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On the spin wave spectrum of manganese fluoride at low temperatures Tam, Wing Gay 1964

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ON THE SPIN WAVE SPECTRUM OF MANGANESE FLUORIDE AT LOV TEMPERATURES by WING GAY TAM B.. Sc. ( S p e c i a l ) , Hong Kong U n i v e r s i t y I960  A THESIS SUBMITTED IN PARTIAL. FULFILMENT OF THE REQUIREMENTS FOR,THE DEGREE OF MASTER OF SCIENCE i n the Department of PHYSICS  We accept t h i s t h e s i s as conforming to the r e q u i r e d standard  THE UNIVERSITY OF BRITISH COLUMBIA August,  1964  In the  r e q u i r e m e n t s f o r an  British  mission  for reference  for extensive  p u r p o s e s may  be  cation  of  written  Department  of  and  by  degree at  the  study*  the  for  the  I further  Head o f my  agree for  that  of •  not  per-  scholarly  Department  shall  of  make i t f r e e l y  or  t h a t , c o p y i n g or  f i n a n c i a l gain  permission-  fulfilment  University  shall  this thesis  The U n i v e r s i t y o f B r i y / i s h C o l u m b i a , Vancouver 8 Canada 5  Library  I t i s understood  this thesis  w i t h o u t my  that  c o p y i n g of  granted  representatives.  this thesis i n partial  advanced  Columbia, I agree  available  his  presenting  be  by publi-  allowed  ABSTRACT I n e l a s t i c neutron s c a t t e r i n g measurements by Okazaki, Tuberfi.eld. and Stevenson (1964) of the antiferromagnetic s p i n wave energy, spectrum i n manganese f l u o r i d e c r y s t a l at low temperatures, agree w i t h the. r e s u l t s p r e d i c t e d by a d i s p e r s i o n r e l a t i o n introduced without proof i n t h e i r paper.  In this  t h e s i s the .question I s considered i n d e t a i l to what extent t h i s dispersion r e l a t i o n i s j u s t i f i e d .  The i n t e r a c t i o n between the  manganese ions i s described by the Heisenberg exchange Hamiltoniaa*  Following. Hoi s t e i n and P.rimakoff s formalism the s p i n 1  d e v i a t i o n operators are introduced.and ton! an  the p a r t of the Hamil-  containing a l l the terms up to those b i l i n e a r i n  the s p i n d e v i a t i o n operators i s d i a g o n a l i s e d by means of the Anderson transformation.  A c o r r e c t i o n i s next obtained by  r e t a i n i n g the diagonal p a r t of those terms which are q u a d r i l i n e a r i n the s p i n d e v i a t i o n operators. t i o n s i t i s shown that, f(  B  Under c e r t a i n condi-  together w i t h the c o r r e c t i o n term  give r i s e to, a d i s p e r s i o n r e l a t i o n which i s i d e n t i c a l w i t h that used by Okazaki et a t . (1964). approximations  F i n a l l y the v a l i d i t y o f the  i s a l s o discussed.  ACKNOWLEDGMENT I. wi sh to thank P r o f e s s o r ¥. Opechowski f o r suggesting t h i s problem... I am indebted to kim f o r h i s guidance and k i n d .encouragement. The f i n a n c i a l support o f the N a t i o n a l Research C o u n c i l of Canada i s also g r a t e f u l l y  acknowledged.  TABLE. OF CONTENTS PAGE X  II III  INTRODUCTION••*»«06«00««0«00 0 0 « 0 0 0 0 « « 0 0 0 « 0 0 0 0 0 0 0 6  I  EXPLICIT FORMULATION OF THE PROBLEM  3  COMPUTATION. OF SPIN WAVE SPECTRUM IN HQLS.TEIN, PRIMAKOFF APPROXIMATION  ...........  IV... HIGHER ORDER CORRECTION. V VI  DISCUSSION AND CONCLUSION ............ 131jESlLjI O.G JlAPMl^  « o » o o o * « » o o Q o a o a » o e o o o o o a o 4 0 4 o o o o Q O »  12 20 42  I.  INTRODUCTION The purpose of t h i s t h e s i s i s to derive the s p i n wave  energy spectrum f o r an antiferromagnetic centred:cubic  c r y s t a l which i s of body-  type and can be d i v i d e d i n t o two s u b l a t t i e e s .  S p e c i f i c a l l y , we are i n t e r e s t e d i n the d i s p e r s i o n r e l a t i o n used by Okazaki., T u b e r f i e l d and Stevenson (1964) without d e r i v a t i o n f o r the spin, wave energy i n manganese f l u o r i d e Mnfi point.  up to Neel  The d i s p e r s i o n r e l a t i o n used by them i s : fa  -2$^zi  ( ^ . o ^ s ) ( ( ^ y - K j  i  (f  -  °  and the meaning of the symbols i s as f o l l o w s : >o> = the frequency of the s p i n wave whose wave vector i s X S  = the s p i n of the magnetic atom ( i n the case of manganese S -  5/2)  = the.number of second nearest neighbours  ...  J, -  -== the. a n t i ferromagnetic exchange i n t e r a c t i o n i n t e g r a l s and  ^>  are functions of  the c r y s t a l s t r u c t u r e .  X  and they a l s o depend on  Their d e f i n i t i o n s are given  by  (3.17) and (3.6') r e s p e c t i v e l y . Experimentally,  .the most d i r e c t method of i n v e s t i g a t i n g  antiferromagnefeLc, e x c i t a t i o n spectrum i n a c r y s t a l i s . b y means of inelastic.neutron d i f f r a c t i o n *  By v i r t u e of t h e i r magnetic  moments,, n e u t r o n s . i n t e r a c t w i t h the s p i n wave e x c i t e d i n the crystal:.and,a. measurement of the energy t r a n s f e r suffered by the neutrons enables the determination spectrum.  of the s p i n wave energy  Recently Okazaki et aL. (196A) reported t h e i r  2  experimental .measurement o f the s p i n wave energy spectrum i n Af«fi „ They compared t h e i r observed r e s u l t s by using i n e l a s t i c neutron d i f f r a c t i o n w i t h those p r e d i c t e d by (1.1), and the two are i n c l o s e agreement a t 4«"2°K. Even a t temperatures 49°5°K and 62 0°K o  the agreement i s s t i l l f a i r l y goodo We w i s h t o i n v e s t i g a t e the t h e o r e t i c a l sion  b a s i s o f the d i s p e r -  r e l a t i o n ( I d ) and the j u s t i f i c a t i o n o f the approximations  involved.  As a s t a r t i n g p o i n t we s h a l l apply Anderson's (1952)  approximate quantum theory o f antlferromagnetism  based on the  formalism introduced by Hoi s t e i n and Primakoff (1940) to the two s u t r l a t t i c e model i n MrvF  2  crystal.  Following the e x p l i c i t formu-  l a t i o n o f the problem i n S e c t i o n I I we s h a l l i n S e c t i o n I I I o b t a i n an approximate d i s p e r s i o n r e l a t i o n by n e g l e c t i n g i n the Hamilt o n i a n terms o f higher order than those l i n e a r i n S„  This d i s p e r -  s i o n r e l a t i o n i s i d e n t i c a l w i t h the term l i n e a r i n S i n (1.1). I n S e c t i o n IV, the c o r r e c t i o n term o f zeroth order i n S w i l l be derived.  Equation (1.1) can then be obtained by u s i n g  certain  additional-approximations which w i l l be discussed i n S e c t i o n V.  3 II.  EXPLICIT FORMULATION OF THE PROBLEM Heisenberg (1926 and 1928) f i r s t pointed out t h a t Coulomb  i n t e r a c t i o n s among e l e c t r o n s together w i t h P a u l i e x c l u s i o n p r i n c i p l e , could l e a d to an exchange e f f e c t s t r o n g l y coupling t h e i r spins which might give r i s e to ferromagnetism favouring p a r a l l e l alignment o f the spins.  F o r a ferromagnetic c r y s t a l the  Heisenberg exchange Hamilton!an i n the l o c a l i s e d s p i n model can then be w r i t t e n ass  where £  Sj, i s the s p i n v e c t o r o f atom i measured i n m u l t i p l e s o f and  0^  i s the exchange i n t e g r a l between atoms i and  The summation runs through a l l p a i r s o f atoms, i n the c r y s t a l . For ferromagnetic i n t e r a c t i o n , the exchange i n t e g r a l s  are  positive. Fo r a n t i ferromagneti c materi a l s whi ch are d i s t i n g u i shed by having a s u s c e p t i b i l i t y which passes through a maximum as the temperature i s r a i s e d , the s i m p l e s t case a r i s e s when the c o n s t i t u ent atoms o f a c r y s t a l can be d i v i d e d i n t o two s u b l a t t i c e s , and the spins o f the atoms i n one s u b l a t t i c e are a l i g n e d i n the oppos i t e d i r e c t i o n to those i n the other s u b l a t t i c e o  F o r the a n t i -  ferromagnetic i n t e r a c t i o n between two atoms i n d i f f e r e n t s u b l a t t i c e s , the exchange i n t e g r a l s Jijj- are then negative. I t turns out,  however, a n t i ferromagnetism i s a much more complicated and  s u b t l e phenomenon than one would expect a t f i r s t s i g h t .  "While the  ground s t a t e o f (2.1) f o r ferromagnet i s given by the completely  order s t a t e where the spins of a l l the atoms are a l i g n e d i n the same d i r e c t i o n , the ground s t a t e o f an antiferromagnet i s unknown I t can he shown that in the case o f two l a t t i c e model the  0  completely  ordered a n t i p a r a l l e l s t a t e , that i s where the spins o f a l l the atoms i n one s u b l a t t i c e p o i n t i n one d i r e c t i o n and the spins o f the atoms o f the other s u b l a t t i c e p o i n t i n the opposite d i r e c t i o n , i s not an eigenstate o f (2.1)•  However Anderson (1951),  Van  Kranendonk and Van V l e c k (1958) have demonstrated t h a t the ground s t a t e does not d i f f e r much from the completely ordered l e l stateo  antiparal-  Thus, Anderson (1952) was able to extend Bloch* s  (1930) s p i n wave theory to describe antiferromagnetism.  Further-  more neutron d i f f r a c t i o n experiments have e s t a b l i s h e d the p h y s i c a l r e a l i t y o f the s u b l a t t i c e model.  Owing to the only approximate  knowledge of the ground s t a t e the theory does not describe a n t i ferromagnet! sn. as w e l l as i n the case o f ferromagnetism.  The  Hamiltonian ( 2 o l ) f o r the s p i n system i s v a l i d only i f we assume there i s n e i t h e r e x t e r n a l f i e l d nor anisotropy o f any kind<>  For  a two s u b l a t t i c e model we take the antiferromagnetic ground s t a t e as one i n which the spins i n one s u b l a t t i c e a l l p o i n t i n one d i r e c t i o n and those i n the other s u b l a t t i c e i n the opposite d i r e c tion.  As the d i r e c t i o n i s s t i l l a r b i t r a r y , the ground s t a t e i s  degenerate^  Though one could s e l e c t a d e f i n i t e one o f these s t a t e s  as the i n i t i a l s t a t e , w i t h Hamiltonian (2.1) t h i s s t a t e would not be s t a b l e against small p e r t u r b a t i o n .  This i n s t a b i l i t y i s removed  by the a n i s o t r o p i c coupling between the spins which p l a y s an t i a l r o l e i n antiferromagnetism.  essen-  Besides s t a b i l i z i n g the ground  s t a t e , i t a l s o removes the degeneracy.  One o f t e n introduces the  5 anisotropy i n the form o f an e f f e c t i v e anisotropy f i e l d seen by each atom and t h i s i s what we s h a l l do.  Ha  as  The anisotropy  f i e l d i s such that the spins on s u b l a t t i c e 1 are p r e f e r e n t i a l l y o r i e n t e d i n the + Z d i r e c t i o n , and those on s u b l a t t i c e 2 i n the - Z direction.  The Hamiltonian i s w r i t t e n as:  o  where atom.  S*  i s the 2 component o f the s p i n v e c t o r o f a p a r t i c u l a r  2jt,2 ^  are summations over the f i r s t and second sub-  l a t t i c e s respectively.  ^  i s Bohr's magneton and g i s the  spectroscopic s p l i t t i n g f a c t o r . For manganese f l u o r i d e , the magnetic s t r u c t u r e has been e s t a b l i s h e d by E r i c k s o n (1952) using neutron d i f f r a c t i o n e x p e r i ment.  The magnetic manganese i o n s have a body-centred cubic  s t r u c t u r e as shown i n Figure 1.  F i g . 1.  Two S u b l a t t i c e Magnetic Ordering o f Mn in  M-w Fj  crystal  ions  R e f e r r i n g to F i g . 1, we i n c l u d e i n our exchange Hamiltonian the ferromagnetic i n t e r a c t i o n between a manganese i o n and i t s 2, f i r s t nearest neighbours i n <o, 0, 1} w i t h exchange i n t e g r a l its  %  1  , I 7., 1, 7>  interaction with i t s  Jl  , a n t i ferromagnetic i n t e r a c t i o n w i t h  ,  - ' y w i t h exchange i n t e g r a l  <i^,o>  directions  second nearest neighbours i n d i r e c t i o n s < '  < \,T, / > , < i,U>  ^''  S,  and <o, d, 1)  /  %  < 7\ 1 , i>  X  1  >, < ~> >>  , < 1., 7, 7 >  ^  and a l s o i t s ferromagnetic  t h i r d nearest neighbours i n d i r e c t i o n s  ^7", o,o>, <o./,o>  ^o,7,o>  w i t h exchange i n t e g r a l  . From now on, we s h a l l adopt the convention of l a b e l l i n g  the manganese i o n s i n the f i r s t s u b l a t t i c e by the l e t t e r I and those o f the second s u b l a t t i c e by m unless otherwise s p e c i f i e d . Hence  The two terms i n the square brackets represent ferromagnetic i n t e r a c t i o n between manganese i o n s i n each o f the f i r s t and second sublattices respectively.  The term containing  JL  i s the a n t i -  ferromagnetic i n t e r a c t i o n between i o n s i n the two s u b l a t t i c e s and  the l a s t term i s the Zeeman energy o f the spins i n an e f f e c t i v e anisotropy f i e l d  Ha  .  i n d i c a t e s summation over a l l  p a i r s o f f i r s t nearest neighbours i n the f i r s t s u b l a t t i c e and ^  i n d i c a t e s summation over a l l p a i r s o f second nearest  neighbours w i t h each i o n o f a p a i r i n a d i f f e r e n t s u b l a t t i c e . We follow. Hoi s t e i n and Primakoff (19-40) and introduce the " s p i n d e v i a t i o n quantum number"  n= S t  r e s p e c t i v e l y f o r the two s u b l a t t i c e s .  and  S i m i l a r l y , f o r the second s u b l a t t i c e ,  +  I n the r e p r e s e n t a t i o n i n  which the s p i n d e v i a t i o n operators are diagonal we have  f o r the f i r s t , s u b l a t t i c e .  - S s£  8  We next d e f i n e c r e a t i o n and a n n i h i l a t i o n operators f o r s p i n deviations  &i, Ox ; fw ,  s;=  S/=  i n the f o l l o w i n g manner,  (2s)Vfifs)  Si-isi-  (2  S--fl*4l'  where ft^S) - ( I -  )*"  ,,  ()  (-2 -12)  and  f ^ S ) - - ( < - £g  9 F o r the operators  at  and  at  , ve have  The commutation r u l e i s given by  •where  i s Kronecker's  For the operators  delta.  , l v we have a corresponding s e t o f  relations  (2.M)  (*2I)  (2-22)  and (2.23) Furthermore,  a^, a*  commute w i t h i v ,  10 I t should be emphasised that the operators  A, ^  at, at  ;  are to be considered as i n f i n i t e dimensional matrices i n order to s a t i s f y commutation r e l a t i o n s (2.19) and (2*23)-« meaningful ranges o f the values o f  The p h y s i c a l l y  and  are given by —i  i m < 2$  0 4  0  This means the components o f  are matrices o f -2S + I  dimensions o n l y .  _ i  $ji and  However t h i s does not  introduce any e r r o r as Kubo (1952) pointed out„  I n the n represen-  t a t i o n , the eigenspaces are completely separated i n t o three p a r t s namely ni,  4 o  . o 4. tvi , ^  4 2S  and  'Hi,  y 2S .  Rewrite the Hamilton!an as  I  y„  >,  z  ^ ^;  i  °* f i  I  i n d i c a t e summations over a l l the .a, and  2j  first  and t h i r d nearest neighbours r e s p e c t i v e l y i n the same s u b l a t t i c e o i n d i c a t e s summation over  second nearest neighbours i n  the other s u b l a t t i c e , t h e r e f o r e r e f e r s to i o n s i n the second sublattice«  i n the term c o n t a i n i n g vX The e x t r a f a c t o r \ f o r  terms i n s i d e the square brackets a r i s e s from the f a c t t h a t we have to count each p a i r o f neighbouring i o n s only once.  ?  11 S u b s t i t u t i n g the c r e a t i o n and a n n i h i l a t i o n operators a , a* L, At t  i n (2.24)  4  f  - 2 J, S X  ji at  bU j A 4 aj  bi*^ fx+f*  ;  12 III,  COMPUTATION OF SPIN WAVE SPECTRUM IN HOLSTEIN PRIMAKOEF.APPROXIMATION As long. as. we w r i t e  ~( )  - ( ' ~ "if)  1  and  2  make no ^expansion the above Hamiltonian (2»25) i s e x a c t  0  However  some form o f approximation i s necessary i f we want to make u s e f u l c a l c u l a t i o n , w i t h , the Hamiltonian,, ..consists, of. putting.. f^sr./...  The s p i n wave approximation 1  and r e t a i n i n g .only those  terms up to the b i l i n e a r ones i n the creation.and a n n i h i l a t i o n operators,.  F o r low temperatures  such t h a t the expectation values  o f the s p i n deviation, quantum, numbers 'ftn, %^  s a t i s f y the  following .inequalities then the approximation i s v a l i d  0  Here the e x p e c t a t i o n value o f  .a dynami.cal. q u a n t i t y A i s defined i n the u s u a l way as (Ay  = W(e"^ /()/+K»«(e-  Let M  7  g  be t h a t p a r t o f the Hamiltonian  JfL  i n g a l l the terms up to those b i l i n e a r i n Al, at; k^^lt*  c.  containthen,  ""-2 X S J ,  a  A bx A + +  + a* & +  ^  (3.1)  where N i s the t o t a l number o f manganese i o n s i n the c r y s t a l The s p i n d e v i a t i o n operators are now defined by the f o l l o w i n g F o u r i e r transformations:  -at?)  ax  0  14  Here  t, ™  are l a t t i c e v e c t o r s , and the summations are over a l l  the l a t t i c e v e c t o r s i n each s u b l a t t i c e .  $  i s the wave v e c t o r .  L e t the dimensions o f the c r y s t a l be L a , L a , L c , where L i s an i n t e g e r and a, a, c are l a t t i c e constants.  Assuming boundary  e f f e c t s to be n e g l i g i b l e , we introduce p e r i o d i c boundary c o n d i t i o n ,  t  then  runs through a l l p o i n t s i n the B r i l l o u i n zone o f the  r e c i p r o c a l l a t t i c e space:  A -—  /U =  x  )  La,  — 7 — f -  7—-  .  I  Lc  .. = _ .L.f,+ /, • • -., -. \ ,. o, . I, . . , u , Uz T  with  L  r  = /z  Also  N  A  z  i s the t o t a l number o f u n i t c e l l s .  The commutation r u l e s f o r the s p i n d e v i a t i o n operators ares  Operators. fl , (X* • commute w i t h >  , b* .  Applying the above F o u r i e r transformations to K  &  , we obtains  15  where  f  ..  J- v ' f t p  A f t e r c o l l e c t i n g terms i n ii'  2  where  /L(^> j>a») + 2 +  ^c^+^tf)  (*7)  /4 = 2S [ J * 0 - <*) + 1 % (i ^ M ] + H>^> -^-Sp* A  Anderson (1952) has shown that the Hamiltonian ^  6  can be  diagonal!sed hy using a s u i t a b l e canonical transformation. The required transformation i s defined by:  16  (a-8)  where  £>  A  s a t i s f i e s the equation  The operators  o<*  commute w i t h  ^*  and obey the  following, r e l a t i o n s ^  *X -Oty<*> =  (3.10)  For a b b r e v i a t i o n , we s h a l l denote ct&kb  y  hy  Hence with  by  C>  and  A^^^a  V  ^ e'»  B  -  V+%P>M«*+fi^)  ^ )  -^s (js,-J&+J 0-H*^SA/-2S^(^ff>+J 2yi>) i  4  J  17  and  = A>(cf+ s;) - 2 &>c>S>  (3J4)  Using (3o9) one f i n d s :  U,ii)  2s; =  where  From F i g . 1  £,=2, 2*~S  and % = 4 . The p o s i t i o n v e c t o r s o f  the two nearest neighbours o f the manganese i o n a t the o r i g i n are  ( o • 0. ± O (3.18)  The p o s i t i o n v e c t o r s o f the e i g h t nearest neighbours are  then  18  .2.  Y2  a.  The p o s i t i o n v e c t o r s of the f o u r t h i r d nearest neighbours are? o,.o)and  (  o,±o.  t  o)  Writing ve o b t a i n the d i s p e r s i o n r e l a t i o n i n t h i s approximation as:  19  Comparing (3o24) w i t h ( l . l ) the expression used by Okazaki e t al» (1964) we see they are i d e n t i c a l except f o r the term o f zeroth order i n S.  This c o r r e c t i o n term i s the subject o f our d i s c u s s i o n  i n the next s e c t i o n .  20 IVo  HIGHER ORDER CORRECTION TERM Instead o f making the approximation  f*(S)=  the binomial expansion and r e t a i n the term i n  f't >-(B  Similarly  I f f - (--? ) s  f.(S)-(l- *§)**(•"  2l)  S"'  I , we use Leo (40 (4.2)  The Hamiltonian (2.25) now becomes  (4i)  Let  H =  Hamiltonian  ^  where  F  (4»3)  fi  f  containing four l i n e a r terms i n  and  +  i s t h a t p a r t of the  4  3  - •'Js ^  AlA. A j ^ j* .> I*  tfi,  l>*  Making use o f the F o u r i e r transformations (3»2) and (3o3)  4  *f  7  U>AV  Y  -l/X-X-X+X)-?  ,(X*X-X)f  Ai AguAjV  V  *„  v *v  i(X+X-X-X)'^ -*X*K i(^t-x-X)-r i(tx-X)-s  A, AAV  -ilM-X+tyr  ,  4  i(XhX-X)a  irf+t-Jri)«r -<x-s  4 1 (*J  I 4-  v,*.*, , -i£C*X-X-Xv* _;(X-x-X)$  2,  v ) ,4,*, aX-XX+Xv** - < (Xi'X^X ~X) ^  1  -c(^i~XVa,  From the symmetry o f the c r y s t a l  f'  where  5\ ^ /  f'  and  jl^ are defined i n (3°6')  ¥ith these r e l a t i o n s , we can s i m p l i f y the expression o f  -  s'X-£-X-X)  At jAj Aj X|  <w>> 4  H  F  X(t*t-il-X)afo&>*P*  2  + uX^X-X-X^a^a^  v/ AW  V 'V\ I \  t (^) ^-2 * ^ &XXX^ k k k v fc(XX-t-X) i j k k k . t sttXXtt)Ckkl>A>> + s (XX-X<XUAkk^ 4  -JJJ;)^\H  X-X+X-X^ tx u k k  +s(i-xxX)itkkk  f  ^^XX-X*X) ^ kkk  27 We may use s l i g h t l y simpler notations without causing confusion,  We now apply Andersonls the diagonal terms only.,  transformation (3o 8) to (4?4)  We s h a l l c a r r y out these e x p l i c i t l y  f o r a.-single term,  •When  X = .x.  Xi - X  7 (*)*I ^ ' '' a:  A,>i  a s  and keep  we have  T  '[^-^" )J^  a  J  28  Dropping o f f the constant term, v h i c h i s o f no i n t e r e s t to us as w e l l as the terms f o u r - l i n e a r i n diagonal terms o f (4-° 5) are  ^  and  ^>  4  the  S i m i l a r manipulation o f the remaining terms i n (4<>4) gives finally:  ^(^  =  ^orU-U^Jr  toy**-  Ai Aj  Ai A  i  4  '  " v  *[n)*2^ '+f^0 c  30  + [non-diagonal terms and terms o f higher order] A f t e r rearrangement and changing i n d i c e s ve obtains  31  $C'- constant term +  K 4  1  2  6 S (4ot ^,)2 1  l  lt  (  2 S  *jT'" ^^->)  + [non-diagonal and higher order terms] Recalling that  2  U»6)  32 hence  ^ ; _  i+^y-  i s an even f u n c t i o n o f  X  Hence the second term i n (4« 7 ) vanishes when summing the odd function of  = 071  ^  X  over the B r i l l o u i n zone»  -  1  -  Consequently  («)  33 As before the term i n v o l v i n g when summed over the B r i l l o u i n zone*  ^/j "^/(i+l./is symmetric I n A 1  2X  A^\^cl  o r Mm,\.^co  vanish  Furthermore, the f a c t o r  and X|  , and  ^  i s an  a b b r e v i a t i o n f o r the t r i p l e summation over a l l p o s s i b l e components o f the wave v e c t o r . X Hence  Consequently,  -2 u.<p  -1wo  2  -2-  2.  Z. /  +  i£*I ^  dfr^ ^  dtrf ^ ' - ^ C i f r J ^ ^  +  "7  Since  j | _ Y^/i^iy  i s  a l l terms containing  3 3 1 e v e n  >w  f u n c t i o n i n Ai*,.)^ ,  , yU*  and A.* , and  (hgtjz)  vanish.  (4.10)  35 X  Use (4 = 8), (4>9) and (4=10) and replace everywhere,  by X  X  by  X  then  (4/i) where  (An)  14-11)  2  (4.14)  and We now n e g l e c t those terms i n (4<>12), (4°13) and (4»14) v h i c h are not summed over the B r i l l o u i n zone as t h e i r c o n t r i b u t i o n s to the d i s p e r s i o n r e l a t i o n s are n e g l i g i b l e when N, the t o t a l number o f manganese i o n s i n the c r y s t a l , i s l a r g e , and obtains  36  A  ~1 -  -.M^-1?\M  4 J  0  38 From, these,  Consider the f o l l o w i n g expression, which occurs i n {1**1*1), and expand i t . keeping terms to the f i r s t order of  V - Jf> 1  a  The above; approximation thus i n v o l v e s l e a v i n g out terms.of order i n  and higher.  We now sum  I-  | -fc^idc^ij4 c^iS l  Z  V2-  3.  (4»20)  over a l l  second )\  :  39 This summation has been evaluated. (Kubo 1952) by transforming i t i n t o an i n t e g r a l , and the r e s u l t i s (4.21)  Now, we t u r n . to. evaluate the approximate, values o f Since; according to. (-4.19) ^ «./ x  [ 2- /  i/'-r>/(.^)  X  For .P  R  and R>  x  7  •2/<luv  V'- F  x  j j -TV  = *., Here we;have s e t .K&=. , x  Q--= ^H  and ) c. » a a  Numeri c a l i n t e g r a t i o n gives  til  (4.22)  A l s o f o r /?>  V' - r*  40  TO  /-  27T  2  (4.2,)  From. (4»2l)',. (4..23) and the i n e q u a l i t i e s :  / -  ('  and  together w i t h the..fact• that  we conclude that we can neglect the c o r r e c t i o n terms i n v o l v i n g Px and  R>  i n (4.18) o  Taking t h i s c o n d i t i o n i n t o account and using (4.17), (4.20) and (4.21) we can s i m p l i f y (4• 18) as f o l l o w s :  11- u f ^  —  2 2S/XJ4  Because o f t h i s we have f i n a l l y  which i s e x a c t l y the same as the d i s p e r s i o n r e l a t i o n (1.1)  42  V.  DISCUSSION AND CONCLUSION I n d e r i v i n g (4.26) we have used the approximation  and . R .. i n (4.18) . I n order  t a s i m p l i f y the .summations i n f>  x  to make t h i s approximation the i n e q u a l i t y *£>Y> must be s a t i s f i e d  4  «  X  R e f e r r i n g to (3.20) and (4.19) when  near the boundary o f .the B r i l l o u i n . zone.  As. \y  . i s s m a l l , the  approximation. in...(4.2 j) i s s t i l l J u s t i f i e d so long as-  £  is  Although the i n e q u a l i t y (4.28)  not i n . t h e r e g i o n near the. o r i g i n . . no longer holds f o r small. X  is  y e t t a k i n g the whole e x p r e s s i o n  under the summation s i g n i n (4*22) we have:  Therefore we ..can .replace the expression. *J ' - ^* j (' t O " +  ^ /- ^  i n .the summation (4.22) f o r a l l v a l u e s o f  b  y  %  Thus equation (4.26) which i s i d e n t i c a l w i t h ( l . l ) i s completely Justified. .^sing the.numerical values chosen for. the parameters, the term, l i n e a r i n S. i n .(4.26) g i v e s a maximum s p i n .wave energy o f 77.8?K corresponding, to. . X = (.^  f,  .%)  I n c l u d i n g the  c o r r e c t i o n term the maximum energy i s . 78.94°K.  Okazaki e t a l .  A3  estimated t h e i r probable e r r o r i n the measurement o f energy t o be ± t'k., hence the c o r r e c t i o n term g i v i n g a maximum c o n t r i b u t i o n of 78.94°K - 77.8°K =. 1.14°K may be i r r e v e l a n t because o f the comparable experimental e r r o r .  u BIBLIOGRAPHY Anderson, P. W.  1952  Bloch, F.  Z e i t s c h r i f t f u r Physik 61, 206.  1930  Erickson,. R. A. Heisenherg, W.  1952  P h y s i c a l Review 86, 694.  P h y s i c a l Review 85, 745.  1926  Z e i t s c h r i f t f u r Physik i 8 , 4L1.  1928  Z e i t s c h r i f t f u r Physik l&,  H o l s t e i n , T. and Primakoff, H. Kubo, R.  1952  1940  619.  P h y s i c a l Review i 8 , 1098.  P h y s i c a l Review 87, 568.  Okazaki, A., T u b e r f i e l d , K. C. and Stevenson, R. V. H. P h y s i c s L e t t e r s 8, 9. Van Kranendonk, J . and Van V l e c k , J . H. Modern Phvsi.cs 30 1. f  1958  1964  Reviews o f  

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