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Macroscopic equations for nuclear spin resonance in density matrix formalism 1960

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MACROSCOPIC EQUATIONS FOR NUCLEAR SPIN RESONANCE IN DENSITY MATRIX FORMALISM SHRIDHAR DATTATRAYA JOG A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of P h y s i c s We accept t h i s t h e s i s as conforming to the s tandard r e q u i r e d from candidates f o r the degree of MASTER OF SCIENCE Members of the Department of P h y s i c s THE UNIVERSITY OF BRITISH COLUMBIA SEPTEMBER 1960 In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree th 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 reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood tha t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n permission. Department of 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 $ f Canada. ABSTRACT Methods of s e t t i n g up g e n e r a l i z e d B l o o h equat ions govern ing the t ime dependence of macroscopic m a g n e t i z a t i o n f o r a system of n u c l e i of 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 the degenerate case "by Bloom, Hahn and Herzog and "by Lure a t , and f o r the non- degenerate case by Bloom, Robinson and V o l k o f f . I n t h i s t h e s i s an attempt i s made to g i v e a u n i f i e d d i s c u s s i o n of these 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 to demonstrate the 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 are not c o n s i d e r e d . The g e n e r a l theory i s developed i n terms of the 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 to the non-degenerate and the degenerate cases . The r e s u l t s are d i s c u s s e d and compared w i t h those of the p r e v i o u s i n v e s t i g a t o r s . TABLE OF CONTENTS ACKN O^VLEDGMENT S ABSTRACT CHAPTER I - INTRODUCTION CHAPTER I I - TIME DEPENDENCE OF THE DENSITY MATRIX CHAPTER I I I - NON-DEGENERATE CASE CHAPTER IV - DEGENERATE CASE REFERENCES ACKNOWLEDGMENT S I w i s h to 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 the problem and f o r h i s constant h e l p i n a l l the phases of development of t h i s work. I a l s o w i s h to thank the N a t i o n a l Research C o u n c i l f o r p r o v i d i n g f i n a n c i a l a s s i s t a n c e throughout the year i n the form of a r e s e a r c h g r a n t . 1 CHAPTER I IHTRODUCTIOff •>"*.• A nuc leus of s p i n I h a v i n g a magnetic moment and ( f o r I $- i ) an e l e c t r i c quadrupole moment eQ has 21 + 1 energy l e v e l s (which 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 combinat ion of a n o n - u n i f o r m e l e c t r i c f i e l d ( c h a r a c t e r i z e d by 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 or w i t h o u t a x i a l symmetry) and a constant u n i f o r m magnetic f i e l d H e . 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 the r a d i o - f requency 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 s u b j e c t i n g 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 to a weak o s c i l l a t i n g magnetic 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 the c o i l p r o d u c i n g i t ) of v a r i a b l e r a d i o f requency ^ ~ , and by o b s e r v i n g as a f u n c t i o n of ^ the resonances i n the a b s o r p t i o n or the 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 types of spec t rometers . Al ternat ive ly , f r e e p r e c e s s i o n or sp in -echo techniques can be u t i l i z e d . The observed 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 to the t ime d e r i v a t i v e of the component a l o n g the a x i s of a r e c e i v e r c o i l of the macroscopio m a g n e t i z a t i o n of the sample. Methods of s e t t i n g up g e n e r a l i z e d B l o c h equat ions govern ing the t ime dependence of t h i s macroscopic 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 which r e l a x a t i o n e f f e c t s are n e g l e c t e d by Bloom, Hahn and H e r z o g t 1 ) , by L u r g a t ^ 4 ) and by Bloom, Robinson and V o l k o f f ( 2 K The object 2 of the present t h e s i s i s to g i v e " a u n i f i e d d i s c u s s i o n of these 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 ( o f . , f o r example, F a n o ^ 3 ) j and to demonstrate the i n t e r r e l a t i o n s h i p between them. An assembly of "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 sub jec ted to 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 21+i rows and columns c o n t a i n i n g (21+ 1) r e a l parameters . The e x p e c t a t i o n v a l u e o f any opera tor A i s g i v e n by < A > = T / f A / o ) = ^ (o ; n in i n n T a k i n g A to be the i d e n t i t y opera tor we have (?) = 4 / L = 1 » l e a v i n g 41 ( I - h l ) independent r e a l parameters to d e s c r i b e 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 are needed to d e s c r i b e comple te ly the macroscopic behaviour of the system. F o r such macroscopic q u a n t i t i e s i t . i s c i n some ; cases convenient to u t i l i z e the t h r e e components of the macroscopic m a g n e t i z a t i o n , the f i v e components of the e l e c t r i c quadrupole moment, d e n s i t y , the seven components o f the magnetic o c t o p o l e moment d e n s i t y , e t c . , each set .expressed 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 ro ta t ions l i k e the s p h e r i c a l harmonics \ w i t h , k b e i n g an i n t e g e r r u n n i n g up to 21 , and, q . r u n n i n g i n i n t e g r a l s teps from - k to k . T h u s r f o r n u c l e i of I - \ the three components of the m a g n e t i z a t i o n d e s c r i b e . t h e system c o m p l e t e l y , f o r 1 = 1 the f i v e components of the quadrupole moment d e n s i t y are i n g e n e r a l needed i n , a d d i t i o n to g i v e a complete d e s c r i p t i o n i n terms of 3 e i g h t q u a n t i t i e s , w h i l e f o r 1 = 3 / 2 the seven oc topole moment d e n s i t y components are a l s o needed to p r o v i d e 15 q u a n t i t i e s , e t c . E q u a t i o n s (1) e s t a b l i s h the connect ion between the d e n s i t y m a t r i x elements and these macroscopic q u a n t i t i e s i f we take A to be i n t u r n equal to each one of the a p p r o p r i a t e 4 1 (1 + 1) m u l t i p o l e moment o p e r a t o r s . S ince the t ime dependence of the d e n s i t y m a t r i x when r e l a x a t i o n processes are n e g l e c t e d i s governed by the e q u a t i o n , where ^ C u j i s the t o t a l H a m i l t o n i a n f o r a s i n g l e n u c l e u s a c t e d upon by the g i v e n magnetic and e l e c t r i c f i e l d s * the t ime dependence of the 4 I ( r + ' l ) macroscopic p h y s i c a l quant i t ies d e s c r i b i n g the system .can i n t h i s , cage be expressed i n the form of 4 1 ( I + 1) s imultaneous f i r s t order d i f f e r e n t i a l . equat ions obta ined by combining (1) . and , (2) ; . - F o r . 1 = \ the complete macitpscopic behaviour of the system i s d e s c r i b e d by, three equat ions in>terms of the t h r e e m a g n e t i z a t i o n components . ' ^ L , M 3 , M s - . , s or- a l t e r n a t i v e l y M 5 can bes e l i m i n a t e d by i n t r o d u c i n g the d i f f e r e n c e n between the. f r a c t i o n a l p o p u l a t i o n s of the m - t e - s t a t e s , • . . Lurg at d i s c u s s e s the s p e c i a l pure quadrupole 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 w i t h . Iv.= l r 3/2 and, 5/2 compl i ca ted se ts of 8 , 15 and 35 s imultaneous equat ions r e s p e c t i v e l y . He then shows how under resonance c o n d i t i o n s • e a c h of these se t s reduces to t h r e e approximate - equat ions analogous to the t h r e e equat ions 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 the case of I -, £ . B loom, Hahn and Herzog and a l s o Bloom, 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 se t s of t h r e e approximate equat ions 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 c e r t a i n s p e c i f i e d c o n d i t i o n s . ' ' I n t h i s t h e s i s we s t a r t w i t h v e q u a t i o n s (2) f o r the A I ( I + 1 ) independent parameters 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 to 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 the method of r e d u c i n g t h i s system of equat ions under resonance c o n d i t i o n s to three approximate equat ions which are then compared t o the r e s u l t s of the p r e v i o u s i n v e s t i g a t i o n s . 5 CHAPTER I I • TIME DEPENDENCE OF THE DENSITY MATRIX The H a m i l t o n i a n - ^ / t o t a f o r a nuc leus of s p i n I i n t e r a c t i n g w i t h a r b i t r a r y s t a t i c 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 '£fa and 'J-j' where -f~fa does not and Ti does depend on the 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 I n (4) x, y 7z- are the p r i n c i p a l axes of the e l e c t r i c f i e l d g r a d i e n t t ensor whose component a l o n g the z - a x i s ( u s u a l l y chosen to be the one of g r e a t e s t a b s o l u t e v a l u e ) i s & , and whose asymmetry parameter i s rj - {$>xx-4r7)/<$ . The t ime dependent p a r t o f the H a m i l t o n i a n (5) i s assumed to be due to a weak a p p l i e d r . f . f i e l d \T(*) of f requency u? which i s u s u a l l y l i n e a r l y p o l a r i z e d . The e igenvalue problem a x = F k % { i ) can be s o l v e d ( n u m e r i c a l l y i f necessary) and l e a d s to 2 I +- 1 e i g e n v a l u e s E k and t ime- independent e i g e n f u n e t i o n s = ? °-i C ^ > k , l =. 1, 2., • - - (ZI. + 1) , where the u x" are e i g e n f u n e t i o n s of' 1̂ ' V Ah example f o r I = 5 / 2 ( A-£ i n Spodumene) i s d i s c u s s e d by Bloom, • Robinson .and V o l k o f f ; • /. ; '. " i : I f twe 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 ' , '^jk/ i i r w j k = ( E j - E k ) / - H =. - «okj. (7) then i n the f . r e p r e s e n t a t i o n equat ions (2) - (7) l e a d to the (2 I +• 1 ) equat ions i s the m a t r i x element of the t ime dependent p e r t u r b i n g o p e r a t o r £f o f equat ion (5) between the s t a t e s V - and <Yk , ' •' • 7* ' . ' * • •" the. • ' and ljk i s the t ime- independent m a t r i x element o f / s p i n o p e r a t o r I between these s t a t e s which can be e x p l i c i t l y e v a l u a t e d i n terms of the known ctk i n f. = ViL u, cik . I n the absence of an a p p l i e d r . f . f i e l d J-f-o, and equat ions (8) have the s o l u t i o n s /% = (f*\ c-"'kt ft) where the (fjk)0 . are constants determined by the i n i t i a l C o n d i t i o n s , so t h a t ; /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 the f r'equenei es ^JV/ZTT: , We are 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 behaviour of the system of n u c l e i when a weak p e r t u r b i n g magnetic f i e l d of f requency i s a p p l i e d to the system w i t h OJ v e r y c l o s e to a p a r t i c u l a r one of the to-k c h a r a c t e r i z i n g the system, say w A s = u ; 0 . 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 the' c~>j< aire d i f f e r e n t ; (we s h a l l speak of t h i s as a non-degenerate system) , or not (we s h a l l speak of a system h a v i n g two or more the same as degenerate even when no energy l e v e l s are degenerate ) . The p e r t u r b i n g e f f e c t of the t ime dependent Ji can be c o n v e n i e n t l y d i s c u s s e d by i n t r o d u c i n g a t r a n s f o r m a t i o n to r e p l a c e the ^ "^V the p o s s i b l y a l s o t ime-dependent t r a n s - formed q u a n t i t i e s f>* r e l a t e d to ^ and t o some a r b i t r a r i l y s e l e c t e d constant f r e q u e n c i e s u > /k/ 2 . - r r ^ by "." • ; v .; JK .. ..- ....... .. - . where, as we s h a l l p r e s e n t l y see, i t w i l l b e convenient, to 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 or v e r y n e a r l y equal to the corresponding ^Pjk . c h a r a c t e r i z i n g the system. I n terms of p* equat ions (8) become 8 ' : ' (11) — c • 2_ The system of (2 1 + 1) s imultaneous equat ions (8) or ( l l ) g i v e s the exact t ime dependence of P or p * f o r "both degenerate and non-degenerate systems ( n e g l e c t i n g a l l r e l a x a t i o n e f f e c t s ) » hut i s t e d i o u s to set up and t o i n t e g r a t e e x a o t l y f o r I > £ . We t h e r e f o r e use an approximate method of s o l v i n g systems of d i f f e r e n t i a l equat ions 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 (as quoted by L u r g a t ( 5 ) ) . Cons ider a system of s imultaneous d i f f e r e n t i a l equat ions - £ w h e r e , £ I s '-a s m a l l parameter , and X V are of the 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 to B o g o l i u b o v and M i t r o p o l ' s k i i the s o l u t i o n of (12) i s g i v e n 9 to a f i r s t approx imat ion by the s o l u t i o n of = £ X , ( t . x , , . . * , , ( 1 4 ) where X u denotes the t ime average of X k . We assume i f ( t ) i n equat ions (5) and (8) to be a weak t ime dependent magnetic f i e l d i n v o l v i n g a s i n g l e f requency ^Vi-rr i n such a way as to make the t ime average o f H * ( t ) e ' ^ equal to a s m a l l constant v e c t o r and the t ime average of H ( t ) e equal to zero f o r any other i o ' ^ to . 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 we h a v e : FT (*) •= Hf co5 u)t • v = 1 H*' (u) E q u a t i o n s (11) where the J~fjk. are g i v e n by (16) can be brought i n t o the form ( 1 2 ) , s i n c e f h*iu { i s assumed to be s m a l l , and we can choose the a r b i t r a r y t r a n s f o r m a t i o n f r e q u e n c i e s u)^ i n equat ion (10) to make ^ - u ^ i n equat ions (11) to be e i t h e r zero or s m a l l . I n the f o l l o w i n g we assume t h a t the t ime average of a*1 i s 1 i f ti>-0t and i s zero i f v d i f f e r s a p p r e c i a b l y from z e r o . To o b t a i n the l a t t e r r e s u l t i t i s necessary f o r v to be l a r g e i n comparison w i t h the r e c i p r o c a l of the t ime over which the 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 to d i s c u s s f i r s t the non-degenerate case ( a l l ^ d i f f e r e n t ) , and then to r e t u r n to the degenerate case when two or more of the to-^ c o i n c i d e . CHAPTER I I I 3ST0N - DEGENERATE CASE I n t h i s chapter f i r s t we s h a l l t r e a t the non- degenerate case ( a l l i o . ^ s d i f f e r e n t ) on the b a s i s of the t h e o r y developed i n Chapter I I . We s h a l l then compare the r e s u l t s w i t h those of B loom, Robinson and V o l k o f f , who a l s o have c o n s i d e r e d the non-degenerate case . I f to i s v e r y c l o s e to some one p a r t i c u l a r u) - UJ > o » then we can choose a l l the u).[ = o., exoept f o r u)^a , whieh i s chosen equal to co , r a t h e r than to toAs = UJO i n order to a v o i d ambigui ty i n e v a l u a t i n g the avers v a l u e of ^ * m Equat ions (11) then become on t a k i n g the t ime averages of the r i g h t hand s idess f o r n o n - d i a g o n a l terms ( j ^ M ( P * ) - o u n l e s s j = r and k = s s i m u l t a n e o u s l y JUL * ! J k ' f o r d i a g o n a l terms ( j = k) cl_ {p*-) - o u n l e s s j = r or j = 8 / 12 where )o denotes the l e a d i n g t ime-Independent term i n the expansion of- r> * i n powers o f IV*I and (u>-uj0) . c) These equat ions are i n the form of equat ions (14) w i t h the r i g h t hand s i d e s comple te ly independent of the t i m e . To the f i r s t a p p r o x i m a t i o n the s o l u t i o n s of these equat ions are the same as the s o l u t i o n s of any o ther set of equat ions whose r i g h t hand s i d e s reduce t o (17) on b e i n g averaged w i t h r e s p e c t to t i m e . I n p a r t i c u l a r equat ions (17) may be r e p l a c e d by a s i m i l a r set w i t h the s u b s c r i p t zero l e f t o f f the /°*~ and the t ime a v e r a g i n g l e f t o f f H ( t ) e + " ^ , and i t i s convenient t o c a r r y out t h i s replacement . Trans forming back to the 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 the a i d of equat ions (10) we then o b t a i n : f o r n o n - d i a g o n a l terms ( j A k) ^- p - -l to-, p u n l e s s j = r and k - s y - s i m u l t a n e o u s l y f o r d i a g o n a l terms (j= k) ?L f - o u n l e s s j = r or j = s -it.'jj 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 the 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 s t a t e b e f o r e the 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 thermal e q u i l i b r i u m which i s d e s c r i b e d b y : " - j The e x p e c t a t i o n v a l u e of any opera tor A f o r a system i n thermal e q u i l i b r i u m i s thus g i v e n by E q u a t i o n s (17) (or (18)) t o g e t h e r w i t h the i n i t i a l c o n d i t i o n s (19) i m p l y t h a t to our degree of a p p r o x i m a t i o n a l l the P. (and consequently a l l the P* ), j ^ k , remain zero except p , and a l l the P (=p*\ • Us • 'jj ^ /jj ) remain constant i n t ime except ^ and ^ whose sum remains constant but whose d i f f e r e n c e v a r i e s . I f we denote t h i s d i f f e r e n c e by " ~ /1a ~ fss - UA ~ fs* - « *" (2.1) the t ime v a r y i n g p r o p e r t i e s of the system which determine the d e v i a t i o n of < A > from < A > , w i l l be g i v e n by ( c f . m o d i f i e d , equat ion (17) ). or e q u i v a l e n t l y by ( c f . equat ion (18) ) A. ( r A s ) = -i"ofKs + c / n Hct; - f • h a; - f f -HCtj) {2 2) (23) We note t h a t (22) or (23) each r e p r e s e n t s a set of three equat ions s i n c e ' p and P * are complex. Thus i f we w r i t e ^ = x + i Y (̂ ) where X and Y are r e a l parameters , then we can r e w r i t e equat ions (23} e x p l i c i t l y i n terms of the three r e a l parameters X , Y and n : d t Y - n V I m H ( ^ ) . ( 2 5 r) <=/t 1 : = I » ( f v H W ) - Y f e ( C- H ft j)J 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 of any opera tor < A > g i v e n by (1) from i t s thermal e q u i l i b r i u m v a l u e < A > ^ g i v e n by (20) i s equal to < A > -<A>^ = AaK A ^ = l x + i Y ) A J , + ( x - Y ) A J T A + J- Re ( A A J + I - ( A * s ) ( 2 6 ) * t ( A , ; A - A „ ) ( » - » . h ) - Thus the t ime dependence of a l l the 4 1 ( I + l ) macroscopic p h y s i c a l q u a n t i t i e s d e s c r i b i n g the system i s expressed i n terms of the t h r e e v a r i a b l e s X , Y , n which can i n t u r n be expressed i n terms o f , f o r example, the t h r e e components of the macroscopic m a g n e t i z a t i o n . To r e l a t e these r e s u l t s to those of B loom, 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 which •j. /As ( 1 7 ) w i t h P , S , T r e a l . Then we have <r.>-<x,\h = v _ p + i ̂ ^ _ ( r o j {ri_ n r k, <T/>-<T>>̂ = 2V5 * i [(r?)^-Cix)J(n-„„) I f we i n t r o d u c e the v a r i a b l e s f . 3 < r . > - < ^ > H , I3 s <*,> - < r y > „ and n e g l e c t i n g the terms c o n t a i n i n g the d i a g o n a l m a t r i x elements i n (28) s u b s t i t u t e x - i * A p , y - iy A 5 and (27) i n t o (25) we o b t a i n equat ions [ 1 3 ] of Bloom, Robinson and V o l k o f f ^ 2 ) . CHAPTER IV DEGENERATE CASE I n the degenerate case where two or more t r a n s i t i o n f r e q u e n c i e s may be equal a c o r r e s p o n d i n g l y g r e a t e r number of m a t r i x elements P may v a r y i n t ime (not j u s t 4 as i n the Jk non-degenerate c a s e ) . The g e n e r a l f o r m u l a t i o n becomes too c o m p l i c a t e d , and hence we d i s c u s s below some s p e c i a l cases . We w i l l f i r s t o u t l i n e the method developed by L u r e a t f o r the degenerate case i n g e n e r a l . Then we s h a l l i n v e s t i g a t e by the method of p r e v i o u s chapters? the s p e c i a l cases t r e a t e d by h i m , and s h a l l compare and d i s c u s s the r e s u l t s . 1 . Lure at'a 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 i n g e n e r a l . The S c h r b d i n g e r ' s t ime dependent equat ion f o r a p h y s i c a l q u a n t i t y A can be w r i t t e n i n the form ^ = f < [ H , A ] > M F o r a nuc leus of s p i n I , p l a c e d i n an e l e c t r i c f i e l d g r a d i e n t of a x i a l symmetry ( ^ o ) and i n a magnetic f i e l d the H a m i l t o n i a n i s total 4 r ( i i - o z . (.31) where ^ - i g the v a l u e of the e l e c t r i c f i e l d g r a d i e n t . L e t he t e n s o r o p e r a t o r s d e f i n e d by the commutation r e l a t i o n s 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 these d e f i n i t i o n s we get X. I 5 , Y „ ± , = t r £ o< fe ^ K - f r ( r + o ] ^ T u r n Over * Y = - ( r 1 = 3 e Q 2. i ( i r - i j (3^) where D t k are the components of the quadrupole moment o p e r a t o r s . Talcing the average v a l u e s of Y k M we get the f o l l o w i n g formulae f o r the components of the macroscopic m a g n e t i z a t i o n M and of the macroscopic quadrupole moment tensor Q, : N y h < Y l 6 > - M v < Y , t l > - + ( M , t ^ j ) < Y i o > = / I Q J J r ' (35") w i t h s i m i l a r express ions f o r Y 3 > J L e t c . . i n terms of oc topole and h i g h e r moments. Making use of the p r o p e r t i e s of t ensor opera tors i t can he shown t h a t S u b s t i t u t i n g (31) i n (30) and making use of equat ion (36) we get 2-k + / V (k+1) 1) \ I k + / - ^ ^ < Y ^ X (37) We note at t h i s p o i n t t h a t t h i s equat ion i s an exact equat ion where a l l the energy l e v e l s are taken i n t o account , 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 of s p i n . U s i n g equat ions (37) and (35) L u r g a t then o b t a i n s f o r the s p e c i a l cases 1 = 1 , 3/2 and 5/2 e q u a t i o n ^ g i v i n g the t ime dependence of the macroscopic p h y s i c a l q u a n t i t i e s M , Q e t c . I n order t o see the b a s i c equiva lence of L u r g a t 1 s method w i t h the one presented i n t h i s t h e s i s we r e c a l l t h a t the mean v a l u e of any opera tor can be expressed i n terms of the d e n s i t y m a t r i x ( c f . equat ion ( l ) ) . Thus the mean v a l u e s of the o p e r a t o r s Yk f t_ and a l s o those of M x •, M y , , Q « » Q«y e t c . , can be w r i t t e n down i n terms of the d e n s i t y m a t r i x /o . To get the v a r i a t i o n i n t ime of the macroscopic p h y s i c a l q u a n t i t i e s M , "Q e t c . , one may u t i l i z e the t ime dependence o f <Yh^> , as L u r g a t has done, or one may use the t ime dependence of the 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 present t h e s i s . 2 . 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 of L u r g a t f o r t h i s case and then s h a l l show how the same equat ions can be o b t a i n e d by the method of p r e v i o u s c h a p t e r s . U s i n g equat ions (37) and d e f i n i t i o n s (35) L u r g a t o b t a i n s the f o l l o w i n g equat ions f o r the t ime dependence of the components of the macroscopic p h y s i c a l q u a n t i t i e s ~ M , Q : dt 5 i ( 4 Q y » ) = ^ ( *4 £ ( ^ C « „ - « „ > ) = - J I u v n O v t v 22 H JLM C38) 7 IE ' IFote t h a t the e i g h t equat ions b r e a i ^ f i h t o three sub s y s t e m s , I t I I and I I I , t h a t i s v a r i a b l e s i n one subsystem do not occur i n any other 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 « C o ; - Q V v Co) = - ± Q > v ( » ; • ' : / • • • - ^ ' • ' • 1 , (3.1) Q j k Co) - o ( t ^ k ) f M , - (o) - o Consequently the q u a n t i t i e s appear ing i n equat ions (38 I I ) are constant and<h(38 I I I ) are zero i n i t i a l l y and remain aero a t any l a t e r t i m e . 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 , namely (38 I ) . I t i s convenient to i n t r o d u c e the t r a n s f o r m a t i o n = n * c o s . t _ ^ a * s i n <_L Q = S'.r, oOt + I ± ft* Co S oot Lungat c o n s i d e r s the case where there i s no s t a t i c magnetic f i e l d and where the t ime v a r y i n g magnetic f i e l d i s H , = H , c « s w t , H y - o , = o A p p l y i n g t r a n s f o r m a t i o n (40) to (38 I ) and n e g l e c t i n g terms i n s i n z^t and cos i « t the t ransformed equat ions are : where u>„ i s the resonance f r e q u e n c y . , As p o i n t e d out by L u r g a t these equat ions become f o r m a l l y i d e n t i c a l w i t h equat ions t23~] of Bloom, Hahn and H e r z o g ^ 1 ) , except f o r a f a c t o r of 2 i n the c o e f f i c i e n t o f H j ( i n the second e q u a t i o n , i f one makes the f o l l o w i n g t r a n s f o r m a t i o n : Bloom , Hahn and Herzog Lure at Of course the d i s c r e p a n c y of the f a c t o r of 2 i s not s u r p r i s i n g . There i s no reason to expect the two sets to 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 Bloom, Hahn and Herzog equat ions 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 the t o t a l number of n u c l e i corresponding to +m s t a t e s . T h i s can not be done f o r the present case of s p i n I - 1 , s i n c e the m = 0 or 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 to +m s t a t e ^ t o - m s t a t e . However, i f we s t a r t w i t h the complete system of n u c l e i as i n chapter I I and then f o l l o w a procedure s i m i l a r to the one f o l l o w e d i n chapter I I I (non-degenerate case) , then we are l e d e x a c t l y to the L u r g a t ' s e q u a t i o n s , as i s shown be low. F o r the degenerate case 0 ; H o - o j two of the three energy l e v e l s , say 1 and 2 ., w i l l c o i n c i d e so t h a t L e t the frequency 0 0 of t h e - a p p l i e d r . f . . f i e l d be c l o s e to the t r a n s i t i o n frequency w_3 - oo,3 = u j e . i e choose the a r b i t r a r y t r a n s f o r m a t i o n f r e q u e n c i e s and ^ to s a t i s f y Then equat ions (11) l e a d , w i t h the approx imat ion d i s c u s s e d i n the p r e v i o u s c h a p t e r s , to the f o l l o w i n g equat ions f o r the t ransformed m a t r i x elements J - * • . . . * t v 2 - / 1 2 . 1 3 ' 2 _ ' 2 - 1 ' 3 A - " i ^ ^ * . T-T — lC. t* • ~h. — O ' U Note t h a t the equat ions i n v o l v e e i g h t independent r e a l parameters ( p * * * *• , b e i n g complex g i v e s i x and / | 3 I 2 3 ' ' 12- ' ^ i i * ' fx* 1 "being r e a l , g i v e t w o ) . I n the 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 x ( t ) =• H," cos 'u)f , , ' H y = = o ) we get %s - HJ = • ^ . a s H, . * *, a s ' ' i s equal t'o ^ f o r a l l n o n - v a n i s h i n g m a t r i x elements of I * w h i c h - a r e i n f a c t the ones "that appear i n ( 4 3 ) . I t i s now p o s s i b l e to choose a complete set of 8 l i n e a r l y independent parameters (which are combinations of the m a t r i x elements P ) such t h a t o n l y t h r e e parameters J* are t ime dependent. The corresponding equat ions f o r these parameters are then enough to d e s c r i b e the behaviour of the system. Thus we d e f i n e where X , Y , Z are r e a l parameters . Moreover l e t (cf.&2-)) u) i 4 - 1-0,3= f o r convenience of n o t a t i o n . Then the equat ions f o r Q* ( o b t a i n e d by adding ( i ) and ( i i ) of (43) ) and Z*" ( o b t a i n e d from ( i i i ) , ( i v ) , (v) o f (43) ) are s = 4 £ ( a ' - ^ ' i n , Gr w r i t i n g i n terms of X , Y , Z these equat ions g i v e : Jt =. - ( i O - u J 0 ) „ Jt Si z_ A p p l y i n g equat ions ( l ) and (35) to the present case of 1 = 1 , 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 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 f o r the components of the macroscopic p h y s i c a l q u a n t i t i e s I f , Q i n terms of the d e n s i t y m a t r i x ; (46) U s i n g (44) and (46) we get z - ^ g g , - Q y ; N << S u b s t i t u t i n g (47) i n (45) we f i n a l l y get These are e x a c t l y the equat ions (41) which are o b t a i n e d by L u r g a t . The apparent d i f f e r e n c e i n the s i g n s of the terms i n v o l v i n g H 1 i s due to the f a c t t h a t whereas we take the t ime-dependent H a m i l t o n i a n J-f to be + v T-^*(P) ( o f . equat ion (5) ) , L u r g a t takes i t to be — v Is I • Vt{1r) . We have thus i l l u s t r a t e d , f o r the case of s p i n 1 = 1 , the complete equiva lence of our treatment w i t h t h a t of L u r g a t , 1 3 . The case of s p i n I = 3/2 : We now c o n s i d e r the case of I = 3/2 as an example of h a l f i n t e g r a l s p i n s . L u r g a t o b t a i n s f o r t h i s case the f o l l o w i n g equat ions where u)„ i s the resonance frequency and <̂> the frequency o f r . f . f i e l d . 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 f i e l d g r a d i e n t i s a x i a l l y symmetric (.*]-<>) and t h a t the s t a t i c magnetic f i e l d H 0 - o . L e t 1 , 2 , 3 , and 4 be the f o u r ( ™ = - 3 / t ) . 4 1 , ( . w = 3 / ^ - ; = " SsJ 3 2. 5<J s t a t e s corresponding to m = 3/2 , , —£ and - 3 / 2 r e s p e c t i v e l y . Then (So) 4 2 Then equat ions (11) l e a d , w i t h the approx imat ion d i s c u s s e d i n p r e v i o u s c h a p t e r s , t o the f o l l o w i n g equat ions f o r the t ransformed m a t r i x elements P' k = i , 2 - , 3 , ^ y i H , >- III Bote t h a t (51) I s made up of t h r e e separate subsets 1 , I I and I I I ( t h a t i s , v a r i a b l e s appear ing i n one subset do not occur i n another s u b s e t ) . I n i t i a l l y when the system i s i n thermal e q u i l i b r i u m we have seen ( c f . equat ions (19) ) t h a t the o f f - d i a g o n a l d e n s i t y m a t r i x elements are z e r o . From (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 appear ing t h e r e i n remain zero at any l a t e r t ime and p l a y no p a r t i n the change of the system i n t i m e . Moreover equat ions (51 I ) and (51 I I ) are f o r m a l l y i d e n t i c a l . ( R e p l a c i n g i n d i c e s 1 and 2 everywhere i n (51 I ) by i n d i c e s 4 and 3 r e s p e c t i v e l y we get equat ion (51 I I ) ) . That i s to say , the n u c l e i i n the s t a t e s 1 and 2 behave i n e x a c t l y the 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 i t i s s u f f i c i e n t to 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 n u c l e i , namely those b e l o n g i n g to the + m s t a t e s . As a matter of f a c t Bloom, Hahn and Herzog use t h i s concept r i g h t a t the s t a r t of t h e i r development of the t h e o r y f o r the h a l f - i n t e g r a l s p i n s ; equat ions (51 I ) , i f w r i t t e n down e x p l i c i t l y i n the parameters ^ e ( /^*) > ^ m (/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 equat ion [23] of Bloom, 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 to 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 e q u i v a l e n t to two oppos i te c i r c u l a r l y p o l a r i z e d r . f . f i e l d s . One of the two components gets e l i m i n a t e d i n the process of approximat ion. Bloom, Hahn and Herzog take c i r c u l a r l y p o l a r i z e d r . f . so t h a t no approx imat ion i s - n e c e s s a r y 4 n t h i s respec t i n t h e i r t rans format ion . ) , - We -have thus demonstrated t h a t out of the sixteen.. . equat ions (51) . o n l y three are u s e f u l , , namely equat ions (51 I ) which govern the behaviour of the system of n u c l e i . We s h a l l now show,- as i n the p r e v i o u s s e c t i o n , t h a t i f s u i t a b l e l i n e a r combinat ions of , . are chosen, . . . . . ijk . t h e n - e q u a t i o n s (51) l e a d to the t h r e e equat ions (49) of L u r g a t . Thus l e t - • v - \\ . ' / a x T ' 4 4 " V 3 ' i • ~ . where X , Y and Z are r e a l parameters . For L u r g a t case which we" are d i s c u s s i n g , we have " ' "~ '' H * ( t ) = — H \ cob wt * *H ' = n\ o ~" ^ E q u a t i o n s (51 I ) and (52 I I ) ; l e a d to the f o l l o w i n g equat ions f o r and Z * : * - • * • ; .... . U-K _ 2 a ; i i L {•: ;* . .•«'•) Jt' x 2- Or e q u i v a l e n t l y , i n terms o f X * .,, Y * -and- Z * we .can. w r i t e 2- 2 . <̂  t As i n the case of s p i n 1 = 1 , 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 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 f o r M x , Q y ^ and Q > y_ ( the parameters appear ing i n (49) ) i n terms of the density- m a t r i x p> ; 2- ^rv 2. ^ / i i i iz U s ' 3 4 - y ( p O U s i n g (52) and (54) and r e c a l l i n g , t h a t y° and /> always '»3 ' / 32-. remain zero ( c f . equat ion 51 I I I and the i n i t i a l c o n d i t i o n s ) we get ^ N r t j S u b s t i t u t i n g (55) i n (53) l e a d s f i n a l l y to the equat ions? E q u a t i o n s (56) are i d e n t i c a l w i t h (49) which are L u r g a t ' s e q u a t i o n s . (The apparent d i f f e r e n c e i n the* s i g n s of the terms i n v o l v i n g H i i s due to the same reason as tha t e x p l a i n e d i n the p r e v i o u s s e c t i o n , namely that whereas we have taken the t ime dependent H a m i l t o n i a n J-$ to he + ^ I • H (t) ( c f . equat ion (5) ) , L u r g a t takes i t to he - vt , J - Hit). I n t h i s s e c t i o n we have thus demonstrated once a g a i n the equivalence of L u r g a t * s method w i t h the one presented i n t h i s t h e s i s . 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 of Bloom, Hahn, H e r z o g . - v ,; 4 . 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 of s p i n w i l l be n a t u r a l l y more compl i ca ted and s h a l l not be d i s c u s s e d here 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 of the g e n e r a l case are conta ined i n the 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 , as a p p l i e d to the two s p e c i a l cases of s p i n I - 1 and 3/2 . 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 1 = 3 / 2 , one need c o n s i d e r those n u c l e i tha t b e l o n g to the +-m - s t a t e s and then the case reduces e s s e n t i a l l y to the non-degenerate case , s i n c e degeneracy occurs i n p a i r s o f ± m s t a t e s . For i n t e g r a l s p i n the t r a n s i t i o n s i n v o l v i n g the m = o s t a t e can he t r e a t e d i n the same way as the 1 = 1 case 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 s t a t e 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 the sense t h a t the t r a n s i t i o n s +m, + m^ and the t r a n s i t i o n s - m t —+ - can he t r e a t e d s e p a r a t e l y . 36 REFERENCES' (1) Bloom M . , Hahn E . L . , and Herzog B . , P h y s . R e v . , 97, 1699 (1955) (2) Bloom M . , Robinson L . B . , and V o l k o f f G. M . , Can. J . P h y s . , 36, 1286 (1958) (3) Fano U . , Rev. Mod. Phys.- , 29, 74 (1957) (4) L u r g a t F . , J . de P h y s . et R a d . , 19, 713 (1958) (5) L u r g a t F . , J. de P h y s . et R a d . , 19, 745 (1958)

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