Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

On the temperature dependence of the shape of magnetica resonance lines McMillan, Malcolm 1959

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1959_A6_7 M18 O5.pdf [ 3.77MB ]
Metadata
JSON: 831-1.0085389.json
JSON-LD: 831-1.0085389-ld.json
RDF/XML (Pretty): 831-1.0085389-rdf.xml
RDF/JSON: 831-1.0085389-rdf.json
Turtle: 831-1.0085389-turtle.txt
N-Triples: 831-1.0085389-rdf-ntriples.txt
Original Record: 831-1.0085389-source.json
Full Text
831-1.0085389-fulltext.txt
Citation
831-1.0085389.ris

Full Text

ON T H E T E M P E R A T U R E OF T H E S H A P E OF MAGNETIC  DEPENDENCE RESONANCE  LINES  by  MALCOLM B.Sc,  University  A THESIS THE  SUBMITTED  of B r i t i s h  in  Columbia,  IN P A R T I A L F U L F I L M E N T  REQUIREMENTS MASTER  MCMILLAN  OF  FOR  THE DEGREE  OF  SCIENCE  the Department of Physics  We  accept to  THE  this  thesis  the required  UNIVERSITY  as  standard  OF B R I T I S H  September,  conforming  1959  COLUMBIA  1958  OF  i i i  ABSTRACT  This  thesis  temperature lines  dependence  i n solids  lattice  which  vibrations  sufficiently resonance was  i s devoted  lines  shape  function  lines  i s defined  (1950)  which  particular,  that  detail.  the temperature  helium also  sample  of nickel  dependence  resonance  temperatures  dependent  and f r o m  lines  and t h a t  on t h e shape  of  This  procedure  also  A  moments o f  this  formula  i s discussed  are applied  fluosilicate special  crystal. case  these of the  noticeable  at  characteristics sample.  i n  t o the case From  i t follows  of the c h a r a c t e r i s t i c s  becomes  line  In  the standard  i s valid  this  by  (1952).  approximations.  extent  of  of the resonance  and s e c o n d  moment  resonance  the shape  a n d Kambe  i n various  that  i s the case a t  a n d was u s e d  t h e shape  of  effect  i s used.  The g e n e r a l f o r m u l a e  general discussions  paramagnetic  (19^+8)  t h e q u e s t i o n t o what  a spherical  the  method"  and t h e f i r s t  This  To d i s c u s s  and U s u i  Van V l e c k f o r the second  great  when t h e d i r e c t  describes  are calculated  study  o f magnetic  c a n be n e g l e c t e d .  by Van V l e c k  and S t e v e n s  of  remain  t h e "moment  Pryce  of  o f t h e shape  low t e m p e r a t u r e s .  introduced  function  to a theoretical  of  liquid a r e then  In presenting  t h i s thesis i n p a r t i a l fulfilment of  the requirements for an advanced degree at the  University  of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference  and  study.  I further  agree that permission for extensive copying of t h i s thesis for scholarly purposes may  be granted by the Head of my  Department or by his representatives.  It i s understood  that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission.  Department of The University of B r i t i s h Columbia, Vancouver Canada. Date  g  ^|,  is, flyj.  ACKNOWLEDGEMENTS  I Professor problem valuable of  this  W.  wish  Opechowski  help  my  advice  gratitude to  f o rsuggesting  and f o rh i s c o n t i n u e d throughout  interest  the  this and  performance  research. I wish  Research  t o express  Council  i n the form  also  t o thank  o f Canada  the National  for financial  of a Bursary.  iv  Table  of Contents Page  ACKNOWLEDGEMENTS ABSTRACT Table  i  of Contents  . . . . . . .  Chapter  I -  Introduction  Chapter  II  Brief System  Chapter  III  and  Second  4 Shape  of i t s F i r s t  Function and  Moments  7  3.1  General  3.2  Truncation of the Hamiltonian, and  3.3  of Chapter  Discussion  IV  . .  of the Interaction  Definition  Derivation Equations Moments  4  Moments  from  and  Chapter  V  VI  Sufficient  . .  18  Equations  •  29  C o n d i t i o n s f o rVan V l e c k ' s  (1948)  E x p r e s s i o n f o r t h e Second  Moment  t o be V a l i d  An A p p l i c a t i o n  and (4.19)  c  Second  (3.3.13) a n d (3-3.15) • Chapter  9  "K.*\  and S t e v e n s ' (nso)  f o r the F i r s t  Directly  ft *  Calculation  and Second  of Pryce  7  Energy,  o f f (v ) a n d  i t s First  i  of the Physical  of the Line  Calculation  i  1  Considered  Definition  i  iv  a n d Summary  Discussion  i  of Equations  4l  (4.18) 49  V  Table  of Contents  (Cont'd)  Page Chapter  VII  * ~~%r „  when  7.2  Application  VIII  Nickel Field IX  . . . •  and  Absorption  78  f o r the Magnetic  by a S p h e r i c a l l y  Fluosilicate Crystal i s i n the D i r e c t i o n  Writing  when  Shaped  the Magnetic  of the Optic  and  general of  the form Kambe  Showing  82  tf  another  field  m = XX  given  by  (195D  that  .  { ? mr.+ f.mp ~\ Ishiguro,  x  96 when  there  and when  is parallel  Usui,  . '  = S field  93  form  the  i s no magnetic  to the Z axis  <  Figures  97  F a c i n i l Page Theoretical dependence shaped  II  .  «K = XX{ £«*p+fimp, j  crystalline  I  Axis  91  of  Rewriting in  C  VI  Conclusions  in B  (7.1.4) t o t h e C a s e  of  Aggendices A  75  i n Chapter  Calculation Resonance  Chapter  of  75  j  t  G e n e r a l Formulae  Chapter  a  ^ _  a  7.1  Considered  anj l^<^ )  f o r ^<t^>  Expressions  of  nickel  Theoretical dependence spherically crystal  curves f o r the temperature #<&v)  fluosilicate  curve of  for a  f o r the  £ <£^> a  shaped  nickel  spherically crystal.  . .  88  temperature for a fluosilicate 88  vi  Table  of,Contents  (Cont'd)  Page Bibliography  99  I  Chapter  Introduction  This of  magnetic  concerned shape we  thesis  of  give  us  magnetic  ions  and  consider, i n the  radiation  width which  both  of  the  temperature effect  of out  very  interactions  low  temperature  of  case  the  lattice  to  lattice thesis  f o r which  ions  of  the  vibrations  we  restrict  this  effect  interactions  temperatures. which  finite  I t i s the i s the  main  para-  field.  The  of  mutual vibrations  the  of  the  topic  of  the  neglected. paramagnetic  temperature  theory of  line.  with  ourselves to be  the  from  lattice  width  can  In  paramagnetic  decreases  between  independent  a  (apart  the  the  lines.  vibrations,  narrow",  and  of  formulae  magnetic  between  "infinitely  practically  effect  The  i s completely negligible).  of  mutual  t o be  effect  i n general,  range  the  the be  In t h i s  static  are  dependence  lines.  the  branch  We  nuclear resonance  of a  paramagnetic  effect  temperature.  at  the  contribute,  However,  turns  would  temperature  or  i n one  resonance.  for definiteness,  of mutual  line  investigation  absorption  presence  resonance  '  magnetic  the  to electron  disregarding  interaction  with  resonance  refer  salt  absence  Summary  theoretical  in particular  can  and  phenomena, namely,  magnetic  Let  the  is a  I  this this  The ions  except  latter thesis.  .3  In we  order to discuss  u s e t h e "moment  (19^8) Kambe  and used  (1952).  calculate  classic  in  paper,  in  b u t when  also  introduce  i s quite shall  now  After  systems  I I I , to define which  then give  describes  we  function.  their  independent  by these  under  Our latter  procedure, authors.  o f the contents o f i n Chapter  called  the " l i n e  o f magnetic  these  I I , we  the  proceed,  shape  resonance  and second  the s o - c a l l e d  lines.  moments o f  expressions,  interaction  H a m i l t o n i a n and o f t h e o p e r a t o r w h i c h moment  their  consideration  Before writing  o f the magnetic  general  into  have  coupling  extent  expressions.  factors.  summary  t h e shape  this  e q u a t i o n s f o r t h e moments  used  a function  In  with the  To what  h o w e v e r , we  the  performed  lines.)  and S t e v e n s  discussing,  have  to find  have  expressions f o r the f i r s t  shape  than  Pryce  the temperature  to a brief  chapters.  of magnetic  line  turn  Is t o  discussed  illustrating  to that  Vleck  one o f t h e t o p i c s  their  temperature  similar  rather  lines  and U s u i and  method  of these  f o r t h e moments v i a B o l t z m a n n  Chapter  this  factors  by'Van  (1950)  on t e m p e r a t u r e .  into  a n d Kambe  function" We  the temperature  remaining  types  lines  the o t h e r hand,  only  We  introduced  t h e moment  i s , i n fact,  give  fact,  the  On  resonance  not concern himself  lines  they  equations in  Van V l e c k does  thesis.  equations  behind  of resonance  i s valid  Boltzmann  Usui  idea  o f resonance  introduced via  was  by P r y c e and S t e v e n s  (Thes  attitude  this  also  o f magnetic  e x p r e s s i o n s f o r the shape  dependence this  method" which  t h e moments  analytical  t h e shape  "truncation"  o f t h e sample  of the  represents to the  3  oscillating we  rewrite  function In with for  magnetic  field.  the f i r s t  i n a  form  Chapter  their  and  t h e moments,  second  involving  I V we  a i d we  are able  and S t e v e n s .  these  formulae  t o a case  Chapter  temperature the  time  the  and  give  the  t h e r e has been  general than  Chapter  moments apply  salt. the  our exact  and  equations  i n the form  V I , we  apply  t h e one c o n s i d e r e d  paper.  conditions (involving  of the i n t e r a c t i o n s )  no e x p l i c i t  shape  assumptions  i n Chapter  sufficient  the magnitude  line  f o r t h e moments  of their  Kambe,  moment  under  i s valid.  discussion  the which  Up t o  of this  point  literature.  In  we  V we  from  Van V l e c k e q u a t i o n f o r the second  this in  i n 'section  and  these operators.  simplifying  Later,  more  Usui  of this  over  to obtain,  by Pryce  In  moments  general formulae  these authors  following  traces  introduce  given  by  Then,  V I I we  than  that  t h e s e more  We  first  From  these  have  been  give  exact  and second  included  a more r e f i n e d  given i n Chapter  graphs  graphs  give  formulae  showing central  some  approximation f o r  IV, and i n C h a p t e r  to a  paramagnetic  the temperature moments  general conclusions  i n the f i n a l ,  very  brief  nickel  variation  of the l i n e  shape  of function.  c a n be d r a w n ; Chapter  VIII  IX.  these  4-  Chapter I I B r i e f Discussion of the P h y s i c a l System Considered  For the purpose of the discussions i n the l a t e r s e c t i o n s , we s h a l l consider the f o l l o w i n g t o be the p h y s i c a l s i t u a t i o n : We have a sample, which may be a c r y s t a l or a powder, containing i d e n t i c a l N " s p i n s " .  For the general c a l c u l a t i o n s  given i n the f o l l o w i n g sections i t i s not necessary t o s p e c i f y whether we are dealing w i t h e l e c t r o n i c systems possessing a magnetic moment or nuclear systems possessing  a magnetic moment,  that i s , the general formulae given can be applied e i t h e r to the case of paramagnetic resonance or nuclear magnetic resonance. Thus, we s h a l l speak of a " s p i n " as f o l l o w s : 1)  t o r e f e r to an e l e c t r o n i c system with a r b i t r a r y but f i x e d spin quantum number S and " g - f a c t o r " , ^ , wherein  jfc--^ g.S B  where u,= magnetic moment of the  e l e c t r o n i c system and p. = Bohr magneton = l ^ T  -  I f the e l e c t r o n i c system i s an atomic d i p o l e and i f no c r y s t a l l i n e f i e l d i s present, then the g-factor w i l l have the form tensor.  g =. ^ EL  where  E  Thus, i n t h i s case, we have  i s the u n i t (JL--^^^^.  • If  the e l e c t r o n i c system i s an atomic d i p o l e i n the presence of a c r y s t a l l i n e f i e l d , the g-factor i n general become a non-unit tensor  3  ,  will  5  2)  to refer fixed  to a  spin  nuclear  quantum  <3 , w h e r e i n of  the  nuclear  magneton We  shall  speak  interacting We a  "spin  that  the  constant magnetic  induces system from  field,  the In  there rest  be  of  to the  physical  field,  and  w h e r e we  consider  on  the  nature  The a  moment  nuclear  is a  tensor).  N  N  of  which  high  high  finds  of N  weakly  the  has  been  placed  frequency  oscillating  frequency magnetic states  that  of  energy  in  the  field  spin  i s absorbed  field.  the  spin  spin  latter  the  spin  within  the  These  lattice  these  spins a  between  the  relationship  procedure  function  g-factor,  p -  system  energy  between  i n the  frequency of  the  one  between  the  included The  the  ^  a  interactions  interactions  magnetic  and  as  and The  world.  between  fields.  .  interactions  field,  not  where  of N  Q  magnetic  magnetic  world  H  between  oscillating  the  consist  system  a-n^t  <3H,  addition  of  and  system"  experimentally  will  nuclear magnetic  (In general  field,  transitions and  I and  but  spins.  assume  magnetic  system  with arbitrary  where  N  .  of a  number  ^ «^,«x  ju.=  N  system  spin  system spin  t o be  between  the  oscillating  system,  and  interactions system the  system  and  or  spin  system  and  that  system  the  part the  of two  absorption magnetic  and  the will  the constant  oscillating the  "lattice",  the  physical  magnetic of  field  energy will  depend  interactions.  i n the  depends  following on  chapter w i l l  the above  be  interactions  to and  define which  describes by  the  the  spin  magnetic  relationship  system  field.  and  the  between  the  frequency of  absorption the  of  ene  oscillating  7  Chapter III D e f i n i t i o n of the Line Shape Function and Calculation of i t s F i r s t and. Second Moments  3.1  General Discussion The contents of this "section w i l l be devoted to f i n d i n g  the f i r s t and second moments of a function which we s h a l l c a l l the " l i n e shape function, f(-v)".  Many functions may be used to  describe the relationship between the frequency of the o s c i l l ating magnetic f i e l d and the power absorbed by the spin system. In section 3.2 we s h a l l define the l i n e shape function such that f(-f) i s proportional to the imaginary part of the high frequency s u s c e p t i b i l i t y , "X."(-») . Furthermore, we s h a l l define  5c© ©  To calculate the moments of f (ri) we s h a l l use a method, f i r s t developed by Van Vleck (1948), which demands a knowledge of the Hamiltonian, Hi , of the spin system. cussion i n Chapter II i t i s apparent that  From the d i s -  must i n general  be written as  where the four terms represent energies:  the following interactions  ft* = ipin-magnetic f i e l d crystalline field  i n t e r a c t i o n plus spin-  interaction when t h i s l a t t e r  i s present. f£  CO  - spin-spin interaction (which may include exchange interactions)  S(*  « spin- l a t t i c e i n t e r a c t i o n  VL**  " spin-high frequency magnetic f i e l d i n t e r a c t i o n .  We have, furthermore, that  fc *,,  Z  1  . V*  In the following, we s h a l l disregard Furthermore, we s h a l l assume that  ti" » V^**  (o)  CO  % »  +  5  completely. i  Ci">  V.  and  so that, to a f i r s t approximation,!^ can  be written as  %^  K  We s h a l l consider as the perturbation  (3.1.1)  < 0  as the unperturbed energy and ti* energy.  Van Vleck (1948) has pointed  out, however, that use of  (3<> 1.1) when finding the moments does not allow us e a s i l y bo compare our results to experimental findings. fact "truncate"  the Hamiltonian.  We must, i n  The reasons behind t h i s are  rather subtle and f o r this reason, we s h a l l devote the next section to this point. Usui and Kambe (for  (1952)  have pointed  example, when a c r y s t a l l i n e f i e l d  out that i n general i s present) a further  truncation i s necessary, namely, truncation of the i n t e r a c t i o n energy section.  ^  c i )  . This point w i l l also be discussed  i n the next  3.2  Truncation and We  to  be  of the Hamiltonian,  of the i n t e r a c t i o n energy, are considering  given  be w r i t t e n  could  be  of  regarded  levels  highly Let  $  us  0 >  , that  of the system  label  could  being  As  induced  proportional  by  .  We  known,  ^  and  3 o (  E^,  i s well  with  of a  the  a  such energy In  elements occur  recall  9 o <  that  the various  between  the states square  For  we  consider  states  (0)  \^y ) >s  that  of the  S*'  WJ  spin  transition  I^^O  and  of the matrix  i s  element '  (3.2.1)  transition  takes  place  the spin  one u s u a l l y  finds  that  system  absorbs  I E ^ - E ^ I .  are non-zero  but rather  f o r certain values  example,  i f  (3.2.1)  that  o f oi were  not a l lsuch  non-zero and  non-zero  ^  matrix  i n addition  1F_E  matrix elements  .  only i f  L-^Ul and  ft  i s the degeneracy  the p r o b a b i l i t y of a  to the absolute  practice  only  number  of  criK-i,,*-') When  , the  large  and e i g e n v a l u e s  ,  t r a n s i t i o n s between  system.  interactions  levels.  a n d w h e r e  induces  compared  consist  system  of the system  i s , i f a l lother  the eigenstates  the eigenvalue  of the spin  I f the Hamiltonian  a s n e g l i g i b l e when  degenerate  °y of  as  ,  .  the Hamiltonian  (3.1.1).  by  could  energy  ft'"  ft^V  1(3.2.2) Inconstant  independent  of *  IO  then  the spin  system  S>  frequency,  would  absorb  , and f ( ^ ) would  0  energy have  at a  single  the simple  form  -- , £ U - s > >  (3.2.3)  0  S(>*-->>)  where  (3.2.3)  (Condition  o  t o the case  when  When t h e H a m i l t o n i a n  of the spin  system  , each  o f t h e N s p i n s h a s t h e same  c,  a  delta  function.  no  crystalline  i s present.)  >  the  Dirac  refers  field  y  i s the familiar  0  case  — a.  L  h.*\ h.  .  energy  (3.2.2)  The c o n d i t i o n  i s written  as  values, say,  corresponds  to  when  =  constant  and,  furthermore,  only  between  f o rk =  when  adjacent  1,2,  transitions levels  ,R-1  can occur  >  (3.2.2)  of the individual  spins . If, _  Q  Q  V  on t h e o t h e r  hand,  ' (3.2.4)  constant  A*i fc. \ (as then  one u s u a l l y the spin  frequencies,  finds  system say,  -o  when a c r y s t a l l i n e will 0  absorb  -»>, -0 }  energy  ~>>  p  i }  •  field  i s present)  at several  In this  case  different  f(-o)  would  p have  the form When  values are  S(v) = T I T .21 i s a d d e d t o Vtp> t h e h i g h l y  split.  g i v e n by f i r s t  T h e new e n e r g i e s order  degenerate  energy  a n d new e i g e n s t a t e s  perturbation theory:  II  C.,W"U,0  E. . ^  +  I X % M r f  (3.2.5)  5a  l O - W V l l l * , , • w h e r e we values  use  *  v  „ to label  +  . Wi)  (3.2.6)  t h e new e i g e n f u n c t i o n s  and  eigen-  and where  (3.2.7)  UM^'U.gX^l^UO  (3.2.8)  U?\v"UO  The  zero  order  states  are  1o(  where  A  i s a unitary matrix and has been chosen so that:  1)  2) This  a  can always  i s never be  done.  infinite.  Now, the probability of ^ between  and  C4>  inducing a t r a n s i t i o n  Is proportional to the absolute square  of the matrix element (3.2.9)  Let us suppose f o r s i m p l i c i t y that ( 3 . 2 . 2 ) holds.  It i s  seen immediately that because of the l a s t two terms i n ( 3 . 2 . 6 ) , (3«2.9) w i l l be d i f f e r e n t from zero f o r many values of and  £  %<  . As a r e s u l t , energy w i l l be absorbed at frequencies  many times larger or smaller than \> . The function f(s>) w i l l 0  now consist of several broad overlapping lines and w i l l have several maxima, the number of which depends on  . As can  e a s i l y be seen, these maxima w i l l occur at multiples of -»  e  when ( 3 . 2 . 2 ) holds.  The following diagrams  will illustrate  the point:  DiflCRnm  x  DlOCftflm  A  o %s"\ *  TI  13  It should be noticed that this phenomenon i s not the result of the simple condition (3.2.2) but that i t i s the result of the fact that to f i r s t order the eigenstates of $ are linear combinations of a l l of the eigenstates of V^* . ;  In general, then, when ft" i s added to ii^ two things happen to f ( ^ ) . F i r s t l y , the i n f i n i t e l y narrow lines represented by the delta functions broaden and secondly, secondary lines occur.  (By secondary lines we mean those lines which  occur because of the presence of the l a s t two terms i n ( 3 « 2 . 6 ) . We s h a l l refer to those lines which result from the broadening of the delta functions as primary l i n e s .  See diagram II.)  Experimentally, one observes one of the primary lines of f ( v ) . We seek to characterize  f(i?) by Its moments, so  that i f we are to compare the moments of the experimental curve with the moments of f ( v ) we must then find some way of eliminating the secondary lines from our function f ( v ) . If a •  and  \> .  could be put equal to zero  the d i f f i c u l t i e s would disappear.  In this case, when con-  d i t i o n (3.2.2) holds, transitions w i l l occur only between those states whose o r i g i n a l separation was o r i g i n a l separation we mean when  W  , where by  i s taken to be zero.  The function f ( v ) w i l l then consist of the broadened  portion  centred about  v  v  , that i s , the primary l i n e about  Q  0  In other words, no subsidiary secondary lines w i l l occur. In general,  a  „  ;  .  M  and  k . -;  c M  a  n  b  e  P  u t  to zero i n only one way, namely, by considering the  equal  Hamiltonian  Vt"  where  o f the system i s that  part  Then,  a s i s w e l l known,  which  are simultaneous  These  functions will  the  there  »j;.  - k  . .„  tf"  which  exists  eigenfunctions  be  a  0,  if"  Thus iltonian  when  of  sC"  and  Iv.O  are zero  d i s c u s s i n g f ( v ) we  of the spin  system  ^ =  spin  . see  i f it"  0>  Because  of  immediately i s replaced  shall  consider  t h e Ham-  t o be  (3.2.10)  0>  refer  to  V  as the "truncated  Hamiltonian"  o f the  system. As  a  result  secondary  lines  will  have  then  of the t r u n c a t i o n of the Hamiltonian, disappear  the  from  f(s>).  the  The f u n c t i o n f(-o)  form  «*1  if  .  w  tf +  shall  the Hamiltonian  condition  (3.2.2)  of the system holds,  <0  set of functions  e i g e n f u n c t i o n s we  b ...„  K + t«  of  commutes w i t h  the f u n c t i o n s  and  instead  vF'  by  We  of  o r t h o g o n a l i t y of these  that  S*° + \F>  t o be  i s given  by  (3.2.10) and  15  or,  if  the form  the H a m i l t o n i a n of the system  condition We the  have  primary  findings and f(v)  (3.2.4) h o l d s  lines.  t h e moments  that  t h e moments  of f ( ^ ) ,  we  must  any o f the primary  lines  For  wish  example, s>  results  then  A  -o. v with  i f we we  v the second  (3.2.10) and  r - 3 .  e x p e r i m e n t a l l y one o b s e r v e s  I f , t h e n , we  by comparing  f ( ) about ^» *  mentioned  and where  i s g i v e n by  must  hope  eliminate  i n which to find  we  from  the second  interested.  moment  the primary our  curve  our function  are not  a r e t o compare  moment  experimental  of the experimental  eliminate  i f we  to interpret  one o f  of  lines  about  theoretical  o f f ( v ) about  found  experimentally. In must  order  to eliminate  introduce another  the unnecessary  artifice.  Instead  primary  lines  of considering  we -Je  111  as inducing transitions we s h a l l suppose that  V  induces  l >  t r a n s i t i o n s , where by the circumflex we mean that we are considering only that part of  which i s relevant to  the p a r t i c u l a r problem under consideration.  For example, i f v  we wish to consider only the broadened portion about  ,  A  then we only wish to consider transitions between the states U.O  and  Ip.k) ' of W  1E^-E \=-JU^  wherein  p  .  In t h i s  A  case, then, we define the operator  V  u >  such that i t has  matrix elements  For any other problem i t i s necessary to redefine  A  H'*'  .  In general, however, the circumflex indicates that the matrix elements  can be non-zero only f o r p a r t i c u l a r values of <* and ^ The following diagram w i l l i l l u s t r a t e the point:  AAA where  and  VP*  induces the t r a n s i t i o n s ,  17  but:  A  where and  M -  where  + V '  and  7  induces  t h e c i r c u m f l e x means, i n t h i s  the  case,  transitions  that  the matrix  element  can  be n o n - z e r o  only  1 E„-€.l = £ v  i f  i  It holds,  should that  necessary consists In with ing 1)  2)  be n o t i c e d  that  i n the simple  i s , no c r y s t a l l i n e  to introduce of only  then,  t h e moments f o u n d  present,  the c i r c u m f l e x over  one p r i m a r y  general,  field  before  {3*2.2)  i t i s not since f ( v )  comparing  t h e moments  e x p e r i m e n t a l l y we  have  t o do  of f ( v ) the f o l l o w -  <  the Hamiltonian  secondary  lines  truncate  S*  from  i n which  f(^>)  where  line.  two t h i n g s : truncate  case  (i>  i n order  to eliminate the  from f ( v ) , i n order we  to e l i m i n a t e any primary  are not  interested.  lines  I *  We its  c a n now  first  s>*  and  second  the frequency  are  interested  considering of  proceed  to defining  moments. about  the primary  the following  the primary  That  line  discussion,  to calculating  For simplicity,  which  i s centered.  f ( v ) and  at we  line  v  Thus,  define  Let of the  ^  so  u s now  shall  define  label  energy  the high  frequency  and  the eigenvalues  of  a function  that  the  purpose  (3.2.11)  of i t s F i r s t  the eigenvalues  eigenstates  probability from  we a r e  that  )  consider  corresponding we  we  and  Moments  .We  If  call  i n which  f o r the  1*  D e f i n i t i o n o f f(s>) a n d C a l c u l a t i o n Second  shall  i s , i n what f o l l o w s ,  1°  3.3  we  i n one  range  by  frequency v  the spin  that  by  E„  g(-o)^-v)  system  oscillating  -\J+A->>  to  %  and  •  g(-*) s u c h  second  eigenfunctions  will  magnetic  i s the absorb field  In  then:  (3.3.D  where PCy")  =  p r o b a b i l i t y that from  the state  the spin In)  system  to the state  undergoes U')  i n one  a  transition second  19  2.  -sum  over a l l of the eigenstates of  it  n  — Z.  = sum  over a l l of those eigenstates of ^  «=sum over a l l of those eigenstates of  wherein  ft  wherein  However,  XZ «  * PU>")  * IT n'  = IZ*PU>^ « «»'  n o  II* !?^ )-^^ 6  so that  Now,  c a  „^  o I  -  (3.3.2)  where  - p r o b a b i l i t y that the system i s i n i t i a l l y i n the  a  state  In}  , that i s , i n the state  high frequency f i e l d =  In)  before  the  i s applied.  p r o b a b i l i t y that i n one second the system undergoes a t r a n s i t i o n to the state  UO  i s i n i t i a l l y i n the state  \<v)  From s t a t i s t i c a l  given that the system  mechanics we have  *  *  r  where  I / k  ^Boltamann's constant  T"  ^ i n i t i a l temperature of the spin system = l a t t i c e temperature Let us now  consider  y^o'  •  From standard  quantum  mechanical theory we have the p r o b a b i l i t y that the energy  perturbing  induces a t r a n s i t i o n between the states  IrO  )r,0  and is  V.  of  the state  i n the time  l«>  t - t  Q  where  at t  Q  the  system  i s  to  where  Now,  i n our problem,  -  -(oscillating  field)•(magnetic  moment) —  -H(t)-M -H(t)M of  *jL  Then, Now,  M  where  M  i s t h e component'  i n the d i r e c t i o n  ofH ( t ) .  -H(t)M  c  we  can  write  If  H(t) i s resolved  where  energy  crossing  unit about  into  area  frequency  range  classical  electrodynamics,  i t sFourier  i n the d i r e c t i o n  the frequency  v  components, the of H(t) per  will  unit  be, a c c o r d i n g t  Thus,  f„ , 0  - 3TTC ' %  (Co I mU')|  Finally,  so  that  In  order  which is  then,  that  our  depends  a  E^,  on  constant,  results  be  independent  experimental  say  U,  Z  g U w W i T V ' u  of  conditions,  for a l l v  .  X ^ ( * " ^ - a ' ^  E  we  v  , a  quantity  assume  that  E  Then:  T  lc.ULOr  )  (3.3.3) i  Now, system range  from \)  E(^ *  i f E(v)/w the  i s the  oscillating  to  power  magnetic  absorbed field  by  the  i n the  • (3.3» as  is well  known  (see  f o r example  Andrew  H(v)= 4TT^> H^-X'TA) where  frequency  then:  4v  But,  spin  i s the  susceptibility,  and  (nss  ))  (3.3.5) imaginary where  SH,  part i s the  of  the  high  amplitude  frequency of  the  v  oscillating  magnetic  Combining  field.  equations  (3-3.3)» (3.3.^)  and  (3.3.5)  we  have  The  area  under  „.«^  t h e c u r v e xl^A  I T  (,-fe-.-^  i s then  k.uuor t  where  ^ We  means s u m m a t i o n shall  w h i c h we  shall  W>»-J^ Our  line  shape  T"(v)  now  define  call  over  a l lstates  a dimensionless  the line  shape  wherein  quantity  ir(: -.^)I^W  function, and  then,  n  £(v)  function:  fe  =  E '>E^  i s proportional  (3.3.6)  t o the f u n c t i o n  furthermore,  O The  first  can  be  and  t h e second  can  be  moment  of f ( v ) which  i s defined  as  written  written  (3-3.7) moment  o f f ( v )which  i s defined  as  33  We  shall  summations a  knowledge  would  task  (19*+3)» could  that  (3.3.8)  (19*+8),  Van V l e c k  and  others  i n doing  this  i s apparent:  obtained  similarity  evaluated  from  traces  using  the functions  require  I s , we =E ln")  SEln)  that  equations  i s , as  c a n be  , that  this  (3.3.7)  over  n  (1932),  by W a l l e r  that  the traces  w  the  would  problem  form,  a  Evaluating stand  of  noticed  by r e w r i t i n g  i s i n v a r i a n t under  c a n be  as they  the eigenvalue  I t has been  be a v o i d e d  advantage  hence  to solve  i n trace  operator  and  two m o m e n t s .  of a l l of the eigenvalues  continuing.  (3*3.8) The  (3.3.7)  i n  be f o r c e d  before Broer  c a l c u l a t e these  arduous and  operators.  The t r a c e  of an  transformation as a basis in)  by a  any  and functions  similarity  transformation. The  (3.3.8) Kambe  i n trace  (1952).  formulae by  procedure  for  these  order  form  shall  and  be  noticed <^y  (3.3.7)  and  by U s u i  and  use t o rewrite  i s identical  It will ^v)  we  to that later  given  that  our  are identical  general  to those  given  authors.  Consider, In  which  now,  to avoid  the denominator  the rather  (3.3.7)  and  summation  2_  of  awkward  (3.3.8). , i t i s  tv  convenient are  defined  to introduce  the operators  ir\  +  and  «\_  which,  as  (3.3.9)  34  Cn I rt\\</)  ii  (3.3.10) L  CnliMnO = CnU \ ') "+CnU_\«0  ThUS  +  We  c a n now  IX  n  write  KT  II.'*"%U  WX«'U,1*0  Cn\*"^\nO=  But, so  o  = 2 /  V  t  CnU m In')  £ ,  that  X,"  ^c«\«.w \ o  KT  XX  Similarily,  f  - T  Then,  »  * j  XX («" - /^OlcoUo'^T^ (/^Lm  where,  +  fe  as i s customary,  C  > tnj]  ~  *V.w\ +  (3.3.11) .  35  Consider, immediately duce  the  be  now,  the  noticed  operators  m  numerator that  here  and  +  of  (3»3»7).  i t i s not  It  will  necessary  to  intro-  since  >"» n' N O W , XX  "  * " ^ C E ^ E ^ k n l m U ' ^ ^ X l * {.E 'C«|AUXnUU') -E^nUU'XnlmU^ FET  0  r,'  w h e r e we  and  have  made u s e  the r e l a t i o n s h i p  n  Since  n  £„ . = Col n  fcloO  X  Z  .  *  "  &  1 n)  C E ^ E ^ I C - U U ' ) ^  -XL/^CEw-E^lcnUUor  have  IZCEW-EX*"^)lcnUio9r  Using in  £  i t s consequences  E ' (o \ m W ) = Cn I m W  we  of  the  form  (3.3.11)  and  - aTr«*(/  (3.3.12)  we  can  fe  rewrite  (3.3.12)  (3.3.7)  5  *  < V >  ' It  and  T  will  Kambe  -s  f  be  »  R  . -A  noticed  (1952)  for  that  (^y  (3.3.13)  -  the expression  given  by  Usui  involves  Tree  However,  T ? « c « ( / ^  L  ^  .  L  as  can e a s i l y  be  to  the functions Consider,  be  necessary  As  before,  3  <\  we  *  ^  seen  now,  =  a  Tro.e  (/^  by  calculating  the numerator  to introduce  of  ft*  £  P  the traces with  (3.3.8).  the operators  ro  +  It will and  write  n  -t-CnlfclrTXnlwln")}  CnU_U'Xn U j i O  respect  -  w_<  again :  27  = Z (nl e  W - a « i ft <V  I n X n | w>  lr>  = Tract  m  + V  m_ ^ 1* I  T  +  -  -»•  - a i f t . i i m 5 t  n\_fl\ v  tt  +  Similarity,  ZZ  =  T~  f  >r ce \ * Q  In-  X  —  *  +  ft  ^  \ ft. j n , n\_ &  Finally, in  CnUjn'Xn'U L>  CE^-O"  ^  t h e n , we  A  .  —  - ^ ( T i ^  ft  .—  (t\J«  can write  •+- m  ft,_,-A +  ^ f>? r0_  the numerator  \  JJ .  of  (3.3.8)  the form:  CCm.^l.CSAll) Using  ^  l  =  (3.3.11)  and  (3.3.1 *) 1  (3.3.1M  we  can write  ^,(.ic[;., U^x) {  as  follows:  <3S  Our equations  (3.3.13)  and  those given by Usui and Kambe  £  S  Dl"(v)Jv  (3.3.15)  (1952)  are i d e n t i c a l with  for  and  respectively.  f x " ( v ) dv  It should be noted that  (3.3.13)  and  (3.3.15)  have been  derived without resorting to any assumptions as to the temperature T.  These equations are quite general and are v a l i d  at a l l temperatures. case of paramagnetic magnetic  resonance.  Furthermore, they can be applied to the resonance or to the case of nuclear  an  Chapter  Derivation  of  and  Moments D i r e c t l y  Second  Another line It  shape  will  be  b  form  found  the  first  Equations  and  g i v e n by  section  equations  g i v e n by  the  (1950)  that  Pryce  and  and  First  (3*3.15)  and  moments  of  Stevens  (1950).  introducing  (3«3»13)  these  a  one  (3»3»15)  can  be  authors.  i t should  be  first  second  and  second  by  f o r the  (3.3.13)  From E q u a t i o n s  been  i n this  proceeding  have  of  has  assumption  i n the  E(v)  Stevens'  derivation  shown  Before Stevens  and  function  simplifying written  Pryce  IV  pointed  out  moments  that of  Pryce  the  and  function  £.V  E(VW  o  where E ( V ) A ^  i s the  the  magnetic  oscillating  power a b s o r b e d field  by  the  spin  system  from  i n the  frequency  range  S) t o V+A\> .  i n the  course  their  discussion  B u t , b y (3.3.5)  Pr£ce  and  have  ca.  that which  Stevens  the  primary  assumed where  line  under  i s the  consideration  of  frequency  about  i s centered.  In  30  other and  words,  second  which  Pryce  and S t e v e n s  moments  should  explicitly instead,  by P r y c e  be n o t e d  and  (3.3.15)  of  ft  shall  define  operator,  operator of  ^  t o >  us  0 .  Stevens  Pryce  to derive  directly  and  of  Stevens m  and  shall  now  from  the  our  i n terms  of  do n o t  ; they  which  introduce  the truncations  with  an operator such  consider  have,  perform  projection  f o r us so t h a t and  1£  these  m  (3.3*13)  instead  v  , which  we  shall  call  a  that  we h a v e  the matrix  respect -1>2,  P  V  o f (4.1)  consider  (  be a b l e  d\  a consequence  Let  We  perform  C v K l p . O - \»  As  that  c a n be w r i t t e n  and  projection  and  then,  projection operators  f o r them. which  first  (3-3.15).  and  introduced  operators  should,  the truncation  operations  We  We  (3.3.13)  equations It  given  c a l c u l a t e the  of the f u n c t i o n  i s our f(v>).  expressions  actually  P.p  vJ  XA  representation  where  (4.2)  v  * A  t o the t o t a l i t y , _n-  * P £ ,  of a  general  o f e i g e n s t a t e s \<s,k) ~n_  <=  t h e number  of  I  31  «0 different is  the  eigenvalues  degeneracy  general  matrix  Now,  the  of  of  V  the  element  matrix  with  respect  written  in  -*£  of  0  elements  of  the  respect  of  w  The  representation  OP^  is  then  are  using  (4.1)  representation  t o t a l i t y of  .)  g^  of  eigenstates  any of  0  operator #  can  be  form  to  the  (*+.3)» we  n  That  Let recall  ^  u  -  we  introduced  tf'+vl"  l<*k")  easily  ,  from  and  write section  the  operator  commutes of  and %  in  is  J  if  with  the  i f and  only  i f  P^f.  M  which V°  with  Vi  writing  #  form  given  that  - o  21  now  which  representations  states  find  is  us  (4.3)  3.2  matrix  CXl.f vf  We  of  ^  matrix  the  this  E  where  for  section  part  Taking  the  in  1 , 2 , . . . . ^  _  In that  to  the  2L X  0=  hy  then,  •  eigenvalue  C ^ l p ^ l p ^ V C/.ilol^,^) Clearly,  ; k  „  (^A)  i n terms 3.2  ^  that  of  projection  the  circumflex  operators. was  introduced  over  V  and hence over M i n order to eliminate from f(-a)  any of the primary lines i n which we are not interested. we are interested only i n the primary l i n e at  Here  so that we  s h a l l define the operator M such that i t has matrix elements  US  m 1^ p  Consider the operator  where  ^  means sum over a l l values of ^  The matrix elements of  MP  0  Thus we can write  From (*+.5) we can immediately write  «v  = ^Z  and  are then:  p.  wherein  33  m »  22  P.^?  2-2-  where  (^.7)  A  means  summation  over  a l l values  of  ^  o f  and  cJ<^  wherein single for  - E L -.=  E  .(The symbol  summation  since  (4.6)  and  large.  speaking,  (4.7)  We  have  (4.7)  i s an exact  the assumption  certainly  that  expression,  S*  i s not too  0J  that  f  KV + tv>  m =.  equations  recall  that  we  (3.3.9)  and  t h e n we  require  V  then  a  +-^r \  H o w e v e r , we  and  denotes  use the notation -E^"  (4.5)  although  involve  2 2 (  and  We  actually  convenience.) Strictly  and  E„=E.-VRV* .  X X  U  E,<Ep  < o >  have  (3.3.10)  defined respectively.  and  m_  by  To w r i t e  that i f  U,0 - E ^ U,0  U,0  -  E  implies  V < ;  W,0  that  E,.<E  p t  f o r a l l i and  k.  (4.6)  3f  If  i s not too large  then  this  i n fact  will  be  true. In shall and  Appendix  show  (1952)  Kambe  be  rather  i s a  (h.h)  to  transformed than  Let  shall  i n Appendix  Using can  A we  B  into  case  of  ,  and  We  Usui,  (M-.5). and  (3.3.15)  It  containing m  form.  of Sshiguro,  (3.3.13)  equations  equations  i n another  (6)  equation  special  * <T\_  ft^m  that  (*+.7),  (^.5)  rewrite  and  m  +  us c o n s i d e r t h e denominator  of  (3-3.13)  and  (3-3*15)•  First,  Now,  L«  L II  P.TOP.  ,H  ^^t'l  P r ^ e r o P - P„rr\P,0(\P \ o  °> < £  Thus, *  «  *  $  f c t  4  Lm_,m i« +  UU  "  by  Of»  from  (if.6)  using  (^.2)  and  p •> "  'P^mP,-, ^  K  ^ P « . P  p  ]  Of.7)  where  we  Using  (^.2)  C  Trace  have  again  again  Consider,  (W.W),  and  ABC=Trace  now,  a n d 0+.5)  we  (W.2).  used  the f a c t  CAB  that  f o r any  c y c l i c a l l y , we  the  numerator  of  matrices  A,B,  have:  (3.3.13).  (W.2),  Using  can write  and  A  Thus,  Trace  a ^  i t i l ^ ^ ]  -  = X T - T * c * [ u . ' ^ ( P , m P V P „ P - P,wf^mP -« ) t  p  r  -ftm  (  o4  •  and  \  N  \  ?  ( l +  *  9 )  Consider,  now, t h e n u m e r a t o r  ( +.+), (*+.6), 1  l  and  ( 4 . 7 ) we  (3.3.15).  of  Using  (4.2),  can write  p"  and  E«,m "1 +  so  that  Thus,  -  = X X1V ^ m P , L E «>-,«3,L« (h J  Trace  I I  E C ^  +  - P ^ P ^ P , ^  l l  T,E R ^ T l )  =  Trac-C P ^ P ^ V P^ ro -  i> vp,mP^-WP m  X4.10)  J  37  We into  have  forms  now  transformed  containing  (3.3.13)  equations  and M r a t h e r  ^  i<  than  m  M  .  +  I t should  introduced By able  i n writing  introducing  to produce  (3)  from  YLP,  We  now  so  that  e  t  This  assume  we  -  T  JL  Usui  (1),  we  (4.10)  be  and  respectively.  (4.11)  we  to a first  have  o  ^  approximation  ^ A  *  ^  ^  a  fcT  —  (4.12)  their  as  shall  (2),  ^  a n d Kambe  be w r i t t e n  point,  been  (4.10).  E ^ p ^  ~1 = o  used  general equations  a t the beginning  When a s s u m p t i o n can  have  and  equations and  > — j  can write  i s e s s e n t i a l l y the assumption  when d e r i v i n g by  -  that  at this  ( 4 . 8 ) , (4.9),  V°%  ? -w'V  (4.8), (4.9),  and S t e v e n s '  -ttP« = ^ \ - » -  Since  no a s s u m p t i o n s  one a s s u m p t i o n  Pryce  V -  that  equations  our equations  Using  p  be noted  m  >  A  and  (3.3*15)  and  (4.12)  follows:  i s used  by Pryce  (1),  (2),  of section  and and  4  Stevens (3)»  of their  and paper.  ( 4 . 8 ) , (4.9) a n d ( 4 . 1 0 )  )ZXr*"  Trace(^U.^l)a(l-*~^  Trace (  A  mtfc.rni) Mi--»-*  ^  T r a c e (*  *  T  It  ,L  r r  )Zl_A , l  K  P^m  r  Troc-  .  ( P.y fcmPjti \  (4.11) and  = EJ-P  rf  Finally, the  then,  following  ( 4 . 2 ) we  +a^P  r t  . 14)  ^  + Pd*P|V P ^ P ^  Using  (^.13)  J (4.15)  can w r i t e , f o r example:  ^\+P,^'^'^  when a p p r o x i m a t i o n relationships  (4.12)  holding:  i s valid  we  have  31  *<*> - If*  If ^  we  and  now d e f i n e  (^^y  -  the second  ^  -  t  n  —  e  first  moment  moment  o f f(s>)  o f f(s>)  about  *—  .  si*  C*.16)  about then  and  so  that  -11*  _ Z Z *~ • " ' T r a c * P ^ m  «•»< ^  r  ( l f - l 8 )  ifo  =r ' r  It  should  give  t h e second  that  i s ,  since If  this  k^^y  be n o t i c e d moment  '  p  that  i s n o t t h e second  *= s>  , then  by  <^o^  <*.i9>  L  our  o f f(->>) a b o u t  quantity i s given  and  does not t h e mean v a l u e central  moment  - (^^V)  t h e second  o f s> , o f f,(v)  •  central  moment  o f f(->>)  are equal. Equations  and  *  Stevens' / ^ E ( ^ v  (W.18)  equations about  and  (W.19)  are identical  f o r the f i r s t the frequency  and second v  to the Pryce moments o f  fl  Chapter Sufficient  V  Conditions f o r Van Vleck's (1948)  Expression for the Second Moment to be Valid  Another derivation of an expression f o r the second moment of a l i n e shape function has been given by Van Vleck (19^8).  Van Vleck has not defined e x p l i c i t l y the l i n e shape  function he i s considering but we s h a l l show i n this  chapter  that by applying suitable assumptions to our expression which i s the second moment of f(%>)= ~C(^)f  (3•3•15)«  ^°  »  o  the generalization of Van Vleck's equation (3) can be found. Van Vleck has considered the s p e c i a l case which we have called  condition  f i e l d i s present. m  to be  (3.2.2)  that i s , the case when no c r y s t a l l i n e  He has considered the operator portion of , and he has written the second moment of  his l i n e shape function as  (5.D  where we use  K.  to denote the truncated Hamiltonian.  As  we have discussed e a r l i e r , i t i s necessary i n general to use M rather than M when discussing the problem.  In the general  case,  then,  t h e Van V l e c k  expression  becomes  A It  will  be shown  equal  when  field  i s parallel Van  before  most the  i s no c r y s t a l l i n e  Vleck s  to the  used  by these  latter  important  point  temperature,  e x p r e s s i o n , Van V l e c k  gave  to the f a c t o r  rise  b e no d o u b t ,  the  Van V l e c k  results. general (that  equations  identical  where  with  Usui,  field  a n d Kambe  a crystalline  i f  given (195D field  case  (3.3.8)  have  \  , then  applied  i s present  the  and which There  seems  temperatures  yields applied  levels  by V a n V l e c k . have  account  high  considered  and e n e r g y  The  not contain  into  (5.2),  (1952)  «  The  In deriving h i s  for sufficiently  i s , equation  that  does  (3.3.15).  n  a n d Kambe  those  i n  ±  to the special  and f o u n d  a n d Kambe.  (5.2)  has not taken  5  that  that  Usui  1  appeared  t o the problem.  i s that  introduced *  i s , no c r y s t a l l i n e  Ishiguro, case  method,  In f a c t ,  equidistant) are  we  however,  the magnetic  moment  (3•3•15) d o e s .  T, w h e r e a s  which  are  and by us i n C h a p t e r I I I ,  approach  to notice  factors  and when  and U s u i  authors,  refined  Boltzmann  field  f o r the second  and Stevens,  i n v o l v e a more  S  and  2-axis.  derivation  1  S,  that  of Pryce  general  to  C  those  methods merely  there  In Appendix  correct their  by Van  Vleck  of each  their  spin  results  Secondly, (5.2)  and have  to the stated  that  the results  Pryce  so obtained  and S t e v e n s '  and a r e r e s t r i c t e d  have  checked From  suitable  possible  to arrive  give  results  such  to sufficiently  found  when t h e  are applied high  (5.2),  section,  (3.3.15) which  There  point then,  appear  that  i t should  i s , i t should  under  moment.  of this  to  that  conditions  i t would  (5.2)  be  yields  h a s , however,  be d e v o t e d  be  possible valid  been  i n the l i t e r a t u r e . will  by  to  no  The finding  t h e p r e s e n t , we  shall  find  under  which  conditions  expression  which  i s equivalent  (3.3.8), be  this  temperatures.  (5.3)  I I tailor  the  to  conditions. For  the  at  discussion of this  (3)  and  approximations  f o r t h e second  remainder  been  independently.  sufficient  explicit  also  the foregoing discussion  applying  to  (1)  equations  case  this  have  which  following a valid  to  (5.2),  i s equivalent  to  two a s s u m p t i o n s  approximation  of  i s a valid  (3*3.15).  approximation of We  are sufficient  (3.3.8):  shall  see  i f (5«3)  that Is to  We  ^  *i  *.  1)  $  the  exponentials i n  W  >  » W  so that  (3.3.8)  to a first  approximation  c a n be r e p l a c e d b y t h e i r  values  when i  2) for  The temperature a l lvalues Let  Chapter  o f <*  u s now  shall  where  and  perform  I I I that  eigenfunctions we  i s high  - E  these  calculations.  We  recall  the f u n c t i o n s  of  E,. ^ E^  IE  y  U _ 0  ft  t e >  and  (-/,; I  \  .  are the  E. N  .  E„  I- •X using  the f i r s t  i=.  assumption  -  Thus,  L luuifcOP *  above.  (3.3.8),  then,  ;  «;|  X  from  simultaneous  In equation  written:  ^=1  \« k~r  p  I n ) = U , 0  put  so that  enough  (3.3.8)  c a n be  15  using  t h e d*s-«nVt.o  of  n  ^  6  »  using  the  Thus,  we  (see Chapter I I I ) .  SL  second have  M  >  assumption  that  when  the  above. above  two  assumptions  are  applied  t o (3.3.8) we g e t  n i l( _n_ n_  1  V>  which  -  is identical We  have  equation should most and  *Jw  shown  (3*3.8),  immediately  unreasonable. the  lowest  except that  f o r the  by  applying  (5.3)  equation be  noticed  The  values of  notation  energy E  .  two  can  that  be  the  equation  assumptions found. second  difference i s at  with  least  to  (5*3).  our  However, i t assumption  between N£V*  the so  is  highest that  the  second  assumption  Since  i t has  valid  results  would  be  by  a  of  this  of  for  Van  to  identical to  It w i l l  be  with find  for  (W.17).  to  1)  given  replace  conditions  compare  equations  convenient  to  rewrite  (5.2)  (k.k) Trace  rather and  0+.5)  & into  Cro?*= T « - o « I Z  and  M,  that  (5-2). ^roP^,™^.  2)  can  done  replaced  the  remainder much  high  validity  for  above.  the  assumed).  is a  (W.17)  from This  2),  valid  and  (3.3.15) is  Therefore,  assumption  i s , we  we  approxi-  (5.2),  i t  containing  shall  substitute  using  in  can  Thus,  •=  i t  temperatures  C+.12).  form  be  in  obtained  (5*2)  in a  temperature,  generally  namely,  to  M  that  been  order  and  be  (W.l^) was  that  assumption  mation  will  out  has  the  what  room  sufficiently  assumption,  conditions  equal  l o  experimentally  assumption  turn  one  under  yields  This  (as  recalled  find  be  that  are  method  simply  In  show  2»  f>>  (5.2)  that  condition.  Vleck  will  that  approximately  temperature  introducing  order  T  desirable  room  the  shown  chapter.  It by  been  much w e a k e r  below  implies  (W.5)  will  47  Thus,  -—  -  —  L  -  Using equations (4.11) and (4.2) we have, f i n a l l y :  4?\^  =  S  X r  -  ( P ^ % m f  IX  T^C<  w  m  -  P,m^  "4- • X X - r ^ c , P mp^ w  We see that equation (5.*+) i s i d e n t i c a l to (4.17) hut with the Boltzmann-factors, last equation.  A  , replaced by unity i n this  As before, i t seems that the procedure can  only be j u s t i f i e d  ifT » ^  . , We s h a l l see that this  is not the case. The procedure now, w i l l be to consider (4.17) f o r the case when we can write:  m=  I  (5.5)  m-  II  Ml  •  (5.6)  4?  where  and,  tC^  then,  the  i s symmetrical  i n our results  temperature  to replace a l l exponentials containing  by u n i t y .  The r e s u l t s i f (5.W)  identical  to those  then  c o n d i t i o n s under  give  tials are  by u n i t y  i s valid.  definitely  that  look upon  then  we  must  so obtained  had been  which  the replacing  I f we  speak  to write  (5.5).  of interest  i n this  field,  forces  o r exchange  forces,  c a n be w r i t t e n  i s , i n the form It  at  this  shall, a  will  valid  that the  this  will  be done  approximation  , a  i  spin  ,  unperturbed  a spin"  f o r the second  (where  R  and  i f m  respectively, ,a valid  have  Then,  if?  then,  energy  approximation  of the system  i fQ pairs  spins",  Secondly,  most  dipole-dipole forces,  f o r t h e second  VI.  (5.2)  which  o f f(s>)i  "energy each  We  We yields suppose  values of  levels  t h e forms  moment  (5.5)  values  , equation  of the  of the N  i s considered  of the energy  separation  the c a l c u l a t i o n s  and a r e l a b e l l e d  values  c a n be w r i t t e n  1  of  physical  i f the energy  the phrase  means t h e e n e r g y  when t h e H a m i l t o n i a n  expon(5.6)  and  I n Chapter  moment  a r e non-degenerate  has  and  i n detail  the c o n d i t i o n s under  (W.12) h o l d s .  unperturbed  4  state  of these  (5»6)X  inconvenient to perform  only  be  shall  a s two-body  very  condition  a, , a  by  f o r example,  prove  time; then,  given  We  (5.5)  the spins as separate  be a b l e  will  of a "system  forces  that  used.  (The r e s t r i c t i o n s  not severe.  i s , i f we  entities,  found  i n i and j  (5.2)  o f f (s>) i f  spins  t o be ^ and  ),  (5.6)  a, , a ^ , & , 3  yields  a  Chapter VI  An  Application  We s h a l l case ..,a  now a p p l y  which  R  labelling The that  energy  i tw i l l  Q pairs  where  K-x<y<  and  f o r any r  for  ^  >  }  a  then  \ ^ \\lS  ) such  i s , i fA  W p^...,  Q>  i f  then  The  following  a  a  Q  Before  method'of spin. spin  I* .  integers  , a^ ,  that  separation  separation  d a second  a, , a  i s such  We c a n  ( l , x , y , . —  )  set of positive  that  , ^  ^ '  Ja^-cx I ^ r  i si ntheset  (l,x,y,....)  and i f b i s n o t i n t h e s e t ( l , x , y , . . . . )  =  convenience  n  to the  suppose  o f theunperturbed  i n the s e t ( ^ + 0 ^ + 5 ,  That  shall  to establish  energy  a  R  values  o f theunperturbed  <£$R-i  jfc .  then  have  energy  a set of Q positive  -  (^rt-a,*+^  have  systems  of levels  construct  labelling  levels  level  always  a  values  We  (W.19)  and  has R energy  be c o n v e n i e n t  t h e energy energy  integers  spin  a n d (W.19)  (W.18)  a r ea l l non-degenerate.  o f these  proceeding  (W.18)  equations  when t h e u n p e r t u r b e d  pairs  that  o f Equations  t h e second example  the energy we s h a l l  call  set i sthe  should  levels  . ( I tshould noil  illustrate  be  noticed  set.) this  of theunperturbed  theset of Q positive  method o f  spin. (For integers  50  (l,x,y,..., level  ) t h e s e t G.)  L e t us suppose  system f o r the unperturbed s p i n  that  the energy  i s given  by Diagram  VII:  Diagram  VII  Diagram  VIII  <x.  R = 10 a n d Q » 4*.  We  have  of  the energy  the  set  levels  c^-o.,-  and  \ a _ < x l ^ -Rv*  i n Diagram  the second  6  r  s  t h e above  method VIII.  the labels We s e e t h a t  s e t i s (8,9,10) a n d  c x ^ a , . - a ^ a = a ^ a ^ X>+  us f i r s t  4  k>r- ^ ^ . . . . , 1 0  of a l levaluate  (W.19) w i t h r e s p e c t ( p.  are given  <7 i s (1,2,W,6)  that  Let  Using  s  r^-e^yo.  the traces  to the eigenfunctions  1,2,.... ,.n_ ; i = 1,2,  i n  (W.18) a n d  of "S^  ,  \|A.>0'  ,g , t h e d e g e n e r a c y o f e„ .)  Trace  = X  P* MP^  M  -  Z  U.LIP^P^U.O  3o<  A - 3 u A.  •»n  ),  -  Wy  CM-.O  H  IJ-M  = 1  X C\>,l\ P ^ m ^ . O -  u>  p-.i  I  I  ^  v  ? \ v.  I  V  S « *3a  Cc/,ilv U k i k U l R {Xa;, U U . O  - X X X  c,,  '  II?.  »-< k = l  i ; i  *  *  «.='  3B SB 3 * 3  9  - X X X X ( i Iv'lp.rtf t|ft1»,0C^lm \ h  n  t  t  ^X^\m\p,0  We m u s t in  t h e above  0+.19). as  how f i n d equations  T h e method  that  used.by  some w a y t o p e r f o r m and t h e summations  which  Pryce  t h e summations  we u s e w i l l  i n  and  be e s s e n t i a l l y  (1950).  and Stevens  (W.18)  This  t h e same  will  now b e  discussed. Let t "  us denote  spin  so  theunperturbed  Ir^  by  eigenfunctions  and t h e corresponding  of  eigenvalue  the by a  r  that  ttjflr^  o. l(\)  =  r  t = 1,2,...,N  t  r - 1,2,  ,R  (6.7) An  eigenfunction  can  then  be w r i t t e n  For  example,  It true where  should  which  fc>  where we  from  AT  c  change  la)  x  x  (6.1)  xlcO  t o (6.6)  since  thefunctions  I u'^)  jl  I  K  , a r e obtained  to  of the form .  a  t h e meaning symbol  remain  U ^ x U ^ l c ^ . - x U ^  + ,-.Eu  now o n , t h i s  1 0 * 10  equations  by  of a l l functions  o +Q + 0 v  shall  that  i sreplaced  diagonalize  combinations  i n the form  + o,- E  + o +Q+ »  X_  be n o t i c e d  i f lj.,0 a  of  o f t h e symbol  will  by taking  linear  I o D . ^ l t ) jcl^") * - K \<*\ Forthis ^fv^)  reason 5  mean  (6.8)  ,  where  This  o + a o . . . . . -•• « o  t+  should Now,  +  lead  t o no  4  (6.9)  , E^,  confusion.  we c a n w r i t e  R  E  -  X  where  o  a  r  (6.10)  r  O*O IN  W  C  R  R and  X nr  where  -  (6.11)  N  r-i Furthermore,  q  Nl  -  If  thedegeneracy  -  r  i s  y 1— o a  when  any eigenfunction  characterized integers  (6.12)  ^ t  function  b y two s e t s a  Xn  3  belongs,  specifies  0 -s n  \  t>$ R  R  that  (a,b,c,  r  set ofR  « N f o r r » 1,2,.... ,R  t o which the  us t h e ^--value.  The second  , d ) , where  l*c«R  We c a n , t h e n ,  The f i r s t  us the eigenvalue  i s , gives  theparticular  c a n be  of integers.  = N, t e l l s  of N Integers ,  of  (n, , n , n ,... . n ), where R  where  U a ^ R  r  4 l  r  Thus,  set  of E  u  «i  r-i  and  g  1-$<A-$R  eigenfunction,  that  i s ,the i-value.  replace  (6.13)  54-  r  where  L  means summation over a l l combinations  of the N integers wherein a -t- a, + a +  + a, = E., .  Further, we can replace  L  Si-  r-  L ^_  by 1  t «  X  where  v  means summation over a l l integers  ,n ,°.H^,n  n  c  (  a  (6.14)  R  wherein 0 ^ n ^ r  N for r =  1,2,...,R such that  •r  For example, we can write  M  R .  Let us now consider the meaning of r e c a l l that  II .We d< p means summation over a l l values of  II •>< <  «*  and  p  wherein E^ - E^ =  . Since E^ and E^  can be labelled by the sets of R integers (n, ,njO,....n ) and R  (n'  (  jn^ ,  ,n' ) respectively, the condition E -E^ = p  and the condition X o - L n ^ N r  imply that some r e s t r i c -  tions must be placed on these 2R integers.  We must, i n f a c t ,  have 1  n><  N  f o r at least one g i n the set ^  0 $ n <<  N  for r ^ g  3  r  (6.15)  55  and  n,-l n  n+1  =  (6.16)  Jfi  5»i n  r\  for  g,g  1.  it  Then  E  ~ 2 ( nl- rO a.  E  a  Z  Thus,  5« -  X  3  a  means  integers  (n, , ^ , . . . . , 0  where  n -N  I  We form.  rather this  Trace  c a n now w r i t e  lo^.c,  than  summation  4  over  a l l values  of the  )  and where- i ^ n . ^ N  r  For simplicity  L,0=  =  (6.1)  Wsome  t o (6.6)  i n w r i t i n g we  g i n the set G of  Q positive  integers.  i n a more  convenient  shall  write  a")  t h e more  cumbersome  form  given  by  (6.8).  Using  and (6.13) we c a n w r i t e  - L*  71  CQ,^,... UU^'X^X..! ml „,(,, c,.l)  (6.17)  = Z"  Trace  H"  (^U,..  l$rL',tY...  * ( « 7 . c \ J mlo > ,cV...'Xa,i> c';...UIo>,c ... ) ,  ,,  Z  Trace P ^ ° f ^ = e  ?  6  p  ,,  >  /  >  Ca.t.c,...  ?  c>,--)Guv,oc<.*L"c';.-') 0  l  > ;  1 t^la,* t V , . 0  57  Now,  l e tus write  (5.5)  and (5.6)  m  and  i n t h e forms  respectively,  given by  that i s ,  M  TH  =  (5.5)  X  1$  TT  where  i s symmetrical i n i  Before (5.5)  and (5.6)  following We  proceeding  shall  Dashes  t o evaluate  i twill  )  and i  (6.17)  be c o n v e n i e n t  to  (6.22)  (5.6)  using  to establish the  convention: use the l e t t e r  will  be used  a to refer  to the states  o f t h e 1st  spin  b  to refer  to the states  o f t h e 2nd  spin  e  to refer  to the states  of the i  t  f  to refer  to the states  of the J  t  g  to refer  to the states  of thek  h  to refer  to the states  of the l * *  to distinguish different  states  h  h  spin spin spin  1  spin  o f t h e same  spin. Substituting elements But  o f the form  m. U.b.c,  SO t h a t  (5.5)  into,  (QW  f o r example, U  (6.17)  l o u ").  IO\XIWXX.~.A  C o X c ' , . . .Jm.U> c ...!)^ra1ayb lb )....(S U.lO--. * ,  i  j  -(s'ma S,A  W  N  ,  yields  matrix  because  of the orthonormality  But,  then  i f  a,' + a , +a.,+  cxod  b  a , b*= b c ' = c  -for  of the eigenfunctions.  E - E.  » +a- + -* Q  a  = a —  (We  shall  i n future  5 - \ Ci  use  t o mean  b  c  -a i£  <Xn«A  Onlij  l£  J  is  in  t o m e a n " f i s i n t h e s e t G"  " f i s n o t i n t h e s e t G".)  Thus, i f  Ca ^ c ^ . . . I mj  then  IQ.WCJ,  ..  o  5-or  (6.23)  S\c  and  Si '  • '  .  >  » *  *  aa  »»b  «_*«•!  Si  (6.2*0  so t h a t  = a \ U ' X & . U - U ) S.  S.  >  m ;  - o Thus  and  +  ; i; \ i Trace  from  (6.17),  S. 5  „ =  Trace  ftnv  =  X  lcelm;U+i^|  3  c€<i.  But  J_  means summation over a l l combinations of  «  the N integers a,b,... ,e,..... Thus, I 1 .  where  0  .  a  +  luU u^l -  +  A  r  q  +  21  3  +  >  v V ' "  and where  2 1 lr*U-l« 0l  a  ;  7-«va -  "'"  r  E  V.^.-Cn-.V.  C  means summation over a l l Q integers i n  the set G. Then, Trace t w P « = X , IG»W U-MT ;  Now,  §  X 21 « *  where  ;  r  p ot  >  (ni  , „  ,  ^  has been explained before.  Because of the term which i s co  «n Pp  , ,  (o-0l  i f o=o  ^values of the integers  i n the denominator  of (6.25)  » we can take the summation over a l l k n ^ t i j , . . . ^ ^  where  o^-n^R  for a l l  to  sz.  Thus,  C o - 0 1 = 00  Since  Trs.Ce Po,ir>:fftro- -  ii- -_ n  we  0  I I  *  R  1 I • n +i> -»"+n t-+(i -.H-l >  1  (  0 $  t  n $. N-I) r  R  1 <  I R  JQ.  have  But  so that  11  *  0 ,  fer  r  r a t e  p p m  n W  .  =  ft  lc«U-U,T/  L  f c T  fit  NOW,  Ce\«\;U+r) =• CilmU + i^  5or i= i, . ..^ N  since we are considering N i d e n t i c a l spins. Then Trace P ^ f l , H T ^ p  a m  ?  p m  .  F i n a l l y , then, using (6.28) we get  Let us now consider Trace (5.5)  and (5.6)  P^^&rrv^ffl  we can write  T r a c e P ^ ' Vm  w h e r e we  Using  P m  write  (6.18),  **C  Thus,  = Z. L I (t^VnA + ( l ^ m-^-Y } +  £  Trace  (6.23),  a n d ( 6 . 2 4 ) we  -S-=.  31H. /fe b  («-*m n \ f l  %  have  "•i", .»»(n-r>l....Cn..'>!... -l ,  ; i m > m  Y-  B  L  i 1 / Ti i  Bll"t Z.  i -°-  Cof-n <N) R  —  )  "-^jT  Ml.  l  i  « «  4 4  '"RJ  i-"»---(n-i). l«-i)'...o.  n 1  0  ^n ^ N-i t  (oi-n $H-j) r  Thus o/<  R  *  =ft*"*I I «  Similarity,  Z Z  l(*UUT G.^L.v)  fcT  CscJ  M  ;  N  Y  *" Z 2_ * " o i m U t . ^ m ^ ^ . ^ i ^ u , ^ ^  •A  a  k T  (6.30)  (6.319  ZZ «*< ^  X X a  and,  '  («;'• m  * *  J  ( «  feT  ZZ  -*^  3  «  C  Z7  -4 * XX  :  i  m . A ^ O  *••*"(*«;•J  m-m\«  * ' .r  *A'*' ^ <  (<r * : i m ; \M >  v  CI  ft  «\= k fe. *  *  +Q  U*.LUX*Jtn|$+v)C«  (« ^ -  IT  M  *• * p  oi< ^  (6.38)  = o. where  we  Using  (6.30)  ^  ill;  use Trace  f e  to  P^y °. P  (6.38)  we  Rm  P,  m  have  (j--/-X>,sVpr" ii;^ A , ^ a  (6.39)  where  (6.40)  (ft(p  In writing the above we have used  »<0  since  tt^  i s symmetrical i n i andj .  Then, using (6.29) and (6.39), we can write (W.18) as follows:  =  «™ *«G  where  (6.W3)  Evaluation of (6.20) to (6.22) involves the same methods as used above.  I t should only be necessary  then to give the  results of our c a l c u l a t i o n s . After using equations we have found that:  ZZ  /^w ( p * V ?  «tG  $*C  ?  (5.6), (6.28), and (6.42)  /  VpWP,Ti\ - a  ?„  ^tt '\^\ + (  V '\  y^) --  ^«-c  4 l l I , where:  (5.5),  vW'.O*  (6>5)  -5 I I  tfeUJa^YBtllmU^C*^! V » | a , t y M I , k | « ? | * * , S )  4-  1  J>5  J  j<*0  ; i(*o  where we use : L  to mean summation over a l l positive integers  Otbr«+4  a and b  wherein  t  »^«*4  a „ + a , ^ a +a  .  mean summation over a l l positive integers  o  a and b wherein  r  a + a - a + a. a  b  and where furthermore  a  «G  to mean summation over a l l positive integers a and b wherein  »+<x. = a +<* .  a  t  Now, using (6.29) and (6.W5) we can write (W.18) as follows:  ILL  l  l  i  -*(-^ - ^ ( - ^ -Vw  .  I-  ^  (  *€(»  :  6  -  ^  )  A  „  (  .  A  f  i  )  n  •70  (W.17)  Equation  c a n now  (te)* + aU*L  <KV>«  where  be w r i t t e n  as f o l l o w s :  (6.1+7)  + Jt <*» > 3  i!<£*>>  a  and  are given  by  (6.W3)  and  (6.W6). (6.W7)  Equation moment  1)  i s a valid  o f f ( v ) whenever  (W.12)  Condition  a  the following  holds,  —  *  -a  expression  ir\ = X «  W)  There has  0.  identical  R energy values  Q pairs  of which  We  pointed  method  yields  containing  are  by  unity  i n  i n i and j  of which  energy  results  unperturbed  i s non-degenerate  separation chapter  as does  spin  (W.17)  the temperature  and  . that when  i n this  the Van  Vleck  a l l the  latter  expression  unity. a l l exponentials  (6.W7),  conditions  and e a c h  out i n the l a s t  t h e same  Replacing  spins  each  have  exponentials replaced  symmetric  ;  are N  have  hold:  that i s ,  and  3)  f o r the second  hold,  which  containing  is identical  yields:  with  the temperature  (W.17)  when  by  the above  7/  NRT.  I  I I I NR*  Let  us  justified.  HflL for  example,  ice i«,  0  Aj^  ,  K M ^ )  I  A U) 0  etc  consider The  A C)  under  which  term  feT  A^G*)  c a n be  written  conditions  this  step  i s  7JL ,  4-  But, fi= 2_ a  -6  -r-  Thus,  so that  fcr  rV  Now, we define  6=  _?«•-««  J L _ <:  f?  QLmax  r  r  f  -tvQT  = L 2 *  — Q.m»«. •  1  J  so that  A  x  _5  «  a  We can then write  etc etc  x.cr.  J-.'  N  N X  I f , now,  *.T  «  -+-  Z  -e€G  6T~  -2a  * S 3  so that  a l l  etc S«i  «.  A c,a 3il  ~i  JL GD 0  then  Similarily, We  found  that  equation  of  have  shown t h a t  s p i n s has R  Rv*  Condition  2)  S4'=  Z Z «V?  3)  m=  1  C  In  (6.48)  i s a  investigated.  valid  the Van V l e c k method,  that i s ,  approximation  second  when  energy  each  f o r the  of the N  values, a.^a^  and Q p a i r s  of which  identical }  Q.  ,  r  have  a l l of energy  holds ; and  symmetrical  i n i ahd j  ^ H-  Chapters  hold,  A.  In  (4.12)  IT  }  c a n be  ,i f :  1)  to  a valid  a r e non-degenerate  separation  (6.47)  (6.47).  (5»2) y i e l d s  unperturbed  i n  then  of f ( i > ) f o r the case  moment  which  terms  i f  approximation T h u s , we  the other  that  — A  V a n d V I we  i s , we  tL  the next  have  •—  c h a p t e r we  have  assumed  considered  condition  (4.12)  that  £L  shall  find  expressions f o r  - R a n d $}<crtj  when  We  shall  these  new  identical  show t h e n  that  the temperature  expressions f o r with  those  found  and by r e p l a c i n g  independent ^*-<a-o*>  terms i n are not  a l l exponentials  containing  the temperature  respectively. regarded Van  Vleck  It will  as n e g l i g i b l e method d o e s  be  when  by u n i t y shown  that  compared  not yield  i n  (4.18) a n d i f  with  a valid  (4.19)  cannot > then  be  the  approximation f o r  Chapter  Expressions f o r  (7;1)  General  The  A <^> l  a n J  M  ^  U  n  Formulae  e x p r e s s i o n s f o r ^<v>  concerned  VII  and  lv^>  with  w h i c h we a r e  a r e g i v e n by  (7.1.D (7.1.2) where  It  B  = right  hand  side  of equation  (*+.8)  C  = right  hand  side  of equation  (^.9)  D  = right  hand  side  of equation  (^.10)  should  then  we  •  be n o t i c e d  first  t h a t w h e n we  put  =  can write: = (right  hand  side  of  everywhere  (7.1.1) except  when  %  i s replaced  by  i n the exponentials i n  (7.1.1)), and, z  (right  hand  side  of  everywhere  (7.1.2) except  when  ^  i s replaced  i n the exponentials i n  (7.1.2)). (see  Chap t e r Let  IV f o r d e f i n i t i o n  u s now  take  of  by  and  fc<  taT).}  •7(i>  A  in  these  (4.2)  «,  W  ^  p  iUw>  expressions f o r  and t h e f a c t  (7.1.3)  ^p;  that  Trace  45*-<e*>*>  ABC = T r a c e  .  Then  using  CAB f o r a n y m a t r i c e s  A , B , C we g e t  i<rw>=  -fr  (7.I. *)  ir  (7.1.5)  1  where 6  =I X(/^- /^ ) ^« r  r  °<<ft +  ~  :  P  ° X  X  ° - XX(  A  *  {  _  a  + £ X X f.  FCT  ^ I X {-n^-c P.*'  -«\P.«\ +  ?  v  T^,« ^tt'f^W'^vnp^m + *  P.^'V  SMP, toP m f  " T r ^ ( ? y ' V v'V.m P«\ - 3 P y V n ? . « V (0 B  " ^  -aP^'Y^wpA^  Trac«( P ^ V ^ ^ W ' ^ W P , ^  4-p v 'V^p^u 'V ^ p m') t  c  ?  +  c 0  o (  w  P X P 0P «'P^ ' P^)} o  w  c  ( (  ?  >  •  ^7  Two  points should be mentioned concerning equations  (7.1.U) and ( 7 . 1 . 5 ) .  F i r s t l y , the expansion indicated by i  Pryce and Stevens (1950) on page h8 of t h e i r paper yields expressions which are s l i g h t l y cruder than (7.1.*+) and a.  Secondly, taking  ~ t- ^  K t  then l e t t i n g  (7.1.5).  T-».OO  i n ( 7 . I . * ) and ( 7 . 1 . 5 ) y i e l d s expressions which are d i f f e r e n t 1  i  from those found when this procedure i s repeated with (*+.l8) and (W. 19) which are the expressions we found f o r £<t>*y .K*0^ >  respectively when  «.  ~« ~  .  lir  and  For example,  E  taking  «  kT  then l e t t i n g -<--*oo  - » - j£-  ^<6v>-  I  I  X X -w« whereas staking  i n (M-.18) y i e l d s :  I  L  (7.1.6)  £mP m p  then l e t t i n g  ™ i-  T**D  i n (7.1.^)  yields  £ Jt* X  - -us  X T ^ C P ^ ' V ^ P , ^  -P,K i>in^(ri) lo  L_!  IS?.1.7)  "Similarly, when this procedure i s carried out on equations (M-.19) and (7.1.5) the r e s u l t i n g expressions w i l l be  temperature  independent i n each case but they w i l l not be eqqal. In other words, i f we apply, f o r example, equations (^.18)  ~7?  and  (7.1.^)  to a special  the  results  f o r high  for  this  i s that  imation  ( a s we  temperatures  do  will  (7.1.6)  i n writing  ^ °" » c  case  we  i n the next not agree. have  chapter) The  employed  reason  the approx-  ft ->  >  <0  *  P «  we  have  9. ~ ^  and the  a  taken  <  - >Y  0  — i  assumption  PU  E P  = a  .  and  .  should using  find  (7.1.6)  Chapter  7.2  that  form given spin  shall  (7.1.7)  show  Application  We  now  of  that  rewrite when m  (5«5)  (5.6)  and  has R energy  degenerate Since  and  VI only  in  case  this  fact,  0 >  o-,,a  a >  then  be  written;  we  obtained i n  i s true.  considered  i n a more  i n Chapter  VI  convenient  i n the forms  and when  each  unperturbed  ,0^ , a l l o f w h i c h a r e n o i j -  3 >  energy  are similar will  used  a i -  » «  c a n be w r i t t e n  a  «.  not  the r e s u l t s  (7.1.^)  o f which have  c a n be  &  t o the case  and «  have  In the a p p l i c a t i o n  in fact  equation  we  taken  between  this  the r e s u l t s  (7.1.^)  have  respectively,  values  Q pairs  n  (7.1.7)  i s small.  the calculations  Chapter  i  P  -jfcf  ; we  i f ,  (7.1.*0  f o r the case by  V  the d i f f e r e n c e  and  V I I I we  ~i-  In w r i t i n g  ^L" ^  that  E  **  separation  t o those  given.  We  .  given i n  have  found  that  where  fcr L  etc  4«  c  3  ^•'Illa;*'.;*).-^^  +A Z Z  Z {(;-&  R  -3  A  4  >  J ^ V * ~ ^ A  H  > ^ ) )  1  R  +«IX I X(  «*c  n  ^, I,-,  'Z Z C * ^ A  J > l  W + « * JLJ^S)):^  +  -2  (7.2.3) 3=1  We  have  before,  AJ*,5>>  given  A„(«\ JL,J*t$)t  (see Chapter  VI).  The  other  and A's  Jl C* i) 1>a  1  a r e as f o l l o w s :  - I I Iccuu^ G,*IS«;?UO  A G*>- I I 4a  { laLU ,TL +  Ma^M*  +  5Xa|mla+iX*4'l»l*) + /  A U,*) 4i  =  II  {laUU*,")P  Z  la*.,*l«*!U,0|  ir  a-n,  A G,s )= 45  A  4 U  l3  I I ^ L ^auu.X4 iUs)C^l^ | * Q +  lauu^LUM^U.^ -M <k$taJ**  UA V- I I I . ,  A^  3  > 3  )-II I  A^G^^^H  U . s V M * ^  ku^ta.*^  Kg)]  H \X. • C . U U ^ L l d U . , i | ^ ! U J . y * l ^ l ^ ) >  1  A„(.,s,,,t).II I Equation  l^wV^-s J ^ l * * " ^ ]  L  (7.1.5)  +  3  l6i^.TtW«"W-(»#l<k*)]( ilt;ijA s>  could also be written i n this, form but  the labour involved would be quite considerable.  In the next  chapter we s h a l l r e s t r i c t ourselves to applying equation to a particular physical system.  (7.2.1)  Chapter  Calculation Magnetic Nickel  the  Magnetic fluosilicate and  resonance  and  expressions  of  absorption  two  (195D  Kambe Van  Vleck  shall That to  have  lines  has  -ft*-<«^>  case.  1  on .  1  a  observed  as  as  The relevance in  the  to  I.U.K.) u s i n g  as  of  question  (1939)  (see  this was also  this  functions  of  recalled  and  nickel  structure  first  Ishiguro,  In  the  (7.2.1)  the  **  *  for  the by  will  not  be  temperature.  (7.2.1)  and  equations  that has  P„« °P,  ^  T  crystal  Becquerel  we  respectively  1  quantum  and  the  -£<*v>  -feW  fluosilicate  discussed  I.U.K.) and  *  Usui,  that  assumption for  theor-  chapter  £*<V->  of  Kittel,  frequencies  by  be  that  and  found  should  Equation  of  (1950),  It  basis  nickel  Holden,  (6.*+3)» ( 6 . ^ 6 ) ,  assumption  structure  by  equations  the  is  of  absorption  -SgS-  derived  crystal  (5.2)).  for  been d e r i v e d  by  Field  Axis.  andStevens  referred  and apply  Magnetic  Optic  been  A<6\>>  (6. +6)  the  mean s q u a r e  equation  and  C J / ^  the  (see  special  Spherically-shaped  been  Penrose  for  shall  a  the  method  i s , we  (6.^3)  by  (hereafter  give  this  by  has  2  f o r the  6  absorption  (NiSiF^-6H 0)  etical  £*< ^>  C r y s t a l when  D i r e c t i o n of  (19^9)  Yager  and  Absorption  Fluosilicate in  <£<&v)  of  Resonance  VIII  \  ). and  mechanical and  repeated  been  the problem  Opechowski here.  The  main  points  1)  each  of interest  each R  3)  N;  i o n has  the energy that  single  Thus, this  case  is The  elements  energy)  ft"* I X  levels  spin  S  of each  =  1 so  of a  that  unperturbed  of each  unperturbed  the s e t G  defined  i n Chapter  ion *  i o n are not VI  consists  3,  equidistant of the  1.  equation  (6.43), (6.46)  and  (7.2.1) t o  take, f o r example,  • itofi  X  reduces  of  >  4  expressions f o r  interaction  effective  levels  when a p p l y i n g  tec ^ r ,  That  of energy  term,  we  i o n i s i n the presence  +  field,  + +  » number  so  N;'*  paramagnetic  crystalline 2)  are  t o one  R<*v>  , where energy.  term and  Z_Z_  If  =  i n this £  a  =  < W }  contain  K  (exchange  case.  , the  matrix  spin-spin  e n e r g y ) -+• ( d i p o l a r  then  + I I -4C( S;. O 3  a  -3r- f, a  S i  .r  1 1  Y,S .r^](8.1) i  In where  | 5  If should then  :  (a  the  X  I <  tf \= not  5JS .+ S,.V  ,  be  .tu  expressions  4 ;  -SS^  M  confused  only matrix  S  3 l |  with  for  6  the  elements  -  of  3 M  J» HS . B  (o>o  4  used  :this  i n Chapter  ft'  VI)  occurring  0  and  are  those  in of  the the  operator  Z I {  f  t  i  S..S.  .  8 S 5 ; j  +  4 ;  4 :  }  (8.2)  where  B ;  i -  and to  -(a  fjj)  w h e r e we the  small.  d o e s -not  a  the d i r e c t i o n  (I.U.K. have .  At  to  high  also  cosine of  9j.= 3„= 3  taken  in  temperatures,  i s dominant  appear  truncated  so  that  (  the  the  +  ^-  relative  their exchange  resulting S^.)  ;  error  this  necessary.)  (8.2)  we  expressions  cs. *  5] as  They have  Using the  use  for  contribution  are  r..* [ I w.*-  i-axis.  expression  is  J»J  fcr  have  found  for, become  that and  (6.*+3)  End  -K <&v > i  l  (6>6), we  respectively:  ft  A  fl  found  which when  (8.2)  Using expression  for  we h a v e  -K<AV>  found  that  we f o u n d  (7.2.1),  when  *  which  i s  the  & * '^C-  >p  *  ^)  AT  '  becomes:  A<»>=-^|-  «f^;(*-SO  (8.5)  kr  where  b=lC.U|^| ft L L  lfl,U  ^ ;^+/^K<VB;U +  V r +  a ^ - / ^ )  In  t h e above  -r-*oo  As the  other  where  fl  hand,  = 2_ A  .  , the r i g h t  hand  as  ,  Q a constant  T - * x >  which  goes  side  6?.3")  of  %- - * o  .  f f -T ^ - *  and  to zero  —•o  as  Q  On  (-£5*)  goes  to zero.  o $hus, The  the right  difference,  (7.1.7) In take  Q  hand  side  then,  o f (8.5)  i n the results  ~~f^+  which  goes  applying  (8.3)  t o (8.5)  i s  the ?-axis  to zero  t o be t h e o p t i c  shajhl a p p r o x i m a t e t h e l a t t i c e cubic  lattice.  cubic  lattice.)  We  shall,  i  JC*i>  furthermore,  found  —co  as  $  to nickel axis,  using  (7.1.6) goes  and  to zero. we  and, f o r s i m p l i c i t y ,  a slightly  approximation  -r->co  fluosilicate  of the n i c k e l  ( I ti s , i n f a c t , This  as  has a l s o  ions  by a  distorted been  shall we  simple simple  made b y I . U . K .  take  (8.6)  S7  This We  h o l d s , as  shall  also  fi i f  i s well-known,  i and  j are nearest has  simple  As  a  spherically  shaped  neighbours  s i x nearest cubic  (Each  neighbours  in a  (8.7)  lattice)  otherwise.  consequence  of  (8.6)  and  (8.7)  we  have  and  Finally,  then,  (6.43), (6.46)  respectively:  e  crystal,  take  ion  o  for a  + a***  4.  5 "  *  and  (7.2.1) b e c o m e  FfKINC  PftCE  88  FIGURE HE. T H E O R E T I C A L  CURVE  D E P E N D E N C E SPHER1COLI.X  Fo*?  THE  ti* <as> ")  FOR  3  S H R P E D  N ; $  ;  T E m P E R O T o R E  F  » CDH^O  « CR><ST«L  L  E Q E N D ;  15  Mo  SO  ci* \»« o  So  ICO  83 .-R<6v> = -<*n  1  V—  +A  + a. 1  L  V  ; t T  yg  T +  - a  —  A 5 g  — A  /  B  %1  -  *  j -t  4  *  0=  KT  — a.  0 w h e r e we (  d  have  i s the  by  are  three  x 3„M-  l  r'^C  lattice  and  -  That taken  energy  a ] * - o. tat N dl  s  c o n s t a n t ) , which  values  of  each  . O  B  (8.8)  equations  a  virtual  XX  is a  result  obtained  I.K.U. The  I  taken  I I ) f o r the  .  (8.10)  to  case  unperturbed  we  have  As  an  plotted  nickel  ion  illustration  graphs  (see  of  Figures  when  + 5,.|* H-& a  i s , we H  s  a  have  taken  constant  =  = + g„ 12  w  + S  kilogauss,  . g„-=a.3(.  .The function  of  experimental  temperature data  of  and  Penrose  we  have  and  taken  Stevens  We  have  ,  quantity  £ is  i t s values  (1950).  The  a  from  the  absolute  value  of the quantity  IftU7.5*.o'%r s  estimated  transition.  by I.K.U. t o be We  have  also  used  value.  eigs  We  have,  and  <(*H)  quantities Figure  i n Figure  I, plotted  i n gauss.  i s  -iWfcv>=  I I , we h a v e  \&H >  i n gauss  9, ^ <AH > b  I I we  have  >  3N  c  moment  As  we  results  mentioned  f o r -R(av)>  ( 7 . 1 A ) ,  and  agree  of f ( v ) .  that  (seeFigure  above  example  I t should  A  plotted  (to> > = ((»- <v>y")> central  as the ordinate  .  B  Figure  i n  t  h  a  these  Similarily,  also  on  i n e r g s and  J r < ^ >  The r e l a t i o n s h i p between  5  a  .  X  i  plotted .  as the ordinate  The r e l a t i o n s h i p between  = <3^(u ( A H )  is  the  has been  f o rthis  3  this  fl  these  quantities  be n o t i c e d  a „ j 4*-<AV >  a  1  i s , (*°*\  I t can easily  i  by t h i s  t  be shown  i n Chapter VII, the high as given  s  h  e  second  that  application of  by ( 6 . 4 3 )  and ( 7 . 2 . 1 ) ,  I ) .  chosen  t o be q u i t e  have  and, as expected,  H  (*»} '-i c  temperature  i s ,as given We  on  where  c  t  that  (4.18) do n o t large i n  the difference i nthe  results f o r JUv> = W is  small For  n  + A<V> a t high  (8.11) temperatures.  T = 100°K,  (8.11)  yield  ( i f , f o r example,  i s negative): -«•»  _o  whereas, example,  = 100°K,  f o rT H  =  -n  (8.11)  and ( 7 . 2 . 1 )  i s negative): _n  & <V)  and ( 6 . 4 3 )  ^.ogXIO  ir-g  -o +O.»aX.l0  -n sr-^ - 3 0 . I 4 / . I O -»rtj  yield  ( i f ,  f o r  The  difference In  the  the  next  conclusions  particular  even  at  100°K  chapter that  physical  can  i s indeed  small.  we  shall  very  briefly  be  drawn  from  this  system.  mention  application  some to  of a  HI  Chapter  In  this  conclusions. the 1)  2)  c h a p t e r we  The f o l l o w i n g  application  i n Chapter  The dependence  magnetic  to  less  The dependence  helium observe of  sample  of our  are apparent  from  of the c h a r a c t e r i s t i c s  lines  i s negligible  of  even  down  1°K.  resonance  temperatures,  paramagnetic  some  VIII of our general formulae:  on t e m p e r a t u r e  a change  outline  g e n e r a l remarks  resonance  than  paramagnetic  very briefly  on temperature  nuclear much  shall  IX  lines  that  of the characteristics should  i s , i t should  i n the resonance resonance  i s lowered  from  be O b s e r v a b l e  lines  room  of  at  be p o s s i b l e  liquid  to  f r e q u e n c y and t h e  as the temperature  temperature  to liquid  shape  of the helium  temperatures. 3)  Both  the resonance  line  depend  We pertain  can also  when  slightly make  i n particular  fluosilicate  which  the energy  temperature,  f r e q u e n c y and t h e shape  we  on t h e shape  a few remarks  from  to the spherically  £  o n e o f s>  sample.  Figures shaped  considered i n Chapter  difference,  keeping  of the  of the resonance  I and sample  VIII:  I I which of  Below  nickel 20°K,  , i s approximately constant  with  or  the  H  c o n s t a n t and v a r y i n g  other,  one s h o u l d  mean v a l u e value  Coupled  Our extreme be  this  will  -  physical  low  temperatures,  take  This  by f i n d i n g  *  That  .  P  i s ppesitive.  of the resonance ^<^v>  i n both find  line.  attains  I ) .  an  I t should  of our  an extreme  c= *. ~* ~  value  and when  F  we  N  )  f o r this  however,  i s not apparent;  i t i s highly  i s valid.  This  (  ^  -  ^  a t these  probable could  the expression i n t h i s  ^  fi  of the v a r i a b l e  that  i s , we  the  coefficient, i s  then  occurs  A  P « ' - >  approximations  * ^  .  take  reason  shown  value  ( i-  The  course,  also  i s lowered, I f t h e mean  1°K a n d 2 ° K ( s e e F i g u r e  f  *  ft  decreases,  i n the width  have  f o r ^»<o.^> w h e n we  these  , the exchange  i n t h e mean v a l u e  the extreme  -5*5.  we  shift  between  that  for  of  fi  calculations  expressions  take  q u a n t i t y changes.  be a d e c r e a s e  value  noted  then  as the temperature  i f t h e mean v a l u e  with  quantity  that  of the v a r i a b l e  increases,  negative;  find  case  that  very neither  be c h e c k e d , f o r  +  would  involve  c o n s i d e r a b l e l a b o u r and h a s n o t b e e n  undertaken  i n this  thesis.  of when  Appendix  Writing  m ^ ~LT.  of  { P^mP. + Pstnp ,) Q  -t < ^  in  Let  us  Chapter energy which in  have  this  where we  values,  p  "  J  another general  suppose  VI, each  A  of  that  equation  the N  (5.5)  identical  a l l of which  energy  form.  holds  .  that,  unperturbed spins  are non-degenerate  separation  and  We  shall  and show  Q  as  has  R  pairs  here  i n  of  that  case  XX  write,  -  I  "and as  is  operator  1)  P.  been  discussed  previously  Xlx P  P«.  i m  P  w l  ) ( ix •--  i s such  * i.  that:  ?• = P« I,  2)  X  3)  p.UO.  P. 4  ^  I  :  --lo. ^  and  customary,  1*1x1*  The  have  (A.2)  , the  identity  W^u^'-a uy  operator  (A.3)  ( A »  where  It and  should  Now, in  i t should  terms  that  f o r some  *e<5  n^x  where  E .=I n a .  be r e c a l l e d  be p o s s i b l e  R operators  of the  , and t h a t  to write  Z.r> -N  where  r  - o,-+ E  the operator  P (ar.,^  .  R  N -» — tt n '. r=i of the d i r e c t product of  In  fact,  1  we  can write  as a  sum o f  terms  5  r  each  of which  i i' ,  ,n «5 •  n  P  s  R  The d i f f e r e n t terms  p  correspond  consists  i n the expression  to the d i f f e r e n t combinations of the P  Similarily  c a n be w r i t t e n  R  P'S  n  as a  sum o f  n P's  for  P^  n. P \ , . . . . , n P ' R  S  .  2L--  terms. Let we  P^m-P^  us c o n s i d e r  P «. Pp  see that  rf  .  consists  ;  From  the above  discussion  of  \/L-  NI  N' T — *- n' l...f..,V.( ^!..^[ n  terms.  n  Because  non-zero.  of  There  (A.2),  however,  can, i n fact,  n o t a l l o f these terms a r e  be a t most  CN-Ol  I-  «..•-,•  etc  non-zero  ...(o-,V.(, ,v... ! +  terms  Thus, Using  (A.3)  ponding form  H  OR  i n P  P*™^ , P  A  i t i s possible  to each  will t o group  of the Q values  of  consist the e  R  into  of  Q R terms  terms corres-  one t e r m o f t h e  Finally,  then  P,  II < f  Ilf.^P^  Similarily  d< p ' Thus  we N  can  write N  N  P« = I P„ . p . ••I • «€<. m  F  Z ltd  j  ? ^.9 *t  *  >  j  Appendix  Rewriting  form  given  Using  A  of  tfWS  m. =  II*  X  p  by I s h i g u r o ,  we  x  have  B  +  P„mpJ  Usui,  from  i nthe  (195D  a n d Kambe  (A.l)  N  S *  I  x  I  { P.S..P ,. P .s,.P,.] +  If,  now,  only  one p a i r  x  V  I*  Equation Ishiguro,  energy the  '  of only  the integer  unity,  , we  from  a i  +  have  P  (B.2)  Usui,  ^  a  ;  p  i  ;  separation 1 and  levels  i s 2.  that  spin  i s ,  wifla  (B.l):  (B.2)  }  i s identical  a n d Kambe  the energy  integers  '  of the unperturbed  x  labelled  '  of levels  { P,S .P  I  (B.l)  fi  "  Q consists  separation  S  +  *  esq  with  (195D.  equation  They  have,  of the unperturbed by  •+  o  (6)  of  however, spin  whose  whereas  we u s e  Appendix  C  A  Showing  that  field  a n d when  the  axis.  Let  Z  us suppose  (1,2,3,....,R-1). (3.2.2) h o l d s , unperturbed only  - 5  spin  between  This  i s no  field  crystalline  i s parallel  the s e t G  i s the s e t of  corresponds  to the case  i s , when  the energy  are equidistant  adjacent  there  levels.  levels  and when  Then,  from  to  integers  when of  condition  each  transitions ( B . l ) we  occur  have:  R-l  A  »• a.=i  (CD  >  If we  when  the magnetic  that  that  X  »  a  A  U \  then,  using  (k.h)  and ( C . l )  have  .C-'ls .U").= G!s,.U,y X, L x  for  It  «:= l , a / i  i s well  eigenfunction  .R-|  known, h o w e v e r ,  of  5^  then:  that  i f UV  denotes  an  S3  Thus,  C  i f  V t  ,  ,  then  c."ii..i.-y..c.-is..i.o  ;  Since equation 1)  Q  we  ( 5 . 2 ) , we  consists  (3.2.2)  [ V-  2)  These by  Van  to  use  are concerned  can replace  of the  the 5*  traces of operators with  s e t (1,2,3,...R-l)  5*  in  i f  le condition  holds  S*>o  two  conditions are satisfied  Vleck S  with  x  (19^8)  instead  and of  hence S  x  f o r the case  i t was  considered  not necessary  f o r him  99  Bibliography  Andrew, E.R., N u c l e a r  Press, Becquerel, Broer,  J . , and Opechowski,  L.J.F.,  Holden,  Ishiguro,  Cambridge  University  (19^9).  E., Usui,  W.,  Physica  6  1039(1939).  10, 801 (1943).  Physica  A.N., K l t t e l ,  1443  Magnetic Resonance,  1955*  C , and Yager,  T . , a n d Kambe,  W.A., P h y s i c a l  K., P h y s i c a  Review  75,  1£ 310 (1951).  P e n r o s e , R . P . , a n d S t e v e n s , K.W.H., P r o c e e d i n g s o f t h e P h y s i c a l S o c i e t y A j £ , 29 (1950). P r y c e , M.H.L., a n d S t e v e n s , K.W.H., P r o c e e d i n g s o f t h e P h y s i c a l S o c i e t y A 6 J , 36 (1950). Usui,  Van  T . , a n d Kambe, 'K., P r o g r e s s  302 (1952).  Vleck,  Waller,  Physics  2it» H68 (1948). f u r P h y s i k 22, 380 (1932).  J.H., P h y s i c a l  I., Zeitschrift  of Theoretical  Review  8,  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0085389/manifest

Comment

Related Items