 4J '- (2o 2  0  - Jj^tc-Q) - 2J '- (2. )  (2u) )  _ 2  2  [4.3.6]  3  0  2  30 (0) 0 0  +  +  ) ( )  )]y (t) 1  j j * " («-<,) 1  + lOJ '- ^) 1  lOjJ'-Vo)  1  - 4J '- (2a 2  2  ) 0  )]y (t) 3  [4.3.7] (d/dt)y (t) 3  = [-0 '  U ) - 2J '" (2co )  _ 1  a  2  o  2  0  -  jj»" (a, )]y (t) 1  0  1  7[-9j]| " («o ) " 3jJ» (u. ) + 1 2 J ' - ( 2 c o ) ] y ( t )  +  ,  1  + [9J '- (co ) a  The  subscript  and  the subscript  ]  - jj*"  0  "a" labels  _1  0  1  2  Q  2  («-<,) - 4 J ' - ( 2 < , ) ] y ( t ) . 2  theautocorrelation  "c" labels  2  0  2  0  spectral  thecross-correlation  3  densities  spectral  [4.3.8]  (c=n),  densities  (c^n ; s e e E q u a t i o n s [ 2 . 3 . 1 ] a n d [ 2 . 3 . 1 1 ] ) . A unique s o l u t i o n be  t o these three coupled d i f f e r e n t i a l  found from t h e i n i t i a l  rf-pulse  directed  conditions  along the y-axis  * I t m i g h t be m e n t i o n e d t h a t identical is  r e s u l t i n g from a p p l i c a t i o n i n the rotating  even i n t h i s  simplest  i s (2 )  o f an  frame, causing t h e  o f systems o f t h r e e  one-half p a r t i c l e s , the straightforward  b y no means t r i v i a l . 3 4  matrix of  spin  e q u a t i o n s may  calculation  T h e number o f e l e m e n t s i n t h e r e l a x a t i o n  a n d e a c h e l e m e n t i s a sum o f 4050 t e r m s .  T r u e , many  t h e e l e m e n t s o f R a r e z e r o o r c a n be d e d u c e d f r o m s y m m e t r y ( s e e  Appendix A ) .  However, t h e t a s k  i sstill  tedious.  -94-  equilibrium magnetization during  t o r o t a t e by e d e g r e e s  the pulse are ignored).  a(t=0) = e x p ( - i e l y ) a  a  1  = exp(-riE/kT)/Tr[exp(--fiE/kT)] - ( %  .9] a n d m a k i n g n o t e  a(t=0) =  1 +  Substituting  exp(-iel) y  kT,  )/Tr[3J]  - (1iE/kT)  of the fact that E =  8kT  0  [4.3.9]  a p p r o x i m a t i o n , TIWQ «  w h e r e 3( d e n o t e s t h e u n i t m a t r i x . [4.3  .  y  Now, f o r t h e h i g h - t e m p e r a t u r e  effects  Therefore,  exp(+iel )  T  (relaxation  Equation  [4.3.10]  [4.3.10]  -Wgl , z  I_ exp(+iel )  [4.3.11]  v  z  into  y  but  exp(-iel^) I  Therefore, a(t=0),  z  exp(+iel ) = I s i n e + I cose  evaluated  y  v  [4.3.12]  •:o  0  '.-0  ° \  0  '..  0  -0  0  0  i  0  /3  0  0  0  /3  cote  2  0 /3  u  .  : o : o  3cote  -cote  2  0 a K Z  z  i n the basis of [4.3.1],  0 If n) 1 .L t o n t i s l n e l ~ ' " 8 "^TOT"!  x  0  -3cote  /3 "  u  -  J  J  C  ' 0 .J c o t e 0 , cote 0 1  0  t  0 0 0  0 0 0  0  0  0  0  0  0  0  0  0 0 0  H  u  0  .0 -  u  u  0  1 1 -cote  0 0 0  0  0  cote  0  0  1  0 0  -cote [4.3.13]  I t then  f o l l o w s from Equations  [ 4 . 3 . 2 - 4 ] and [4.3.13] t h a t  y^O)  = (cose - 1 ) < I >  y (0)  = (1/6^(0)  2  Z  T  [4.3.14a] [4.3.14b]  -95-  y (0)  = (V4)y (0)  3  In order it  t o solve Equations  i s necessary  rotational  [4.3.14c]  [4.3.14],  [4.3.6-8] s u b j e c t t o c o n d i t i o n s  t o choose a m o l e c u l a r  dynamics.  .  1  s y s t e m a n d d e f i n e a model  For the general  (but s t i l l  for its  moderately t r a c t a b l e )  system o f a s y m m e t r i c 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 attached  a t an a n g l e  of t h e top, t h e appropriate Transformation  k J a  '- (S) k  1  k  0  V r  J  3  \  3sin 6cos B(5D, (5D^ + D„ )  (6D  L  + 4D.)  (2D  2  + (3/8)[(l (2D  + (kco ) 0  + cos g) 2  X  2  2  (3/4)sin B(2D,  +  + 4D )  L  4 |  2  + 4D ) |(  + (ko> ) Q  2  2  1  +  (5Dj_ + D„ + 4 D )  2  i  + 4cos 3](2D 2  + 4(D„ + D ^ )  2  2  ( 3 / 2 ) s i n B ( l + c o s B ) (5D +D  4  2  ^  4  t|  ( 9 / 8 ) s i n B ( 6 D j _ +-4D.)  +  (k^)  +  2  X  0  -f  o A 2  (6D )  + (ko) )  2  ( 1 / 4 ) ( 3 c  + p )  2  2  +  by F o u r i e r  [2.3.11],  = (-D |o(4-f 4  t o t h e symmetry a x i s  spectral d e n s i t i e s are obtained  o f Equation  °  9  6 with respect  1  + (k^)  u  +4D .) n  + (ku> )  2  Q  + 4 ( D + D.)) U  2  2  )  J  [4.3.15]  J c  k c  - ( S ) - J '~ H> k  k  \ ^ ^ % - ± ^ ^  k  a  0  a  0  8  + (3/2)sin B(l  + 0/8)[(l  X  J  2  0  2  2  n  2  +  (k  U Q  )  2  2  + D,, + 4D.)  1  + 4 D . ) + (ko) )  u  + cos B) ±  (6D _+ 4 D . )  (  2  + D  (2D  /  3  + cos B)(5D  2  (5D  V r  0  2  + 4cos B](2D 2  + 4(D„ + D ^ )  2  1  + 4(D„ + D . ) ) \  + (k^)  2  J .  [4.3.16]  -96The  magnitude of the 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 , which 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 d e p e n d on tion and  the r e l a t i v e o r i e n t a t i o n s of the  [ 4 . 3 . 1 6 ] was 6  = 6^  rotational  be  p o s i t i v e or  internuclear vectors.  \$  d e r i v e d from [2.3.11] assuming  = TT/2.  In t h e s e  equations,  M  the d i f f u s i o n  f o r the  constant  internal  diffusion  be  equations  ([4.3.6-8])  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  for  an  z  > - <I  z  initial  >  T  = <I  z  T > ' ( c o s e - 1)  3  E i = 1  = 1.  o r s w o u l d be e x c e e d i n g l y  bulky to l i s t  magnetization  for  symmetry a x i s  c o n s t a n t , and  first  is  group  The  order,  with constant  i s given  The  and  form  by,  A.exp(-x.t) i 1  pre-exponential  linear  coefficients  Transform methods.  [4.3.17]  p r e p a r a t i o n o f t h e s y s t e m by a e - p u l s e .  tions dictate that E  tudinal  constant  frame.  of the s o l u t i o n f o r the e x p e c t a t i o n value o f I  <I  ^3  r o t a t i o n of the methyl  By t h e u s u a l m e t h o d s , t h e t h r e e s i m u l t a n e o u s , homogeneous d i f f e r e n t i a l  Equa-  =  Dj_ i s t h e d i f f u s i o n  i s the p a r a l l e l  w i t h respect to the m o l e c u l a r  may  - 'f'J^  d i f f u s i o n a b o u t an a x i s p e r p i n d i c u l a r t o t h e  of the symmetric top, D  negative,  Initial  condi-  exponential  i n c l o s e d f o r m , but  the  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  fact-  longi-  (see  Figures  4.1-4). S i n c e <I  A  >^  = 0 ( t h e r e i s no  equilibrium magnetization  i n the  x-y  plane),  <I (t)> x  and  i t suffices  = Tr[ (t)I ] = Tr[a(t)I ] x  x  to determine the time  the d e n s i t y m a t r i x  equations  x  ,  evolution of a ( t ) .  [4.3.18]  On  setting  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  up  the  -97-  longitudinal metrization  calculation,  i s i n terms o f t h r e e l i n e a r l y independent  m a t r i x elements  q^t)  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 combinations o f  o f a, c o n v e n i e n t l y c h o s e n a s ,  = T r [ a ( t ) I ] = /3 R e [ a x  ] 2  + o^]  + 2Re[a ] + Re[a 2 3  5 g  + a  y 8  ] [4.3.19a]  q (t)  = Re[a  q (t)  = 2Re[a ] + q ( t ) = q ^ t )  2  3  + c  5 6  ? 8  2 3  ]  [4.3.19b] - /3 R e [ a  2  w h e r e Re i s s h o r t h a n d f o r " r e a l With these d e f i n i t i o n s ,  + a^]  ] 2  [4.3.19c]  part of".  and the help o f Equations [2.2.11-15], 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 , and q 2  (d/dt)  q i  ( t ) = [-3(J°°(0) + J°°(0)) + B J ^ U Q ) + 2J '- (2o 2  2  J o  + [6(J°°(0)  ) ] q ( t ) + [6(-jJ-- ( 1  1  U o  may be o b t a i n e d ,  3  +  jj' (u) ) - 2 J ' - ( 2 u )  )  + J '- (2a) ))]q (t)  _1  2  0  2  2  0  2  n  [4.3.20]  3  + [2(J '- (co )  (d/dt)q (t) = C J a ' ' ^ ) - J ^ ' ^ ^ ^ d ) 1  a  2  n  2  - J '" (2a> ))]q (t) 2  2  1  Q  jJ'~V ))  -  n  - 2 ( J ' - ( 2 u ) ) - J ' " ( 2 c o ) ) ] q ( t ) + [-(J°°(0) - 0 ° ° ( 0 ) ) 2  2  2  0  2  0  2  + (-J '~V ) + jJ»" (o. ))]q (t) a  (d/dt)q (t) = 3  1  n  [J ' (u> ) - J c ' ~ _1  a  n  1  (  w  + 6J '- (2co )]q (t) 2  + 0j'  2  n  _ 1  0  0  )  3  q  l  [4.3.21]  3  (  t  )  +  [ 3 (  - a'" J  1 ( a ,  0  )  + [-3(J°°(0) - J ° ° ( 0 ) ) +  2  ^ ' ^ S ^  "  2(2J '- (a ) 1  a  )()  U ) ) - 2(J '- (2a) ) + 2 J ' - ( 2 u ) ) ) ] q ( t ) . 0  2  2  Q  2  2  0  3  [4.3.22]  The  initial  [4.3.13],  boundary  conditions can e a s i l y  be f o u n d f r o m  Equation  -98-  q-|(0) =  sine<I >  [4.3.23a]  q (0)  =  (l/6)q (0)  [4.3.23b]  q (0)  =  (l/2)q-, (0)  [4.3.23c]  2  3  The initial  z  n  system of d i f f e r e n t i a l conditions  equations  ([4.3.23]), yields  ([4.3.20-22]) subject to  a solution  of the  the  form,  3 <I  (t)> = sine<I A  Again the solution  B's  i s adapted to the  equations  yields  is multiplied phase f a c t o r ) . the  graphical  E  Y  rotating  been s u p p r e s s e d  a r e s u l t where the  by  the As  sum  being  equal  to u n i t y .  frame i n which the ( i . e . the  rapid  true solution  This oscillatory to  r i g h t hand s i d e o f E q u a t i o n  t e r m , e x p ( - i o > Q ( t + t ' ) ) , w h e r e t ' i s an  with  the  pre-exponential  results  [4.3.24]  B.exp(- ,t).  i=l  are c o n s t r a i n e d , the  t i m e d e p e n d e n c e has  listing  L.  >T  longitudinal and  are analyzed  relaxation,  exponential i n the  factors  Discussion.  i t i s not  Redfield's [4.3.24]  initial worth  a n a l y t i c a l l y ; the  -994.4  RESULTS AND DISCUSSION Before proceeding  it will ation  t o a general d i s c u s s i o n o f the numerical  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  limits o f the relax-  behavior. 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  it  results,  c a n be s e e n  from  Equations  (i.e.,  s e t t i n g a l l J ( ) = 0), w  c  [4.3.6-8] and [4.3.20-22] r e s p e c t i v e l y  that  (<I (t)> z  - <I > )[(cose-l)<I > ]T  T  z  = exp[2(J '~ (a) ) - 4J?r (2u) ))t]  1  1  a  z  2  0  0  [4.4.1] and  <I (t)>(sine<I > r z  x  T  1  = exp[(-3J°°(0) + S j J ' " ^ ) 1  - 2J '- (2co ))t] . 2  2  Q  [4.4.2]  It  i s noted  from  t h a t these expressions a r e i d e n t i c a l  intuition  i n the last  For t h e f a s t motion  considered previously, In  t h i s case, Equations  (d/dt)  y i  (t) = 2[-5J°°(0)  chapter.  limit,  1g  with those introduced  6 D . » 2u> ( J ' " ( k u ) ) -*• u a , c u n  k  k  n  = (-1 ) J ? ° ( 0 ) ) , a,c k  y - | ( t ) a n d y ( t ) a r e no l o n g e r c o u p l e d t o Y 3 ( t ) . 2  [4.3.6] and [4.3.7] reduce t o - J°°(0)]  y i  ( t ) + [12J°°(0)]y (t)  [4.4.3]  2  ( d / d t ) y ( t ) = [ - j f ( O ) + J ° ° ( 0 ) ] ( t ) + 4[-J°°(0) + J ° ° ( 0 ) ] y ( t ) . 2  y i  2  [4.4.4]  Furthermore,  i nthis  limit,  q-,(t) c y ^ t )  [4.4.5]  -100-  and q (t) - y (t) . 2  That i s , the l o n g i t u d i n a l identical  manner.  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  However, i n t h i s  limit,  to define t h e f o l l o w i n g combinations  P^t)  -cc y ^ t )  P (t) 2  Equations motion  (d/dt)  [4.4.6]  2  - y ( t ) -cc 2  q  i tproves  of matrix  ]  d e c a y i n an  more  instructive  elements,  ( t ) - q (t)  [4.4.7a]  2  s y (t) « q (t). 2  [4.4.7b]  2  [ 4 . 3 . 6 ] a n d [ 4 . 3 . 7 ] c a n be u s e d t o d e t e r m i n e  the equations o f  of thep's,  P l  ( t ) = 3[-3J°°(0) - J ° ° ( 0 ) ] ( t ) + 5[-J°°(0) + J ° ° ( 0 ) ] p ( t ) P l  2  [4.4.8] ( d / d t ) p ( t ) = [-J°°(0) + J ° ° ( 0 ) ] ( t ) + 5[-J°°(0) + J ° ° ( 0 ) ] p ( t ) 2  P l  2  [4.4.9]  w h e r e ( < I ( t ) > - <I > ) [ ( c o s e T  7  This combination A s i d e from combination  -1)<I > ] T  _ 1  =<I (t)>[sine<I > ] T  i s much more i n f o r m a t i v e t h a n  E p,(t)+p (t). ?  t h o s e c h o s e n by H u b b a r d .  being a e s t h e t i c a l l y p l e a s i n g (orthogonal combinations), emphasizes t h e p h y s i c a l  f a s h i o n from  i n a physically  t h e d o u b l e t s t a t e s ( p ' s ) and hence, t h e r e s u l t a n t 2  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 , n o t be c h a r a c t e r i z e d by a u n i q u e I f D. > D > ±  to , then j£'~ (! k  Q  this  basis of the resulting biexponential  d e c a y 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 distinct  - 1  v  time ka) n  ) ~  constant. J  a'~ ( V k  k(  asseenfrom  E  c  1  u  a  t  i  o  n  can-  -101-  [4.3.16].  In t h i s  limit,  E q u a t i o n s [ 4 . 4 . 8 ] and  further  simplifications  and  [4.4.9] reduce to  ( d / d t j p ^ t ) = -~\2J {0)p^{t)  ; p^O)  00  (d/dt)p (t) = 0 2  with the  are possible  = 5/6  [4.4.10a]  ; p ( 0 ) = 1/6  [4.4.10b]  2  solution  p ^ t ) + p ( t ) = (1 + 5 e x p ( - 1 2 J  0 0  2  (0)t))/6  .  [4.4.11]  T h u s , t h e q u a r t e t s y s t e m and d o u b l e t s y s t e m s a r e c o m p l e t e l y i s o l a t e d from mutual lose their  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 s y s t e m s residual  interactions.  Note  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 that this  conclusion  lation  magic  a n g l e ) , and  and a u t o c o r r e l a t i o n  Although  limiting  i s dependent spectral  s i t u a t i o n where B e q u a l s  o n l y on w h e t h e r  densities  the  are i d e n t i c a l  cross-correi n magnitude.  e x p r e s s i o n s c a n be a n a l y z e d q u i t e r e a d i l y ,  have chosen t o e x p r e s s a l l g e n e r a l For these p l o t s ,  dipolar  i s independent of the angle of  attachment, B (aside from the e x t r e m e l y u n l i k e l y t h e NMR  never  i t was  results graphically  i n Figures  assumed t h a t t h e m e t h y l g r o u p a x i s was  we 4.1-4.  coinci-  d e n t 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 attached  (i.e.  in general  B=  0° i n E q u a t i o n s [ 4 . 3 . 1 5 ] and  [ 4 . 3 . 1 6 ] ) , a s i t was  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  exaggerated f o r t h i s case. angle-dependence  would  m a g n e t i c f i e l d was  d i s t a n c e o f 1.8  instance, e x p l i c i t display of  be u n p r o f i t a b l y l e n g t h y .  chosen as t y p i c a l 8  B Q = 2.35T (cog  I n any  = 2TT x 10 A was  For each p l o t ,  f o r h i g h r e s o l u t i o n NMR  was found most  this the  experiments:  -1 s e c " ).  Finally,  f o r e a c h p l o t , an  a s s u m e d s i n c e X - r a y and m i c r o w a v e  interproton  v a l u e s average  -102-  close to t h i s  value.  F i g u r e 4.1 for  three values  a b o u t an The  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 ) of D ,  the d i f f u s i o n  x  limit  the  t o an  t r a n s v e r s e and  same t i m e - d e p e n d e n c e so o n l y one show t h e  longitudinal  magnetization longitudinal izations  g r o u p on  i n very viscous  ( F i g . 4.1a)  longitudinal  the  = LOQ;  ±  transverse  < io .  t o Hubbard's  ( b o t t o m two according  p l o t s ) , the  internal  longitudinal  the  in solution  (or  extreme-nar-  F i g . 1 except  From c o m p a r i s o n o f  for a  curves  s i z e , r e l a x a t i o n (whether  i s most n o n e x p o n e n t i a l  p r o l a t e and/or e x h i b i t s f a s t e s t  (right-hand plot)  The  n  X  f o r e a c h g r a p h , f o r a s y m m e t r i c t o p o f any or transverse)  plots  ( r i g h t - h a n d p l o t ) magnet-  large molecule  s o l u t i o n ) , 6D  is identical  have  the bottom p l o t s g i v e  transverse  a very  magnetizations  i s r e q u i r e d ; the middle  change i n n o t a t i o n o f the a b s c i s s a s c a l e .  longitudinal  top. and  ( l e f t - h a n d p l o t ) and  ( l e f t - h a n d p l o t ) and  f o r a methyl  rowed c a s e  ly  diffusion  e x t r e m e - n a r r o w e d c a s e , 6Dj_>> lo^,  plot  f o r a case where 6 D  small molecule  P-V  for rotational  time  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 s y m m e t r i c  upper p l o t corresponds  in t h i s  constant  versus  motion.  magnetization  when t h e For  large  top  i s most  molecules  decays very  nearly  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 the t r a n s v e r s e decay i s marked-  nonexponential. 27 Two  general  F i g u r e 4.1. tards the  p r o p e r t i e s commented upon by  First,  the  correlation  second, the  initial  slope of the s i n g l e exponential i s neglected  are manifest  i n c l u s i o n of c r o s s - c o r r e l a t i o n terms always  r e l a x a t i o n , and  proaches the  Runnels  (see F i g u r e  4.2).  s l o p e of each curve  in re-  ap-  d e c a y o b t a i n e d when c r o s s -  -103-  This  l a t t e r f a c t c a n be e a s i l y  [4.3.14],  [4.3.20],  ficiently  small  and [ 4 . 3 . 2 3 ] .  be d e r i v e d f r o m E q u a t i o n s I t follows directly  T, (d/dtjy^x) - y ( 0 ) [ 2 J ' ]  - 1  a  a  that the i n i t i a l  pairwise interactions. great  2  2  ) ( )  practical  1  of the l i m i t usefulness  Finally,  2 4  "reduced time"  This  assuming a d d i t i v e  i t should  that i n  u n i t s o f seconds  i s because, unless working  6Dj_ >>WQ, t h e e x p l i c i t f i e l d  rather  within the confines  dependence d e t r a c t s from t h e Although  t h e s c a l e a s 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 a n i s o l a t e d t h e graphs a r e s t i l l  be n o t e d  justified,  v a r i a b l e a s seems t o be t h e common  o f t h i s form o f p r e s e n t a t i o n .  2.35T f i e l d ,  I t i s seen  1  This observation which i s a n a l y t i c a l l y  implications.  t h e more g e n e r a l  convention. ^'  Q  )]q (0).  to the r e s u l t obtained  F i g u r e 4.1 a n d 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 than  2  decay i s independent 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  has  2  0  that f o r suf-  (COQ) - 8 J ' " ( 2 t o ) ] a n d  ( d / d t ) ( x ) - [-3J°°(-0) + 5 J ' ~ V ) - 2 J ' " ( 2 a q i  [4.3.6],  rather general  -CH^ g r o u p i n a  in their  presentation. 19  For example, i f t h e gyromagnetic r a t i o were s m a l l e r shapes o f a l l curves contracted curves  For d i f f e r e n t values  t h e same s h a p e s , p r o v i d e d  a r e c h a n g e d by t h e a p p r o p r i a t e In F i g u r e 4.2, c u r v e and  resolved into  F), the  s c a l e w o u l d be  I f r were s m a l l e r , t h e shapes o f t h e  r e m a i n t h e same, a n d t h e t i m e  (faster relaxation). retain  w o u l d r e m a i n t h e same, b u t t h e t i m e  (slower r e l a x a t i o n ) .  would again  (as f o r  s c a l e w o u l d be e x p a n d e d  o f WQ, t h e c u r v e s  that the rotational  will  diffusion  again  constants  factor.  V o f F i g u r e 4.1b i s r e p r o d u c e d  i t s t h r e e component e x p o n e n t i a l s  (curved  (the three  line),  solid  l i n e s ) , a n d c o m p a r e d 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-  neglected  (<I (t)> z  (dotted l i n e ) .  - <I > )/(cose T  z  The a c t u a l e q u a t i o n  - l)<I >  of the curve  is,  = 0.125exp(-0.678t) + 0.171exp(-12.5t)  T  z  + 0.704exp(-5.34t).  [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 1 o g ( 0 . 1 2 5 + 0.704) a n d l o g ( l ) ; show t h e l i m i t i n g clear  behavior  that the i n i t i a l  this  lines are: log(0.125),  seems t o be  a t long times.  slope o f the curve  t h e m o s t u s e f u l way t o  From t h i s  typical  plot,  i t is  approaches t h a t o f t h e dotted  line. The m o s t g e n e r a l  d i s p l a y of the present  c a l c u l a t i o n s i s provided  by F i g u r e s 4.3 and 4.4, w h 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 a n d constants  of the three exponentials f o r the longitudinal  transverse  are  shown i n t h e u p p e r t h r e e p l o t s o f e a c h f i g u r e , a n d t h e r e l a t i v e  the  i n the lower  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  same p u r p o s e a s a n e x t e n s i v e d i g i t a l  p l o t d e p i c t i n g t h e mutual  on t h e r e l a x a t i o n b e h a v i o r  simply  contour  map.  table or a  of the spins.  i n solution.  T h e s e two f i g u r e s a r e  group attached  i s roughly  sec'J a n d  = 10  1 0  t o t h e backbone  s p h e r i c a l i n s h a p e and  t h e s y s t e m c a n 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 9  flexi-  F o r t h e p u r p o s e o f s e r v i n g a s an e x -  a m p l e , assume t h a t t h e l a r g e m o l e c u l e  = 10  serves  F o r e x a m p l e , s a y 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 b e -  of a large molecule  (l  time-  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 -  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 m e t h y l  D  factors  three-dimensional  e f f e c t s o f s i z e , s h a p e , and i n t e r n a l  bility  ical  The p r e - e x p o n e n t i a l  4.3)  and  constants  ( F i g u r e 4.4) m a g n e t i z a t i o n s .  (Figure  time  secT  1  constants, d  ±  R e f e r r i n g t o F i g u r e 4.3, i t i s  =  -105-  found t h a t A  2  terms a r e ( A  2  - *  + A^) - 0.92 a n d A^ - 0.08. i f l O D ^ = D„ = 1 0  w o u l d be e x p e c t e d due  - 4 A ^ and t h e p r e - e x p o n e n t i a l s  3  t o the f a c t that f o r these  effectively  1 0  plots,  a single variable).  Of c o u r s e ,  sec  - 1  and  order  = 10  i  2  By r e f e r r i n g  t h e r a t i o o f A-|/x  (and/or  relative,  (|  + D.) i s  t o F i g u r e 4.1, i t 2  (and/or  A  i s depicted  c h o i c e s o f D.,  two  interpolations or extrapolations.  behavior  t o be m a n i f e s t ,  of the pre-exponential  i t i s necessary  exponential internal  D , n  a n d Dj_.  (upper r i g h t  x A 3  2  pre-  I n o r d e r f o r nonboth  that at least  exponentials  differ  From F i g u r e s 4.3 a n d 4.4, i t i s s e e n t h a t n o n -  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  rotation  )  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  m a g n i t u d e 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 a p p r e c i a b l y from u n i t y .  2  Again,  W i t h t h e a i d o f F i g u r e s 4.3 a n d 4 . 4 , i t i s p o s s i b l e t o make q u i t e  exponential  / A  Therefore,  i n these 'figures.  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  cise qualitative  3  Also, only the ratios  f a c t o r s a r e shown i n F i g u r e s 4.3 a n d 4.4.  not absolute behavior  (this i s  1  A^) m u s t be i n m a g n i t u d e i n  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 .  of the exponential  sec"  9  F i g u r e s 4.3 a n d 4.4 a r e m o s t i n f o r m a t i v e  how s m a l l  a l s o how c o m p a r a b l e A^ a n d A  and D  t h e same r e l a x a t i o n  8 = 0 ° , t h e r e f o r e , (D  when u s e d 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. i s now p o s s i b l e t o c o n c l u d e  f o r t h e two d i s t i n c t i v e  i s f a s t c o m p a r e d t o D^.  hand r e g i o n o f e a c h p l o t ) , A  - 1, i n e i t h e r e v e n t ,  p r o l a t e tops  a n d / o r when  In t h e extreme-narrowing 3  and B  3  approach zero  limit  and/or  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 obtained the  by H u b b a r d .  p l o t s where  f o r an e x t r e m e l y  > (D.  I t should  a l s o be m e n t i o n e d t h a t t h e r e g i o n s o f  + D ) have l i t t l e ()  flattened oblate top)  p h y s i c a l meaning s i n c e doesn't d i f f e r  (even  appreciably  -106the value f o r D,  from  A final for  r  intriguing  feature of the transverse magnetization  a [(D„ + D )/D _] r a t i o o f 7 / 4 , t h e r e l a x a t i o n i  b e h a v i o r h a s been n o t e d  more n o v e l  predictions  (J?°(0) »  -jl'ZVn) a,C  tudinal  as seen i n F i g u r e 4 . 4 .  nonextreme-narrowed case  concerning 2  behavior  the simplification  n  a ,C  This Examina-  p r o v i d e s a few  the magnetization decays.  * J?'^ (2w )),  U  relaxation  1.00  2  f o r t h e extreme-narrowed c a l c u l a t i o n .  t i o n o f i t s source f o rt h e  a,C  approaches t h a t o f a  J  single exponential, that i s , B  i s that  If D  (  < to  n  of the longi-  U  (Equations  [4.3.6-8]) i s not r e a d i l y  achieved.  1/2 However, u n l e s s  (24D^D )  mated v e r y w e l l  by a u n i q u e e x p o n e n t i a l .  In motion  > COQ, t h e r e l a x a t i o n  1  i s obviously approxi-  stark contrast, f o r the transverse relaxation limit,  (d/dt)  i  q i  Equations  (t) = -3[J°°(0)  (d/dt)q (t) = 3[-J°°(0) 3  from which <I (t)> x  slow  reduce t o ,  + 3J°°(0)] (t) + 6 J ° ° ( 0 ) q ( t )  [4.4.12]  + J°°(0)]q (t)  [4.4.13]  qi  3  3  i tfollows that  <V°)>  =  [4.3.20-22]  i n this  {exp(-3(J  a  0 0  (0)  +  J°°(0))t)  +  exp(-3(J  00 a  C0) - J ° ° ( 0 ) ) t ) } . [4.4.14]  If J ^ ° ( 0 ) = 0 (cross-correlation p r e s s i o n reduces Equation equal  spectral  to a single exponential.  d e n s i t i e s v a n i s h ) , t h i s exFor 6 = 0 ° , examination of  [ 4 . 3 . 1 6 ] shows t h a t i f [ ( D \ + D ^ / D j  zero.  there w i l l  I t s h o u l d be n o t e d i n general  = 7/4, J ° ° ( 0 )  does  indeed  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,  e x i s t a unique  ratio  of rotational  constants f o r  -107-  which  a l l cross-terms  4.3,  this  identically  i s not t r u e f o r l o n g i t u d i n a l  From E q u a t i o n v e r s e decay i f D  As o b v i o u s l y shown by F i g u r e  relaxation.  [4.4.12-13] note 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 < 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 ,  x  make no s t i p u l a t i o n  >> COQ o r t h a t B =  that  (t)  q ]  -  q (t)  \ exp(-6J  0 0  D.. >> Dj_ ( n o t e we  0°!).  In t h i s  case,  (0)t)  [4.4.15]  - q ( 0 ) = 1/2 .  3  Indeed,  vanish.  [4.4.16]  3  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 the t h e o r e t i c a l  plot  decay.  This behavior  i n F i g u r e 4.1e.  i s q u i t e apparent i n  I t i s interesting  t o comment o n  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  amplitude and  the  i n f l u e n c e o f these terms.  for  J  tral  c  = 0, no e f f e c t  densities  effect will  fall  i s seen.  When J , = J , a m a x i m a l a c  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  l i e between t h e c o r r e s p o n d i n g  2.21+0.04.^, a n d t h e  F  The  magnetogyric  i s ( 2 5 1 7 9 / 2 6 7 5 3 ) 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  identical  will  spec-  extremities of behavior.  t o -CF^ groups f o l l o w s d i r e c t l y .  inter-fluorine distance i s typically  and  i s seen;  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  Extension o f the treatment  ratio  effect  molecular dynamics, t h e  be down by a f a c t o r o f ( 2 . 5 2 / 2 . 6 7 )  a proton methyl  F dipole-dipole relaxation 4  (1.77/2.21)  6  rate  - (1/5) t h a t f o r  group.  From F i g u r e s 4.3 a n d 4.4, i t i s e v i d e n t t h a t t h e b e s t s y s t e m i n which  t o observe  nonexponential  c u l e possessing a methyl  d e c a y i n m a g n e t i z a t i o n s h o u l d be a m o l e -  g r o u p whose i n t e r n a l  rotation  rate i s fast  -108-  compared t o r e o r i e n t a t i o n cules  in liquid  tions  t o be n o t i c e a b l e ,  group  r o t a t i o n approaches  o r d e r o f (kT/1 for  acetonitrile fested times  s t a t e , t h i s means t h a t  + u  D. w i l l  1/2  , ) metnyI  the methyl  o f t h e m o l e c u l e as a w h o l e .  group).  Just  13  f o r the e f f e c t of  state  i n CH^CN, i t may  cross-correla-  d i f f u s i o n " l i m i t when  -1 sec, (I  n  methyl  i s on  i s t h e moment o f  s u c h an e x a m p l e i s p r o v i d e d by  (the molecule f o r which c r o s s - c o r r e l a t i o n  i n the s o l i d  mole-  be r e s t r i c t e d t o c a s e s w h e r e t h e m e t h y l  the "free =10  n  For small  ) ; by c o m p a r i n g  be shown t h a t  was  inertia  liquid first  p r o t o n and n i t r o g e n  the r o t a t i o n  the  mani-  relaxation  a b o u t t h e C-N  axis i s  43 a b o u t t e n t i m e s a s f a s t as r e o r i e n t a t i o n the is  p r o t o n r e l a x a t i o n was resolved  that  of that  axis  o b s e r v e d t o be e x p o n e n t i a l .  itself. This  However, incongruity  by a s t u d y o f t h e t e m p e r a t u r e d e p e n d e n c e o f T - j , w h i c h  a spin-rotation  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  shows  dominant  intramolecular cently to  r e l a x a t i o n m e c h a n i s m a b o v e 25°C. M o r e o v e r , i t h a s r e 44 45 46 been a r g u e d ' 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  internal  rotation  even f o r methyl ly  free  internal  rotation  The  rotation  a  contribution  to dipole-dipole  spin-internal  i n t e r a c t i o n ; thus  internal  spin-  interactions.  rotation  rotation  (exponential)  or near-  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  =  interaction generally  ! *~ Vo?SR  i s t h e moment o f i n e r t i a  responding  a spin-rotation  should introduce a large  (^spin-rotation where I  produce  g r o u p s 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  relaxation  e f f e c t s due  can a l s o  kT  of the internal  spin-rotation  has t h e f o r m  >  47  £4.4.17] rotor, C  c o n s t a n t , and x  a C D  i s the cori s the  -109-  correlation  t i m e f o r changes  T<.p i s d i s t i n g u i s h e d  action. time,  f o r changes  First,  may  of the spin-rotation  i n two ways f r o m t h e a v e r a g e  i n magnitude  of the dipole-dipole  the p r i n c i p a l axis  r o t a t i o n , so t h a t  becomes more h i n d e r e d ( t h i s may  of the i n -  as t h e r o t a t i o n a l  be c o n t r a s t e d  with  the order of a radian).  axis  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  rotation  interactions  R  i s i n general shorter  than  spin-  difficult  their relative  contributions.  T^.  While the importance of s p i n - r o t a t i o n notoriously  and  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 x<.  to re-  T h e r e f o r e , T 2 becomes l o n g e r a s t h e  rotation  Second,  motion  X2» w h i c h may be r e -  g a r d e d as t h e a v e r a g e 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 orient  correlation  "collisions"  direction or rate  becomes s h o r t e r  inter-  interaction.  be t h o u g h t o f a s t h e a v e r a g e t i m e b e t w e e n  which change e i t h e r ternal  i n magnitude  contribution  to relaxation i s  t o q u a n t i t a t i v e l y assess i n advance,  47  t h e form o f 48 49  the  relaxation  i s well  established  f o r rotational diffusion of spherical,  '  50 51 52 ' ' t o p s , a n d 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 m o d e l 53 f o r symmetric tops. I n 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^ and  symmetric  groups  48 54 46 58 ' and -CF^ groups ' on s m a l l  however, one s i t u a t i o n i n w h i c h in  liquids.  to the spin motion  nonexponential decay  In o r d e r t o emphasize internal  (yet rapid  molecules i n l i q u i d s .  the dipole-dipole  r o t a t i o n , systems w i t h  provides promising subjects.  s h o u l d be o b s e r v a b l e interaction  relatively  compared t o t h e r e o r i e n t a t i o n  There i s ,  compared  hindered internal  o f t h e m o l e c u l e as a  Furthermore, i f the o v e r a l l motion  whole)  i s on t h e  -noo r d e r o f t h e L a r m o r 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 very 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 (in  t h e same l i m i t ,  the longitudinal  relaxation  cross-correlation  terms  i s almost independent o f  t h e s e terms) and might 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  studies.  -111-  FIGURE 4 . 1 :  Plots o f magnetization  (deviation  versus  a e-degree p u l s e d i r e c t e d  time, following  from thermal  the y - a x i s o f a r o t a t i n g frame, f o r a methyl along  equilibrium) along  group  attached  t h e symmetry a x i s o f a s y m m e t r i c t o p m o l e c u l e .  For plots  (b) and ( d ) , <I (t)>  -  z  C(t)  - 1) <I >  =  <  I x  sine plot  (a), longitudinal  F o r a l l p l o t s , io D  ±  = 10  plots  1  0  n  (  t  )  Z  = transverse magnetization  sec" ; f o r plots = 10  7  (c) ( o r ( d ) and ( e ) ) , c u r v e s ((  explanation''  x  8  For plot ( a ) ,  sec" ;f o r 1  ( a ) , curves  v a l u e s o f [ D . + D ] o f 1, (J  1  0  sec" ; f o r plots 1  P,Q,...V c o r r e s p o n d  as f o r p l o t  and d i s c u s s i o n .  For plot  1  4, 6, 1 0 , 2 0 , a n d 1 0 0 x 1 0  [9D.j + D ) / D ] - r a t i o s  , r ^ _ ^ = 1.8A .  sec" .  to respective  =?(t).  0  ( b ) a n d ( c ) , Dj_ = 1 0  1  (d) and ( e ) ,  >  <I > ^  8 - 1 = 2TT X 10 s e c  P,Q,...V c o r r e s p o n d (7/4),  T  z  ( c ) and ( e ) .  5(t)  for  T  Z  = (cose  for plots  <I >  (a).  ( b ) and  t o t h e same  See t e x t f o r f u r t h e r  -1 1 2 -  -113-  FIGURE 4.2:  R e s o l u t i o n of a p l o t of the versus  time  into  longitudinal  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 of F i g u r e 4.1(b). linear  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  effects.  The  three s o l i d  slope) are plots  Equation  lines  Dotted  and  [ 4 . 4 . 1 2 ] and  line  ( i n order of 5  increasing log((0.125  log(exp[-12.5t])versus t . accompanying  denotes  cross-correlation  of log (0.125exp[-0.678t])  + 0.704)exp[-5.34t]), See  magnetization  discussion.  -115-  FIGURE 4.3:  Non-exponentiality magnetization, contour  of the  time-decay f o r the  d i s p l a y e d as c o n t o u r  i s a l i n e of constant  exponential  f a c t o r s (A.)  longitudinal  p l o t s , where each  magnitude f o r e i t h e r  or exponential  ratios  I  from Equation  [4.3.17],  pre-  A./X. K  J  as  a f u n c t i o n o f D,, and  (D  (|  +  D..).  -117-  FIGURE 4.4:  Non-exponentiality magnetization, contour  of the time-decay f o r the transverse  d i s p l a y e d as c o n t o u r  p l o t s , where each  i s a l i n e of constant magnitude f o r e i t h e r p r e -  exponential  f a c t o r s (B.) o r e x p o n e n t i a l  from Equation  [4.3.24],  ratios  Y-/Y|, J  I  as a f u n c t i o n o f  a n d (D  K ((  + D.. ) .  -119-  4.5  SUMMARY It  ation  h a s b e e n shown t h a t  f o r i n t e r n a l l y r o t a t i n g methyl  medium s i z e d m o l e c u l e s b e e n shown t h a t  in solution.  induced  relax-  itudinal  (or trifluoromethyl)  For large molecules  g r o u p s on  in solution, i 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 n o n -  e x p o n e n t i a l , whereas a unique  may be u s e d t o c h a r a c t e r i z e  the long-  relaxation.  However, u n l e s s t h e i n t e r n a l m o t i o n the  dipole-dipole  f o r an i s o l a t e d m e t h y l g r o u p c a n 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 -  pecially  has  intramolecular  spin-rotation  exponential  relaxation will  contribute  e f f e c t experimentally.  in m a g n e t i z a t i o n , t h e methyl  itself  i s appreciably  hindered,  a n d p r o b a b l y mask a n y n o n -  Thus, t o detect nonexponential  group i n t e r n a l  rotation  should  decay  be f a s t  com-  p a r e d 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 , b u t s l o w compared t o a "free be  d i f f u s i o n " model.  observable  F o r -CF^ g r o u p s , t h e e f f e c t w i l l  in liquids,  because t h e d i p o l e - d i p o l e  s l o w e r by a f a c t o r o f f i v e , a n d t h e s p i n - r o t a t i o n than f o r protons.  F o r -CF^, n o n e x p o n e n t i a l  o n l y when t h e i n t e r n a l the  rotational  relaxation  coupling  never  rate i s  i s much  larger  r e l a x a t i o n w o u l d be e x p e c t e d  r o t a t i o n o f t h e -CF^ g r o u p i s v e r y  anisotropy differs  probably  appreciably  from u n i t y .  "hindered" y e t T h e r e i s one  o t h e r way i n w h 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  that  i s due t o p o s s i b l e  relaxation  processes.  calculation  and w i l l  Finally, longitudinal  cross-correlation  T h i s problem be t r e a t e d  i s formally  i n the next  i t has b e e n shown t h a t  e f f e c t s between  competing  s i m i l a r to the present  chapter.  a plot of either  the transverse or  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  slope which i s  -120-  the  same a s w o u l d be o b t a i n e d  further  compounding t h e e x p e r i m e n t a l  entiality an  r o t a t i n g methyl  show n o n e x p o n e n t i a l  relaxation The  difficulty  of detecting  predicted,  relaxation.  although quite  e a s i l y q u e n c h e d by s p i n - i n t e r n a l T h i s b e i n g s o , one  or absence o f these e f f e c t s  effects,  nonexponNevertheless,  g r o u p 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 The f a c t t h a t  non-exponential  has n o t b e e n r e p o r t e d f o r s u c h s y s t e m s i s v e r y  effects  actions.  a l l cross-correlation  ( a l t h o u g h making a simple i n t e r p r e t a t i o n e a s i e r ) .  internally  should  by n e g l e c t i n g  understandable.  d r a m a t i c , a r e v e r y t e n u o u s and a r e  rotation or intermolecular  dipolar  inter-  immediate source o f knowledge t h e presence , p r o v i d e s c a n be l i k e n e d  to the  information  p r o v i d e d by O v e r h a u s e r e n h a n c e m e n t s ; a means t o d e t e r m i n e t h e m a g n i t u d e of  the intramolecular  overall  relaxation  dipolar  rate.  relaxation  pathways i n comparison w i t h t h e  -121-  REFERENCES: CHAPTER IV  1.  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B l o o m , C a n . J . P h y s .  5 7 , 2641 ( 1 9 7 2 ) .  L e t t . 18,  553 ( 1 9 7 3 ) .  4 7 , 1195 ( 1 9 6 9 ) .  13. M. M e h r i n g  a n d H. R a b e r , J . Chem. P h y s .  5 9 , 1116 ( 1 9 7 3 ) .  14. J . C u t n e l l  a n d L. V e r d u i n , J . Chem. P h y s .  5 9 , 259 ( 1 9 7 3 ) .  15.  P. S. A l l e n , A. W. K. K h a z a d a , a n d C. A. M c D o w e l l , M o l . P h y s . 2 5 , 1273 ( 1 9 7 3 ) .  16.  P. S. H u b b a r d , J . Chem. P h y s .  51_, 1647 ( 1 9 6 9 ) .  17.  P. S. H u b b a r d , J . Chem. P h y s .  52, 563 (1970).  18.  P. S. H u b b a r d , P h y s . R e v . 1 0 9 , 1 1 5 3 ( 1 9 5 8 ) ; i b i d . , JJJ_, i b i d . , P h . D. T h e s i s , H a r v a r d , 1 9 5 8 .  19.  I . V. A l e k s a n d r o v , S o v i e t P h y s .  20.  P. S. H u b b a r d , P h y s .  21.  Prof.  22.  P r o f . M. B l o o m , p r i v a t e  Doklady  3_, 110 ( 1 9 5 8 ) .  Rev. 1 2 8 , 650 ( 1 9 6 2 ) .  S. E m i d , p r i v a t e  communication. communication  1746 ( 1 9 5 8 ) ;  -122-  23. A. A b r a g a m , The P r i n c i p l e s o f N u c l e a r M a g n e t i s m , C l a r e n d o n O x f o r d , 1 9 6 1 ; page 293. 24. G. W.  Kattawar  a n d M. E i s n e r , P h y s .  Rev. 1 2 6 , 1054  (1962).  25. M. D. Z e i d l e r , B e r . B u n s e n g e s . p h y s i k . C h e m . 7 2 . 481 27  (1963).  Rev. 1 3 4 , A 2 8  (1964).  (1968).  26.  P. M. R i c h a r d s , P h y s .  27.  L. K. R u n n e l s ,  28.  D. F e n z k e , A n n . P h y s i k ' 1 6 281 ( 1 9 6 5 ) ; D. F e n z k e and H. Ann. P h y s i k 19., 321 ( 1 9 6 7 ) .  29.  N. C. P y p e r , M o l . P h y s .  21_, 1  30.  N. C. P y p e r , M o l . P h y s .  22, 433  Phys.  Rev. ]3Z,  (1971).  B u c h n e r , J . Magn. R e s . 11_, 46  (1973).  32.  W.  B u c h n e r , J . Magn. R e s . J_2, 8 2  (1973).  33.  H. S c h n e i d e r , A n n . P h y s i k 1 3 , 313  (1963).  34.  H. S c h n e i d e r , A n n . 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(1971).  52,  -123-  46.  T. E. B u r k e a n d S. I . C h a n , J . Magn. R e s . 2, 120 ( 1 9 7 0 ) .  47.  C. D e v e r e l l ,  48.  D. K. G r e e n a n d J . G. P o w l e s ,  49.  P. S. H u b b a r d , P h y s .  50.  H. J . B e n d e r a n d M. D. Z e i d l e r , B e r . B u n s e n g e s . p h y s i k Chem. 75.* 236 (1971 ) .  M o l . Phys.  1 8 , 319 ( 1 9 7 0 ) . P r o c . Phys.  S o c . 8 5 , 87  (1965).  R e v . 131_, 1 1 5 5 ( 1 9 6 3 ) .  5 1 . C. H. Wang, D. M. G r a n t , a n d J . R. L y e r l a , J . Chem. P h y s 5 5 , 4 6 7 4 (1971 ). 5 2 . C. H. Wang, J . Magn. R e s . 9, 75  (1973).  53.  R. E. D. M c C l u n g , J . Chem. P h y s .  5 7 , 5478 ( 1 9 7 2 ) .  54.  R. G. P a r k e r a n d J . J o n a s , J . Magn. R e s . 6_, 106 ( 1 9 7 2 ) .  -124-  CHAPTER V INFLUENCE OF F I N I T E CROSS-CORRELATION TERMS BETWEEN P H Y S I C A L L Y D I S T I N C T RELAXATION MECHANISMS  5.1  1  INTRODUCTION For  nuclear  invariably  magnetic r e l a x a t i o n studies  observed 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  magnetization of a perturbed respective  thermal  acterized  spin  exception, ponential  truly  exponential  not the r u l e . decay o n l y  =  to their  a n d x^  l.A-| 2)'  *  n  c  o  n  t  r  a  s  t  ( o r by t h e t  experiment,  o  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  A priori,  i ti s possible  t o p r e d i c t true_ e x -  i n t h e c a s e w h e r e t h e s t a t i c Zeeman H a m i l t o n i a n  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 spin-space i s  fully  s p a n n e d by two e i g e n k e t s ) ,  ist.  For example, consider  * A novel  a disconcerting  even t h i s  philosophical  (1967).  fashion  unique paper:  nucleus  I t has  i n - quantum mech-  1/2 n u c l e u s t o a p p r o a c h  (however s l i g h t ) .  and e p i s t e m o l o g i c a l  make n o t e o f t h i s  simple statement.  the presence o f hidden v a r i a b l e s  would cause a r e l a x i n g s p i n  in a nonexponential  fact f o rthe perfection-  the quadrupolar r e l a x a t i o n of a s i n g l e  calculation jeopardizes  been s u g g e s t e d t h a t anics  components o f  T h e s e d e c a y s c a n t h u s be c h a r -  relaxation constants  commonly q u o t e d t i m e c o n s t a n t s T^ ^ theory predicts  state, i t i s  system decay e x p o n e n t i a l l y  equilibrium values.  by t h e r e s p e c t i v e  consists  i n the 1iquid  equilibrium  Wishing t o avoid t h e  implications of this  t o p i c , we  only  R.K. W a n g s n e s s , P h y s . R e v . 1 6 0 , 1190  -125-  with  spin  that  a unique  o f two  I > 1.  I f the  extreme-narrowed l i m i t  ^ does not  e x i s t and  o r more e x p o n e n t i a l s  chemical  shift  anisotropy  (for nuclei containing shift  anisotropy  e x a m p l e i s by  may  be  examples i n t h i s of  limited pairwise  spin  a two-spin system, the examples.  I f the  (or m u l t i - )  is either a  that  one  Again, multiexponential  the  behavior.  rare;  to point  discussion  spins  chemical  are  chemical  out  s h i f t anisotropy  confined  "sample" the  simplest  to  1/2  such nuclei,  indirect dipolar  interactions  n u c l e i are  scalar  as  possibility.  i s not  two  included  even i n the  isochronous spin  I f the  this  relaxation  they are  terms between the  ( d i r e c t ) d i p o l a r or  n u c l e u s may  spin  relaxation  between t h e s e competing  relaxation is a  interference  r i s e to nonexponential there  two  quadrupolar  n u c l e o n s , where the  systems are  sum  As  o r d e r o f t e n s o f t h o u s a n d s o f ppm,  introduction only  d i r e c t d i p o l a r , and  so  the  spin systems, nonexponential For  a  details).  s i t u a t i o n where both  number o f  interference  I s o l a t e d one  by  interactions relax a single nuclear  large on  the  no means c o n t r i v e d ) .  would r e s u l t from the mechanisms.  a  relaxation i s given  (see Appendix C f o r f u l l  another case i n p o i n t , consider and  the  i s v i o l a t e d , i t i s known  give  "unlike"  (indirect dipolar)  s t a t e of the  each s p i n  and coupling  other nucleus,  c r o s s - r e l a x a t i o n causes the m a g n e t i z a t i o n decay of  or  then  t o be  bi-  4 exponential For  even i n the  a three-spin  u n m a n a g e a b l e and grouping of instances,  the  absence of a l l i n t e r f e r e n c e  system, i n c l u s i v e c a l c u l a t i o n s begin to p o s s i b i l i t y of  indistinguishable spins the  effects.  deviation  true  r e l a x a t i o n of  becomes r e m o t e , a l t h o u g h  i s experimentally  p l e s t example of a t h r e e - s p i n  exponential  unobservable.  system, consider  three  become  As  any  i n most the  identical  sim-  spin  1/2  -126-  nuclei.  Even i f o n l y  the dipolar i n t e r a c t i o n i s operative,  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 , a s 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 course, i fother netization the  spins  i n a methyl  r e l a x a t i o n mechanisms c o n t r i b u t e  and/or  the three  spins  are not " a l i k e "  group.  Of  t o t h e d e c a y o f magand/or  o n e o r more o f  h a v e 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  consideration  m u s t be g i v e n  terms a r i s i n g  f r o m 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  two  interference  t o such obscure p o s s i b i l i t i e s  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  and  as i n t e r f e r e n c e  o f t h e 2nd k i n d f o r  a nonvanishing  quadrupole  moment.^ To s u m m a r i z e , nonexchanging  nonexponential  systems w i l l  ( i . e . multiexponential)  i n general,  be e x p e c t e d when t h e r e  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  limit,  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 Interference called  terms o f t h e f i r s t  i s , (1)  (2) c r o s s -  o r second  kind.  k i n d , which might a p p r o p r i a t e l y  be  "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 the  relaxation f o r  second  on d i f f e r e n t n u c l e a r  k i n d , which might  spins.  Interference  be c a l l e d " B l i c h a r s k i t e r m s " o r  terms o f "hetero-  t e r m s " , a r i s e f r o m two o r more d i f f e r e n t r e l a x a t i o n m e c h a n i s m s a t t h e same n u c l e u s .  Interference  terms  of the "third  r e l a x a t i o n m e c h a n i s m s on d i f f e r e n t c e n t e r s ference  with  used t o denote changes poration This  that  further consideration  interas  the term c r o s s - r e l a x a t i o n i s  i n r e l a x a t i o n times r e s u l t i n g from t h e i n c o r -  of cross-correlation functions  i s unfortunate  kind", different  (e.g. i n d i r e c t dipolar  d i r e c t d i p o l a r ) , are not given  * Sometimes i t i s found i n l i t e r a t u r e  operative  into the relaxation  a s t h e two t e r m s s h a r e no r e l a t i o n  equations.  t o each  other.  Ft  -127-  t h e y a p p e a r t o be o f s o l i t t l e  importance  a t present.  Hubbard  terms  c o u l d a l s o be d e f i n e d a s i n t e r f e r e n c e t e r m s b e t w e e n r e l a x a t i o n  path-  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 form whereas B l i c h a r s k i  o f t h e same  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  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 t e r m s i n t o two s u b c l a s s e s arbitrary  s i n c e both  functional  share  a common h e r i t a g e , t h e y  form.  i s completely  both  result  zero c r o s s - c o r r e l a t i o n  t e r m s 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  magnetic 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  directions  cross-correlation  experimental  fixed  r a r i t y o f nonexponential  propriate cross-terms i n such  s p i n s o r has a n u c l e u s  relaxation  are vanishingly small or enter  approach t o r e l a x a t i o n  terms which r e s u l t  unfortunate  that nature  a s s u m p t i o n s may v e r y  description will The  i s to simply I ti s  t o t h e experimen-  observation of interference effects of the relaxation  avenue t o o b t a i n chemical  f o r m a t i o n from r e l a x a t i o n measurements. such  calculations  importance.  i n multiexponential behavior.  would g r e a t l y enhance o u r u n d e r s t a n d i n g a l s o p r o v i d e an a d d i t i o n a l  time", the  into the calcula-  their  has b e e n s o b e n e v o l e n t  because the experimental  being r e -  i m p l i e s t h a t t h e ap-  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  the usual  neglect these  talist  another  be p r e s e n t w h e n e v e r t h e s p i n  l a x e d by two d i f f e r e n t m e c h a n i s m s " o f t h e same c o r r e l a t i o n  truly  non-  i s h i s t o r i c a l , due t o two  terms w i l l  s y s t e m has t h r e e o r more r i g i d l y  Therefore,  from  of concentration i n the l i t e r a t u r e .  Although  tions  Actually,  likely  process  and  and p h y s i c a l i n -  However, i n f u t u r e s t u d i e s ,  p r o v e u n w a r r a n t e d and. t h e more c o r r e c t  n e e d be a p p l i e d .  purpose o f t h i s  chapter  i s t o examine t h e e f f e c t and  importance  -128-  of these interference  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  C h a p t e r I V ) and t o j u s t i f y present experimental  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  oddity  o f such e f f e c t s .  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 f i e l d s with  basis.  laxation no  fields the  studies, the majority  means p r e j u d i c e  laxation  behavior.  i s that  future  on a  variety of  t e c h n i q u e , and p r e v i o u s r e -  o f w h i c h w e r e d o n e o n p r o t o n s , s h o u l d by  S e c o n d l y , an o f t e n  the chemical  shift  discussed  anisotropy  field  so t h a t  consequence o f l a r g e  i n t e r a c t i o n increases  as  peculiarities relating  calculations dealing  in validity  on two g r o u n d s :  with  the e f f e c t of cross-terms are  They t r e a t o n l y  i s o t r o p i c molecular  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 .  ever, the motion o f a spin or a s e t o f i s o l a t e d spins framework  anisotropic narrowed  sensi-  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 -  r e o r i e n t a t i o n and a r e o f t e n  bulky  (Fourier  p a t h w a y o f r e l a x a t i o n may become m a n i f e s t .  Existing limited  static  ( b i o m o l e c u l e s ) c a n be s t u d i e d  by t h e NMR  s e c o n d power o f t h e a p p l i e d  to t h i s  toward usage o f l a r g e  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  c a n be f e a s i b l y s t u d i e d  to the  justification  The i m m e d i a t e c o n s e q u e n c e i s i n c r e a s e d  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  nuclei  An i m m e d i a t e  d a t a a q u i s i t i o n by way o f a t i m e d o m a i n s i g n a l  Transform techniques).  routine  begun i n  (such as a small  biopolymer) i s l i k e l y  (due t o i n t e r n a l m o t i o n s ) and/or f a i l  attached  How-  to a  t o be h i g h l y  to satisfy  the extreme-  approximation.  In p a r t i c u l a r , t h e remainder o f t h i s  chapter w i l l  examining the e f f e c t o f a n i s o t r o p i c motion o f t h e spin  be c o n c e r n e d  with  system and t h e  f a i l u r e o f t h e e x t r e m e - n a r r o w e d 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 are  i n c l u d e d i n the r e l a x a t i o n  terms"  will  be d i s c u s s e d , s i n c e t h e  terms"  t r e a t e d i n t h e same m o t i o n a l  h i g h l y a n i s o t r o p i c m o t i o n has H u b b a r d t e r m s and To useful  calculations. previous chapter limits.  7  "Blicharski  discussed  I t i s well  pronounced e f f e c t s  hence the p r e s e n t  Only  on  the  known t h a t importance  extension should prove  lations.*  papers concerning  I n s e c t i o n t h r e e and  the framework o f the s e m i - c l a s s i c a l calculations in Section  are presented,  cross-correlation  f o u r , the theory  of  interesting.  help o r i e n t the r e a d e r , the second s e c t i o n i s devoted resume o f p r e v i o u s  "Hubbard  to a calcu-  i s formulated w i t h i n  r e l a x a t i o n m a t r i x and  the r e s u l t a n t  f o l l o w e d by a d i s c u s s i o n o f m a j o r  conclusions  5.  * Unfortunately this  a p p r o a c h i s n e c e s s i t a t e d as  literature  p r e v i o u s w o r k 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 glaring omission  from p u b l i s h e d work a l t h o u g h  p l a n n i n g ; J.S.  Blicharski  i n Advan. Mol.  Relax.  i n A d v a n . Magn. Res.  Processes.  accounts  of  t e r m s compose a  two and  reviews L.G.  are i n the  Werbelow i n  -130-  5.2 RESUME OF PREVIOUS STUDIES In t h i s  section,  previous expositions  c o r r e l a t i o n terms a r i s i n g netic relaxation be  i n the density  are reviewed.  exhaustive, but rather  and  peripheral As  will  equations.  background  later  first  t h e "same  terms w i l l  relaxation correlation  appear i n the r e l a x -  w h a t i s m e a n t by t h e "same c o r r e l a t i o n  i n the text.  i s t o cause a perturbed  nonexponential  i fthe nuclear  interactions with  (interference)  Exactly  be c l a r i f i e d  The  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  i t may be s a i d t h a t  i s d o m i n a t e d by v a r i o u s  cross-terms  mag-  material.  time", cross-correlation ation  cross-  matrix theory o f nuclear  t o s e r v e as a g u i d e t o r e l e v a n t  an i n t r o d u c t i o n ,  process  o f t h e problem o f  The p h y s i c a l nuclear spin  time"  e f f e c t of these  system t o r e l a x  in a  fashion. interest  c e r t a i n experimental  in cross-correlation  effects  questions connected with  ESR, t h e rrij d e p e n d e n c e o f l i n e w i d t h s  arose i n answering  electronic  was e x p l a i n e d  relaxation.  by n o t i n g  that  t e r m s 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  In  cross-  dipolar  o  interactions  could  lead  t o such behavior.  subject of cross-correlation are  the a r t i c l e s  by F r e e d  discusses at great length ation  9  V e r y good summaries on t h e  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  and F r a e n k e l .  9 10 '  the applications  i n general  R e f e r e n c e 10 i n p a r t i c u l a r  of cross-terms  i n ESR  relax-  studies. The  examined utilize  importance of.cross-terms  i n n u c l e a r r e l a x a t i o n was  i n d e p t h when i t was r e a l i z e d t h a t  first  i t m i g h t be p o s s i b l 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  tool.  Just  to as  -131-  the  cross-correlation  electron-nuclear of  the  sign  anisotropy  dipolar  of the i n the  chemical  should allow  coupling  constant.  one  g-tensor anisotropy  i n t e r a c t i o n s permit experimental  hyperfine  actions  namely the  terms between the  splitting, shift  1 0  the  determination  absolute sign  s y s t e m composed o f  the  nuclear dipolar  A number o f p a p e r s h a v e t r e a t e d  scalar-coupled  the  a c r o s s - t e r m between  t e n s o r and  to determine the  and  two  inter-  of J , the  an AX  scalar  spectrum,  non-identical,  spin  1/2  nucleus  is  11-13 nuclei. relaxed  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 s o l e l y by  additionally time"  (e.g.  manifested (i.e.  relaxed chemical  i n the  dipolar by  i n t e r a c t i o n and  shift  anisotropy).  dom  observable  changes are  magnetic f i e l d  "X"  nucleus  linewidths  terms  within  the  not  c h a n g e s due  be  to the  result  equations.  i n these  confused with  MacLean  this  treatments  interference), pairwise to  are X-multiplet  M a c k o r and  relaxation  considered  is  "same-correlation  interference  terms i n the  a c e r t a i n c r o s s - t e r m was  linewidth  "A"  "anomalous r e l a x a t i o n " t o d e s c r i b e  (deemed t h e m o s t l i k e l y  discussed  the  the  for different transitions).*  phrase  * These l i n e w i d t h  The  unequal  i n c l u s i o n of c r o s s - c o r r e l a t i o n  Although only  that  a second mechanism w i t h  appearance of  d i f f e r e n t T^'s  have c o i n e d t h e of  the  the  interference  other  frequently  c o r r e l a t i o n of a c l a s s i c a l  ( s t o c h a s t i c f i e l d model) produced at  two  ran-  different  14-19 nuclear s i t e s . width within  T h i s phenomenon a l s o  e a c h m u l t i p l e t o f a two  a l t h o u g h f o r an analysis  of  AX  system  ( J ^ «  cross-correlation  ^^y)>  directly  on  be  fruitful  the  present  (see  system  this  terms using  proximation, especially i n conjunction m e n t s , can  spin  causes v a r i a t i o n s of  random l o c a l  AB  but  does not  system) The  field  double resonance  above r e f e r e n c e s ) ,  discussion.  i n an  e f f e c t vanishes.  the  with  (e.g.  line-  ap-  experibear  -132-  effects  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 4  could that  also y i e l d t h i s i s not  chapter  (see  the  same i n f o r m a t i o n .  true  f o r the  A p p e n d i x D).  coupled  systems  20 '  However, i t can  interference  be  terms c o n s i d e r e d  shown  i n the  last  A paper which g i v e s a v e r y good i n s i g h t  into  the e f f e c t of 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 the 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 . Other relevant  papers d e a l i n g  with  the  general  topic  are:  Void  20  and  21 Gutowsky's  analysis  of  a spin  1/2  - spin  3/2  scalar coupled  system,  22 Sykora's  interesting discussion 23  c r o s s - t e r m s , and  Hoffman's  cellent exposition Up  terse  of magnetic  to t h i s p o i n t ,  of  the  the  o c c u r r e n c e and  discussion  of cross-terms  enlightening  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 m e c h a n i s m s can  In a s e r i e s  the of  extraction  fundamental, yet  with  t e r m s i n an  or d i s c u s s e d  of  Blicharski,*  Effects  in a quantitative  systems of  unlike  anisotropy, The  first  being  quadrupolar  included  i n the  spins  and  * An  l i m i t of  Jadrowej U  from these  by  and  treatments  K r a k o w i e " R e p o r t INP  No.  792/PL  of  spin-rotation  terms.  Both  appears  (1972).  shift  interactions.  i n most c a s e s , i t w i l l  (in Polish)  Relaxation  consider  spin-rotation  isotropic reorientation.  t h i s work  terms.  d i p o l a r , chemical  i n t e r f e r e , the  interference  They  interfere,  effect  mechanism be  the  longitudinal  b e h a v i o r i s examined f o r 1,2,3,4, o r  rapid  i n c l u s i v e summary o f  the  His  relaxed  calculation, since,  transverse relaxation  systems i n the  being  mutually  dominating masking e f f e c t of  mutually  i n Nuclear Magnetic  manner.  (where a p p l i c a b l e ) ,  t h r e e t e r m s can  ex-  cross-correlation  t h o r o u g h l y e x p l o r e s the  0 1  cross-correlations l i k e or  in his  very l i m i t e d context.  practical information  papers " I n t e r f e r e n c e  I - V I I I " , J . S.  of  relaxation.  p a p e r s c i t e d have d e a l t  and  importance  Central  6-spin to  in "Instytut  Fizyki  -133-  Blicharski's it  treatments  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 s w e l l known t h a t when t h e s p e c t r u m 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 , the lineshape  i s i n general  a sum o f L o r e n t z i a n s w i t h  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 exponential  z  - <I >  x  P l o t s o f t h e a . j , B., ing constants  T  Z  <I (t)>  =  = Ea exp(-\ t)  [5.2.1]  3 exp(-y t) .  [5.2.2]  i  £  i  i  i  A . . , a n d t h e y^ a s 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 -  o f t h e i n t e r f e r i n g m e c h a n i s m s c a n be f o u n d  papers i n t h i s  s e r i e s by B l i c h a r s k i .  i n various  These papers p r o v i d 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  magnetic  i s multi-  (nonexponential),  <I (t)>  hensive  various  t h e most compre-  i n f l u e n c e on n u c l e a r  relaxation.  Although  the basic theory  i s well defined, experimental  behind  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  observations  of the resultant  nuclear r e l a x a t i o n are rather scant  (excepting of course  experimental  This  s t u d i e s i n ESR w o r k ) .  shortage  masking and q u e n c h i n g e f f e c t s , and b e c a u s e , u n t i l  nonexponential  the wealth o f  c a n be a t t r i b u t e d t o very  recently, only  19 proton  (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 times  o f small  molecules  were r o u t i n e l y and r e l i a b l y m e a s u r e d ; and i n such c a s e s , one w o u l d n o t expect  nonexponential  e f f e c t s t o be  Relative line-broadening  two-spin fact  observable.  o f v a r i o u s components o f t h e s c a l a r 12  s y s t e m C F H C ^ has b e e n o b s e r v e d ,  i s directly  attributed  coupled  and as m e n t i o n e d b e f o r e ,  this  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.  Interference effects are d e f i n i t e l y  responsible f o r nonexponential  nuclear  -134-  relaxation  f o r the f l u o r i n e  nuclei, i n C F g C ^ ^ ' ( f r e o n 12) and  E a c h 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 temperatures  a b o v e 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. atures, the effect interaction.  i s obscured  In c o n t r a s t t o the e x p e r i m e n t a l l y observed  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  by  rationalized  the strong s t a t i c 32  lation  i n a g a s e o u s s a m p l e o f HD.  physical trast,  state studies.  This  a r e quenched  34  theory.  i s n o t t h e o n l y mathe-  I n some o t h e r a p p r o a c h e s , c r o s s -  t e r m s c a n be i n c o r p o r a t e d i n t o  reasons  t e r m s has t o o u r  in solid  density matrix theory  formulation of relaxation  correlation  temper-  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 -  course, the standard  matical  just  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  33 '  effects  .  effect of  by a s s u m i n g t h a t i n t e r f e r e n c e e f f e c t s  temperature.*  Of  At higher  3 0  by t h e dominance o f t h e s p i n - r o t a t i o n  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  is easily  ^BF^  the formalism  f o r t h e same  w i t h many o f t h e same p h y s i c a l c o n s e q u e n c e s .  Kubo's l i n e a r r e s p o n s e  theory  i s inadequate  to treat  I n con-  general  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 degenerate t r a n s i t i o n s as it  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 presence  Thus l i n e a r r e s p o n s e it  fails  theory  is ill-suited  of cross-terms.  f o r t h e p r o b l e m a t hand a 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 closer  look a t  35 this  p r o b l e m shows some o f t h e c r i t i c i s m  we c i t e  i s unjustified.  the formulation o f Redfield's theory  * Indeed, t h i s  e x p l a n a t i o n suggests  terms appear s u p e r f i c i a l l y  Finally,  i n L i o u v i l l e space.  In a  t h a t w h i l e Hubbard and B l i c h a r s k i  a l i k e , there are radical 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 n o t c o m p l e t e l y  quenched i n s o l i d s .  -135-  pair of papers,  Pyper  J D  thoroughly  discusses relaxation  in this  "super"  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 ,  only f l e e t i n g  comments on  Blicharski  terms  p a r t , t h e o n l y d i f f e r e n c e b e t w e e n t h e s e was by P y p e r a r e and  to the  t o be v e r y  r e c o m m e n d e d , as t h i s  t h a t the term  brief  semantics).  These papers in general,  terms i n p a r t i c u l a r ,  i n t r o d u c t i o n t o the problem a t hand, i t i s  " 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  sense.  correlation  f o r t h e most  appears  promising.  From t h i s  stricted  (but r e c a l l ,  approach to r e l a x a t i o n  influence of cross-correlation  makes  to the p r a c t i c i n g  in a rather re-  A l s o , the 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 the term  touches  on  NMR  other related spectroscopist.  (and  apparent  cross-  unrelated) topics familiar  -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  € &Q =  iently  be d i v i d e d  tonian  ( c o m p o s e d o f t h e Zeeman t e r m a n d f i r s t o r d e r c o r r e c t i o n s ) , £ Q ,  determines l i n e  i n t o two p a r t s ,  s p i n H a m i l t o n i a n can conven-  + £ ( t ) , where t h e s t a t i c  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 ) , determines the l i n e w i d t h s ! l a t t i c e coupling,  may  in itself  w h i c h f o r e m p h a s i s , may  I  - II  1  be composed o f many s e p a r a b l e  be w r i t t e n  1  (t)-I  as b i l i n e a r c o u p l i n g s .  + I - - ( ^ . ( t ) - a.i  i  df(t), the spin-  n  )-B  n  +  Q  couplings,  For example;  I - t ^ t J - J ^ t )  + I.j-B.ft)  [5.3.1]  E as t h e second rank u n i t t e n s o r .  quency u n i t s .  The r e s p e c t i v e  d i r e c t dipole or anisotropic (Q), field  chemical  shift  The c o u p l i n g s  couplings spin-spin  anisotropy  (RF) c o u p l i n g s .  r e l a x a t i o n , chemical  a = T r [ # ] / 3 , and  r e f e r to n u c l e i , J E T r [ j ]/3,  where t h e s u b s c r i p t s I  Hamil-  are i n angular  are the d i r e c t dipole (ID),scalar  (CSA), s p i n r o t a t i o n  I t i s assumed t h a t e x c h a n g e , and f i e l d  fre-  (D), i n -  (SC), quadrupolar ( S R ) , a n d t h e random  the e f f e c t s o f inhomogenities  intermolecular comprise  Of c o u r s e , u s u a l l y one o r two m e c h a n i s m s d o m i n a t e t h e r e l a x a t i o n o f a given nucleus in  ( a n d o t h e r m e c h a n i s m s may v a n i s h  c e r t a i n environments).  considered,  and t h e r e f o r e  In t h i s chapter, only 6?g m u s t v a n i s h .  completely f o r nuclei spin  1/2 n u c l e i w i l l  Furthermore, i t shall  be  be  -137-  assumed t h a t  the  spin  system  is sufficiently  d i l u t e i n an  free  from paramagnetic i m p u r i t i e s , extraneous f i e l d  ical  e x c h a n g e so  that  can  be  ignored.  of a time dependence of e i t h e r the (scalar coupling one  of  the  of  coupled chemical  relaxed  the  ignored.  1st  spins  absence of by  the  The  the  of  i f neither  the  anisotropic  the of  quadrupolar i n t e r a c t i o n this  Modulation of  term c ? ^  projection  (scalar coupling  coupling  fashion  analogous to r e l a x a t i o n  constant  kind).  In  of  the is  r e l a x a t i o n m e c h a n i s m may  portion  i n d u c e d by  because  nuclear spins  of  the  t e n s o r , J , by m o l e c u l a r m o t i o n c a n  dipolar  arises  chem-  quantum number  2nd  the  solvent,  g r a d i e n t s and  i s o t r o p i c scalar coupling  kind) or  e x c h a n g e and  inert  spin-spin  be  indirect  cause r e l a x a t i o n  the  direct dipolar  may  be  in  coupling.  37 T h i s seldom d i s c u s s e d form of r e l a x a t i o n nuclei  with  a large  nored  i n any  number o f  further  n u c l e o n s , but  discussions  in this  assumptions, i t i s only necessary to actions  i n the  The  relaxation  t h e o r y of S^{t)  sequences, i s well ters.  d o c u m e n t e d and  In c o n t r a s t , ^ ^ ( t ) ,  recently  escaped the  practical routine  c a l c u l a t i o n s , 8{t)  purposes).  w o r k , and  the  well  R  although  with  coupling  role  i t may  not  i n the  the  + <  D  as  the  relaxation  ^SR^^' con-  in previous  now  until  chap-  very  (for a l l  available  studies ( t ) may  relaxation  ig-  inter-  experimental  l a r g e r magnetic f i e l d s  m a g n e t i z a t i o n decay.  be  these  = c? (t) + c^^U)  e f f o r t s of d e t e c t i o n  i n t e r p r e t a t i o n of future  for  three remaining  l o n g d i s c u s s e d , has  r e c e n t e m p h a s i s on  dominate the  will  Thus, with  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 that.dt^ larger  importance  have been d i s c u s s e d  experimentalist's But  this  paper.  include  a n d c ? < - ( t ) , as  of  of  for  nuclei  play  d a t a , even  Indeed, i n a d d i t i o n  a though to  a  -138-  recent experimental  studies confirming  r e m a r k a b l e "7/6" e f f e c t u n i q u e l y  this interaction,  3 8 39 ' even t h e  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 apparently  been  observed.  Each o f t h e s e products most  o f spin  interactions  and l a t t i c e  i nturn,  c a n be w r i t t e n  operators.  The f o l l o w i n g  a s t h e sum o f expansions a r e  useful: n  f (t) £ D  +  C S A  Z  (t) £ (t)=§ +  2  S R  E  uU.jvU.jfi-Sij)  _  n  A=SR,CSA where t h e l a t t i c e  1  i = l k=-l  ^  parameters f o r t h e two, one-spin  A  ;  i  l  M  J  l*-*-'!  i  interactions are  given by,  U  u  (s  R ) i  (CSA)  H) MC]oi  =  ? C S A / -  = i  k  k  +  (^)  1  /  2  AC  1  )  k  k  ^ i  E  (  t  )  ^ - -  )  5  S j k ' k l )  J „ k  Y  2 '( k  3  f  i  3 ]  i^'  [5.3.4] where £ nuclei.  C  S  A  = (2TT/1 5 )  -  The s p i n  V  V  (CSA)  (S  The  expansion  The  time  axis  R ) i  1 / / 2 Y  -JBQ(AO)  operators  i  = <8/3)  " z !  •  f o r these  1/2  lj ;  (SR),. "  V  f o r the dipolar  dependent a n g l e s ,  o f the i  chemical  and J Q = J.J.  Script  interactions  i and j r e f e r t o  are given by,  v f ^ , . = +1*  [5.3.5]  K  [5  IJ1  •  interaction i s given  -'  by E q u a t i o n s  ( t ) ,define the orientation  shift  (orspin-rotation)  3 6]  [3.1.3-4],  o f the principal  coupling tensor  with  -139-  respect to It written scalar  the  lab  frame.  The  is usually  stated  that  as  a product of  contraction  fruitful,  as  the  of  the  analysis  cients for  (the  of  the  as  a scalar  whereas the spherical  used  problem.  of  coupling  are  (by  i t c a n n o t be  written  two  spherical  tensors  ( a n o t h e r way  of  o f T-|  and  T  be  as  a scalar  saying  i s chosen is  arec2g, of  the  rise  say to  CSA  that the  i n the  @JQ)> first  contraction  t h i s i s to  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  b e e n made.  The  spin-rotation  izable with  two  i n d e p e n d e n t c o m p o n e n t s , C„  and  to the  a n g u l a r momentum o p e r a t o r s  ( i . e . the  [5.3.4]),  tion  coeffi-  couplof  the  inter-  intrigu-  extreme-  approximation.  In w r i t i n g  addition  under  coupling  note that  .  simplifies  a contraction  even f o r a s i n g l e n u c l e u s  2  tensors  (as  a  extremely  [5.3.3-6]  dipolar  r o t a t i o n a l l y i n v a r i a n t ) , thus g i v i n g  inequality  narrowed  the  be  to  is  d e f i n i t i o n ! ) and  F i n a l l y , we  in that  ing  reduced  rank t e n s o r s  i s unique  i s not  be  [ 3 . 1 . 3 - 4 ] and  ing  action  should best  spherical  i s seen to  R[  standard.  A s i d e from n o r m a l i z i n g  second  <£ -).  as  This approach  of  i t i s seen t h a t  contraction  (as  known  i n Equations  spin-rotation  tensors  properties  i s very well  simplicity-not rigor),  written  t e n s o r s w h i c h can  i n d i v i d u a l components.  relaxation  notation  i s regarded  magnetic perturbations  spherical  transformation  (molecular) rotations  other notation  the  lattice  interaction tensor  functions  may  also  necessarily  i s assumed t o  C J A C  be  diagon-  = C,,-^).  Also,  first  in  d e p e n d on  term  the  in  Equa-  orientation 41  of if  the t  molecule through can  be  written  dependence v a n i s h e s . has  been w r i t t e n  as  as  the  second rank s p h e r i c a l  a multiple  The  of  the  harmonics.  unit matrix, this  chemical  Of  course  orientation  s h i f t anisotropy i n t e r a c t i o n constant 1/2 1/2 = ( 2 i r / 1 5 ) ' Y -' g(o, -a ) = (2TT/15) u^bo.. B  1  l  L  -140-  Implicit screening cribed  i nthis  assumption  i s t h e f a c t t h a t t h e asymmetic second  tensor, £ , i s isotropic  parallel  perpendicular  components w i l l  t e n s o r and t h es h i e l d i n g  A l t h o u g h c a l c u l a t i o n s c a n be c a r r i e d o u t  «v  the general  ( l  to theaxis of the shift  to this axis.  des-  and a ; t h e e l e c t r o n i c  by o n l y two i n d e p e n d e n t c o m p o n e n t s  screening  using  i n o n e p l a n e a n d c a n be f u l l y  rank  42  f o r m o f 3,  the simplification  o f two i n d e p e n d e n t  be made, e q u i v a l e n t t o t h e a s s u m p t i o n  that the spin  lies  43 at a s i t e  C  3 v  o r higher  interaction vanishes.) ing  tensor,  (2a +a^)/3, x  symmetry.  I t i s a l s o assumed i s included  Furthermore, t o avoid  attention  symmetry,this  that the trace of the shield-  in £ .  complications  s y s t e m o f two i s o c h r o n o u s s p i n s is considered.  (However, f o r c u b i c  Q  i n v o l v i n g Hubbard t e r m s , a  (with n e c e s s a r i l y degenerate t r a n s i t i o n s )  W i t h 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  to the calculation of the spin  behavior.  -141-  5.4 SOLUTION OF THE RELAXATION MATRIX As  eigenstates  we s h a l l  o f t h e Zeeman  Hamiltonian,  choose t h e s i m p l e uncoupled  a-j > - |1> =  simply  erate For  a >  =  |3> =  a >  =  |4> , l " >  [5.4.2]  so c h o s e n , t h e m a t r i x  evaluated.  As s t a t e d  a l l off-diagonal  ^  3  elements o f Equation  so o f t e n , o n l y contributions)  is  now q u i t e  lists 10  vector  = <a|x(t)j3>).  (t)  terms which connect n e e d t o be  metry incorporated  a a  = R gg'aa'  x  2 2  (t), x  n o t a t i o n , \p i s t h e s p i n  e n a b l e s one t o except x  a  d  n  2 3  x U),  (t),  3 2  [5.3.6],  a table  x  3 3  (t),  x  4 4  (t)];  and [ 3 . 1 . 4 ] , i t  such as T a b l e  5.1 w h i c h  Although  only  26 t e r m s c a n be d e d u c e d f r o m t h e sym-  into the construction  i n w h i c h c a s e R, .  matrix  degen-  considered.  to the relaxation matrix.  theother  = R V a 3 ' 3  2 3  [5.3.5],  to construct  - R " 3'3a'a  'gg. except f o r the s h i f t  tions  i n thedensity  Equations  contributions  elements a r e l i s t e d ,  K  Using  [2.2.13] can  o f these values o f x ( t ) as d e f i n i n g a  (Lx^U),  straightforward  thevarious  R aa'33'  elements  i sconvenient to think  six-dimensional  =  |-+>  thecalculation of the longitudinal relaxation, this  x 2-  a g  +  transitions (the secular  neglect  X  + +  |2> = | ->  4  be  | >  a^> = 3  With these s t a t e s  basis.  ( i . e . d e f i n i t i o n ) o f R;  and R 5-a,5-a',5-3,5-3'  anisotropy-dipolar = -R  ,  O D  cross  term  , ( s e e Appendix A ) .  inversion operator  - p " ijwjja >34<B  contribuIn t h i s  ( e . g . ^|1> = |4>, \|J|2> = |3>).  1  -142-  Furthermore,  t h e t e r m s R-j-|-|-| and  these terms g i v e the t o t a l a r e d e p l e t e d by conditions  (i.e. zero.  i n  t  h  to other  i  s  T  a  b  l  e  a  r  redundant s i n c e  e  s t a t e s 1 and  levels.  2  [respectively)  N o t e t h a t t h e two  i m p o s e d upon R i m p l y t h a t R - j , ^ 2 2 3 3 ' ^ 2 3 2 3 '  t h e 25  R  2222  r a t e at which  transitions  h a v e no d i p o l a r - s h i f t Of  R  ,  E  r ? T 1  'R ' R  c >n  t o e a c h o f t h e 36  ^2332  elements  of  )» 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  F i v e are the conventional "auto"  contributions,  6  R Cn  terms a r i s e from  '  c  a  the cross-terms  b e t w e e n CSA-j  a n c 2  '  ,„„,, aa  R nonfour  66  the d i p o l a r  mechanism, CSA  and  the remaining CD  or R  f o u r terms a r e pseudo c r o s s - t e r m s  i  j .  acterized  Cross-terms  by d i f f e r e n t  correlation  b e t w e e n CSA^  orientation  orientations  are  i  j  j  2  2  k  a l l j , k, a n d  elements  parameters  be  2  charThe  <d. ( t ) Y ( t ) Y ( 0 ) > 2  2  a s s u m e d t h a t t h e m o l e c u l a r a n g u l a r momenand  furthermore, that a l l i n i t i a l  = <Jj(t)><Y (t)Y (0)> 2  k  j  2  [2.2.13], C ^ ( a l l i n s p e c t i o n , i t can  = 0 and  =  these  correla-  <J^(t)>  k  [5.4.3]  0  t ) = 0 -* J J-(u) = 0 + be  = 0  2  = <J (t)><Y (0)>  i since <Y (t)>  o f x are not  are  SR-j  to  <0j(t)Y (0)>  By  t h e d i p o l a r mechanism and  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  f u n c t i o n s reduce  Equation  r  dependent l a t t i c e  independent,  <J .(t)Y (t)Y (0)>  for  -CSA  R  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 .  I f i t can  2  and  o 2  f u n c t i o n s between t h e s e terms a r e o f the form  or < J j ( t ) Y ( 0 ) > .  tion  o f the form  CD  n e c e s s a r i l y vanish s i n c e the time  tum  n  anisotropy cross-terms.  possible contributions  , =  a n c  symmetry  = 0. R  C n  Therefore, in  = 0 f o r a l l k and  seen t h a t the time dependency o f the s i x  independent.  I t i s now  convenient  to  introduce  a.  -143-  the  f o l l o w i n g four orthogonal  y-,(t) =  (t') - x  X l l  y (t)  = x  2 2  y (t)  = -  X  y (t)  - x  3 2  2  3  4  linear  ]  (t)  = x  n  ( t ) - x  (t)  4 4  elements,  (t)  [5.4.4]  (t)  [5.4.5]  3 3  - x  (t) + x  N o t e t h a t y-j ( t ) has b e e n d e f i n e d  y^t)  4 4  (t) - x  1  combinations of matrix  4 4  (t) + x  (t) + x  2 2  (t)  3 3  [5.4.6]  (t)  2 3  such  [5.4.7]  that  = Tr[ (t)I ] x  = <I (t)>  z  -  z  <I >  T  Z  [5.4.8] I t t h e n f o l l o w s f r o m T a b l e 5.1  (d/dt)  y i  ( t ) = [A-,- 4 A  + s]  1  - *  + 2(A]  2  2 2  that  ]y (t) 2  ]  + A  2  2  )+ $j  + [2(r]  + s f j y ^ t )  ]  + r ')]y (t)  + [2(A]  + [2(r]  2  3  - A  ]  + r  2  )  2  )]y (t)  2  4  [5.4.9] (d/dt)y (t) 2  =  [2(A] -  +  + $  ]  O]  A ?)+ 2  2 2  1  - $  2 2  ]  y i  (t)  ] y ( t ) + [2(r]' - r  +  2 : :  2  [-§  A  )]y (t) 3  +  Q  A  ]  2(A] + ]  +  A  2  )  2  + [j(rj - r ' )]y (t) 2  4  [5.4.10]  (d/dt)y (t) 3  = [4(r] + r ) ] 2  +  (d/dt)y (t) 4  2($]  ]  Y  l  + * f )]y (t) 3  = [2(r] + r ) ] 2  +  (t)  " fCAfj  1  + [4(r] - r ) ] y ( t ) 2  2  + [2A-, + 8 A ]  + 4$] ]y (t)  2  4  ( t ) + [|(rj - r ) ] y ( t )  + A  2  2 2  -2Aj ) 2  (.Jl  2 +  $  2  +  ]  + [>, + 4 A ] + 2  Q  +  2 2  2$] ]y (t)  2  - * )  A )  [5.4.11]  2  2  y i  + [2A-, + 4 ( A ]  2 ( A ^  2  3  +  A  2  2  )  -144-  + o] + $22]y4Ct)  £5.4.12]  1  k -k where t h e f o l l o w i n g A  k  shorthand  ~ CSA -CSA. S  J  J  (  k  The  general  ) ;  r  k ~ °CSA -D k  solution  linear differential  n o t a t i o n has been i n t r o d u c e d : ( k u  0  ) ;  E  J  SR.-SR.  f o r a system o f n f i r s t  equations  with constant  ( k c o  order  coefficients  (kcjg);  = J^'  0 ' }  homogeneous i s given by,  n  y,-(t) =  S  [5.4.13]  a,,exp(x.t)  j=l  1  1 J  J  w h e r e t h e a., a r e d e t e r m i n e d b y t h e a p p r o p r i a t e b o u n d a r y c o n d i t i o n s . s o l u t i o n c a n be o b t a i n e d Laplace  but is  b y m a t r i x m e t h o d s , a l g e b r a i c m e t h o d s , o r by  T r a n s f o r m methods, b u t i n any c a s e , w i l l  of the roots o f a n  used t o s o l v e t h e present  toexp(b)to y(0) _1  w h e r e 111 i s a n o n s i n g u l a r i s the diagonal  ^2^'  y  3^)'  y^tJL  form o f  perusal  ;  b ^ a t t l  [5.4.14] 1  [y-|(t),  a n d 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 . The d i a g o n a l  o f Equations  the decay o f m a g n e t i z a t i o n  expect  thefollowing  approach  t r a n s f o r m a t i o n m a t r i x , fat) i s i t s i n v e r s e ,  o f Sl a n d t h e c o l u m n s o f t D a r e  although  f o r any  The m a t r i x  s i m i l a r i t y m a t r i x o f 91 , y ( t ) i s t h e v e c t o r  ( d / d t ) y ( t ) = £1y(t)).  By  i s tedious  equations.  problem and y i e l d s  determination  44  y(t) =  h  involve  degree e q u a t i o n , a t a s k which  a s y s t e m composed o f o n l y two c o u p l e d  solution,  The  i n t h e absence  elements o fb a r e t h e eigenvalues  the corresponding  eigenvectors.  [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 ,  i s nonexponential  ( t h e sum o f f o u r  exponentials)  ( 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  t h e d e c a y t o be d e s c r i b e d a s t h e sum o f o n l y two e x p o n e n t i a l s .  -145-  If  the spectral  spin-rotation  y^t)  d e n s i t y a t u> i s i d e n t i c a l  f o rboth  Q  i n t e r a c t i o n s , then  = y (0')exp[(A 1  - 4A  1  a unique exponential w i l l  + 4A  2  chemical  ]  s h i f t and suffice,  = y^OexpC t / T ) .  + 2^)t  Q  [5.4.15] Other l i m i t i n g  cases  areir^, A^  -> 0 w h e r e t h e d e c a y i s g i v e n b y  J  y - ] ( t ) = y-j ( 0 ) e x p ( ( - - | - 4 A ) t ) a n d A  2  is given  are very f a m i l i a r from find  i t convenient  [5.4.9]  t h e decay  standard  treatments.  M o r e g e n e r a l l y , we  values will  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  1  solve the general  specified  (2)  ->• 0 i n w h i c h c a s e  as T" .  To  rices  K  22 t ) + exp(4A^ t ) ] . These l i m i t i n g  I ' l l ( t ) = ^{exp^A-j  by  , A  as w e l l  relating  as t h e t i m e  these  the molecular  p r o b l e m , s e v e n c o o r d i n a t e s y s t e m s m u s t be (in)dependent  various coordinates.  nature o f the r o t a t i o n These a r e ;  frame i n which t h e d i f f u s i o n  tensor  mat-  (1) t h e l a b frame, i s d i a g o n a l , (3-6)  c o o r d i n a t e f r a m e s w h i c h d i a g o n a l i z e e a c h o f t h e two s p i n - r o t a t i o n  coupling  t e n s o r s a n d t h e f r a m e s w h i c h d i a g o n a l i z e e a c h o f t h e two c h e m i c a l  shift  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 along  the i n t e r n u c l e a r vector. As m e n t i o n e d 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 a n d c h e m i c a l  shift  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  included  i n the l a t t i c e  However, t h e s t a t i c  parameters  to the molecular  spectral  densities  in detail.  [ 3 . 1 . 3 ] and [ 5 . 3 . 3 ] ) .  o r i e n t a t i o n o f these various coordinate  relative  amined  (Equations  explicitly  systems  frame i n f l u e n c e s t h e magnitudes o f t h e v a r i o u s  appearing Consider  i n Equation first  [ 2 . 2 . 1 3 ] a n d m u s t now be e x -  the correlation  f u n c t i o n s A^, A ^ , and J  -146We w i l l assume the c o r r e l a t i o n f u n c t i o n s are a c c u r a t e l y  r^.  by E q u a t i o n [ 2 . 3 . 1 0 ]  (i.e.  the m o l e c u l a r frame dynamics are f u l l y d e s -  c r i b e d by o n l y two c l a s s i c a l d i f f u s i o n c o n s t a n t s , D a symmetry a x i s and major a x i s ) .  described  f o r motions about  (1  f o r motions about an a x i s p e r p e n d i c u l a r t o t h i s  In t h e s e e q u a t i o n s , i t i s assumed t h a t the s p i n s are  r i g i d l y a t t a c h e d t o t h i s symmetric body.  S i n c e the p r i n c i p a l concern  o f t h i s paper i s w i t h l a r g e m o l e c u l e s , and o f t e n l a r g e m o l e c u l e s  are  g l o b u l a r i n n a t u r e , t h i s e q u a t i o n and a l l f o l l o w i n g e q u a t i o n s w i l l h o l d t r u e i f we assume D E D  ±  and D..^ = D  ()  -  also  where D. ^ i s the c l a s -  s i c a l 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 s t i c o f an i n t e r n a l r o t o r a t t a c h e d to t h i s b u l k y framework.  The a r b i t r a r y attachment o f an i n t e r n a l  t o an asymmetric top cannot be t r e a t e d as s i m p l y as shown here Equation [4.3.15-16]). and  0  r  k £  ?n  It  rotor  (see  then f o l l o w s from E q u a t i o n s [ 2 . 3 . 1 0 ] ,  [3.1.3],  [5.3.3.],  /t) T 7 r ( 3 c o s e '  ( k a ) ) = ( - l ) £ ? 6. k  n  0  c n k,-£\5/ 16rr  +  30  c  ?  C  n  C  n  C  n  Z  (  6  D  j2  n * (2D, + 4 D . / + ( k . ) 2D  n  30 + n-cose'cose'sine'sine'cosU'  .  2  2 2 s i n e ' s i n e'cos(2(d)' - *'))  3 2 7 1  8T  - D(3cos e' - 1 ) — ^ —  2  J.  +  4 D  n  - <$>)' n  5 D  (5D _ J  i  +  + DJ  D  2  ^  +  2  ^ 2  u + (kco ) Q  2  •  [5.4.16] ( R e c a l l t h a t the primed a n g l e s r e f e r t o m o l e c u l e - f i x e d a n g l e s . )  It  is  apparent t h a t the magnitude o f the s p e c t r a l d e n s i t i e s are dependent o n l y upon the rank o f the s p h e r i c a l harmonic and not on i t s component.  J  Cn  = J . nc  45  Also.  -147-  For  the calculation of  a n g u l a r momentum tions.  This  , i t i s necessary to evaluate the various  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  i s by no means a t r i v i a l  i n contempory  studies.  p r o b l e m a n d a t t e n d s much  An e x p r e s s i o n f o r ( T ^ )  S  R  assuming  func-  attention  anisotropic  46 molecular reorientation of  t h i s expression  i s i t s e l f quite  complex,  and t h e  incorporation  i n t o t h e present c a l c u l a t i o n s would n e c e s s i t a t 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 tensor,  ( 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  to separate t h e angular p o s i t i o n a l  reorientational  the  coupling  ( i fi ti s not possible and a n g u l a r  momentum  reorientational  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  reorientation.  T h e r e f o r e , we s h a l l  make t h e t a c i t  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 , action  could  be r e t a i n e d  approach would lend to t h i s  point  The s p i n - r o t a t i o n  inter-  as a phenomenological masking p a r a m e t e r , b u t t h i s  little  t o the present treatment.  We s h a l l  return  later.  Substitution the  = 0.  assumption that the  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  subsequent s o l u t i o n  ( b y means o f E q u a t i o n s  coupled e q u a t i o n s s u b j e c t t o t h e boundary y^O)  = (cose -  [5.4. 14]) o f these  conditions,  1)<I >  T  Z  y (o) = 0 2  y (o) = o 3  y (0) = 0  [5.4.17]  4  f o r v a r i o u s ranges o f geometries, 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.  -148TABLE 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 relaxation matrix  R  A  'llll  1122  '1123  M l 33  k  1144  D  -2A  ]  l  2  11 77 2A-] ' + 2 A " 22 1  -A-,/2  -2A  12 1  r 1  -y2  -2A  11 1  2r.  2A,  0  V  '2323  A /3+A  r  k  r  +  $  j  1  12 1  r l  .11 1 ?  *  - 2r'  o ]  2  Q  N  2 2  2  ]  +  superscripts refer  CSA - (S)' D  2 2  2A  -2A  2 0  )  11  0  22  ,o  >0 "*0 o  2 2  +  J  1 +  +2$  *  12  2 2  0  Q  k  >1  2r /3 -2r /3  -8/3(A +A  1  A /3  s J  i  3  Q  Notation:  SR -SR  2  2  +  2r  12  +2A]  u  - 2r  R  22 1  1  1  1  2A  CSA.-D  2r!  11 77 2A-J + 2 A "  1  A-,/2  2332  -2r]  -2A  7233  v  R  J  -A-,/2  Q  2223  CSA,-CSA.  K  •A /3+A  '2222  l  D  a n d  to nuclei;  *k S^R.<V" SJ  A  Ej '~ (k k  K  k  U ( )  )  ,  AJ " E J  j£sJJ _ k  CSA  (ko» ), 0  -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  [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  a simultaneous analysis zation.  Still,  of various effects  a few exemplory  densities  i n Equation  ([5.4.9-12])  precludes  on t h e l o n g i t u d i n a l  magneti-  c o n c l u s i o n s c a n be d e d u c e d f r o m  Equation  [5.4.16]. If  6p = c o s  _ 1  (l//3),  then n e i t h e r  the cross-terms nor the d i p o l a r 2  spectral D  u  >  >  densities  x ' ? L ~ ^0 ^  D  6 C  depend on t h e 6D /((6D _) 1  f o r  k  =  0  t  e  r  m  s  t  n  i  i  latter  s  2 + (kco ) ) t e r m .  s t i p u l a t i o n i s super-  fluous),  i s approximately (but not i d e n t i c a l l y ) equal  and  a r e much d i f f e r e n t f r o m c o s  e^  T h u s , when  Q  to c o s  - 1  (l//3),  ( l / / 3 ) , then the terms w i l l 1»2 assume l a r g e r m a g n i t u d e s t h a n t h e A ^ t e r m s e v e n i f t h e s t r e n g t h o f t h e SA  CSA i n t e r a c t i o n »  i s much l e s s  _ 1  than t h a t o f t h e d i p o l a r  interaction  ( ?  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 s2 c o o r d i n a t e s y s t e m s may h a v e a n 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 cross-terms.  F o r a n o t h e r example, f o r t h e geometry where  and  -1 - cos  1  2  ( 1 / / 3 ) , a l l c r o s s - t e r m s v a n i s h e v e n i f t h e two r e l a x a t i o n m e c h a n -  isms a r e o f comparable  magnitude  these e q u a t i o n s i s a seldom the slow motion (say  '  regime,  ~ 1 »2  emphasized  internal  A l s o apparent  result of diagnostic  rotation  absence o f i n t e r n a l  flexibility.  S i n c e each m o l e c u l e would impractical  value: i n  leads to spectral  a t t h e Larmor f r e q u e n c y ) which a r e l a r g e r 47 4 8  from  densities  i n magnitude than  i nthe  '  present a v i r t u a l l y  unique  to exhausively analyze the general r e s u l t s  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  case, i t i s  outlined  f o rdiscussion.  thus f a r . The  -150-  first  corresponds t o that  implied  i n the pioneering  work o f B l i c h a r s k i ,  g e o m e t r y J_, a n d t h e s e c o n d 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 g e o m e t r y f o r two J_,  1/2 n u c l e i  spin  (1/3),  II,  - <f)^  (cj>'  l i n e a r with  t h e c o v a l e n t bond a x i s  is directed  along  and t h a t  tensor  the principal diffusion  a grounds f o r a numerical  experimentally,  anisotropics  t o t h e m o l e c u l a r frame.  i t proves q u i t e  carbon  i t i s assumed t h a t 5.1-3 g i v e  yB  decay, i n g e n e r a l ,  therelaxation  i s t h e sum o f f o u r  1  :  to relate  shift calcu-  -1  parameters which  exponentials, 11  tensor principal  I t m i g h t pay  sec" .  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  shift  axis  (say) t o  F i n a l l y , i n the numerical  = 6 x 10  n  example.  difficult  8  the  i s t a k e n as c o l -  By no means do we w i s h t o i m p l y a n y  to note t h a t  Figures  F o r geometry  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  lations,  ) = 5TT/6.  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  r e s t o f t h e m o l e c u l a r frame.  generalizations  shift  In  rc  SA  o f each chemical  I_I_.  = T T / 2 ,e n  AS ) = T T / 6 , a n d (<j>p - <j>^  D  the principal axis  the  e'  v e c t o r s a r e c o l l i n e a r ; i n Jl,  a l l relaxation  cos"  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  a x e s a r e c o l l i n e a r , A^  a  characterize  e-pulse.  The  b u t when t h e t w o 22  12  = A^  = A^  = A^ a n d  2 =  = r^.  nentials  I n t h i s c a s e , t h e d e c a y r e d u c e s t o t h e sum o f t w o e x p o -  because o n l y Equation  [5.4.12] remain (d/dt)  [ 5 . 4 . 9 ] a n d t h e sum o f [ 5 . 4 . 1 1 ] a n d  coupled, y ]  (t)  = [ A - 4A 1  2  + 4A ]y (t) 1  1  + [4  r i  ](y (t) 3  + y (t)) 4  [5.5.1] (d/dti)(y (t) + y ( t ) ) 3  4  = [ 1 2 1 ^ ( t ) + [3A  1  + 12A ](y (t) + y (t)) ]  3  4  .  [5.5.2]  -151-  It  i s convenient to  <I  (t)>  - <I >  write  ^  T  z  f ( c o s e - 1)<I  where  £  a.  11 +  2A-J  and of  1  c.  = X.[A  - 4A  1  2  +  =  2A^  a.exp(-e.t')  JL, i=l  1  1  1  [5.5.3]  + 2A"]"  , and  t' = t[A  -  1  4A  2  22 +  2A-|  four X-|,  2~i *.exp(-\.t) i=l  =  >  = 1,  ^  ].  For  g e o m e t r y J_ and  terms r e s p e c t i v e l y .  A , 2  a - j ' ( a l s o X^,  and  the  Figures X^,  a , 2  5.1-3  and  summations e x t e n d o v e r g i v e the  normalized  i f applicable)  two  magnitudes  under  various  conditions. F i g u r e 5.1  g i v e s the  relaxation  parameters assuming the  all  c o n c e i v a b l e s i t u a t i o n s : G e o m e t r y J_; i s o t r o p i c m o b i l i t y ;  The  relaxation  parameters are  plotted  as  a function  of the  simplest  io x << n  of 1.  2  interaction 24  constants magnitudes. and  provides a basis  tial  This p l o t corresponds to for  shows t h e  relaxation  and  r e l a t i o n s h i p between the  r e l a t i v e magnitudes of  the  two  case, dipole-dipole  tropy).  5.2A  B c o r r e s p o n d t o g e o m e t r y J_, 5.2C  geometry  Plots  5.2A  nonexponen-  correlated  (in this and  and  degree of  a t i o n mechanisms Figures  calculation  comparison.  F i g u r e 5.2  the  Blicharski's  chemical  and  relax-  shift  aniso-  and  to  D  C represent solutions for D = D = 10 sec"^ 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 p r e - e x p o n e n t i a l f a c t o r s are p l o t t e d versus the d i m e n s i o n l e s s r a t i o of i n t e r a c t i o n -1 = 2 * Aar 3 c o n s t a n t s , ?Q5A?[) )/(3^>Y )• As e x p e c t e d , t h e d e c a y i s d e s c r i b e d 7  x  u  (<°o  by  a u n i q u e T-j when 5 ^  >>  o r when  «  4 * Assuming ?  D  = 2 x 10  The  greatest  devi-  -1 Hz,  a ratio  £  C S A  ?  D  - 1,  implies  that  Aa  - 300  ppm.  -152-  ation is  d e c a y o c c u r s when 4 > S Q ^ Q J ^  from exponential  evident that  deviations  f o r b o t h s l o w m o t i o n and the  from exponential  only  subtle  d i f f e r e n t geometries i n v e s t i g a t e d ; f a c t o r as  dicussed 5.3  Figure tance of  the  2  -  decay are  f a s t motion p l o t s  slow motion l i m i t ) , with  >  However, i t  rather  ( s l i g h t l y more p r o n o u n c e d  differences  a r i s i n g from  h o w e v e r , g e o m e t r y can  presents the  cross-terms.  e f f e c t of  " s h a p e " and  on  b e e n a s s u m e d , ?rj?cSA  r e l a x a t i o n parameters are P l o t s 5.3C  " s h a p e " v a r i a b l e , D„. a c o n s t a n t "shape"  (Dj_ = D^;  =  3  "  ^  o  t  i n t e r n a l geometries  the  relaxation  may  change r a d i c a l l y over q u i t e  (J_ and  (  and  of the  Z^^Q  a partial  = 10  appears that  small  and  10  D on top  the  B  -1 sec"  respec-  a function  o t h e r hand, are  a p p r o x i m a t i o n ) , but  c o m p l e x as m i g h t be variations  i n t e r p l a y between  drawn f r o m any  quantitative  e^, e£<-  one  A  ,  a)^,  1 >2  differ6D^,  =  and  I t must ,  fy^^  the  of  based  expected,  in "size".  figure necessarily  spin  be  D^,  i»2  generality  provides  only  b e h a v i o r shown i n F i g u r e s 5.1-3, i t  f o r a l l p r a c t i c a l purposes, rather  exponential  decay are  exponential  factors  are  or  of  picture.  From t h e  usually  impor-  and  8  » f o r m s a m o s t i n t r i c a t e a r r a n g e m e n t , and  conclusions  the  J_I_ r e s p e c t i v e l y ) . N o t e t h a t when  b e h a v i o r becomes q u i t e  the  x  _1 D,  dominating  5-3A  s  t h e n d e t e r m i n e d as  spherical  ent  kept i n mind t h a t  the  In each of t h e s e p l o t s , a s i n g l e r a t i o  denote a constant " s i z e " corresponding to D  the  a  " s i z e " on  7  The  be  in  previously.  i n t e r a c t i o n c o n s t a n t s has  tively.  similar  d i f f e r by  generally  are  t o be  expected.  a p p r o x i m a t e l y e q u a l , the  a f a c t o r of  l e s s t h a n two;  s i g n i f i c a n t l y d i f f e r e n t , one  minor d e v i a t i o n s E v e n when t h e exponential  when t h e  pre-exponential  from pre-  factors  exponential  factor usually  factors dominates  -153-  the others.  One f e a t u r e  a marked a n i s o t r o p y  i s c l e a r - i n the instances  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  in magnifying the nonexponentiality An  o ft h e decay.  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 exhibited i n Figure  5.4.  The s o l i d 8  f o r g e o m e t r y J_ when  = D  u  = 10  l i n e shows t h e r e l a x a t i o n  1  represent  d  the constituent exponentials  broken l i n e  ignored.  5.3.  S A  The important  = 3.  The v a r i o u s  T h e two t h i n  solid  o f t h e composite decay.  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  u n d e r t h e same c i r c u m s t a n c e s  behavior  1  s e c " and ? 5 Q  p a r a m e t e r s may be d e t e r m i n e d f r o m F i g u r e  the  examined h e r e , even  decay  lines  Finally,  expected  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  conclusion  t o be d r a w n 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 which i s l e s s efficient is  ( 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  independent o f whether cross-terms a r e included o r not.  c l u s i o n s c a n r e a d i l y be d e r i v e d [5.4.17])  and E q u a t i o n s  Since y ( 0 ) 2  = T~^, a n d t h a t  = y (0) 3  from t h e boundary c o n d i t i o n s  [5.4.9-12],  = y (0) 4  [5.5.3]  and  These  con-  (Equations  [5.4.15].  = 0, i t d i r e c t l y follows that  i a.\. = T ^ , w h e r e T ^ i s d e f i n e d  of y-i(t) i n Equation [ 5 . 4 . 9 ] . (T \. « 1 f o r a l l i ) ,  decay  (d/dtJy^O)  as the c o e f f i c i e n t  Furthermore, a t a short time l a t e r , T  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/dt)  I t then f o l l o w s  y ]  (t) -  that  T"  1  +T" x 2  [5.5.5]  -154-  ^ and  a^ !  hence the  that  the  while  >_ ( X l c i a . ) !  2  2  slope  decay a t  absence of  [5.5.6]  2  i s a l w a y s more p o s i t i v e .  presence of  the  T~ T  =  cross-correlation  short  T h i s argument  illustrates  terms always r e t a r d s  times approaches the  relaxation,  decay o b t a i n e d  in  the  cross-terms.  I n no  instance  been c o n s i d e r e d .  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  ^ = 10^  ±  Owing t o  the  close  r e l a t i o n s h i p between the  sec"^  origin  of  49-51 the  chemical  shift  expected that  the  f o r mobile spin rotational the  tensor  and  the  spin-rotation  motion characterized  experimental  resulting  from the  [ 5 . 4 . 9 - 1 2 ] , i t may densities, "special  y^(t)).  the  be  seen t h a t  in'the  then  Qualitatively,  of  the  the  i t i s obvious that  spin-rotation  observed  (containing  an  nuclei  with  the  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  ^>  be  enough  10^  sec~\ to  relaxation From of  Equations  the  the  y (t)  law  shift  relaxation  finding  from y ( t )  2  exponential.  a spin  straighten  population out  Spin-rotation o b e y e d by  small  anisotropies)  behavior.  the and  3  decay i s  thus tends to  large  spectral  which would minimize  mechanism r e l a x e s  exponential  inherently  rapid  u  incorporation  time-domain r e l a x a t i o n .  enforcer of  D  For  chapter.  ( d e c o u p l i n g y - | ( t ) and  curvature  as  by  (nonexponential)  present formulation  o t h e r m e c h a n i s m and  thought of  ratios.  should  in this  relaxation  c a l c u l a t i o n s w o u l d amount t o  independently of i n the  arbitrarily  quantitative  i t is  dominate the  spin-rotation  relaxation  cross-terms  If,$^=  of  detection  , i n t o the  of  rather  cross-terms discussed  geometries"  influence  mechanism w i l l  tensor,  have s i z a b l e C Q S A ? ^  systems which  (exponential) contribution  mask any  spin-rotation  any may  be  molecules  which  might  -155Although o n l y any d e t a i l ,  the longitudinal  calculations  verse relaxation  also.  orthogonal  In general,  = Re(x  q (t)  = Re(x-, -  q (t)  £  (d/dt) (t) qi  R e  1  2  1  2  2  + [-r]  = [|(rj  2  +f ( - A  2  - 3r] + f ( r  = [A] - A  A  x  x  - fuj - r  2  24  +  x  W  +  13  +  x  =  T r  x  24  [5.5.8]  }  [5.5.9]  34>  2 4  )  [5.5.10]  f  +  A  - |r  2  + A  N  2 2  section,  are operative),  C S A  + A  1  i nthe last  2  - | ( A J  2  + [r]  )]q (t) 2  ]  +  - 3r  A  2  2  2  )]  q  ]  (t)  + |(rj  + A  U Q  2 2  )]q (t)  2 2  2  +  )]  q ]  [r]  (t)  +  [ A / 6 - A.,/2 +  +  r  2  Q  +  2  2  q i  3  A  + A ]  2  1  ( t ) / 2 + [|(rj  3  [5.5.12]  4  -3r ]  2  | r j - 2r ]q (t)  + 2r ]q (t) Q  + 5r )]q (t) [5.5.11]  4  + 5 r ) + r] 2  + A ]  2  [5.5.7]  ^(t)l ]  + 5rJ)]q (t)  2  + ^(-A  2  1  2  elements,  proceedures as i n t r o d u c e d  q  +  3  x  = [-3A /2 + 5A /2 - A  + [r  (d/dt)q (t)  •• 3 4  1 2  - x  1 3  +  D  1  2  (x  13  (assuming only c? and c 3  that  + [A] - A  (d/dt)q (t)  x  2  from w h i c h , by a n a l o g o u s follows  +  1 2  = Re(-x  4  apply.  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  q (t) 3  it  t h e same c o n c l u s i o n s  combinations o f matrix  2  h a s been d e s c r i b e d i n  h a v e been e x t e n d e d a n d e x a m i n e d f o r t h e t r a n s -  To c a l c u l a t e t h e t r a n s v e r s e following  relaxation  - 3 r ) + r] + r ] q ( t ) / 2 2  2  2  -156-  + [-5A /6 +  - A  Q  0  + 4AJ  + 2A  1  2 2  - |A  2 2  ] q ( t ) + [-|A 3  + 4AJ ]q (t)  [5.5.13]  2  4  (d/dt)q (t) = [|(r 4  + 5 r J ) + v\ - 3 r ] ]  2  ( t ) / 2 + [-|(r  q i  - 3 r ) - r] - r ]  2  2  Q  x q ( t ) / 2 + [ - | A + 4 A J ] q ( t ) + [-5A /6 + L /Z - A 2  2  Q  + 2A]  The *  1  Q  3  - §A ]q (t)  Inthis  4  case  }  £  + 4A  2 2  [5.5.14]  .  n  simplest reduction o fthese equations = r^.  Q  i s realized  i f A ^ = A ^ and J  24 25 ' ), the  ( 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  equations reduce t o  (d/dt)  q i  ( t ) = [-3A /2 + 5 A / 2 - A Q  x  1  + 2A - 8 A / 3 ]  2  0  1  3  [5.5.15]  1  4  x  Q  ]  4  A ^ 2- A  :  (The  + [-2r + 4 r ]  (t)  (q (t) + q (t))  ( d / d t ) ( q ( t ) + q ( t ) ) = [-2r +4r-]qj(t)ii [r3A /2+ 3  q i  0  + 10A-, - 8 A / 3 Q  2  [5.5.16]  (q (t) + q (t)) . 3  4  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 Blicharski's  25 paper  c a n be t r a c e d t o s i g n e r r o r s Before leaving  this  i n h i sequations.  )  p o i n t b e h i n d , o n e comment u n i q u e  to the trans-  v e r s e decay i s i n o r d e r .  I f 6 D < GUQ, t h e n t h e a p p r o x i m a t e  of motion  3  (d/dt)  q ]  L  equations  f o r q-,(t) a n d q ( t ) + q ( t ) a r e 4  ( t ) - [-3A /2 - 8 A / 3 ] Q  (d/dt)(q (t) + q (t)) = 4 r 3  4  0  Q q i  q i  (t)  + 4r (q (t) + q (t)) Q  3  [5.5.17]  4  ( t ) + [-3A /2 - 8 A / 3 ] ( q ( t ) + q ( t ) ) . Q  Q  3  4  [5.5.18]  -157-  Since q^O)  = <I (0)>  <I (t)>/<I (0)> x  where ^  x  =  z  3a  2  8A  be e x t r e m e l y  4  /3 ±  ^o* ^ r  A  [5.5.19]  2  1  0  nonexponential  ~ 0 A  ~  r  0'  n  o  t  l  the other  decay.  approximation  c  e  t  n  a  t  t  w i t h one t i m e c o n s t a n t (with equal  T h i s one e x a m p l e p o i n t s o u t q u i t e c l e a r l y  an e x t r e m e - n a r r o w i n g  transverse  exp.(x t)|  l{exp(A t)+  orders o f magnitude l a r g e r than  factors!). of  3  ^ ~ (/ " o  decay w i l l two  a n d q ( 0 ) + q ( 0 ) = 0,  X  n  e  almost  pre-exponential  that the f a i l u r e  may h a v e d r a m a t i c  effects  on t h e  -158-  FIGURE 5 . 1 :  Plots  o f the decay c o n s t a n t s  magnitudes  o f t h e two i n t e r f e r i n g  (dipolar-shift of  (unitless)  the s h i f t z  T  z  w h e r e e. =*.JTQ a n d t ' = t / T . Q  '•>• • f i n , E q u a t i o n [ 5 . 4 . 9 ] .  _ 1  the r e l a t i v e  mechanisms  that the p r i n c i p a l  tensors are c o l l i n e a r . T  z  relaxation  a n i s o t r o p y ) assuming  [<I (t)>-<I > ][<I > (cose-l)]  versus  The d e c a y  i s given  = a exp(-e t') + 1  1  axes  (l-a  T ^ i s the c o e f f i c i e n t  1  by )exp(-e t')  of y-j(t)  2  -160-  FIGURE 5.2:  Plots of the normalized  decay c o n s t a n t s  ( u n i t l e s s ) versus the  r e l a t i v e m a g n i t u d e s o f t h e two i n t e r f e r i n g isms  (dipolar-shift anisotropy).  The c a l c u l a t e d  acterize  the longitudinal  [<I (t)>  - <I > ][(cose - 1)<I > ] T  z  mechan-  values  char-  decay o f m a g n e t i z a t i o n ,  T  z  relaxation  _ 1  Z  =  z a.expf-e.t'), _i  1  w h e r e J a . = 1 , I a.,^. = 1, e.. = X ^ T Q , a n d t ' = t / T .  TQ i s  Q  the 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  S ^ S Q  3  plotted  as t h e l o g a r i t h m o f t h e r a t i o ,  Figures  ( A ) a n d ( B ) a r e f o r g e o m e t r y J_, ( C ) a n d (D) a r e f o r  g e o m e t r y J_J_ ( s e e t e x t ) . 10  7  Figures  (A) and (C) a r e f o r D  s e c " , a n d ( B ) a n d (D) a r e f o r D„ = D 1  (=BQAar / 3 y f i ) .  = 10  8  sec"  (|  1 .  = D  L  =  -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 ) versus  hydrodynamic  " e f f e c t i v e " s h a p e s and s i z e s  frameworks.  The c a l c u l a t e d  tudinal  decay o f m a g n e t i z a t i o n , T  z  T  z  z a.  In F i g u r e s  = 1, E  c^.e.. = 1,  =  E a.expf-e.f) ,  = A ^ T Q , and t  1  = t/T  (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  respectively.  is  1  z  f u n c t i o n o f D,, a s s u m i n g  plotted  of molecular  values c h a r a c t e r i z e the l o n g i -  [<I (t)>-<I > ][(cose-l)<I > ]" where  different  In F i g u r e s  as a f u n c t i o n o f D  = 10  7 - 1 sec"  and  = 10  ( C ) and ( D ) , t h e s e |(  assuming  D^/D^  n  as a 8 - 1 sec"  parameters are  = 1.  Figure  (D)  t h e c a l c u l a t i o n p e r f o r m e d w i t h g e o m e t r y J_, a n d F i g u r e s ( A ) ,  ( B ) , and (C) a r e done a s s u m i n g geometry plots  assume  ? /?^Q = 3. n  fl  II_ (see t e x t ) . A l l  -1 6 3 -  -164-  FIGURE 5.4:  The l o n g i t u d i n a l d e c a y o f m a g n e t i z a t i o n 8 in  the l i m i t :  D,, = D = 1 0 L  f o r a p a i r of spins  -1 s e c " , geometry  I_, a n d S p / ^ C S A  =  The t h i c k s o l i d  upper curve 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 ,  the  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 -  broken l i n e  correlation curve the  terms have  been  s e t equal  to zero.  The  3.  solid  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 d e c a y s a r e shown i n  lighter  inverse  lines.  The a b s c i s s a  i s i n u n i t s o f TQ ( t h 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 a n d t h e p r e c e e d i n g one h a v e a n a l y z e d t h e s i g n i f i c a n c e of  cross-correlations  i n the analysis  data, p a r t i c u l a r l y f o r large  o f nuclear magnetic  molecules  i n solution.  relaxation  The m a j o r  points  may now be s u m m a r i z e d . (1)  Whereas a n i s o t r o p i c  reorientation  H u b b a r d t e r m s , t h i s p a p e r h a s shown t h a t erally  magnifies the effect of  nonspherical  h a s 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  be  noted that  when r e o r i e n t a t i o n  of  Hubbard terms i s n e g l i g i b l e , b u t B l i c h a r s k i  terms.  diffusion  gen-  I t should  i s i s o t r o p i c , the experimental  also effect  t e r m s may s t i l l  be s i g -  nificant. (2)  For cross-correlation  we have n o t e d t h a t ation it  certain  terms o f t h e second  relationships  (Blicharski)  between t h e i n t e r n a l  c o o r d i n a t e s y s t e m s may s e r v e t o m a g n i f y t h e s e t e r m s .  i s possible  chemical served  that  nonexponential  s h i f t anisotropy  In f a c t ,  ( C S A ) m e c h a n i s m i t s e l f may d o m i n a t e t h e o b -  of the dipolar  ly  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  further  peculiarities arise  magnitudes o f t h e p r i n c i p a l quency.  to different  limit,  holds,  between t h e fre-  c o n s i d e r a t i o n s a p p l y t o Hubbard  variability  orientations.  extension of Blicharski's  com-  sufficient-  d i f f u s i o n c o n s t a n t s and t h e Larmor  case l e s s  internal  For molecules  from p a r t i c u l a r r e l a t i o n s h i p s  Although s i m i l a r geometrical  terms, there i s i n that ing  (D) i n t e r a c t i o n .  constant i s small  pared t o that that  relax-  d e c a y o r d e c a y by means o f t h e  r e l a x a t i o n , e v e n when t h e CSA i n t e r a c t i o n  large  kind,  i n relaxation  correspond-  Furthermore, i n t h e slow motion  original calculation  shows t h a t t h e  -167-  transverse  r e l a x a t i o n may  ( 3 ) The results  i s , the  d e c a y c u r v e a t any  or  given  several  circumstances which w i l l  r e l a x a t i o n of the  molecular relaxation Spin-rotation  kind  i n t e r a c t i o n s may  intramolecular r e l a x a t i o n and (5) A l t h o u g h  C h a p t e r IV  decay in-  are  implies  not  a large  SR  contribute  nonexponential  discussed  a  extended  intereffect. large  i n t e r a c t i o n constant.  a l s o mask  here.  system, the  while  conclusions  (at l e a s t q u a l i t a t i v e l y ) to  systems c o n t a i n i n g  Finally,  (or c o m p l i c a t e ) nonexponen-  further  treats a three-spin  e a s i l y be to  will  chapters.  present paper t r e a t s a two-spin system,  necessarily  s y s t e m s and  spins  observation  i n these  mask B l i c h a r s k i t e r m s , s i n c e  m e c h a n i s m s may  the  e i t h e r case could spin  initial  preclude  discussed  ( e x p o n e n t i a l ) , m a s k i n g any  i n t e r a c t i o n constant  tial  The  ( i n magni-  same, w h e t h e r c r o s s - c o r r e l a t i o n s a r e  Paramagnetic i m p u r i t i e s or nearby nuclear  other  functions,  t i m e shows a s m a l l e r included.  cal-  not.  nonexponential  CSA  function  found from a  auto-correlation  when c r o s s - c o r r e l a t i o n s a r e  (4) There are of  appropriate  t i m e goes t o z e r o ) i s the  cluded  nonexponential.  i n c l u s i o n of e i t h e r type of c r o s s - c o r r e l a t i o n  i n v o l v i n g j u s t the  tude) slope (as  extremely  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 t h a n w o u l d be  culation that  be  chemically  in larger  different nuclei  (J'hetero"-nuclei). In c o n c l u s i o n , is  a t p r e s e n t an  that  for future  ternal  NMR  studies  magnetic f i e l d s ,  confusion,  nonexponential  experimental  dent e x p e r i m e n t a l l y , and  although  and  about the  c u r i o s i t y , the  of  provide  magnetic  present theory  large molecules  nonexponential can  nuclear  suggests  i n s o l u t i o n at  r e l a x a t i o n may  appreciable  s p i n e n v i r o n m e n t and  relaxation  well  additional  motional  high  become  exevi-  information,  dynamics.  •168TABLE 5 . 2  11  SUMMARY: EFFECT OF INTERFERENCE TERMS HUBBARD Criteria  ( F I R S T KIND)  BLICHARSKI(SECOND  f(c ) / f(c )  of definition:  5  n  example:  dipole-dipole  dipole-shift  sign reversal of r e l a x a t i o n terms under spin inversion?  no  yes  observed  i n solids?  yes  no  observed  i n liquids?  CD C0CD CH  lis \To  •r-  2  3  BF ,  •1/Ti  CF C1  (d/dt)M  2  2  anisotropy  , CHFC1  2  -1/T,  -1/T,  A  -1/T  rapid, isotropic  2  nil marginal  ts  c  3  3  w^..  KIND)  •r—  £  S- •I—  CD  •1—  o  o  Q.  E  pronounced  marginal  marginal slow, i s o t r o p i c  nil  o  •r—  nil slow,  anisotropic  (longitudinal)  pronounced(transverse)  to  var  so  anisotropic  +J  (O B  +J  rapid,  re  o  on  +->  1  (longitudinal)  pronounced(transverse)  marginal  (longitudinal)  pronounced(transverse)  n T h e 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 a n d a r e n o t 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 explanation. * A l l s t a t e m e n t s a r e b a s e d on t h e a s s u m p t i o n identical nuclei.  of three,  equally  spaced  ,  -169-  REFERENCES: CHAPTER V  1.  L. G. W e r b e l o w a n d A. G. M a r s h a l l , M o l . P h y s .  ,  (1974).  2.  A. 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D e u t c h a n d J . S. Waugh, J . Chem. P h y s .  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 , Magn. R e s . 5_, 101 ( 1 9 7 1 ) .  a n d A. A l l e r h a n d , J . Chem. P h y s . 5 6 , ~~~  5 1 . R. V. R e i d a n d A. C h u , P h y s .  4 3 , 1914 ( 1 9 6 5 ) .  a n d K. H. H a u s s e r , J .  R e v . A 9 , 609 ( 1 9 7 4 ) .  -172-  CHAPTER V I EFFECT OF MOLECULAR SHAPE AND F L E X I B I L I T Y GAMMA-RAY DIRECTIONAL CORRELATIONS  6.1.  ON  1  INTRODUCTION Nuclear Magnetic  the physical  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  date macromolecular lation  Resonance i s n o t 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  dynamics.  (PAC) e x p e r i m e n t s  The p o t e n t i a l  a s a means t o w a r d s  c a n be u s e d  of Perturbed Angular Correm o t i o n a l and s t r u c t u r a l  information concerning b i o l o g i c a l l y i n t e r e s t i n g molecules  h a s been d i s -  2-9 cussed  i n several  recent publications.  to eluci-  •  •  I n complete, analogy w i t h i t s  n e a r > e l a t i v e , NMR;* a t h o r o u g h  understanding of the effects  motion  i s necessary i n order to apply t h i s tech-  ;  on a n g u l a r c o r r e l a t i o n s  nique e f f e c t i v e l y t o t h e study o f b i o l o g i c a l 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 of  low m o l e c u l a r weight d i s s o l v e d  Recently, several correlations isotropic  calculations  o f slow  rotational 1g  dimension.  of molecular  systems.  f o ra solid  sample o r f o r s p e c i e s  i n isotropic liquids i s well  have p o i n t e d o u t t h e e f f e c t s  known.  on a n g u l a r  ( c o r r e l a t i o n times g r e a t e r than a few nanoseconds) 13-17 diffusion a n d 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  However, t h e e f f e c t o f a n i s o t r o p i c  motions  has n o t been  commented upon i n t h e l i t e r a t u r e . The strated  importance  of anisotropic  rotational  f o r e x p e r i m e n t a l l y observed q u a n t i t i e s  Nuclear Magnetic  Resonance ( s e e Chapters  d i f f u s i o n h a s b e e n demoni n such  t e c h n i q u e s as  I I - V ) and f l u o r e s c e n c e d e p o l a r -  -173-  ization. motions (see  ~  F u r t h e r m o r e , i t has b e e n shown t h a t  can markedly  Chapters  internal  rotational  a f f e c t the r e s u l t s o f Nuclear Magnetic  II-V), fluorescence depolarization,  Resonance  and p a r a m a g n e t i c  "spin  22 label"  experiments.  diffusion  and a l s o  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 of internal  rotation  on gamma-ray  angular  rotational correlations  is thus of great i n t e r e s t . The c a l c u l a t i o n s extreme-narrowing  presented i n t h i s  limit  (rapid motion)  m o t i o n ) ; t h e two l i m i t i n g of  rotational  c a s e s span  reorientation.  section  are adapted  and an a d i a b a t i c  nearly  t o b o t h an  limit  t h e whole range o f  (slow rates  -174-  6.2.  PERTURBED ANGULAR CORRELATIONS In  the realm  correlations  of conventional  i s probably  o f n u c l e a r phenomena due symmetry p r i n c i p l e s . with  o f t h e b e s t and  reason  excitation  general space  f o r the e x i s t e n c e of a n i s o t r o p i c angular c o r r e l a t i o n s  c a s c a d e and  propagation relation  detected  r a t e may  vectors,  may  be  d e p e n d s t r o n g l y on  and  perturbed  by a c o i n c i d e n c e  o f t h e two (attenuated)  are emitted  spectrometer,  the angle  gamma-rays.  by  the  F i g u r e 6.1  scheme f o r a y-y  process.  The  an e n e r g y - l e v e l  t r a n s i t i o n from the  initial  m e d i a t e s t a t e i s a c c o m p a n i e d by e m i s s i o n l i f e t i m e of the  cause the  a transition  * The  from the  term a n g u l a r  polarization serves  some t i m e  intermediate  the c o u p l i n g of the  t a f t e r emission  s t a t e to the f i n a l  the d i r e c t i o n a l  The  e x p e r i m e n t we  correlation  H o w e v e r , bound by c o n v e n t i o n ,  and  this  wish  title  the term a n g u l a r  interIf  nucleus rotation respect  to  there  is  o f y-|,  state, f,  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  correlations.  decay  p h o t o n , y-j.  proper magnitude, then molecular  At  cor-  cascade.  nuclear  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  a l a b o r a t o r y - f i x e d system.  co-  extranuclear  s t a t e of the  o f the f i r s t  de-  the  nuclear s t a t e i to the  i n t e r m e d i a t e s t a t e and  i t s environment are o f the  the  between  interaction of  intermediate  illustrates  in a  This angular  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  can  very  isotropy of three-dimensional  gamma-rays f r o m a r a d i o a c t i v e n u c l e u s  incidence counting  to  angular  most comprehensive t h e o r i e s  t o t h e f a c t t h a t i t i s b a s e d on  I t i s the  of  distributions. When two  the  theory  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 and  one  nuclear s t u d i e s , the  accompanied  directional  to discuss only  and ob-  w o u l d be more a p p r o p r i a t e .  correlation will  be  used.  -175-  o f a second photon, Y -  by t h e e m i s s i o n  2  The phenomena o f e n t r y i n t o , e x i t f r o m , a n d p e r t u r b a t i o n w h i l e in the intermediate k-|, and Y  i s emitted  2  function,38^(kp o f Y-| a n d Y  &(k,,  in direction  time  in direction  correlation  between  observation  by  ?  L  i s emitted  the angular  2  2  k ,t)^ e x p t - t / x j  1  k , then  k , t ) , f o r a p a r t i c u l a r delay  i s given  2  I f Y]  state are separable.  IN  H  SS k k N N 1  x GJjljj2(t) [ ( 2 k  2  ]  [(2k,+ 1 ) "  /  Y^KfiJA. l l  2 K  1  2  + 1)~  2  1  1  1  /  2  YJj2(n )A 2  k  (1)]  K  (2)].  [6.2.1]  The s u b s c r i p t s 1 a n d 2 r e f e r t o r a d i a t i o n s 1 and 2, a n d t h e a r g u m e n t s o f each s p h e r i c a l harmonic r e f e r t o the angles  b e t w e e n an  coordinate  o f t h e i n d i c a t e d gamma-  f r a m e and t h e d i r e c t i o n o f e m i s s i o n  r a y , a s 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.  o n l y on t h e s p i n s a n d m u l t i p o l a r i t i e s and is  described  with  transition,  i t s environment  N N 11 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 ) , 1 2 K  G^2(t) -  ( i ) depend  associated with the i  the coupling of the i n t e r m e d i a t e - s t a t e nucleus completely  arbitrary  [(2k,  g  • D(2k  2  K  • 1 ) ] V 2 ^  ^ jJ k  a b  x  (i-™ X) ' <n  b  Im > and |m'>:are i n t e r m e d i a t e a a 1  1  emission of Y  2  ,  A N D  o f y-j > \\>  a  n  d  h> b  '  biA(t>iv<rai  *•  nuclear substates are substates  A ( t ) i s the evolution operator  ''  iA(t)ira;>  [6  immediately  immediately  before  2  2]  after the emission  f o r the intermediate state.  -176-  I \ m m, m  The  m  bols)  m 9  / a r e the familiar m~/ '  vector coupling coefficients  ( 3 J sym-  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  s t a t e m u s t h a v e n u c l e a r s p i n >_ 1.  correlation, the intermediate  Such a n u c l e u s  may h a v e a q u a d r u p o l e  moment, a n d i n m o s t s t u d i e s o f m a t e r i a l s i n a c o n d e n s e d p h a s e , t h e q u a d rupolar interaction angular  represents  correlation.  a computational  t h e m a j o r p e r t u r b i n g i n f l u e n c e on t h e  I t i s o f t e n a good a p p r o x i m a t i o n  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  responsible f o r the quadrupolar this in  k  interaction  has a x i a l  a coordinate  ( t > =X)  N k  1 2  s y s t e m whose z - a x i s i s p a r a l l e l  [ m  K  symmetry.  to the principal  ( 2 k l  +  i )  (  1  2  k  2  D :  +  1  ^  m  - E .)t) m  1  1  \m'  c  A( -m N / \m' k  l  _ 1  principal  value  -m  axis  [6.2.3]  m  2  = (e qQ/fi)[3m  - 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, ( e q ) i s t h e of the e l e c t r i c  field  g r a d i e n t t e n s o r , and m i s t h e  q u a n t u m 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 . 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 expansion tion  evaluated  k2  .  following definitions-are"employed: E (41(21+1)) ,  When  ) N /  1  2  x  gradient  i s g i v e n by  x exp(-i(E  The  field  i s the case, the perturbation f a c t o r f o ra s i n g l e nucleus,  of the i n t e r a c t i o n ,  G  - and c e r t a i n l y  series  In prac-  forl^  (Equa-  [ 6 . 2 . 1 ] ) , a n d o f t h e s e , o f t e n o n l y o n e , o r two a t t h e m o s t , a r e  significant  (other than  the zero order  term).  -177-  FIGURE 6.1:  Schematic energy-level from an e x c i t e d of  spin  I  f  nuclear  d i a g r a m f o r a gamma-gamma state of spin  t h r o u g h an i n t e r m e d i a t e  cascade  I . t o a ground  state of spin I.  state  Y  2  -179-  FIGURE 6.2:  Angular 1<2>  coordinates  of the  propagation  directions,  o f s u c c e s s i v e gamma r a y s f r o m a c a s c a d e , w i t h  t o an  a r b i t r a r y frame of r e f e r e n c e  (see Equation  k-j  and  respect [6.2.1]).  -181 -  6.3  ANISOTROPIC MOTION IN THE In  their  orienting to  ABRAGAM-POUND REGIME  d e r i v a t i o n of the perturbation  n u c l e a r s p i n , Abragam and P o u n d ^ ' ^  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  When t h e a n g u l a r c o r r e l a t i o n any a p p l i e d tion  static  (WQTQ <<  the  factor  (external) f i e l d ,  1) w i l l  mean l i f e t i m e  experiment  pertain.  employed.methods  Resonance  (NMR)  i s conducted  f o r a l l cascades  times.  i n the absence  i f T Q << x^  In a d d i t i o n ,  analogous  relaxation  the "extreme-narrowing"  of the intermediate state; t h i s  10-100 nanoseconds  for a rapidly re-  i s on  of  approxima-  (where  x^ i s  the order of  o f p r a c t i c a l i m p o r t a n c e , ) , and  x  «  Q  p 1/CJQXQ ( w h e r e  i s the fundamental  q u a d r u p o l e f r e q u e n c y ) , Abragam  and  P o u n d 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 , and-the-angular c o r r e l a t i o n  Wtf  l<2> *)  v  =  function  xjj exp(-t/T ) 1  N  S  reduces t o the f o l l o w i n g  form,  A (l)A (2)exp(-X t)P (cosn).. k  k  k  k  [6.3.1]  k  The  e n c l o s e d a n g l e b e t w e e n 1<-| and "k^  for  an a x i a l l y  in  In  particular,  quadrupole i n t e r a c t i o n , the exponential  .  f j  ( < k  kt1  factor  >WW) - " ^ - ^ ( e W v  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 In t h i s  correlation  time reduces to the f a m i l i a r  duced  c a s e , i t c a n be r a t i o n a l i z e d NMR  that the  diffusion  is  rotational  time constant f i r s t  intro-  i n Chapter I I . A comparison o f the d e r i v a t i o n  s i o n s o b t a i n e d f o r t h e NMR A  [6.3.2]  2  isotropic.  and  by Q.  E q u a t i o n [ 6 . 3 . 1 ] i s g i v e n by  X | (  In  symmetric  denoted  1 S  r  of Equation [6.3.2] w i t h  relaxation  are simply proportional  time, T ,  to a spectral  ]  the expres-  shows t h a t b o t h  density  (1/T^  at zero frequency  -182-  of a c o r r e l a t i o n function Equation  [ 6 . 3 . 2 ] may  diffusion  simply  2  x  +  In the p r e s e n t case densities), this f(D ,  D,  t  ,6)  | (  2  3sin 0cos e(5D 2  +  1  + 4D„ )  _  2  x  [3.2.5];  + D j "  1  .  1  [6.3.3]  (no f r e q u e n c y d e p e n d e n c e i n t h e p e r t i n e n t  c a n i n f o r m a t i v e l y be w r i t t e n  = (eoj-^i+  M^ We+ ()  5D +D L  may  L  ) (  form o f Equation  i  4  Thus  w h e r e f ( D , U ,B)  by f ( D , D ^ e )  - l) (24D )"  (3/4)sin e(2Dj_  harmonic.  to account f o r a n i s o t r o p i c r o t a t i o n a l  from a m o d i f i e d  (3cos e  D ,e) =  n  be m o d i f i e d  by r e p l a c i n g  be r e a d i l y o b t a i n e d  f(D ,  o f a second rank s p h e r i c a l  spectral  as  •9iD _ _q|) sin 9-2  1  2(5D  n  4  z  L  L  + D,,)^ +  2D ) |(  [6.3.4] Theta i s the p o l a r angle which r e l a t e s the e l e c t r i c cipal  axis to the p r i n c i p a l (molecule-fixed)  diffusion well  tensor.  I t i s assumed t h a t  a p p r o x i m a t e d by a s y m m e t r i c t o p .  where D  ()  = D= L  (6D)~^  = x2•  static  fields,  axis  , D"  1  gradient  Finally, f o r a spherical  << T ^ , ^  prin-  system o f the r o t a t i o n a l  the molecule i s dynamically  D, one r e c o v e r s t h e A b r a g a m - P o u n d Thus, whenever  field  quite rotor  r e s u l t where f ( D ) = and t h e r e  a r e no  applied  t h e d e p e n d e n c e 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  s y m m e t r y c a n be f o u n d f r o m 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 c a s e i n w h i c h t h e two d i f f u s i o n r o t o r are each small quadrupole  compared t o t h e fundamental  constants of a  symmetric  f r e q u e n c y , cog, o f t h e  interaction  [6.4.1]  For  a single nucleus f i x e d  to a symmetric r o t o r m o l e c u l e , the "atomic  f r a m e " c a n be d e f i n e d as a c o o r d i n a t e s y s t e m whose z - a x i s the  principal  axis of the e l e c t r i c  field  t o k-,.  The  rotations  t h e a t o m i c f r a m e and t h e l a b f r a m e a r e c o n v e n i e n t l y d i v i d e d  a rotation  from the atomic frame t o the d i f f u s i o n  s y s t e m f o l l o w e d by a r o t a t i o n  to  g r a d i e n t t e n s o r and a " l a b  f r a m e " c a n be c h o s e n s u c h t h a t i t s z - a x i s i s p a r a l l e l between  i s parallel  from the d i f f u s i o n  tensor principal  into axis  t e n s o r frame t o the l a b  frame. Now  i f the d i f f u s i o n  t e n s o r frame changes  i t s orientation  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 state,  there 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  slowly  intermediate  the various  substates  * of  the intermediate state.  [ 6 . 2 . 1 ] and spherical of  In t h i s a d i a b a t i c  limit,  the forms o f E q u a t i o n s  [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  harmonic  i s now  time-dependent.  Thus,  the o n l y cause 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 v i e w e d f r o m t h e  * This adiabatic approximation y i e l d s proaches.  A discussion of this  results  second decay  laboratory  d i f f e r e n t from o t h e r ap-  p r o b l e m has b e e n p r e s e n t e d by  Lynden-  23 Bell. beyond  However, t h e d i f f e r e n c e s experimental v e r i f i c a t i o n  f o r PAC and n e e d  s t u d i e s a r e , on t h e m o s t n o t c o n c e r n us h e r e .  part,  -184-  axes i s the  randomization  of molecular  Taking advantage of the after  <  a few  steps  £g(k k , r  to the  p r o p e r t i e s of the  expression  S  t)>=  2  orientation. D matrices,  (irrelevant  factors  A (l)A (2)G (t)P (cosn)X] k  x exp(-E  k p  k  k k  24  are  4  k  one  is led  omitted)  ^*(e)YP(e)  t)  [6.4.2]  2 w h e r e E, "kp  = k(k+l)D  aixx e s o f t h e  cipal  E «i/ ii  I k\2 i \\ n n' -n -n N/ N/ noted at t h i s  + p  x  ( D - D J , e defines  diffusion  separately isotropic <<  as  n  stage:  the  1  which  is just  of m o t i o n .  1 1  and  Two  side of  features  k  the  k  familiar  k k  by  k k  (t)  detectors  = may  as  appears  i s the  becomes v e r y  [6.4.2] reduces  case slow  (D , h  [6.4.3]  k  polycrystalline  rotational  for  to  (t)P (cosn)  s i g n i f i c a n t feature of  i s unaffected  G  prin-  of t h i s expression  diffusion  Equation  "angle of attachment", t h e t a , i s zero relation  t e n s o r s , and  a n g l e between the  ( 2 ) when t h e  A (l)A (2)G  A  ,) t ] .  gradient  between the  n  ( 1 ) The  right-hand  S  field  angle  argument o f a Legendre p o l y n o m i a l ,  diffusion,  T ^ ) , the  and  e x p [ - i ( E -E  1  be  the  H  result  Equation  expected  absence  [ 6 . 4 . 2 ] i s t h a t when  degrees, the diffusion  i n the  observed angular  a t any  r a t e about  the cor-  the  25 * More g e n e r a l l y  i t has  such t h a t i t does not and be  polycrystalline invariant  only  factor.  diminished  coefficient.  introduce  sources),  correlation Hence, the  The  the  result  angular  i f the  perturbation  direction  perturbation  function will M  and  that  a privileged  then the  under r o t a t i o n s .  that the angular Pi,(cosft)  been r a t i o n a l i z e d  i n space k  o f s u c h a demand i s £he always c o n t a i n  distribution  (liquid  term, G l 2 ( t ) k  i s not  a  is  must fact  separable  altered,  but  M  term G.1.2(t) i s r e f e r r e d 12  t o as  an  attenuation  -185-  symmetry  axis.  Finally,  f o r the s p e r i c a l r o t o r , f o r which D  harmonic a d d i t i o n  theorem  |(  = D^,  the spherical  reduces Equation [6.4.2] t o the  previously  13 obtained  result,  < & ( k  k , t)> =  r  Equation for  ^  2  A (l)A (2)G k  k  [6.4.2] thus gives  k k  (t)P (cosn)exp(-k(k+l)Dt).  [6.4.4]  k  the predicted  Perturbed Angular  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  Correlation  be v a l i d  when-  ever the 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 of the molecule i s s u f f i ciently  slow that  D^,  D^<<  Wq.  Extending these r e s u l t s t o the problem o f i n t e r n a l motions i s straightforward.  When t h e m o t i o n o f t h e m o l e c u l e a s a w h o l e  ( A b r a g a m - P o u n d l i m i t ) one may make t h e f o l l o w i n g D^-D^E D.  Equation [6.3.3] o r [6.3.4]; overall  isotropic mobility  is  identification in  w h e r e Dj_ now c h a r a c t e r i z e s  (Dj_ ->• D) and D- ^  degree o f i n t e r n a l m o b i l i t y .  i s rapid  n  characterizes  the single  When t h e m o t i o n o f t h e m o l e c u l e a s a w h o l e  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 rapid  compared t o t h e o v e r a l l  m o t i o n , b u t 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  t h e n t h e same i n t e r p r e t a t i o n c a n be a p p l i e d intermediate  case, fast internal mobility  a number o f p r o b l e m s  the  < < n t  to Equation [6.3.6].  and s l o w o v e r a l l m o t i o n  a n d has n o t b e e n s a t i s f a c t o r i l y  solved.  "Q)» The poses  -186-  6.5.  DISCUSSION It  is instructive  correlation is  the  to  illustrate  ' S^ »") n  angular  =  Tjyi  i s t h e mean l i f e t i m e o f  is  the  Figures the  rotation. 1 8 1  Ta,  * The  1 9 9 m  H g , or  be  pletely  Pb*  has  attenuated.  perturb  an  Of  t h a n a few  No  angular tens  correlations  can  s h i p between the  on  the  symmetry o r  p o s s i b l y be  long  s h o r t e n o u g h so  ^Co,  ^ Zn, 2  u s e d i n PAC  extranuclear  of the  fields  are  c o r r e l a t i o n i n which the  of picoseconds.  two  i f one  state lifetime.  For  coincidences  outweigh true coincidences  9  m  Sn,  ^  t h a t the  that  integral  enough t o  nuclear  lifetime  fact that  and  >_ 10 the  upper l i m i t  the  attenuation com-  attenuation, duration measurably is less angular  establishes a generic  lifetimes  Ba,  the  i s not  i n t e r a c t i o n and strong  3 3  s t u d i e s , most  anisotropy  radiations sets a practical  intermediate  ^  attach-  Furthermore,  In a d d i t i o n , the  be m e a s u r e d o n l y  n  internal  I >_ 1 and  e n o u g h so  t h a t the  angle of  rate of  course the magnitude of the strength  x^£n(2)).  =  k^.  suitable nuclei i s that  <  interaction.  ( x-, , ^  state  as  [6.5.2]  N  a h a l f - l i f e o f hours or days.  yet  [6.5.1]  t)exp(-t/x ).  2>  s u c h as  s t a t e l i f e t i m e m u s t be  manifest,  k  k-j and  isotopes  could  on  -1  d e p e n d e n c e o f <^l>  G j | ( ) d e p e n d s upon b o t h t h e of  k  intermediate by  00  <  (r  f°%&  ]  the  several  2 0 7 m  angular  quantity  c o r r e l a t i o n function i s defined  e i t h e r molecular  limitations  parent nucleus  will  and  Although  intermediate  N  show t h e  label  primary  T  angle defined  6.3-5  convenient  = (W(^,~)/W(^/2,-))  time-integrated  enclosed  One  of the  (or i n t e g r a l ) a n i s o t r o p y ,  <&>  ment o f  c a l c u l a t e d behavior  i n terms of measureable q u a n t i t i e s .  time-integrated  where the  the  relation-  on  the  seconds, accidental PAC  technique  loses i t s  -187-  labeling  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  d e c a y 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.  Cd.  The  The v a r i o u s  plots  ( F i g u r e s 6.3-5) h a v e been g e n e r a t e d f r o m t h e p a r a m e t e r s a s s o c i a t e d V 2 111m 11/2 >>>>>>>>> 5/2 >>>>>>>> 1/2 d e c a y o f Cd. E3)  the  Y  (  E  2  )  these  values  A (2)  = - 0 . 3 3 4 , A ( l ) = 0 . 6 1 7 , a n d A ( 2 ) = 0.007.  2  a r e reproduced here:  Q  considerations,  Q  2  A l l other  4  are identically  For convenience,  A ( 1 ) = A ( 2 ) = 1, A ( l ) = -0.535,  4  coefficients  with  equal  t o zero.  cascade  I n g e n e r a l , due t o symmetry  i t c a n be shown t h a t f o r d i r e c t i o n a l  y-Y PAC  experiments,  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].  Further-  m o r e , s i n c e A ( 1 ) A ( 2 ) >> A ( 1 ) A ( 2 ) ,  purposes,  only  2  one  2  nontrivial  4  4  f o ra l l practical  term c o n t r i b u t e s t o t h i s  summation.  Finally,  the factor  FOOTNOTE CONTINUED usefulness. 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 w h e r e 10 s e c <_ ^ <_ 10 s e c w h i c h u s u a l l y i m p l i e s Ml o r E2 d e c a y o f T  the  intermediate  typical gral is  range f o r usable  ( I t i s interesting  p l o t t e d versus For T  2  interpolated  correlation  < CJQ^ , t h e A b r a g a m - P o u n d r e s u l t  (this  In Figure  a p p r o x i m a t e l y , e q u a l s . . $3  an i s o t r o p i c  adiabatic limit  t o note t h a t t h i s  Mossbauer i s o t o p e s . )  a t t e n u a t i o n f a c t o r (which  T^. the  state.  i s applied. i sjustified  6.7, t h e i n t e 2/(3A (l)A (2))) 2  time f o r various  times,  2  values o f  i s applied; f o rx  For intermediate  i s the  2  > co^ ,  the curve i s  a s i t h a s b e e n 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 : s e e reference helps  14).  Note t h a t a c o n s t a n t  t o i n d i c a t e why, i n o r d e r  lifetime  to obtain motional ways, t h i s T  9  ratios  falls  f o r Wq i s assumed.  t o obtain chemically  mation, i t i s d e s i r a b l e (necessary) If the nuclear  value  interesting  plot  infor-  f o r 10 n a n o s e c < x ^ < 1 0 0 n a n o s e c .  outside o f this  information.  This  range, i t i s d i f f i c u l t  I t i s interesting  t o n o t e t h a t i n some  i s q u i t e a n a l o g o u s t o t h e s i t u a t i o n when o n e e m p l o y s T-j o r f o r motional  information  (see section 3.3).  -188-  G  k k  ( t ) f o ra polycrystalline  N.n  =  G  I  k \  n'  -n N J  constant  i s chosen,  k k  =  n  + 5cos3u)gt)/35  G  [6.5.3]  f o r the quadrupole  kk^ ^ t  a l s o be i n t e r -  attenuation coefficient, G ^  0  0  ),  where  and  N  k k ^  results  n  purposes, the ordinate a x i s could  (») = xjJ G (t)exp(-t/T )dt  This  P  (yet a r b i t r a r y ) value 2 9 |e qQ/fi]2TT = 10 Hz.  i n terms o f t h e i n t e g r a l  G  by  ex (-i(E -E ,)t)  A typical  1  k k  [6.4.3]) i s given  2  ( 7 + 1 3 c o s w p t + 10cos2ojpt  For a l l p r a c t i c a l preted  I  (see Equation  \  2 w h e r e cog=3e qQ/20fi. coupling  sample  4 7 T  YP  from t h e f a c t  that  (e)YP(e)exp(-E t).  [6.5.4]  k p  <iH>  as d e f i n e d  i n Equation  [6.4.1] can  be w r i t t e n a s  = [1 + A ( 1 ) A ( 2 ) G ( » ) ] [ 1 / ( 1 2  Since  2  22  A ( 1 ) A ( 2 ) G ( ° ° ) < 1, a s t a n d a r d 2  2  22  - \ A (1)A (2)G 2  2  2 2  (~)]  - 1. [ 6 . 5 . 5 ]  expansion of the f r a c t i o n a l  term  yields <9>  and  h e n c e , <!H> Figure  to  of  2  2  22  6.3 shows t h e e f f e c t  by s e t t i n g  of attaching a label  at various  The s h a p e o f t h e m o l e c u l e  angles  has b e e n  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  varied continuously  <iU>  [6.5.6]  2 2  <= G ( ° ° ) .  a p r o l a t e symmetric t o p molecule.  fixed is  = ! A (1)A (2)G (») ,  on t h e a n g l e  by v a r y i n g  D. x  I t i s evident  o f attachment o f the label  t h a t t h e dependence  to this quite  asymmetric  -189-  molecule  i s weak, b u t m e a s u r a b l e .  r a p i d m o t i o n makes <^i>  f a s t motion l i m i t , ation  constant i s inversely  while  i n the adiabatic  the  relevant  as  proportional  limit,  relaxation  coefficients.  I t should also  be n o t e d t h a t  larger  because the r e l a x -  to the diffusion  m o t i o n makes <!H>  rapid  constants are proportional  f o r a spherical  rotation varies  bound t o a s p h e r i c a l  molecule with  D  (1  according  the  = Dj_. rotation  The f i g u r e s  Several  field  rotation  rate  6.4, i t a p p e a r s t h a t  d e s e r v e comment.  even v e r y f a s t  attenuation factor only  internal  internal  rupolar  interaction  First,  from  effect  axis  of  axis. Figure  affect the  i s f o rthe  P h y s i c a l l y , we h a v e n o t e d t h a t  very  " e f f e c t i v e l y " reduces t h e magnitude o f t h e quad-  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  the  perturbation).  the  o b s e r v e d a n i s o t r o p y when b o t h t h e i n t e r n a l  the  m o l e c u l e a s a w h o l e a r e s l o w , as s e e n i n F i g u r e 6.5.  In contrast,  s p e a k i n g , F i g u r e 6.5 shows t h a t as  >  i f the angle o f attachment i s s i g n i f i c a n t l y  e - cos~^(l//3).  rotation  rotation  rotation will  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  rapid  an i n t e r n a l  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  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  features o f these plots  NMR m a g i c a n g l e ,  >  show how < !H  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 electric  on  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  choices f o r internal of  i s t h e same  m o l e c u l e i n s u c h a way t h a t  a b o u t j u s t one bond c a n o c c u r .  because  to the d i f f u s i o n  F i g u r e s 6.4 a n d 6.5 show t h e e f f e c t o f i n t e r n a l for a label  constant;  smaller  T h e 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  w o u l d be o b s e r v e d  i n the  a whole i s slow, i n t e r n a l  internal  rotation  acts t o decrease  rotation  e v e n when r e o r i e n t a t i o n  rotation  results  (reduces  in little  and r o t a t i o n  of  Practically of the molecule change i n t h e  -190-  observed anisotropy unless the rate ably faster For  than t h e r a t e  t h e most p a r t ,  of the internal  of reorientation  J  > : >  1  T  a n c  nt  follows  9  [ 6 . 3 . 4 ] , [ 6 . 4 . 2 ] , and [6.5.1]  <&>  -  T ) 2  2  1  the  It  metrical no  effect  zero  factors.  2  t o note that  pendent o f t h e r a t e  N int D  I fT^D.^ »  2  i s interesting  T  +  1  +  )  V i n t  '  1, t h i s r e d u c e s t o t h e r e s u l t o b t a i n e d i n  > = |Q A ( l ) A ( 2 ) ( 3 c o s 6 2  +  3sln%  2  [6.5.7]  absence o f i n t e r n a l motions.  < H  Equations  .  2  F u r t h e r m o r e , i f T^D^^. <<  d i r e c t l y from  12cos 6sin 9  +  * x A (1)A (2)  Making t h e assumptions,  that  * "3 j ( 3 c o s 2 e  2  apparent from t h e a s y m p t o t i c  attenuation factor.  cog T^, 0, and Q » ~\\ ' ,^ iDn.t '; ^ D* ->®' ' ^e f 0 °^,  i s appreci-  o f t h e m o l e c u l e as a whole.  the behavior i s readily  values o f the time-integral  rotation  - l )  i n either  2  1,  .  limit,  [6.5.8]  the anisotropy  i s inde-  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 geoFinally, internal  r o t a t i o n , w h e t h e r f a s t o r s l o w , has  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 a t t a c h m e n t i s  degrees. 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  exposition  h a v e shown t h a t  particularly molecules  and p l o t t e d  results  i nthis  gamma-gamma a n g u l a r c o r r e l a t i o n s  afford  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  i n dilute solution,  since  t h e observed parameter  short a  large  (time-inte-  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 molecule the  ( a s shown i n F i g u r e 6.3) b u t c a n be c h a n g e d q u i t e  presence of l o c a l f l e x i b i l i t y  (internal  rotation)  markedly i n  a t the site of  -191-  attachment of the contain the  r a d i o a c t i v e t r a c e r to the  information  geometry  (angle  combination with  the  on  both the  of  attachment of the  previously  rate of  large molecule; these  internal  m o t i o n as w e l l  t r a c e r ) f o r the  established  changes  as  complex.  advantages that  the  on In  concen-  -12 tration  s e n s i t i v i t y a p p r o a c h e s 10  simple,  highly  s t u d i e s , and in  this  the  discussion NMR  cerning  the  and  i s not this  e x p e r i m e n t a l measurement ideally  suited for in  apparatus i s a l l commercially a v a i l a b l e , the enhance the appeal  a t t e n t i v e e y e s may  note the  that of Chapter I I I . by  the  gamma-rays a r e  chapter considerably  Finally,  and  penetrating  M,  coincidence,  rapport  are  now  and  The  of such  are  vivo  calculations  studies.  extreme s i m i l a r i t y of  this  o b v i o u s r e l a t i o n s h i p between  PAC  some t h o u g h t p r o v o k i n g comments c o n -  discussed.  -192-  FIGURE 6.3:  Plot of the time-integrated anisotropy a p r o l a t e symmetric t o p having f a s t motion l i m i t  o  D^/D^  o f a t t a c h m e n t , 6,  In the a d i a b a t i c l i m i t  x  = 8.  In the  i n the order,  (right-hand set  anisotropy  varies with  9 i n the order,  In both l i m i t s ,  the various  parameters  of c u r v e s ) , t h e i n t e g r a l 0°> 30°> 60°> 9 0 ° .  the ratio  log(6D ) f o r  (left-hand set of curves), the integral  anisotropy v a r i e s with angle 60°:> 9 0 . > 3 0 ° , > 0 ° .  versus  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:  Plot  of the  versus  integral  l o g ( 6 D ) , where D i s t h e r o t a t i o n a l  for a spherical to  a n i s o t r o p y i n the f a s t motion  a fixed  figure.  molecule.  choice of  For  diffusion  Each f a m i l y o f c u r v e s  " a t t a c h m e n t a n g l e " as  individual  curves  limit  (A) t h r o u g h  listed  constant  corresponds i n the  (D), the  internal 12  rotational 10^, set  and  diffusion 10  (dashed  of internal  9  sec"^  line)  c o n s t a n t , Drespectively.  i s the  integral  ^,  The  needed t o g e n e r a t e  lowest  t o 10  i n each  a n i s o t r o p y i n the  absence  (see t e x t ) , f o r  For a l l c u r v e s , the v a r i o u s these  plots  ,10  curve  r o t a t i o n , or equivalently  "attachment angle".  i s equal  11  are given  zero  parameters  in section  6.5.  ,  ANISOTROPY  ANISOTROPY  ANISOTROPY  -196-  FIGURE 6.5:  Plots is  of the integral  the rotational  (in  the adiabatic  anisotropy versus  diffusion limit).  constant f o r a sperical The two s e t s o f c u r v e s  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 constant, varies 30°,  n  t  5  rotational  shown on t h e f i g u r e .  with different  t o c u r v e s a-d  needed t o g e n e r a t e  these plots  correspond  anisotropy  the values 0°, respectively.  f o r e= 0° i s t h e same a s f o r no i n t e r n a l  a l l and p r o v i d e s a c o m p a r i s o n .  molecule  diffusion  The i n t e g r a l  angles o f attachment;  6 0 ° , a n d 90° c o r r e s p o n d  The r e s u l t at  D ^  l o g ( D ) , where D  The v a r i o u s  rotation  parameters  are given i n section  6.5.  - 1 9 7-  A  ID  A  ^  A  A  CO  CM  OOLx]AdOH10SINV  i  A i  -  Q  -198-  FIGURE 6.6:  D e c a y scheme o f ^ I n a n d  1 1 1 m  Cd  relevant f o r angular correlation  showing t h e photon cascades studies.  -19 9-  111  t, =2.8 days  In  2  ENERGY (KeV)  ti, =0.12 ns  SPIN  ti/, = 49min  420 397  7/2 11/2  t, =85 ns  247  5/2  o  1/2  2  /2  stable  4  -200-  FIGURE 6.7:  P l o t of the integral  attenuation coefficient  isotropic correlation  time f o r v a r i o u s  versus  intermediate 8  lifetimes left  (assuming  and r i g h t - h a n d  I = 5/2, u>g = 1.5 x 10 p o r t i o n s o f each curve  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 intermediate  region  i s the i n t e r p o l a t e d  an state  -1 sec  ).  are given  The by t h e  respectively.  behavior.  The  -201-  -202-  6.6.  CONNECTION BETWEEN NMR AND PAC Before concluding t h i s  c h a p t e r , we w i s h  t o emphasize t h e chapter's  r e l a t i o n s h i p with the previous material presented 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 Nuclear Magnetic  i n this  thesis  which  studies.  R e s o n a n c e 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  * information gathered In  from  transient nuclear polarization  NMR, a b r u t e f o r c e m e t h o d i s a d o p t e d  magnetic f i e l d periments  Another  i n which  alternative  on t h e p r e s e n c e  to this  approach,  which  Two e x p e r i m e n t a l do j u s t  this.  does n o t r e l y  techniques  by c h e m i s t s  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 .  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  which  i s to simply l e t "nature"  performed  and s p e c i f i c i t y .  b e t t e r t h a n t h e muon e x p e r i m e n t manipulate far  radioactive  '  the application of this  common, t h e m a i n d i f f e r e n c e b e i n g  tions  induced.  p o s i t i v e muons.  t h a t cannot  involved are precisely  term  being a general i s one s t e p  I n any c a s e , as s h a r e much i n  those o f the t r a n s i -  t e c h n i q u e s , e n e r g e t i c photons o r  polarization  t o s i g n i f y any s p i n  be c h a r a c t e r i z e d b y a d e n s i t y m a t r i x w h i c h  the unit matrix.  technique  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  In these a l t e r n a t i v e  * H e r e we u s e t h e g e n e r a l  of  T h e PAC e x p e r i m e n t  t h e muon  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  n u c l i d e s than  that the energies  experiment,  Although  as t h e n o n e x p e r i m e n t a l i s t goes, t h e t h r e e t e c h n i q u e s  the f a c t  a r e now b e i n g  One i s t h e PAC 26 27  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  lack of s e l e c t i v i t y  A strong  the magnetic d i p o l a r p o l a r i z a t i o n i s  o f t h e Zeeman f i e l d ,  p o l a r i z e t h e s p i n ensemble.  is  this goal.  t h e d e g e n e r a c y o f t h e s p i n s t a t e s and v a r i o u s e x -  a r e performed  monitored. directly  lifts  t o achieve  experiments.  system  i s a multiple  -2036 e l e c t r o n s are the experimental increase i n s e n s i t i v i t y . NMR  and  PAC  in greater  At f i r s t  L e t us now  always  thought,  valid.  w h a t more s u b t l e and operator formalism (k  experiment  c o n n e c t i o n b e t w e e n NMR can  k ,  probably  be  and  PAC  10  between  2  is really  approximation  techniques  seen most r e a d i l y  i s some-  i f a density  the angular c o r r e l a t i o n  t) = Tr[a(k )a(k t)]  2  technique  where the z e r o f i e l d  i s used t o express  r  -  consider the s i m i l a r i t i e s  i t m i g h t seem t h a t t h e PAC  The  10  detail.  n o t h i n g more t h a n a NMR is  observable with a resultant  10  17 28function, '  .  r  [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 s y s t e m  a f t e r the e m i s s i o n of the f i r s t t at  = 0.  The  density matrix a(k ) Due  i n the d i r e c t i o n  corresponds  2  a l a t e r time t .  to the second  k^  a t t i me  transition  to the i n t e r a c t i o n with extranuclear 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 t h e von  radiation  immediately  Neumann E q u a t i o n  i t s v a l u e a t t = 0 and (Equation [2.2.2]).  The  can  be d e t e r m i n e d  standard  from  approach  then r e s o l v e s the d e n s i t y operator i n t o a m u l t i p o l e expansion  (Fano's  statistical  environ-  tensors or state m u l t i p o l e s ) .  I f the macroscopic  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 e a c h t e r m sion evolves  independently  °k The  ( t )  =  i n time,  G  kk  ( t ) a  index k i s the pole order  components o f ^ ( t ) . or relaxation identically  k  ( 0  >-  ( a 2 - p o l e ) and  T h e s e G's  c o e f f i c i e n t s , and  equal  i n t h i s m u l t i p o l e expan-  may  be c a l l e d  C6.6.2] N l a b e l s each o f the  perturbation,attenuation,  as t h e n o t a t i o n r e a d i l y  to those a t t e n u a t i o n f a c t o r s  2k+l  implies,  introduced e a r l i e r  are i n the  -204-  chapter. e v e n k.  I n p a r t i c u l a r , t h e PAC e x p e r i m e n t p r o b e s I n NMR,  this  same f o r m a l i s m c a n be a p p l i e d .  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 Thus, i n t h i s  only those poles with  a p p r o a c h , we s e e t h a t NMR  cases o f a h i e r a r c h i c a l  i n NMR  However, o n l y t h e relaxation  a n d PAC a r e o n l y two  structure of nuclear relaxation. ^  isolated Relaxation  1  t i m e s m e a s u r e d by c o n v e n t i o n a l NMR  experiments.  techniques characterize the i r r e v e r s i b l e  behavior o f only the (magnetic) d i p o l e p o l a r i z a t i o n .  In a conventional  PAC e x p e r i m e n t , o n e o b s e r v e s u n d e r s u i t a b l e c o n d i t i o n s , d a m p i n g c o n s t a n t s belonging 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 and  hexadecapole  polarizations).  izations  and u n i f y i n g  approach  to nuclear spin  It approach  relaxation  iment.  w i t h another broad M o s t common f o r m s  dynamics o f f l u i d of which  along these  thorough  lines.  to note the apparent s i m i l a r i t y i n t e r p r e t a t i o n o f t h e NMR  o f spectroscopy which  harmonic  thought t o a r i s e from assuming  of this  relaxation  probe  IR a b s o p t i o n p r o b e  and  Rayleigh scattering Y - ^ f t ) , and so on.  i n terms  ( t h i s may be  a multipole expansion of the s o l u t i o n to  Y ^ ( f i ) , NMR, probe  exper-  the reorientational  the lineshape r e f l e c t s  diffusion equation).  and  probes  be d e r i v e d f r o m a more  general-  s o l u t i o n s c a n 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  rank s p h e r i c a l  the r o t a t i o n a l  quadrupole  I t i s obvious that e x c i t i n g  thoughts might  i s very i n t e r e s t i n g  (e.g. ( e l e t r i c )  For example, d i e l e c t r i c  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 , a n d Raman  Y,,(ft), s e c o n d  harmonic  light  The u s e f u l n e s s o f v i e w i n g NMR  c a s e o f more g e n e r a l p r i n c i p l e s  relaxation  has been e x p l o i t e d  scattering  as a s p e c i a l  r e c e n t l y and has  g r e a t l y aided our understanding of seemingly unrelated  spectroscopic  + u • 32,33 techniques. ' I n a comment, we w i l l  briefly  v i e w 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 . result  (Equation -  i A  [6.3.2]) a s , k(k + 2Tp-  k  1)  T-| i s n o t h i n g more t h a n common t o NMR in this  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  j  j  1  , ~  discussions.  Hence, f o r t h e s p i n  instance, the q u a n t i t a t i v e connection  not  i n general  ing  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 variables.  expect  t o be a b l e t o e x p r e s s  5/2  Cd,  x  i s complete.  i n f o r m a t i o n on t h i s  novel  numerous r e f e r e n c e s c i t e d  theoretical  technique in this  o-i  - 6  3 ]  ^/ -]» 8T  a n d  However, as one must  NMR  appearrelaxa-  relation-  '  background.  c a n be r e a d i l y chapter.  2  ,  i  OR  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  and t h e n e c e s s a r y  f  so  t h e damping c o n s t a n t s  i n terms o f c o n v e n t i o n a l  i n Gabriel's articles.  time =  2  Further d i s c u s s i o n s p e r t a i n i n g to numerical  s h i p s c a n be f o u n d  in detail  relaxation  p h y s i c a l parameters,  17  This b r i e f  r  [ 6  the f a m i l i a r s p i n - l a t t i c e  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  tion  I  -2  k ( k + 1) 4 1 ( 1 + 1) - 3 " ( *  i s o f course  lacking  However, f u r t h e r obtained  from the  -206REFERENCES: CHAPTER V I  1.  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Rev. 1 8 8 ,  -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 developed b i o c h e m i s t who r e l i e s for  clues t o unravel The p r i m a r y  on n u c l e a r s p i n r e l a x a t i o n the complexity  theme d e v e l o p e d ,  a b a n d o n many c o n v e n t i o n a l Treading is  on v i r g i n  soil  isotropic  i d e a s one a t t r i b u t e s  f o rgeneral  Adopting  o n l y w i t h extreme c a u t i o n ; t h i s where s i m p l e  as an a i d i n h i s s e a r c h  thesis clearly  problem f o r t h e e x p e r i m e n t a l i s t .  small, motionally undertaken  has shown many i n s t a n c e s  conceptual  difficulties.  a highly  underdetermined  H o w e v e r , on a more o p t i m i s t i c  was s e e n 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  relaxation  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 addition to static  geometrical  constants) which g r e a t l y  reduced  pretation.  Indeed, i ti s f e l t  t h a t NMR w i l l  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  continue  The o b v i o u s  extension  were  molecular of misinter-  t o extend i t s  studies, especially  tracer  techniques.  As f o r e x t e n s i o n s o f t h i s w o r k , t h e r e a r e numerous p a t h s explore.  note, i t  times  the possibility  f o rbiological  when u s e d 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  It  relaxation  a n a p p r o a c h s h o u l d be  t h a t NMR p r e s e n t s  behavior.  as i t i s r e w a r d i n g .  from r e l a t i v e l y  such  must  to transient spin  a n a l o g i e s from proton  i n d u c t i o n may l e a d t o g r a v e  A l s o , i t was commented  a i d the  emphasized t h a t t h e biochemist  o r from r e s u l t s o b t a i n e d s p i n systems.  III-VI will  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 .  c a n be a s e m b a r r a s i n g  a l l t o o easy t o search  experiments  i n Chapters  one c o u l d  i s a complimentary experimental  study  -209-  on m o d e l s y s t e m s w h i c h w o u l d 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 ested s o l e l y i n the pleasure of understanding practical chemical  extensions of applied  relaxation  i n t e r e s t , t h e most f r u i t f u l  t h r o u g h l y examine case o f extreme  relaxation  anisotropic  relaxation  and  As f o r  theory to problems o f  systems.  These  bio-  attempt to systems are  m o t i o n w h i c h c a n n o t be c o r r e c t l y t r e a t e d  i n extending  relaxation  relaxation.  c a l c u l a t i o n s would  i n ordered  t h e methods used i n t h i s t h e s i s . be m o s t i n t e r e s t e d  spin  Finally, the general  theories  t h e t h e o r e t i c a l minded theory of nuclear  which are a p p l i c a b l e  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  lattice  linewidth.  one by  would  multipole  when m o l e c u l a r  -210-  APPENDIX A THE RELAXATION  All  c a l c u l a t i o n s presented  i n this  t i o n t o the equation o f motion elements ential  o f a ( o r x = cr - a )  , = iu  l X  aa  aa  . +  •i  terms i n t h i s e q u a t i o n  R  i ,, m x  • • •  aa a  and c o m p u t a t i o n a l  elements  Presuming t h e e i g e n b a s i s  differ-  a  . •  [A.l]  aa  II.  In t h i s Appendix, a  implications  of various  o f x have a  "physical"  |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  o f x c a n be s i m p l y i n t e r p r e t e d  more c o r r e c t l y , a s t h e r m a l  S  of spins i s constant,  order  i s presented.  F i r s t , we may a s k i f a n y o f t h e e l e m e n t s meaning.  first  a  d i s c u s s e d i n Chapter  discussion of the physical  the solu-  coefficients,  a T h i s e q u a t i o n was f u l l y  r e s o r t to determing  o f the d e n s i t y operator i n which the  equations with constant  aa  thesis  a r e d e s c r i b e d by c o u p l e d  T  (d/dt)x  MATRIX  as s t a t e p s e u d o p o p u l a t i o n s ,  deviation populations.  x  t * "aa  As t h e t o t a l  or  number  = 0  [A.2]  a  (d/dt) _ = 0 . aa  [A.3]  x  I f degenerate  energy  levels  f i c a n c e i n the concept degenerate  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 -  o f t h e number o f s p i n s a s b e i n g  e i g e n k e t s ; only spins having  guishable i n this no u n i v e r s a l  problem.  i n one o f t h e  d i f f e r e n t energies are d i s t i n -  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 ,  i n t e r p r e t a t i o n , although  f o r s p i n problems,  there i s  i n a basis  which  -211-  d i a g o n a l i z e s C^Q,  these  can  coherence of the spins  be  i n t e r p r e t e d as a m e a s u r e o f t h e  phase  (the 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 the absence of r e l a x a t i o n , the elements o f the t e t r a d i c are zero.  I t i s seen t h a t i n t h i s  undergo a s i m p l e  oscillatory  *aa"  4  e  i n s t a n c e , the d e n s i t y m a t r i x  R, elements  motion  ^ a a ' V  £A.4]  co ,. I t i s r a t i o n a l i z e d t h a t elements of R aa 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  at the n a t u r a l frequency, connecting  to average out over  is  true only  interval.  a period o f time  In o t h e r  aa  aa  , - co ,, ,,, a a  .  ( t h e so c a l l e d  , a  .I  a  •(• < < 1  nonsecular  purposes, R r  i  aa  - co , , a a  this  I .  111  [A. 5]  1  L  d i f f e r by more t h a n  t e r m s ) can  s a f e l y be  a few  linewidths  i g n o r e d , and  ^ c o , , , , , . a a a a  J  f o r most  This fact  allows  i n t h e number o f e l e m e n t s o f x t h a t n e e d t o  in applications.  o n l y t e r m s w h e r e co  I to  , i , i , i = 0 i f c o a a a a  a great s i m p l i f i c a t i o n  Furthermore, f o r longitudinal  , - 0 n e e d be  considered.  The  be  relaxation,  elements  involved  aa  J  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 cannot couple relax  This  words,  Terms w h o s e n a t u r a l f r e q u e n c i e s  calculated  (co  to  i f t h e e l e m e n t s o f x u n d e r g o undamped m o t i o n d u r i n g  R  practical  equal  w i t h the  independently.  longitudinal The  elements, the  r e l a x a t i o n Hamiltonian  elements of the d e n s i t y m a t r i x  having  two  elements  different natural  (longitudinal  elements  cannot d i r e c t l y  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 i m p l i f i e d : A l l diagonal  sets of  hence  couple  frequencies.  statements  may  be  r e l a x a t i o n ) r e l a x inde-  -212-  pendently  o f a l l o f f - d i a g o n a l e l e m e n t s and v i c e - v e r s a .  of overlapping resonances,  t h i s allows a simple  a t i o n elements o f the form R  , ',.  . = (iw , + R  (d/dt)x  aa  aa  centered  a t to  , )  x  interpretation of relax-  [ A . l ] becomes  .  [A.6]  aa aa ' aa A  x i  where t h e time dependence o f A line  ,  Equation  In the absence  i s o f t h e f o r m o f a damped  a a  oscillation.  , 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 and  ,  , (_"|)  aa aa  '  v  the identification  -R  .••,  (T: )  =  d.  aa aa is  r e a d i l y made.  affected sition  by o t h e r  I t i s noted  [A.7]  a-Hx  that the transverse relaxation  l e v e l s which are not d i r e c t l y  concerned  i s not  with the tran-  under c o n s i d e r a t i o n . The  longitudinal  relaxation of individual  simple correspondence. any  ,  1  given t r a n s i t i o n  a related  quantity,  The l o n g i t u d i n a l  ^  a  i  a  >  a  c  L  >  does have a s i m p l e  interactions with the molecular transition  i s the total  corresponding  interpretation  from s t a t e a  surroundings.  Using  1  such to  However, as t h e  to state a  the standard  through notation  probabilities, R  -R  magnetization  h a v e no  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 .  p r o b a b i l i t y per u n i t time o f a t r a n s i t i o n  of c l a s s i c a l  transitions  i i = W , . a a aa aa  [A.8]  r a t e a t w h i c h 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  aaaa  =  Z—• a ifa,  R  a i a iaa  [A.9]  -213-  w h i c h c a n be w r i t t e n  as  £ _  R . , a a aa  i  = 0 .  [A.10]  a I n t h e BPP t h e o r y , one i m p l i c i t l y e l e m e n t s o f x)> a n d h e n c e t h i s case  (often  inapplicable)  treats  only state  populations  a p p r o a c h c a n be s e e n t o be a  o f t h e more g e n e r a l  density  (diagonal  specialized  operator  approach.  m a t r i x o f t h e f o r m R , , c a n be a aa a terms, the 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 ,  To s u m m a r i z e , t e r m s o f t h e r e l a x a t i o n d e s c r i b e d as l i n e w i d t h  a a aa  r  J  and a l l o t h e r s s i m p l y a s " i n t e r a c t i o n " t e r m s . I t s h o u l d be e m p h a s i z e d that the relaxation matrix i s not a t r a n s i t i o n matrix. I t i s seen t h a t terms o f t h e form R , , m u s t be r e a l n e g a t i v e q u a n t i t i e s a n d t h a t t h e aa aa R ,', m u s t 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 n  The An  secular  approximations mentioned, s i m p l i f y c a l c u l a t i o n s  examination of the construction  the  relaxing  acheived.  interactions  Likewise, (  k V  =  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  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  I t i s obvious R  immensely.  that  , , , , , , = R ,,,,, , • aa a a a a a a  due t o t h e f a c t t h a t  the spin  [A.11]  operators occur i n adjoint  pairs  (-l) V~ ), k  k  R  i  II  aa a Therefore,  R  =  a a a a Additional  111  =  R ii  •••  a a a a  terms under s p i n  inversion.  polar  the spin  i i  [A. 1 2 ]  a aa  relations are obtained  couplings,  R  a  I  =  a  R I  a a a a  it  =  R  •••  •!  •  a a a a .  i f we n o t e t h e b e h a v i o r o f c e r t a i n  For i n d i r e c t or direct dipolar  and q u a d r u -  o p e r a t o r s a r e components o f a second  rank  -214-  spherical  t e n s o r , and f o r s h i f t  operators  a r e components o f a f i r s t  calculation  shows  anisotropy  and s p i n - r o t a t i o n , t h e s p i n  rank s p h e r i c a l t e n s o r .  A  direct  that  <^a|V |^a'> = ( - l ) < a | V ^ | a ' > k  [A.13]  L  L  where is  i s the spin inversion operator  the tensorial  with  rank o f t h e s p i n o p e r a t o r s .  the appropriate  For  cross-terms,  L f  V , the relation  Hence a n y t e r m s  auto-correlation functions  i I I I I I ~ Ri  R  (c=n  - i, i i , I I I v i  i f L = L', r e l a t i o n s h i p [ A . 1 4 ] s t i l l  R,  , I , I I , I I I  ijjaipa i|;a is  ( r o t a t e s a l l s p i n s by T T ) a n d L  ya  =  i ii  - R  aa a  appearing  i n Equation  s  „  [2.2.13]),  [A.14J  h o l d s , whereas i f  iii  LA.15J  a  valid. To  imations  d e m o n s t r a t e how d r a s t i c a l l y  these  various  s y m m e t r i e s and a p p r o x -  reduce the magnitude o f t h e c a l c u l a t i o n , consider  the following  2  example.  2  F o r t w o 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 h a s ( 2 )  = 1 6 elements  2 2 2  and  R has ( ( 2 ) )  only Of  or 256 elements.  For the l o n g i t u d i n a l r e l a x a t i o n ,  s i x elements o f x a r e r e l e v a n t  (four f o rtransverse r e l a x a t i o n ) .  t h e 3 6 e l e m e n t s o f R t h a t n e e d be e v a l u a t e d , o n l y f i v e  c a l c u l a t e d , the others [A.14],  or  [A.15].  being  found from Equations  n e e d t o be  [A.10], [A.11],  [A.12],  -215-  APPENDIX B INFLUENCE OF THE SECOND ORDER FREQUENCY S H I F T TERMS  As was m e n t i o n e d imation  fails,  i n Chapter I I , i fthe extreme-narrowing  certain  imaginary c o r r e c t i v e  t e r m s m u s t be  approx-  incorporated 1-4  into the equations governing the time evolution  of the density operator.  I n 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 a n y e l e m e n t o f x c a n be w r i t t e n a s (d/dt)x  aa  i = Natural  frequency term + R e l a x a t i o n  + Second  In a l l c a l c u l a t i o n s l a t t e r terms terms  o r d e r c o r r e c t i v e terms  h a s been i g n o r e d .  identical  simplest spin  extensively  relaxed  i n Chapter I I I ) .  example.  '  *  calculations  by d i p o l a r c o m m u n i c a t i o n  The i n f l u e n c e  N(a"a"')[6  A  ,  a  I I  ,  V  a  , ,,,x . . - S a a aa aa  a  a  a  1V  k  1  1  k  1v+  , ,, a a  l X  aa  k  k  - ^  , , - 6 aa  * J '~ (w) . k  k  correc-  J  'lvV"> l  l  X  . . . . ] a a  k -k k -k w h e r e Q ' (CD) a n d J ' (w) f o r m a H i l b e r t T r a n s f o r m  Q '" (u>) =  . .X >>• a a  ,k  X <a |V" |a'''>[6  treated  A  «'"<K"„ k  (a case  as ( s e e E q u a t i o n [ 2 . 2 . 1 0 - 1 1 ] ) ,  A  •  t r e a t s two  o f these second o r d e r  t i o n s on a n y e l e m e n t , Y ,, c a n be w r i t t e n aa  x  o f these  In t h i s Appendix, the importance o f these  of a l l (nontrivial) relaxation  1/2 n u c l e i  (d/dt)x . = i ' ' aa  [B.l]  presented i n t h i s t h e s i s , the influence  i s e x a m i n e d by means o f a s i m p l e The  terms  [B.2]  pair,  [B.3]  -216-  Choosing  as b a s i s  states,  |1>  = |++>  |2> =  (|+-> + |-+>)//2  |3> =  (|+-> - |-+>)//2  |4> =  it  |->  [B.4]  ,  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  <I (t)>  - <I >  z  where R  ]  ]  = (<I (0)> - < I > ) e x p { ( - R  T  T  Z  2  4  4  relaxation  = 2J '~ (2OJ ) and R 2  2  0  result  i s obtained  from  [B.l],  but i s identical  terms a r e i g n o r e d .  1 1 2 2  = - J  1  '  -  1  ^ )  t h ecomplete r e l a x a t i o n to thecalculation  - R  1 1 4 4  2  i s g i v e n by  ) }  t  t  1  1  2  2  + 2J '" (2OJ ). 2  2  ^  of Equation  where t h e second o r d e r  T h i s c a n be s e e n by e x a m i n a t i o n  5  This  0  description  -  B  o f Equation  shift  [B.2].  The o n l y n o n z e r o t e r m s o f t h e m a t r i x N a r e ,  N(ll)  = Q ° ° ( 0 ) / 6 - Q '" (co )/2 + Q '" (2o) )  [B.6]  N(22)  = 2Q°°(0)/3 - Q  [B.7]  1  1  2  0  1>_1  2  (u)J)  [B.8]  N(44) = N ( l l ) .  Hence, f o r t h e l o n g i t u d i n a l x  2 2  relaxation  a n d X44 n e e d t o be c o n s i d e r e d ; a ' '  = 1, 2, o r 4.  Therefore  are o f t h e form  [6  ,,x  aa  11 - 5  be g e n e r a l i z e d a n d i t f o l l o w s these  corrective  terms  w h e r e o n l y t h e c o u p l i n g o f X-J-JJ = a'  1 1  = 1, 2, o r 4 , a n d a '  t h eo n l y terms t h a t a r i s e  aa  11X  aa  ,,]  = 0!  =a  i n Equation [B.2]  Indeed, t h i s  r e s u l t can  aa  that i t i s completely  i n any l o n g i t u d i n a l  For t h e t r a n s v e r s e r e l a x a t i o n  0  justified  relaxation  t o ignore  calculation.  c a l c u l a t i o n , o n l y two elements o f x  -217-  need t o be t r e a t e d , x - ]  (d/dt)x  = iu  1 2  1 2  x  +  R  1 2  a  d x  n  2  2  1212 12 X  -  4  +  R  ^  1  s i m p l y shown  S  1224 24 x  i ^  +  2  )  2  that  " N(ll))x  1 2  [B.9] (d/dt)x  = iu 4 24 x  +  R  2  2 4  2412 12 x  +  R  2424 24 x  +  i  ( ( N  4  )  4  "  N(22))x 24  [B.10] S i n c e N ( l l ) = N ( 4 4 ) , co-| = 2  equations  OO  2  4  -WQ, a n d R-| -|  E  2  =  2424'  R  2  t n  isc.pair ° f  c a n be r e g r o u p e d a n d r e w r i t t e n a s ,  (d/dt)<I (t)> = +  (d/dt)(x  +x )  = (d/dt) (t)  2 4  1 2  = ^ ( t )  y i  + ivy (t) 2  [B.ll] (d/dt)y (t) 2  = (d/dt)(x  w h e r e $ = -iwg + R-| -| 2  the r a p i d  +  where R  +  ] 2 ] 2  modulation  +R  4  a 2  2  ) = i y y ^ t ) + ipy (t)  n  4  ] 2 1 2  d v = (N(22') - N ( l l ) ) .  +R  1 2 2 4  ]  effect  k  = (-l) 5 T (l k  2  T h i s a l l o w s one t o w r i t e  <I (t)>/<I (0)> +  2  + LO T 2  (u> ) 0  - 2J '~ (2u> ). 2  2  For simplicity,  [B.13]  I t i s clearly  0  2  toa  i f we f u r t h e r  assume  )~\ t h e n Q " ( u > ) = u>T J '" (u)).  2  k,  k  k  k  2  [B.13] a s ,  = exp|(-3 -5(1 + x  + x  1  + exp(-iyt)}  o f t h e imaginary terms g i v e r i s e  Equation  x cos|(x(l  _  Suppressing  d e c a y c a n be w r i t t e n a s ,  )t]|exp(iyt)  = - | o ( 0 ) + |o ' 0 0  ] 2 2 4  [B.12]  2  o f t h e decay envelope.  t h a t J '" (u>)  +  ]  = ^exp[(R  seen t h a t t h e p h y s i c a l  k  R  2  2  Larmor p r e c e s s i o n , t h e t r a n s v e r s e  <I (t)>/<I (0)> +  - x  1 2  2  ) "  1  - 2(1 + 4 x ) 2  ) " + 4x(l + 1  x = W  Q  T  2  .  - 1  )tc x /2 2  2  }  4x )" )U x /2J; 2  1  2  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  <I (t)>/<I (0)> +  = Re[((-1/T  +  w h e r e T-j a n d T  2  1-0)^2/2^ ) t ) ]  -  2  r e w r i t t e n as  are conventionally defined.  In F i g u r e B . l , l o g ( < I ( t ) > / < I ( 0 ) > ) +  variable, tg x  the u n i t l e s s 10  ±  2  COQT  ^,  2  l O  1  +  2  , for five  i splotted  different  as a f u n c t i o n o f t h e  The upper  correspond t o the l i m i t s open s q u a r e s  d e p i c t t h e p r e d i c t e d decay  ( f o r thecase, UQT  =1).  2  COQX  to  =  curves  << 1 r e s p e c t i v e l y .  2  T  values o f  and lower t r i a n g u l a r l a b e l e d  o f togx,, >> 1 a n d  ( o2  v a l u e s o f togx,,  ^ , and 10^; curves p - t correspond t o i n c r e a s i n g  respectively).  ignored  [B.15]  The  when t h e i m a g i n a r y t e r m s a r e  I t i sobvious  t h a t even i n t h i s  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  optimum  b o r d e r s on b e i n g  neglible. A diagram which similarity  summarizes t h e s e f a c t s  t o F i g u r e 3.2).  i s F i g u r e B.2 ( n o t e t h e  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  spectral  2 2-1 2 2 2 2-1 d e n s i t i e s , u)j(to) = -r co(l + to x ) ~ a n d toq(to) = to x ( l + to x ) as w e l l 2  as w j ( 0 )  = UJT  of magnitude,  2  2  areplotted  2  2  as a f u n c t i o n o f tox . 2  Dealing with orders  j ( 0 ) i s a measure o f t h e l i n e w i d t h , j ( t o ) i s a measure o f Q  ( 1 / T . j ) , a n d q(tog) i s a m e a s u r e o f t h e m a g n i t u d e o f t h e s e c o n d shift  terms.  T h i s p l o t shows v e r y e x p l i c i t l y  s i g n i f i c a n c e o n l y when  J(U)Q)  Introduction of internal as  i n Chapter  III.  - q(tog) - j ( 0 ) , t h a t  motions  i s , when  be o f  COQT  2  - 1.  c a n be d e s c r i b e d i n t h e same f a s h i o n  I t c a n be r a t i o n a l i z e d  m o b i l i t y only minimizes  t h a t q(cog) w i l l  order  t h e importance  that theeffect of anisotropic  o f these second  order  static  corrections. Although only 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  treated  i n this  -219-  section, tedious generalizations follow d i r e c t l y .  However,  weighing  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 , an a p p r o a c h  has n o t been  adopted.  1.  H. G a b r i e l , P h y s .  Stat.  S o l . 2 3 , 195  2.  R. A.  3.  H. P f e i f e r , Ann. P h y s i k V 3 , 174  4.  G.  (1967).  H o f f m a n , A d v a n . Magn. R e s . 4_, 87  K. F r a e n k e l , J . Chem. P h y s .  (1970).  (1964). 4 2 , 4275  (1965).  such  -220-  FIGURE B . l :  The t r a n s v e r s e d e c a y i s p l o t t e d a s a f u n c t i o n  of the unitless  v a r i a b l e , t£ x^  CJQT2  for five  d i f f e r e n t values o f  +2/3 +1/3 0 10— ' , 10— ' , a n d 10 ; c u r v e s p - t c o r r e s p o n d values o f  COQT^  ).  curves correspond respectively.  The upper and l o w e r to the limits  The o p e n s q u a r e s  of depict  b e h a v i o r when t h e i m a g i n a r y ( s e c o n d corrections  are ignored( u) T n  9  l  a n c  =  to increasing  triangular > > -  (WQT£  l Q 2 w  t  labeled <  k  1  the predicted(linear)  order s h i f t  = 1 i s assumed).  terms)  -2 2 1-  -222-  FIGURE B . 2 :  Plots  o freduced, frequency weighted, spectral  2 2-1 2 coj(w) = 12^(1+ co T^) a n d toq(<o) = co TgjCco) <oj(0) Etox a r e p l o t t e d 9  as a function of  densities,  as well as  IDT . 9  -2 2 3-  -224-  APPENDIX C QUADRUPOLE RELAXATION OF I = 3/2 NUCLEI  In volved  recent years, nuclear relaxation i n chemical  e x c h a n g e have p r o v e n  s t u d i e s o f s p i n 3/2 n u c l e i i n very f r u i t f u l  biomolecules  i n solution.^  at  i s c h a r a c t e r i z e d by a u n i q u e  each s i t e  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  i t was m e n t i o n e d t h a t t h i s a s s u m p t i o n relaxed nucleus for  f o r t h e study o f  time constant.  will  n o t be v a l i d  ( w i t h I >_ 3/2) when e x t r e m e - n a r r o w i n g  most b i o m o l e c u l e T ' S ) . C  In Chapter for a  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 motions and n o n - e x p o n e n t i a l  Although since Bloch  t h i s f a c e t o f quadrupolar  2  f i r s t commented on t h i s  quadrupolar  i sviolated ( i . e .  AS a l a r g e p o r t i o n o f t h i s  a b r i e f mention o f the n o v e l i t y o f t h i s  V,  relaxation,  phenomenon i s i n o r d e r . relaxation  h a s been known  ever  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 numerical  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 o n t h e p e r t u r b a t i o n h a s n o t been commented u p o n .  This i s a very real  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 behavior o f a s p e c i e s such as C / ^ - H g / CI n u c l e u s The following  r o t a t e s r a p i d l y about t h e macromolecule-Hg  quadrupolar  where t h e  bond.  H a m i l t o n i a n c a n be c o n v e n i e n t l y e x p r e s s e d  i n the  form, £ (t) ^ n  =  if k=-2  U (t)V k  k  [C.l] .  where  U (t) = (-l) U/180) k  K  1 / 2  (^-f-j  r (ti(t)) k  2  [C2]  -225-  = 3I  V°  For  7  L i ,  1 ]  nuclear  B,  ± ]  = +[I ,  V  ± 2  Z  2 3  field  I ] /2/6  [C4]  .  [C.5]  ±  +  = I /2/6 2  Na,  3 3  S,  3 5  C1,  CI, Br,  3 7  7 9  environment i s completely  electric  [C.3]  2  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 h a s I = 3/2  simplicity,  (e.g.  V  - I  2  gradient  Upon c h o o s i n g  8 1  coupled  (d/dt)  e t c . ) and t h a t t h e  c h a r a c t e r i z e d b y an a x i a l l y  symmetric  (n=0).  the basis functions  |1>  = |3/2 3/2>  |2>  =  13/2  |3>  =  13/2  -l/2>  |4>  =  13/2  -3/2>  |IM>,  l/2>  [C.6]  ,  a rather straightforward c a l c u l a t i o n using density operator  Br,  theory  t h e s e m i c l a s s i c a l form o f t h e  of relaxation yields the following pair of  equations,  y ]  (t) = seco '- ^) 1  1  - J '" (2w )] 2  2  0  y i  ( t ) + 36EJ  , _ 1  1  (a3 ) 0  + J '- (2w )]y (t) 2  2  0  [C.7]  2  ( d / d t ) y ( t ) = SGLJ '"^^) + J ~ ( 2 a > ) ] y ( t ) + 1  2 ,  2  -  J '" (2 2  2  W o  2  0  1  seEJ '"' ^) 1  1  [C.8]  )]y (t) 2  where  y^t)  = Tr[ l ] x  z  = <I (t)>-<I > z  z  T  = |(  Xll  - x  4 4  )  +  j(x  22  - x  3 3  )  [C9]  -226-  1 y ( t ) - "2"(x44 " x - | )  3 2"(x  +  2  The  spectral  - x  33  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  2 2  ) •  F o u r i e r Transform o f  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 (composed f r o m 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 relations  5  frequency  k k -k  , (-1) J '  = Tr[I  + x  ( t ) ] = <I (t)>  q ]  (t) = 36[-J  t h e Hubbard order  Similiarily,  = /3(x  +  A short calculation  Furthermore,  the l a t t i c e  00 (ktjg), a r e v a l i d , and t h e second  shifts vanish identically.  q^t)  (d/dt)  (ku>g) = J  [C.2]).  [C.10]  + x  1 2  ) + 2x  3 4  [C.ll]  2 3  yields,  0 0  ( 0 ) + J » - ( o ) ) - 2 Q ' - ( a , ) ] q ( t ) + 3 6 [ J°°(0) 1  1  1  ( J  1  - J ' ' ( 2 u ) + 2(2Q '- ( , ) + 2  1  2  Q '  1  0  (  0  2  0  _ 2  1  (2a  ) ( )  ))]q (t) 2  [C12] ( d / d t ) q ( t ) = 36[ J 2  1  '  -  1  - J '" (2a) ) +  ^ )  2  2  2(Q '- (a) )+Q 1  0  1  0  2 7 2  U ))^2  ( t )  0  [C.13] where q (t) = 2x 2  .  2 3  Although  previous c a l c u l a t i o n s d i d not mention  rections  ( i . e . f i n i t e Q ' s ) , i t c a n be s e e n  importance  and a r e h e n c e f o r t h i g n o r e d .  subject to the i n i t i a l  [C.14]  t h e second  order cor-  that they a r e of l i t t l e  The s o l u t i o n s f o r y - | ( t ) a n d q - j ( t )  p r e p a r a t i o n o f t h e s p i n system  y ^ O ) = (cose - 1 ) < I >  by a e p u l s e ,  T  Z  y (0) 2  =  q-j ( 0 ) = q (0) 2  -(3/5^(0) sine<I >  = (2/5)  T  z  q i  (0)  [C15]  -227-  are found  t o be, [C.16]  y-,(t)  q-,(t)  [C17] that i fJ  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 then  both  general It  2,-2  1,-1  d e c a y s 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 .  (2co  0  However, i n  both decays a r e b i e x p o n e n t i a l . i s interesting  t o note  the t r a n s v e r s e o r l o n g i t u d i n a l in  ( 0 ) = -J  t h e form  of the i n i t i a l  relaxation.  decay o f e i t h e r  Abragam's t r e a t m e n t  the present notation) gives the following  (recast  expressions f o rthe relax-  ation , -J '" ^) 1  1  [C.18]  + 4J '" (2(o )} 2  2  0  [C.19]  Unfortunately, the v a l i d i t y o f these expressions nuclei  (ora white  spectral  s u b s t i t u t i o n o f Equation that Equations of  The p r e s e n c e  = 4.  density approximation).  [C.15]  into  [ C . 7 ] and [ C . 1 2 ]  t o s p i n 1=1  However, as t-K), shows v e r y  clearly  [ 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  the magnetizations  ation behavior.  i s limited  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  decay  assumptions!  o f a n i s o t r o p i c m o t i o n s h a s l i t t l e e f f e c t on t h e r e l a x F o r example, f o r slow, i s o t r o p i c motions,  I f an a d d i t i o n a l  degree o f motional  J  0 0  (to )/J 0  freedom i s assumed, o f t e n  0 0  (2u>  this  0  -228-  ratio will  approach  u n i t y even though t h e motions  perpendicular tothe  s y m m e t r y 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 s l o w on t h e t i 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 Chapter 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 motion may s t i l l  be a p p r o x i m a t e d  more, i t w i l l  quite well  be r e c a l l e d  s e n s i t i v e t o motions of  the diffusion  in  t h i s problem  i s assumed, t h e l o n g i t u d i n a l by a s i n g l e e x p o n e n t i a l .  that J ^ ( 0 ) ,  t o a good a p p r o x i m a t i o n , i s i n -  Therefore, a discussion of anisotropic  holds l i t t l e  interest.  Before d i s m i s s i n g t h i s  however, i t i s w o r t h w h i l e t o n o t e t h a t t h e most u n i q u e h a v e t o o f f e r seems t o have n e v e r  l i g h t o f the preceeding  that the r e o r i e n t a t i o n process case, J  nn  9  motions  topic  prediction  been n o t e d  that  i n pre-  d i s c u s s i o n , a n d f o r s i m p l i c i t y , we assume i s c h a r a c t e r i z e d by a u n i q u e  9  99 i  (co) = ( e qQ/fi) x / 7 2 0 (1 + w T ) 2  [C.16] and [C.17] reduce  axis  3 7 problem.'  vious considerations o f t h i s In  Further-  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  tensor.  these c a l c u l a t i o n s  relaxation  2  t o the asymptotic  .  F o r CO T Q  2  x,,.  In t h i s  > 1, E q u a t i o n s  form  [C.20]  q,(t)  = q ^ O ) |3exp(-2a(co x ) t) 0  2  where a E ^ q Q / f O S ^ O ^ g - r , , ) ^ )  [C21]  + 2 e x p ( - 5 a t / 2 ) \ /5  2  - 1  .  T h i s shows v e r y e x p l i c i t l y t h a t u n d e r  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  partial  l o s s o f x-y p o l a r -  i z a t i o n , < I ( t ) > d e c a y s on t h e same t i m e s c a l e a s t h e l o n g i t u d i n a l  magneti-  zation.  i n the  +  In other words, a decreasing m o b i l i t y  observable linewidth  (this  statement  component o f m a g n e t i z a t i o n w i l l  leads to a decrease  assumes t h a t t h e r a p i d l y  decaying  be b r o a d e n e d t o t h e p o i n t o f t o t a l  -229-  obscurity).. prediction.  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  Furthermore,  v a l i d i t y of t h i s  2  is limited  and  this  2  thought  for nuclei  and  such  by t h e f a c t t h a t  p u t s a s t r i n g e n t l i m i t on  > 2 T  <  should lend i t s e l f  as \ i  which  have s m a l l Q and  S c o t t , Ann.  R.K.  W a n g s n e s s and  3.  P.S.  H u b b a r d , J . Chem. P h y s .  4.  N.C.  P y p e r , Mol. Phys.  5.  P.S.  Hubbard, Phys.  6.  A. A b r a g a m , P r i n c i p l e s o f N u c l e a r M a g n e t i s m , C l a r e n d o n 1 9 6 1 ; pg. 3 1 1 .  7.  T.E.  F.  Rev.  B l o c h , Phys.  B u l l , J . Magn. R e s .  728  1_, 27  (1972).  (1953).  (1970).  319  344  (1969). Press, Oxford,  (1972).  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  correlated  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 found  the t r a n s v e r s e decay i s w e l l  approximated  i r r e g a r d l e s s of the magnitude of o v e r a l l If the o v e r a l l the motion  m o b i l i t y of the t r i a d i s i n c r e a s e d , one  other decreases. n i t u d e , then in  89,  Bioeng.  verification  (1971).  180,  8,  Rev.  5 3 , 985  2]_, 1  Rev.  Biophys.  intri-  y.  large  2.  in  maximal  r e a d i l y to experimental  S y k e s and  analogous  (relax-  H o w e v e r , ^ i s i s an  B.D.  motions  M.D.  or  o^"  u  1.  * An  <<  of  - 1 - 1  v a l u e s o f e qQ/fi ( e qQ/fi « guing  this  i t s h o u l d be b o r n e i n m i n d t h a t t h e r a n g e  prediction  a t i o n matrix element)"^  consequences of  a s a sum mobility  i s reduced  o f two  that  exponentials  (see Chapter  IV).  w h i l e the a n i s o t r o p y  of the time constants i n c r e a s e s , the  I f the r e l a t i v e magnitude d i f f e r  i t i s v e r y c o n c e i v a b l e t h a t one  exact analogy w i t h the present  case.  by o r d e r s o f mag-  could conclude  a  T^,  -230-  APPENDIX D NUCLEAR MAGNETIC RELAXATION FOR INDIVIDUAL TRANSITIONS OF AN AMX USE OF INTERFERENCE TERMS TO DETERMINE  It  has b e e n shown t h a t  b e t w e e n two p h y s i c a l 1 y  SIGNS OF SCALAR COUPLING  the presence o f f i n i t e  CONSTANTS  interference  terms  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 m e c h a n i s m s , s u c h as  b e t w e e n d i p o l a r and s h i f t mination of the absolute the  SPECTRUM:  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 sign of the scalar coupling  i n d i r e c t dipolar coupling  f o r deter-  constant  (trace of  t e n s o r ) from comparison o f t h e r e l a t i v e 2-4  linewidths  i n an AX s p e c t r u m .  F o r s p i n systems c o n t a i n i n g  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  interference  three  terms  or  arise  f r o m c r o s s - c o r r e l a t i o n e f f e c t s b e t w e e n p h y s i c a l l y 1 i k e r e l a x a t i o n mech5 a n i s m s , a s b e t w e e n two d i p o l e - d i p o l e  interactions,  i n q u i r e w h e t h e r t h e s e e f f e c t s c a n be u s e d t o f i n d constants. into  F i n a l l y , t h e answer t o t h i s  t h e r e l a t i o n b e t w e e n t h e s e two d i s t i n c t  effects,  5 6 ' this  The s i m p l e s t "AMX"  aspect providing  A  M  «  ft  A schematic energy l e v e l  definiteness  direction  - v ); J M  diagram  i s that  than f o r p a r a l l e l  i s w e a k , an u n c o u p l e d b a s i s order:  should provide  insight  X  «  (v  A  spins.  of  interference discussion.  x  | J  A  Figure M  | = 2|J  applied  constant lead  the s t a t i c  M  X  «  (v  M  D.l. For A X  [  =  4|J  M X  |;  i n t h e z-  t o a lower  Because t h e s c a l a r  set diagonalizes  i s the  - v ) , and J  i s shown i n  a s t a t i c magnetic f i e l d  and a p o s i t i v e s c a l a r c o u p l i n g  for antiparallel  coupling  spin one-half nuclei A  to  of  interest of this  i n t h e diagram, i t i s supposed t h a t  sign convention  first  (v  the signs  classes  the primary  s u i t a b l e system o f three  case, f o r which J  - v^).  the  question  so i t i s n a t u r a l  energy  coupling  Hamiltonian to  -231-  |1> = |+++> |2> =  |++->  |3> H  |+-+>  |4> =  |-++>  |5> E  |+—>  |6> =  |-+->  |7> = I—+> |8> = | — > ; The  |i>=|I^lV.  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  magnitudes o f the s c a l a r couplings A t w e l v e - l i n e spectrum corresponds  [D.l]  a r b i t r a r y , and t h e r e l a t i v e  have been e x a g g e r a t e d  (three quartets) i s observed,  t o one o f t h e t w e l v e  transitions  for visibility.  f o r which each  line  shown i n t h e r i g h t - h a n d  diagram f o r Figure D . l . The  p r o b l e m w i t h t h e AMX s p e c t r u m i s t h a t f o r g i v e n m a g n i t u d e s o f  the three c o u p l i n g c o n s t a n t s , there a r e eight, .possible energy which would y i e l d diagrams d i f f e r  identical  according  three coupling constants  spectrum  all  each q u a r t e t correspond  tions.  lines  coupling  constants  following  to  obtained  now c o r r e s p o n d through  means f o r o b t a i n i n g r e l a t i v e  to the high-  a l l eight  combina-  and a b s o l u t e  signs of  a r e f r o m 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 or orientation  i n an e l e c t r i c  field  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 c a n be  ( 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  individual  lines  r e s p e c t i v e A-,, M-,, a n d X-, t r a n s i t i o n s ; i f  e i t h e r double i r r a d i a t i o n  or i n a l i q u i d  For  a r e a l l p o s i t i v e , the low-frequency  i n each q u a r t e t , and so f o r t h  The p r i n c i p a l  positions); the  (see left-hand p o r t i o n o f Figure D . 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  frequency  line  t o t h e c h o i c e o f absolute s i g n o f each o f t h e  example, i f J ^ , J ^ , and in  (identical  diagrams  transitions  o f an AMX  spectrum.  This  linewidths of  present  exposition i s  -232-  closely  akin  coupling As  t o a s i m i l a r problem o f s i g n  constants  i n Electron  determination of hyperfine  S p i n Resonance  spectra.  7  shown i n A p p e n d i x A, o n e c a n make t h e i d e n t i f i c a t i o n  - iJu • 2 > ^ • R  -  ( T  CD  Therefore, t o determine the complete s p e c t r a l of  12 r e l a x a t i o n m a t r i x e l e m e n t s  corresponding For  of  are  Dipolar  contibutions  t h e AMX s p e c t r u m a r e l i s t e d  D.2 w i t h  the notation  defined  (of the possible  4 0 9 6 ) m u s t be  listed  by E q u a t i o n s  we assume o n l y d i p o l a r to the linewidth  beneath the Tables  w h e r e \\i i s t h e s p i n ip|4> = |5>.  inversion  Relation  different dipole-dipole and T a b l e D . l ,  tions for  (the spectral  [D.3] s t i l l  interactions  a given s e t of r e l a t i v e scalar  cally  be b r o a d e r  of cross-correlation  Only s i x t r a n s i t i o n s are elements:  -'-  l  h o l d s when c r o s s - c o r r e l a t i o n are included.  D  3  1  within  that  |6>,  between  With t h e a i d o f F i g u r e e v e n when  cross-correla-  a given m u l t i p l e t  are different;  coupling  signs,  the inner  two l i n e s o f  ( o r n a r r o w e r ) t h a n t h e o u t e r two l i n e s . terms m a g n i f i e s t h i s d i f f e r e n c e .  speaking, i fa white spectral  largest molecules  densities  o p e r a t o r ; ij>|l> = |8>, ip|2> = |7>, ii\3> =  are neglected, the linewidths  Inclusion  column o f T a b l e  %i>j>k^Ji  =  i t c a n be e a s i l y j u s t i f i e d  a given quartet w i l l  the  ijk*  interactions are  i n T a b l e D.l and t h e f i r s t  [ 2 . 2 . 1 4 ] and [ 3 . 1 . 3 ] ) .  R  D.l  calculated,  f o r each t r a n s i t i o n  g i v e n due t o t h e s y m m e t r y o f t h e r e l a x a t i o n m a t r i x  and  a total  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 .  the present discussion,  operative.  lineshape function,  2]  i n ordinary  density  solvents),  i s assumed  (valid  and a l l d i p o l a r  Practi-  f o r a l l but interaction  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 , t h e 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-  b e t w e e n members o f a m u l t i p l e t s h o u l d if  be o f t h e o r d e r  the spectral d e n s i t i e s f o r cross-terms  auto-correlation creases  terms  (a " t y p i c a l "  the by J's  signs  "A" q u a r t e t ,  the relative  are determined  stances  r e l a x a t i o n mechanism  o f the J's are a v a i l a b l e . signs  of  quartets  r a t h e r than j u s t  from  a b o v e , as when t h e e x t r e m e - n a r r o w e d a p p r o x i m a t i o n  very  tropic  m e d i a ) , o r when t h e p a i r w i s e  examination  different  vectors  satisfy  ( i . e . y y r „ >> o r << y . y n  sion axis.  We s h a l l  not discuss  appropriate  correlation functions  could  difficulty  a unique  i s t h a t each t r a n s i t i o n w i l l information  coupled  diffu-  limits,  o f i n t e r f e r e n c e terms. here, a  no l o n g e r  similar transition.  be c h a r a c t e r i z e d b y  pertaining to longitudinal  f r o m knowledge o f any i n d i v i d u a l  o n l y one o f t h e t h r e e  i s aniSO-  to contrast the qualitative  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 e a c h  c a n be o b t a i n e d  constants  evaluation of the  o r elements of R i n various  relaxation i s discussed  "T^', a n d no p r a c t i c a l  relaxation  interaction  to the principal  further the specific  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  analysis  indi-  (large  r . ) , o r when t h e r e  respect  the o b j e c t i v e o f t h i s Appendix i s merely  transverse  than  orientations of the internuclear  unique r e l a t i o n s h i p s with  Although only  fails  circum-  -3  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  If  of a l l three  two). There a r e s p e c i a l  -3  The  derives  signs  f o r w h 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  molecules and/or viscous  since  may be e s t a b l i s h e d ;  the r e l a t i v e  (no a d d i t i o n a l i n f o r m a t i o n  present,  F o r e x a m p l e , by e x a m i n i n g  and  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 ,  of a l l three  are  ( 1 / 4 ) as l a r g e a s f o r  t o about 30%.  relative  cated  Further,  value), the linewidth variation i n -  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 only  are only  o f 20%.  nuclei i s relaxed  e l e m e n t o f R.  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 chemical quite  shift  anisotropy  straightforward  couplings the  (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 f o ra nucleus other  t o show t h a t a b s o l u t e  various  quartets.  This  i s evident  shift anisotropy-dipolar  that only spectral The  signs  then  of a l l three  i t is  scalar  c a n be d e t e r m i n e d f r o m 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  c o l u m n s i n T a b l e D.2 c o u p l e d w i t h the  than a p r o t o n ) ,  densities are defined sign determination  sign o f t h e chemical  shift  t h e u s e o f t h e l a s t two  t h e f a c t t h a t R.., „ = -R,., .,, ,„ f o r  cross-terms  n u c l e u s "M" i s r e l a x e d  absolute  with  ( i n t h i s T a b l e , i t i s assumed  by s h i f t a n i s o t r o p y  by E q u a t i o n s  interactions: the  [2.2.14], [3.1.3],  and  [5.3.3]).  o f c o u r s e d e p e n d s on k n o w i n g t h e a b s o l u t e  anisotropy.  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 scalar coupling  constants,  of the absolute  i n t e r f e r e n c e terms between l i k e  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  signs  the  i n linewidths  experimental  detection  o f a p a r t i c u l a r AMX q u a r t e t genity,  intermolecular  of differences  o f J's.  signs of  relaxation  In order  that  b e t w e e n members  n o t be m a s k e d b y t h e p r e s e n c e o f f i e l d  inhomo-  relaxation, or spin-rotation e f f e c t s , the linewidth  m e a s u r e m e n t w o u l d 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 of  interest,irv-acdeuterated solvent  (such  as g l y c e r o l - d g ) .  1.  L. G. W e r b e l o w a n d A. G. M a r s h a l l , Chem. P h y s . L e t t . 2 2 , 568 ( 1 9 7 3 ) .  2.  H. S h i m i z u , J . Chem. P h y s . 4 0 , 3357  3.  J . M. A n d e r s o n , M o l . P h y s . 8, 505 ( 1 9 6 4 ) .  4.  E. L. M a c k o r a n d C. M a c L e a n , J . Chem. P h y s . 4 4 , 64 ( 1 9 6 6 ) .  5.  L. G. W e r b e l o w a n d A. G. M a r s h a l l , J . Magn. R e s . 1 1 , 299 ( 1 9 7 3 ) .  6.  L. G. W e r b e l o w a n d A. G. M a r s h a l 1 , M o l . P h y s .  7.  J . H. F r e e d a n d G. K. F r a e n k e l ,  (1964).  ,  (1974).  J . Chem. P h y s . 4 0 , 1 8 1 5 ( 1 9 6 4 ) .  TABLE D . l Linewidth contributions  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 a n AMX t h r e e s p i n s y s t e m : P a r t 1.  Linewidth element  Dipolar  - J  3535  R  R  '313  E ( ^  2  1212 " ' (  - "  C  6  4  )  ?  contributions  • 1/2 [ J ^ U 0  / 3  J  > r, J  > >  1 1  n  )  J ^ U  +  {  ) • jl („ ) + ?  •  n  +  4  > >  J  +  J  {  e  B  0 R  2525  R  1313 "  t  l  ''  - n« <V\> J  R  2  6  2  R  6  (  3  )  J  ( n  +  >  n  -  5  ) ]  >  +  J  « < V »  S  >  ]  "  4 W " „VvV  V E  )  t  + 0  e  n  +  +  jl  ? ( U {  , j j ^ , ] +  2 ^ns'W  +  t  {  J  n  1  ^ ) + J j ( . ) • o; (. ) • 0 ?  n  {  5  ^  ) •  ]  - 0 / 3 ) J ° ( » ) + 2J („ .+ „ ) 2  1  4  1  4  5  v  5  5  (  l  {  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),  Greek s u b s c r i p t s  subscripts  refer t ostationary states  density  ?  u  +  VZC J j ( » ) - J  ^Notation:  spectral  ^  ^nc'VV  +  2  5 +  j  2  V"t'  ( c  - J U » ) + l / 2 [ j l ( . ) jj („ ) J ( „ ) j j ^ ) ?  • j  a n d n u c l e u s X (c).  ( s e e Eq. [ D . l ] ) f o r a n u n c o u p l e d r e p r e s e n t a t i o n .  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 b e t w e e n n u c l e i a and g .  Arabic  subscripts  J g( ) k  w  1  S  the  TABLE D.2 Linewidth  Linewidth  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 t e m : P a r t 2.  Dipole-dipole cross-term contributions  element  •(4/3)J°  '1212  (0) +  2J (  J ^ U J  1  (4/3)JnW(0) - J n ^ ( a > )  ^3535  Dipole-shift anisotropy cross-term contributions  Shift anisotropy contributions )  U  -2J  2 ^ )  ?  (8/3)J°  -(8/3)J°(0) + 2jJ(^)  '1313  - 2J  ,1  -(4/3)J° (4/3)J°  '2626 ^Notation: to  c  (0) + J  AO)  + J  )  2J'U  (a> )  2JJ(  U  1  1  c  Same a s f o r p r e c e e d i n g T a b l e w i t h  ) W ?  1  -2J  anisotropy  a  (to)  k  i n t e r a c t i o n , Jv  ct py  a t nucleus a)and the d i p o l a r ( operative  terms, and J g  ) 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  (  u  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 a n d  y-s  .  1  2J  -  Jto  r  (aj  )  ,0  -  (a) ) + 2 J  1  (8/3)^(0) 1  U )  )  (to) i s t h e s p e c t r a l  anisotropy(operative  y ( 5  )  (8/3)^(0)  i s the spectral density at  a t to f o r t h e s h i f t a  (to  1  +  )  the following additions; J  f o r the autocorrelation function of the shift  (0)  (8/3)^(0) -2J  '1414  n C  0  (8/3)J°(0) + 2 J ^ )  '2525  (a) )  1  density  a t n u c l e i e a n d y) c o r r e l a t i o n terms  cross  -237-  FIGURE D J :  The e n e r g y l e v e l d i a g r a m f o r an AMX right-hand  parallelogram  three  shows t h e v a r i o u s  f o r m t h e A , M, a n d X q u a r t e t s .  X-|, X , X^, X^. 2  (scalar  coupling  of t h i s  figure.  The  transitions  which  Scalar coupling  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^, and  s p i n system.  removes t h e  A^, A^; M-j, M^, M^,  The e i g h t p o s s i b l e e n e r g y l e v e l  included)  a r e shown i n t h e l e f t  diagrams  hand p o r t i o n  M^;  PUBLICATIONS A.  Refereed Journals : 1.  L.G. Werbelow, "Homonuclear Overhauser Enhancements as Probes of  2.  L.G. Werbelow and A.G. M a r s h a l l , " A n i s o t r o p i c Reorientation and Non-exponential Nuclear Magnetic R e l a x a t i o n " , Mol. Phys., accepted and i n press.  3.  L.G. Werbelow and A.G. M a r s h a l l , "Nuclear Magnetic Relaxation for Individual T r a n s i t i o n s of an AMX Spectrum; Use of Interference Terms to Determine Signs of Scalar Coupling Constants", Chem. Phys. L e t t . 22, 568-571 (1973).  4.  L.G. Werbelow and A.G. M a r s h a 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. M a r s h a l l , and L.G. Werbelow, "Determination of Molecular C o n f i g u r a t i o n from Lanthanide Induced Proton NMR Chemi c a l S h i f t s " , in Nuclear Magnetic Resonance S h i f t Reagents, Academic Press I n c . , 1973, pg. 313-339.  6.  L.G. Werbelow and A.G. M a r s h a l l , "Internal Rotation and Methyl Proton Magnetic Relaxation f o r Macromolecules", J . Amer. Chem. Soc. 95, 5132-5134 (1973).  7.  I.M. Armitage, L.D. H a l l , A.G. M a r s h a l l , and L.G. Werbelow, "Use of Lanthanide Nuclear Magnetic S h i f t Reagents i n Determination of Molecular C o n f i g u r a t i o n s " , J . Amer. Chem. Soc. 95, 1437-1443 (1973).  8.  M.P. Hanson and L.G. Werbelow, " C o l l i n e a r C o l l i s i o n s of an Atom and  9.  A.G. M a r s h a l l , L.G. Werbelow, and C F . Meares, " E f f e c t of Molecular Shape and F l e x i b i l i t y on Gamma-Ray D i r e c t i o n a l C o r r e l a t i o n s " , J . Chem. Phys. 57_, 364-370 (1972); J . Chem. Phys. 57, 4508 (1972).  Molecular M o b i l i t y " , J . Amer. Chem.  S o c , submitted.  S t r i n g O s c i l l a t o r " , J . Chem. Phys. 58, 3669-3675 (1973).  10. L.G. Werbelow, "Interference Effects in Nuclear Magnetic R e l a x a t i o n " , Adv. Mol. Relax. Processes, in progress.  

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