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Application of Green's function technique to paramagnetic resonance Frank, Barry 1965

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The  University  of B r i t i s h  Columbia  FACULTY OF GRADUATE STUDIES  PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE  DEGREE OF  DOCTOR OF PHILOSOPHY  of  BARRY  B.Sc,  FRANK  M c G i l l U n i v e r s i t y , 1961  M.Sc,,. M c G i l l U n i v e r s i t y , 1962  THURSDAY, JULY 8th,  1965, AT 9:30 A „ M  0  IN ROOM 301, HENNINGS BUILDING  COMMITTEE IN CHARGE Chairman: R, B a r r i e J . A. R, Coope J . M. M c M i l l a n External  W. Ho Gage  Co  Examiner:  P. R a s t a l l F, Schwerdtfeger R. F. Snider D„ D. B e t t s  Department of P h y s i c s U n i v e r s i t y of A l b e r t a Edmonton, A l b e r t a  APPLICATION OF GREEN'S FUNCTION TECHNIQUE TO PARAMAGNETIC RESONANCE ABSTRACT T h i s t h e s i s c o n t a i n s d i s c u s s i o n s of a number of p o i n t s which arose when the author was s t u d y i n g the "paramagnetic  resonance  l i n e shape problem".  The so-  c a l l e d moment method i s d i s c u s s e d , and a new d e r i v a t i o n of t h e moments of the l i n e shape f u n c t i o n i s given.  S i n g l e - s p i n o p e r a t o r s a r e i n t r o d u c e d which  simplify  the c a l c u l a t i o n of these moments.  The  Green's f u n c t i o n technique,, as a p p l i e d t o t h i s lem,  and the d e c o u p l i n g approximations  w i t h the technique,  a r e looked at from  view of r e l i a b i l i t y  and c o m p l e x i t y .  reliability  associated the p o i n t of  As a t e s t  of the  of any d e c o u p l i n g , a theorem c o n c e r n i n g  the moments of a l i n e shape a r i s i n g d e c o u p l i n g i s d i s c u s s e d and proved. f u n c t i o n technique dimensional  prob-  from  such a  The Green's  i s a p p l i e d t o the case of t h e one-  I s i n g model w i t h s p i n 1/2, where no de-  c o u p l i n g of the h i e r a r c h y of Green's f u n c t i o n equations i s necessary. averages is given.  A method of c a l c u l a t i n g  f o r t h i s case, u s i n g d i f f e r e n c e  thermal  equations,  GRADUATE STUDIES  Field  of Study;  Physics  Quantum Theory of S o l i s Plasma  Physics  Advanced  Magnetism  Statistical  Mechanics  Advanced Quantum Mechanics  R.  Barrie  L. Go de Sobrino M. Bloom Ro  Barrie  : F. A, Kaempffer  APPLICATION OF GREEN'S FUNCTION TECHNIQUE TO PARAMAGNETIC RESONANCE by BARRY FRANK B.Sc., M c G i l l U n i v e r s i t y , 1961 M . S c , M c G i l l U n i v e r s i t y , 1962  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  i n the Department of PHYSICS  We accept t h i s thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA June, 1965  In p r e s e n t i n g the  this  thesis  Columbia,  I agree that  the Library  a v a i l a b l e f o r r e f e r e n c e and s t u d y . mission  f o r extensive  representatives,,  cation  of this  w i t h o u t my w r i t t e n  for financial  permission.  Department o f  Physics  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a T ^ L  8  i t freely  thesis  per-  f o r scholarly  by t h e Head o f my D e p a r t m e n t o r by  I t i s understood  thesis  s h a l l make  I f u r t h e r agree that  copying o f t h i s  p u r p o s e s may be g r a n t e d  Date  fulfilment of  r e q u i r e m e n t s f o r an advanced d e g r e e a t t h e U n i v e r s i t y o f  British  his  in partial  Columbia  ^lQL£  that  gain  copying o r p u b l i -  shall  n o t be a l l o w e d  ii  ABSTRACT This thesis contains discussions of a number of points which arose when the author was studying the "paramagnetic resonance line shape problem". The so-called moment method is discussed, and a new derivation of the moments of the line shape function is given.  Single-spin operators are  introduced which simplify the calculation of these moments.  The Green's  function technique, as applied to this problem, and the decoupling approximations associated with the technique, are looked at from the point of view of reliability and complexity.  As a test of the reliability of any  decoupling, a theorem concerning the moments of a line shape arising from such a decoupling is discussed and proved.  The Green's function technique  is applied to the case of the one-dimensional Ising model with spin ht where no decoupling of the hierarchy of Green's function equations is necessary.  A method of calculating thermal averages for this case, using  difference equations, is given,  i ii  TABLE OF CONTENTS  Page Abstract . . . . . . . . . .  ii  Table of Contents  iii  Acknowledgements  .  v  Chapter I - Introduction and Summary Chapter II  .. ..  - General Formalism  . .  5  Section 1: System Hamiltonian  5  Section 2: General Formalism of Absorption Line Shape Chapter III  1  12  - Green's Functions and the Ising Model  17  Chapter IV - The Moment Method  22  Section 1: General Discussion . . . . . . . . . . . . . .  . . .  22  Section 2: Application to Paramagnetic Resonance Line Shape Problem  . . . .  26  Chapter V - The Moment Theorem . .•  35  Section 1: Relation of Moments to Terms in Hierarchy of Equations Section-2: Decoupling Procedures  35 . . . . . . . . .  36  Section:3: Statement and Proof of Moment Theorem  43  Section 4: Summary on Use of Theorem  50  Chapter VI - Resonance Line Shape Problem  51  Section 1: Decoupling Methods  51  Section 2: Solution of Equations  61  Section 3: Complexity Requirement Section 4: Application to Case of Completely Single-Spin Energy Levels Section 5: Summary of Chapter VI  . .  67  Non-equidistant 71 .  75  iv Page. Appendix I: Mathematical Appendix to Chapter III Appendix II:  . . . .  76  Thermal Averages with Ising Model'Hamiltonian .' . . . . 78 :  i  Appendix III:  Proof of (5-14) from (5-2a) and (5-3) p Appendix IV: Reality of A^ of Equation (5-47) Bibliography . . . . . . . . . . . . . . . .  .84 .87 . . . . . . . . . 89  V  ACKNOWLEDGEMENTS  I wish to express my gratitude to Professor R. Barrie for suggesting this problem and for his continued interest and valuable advice throughout the performance of this research. I wish also to thank the National Research Council of Canada for financial help in the form of a Studentship.  1 CHAPTER I  Introduction and Summary This thesis contains discussions of a number of points which arose when the author was studying the "paramagnetic resonance line shape problem."  This  problem is one which has long defied attempts at solution and we do not succeed in solving i t .  However, some points arose which i t was thought were, in  themselves, worthy of discussion.  The motivation for tackling the problem  was that with the recently introduced techniques of double-time temperaturedependent Green's functions, some hope of solution might be entertained. Let us f i r s t briefly review the problem (details are given in Chapter  II).  The physical system consists of a crystalline sample containing N identical paramagnetic unit systems or "spins". magnetic field H . Q  This system is acted on by a constant  As a result the system has a certain number of eigenstates,  and transitions between these eigenstates can be induced by a high-frequency oscillating magnetic field H-j(t) = H^ cos wt.  The resulting absorption of  the high-frequency radiation is referred to as paramagnetic resonance absorption and the problem is to calculate, from f i r s t principles, the shape of one of the absorption lines.  In particular one wishes to know how this shape changes -  with temperature. The problem that has been studied is that in which the temperaturedependence of the line shape arises solely from the average occupation numbers of the various energy levels of the system.  In the absence of interaction  between the unit spin systems, these energy levels would be, except for the lowest and the highest energy levels, highly degenerate.  The spin-spin  interactions then remove these degeneracies to a certain degree.  There  exists, therefore, no natural relaxation mechanism that would yield a continuous line shape: the line shape function consists of a series of delta-  2 functions.  One has basically to carry out a quantum-mechanical calculation to  find the energy levels, and then to follow this with a statistical-mechanical treatment to find occupation probabilities.  With the inclusion of other  thermal effects (e.g. lattice vibrations), one would be led to broadened temperature-dependent energy levels and perhaps a continuous line shape. We do not discuss this aspect of the problem. The Hamiltonian of the system consists of three terms: (a) a spin Hamiltonian which is the sum of the Hamiltonians of the individual paramagnetic units in the presence of the applied constant magnetic field and the crystalline f i e l d ;  (b) a spin-spin interaction which consists of a dipole-  dipole interaction and an exchange interaction; applied oscillating magnetic f i e l d .  (c) the interaction with the  With no crystalline field present, the  energy levels of the individual spins are equally spaced, with spacing "tvtJ say.  0  The absorption spectrum then consists of a set of lines at frequencies  0, W , 2W , 3W , . . . Q  Q  Q  and to single out the primary line at W one does a Q  truncation of the spin-spin interaction (Van Vleck, 1948).  If a crystalline  field is present and the effective spin is greater than one-half, then in general the individual spin levels for part (a) of the Hamiltonian will not be equally spaced.  One then obtains a set of primary lines (with their  associated secondary ones).  A truncation of part (c) of the Hamiltonian is  then required (Kambe and Usui, 1952) to single out a particular primary line. This latter truncation has been performed (Pryce and Stevens, 1950); McMillan and Opechowski, 1960) using projection operators which pick out eigenstates of the f u l l Hamiltonian of part (a).  It is possible to carry out the second  truncation (Kambe and Usui) using operators defined in the representation of single-spin eigenfunctions. Chapter II  of this thesis.  These latter operators are introduced in  3  The authors mentioned above (Van Vleck, Kambe and Usui, McMillan and Opechowski) did not attempt to calculate the line shape but contented themselves with calculating its low moments ( f i r s t and second).  This is referred to as  the moment method, and l i t t l e can be said from these moments concerning the line shape i t s e l f .  We show in Chapter IV that our single-spin operators  simplify the calculation of these moments. In recent years, the technique of double-time temperature-dependent Green's functions has been used to calculate line shapes directly, and the motivation of this work was to find i f this technique was of value in the paramagnetic resonance line shape problem.  As a preliminary exercise, the  line shape function for the one-dimensional Ising model with spin H and nearest^neighbour interactions was studied and is discussed in Chapter  III.  In this case no decoupling of the hierarchy of Green's function equations is necessary. The technique has been applied by Tcmita and Tanaka (1963) to the case in which no crystalline field is present.  Their decoupling procedure is  designed to make the resulting closed set of equations as tractable as possible.  Our attempt at decoupling is designed to be as reliable as possible,  but unfortunately gives rise to a set of equations which we have not been able to solve.  As a test of the r e l i a b i l i t y of any decoupling, a theorem  concerning the moments of a line shape arising from such a decoupling was used, and is discussed and proved in Chapter V. also discussed in Chapter V.  Decoupling approximations are  The result of our decoupling is given in Chapter  VI and compared with that of Tomita and Tanaka (1963). These decouplings of course lead to a set of delta-functions for the line shape, and Tomita and Tanaka derived a continuous line shape by introducing rather arbitrarily a smearing of delta-functions.  This aspect  4  of the problem is also discussed in Chapter VI where the general Ising model is used as an illustrative example. For a treatment of the problem in which the individual spin energy levels are not equidistant, the individual spin operators previously mentioned were used; i t was for this problem that they were originally introduced.  This  problem proved intractable and the reasons for this are discussed, also in Chapter VI.  5  CHAPTER  II  General Formalism  1.  System Hamiltonian Our physical system consists of a c r y s t a l l i n e sample containing N  identical paramagnetic electronic unit systems, or " s p i n s " , possessing magnetic moments. f i e l d \j  Q  It  i s assumed that the sample has been placed in a constant magnetic  and a high frequency o s c i l l a t i n g magnetic f i e l d H-|(t) = H-j cos fe?t)  perpendicular to H . Q  Each spin is defined by a "spin-Hamiltonian" H( ^ 0  which incorporates the e f f e c t of the external constant magnetic f i e l d and of any arbitrary c r y s t a l l i n e e l e c t r i c f i e l d that may be present.  The o s c i l l a t i n g  magnetic f i e l d induces transitions between the energy states of the spin system, and experimentally, we find that energy i s absorbed from the o s c i l l a t i n g field.  The "line shape problem" cons.ists of finding the power absorbed by  the spin system as an analytic function of <0 , the frequency of the oscillating field. In addition to the interaction of the spins with the two magnetic f i e l d s and with the c r y s t a l l i n e e l e c t r o s t a t i c f i e l d , there are weak interactions within the spin system.  Were these l a t t e r interactions absent, the line^shape  would consist only of widely separated delta-function peaks; these "spin-spin interactions" serve to "broaden" the l i n e s , giving them their shape ( i . e . , closely spaced delta-functions of d i f f e r i n g strengths occur, and they are not experimentally resolvable). Mention must also be made of interactions between the spin system and the l a t t i c e .  We take i t  that the temperature is low enough so that the  effects of l a t t i c e vibrations can always be neglected. The Hamiltonian describing our paramagnetic system may be written  6  Hr- -  (2-1)  where  ^  T  ^  n  '14^ =  4  ^  V  »  ^  °  '  »  ^  ( 2  2 )  (2-3)  ^  (  2-4)  is the Zeeman, or unperturbed  2ZT4-;°^  Hamiltonian with i l a b e l l i n g the lattice  (2-5)  sites  a-ZSl W!) 1 s the spinr-spin interaction il  _  Hamiltonian  including dipole-dipole and i s o t r o p i c exchange including di  (2-6)  interactions - M K c e a o t i s the external  perturbation, with M the magnetic  moment of the spin system. (2-3)  and (2-4)  are weak-interaction assumptions.  We assume that of  *  1*^  possesses axial symmetry, the axis being the direction  L"L> , which we choose to be the z - a x i s ; and that the eigenstates of  14^ are  also eigenstates of the operator corresponding to the z-component of the spin. Before discussing the line-shape function must modify the Hamiltonian in two ways.  cr  The term  or i t s moments, we 04-^ in the Hamiltonian  must be "truncated" in order to eliminate the weak "secondary" lines from o~ <A>). The argument for this has been given by Van Vleck (1948). consists in keeping only that part of commutes with  "H^  ;  T-r^  This truncation  (we w i l l c a l l i t  14^  i.e.  f V ^ ]  - O  (.) 2 8  ) which  7  We further define  Then  -  (2-10)  O  A l s o , when a c r y s t a l l i n e e l e c t r i c f i e l d i s present, and, thus, the energy levels of an unperturbed spin are not in general equidistant,  must be  modified in such a way as to cause transitions between only two levels of a spin.  This must be done in order to ensure that other "primary" lines of  comparable intensity to the one under consideration, are not included in the calculations*  The necessity f o r this further truncation was pointed out by  A Kambe and Usui (1952).  We l e t  represent the modified >f^, and we  put a circumflex over i t s associated operators. When no c r y s t a l l i n e f i e l d i s present, and 'W'^c-f.  the Hamiltonian  1 S  i s in the  H  e  (-Z)-direction,  Qiven by equations (2-9), (2-5), and (2-6)  with J  (2-11)  d  fail. or  ..f. -  t  <\  s~s+)  +  where  C..= A + B., I  J  y  VY fa  1  J  +±v V^4"'^  "  a/l  i s the direction cosine of the line of length r  (2  ;  13)  j o i n i n g spin i to  c  : j  s * s  8  spin j  r e l a t i v e t o the z - a x i s ; A .  g^and g {!>  ;  1 (  are the g - f a c t o r s  are the exchange i n t e r a c t i o n  p e r p e n d i c u l a r and p a r a l l e l  t o the symmetry  axis;  i s the Bohr magneton; |. j  and  0 is  constants;  o  frf  -  (2-14)  s  I >  0  the unperturbed a b s o r p t i o n f r e q u e n c y . Where a c r y s t a l l i n e e l e c t r i c  i n g e n e r a l , non-requidistant energy l e v e l s . s i n g l e - s p i n Hamiltonians  the unperturbed s p i n s  of  Si  and  have,  We c o n s i d e r here o n l y those s p e c i a l  which have the p r o p e r t y  the simultaneous e i g e n s t a t e s respectively,  present,  04  t  that, i f  with e i g e n v a l u e s  are /\  and  --Is  i.e. (2-15)  then 1 1 1  A  (2-17)  E q u a t i o n (2-17) s i g n i f i e s t h a t i n the e x t e n s i o n from the n o - c r y s t a l l i n e  field  case t o the c r y s t a l l i n e  levels,  field  c a s e , the o r d e r o f the unperturbed energy  which i n both cases are l a b e l l e d by the e i g e n v a l u e s o f The statement about n o n - e q u i d i s t a n t energy l e v e l s generally  \ k  when The >T  true  =  14^' L  than means t h a t i t  i s not  that  ^  V-  remains unchanged.  . g i v e n by e q u a t i o n where > K  (2-12)  does not any l o n g e r commute with  i s d e f i n e d by ( 2 . 1 6 ) .  It  must be t r u n c a t e d  9  still  further.  The truncation can e a s i l y be done i f one defines operators  by the relation  From d e f i n i t i o n (2-19), equation (2-16), and.the well-known commutation relations s a t i s f i e d by the ordinary ladder operators  J>j  , the following  equalities can be derived and are found to be u s e f u l :  LAl  t  Sj^J  7 R  s  Ax^I'  =  AA4,'>'  A> 4 v '  A*  (2-20)  AlA'A+i S  =  FO-I  (2-22a)  A'> A'A^I  (2-22b)  ^A+IAAAA-H  (2-22c)  A> At- = £V< A:  (2  -  22D)  With the use of equations (2-20) to (2-22), we find f o r the c r y s t a l l i n e case  _  + where  field  iV-*  means summation over a l l ordered pairs  "  (2  23)  (X"^')  # I am indebted to Dr. K. Nishikawa f o r the suggested use of these operators.  10  satisfying Iy  H  -TJ/  -  / A">I  '  (2-24)  1  Using  (/>$)* = A M  (2-25)  which follows from definition ( 2 - 1 9 ) , one can verify that Hermitian.  ^  i'&Lj^c-p,  s  (The asterisk will always be used to denote the Hermitian  conjugate). A Hamiltonian which we consider at some length in a later chapter is that of the Ising model.  This Hamiltonian was f i r s t proposed by Ising (1925)  to describe certain ferromagnetic systems.  It turned out not to be valid  from a physical point of view, in that i t does not describe, with any accuracy, the interactions taking place within any known spin system.  However,  i t is invaluable from a mathematical point of view, in that i t is a specialization, of equation (2-12) which is easy to handle. >  The model Hamiltonian is given by ( 2 - 1 ) , ( 2 - 2 ) , (.2-5) and (2-6) with ^  = -  O  (2-26)  S ^  0  '-  (2 27)  ^x,M>-^^r where the  cfy  are the exchange interaction constants.  It is seen that  (2-27) is just the general paramagnetic resonance Hamiltonian (2-23) with the A ;j and the  C\\ formally put equal to zero and — n  respectively.  In many instances a further specialization is useful, namely that of putting £ / ^ Q R  £  tj  -  L, \ nearest neighbours » otherwise  (2-28)  11  The Hamiltonian with this restriction on the "Ising model Hamiltonian".  £y  will be called simply the  When this restriction is removed, we will  specifically refer to the "general" Ising model Hamiltonian. These are the only Hamiltonians that will be considered in this thesis. A  It remains now to exhibit  ^  x  for these various cases.  We have, from.equation (2-7), A.  (2-29) as the oscillating magnetic field is taken to be in the x-direction. No truncation of  M x is needed for the case of equidistant unperturbed  energy levels, and so in the absence of a crystalline, electric field (and this includes the Ising model cases),  However, for non-equidistant  energy levels, i.e. in the presence of a crystalline field, one must use  2.  where  ^—  means summation over all / A-  A  -T7-U),  A + 1  such that (\6^  •  (2-31)  and u), is the unperturbed frequency of the primary line under consideration, not necessarily given by (2-14), but always positive. It is of practical interest to know that (H ) °f (H) ,jf, , and that (At)^ .-f. c  i s  M  ^4 is a special case  a special case of ( f^\ ) , ^ x c i  c  i  .  To show  this, one needs the help of the relations (2-32)  X t C ^ d l  X  (W»  (2  . ) 33  12  where in (2-34) i t i s assumed, n a t u r a l l y , equation (2-24)). sets  <3i  and,6,  that  i^,  and / ^  (  exist (see  (2-34) i s a consequence of (2-33) and the d e f i n i t i o n s of the  ..  Equation (2-32) follows immediately from the d e f i n i t i o n  (2-19) of the <d.J . In developing our formalism f o r the l i n e shape f u n c t i o n , then, only the - ,  i  Hamiltonian, describing the c r y s t a l l i n e f i e l d case need be taken into consideration.  2.  General Formalism of Absorption Line Shape. We s t a r t from the well-known relation between the absorption constant  <T"(iS) and the (T(S)  ?£(<S):  magnetic s u s c e p t i b i l i t y  coyest.  -  O  l  ^ (u)  (^>0)  t  ,  (2-35)  In the Kubo-type formulation (Kubo, 1957; O'Rourke, 1957) "X (t^)  t  o  -  ^  '  <  [  >  j  x  (t)  >  i s written  < 2  "  3 6  '  where we have used the following notation: M  x  :  the x-component of the total magnetization  O f t 1 f 1+ -  t  * O f  l  f  t  operator  fli-i),  i  (2-37)  w i l l henceforth be c a l l e d the Hamiltonian of the system  <<- > * a u (e. k  a  T  -J  ;  a= u  :  Boltzmann's constant,  :  temperature of the system at the time t = - *0 .  /  (2-38)  The following assumptions were involved in these formulas: (1)  The external magnetic f i e l d H^(t) has been applied in the x - d i r e c t i o n .  13  (2)  The response i s looked at in the x - d i r e c t i o n .  (3)  H-j(t) has been applied a d i a b a t i c a l l y starting from the distant past: lim H^t)  =0.  (4)  The system was in thermal equilibrium at t = - oo .  (5)  The Hamiltonian associated with the o s c i l l a t i n g f i e l d i s of the form  - £.H,£) (6)  .  (2.39)  The o s c i l l a t i n g . f i e l d i s s u f f i c i e n t l y weak that i t s e f f e c t can be treated in f i r s j : order perturbation  -  theory.  One normally defines the paramagnetic resonance line shape function to be not  V~(\S) g  , as given by (2-35), but  6JY* cow/f. Z"*,  y.(iS)  g (u)  (lJ>o)  (  where the constant i s such that ^^j)  as given by  (2-35a)  i s normalized to unity.  (Cf.McMillan  and Opechowski (I960)). It  is of advantage to keep only that part of  which vanishes for |*)<0  •  =  identity  Z L [.„ [ F Q  where  (2-40)  fcrf fi]  fi  ]  f F  p"  one has  ft* f t ) . f  i  W  | ! l t * .  as defined by (2-36)  Employing equations (2-8), (2-'9), (2-37), (2-30),  (2-21), and (2-31), and the well-known  c V c *  X6*0  ^(V'.WE^gJ'^fl.)*  (2-41)  14  With the use.of the identity (2-43) which follows from the.cyclic properties of the trace for.any operators A and B  <t^-0,*>'  <C^B(t^>-  (2-43)  j  and the relation  . ^  M*7*^ ^  '  (2  "  44)  equation u - o o ; e a n D e rewritten as TO MOWS:  + L ^ , Wt W  j  /  ^fS;,^^ - 4 * _ /  ft  4  / S ^ C ^ I  ^  (2.45,  First, notice that in the second integral, Sj = £1 ^ M V I replaced by Z L  J  may be  (see Appendix I).  As the system is definitely quantized along the z-axis, the f i r s t and fourth integrals vanish automatically. a negligible contribution to % (\S)  Moreover, the third integral provides  for u) ^ 0; this follows from the weak- -  interaction assumption:(2-r3)° i f one puts delta-function peak at  u> - - t0  0  ^  = O , the integral gives a  . Inclusion of >r  c0  serves to  introduce many,other delta-function paaks into the region around - U^> very l i t t l e overlap into the region of positive  u)- .  By a similar argument,  the second integral is seen to provide a negligible contribution to the region of negative  6J  , with  in  .#  # This argument is_ equivalent to that of McMillan and Opechowski (1960). They argue that i f >}»> is not too large, < E implies E^,: < E^k except for a negligible number of pairs of states l*,l) ftjk) where >|JW //*,kj = /O and -4 l^ k) (k = 1,2,...^, the degeneracy of E^ ). A  t  t  15 Thus, i f the magnetic susceptibility is redefined in terms of the second integral alone, i.e.  then (2-35) may be rewritten without, the condition " t*>><? " . F u r t h e r , one always has (2-47) where  is any function of u)  The function  /(<S)  as given by (2-46) is closely related to the  temperature-dependent double-time Green's function.  These Green's functions  were f i r s t introduced by Bogoliubov et al. (Bogoliubov and Tyablikov, 1959; Zubarev, 1960).  The Green's function  <t"A I B^.  of the operators A and B is  a two-branch analytic function of E defined everywhere outside the r»al axis C . IU  U t  <[*ft), Q >  ; X,  £  < o  By comparison of (2-46) with (2-48) one.can immediately see the relation  *  (2-49)  where we have used only the.identity (2-43). Note that the above Green's function is of the form ^ A | A ^ One can show that the Green's function  ^AlB^  (by (2-25)).  as defined by (2-48)  is analytic both in the upper and lower half-planes and that i t is found from the hierarchy of equations  e « f l i ^ =  , ,v ( 2  _  5 0 )  16 Our problem is then to solve this hierarchy with the condition that the solution be analytic in the domain for which the Green's function is defined, that i s , outside the real axis.  The actual procedure of solving this equation  approximately will be discussed in Chapter V.  For further details of the  Green's function method, we refer to Zubarev's article.  17  CHAPTER III  #  Green,'s. Functi.Qtis....and--th.e. Isi ng - Mode-1 ...  In some simple.cases i t is found that the hierarchy of Green's Function equations (2-50).decouples automatically after,a finite number of steps. One such case is the.problem of finding.the amplitudes and frequencies of the three component the absorption band.of the one-dimensional Ising model, consisting of an arbitrary number N of identical.spins arranged in a ring. It is assumed that each spin has spin quantum number h, and that only nearestneighbour interactions occur. The Hamiltonian of this system is given by equations (2-26), (2-27), and (2-28), i.e. .  14 =  - U  A/  .2:  4"  r  *  - c l j^y J k 3 k+i  (3  _D  where h = k)o as equation (2-14) £=  exchange interaction constant  I and k , label.the lattice sites, with the (N+l)  st  site identical with  the 1st. The line shape function  ^ (iS)  is again defined by equation (2-35a),  which, through equation (2-49) specialized to the no-crystalline field case, leads to  A(U))OL  &W  X*  tfeta-tg.  (3-2)  # I am indebted to Professor W. Opechowski for his suggestion that this problem be considered.  18  where  c.  (s 3  ^  0-3)  <i^/i-;^  The f i r s t three Green's function equations are then  (£-U W  ^  a  <  ^  >  ^  W  * «tf  (3  .4)  (3-6)  The last term in equation (3-6) arises since  (3-7) is a consequence of the relation  It is now seen that equations (3-4), (3-5), and (3-6) form a closed set of simultaneous equations which can easily be solved for Cj j 0  is no need to write any higher-order equations.  s  ; that i s , there  Since this state of affairs  appeared without .any.decoupling.becoming.necessary, we say that the hierarchy decouples automatically at the tlfird stage.  The solution is  (3-9  1  where we have introduced the notation  =  <S&  (3-10)  noting that ^ l " ^ » ^ 2 ^ ,  and  3^  are, independent of i because of  translational invariance. The line shape is then given, with the use of (3-2) and (3-3), by  =  +  -^[(<»-*<*>)r(>>-K)  (±<p+a<&  + *<ii)P(u-^t)]  (3_11)  which is a sum of three delta-functions at specified frequencies, with amplitudes given by linear combinations of thermal averages. It now remains to calculate these thermal averages.  Since several  authors (e.g. Huang, 1963) have derived explicit representations for the partition function Z, we will take i t that Z is known, and use Huang's result  Z:  =  ( A 4.)  *  (/-/  (3-12)  20  where  ]/cosk^l) -^'^^k(i^J  A± We immediately get <1>  and <2>  by direct differentiation from the  formulas '  (3-14)  "^S4)  (3-15)  1 =  ( 3 ^ is obtained as a linear combination of 0 ^ and  The thermal average ^2^  A/-2:  by the use of the following.general identity (see, e.g., Zubarev, 1960)  (3-16)  Putting  t = t' = 0, A = S. , and.B..= +  in equation (3-16), and using  the result (3-9), we get -J-  <i7  -  <y-^< >. 3  - i (<•> -  ^6->  - p ^ ^ - l  +  4<3>) ( 3  _  1 7 )  or  (3-18) where DQ, D-J , and D are simple functions of h and & . In the derivation 2  of (3-17), use was made of the relation  (3-19)  (  21  Equations (3-14), (3-15), and (3-18), when combined with equations (3-12), (3-13), and (3-17), give us, when substituted into equation (3-11), an explicit,representation of the absorption band of the one-dimensional Ising model.. The result corresponds exactly with that found by Y.Y. Lee (1961), who had quite complicated summations to perform.  The latter arose  from his combinatorial approach to the problem, and necessitated the finding of suitable.generating functions. He simplified the combinatorial problem by specializing to the case in which N = 4P -1 (P=l,2,...) and then considered his results valid for large N.  In our approach there is no  need for this specialization, as the complicated combinatorial problem is avoided.  The advantage of our method lies in the identification of the  amplitudes of the resonance lines with easily recognizable thermal averages, and in the ease with which these averages can be calculated, from a knowledge of the partition function, through differentiation and further application of the Green's function technique. The thermal average  <3^ cannot be obtained directly from Z; nor can  higher-order thermal averages.  A direct method of finding thermal averages,  using difference equations, is given in Appendix II.  22  CHAPTER IV The Moment Method  1.  General Discussion In the problem of the general paramagnetic resonance line shape,  unlike that of the one-dimensional Ising Model, the hierarchy of Green's function equations (2-50) does not decouple automatically at an early stage. The solution of this problem by the Green's function technique then involves looking for a decoupling procedure which gives a closed set of equations satisfying the following requirements:  i t must be (a) sample  enough to be tractable, (b) complicated enough to yield far more than the small number of isolated delta-function peaks obtained for the Ising model, and (c) "faithful" enough so that i t furnishes a line shape function that is resonably close, in some approximation at least, to the experimentallyobserved line shape. The difficulties inherent in the search for such a suitable decoupling procedure are representative of those encountered in other approaches to the line shape problem.  Consequently, a method has been evolved which  enables one to discuss the shape of the resonance lines without actually having to find an analytic expression for the line shape function.  Known  as the "moment method", this procedure consists in defining a line shape function in a formal way, and in calculating its f i r s t few moments in various approximations.  It was introduced by Van Vleck (1948), and used  by Pryce and Stevens (1950), Kambe and Usui (1952) and McMillan and Opechowski (1960).  Kambe and Usui derived quite general exact expressions  for the moments, and these were applied by McMillan and Opechowski to the case in which a crystalline electrostatic field is present.  In this  23  section, a derivation, different from that of Kambe and Usui, of the expressions for the moments is given. We shall show that i f a normalized line shape function  g  r t A  £ud)  ,  defined by  (4-1) with  satisfies the conditions that A* = B  (4-3)  and that  a  ye,*  *t h  LiS)  -  O  >  u)<0  then its p- moment about the o r i g i n , ,  /y- <f{^K 3>/<f",Bl> B  (4-4)  *  is given by  p=0,l ,... >2  (4-5)  With the use.of (2-40), (2-43), (4-3), and the identity  <£»ct)0 = <[B*,fl*fc\]> we can immediately write, from (4-2),  (4  -  6)  x  (4-7)  24  We now make-use of the characteristic function M(t), defined by  M(t) =  / ^  e.* —  ^  B  A  M  (4-8)  which possesses the inverse Fourier transform  Repeated differentiation of (4-8) with respect to t, i.e.  gives, simply, the Taylor coefficients in the expansion of M(t) about t=0, i.e.  M(t).  •  (4-H)  Substituting from (4-11) into (4-9), we get  -  .;jt  c.t) ,  ,  f  '  °  ,  (4-12)  Comparison of (4-12) with the alternate expression for g  (iS)  given by  (4-1) and (4-7) yields  = (constant, independent of p) X <C£ (/9 'H} ;  P )  Bjj^ ,  The constant is determined by the normalization requirement on g  (4-13)  (^)  resulting in the required expression, (4-5), for the moments about the origin.  ,  25  If, further^ ^  can be expressed according to (2-9) as  where (2-10) holds, and also [ f K ^ ^ J  ~  (4-14)  ^>o?\  then  <I1^^> . (4  are the moments about the unperturbed frequency  15)  . (We call the M  p  "central moments".) For  T =O  (4-16)  Also, by induction i t can be shown, using (4-14) and (2-10), that  fi\ft} = p  i i  t  r  <  o  /  "  7  ,  (4-17)  Combining with (4-5), we get  t. c^r«,- £ %»:KCI«W\tf> . f  T 2 0  (4  26  Taking p=0,l,2,  2.  successively, we see that formula (4-15) follows.  Application to Paramagnetic Resonance Line Shape Problem. For the paramagnetic resonance .line shape problem, one can immediately  proceed to calculate moments in various approximations by substituting  f\  -  «s£—-  X  (4-19)  and 1 3 * * * *  (4-20)  into the expressions (4-15) for the central moments. The expressions (4-19) and (4-20).for A and B come from equation (2-46) for ^C(^). commutators with  One takes  (equation (2-23)) according to the rules (2-20)  and (2-22), and takes traces in the most convenient representation. This procedure is straightforward. McMillan and Opechowski (1960), starting from formulas f i r s t derived by Kambe and Usui, were able to cast expressions for the f i r s t and second central moments into a form suitable for computation.  They then applied  their results to the specialized problem that we have been considering. Their only i n i t i a l assumptions are given by our inequalities (2-3) and (2-4). The procedure for their calculation of the f i r s t central moment, M.|, will here be given in outline.  H,  -  _  They start with  .  ——r-  (4-21)  27 where  ^  A  A  M = M  +M_  +  is the component of the magnetic moment of the spin  system in the direction of the oscillating f i e l d , truncated so as to single out a particular primary line. A  A  M  and M_  +  '^,ilM.I/j,kV  where  /=^,0  are defined by  l  0  0  *  (  4  .  2  3  )  satisfies the eigenvalue equations  (4-25)  They then rewrite equation (4-21) in terms of  A A  Q  and M rather than tr ,  A  M, M and M_ . This is done with the use of projection operators P^ defined such that +  (4-26)  "H "^ is not too large, then  With the use of the observation that i f E-<A ^  implies  ^  number of pairs of states W^'O (4-23) of  and M_  ^ IA  are changed to  t  1  k  1%)  except for a negligible »  t n e  definitions (4-22) and  28  o'the.-ruj!Se.  Q  ^.ilA-lAkV  (4-27)  /KWMl^k) '  ;  0  (4-28)  oik  Then *  = rt+  A  where  ^  =  N  A  P  (  ° <  2 L  (4-21)  3  -  « , ,  (4-30)  (4-3!)  means summation over all values of <?< and A  ^  =  U)  wherein (4-32)  0  becomes M,  " B =^ ^ h  P  ^  =  T Equation  *  =  d"'  (4-33)  / S  (t^S.MP.M-t^^P^MP.M/  (4  .  34)  McMillan and Opechowski make use of an approximation technique used by Pryce and Stevens, and also by Kambe and Usui. This technique depends on the weak-interaction assumption involving  ^  as follows  (2-3).  One expands a l l exponentials  29  [ c , ^ ^  provided  )  ^ O  C , ^ c J ]  '  and  k e e p s  o n l y  the  lowest-order terms in the square brackets. Here Cj and Cr, are any operators such that the above provision is- satisfied. which consists in keeping only the f i r s t  We call the approximation  £ terms in the brackets, the  th "  approximation of McMillan and Opechowski", and denote i t by (MO)^ Evaluation of formula (4-33) in (M0)j b  by  *  consists then in replacing  • The analysis is done by McMillan and  Opechowski for the case when the unperturbed spin has R energy values ap  ag, ..., a^  degenerate. T  (the equivalent of our  In this connection, a set  is in 6>  is defined such that i f  , there exists an integer T  4  Q-r' "  In the f i r s t approximation the term  ) which are all non-  /  such that  ~  ^°  (4-36)  Q  (4-34) can then easily be  reduced to the form  |< ft. . .  d - t ^ S L t -S-f  6-  In the above form, the calculation of  lUHMl^kjl 8  .  (4-37)  becomes a combinatorial problem,  and involves doing summations like  +  hR  hi  T6.  where  are the unperturbed eigenfunctions of the i  eigenvalues Qy.  so that  t  h  spin, with  .  30  (4-39)  i  1  The cause of the combinatorial problem is in the degeneracy of the E ^ The degree of complexity increases when expressions like  .  and those  entering the higher moments are considered, and when approximations higher than the f i r s t are used. The combinatorial problem is avoided for the special case that we consider by the introduction of the operators  defined by (2-19).  To see how these operators simplify the calculations for our special case, we show how the more general case of McMillan and Opechowski is simplified by the use of a different kind of projection operator; these projection operators, f i r s t introduced by Kambe and Usui (1952), are then related,for the special case, to the operators  'v  We define projection operators rather than the operators P  v  which project onto eigenstates of ?*•  which project onto eigenstates of the full  Zeeman Hamiltonian, by  (4-40) The relations (4-41) and  0 follow from the definition (4-40).  (4-42)  31 X  Note that in these equations  no longer labels eigenvalues of  ,  ..Co)  but only eigenvalues-of  according to equation (4-39). A  A  One can then show that the following expressions for M  +  and M  satisfy  the definitions (4-27) and (4-28):  M. where  -  P/M,  =  p;  ( 4  M j  .  4 4 )  (4-45)  One also has  - JIS:  ^ where  z=_  p,i  p/ p ;.  ^  5  means summation over a l l A j ,  ^>  (4-46)  A  /iy  such that  (  4  "  4  7  )  A  One can prove the above statements by considering matrix elements of M , +  A  —,  M_, and  (as given above) between the orthonormal product eigenstates  I^X^\*"*I »)* X  \KkK...L)  -  Here  , for example, is an eigenstate of  /^^  Q^  (4-48) with eigenvalue  . The product eigenstates (4-48) are themselves eigenstates of  <y^{9)  =  >f;  linear combinations of the Let  with eigenvalues a^ l  M /r )  i)  + a^  +...+  a^  =  , and are  defined by (4-24) and (4-25) for 3  f i x e d .  us now evaluate (4-49)  32  the f i r s t approximation part of .the...denominator of  given by equation (4-33)  with the use of the PJV . Using (4-43) and (4-44),  3 •  (4-50)  From (4-41) and translational invariance,  s = ifH.  i x  [b  (?;  5  H  pj  :  M-  p j  (4-51)  Let us consider only the f i r s t term of (4-51), and expand the trace in the orthonormal product eigenstates (4-48), ~rec«^  to)  >, A, .., A v )  )  )  /  (P H Pj.M Pj)lK^.-.^)} :  r  i  T 6 . Gi  i  v  (4-52) where  a  a.,  4-  I-.  + -  (4-53)  One derives B (second term) in a similar fashion to get  6  = A/Z^'  d-^OZL- i M M ^ M P  (4-54)  33  where (4-55) and use has been made of equation (4-36). There was no need to solve a combinatorial problem, and result (4-54) agrees with McMillan (1959).  AIT other traces can be done in the same manner,  even in higher approximations. For the special case when (a) }4 •  has axial symmetry; (b) the external  L  magnetic field  tl° is directed along the axis of symmetry (Z-axis); (c) the  order of unperturbed energy levels is preserved in the transition to the no-crystalline field case (equation (2-17)); and (d) the relation ~  [S-J^  °  holds, we may write (4-56)  M_  ^  =  /15a.  _  where X , <&±y ,  V  (4-57)  c  j_ * P » and  a s  G  t  9  i v e n b  y (2~ ) 23  are as defined in Chapter II. A  direct comparison between equations (4-56), (4-57) and equations (4-43), (4-44) cannot be.made because of the differences in the definitions of G^ and G^* and in the labelling procedures.  However, -it may easily be verified that  ^  A  M , as given by equation (4-56), conforms to the definition (4-27), and that M , +  as given by equation (4-57), conforms to definition (4-28). It has been checked, for the specialized problem wherein conditions (a) to (d) (above) hold, that formula (4-15), in the case of an absorption line from the ground state to the f i r s t excited state of the unperturbed spin, yields the same final results for the f i r s t and second central moments as were obtained by  34 McMillan and Opechowski (1960); the calculations were done in the f i r s t two approximations of McMillan and Opechowski, for the cases of effective spin S = % and S = 1.  35 CHAPTER V The Moment Theorem 1. Relation of Moments to Terms in Hierarchy of Equations. It is interesting to take note of the identification between the raw moments (moments about the origin) and the f i r s t terms on the R.H.S. of the Green's function hierarchy of equations (2-50).  The latter may be written  P = 0,1,2,... Comparing with the expression (4-5) for yt^ , one sees that, apart from a p  factor 3TT  the identification is complete.  It is assumed that  conditions (4-3) and (4-4) are f u l f i l l e d . Moreover, i f condition (4-14) holds,the hierarchy of equations (5-1) is easily cast into the form  ?  IB>  £  • £ <a^ifB7>+i(f>mns%  (  5  .  2  )  p = 0,1 ,2 It is in this form that the Green's function hierarchy of equations will be used in the remainder of this thesis.  Apart from the same factor  /^A.B]^,  one identifies the f i r s t terms on the R.H.S. of the latter equations with the central moments, M  (equation (4-15)).  p  One may ask whether this identification can be made directly from equation (5-2) and the relation  -  I—I  r  „  (5-3,  36  between the Green's function <^/9)B~/^  and the line shape function  ^  proof that this can be given in Appendix III. This identification i s , of course, valueless i f one wants only to calculate exact moments; the Green's function technique need not be considered at a l l for this purpose.  However, very often the Green's function technique, coupled  with a certain decoupling procedure, is used to derive an approximate analytic expression for the line shape function, and i t is asked what the associated moments are and how closely they approximate the true moments. The integrations involved are often extremely d i f f i c u l t to carry out.  A theorem, closely  related to the aforementioned identification, can be proved which provides a relatively simple method for finding the approximate moments, and eliminates the necessity for performing any integrations.  Before this theorem is stated  and proved, however, an outline of decoupling procedures as they are normally followed, will be given. 2.  Decoupling Procedures First, rewrite (5-2)'.as follows: e.  Gn  p  ~  where we have replaced  '  p  +•  E ^ U)„  G»  p  <[{Mf  (5-2a)  |  by  $Cfi^f\3%-by and  +  B ] ) > by  £_  (5-4a)  a  (5-4b)  T"  ?  p  (5-4c)  A method one might follow would be to write the f i r s t q equations as they stand, and in.the q+lst to replace  by a linear combination of the q+1  preceding Green's functions, the coefficients being independent of E; i.e.  (usually thermal averages)  37  e. £>,  -  T  +•  (5-5a ) 2  i  £  f  -  i  i  A ^  ~T" +  1  (5-5a ) q  We call this "decoupling at the (q+l)st stage".  A closed set of linear  equations is obtained which must be solved'for C) to G  0  , which is simply related Oi stands for the exact  (to) .. (It will be obvious throughout when 0  0  and when i t is used to mean the approximate value obtained on solution  of the decoupled set of equations.) This method is fouffd" to be unsatisfactory in most cases. made up of more elementary Green's functions  Q  Usually  G> is 9  such that  0  (5-6) e.g. a  0  =  ^  I  L  ^  L  A  ^  {  ^  ^  -  ^  1  %  (5-7a)  and  Let us write a separate hierarchy of equations for each of the elementary Green's functions Ln  r - v a.^ (»0  6i)  £  ^  U,  6 ^  -  =  ',  T ^ *  ^  ^  (5-83^ (5-8a ) 2  (5-8a ) q+1  38  Equations (5-8) represent by a different  ^  Tl hierarchies of equations, each hierarchy labelled  . They differ from equations (5-2a) in that different A  and B have been used in the defining equation (5-2).  The T  thermal averages associated with the Green's functions  V  are the . The  CL U r  (V^o)  are not meant to be elementary Green's functions, but are defined by equations (5-8), through the given  G  and equations (5-2). (There is no difficulty  0  (<*)  in picking out which terms are to be grouped with the T^_ the  Cv  C—>  in writing equations (5-8); a l l the °°  , whereas the T ^  Cv  and which with  vanish in the limit as  are independent of £. .)  One might want to decouple each hierarchy as follows:  where independent of  o<  (5-9a)  o4 ) of (5-8) is now closed, and can be solved for  Each set (labelled by  Then the sum (5-6) can be taken over a l l the solutions.  Gi  g  The above procedure  is not equivalent to that leading to equations (5-5), for i f we sum equations (5-8) over ©4  , we get the set of equations  ^ a, -  T  !  ^  +  i  I  (5-10)  Because of condition (5-9a), the set (5-10) is not closed, as was the set (5-5). This method of decoupling is also unsatisfactory in most cases. usually happens is that lly h,  tf\  G>,  What  is also made up of elementary Green's functions  Z -  (5-1,)  39  and one might want to decompose  C ^ , in terms of the u,  coefficients for each 60^6) pair.  '  with different  So separate equations for  are needed.  Continuing on in this way, we find that our most general set of equations can be written /*  (5-12 ) ai  r  where Gi  'a.  (5-12a ) 2  each Green's function that appears is elementary. , the result of some specified  Here  decomposition  (5-12b)  is some linear combination of a l l the other elementary Green's functions. Equation (5-12a,) represents T"L  equations, one for each value of oi  ;  equation (5-12a ) represents ( TL TTL* ) equations, where T T l is the number 2  of ^3 -values; and so on.  The number of equations in the set (5-12) is equal  to the total number n of elementary Green's functions that appear.  We call  equation (5-12) our "closed set of decoupled equations". For convenience, we relabel a l l the elementary Green's functions in (5-12) by the single subscript  (L G)^ *  -r-  . Equation (5-12) is then rewritten  J>Z_  G,  o**!,!,...,*  (5-13)  40  where T,^  (  —  are coefficients which may be zero, and are independent of E.  are the thermal averages associated with the  &^  The  , in accordance with  equations (2-50). 0)  In order to identify  S o the set of all values of o <  , we call  0  such that A  °  S T .  =  Cn,  It is then possible to identify in terms of the e- J E L  and the Cf . =  e.  G = 0  ^  • £>  i  v  ,,,  (5-i4)  t  A|  i  For, summing (5-13) over o<£ «T , we get e  -r- 2 2  2 1 7 7  £f  6*,  (5-15)  where the summation convention (repeated indices are summed over all possible values) has been used, and will be used throughout the rest of the chapter. Comparing with (5-2a) for p=0, we get  <^6.S«  a  >  f r z  =  (5-16)  *  &  ( 5  -  1 7  »  We also have, s t i l l using equations (5-13),  Comparison with (5-2a) for p=l gives  4  G  v  =  X L  I  C  ^  U  (5-19)  ( " °) 5  2  41  In general, for ^ - ^  r  -  ^  >  w e  c a n  write  (5-21)  rf" a  —  c+t-So  f '  !  f  ~  (5-22)  as up to this point the decomposition (5-12b), which represents our (q+l)st state decoupling, does not affect the original hierarchy of equations (5-2a) For f)>^  » the above identifications ((5-21) and (5-22)) cannot be made.  However, defining primed quantities 7 7 and Q  by  A ^L£  (5-23)  9  7Tr.  for a l l i  C  ^  T<  (5-24)  , we can continue equations (5-15) and (5-18) into the infinite set  + Ht*. GL' , » » e- \ = '/ U + ail  -  l L  <5  which is based on the closed set of decoupled equations (5-13).  1 -  -J2'=  (4  if)  25)  Note that  (5.26a)  and 0>[  ~  .  ( -26b) 5  42  It is of interest to note that one can equivalently define  and G  e  by equations (5-25), (5-26a), (5-26b), and the decoupled set of equations in which one is interested, e.g. (5-5) or (5-13). our previous definitons of  Q,  and  Equations (5-23) and (5-24),  , may then be deduced through the  use of equations (5-13), (5-21), and (5-22). Let us consider, as an example of this alternate definition, the special case of decoupled equations (5-5). Looking at equation (5-5a ,,), we define / q+i by  f  |TF  *  (5-27)  and write a Green's function equation for i t , using equations (5-5) to do so:  e~  ^+» - e.  ^  +  ol  t  e.  ^  - o  +- ^  +-  d^ 2^  alt (5-28)  J L I^  A  + X*  +d<di)G,{  6^  ls0 _£ = 0 where in (5-28) we define d_j, d_2,.to be identically zero. 5  We then define  by  ' 5+1 A N D  -  J>  1  (5-29)  1  G<J+SL by ^  (5-30)  We next write an equation for  6^1  =  and define  6^ + 3.  /  to get, in the same way as above.  + cLdf*) Ji l  (fereen's functions)  (5  by  / ' = >~ (M.,-h 7  *  °  d*<J4)Tj  (5-32)  and so on. /  It is this latter definition of the  ^  that is useful in practical  applications of the Moment Theorem, which we are now prepared to state. 3.  Statement and Proof of Moment Theorem Theorem: The ^  moment M^  associated with the line shape function  derived from the closed set of decoupled equations (5-13) is given by  H for a l l f? .  - "Tr'/77'  (5-33)  44  Corollary I For (q+l)strstage decoupling, the ..first q+1 central moments M , M^, Q  Mq  are exact. The proof follows immediately from the definition (5-23) of  the identification (5-21) for .( ^  and  .  Corollary II If one wants a line shape function which gives the first.q+1 moments exactly, one cannot afford to decouple before the (q+1)  stage.  The proof is analogous to that for Corollary I, with the additional remark that a thermal average cannot in general be expressed exactly in terms of lowerorder thermal averages.  Proof of Theorem The proof will be part for a simple case f i r s t , and then in full for the general case. Case (a): Closed set of equations given by (5-5). Equations (5-5) form a special case of the set of equations (5-13). For proving this case i t is more instructive to use the language of the special case, which means using the definitions of  and G\  €  given at the end  of Section 2. Since equations (5-5) represent q+1 linear equations, each  VK  (Oittj)  can be written, by Cramer's Rule, as a quotient of determinants, the denominator being a polynomial one degree higher in accidental degeneracies, then, each  '  ~  ^ p  than the numerator. Barring can be written as  ~f*7 S3  |  C » "* fe» p  (5-34)  45  which results from the decomposition solution.  into partial fractions of the quotient  are certain coefficients independent of E, some of which f  may be zero. case.  The  The A^  are real as is shown in Appendix Ift.for the general  Ep are real, distinct, and independent of t . The reality  of the £p follows from the analyticity conditions on the Green's functions. Their independence of i  follows from the observation that a l l the Gp have  the same denominators in the quotient of determinants. Substitution of equation (5-34) into equation (5-5) yields P  Zl  P  fe-V e? / £  = 17 +  ff  >  ( 5 36  Equating coefficients of  results in three conditional  t  £, A j -  5  >  Kj,  0  * 4<  f  Now, using equations (5-3), (5-34), (5-4a) and the asymptotic  i •  __J  PfA*  L T ? C £ )  where P denotes the Cauchy principal value, we find  equations  (5-37)  ( 5  formula  .  3 3 )  46  +  -IT^V,  dole  ? r  Jul  1  p  p  From (5-38) and (5-39), i f  (5-41)  0  ^  ^ ^  , (5-42)  and from (5-37) from (5-26a)  (5-43)  47  So, for T ^ ^ , equation (5-41) yields  r  "  ^  '  °  •  (5-44)  If T= q+1, we have from (5-38) P  from (5-39) f-  Jj.T^  from (5-37)  £=0  from (5-29)  + i  (5-45)  and so (5-46)  1* from (5-41), (5-44), and (5-45). q+1 (equation (5-33)).  The theorem is thus verified for ( = 0,1...,  The proof is not continued for higher values of f for  this special case, as the notation becomes quite unwieldy. Case (b): General closed set of equations (5-13). For this case we make use of the definitions (5-23) and (5-24) of \JL  and  From the theory of linear equations, the solution of equation (5-13) is given as a quotient of determinants, the denominator being a polynomial one degree higher in £. than the numerator. Barring accidental degeneracies then, each Ci. can be written as ^  p  48  which results from the decomposition into partial fractions of the quotient solution.  n» the number of elementary Green's functions, is the degree of  the polynomial comprising the denominator;. independent of £• and independent of  are certain coefficients  , some of which may be zero, and the  are real,, distinct,  . The reality of the £^a follows from the condition  U  that the Green's functions be analytic on the upper and lower half-planes. Their independence of  <?< follows from the observation that all the  Cj*.  have the same denominators in the above-mentioned quotient of determinants. Substitution of (5-47) into (5-13) yields  where the summation convention is s t i l l in use.  Equating coefficients of  results in two conditional equations: •  *  ^  all  o4  '  (5-49)  A . be shown to be real.  It is important for the proof that the is done in Appendix IV.  .a  This  Multiplying equation (5-50) by 1, £p , f ^ ) , . . . successively, ancj using (  (5-49), one finds  P  a  fl  A  49  From equations. (5-3), (5-4a), (5-4b) for p=0, (5-47), (5-40), (5-14), and the reality of the  ^ £ ,' the moments of  are given by  M; = ^"^  &>ol£*^~i,  L  ^  1^  p  by (5-51)  ' / 7 T '  This completes our proof.  b  Y definition  (5-29)  50  4. Summary on Use of Theorem.. It may be worthwhile here to summarize this chapter indicating how the theorem would be used.  Examples of i t s use will be given in the following  chapter. The line shape function can be expressed in terms of a Green's function G  0  and the equation for G leads to a hierarchy of equations involving o  higher-order Green's functions  C,^ . This hierarchy is decoupled in some  manner leading to a closed set of equations which can be solved for the approximate  G) \ 0  One could then find the moments of the line shape by  suitable integrations with the approximate line shape function.  The theorem  shows that this integration is not necessary. The approximate moments up to the stage q+1 at which the decoupling was done are just the usual thermal averages appearing in the equations and agree with the exact moments. To find the higher moments of the approximate line shape one continues the closed hierarchy of equations. appear;  each time a  In this continuation no new Green's functions  Green's function appears in an equation i t is  decomposed with the rule f i r s t used.  The thermal averages appearing in these  higher equations are the appropriate higher moments of the approximate line shape function. This procedure for finding moments has the advantage that i t does not require one to solve equation (5-13) explicitly; one need specify only the decomposition (5-12b) that is being used.  CHAPTER VI Resonance Line..Shape Problem 1.  Decoupling Metftods At the beginning of Chapter IV, the properties demanded of a suitable  decoupling technique are outlined.  These are the tractability, degree of  complexity, and faithfulness of the resulting closed set of linear equations. It i s , in any specific case, a simple matter to determine whether a set of equations is complex or tractable.  It is the question of faithfulness that  will be dealt with in this chapter. We postulate a faithfulness criterion: one decoupling procedure is said to be more faithful than another i f the f i r s t few moments of its consequent line shape are closer to the true moments in some valid approximation. The phrase "in some valid approximation" is inserted, as expressions for the moments are invariably given in terms of thermal averages; and even the thermal average of  ->i has never been given exactly.  The criterion is practical, for the moment theorem of the last chapter immediately enables one to write down expressions for the moments which are to be compared with the true moments.  In line with this practical approach,  only the f i r s t few moments are usually asked for, as higher moments become increasingly d i f f i c u l t to calculate in any approximation. Our criterion will now be used to investigate the relative faithfulness of two specific decoupling methods, when these are applied to the problem of finding the paramagnetic resonance line shape function associated with the physical system described by the no-crystalline field Hamiltonian (2-9), where 1^: and  are given by equations (2-11) and (2-12a) respectively.  One method we shall call the excess-over-random  (EOR) method; the other, due  to Tomita and Tanaka (1963), which is a partial excess-over-random.method,.we shall refer to as the T-T.method.  52  We specialize our physical system s t i l l further by requiring that i t consist of spins having S = .  For this case, from  and the well-known commutation relations for spin operators, the following relations can be proved:  JVJT-  s  :  ' s r =  o  The significance of the above relations is that any product of two spin operators at one lattice site can be reduced to a linear combination of single spin operators. The EOR method, which takes full advantage of relations (6-2), proceeds as follows:  One rewrites the Green's function equations (2-50) for the  "ordinary" Green's functions  [k\ =  f A/B"] =  A/B^-  in terms of  comulant Green's functions (A/ = (A/B) which are defined by  01  -  ('1 (6-3a)  rial * < / > d l +- <*>('! +•  0*1 (6-3b)  etc., where the numerals "1", "2",... represent various operators.  After  53  the above decompositions, one stops at a certain order and discards the highest-order cumulant Green's function.  For example, in second-stage  decoupling, one stops after equation (6-3c) and neglects the cumulant Green's function (l23"|, supposing i t to be small.  One of course v e r i f i e s that the  operators "1" and "2" i n , for instance,Cl23l lattice sites,  , are associated with  Otherwise, the product of operators "1." and "2" is to be  treated as a single operator "12" in the above decompositions. the Green's function [Sbe changed to  different  Si  -^p [.Sj^j  Sj  j  As an example,  for the case S =  must  before any decompositions are made.  The T-T method proceeds in the same manner, with the one essential difference that products of operators associated with the same l a t t i c e s i t e are not treated as a single product operator in the decompositions (6-3). The f i r s t . t w o Green's function equations, written for the  elementary  Green's functions, are  4 -  S L C ^ S + l ^ Z  "  (6  4a)  Note that we in fact want to find  x  These equations hold for any e f f e c t i v e  (6-5) spin S.  54  Second-stage  decoupling by the EOR method, f o r the case S=T/2, consists  in making the following replacements:  where  ^A.5  (^J^  1 S  independent of  i  by t r a n s ! a t i o n a l invariance.  When any two of p , i , and j are equal, use of equations (6-2) ensures that no new Green's functions are present, and hence that the r e s u l t i n g set of decoupled equations i s closed. Second-stage  decoupling by the T-T method consists i n making the  replacements  +  (<f,V>-vVf^  "  (6 7a>  ^<s-rr><^i^  -  (6 7b)  f o r al1 p , j , and i . I t may be seen.that i n the EOR method c e r t a i n c o r r e l a t i o n s , namely autoc o r r e l a t i o n s , are retained i n t h e i r exact f o r m . w h i l e i n the T-T method they are approximated.  Thus, i f we take p - i ^ j i n equation (6-7a), the  replacement  *  /  /  £  (6-7c)  55  is effectively performed in the T-f method; in the EOR method, the left-hand side of (6-7c) is retained without approximation. Substituting equations (6-6). and (6-7) into equation (6-4), the EORdecoupled equations are  (6-8a)  and  of  w  2»  where we have used the notation  (6-9)  56  Other types of seconcUstage decoupling may certainly be tried. that seems worthy of consideration is due to Callen (1963). the fact that £ •  may be expressed in two ways in terms of  for S =  One type  It makes use of £ ~ a n d  i.e.  and  f,-* 't-S^'f* s  -\'  (6  "  11b)  is replaced by  L(?s:-rx) o-A(±-rrs:)  ta  +  (  ,  l l c )  57  Here &<C  a f t e r which decoupling proceeds by the usual EOR method. arbitrary  i s an  parameter which must be fixed by some suitable physical c r i t e r i o n .  The Callen method w i l l not be considered in this t h e s i s , as the calculations become very long and involved. Other possible methods have been looked at and tested on the Ising model Hamiltonian, but they were found to be of far less value than the methods being considered here. From Corollary I to the Moment Theorem, the f i r s t moment, derived from both the EOR- and the T-T-decoupled equations (6-8) since q = 1.  and (6-10), is exact,  Let us look,then, at the second moment.  By use of the procedure  suggested by the Moment Theorem, one arrives at the following expressions for M (T-T) and M (E0R) : 2  2  (6-12)  M  2 ( E  O ) . ( T - T > ;• • ; R  M2  m*t«$T>  ,3£  N  )  (6-13)  The correct expression for M , denoted here by M (C0RRECT), i s derived from 2  2  equation (4-15) with  (K = SZ-  S>  IB  X  -  (. ) 6  ,  14a  (6-i4b)  58  It  is given by  ^(CORRECT).  = -  ^ C ; < f / f y ( a < ^ V * >  +<^;*:>-<fs;s,9  ( 6  ,  5 )  To compare these formulae for the second central moment, we use the McMillan and Opechowski approximation technique which was described in Chapter IV in the section after equation (4-35). v a l i d to (M0)  2  and for S =  The following r e l a t i o n s ,  are needed for our proposed comparison.  (6-16)  is the first-approximation part, of  » and is independent of  i,  59  is independent of i by translational invariance,  Substituting the above.expressions into (6-12), (6-13), and (6-15) the expressions for M become, in (MO).^ , 2  -  ^  {V  £ ^  +  A  A  (  B  ^ <  (  6-i ) 7  in (M0)  2  M , (T-f) = - R X £  */ «-/?) - H i r  ^  '  / 4 )  M (E0R) = M (C0RRECT) = R.H.S. of (6-18) -f 3 / y 3 2  2  ^ ^  (u^ ./^ * m  and +•  *  ( / ^ ) ]  t  (6-19)  And so the second-approximation part of M (T-T~) is found to be in 2  error. part.  and  Moreover, this error is not 0(1/N) of the correct second-approximation This may be seen by comparing, for example, the terms  60  The f i r s t  term i s not 0(1/N) of the second even though the l a t t e r involves  an extra summation. as l / l ^ - ^ e j  This i s so because  and Cj^  f a l l o f f as rapidly  with increasing distance between l a t t i c e s i t e s .  3  O ' R e i l l y and Tsang (1962) have done summations l i k e the above f o r the special case of Calcium Fluoride c r y s t a l s , bearing out this Note, however,  the e r r o r . i s n i l when T = 0, as in this  statement.  limit  and also when T = ° ° , as then  -  O  We now look at third c e n t r a ! moments.  With considerable work i t can  be shown that, to (M0)^ , M (E0R) = M (C0RRECT) 3  3  a T T  '°  _  C  _  3 5" §8 - A 133 f-trZl% i  ^-,)  (6-20)  l3n}}  i  where the notation  ABCD~  =  ZZ.  iQuL^itCteb;?"  has been used. M (T-T) i s more d i f f i c u l t 3  to compute, even to (MOjj, but i t can be shown to  be in error generally by showing i t to be so f o r the Ising model.  For this  case M (CORRECT) = 3  "^-J-*  4/^o  )  [K /~ fo~)  (6-21)  ^)  (6-22)  z  H  and M (T-T) = M (C0RRECT) 3  3  £^jf  61 Again i t  is seen that the error is not of 0(1/N), and that i t  vanishes only  in the limits T=0 and T = The following table is drawn up in order to highlight the differences in moments calculated by the EOR and T-T methods.  The ticks represent areas  of agreement with the correct values, the. crosses, disagreement.  M  (MO^  (M0)  2  l  M  "2  (M0)  3  (M0)j  (M0)  (M0)  2  X  T-T EOR  By our c r i t e r i o n , i t  X  X  1/  X  follows that EOR second-stage decoupling should y i e l d  a more f a i t h f u l line shape function than the method of Tomita and Tanaka.  2.  Solution of Equations. When one comes to consider the s o l u t i o n . o f the EOR and T-T-decoupled  equations (6-8) importance.  and (6-10), the t r a c t a b i l i t y requirement assumes extreme  The T-T equations turn out to be completely admissible of  an analytic solution in Fourier-transformed space, while the EOR equations/ which are almost i d e n t i c a l , have some added terms which increase the number of simultaneous equations which must be solved for.  Cn  0  to  something of the order of N. Moreover, the T-T method y i e l d s a set of tractable equations for spin values higher than J^, and can e a s i l y be extended to third-stage decoupling. The EOR method, on the other hand, becomes p r o h i b i t i v e l y complicated when higher spin values and higher-stage decoupling are used.  2  62  For the reason that T-T second-stage decoupling does not conform sufficiently to the complexity requirement at high temperatures(i.e., temperatures in the paramagnetic range), Tomita and Tanaka extended their calculations to include third-stage decoupling.  Now, T-T third-stage  decoupling leads to equations which,;according to our criterion, are even more faithful than EOR second-stage decoupling.  This follows  from Corollary I to the moment theorem, which tells us that the N^, as well as the Mp resulting from a third-stage decoupling, is exact.  For the  above reasons, no serious attempt will be made to solve the second-stage EOR-decoupled equations other than formally.  A simple case will be  considered, however, to show that these equations satisfy the complexity requirement, even at high temperatures. The f i r s t two Fourier-transformed second-stage T-T-:decoupled equations, derived from the spatial equations (6-10) are  (6-23a)  and (6-23b)  where  (6-24a)  (6-24b)  (6-24c)  (6-24d)  63  and the notation of Tomita and Tanaka  J •4  ,i !  l-k  =  (6-24g)  has been employed.  <-J|^ J J ^ . ^ I ^ 0\.<^ cj-h ^  and  are Fourier transforms of  0-^,$  certain linear combinations of thermal averages. Note that G = 0" n  '+•1= /fl [o\-- u k =  (6-25)  Equations (6-23) are easily solved for  (t\  .  The solution is  (6-26)  where  f  0  «f-  (6-27a)  (6-27b)  Equation (6-26) does not satisfy the complexity requirement for high temperatures,.as  64  and  reduces to a sum of two delta-functions. The f i r s t two Fourier-transformed second^stage EOR-decoupled equations,  derived from the spatial equations (6-8)  are  (6-29a)  + f('M P(fc,t)(tl+ 0 / / f c < ^ , t M ( u < £ | ] where  «k,^ - (ft) r_ {c ( U ' - , t-t - k ) f  y-v  t  (8.30.,  - / ^ ( k - k ' - ; , k-k'), To solve equations (6-29) formally for  ^ |  (6-295)  <6-30b)  , we treat the terms in  the square bracket of equation (6-29b) as sufficiently small, compared to (i/VT) {{(k f)  s o  t  n  a  t  a n  iteration procedure can be used; this  is equivalent to the recognition that the EOR and T-T methods do not diverge widely from one another.  Letting (6-31)  the second EOR equation, (6-29b), may be written  65  +  o M ^ M ^ . , t<| , k  or, with the use of the iteration procedure on 0  (  32)  f j^.  3 ?  +• ' (6-33)  where  (6-34)  S~  We note that  (k  s cud- 3L : s  Similarly,  ^  e  satisfies the "integral equation"  +  mzL  ^ > '. (L  L ]  (6-35)  ( | ? ^ satisfies the "integral equation" (  Equations (6-29), solved simultaneously, yield the formal expression  =  £ - A dcV-*T V a  1  -  DC^M^M^  "  (6-37)  66  One would like to solve equations (6-35) and (6-36) exactly for ^  ^  a  n  lO  d  respectively.  Unfortunately, this  cannot readily be done, as the coefficients in these equations do not possess convenient symmetry properties.  Let us, however, consider the simple  case when  (6-38)  Equations (6-35) and (6-36) then have the solutions  (6-39)  J f i  (6-40)  N  Substituting (6-39) and (6-40) into equation (6-37), we get the following expression for  jk  :  '  —- ,  (6-41)  67  Equation (6-41) satisfies the complexity requirement even in the high-temperature limit for which (6-28) holds.  We note that this requirement is also satisfied  when equation (6-41) is specialized to the Ising model case, when (see equation (6-47))  independent of q  (6-42)  The latter statement cannot be made about equation (6-26), the equivalent T-T solution.  It will be demonstrated, in the next section, that  also 3rd-stage T-T decoupling is not "complex" enough for the Ising model case.  3.  Comp 1 exity Requirement We have seen above that at second-stage decoupling the EOR method  satisfies the complexity requirement that the line shape consist of many delta-functions.  In order to have this requirement satisfied with Tomita-  Tanaka decoupling, these authors had to go to a third-stage decoupling. this section, we shall discuss this point. Using third-stage decoupling, Tomita and Tanaka arrived at a closed set of equations which proved to be quite tractable, having as a solution  (6-43)  with  r  In  68  where  ' ^ a * (^.^  and C. ^,^') 3  involve Fourier transforms of A^,  C.., and of certain linear combinations of thermal averages, and  (6-45)  Equation (6-43) can, by use of the method of partial fractions, be cast into a form which leads to a sum of delta-functions for the line shape. These must be smeared in some way so that a continuous line shape function results.  One can perform this smearing only i f the delta-functions are  sufficiently large in number, and their arguments sufficiently dense along the <0-axis.  It is this condition that has been referred to as  the complexity requirement.# At f i r s t sight, Tomita and Tanaka's expression (6-43) for (^j to satisfy the complexity requirement. general paramagnetic case.  seems  It especially seems to do so for the  However, i t demonstrably does not for the case  of the general Ising model, at any temperature and for any spin.  This will  now be shown. # Grant and Strandberg (1964) satisfied this complexity requirement by treating the distribution of spins well-removed from the one of interest as continuous.  69  Recall that for the general Ising model (2-27)  = o  A^M  (6-46a) i.e.,  f\  O  5  As a consequence, the quantities L  0  equation  £  all  (6-46b)  (fi and L*(k% %') t  involved in  t  (6-43) become  e  (j)alJ  -r / u C W =  6  A  independent of q  e  (6-47)  and  iXwO-C^fert ^.^- " 3  Then J) " 1  3J  =  and Y[?~ (equations r  -  <6  48)  (6-44a) and <(6-44b) ) become  /  ^  l(6-49a)  V --^o-^ vM-r)7 . t  6  4 9 b )  Now  t^fc.  »Z l ^ ^ , ^ . 3  f  t> (^ V  (6-50)  ^  by translational invariance. as follows  i.  So, i f we define  Q  ^  C (IA) g  3  C ik) and  70  (6-51)  8  C -  3  M s  ZlIU  we may write  (6-52)  This yields, via (6-25 ) and (5-40 )., a line shape function which is expressed as a sum of three delta-functions.  It i s , of course, wrong to extrapolate  from these to a continuous line shape function. Now, the general Ising model admits interaction of any spin with a l l its neighbours, no matter how far away. The most elementary considerations indicate that unless one is dealing with the one-dimensional Ising model with S = %, nearest-neighbour interactions, many more than three delta functions  71  are required to describe the line shape.  One must conclude that Tomita  and Tanaka's third-stage decoupling,cannot .be.used on the Ising model with any success.  4,  Application to .Case of Completely Non'Teqwdlst-ant" Si.rjgle-Spin Energy Levels-. This chapter so far has considered the case in which the single-spin  energy levels are equally spaced.  The most fruitful approach to the case  of completely non-equidistant energy levels would appear to be through £  the use of the operators  introduced in Chapter II.  This, however, proves sufficiently complicated, even with T-T decoupling, that i t is unlikely to yield a line shape, and one has to be content with calculations of moments as in Chapter IV.  To see this, let us discuss  first-stage decoupling and indicate! what would,happen in second-stage decoupling. The statement that the unperturbed energy levels are completely nonequidistant implies that whenever  one has V  =  A  This is a restriction on the elements of the set 67 The Hamiltonian  14= ^-Wf where  M  +t  (6-53) defined in Chapter II.  can then be written as.(see equation (2-23))  ^ [ f y ^ . ^ ' ) * ^ * /  is defined by equation (2-16).  (6-54)  72 We will take the unperturbed frequency, W)  a  ,of the primary line under  consideration to be given by  (6-55) We then look for  (6-56) in order to find the line shape function  (6-57)  ft The f i r s t Green's function equation is  +  (6-58)  where we have used the relation  (6-59)  73  which follows from  combined with (2-32) and (2-22). T-T decoupling in the f i r s t stage consists, here, in performing the decomposition (6-3c) and neglecting higher-order cumulant Green's functions. This is equivalent to making replacements like the following:  With this decoupling, equation (6-58) becomes  ^* ('^mX  (  air  =  +  IZ.  -  A  i <AJA  ZZ- A*j  ^,>(/U!\  <C^>2-  A^^UmX  J  (6-60)  74  Taking Fourier transforms as in Section 2 and using (6-56) and (6-57), one gets a single delta-function for the line shape:  g(iS) =  F [ oo  - u  A  0  (0  c(o)  k'  where  yti^" ^  _  ^Cl^^  (6-61)  independent of i  ^S^y  >  independent of i j  In deriving equation (6-61), one had simply to solve one equation for  To see the difficulties that arise when second-stage decoupling is attempted, consider the simplest case, that of effective spin S = 1 (with A  = -1, say, in (6-55)).  where there are two  ) S  One must keep a l l Green's functions of the form  and one s(L  in each Green's function.  are nine of these, seven of which are independent.  There  A separate Green's function  equation must be written for each of these seven, a decomposition made, on the right hand side of each equation, of higher-order Green's functions into lower-order ones, according to equations (6-3), and Fourier transforms taken. One then has, optimistically, seven coupled simultaneous linear equations to  75  solve.  The latter number increases quickly when higher spins are used.  This  process is quite unwieldy, and so the analysis has not been pursued any further.  S  Summary of Chapter VI. We have,,in this chapter, applied.a moment theorem to compare the  faithfulness of two decoupling techniques.  The results of the comparison  are summarized in the table on page 61 . The EOR decoupling proves more faithful than T-T decoupling, but unfortunately, less tractable. It satisfies the complexity requirement at second-stage decoupling, whereas T-T decoupling satisfies this requirement only with a third-stage decoupling. For the Ising model, this complexity is not satisfied even with 3rd-stage T-T decoupling. The argument in favour of Tomita and Tanaka's approach to the general paramagnetic resonance line shape problem is that i t does, at least in the case of equally-spaced unperturbed energy levels, provide an analytic line shape function. moments.  This may be more important than its providing correct  76  APPENDIX I  To show that in  (A-l)  ^ Z L ^ ' f i  m  a  y  b e  r e  P  1 a c e d  b  y  AJA'+I  ,  Now,  where we have expanded the trace in terms of eigenstates with eigenvalues  of  ?^  .  Let  Then  Using (2-8), (2-21), and the hermiticity o f  - £ M/A  1 4 ^ , we have  - (T^Kf  - (77*i-T') A  ^ (A-5)  .  77  Therefore  But, for A£ ^ ,  -~T^  =  tJ  e  (see equation (2-31)),  So  CXec,)  i.e.,  "<u  { A 6 6,)  is zero unless  ( a  \ £  C  .  7 )  h  Similarly,  where  (A-9)  From (A-8), (A-7), (A-9), (A-3), and (A-2)  •  xea,  APPENDIX II Calculation of Thermal Averages for the One-Diimensional Ising Model using a Difference Equation Technique.  As an illustration of the technique, the derivation of the thermal average £ 0 7  will be given.  independent of i  where  =  'A  J7)^  71 j.  i ^ / ^  S ^)  where we have taken i = N  are eigenstates and eigenvalues of  (A-12)  7?i/^>  S  and 71 ^ can assume the values ±  5 ^ » i«e.  j  - 1 /l£T~  ; and the  following notation is used:  (A-13)  («,>U - ^ K ^ l ^ ' ' / O l ^ j o e? , 4  A  (A-i4)  We will also use  <*dc* *hfy s  .  (A 15a)  (A16a)  (A-16c)  (A-16d)  Performing the summation over 01^ ,  (A-17)  Looking at the f i r s t term on the R.H.S. of (A-17), we write i t in the following way:  (A-18)  Noting that (A-19a) 1L and  <^^VTI,.XV^  ^<^t>+ku;,>  _  (A  i t follows that equation (A-18), after i t is summed overtf,^  igb)  »  As ^ »  80  may be written in the form  (A-20)  where  "P*  #p  and  bp  P  1  ~  L.,  satisfy the recursion relations  '  (A-21a)  P  <f#p +  ( A  .  2 1 b )  with the i n i t i a l conditions b-j = 0, a-j = 1. Equations (A-21) can easily be cast into the form of a pair of second-order difference equations:  (A-22a)  (A-22b)  Equations (A-22) possess the solutions a  f  L p  r W  -  f  t  t  u«  f  F^WGifA-Y  ,  where F , F^, G ,.G^ a  ,  ( 23a)  (A  .  24b)  are independent of p and are determined from the  i n i t i a l conditions, and  A±  are given by equation (3-13).  81  In the calculation of the second term of equation (A-17), one has occasion to look at the equations  £  5 M  (A  4+1  ^ r*  (- )  c  =  _25a)  A  25b  with i n i t i a l conditions C| = 0, d^ = 1. They possess solutions  T  c/p  6\+V  F;  »  F*/  where F , G , F , G c  c  d  +•  Cc  4  ^  ( . A  2 6 a  )  (A-26b)  are again determined from the i n i t i a l conditions.  d  Whatever thermal averages are done, one can always arrange the terms in the appropriate summations in such a way that either (A-21) or (A-25) need be used for each term.  A l i s t of the values of the F's and G's  follows, where the notation ]f is used.  =  ]/  U-  +  (A-27)  82  6w r )  From the above table, the following relations can constructed and are found to be useful: (A-28a)  (A-28b) Continuing from equation (A-20), performing the final summations, and using equations (A-21), we have  (A-29)  2. ^rt+l . Similarly, the second term of (A-17), which we call  fetii)^  , is given by  (A-30)  83  and so  -  a  -  y  '  (A-3D  One similarly finds that the partition function Z is given by  as given by Huang (1963). All other thermal averages can be found in the same way, when this is tractable.  For example, the thermal average  S^  9  ^  cumbersometo find by this method, as eleven summations must be done explicitly before the difference equation solution can be applied.  becomes  84  APPENDIX III Proof of (4-15) directly from (5-2a) and (5-3)  Let  ,  tf>  i N  ^j (A-33)  (A-34) Then (A-35) Multiplying both sides of equation (5-2a) by fit)>  from 0 to «Q  and summing over n  , one derives the equation  which has as solution  B  (A-37)  J  A boundary condition is needed to fix t t h e constant of integration. e?  make use of the fact that when  74  w  - 0  , the hierarchy of equations  (5-2a) reduces to the single equation e. Gi  9  ~ T  (A-38)  9  and moreover, "  "  '•  •  (A-39)  We  Substituting (A-39) into (A-37) and using (A-35),  C when  14  1  0  &AE)  =  ~T~ O - e / " ' ^ ' )  (A  _4o)  = 0 . To make (A-40) (A-38), one needs  The assumption is then made that (A-41) holds also when  7/")  ^ o  Now  (A-42)  86  where we have used the relation  -r Vr)  - ~i~(-r)  which follows from (A-34) and the reality of the "77,  { M 3  ,  . Equation (A-42)  is equivalent to the stage reached in Chapter IV, equation (4-7), where a similar problem was considered.  The proof is completed in essentially  the same way, i.e., via the definition of a suitable characteristic function.  87 APPENDIX IV Real i ty of A^ of •equa.t.ion /3 and C'.,  It is assumed that  (5-4-7-)  are real, as they are invariably simple  known combinations of thermal averages.  (5-51), 2L (  From equation  A [ [fC]  Let  ^  are real for all  c<  Real part of Aok Imaginary part of A  Then  {  (A-44) i.e.  r (A-45) Writing equation  0,1,2  (A-45) in matrix form for  having the values  n-1, i ,e.  £  f f  A  (A-46)  88 i t is seen that the " all the £ p  £.• matrix", being a Vandermonde matrix and having  distinct, is non-singular, leading to-.the conclusion that  (A-47)  all  o(  . This completes the proof.  89 BIBLIOGRAPHY  Bogoliubov, N.N. and Tyablikov, S.V. .1959.  Dokl.Akad.Nauk.(USSR),  126, 53; Translation Soviet Phys. D#klady, 4, 604. Call en, H.B. 1963. Phys.Rev.230, 890. Grant, W.J.C. and Strandberg, M.W.P. 1964. Phys.Rev.135, A715. Huang, K. 1963. "Statistical Mechanics" (John Wiley & Sons, Inc., New York). Ising, E. 1925. Z.Physik.3J_, 253.  Kambe, K. and Usui, T. 1952. Progr.Theor.Phys.8, 302. Kubo, R. 1957. J.Phys.Soc. (Japan), 12^, 570. Lee, Y.Y. 1961. Can.J.Phys.39, 1733. McMillan, M. 1959. M.Sc. Thesis, University of British Columbia, Vancouver, B.C. McMillan, M. and Opechowski, W. 1960. Can.J.Phys.38, 1168. O'Reilly, D.E. and Tung Tsang. 1962. Phys.Rev.128, 2639. O'Rourke, R.C. 1957. U.S. Naval Research Laboratory Report No.4975, p.207. Pryce, M^H.L. and Stevens', K.W.H. 1950. Proc.Phys.Soc.A,63_, 36. Tomita, K. and Tanaka, M. 1963. Progr.Theor.Phys.29_, 528. Van Vleck, J.H. 1948. Phys.Rev.74, 1168. Zubarev, D.N. 1960. Usp. Fiz.Nauk. (USSR) ,71_, 71; translation: Soviet Phys.USPEKHI 3, 320.  


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