UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Macroscopic equations for nuclear spin resonance in density matrix formalism Jog, Shridhar Dattatraya 1960-01-31

You don't seem to have a PDF reader installed, try download the pdf

Item Metadata

Download

Media
831-UBC_1960_A6_7 J6 M2.pdf [ 1.89MB ]
Metadata
JSON: 831-1.0103752.json
JSON-LD: 831-1.0103752-ld.json
RDF/XML (Pretty): 831-1.0103752-rdf.xml
RDF/JSON: 831-1.0103752-rdf.json
Turtle: 831-1.0103752-turtle.txt
N-Triples: 831-1.0103752-rdf-ntriples.txt
Original Record: 831-1.0103752-source.json
Full Text
831-1.0103752-fulltext.txt
Citation
831-1.0103752.ris

Full Text

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' 1^ ' 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) 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0103752/manifest

Comment

Related Items