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Some applicatins of the quantum theory of magnetism Paquette, Guy 1953

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SOME APPLICATIONS OF THE QUANTUM THEORY OF MAGNETISM  by GUY PAQUETTE  A Thesis Submitted i n P a r t i a l Fulfilment ®f the Requirements f o r the Degree of MASTER OF ARTS i n the Department of Physics  We accept t h i s thesis as conforming to the standard required f o r the degree of MASTER OF ARTS  Members of the Department of Physics THE UNIVERSITY OF BRITISH COLUMBIA September, 1953-  ABSTRACT  -  Part I -  The process of the establishment of the equilibrium value of the magnetic moment, when the i n t e n s i t y of the magnetic f i e l d i s suddenly changed, i s discussed f o r a system of two i n t e r acting spins,.  F i r s t , some of the results obtained by previous  writers (Waller, Eisenstein) f o r the case i n which the distance between the two spins remains constant, are summarised.  Then,  thermal o s c i l l a t i o n s of the spins are introduced into the c a l c u l a t i o n s , not as such, but as a modification of the equation s a t i s f i e d by the density matrix of the system.  In t h i s way, an expression  for the time dependence of the average value of the magnetic moment i s obtained, and i t i s shown that the result includes Eisenstein*s expression as a special case.  I t s asymptotic value (t-»-oo) i s  i d e n t i c a l with the s t a t i c value o f the magnetic moment corresponding  to the new value of the suddenly changed magnetic f i e l d . -  Part I I -  A theory, proposed by Opechowski, gives the change of the paramagnetic rotation caused by introducing a high frequency magnetic f i e l d perpendicular to the steady magnetic f i e l d .  To  get a more detailed information about t h i s change, the theory was applied to the case of Ni S i F  6  6H 0, f o r 9G°K, 9° K and 1.9°K, 2  making reasonable assumptions about the s p i n - l a t t i c e relaxation constant.  ACKNOWLEDGEMENT  The author wishes to express his gratitude to Professor W. Opechowski f o r suggesting the problems and for many h e l p f u l discussions and c r i t i c i s m s . The author i s also much indebted to the National Research Council of Canada f o r generously granting him a Bursary and a Studentship without which the present work could not have been completed.  FOREWORD  The present thesis deals with two problems pertaining to the quantum theory of magnetism.  Both problems have to do  with the s p i n - l a t t i c e relaxation time.  In both cases, the  basic equations are written i n terms of the quantum mechanical formalism of the density matrix. In Part I (the " f i r s t " problem), the process of the establishment of the equilibrium value of the magnetic moment, when the i n t e n s i t y of the magnetic f i e l d i s suddenly changed, i s discussed f o r a simple quantum mechanical system. In Part II (the "second" problem), Opechowski's theory of the influence of paramagnetic  resonance on the magneto-  o p t i c a l effects i s applied to the case of a c r y s t a l of Ni S i F  6  .6 H 0 . 2  These two problems are d i f f e r e n t i n t h e i r physical nature, but they have some connection through the mathematical way they are treated.  In both cases, a modification of the  time v a r i a t i o n of the density matrix i s employed i n order to introduce i n an i n d i r e c t way into the calculations such propert i e s of the systems studied that would be too d i f f i c u l t to take into account d i r e c t l y i n the Hamiltonian.  CONTENTS  Page ABSTRACT  i  ACKNOWLEDGMENTS  i i  FOREWORD  i i i PART I - Establishment of the S t a t i s t i c a l Equilibrium i n a System of two Spins when an External Magnetic F i e l d i s suddenly changed. 1  INTRODUCTION SECTION I -  System of Two Interacting Spins at a Fixed Distance from Each Other  7  A - Energy eigenvalues of the System......  7  B - Average value of the Magnetic Moment of the System SECTION I I - System of Two Interacting Spins Undergoing Thermal O s c i l l a t i o n s (  12 20  A - Introduction  20  B - Case of « =f  23  a) Magnetic moment matrix and s t a t i c value of the magnetic moment  23  b) Density Matrix p and average value "Mz of the magnetic moment.... 28 1 ° Calculation of_ f> 29 2° Average value M 32 3° Discussion of the expression f o r Hz 39 0  z  C - Case of 6 = 0  43  D - Conclusion  48  CONTENTS  CONTINUED  Page  PART I I - Influence of the Paramagnetic Resonance on the Faraday Rotation; Discussion of an Example.  1 - Introduction  49  2 - Formulae f o r the Case of Ni S i F 6H 0  54  3 - Numerical Values and Conclusion ...  59  6  REFERENCES  2  62  - PART I -  Establishment of the S t a t i s t i c a l Equilibrium i n a System of Two Spins when an External Magnetic F i e l d i s Suddenly Changed  INTRODUCTION  A given magnetic substance has, f o r a given constant external magnetic f i e l d and a given temperature, a well defined value of the magnetic moment i n the d i r e c t i o n of the f i e l d .  If  the f i e l d i s changed suddenly to another constant value, the value of the magnetic moment corresponding to the new value of the f i e l d establishes i t s e l f , under normal circumstances, i n an extremely short time.  These are the elementary experimental  facts. However, t h i s extremely short "relaxation time" i s f i n i t e , and i t plays an important role i n experiments with high frequency f i e l d s .  Suppose that one applies a high frequency  f i e l d to a magnetic substance.  One might thus induce transitions  between the energy l e v e l s of the atoms i n the substance, i f those l e v e l s have the appropriate separation.  Every allowed t r a n s i -  t i o n between two l e v e l s w i l l give r i s e , then, to an absorption l i n e , and i t happens that f o r many substances the shape of those l i n e s i s determined mainly by the "relaxation time" or time i n t e r v a l between the v a r i a t i o n of an external steady f i e l d and the subsequent establishment of the magnetic moment corresponding to the new value of the f i e l d .  Such a s i t u a t i o n takes place,  f o r instance, i n the "paramagnetic resonance method" of studying  2  the  magnetic properties of a c r y s t a l .  This method w i l l be  described shortly i n Part II of t h i s t h e s i s . It i s p r a c t i c a l l y necessary, i n calculations involving  a magnetic substance, to replace the complicated system of  interacting atoms forming the substance by an equivalent, or nearly equivalent, simple model.  As a model f o r a system of  atoms, one can think of a system of p a r t i c l e s possessing an i n t r i n s i c magnetic moment, or "spin", so that the magnetic i n t e r a c t i o n between the atoms of the substance i s represented i n the model by the magnetic i n t e r a c t i o n between the "spins". The atoms of a substance also undergo thermal motions.  If  one wishes to take these motions into account i n a model, one can assume that the p a r t i c l e s forming a "system of spins" also o s c i l l a t e with time.  When a magnetic f i e l d applied to the  system i s suddenly changed, an exchange of energy between the system of atoms and the thermal motion must take place i n order that the magnetic moment of the system reach an equilibrium value.  Therefore, i f the thermal motions are not taken into  account, one can expect that, when the external f i e l d i s suddenl y changed, no establishment of the equilibrium can take place i n the system.  Because of t h i s role of the energy exchange,  i t would seem that f o r a system of spins whose distance from each other does not depend on the time (no thermal motion), the  relaxation time would be i n f i n i t e ;  i n other words, f o r  the  new value of the suddenly changed external f i e l d , no well  3 defined value of the magnetic moment would be obtained; i t s calculated expression would be, f o r instance, an o s c i l l a t o r y function of the time. The f i r s t t h e o r e t i c a l investigations concerning obvious point were made by Waller  (W) i n 1932,  that  on a system of  N interacting spins, assuming that t h e i r distance from each other remained constant.  Waller applied an external magnetic  f i e l d to the system, and, assuming that the energy of the spins i n the f i e l d i s small compared with t h e i r magnetic i n t e r a c t i o n , he could calculate approximately the quantum mechanical expectat i o n value of the magnetic moment f o r three d i f f e r e n t cases of varying external f i e l d .  For the case i n which we s h a l l be  interested - that i s , when the f i e l d i s suddenly changed - (stepfunction of the time), Waller calculated a root mean square value f o r the o s c i l l a t i n g part of the magnetic moment of the system of N spins.  He showed that this average value i s small  i n comparison with the s t a t i c part.  With the help of these  approximations, he could conclude that a "spin - spin relaxation" process (that i s , an establishment  of the magnetic moment) exists  for not too low temperature and f o r an external f i e l d small i n comparison with the i n t e r n a l f i e l d .  Therefore his investiga-  tions, from the point of view of "physical i n t u i t i o n " , did not lead to clear-cut conclusions  ( c f . also Broer (B5) )•  In a recent paper (E), Eisenstein considered, as a very s i m p l i f i e d model of a paramagnetic c r y s t a l , a system of  ,  4  two interacting spins, f o r which he could calculate exactly the  eigenvalues and eigenfunctions, and determine the exact  time dependence of the magnetic moment f o r some of the cases studied by Waller.  The distance between the two spins was  also assumed to remain constant.  The expressions Eisenstein  obtains f o r the magnetic moment, when the external f i e l d i s varied, are o s c i l l a t o r y functions of the time i n every case, and they agree with Waller's approximate formulae i f we r e t a i n only the terms proportional to the f i r s t power of the f i e l d . However, the o s c i l l a t i o n s of the magnetic moment cannot be neglected, and Eisenstein concludes that no relaxation process takes place i n a system of two interacting spins and probably also i n a system of N spins.  As he points out, the procedure  used by Waller f o r the case of a suddenly turned on external constant f i e l d i s misleading;  i t may happen that the amplitude  of the o s c i l l a t o r y part of the magnetic moment cannot be neglected i n comparison with the s t a t i c part, while the root mean square value as calculated by Waller can be neglected as Waller did. The conclusions Eisenstein draws from the comparison of his e x p l i c i t calculations f o r the system of two spins with Waller's approximate formulae are as follows: " ( 1 ) I f the external f i e l d i s suddenly turned on, the magnetic moment of a system of interacting spins w i l l never reach an equilibrium value. ( 2 ) The high degree of degeneracy of such a system makes the application of Waller's formulae somewhat uncertain.  (3) In large f i e l d s , no spin - spin relaxation can take place i n a r i g i d l a t t i c e . " The obvious way of modifying Eisenstein's assumptions, so as to obtain a f i n i t e relaxation time, would be to introduce the thermal o s c i l l a t i o n s into the c a l c u l a t i o n s .  This could be  done as follows: a.  In a semi-classical way, by considering the d i s -  tance between the two spins as time-varying. b.  In a quantum mechanical way, by assuming the two  spins to behave l i k e quantum mechanical harmonic o s c i l l a t o r s . However, both methods lead to rather complicated mathematical problems.  To avoid t h i s complication, we have carried out a  more modest programme:  we have modified E i s e n s t e i n s equations 1  i n a phenomenological way (introducing from the very beginning a " s p i n - l a t t i c e relaxation constant" solved them exactly.  as a given parameter) and  We obtain i n t h i s way an expression f o r  the magnetic moment, which, f o r t-*oo , becomes equal to the expression f o r the s t a t i c value of the magnetic moment c a l c u l a ted with the new value of the external f i e l d .  (By " s t a t i c  value" of the magnetic moment of a substance i n the presence of a f i e l d , we s h a l l mean the value calculated under the assumption that the constant  f i e l d has been applied to the system for a  time i n f i n i t e l y long, and has never been changed).  In other  words, the magnetic moment reaches i t s correct equilibrium value.  In Section I, E i s e n s t e i n s calculations are 1  ummarised to the extent necessary for our own escribed i n Section I I .  calculations  SECTION I  Magnetic Moment of a System of Two Spins when a Constant External F i e l d i s Suddenly Applied  A - C a l c u l a t i o n of the energy eigenvalues two spins.  o f a system o f  The Hamiltonlan 2f f o r a system of two p a r t i c l e s of spin 2 i *  1  a  n  external f i e l d H p a r a l l e l to the z-axis i s composed  of two parts:  (1)  The f i r s t part, 3 f , i s the Hamlltonian for the spin-spin $s  interaction:  ss where  *  * V *  0*  5  - 3  (of . 2 ) 1 5 1 • * ) )  ,  (2)  g  i s the Lande factor f o r the system,  £  i s the Bohr magneton,  To  i s the vector from p a r t i c l e 1 to p a r t i e l e  d~  and C are the two spin matrices two p a r t i c l e s , i n units of M, (2TTJJ = h, the Planck constant). t  for the  8  In the representation i n which the z-components o j and  0£  x  of  and  b e i n g i l ) , 5^ and  o| are taken to be diagonal (the eigenvalues  o| can be written as follows, i f the rows and  columns are labelled C72  =  a  n  q = l,Cg 1;  -1, <£  d  x  =  t  Ol - -1,  -i<  I  0  o  o  o  k  o  o  o  o  0  o O  f  0  -K  0  0  s.  =  t  -1?  0 =  07 l»  -1 respectively: 1 - 7  I  <£ = 1;  x  <  CT;  t  t  *» lit  #•  o -1  where i , j and k are the three unit vectors along the three axes Ox, Oy and Oz. The t o t a l spin and the z-component of the t o t a l spin of the system can be diagonalized by the canonical transformation  0  0  0  0  2-'  0  0  - c  1  Q  1 Si  0  0  0  .'-'V  (3)  9  When S-L i s applied to <^ > we obtain the result ( c f also (PjlJ ss  and  (A) ):  0 0 0 0  0  0  0  This matrix can be diagonalized with the help of a transformaIn polar coordinates, S2 i s :  d  St-  0  ~ C ^aUt 0  0  0 0  0  0  i  where 0 i s the angle between the Ox axis ( d i r e c t i o n of the f i e l d H) and the l i n e joining the two s p i n s 5  and  (a i s the angle between the yz plane and the plane containing Oz and the two spins.  (4)  The result i s :  0  0  0  S> Sj 3f Sj Sj, = 0  4A  0  0  0  0  0  0  -2A ss  where  0 0  o  -3  A =  9  We see that when the external f i e l d i s zero, the possible energies of the two - spin system correspond to one doubly degenerate l e v e l and two non-degenerate ones. The Zeeman part of the Hamiltonian ?f i s  for a f i e l d H p a r a l l e l to the z-axis.  I t i s diagonal i n the  o r i g i n a l representation and becomes, after  and S are 2  applied to i t ,  o 1  0  o  O  o  o  o  o  o  o  o  o )  11  The t o t a l Hamiltonian 2{ , (Eq. ( 1 ) ), i n this representation, leads to the secular equation:  E[E 3  -12A*E -16A 3 - E(jPH)* •aA(^H)(3«^0-i)] =0  ( 8 )  which can be solved exactly. For the simple case 6 = 0 (that i s , when the l i n e joining the two spins i s p a r a l l e l to the d i r e c t i o n of the external f i e l d ) , the solutions of Eq. (8) give, f o r the diagonalized t o t a l Hamiltonian,  0  0  For 0 = ~  0  0  0  0  0  0^  0  0 (9)  0 0  01  , one finds from Eq. (8):  o  A+VSAM^M*  0  0'  0  0  -2A  0  (10)  12 For the case  9  = TT ( i . e . when the l i n e joining  2 the two spins i s perpendicular to the d i r e c t i o n of the f i e l d ) , the application of a f i e l d H changes the f i r s t two of W$s  given by Eq. (5).  eigenvalues  Because the ease 0 = E  w i l l be  considered i n d e t a i l , and because i t w i l l be convenient to have only the upper l e f t part of the matrices involved i n the calculations, we s h a l l keep the l a b e l l i n g of the rows and columns used by Eisenstein,-and write the eigenvalues i n the order shown i n Eq. (5)*  B - Average value of the magnetic moment of the system of two spins.  The quantum mechanical average value of the z-component of the magnetic moment was  calculated by both Waller and  Eisenstein, with the help of the density matrix formalism, (T), p 327).  (cf.  A short survey of the fundamentals of t h i s  formalism i s given below. In c l a s s i c a l mechanics, one can represent the state of a system of  f  degrees of freedom by the p o s i t i o n of a point  i n a position - momentum (q - p) phase space of 2f dimensions. An ensemble of such systems can be represented by a "cloud" of  13 s u c h p o i n t s d i s t r i b u t e d w i t h t h e d e n s i t y j? .  j> i s n o r m a l i z e d  to unity i f  ^•••^p dLcji-.. dcj|i d p i ••• «ip^  The F  mean v a l u e  1 •  =  (11)  f o r t h e systems i n t h e ensemble o f any f u n c t i o n  ( q , p ) o f t h e p o s i t i o n and t h e momentum c a n be c a l c u l a t e d  S i m i l a r l y (N), a q u a n t i t y c a l l e d w i t h components  c a n be i n t r o d u c e d  p l a y a r o l e s i m i l a r t o t h e one p l a y e d mechanics.  the density matrix p,  i n quantum m e c h a n i c s t o b y t h e d e n s i t y j> i n c l a s s i c a l  The quantum m e c h a n i c a l e q u i v a l e n t  t h a t t h e sum o f t h e d i a g o n a l t r a c e , i s equal  thus:  o f Eq. (11) i s  elements o f the d e n s i t y m a t r i x , o r  t o 1*.  I n a s i m i l a r way, t h e quantum m e c h a n i c a l e q u i v a l e n t I s , f o rany operator  7  o f Eq. (12)  F,  * £tfF)™n  = Tn.(fF)  ,  w h e r e f> and F a r e w r i t t e n i n t h e same r e p r e s e n t a t i o n . For  a c a n o n i c a l ensemble, t h e d e n s i t y m a t r i x i s g i v e n  (14)  14  ( i n a system o f r e p r e s e n t a t i o n i n which t h e energy - operator i s diagonal) by:  pm* - f«  C  i  (15)  - M T )  (  1  6  )  The E ' s a r e t h e e n e r g y e i g e n v a l u e s o f t h e s y s t e m . n  The d i s t r i b u t i o n ponds t o t h e t h e r m a l  g i v e n b y E q s . (15) a n d (16) c o r r e s -  e q u i l i b r i u m o f t h e system.  I n our case,  where a n e x t e r n a l m a g n e t i c f i e l d H i s s u d d e n l y a p p l i e d , s u c h a d i s t r i b u t i o n w i l l n o t be p r e s e r v e d , and we need t h e e q u a t i o n w h i c h d e s c r i b e s t h e time dependence o f t h e d e n s i t y m a t r i x i n order to calculate i t .  This equation i s , according t othe  f u n d a m e n t a l p r i n c i p l e s o f quantum s t a t i s t i c a l  =  )£f - fX  mechanics:  >  (17)  where # i s t h e t o t a l H a m i l t o n i a n o f t h e s y s t e m ( E q . (11) ). E q . (17) i s t h e quantum a n a l o g u e o f L i o u v i l l e ' s classical statistical  theorem i n  m e c h a n i c s ( ( T ) , p 335).  When t h e d e n s i t y m a t r i x i s known b y s o l v i n g E q . (17)? t h e a v e r a g e v a l u e S z o f t h e z-component o f t h e m a g n e t i c moment c a n be c a l c u l a t e d  b y a p p l y i n g E q . ( 1 4 ) t o t h e o p e r a t o r Mz:  h  t  = Tt CflO 7  US)  15 0  w h e r e Mz i s t h e o p e r a t o r According  t o Eq.  (7),  o f t h e m a g n e t i c moment  when  9U  i s diagonal,  -<a«*\0  o  Mz  SS  =  0  O  O  0  0  O  O  o  o  0  c  so t h a t E q . ( 1 8 ) g i v e s , i n t e r m s o f t h e d e n s i t y  (19)  matrix  elements,  (20)  I t can be shown that Mz i s i r P v a r i a n t with respect to the representations i n whichp and M can be written. z  The  process  of applying the external f i e l d  i s des-  c r i b e d by:  The  H  =  0  H  =  Ho  (t  <0J  (t  »0)  i n i t i a l d e n s i t y m a t r i x elements a r e chosen so as t o  describe the thermal (  = 0).  (Eq.  (5) )•> t h e y  e q u i l i b r i u m i n a zero  external  field  I n the representation i n which J ^ i s diagonal c a n be w r i t t e n a s f o l l o w s :  16  o  where  _  © r\  ^-(Wkr)  6  P  -  -  (21)  E  J. s 1 The E  f n  s are, here,  the eigenvalues  given by Eq.  (5).  I f one w r i t e s E q . ( 1 ? ) e x p l i c i t l y i n t h e r e p r e s e n t a t i o n i n w h i c h ?€  i s d i a g o n a l , one o b t a i n s , f o l l o w i n g E i s e n s t e i n ,  ss  a system o f simultaneous  equations  f o r the d e n s i t y matrix  elements:  = ( f i t - ?u)  R  0  ^ d  -Ltt f i t » ( f i t - f a t )  ^  i t \ ?ia • .( f « - fj»)  ft.  +  6  ( f « ~ ?n)R<,«"*  -  - ?u  1) ,  - f>tiK '^  2) 3)  d  0  (22)4) HX^H  -  - C ?*2 - f i t )  ilifta  -  *fta  It* fat  *  -  i-ti f s i  -  - * ?3* +  ^»  a  + f i ft a  5)  Ro 0  <i^e  -  " fas) fto<~*d +fjtRo^e  ' C ? u - f a i ) ft© -"^  where  The m a t r i x e l e m e n t s c a l c u l a t i o n of" Mz b y E q . ( 2 0 ) .  -  fji R fa  7) 0  8)  f\ «*+>* a  d  R<> -  *  6)  Pi©  H  = 6A  and p^-  0  -  do n o t e n t e r i n t o t h e  Therefore,  one d o e s n o t n e e d  ^  8  9)  17 to  evaluate  them.  E q s . (22) Laplace for  c a n be s o l v e d e x a c t l y w i t h t h e h e l p o f a  transform solution.  F o r 0 = 0 and 0 = -jr » t h e e x p r e s s i o n s  t h e a v e r a g e v a l u e o f t h e m a g n e t i c moment, a s c a l c u l a t e d  Eqs.  (20) and (22), a r e p a r t i c u l a r l y  from  simple.  When 0 = 0, one f i n d s :  M,  =  -?P( f« * fw) = 0 •  The v a l u e 0 f o r t h e m a g n e t i c moment Mz a t 0 f a c t , f o r any time-dependent initial  c o n d i t i o n (21) i s u s e d .  show t h a t , f o r 0 = 0, p  i3  R  31  0  field  E q s . (22)  - 3) and (22) -  31  =0.  , one i s l e / ( d t o t h e e x p r e s s i o n :  St  -aocqpR.S'HfrfjU 2  where S = ( 4 R given by Eq. It  7)  of the value of  o *  arises, i n  a p p l i e d a t t = 0, when t h e  = - 0 , independently  a s l o n g a s fc{0) = f (0) When 0 =  =0  Q  2 V? + cL )  and  o  • t  1  ' " ~ T \ ,  and  «> ^  (2 ) 3  are  (21). c a n be shown t h a t t h e v a l u e g i v e n b y E q . (23)  will  remain o s c i l l a t o r y , independently o f the choice o f the i n i t i a l c o n d i t i o n on t h e m a t r i x ^ .  One c a n i n f e r a l s o t h a t t h e  18 e x p r e s s i o n f o r Mz w i l l r e m a i n o s c i l l a t o r y f o r t h e c a s e o f a general angle 0  between the d i r e c t i o n o f the e x t e r n a l f i e l d  and t h e l i n e j o i n i n g t h e two s p i n s .  T h e r e f o r e , no  o f t h e s t a t i c v a l u e o f t h e m a g n e t i c moment o c c u r s two i n t e r a c t i n g stant f i e l d , constant w i t h  spins a f t e r the application  establishment  i n a system of  o f an e x t e r n a l con-  i f t h e d i s t a n c e b e t w e e n t h e two s p i n s i s k e p t time.  0  SECTION I I  M o d i f i c a t i o n of the equation f o r the Density M a t r i x , a n d t h e Time D e p e n d e n c e o f t h e M a g n e t i c Moment f o r a s y s t e m o f Two S p i n s  A  In  -  Introduction  order t o d e s c r i b e t h e passage from t h e s t a t i e  v a l u e o f t h e m a g n e t i c moment c o r r e s p o n d i n g  t o a given value, o f  the f i e l d ,  t o the s t a t i c value corresponding  the f i e l d ,  i ti s necessary,  to  t o a new v a l u e o f  a s we h a v e s e e n i n t h e I n t r o d u c t i o n ,  make t h e d i s t a n c e b e t w e e n t h e two s p i n s ffi v a r y w i t h  I f we c o u l d e x a c t l y t a k e i n t o a c c o u n t Hamiltonian  time.  these o s c i l l a t i o n s i n the  , then t h e d e n s i t y m a t r i x f o r t h e o s c i l l a t o r y two-  s p i n s y s t e m would have t o s a t i s f y t h e e q u a t i o n :  • i>*f' >  #/> -  ".  (24)  H o w e v e r , t h e m o d i f i c a t i o n o f 2£ ( E q . ( 1 ) ) , e i t h e r i n a c l a s s i c a l o r a quantum m e c h a n i c a l mathematical  semi-  way, l e a d s t o c o m p l i c a t e d  problems i n s o l v i n g Eq. (24).  A n i n d i r e c t way o f  i n t r o d u c i n g " l a t t i c e " v i b r a t i o n s would be t o m o d i f y ,  rather,  the time v a r i a t i o n o f the d e n s i t y m a t r i x , t h a t i s , t o modify Eq.  (24) i t s e l f .  The m o d i f i c a t i o n o f t h e e q u a t i o n f o r j>  should take i n t o account  t h e f a c t t h a t , when t h e e x t e r n a l f i e l d  2G  i s v a r i e d , the instantaneous Suppose t h a t i m m e d i a t e l y magnetic f i e l d  H  e q u i l i b r i u m c a n n o t be  f o l l o w i n g the v a r i a t i o n of the e x t e r n a l  ( t ) , at time t  equilibrium occurs.  The  , the establishment of  e  u) =  because the magnetic f i e l d We  ( 2 5 )  depends e x p l i c i t l y  s a t i s f i e s the modified  -  -i(Xf  f  H where  #  Xz a  =  #SS  i n Eq.  i s t o assume t h a t  *  i  - * < r f - >  -  ( 2 6 )  (1):  >  +  < 7) 2  « ^ V [ A o \ - o \ - 3^)1%-/?)] , f  s s  The  equation:  w h e r e o f i s the t o t a l H a m i l t o n i a n g i v e n by Eq.  constant  on  switching-on  instantaneously.  of t a k i n g t h i s f a c t i n t o account  $  The  I-B)t  know t h a t t h e a c t u a l d e n s i t y m a t r i x j> f o r o u r 0  p  para-  (cf. Section  changes d u r i n g the  p r o b l e m w i l l n e v e r r e a c h t h e v a l u e j> s i m p l e s t way  be  -(«lt)/KTJ  w h e r e £f ( t ) i s t h e H a m i l t o n i a n , w h i c h now  process.  will  i t w i l l depend  m e t r i c a l l y o n t h e t i m e t , i n t h e f o l l o w i n g way  t,  the  d e n s i t y m a t r i x , i n t h a t case,  c a l l e d the instantaneous d e n s i t y m a t r i x ;  P  reached.  = $ (26)  H  a  U  i f  H  l s  a  l  o  n  S  t h e  c a n be i n t e r p r e t e d , a s we  z-axis. shall  see  21  a t t h e end o f t h i s S e c t i o n , a s t h e r e c i p r o c a l v a l u e o f t h e (spin-lattice) relaxation time.  I t must be m e n t i o n e d  also,  t h a t t h e p s a t i s f y i n g E q . (26) i s n o t a " d e n s i t y m a t r i x " i n t h e u s u a l sense o f t h e word.  I n fact, a "true" density matrix  s a t i s f i e s b y d e f i n i t i o n E q . ( 2 4 ) , and n o t E q . (26).  However,  we s h a l l assume t h a t t h e ^ s a t i s f y i n g E q . (26) p o s s e s s e s t h e p r o p e r t i e s o f t h e d e n s i t y m a t r i x o f t h e s y s t e m o f two s p i n s with thermal o s c i l l a t i o n s ;  we s h a l l u s e t h a t p a s i f i t w e r e  a d e n s i t y m a t r i x and. s h a l l c o n t i n u e t o c a l l  i t b y t h a t name.  The p r o b l e m o f d e t e r m i n i n g t h e t i m e d e p e n d e n c e o f t h e a v e r a g e v a l u e o f t h e m a g n e t i c moment Mz, i n t h e c a s e when t h e two s p i n s u n d e r g o  thermal o s c i l l a t i o n s , i s thus replaced  b y t h e one o f s o l v i n g E q . (26) a n d c a l c u l a t i n g Mz f r o m Eq\.(18):  F l  Eq.  z  (pMj  =  .  (28)  (26) was p r e v i o u s l y u s e d b y s e v e r a l a u t h o r s i n  connection w i t h various problems.  K a r p l u s and Schwinger ( K l )  used i t f o r t h e c o m p u t a t i o n o f t h e shape o f microwave l i n e s i n a gas.  Frohlich  absorption  ( F ) , b e f o r e them, used i t i n i t s  c l a s s i c a l f o r m , and l a t e r B i j l  ( B 4 ) u s e d E q . (26) i n c o n n e c t i o n  w i t h the paramagnetic resonance a b s o r p t i o n .  Lately,  Opechowski  (02) made t h e same a s s u m p t i o n r e g a r d i n g t h e v a r i a t i o n o f £ i n a t h e o r y on t h e i n f l u e n c e o f the paramagnetic resonance on t h e magneto-optical e f f e c t s ,  ( c f . Part I I )  22  In this field  t h e s i s , we s h a l l assume t h a t t h e e x t e r n a l  H, a p p l i e d a l o n g  the z-axis, varies i n the following  ways  and s h a l l e v a l u a t e  H  =  Hi  H  = H * H* t  ( t < 0)  ,  (t *0)  ,  the exact average value o f Eqs. (26)  moment w i t h t h e h e l p  o f t h e magnetic  and ( 2 8 ) .  The a s y m p t o t i c  s o l u t i o n ( t - f o o ) f o r I z w i l l be compared t o t h e s t a t i c  value  o f t h e m a g n e t i c moment c a l c u l a t e d i n t h e p r e s e n c e o f a c o n stant  field  %  = %  + R"2  applied  t o t h e s y s t e m o f two s p i n s .  The c a s e i n w h i c h t h e l i n e j o i n i n g t h e two s p i n s ular to the z - axis ( 0  =  will  be c o n s i d e r e d  will  be s h o r t l y o u t l i n e d .  7r)>  b e c a u s e i t i s more i n t e r e s t i n g ,  i ndetail.  The more t r i v i a l c a s e 0 = 0  I n o u r n o t a t i o n , f> i s t h e i n s t a n t a n e o u s o  matrix  ( c f . E q . (25) ) and o° w i l l  of the d e n s i t y matrix The r e p r e s e n t a t i o n s  i s perpendic-  represent  the i n i t i a l  ( i . e . the density matrix  i n which the matrices  density  will  form  f o r t < £>>). be w r i t t e n  H be i n d i c a t e d b y a s u p e r s c r i p t . written i n the representation is  diagonal.  T h u s , p ° means t h a t p i s i n which  will  23 TT B  In Eq.  Case o f o  -  =  o r d e r t© c a l c u l a t e t h e m a g n e t i c moment u s i n g t h e  (28), we must  evaluate:  a)  The m a t r i x e l e m e n t s o f t h e m a g n e t i c moment i n some r e p r e s e n t a t i o n  b)  The d e n s i t y m a t r i x i n t h e same r e p r e s e n t a t i o n , b y s o l v i n g E q . (26).  I t w i l l be c o n v e n i e n t  t o use t h e r e p r e s e n t a t i o n i n which t h e  Hamiltonian  W*.  i s diagonal. of a constant  a)  =  X  S  5  *  T h e Zeeman e f f e c t i s d u e h e r e t o t h e a p p l i c a t i o n field  H  Q  t o t h e system.  M a g n e t i c moment m a t r i x a n d s t a t i c v a l u e o f t h e m a g n e t i c moment f o r a c o n s t a n t f i e l d H ~ H^ + H Q  The Eq.  (30)  •».>  eigenvalues  E^° o f <W  Ho  ( 1 0 ) where H i s r e p l a c e d b y H ;  E";  --  e:  -  , f o r &- ^ , a r e g i v e n b y  they  Q  A - O A * R') 1  o  ,  2  are: V I  ,  24  where  The d e n s i t y m a t r i x which  3£  H  p  A  i s diagonal  i s diagonal;  =  ^l*^****  i n the representation i n  i thas, t h e r e f o r e , the elements:  (32)  where  ^  =  ^ £  To f i n d  are the Boltzmann Actors.  -( "7KT) e  e  Mz i n t h a t r e p r e s e n t a t i o n , we h a v e t o c a l c u -  l a t e the eigenfunctions  corresponding  to the  eigenvalues  g i v e n "by E q . (31).  2£  In the representation i n which is  g i v e n b y E q . ( 1 9 ) where & i s p u t e q u a l  ' o  -1  o  - i  0  0  0  o  o  o  o  O  o  o  c»  M, S  L e t us c a l l  <p , A  c p ^ , Cp  3  and  5S  i s diagonal,  to ~  :  c p , t h e f o u r wave 4  Mz  (33)  functions  25 c o r r e s p o n d i n g t o t h e f o u r e i g e n v a l u e s o f 3^ ,  -2A, 4A,  s5  and  0, r e s p e c t i v e l y  ( c f . E q . (5) ) .  T h e n t h e wave f u n c t i o n s  v|; , c o r r e s p o n d i n g t o t h e e i g e n v a l u e s E ° K  K  e a s i l y b y w r i t i n g them a s f o u r e x p a n s i o n s  +K  f?t « C  =  L  In the r e p r e s e n t a t i o n i n which  ^  <^  tc  i  of ^  H e  c a n be  found  o f t h e c^'s:  K«  1,2., 3,4-.  i s diagonal,  '-ZA  -Ro  o  o  -ft.  4A  0  0  0  o  -Zf\  O  o  0  o  o.  x  -2A  (34)  is  N  (35)  t o o b t a i n t h e ^ ' s , we s o l v e t h e e q u a t i o n s o b t a i n e d b y w r i t i n g  R e p l a c i n g *p b y i t s e x p a n s i o n  i n t e r m s o f t h e cj>'s, and u s i n g  t h e v a l u e s g i v e n i n E q . (3D»  we f i n d , f o r E q . ( 3 4 ) , t h e f o l l o w -  K  ing  expansions:  ViC9A * Hi)* l  v  (36)  26  The t r a n s f o r m a t i o n m a t r i x f r o m t h e r e p r e s e n t a t i o n i n w h i c h 2$si i s d i a g o n a l t o t h e o n e i n w h i c h £(L i s d i a g o n a l , i s , t h e r e f o r e :  /(9A^Ro) -3/4 * Vt  2(9A*  +  _  /<9A * 2  li) ** 1  R?) -3A 1/z  (9A* + ft*) S~3A V  2(9A +  2.(9A +ftJ)  2  2  ft*)*'*  (37)  1  °  o  1  W i t h t h e h e l p o f E q s . (36) a n d (33)? o n e c a n e a s i l y f i n d the only non-vanishing  (*|M.|*o -  m a t r i x e l e m e n t s f o r Mz w i l l b e s  U h J + 0  = - ^ y v r  that  27  The m a g n e t i c moment m a t r i x i n t h e r e p r e s e n t a t i o n i n w h i c h  3£  H o  i s diagonal i s , therefore:  -3A  At t h i s  0  0  0  0  3A  K  0  0  0  0  0  0  0  0  0  (38)  p o i n t , we c a n a p p l y E q . (28) and u s e E q s . (32) a n d (38)  f o r p and Mz, t o e v a l u a t e t h e s t a t i c moment M z system i n a constant  field H . 0  We  s t  t  -  t  o fthe  obtain  (39)  statu where  p^ °  and  ( 9 A + .*S) a  %  p^* a r e g i v e n b y E q . (32).  28  b)  Density matrix o moment '  and a v e r a g e v a l u e Mz o f t h e m a g n e t i c  In t h e s p e c i a l case considered here, t h e f i e l d is  In  H  g i v e n b y E q . (29):  H  = H  H  = Hi + Hi (t *o) ,  (t<0)  t  ,  s o l v i n g E q . (26), i t w i l l be c o n v e n i e n t  t o use t h e r e p r e s e n -  t a t i o n i n w h i c h t h e H a m i l t o n i a n f o r t <0 i s d i a g o n a l .  This  Hamiltonian i s K  and  H a  .  -  +  ^  H  l ( ° l r * ,  i s i d e n t i c a l t o t h e H a m i l t o n i a n 3^  the value o f the f i e l d . in  #SS  i  «iz)  (40)  o f Eq. ( 3 0 ) , except f o r  The m a t r i x f o r t h e m a g n e t i c moment,  t h a t r e p r e s e n t a t i o n , i s g i v e n b y E q . (38) where RQ i s  replaced by  (%  =  g j % ) :  0  0  ft*.  0  0  0  0  0  0  0  0  0  0  -3A  -3A  N  (41)  29 1  In  o  Calculation of o  o r d e r t o s o l v e E q . (26), we s t i l l  evaluate the instantaneous d e n s i t y matrix t i o n i n which  ?£  Hi  f o r m u l a E q . (25).  i s diagonal.  and  i s g i v e n by t h e g e n e r a l  o  .Ht +  -  9% ( t ) i s  c a n be w r i t t e n r e a d i l y i n a  diagonal representation.  rti  i n the representa-  I n our case, the Hamiltonian  c o n s t a n t f o r t > .0, s o t h a t  W  p  need t o  ( t ) i s , f o r t > 0:  #(Hi  + H  * ) U z «fci) + ^ss .  (42)  +  i s i d e n t i c a l t o t h e H a m i l t o n i a n E q . (30), e x c e p t f o r t h e  value of the f i e l d . where R  Q  I t s e i g e n v a l u e s a r e g i v e n by Eq.  i s replaced by  + R  2  (R  = g(JH ).  2  2  (3D  Therefore,  (43) ej ** 1  and  =  -2A  ,  t h e i n s t a n t a n e o u s d e n s i t y m a t r i x e l e m e n t s c a n be w r i t t e n  (44) where - (£*7»T)  30 The i n s t a n t a n e o u s d e n s i t y m a t r i x i s , t h e r e f o r e , d i a g o n a l :  f. o  o  o  0  o  o  o  o  (45)  We s h a l l h a v e t© t r a n s f o r m E q . ( 4 5 ) i n t o t h e r e p r e s e n t a t i o n i n which  2^  i s diagonal.  transformation matrix c^sj  F o r t h i s , we n o t e f i r s t C  1  that the  from the r e p r e s e n t a t i o n i n which 2£ ^  I s d i a g o n a l t o t h e one i n w h i c h  i s diagonal i s given  H  b y E q . (37) w h e r e R  0  i s r e p l a c e d b y R-^.  formation matrix  C *- - f r o m t h e r e p r e s e n t a t i o n i n w h i c h H  +H2  i s d i a g o n a l t o t h e one i n w h i c h Eq.  (37) w h e r e R  Q  Similarly, the trans-  ^m+jU  i s replaced by  5  'Hi  to  i  s  d  + Rg.  ^ S a  o  n  a  l  i  s  S  i  v  e  n  ^7  Therefore, the  be a p p l i e d t o E q . ( 4 5 ) i s g i v e n  by -1  Its  explicit  form i s  0  0  0  0  0  i  0  0  0  1  -Si  fa  S. 0  s°  (46a)  31  where  Vt(9Aft?)  3AT  \ 3 h M ^ H R ^ f ^  (46b)  (46c)  I f the t r a n s f o r m a t i o n Eq. (42) i s a p p l i e d to' the m a t r i x Eq.  ii  ( 4 1 ) , one f i n d s f o r ^  , the instantaneous density  matrix w r i t t e n i n the representation i n which ^  i s diagonal:  H  i  It  where  (a  (C)  - tin + tin, /  (47a)  (47b)  32  (47d)  2°) A v e r a g e v a l u e o f t h e m a g n e t i c moment  We a r e now r e a d y t o s o l v e E q . (26) f o r t h e d e n s i t y matrix p Mz.  a n d f i n d t h e t i m e d e p e n d e n c e o f t h e m a g n e t i c moment  The i n i t i a l  c o n d i t i o n on t h e d e n s i t y m a t r i x p i s taken  t o be •Hi ?  When  c  =  -U  i sdiagonal,  \)  -(**/hT) C  -(**/HT)  '  c a n be w r i t t e n a s f o l l o w s :  /mv/n  \»  I™ °  ,  (48)  . -(E"7KT)  where £  e-^'/KT)  H The E s a r e t h e e i g e n v a l u e s o f t h e H a m i l t o n i a n i,  Ku  ( E q . ( 4 0 ) )'.  (49)  33  The  e x p l i c i t dependence o f t h e average v a l u e o f t h e magnetic  moment o n t h e d e n s i t y m a t r i x e l e m e n t s i s , b y E q s . ( 4 1 ) and (28),  One n e e d s t h e r e f o r e t o s o l v e o n l y f o u r e q u a t i o n s Eq.  of the set  (26) i n o r d e r t o e v a l u a t e Mz. If,  f o l l o w i n g K a r p l u s a n d S c h w i n g e r ( K l ) , we make  t h e change o f v a r i a b l e  (51)  Eq.  (26) becomes  AD = . i f X D - D « ] - aD where K  i s the t o t a l Hamiltonian.  independent o f t h e time  -  I n our case,  the  ^  o  ( 5 2 )  p  o  i s  ( c f . E q . (47) ) , so t h a t t h e l a s t  o f t h e r i g h t h a n d s i d e o f E q . (52) i s z e r o . t u t i o n E q . (5D?  Irfo,  term  With the substi-  does n o t t h e r e f o r e appear e x p l i c i t l y i n  equation.  VL  c a n be w r i t t e n , when t £ 0 , i n t h e f o l l o w i n g way:  K  =  MH±  +  ^ ^ ( q . ^ a )  (53)  34  Therefore, i n the representation i n which  $6  H]L  i s diagonal,  Eq. (52) becomes:  (54)  where  -  7  and the D 's are the elements of the matrix D. equations  f o r the four matrix elements  The four  that we s h a l l need  are, using Eq. (41),  3ARi  where  P  (55a).  " i pHi  w ^ ^ W t  35  By the d e f i n i t i o n of the D 's,  P^z  = fti  - ( ?o*)«t 7  Dai  = ?«i  -  (56)  (tf'L.  Therefore, by Eq. (48), the i n i t i a l conditions on the D^^'s are  (57)  I f we add Eqs (55a) and (55c) we r e a d i l y obtain, with the help of Eqs. (53) ,  o = {[ i n * (n\ M  -  • i c u ) e*  at  It can be shown, with the help of Eqs. (46b), (46c), (47b) and (47c), that  These l a s t two matrix elements are known from Eq. (44).  ( 8) 5  36  The set of Eqs. (55) can be reduced to a set of three equations by adding and subtracting.  D  Z 1  0.1  + Oxz  =  x  ,  Dtt  =  )J  »  -  Let us write  (59)  The i n i t i a l conditions f o r these new functions are:  x(o) 7(0)  0 ,  (60)  zto)  - [<a- t o ] by Eqs. (57).  I f , f o r s i m p l i c i t y , we introduce the notation  (61) 1  * (9A *  tfl*  2  then, the set Eqs. (55) reduces to the following system of equations, i f we remember that  co  al  =. - u> : 1JL  (62)  37  A L a p l a c e t r a n s f o r m s o l u t i o n c a n be p e r f o r m e d s e t E q . (62).  e a s i l y on t h e  I f one w r i t e s :  *<s)  =  (V  x(tut  51s)  = j y ^ y M x  z(s)  = f e  s t  ,  z(t)at  ,  rt  t h e s e t E q s . (62) becomes f o r t h e t r a n s f o r m e d  (s,a)x  5c  -  -z(0)  - x(0)  t iff  -  ( S + *)$  ^2  y  ~ ^  +  functions:  -  "  £  * *  . (S+3)  T h i s l i n e a r s y s t e m o f e q u a t i o n s c a n be s o l v e d (63a)  0 ,  0  >  0  ,  ( 6 3 a )  (63b)  (  6  3  readily;  gives  5 = - j - [ ( s + a)x  s o t h a t (63c)  -  x(o)J  +  1  j  becomes z(Q)  ±  5  x  (  Q  i  I f b o t h t h e s e v a l u e s a r e s u b s t i t u t e d i n E q . (63b), one o b t a i n s f o r  x  :  ~  =  ?(0) + »%(<>)) ~ (sta)(cs*a)H/*V]  x(o)(s+a) *  (sta) + ^ + i a  z  c  )  38  The inverse transformation can be performed e a s i l y ; x\ becomes a f t e r some rearrangements:  f *  •'M'  (64)  In a similar way, one obtains f o r the other two functions:  *> <* +  J>*M  e  (66)  l  To write down the average value of the magnetic moment, we use now Eq. (50).  We know, from Eqs. (56) and  (57), that  f«"f« - Z - [ t f V ( f t ] = and  +  s  V ^ ' A ] - (67)  X - x(o) .  Then, the average value Mz of the magnetic moment becomes, (Eq. (50) ):  M..^J^[«^r-^^3A(x-^)tftl(ri-(flj]  • (68)  39 More e x p l i c i t l y , Eq. (68) can be written (introducing the values of x and z given by Eqs. (64) and (66) ):  Mr-  z(o)  9i>  *w  *3A  e  "  a t  r » . * * « ^ ( * f + » ) * t ) - x(<» l  ,  (69)  3 ) Discussion of the Eq. (69) f o r Mz.  We consider the following three special cases of Eq. (69)* <*)  t = 0  (Value at the i n i t i a l moment)  £)  R^=  and  y)  t j - * 00 (asymptotic value)  0  a-* 0 (Eisenstein's case)  40  c<)  When t = 0, z - z(0) and x = x(0), so that  Eq. (69) becomes:  jvj . _ _ # * v *  (9A  __f/.-^  -<v*)1  (70)  Z  which agrees (as i t should) with the value of the s t a t i c magnetic moment i n the presence of the magnetic f i e l d H i (compare Eq. (39) )• ^) When 1^=0 found by E i s e n s t e i n .  Mi  and a = 0, we should get the value In f a c t , i n that case, Eq. (69) i s  =  -3i*(  x  -  x ( 0 )  ) •  (7D  In x, as given by Eq. (64), we have to replace the following  {  values:  a = 0, h  a co,. = —  % = s© that  f+tff  =  -^T ^" *  5  from Eqs (61) and (55e),  from Eq. (61), s, ~  i n Eisenstein's notation.  Also, from Eqs. (60) and (47d)  x(o) - -HOa  = - 2 S & & f ? H £ ) J , ix H - 0 . X  41  Therefore, from Eqs. (46b) and (46c),  Furthermore, from Eqs. (60) and (47),  from Eqs. (46b) and (46c).  I f these values are substituted i n the expression x - x(0), we get, f i n a l l y : R«  = -2s?«R»S'*(jH;Xi  '  (72)  which i s the value found by Eisenstein. y ) When. t~»oo , z-* 0  and x-*0, so that Eq. (69)  becomes  from Eqs. (60) and  (4?)  42  from Eqs. (46b) and (46c).  Therefore, the asymptotic value f o r Mz i s  This value i s i d e n t i c a l to the expression f o r the s t a t i c moment, Eq. (39)» i f we write  R  i +  R  = 2  R  o*  I  n  o  t  n  e  r  words, the  asymptotic s o l u t i o n Eq. (73) does give the correct value f o r the s t a t i c magnetic moment.  43  C - Case of 6 = 0  When the l i n e joining the two spins i s p a r a l l e l to the z-axis ( 6 = 0 ) the s o l u t i o n of the problem i s similar to the one discussed i n the case o f 6 a | , but i t i s much simpler. I f the f i e l d i s applied i n the following way:  H « Hi  (t <o)  H - Hi+ H  1  (,t >, o) ,  t  i t can be shown, with the help of Eq. ( 2 6 ) , that the asymptotic s o l u t i o n f o r Mz i s also equal to the s t a t i c value of the moment calculated f o r the case of a constant f i e l d H = H + H . o 1 2 In the representation i n which  ^Ho ^ Xss * #H (<r *<r ) , 0  u  2x  ( H constant), 0  i s diagonal, the matrix f o r the magnetic moment, corresponding to Eq. (38) i s  i  o  o  o  0  0  0  0  o  o  -a  o  (74)  44  The eigenvalues  of «  , f o r 0 = 0, are  H  =  4  A  (75)  1  where R  Q  = gj£R"  c  The s t a t i c moment i n the presence of ET i s , therefore,  M  w < =  - s ' t ^ - r f - J ,  (  7  6  )  „"( "%T)  H.  £  where  It can be shown also that, f o r the case 6s§, the transformation matrix The instantaneous  C  (Eq. (46) ), i s a unit matrix.  H l  density matrix p can therefore be written o  r e a d i l y i n the representation i n which %  u  i s diagonal.  It i s  of the usual form:  (77)  where  The  £  1  1  by R-, + R« •  =  ^  ^  /  h  T  '  ,  's are given by Eq. (75), where R^ i s replaced  45  In the solution of Eq. (26), one uses the i n i t i a l condition  U  where  /« O^r*. ,  VI  I mm  "(£/«•)  e  if 1  -  £  (78)  e  -( "VhT) £  H  The  8^  's are given by Eq. (75), where R5 i s replaced by R i . The magnetic moment w i l l be, i n terms of the density  matrix elements:  ~  ~ 3@ C ? i i ~ ?3s) j  from Eq. (74). (79)  Therefore, we need to solve only two of the equations (26) i n order to evaluate Mz.  These equations are, i f one makes the  substitution:  D  =ft,- Crf **)! , 1  tt  and uses Eq. (74):  (ft • ) °* = a  (^  +  so that the solutions are  a)D  s3  0  = 0 •,  ° CoVe u  Ptl(O)  =  Co)  =  o  n  -at -at  Now,  so that Eq. (74) gives, explicitly'.  rat  For t = 0, we  [[rt-(fU]-f(rt(rtl]Hr|-(r-i obtain  which i s the expected value.  For  = 0 and  a  = 0, (Eisenstein's case), we  obtain  because of the zero f i e l d degeneracy (levels 1) and 3) coincide ( c f . Eq. (75) ) ).  47  For t -»oo,  which i s the s t a t i c value of the moment, as given by Eq. (76), I f one writes  + H  2  = H . 0  Therefore, l i k e Eq. (69) f o r the case 6 = X , Eq. (75) gives the correct time dependence of the magnetic moment f o r 6 =• 0 .  48  D  -  CONCLUSION  Eqs. (69) and (80) show i n a l l d e t a i l s how the average magnetic moment of the system of two spins changes from the s t a t i c value corresponding to the f i e l d  to the  s t a t i c value corresponding to the f i e l d H 2 , when the change of the f i e l d i s sudden.  The significance of the parameter a,  introduced into the equation s a t i s f i e d by the density matrix, appears now c l e a r l y .  A glance at Eqs. (68) and (80) shows - at  that the constant a, which appears i n the factors &  , can  be interpreted as the r e c i p r o c a l value of the relaxation time for the process of the establishment of the equilibrium. In the case where the l i n e joining the two spins makes an a r b i t r a r y angle with the d i r e c t i o n of the applied magnetic f i e l d , the problem could be formulated as i n the l a s t Section, and a Laplace transform solution could be found f o r the equations that would arise from Eq. (26). computations  However, the  would be tedious, since the expressions f o r the  energy eigenvalues are complicated i n the general case ( c f . Eq. (8) ).  -  PART  II -  Influence of the Paramagnetic Resonance on the Faraday Rotation; Discussion of an Example  49  1  - Introduction  As pointed out by K a s t l e r , (K2) i n a b r i e f q u a l i t a t i v e discussion, the paramagnetic Faraday r o t a t i o n of v i s i b l e l i g h t , as observed i n a paramagnetic s a l t , changes considerably under the influence of paramagnetic resonance.  In other words,  the rotation angle w i l l change i f the paramagnetic s a l t i n question i s subjected to microwave radiation the frequency of which corresponds to an allowed magnetic dipole t r a n s i t i o n between the energy levels of paramagnetic ions responsible f o r i the Faraday r o t a t i o n .  A necessary condition f o r this effect to  take place i s that the occupation numbers of the lowest energy l e v e l s of these ions must be d i f f e r e n t ;  this situation arises,  f o r example, i f the temperature of the medium i s low enough. This s i t u a t i o n was discussed q u a l i t a t i v e l y by Kastler f o r the case of n i c k e l f l u o s i l i e a t e Ni S i F$ 6H 0. ( K 2 ) . 2  He pointed  out that the r o t a t i o n angle f o r the polarized l i g h t (Faraday e f f e c t ) i n t h i s paramagnetic s a l t should change considerably i f a microwave r a d i a t i o n i s applied to the c r y s t a l of Ni S i F  &  6H2O, such as to induce transitions between the energy levels of  the paramagnetic ion N i  + +  responsible f o r the Faraday rotation.  A general theory of t h i s effect has recently been given by Opechowski ( 0 2 ) , who considers the general case of a medium composed of unit systems, such as paramagnetic ions, i n which two electromagnetic radiations of widely d i f f e r e n t frequen c i e s , such as l i g h t and a high frequency magnetic f i e l d , are  50 propagated.  Opechowski bases h i s theory on the equation  *J  -. - i ( W - H ) P  P  - a( -f.) ,  (1  P  which was discussed i n Part I of t h i s t h e s i s .  >  The meaning of  the symbols i s very similar to what i t was before, except, of course, f o r the Hamiltonian VI ,  The Hamiltonian i s , this time,  given by the following expression:  (2)  where "jp and *n are the e l e c t r i c and magnetic dipole moment operators; E and u> are the amplitude of the e l e c t r i c vector and the angular frequency of l i g h t ; p  -»  Hand cO^are, s i m i l a r l y , the amplitude of the highfrequency magnetic f i e l d and i t s angular frequency. $  0  i s the Hamiltonian f o r a system i n the absence of both radia-  tions.  I t contains the homogeneous external magnetic f i e l d ,  and also terms such as the c r y s t a l l i n e f i e l d and other possible time independent Ejj..  perturbations.  We denote i t s eigenvalues by  In the absence of both radiations, the elements of the  density matrix p can therefore be written i n the form of the Boltzmann population factors f o r the energy levels E^: 0  pv**.  ~  ° ST > -( "/KT) £  e  where  <?  VY>  -  * —-  ;  (3)  51  Opechowski assumes that the i n t e n s i t y of the high-frequency magnetic f i e l d i s not too high, and solves Eq. (1) f o r the case when the frequency of the high-frequency f i e l d i s i n resonance with the Bohr frequency corresponding to an allowed t r a n s i t i o n between two energy levels E . k  The solution p  o f Eq. (1) i s  the density matrix of a unit system of the medium. The three magneto-optical effects that can be observed on the l i g h t , that i s , the Faraday rotation, the birefringence and the absorption, depend on the average p o l a r i z a b i l i t y tensor of the unit systems of which the medium consists.  The p o l a r i z -  a b i l i t y tensor i s , by d e f i n i t i o n , given by the tensor r e l a t i o n between the e l e c t r i c dipole moment ^ of a unit system and the e l e c t r i c vector of the l i g h t wave.  The average value "ft  of  the e l e c t r i c dipole moment can be written as follows ( c f . Part I ) :  ~f  =  T A ( ? ? )  ,  (4)  where p i s the density matrix of the unit system, as obtained from Eq. (1). matrix p  In the approximation used by Opechowski, the  w i l l depend l i n e a r l y on the e l e c t r i c vector of the  l i g h t wave, so that Eq. (4) i s a c t u a l l y the tensor r e l a t i o n defining the p o l a r i z a b i l i t y tensor.  When a high-frequency  magnetic f i e l d i s also applied, additional terms i n the expression f o r the p o l a r i z a b i l i t y tensor f o r l i g h t w i l l appear.  These  terms, calculated by Opechowski, give the influence of the highfrequency f i e l d on the magneto-optical e f f e c t s .  52 I f we due  -fa  c a l l A11^  the correction to the Faraday rotation  to the transitions induced by the high-frequency f i e l d  H cos cO t between two m  l e v e l s a and  b of the ion responsible f o r  the Faraday e f f e c t , then the t o t a l rotation fl^ w i l l be given by  = A +  ill A.  •  (5)  i s the Faraday rotation i n the absence of the high-frequency  field.  where f L  I t i s given by the sum  over a l l energy l e v e l s  i s the contribution of the l e v e l n to the  and p* i s i t s population factor, given by Eq.  rotation  (3).  _ b  The  correction / i u  f t  should be proportional  ference i n population of the two  o  l e v e l s (^  ©  to the d i f -  - p ), and b  the difference between t h e i r contribution to the  also to  rotation  (7)  In the case of the Faraday e f f e c t ,  i s given, according to  Opechowski, by:  H,  co^  and  a  have already been defined,  ( c f . Eqs.  (1)  and  53  (2) ).  m  aD  i s the matrix element of the magnetic moment  between the l e v e l s  a  and  b;  and  co  ab  J;^'  £  B  -  In order to get an idea about the order of magnitude and other c h a r a c t e r i s t i c s of the change i n the Faraday rotation, the above formula f o r Afl^ w i l l be applied to the case of the Faraday effect i n Ni S i Fg 6R2O.  The reason f o r the choice of  t h i s case was the fact that the Faraday effect i n Ni S i F$ 6H2G has been extensively investigated experimentally and f u l l y understood from a t h e o r e t i c a l point of view ( B l , B2, P2, 01, I ) .  On  the other hand, i t must be emphasized that the shape of paramagn e t i c resonance l i n e s i n Ni S i F^ 6H2O appears to be  determined  mainly by the magnetic and exchange interactions, and not by the s p i n - l a t t i c e relaxation, the formula f o r A A ^  .  as assumed i n the theory underlying  Consequently,  the considerations and  conclusions given below should not be taken too l i t e r a l l y .  0  54  2 - Formulae f o r the Case of Ni S i F^ 6H 0 2  P r i o r to the introduction of the paramagnetic  resonance  method i n experimental physics, some of the properties of the Ni++ ion i n a c r y s t a l of Ni S i F^  6H 0 had been investigated 2  by the measurement of the v a r i a t i o n of the Verdet constant with temperature.  The Verdet constant i s the proportionality con-  stant i n the Faraday law.  In a paper by Becquerel and Opechow-  s k i (B2), i t was found t h e o r e t i c a l l y that the lowest levels of Ni  are a doubly degenerate l e v e l and a single l e v e l separated  by a distance £  ;  the doubly degenerate l e v e l is. the lower  one of the two, and the other energy levels of the ion are much higher.  The component of the magnetic moment of the degenerate *  l e v e l i n the d i r e c t i o n of the o p t i c a l axis (z-axis) of the c r y s t a l i s  where (3 i s the Bohr magneton.  With the help of the wave func-  tions given by Gpechowski (B2), i t can be shown furthermore that the x - component of the magnetic moment between the levels and  b  i s equal to that between the levels  a  and  c,  i.e.:  This result i s needed i n the application of Eq. (7) to the present case.  a  55  In the paramagnetic resonance method of studying the magnetic properties of N i * * (P2), the ion i s subjected  to an  external magnetic f i e l d H , applied p a r a l l e l to the o p t i c a l axis c  (z - a x i s ) , so that the lowest energy l e v e l undergoes a Zeeman effect of magnitude  ^  H  fe  .  A high frequency magnetic f i e l d  of amplitude H and angular frequency co^ i s applied at right angles to H between  (say, along the x - a x i s ) , so that t r a n s i t i o n s  c  a  and  b, and  a  and  c (they are the only allowed  ones) may be induced when the energies of the l e v e l s involved have the appropriate values.  Then, corrections  to the Faraday rotation w i l l a r i s e .  and a f l  a  For d i f f e r e n t values o f  the frequency of the f i e l d H, i t i s possible to study the behavi o r of M l  and 4 i l  a  ft  when the steady f i e l d H  i s varied.  c  If we c a l l A the contribution of l e v e l Faraday r o t a t i o n , then „  A w i l l be that of l e v e l b.  to the The l e v e l  a, being non-degenerate, i s non-  0 t  -A  c  magnetic and i t s contribution to the r o t a t i o n i s 0.  A  ft"* "V  I t i s r e a d i l y shown,  then, that the Faraday rotation can be written as follows, using Eqs. (6) and (3),  where  X  (10)  = kT  I f we take into account the diamagnetic connection, to B\  we write:  proportional  56  where V<j, the diamagnetlc Verdet constant, i s a small factor. I f H Is applied p a r a l l e l to the x - axis:  Also, from the diagram,  Using the r e l a t i o n s given above and the dimensionless quantities:  a  *  )  KT  KT  KT we  can write the corrections to the rotation given by Eq.  as  follows:  (7)  57  *a = - A,  e - e-  b  ft  x  (12)  2 f o r p„H >S :*, t  also, AJX • C  A  e  a  -  e  (13)  ^  2 It i s somewhat more convenient to study the v a r i a t i o n and  of the ratios H  c  (x).  as functions of the external f i e l d  I f one neglects V^, some information can be obtained  from the a n a l y t i c a l study of the behavior of those r a t i o s . Their maximum i s very close to the resonance point ( CJ b - °^t\. A  and occurs f o r a s l i g h t l y smaller value of the f i e l d . b always negative. aAa. i s negative for JU„ H > S  is /  and  t  p o s i t i v e f o r j* H n  c  <%  •  But i t i s not zero when ^ „ H  i t has there a peculiar small f i n i t e d i s c o n t i n u i t y . ance peaks w i l l f l a t t e n and widen as When the diamagnetic  yu„ tf approaches c  £  C  The resonS"  constant V<j and the effect of  exchange i n t e r a c t i o n are taken into account, the study of and  ^^Lj£\  c a n  o n  ly  D e  ;  done i n a numerical way.  A».J^  A  Explicit  )  58  calculations have shown that the introduction of these corrections does not change the above ratios very much.  The effect  of exchange i n t e r a c t i o n between the ions can be taken into account i n a way suggested by Ollom and Van Vleck (0 1), who introduced a "molecular f i e l d " y(H-\^H ) added to H . c  The  Q  rotation XI i s given, i n that case, by the following i m p l i c i t relation:  where  KT or  Similarly, i n AA ratios each x  £1 1  and  fc  SI  X  1  - X +  and A U  A  L  /*«yA / KT  M x  \c^x  1  "\  1  + -| e  , x i s replaced by x .  -(3  Then, the  are made to vary numerically with x ;  used, the corresponding H  c  (15)  for  i s found from Eq. (15)? and  the functions are plotted as functions of H . c  59  3 -  Numerical Values and Conclusions  As we see from Eq. ( 8 ) , the values of the corrections AfX  a  and 4 f l  t  at resonance (c*> = c*^ ) w i n depend on the magab  netic moment m^. (Eq. ( 9 ) ) and on the r a t i o a / H.  On the other  hand, i n order to be able to observe the effect experimentally, Hm  must be at least of the order of 4Sa, that i s  H  >  1  (16)  ~  i s constant, and i s given by Eq. ( 9 ) .  I f we substitute i t s  numerical value i n Eq. ( 1 6 ) , we obtain the condition — % H For instance, i f a  = 1.2 x 1 0 ' gauss  (17)  sec „  -K  9 -1 i s taken to be 10 sec , H must be of 7 -1  the order of 100 gauss.  If a  i s 10  sec  , we may choose H  to be of the order of 1 gauss. The values of  which varies with temperature, are  given by Penrose and Stevens (P2).  The constants V , A and y d  are given by Becquerel (B3).  0  9  The two r a t i o s were studied f o r the temperatures 0  K and 1.9  quency f i e l d .  i  K, and f o r various wave lengths  o 90 K,  of the high f r e -  The case i n which T - 1 . 9 ° K i s more interesting  Figure  1  e.ii7  facing  IToIa  7.70  $0  9.93  Ft jure  2.  facing  VALUE 5  NEAR  paye  6*0  Of  R E S O N A N C E .  (T- 13" K  \-5^)  Jooo  +SOO  5ooo  He  (s««s ) s  60  since the influence of the paramagnetic resonance i s then more noticeable.  The results f o r that case are presented i n Figure  1, 2 and 3. Figure 1 shows the values of ^/si  and ^/si  at  resonance, as functions of the f i e l d H , f o r the value of a / H given by Eq. (17)*  The appropriate values f o r  are indicated,  and, also, the corresponding Faraday rotation Si. .  We see that  AA»  /U< H  (top curve) w i l l increase the rotation  and decrease i t f o r fx H^>% • n  for  c  5200 gauss. AA^  to A  T  < &  The minimum of the curve (minimum 2.23 cm. at  of the t o t a l Faraday rotation) occurs f o r H  (  The curve i s asymptotic to the H  axis.  c  always decreases A ^ and i s equal and opposite  (no Faraday rotation) f o r H. ^ 150 gauss and X  The curve i s also slowly asymptotic to the H  c  axis.  7«43 cm. A l l these  results agree with K a s t l e r s q u a l i t a t i v e predictions ( K 2 ) , except 1  that the inversion i n sign of A + Af£ does not happen at saturat i o n , but f o r a rather small f i e l d of about 150 gauss.  In the  region where the rotation fL can be measured, say, around 3000 gauss, the value of A A ^ , f o r instance, would be about 20 per o cent of the t o t a l rotation, i . e . , about 0.7 . Figures 2 and 3 show the values of ttlt A  respectively, near resonance, and f o r case, a  was chosen to be 10^ sec  equal to 100 gauss; 300  = 3 cm.  and  ± ^ A  In this  so that H may be chosen  the width of the peaks i s then about  gauss f o r both r a t i o s .  The value H = 100 gauss i s probably  61  much beyond experimental p o s s i b i l i t i e s . 7 - 1 10  sec  gauss.  If a  i s taken to be  , H may be chosen much smaller, of the order of 1 In that case, the values of the functions at resonance  would not be changed, but the peaks would be wider.  62  REFERENCES  (A)  - E. R. Andrew and R. Bersohn, J . Chem. Phys., 18, 159, (1949).  (B I) - J . Becquerel and J . van den Handel, Physica, 6, 1034, (1939). (B 2) - J . Becquerel and W. Opechowski, Physica, 6, 1039? (1939). (B 3) - J . Becquerel, Physica, 18, 183, (1952). (B 4) - D. B i j l , Thesis, Leiden 1950, p. 77 (B 5) - L. J . F. Broer, Physica, 10, 801, (1943). (E)  - J . Eisenstein, Phys. Rev., 8£, 603, (1952).  (F)  - H. F r o h l i c h , Nature, 1£Z, 478, (1946).  (I)  - E. Ishiguro, K. Kambe and T. Usui, Physica, 17_, 310, (1951).  (K 1) - R. Karplus and J . Schwinger, Phys. Rev. 2£» 1020, (1948). (K 2) - A. K a s t l e r , Comptes rendus, 232, 953, (195D* (N)  - von Neumann, Gottinger Nachrichten, (1927).  (0 1) - J . F. Ollom and J . H. Van Vleck, Physica, 12, 205, (1951). (0 2) - W. Opechowski, Rev. Mod. Phys., 2£, 264, (1953). (P 1) - J . Pake, J . Chem. Phys., 16, 333, (1948). (P 2) - R. P. Penrose and K. W. S. Stevens, Proc. Roy. S o c , M i , 29, (1950). (T)  - R. C. Tolman, P r i n c i p l e s of S t a t i s t i c a l Mechanics, Oxford University Press, (1938).  (W)  - I . Waller,  Z. Physik, 22, 370, (1932).  

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