UBC Theses and Dissertations

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UBC Theses and Dissertations

Macroscopic equations for nuclear spin resonance in density matrix formalism Jog, Shridhar Dattatraya 1960

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MACROSCOPIC EQUATIONS FOR NUCLEAR SPIN RESONANCE IN DENSITY MATRIX FORMALISM  SHRIDHAR DATTATRAYA JOG  A THESIS SUBMITTED I N PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of Physics  We a c c e p t t h i s t h e s i s as conforming  to  s t a n d a r d r e q u i r e d from c a n d i d a t e s f o r  the the  degree of MASTER OF SCIENCE  Members o f the Department of THE UNIVERSITY OF BRITISH COLUMBIA SEPTEMBER  1960  Physics  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 f o r an advanced degree a t the  University  o f B r i t i s h C o l u m b i a , I agree t h a t the L i b r a r y s h a l l make it  freely  a v a i l a b l e f o r r e f e r e n c e and  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  study.  I further  copying of t h i s  be g r a n t e d by the Head o f  Department o r by h i s r e p r e s e n t a t i v e s .  Department o f  be a l l o w e d w i t h o u t my w r i t t e n  P h y £ i <l£  The U n i v e r s i t y of B r i t i s h Columbia, Vancouver $ Canada. f  my  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 g a i n s h a l l not  thesis  financial  permission.  ABSTRACT Methods o f s e t t i n g up g e n e r a l i z e d B l o o h g o v e r n i n g t h e time dependence o f m a c r o s c o p i c  equations  magnetization  f o r a system of n u c l e i o f s p i n I , • i n g i v e n magnetic and e l e c t r i c f i e l d s , have been proposed f o r t h e degenerate case "by B l o o m , Hahn and H e r z o g and "by Lure a t ,  and f o r the n o n -  degenerate case by B l o o m , R o b i n s o n and V o l k o f f . I n t h i s  thesis  an attempt i s made t o g i v e a u n i f i e d d i s c u s s i o n of t h e s e methods "by u t i l i z i n g the d e n s i t y m a t r i x f o r m a l i s m and t o demonstrate  t h e i n t e r r e l a t i o n s h i p "between them. R e l a x a t i o n  e f f e c t s a r e not  considered.  The g e n e r a l t h e o r y i s d e v e l o p e d i n terms of d e n s i t y m a t r i x f o r m a l i s m and i s a p p l i e d t o the  the  non-degenerate  and the degenerate c a s e s . The r e s u l t s a r e d i s c u s s e d and compared w i t h t h o s e of the p r e v i o u s  investigators.  TABLE OF CONTENTS  ACKN O^VLEDGMENT S ABSTRACT CHAPTER I CHAPTER I I  INTRODUCTION - TIME DEPENDENCE OF THE DENSITY MATRIX  CHAPTER I I I CHAPTER  - NON-DEGENERATE CASE  IV - DEGENERATE CASE  REFERENCES  ACKNOWLEDGMENT S  I w i s h t o thank P r o f e s s o r G . M . V o l k o f f f o r s u g g e s t i n g t h e problem and f o r h i s c o n s t a n t h e l p i n a l l t h e phases of development o f t h i s w o r k . I a l s o w i s h t o thank the N a t i o n a l R e s e a r c h f o r providing f i n a n c i a l assistance t h e f o r m of a r e s e a r c h  grant.  Council  t h r o u g h o u t the y e a r i n  1  CHAPTER I  IHTRODUCTIOff  •>"*.•  A n u c l e u s of s p i n I h a v i n g a m a g n e t i c moment ( f o r I $- i ) an e l e c t r i c q u a d r u p o l e moment eQ has 21 + 1  and energy  l e v e l s ( w h i c h i n some s p e c i a l cases become degenerate i n p a i r s ) when p l a c e d i n a c o m b i n a t i o n o f a n o n - u n i f o r m e l e c t r i c  field  ( c h a r a c t e r i z e d b y an e l e c t r i c f i e l d g r a d i e n t t e n s o r w i t h o r w i t h o u t a x i a l symmetry) and a c o n s t a n t u n i f o r m magnetic  field  H . T r a n s i t i o n s between such l e v e l s u s u a l l y f a l l i n t h e r a d i o e  f r e q u e n c y r e g i o n , and can be s t u d i e d e x p e r i m e n t a l l y by subjecting the s i m i l a r l y s i t u a t e d n u c l e i i n a s i n g l e c r y s t a l t o a weak o s c i l l a t i n g m a g n e t i c f i e l d ( u s u a l l y l i n e a r l y p o l a r i z e d a l o n g the a x i s of t h e c o i l p r o d u c i n g i t ) r a d i o frequency  ^~  of v a r i a b l e  , and by o b s e r v i n g as a f u n c t i o n o f ^ t h e  r e s o n a n c e s i n t h e a b s o r p t i o n o r t h e i n d u c t i o n s i g n a l s i n some of the by now c o n v e n t i o n a l t y p e s o f s p e c t r o m e t e r s . A l t e r n a t i v e l y , f r e e p r e c e s s i o n o r s p i n - e c h o t e c h n i q u e s can be u t i l i z e d . The o b s e r v e d s i g n a l s t r e n g t h i n a l l such experiments i s p r o p o r t i o n a l t o the t i m e d e r i v a t i v e of the component a l o n g t h e a x i s o f a r e c e i v e r c o i l of the m a c r o s c o p i o m a g n e t i z a t i o n o f t h e sample. Methods o f s e t t i n g up g e n e r a l i z e d B l o c h e q u a t i o n s g o v e r n i n g t h e t i m e dependence of t h i s m a c r o s c o p i c m a g n e t i z a t i o n have been proposed f o r d i f f e r e n t cases i n w h i c h r e l a x a t i o n e f f e c t s a r e n e g l e c t e d by B l o o m , Hahn and H e r z o g t ) , 1  by L u r g a t ^ ) and by B l o o m , R o b i n s o n and V o l k o f f ( K 4  2  The object  2  of t h e p r e s e n t t h e s i s i s t o g i v e " a u n i f i e d d i s c u s s i o n of t h e s e methods by u t i l i z i n g 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 (of.,  f o r example, F a n o ^ ) 3  j and t o demonstrate  the  i n t e r r e l a t i o n s h i p between them. An assembly o f "8 m u t u a l l y n o n - i n t e r a c t i n g n u c l e i of s p i n I  s u b j e c t e d t o i d e n t i c a l e l e c t r i c and magnetic  f i e l d s can be d e s c r i b e d by a H e r m i t i a n d e n s i t y m a t r i x p of rows and columns c o n t a i n i n g (21+ 1) e x p e c t a t i o n v a l u e o f any o p e r a t o r <  A  >  T/fA/o) = ^  =  ;  Taking  A  leaving  r e a l parameters. A  21+i  The  i s g i v e n by  (o n in  i n n  t o be t h e i d e n t i t y o p e r a t o r we have  (?) = 4 / L  41 ( I - h l ) independent r e a l parameters t o d e s c r i b e  =  1  »  the  system.. T h i s means t h a t 41 (I-+-1) independent p h y s i c a l q u a n t i t i e s a r e needed t o d e s c r i b e c o m p l e t e l y t h e m a c r o s c o p i c behaviour of t h e s y s t e m . F o r such m a c r o s c o p i c q u a n t i t i e s i t . i s c  i n some c a s e s c o n v e n i e n t t o u t i l i z e t h e t h r e e components of ;  t h e m a c r o s c o p i c m a g n e t i z a t i o n , the f i v e components of t h e e l e c t r i c q u a d r u p o l e moment, d e n s i t y , the seven components o f t h e m a g n e t i c o c t o p o l e moment d e n s i t y , e t c . , i n t e n s o r i a l form  \Jhi t r a n s f o r m i n g under c o - o r d i n a t e r o t a t i o n s  l i k e the s p h e r i c a l harmonics r u n n i n g up t o - k to k .  each set .expressed  \  with, k  b e i n g an i n t e g e r  2 1 , and, q . r u n n i n g i n i n t e g r a l s t e p s from  Thus  r  f o r n u c l e i of I - \  the magnetization d e s c r i b e . t h e  the t h r e e components o f  system c o m p l e t e l y , f o r  1=1  the  f i v e components of the q u a d r u p o l e moment d e n s i t y a r e i n g e n e r a l needed i n , a d d i t i o n t o g i v e a complete d e s c r i p t i o n i n terms o f  3  eight q u a n t i t i e s , while f o r  1=3/2  t h e seven  octopole  moment d e n s i t y components a r e a l s o needed t o p r o v i d e 15 quantities,  etc.  Equations  (1)  e s t a b l i s h the  connection  between t h e d e n s i t y m a t r i x elements and t h e s e m a c r o s c o p i c q u a n t i t i e s i f we t a k e of t h e  appropriate  A 4 1  t o be i n t u r n e q u a l t o each one (1 + 1)  m u l t i p o l e moment  operators.  S i n c e the t i m e dependence of t h e d e n s i t y m a t r i x when r e l a x a t i o n p r o c e s s e s a r e n e g l e c t e d i s governed by the equation  where  ,  ^ C u j i s the t o t a l Hamiltonian f o r a s i n g l e nucleus  a c t e d upon by t h e g i v e n magnetic and e l e c t r i c f i e l d s * t h e t i m e dependence o f t h e  4 I  (r+'l)  macroscopic p h y s i c a l quantities  d e s c r i b i n g t h e system .can i n t h i s , cage be e x p r e s s e d i n the form of  4 1  ( I + 1)  simultaneous f i r s t order d i f f e r e n t i a l .  e q u a t i o n s o b t a i n e d by combining - F o r .1 = \  (1)  . and , (2)  ;  .  t h e complete macitpscopic b e h a v i o u r o f  t h e system i s d e s c r i b e d by, t h r e e e q u a t i o n s in>terms of  the  t h r e e m a g n e t i z a t i o n components  or-  alternatively difference  n  M  5  .'^L ,  M  3  ,  M - . s  ,  s  can bes e l i m i n a t e d by i n t r o d u c i n g t h e  between t h e . f r a c t i o n a l p o p u l a t i o n s of t h e  states, • .  m-te-  . Lurg a t d i s c u s s e s t h e s p e c i a l pure q u a d r u p o l e case w i t h  an a x i a l l y symmetric f i e l d g r a d i e n t and f i r s t . o b t a i n s f o r n u c l e i with.  Iv.= l  r  3/2  and, 5/2  c o m p l i c a t e d s e t s of  simultaneous equations r e s p e c t i v e l y .  8,  15  and  He t h e n shows how under  35  r e s o n a n c e c o n d i t i o n s • e a c h of t h e s e s e t s reduces t o t h r e e a p p r o x i m a t e - e q u a t i o n s analogous t o t h e t h r e e  equations  d e t e r m i n i n g the m a g n e t i z a t i o n i n t h e case o f  I -, £  .  B l o o m , Hahn and H e r z o g and a l s o B l o o m , Robinson and V o l k o f f a r r i v e more d i r e c t l y at s e t s of t h r e e approximate e q u a t i o n s f o r n u c l e i of a r b i t r a r y s p i n I  under  certain  specified conditions.'' In t h i s t h e s i s the  A I  (I + 1 )  we s t a r t w i t h e q u a t i o n s v  independent parameters  (2)  for  of the d e n s i t y m a t r i x  c o r r e s p o n d i n g t o a. f a i r l y g e n e r a l H a m i l t o n i a n and examine method o f r e d u c i n g t h i s system of e q u a t i o n s under c o n d i t i o n s t o t h r e e approximate e q u a t i o n s w h i c h a r e compared t o t h e r e s u l t s o f the p r e v i o u s  the  resonance then  investigations.  5  CHAPTER •  II  TIME DEPENDENCE OF THE DENSITY MATRIX The H a m i l t o n i a n  -^/  f o r a nucleus of spin  tota  interacting with arbitrary static  I  e l e c t r i c and magnetic f i e l d s  and a weak o s c i l l a t i n g m a g n e t i c f i e l d can be s p l i t i n t o two part's  '£f  a  and 'J-j' where  -f~f  does n o t and Ti  a  does depend  on t h e t i m e , so t h a t  where, f o r example, ( c £ . B l o o m , , R o b i n s o n and V o l k o f f )  and  In  (4) x, y z 7  a r e t h e p r i n c i p a l axes o f t h e e l e c t r i c  g r a d i e n t t e n s o r whose component a l o n g t h e  z - axis  (usually  chosen t o be t h e one o f g r e a t e s t a b s o l u t e v a l u e ) i s whose asymmetry parameter i s  rj -  dependent p a r t o f t h e H a m i l t o n i a n a weak a p p l i e d  r . f .  field  field  &  , and  {$> -4 )/<$ . The time xx  (5)  r7  i s assumed t o be due t o  \T(*) o f f r e q u e n c y  u?  which  i s usually linearly polarized. The e i g e n v a l u e problem a  x  =  F  k  %  can be s o l v e d ( n u m e r i c a l l y i f n e c e s s a r y )  {  and l e a d s t o  i  )  2 I +- 1  eigenvalues =  ?  °-i  where the I  E ^  C  and t i m e - i n d e p e n d e n t >  k , l  =.  1,  x  ( A-£  i n Spodumene)  eigenfunetions  2., • - - (ZI. + 1) ,  a r e e i g e n f u n e t i o n s of'  u"  =5/2  k  Ah example f o r  1^' V  i s d i s c u s s e d by B l o o m , •  R o b i n s o n .and V o l k o f f ; "  /. ;  t  then i n the  =  j k  f.  (Ej-E )/-H k  =. - «o .  (7)  kj  r e p r e s e n t a t i o n equations  (2 I +• 1 )  '.  '^jk/iir  : I f we d e f i n e t h e - t r a n s i t i o n f r e q u e n c i e s ' ,  i  w  the  •  (2)  - (7)  lead to  equations  i s t h e m a t r i x element of t h e t i m e dependent p e r t u r b i n g operator ' •' •  and  £f  ' .  7*  l  jk  o f e q u a t i o n (5) between t h e s t a t e s '  *  •  •"  V-  and  <  Y  k  ,  • '  the.  i s t h e t i m e - i n d e p e n d e n t m a t r i x element o f / s p i n  operator I  between t h e s e s t a t e s w h i c h can be e x p l i c i t l y  e v a l u a t e d i n terms o f t h e known  c  tk  in  I n the absence o f an a p p l i e d e q u a t i o n s (8) have the s o l u t i o n s  f. = ViL u, c . ik  r. f.  field  J-f-o,  and  /% where t h e (f ) jk  =  (f*\  ft)  kt  . are constants determined by t h e i n i t i a l  0  C o n d i t i o n s , so t h a t ; f r'equenei es  c-"'  /o-, o s c i l l a t e h a r m o n i c a l l y w i t h t h e  ^JV/ZTT: ,  We a r e i n t e r e s t e d i n i n v e s t i g a t i n g - t h e b e h a v i o u r of t h e system o f n u c l e i when a weak p e r t u r b i n g magnetic of f r e q u e n c y  i s a p p l i e d t o t h e system w i t h  c l o s e t o a p a r t i c u l a r one o f t h e s y s t e m , say w = u ; . As  0  to-  k  OJ v e r y  characterizing the  The d i s c u s s i o n w i l l d i f f e r somewhat  depending on whether a l l t h e ' c~>j< speak o f t h i s as a non-degenerate  aire d i f f e r e n t system)  ,  speak o f a system h a v i n g two o r more degenerate  field  ;  (we s h a l l  or not  (we s h a l l  t h e same as  even when no energy l e v e l s a r e d e g e n e r a t e ) . The  p e r t u r b i n g e f f e c t o f t h e time dependent  Ji  can be  conveniently discussed by i n t r o d u c i n g a transformation to replace the ^  "^V the p o s s i b l y a l s o time-dependent  formed q u a n t i t i e s  f>* r e l a t e d t o ^  s e l e c t e d constant frequencies  "." •  ;  v .;  J  K  u>  /k/ .- ^ 2  rr  ..  trans-  and t o some a r b i t r a r i l y by  ..-  .......  .. -  .  where, a s we s h a l l p r e s e n t l y s e e , i t w i l l b e convenient, t o choose the" a r b i t r a r y q u a n t i t i e s  ^J'K t o be e i t h e r e x a c t l y o r  very n e a r l y equal to the corresponding the system.  I n terms o f  p*  ^Pjk . c h a r a c t e r i z i n g  equations  (8)  become  8  '  :  '  (11)  — c•  2_  The system o f (8)  or p*  (ll)  (2 1 +  1)  simultaneous equations  g i v e s t h e exact t i m e dependence o f  f o r "both degenerate and non-degenerate  (neglecting a l l relaxation effects) up and t o i n t e g r a t e e x a o t l y f o r  I  » >  or  P  systems  hut i s t e d i o u s to £ .  set  We t h e r e f o r e use an  a p p r o x i m a t e method o f s o l v i n g systems o f d i f f e r e n t i a l e q u a t i o n s d e s c r i b e d by B o g o l i u b o v and M i t r o p o l ' s k i i by L u r g a t ( ) 5  (as quoted  ) .  C o n s i d e r a system of s i m u l t a n e o u s d i f f e r e n t i a l equations  where, £  £  I s '-a s m a l l p a r a m e t e r , and  X V a r e o f t h e form  i'vt  V the  v's  b e i n g any f i x e d f r e q u e n c i e s .  Then a c c o r d i n g t o  B o g o l i u b o v and M i t r o p o l ' s k i i t h e s o l u t i o n of  (12)  i s given  9  t o a f i r s t a p p r o x i m a t i o n by t h e s o l u t i o n of  =  where  X u  £  X ,  (t.x,,..*,,  (14)  denotes t h e t i m e average o f  We assume  i f (t)  X  i n equations  .  k  (5)  and  (8)  t o be  a weak t i m e dependent magnetic f i e l d i n v o l v i n g a s i n g l e frequency  ^Vi-rr i n such a way as t o make the t i m e average  H*(t) e'^  and  t h e t i m e average of  other we  equal to a small constant  i o ' ^ to .  H (t)  of  vector  e  e q u a l t o zero f o r any  F o r example, f o r a l i n e a r l y p o l a r i z e d f i e l d  have:  FT (*) •=  H  f  Equations  co5 u)t  (11)  • v  =  1  where the  can be b r o u g h t i n t o t h e form  (u)  H*'  J~f . a r e g i v e n by  (16)  jk  (12),  since  f h * i u { i s assumed t o  be s m a l l , and we can choose t h e a r b i t r a r y t r a n s f o r m a t i o n frequencies equations  u)^ (11)  i n equation  (10)  t o be e i t h e r z e r o o r s m a l l .  we  assume t h a t the t i m e average o f  is  zero i f  v  t o make  a*  1  is  ^ - u ^ I n the  1  d i f f e r s a p p r e c i a b l y from z e r o .  l a t t e r r e s u l t i t i s necessary f o r  v  if  in  following ti>-0  t  and  To o b t a i n the  t o be l a r g e i n  comparison w i t h t h e r e c i p r o c a l o f the t i m e over w h i c h t h e t i m e average i s t a k e n .  At t h i s ' p o i n t i t i s convenient non-degenerate case  (all  ^  to discuss f i r s t  different),  and then  r e t u r n t o the degenerate case when two or more o f the coincide.  to to-^  the  CHAPTER  III  3ST0N - DEGENERATE  CASE  I n t h i s c h a p t e r f i r s t we s h a l l t r e a t t h e n o n degenerate case  (all  io.^s different)  t h e o r y d e v e l o p e d i n Chapter I I .  on t h e b a s i s of  the  We s h a l l t h e n compare  the  r e s u l t s w i t h t h o s e of B l o o m , R o b i n s o n and V o l k o f f , who a l s o have c o n s i d e r e d t h e n o n - d e g e n e r a t e c a s e . If  to  » t h e n we can choose a l l the  u) - UJ > o for to  As  i s v e r y c l o s e t o some one p a r t i c u l a r  u)^ , whieh i s chosen e q u a l t o UJ  i  O  n  v a l u e of  exoept  co , r a t h e r t h a n to  a  =  u).[ = o.,  order to a v o i d ambiguity i n e v a l u a t i n g the ^  *  avers  E q u a t i o n s (11) t h e n become on  m  t a k i n g t h e t i m e averages of the r i g h t hand  sidess  f o r n o n - d i a g o n a l terms ( j ^ M ( P* )JUL * J ' !  o  unless  j = r  and  k = s  simultaneously  k  f o r d i a g o n a l terms cl_ {p*-)  - o  (j = unless  k) j = r  or  j = 8  /  12  where  )  o  denotes the l e a d i n g time-Independent  t h e e x p a n s i o n of-  r> *  i n powers o f  IV*I  and  term i n  (u>-uj )  .  0  )  c  These e q u a t i o n s a r e i n t h e f o r m of e q u a t i o n s  (14)  w i t h t h e r i g h t hand s i d e s c o m p l e t e l y independent o f the  time.  To t h e f i r s t a p p r o x i m a t i o n t h e s o l u t i o n s of t h e s e  equations  a r e the same as the s o l u t i o n s of any o t h e r set o f  equations  whose r i g h t hand s i d e s reduce t o respect to t i m e .  (17)  on b e i n g averaged w i t h  I n p a r t i c u l a r equations  (17)  by a s i m i l a r set w i t h t h e s u b s c r i p t zero l e f t the time a v e r a g i n g l e f t  off  H (t)  c o n v e n i e n t t o c a r r y out t h i s  e "^  may be  o f f the  , and i t  +  replaced /°*~ and  is  replacement.  T r a n s f o r m i n g back t o t h e o r i g i n a l d e n s i t y m a t r i x elements w i t h t h e a i d o f e q u a t i o n s (10) f o r n o n - d i a g o n a l terms ^- p  - - l to-, p  f o r d i a g o n a l terms ?L f  -it.'jj  -o  unless  (j A k)  unless  j = r  and  (j= k) j = r  we t h e n o b t a i n :  or  j  = s  k -  s  y  -simultaneously  A l t h o u g h no r e l a x a t i o n mechanisms have been i n c o r p o r a t e d i n t o t h e above f o r m a l i s m such mechanisms do i n f a c t e x i s t i n an a c t u a l system,' and the i n i t i a l  state before  t h e r a d i o - f r e q u e n c y f i e l d i s a p p l i e d w i l l be one of t h e r m a l e q u i l i b r i u m which i s described b y :  "  -  j  The e x p e c t a t i o n v a l u e of any o p e r a t o r  A  f o r a system i n  t h e r m a l e q u i l i b r i u m i s t h u s g i v e n by  Equations initial  conditions  (17) (19)  approximation a l l the j ^ k ,  ( o r (18))  together w i t h the  i m p l y t h a t t o our degree  of  (and c o n s e q u e n t l y a l l the  P.  r e m a i n z e r o except •  ,  p Us  remain c o n s t a n t i n t i m e except  and a l l the •  ^  and ^  c o n s t a n t b u t whose d i f f e r e n c e v a r i e s .  P*  ),  P (=p*\ 'jj ^ /jj )  whose sum remains  I f we denote  this  d i f f e r e n c e by "  ~  /1a  ~ f  ss  -  UA  ~ f* s  -  « *"  (2.1)  t h e t i m e v a r y i n g p r o p e r t i e s of the system w h i c h determine t h e d e v i a t i o n of  <A >  from  <A >,  w i l l be g i v e n by  (cf.  modified, equation  (17)  ).  {2 2)  o r e q u i v a l e n t l y by A. ( r  (cf.  ) = -i"of  A s  equation  + c/n  Ks  (18)  )  Hct; (23)  -  f • h a; - f  We n o t e t h a t  (22)  equations since ' p  o r (23)  each r e p r e s e n t s  and  P * a r e complex.  ^ where  X  and  equations  Y  (23}  parameters  dt  f -HCtj)  =  x  a set of  Thus i f we w r i t e  + iY  are r e a l parameters,  (^) t h e n we can r e w r i t e  e x p l i c i t l y i n terms o f the t h r e e  X ,  Y  and  Y  -  nV  three  real  n: Im  H(^)  . <=/t  1:  =  I » ( f v H W ) - Y f e ( C - Hftj)J  (  25r  )  The d e v i a t i o n of the e x p e c t a t i o n v a l u e o f any operator value <A>  <A>  <A>^ -<A>^  g i v e n by given by =  =  from i t s t h e r m a l e q u i l i b r i u m  (1)  i s equal to  (20)  A  A  aK  l  x  +  i  Y ) A  ,  J  ^  ( x - Y ) A  +  J  T  A  + J-  Re ( A J A  +  I -  (A* ) s  (  2  6  *t(A,;A-A„)(»-».h)-  Thus the t i m e dependence o f a l l the physical quantities  4 1  (I + l ) macroscopic  d e s c r i b i n g t h e system i s e x p r e s s e d i n  terms of the t h r e e v a r i a b l e s be e x p r e s s e d i n terms o f ,  X ,  Y ,  n  f o r example,  w h i c h can i n t u r n the three  components  of the macroscopic m a g n e t i z a t i o n . To r e l a t e t h e s e r e s u l t s t o t h o s e o f B l o o m , Robinson and V o l k o f f we c o n s i d e r t h e i r H a m i l t o n i a n f o r w h i c h  •j. /As  with  P ,  S ,  T  real.  Then we have  (17)  )  <r.>-<x,\ = _ p i ^ h  v  ^ _  +  (  r  o  j _ {ri  ,  n r k  </>-<>>^= V5 * i [(r )^-Ci )J -„„) T  T  2  ?  x  (n  I f we i n t r o d u c e t h e v a r i a b l e s  f.  3  < r . > - < ^ >  H  , I  s 3  <*,>  -<r >„ y  and n e g l e c t i n g t h e terms c o n t a i n i n g t h e d i a g o n a l m a t r i x elements i n x  and  (27)  (28) -  into  substitute i *  A  (25)  p  ,  y  -  i  y  A  we o b t a i n e q u a t i o n s  R o b i n s o n and V o l k o f f ^ ) . 2  5  [13]  of Bloom,  CHAPTER  IV  DEGENERATE  CASE  I n t h e degenerate case where two o r more t r a n s i t i o n frequencies  may be e q u a l a c o r r e s p o n d i n g l y g r e a t e r number o f  m a t r i x elements  may v a r y i n t i m e  P J  ( n o t j u s t 4 as i n t h e  k  non-degenerate c a s e ) . complicated,  The g e n e r a l f o r m u l a t i o n becomes too  and hence we d i s c u s s b e l o w some s p e c i a l  We w i l l f i r s t  o u t l i n e t h e method d e v e l o p e d by L u r e a t  f o r t h e degenerate case i n g e n e r a l .  Then we s h a l l  investigate  by t h e method of p r e v i o u s chapters? t h e s p e c i a l cases b y h i m , and s h a l l compare and d i s c u s s t h e  1 .  Lure a t ' a  cases.  treated  results.  Method  We s h a l l o u t l i n e i n t h i s s e c t i o n L u r g a t ' s method in  general. The S c h r b d i n g e r ' s  physical quantity  ^  A  t i m e dependent  equation f o r a  can be w r i t t e n i n t h e form  =  f  < [ H , A ] >  M  For a nucleus of s p i n I ,  p l a c e d i n an e l e c t r i c  of a x i a l symmetry ( ^ o )  and i n a magnetic f i e l d  Hamiltonian  is  field  gradient the  total  4 r ( i i - o  where  ^-  z  . (.31)  i g the v a l u e of the e l e c t r i c  Let commutation  he t e n s o r o p e r a t o r s  field  gradient.  d e f i n e d by the  relations  w i t h the n o r m a l i z a t i o n c o n d i t i o n  As a r e s u l t  of t h e s e d e f i n i t i o n s we get I  X.  fe  ^  ,  5  K  -  Y „  f  r  ±  (  = t r  ,  r  +  o  £  ]  ^  o<  Turn  Over  *  Y  =  -  (r  =  1  (3^)  3 e Q 2. i  where  D  ( i r - i j  a r e the components of t h e q u a d r u p o l e moment  t k  operators.  Talcing the average v a l u e s o f  Y  k M  we get  the  f o l l o w i n g formulae f o r t h e components of t h e m a c r o s c o p i c magnetization tensor  M  and of t h e m a c r o s c o p i c q u a d r u p o l e moment  Q, : Nyh  < Yl6 > -  <  Y ,  <  Y  t  l  i o  M  >  -  >  = / I  v  + (M, t ^ j )  with s i m i l a r expressions f o r  Q  Y  3>JL  J J r  '  etc..  (35")  i n terms of  octopole  and h i g h e r moments. Making use o f the p r o p e r t i e s of t e n s o r o p e r a t o r s can he shown t h a t  it  Substituting  (31)  in  (30)  and making use of e q u a t i o n  (36)  we get  2-k + /  V  - ^  ^  (k+1)  <  1)  Y  ^  \  I k+  /  X  (37)  We n o t e a t t h i s p o i n t t h a t t h i s e q u a t i o n i s an e x a c t e q u a t i o n where a l l the energy l e v e l s a r e t a k e n i n t o a c c o u n t ,  and h o l d s  f o r i n t e g r a l as w e l l as f o r h a l f - i n t e g r a l v a l u e s o f  spin.  Using equations s p e c i a l cases  (37) 1=1,  and  (35)  3/2  and  Lurgat then obtains f o r 5/2  the  equation^ g i v i n g the time  dependence of t h e m a c r o s c o p i c p h y s i c a l q u a n t i t i e s  M,  Q  etc.  I n o r d e r t o see the b a s i c e q u i v a l e n c e o f L u r g a t s 1  method w i t h the one p r e s e n t e d i n t h i s t h e s i s we r e c a l l  that  t h e mean v a l u e of any o p e r a t o r can be e x p r e s s e d i n terms o f the d e n s i t y m a t r i x of t h e o p e r a t o r s Q«  »  matrix  Q«  (cf. Y _  /o .  Thus the mean v a l u e s M •,  M  x  y  ,  ,  can be w r i t t e n down i n terms of the d e n s i t y  To get t h e v a r i a t i o n i n t i m e of the m a c r o s c o p i c  physical quantities dependence o f  ).  and a l s o t h o s e of  kft  etc.,  y  equation ( l )  M , "Q  etc.,  one may u t i l i z e the t i m e  , as L u r g a t has done, o r one may use  <Y ^> h  t h e t i m e dependence of t h e d e n s i t y m a t r i x d i r e c t l y , as i s done i n the p r e s e n t  2 .  thesis.  Case of 3T>in  1 = 1:  We s h a l l f i r s t  s t a t e the r e s u l t s o f L u r g a t f o r  this  case and t h e n s h a l l show how t h e same e q u a t i o n s can be o b t a i n e d by the method of p r e v i o u s Using equations  (37)  chapters.  and d e f i n i t i o n s  (35)  Lurgat  o b t a i n s the f o l l o w i n g e q u a t i o n s f o r the t i m e dependence o f components o f the m a c r o s c o p i c p h y s i c a l q u a n t i t i e s  ~M, Q :  dt  5i  ( 4  Qy») =  £ ( ^ C « „ - « „ > ) =  ^  ( *4  -  J I u v n  O v t v  the  22  H  JLM C38)  7  IE'  IFote t h a t the e i g h t e q u a t i o n s b r e a i ^ f i h t o t h r e e sub s y s t e m s , I  t  II  and  III ,  t h a t i s v a r i a b l e s i n one subsystem do not  o c c u r i n any o t h e r subsystem.  I f i n i t i a l l y the system i s i n  thermal e q u i l i b r i u m , then the i n i t i a l c o n d i t i o n s are. ; Q«Co; • ^ Qj  k  -  Q v Co) = ' • Co) - o (t^ ) k  - ± Q (»;•' ' • M , - (o) - o  f  Consequently t h e q u a n t i t i e s a p p e a r i n g i n e q u a t i o n s and<h(38 I I I ) time. namely  :  >v  V  1 ,  /• • (3.1)  (38 I I ) are constant  are zero i n i t i a l l y and r e m a i n aero a t any l a t e r  Thus we are l e f t w i t h o n l y t h r e e u s e f u l e q u a t i o n s , (38 I ) .  transformation  It  i s c o n v e n i e n t t o i n t r o d u c e the  n *  =  <_L Q  =  _  cos . t  S'.r, oOt  ^  +  a  *  I±  s i n  ft*  Co S  oot  Lungat c o n s i d e r s t h e case where t h e r e i s no s t a t i c f i e l d and where the t i m e v a r y i n g m a g n e t i c f i e l d H , = H, c«s wt Applying transformation terms i n  s i n z^t  and  ,  (40)  H  y  - o  to  cos i « t  ,  (38 I )  magnetic  is  = o and n e g l e c t i n g  the transformed equations  are :  where  u>„  i s t h e resonance  frequency.,  As p o i n t e d out by L u r g a t t h e s e e q u a t i o n s become formally  i d e n t i c a l with equations  and H e r z o g ^ ) , 1  of  H  j (  t23~]  o f B l o o m , Hahn  except f o r a f a c t o r of 2 i n the  i n the second e q u a t i o n , i f one makes the  coefficient following  transformation : Bloom ,  Of course  Hahn  and  Herzog  Lure a t  the d i s c r e p a n c y of t h e f a c t o r  surprising.  of 2  i s not  There i s no r e a s o n t o expect the two s e t s  t o be i d e n t i c a l  (even f o r m a l l y )  s i n c e the B l o o m , Hahn and  H e r z o g e q u a t i o n s h o l d o n l y f o r h a l f i n t e g r a l s p i n s where i t was p o s s i b l e t o c o n s i d e r o n l y h a l f of t h e t o t a l number of n u c l e i corresponding to done f o r t h e p r e s e n t  +m  states.  case of s p i n  T h i s can not be  I - 1,  s i n c e the  s t a t e can not be c l a s s i f i e d as b e l o n g i n g t o  +m  m = 0 or  state^to  - m state. However, i f we s t a r t w i t h the complete system o f n u c l e i as i n c h a p t e r  II  and then f o l l o w a procedure s i m i l a r  t o the one f o l l o w e d i n c h a p t e r  III  (non-degenerate case) ,  t h e n we a r e l e d e x a c t l y t o t h e L u r g a t ' s  equations,  as  is  shown b e l o w . F o r the degenerate case t h r e e energy l e v e l s ,  L e t the f r e q u e n c y  0 0  say  1  and  0 ;  H -oj o  two o f  2 ., w i l l c o i n c i d e so  of t h e - a p p l i e d  r. f.  . f i e l d be  the that  close  t o the t r a n s i t i o n f r e q u e n c y  w_ - oo, = u j . i e choose the 3  3  e  a r b i t r a r y transformation frequencies Then e q u a t i o n s  and  ^  lead,  (11)  to with  approximation discussed i n the previous chapters, f o l l o w i n g e q u a t i o n s f o r the t r a n s f o r m e d m a t r i x  - *  J  • .  .  .  *  t  2_  i ^ ^ *  - "  . T-T  to  ( p * /  ^ i i * ' fx*  1  |  3  * * I  2  3  *• '  — lC.  ,  / 12.  "being r e a l , g i v e  '  1 3  '2-1  '  3  t* • ~h.  b e i n g complex  ' 12-  the  elements  Note t h a t t h e e q u a t i o n s i n v o l v e e i g h t independent parameters  the  v  2-  A  satisfy  —O'U  real  g i v e s i x and '  two).  I n t h e s p e c i a l case d i s c u s s e d by L u r g a t  26  ( H (t) x  •^.as  H  =• H," cos'u)f * *, a s ' '  , .  m a t r i x elements o f in  (43). It  ,,'H =  = o )  y  i s e q u a l t'o  I*  ^  we get  %s  - HJ =  for a l l non-vanishing  w h i c h - a r e i n f a c t the ones "that  appear  i s now p o s s i b l e t o choose a complete set o f 8  l i n e a r l y independent parameters  (which are combinations of  t h e m a t r i x elements  P ) J*  such t h a t o n l y  three  parameters  a r e t i m e dependent.  The c o r r e s p o n d i n g e q u a t i o n s f o r t h e s e  parameters a r e then enough t o d e s c r i b e the b e h a v i o u r of  the  system. Thus we d e f i n e  where u)  i4  X, Y, Z  - 1-0,3=  and  Moreover l e t  f o r convenience of n o t a t i o n .  equations f o r (43) )  are r e a l parameters.  Q*  ( o b t a i n e d by a d d i n g  Z*" ( o b t a i n e d from  (iii),  are s  = 4 £  (a'-^'in,  (i)  (cf.&2-))  Then t h e and  (ii)  of  ( i v ) , (v) o f ( 4 3 ) )  Gr w r i t i n g i n terms of  =.  Jt  -  ( iO -  Jt  X ,  Y ,  0  z_  Si  1=1,  these equations g i v e :  uJ )„  Applying equations case o f  Z  (l)  and  (35)  t o the p r e s e n t  a s t r a i g h t f o r w a r d c a l c u l a t i o n leads to the  f o l l o w i n g e x p r e s s i o n s f o r the components o f the m a c r o s c o p i c physical quantities  If,  Q  i n terms o f t h e d e n s i t y m a t r i x ;  (46)  Using  (44)  and  z  (46)  we get  ^  gg, - Q ;  -  y  N <<  Substituting  (47)  in  (45)  we f i n a l l y  get  These a r e e x a c t l y t h e e q u a t i o n s  (41)  which are  obtained by L u r g a t .  The apparent d i f f e r e n c e i n t h e s i g n s o f  t h e terms i n v o l v i n g  H  1  i s due t o t h e f a c t t h a t whereas  we t a k e t h e time-dependent H a m i l t o n i a n + v  T-^*(P)  (of.  — v Is I • Vt{1r) .  e q u a t i o n (5)  ) ,  J-f  t o be  L u r g a t t a k e s i t t o be  We have t h u s i l l u s t r a t e d , f o r t h e case o f 1=1,  the complete e q u i v a l e n c e of our t r e a t m e n t w i t h t h a t  of Lurgat,  3 .  1  The case o f s p i n  I = 3/2 :  We now c o n s i d e r t h e case of of h a l f i n t e g r a l s p i n s . following  I = 3/2  as an example  L u r g a t o b t a i n s f o r t h i s case the  equations  where u)„ i s the resonance f r e q u e n c y and <^> of  spin  r . f.  the f r e q u e n c y  field. We s h a l l assume w i t h L u r g a t t h a t the e l e c t r i c  g r a d i e n t i s a x i a l l y symmetric  (.*]-<>) and t h a t the  magnetic f i e l d  1, 2, 3,  ( ™ = - 3 /  t  )  = " SsJ  H 0  o.  . 4  1  ,  and  ( . w  2.  3  states corresponding to respectively.  Let  Then  m = 3/2 ,  ,  =  static  4 be the f o u r  3/^-;  5<J  —£  field  and  -3/2  (So) 4 2  Then e q u a t i o n s  (11)  i n previous chapters,  l e a d , w i t h the approximation discussed to the  t r a n s f o r m e d m a t r i x elements  f o l l o w i n g equations f o r P'  k  =  i , 2 - , 3 , ^  y  i  H,  >-  III  the  Bote that 1 ,  II  (51)  and  I s made up of t h r e e s e p a r a t e s u b s e t s  III  ( t h a t i s , v a r i a b l e s a p p e a r i n g i n one  subset do not o c c u r i n another Initially we have seen  (cf.  subset).  when the system i s i n t h e r m a l e q u i l i b r i u m e q u a t i o n s (19)  )  d e n s i t y m a t r i x elements a r e z e r o .  t h a t the From  off-diagonal  (51 I I I )  we then  see t h a t a l l the d e n s i t y m a t r i x elements a p p e a r i n g t h e r e i n remain z e r o a t any l a t e r t i m e and p l a y no p a r t i n the o f the system i n t i m e . (51 I I ) 2  Moreover  equations  are f o r m a l l y i d e n t i c a l .  everywhere i n  we get e q u a t i o n the s t a t e s  1  (51 I ) (51 I I )  and  2  and  (Replacing indices  by i n d i c e s ).  (51 I )  4  and  3  change  1  and  respectively  That i s t o s a y , the n u c l e i i n  behave i n e x a c t l y t h e same way as  would the n u c l e i i n the s t a t e s  4  and  3  and t h e r e f o r e  it  i s s u f f i c i e n t t o c o n s i d e r o n l y h a l f of the t o t a l number of nuclei,  namely  matter of f a c t  t h o s e b e l o n g i n g t o the + m  states.  As a  B l o o m , Hahn and H e r z o g use t h i s concept  right  a t the s t a r t of t h e i r development of t h e t h e o r y f o r the  half-  integral spins; i n the parameters  equations  (51 I ) ,  ^ (/^*) > ^ e  m  i f w r i t t e n down e x p l i c i t l y  (/J*J  a n <  become i d e n t i c a l f o r m a l l y w i t h e q u a t i o n  *  " (=  [23]  )  of B l o o m , Hahn  and H e r z o g ^ ) ' . 2  (We choose f o r t h i s purpose the  r. f.  f i e l d t o be  l i n e a r l y p o l a r i z e d i n the x - y p l a n e , which i s equivalent two o p p o s i t e c i r c u l a r l y p o l a r i z e d  r. f.  fields.  One of  to the  two components g e t s e l i m i n a t e d i n t h e p r o c e s s of a p p r o x i m a t i o n .  B l o o m , Hahn and H e r z o g t a k e c i r c u l a r l y p o l a r i z e d  r. f.  t h a t no a p p r o x i m a t i o n i s - n e c e s s a r y 4 n t h i s r e s p e c t transformation.)  ,  i n their  -  We -have t h u s demonstrated t h a t out o f the equations (51 I )  (51)  so  . o n l y three are u s e f u l , ,  namely  sixteen...  equations  w h i c h govern the b e h a v i o u r of the system o f n u c l e i . We s h a l l now show,- as i n t h e p r e v i o u s s e c t i o n ,  that i f s u i t a b l e l i n e a r combinations of ,  i  . . . . . then-equations Lurgat.  (51)  Thus l e t  \\  where  X  ,  Y  jk  -  Equations  =  equations f o r  .... . -K U  Jt'  of  v  ' 4 4  T  and  "  Z  —H\  V  3'i  • ~  .  are r e a l p a r a m e t e r s .  cob w t  and and  _  (49)  • -  . ' / a x  (51 I )  chosen,  .  l e a d to the three equations  case which we" a r e d i s c u s s i n g , we have ' H * (t)  . are  2  (52  II)  Z* :  a x  ;  Or e q u i v a l e n t l y , i n terms o f  *  i i L  " ' "~  * *H ' = ;  For Lurgat  n\  o  ~"  ^  l e a d to the f o l l o w i n g - • * • ;  {•:;*..•«'•)  2-  X * .,, Y * -and- Z * we .can. w r i t e  2-  <^ t  As i n t h e case o f s p i n  2.  1=1,  a straightforward  l e a d s to the f o l l o w i n g e x p r e s s i o n s ( t h e parameters a p p e a r i n g i n (49) matrix  2-  )  M  x  ,  Q ^ and  Q _  y  i n terms o f the  >y  density-  p> ;  ^rv  Using  for  calculation  2.  (52)  ^  and  i iz  / i i  (54)  U  s  (pO  ' 3 4 - y  and r e c a l l i n g , t h a t  y°  and />  '»3  remain zero ( c f .  e q u a t i o n 51 I I I  and the i n i t i a l  always  ' / 32-.  conditions)  we get ^  Substituting  (55)  in  Nrtj  (53)  l e a d s f i n a l l y t o the  equations?  Equations  (56)  equations. involving  are i d e n t i c a l w i t h  w h i c h are L u r g a t ' s  (The apparent d i f f e r e n c e i n the* s i g n s of the terms H  i  i s due t o the same r e a s o n as t h a t  t h e p r e v i o u s s e c t i o n , namely time dependent H a m i l t o n i a n (cf.  (49)  e q u a t i o n (5)  that J-$  explained i n  whereas we have t a k e n the  t o he  + ^  ) , L u r g a t t a k e s i t t o he  I • H (t) - vt, J -  Hit).  I n t h i s s e c t i o n we have t h u s demonstrated once a g a i n the e q u i v a l e n c e of L u r g a t * s this thesis.  method w i t h t h e one p r e s e n t e d i n  The treatment a l s o b r i n g s out the  i n t e r r e l a t i o n s h i p between the t h e o r i e s of L u r g a t and o f B l o o m , Hahn, H e r z o g . 4 .  -  v,  ;  The case of h i g h e r s p i n s : The cases f o r h i g h e r v a l u e s o f s p i n w i l l be n a t u r a l l y  more c o m p l i c a t e d and s h a l l not be d i s c u s s e d h e r e i n d e t a i l , because the e s s e n t i a l f e a t u r e s o f t h e g e n e r a l case are c o n t a i n e d i n t h e f o r m a l i s m of the p r e v i o u s two s e c t i o n s , a p p l i e d t o the two s p e c i a l cases o f s p i n  I - 1  and  F o r h a l f i n t e g r a l s p i n s , as i l l u s t r a t e d f o r one need c o n s i d e r those n u c l e i t h a t b e l o n g t o the  as  3/2 . 1=3/2,  +-m - s t a t e s  and then t h e case r e d u c e s e s s e n t i a l l y t o the c a s e , s i n c e degeneracy o c c u r s i n p a i r s o f  ±m  non-degenerate states.  For i n t e g r a l spin the t r a n s i t i o n s i n v o l v i n g the m = o case  s t a t e can he t r e a t e d i n the same way as the was t r e a t e d .  T r a n s i t i o n s not i n v o l v i n g  m = o  1=1 state  can he t r e a t e d l i k e the h a l f - i n t e g r a l case i n t h e sense t h a t the t r a n s i t i o n s  +m,  - m  can he t r e a t e d  t  —+  -  +  m^  and the separately.  transitions  36  REFERENCES'  (1)  Bloom  M.,  Hahn  E. L . ,  Phys. R e v . , 97, (2)  Bloom Can.  M., J. U.,  B.,  1699 (1955)  Robinson Phys.,  and Herzog  L. B.,  36,  R e v . Mod.  1286 Phys.-,  and V o l k o f f  G. M . ,  (1958)  (3)  Fano  29,  74  (1957)  (4)  Lurgat  F.,  J . de  P h y s . et  Rad.,  19,  713  (1958)  (5)  Lurgat  F.,  J. de  P h y s . et  Rad.,  19,  745  (1958)  

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