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Some experimental and theoretical studies of magnetic properties.of solids Wong, Samuel Kim Po 1970

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SOME EXPERIMENTAL AND THEORETICAL STUDIES OF MAGNETIC PROPERTIES OF SOLIDS  by  SAMUEL KIM PO WONG  B.Sc,  (Sp. Hon.)  M.Sc,  A THESIS SUBMITTED  University  o f Hong Kong, 1965  University  of B r i t i s h  Columbia, 1967  IN PARTIAL FULFILMENT OF  THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  i n the Department  of  Physics  We a c c e p t required  this  t h e s i s as conforming  t o the  standard  THE UNIVERSITY OF BRITISH COLUMBIA May, 1970  In p r e s e n t i n g  this  thesis  an a d v a n c e d d e g r e e a t the L i b r a r y I  further  for  agree  scholarly  by h i s of  shall  the U n i v e r s i t y  make  it  written  thesis  freely  that permission  for  It  financial  Y^j,  X4>  ;  111*  of  Columbia,  British for  gain  Columbia  the  requirements  reference copying o f  I agree and this  shall  that  not  copying or  for  that  study. thesis  by t h e Head o f my D e p a r t m e n t  is understood  Department  Date  of  for extensive  permission.  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada  fulfilment  available  p u r p o s e s may be g r a n t e d  representatives.  this  in p a r t i a l  or  publication  be a l l o w e d w i t h o u t my  i  Abstract  This thesis i s concerned with three aspects of magnetic properties in solids.  1.  The mean square value  <I(I + 1)>  molecule has been measured to be The experiment was  3.73  of the proton angular momentum per ± 0.18  in solid  CH^  at 4.2 K .  done by measuring the r a t i o of the proton magnetic 13  resonance free induction decay s i g n a l to that of 13 in a sample containing  53%  CH^  .  C  at the same frequency  The measured value of  d i f f e r s appreciably from the high temperature value of from the value of  <I(I + 1)> = 6  into the A spin symmetry species. a sample containing  0.002%  of  CH4  of  <I(I + 1)>  0.05%  <I(I + 1)>  when a l l the molecules  =3  and  are converted  Experiments at lower temperatures  using  O 2 impurity indicated that there may be a  s l i g h t increase i n the value of The presence of  <I(I + 1)>  <I(I + 1)>  between  4.2 K  and  2.45  K .  O 2 impurity shortened the time constant f o r conversion  between d i f f e r e n t symmetry species.  The i n t e r p r e t a t i o n of measurements  i n the presence of large amounts of  complicated by changes i n the NMR  l i n e shapes.  0  2  impurities i s  These complications are  discussed i n terms of the t h e o r e t i c a l results obtained i n the second part of the t h e s i s . 2.  The NMR  l i n e shape inhomogeneously broadened by paramagnetic impurities  in s o l i d s  was investigated t h e o r e t i c a l l y using the s t a t i s t i c a l method of  Margenau.  The general expression f o r the Fourier transform of the l i n e  ii  shape f u n c t i o n was  derived.  c u t - o f f r e g i o n and  i t s complement.  give r i s e  to s a t e l l i t e  the main l i n e and  The  l i n e s and  satellite  lattice  can be  Impurities  d i v i d e d i n t o an i n n e r s p h e r i c a l i n s i d e the c u t - o f f r e g i o n  i m p u r i t i e s o u t s i d e determine the shape of  lines.  D e t a i l e d numerical  c a l c u l a t i o n s were  performed f o r a f . c . c . l a t t i c e when the i m p u r i t i e s are magnetic d i p o l e s ,  (b)  s p i n 1/2  systems and  (c)  asymmetry i n the l i n e shape i s p r e d i c t e d a t f i n i t e peak i n t e n s i t y  is shifted  c o n c e n t r a t i o n and  about the c e n t r e .  The  i n such a way  classical  s p i n 1 systems. temperatures  An  .  The  an amount l i n e a r i n the  i n the magnitude of the magnetic moment and  l i n e shape a d j u s t s i t s e l f  the  I t depends on  the main l i n e and  which i s p r e d o m i n a n t l y L o r e n t z i a n .  impurity  overall  to g i v e a v a n i s h i n g f i r s t moment  l i n e shape i s always L o r e n t z i a n at the c e n t r e  G a u s s i a n i n the wings. concentrations,  from the c e n t r e by  (a)  the i m p u r i t y  the s a t e l l i t e The  concentration.  and  At  low  l i n e s have the same shape  shape of a powdered sample i s a l s o  discussed.  3. of  General  expressions  are d e r i v e d f o r the f i r s t  the s p e c i f i c heat of a paramagnetic s a l t  two  terms i n the  i n powers of  T  .  expansion  It is  •assumed t h a t the i n t e r a c t i o n between the paramagnetic i o n s c o n s i s t s o f magnetic d i p o l e and of  i s o t r o p i c exchange i n t e r a c t i o n s , and  each i o n i s a x i a l and has  the same o r i e n t a t i o n f o r a l l i o n s .  formulae are e v a l u a t e d n u m e r i c a l l y with  the e x p e r i m e n t a l  t h a t the  f o r cerium magnesium n i t r a t e .  d a t a of Mess and  g-tensor The  general  Comparison  coworkers i n d i c a t e d t h a t the exchange  parameter corresponds to a f e r r o m a g n e t i c  interaction.  iii  LIST OF TABLES  Table 1.  2.  Page Experimental r e s u l t s f o r an oxygen-free sample of ' containing 53% 13 CH^ Experimental r e s u l t s f o r a sample of 53%  3.  1 3  CH  4  Values of  and  0.05%  0  <I(I + 1)>  CH^  CH, 23  containing 34  2  and the r e s i d u a l entropy  S  Q  f o r d i f f e r e n t l i m i t i n g cases of the energy l e v e l s of Fig.  4.  13  Values of  46 £k , P^ ,  and  f o r a face-centred  cubic l a t t i c e 5.  Values of  b  2  107 and  b  3  f o r s i n g l e c r y s t a l 6MN  .  157  iv  LIST OF ILLUSTRATIONS  Figure  Page  1.  A block diagram of the pulsed  r f spectrometer  2.  Plot of proton f.i.d. signal in arbitrary units  11  in a sample of solid methane containing 53% 13  3.  CH  4  at 4.2°K 13 C  Plot of  12  f . i . d . signal in arbitrary units  for the same sample as shown in Fig. 2  13  4.  Circuit diagram of the transmitter  14  5.  A diagram showing the setup in the helium inner Dewar  17  6.  7.  Plot of log[btV^(t)/sin bt] in arbitrary units 2 *• versus t for the proton f.i.d. of Fig.2  t 8.  Plot of  Sp  and  S  £  53%  13  CH  Plot of  CH^  containing  had been cooled to 4.2°K .  4  S  in arbitrary units versus 13 time after a sample containing 53% CH^ and 0.05% 0 had been cooled to 4.2©K . o  and  22  in arbitrary units versus  time after an 02~free sample of  9.  21  Plot of 2  log[V^(t)] in arbitrary units versus 13 for the C f . i . d . of Fig. 3  0  26  S  27  V  10.  P l o t of  Sp  ..a sample o f  11.  P l o t of  S  a sample o f 0.05%  12.  CH^  containing  0.002%  30  2  i n a r b i t r a r y u n i t s v e r s u s time f o r 13 CH^ c o n t a i n i n g 53% CH^ and 32  2  P l o t of calculated B/kT  < I ( I + 1)> , S ,  and  C  v  f o r a spherical free-rotor  43  Energy l e v e l s and d e g e n e r a c i e s o f the l o w - l y i n g states of  o f the  CH^  A, T,  and  E s p i n symmetry  p e r t u r b e d by a t r i g o n a l  species  crystalline  electric field  14.  0  0 . .  versus  13.  i n a r b i t r a r y u n i t s v e r s u s time f o r  -.  •- 45  P l o t o f c a l c u l a t e d v a l u e s o f • < I ( I + 1)> , S , and  C  v  versus  T  f o r CH^  in a  tetrahedral  crystal field  15.  48  P l o t o f the c a l c u l a t e d and field  C  v  versus  T  values of  f o r CH^  < I ( I + 1)> , S ,  i n a trigonal  crystal  .  49  16a.  P l o t o f R e a l ( J ( u ) ) versus u .  77  16b.  P l o t o f -Imag(J(u)) v e r s u s u .  78  17.  L i n e shape f u n c t i o n  I^(CLI)  when  pMp(q) = 0 . 1  81  18.  L i n e shape f u n c t i o n  I^(co)  when  pMp(q) = 0.5  82  19.  L i n e shape f u n c t i o n  1^ (u) when  pMp(q) = 2  83  vi  20.  Plot n =  21.  Plot n =  22.  Plot n =  23.  Plot n =  24.  Plot n =  25.  Plot n =  26.  Plot  o f R e a l ( J ( u ) ) and R e a l ( J ^ ( u ) ) versus  (0,0)  Plot  86  o f - I m a g ( J ( u ) ) and - I m a g ( J ( u ) ) v e r s u s 1  u when  (0,0)  87  o f R e a l ( J ( u ) ) and R e a l d ^ u ) ) (rr/18,  versus  u when 88  TT/36)  o f - I m a g ( J ( u ) ) and - I m a g ( J ( u ) ) v e r s u s 1  u when 89  (TT/18, TT/36)  o f R e a l ( J ( u ) ) and R e a l ( J ( u ) ) v e r s u s 1  2TT/9,  u when  90  7TT/18)  o f - I m a g ( J ( u ) ) a n d -Imag (J-^ ( u ) ) v e r s u s  (2TT/9,  of  b = 10, 27.  u when  7TT/18)  f(w) and  u when  .  and  91 g(to)  versus  to when  a =  1,  c = 0.27 . ..  of Real(G(v))  104  and R e a l ( G ( v ) ) 1  versus  v when  n = (0,0) 28.  Plot  113  o f -Imag(G(v)) and - I m a g ( G ( v ) ) v e r s u s 1  v when  n = (0,0) 29.  Plot  of Real(G(v)  114 and R e a l ( G ( v ) ) 1  versus  v when  n = (TT/4, 0) .  30.  Plot when  115  o f -Imag(G(v)) and -Imag(G- (v)) v e r s u s L  n = (TT/4, 0)  v  116  P l o t of Real(G(v)) n  =  (IT/8, 0)  and R e a l C G ^ v ) ) v e r s u s v when  .  P l o t o f -Imag(G(v)) and -Imag(G^(v)) v e r s u s v when  n = (TT/8, 0)  Values o f  -w  o b t a i n e d by f i t t i n g Eq. [3.19]  to e x p e r i m e n t a l d a t a o f Mess e t a l . (59) at v a r i o u s temperatures  S p e c i f i c h e a t o f CMN as f u n c t i o n o f  temperature  viii  Acknowledgement  The of  Professor  my  sincere  J.  Lees  M.  B l o o m and  gratitude  I fruitful  work d e s c r i b e d  am  indebted  discussions, for their  Committee  their  financial  and  by  the  t h e s i s was W.  t o Dr.  support  National  special  and  Dr.  S.T.  the  supervision  them I w i s h  to  express  guidance.  f o r many s t i m u l a t i n g Dembinski  , and  and  Mr.  work.  Research  are  To  Alexander  provided  thanks  encouragement  S.  interest  J . Noble,  in this  done under  Opechowski.  continued  to P r o f e s s o r  and  The  and  Professor  assistance  Finally, sacrifices  for  i n this  by  the  Council  due  throughout  t o my  the  Commonwealth is gratefully  wife,  Scholarship acknowledged.  Eleanor,  preparation  of  this  for  her  thesis.  T A B L E OF  Page  CONTENTS  ABSTRACT  i  L I S T OF T A B L E S  i i i  L I S T OF I L L U S T R A T I O N S ACKNOWLEDGEMENTS  CHAPTER  I  i v . .  viii  NUCLEAR MAGNETIC  SUSCEPTIBILITY  I N SOLID  CH^  AT 4.2°K .  1  1.  INTRODUCTION  2.  THEORY  3.  EXPERIMENTAL METHODS  9  4.  D I S C U S S I O N OF THE EXPERIMENTAL RESULTS  18  A.  ....  '..  OF THE EXPERIMENTAL METHOD  Measurement o f  < I ( I + 1)>_  5.  THE MAGNETIC  6.  COMPARISON  CHAPTER  2  CH^  frV. ....... .i-  Dependence o f  Sp  on Time and T e m p e r a t u r e  P R O P E R T I E S OF  WITH OTHER  0  2  IN SOLID  CH^  EXPERIMENTS  NUCLEAR MAGNETIC  RESONANCE  BROADENED BY PARAMAGNETIC  1 4  f o r Oxygen-free  a t 4.2°K B.  ...  19 ^  25  33 40  LINE  SHAPES  IMPURITIES  IN SOLIDS  54  1.  INTRODUCTION  54  2.  D E R I V A T I O N OF THE L I N E SHAPE FUNCTION  57  3.  LINE  SHAPE FUNCTION DUE TO A SYSTEM OF  C L A S S I C A L MAGNETIC  DIPOLES  .  70  4.  LINE SHAPE FUNCTION DUE TO A SYSTEM OF SPINS, S = 1/2  5.  AND  S = 1  95  121  CONCLUSION  125  APPENDIX  CHAPTER 3  EFFECT OF THE EXCHANGE AND MAGNETIC DIPOLE-DIPOLE INTERACTIONS ON THE SPECIFIC HEAT OF PARAMAGNETIC AT VERY LOW TEMPERATURES  SALTS 136  1.  INTRODUCTION  136  2.  GENERAL THEORY  141  3.  THE CASE OF CERIUM MAGNESIUM NITRATE  147  APPENDIX  159  BIBLIOGRAPHY  162  CHAPTER 1  NUCLEAR MAGNETIC SUSCEPTIBILITY IN SOLID CH.  AT  4.2°K  4  1.  INTRODUCTION  Each of the f i v e i s o t o p i c m o d i f i c a t i o n s of methane  CH  7  D ,  4-n  n = 0,1,2,3, and 4, e x h i b i t s two temperatures CH^,  (1).  With  s p e c i f i c heat anomalies  n  i n the s o l i d  at  low  the e x c e p t i o n o f the lower s p e c i f i c heat anomaly o f  these anomalies have been demonstrated  t r a n s i t i o n s i n the s o l i d s .  To understand  James and Keenan (2) proposed  to be a s s o c i a t e d w i t h phase  these anomalies  theoretically,  a model which assumes an e l e c t r o s t a t i c o c t o p o l e  o c t o p o l i n t e r a c t i o n of neighbouring molecules i n a f . c . c . l a t t i c e .  Although  the James-Keenan model (2) and i t s quantum m e c h a n i c a l e x t e n s i o n by Yamamoto et  a l . (3,4,5,6) i s s u c c e s s f u l i n r e p r o d u c i n g these double t r a n s i t i o n s  s p e c i f i c heat anomalies  i n a s e m i - q u a n t i t a t i v e manner, the c o n t r o v e r s y  developed over the i n t e r p r e t a t i o n of the s p e c i f i c heat anomaly w i t h the lower o f the two (7,8,9).  the v i c i n i t y CH^,  o f the anomaly  (7).  The  apparent  has been i n t e r p r e t e d by C o l w e l l , G i l l ,  symmetry s p e c i e s o f and  E  CH^  remains  unsettled  to d e t e c t because  zero temperature  o f the  entropy of  on s p e c i f i c heat measurements e x t e n d i n g down to  Nagamiya's c a l c u l a t i o n  T,  difficult  of  associated  are r e q u i r e d f o r the e s t a b l i s h m e n t o f t h e r m a l e q u i l i b r i u m i n  which i s based  2.5°K  s p e c i f i c heat anomalies  T h i s s p e c i f i c heat anomaly was  f a c t t h a t hours  and  (10) o f the energy  CH^.  s p e c i e s of  CH^  The  about  and M o r r i s o n (8) i n terms o f  l e v e l s o f the  d e g e n e r a c i e s o f the ground  i n a tetragonal c r y s t a l l i n e  A, T,  and  E  spin  s t a t e s o f the field  a r e i n the  A,  2  ratio  5:9:2,  nuclear  r e s p e c t i v e l y , which are  s p i n s t a t i s t i c a l weights  the same as  (11).  5:6:2  f i e l d parameter.  species  depending on  solid  by Hopkins, Donoho, and  zero-point  that  P i t z e r (9) who  pure  CH^.  their results  changes had  been r e p o r t e d  ,  p r e v i o u s l y by Wolf and  measurements o f  r e l a t i v e number of states.  CH^  amounts of  0  Sp 2  , Sp  v e r y much (1%) 0  2  0  can be  impurity  no  as a  are  for  (12). 0 S^  CH^  Such  Since  f o r the  the A,  T,  i s proportional  used to d e t e c t  of s o l i d  function  impurity.  2  and  r e s p e c t i v e l y , and  samples  changes i n  the  s p i n symmetry to 4.2°K  quickly  They found t h a t f o r moderate  approximately  24 hours, w h i l e f o r samples w i t h v e r y  molecules  CH^  Whitney  1 = 2 , 1 ,  as a f u n c t i o n of time. i n c r e a s e d by  has  i s extremely long  m o l e c u l e s i n the d i f f e r e n t n u c l e a r  Hopkins e t a l . (9) c o o l e d  monitored  Sp  CH^  species.  t h a t the r e l a x a t i o n time  d i f f e r e n t amounts of  s p i n symmetry m o d i f i c a t i o n s ,  <I(I + 1)>  CH^  in solid  p r o t o n a n g u l a r momentum per m o l e c u l e i s E  T  on measurements of the v a r i a t i o n o f  the p r o t o n magnetic resonance i n t e n s i t y of time f o r samples c o n t a i n i n g  between  suggest t h a t when  of e q u i l i b r i u m among the s p e c i e s  They base t h i s s u g g e s t i o n  the  crystal  takes p l a c e  e n t r o p y of  i s i n e q u i l i b r i u m a t 4.2°K, almost a l l the  f o r the e s t a b l i s h m e n t  and  the s i g n of the  conversion  i n the s t a t e of the A s p i n symmetry s p e c i e s , but  to  the  i s a f f e c t e d , changing  spin species, Morrison et a l . f i n d  above i n t e r p r e t a t i o n of the  been c h a l l e n g e d  and  only  c o n s i s t e n t w i t h a 6 - f o l d degeneracy of the ground s t a t e of the  The  CH^  5:3:2  T  With the assumption t h a t no  the d i f f e r e n t n u c l e a r are  or  temperature  In a t r i g o n a l f i e l d  degeneracy of the ground s t a t e of the above r a t i o s to  the h i g h  little  60%  (a few  over a p e r i o d of about p a r t s per m i l l i o n ) o r  changes w i t h time o c c u r r e d .  Hopkins e t a l .  3  suggest  that f o r very l i t t l e  i s much l o n g e r than  i m p u r i t y , the r e l a x a t i o n time f o r c o n v e r s i o n  the time o f t h e experiment  the r e l a x a t i o n time i s s h o r t e r than ment (tens o f m i n u t e s ) .  the time taken  t o make the f i r s t  anomaly r e p o r t e d by M o r r i s o n  a s s o c i a t e d w i t h s p i n c o n v e r s i o n induced i n their  f o r 1% 0£ measure-  The i n f e r e n c e drawn from these r e s u l t s by Hopkins  et a l . i s t h a t the s p e c i f i c heat  (0.005%)  (24 h r ) , w h i l e  by a s m a l l amount o f  et a l .i s 0  impurity  2  sample.  A v e r y e x t e n s i v e s e r i e s o f n u c l e a r magnetic resonance measurements was c a r r i e d out on  CH^  (13, 14, 1 5 ) . D u r i n g with  time i n s o l i d  ment w i t h  and i t s d e u t e r a t e d  the course  CH^  m o d i f i c a t i o n s by G.A. de Wit  o f these measurements, an i n c r e a s e o f  c o o l e d t o 4.2°K was observed,  the " o b s e r v a t i o n o f Wolf and Whitney  (12).  i n qualitative  The  i n t o the A s p i n symmetry s p e c i e s was never s e r i o u s l y  main reason  r e l a x a t i o n time 1.2° with  f o r adopting  considered.  T-j_ i s a v e r y s l o w l y v a r y i n g f u n c t i o n o f temperature between  and 4.2°K and i s f a i r l y  s h o r t , which seems c o m p l e t e l y i n c o n s i s t e n t  the i n t e r p r e t a t i o n o f Hopkins e t a l (9, 1 5 ) . We s h a l l r e t u r n t o t h e  reason  f o r b e i n g s k e p t i c a l about e x p e r i m e n t a l  There i s another  demonstrations o f conversion  between n u c l e a r s p i n symmetry s p e c i e s s o l e l y on the b a s i s o f a l o n g constant  the  CH^  t h i s p o i n t o f view was t h a t the s p i n - l a t t i c e  s i g n i f i c a n c e o f the •T-^ measurements l a t e r i n t h i s c h a p t e r .  years  agree-  However, the p o s s i b i l i t y  t h a t such changes c o u l d be due to almost complete c o n v e r s i o n o f the molecules  Sp  associated with a p h y s i c a l observable  ago B o r s t e t a l . (16) concluded H2O  molecules  t h a t both  i n gumdrops c o n v e r t e d  a t low temperatures. solid  acetylene  t o the p a r a s p e c i e s .  time A few  (C2H2) and These c o n c l u -  s i o n s were based on the time dependence o f the t o t a l n e u t r o n c r o s s - s e c t i o n  4  i n these s o l i d s a f t e r b e i n g c o o l e d to 4.2°K. resonance measurements c o n c l u s i o n s were  n u c l e a r magnetic  (17) between 4.2°K and 1.2°K demonstrated  that  these  incorrect.  There i s an obvious need <I(I + 1)>  Subsequent  f o r the p r o t o n s i n  f o r a more d e f i n i t i v e measurement o f  CH^  s o l i d a t low temperatures.  r e q u i r e d i s a r e l i a b l e measurement of has e s t a b l i s h e d e q u i l i b r i u m .  <I(I + 1)>  What i s  a t a time a f t e r the system  We r e p o r t h e r e the r e s u l t s o f such an experiment.  As w i l l be d e s c r i b e d i n more d e t a i l i n the next two s e c t i o n s , we used a sample 13 of  CH^  containing  53%  o f the  C  isotope.  By measuring  the r a t i o o f  13 S to the C magnetic resonance i n t e n s i t y S a t the same f r e q u e n c y , we P o b t a i n the v a l u e o f <I(I + 1)> f o r the p r o t o n s p e r CH^ m o l e c u l e i n terms 13 t  of the known v a l u e of  <I(I + 1)> = 3/4  f o r the  C  spins.  of our measurements f o r pure methane and methane doped w i t h d i s c u s s e d i n S e c t i o n s 4 and 5.  The r e s u l t s 0.05% 0  The r e l a t i o n s h i p between the NMR  in  CH^  2.  THEORY OF THE EXPERIMENTAL METHOD  are  2  measurements  and s p e c i f i c heat and i n f r a r e d s t u d i e s i s developed i n S e c t i o n  6.  In t h i s s e c t i o n , we review the p r i n c i p l e of the n u c l e a r magnetic resonance  (NMR)  experiment  f o r d e t e r m i n i n g c o n v e r s i o n between n u c l e a r s p i n  species.  The s t a t i c n u c l e a r s u s c e p t i b i l i t y  x  Q  obeys C u r i e ' s law, which  we w r i t e as f o l l o w s :  X  where  M  0  M  _ Q  R  q  2 2 o __ NY ft <I(I + 1)> 3  k  T  ,  , L-L.-l-J r  i s the e q u i l i b r i u m n u c l e a r m a g n e t i z a t i o n per u n i t volume i n the  5  the  field  H  and temperature  the  n u c l e a r gyromagnetic  the  n u c l e a r a n g u l a r momentum p e r m o l e c u l e .  Q  T, N  r a t i o and  i s number o f m o l e c u l e s p e r cm^, <I(1 + 1)>  y is  i s t h e mean square v a l u e o f  F o r t h e case o f  CH^,  each  molecule c o n t a i n s f o u r p r o t o n s o f s p i n 1/2.  I f t h e a n g u l a r momentum v e c t o r s  of  <I(I + 1)> = 4(1/2)(3/2) = 3.  t h e f o u r p r o t o n s were u n c o r r e l a t e d , then  On t h e o t h e r hand, i f each E  CH^  m o l e c u l e belonged  s p i n symmetry s p e c i e s , then we would have  and l e t t i n g  P^  to one o f the A, T,  I = 2, 1, o r  0,  or  respectively,  be the p r o b a b i l i t y t h a t a m o l e c u l e belongs t o t h e  i  spin  symmetry s p e c i e s , we o b t a i n  <I(I + 1)> = 6 P  At  A  + 2P  T  .  h i g h temperatures, t h e A, T,  [1.2]  and  E  s p e s c i e s s h o u l d be  r e l a t i v e l y p o p u l a t e d a c c o r d i n g to t h e i r s t a t i s t i c a l respectively, giving  < I ( I + 1)> = 3,  as e x p e c t e d .  s t a t e o f t h e A s p e c i e s were lower than the ground by an energy much g r e a t e r than then is  <I(I + 1)> = 6,  kT, where  k  5,9,  and  s t a t e of any o t h e r s p e c i e s  x T Q  constant,  by a f a c t o r o f 2. I t  e t a l . (9) c l a i m t o have o b s e r v e d u s i n g NMR  techniques.  If  a n u c l e a r s p i n system i n i t i a l l y  a r o t a t i n g magnetic H-^  field  and a n g u l a r f r e q u e n c y  i n e q u i l i b r i u m i s subjected to  i n the plane p e r p e n d i c u l a r to U3 = Y H Q  Q  f o r a time  2,  I f t h e ground m o l e c u l a r  i s t h e Boltzmann's  l e a d i n g t o an i n c r e a s e i n  t h i s i n c r e a s e which Hopkins  weights o f  H  T ^ , the  m a g n e t i z a t i o n f o l l o w i n g the p u l s e i s g i v e n by (18, 19) :  o f magnitude x  component o f  6  sin e^osCcOot + <J))G(t) ,  M (t) = M  Q  where  0-^ = ^x l  ,  T  [1.3]  <J> i s a phase angle which depends on the i n i t i a l direction  of the rotating f i e l d and  G(t)  is the free induction decay (f.i.d.) function.  For a proper choice of the origin of the time parameter  t,  example, in the limit that  corresponds to the  T-^ approaches zero,  t = 0  G(0) = 1.  For  time at which the pulse is applied. More generally for f i n i t e  t = t' + fx  where  t'  ±  ,  ^  0 <_ f <_ 1 ,  i s the time measured from the end of the pulse.  [1.4]  It has been  demonstrated experimentally and theoretically (19) that under quite general conditions,  f = 1/2  represents a proper choice of the origin of  t.  It has  been found experimentally (19) that with this choice of origin, the f . i . d . shape G(t) where  G(t)  is undistorted even for quite large values of  T^,  so that  is the Fourier transform of the absorption line shape function g(u), u = UJ - t o ,  and we can write (18)  o  •00  G(t) =  g(u)e~  1Ut  du  n=2 where  Mj^  i s the nth moment of the line shape.  We are interested here in the experimental determination of Th e effect of  M (t)  is to induce a voltage  V.(t)sin(co t + $)  M  Q  in a coil  .  7  having  i t s axis p a r a l l e l  detected  i n u s u a l way  to the x - a x i s .  T h i s v o l t a g e can be a m p l i f i e d and  (20) to g i v e a f . i . d .  voltage  V ( t ) g i v e n by Q  V ( t ) = A ( t ) B M s i n 0 G(t) Q  o  1  = A(t)V (t) ,  [1.6]  i  where coil  B  i s a parameter which depends on  and the sample, and  voltages.  the measured  to the i n p u t  time r e q u i r e d by the r e c e i v e r t o r e c o v e r  instrumental  O  Q  appreciable systematic  t .  F o rm t < , m t '  problem i s s t i l l  many  essentially  the i n f l u e n c e  p r o c e d u r e may  introduce  errors.  G ( t ) were known, the measurements o f V ^ ( t )  c o u l d be e x p t r a p o l a t e d back t o the o r i g i n , where G(t)  Therefore,  e l i m i n a t e d and we  so l a r g e t h a t t h i s e x p e r i m e n t a l  Now, i f the form o f t > t  voltage  I n p r a c t i c e t h i s statement i s c o r r e c t  g r e a t e r than a c e r t a i n time  of the p u l s e i s s t i l l  a stable reference  through a c a l i b r a t e d a t t e n u a t o r .  V (t)/A(t).  from the  be d i s c u s s e d i n the next s e c t i o n ,  d i s t o r t i o n s were, i n p r i n c i p l e ,  measured the q u a n t i t y only f o r t  As w i l l  V ( t ) was compared d i r e c t l y w i t h  a p p l i e d a t the sample c o i l  practice,  and t h e geometry o f t h e  A ( t ) i s the r a t i o o f the output  i n f l u e n c e o f the l a r g e r f p u l s e .  for  0  I t i s time dependent because the d e t e c t o r i s n o n l i n e a r , i n g e n e r a l ,  and because o f the f i n i t e  J  LO , T  i s never known t h e o r e t i c a l l y i n NMR, unsolved.  If  t  G(0) = 1.  In  s i n c e the l i n e - s h a p e  i s n o t too l a r g e , however, i t i s  p o s s i b l e t o make a m e a n i n g f u l e m p i r i c a l e x t r a p o l a t i o n t o the o r i g i n o f  t,  a i d e d by the f a c t  If  t h a t the s l o p e o f  G ( t ) must be zero a t the o r i g i n .  8  independent  knowledge o f the second moment 2  Eq. of  [1.5] t h a t the b e h a v i o r o f  M  i s a v a i l a b l e , we see from  2  2  d G/dt  near t h e o r i g i n p r o v i d e s a check  the c o n s i s t e n c y o f the e x t r a p o l a t i o n p r o c e d u r e , and enables one t o e s t i m a t e  the e r r o r i n v o l v e d i n such an e s t i m a t e o f V ^ ( 0 ) . The e x p l i c i t e m p i r i c a l e x t r a p o l a t i o n p r o c e d u r e s we have f o l l o w e d f o r o b t a i n i n g t h e v a l u e s o f V^(0) 13 for  the p r o t o n s and the  Sp  and  S  c  ,  C  n u c l e i , which we denote  respectively, w i l l  f o r s i n 0^ = 1  by  be d e s c r i b e d i n S e c t i o n 4. 13  S i n c e both the p r o t o n and same f r e q u e n c y and temperature the q u a n t i t y  B  C  NMR  experiments were done a t " t h e  and w i t h no change i n the sample geometry,  was t h e same f o r each and we o b t a i n , u s i n g Eqs. [1.1] and  [1.6] and the d e f i n i t i o n o f  S  and  S ,  P  c  Y S  <I(I + 1)> = q  I^E-  [1.7]  p c  where  q  i s the f r a c t i o n o f  we have used the f a c t  CH^  13  molecules having the  t h a t the f i e l d  was l a r g e r than the p r o t o n resonance  a t which the  13  C  and the v a l u e  Y  f i e l d by a f a c t o r  f o r the p r o t o n and  q = 0.53  C ,  i s o t o p e and  resonance was o b s e r v e d Y /Y . p c  13 known v a l u e s o f  C  which g i v e  ^^/Y^  U s i n g the  = 0.25,  f o r our sample, we have  <I(I + 1)> = 0.099 S / S . \ * p ' c  Comments on the use o f S t e a d y - S t a t e  Another way t o measure  x  [1.8]  Techniques.  D  u s i n g NMR  phase" component o f the n u c l e a r s u s c e p t i b i l i t y  i s t o measure t h e "out o f x"( ) u  a  s  a  function of  9  u =. W - ai that  .  x"( ) u  I f the amplitude of the r f f i e l d i s i n d e p e n d e n t o f H-^,  i s sufficiently  i t may be shown r i g o r o u s l y  small  (16) t h a t  oo  f  Xo =  C  X"(u)  du ,  [1.9]  —00  where -oo  C  i s a constant.  t o +oo  The p u l s e experiment a n a l o g o f t h e i n t e g r a t i o n from  i n Eq. [ 1 . 9 ] , w h i c h i n v o l v e s an e x t r a p o l a t i o n o f t h e measured  v a l u e o f x"( )  o v e r a f i n i t e range o f u  u  t o ±oo ,  i s that the detector  i n t h e p u l s e e x p e r i m e n t must have a s u f f i c i e n t l y wide bandwidth and t h e measurement must be made s u f f i c i e n t l y soon a f t e r t h e p u l s e as compared w i t h a time w h i c h i s r o u g h l y t h e i n v e r s e o f t h e w i d t h o f t h e resonance o f x"( )» u  H o p k i n s , Donoho, and P i t z e r of  u.  where  (9) measured  dx"(u)/du  as a f u n c t i o n  They d e f i n e d t h e i n t e n s i t y o f t h e NMR l i n e as t h e p r o d u c t x"(0)  w  a  s  l i n e t o u = 0,  o b t a i n e d by i n t e g r a t i n g and 6  dx"(u)/du  x"(0)<5  not change.  I  s  from t h e wings o f t h e  proportional to x  o  versus  u.  o n l y i f t h e l i n e shape  Presumably, t h i s c o n d i t i o n was s a t i s f i e d  Strictly x"( ) u  does  i n t h e experiments o f  Hopkins e t a l . , b u t they have n o t commented on t h i s p o i n t i n t h e i r  3.  u  i s t h e s e p a r a t i o n i n f r e q u e n c y u n i t s between t h e  maximum and minimum p o i n t s i n t h e p l o t o f dx"(u)/du speaking  x"( )^ >  paper.  EXPERIMENTAL METHODS  The NMR measurements r e p o r t e d here were c a r r i e d o u t a t a f r e q u e n c y of 22.1 MHz c o r r e s p o n d i n g t o a resonance f i e l d o f 5180 gauss f o r p r o t o n s and 13 of 20500 gauss f o r  C.  The magnet was a Magnion 1 5 - i n c h e l e c t r o m a g n e t  10  (Model L-158) w i t h a 2.25-inch gap. resonance by s e t t i n g digitally  the f . i . d . a f.i.d.  the NMR  signal.  c o u l d be s e t q u i t e c l o s e to  r e g u l a t o r (Magnion  HS-10200)  F i n a l adjustment o f the f i e l d was made  A 90° p u l s e  ( 0 - ^ = TT/2) was s e t up by maximizing  s i g n a l f o l l o w i n g the p u l s e and n o t i n g t h a t t h i s corresponded  s i g n a l o f amplitude  millisecond  The  the magnetic f i e l d  to the e s t i m a t e d v a l u e .  by m o n i t o r i n g  The f i e l d  Pulse  c l o s e t o zero f o l l o w i n g a p u l s e a p p l i e d a few  later.  Spectrometer  A b l o c k diagram o f the p u l s e s p e c t r o m e t e r  i s shown i n F i g . 1.  t i m i n g u n i t c o n s i s t e d o f a T e k t r o n i x 162 wave form g e n e r a t o r , which the p u l s e r e p e t i t i o n r a t e , and a T e k t r o n i x 163 p u l s e g e n e r a t o r width had  to  o f the r f p u l s e .  a width  The  controlled  t o c o n t r o l the  Under o p t i m a l c o n d i t i o n s o f o p e r a t i o n , a 90° p u l s e  o f about 5 usee f o r p r o t o n s  and about 13 usee f o r  13  C.  I f the  13 value of  f o r the p r o t o n and  C  s t u d i e s were t h e same, the r a t i o o f  the p u l s e w i d t h s would be about 4 s i n c e resistor to  R^  i n the sample c o i l  tuned  Y^/Y  c  circuit  give a n e g l i g i b l e d i s t o r t i o n of the f . i . d .  for  the p r o t o n s  decayed more r a p i d l y  and  F i g . 3, a s m a l l e r v a l u e o f  r e s u l t i n g i n a smaller value of  R^  = 4.  was a d j u s t e d f o r each resonance shape. 13  than the  WA-600D.  C  Since the f . i . d .  H-^.  Its circuit  h a l f wave diode d e t e c t o r .  thus  The bandwidth o f the p r e a m p l i f i e r The wide band  amplifier  The t r a n s m i t t e r was a p u l s e d e l e c t r o n - c o u p l e d  H a r t l e y o s c i l l a t o r f o l l o w e d by one c l a s s A a m p l i f i e r and t h r e e c l a s s amplifiers.  signal  s i g n a l , as shown i n F i g . 2  had t o be chosen f o r t h e p r o t o n s ,  (Arenberg PA620-SN) c o u l d be a d j u s t e d s e p a r a t e l y . was an Arenberg  However, t h e damping  diagram i s shown i n F i g . 4.  C  The d e t e c t o r was a  Yo OSCILLOSCOPE  DETECTOR  SYNCH?  WIDE BAND AMPLIFIER  PREAMPLIFIER FD100  REFERENCE SIGNAL GENERATOR  ATTENUATOR  TIMING UNITS  Fig.  1.  A block  diagram of the pulsed  r f spectormeter.  10Pf  t  ( uS)  13 Fig. 3. Plot of C f . i . d . signal in arbitrary units for the same sample as shown in Fig. 2. The shaded area represents rf pulse. 1  6DK6  12AU7A  6L6  6L6  100uh 220pf  39K'  .  l l H  ->+300v  t  J  0 0  P ^  -02  h  pulse input 47k  r.f. 25pf o u t p u t f e H I—f  6L6  829B  >+500v  >1-5k  •05  + 1 2 0 0 V  +  750v  Fig. 4 .  -150V  C i r c u i t diagram of the transmitter.  -50v  ^  15  Measurement o f the S i g n a l  The r e c t i f i e d its  amplitude  pulse.  V (t) Q  f.i.d.  s i g n a l was  d i s p l a y e d on an o s c i l l o s c o p e and  was measured a t a p r e s e l e c t e d time f o l l o w i n g the  Immediately b e f o r e and immediately  a f t e r t h i s measurement a r e f e r e n c e  s i g n a l generated by a T e k t r o n i x 191 c o n s t a n t amplitude measured, the average f.i.d.  signal.  attenuator  s i g n a l generator  of these two measurements b e i n g used  Packard  (Model 355A, B) was a d j u s t e d so t h a t the a m p l i f i e d  o f f s e t f e a t u r e o f the Type  The p r o c e d u r e  Z  V (t). Q  T h i s was  VHF  reference -  done u s i n g the zero  p l u g - i n o f the T e k t o n i x 531A  d e s c r i b e d above enabled  was  to c a l i b r a t e the  B e f o r e each s e t o f measurements the Hewlett  s i g n a l was w i t h i n a few p e r c e n t o f  rf  oscilloscope.  us to measure  V^(t) i n  terms o f the a t t e n u a t o r s e t t i n g and the output v o l t a g e o f the r e f e r e n c e s i g n a l g e n e r a t o r , which was v e r y s t a b l e over l o n g p e r i o d s o f time. the c o r r e c t i o n due t o the d i f f e r e n c e i n the amplitudes s i g n a l and did  V (t) Q  Since  o f the r e f e r e n c e  was o n l y a few p e r c e n t , the p r o p e r t i e s o f the a m p l i f i e r  not e n t e r i n t o the measurement  i n an important way.  e m p i r i c a l l y , however, t h a t t h i s p r o c e d u r e i n d i c a t e d i n F i g s . 2, and 3.  broke down f o r  I t was  found  t < t ,  as  F o r these s h o r t times, the d i s t o r t i o n o f the  s i g n a l by the a m p l i f i e r was found  to be d i f f e r e n t  f o r the r e f e r e n c e s i g n a l ,  which was p r e s e n t b o t h b e f o r e and a f t e r the p u l s e , and the f . i . d . which o r i g i n a t e s at the time o f the p u l s e .  Because t h i s d i f f e r e n c e depended  on the t u n i n g o f the c o i l  and on  t > t  The a n a l y s i s g i v e n i n the next s e c t i o n i s based  m  were m e a n i n g f u l .  c o m p l e t e l y on the d a t a f o r  R^,  signal,  t > t ,  i t was d e c i d e d t h a t o n l y d a t a f o r  by which time  the i n f l u e n c e o f the  rf  16  p u l s e on the a m p l i f i e r had become n e g l i g i b l e .  Samples.  The samples o f methane c o n t a i n i n g 53%  13  CH^  were 13  s u p p l i e d by Merck, Sharpe, and Dohme of Canada, who concentration.  a l s o measured the  C  The oxygen-free samples were p r e p a r e d u s i n g a g e t t e r i n g  t e c h n i q u e (21) and then s e a l e d i n a g l a s s tube o f l e n g t h 21 cm and diameter 0.9 of  cm f o r the bottom 13 cm and 1.8 the  that  0  c o n c e n t r a t i o n i n the doped  2  1/T-^  sample  was  samples was  a l i n e a r f u n c t i o n o f the  containing  T-^ = 6.3  cm f o r the top 8 cm.  0.002%  sec a t 105°K  0  2  and was  0  based on the assumption  concentration  2  0  T±  had a v a l u e of  2  Cryogenics. Dewar system.  (21, 22).  taken d i r e c t l y  from the c y l i n d e r o f r e s e a r c h  = 0.25  sec at  The sample  containing  '77°K.  The low-temperature system was  a c o n v e n t i o n a l double  The setup of the i n n e r h e l i u m Dewar and the sample  shown i n F i g . 5.  The samples were f i r s t  c o o l e d s l o w l y to  77°K  p e r i o d of s e v e r a l hours by f i l l i n g the o u t e r Dewar w i t h l i q u i d Then l i q u i d h e l i u m was  tube i s over a  nitrogen.  t r a n s f e r r e d to the i n n e r Dewar where i t was  c o n t a c t w i t h the sample.  The  a c c o r d i n g to t h i s c r i t e r i o n had a v a l u e o f  grade methane s u p p l i e d by P h i l l i p s P e t r o l e u m Company. 0.05%  The measurement  The upper p a r t o f the sample  tube was  i n direct  wrapped w i t h  foam rubber padding to ensure t h a t , d u r i n g the h e l i u m t r a n f e r , the s o l i d CH^  sample  CH^  vapour was  solid to  CH^  condense  a t the bottom o f the tube was  at  cooled f i r s t .  first  c o o l e d and condensed,  77°K  would  the h i g h vapour p r e s s u r e o f the  cause most o f the sample  i n the upper r e g i o n away from the sample  g r e a t l y reduced f . i . d .  signal.  Otherwise i f the  The temperature was  t o be "pumped" up coil,  and  resulting i n a  determined from the  17  HE TRANFER SIPHON .15 FITTING—I HE PUMPING AND RETURN 4; LINE  R F INPUT TERMINAL  h KOVAR SEALS FOR ELECTRICAL LEADS  RADIATION  SAMPLE TUBE HOLDER  METAL  TEFLON SPACER  5.  SHIELD  FOAM RUBBER PADDING  SAMPLE  F i g .  SHIELDS  A  diagram  showing  the  setup  i n  the  helium  inner  COIL  Dewar.  18  h e l i u m vapour p r e s s u r e .  Of c o u r s e , an e s s e n t i a l f e a t u r e o f the t e c h n i q u e  used here to measure the p r o t o n  <I(I + 1)>  as a f u n c t i o n o f time  and  13 temperature was which had 4.  t h a t the  s p i n s s e r v e d as a secondary  thermometer  the same s p a t i a l d i s t r i b u t i o n i n the sample as the p r o t o n s p i n s .  DISCUSSION OF THE  We type was  C  EXPERIMENTAL RESULTS  s h a l l d i s c u s s two  performed  The o t h e r type was  on an  types o f e x p e r i m e n t a l r e s u l t s .  0 2 ~ f r e e sample of  performed  on a pure  CH^  CH^  containing  The  first 13  53%  sample c o n t a i n i n g  CH^.  0.002%  0  2  13 and on a  CH^  ments o f  Sp  sample c o n t a i n i n g and  S^  p o s s i b l e to determine <I(I + 1)>  53%  CH^  and  0.05%  <I(I + 1)>  as a f u n c t i o n o f time.  <I(I + 1)>  within experimental error, On  <I(I + 1)>  two o r t h r e e hours.  the sample c o n t a i n i n g  change s i g n i f i c a n t l y used to determine  was  justified.  reliable  independent  0.05%  O2.  S^  We  found  that,  of time under t h e s e were n o r m a l l y  Assuming t h a t the l i n e shape does n o t  d u r i n g the course of such measurements, they can  the time-dependence o f  f o r a few v a l u e s o f  observed  These i n c r e a s e s were much more r a p i d  F  V^(t)  as  were o n l y made f o r times g r e a t e r than o r o f the  the o t h e r hand, l a r g e i n c r e a s e s i n  d u r i n g the f i r s t  G(t),  As a consequence,  o r d e r of a few hours a f t e r c o o l i n g the sample to 4.2°K.  values of  Measurement o f  r e q u i r e d a d e t e r m i n a t i o n o f the f . i . d . shape f u n c t i o n  measurements o f  for  From measure-  made over p e r i o d s o f a p p r o x i m a t e l y 24 h r , i t was  w i l l be d e s c r i b e d below, which took a l o n g time.  conditions.  O2.  t  S . p  Occasional  checks  of  be relative  i n d i c a t e d t h a t t h i s assumption  was  19  A.  Measurement o f  We p l o t  < I ( I + 1)>  f o r Oxygen-free  t y p i c a l measurements o f  CH^  a t 4.2°K  13  V . ( t ) f o r the p r o t o n and  C  13 resonances  f o r CH^  with  53%  CH^  i n F i g s . 2 and 3, r e s p e c t i v e l y .  d i s c u s s e d i n S e c t i o n s 2 and 3, we f o u n d , t h a t correction factor t  for t < t ,  A ( t ) c o u l d n o t be determined  As  the e x p e r i m e n t a l  reliably.  The v a l u e s o f  which depends on the p u l s e w i d t h  and t h e v a l u e o f R^, are indicated 13 i n F i g s . 2 and 3 f o r the p r o t o n and C, r e s p e c t i v e l y . Our problem i s t o extrapolate V,- ( t ) to t = 0 f o r each case t o o b t a i n S and S . An p c m >  v  1  adequate f i t o f the s h o r t time b e h a v i o r o f the p r o t o n f . i . d . by  shape i s g i v e n  (18, 23)  2 2 . „, \ , a t s in bt G(t) = exp(- ~ 2 — ) — ^ — , N  where  b  form o f M  i s found  from the f i r s t  G ( t ) has been found  f o r the case o f  2  ally  CaF  zero a t  to y i e l d  even though the f . i . d .  2  t o be u n e q u a l l y spaced,  the b e h a v i o r o f  a  and  G(t) a t long  b,  M  = a  2  b t = TT .  z e r o s a r e found  This  moment  experiment-  + b  2  M  2  i s not influenced appreciably  times.  which y i e l d  2  by p u t t i n g  i^t [1.10]  c o n t r a r y to the p r e d i c t i o n s o f Eq. [1.10] ( 2 3 ) .  We f o l l o w e d the procedure values of  Q  a c c u r a t e v a l u e s o f the second  T h i s i s n o t s u r p r i s i n g as the v a l u e o f by  t = t  r  o f B a r n a a l and Lowe (23) t o o b t a i n t h e the second  / 3.  moment  [1.11]  20  The parameter a was log  [btV^Ct).sin bt]  was  f o l l o w e d f o r the  s i m p l e r case s i n c e for  t m < ot < t-,  o b t a i n e d from versus 13  C  V^(t) and  t  the s l o p e o f the p l o t as shown i n F i g . 6.  2  resonance,  of The  same  procedure  as shown i n F i g . 7, though t h i s i s a  i s gaussia'n, i . e .  the measured v a l u e s of  b = 0.  U s i n g the  a  b,»  and  V^(t)  values of  date  S =  V.(0) l  13 were found of  f o r the p r o t o n and  <I(I + 1)>  u s i n g Eq.  C  resonances,  respectively.  The  values  l i s t e d i n T a b l e 1 were o b t a i n e d from the e x p e r i m e n t a l  results  [1.8]. We  postpone a d e t a i l e d d i s c u s s i o n of the s i g n i f i c a n c e of  e x p e r i m e n t a l v a l u e of  <I(I + 1)> = 3.73  ± 0.18  u n t i l S e c t i o n 6.  the For  the  p r e s e n t d i s c u s s i o n , r e c a l l t h a t i f no c o n v e r s i o n between the n u c l e a r s p i n symmetry s p e c i e s takes p l a c e a t low  temperatures,  w h i l e complete c o n v e r s i o n to the A s p e c i e s g i v e s these a l t e r n a t i v e s are h i g h l y improbable,  Accuracy  of the Measurements of  then  <I(I + 1)> =  <I(I + 1)> = 6.  a c c o r d i n g to our  <I(I + 1)>  3,  Both  results.  and S y s t e m a t i c E r r o r s  The maximum e r r o r i n each of the i n d i v i d u a l d e t e r m i n a t i o n s o f <I(I + 1)> lines  was  through  about  ri• S corresponding  r -j i . range of v a l u e s between  u n c e r t a i n t y of  T h i s e s t i m a t e was  made by drawing extreme  the data of the type shown i n F i g s . 6 and  i . . i r and minimum v a l u e s of The  ± 10%.  ± 10%.  ' max ,min S^ /S /f  c  to , and  .max S  . and  _min._max /S c  7 to o b t a i n maximum „min S ,  . respectively.  , corresponds  to an  t  The s c a t t e r i n the most p r o b a b l e v a l u e s o b t a i n e d i n  the t h r e e runs i s q u i t e c o n s i s t e n t w i t h t h i s "maximum u n c e r t a i n t y " .  o o  c CO  O  m  4  0 t  2  6 (xlOO(uS) ) 2  Fig.  7.  Plot  of l o g [V (t)] i  i n arbitrary  8  2  units  versus  t  • 13 f o r the  C  f.i.d.  of Fig.  3  23  < I ( I + 1> )  Run  M°  2  __8 -'2 10 sec  M  2  / M  2  8 -2 10 s e c  1 r i  Nov.  14  3.75  39.1  1.21  32.4  Nov.  17  3.50  36.7  1.17  31.4  Nov.  20  3.93  31.5  0.90  35.0  35.8  1.09  33.0  Average  T a b l e 1.  3.73  ± 0.18  E x p e r i m e n t a l r e s u l t s f o r an oxygen-free 53%  13  CH  4  sample o f  CH^  containing  24  A l l o f the p o s s i b l e s y s t e m a t i c e r r o r s we have so f a r c o n s i d e r e d are s m a l l e r than the s t a t i s t i c a l the b a s i s o f t h r e e r u n s .  e r r o r o f about  5%  Our e x p e r i m e n t a l procedure  s i g n a l d i r e c t l y w i t h a r e f e r e n c e v o l t a g e f e d through  g i v e n i n T a b l e 1 on o f comparing the f . i . d . the same a m p l i f i e r  s e r v e s to e l i m i n a t e many t y p i c a l s y s t e m a t i c e r r o r s a s s o c i a t e d w i t h n o n l i n e a r i t y and d r i f t .  The remaining  i n v o l v e the e x t r a p o l a t i o n o f  amplifier  s y s t e m a t i c e r r o r s t o be c o n s i d e r e d  V ^ ( t ) to  t = 0.  The f i r s t  p o i n t to note i s  t h a t the c o r r e c t i o n f a c t o r i n v o l v e d i n such an e x t r a p o l a t i o n i s o f the o r d e r of  ? exp [M2t^/2] = 1.25  error i n M of  and  1.05  o r the o r i g i n o f  2  o n l y a few p e r c e n t .  t  f o r protons of  20%  and  for t < t  m  .  respectively.  g i v e s a change i n  There remains the p o s s i b i l i t y  does something unexpected  13 C,  S  and  t h a t the f . i . d .  An c  shape  I f t h i s occurred, the value o f  M  2  would have to be much g r e a t e r than the v a l u e we o b t a i n e d , as may be seen from Eq. [ 1 . 5 ] . our v a l u e o f  This p o s s i b i l i t y  i s p r e t t y w e l l e l i m i n a t e d by the f a c t t h a t  agrees w i t h the v a l u e o b t a i n e d by J . E . P i o t t o f the  U n i v e r s i t y o f Washington Wolf and Whitney  (24) and i s o n l y  (12) u s i n g d i f f e r e n t  15%  lower  techniques.  than t h a t o b t a i n e d by  A l s o , the r a t i o o f the  13 p r o t o n and  C  second  moments i s v e r y c l o s e t o the t h e o r e t i c a l  value  assuming t h a t the dominant c o n t r i b u t i o n a r i s e s from i n t e r m o l e c u l a r d i p o l a r i n t e r a c t i o n with protons  on n e i g h b o r i n g  CH^  molecules.  In t h i s  case (18) [1.12]  which i s t o be compared w i t h the average  v a l u e o f 33 g i v e n i n T a b l e 1.  25  B.  Dependence o f  Sp  on Time and  The v a r i a t i o n of to  4.2°K  was  Sp  not c o m p l e t e l y  Temperature  w i t h time a f t e r  the methane had been c o o l e d  r e p r o d u c i b l e , as can be seen  from one  c o o l i n g curve which w i l l be d i s c u s s e d l a t e r i n t h i s s u b s e c t i o n . t h a t s t r a n g e r e s u l t , we in  S  p  or  S  w i t h time a f t e r the f i r s t  c  v a r i a t i o n s of oxygen-free  can say t h a t n o r m a l l y  Sp  and  sample  S  Sp  £  t h e r e was  i n c r e a s e d r a p i d l y d u r i n g the f i r s t 25%  i n c r e a s e i n s i g n a l was  (12).  We  primarily,  h a l f hour o r so  d u r i n g the next  time-independent  value i s nonexponential.  I f the a p p r o x i m a t e l y  t o an i n c r e a s e i n  a f a c t o r of about  25%  b e l i e v e t h a t the i n i t i a l  1.25.  measurements of J . E . 0.002%  rapid  The  approach o f  Sp  Piott  T h i s i s not s u r p r i s i n g s i n c e we  the sample c o n t a i n i n g  over i t s h i g h temperature  This i n d i c a t e s a value of  oxygen was  symmetry  by Hopkins e t a l . ( 9 ) , t h i s would  <I(I + 1)>  (24).  to a  i n c r e a s e i n s i g n a l were a s s o c i a t e d w i t h  agreement w i t h our measured v a l u e s of  for  or  The magnitude of  w i t h c o n v e r s i o n among t h r e e d i f f e r e n t n u c l e a r s p i n  a s p i n c o n v e r s i o n a t 4.2°K as suggested  containing  two  though not n e c e s s a r i l y c o m p l e t e l y , a s s o c i a t e d  w i t h c o n v e r s i o n between n u c l e a r s p i n s p e c i e s .  correspond  the  a s s o c i a t e d w i t h c o o l i n g of the sample to 4.2°K w h i l e  the slower i n c r e a s e was  species.  For  the time c o n s t a n t a s s o c i a t e d w i t h i t a r e i n agreement w i t h  the o b s e r v a t i o n s o f Wolf and Whitney  are concerned  from  Some t y p i c a l  w i t h time a r e shown i n F i g s . 8 and 9.  t h r e e hours to a n e a r l y c o n s t a n t v a l u e as shown i n F i g . 8. t h i s change and  Apart  v e r y l i t t l e change  few hours a t 4.2°K.  f o l l o w e d by a f u r t h e r slower i n c r e a s e of about  strange  The  <I(I + 1)>  <I(I + 1)> from  time-dependence of  Sp/S , c  Sp  value  ~ 3.75, and  0  9  and  53%  CH,  the  in  with  f o r the sample  s i m i l a r to t h a t of the p r e v i o u s sample.  0.05%  by  corresponding  As  M 0  UJ Q Z)  •  1-00  6b  CL  <  CP  o  < 0  0-90  ° -  to  A -  0-80 0  4  8  12 TIME  Fig. 8.  Plot of  S  containing  and 53%  16  20  24  s  p  s  c  26  ( HOUR )  S in arbitrary units versus time after an 0 -free sample of 13CH^ had been cooled to 4.2°K 2  CH^  1-1 0  Li Q H  o •o°  1-00  A  oo  _l Q_  o  A  < <  o  o - Sr 0-90  l  8  0  12 TIME  16  20  24  ( HOUR ) 13  Fig. 9.  Plot of and  Sp  0.05%  and 0  2  S  £  in arbitrary units versus time after a sample containing  53%  CH^  had been cooled to 4.2°K . N5  28  increase i n  Sp  o c c u r r e d much more r a p i d l y as shown i n F i g . 9, i n agreement  w i t h Hopkins e t a l ( 9 ) . w i t h i n an hour or s o .  Here  Sp  i n c r e a s e d to i t s n e a r l y c o n s t a n t  A l s o , the measured v a l u e o f  <I(I + 1)>  value  from  S /S P  was  h i g h e r than t h a t o f the  w i l l be  C ^ - f r e e sample.  The  s i g n i f i c a n c e of t h i s  have a n a l y z e d the time-dependence o f the r a t i o  as a f u n c t i o n o f time  T  f i v e s e p a r a t e runs  R(x)  a f t e r c o o l i n g the sample to 4.2°K and  i n c r e a s e i n s i g n a l d i s c u s s e d above had been completed. to the  We  =  f^p/^ ^ c  after  fitted  the d a t a  formula  ±  x > x^,  where  x^  was  to  <I(I + 1)>  (i.e.  r ( x - x^)  w i t h a time  typically  choosen to be  r <_ 0.0012 h r  assume t h a t  <<  r e l a x i n g towards an e q u i l i b r i u m v a l u e o f  1)  constant  x-p  <I(I + 1)>  then i t i s easy  <  the h y p o t h e s i s i s made t h a t  c o n v e r s i o n to the A s p e c i e s and  gives  typical  9.  The  i s very slowly <  ^^ -'-^ +  > e  q ii U  to show t h a t  .-, - <I(I+1)> equil  I (  i i)> +  x. l  x  •  I  <I(I + l ^ g q y - Q <I(I + 1)> T  r < 0.0012 h r  a  I f the s i g n a l i s p r o p o r t i o n a l  and we  r=  of  about 4 h r , and  over 24 h r , as shown i n F i g s . 8 and  <I(I+1)>  If  [1.13]  x  s e r i e s o f measurements extended r e s u l t s are c o n s i s t e n t with  T  the  R(T) = R ( T ) [ 1 + r(x - T ) ]  for  result  d i s c u s s e d i n more d e t a i l i n S e c t i o n 5.  We  of  -  t  > 600 h r .  i  =  ^>  [  corresponding  = 3.73,  t h a t the e q u i l i b r i u m s t a t e a t 4.2°K corresponds  to most  CH^  the  >  1  4  ]  to complete  then the upper  In o r d e r to s u p p o r t  1  limit  hypothesis  molecules  29  b e i n g i n the A s p i n symmetry s p e c i e s , one would have to f i n d a mechanism for allowing p a r t i a l being i n h i b i t e d  c o n v e r s i o n over a few hours, f u r t h e r c o n v e r s i o n then  f o r p e r i o d s o f more than 2 weeks.  This p o s s i b i l i t y sample c o n t a i n i n g  O2.  0.05%  o n l y p l a c e an upper l i m i t T h i s upper l i m i t of 6 f o r  seems to be e x c l u d e d by our experiments on the  on  As w i l l be d i s s c u s s e d i n S e c t i o n 5, we <I(I + 1)>  from our experiments on t h a t  ., equil  s i n c e c o n v e r s i o n i s presumably  Furthermore, measurements of  Sp  i n d i c a t e that  <I(I + 1)>  we  found t h a t  SpT  is  q u a l i f i e d i n an i m p o r t a n t way  was  about c o n s t a n t between 4.2  temperature o f the sample. Sp  i n Section  sample  S^ We  and 1.05°K.  This  since  statement  6.  t h a t the time c o n s t a n t o f a few  hours  a t 4.2°K i s not due to a v a r i a t i o n of the c o o l e d a sample of  CH^  with  0.002%  0  to  2  as a f u n c t i o n of time f o r a p e r i o d o f almost 6 h r .  A f t e r t h i s time, the temperature o f the system was  reduced to 2.45°K.  shown i n F i g . 10, the s i g n a l then i n c r e a s e d by a f a c t o r of w i t h i n experimental e r r o r .  The q u a n t i t y  SpT  F i g . 10 c o r r e s p o n d s to the smooth curve drawn Wong e t a l . (25) u s i n g the same sample.  4.22/2.45 =  The  1.78,  curve shown i n  through the d a t a  The data was  fitted  o b t a i n e d by  to a s i n g l e  The purpose of t h i s i s to demonstrate whether t h e r e i s any <I(I + 1)>  As  i s seen to be unchanged w i t h i n  the e x p e r i m e n t a l a c c u r a c y when the sample i s c o o l e d .  of a change o f  sample  does not change a p p r e c i a b l y w i t h temperature  change i n  4.2°K and m o n i t o r e d  rapid i n this  as a f u n c t i o n o f temperature i n t h i s  We have a l s o demonstrated  point.  sample.  i s a v a l u e of about 4.4 which seems to r u l e out the v a l u e  <I(I + 1)>  f o r the i n i t i a l  can  indication  w i t h d e c r e a s i n g temperature, as r e v e a l e d by an  0  2 4 5 °K  Q  OOo° o  oo  ooo  o  Q  oo o 0  o  0  o oooo O  0  q  0  0  <  0  o  v  00  1-1 o —  o o° Po°o  0  oo  Oo°  °  1-00  0-90  6 Fig.  10.  P l o t of  S  p  8  10 12 TIME  i n a r b i t r a r y u n i t s v e r s u s time f o r a sample o f 2.45°K.  Then  containing  reduced from  2.45°K.  curve c o r r e s p o n d s to the d a t a o f Wong e t a l . ( 2 5 ) .  the time i n d i c a t e d by the shaded a r e a .  to  CH^  the temperature was The smooth  4.22°K  14 16 ( HOUR ) (2.45/4.22)S  p  20  18 0.002%  02  22  A f t e r almost 6 h r ,  i s p l o t t e d f o r the d a t a taken a t  The apparatus was b e i n g r e p a i r e d  during  31  i n c r e a s e i n the a s y m p t o t i c about  6%  v a l u e of  i s , i n f a c t , observed.  Sp  a t l o n g times.  I t s h o u l d be  the o r i g i n of time f o r the data o f " F i g . 10 and curve  value  c o u l d be  cooled  i n c r e a s e d by  small systematic  NMR  emphasized, however, t h a t that represented  by  the  to 4.2°K.  By a d j u s t i n g the o r i g i n ,  s e v e r a l percent.  the  fitted  asymptotic  A l s o , t h e r e are some p o s s i b l e  e r r o r s a r i s i n g i n t h i s experiment a s s o c i a t e d w i t h  the  variation  time of the l e v e l of l i q u i d h e l i u m , s i n c e the r e f e r e n c e s i g n a l and s i g n a l are g e n e r a t e d at o p p o s i t e  \ of experiment s h o u l d be i n which such s y s t e m a t i c  In one  repeated  4.2°K. steady  The  ends of the h a l f - w a v e l i n e .  c a r e f u l l y with  e r r o r s are  Sp  Sp  r e p r o d u c i b l e , we  was  have no  when the sample was temperature.  0.05%  O2,  The Since  we  t h i s b e h a v i o r was  observed to  not time-dependence.  c o o l e d q u i c k l y from l i q u i d n i t r o g e n to l i q u i d  helium  Perhaps, i n t h i s experiment, the sample p u l l e d away from  t r a n s f e r of heat to the h e l i u m b a t h by  conduction.  In o r d e r  s h o u l d be v e r y  to e s t a b l i s h  cautious i n analyzing  of the time-dependence o f e x p e r i m e n t a l to l i q u i d h e l i u m  temperatures.  the  f o r the  the v a l i d i t y o f t h i s i n t e r p r e t a t i o n , more experiments a r e r e q u i r e d .  quickly cooled  a  to the c o n t r a c t i o n of the s o l i d methane  w a l l s of the sample tube l e a v i n g o n l y a s m a l l a r e a of c o n t a c t  implications  CH^  time r e q u i r e d to a c h i e v e  c o n c l u s i v e e x p l a n a t i o n of t h i s s t r a n g e  e x p l a n a t i o n i s c o r r e c t , one  type  13  a f t e r the sample had been c o o l e d  more than 8 h r .  t e n t a t i v e l y a s c r i b e the r e s u l t  the  eliminated.  r e s u l t s are shown i n F i g . 11.  value of  This  a sample c o n t a i n i n g  experiment on the sample doped w i t h  an anomalous time-dependence of  We  i n c r e a s e of  i s not n e c e s s a r i l y the same because of the u n c e r t a i n t y i n the time a t  which the sample was  with  An  I f the  the  q u a n t i t i e s i n samples  P-0  o  o_  o  1-00  o-  0-80  0-60 U  0 4 0  0-20 8  0  10  TIME F i g . 11.  Plot of 0.05%  Sp O2.  12  S  18  20  22  (HOUR)  i n a r b i t r a r y units versus time for a sample of The behavior of  16  14 CH^  containing  53%  13, CH^  shown i n this graph was not reproduced i n subsequent  and  experiments.  u>  33  Summary. Sp/S  We  summarize our r e s u l t s as f o l l o w s :  i n d i c a t e t h a t the e q u i l i b r i u m v a l u e of  c  Measurements o f the  <I(I + 1)>  i n pure  CH  ratio with  4  13 53%  CH  from  i s within  4  5%  of the v a l u e of  the h i g h temperature  observed and  CH  i n c r e a s e of with  4  S^  0.002%  0  i s reduced  2  (25).  at 4.2°K.  <1(I + 1)> = 3  w i t h time by about  the e q u i l i b r i u m v a l u e of temperature  v a l u e of  3.73  25%  This  i s consistent with  the  f o r samples o f pure  There i s some e x p e r i m e n t a l  <I(I + 1)>  departure  CH^  i n d i c a t i o n that  i n c r e a s e s by a few p e r c e n t as  from 4.2°K to 2.45°K, but these experiments  the a r e not  conclusive.  5.  THE  MAGNETIC PROPERTIES OF  The measurements of 0  2  0  S^  were a n a l y z e d i n the same way  the r e s u l t s a r e p r e s e n t e d  IN SOLID  o  2  and  S  c  CH.  4  f o r a sample c o n t a i n i n g  as the d a t a f o r the oxygen-free  i n T a b l e 2.  The v a l u e s of  Sp/S  l a r g e r than the c o r r e s p o n d i n g v a l u e s f o r the oxygen-free however, t h a t these r e s u l t s s h o u l d o n l y be used <1(I + 1)>  .  The  are  c  sample.  0.05%  sample,  significantly We  believe,  to deduce an upper l i m i t  reason f o r t h i s b e l i e f i s t h a t whereas  H tl^  and  to  i s the same Q  f o r t h i s sample as f o r the oxygen-free  sample, the v a l u e of  M^ H  icantly greater.  The  o r i g i n of t h i s  change i n the r a t i o  c e r t a i n l y a s s o c i a t e d w i t h the magnetic p r o p e r t i e s of the solid  lattice.  C  M2/M2 0  2  is signif-  i s almost  molecule  in a  I f t h i s i s so, i t can be shown i n the f o l l o w i n g paragraphs  t h a t we have s i g n i f i c a n t l y underestimated S i n extrapolating V^(t) v a l u e s bf t < t thus r e s u l t i n g i n an o v e r e s t i m a t e of <I(I + 1)> . m c  to  &  N e v e r t h e l e s s , as we  have p o i n t e d out i n the p r e v i o u s s e c t i o n ,  the upper •  34  S /S p' c  T  Run  M  <I(I+1)>  •  °K  H  2  10 sec~ 8  2  2  o _o 10 s e c  Dec. 8  4.22  42.3  4.18  33.4  2.02  16.5  Dec.  12  4.22  44.6  4.42  32.2  1.59  20.3  Jan.  18  4.22  45.7  4.52  33.6  1.56  21.6  Average  4.22  44.2  4.37  33.1  1.72  19.5  Jan.  18  1.78  Jan.  18  1.08  1.24  28.7  T a b l e 2.  35.7 51.0  35'.4  5.05  E x p e r i m e n t a l r e s u l t s f o r a sample of 53%  1 3  CH  4  and  0.05%  0^  CH^  containing  35  l i m i t s obtained f o r  We  <I(I + 1)>  f o r these samples are v e r y  useful.  can i n f a c t e s t i m a t e the e r r o r i n t r o d u c e d i n t o the e x t r a p o l a t e d  v a l u e of  V^(0)  f o r the oxygen doped sample u s i n g the r e s u l t s of the  developed  i n chapter I I .  by paramagnetic  In a d d i t i o n to u s i n g the t h e o r y o f NMR  i m p u r i t i e s , we  line  theory broadening  assume t h a t the c o r r e l a t i o n f u n c t i o n of  the  13 NMR  spectrum  of  0  of the r e s o n a t i n g n u c l e a r s p i n s (protons o r  molecules  2  i s the p r o d u c t  t h a t o b t a i n e d when the i n f l u e n c e of  0-> 0  molecules molecules  2  of two  c o r r e l a t i o n f u n c t i o n s , one  are absent  alone.  and  presence  being  the o t h e r b e i n g t h a t due  to  T h i s i s e q u i v a l e n t to assuming t h a t  the l i n e shape f u n c t i o n i n the presence of the o r i g i n a l  C) i n the  of  0  2  molecules  0 - f r e e l i n e shape f u n c t i o n and 2  i s the c o n v o l u t i o n  the l i n e shape f u n c t i o n due  to the i n f l u e n c e of 0 molecules alone. I f we now choose a s e t of d a t a , V- ( t ) , (where V-.- ( t ) was d e f i n e d i n page 7 ) o b t a i n e d from the 1 oxygen r & / 2  1  sample c o n t a i n i n g  0.05%  0  o b t a i n e d from the  0 ~ f r e e sample, and  2  and  a c o r r e s p o n d i n g s e t of d a t a ,  2  V.(t) /V.(t).. , 1 oxygen 1 -free'  i f we  then the f u n c t i o n  to the c o r r e l a t i o n f u n c t i o n due  to the  0  plot  F(t)  the s e t of p o i n t s so o b t a i n e d i s p r o p o r t i o n a l ^ ^  molecules  2  the e r r o r i n our e x t r a p o l a t i o n p r o c e s s i n t r o d u c e d by shape a s s o c i a t e d w i t h of d a t a  ^i^*-)  F(t)  &  O X  yg  e n  to g i v e a new  extrapolation.  0  2  i m p u r i t i e s , we  F(t)  To  correct for  the change of  f.i.d.  need to d i v i d e the v a l u e of a s e t  V.(t),. , 1 free  U n f o r t u n a t e l y , due  r e s u l t s , d i f f e r e n t p a i r s of s e t s of ' different  alone.  e x p e r i m e n t a l l y measured by the c o r r e s p o n d i n g v a l u e of s e t of d a t a  r  ^i^^free'  and  then use  V.(t),. 1 free  to the l a r g e s c a t t e r of the V,- ( t ) oxygen 1  and  V-(t), 1 free  for  experimental J  yield  w i t h w i d e l y d i f f e r e n t decay times, f o r c i n g us to a n a l y z e  the e r r o r i n the f o l l o w i n g l e s s a c c u r a t e  way.  36  We f i r s t consider the case of the proton resonance. f u n c t i o n due to 0.05% .0^  The c o r r e l a t i o n  i m p u r i t i e s alone i s , to a very good approximation,  represented by an exponential f u n c t i o n i n time with a decay constant (chapter I I ) .  At 4.2°K the value of . t ^ D  the c r y s t a l f i e l d parameter  i s not s e n s i t i v e to the value of  i n the s p i n hamiltonian of the 0  This w i l l be shown i n Chapter I I . By s e t t i n g expression f o r  t ^ (from Eq.  t^  2  spins.  D = 0, we obtain the f o l l o w i n g  [2.78.])  9/3 a [ l + 2 cosh(ggH/kT)] J  t, =  pY3g sinh(3gh/kT)  64TT  where a i s the l a t t i c e  constant of the f . c . c . l a t t i c e ,  the e x t e r n a l magnetic f i e l d ,  p  g  i s the g-factor of the 0  g  of value 2.  we obtain  ,  :  2  i s the f r a c t i o n a l  0  Ll-15 J n  H 2  i s the value of  concentration, and  molecule which i s assumed to have an i s o t r o p i c  S u b s t i t u t i n g H = 5180 gauss and p = 0.005  t ^ = 301 usee.  i n t o Eq. [1.15],  To estimate the e r r o r introduced by the e x t r a p o l a t i o n  process, we note that both sets of data ' v  V.(t).. l free  and V.(t) l oxygen  were  f i t t e d to the same expression  Ce  -M t 12  (sin bt / bt)  i n a semi-logarithmic s c a l e , whereas  V.(t) I oxygen  &  -M°t /2 - t / t  should have the form  2  Ce 1  where  C  and C^  d  ( s i n b t / bt) ,  are a r b i t r a r y constants, and M°  i s the second moment of  37  the oxygen-free f . i . d . s i g n a l . n e g l i g i b l y small when  0  Since the change of the value of b i s  molecules are introduced, our e x t r a p o l a t i o n  2  process i s equivalent to f i t t i n g the curve  f ( s ) = M°s/2 + s ^ / t , I d  [1.16]  g(s) = M s/2 + P ,  [1.17]  by the f u n c t i o n  2  o  where  s = t , M  2  i s the "measured" second moment, and P  determined by the e x t r a p o l a t i o n .  i s the y - i n t e r c e p t  For a set of points centering around  2  s  Q  = t  , i f we take the slope of f ( s ) at- s  as that of g(s) at s  Q  Q  ,  and use g(s) to c a l c u l a t e the second moment, we get an apparent increase i n the second moment given by M -M°=l/ 2  (s*t ) - l / ( t t ) d  D  d  [1.18]  If  t = 1 2 usee i s chosen near t , the increase i n "measured" second o m ^ g 2 moment i s M - M„ = 2.31 x 10 sec I f t = 28 usee i s chosen near the £ L o 0  t a i l of the f . i . d . s i g n a l , then the increase i s lowered to  M  - M° = 9.89 x 1 0 sec 7  2  2  These changes are consistent w i t h the experimental values as shown i n Tables 1 and 2.  The value of P  i s Eq. [1.17] i s given by  38  p = t\ (M° - M )  / 2  2  t  +  c  /  t  .  = ' ^  d  a  Therefore t  Q  .  the y - i n t e r c e p t of  Since i n our e x t r a p o l a t i o n process, p o i n t s i n s i d e the i n t e r v a l  12 usee _< t  _< 20 usee  corrected value of from  g ( s ) has a p o s i t i v e v a l u e w h i c h depends on  e  P  = 1.016  r e c e i v e d much more weight  Sp  to  than p o i n t s o u t s i d e , t h e  s h o u l d be i n c r e a s e d by a f a c t o r r a n g i n g i n v a l u e 1.027.  13 I n t h e case o f f u n c t i o n due t o 0 magnetic f i e l d o f  C  resonance,  molecules  2  20500  t h e decay time o f t h e c o r r e l a t i o n  a l o n e i s t j = 389  gauss.  usee  The c o r r e s p o n d i n g i n c r e a s e s i n t h e o 7  "measured" second moment a r e M when  t  - M  2  assumes' the v a l u e s o f  = 8.6 x 10  2  30 usee  and  ^  sec  The o v e r e s t i m a t e i n t h e v a l u e o f 1%  to  i n t h e "measured" v a l u e o f sample t o  4.37  ~j  _  2  2.4 x 10 s e c  respectively. 30 y s e c  t  Q  <_ 110 ys  than p o i n t s o u s i d e , t h e c o r r e c t e d v a l u e o f  s h o u l d be i n c r e a s e d by a f a c t o r r a n g i n g i n v a l u e from  t h e r e f o r e range from  and  110 usee,  Again, i n our e x t r a p o l a t i o n , p o i n t s i n s i d e the i n t e r v a l r e c e i v e d much more w e i g h t  i n an e x t e r n a l  < I ( I + 1)>  14% .  1.039  to  1.152.  i n t h e oxygen doped sample can  This i s consistent with the increase  < I ( I + 1)>  from  f o r t h e sample c o n t a i n i n g  3.73 0.05%  f o r the oxygen-free oxygen.  There i s a l s o a d d i t i o n a l e x p e r i m e n t a l e v i d e n c e t h a t o u r measurement of  S /S  experiment,  i n the oxygen-containing measurements o f  measurements o f  S  Sp  sample i s r e a l l y an o v e r e s t i m a t e .  were made a t  were made a t 4.2  c i n d e p e n d e n t o f t h e temperature  T  and  4.2, 1.78,  1.08°K.  and  The p r o d u c t  1.08°K  I n one while  S T was P to w i t h i n the experimental accuracy of  39  about  5%  which  i s c o n s i s t e n t w i t h no change i n the p o p u l a t i o n s o f the  d i f f e r e n t n u c l e a r s p i n symmetry s p e c i e s o c c u r r i n g as the temperature The v a l u e o f  S T  at  C  1.08°K  S i n c e an u n d e r e s t i m a t e o f  S  was .about  30%  lower than the v a l u e o f  0  0  ,  2  and s i n c e t h e  i n c r e a s e s w i t h d e c r e a s i n g temperature,  2  4.2°K.  o f t h e , t y p e we suggested above would be more  c  i m p o r t a n t , the l a r g e r t h e i n d u c e d paramagnetism o f i n d u c e d paramagnetism o f  i s changed.  this  change i s c o n s i s t e n t w i t h our p i c t u r e .  0^  The i n d u c e d the  evaluation of  paramagnetism a l s o causes s y s t e m a t i c e r r o r s i n  Sp ,  though  i n v o l v e d i n the e v a l u a t i o n o f be independent of 0  2  the i n c r e a s e i n  .  c  Thus the f a c t t h a t  SpT  was found t o  SpT  with decreasing  T  cancellation  by the i n f l u e n c e o f the i n d u c e d  paramagnetism on o u r measurement, s i n c e the i n d u c e d magnetic moment, and  around  1°K.  moment  ^  <  In f a c t , i-  > z  nt  n  e  o r i e n t a t i o n o f the field  3°K  0  T = 1°K 5.7k, D.  when  V  <  L > Z  <  D = 0.  1  magnetic  averaged over the  i s l e s s than about <  l  i  > z  J  > Z  5.7k,  where  k  i - approximately l i n e a r i n s  is still  i s g r e a t e r than  However, when  D  i s the Boltzmann  1/T  5  approximately l i n e a r i n  for T  complicated increase of  1/T  depending  1 / t ^ a t lower  we c o n c l u d e t h a t we cannot r u l e out a change o f  ST  down  K i l o g a u s s . At down t o  has some f i n i t e v a l u e l e s s  becomes s a t u r a t e d a t h i g h e r temperatures  Due to t h i s  a t such low temperatures  f i g u r e - a x i s v a r i e s o n l y s l i g h t l y w i t h the c r y s t a l  2  when the e x t e r n a l f i e l d  lower temperatures,  1/T  d i r e c t i o n o f the e x t e r n a l f i e l d  The v a l u e o f  about  1 / t ^ i s not l i n e a r i n  the v a l u e o f the component o f the i n d u c e d  D-parameter which  constant.  of  S  o f temperature may p o s s i b l y be due t o an a c c i d e n t a l  consequently the value of  to  these e r r o r s a r e much s m a l l e r than the e r r o r s  than  on the v a l u e  temperatures,  as g r e a t as  10%  as  AO  A.2  the temperature i s changed from  1.08 K.  to  It would be i n t e r e s t i n g to make a more thorough analysis of the influence of induced  0  2  paramagnetism on NMR second moments and s i g n a l  i n t e n s i t i e s but this would be better done using data obtained from a more suitable experimental set-up;  we require either a pulse spectrometer with  a much shorter recovery time or a c a r e f u l experiment with a steady state spectrometer.  6.  COMPARISON WITH OTHER EXPERIMENTS.  Heat capacity r e s u l t s .  Colwell, G i l l , and Morrison (8) have  measured the heat capacity of s o l i d  CH^  down to about  of t h e i r heat capacity data smoothly down to of  S  Q  0°K  2.5°K.  Extrapolation  gave a r e s i d u a l entropy  = A.93 ± 0.10 cal/mole/°K . The r e s i d u a l entropy can be expressed i n  terms of the apparent degeneracies P., P„, and P A 1 respectively,  Q^, Q^,,  and  and the p r o b a b i l i t y  f o r the molecules being i n the A, T,  and  E  species,  £J  S  where  0  = R( P  R = 1.987 cal/mole/°K  S  mix - "  logQ  A  A  + P  T  logQ  4- P  T  E  logQ ) + S . E  and the entropy of mixing  R ( P  A  l 0  § A P  +  P  T ^ T  +  P  E  l o  § E P  S . mix  )  '  m  x  ,  [1.19]  i s given by  [ 1  ' °  When the d i f f e r e n t species are populated according to t h e i r r e l a t i v e  2  ]  41  s t a t i s t i c a l weights, i . e . entropy  P^ : P  : P  :: Q  E  : Q  A  : Q  T  Q  = R log(Q  above c o n d i t i o n i s c e r t a i n l y  A  + Q  T  + Q )  then  the r e s i d u a l  .  E  [1.21]  t r u e i f the energy d i f f e r e n c e s between t h e  ground l e v e l s o f the t h r e e s p e c i e s a r e much l e s s than lowest  ,  £  i s simply  S  The  T  temperature reached  f o r the e x t r a p o l a t i o n .  kT  where  Morrison  T  i s the  e t a l . (8)  assumed t h a t no c o n v e r s i o n o c c u r r e d between the d i f f e r e n t n u c l e a r s p i n s p e c i e s d u r i n g the time o f t h e i r experiments helium  temperature range.  I f t h i s i s t r u e then  They then took Nagamiya's v a l u e s o f crystalline field determination, 0 = 2 E  value of  S  Q  Q , Q ,  symmetries and found  only a t r i g o n a l f i e l d ,  giving .  S  O  =5.06  (many hours) i n t h e l i q u i d P^ : P^ : P^ = 5 : 9 : 2 .  and  taking place.  corresponding  cal/°K/mole,  The a n a l y s i s o f M o r r i s o n  to  Q  of t h e i r  = 5,  Q  = 6,  and  gave an adequate f i t t o the e x p e r i m e n t a l e t a l . (8) b r i n g s o u t a s u b t l e p o i n t , associated with  nuclear  any change i n the r e l a t i v e p o p u l a t i o n o f the n u c l e a r s p i n s p e c i e s I f the r e s i d u a l entropy was a s s o c i a t e d w i t h  spin disorder i n To understand  for different  t h a t w i t h i n the a c c u r a c y  t h a t i t i s p o s s i b l e to remove some o f the entropy s p i n without  Q  CH^ ,  how t h i s  we note t h a t the h i g h  then we would have  decrease  in  S  Q  S  0  = R  complete n u c l e a r  l o g 16 = 5.51 cal/mole/°K.  can take p l a c e w i t h o u t  conversion,  temperature r e l a t i v e s t a t i s t i c a l w e i g h t s o f  5 : 9 : 2  are a s s o c i a t e d w i t h s p i n s t a t i s t i c a l weights o n l y , i . e . t h e A s p i n s p e c i e s has  one s p i n q u i n t e t  (I = 1 ) ,  and the  E  (1=2),  the  T  s p i n s p e c i e s has  s p i n s p e c i e s has two s i n g l e t s  3  spin  (1=0).  the ground f r e e r o t a t i o n a l s t a t e s o f d i f f e r e n t s p e c i e s s l i g h t l y  triplets  Now,  consider  perturbed  42  by a t e t r a h e d r a l c r y s t a l l i n e f i e l d . q u i n t e t because the  T  1=2.  to  t h e r e i s o n l y one  degeneracy o f the five-fold  M  ground  J = 0  s t a t e remains a  The n i n e - f o l d degeneracy of the  species i s also unaffected.  t h i s case corresponds i.e.  The  3  s t a t e of  However, the n i n e - f o l d degeneracy i n  values of  spin t r i p l e t ,  J = 2  J = 1  M  combined w i t h  3  values of  so t o speak, i n t h i s c a s e .  s t a t e i s reduced  degeneracy b e i n g s p l i t  The  M ,  ten-fold  to a t w o - f o l d s p i n degeneracy,  i n t o a d o u b l e t and a t r i p l e t .  the  I f one  «J  now  f o l l o w s Nagamiya's argument  (10) to assume t h a t the number of ground  r o t a t i o n a l l e v e l s i s e q u a l to the o r d e r of the symmetry group of the or  of the c r y s t a l l i n e f i e l d , whichever i s the l a r g e r ,  molecule  then the number of  ground l e v e l s i s 12, which i s the o r d e r of the t e t r a h e d r a l group of the molecule,  and  from the  J = 2  field  the  E  s p i n symmetry s p e c i e s has  rotational states.  a f f e c t s o n l y the  M  T h e r e f o r e the i m p o s i t i o n of a  degeneracy of the  t h r e e - f o l d and a s i x - f o l d degenerate o c c u p i e d , we lowered  see from Eqs.  below  Our  R l o g 16  experiments  state.  [1.19] and  even i f no  a d o u b l e t of ground  T  t h i s would g i v e  <I(I + 1)> = 3.73 s t i l l be used now  ± 0.18  c o n v e r s i o n takes p l a c e .  <I(I + 1)> = 3.00  to i n t e r p r e t  The  t h a t no s p i n  i n disagreement  conversion  the v a l u e s of  <I(I + 1)> The  and  w i t h our v a l u e of  dependence of the v a l u e s of  <I(I + 1)>  S  i t must  i n f l u e n c e of the  crystal  ,  Q  ,  can  but  on the l o w - l y i n g s t a t e s must a l s o be s i g n i f i c a n t .  temperature  electric  framework used by M o r r i s o n e t a l . (8)  be assumed t h a t c o n v e r s i o n can take p l a c e .  field  to a  [1.20] t h a t the r e s i d u a l entropy i s  r u l e out the h y p o t h e s i s  .  trigonal  of t h e s e s t a t e s i s  takes p l a c e s i n c e , r e g a r d l e s s o f the symmetry of the c r y s t a l l i n e field,  levels  species giving r i s e  I f o n l y one  CH^  F i g . 12 shows the entropy  S,  and  B / K T Fig.  12.  Plot  of  calculated  <I(I  +  1)>  ,  S,  and  C  versus  B/kT  for a  spherical  free-rotor.  44  s p e c i f i c heat  C  a s s o c i a t e d w i t h the r o t a t i o n a l degree of freedom f o r a  v  l e v e l arrangement r e s e m b l i n g  t h a t of a f r e e - r o t o r w i t h r o t a t i o n a l  constant  13 B.  Although  our sample c o n t a i n s  53%  CH^  ,  i t has been demonstrated by  McDowell (26) t h a t the i s o t o p i c s h i f t of the v a l u e of we  can take  of  B = 5.24  <I(I + 1)>  cm  =5.81  ,  which i s e q u i v a l e n t t o  i n disagreement  There i s a p o s s i b i l i t y  B  is negligible,  7.54°K,  giving a value  w i t h the e x p e r i m e n t a l of  t h a t the e f f e c t of the c r y s t a l l i n e f i e l d  l y i n g l e v e l s i s to l e a v e the l e v e l arrangement unchanged, b u t v a l u e of  B  only.  Then a v a l u e of  However, t h i s w i l l is  B = 1.8°K  also give a value of  of the l o w - l y i n g s t a t e s of the  s p e c i e s p e r t u r b e d by a t r i g o n a l f i e l d field for  e  i  =  •  e 2  2  ^  v a l u e s of  e  13 a r e g i v e n i n T a b l e The  £  T  <<  kT,  values of  e^,  (c)  <I(I + 1)>  and  (a)  e , 2  e  3  >>  E j <<  kT,  to  f i t the assumption  E  >>  3  kT,  two  cases  >>kT,  (d)  e  »  e  the  3.73.  2.45°K  levels spin  which  and  symmetry  For a t e t r a h e d r a l S  Q  corresponding  << kT,  3  can t h e r e f o r e be e l i m i n a t e d .  kT,  where  <I(I + 1)>  T = 4.2°K.  dependence of kT,  =  e  = 0;  (b)  o b v i o u s l y do not f i t the  p r o v i d i n g t h a t the r e s t r i c t i o n  :  E  levels  2  of a t e t r a h e d r a l f i e l d ,  c o n s i d e r the temperature  low-  the r e s i d u a l e n t r o p y  e , £ ,  which f i t s b e s t the e x p e r i m e n t a l v a l u e s o f to  on the  to  3.  e^, and  at  energy  and  cases of the energy  three configurations :  and  The  A, T,  ±0.18.  to modify  are shown i n F i g . 13.  <I(I + 1)>  a l l the d i f f e r e n t l i m i t i n g  Fig.  value.  3.73  < I ( I + 1)>  <I(I + 1)> = 4.68  too h i g h as compared w i t h the observed  degeneracies  gives  and  £  1  experimental  The c o n f i g u r a t i o n and  i n which case <<  e = e  »  3  S  corresponds  q  However, the d a t a a l s o seem  kT  <I(I + 1)> and  e^, £  2  was ,  S,  = 7.04k  E  i  =  e  lifted. and  C  2  '  with  L e t us v  now  f o r these  f o r the case o f  kT,  45  T  e,  Energy  Fig.  13.  —  —  —  Degeneracy  -  T -  Spin Species  Energy l e v e l s and d e g e n e r a c i e s of the l o w - l y i n g s t a t e s o f the A, • T, and E s p i n symmetry s p e c i e s of CH^ p e r t u r b e d by a t r i g o n a l ; ' crystalline electric field.  46  <I(I +  Case  e  l  e  3  £  2  1  £  £  , e ,  e  2  »  «  3  kT;  e  e  2  S (cal/°K/mole) 0  3.00  5.51  << kT  3.42  5.24  kT  1 (  1)>  , e  3  »  kT;  e _ << kT  3.82  4.76  , e  3  »  kT;  e  << kT  4.50  4.31  1 '  e  2>  £  T a b l e 3.  3  >  >  ]  k  2  T  V a l u e s of different  3.73  <I(I + 1)>  ±  0.18  and the r e s i d u a l  l i m i t i n g cases of the energy  4.93  ±  0.10  entropy levels  S  Q  for  o f F i g . 13.  47  the t r i g o n a l f i e l d ; of  and  (e)  the t e t r a h e d r a l f i e l d .  e  The  3  >>  and  15,  3.73  increases very r a p i d l y to  1°K,  down to about  to the v a l u e of  3.8  1°K  a t 4.2°K  < I ( I + 1)>  A  ± 0.3 0.1%  4.7  < I ( I + 1)> 4.0  quoted  results v a l u e and  On  conclusions.  The  and decreases  goes  the o t h e r hand, < I ( I + 1)>  the ground  CH^  at  1.06°K  1=2.  levels.  a v a l u e of 1.06°K  the  CH^  Although  this  and molecules result  4.2°K  t h e r e are doubts about  h i g h e r than t h a t expected  e x p l a i n e d away t h e o r e t i c a l l y .  doubts a r i s e from t h a t u s i n g the same  f o r f r e e p r o t o n s , and  4.7  at  4.2°K  and  1.1°K  Most r e c e n t l y , H. G l a t t l i  ,  HD  t h i s anomaly cannot  ( p r i v a t e communication) o b t a i n e d p r e l i m i n a r y v a l u e s o f and  and their  method of measurement they o b t a i n e d a v a l u e f o r the p o l a r i s a t i o n of 50%  very  c o n s i d e r e d , t h e r e a r e numerous  than ours which i s a t  cannot be compared d i r e c t l y ,  1)>  increases only  (24).  suggesting that at  much lower  results  unlikely,  as the temperature  schemes to arrange  s p i n symmetry s p e c i e s w i t h  The  i s very  as the temperature  from t h e i r measurement of O2,  case  <I(I +  a t 2°K  f i t s the e x p e r i m e n t a l v a l u e s of  i s o b t a i n e d a t a temperature the two  4.2°K.  = 4.08  as demonstrated by J.E. P i o t t  Kilogauss with to the  at  (27), R u n o l f s s o n , Mango and B o r g h i n i quoted  <I(I + 1)> = 5.88  belong  f o r the  The b e h a v i o r of  In a d d i t i o n to the two p o s s i b i l i t i e s j u s t  a r e c e n t paper  = 1.33k  <I(I + 1)>  to about  o t h e r p o s s i b l e , though l e s s l i k e l y ,  25  =  of a t e t r a h e d r a l f i e l d  the v a l u e of  the case of t r i g o n a l f i e l d  In  to  1  i n each case i s so chosen t h a t  whereas the e x p e r i m e n t a l v a l u e of  s l o w l y from about  well.  e  respectively.  a l o n e i n d i c a t e s t h a t the p o s s i b i l i t y because f o r such a f i e l d  e = e  v a l u e of  g i v e s the e x p e r i m e n t a l v a l u e of are shown i n F i g s . 14,  kT,  at Baclay, <I(I + 1)>  r e s p e c t i v e l y , w i t h a nominal  be  France, =  4.1  e r r o r of  (  T Fig.  14.  P l o t of c a l c u l a t e d values of tetrahedral crystal field.  <I(I  +  1)>  ,  S,  and  °K C  v  ) versus  T  for  CH^  in  a  o 2 5.2  MOLE ;  •—s  o  ^  LU  /  3-8 0-2  4-8  ( CAL  _J < u w  " < 1(1+1 )>  ;  1  LO  >  U  S  _ ^ ^ [  3-7 7  0.1 NX  4-4  3.6 0  . •  J  0  Fig.  15.  Plot in  of  the  calculated  a trigonal  crystal  iI  2  values field.  of  1  I  •  T  <I(I +  1)>  ( ,  4 °K  S,  I  and  I  6  ) C  v  versus  T  for  CH^  4 N  VD  50  5% .  The v a l u e a t  4.2°K  i s c o n s i s t e n t w i t h our v a l u e o f  while  the v a l u e a t  1.1°K  disagrees with  the r e s u l t s o f G l a t t l i crystal field of  over  J.E. P i o t t .  that of Runolfsson  f a v o u r a l e v e l scheme c o r r e s p o n d i n g  0.18,  et a l .  However,  to a t e t r a h e d r a l  t h a t of a t r i g o n a l f i e l d which i s i n d i c a t e d by the r e s u l t s  In o r d e r to deduce unambiguously the c o r r e c t l e v e l  a more p r e c i s e measurement or  3.73 ±  of the temperature dependence o f  a t low temperatures,  e s p e c i a l l y below  Infra-red studies. n u c l e a r s p i n symmetry  1°K,  scheme,  <I(I + 1)>,  i s needed.  The most d e f i n i t i v e study o f c o n v e r s i o n  species of  CH^  S,  between  has been done by F r a y e r and Ewing ( 2 8 ) .  A s i m i l a r study has a l s o been c a r r i e d out on s e v e r a l o t h e r molecules  (29).  These experiments were not c a r r i e d out i n s o l i d methane, b u t r a t h e r on s m a l l c o n c e n t r a t i o n s of  CH^  i n s o l i d Argon.  Under these  c o n d i t i o n s i t proved  to be p o s s i b l e to r e l a t e d i f f e r e n t f e a t u r e s o f the v i b r a t i o n - r o t a t i o n a b s o r p t i o n s p e c t r a to d i f f e r e n t n u c l e a r s p i n symmetry  states of  CH^.  This  enabled  F r a y e r and Ewing to measure the dependence o f the p o p u l a t i o n s o f  the  and  A  T  s p i n s p e c i e s on time a f t e r the sample was  In the case o f the Argon l a t t i c e , b e l i e v i n g t h a t the field  CH^  of c u b i c symmetry,  the energy l e v e l s of by K i n g and H o r n i g  CH^  molecules  4.2°K.  t h e r e i s a good t h e o r e c t i c a l b a s i s f o r a r e s u b j e c t e d to a c r y s t a l l i n e  and F r a y e r and Ewing u t i l i z e d i n a c u b i c f i e l d by H.F.  (30) to i n t e r p r e t  their results.  f o r spontaneous c o n v e r s i o n of  CH^  electric  some c a l c u l a t i o n s o f  King  (unpublished)  Some a s p e c t s  i n f r a - r e d s t u d i e s r e l a t e d to our r e s u l t s a r e as f o l l o w s constant  cooled to  between the  :  (1) A  and  o f the  The time  and  T  species  due to i n t r a - m o l e c u l e d i p o l a r and s p i n r o t a t i o n i n t e r a c t i o n s (31 , 32) was found  to be 90 min.  T h i s i s v e r y s i m i l a r to the b e h a v i o r we  found  in solid  51  CH^, two  as g i v e n  i n F i g . 8, b u t t h e agreement i s p r o b a b l y f o r t u i t o u s as the  s t u d i e s were made on d i f f e r e n t s o l i d s and t h e d i f f e r e n c e i n energy  between t h e A two  cases.  (3)  and  (2)  ground s t a t e s i s a p p a r e n t l y  The i n f l u e n c e o f  The e q u i l i b r i u m c o n c e n t r a t i o n  found t o be about CH^  according  octahedral constant by  T  90%,  o f the  This  The  eigenfunction where  orientation.  greater  or  between  20  and  J = 0  J = 1  and  J = 1  i s a s p h e r i c a l harmonic and  and  states  they o b s e r v e d .  J  that the  t r a n s f o r m s as  denotes t h e m o l e c u l a r  p o t e n t i a l can be w r i t t e n i n terms o f s p h e r i c a l  t h e energy l e v e l s o f the  J = 4  solid  energy s e p a r a t i o n t o  o f a s t a t e o f r o t a t i o n a l a n g u l a r momentum  An o c t a h e d r a l  was  and Ewing to u n d e r s t a n d , a t  can be e a s i l y u n d e r s t o o d i n terms o f t h e f a c t  Y^(fi)  4.2°K  times t h e r o t a t i o n a l  J = 0  Y^  .  J = 0  Therefore, and  J = 1  t h e lowest unperturbed s t a t e s a f f e c t i n g t h e the  at  than t h a t f o r  t h e l a r g e amount o f c o n v e r s i o n  harmonics, t h e lowest term i n v o l v i n g  being  spin species  10  f a c t enable F r a y e r  i n s e n s i t i v i t y o f the  the c u b i c f i e l d  s i m i l a r i n t h e two c a s e s .  I t was found t h e o r e c t i c a l l y t h a t even  do n o t change t h e s p a c i n g  least semi-quantitatively,  order,  A  c r y s t a l l i n e f i e l d s as l a r g e as B  influences  i s very  which i s c o n s i d e r a b l y  to o u r r e s u l t s .  a l a r g e amount.  Y ^(fi),  0^  q u i t e d i f f e r e n t f o r the  J = 3  states, respectively.  the f i e l d  s t a t e s i n second  J = 0  and  should  be p r e s e n t ,  or t r i g o n a l f i e l d  a r i s i n g from t h e o r d e r i n g  J = 1  states  I t would be i n t e r e s t i n g  to i n v e s t i g a t e t h e c o r r e s p o n d i n g t h e o r e t i c a l problem i n s o l i d tetrahedral f i e l d  only  CH^  where a  and where we might imagine a l s o an a x i a l o f t h e o r i e n t a t i o n s o f t h e CH^  m o l e c u l e s a s s o c i a t e d w i t h the low-temperature phase t r a n s i t i o n .  I t would  appear t h a t i n t h i s case too t h e e f f e c t o f t h e c r y s t a l f i e l d would o n l y  appear  52  in  second o r d e r .  The  t e t r a h e d r a l molecule  lowest permanent e l e c t r i c m u l t i p l e moment o f a i s an o c t o p o l e moment.  as the t e t r a h e d r a l , e l e c t r i c in  f i e l d would m a n i f e s t  terms o f a s p h e r i c a l harmonic of o r d e r  T h i s means t h a t the J = 3  J = 0  and  J = 2  and  1  the  to  examine t h e o r e t i c a l l y whether the f a c t J = 1  limiting  case  kT,  vicinity  of  e  >>  2  e^^ <<  for  t h a t both  kT,  We  T-^  order  interesting  with  the  which f i t s the data so w e l l i n the  The most comprehensive NMR de Wit  (13, 14,  i t s deuterated m o d i f i c a t i o n s .  s h a l l comment h e r e  I t was  found  15) who  T-^  CH^  minimum o c c u r r e d near ^ i ^  m  w h i l e the v a l u e of  i  n  6°K  d i r e c t l y r e l a t e d to  ~ -^0 1.2°K  msec was  versus  temperature p l o t s f o r  very shallow.  The  10-20%  t h e o r y o f s p i n r e l a x a t i o n p r e d i c t s a v a l u e of  (T^)  higher. .  a  the  CH^  and  characteristic  For  v a l u e of  f o r a resonance frequency o n l y about  area  .  temperature s o l i d phase.  and was  1.2°K  reorientation in solid  a l l of the d e u t e r a t e d m o d i f i c a t i o n s of methane went through minimum i n the r e g i o n of the low  of  s t u d i e d the  They r e p r e s e n t a f r u i t f u l  on a few matters  t h a t the  study  as a f u n c t i o n of temperature down to  problem of s p i n c o n v e r s i o n i n s o l i d  minimum was  orientation.  the t r i g o n a l and t e t r a h e d r a l  t e s t i n g of d i f f e r e n t m o l e c u l a r models o f m o l e c u l a r  methane.  order  4.2°K.  s p i n - l a t t i c e r e l a x a t i o n time and  I t would be  s t a t e i n second o r d e r i s compatible  s o l i d methane has been c a r r i e d out by G.  CH^  i n the lowest  i n the m o l e c u l a r  states, respectively.  N u c l e a r magnetic resonance.  for  Y„ 3m  itself  s t a t e s are i n f l u e n c e d i n lowest  by  terms i n f l u e n c e the  T h e r e f o r e , the t r i g o n a l , as w e l l  of The  CH^ T^ 28.5  the at  T^ the  MHz,  conventional  = 8 msec (15) .  The  53  f a c t that  (T,) . 1 mm  i s a c t u a l l y l a r g e r than t h i s can be u n d e r s t o o d i n terms J  of the f a c t t h a t o n l y  the  T  G  s p i n symmetry s p e c i e s  m o l e c u l a r d i p o l a r i n t e r a c t i o n (32). belonging  t o the  T  can r e l a x by i n t r a -  I f the r e l a x a t i o n r a t e o f the s p i n s  s p i n symmetry s p e c i e s  is  1 / (T^)^ ,  then the  r e l a x a t i o n r a t e o f the e n t i r e system i s (15)  c  1 T  where energies  and  C^,  o f the  A  l  C  A  +  C  T  VP  1 -(—)  T T  l  T  = 6 P  A  +  and  T  spin species,  little  change i n  4.2°K  to  l  T  Tl 221  '  P /P. T  respectively;  The e x p e r i m e n t a l f a c t t h a t  as the temperature i s reduced from  from  T T  little  <I(I + 1)>  P  a r e the heat c a p a c i t i e s a s s o c i a t e d w i t h the Zeeman  been d e f i n e d p r e v i o u s l y .  that  2  1 (—)  T  takes p l a c e .  G  to  T^  1.2°K  P  and  P  increases  have very  i n d i c a t e s that  T h i s i s c o n s i s t e n t w i t h our  observations  does not change v e r y much as the temperature i s v a r i e d 1.08°K .  very  54  CHAPTER 2  NUCLEAR MAGNETIC  RESONANCE  PARAMAGNETIC  1.  L I N E SHAPES BROADENED BY  I M P U R I T I E S I N SOLIDS  INTRODUCTION  In  C h a p t e r 1 we  reported  experiments  nuclear magnetic  susceptibility  of solid  sample  CH^  with a small  of solid  was  doped  performed  CH^.  t o measure t h e  I t was n o t i c e d  concentration  t h a t when t h e  of  0  molecules,  2  13 which  a r e p a r a m a g n e t i c , t h e s e c o n d moment  resonance  increased  significantly,  r e s o n a n c e , b e i n g more t h a n within to  experimental errors.  arise  from  responsible moment  of the  individual  0  problem  exact solution  light  remained  magnetic rf  the microscopic  i s of great  to the general problem  i t a p p r o x i m a t e l y have been conditions  shape  of the  on t h e n a t u r e and magnitude  problem.  introduced  Basically  there  unchanged  l i n e was  field.  interactions  among a  i n resonance  studies. Although  v a r i o u s ways t o  f o r a given  useful  system  observed spectrum o f  lacking,  so t h a t  thought  and t h e magnetic  of these i n t e r a c t i o n s .  is still  C  moment, w h i c h i s  energy,  importance  and e x p e r i m e n t a l p r e c i s i o n ,  one o f t h e s e ways.  line  the nuclear  absorption  of relating  uncoupled p a r t i c l e s  an  in  i n value,  to the broadening of the macroscopically  sheds  physical  larger  of  of the proton  molecules i n d u c e d by t h e e x t e r n a l magnetic  2  It  solve  times  spectrum  The b r o a d e n i n g o f t h e r e s o n a n c e  f o r the resonance  particles  w h i l e t h e s e c o n d moment  the i n t e r a c t i o n between  The of  20  of the absorption  results  range o f can be  a r e t h r e e methods o f a p p r o a c h  derived to this  55  The and  most  w e l l - k n o w n method  elegant.  expansion  For  knowing  shapes,  a Lorentzian line moments v a n i s h since  behavior be  equivalent the  However,  the Maclaurin  there  from  of the l i n e  a finite  series  shape near  to the s i x t h  If  the q u a l i t a t i v e  of  the f i r s t  f e w moments i s v e r y  function.  o n e t o know t h e l i n e  a cusp  i n this  long  at the o r i g i n ;  times,  this  i t s odd  diverge.  Also,  by t h e  information  cannot  are mathematically  a t zero  i s already very shape  approach.  F o r example,  t h e z e r o t h moment,  of the transform  feature of the l i n e  shape  the centre i s r e f l e c t e d  at very  moment  of the Maclaurin  may n o t c o n v e r g e .  number o f moments, w h i c h  to the d e r i v a t i v e s  calculation  enable  i s the simplest  a r e two d r a w b a c k s  a n d a l l e v e n moments, e x c e p t  of i t s Fourier transform  obtained  coefficients  has a F o u r i e r transform w i t h  the behavior  (33)  of the absorption l i n e  a l l t h e moments s h o u l d  function completely.  some l i n e  expansion  T h e moments a r e e s s e n t i a l l y  of the F o u r i e r transform  In p r i n c i p l e , shape  o f t h e moment  time.  I n most  cases,  lengthy  a n d cumbersome.  i s known, t h e n  the c a l c u l a t i o n  valuable to f i x the l i n e  shape f u n c t i o n  quantitatively.  The function a  broadening  of the  fluctuating  (34, 35)  operator  aims  at calculating  S ( t ) by t r e a t i n g x  p e r t u r b a t i o n to the resonance  i s observed.  effects  I t has wide  applications,  as "exchange and m o t i o n a l  f u n c t i o n to concrete  cases  to i n t r o d u c e assumptions  Although  simplify  the c a l c u l a t i o n s .  stage  i s generally too complicated.  Very  the nature  o f t e n these  a  for investigating  the derivation of specializing One i s t h e n  o f t h e random  assumptions  as  As a r e s u l t ,  particularly  narrowings".  concerning  the a u t o c o r r e l a t i o n  the interactions  line.  the a u t o c o r r e l a t i o n f u n c t i o n i s rigorous, the f i n a l  forced to  theory  of the t r a n s i t i o n  randomly  such  stochastic  process  are s t a t i s t i c a l  56  in  nature  study  and  c a n n o t be  The  statistical  pressure  have  recently  (39).  This  broadened due  to  been  i s the  line  given  density  functions,  expressed  in a  involved.  By  be  The  be  the  closed  This  should  can  (38,  approximation  so  a  shape  derived  the  works  the  in  that  transform  the  of  the  t h e n be  numerical the  of  the  broadening  function  evaluation  of  the  can  be  continuum lattice  assumptions  are  c a l c u l a t i o n of  the  sums valid,  d i l u t e medium.  due  to  the  inhomogeneous b r o a d e n i n g by  the  line  function  for  following  in detail S =  1/2  shape  section.  i n which  section,  discussed.  shape a  two  Strandberg  many p a r t i c l e p r o b a b i l i t y  by  on  spin  to  evaluated  above  and  method  single particle probability  line  assumption  a system,  concluding is  and  of  this  to  inhomogeneously  additive  a product  on  Grant  contributions  the  the  by  the  problem,  i s discussed are  calculate  statistical  expressions  impurities  to  the  The  dipoles,  earlier  the  Margenau  reviews  and  using  of  by  that  I t can  function  formulation  to  (38)  initiated  Extensive  p r e s e n t work i s c o n c e r n e d w i t h  in a solid  expressions  was  gases.  factorized into  o r by  37)  approach  assuming  form.  for  impurities  be  of  method works whenever  line  general  (36,  Stoneham  Fourier  39)  absorption  In  by  i n d i v i d u a l p a r t i c l e s are function  will  spectra  most p o w e r f u l  shapes.  rigorously.  approach  e f f e c t s on  density  which  justified  the  the  The  3.  S =  1  factorisability  The are  r e l a t i o n s h i p between  and  are  the is  i t s Fourier  classical  cases  relaxed. transform  these  magnetic  i n which  considered this  paramagnetic  However, i n  s p e c i a l i z a t i o n of  impurities  i n Section and  approach.  NMR  the  i n Section  c a l c u l a t i o n and  4.  other  57  2.  DERIVATION OF THE L I N E SHAPE  We  shall  make t h r e e  FUNCTION  assumptions  t h a t w i l l be t h e b a s i s  of our  calculations.  1.  The f r e q u e n c y frequency  shifts  shift  of a given  produced by each This  2.  to assuming  that  be  neglected.  If  q  i s a s e t of v a r i a b l e s which,  are  necessary  f o r the impurity of  The c o n c e n t r a t i o n  The  "r  q  of the second  with  impurities.  the impurities  the p o s i t i o n vector at  Y,  Y,  then the  s e t of values  of a l l other  of  q  impurities.  a s s u m p t i o n w i l l become o b v i o u s  In general,  of sufficiently  e x t e r n a l magnetic  hold  t o a good  "resonating  together  the f i r s t  the impurities usually interact  high  p  i n t e r a c t i o n s between  t o assume a p a r t i c u l a r  are specified.  However, i n t h e c a s e  shall  of a l l other  shifts  of impurities i s small.  c o r r e c t because  concentration  i n the absence  and o f t h e p r e s e n c e  validity  when v a r i a b l e s  We  i s t h e sum o f f r e q u e n c y  t o s p e c i f y t h e s t a t e o f an i m p u r i t y  independent  not  centre  impurity  can  is  the  resonant  individual  i s equivalent  probability  3.  p r o d u c e d by t h e i m p u r i t i e s a r e a d d i t i v e , i . e . t h e  field  low i m p u r i t y  ( a few K i l o g a u s s )  later  assumption i s  among  themselves.  concentrations,  the assumption  and  should  approximation.  now  derive  the l i n e  which i s expressed centre"  whose  shape  i n molar  resonance  f u n c t i o n due t o a fractions.  given  The p o s i t i o n o f  i s to be observed  i s chosen  to be  58  the o r i g i n o f the c o o r d i n a t e among  N  system.  available lattice sites.  The Let  D  n  impurities  are d i s t r i b u t e d  denote a c o n f i g u r a t i o n o f such  distributions,i . e .  D = (r , q; 1  where  r_^, q^  represent  is  • ..;  r , q^} n  ,  r e s p e c t i v e l y the p o s i t i o n v e c t o r  s t a t e o f the i t h i m p u r i t y . shift  ±  From the f i r s t  o f the r e s o n a n t c e n t r e  [2.1]  and the i n t e r n a l  assumption, t h e a n g u l a r f r e q u e n c y  due t o the i m p u r i t i e s h a v i n g c o n f i g u r a t i o n  D  g i v e n by  u  where  co(r\,q^.)  D  =  n I u ( r . ,q)  '  i s the a n g u l a r frequency s h i f t  •when a l l o t h e r i m p u r i t i e s  a r e absent.  p r o b a b i l i t y f o r the c o n f i g u r a t i o n  D  [2.2]  caused by the j t h i m p u r i t y  From the second assumption, t h e can be f a c t o r i s e d i n t o the f o l l o w i n g  factors : n P(D) = P(d) n p(q.) ,  where  P(d)  i s the p r o b a b i l i t y f o r the i m p u r i t i e s  [2.3]  to be found i n t h e  configuration  d = {r. , • • •, r } 1 n  and  p(q.)  i s the p r o b a b i l i t y f o r the j t h i m p u r i t y  to be i n s t a t e  q..  The  59  two p r o b a b i l i t y d e n s i t y  functions  a r e so d e f i n e d  t h a t they a r e s e p a r a t e l y  n o r m a l i s e d to u n i t y ,  Ip(q)  = 1 , '  [2.4a]  q  and  £p(d) = 1 . d  [2.4b]  From these n o r m a l i s a t i o n  conditions, i t follows  I  )>(D) = D  I P ( d ) II p ( q )  l ' " " '  q  that  q  J  d n  =  1  n  I q  =  r  n  n  - " , q  (I  n  p(q  i=lq.  P(q.)  j = l  ) J  [2.4c]  I f we by  I (co), 0  I(co) ,  denote the l i n e shape f u n c t i o n i n the absence o f i m p u r i t i e s  and the l i n e shape f u n c t i o n i n the p r e s e n c e o f i m p u r i t i e s by  then  1(a)) = £P(D) I (to - u> ) 0  Here we have assumed t h a t an i m p u r i t y resonance l i n e a l o n g  has o n l y  D  .  the e f f e c t  [2.5]  of s h i f t i n g  the cu-axis, w i t h o u t changing i t s shape.  a  60  U s i n g the two f o l l o w i n g  identities  I (to - co ) = Q  I  D  J  and  Eq.  6(U  "  V  =  —oo  O  (to -  e  "27  y)5(u -  to  Ll  )dy  i(u-co ) t dt  [2.5] can be r e w r i t t e n i n t o the f o l l o w i n g form  Kco) =  dt I (to -  dy  2TT  y)e  l y t  [2.6]  F(t)  -ito t  F ( t ) = J>(D)e D  where  I f we now  D  denote  T,/*.\J4-  itOt  e  [2.7]  F(t)dt ,  then we get  Kco) =  Eq.  I (to Q  " unperturbed "  I^(to).  Since  l i n e shape f u n c t i o n  the f u n c t i o n  Ij(to)  contains  the b r o a d e n i n g by the i m p u r i t i e s , and when  Q  [2.8] shows t h a t the t o t a l l i n e shape f u n c t i o n  of the  I ( ) w  0  " unperturbed  i  s  "  [2.8]  y)I (y)dy  I (to)  l (w) , Q  i s the c o n v o l u t i o n  and the f u n c t i o n  a l l the i n f o r m a t i o n  I^(to)  concerning  becomes i d e n t i c a l w i t h  I (to)  i n f i n i t e l y s h a r p , we need no l o n g e r be concerned w i t h the l i n e shape f u n c t i o n  I (to). Q  Henceforward, we s h a l l  thus  61  assume  I (w) 0  to be i n f i n i t e l y  sharp.  We s h a l l c o r r e s p o n d i n g l y  the l i n e shape f u n c t i o n and denote i t s i m p l y  1(0))  e  = 2TT  i W t  -im and  F ( t ) = £P(D)e D  Eq.  by  o f the l i n e shape f u n c t i o n  expression  f o r F ( t ) . We f i r s t  I(o)), i . e .  [2.9a]  .  [2.9b]  t .  F ( t ) , which i s the f o u r i e r  I(o)) ,  of the b r o a d e n i n g by the i m p u r i t i e s .  I^(w)  F(t)dt  [2.9] shows t h a t the c o r r e l a t i o n f u n c t i o n  transform  call  contains  a l l the information  Our t a s k now i s t o f i n d an e x p l i c i t  denote the average  £p(q)e-  i u ( Y  '  q ) t  = < e  -  ^  t  >  e  .  f  (  ?  j  t  )  .  [ 2  .  1 0  ]  q  The  correlation function  F ( t ) can be r e w r i t t e n i n a more c o n v e n i e n t  n F(t) = )>(d) n f ( r \ , t ) . d j=l  form  [2.11]  2  Eq.  [2.11] shows t h a t the c o r r e l a t i o n f u n c t i o n  average o f a p r o d u c t o f of  N  functions,  j,  the f u n c t i o n  n  f(f\,t) f(r\,t)  f u n c t i o n s , each o f which i s chosen from the s e t ,  j = 1,  i s chosen.  occupy the same s i t e and i n t e r c h a n g i n g the d  i d e n t i c a l p r o d u c t o f the  is  F ( t ) i s mathematically the  n  N. Since  I f an i m p u r i t y  no more than one i m p u r i t y can  impurities i n sites  functions,  i s found a t s i t e  i  and  j  gives  the t o t a l number o f c o n f i g u r a t i o n s  62  /N\  =  _  We now  NI  n !(N - n)!  In'  assume t h a t the d i s t r i b u t i o n i s random such t h a t a l l c o n f i g u r a t i o n s  are e q u a l l y p r o b a b l e .  Then we  have  P(d) =  and Eq.  0 ' 1  [2.11] becomes  F ( t ) =  7NT U  To e v a l u a t e  the summation  by Darwin and Fowler  Acr^t)..--  E n  i+--n.=0,l l + n  i n the above e x p r e s s i o n ,  (40).  define a generating  the summation  by the method o f s t e e p e s t  introduced into  descent.  an We  function  N _ n [1 + z f ( r . , t ) ] j=l  <j>(z)  we use a method  The i d e a i s to t r a n s f o r m  cb(z) =  If  [2.12]  N  i n t e g r a l which can then be e v a l u a t e d first  f^Cr^t).  = n  .  [2.13]  2  i s expanded as a p o l y n o m i a l i n  z ,  N ,(z) =  then the sum i n Eq. Eq.  [2.14], i . e .  I c z 1=0  %  ,  [2.12] i s e x a c t l y e q u a l to the c o e f f i c i e n t  [2.14]  c  n  in  63  I n  f  l *"• +  + n  N  n  i  ( r , t ) . . . .  f ^ C r ^ t )  1  = n  ii.=0,1  1  The  coefficients  c  i s then calculated from  n  L  1 2TT±  ~n  Since  cj>(z)  origin.  i s an entire  Since both  can be evaluated by  n  -n-1  0 z  <j>(z)dz .  function, C  and  N  can be any closed contour around the  are very large , the i n t e g r a l i n E q .  by the method of steepest  (see for example, page 437 of Ref.  c  where the function  g(z)  n  [2.15 ]  = z-  i s the  U  1  Q  descent, and  [2.15 ]  the value i s given  41)  <J)(z )//2Tr|g"(z )| 0  logarithmic  0  ,  function of  [2.16]  z  ^<f>(z)>  n  i.e.  N  ;(z) = -(n + l ) l n z +  J l n [ l + zf (r. ,t) ] , 1=1 3  and  z  Q  i s the saddle point at which  g'OO  D i f f e r e n t i a t i n g the function  = o .  g(z)  twice, we  get  [2.17]  64  g  >  (  z  , --1L±1  )  N J  +  J  1  +  z  f(r.,t) ; (  >  t  .  )  J  = 1  . _ N f (r\,t) l L ± A . . J J z^ j = l [1 + z f C r t ) ]  [2.18a]  2  and  "  g  (  z  )  e  .  [ .l8b] 2  2  j 5  Z  The v a l u e of  q  i s then o b t a i n e d by p u t t i n g  then s o l v i n g the e q u a t i o n  n  +  g'(z ) = 0 , Q  N  1  ~z~  E  =  With the u s u a l assumption  that  n  and  sufficiently  N  a r e , we  [2.18a] and  i.e.  1 + z f(r.,t)  '  n  be regarded  and  N  p = ^  may  [ 2  '  1 9 ]  as t e n d i n g to  i s kept c o n s t a n t whatever the v a l u e s  show i n the Appendix to t h i s c h a p t e r t h a t when  N  is  large,  z  o = r^7  >  [2  n  and  i n t o Eq.  Q  f(r.,t)  i n f i n i t y w h i l e the c o n c e n t r a t i o n of  z = z  c  n  I  = n  l  +  '  f  l  n  (r-j^t)  f  N  -  20A]  Cr" ,t) N  •'%  n.=0,l  >"  pN  " \  (1 -  P  )  p  N  " \  J - J l  + j^-  f(r.,t)] :  /2TT¥  p r o v i d e d t h a t the frequency  s h i f t by i m p u r i t y at  r  is  ,  |2.20b ]  65  u(r,q)  and t h a t the v a l u e of  t  o C T  (A 1 3 ) ,  A  lies within a f i n i t e  [2.20c]  time i n t e r v a l , i . e .  -T <_ t <_ T ,  f o r some p e r - a s s i g n e d v a l u e of  [2.20d]  T.  We note here t h a t Eqs. [2.20c] and on our subsequent c a l c u l a t i o n s .  S i n c e we  [2.20d] impose no r e a l  s h a l l a p p l y our r e s u l t s  restriction  to b r o a d e n i n g  by magnetic d i p o l a r i n t e r a c t i o n s , where  . -3 u)(r,q) oC r  Eq. [2.20c] w i l l be s a t i s f i e d . very l a r g e  for  T,  say  ;  As to Eq.[2.20d], we  10"^ s e c , such t h a t  F(t)l  «  tI >  T ,  F(0) = 1  then a l l t h e i m p o r t a n t i n f o r m a t i o n c o n c e r n i n g interval  can always choose a  F ( t ) i s contained i n the  )  66  A naive, non-rigorous  and y e t c o n v e n i e n t  i n Eq. [2.20] a r e o b t a i n e d In the l i m i t when  way t o see how the r e s u l t s  from Eqs.[2.16],  N -* oo ,  [2.18b] and [2.19] i s as f o l l o w s .  the main c o n t r i b u t i o n to the sums i n Eq. [2.19]  _ is  from terms i n which  j  i s very l a r g e .  the v a l u e o f the f u n c t i o n s By p u t t i n g a l l  expressed  f(r^.,t)  f(r"^,t) = 1  and  Since  -SL  co(r,t)  (£ >_ 3 ) ,  r  i s e s s e n t i a l l y u n i t y f o r very l a r g e j .  n + 1 = pN  i n Eq. [2.19], and s o l v i n g the  e q u a t i o n , we g e t J  Substituting  o  t h i s value of  f(r\,t) = 1 ,  1 - p  z  Q  '  into  Eq. [2.18b] and a g a i n p u t t i n g a l l  we o b t a i n  j " ( z ) = N ( l - p) o 3  1  Q  Substituting Eq.  z  Q  = ^  P  and  g"(z )  i n t o Eq. [2.16], we o b t a i n  0  again  [2.20b].  Using  Sterling's  \n'  formula,  f o r large  N  and  n = pN,  N! n ! (N - n) ! N -N N e n -n. N-n, ,N-n -N+n n e N (1-p) e M  T  'p-  p N  ~ \  n  (1 - p )  p  N  N  /„ ... v /2Trn(N-n)  ~ \  we have  67  From Eqs.  [2.11],  [2.20b] and  correlation function  F(t)  i s given  F(t) =  The  correlation function  value  of i m p u r i t y  Eq.  [2.21] by  (1 - p )  N  n [1 + -r-Z— f(F,,t)] 1=1 1 " P J  n {(1 j=l  =  oo -io)("r.)t n {1 - p [ l - <e >]} 3=1  provided,  concentrations  an e x p o n e n t i a l  of  p,  we  J  for  any  can  approximate each f a c t o r i n  -iu)(r".)t -p [1 - <e > ] 2  > ] -  e  ,  [2.21] becomes oo F(t) = e  by  [2.21]  function,  -ico(r.)t  -p E [1 - <e  Here we  .  the i m p u r i t i e s are a d d i t i v e .  1 - p [1 - < e  Eq.  oo)  of c o u r s e , that- the b r o a d e n i n g  -iw(r\)t  and  (N +  2  as e x p r e s s e d i n E q . [ 2 . 2 1 ] , i s v a l i d p  the  (N -> oo )  p ) [ l + r - - f(r".,t)]} 1 ~ P j  2  F(t),  that  by  =  concentration  e f f e c t s produced by  For low  the above r e s u l t , i t r e a d i l y f o l l o w s  n o t e t h a t i f one  >]  J=l  f o l l o w s Grant and  .  Strandberg  assuming t h a t the many p a r t i c l e p r o b a b i l i t y d e n s i t y  i n t o a p r o d u c t of s i n g l e p a r t i c l e p r o b a b i l i t y d e n s i t y  [2.22]  (39)  and  Stoneham  (38)  function i s factorisable f u n c t i o n s , one  arrives  68  a t Eq.  [2.22] d i r e c t l y .  In c a l c u l a t i n g function I and  II.  F(t),  i t i s more convenient  Region I c o n t a i n s  Region I I c o n t a i n s sites  the summation i n the exponent of the  M  to d i v i d e the l a t t i c e i n t o two  sites  the remaining  t h a t are n e a r e s t  sites.  region I i s retained.  site  assumed to be  In r e g i o n I I , the f u n c t i o n  the summation i n Eq.  [2.22], we  -pM  I I  M p E J  +  =  co Z  -p and  define  i ( ) w  a  impurities i n region  a  II  (  t  )  a )  ^  ^ ^ ^ > w  r  t  t >  changes more  the l a t t i c e  can  be  the summation over  sites  Rewriting  (t) ,  F-^t) = e  F  ^-  the  have  I  where  <e  an i n t e g r a l over r e g i o n I I .  F(t) = F ( t ) F  now  <e  to a good a p p r o x i m a t i o n ,  approximated by  origin.  the summation over s i t e s i n s i d e  an i s o t r o p i c , homogeneous continuum and  i n r e g i o n I I can be  I f we  the f u n c t i o n  to a n o t h e r , and  smoothly over the s i t e s and,  to the  regions,  In r e g i o n I, the d i s t a n c e o f  to the r e s o n a t i n g c e n t r e i s s m a l l , and  changes r a p i d l y from one  correlation  =  e  J  (a = I , I I )  =  M  +  [2.23] -ia)(r.)t <e  >  ,  1  -ico(r".)t [1 - <e  3  >]  1  as the l i n e shape f u n c t i o n due  to  alone, i . e .  I  (io) = or  ±-  2TT  e  i t 0 t  F (t)dt a  (a = I , I I ) ,  [2.24]  69  then as a consequence o f the f a c t o r i s a b i l i t y o f the t o t a l c o r r e l a t i o n function  F ( t ) , the t o t a l l i n e shape f u n c t i o n can be e x p r e s s e d as the  convolution  of  I-j.(u>)  and  1^.^.(0)),  Ku)  I K U H J K U ) - y)dy .  =  [2.25]  —CO  T h i s r e s u l t f o l l o w s as a s p e c i a l case from a g e n e r a l theory  t h a t the j o i n t p r o b a b i l i t y d e n s i t y  independent random v a r i a b l e s i s o b t a i n e d probability density are  functions  the c o n t r i b u t i o n s  to  w  (42).  theorem i n p r o b a b i l i t y  f u n c t i o n o f the sum o f a number o f by c o n v o l u t i n g  a l l the i n d i v i d u a l  I n t h i s case, the independent v a r i a b l e s  due t o i m p u r i t i e s i n r e g i o n  I  and r e g i o n I I .  In another case when the l i n e shape i s broadened by s e v e r a l of n o n - i n t e r a c t i n g  i m p u r i t i e s i n such a way t h a t each s p e c i e s  can be d i s t r i b u t e d as i f a l l o t h e r t r u e when the t o t a l c o n c e n t r a t i o n present  species  a r e absent  of impurities  ( t h i s i s approximately  of i m p u r i t i e s i n c l u d i n g a l l the species  i s l o w ) , then the t o t a l l i n e shape f u n c t i o n i s simply  the convolution  of the i n d i v i d u a l l i n e shape f u n c t i o n s when a p a r t i c u l a r s p e c i e s a l o n e causes the b r o a d e n i n g  species  of i m p u r i t i e s  (38, 39).  U s i n g E q s . [2.10] and [2.22],  the c o r r e l a t i o n f u n c t i o n  F ( t ) can  a l s o be r e w r i t t e n as  F(t) = n F(q,t) q where  ,  [2.26]  70  - p p ( q ) Z [1 - e F(q,t) = e J " m  -ito(r.,q)t ] J  1  T h e r e f o r e when t h e c o n c e n t r a t i o n be  i s low, the c o r r e l a t i o n function  F(t),  f a c t o r i s e d into a product of i n d i v i d u a l c o r r e l a t i o n functions,  such that  each  impurities can  F(q,t)  F(q,t);  c o r r e s p o n d s t o a c o r r e l a t i o n f u n c t i o n when a l l t h e  a r e i n t h e same s t a t e  q.  The t o t a l  thus be e x p r e s s e d as a c o n v o l u t i o n  corresponding  can  to different states  I(q,to)  q,  = ~^  e  l u ) t  line  shape f u n c t i o n  of a l l the l i n e  I (to)  shape f u n c t i o n s  I(q,to)  where  F(q,t)dt  .  [2.27]  J _o  3.  L I N E SHAPE FUNCTION DUE TO A SYSTEM OF C L A S S I C A L MAGNETIC  We s h a l l now s p e c i a l i z e t h e e x p r e s s i o n s line  shape broadened by a system o f d i l u t e c l a s s i c a l  having y~(q) ,  a  internal states.  here that  q  i s completely general.  a quantum s t a t e o f t h e i m p u r i t y , of the impurity,  are present.  resonating  centre,  to the  magnetic d i p o l e s ,  q = 1,  each  denoted by  • • • ,o .  It i s  F o r example, i t can r e p r e s e n t  an o r i e n t a t i o n i n space o f t h e m o l e c u l a r  or a species  impurities  earlier  T h e m a g n e t i c moment o f a d i p o l e ,  i s determined u n i q u e l y by t h e s t a t e v a r i a b l e  stressed  axis  derived  DIPOLES  o f i m p u r i t i e s when s e v e r a l  The l o c a l m a g n e t i c f i e l d  which i s a t the o r i g i n ,  produced by  i s g i v e n by  species TT(q)  of at the  71  HCr.q)  = - ±r  [y(q) -  •  S i n c e we a r e r e s t r i c t i n g H(r,q) << H  where  H  [2.29]  r  3  our c a l c u l a t i o n s to the h i g h f i e l d  i s the e x t e r n a l magnetic f i e l d  l i m i t when  and t o t h e case i n  which the i m p u r i t i e s and the h o s t c e n t r e s a r e u n l i k e s p i n s , o n l y t h e component o f  H(r,q)  which i s p a r a l l e l  e f f e c t i v e i n broadening  t o the e x t e r n a l f i e l d  the resonance l i n e .  The frequency  H  is  s h i f t i s then  g i v e n by  r~ \ oi(r,q) =  Y  r—f ^ — -[u(q)-n  3(y"(q) -T) (n-7), -] • r 2  r  where  n  i s a u n i t v e c t o r a l o n g the d i r e c t i o n o f  gyromagnetic r a t i o  o f the r e s o n a t i n g c e n t r e .  H  and  ,  Y  [2.30]  r o  i s the  When the e x t e r n a l f i e l d  s u f f i c i e n t l y h i g h , which i s u s u a l l y t r u e f o r most n u c l e a r magnetic ll(q)  precesses  averaged  "jT(q)  r a p i d l y about  H  and o n l y t h e l o c a l f i e l d  i s e f f e c t i v e i n broadening  d i r e c t i o n o f the e x t e r n a l f i e l d  W  H .  (r\q) = -  ]T(q)  is  resonances,  produced by the  the NMR l i n e shape.  words we assume t h a t the magnetic d i p o l e moments  H  In other  are p a r a l l e l  to the  Eq. [2.30] then becomes  (1 - 3 c o s 0 ) , 2  [2.31]  r  where  9  i s the angle between  H  and  d i p o l e moment i s n o t e x a c t l y p a r a l l e l  r .  A case  i n w h i c h the magnetic  t o the e x t e r n a l f i e l d w i l l be d i s c u s s e d  72  i n the next s e c t i o n .  From Eqs.  [2.23[ and  F(t) =  [2.24], the c o r r e l a t i o n f u n c t i o n becomes  a n F (t) , q=l  '  q  [2.32]  where F (t) = F^(t)Fj (t),  [2.32a]  q  I  r  F?(t) = e " i  p M  P  \ co  ( q )  E £(pp(q) ? k=0 j=l  =  q. a  p(q)  The t o t a l c o r r e l a t i o n f u n c t i o n F (t),  M  e  3  [2.32b]  ,  1  F(t)  [2.32c]  i s f a c t o r i z e d i n t o a product of and each  and i t s F o u r i e r  s h a l l consider  F (t) q  i s the c o r r e l a t i o n  a r e found i n the s t a t e  only  convoluting  I (to) q  ,  t r a n s f o r m , which i s the l i n e shape f u n c t i o n  From now  by  I (co). q  successively  q = 1, • • • ,a  L e t us f i r s t II.  q.  the c o r r e l a t i o n f u n c t i o n  The t o t a l l i n e shape f u n c t i o n i s o b t a i n e d , as mentioned b e f o r e ,  region  , ) \  J  +  q = 1, • • • ,a ,  q  the r e s t o f t h i s s e c t i o n we  q  R  • -itjj(r.,q)t (1 - e )  f u n c t i o n r e s u l t i n g when a l l the magnetic d i p o l e s  F (t)  ( . >q) t  i s the p r o b a b i l i t y t h a t a magnetic d i p o l e i s found i n the s t a t e  functions,  For  0 0  K !  oo -pp(q). S F^Ct) = e J  and  ""i  M  -,  calculate  Fj-j-(t)  which i s due to i m p u r i t i e s i n  on we s h a l l always choose a s p h e r i c a l volume as r e g i o n I .  73  I t w i l l be r e f e r r e d radius.  to as the c u t - o f f r e g i o n  I f the c u t - o f f  radius  r  Q  and i t s r a d i u s  i s s u f f i c i e n t l y large,  i n the f r e q u e n c y s h i f t s produced by i m p u r i t i e s  that  as the  cut-off  the v a r i a t i o n  are neighbours t o each  o t h e r w i l l be s u f f i c i e n t l y s m a l l t h a t , t h e summation can be approximated an i n t e g r a l . lattice,  a  Denoting  v  as the volume o f a u n i t  as one o f the l a t t i c e c o n s t a n t s and  s i t e s per u n i t c e l l  the summation i n Eq. [2.32c]  cell f  by  i n the c r y s t a l  as the number o f  can be approximated  lattice by  00  - e  r  2  dr[l  [2.33]  o  Substituting  y = cos 8 ,  i n t o Eq. [2.33] , we  x =  (r /r) D  3  ,  a(q) = Y y ( q ) / r  0  ,  and  u = a(q)t  get  =  q  = MJ(u),  [2.34]  where region.  M = 4irfr /(3v) o  The e v a l u a t i o n  i s the number of l a t t i c e s i t e s i n s i d e the  cut-off  o f the i n t e g r a l has been done by Grant and S t r a n d b e r g  74  (39), although one of their asymptotic series (Eq. [63a] of (39)) i s incorrect.  The value of the integral is given by  J(u) = \ -1 - 2iu(l +  —  log ^  1 U  [  •  "  where the  function  Our J(u).  | + | M(l,3/2, 3iu) + | M ( l , l / 2 ,  _9-  /o  T/3  00  r  M(a,b,z)  I  ( 0  "  1 / 2 ) r  i s a confluent  *  have the  represents  the  ^  M(l,-r+l/2,2iu)|,  hypergeometric f u n c t i o n  i s to i n v e s t i g a t e the b e h a v i o r of the  ,  (43),  function  [2.36]  complex c o n j u g a t i o n .  d i r e c t l y from the d e f i n i t i o n [2.34] of  The  J(u);  above r e l a t i o n f o l l o w s  or from a p p l y i n g  M * ( a , b , i c ) = M(a,b,-ic)  to Eq.  [2.35]  f o l l o w i n g important r e l a t i o n  J*(u) = j(-u)  where  (l/2) r ' ( r + 1/2)  3iu) ]  r  6  problem now  F i r s t l y we  )/3  1  /T~ + 1  /T~ + e  ~  the i d e n t i t y  (c r e a l )  [2.35].  For s m a l l and  large values of  u,  we  expand  J(u)  i n t o a power  75  series and an asymptotic series, respectively (a).  Power series expansion  n=0  P = j n ^  where  k  (b).  Q  (n-k)!k!(2k+l)  Asymptotic series expansion  J(u) = -1 -4  i u ( l + ±- l o g ^ J , ) /3 /3 + 1  +  2^| | 3/3 u  (1/2) _ 1„ Ar .IU s+1 y n 3/1 S,. sn-% n=2 (xu) y  r  6  -2iu  s (1/2) (2n+l)  =  n  and  1  n=l  + 0(|u|  c  J  f  9  where  s (1/2)  1  S  1  )  n - l _ . .n (3iu)  / 0  +  l  n+1 ^ . n n=l (2iu)  ,  n  J  [2.38]  y ,—..,. , ; . k=l fa-k)!k!(k-M?)  ( a ) = a(a + 1) n  For intermediate values of u,  n  (a+n-1) .  i t i s most convenient to represent  76  J(u)  graphically.  u s i n g the IBM expansion, u 210  .  performed  360/67 computer.  up to the term  We  p a r t by  We  u  -7/2  the i n t e g r a l i n Eq. The  function  ,  For  J(u)  u < 0,  by R e a l  the b e h a v i o r o f  from the graphs u s i n g the r e l a t i o n g i v e n Eq. s e r i e s expansions, for  and i t s a s y m p t o t i c  are p l o t t e d i n F i g s . 16a and 16b  denote the r e a l p a r t of  Imag ( J ( u ) ) .  J(u)  [2.34] n u m e r i c a l l y  [2.36].  (J(u)) J(u)  and the can be  imaginary  obtained  From the graphs  and  i t i s clear that :  |u| <_ 1 ,  J ( u ) = 2u /5 + 0 ( u ) 2  and f o r  ;  3  [2.39]  | u| >_ 4  J(u) = — | u f - 1 - 4 3/3  iu(l + — l o g ^ ~ 1 /3 /3 + 1  From the p r o p e r t i e s of the f u n c t i o n  J(u)  b e h a v i o r of the l i n e shape f u n c t i o n  1^(10)  ,  i n r e g i o n I I and b e i n g i n the magnetic s t a t e  The  first  i s not symmetrical frequency  shift.  c o n c l u s i o n one  one ,  ) + OClul" ). 1  which i s due  to i m p u r i t i e s  q.  can draw i s t h a t the l i n e shape f u n c t i o n  about i t s c e n t r e , which by d e f i n i t i o n corresponds S i n c e the l i n e shape f u n c t i o n  I^(u>)  i s obtained Fjj(t),  l i n e shape near the c e n t r e i s determined  F^(t)  t.  Since  [2.40]  can deduce the g e n e r a l  F o u r i e r t r a n s f o r m i n g the c o r r e s p o n d i n g c o r r e l a t i o n f u n c t i o n  large  for  by the b e h a v i o r of  t o no by the at  77  Fig.  16a.  P l o t of R e a l ( J ( u ) ) v e r s u s u. The s o l i d l i n e r e p r e s e n t s Real(J(u)), w h i l e i t s a s y m p t o t i c expansion up to the -7/2 term u i s r e p r e s e n t e d by p o i n t s 0.  79  F^Ct)  it  i s determined  from  Eq.  [2.40]  Consequently shitted  that  (  c  shift  p  the  cut-off  the  behavior of  q  =  _ 8  T r  fYpp( )p(q)  1  q  9  a  (  q  )  of  )  ,  [2.41]  .  J(ta(q)).  I t can be  seen  J(ta(q))  i s linear  i n  of  t h e peak  t.  intensity  The b e h a v i o r o f  J(ta(q)) of  region  i n spite  f o r small  J(ta(q))  of the s h i f t  shape v a n i s h e s . has a f i n i t e  This rather  product q  t.  correlation  function  to both  t,  term  the wings  o f the term which  of the l i n e  the cut-off  decreases  very  2  ]  temperature o f moment  function number  volume. repidly  are predominantly  moment  t, a way  of the  i s e x p e c t e d when t h e  as has been  expansions  F ^(t) q  demonstated  (46).  i s linear  of impurities  t  i n  i n such  which  For large values of as  by  i n t h e power  i s cubic  the f i r s t  shape  of  i s determined  Since the leading  i n t h e peak f r e q u e n c y ,  infinite  4  the impurity  i s n o t too f a r out i s asymmetric  than  >  /3 + 1  i s quadratic i n  asymmetry  2  a t the wings  I t r e p r e s e n t s the average  contained inside  [  l!jl (u>)  logarithm of the correlation  pMp(q).  _  u ( q ) , but i s independent  M c M i l l a n and Opechowski by t h e method  The  /3_-±  moment  o f t h e wings which  the  t  i s proportional  the  state  (  /3  However, due t o t h e i n f l u e n c e  by  J  1_  [ 1 +  V  and t o t h e magnetic  volume.  expansion  system  M  the centre i s Lorentzian with  Gaussian.  line  )  the asymptotic behaviour  o f t h e peak i n t e n s i t y  concentration  that,  l  b y a n amount  1  series  P  p  by the asymptotic behaviour  t h e shape near  A  This  = e "  increases.  i nthe i s i n  pMp(q), Consequently  80  the l i n e shape i s m a i n l y determined by the s h o r t time b e h a v i o r and  looks  l i k e a Gaussian.  of  On the o t h e r hand, when the v a l u e o f  J(ta(q)), pMp(q)  is  s m a l l , the l i n e shape w i l l be determined by the l o n g time b e h a v i o r o f J(ta(q))  and w i l l be p r e d o m i n a n t l y L o r e n t z i a n .  the l i n e shape a r e n e i t h e r simply  the c o n c e n t r a t i o n o f the i m p u r i t i e s n o r  the c u t - o f f volume, b u t i n v o l v e s the p r o d u c t = 0.01, and  2.0,  The parameters t h a t i n f l u e n c e  the l i n e shape i s p r a c t i c a l l y  of both f a c t o r s .  pMp(q)  L o r e n t z i a n ; f o r pMp(q) = 0.1,  the l i n e shape f u n c t i o n and the c o r r e s p o n d i n g  Lorentzian lines  For  0.5,  G a u s s i a n and  f i t t e d t o the same peak i n t e n s i t y a r e shown, r e s p e c t i v e l y ,  i n F i g s . 17, 18, and 19.  From the graphs, we see t h a t , t o a good  the l i n e shape can be r e p r e s e n t e d and by a G a u s s i a n one, when  by a L o r e n t z i a n curve when  pMp(q) > 2.  In.the  shape i s c h a r a c t e r i z e d by the h a l f - w i d t h from the c e n t r e o f the l i n e  q  u,  approximation,  pMp(q) < 0.1,  Lorentzian l i m i t ,  the l i n e  which i s d e f i n e d as the d i s t a n c e  t o the p o i n t o f h a l f  intensity  and i s g i v e n by  8Tr pfYy(q)p(q) 2  =  [2.43]  9/3 v  %  This h a l f - w i d t h i s p r o p o r t i o n a l to the impurity magnitude o f the magnetic moment  concentration  T h i s i s i n agreement w i t h  Anderson  (44) and Grant and Strandberg  the c a l c u l a t i o n s and o b s e r v a t i o n s o f (45). I n t h e G a u s s i a n l i m i t , t h e  l i n e shape i s c h a r a c t e r i z e d by the second moment  16.f Y y (q) 2  ( q ) >  and the  u ( q ) , b u t i s independent o f the c u t - o f f  volume.  *  p  =  P  15rjv  2  p ( q )  2 <OJ (q)>  .  which i s g i v e n by  [2.44]  81  F i g . 17.  L i n e shape f u n c t i o n  I^(w)  when  and  Gaussian l i n e s , b o t h f i t t e d  as  I (w), q  are represented  pMp(q) = 0.1.  The L o r e n t z i a n  t o have the same peak  by p o i n t s  x  and  0,  intensity  respectively.  82  Fig.  18.  Line  shape f u n c t i o n  I  q  and  Gaussian  lines,  as  I ^(w),  are represented  q  both  (UJ) when fitted  p M p ( q ) = 0.5.  t o have  by p o i n t s  The  t h e same p e a k x  and  o.  Lorentzian intensity  83  <*> / oc ( q ) Fig.  19.  Line  shape  function  and  Gaussian  as  I  q  (to) ,  lines,  1^(10) both  when  fitted  are represented  p M p ( q ) = 2.  t o have  by p o i n t s  The  Lorentzian  t h e same p e a k x  and  o,  intensity  respectively.  84  The  second moment i s p r o p o r t i o n a l to the i m p u r i t y c o n c e n t r a t i o n  square of the magnetic moment, but volume.  The  46).  s h a l l now  estimate  the e r r o r i n t r o d u c e d by  [2.32c] by  note t h a t when each p o s i t i o n v e c t o r  approximating  the i n t e g r a l of Eq.  r\  changes by  f r a c t i o n a l change i n the c o r r e l a t i o n f u n c t i o n  [2.33].  an amount  F ^(t)  the We  <5r\ ,  the  i s , from Eqs.  q  first  [2.3l]  [2.32],  -iw (r , q ) t  rjl  II  r e  ;  -9  2  F . ( t ) = 3 i t p Y y ( q ) j=M+l I -  r  q  ]  Due  the  same f u n c t i o n a l r e l a t i o n i s o b t a i n e d i f i t i s c a l c u l a t e d by  d i s c r e t e l a t t i c e sums i n Eq.  and  and  i s i n v e r s e l y p r o p o r t i o n a l to the c u t - o f f  means of the method of moments (33,  We  p  [(3cos 9. - l ) 6 r . + r . s i n  26.69.]  to d e s t r u c i v e i n t e r f e r e n c e , the a b s o l u t e v a l u e of the i n f i n i t e sum  assumed to be about the same o r l e s s than term i n the i n f i n i t e sum.  Since  Sr.  the a b s o l u t e v a l u e of the  and  r.SO . < a  J  '  5  F  l l  |F  q x  (  t  )  '  J  3t Yy  K  P  ( q )  3  ,  we  extremal  have  ~  a  3  (t)|  <  27/3 — 2 ~ 8TT  ,v ,a t ( — ) — — r o c  , . (when  T  Q  < Tr,  r^ x O  C  pM «  is  1 N  1)  85  where  x  i s the time a t which the v a l u e of the c o r r e l a t i o n f u n c t i o n i s  c  decreased  to  — e  of i t s v a l u e a t  F  When  p < 1%  and  | t | < 5x  the c u t - o f f r a d i u s  To we  II  r  £  t = 0, i . e .  c)  ( T  ,  = F (0)/e  .  q  i  the f r a c t i o n a l change i s l e s s  i s g r e a t e r than a few  Q  lattice  compare the continuum i n t e g r a l w i t h  calculate numerically  the  x  for r  D  a face-centred =  >  = i  I j=M+l  3, and  iur^(l-3cos 6 [1 - e ° '  )/r ]  6.  The  which i s one  25.  behavior  J-^(u)  of the h i g h e s t  directions,  In a l l c a s e s ,  i n c r e a s e s from centred  of  sums,  lattice H  sum ,  3  [2.45]  J  cut-off radius :  J-^(u)  as d e f i n e d h e r e depends  because the a n g l e s  8^  depend  a//2  cubic l a t t i c e  when  21.  (TT/18, TT/36) J-^(u) to  n  l i e s i n the d i r e c t i o n  (0,0)  symmetry d i r e c t i o n s i n the f a c e - c e n t r e d  i s shown i n F i g s . 20 and  arbitrary to  i n length.  latter.  The  lattice  J  for different  on the d i r e c t i o n of the e x t e r n a l f i e l d on the  when  sum  c u b i c l a t t i c e and  m = 1,  constant  -—^  the d i s c r e t e l a t t i c e  2  J (u)  than  /3a.  The and  case when  rT  l i e s along  (2TT/9, TT/18) ,  converges q u i t e r a p i d l y to I t can s a f e l y be  cubic two  other  i s shown i n F i g s . J(u)  concluded  the continuum a p p r o x i m a t i o n s h o u l d be  when  r  Q  that f o r a facegood whenever  22  o a  0  10  20  U . 20.  P l o t of The  solid  Real(J(u)) line  r /a = 1//2  ,  respectively.  and  RealCJ^u))  versus  u  when  (0,  represents  Real(J(u)).  /3j2  i s r e p r e s e n t e d by p o i n t s . , x, • and  ,  /3  RealCJ^u))  n =  for  87  •  X  K  CD <  2  0  -2 0  10  20  u Fig.  21.  Plot of  ~Imag(J(u))  rT = ( 0 , 0 ) . -Imag(J  The s o l i d  (u))  for  and  -ImagCJ^u))  line  represents  r /a =  ,  Q  r e p r e s e n t e d by p o i n t s . ,  x,  and  /372 o,  versus  u  when  -Imag(J(u)). ,  and  /3  is  respectively.  88  Fig.  22.  Plot n  =  of  Real(J(u))  (ir/18, n'/36).  ReaKJjCu)) represented  for by  and The  Real(J  solid  line  r /a = lf/2 , Q  points.,  x,  and  (u))  versus  represents  /3]~2 , o,  and  u  when  Real(J(u)).  /3  is  respectively.  solid is  line  represents  represented  by  -Imag(J(u)). points.,  x,  -Imag ( J ^ (u) ) and  o,  for  r / a = 1//2, Q  respectively.  fs~l2,  and  90  Real(J  (u))  for  r / a = 1//2 Q  r e p r e s e n t e d by p o i n t s . ,  x,  ,  /3/2  and  o,  ,  and  /J  is  respectively.  10 Fig. 25.  Plot of  -Imag(J(u))  solid line represents  V3>  and  20  U  -ImagCJ^u))  -Imag(J(u)).  i s represented by points . , x,  versus  u when n =  -Imag ( J (u)) 1  and  o,  (2TT/9, 7TT/18). The  for r / a = l/7l, Q  respectively.  /3/2, and  92  the c u t - o f f r a d i u s i s as s m a l l as our  c a l c u l a t i o n to low i m p u r i t y  Q  = /3a )  such t h a t  pM << 1  or l a r g e r .  concentrations  choose a s u i t a b l e c u t - o f f r a d i u s r  /Ja  r  Q  S i n c e we a r e r e s t r i c t i n g  (p <_0.1%),  we can always  ( f o r example, we can always  i s always t r u e and the l i n e  choose  shape  is Lorentzian i n this l i m i t , i . e .  I  where  to?  and  q T  pMp(q) (o3)=-^-  Aw?.  not w i t h  ,  [2.46]  are g i v e n i n Eqs. [2.42] and [2.43],  we n o t e t h a t a l t h o u g h we i d e n t i f y  uj — — — (to|) + (to - Ato^)  r i g o r o u s l y the f i r s t moment o f  the c e n t r e o f the L o r e n t z i a n l i n e w i t h  respectively.  I^u)  always  the peaks o f  Here  vanishes,  I^O*)),-  i t s c e n t r e , s i n c e the s h i f t of the peaks from the c e n t r e i s (from  Eqs. [2.42] and [2.43]). o n l y about 0.13  o f the v a l u e  o f the h a l f - w i d t h o f  'the l i n e .  To e v a l u a t e  the c o r r e l a t i o n f u n c t i o n  F (t) q  due to i m p u r i t i e s  i n s i d e the c u t - o f f r e g i o n , we n o t e t h a t the sum (see Eq. [2.32b])  M  has a magnitude  iYy(q)t(l -  3cos 6.)/R 2  of n o t e x c e e d i n g  M  3  and we may choose  pM << 1.  then approximate Eq. [2.32b] by r e t a i n i n g o n l y the f i r s t  „ F (t) = e " q  p M  two terms, i . e .  , N M iYy(q)t(l-3cos 0.)/R P [ l + pp(q) I e j j]. 2  ( q )  3=1  We can  3  [2.49]  93  The  l i n e shape f u n c t i o n  I (to)  due t o i m p u r i t i e s  q  i n the cut-off region i s  thus t h e sum o f a s e t o f 6 - f u n c t i o n s , one r e p r e s e n t s others represent  the main l i n e and t h e  side-bands whose w e i g h t s a r e p r o p o r t i o n a l  to the concentration  P :  M  l (to) = q  e~  p  M  p  (  q  )  + pp(q)  [6(co)  I 6(10 - co.)]  3=1  where  co. = -  Y  y  ^  q  (1 - 3 c o s 6 . ) .  )  2  R.  3  3  ( I t may happen t h a t some o f t h e d e l t a f u n c t i o n s  i n t h e second term may  c o i n c i d e , hence t h e number o f s i d e - b a n d s may be s m a l l e r  Now t h e l i n e shape due t o c o n t r i b u t i o n s region  [2.47]  J  I and I I when t h e i m p u r i t i e s  are i n state  than  M).  of impurities q  i n both  i s , from Eq. [2.32],  r°° 1 % )  l (co -  y)I (y)dy  q  =  q  I  In t h e L o r e n t z l i m i t , from Eqs. [2.46] and [2.47], we have  *  L (a,J) + (to - Aco^) 2  2  M + Pp(q)j =I l ~(toj) — *2  20  + (~ w - (o. - A t~o —) r \ • q  2  T  3  [2.48]  94  Therefore, the l i n e shape due to contribution of impurities i n both regions consists of a main l i n e centred at  Aw!*  and side-bands at  II  co. + Aw!* J  >  II  and a l l the l i n e s , main l i n e and side-bands, are Lorentzian, since  pM << 1.  The above considerations should also be v a l i d f o r other l a t t i c e systems.  For l a t t i c e s which are less symmetric than the face-centred  l a t t i c e , the value of  r Q  /a  cubic  may have to be somewhat larger f o r the continuum  approximation to be s u f f i c i e n t l y close to the discrete l a t t i c e summation. However, i n these cases, the number of s i t e s i n a unit c e l l i s smaller, and a cut-off radius  r  can usually be chosen such that  Q  pM << 1  l a t t i c e sums can be approximated by the continuum i n t e g r a l .  and the  Therefore, we  can conclude that the c o r r e l a t i o n function of a system of c l a s s i c a l magnetic dipoles  u (q),  q = 1, . . . ,0 ,  as a product of a  F  q  ™  ( t )  =  e  /  can be expressed, to a good approximation,  functions  WI-LT/^  t  M  \ \  -pMp(q)(l J(ta(q)) +  [ 1 + p p ( q )  iYy (q)t(l-3cos 6 .)/R j 2  ^  e  3  ]  3 >  [  2  5  Q  ]  3=1  The Fourier transform  l (co) q  centre and side-bands at  -  of  F ( t ) consists of a main l i n e at the q  (1 - 3 c o s G . ) . 2  RT  As before, a l l the l i n e s ,  3  3  main l i n e and side-bands, are Lorentzian, since  pM << 1.  shape function i s the convolution of the functions  l (co), q  The t o t a l l i n e q = 1, • • • ,o .  95  4.  LINE SHAPE FUNCTION DUE TO A SYSTEM'OF SPINS,  We s h a l l now c o n s i d e r  two examples.  shape f u n c t i o n o f a powdered sample : and  the o t h e r  SPIN  AND  S = 1  We s h a l l c a l c u l a t e the l i n e  one due to a system o f s p i n s  S = 1/2,  S = 1.  1/2. The  i s located at  d e r i v a t i o n o f the frequency s h i f t r" and i n the magnetic s t a t e  For s i m p l i c i t y we s h a l l assume t h a t and  that  The  p a i r hamiltonian  at  Y  q  Or,q)  due to a s p i n which  i s s i m p l e and  straight-forward.  the s p i n s have an i s o t r o p i c g - f a c t o r  they i n t e r a c t w i t h the n u c l e i by means o f the d i p o l a r i n t e r a c t i o n . f o r a nuclear  then c o n s i s t s o f t h r e e  H = H  and  gyromagnetic r a t i o  z  + H + H e d '  H  = 3gS-H  d  ,  [2.51]  ,  = M . [s.I - 3(g-r)(I-r) 3 2 r r  J  the Zeeman energy o f the n u c l e a r Y,  S = 1/2  terms :  = - YI-H , '  H  represents  s p i n I a t the o r i g i n and the s p i n  Hz  &  Here  S = 1/2  i n an e x t e r n a l magnetic f i e l d  the Zeeman energy o f the i m p u r i t y  s p i n S.  H  s p i n I , which has H.  represents  represents the magnetic  96  d i p o l a r i n t e r a c t i o n between the nuclear s p i n I and the impurity s p i n S. f3 i s the Bohr magneton.  I f the magnitude of H  i s s u f f i c i e n t l y large .  ( >_ 1 k i l o g a u s s ) , as i s g e n e r a l l y found i n most of the resonance  experiments,  then the f o l l o w i n g i n e q u a l i t y holds :  tf >> tf » tf, . e z d  [2.52]  We can evaluate the frequency s h i f t by means of the p e r t u r b a t i o n method. We t r e a t tf + tf as the zeroth order term and tf , as the p e r t u r b a t i o n , e z d r  In a coordinate system i n which the z-axis i s i n the d i r e c t i o n of the e x t e r n a l field  H,  we have  tf = -YHI , z z  and  tf  = ggHS e  The eigenstates of (m  = ± 1/2)  tf  and  . z  tf  are denoted by  corresponding to eigenvalues  <J> (m )  E(m ) = - m'YH JL  o  = ggm H , r e s p e c t i v e l y . (j)(m )^(m ) T  and ij> (m ) and E (ra ) 6  J.  L>  The eigenstates of tf + tf are then given by  corresponding to eigenvalues  (-m Y + m Bg)H . The f i r s t order  d i p o l a r energy i s then  E  (m ,r; ir^) = ^  m^d  s  - 3cos 6), 2  [2.53]  r  where  0  i s the angle between  r  and the z - a x i s .  Since nuclear magnetic  97  resonance i s observed f o r the t r a n s i t i o n shift  io("r,m )  m^.  > m^. - 1 ,  the frequency  of the resonance i s then given by  co(r,m ) = ^ | m ( l - 3 c o s e ) . r  [2.54]  2  s  s  This equation i s the same as Eq. [2.31] i n the case of c l a s s i c a l magnetic dipoles.  I n f a c t , the induced magnetic moment of the s p i n S i n s t a t e  m^  is -Bg<iKm ) |s|i (m )> = -egm n s  where  n  r  s  g  ,  i s a u n i t vector along the d i r e c t i o n of H . With  q = ± 1/2 ,  the case of  S = 1/2  u(q) = -$gq ,  i s the same as the p r e v i o u s l y discussed  case of c l a s s i c a l dipoles each of which has two s t a t e s . w i t h which a s p i n S i s found i n s t a t e  q  The p r o b a b i l i t y  i s given by the Boltzmann f a c t o r  -BgqH/kT p ( q )  where  T  2cosh(BgH/2kT)  i s the temperature and k  i s the Boltzmann's constant. The l i n e  shape f u n c t i o n i s the convolution of two l i n e s ,  I (o>), q  q = ± 1/2,  of which i s a Lorentzian l i n e w i t h side-bands i n the wings.  each  Since the  convolution of two L o r e n t z i a n l i n e s i s again a Lorentzian l i n e s which has a s h i f t and h a l f - w i d t h equal to the sum of the s h i f t s and h a l f - w i d t h s , r e s p e c t i v e l y , by the two convoluted l i n e s , n e g l e c t i n g the side-bands, from Eqs. [2.42] and [2.43] the l i n e shape width  I(o>) i s again L o r e n t z i a n w i t h h a l f  98  2  = 2  and  4TT pfYBg 9/3v  t a n h (  g  g H  /  ( 2 k T  ))  [2.55]  >  shift AM  = - —  co, [1 + —  17  2  log^  /3  "  ]  1  [2.56]  /3 + 1  In the case when the temperature i s i n f i n i t e l y h i g h and t h e side-bands can be n e g l e c t e d , which i s e q u i v a l e n t cut-off radius  r  Q  to p u t t i n g  o r — - [J(Y3gt/(2r )) P  3  lime o-K)  J  3  3  +  J(-Yegt/(2r ))]  expansion, E q . [ 2 . 3 8 ] , o f the f u n c t i o n  J ( u ) + J ( - u ) = -2 + 4TT | u ] / (3/3)  The  3  V  r  From the a s s y m p t o t i c  and  and making the  s h r i n k t o z e r o , the c o r r e l a t i o n f u n c t i o n becomes  2 7 T f r  F(t) =  p(q) = 1/2  F(t)  = -^ fpY3g|t|/(9/3v)  + Odul" ) 1  ,  2  e  m  [  l i n e shape f u n c t i o n i s t h e r e f o r e a L o r e n t z i a n w i t h  =  J ( u ) , we have  a half-width  4Tr fpY3g/(9/3v) 2  T h i s r e s u l t i s the same as t h a t o b t a i n e d  by Abragam  (18)  2  >  5  8  ]  99  In the above c o n s i d e r a t i o n s , i t has been assumed t h a t t h e impuritys t a t e s have a l i f t  time which i s l o n g compared t o the p r e c e s s i o n p e r i o d o f  the n u c l e a r s p i n s .  T h i s happens when the c o u p l i n g between t h e i m p u r i t y  s p i n s and t h e r m a l b a t h spins. for  In t h i s  i s much s m a l l e r than the Zeeman energy o f t h e n u c l e a r  "long l i f e - t i m e c o r r e l a t i o n l i m i t " ,  the c o r r e l a t i o n f u n c t i o n i s r e p r e s e n t e d  £  P  where  E(q)  (  q  =  )  Z e  the g e n e r a l  by Eq. [2.26],  expression  i n which  -E(q)/kT -E(q)/kT  '  [ 2  i s the energy o f the i m p u r i t y i n s t a t e  q.  '  5 7 ]  I n t h e case o f t h e  " s h o r t l i f e - t i m e c o r r e l a t i o n l i m i t " , when the t r a n s i t i o n r a t e s between d i f f e r e n t s t a t e s o f the i m p u r i t y s p i n s a r e v e r y much l a r g e r than the p r e c e s s i o n frequency the i m p u r i t y  o f the n u c l e a r s p i n s i n the l o c a l f i e l d produced by  s p i n s , the l o c a l f i e l d s  the n u c l e a r s p i n e x p e r i e n c e s  f l u c t u a t e s so r a p i d l y t h a t , i n e f f e c t ,  an averaged l o c a l f i e l d produced by an averaged  magnetic moment -tf /kT true  ^  <y>  =  -tf /kT tre  For  S = 1/2,  the observed s h i f t  •  '  5 9 ]  6  i n resonance frequency  w(r) = - Y u ( l - 3 c o s 9 ) / r , 2  where  [ 2  3  u = -pg tanh(Y3gH/(2kt))/2  i s then  [2.60)  100  In t h e L o r e n t z i a n l i m i t , u  i n t o Eqs. [2.42] and [2.43],  those i n t h e case o f t h e  s u b s t i t u t i n g the value  and comparing t h e s h i f t  "long l i f e - t i m e  t h a t b o t h cases g i v e t h e same s h i f t t r u e f o r S = -|- , be  and h a l f - w i d t h  correlation limit",  and h a l f - w i d t h .  fashion.  correlation limit",  From Eqs.  with  i t can be seen  This r e s u l t  i s not only  b u t i s t r u e g e n e r a l l y i n the L o r e n t z i a n l i m i t .  shown i n a s t r a i g h t forward  "long l i f e - t i m e  o f magnetic moment  T h i s can  [2.42] and [2.43],  i n the  t h e t o t a l h a l f - w i d t h and t h e t o t a l  shift  are g i v e n by, r e s p e c t i v e l y ,  8Tr pfY 2  ^  =  9/3v  2  and  Aco = -  then  E p(q)p(q) q  case o f t h e  '  q  i s the average magnetic moment o f the i m p u r i t y  "short l i f t - t i m e  g i v e the same l i n e  . . ••  Ep(q)y(q) , q  Sp(q)y (q) . 9v  But  „  shape.  correlation limit".  However, when t h e i m p u r i t y  the L o r e n t z i a n l i m i t  cannot be used, and t h e l i n e  the two c a s e s .  "short l i f e - t i m e  a narrower  The  Therefore,  both  concentration  i n the  limits i s high ,  shape i s d i f f e r e n t f o r  correlation limit"  then u s u a l l y  gives  line.  We s h a l l  now i n v e s t i g a t e t h e l i n e  shape due to i m p u r i t i e s o f s p i n  i n a powder sample.  From Eq. [2.50], o r i e n t a t i o n of  H,  the c o r r e l a t i o n f u n c t i o n , f o r a p a r t i c u l a r •  can be expressed  as  101  102  where  u  8TT fp .  >2 =  Z~  5  9/3  »  /3 + 1  /J •  .  P  v R. J  The  .  6 = Yu/v  and  l i n e shape f u n c t i o n c o n s i s t s of a c e n t r a l main l i n e  I  which i s L o r e n t z i a n  . (to) = main  with  2  TT r [ t O ,  +  (tO  +  CO  P  )  ]  half-width  SrTfpYu 9/3v and  s  side-bands  pN(k)owl T  ( ) U  k  =:  .dv_  2  TT  o to  2  +  k = 1,•••,s,  (to + (Op + t 0 j ( l - 3 y ) ] 2  each of which corresponds to the c o n t r i b u t i o n due shell.  The  shape of the s i d e - b a n d has  a general  2  to i m p u r i t i e s form g i v e n by  i n a nearby the i n t e g r a l  103  [2.64]  f.(o>) = o  To s e e how  f(to)  behave,  a  +  2  [to + b ( l -  3y )] 2  f(to)  the f u n c t i o n s  and  2  g(to),  which  i s defined  as,  c(to + b ) g(co)  2b  2  > to >  -b  = otherwise  are p l o t t e d i n F i g . 26 c = 0.27. to the  f o r the  following values  From the p l o t i t i s seen t h a t  smearing out  sample due  The  Knight s h i f t  i n t e g r a l i n Eq.  a = 1,  the b e h a v i o r of  of the well-known l i n e shape  to a n i s o t r o p i c  :  (47,  [2.64] can be  g(to)  b = 10, f(to)  and  is similar  of a p o l y c r y s t a l l i n e  48).  evaluated  in a  straight-forward  manner, g i v i n g  n  +  3b + n/6b( coscj) + sin<|>)  cos (7- + <f>) l o g  f(u>)  n +  4a/3bri  , +  2  3b  . , I T , ,. , . -l/6b~ n (sind) - cbscb) 2 s i n ( r + <j)) [ TT + t a n — ^  0  r  3b  where  - r|/6b"(cos<j> + sincj>)  ri =  [a  2  +  (to +  - n  b) ] 2  1  '  2  /  -, \ ] V ,  4  [2.65]  o  104  OJ  26. Plot of f ( c o ) and and c = 0.2 7.  g(co)  versus co .-  when  a = 1, b = 10,  105  , and  n=  1 _ -l,co •+ b tan ( ) z. a N  —  I t would be u s e f u l t o know the p o s i t i o n and peak i n t e n s i t y o f t h e s e s i d e bands.  To do t h i s , we f i r s t  define  the f u n c t i o n  °°  A  f  [2.66]  o a + [co + b ( l - 3 y ) ] 2  Then f o r a l l v a l u e s o f  2  2  co, we have  f^co) > f(co) > 0 .  Th e i n t e g r a l i n  f (to)  can be e v a l u a t e d e a s i l y , g i v i n g  f (co) = —  sinCr + <j>)  4a/3bn  When  co = co = - b + a//3 , m  f (co) a t t a i n s i t s a b s o l u t e maximum v a l u e  max  .  .  (3) 4a/2ab  We now expand the shape f u n c t i o n co„ m .  Let e  f(co) o f the s i d e - b a n d about the p o i n t  be an i n f i n i t e s i m a l d e v i a t i o n from  f ( % + e )  =  [ 1 4a/2a¥  ?,,#) 27TT3 1 0 v  1 / 4  b  ui m .  3 / 2  Then  + 0(e ) + . , . ] . 2  [2.67]  106  T h e r e f o r e , whenever t h e maximum o f centre  than  f(to) i s l o c a t e d f a r t h e r away from t h e  the h a l f - w i d t h o f the main l i n e , i . e .  b > a ,  the maximum o f  f(to) o c c u r s  a t about t h e same p o s i t i o n l/4 — — 4a/2ab  to  ,  and the maximum  3  can be approximated by the e x p r e s s i o n  lattice  and f o r low i m p u r i t y  concentration  r e g i o n i n c l u d e s a few n e a r e s t n e a r l y exact  shells,  the p o s i t i o n and peak i n t e n s i t y and  the peak i n t e n s i t y  I  Q  f a r away.  o f the main l i n e .  kth  s h e l l , we d e f i n e the f o l l o w i n g q u a n t i t i e s :  /  „ 2 3 32TT pRT  ,  a n  d  The v a l u e s  Q  of  V  X  o -  n - T / T k " k main I  E, , k  cubic l a t t i c e with  / I  P. , k  Ik(u) corresponding  to the  '  D  3l/  k  to ^. as the p o s i t i o n  9/3v  %  P ' - 1 / i - W>u v P  I t i s u s e f u l to r e l a t e  Denoting  as the peak i n t e n s i t y o f t h e s i d e - b a n d  mk  i s indeed  o f the s i d e - b a n d s t o the l i n e - w i d t h to,  1^.  k  cubic  i f the c u t - o f f  the above a p p r o x i m a t i o n  and  a  For a face-centred  (p <_0.1%),  then  as the s i d e - b a n d s a r e r e a l l y  .  4 / 2  ( w  ( mk  and  lattice  )  Q, k  -  h  I5 l k  . ~ P ~ k P  |  3  +  1 / 4  pN(k). l k>  f o r the f i r s t  constant  r  13/4  ?  a are l i s t e d  20  shells  i n Table  4.  f o r a face-centred From the t a b l e ,  we s e e t h a t f o r low i m p u r i t y c o n c e n t r a t i o n s , the s i d e - b a n d s a r e d i s t i n c t i v e l y resolvable.  F o r example, when  p = 0.05%,  the p o s i t i o n of the peak  intensity  107  Shell  k  N(k)  i y a  -p§  1  12  0.707  1.396  X  io"  2  6  1.000  4.936  X  io"  3  24  1.225  2.687  X  IO"  4  12  1.414  1.745  X  io"  5  24  1.581  1.249  X  6  8  1.732  9.499  7  48  1.871  8  6  9  P  Q  1  23.47  4.575  X  io  2  19.74  4.809  X  io"  2  107.0  7.725  X  io"  2  2  66.39  2.022  X  io"  2  io"  2  157.0  2.448  X  io"  2  X  io"  3  59.99  5.413  X  io"  3  7.538  X  io"  3  404.1  2.296  X  io"  2  2.000  6.170  X  io"  3  55.83  2.125  X  io"  3  36  2.121  5.171  X  io"  3  365.9  9.783  X  _3 10  10  24  2.236  4.415  X  io"  3  264.0  5.145  X  io"  3  11  24  2.345  3.827  X  io"  3  283.6  4.152  X  io"  3  12  24  2.449  3.358  X  io"  3  302.7  3.414  X  io"  3  13  72  2.550  2.978  X  io"  3  964.3  8.554  X  io"  3  14  48  2.739  2.403  X  io"  3  715.7  4.133  X  io"  3  15  12  2.828  2.181  X  io"  3  187.8  8.935  X  io"  4  16  48  2.915  1.992  X  io"  3  786.1  3.118  X  io"  3  17  30  3.000  1.828  X  io"  3  512.8  1.714  X  io"  3  18  72  3.082  1.686  X  io"  3  1282  3.642  X  io"  3  19  24  3.162  1.561  X  io"  3  444.0  1.082  X  io"  3  20  48  3.240  1.451  X  io"  3  921.1  1.938  X  io"  3  4.  Values  of  £, , P, ,  and  k  2  0.  _  3  /  k  2  Table  P k  f o r a face-centred  p  l  /  2  - 1  cubic  lattice.  108  of t h e s i d e - b a n d due t o the s h e l l as f a r away as t h e about  10  i s about  Spin  times the h a l f - w i d t h 0.4  shell i s s t i l l  o f t h e main l i n e , w h i l e t h i s peak i n t e n s i t y  o f the i n t e n s i t y o f the main l i n e  there.  S = 1  The spin  9th  p a i r hamiltonian f o r spin  S = 1/2  to be added t o  (Eq. [ 2 . 5 1 ] ) , H  e  .  S = 1  except t h a t  i s the same as t h a t o f  terms due t o t h e c r y s t a l f i e l d a r e  These a d d i t i o n a l terms make t h e c a l c u l a t i o n o f t h e  magnetic moment and c o n s e q u e n t l y t h e e x p r e s s i o n f o r the d i p o l a r of the r e s o n a t i n g  centre very complicated.  o f t h e e x t e r n a l magnetic f i e l d , i n d i v i d u a l cases.  i n the p r e v i o u s  section.  Except f o r some s p e c i a l  directions  one has t o a p p l y n u m e r i c a l s o l u t i o n s t o  Once the magnetic moment i s known, the r e s t o f t h e  c a l c u l a t i o n and d i s c u s s i o n s treated  broadening  o f the shape f u n c t i o n w i l l be t h e same as t h a t  s e c t i o n and i n the case when  F o r our example, we s h a l l c o n s i d e r  S = 1/2  of t h i s  the b r o a d e n i n g by oxygen 3  m o l e c u l e s a t low temperatures.  The m o l e c u l e i s i n the ground  s t a t e w i t h no n e t e l e c t r o n i c o r b i t a l a n g u l a r momentum. centred  c u b i c . We s h a l l c o n s i d e r  only  the ."short  electronic  The l a t t i c e i s a g a i n f a c e -  life-time correlation  limit".  In o r d e r t o reduce the mathematics o f t h e problem b u t t o r e t a i n the e s s e n t i a l physics,  we assume t h a t  the m o l e c u l e has an i s o t r o p i c g - f a c t o r  of a f r e e e l e c t r o n and t h a t  equal to that  t h e s p i n h a m i l t o n i a n i s a x i a l l y symmetric (49,  50,51) : ' H  = ggS-H + D[(S.n, )  2  - S(S + l ) / 3 ] ,  [2.68]  109  where along  D  i s the c r y s t a l f i e l d  parameter and  i s a unit vector directed  the oxygen m o l e c u l e f i g u r e a x i s .  When S = 1/2,  D = 0,  the r e s u l t  n  i s ' e x a c t l y the same as the case of s p i n  except t h a t the v a l u e of the magnetic moment i s now  y(q)  where  n^  the average induced  H  Except f o r the two the e x p r e s s i o n s  the e i g e n v a l u e  cases when  n^  f o r the energy and  H  e  system t h a t i s f i x e d w i t h and  n^ =  (a,g).  H  e  Eq.  r  = BgH[  + D{[  r e s p e c t to the can be  n 1 ,  /  U  J  i s much more i n v o l v e d .  the magnetic moment a r e too We  „ U  problem f o r  form, we  choose a  crystal lattice.  to  n,  complicated  s h a l l s o l v e the problem  i n a more convenient  [2.68]  "short l i f e - t i m e  i s e i t h e r p a r a l l e l or p e r p e n d i c u l a r  to be u s e f u l i n a c t u a l c a l c u l a t i o n s .  To express  In the  *  y  D ^ 0,  .  [2.69]  magnetic moment becomes  _ _ 2gg s i n h ( g g H / k T ) n 1 + 2 cosh(ggH/"kT)  When  by  q = 0, ± 1 ,  = -ggqE  i s the u n i t v e c t o r i n the d i r e c t i o n of  correlation limit",  given  Let  numerically.  coordinate n = (0,<f>)  r e w r i t t e n as  s i n e (S cosd) + S s i n * ) + S cos0 ] x y T  2  s i n a ( S cosg + S s i n g ) + S cosa x y z  ]  2  - 2/3}  .  [2.71]  110  At t h i s p o i n t , we  note that  i s not known a p r i o r i  although  t h e r e must  be  some more f a v o r a b l e o r i e n t a t i o n s i n which the energy o f the m o l e c u l e i s  at  a minimum.  One  expects t h a t these  d i r e c t i o n s o f the c r y s t a l l a t t i c e , for  a face-centred  to be  found a l o n g  cubic l a t t i c e , equivalent  o r i e n t a t i o n s to be a l o n g  e.g. and  (1,0,0),  (1,1,0),  d i r e c t i o n s i n the f a c e - c e n t r e d  cubic l a t t i c e  have the h i g h e s t  assume t h a t the m o l e c u l e i s e q u a l l y l i k e l y  of the s i x  (1,0,0)  (D, H,  n", n j ^ ) >  the induced  1  ""*>  _ n  k ±  ^(  correlation function  symmetry, found i n  i s diagonalised numerically n k  ^)•  _ n) = - — [ y r  ,  (1,0,0)  Then f o r a g i v e n s e t of v a l u e s  6,  magnetic moment  w(r,  The  1>  =  to be  F(t,rT)  The  frequency  _ (n" )-n  line  we  any  of to o b t a i n  produced i s then  3(,T(E ).r)(n.r) ] . r k i  k±  and  shift  etc  f o r the m o l e c u l e  S i n c e the  shall first  directions.  (1,1,1)  t h a t the p r o b a b i l i t i e s  d i r e c t i o n s are equal.  some symmetry  shape f u n c t i o n  [2.72]  I(co,n)  g i v e n e x t e r n a l f i e l d o r i e n t a t i o n i s then c a l c u l a t e d by means of the  for a average  value  <  and  Eq.  Ti^ — >  e  [2.26]. - rT  k  ,  . , . i (r,n)t W  >  =  1  , 6  6  £ L p=l e  iw(r,n^ \p -  ,n)t C2.73]  S i n c e the h a m i l t o m i a n i s i n v a r i a n t under the i n a c t u a l c a l c u l a t i o n s Eq.  • / <  l w ( F e  \». '  n ) t  i  icoCFjn,  3  > =\l  e p=l  [2.73] can be  k  P  transformation  replaced  by  ,n)t ,  [2.74]  Ill  where  i s one o f the f o l l o w i n g o r i e n t a t i o n s  (0,0,1).  The  correlation function  • 3 F  I l (  t,H) =e  P  »=  1  J=  M  +  ivr  r  l  Fjj(t,n")  (0,1,0),  i s then s i m p l y  g i v e n by  2  \  [2.75]  3  and  S(p) = y ( i ^ )/(Bg)  For s u f f i c i e n t l y l o g a r i t h m of do t h i s , we ri  F(t,n)  large  r  0  Q  (/3a),  the summation i n v o l v e d i n the  can a g a i n be approximated  a g a i n assume t h a t the average  spin  by a sum S(p)  of i n t e g r a l s .  p o i n t s i n the  To  direction  approximately, i . e .  S(p)-n-  8  3(S(p) -T) ( r T . r ) / r  i s the a n g l e between  This approximation  r  and  2  * S(p)-n(l -  3cos 6)  be shown l a t e r ,  i s good o n l y i f the c r y s t a l f i e l d  t h i s i s a good a p p r o x i m a t i o n  T = 4.2°K .  2  n .  h a m i l t o n i a n i s not too much g r e a t e r than the Zeeman term.  and  and  3  t  v = gYgt/r  where  (1,0,0),  [S(p)-n-3(S-(p).r ) ( n . r . ) / r / r . ] >i  3  where  of  :  for  term i n the  However, as  D=5.6k,  will  H = 2 0 kilogauss,  S i n c e we want to apply the t h e o r e t i c a l r e s u l t s  to i n t e r p r e t  13 the b r o a d e n i n g  of the  C  and p r o t o n resonances  the above p a r t i c u l a r v a l u e s of  H  and  T  l i n e s by oxygen  are chosen so t h a t they  impurities, correspond  112  to e x p e r i m e n t a l D = 5.6 k  values already discussed i n chapter I .  i s chosen u s i n g some p l a u s i b i l i t y  The v a l u e o f  c o n s i d e r a t i o n s (50, 51) as i t s  a c t u a l v a l u e f o r oxygen m o l e c u l e s i n s o l i d methane l a t t i c e i s unknown.  We now denote the summation i n Eq. [2.75] by  ,  3  »  ivr ["s(p).n-3(S(p).T.)(rI.T.)/r ]/r .  ,  3  2  3  [2.76] p=l j=M+l  From Eqs. of t h r e e  [2.33], and [2.34], t h e summation can be r e p r e s e n t e d by t h e average integrals  1 G(v) = ± . I 3  When  T = 4.2°K  and  represented i n Figs. directions n  ,  and  of  n ,  D = 5.6 k , 27 t o 32.  numerical values of 4  o  when  n  r  (|"> ); 0  o  changes.  such as the  o  (l,l,0)'s  again n e g l i g i b l e .  r  Q  n  d  r  o f n",  = /2a .  n^  4  4  ),  i  Q  n  G(v) a r e  each d i r e c t i o n o f  = (p/2)  5  ,  p = 1, 2,  G(v) i s a l r e a d y q u i t e A l s o i f one compares t h e  one f i n d s v e r y l i t t l e  n :  and the  (1,1,1)'s,  (0,0),  v a r i a t i o n i n G(v)  i s a l o n g some o t h e r s e t s o f e q u i v a l e n t  This i s consistent with  centred cubic l a t t i c e .  G (v) and  f o r the f i v e d i f f e r e n t d i r e c t i o n of  and  When  a  a r e chosen,  Q  G^(v) when  G(v)  (T-,0), & , \ ) ,  (77,0),  and  5  For a l l three d i f f e r e n t d i r e c t i o n s  a good a p p r o x i m a t i o n o f  of  [2.77]  The f u n c t i o n s a r e p l o t t e d f o r t h r e e d i f f e r e n t  three d i f f e r e n t values of 4.  the behavior  O|- 0),  (0,0),  • J((S(p)-n)v) .  the v a r i a t i o n o f  directions,  G(v) i s  t h e h i g h symmetry o f t h e f a c e -  T h e r e f o r e , f o r low c o n c e n t r a t i o n s o f i m p u r i t i e s ,  27.  Plot of  Real(G(v))  represents by points  and  Real(G(v)). o, .,  and  RealCG^v)) ReaKG^v))  versus for  x, respectively.  v  when n = (0,0).  r / a = 1//2, Q  1,  and  /l  The solid line is represented  e  e  > CD  b < ~  0  -1  -2  0  Fig.  28.  10  P l o t of The s o l i d r /a D  x,  20  -Im(G(v)) line  = 1//2,  and  represents  1,  respectively.  and  /2  30  -ImCG^v)) -Im(G(v)).  versus  U0  v  when  -InKG^v))  i s r e p r e s e n t e d by p o i n t s  n =  (0,0)  for o,  .,  and  9  Fig. 29.  Plot of  Real(G(v))  represents points  o,  and  Real(G(v)). and  RealCG^v)) RealCG^v))  x, -respectively.  versus for  v  when  n = (TT/4, 0). The solid line  r / a = 1//2, 1, D  and  Jl  is represented by  V Fig. 30. Plot of -Imag(G(v)) represents points  and  -Imag(G(v)).  o, ., and  x,  -ImagCG^v)) -ImagCG^v))  respectively.  versus  v when n = (TT/4, 0). The solid line  for r / a = 1//2, 1, Q  and  /2  i s represented by  o O  20-  o O  v*"®  >  O  o  <  O  o  LY.  X''©  _*  X^o  S-"©  10 o  0  10  0  20  50  30  V Fig. 31.  Plot of  Real(G(v))  represents points  o,  and  RealCG^v))  Real(G(v)). RealCG^v)) and  x,  versus  v  when n = (TT/8, 0). The solid line  for r / a = 1//2, 1, Q  and  /2  i s represented by  respectively. r-  1  Fig.  32.  Plot  of  -Imag(G(v))  represents points  o,  and  -Imag(G(v)). .,  and  x,  -Imag(G (v)) 1  '-Imag (G  l  (v))  versus for  v  r /a = D  when  n =  (ir/8,  1//2,  1,  and  0). /2~  The  solid  line  i s represented  by  respectively. oo  119  when the side-bands at  the wings and  corresponding  to the f i r s t  in  t  (0,0).  n" =  different  pK  o f the a s y m p t o t i c  the f o l l o w i n g way.  i.e.  The  We  where  expansion  = S(3)  (0,1,0)  =  (0,  and  (1,0,0)  to be  K  can be  the  calculated  a l o n g the z - a x i s ,  calculated axis.  of  Along  f o r the the  three  (0,0,1)  2sinh(BgH/kT)  + A  2w(w 2  D/kT  +  2c  osh(3gH/kT)  d i r e c t i o n s , we have  0,  where  + w/l + w ) ( e 2  + w  2  )sinh(D/l + w  _ D / ( 2 k T )  [2.75] and  [2.43], the h a l f - w i d t h i s g i v e n by  ? =  (1) + 2S Z  ^2  8Tr fYBg/(9/3a ) 2  (2))/3 Z  3  2  /(2kT))  + 2cosh(D/l + w  w = 2ggH/D  co, = pK = pC(S  where  G(v).  Lorent-  have  (1 + w  From Eqs.  of  average s p i n v a l u e can be  e  Along the  i s the c o e f f i c i e n t  take the e x t e r n a l f i e l d  S(l) = (0, 0, -  S(2)  K  o r i e n t a t i o n s of the oxygen m o l e c u l a r  d i r e c t i o n , we  a r e sp f a r away  oxygen i m p u r i t i e s can a g a i n be approximated by a  z i a n l i n e w i t h h a l f - w i d t h g i v e n by term i n  shells  can be n e g l e c t e d , the l i n e shape f u n c t i o n o f a powder  sample broadened by  linear  few  2  /(2kT))  ^  120  At  T = 4.2°K ,  the h a l f - w i d t h when  D = 5.6  k  is  to, = 0.359p<; •  At t h i s  temperature, the h a l f - w i d t h i s r a t h e r i n s e n s i t i v e to the change i n  the D-parameter.  When  D = 0 ,  =  ^  Eqs. [2.43] and  [2.70] g i v e  16TT fYgg s i n h (3 gH/kT) P  9 / 3 a ( l + 2cosh(3gH/kT)) 3  = 0.399p? ,  which i s o n l y about  11%  h i g h e r than t h a t when  D = 5.6 k .  As the l i n e shape i s r a t h e r i n s e n s i t i v e t o the change of the p a r a meter D, Eq. [2.78] i s used i n c h a p t e r I to e s t i m a t e the b r o a d e n i n g o f the13 C  and p r o t o n resonance l i n e s i n s o l i d methane by  0  2  impurities.  121  5.  CONCLUSION.  The  l i n e shape f u n c t i o n f o r the resonance a b s o r p t i o n o f a  r e s o n a t i n g c e n t r e due t o inhomogeneous b r o a d e n i n g by a d i l u t e system o f p e r t u r b i n g i m p u r i t i e s was c a l c u l a t e d as the F o u r i e r t r a n s f o r m function  F ( t ) by the s t a t i s t i c a l method.  of a c o r r e l a t i o n  I f the i n t e r a c t i o n  responsible  f o r the b r o a d e n i n g i s made up o f a number o f i n t e r a c t i o n s , each o f which broadens the l i n e and i s u n c o r r e l a t e d w i t h product  the o t h e r s ,  of c o r r e l a t i o n functions corresponding  In the frequency  then  F ( t ) i s the  t o the i n d i v i d u a l i n t e r a c t i o n s .  space, the t o t a l l i n e shape f u n c t i o n i s o b t a i n e d  by  convoluting  l i n e shape f u n c t i o n s a r i s i n g from the u n c o r r e l a t e d b r o a d e n i n g i n t e r a c t i o n s . T h i s i s a u s e f u l well-known r e s u l t .  In g e n e r a l , one can s e p a r a t e t h e  b r o a d e n i n g e f f e c t i n t o a s e t o f u n c o r r e l a t e d e f f e c t s which a r e c o n v e n i e n t f o r the p a r t i c u l a r problem under i n v e s t i g a t i o n .  The l i n e shape f u n c t i o n due  to two o r more d i f f e r e n t s p e c i e s o f i m p u r i t i e s can be o b t a i n e d by c o n v o l u t i n g the l i n e shape f u n c t i o n s due t o each s p e c i e s o f i m p u r i t i e s , as p o i n t e d out by  Grant and S t r a n d b e r g  impurities,  ( 3 9 ) , and by Stoneham (38).  For a given species of  the t o t a l l i n e shape f u n c t i o n can be expressed  as the c o n v o l u t i o n  o f f u n c t i o n s , each r e p r e s e n t i n g the l i n e shape when the i m p u r i t i e s a r e a l l i n a p a r t i c u l a r i n t e r n a l s t a t e o r when the i m p u r i t i e s a r e d i s t r i b u t e d i n one of the s p a t i a l p a r t i t i o n s o f the l a t t i c e .  In g e n e r a l , i t i s more convenient two  r e g i o n s , an i n n e r  "cut-off region",  r e s o n a t i n g c e n t r e , and i t s complement.  to p a r t i t i o n the l a t t i c e i n t o  which i s a sphere around the We have shown t h a t i m p u r i t i e s i n s i d e  the c u t - o f f r e g i o n g i v e r i s e t o a side-band  s t r u c t u r e t o the resonance  line  122  and  i m p u r i t i e s o u t s i d e g i v e r i s e t o a g e n e r a l b r o a d e n i n g o f the l i n e  function. by  T h i s has a l s o been p o i n t e d o u t by Grant and S t r a n d b e r g  Stoneham (38).  We have performed d e t a i l e d n u m e r i c a l  shape  (39) and  calculations for  the case when the p e r t u r b a t i o n i s the magnetic d i p o l a r i n t e r a c t i o n and when all  the p e r t u r b i n g d i p o l e s a r e time-independent w i t h  and  a r e d i s t r i b u t e d randomly i n the l a t t i c e .  here would c o r r e s p o n d  t h e same magnetic moment  The v a l u e o f the magnetic moment  t o the time averaged moment o f the i m p u r i t i e s i n the  s h o r t c o r r e l a t i o n time l i m i t as w i l l be d i s c u s s e d l a t e r .  The n u m e r i c a l  c a l c u l a t i o n s u s i n g a f . c . c . l a t t i c e show t h a t i n t e g r a l s a r e i n t h i s  case  a reasonable  a p p r o x i m a t i o n to the l a t t i c e sums when the c u t - o f f r a d i u s i s  sufficiently  l a r g e f o r the l a t t i c e a n i s o t r o p y  out  to be averaged o u t .  I t turns  t h a t the continuum a p p r o x i m a t i o n i s v e r y good even when t h e c u t - o f f  r a d i u s i s o n l y a few l a t t i c e c o n s t a n t s . of the side-bands changes as the f i e l d  As a consequence, w h i l e  the p o s i t i o n  o r i e n t a t i o n varies with respect to  the c r y s t a l a x i s , the shape o f the main l i n e remains t h e same.  The l i n e  shape due to i m p u r i t i e s o u t s i d e the c u t - o f f volume depends on the p r o d u c t of i m p u r i t y  c o n c e n t r a t i o n and the volume i n s i d e the c u t - o f f r e g i o n , i . e .  depends on the average number o f i m p u r i t i e s i n s i d e the c u t - o f f volume. the number i s l a r g e , the shape i s p r e d o m i n a n t l y G a u s s i a n ; small,  the shape i s a t r u n c a t e d L o r e n t z i a n .  been o b t a i n e d by Grant and Strandberg  (39).  b u t when i t i s  T h i s dependence has a l r e a d y F o r a g i v e n c u t - o f f volume, the  l i n e shape i s t h e r e f o r e L o r e n t z i a n when the i m p u r i t y low  concentration i s very  and becomes more and more G a u s s i a n when the c o n c e n t r a t i o n  T h i s i s i n agreement w i t h  increases.  the r e s u l t o b t a i n e d by K i t t e l and Abrahams (52)  u s i n g the method of moment e x p a n s i o n s . to be v a l i d ,  When  However, f o r the s t a t i s t i c a l method  the c o n c e n t r a t i o n cannot be too l a r g e and o n l y the case when  123  the of  L o r e n t z i a n shape predominates  i s important.  In t h i s case, the side-bands  a s i n g l e c r y s t a l have i n t e n s i t y p r o p o r t i o n a l t o the i m p u r i t y c o n c e n t r a t i o n  and have the same shape as the main l i n e .  I n g e n e r a l , the l i n e  L o r e n t z i a n a t the c e n t r e and i s G a u s s i a n a t the wings.  shape i s  As can be a n t i c i p a t e d  from t h e r e s u l t s o b t a i n e d by M c M i l l a n and Opechowski (46) from moment expansions a t f i n i t e temperatures, t h e l i n e shape i s n o t s y m m e t r i c a l the  c e n t r e o f the u n p e r t u r b e d l i n e .  about  Rather, the p o s i t i o n o f i t s peak  i n t e n s i t y i s s h i f t e d by an amount t h a t i s l i n e a r i n t h e i m p u r i t y c o n c e n t r a t i o n and i n the magnetic moment o f the p e r t u r b i n g i m p u r i t i e s .  The whole  line  shape i s asymmetrical i n such a way as to make the f i r s t moment about the c e n t r e o f the unperturbed l i n e v a n i s h .  The s h i f t  be observed e x p e r i m e n t a l l y f o r systems  i n which  by the unperturbed l i n e . us i s 0 13  impurities  2  T = 4.2°K to  0  2  and  13  i n solid  concentration of shift  From Chapter 1, the r.m.s. w i d t h o f containing  >  = 1.6 gauss.  < M p  0  2  .  Since  < S  z  >  z  gauss, where  53%  CH4  at  The s h i f t due p  i s the molar  has a magnitude o f about u n i t y , the  to the r.m.s. w i d t h o f the u n p e r t u r b e d l i n e when  and s h o u l d be o b s e r v a b l e .  The  case i n which  the p e r t u r b i n g s p i n s have v a l u e  gated by n u m e r i c a l c a l c u l a t i o n s . a broadened  13 CH^  i s 6H = 2.5 x 102 <S >  i s comparable  p > 0.2% ,  at  CH^.  H = 20 K i l o g a u s s i s  impurities  i t i s n o t too much broadened  One system which has been s t u d i e d e x p e r i m e n t a l l y by  ' resonance f o r an 0 2 ~ f r e e sample o f  C  o f the peak i n t e n s i t y can  1/2  was  In the " l o n g l i f e - t i m e c o r r e l a t i o n  l i n e w i t h a s h i f t e d peak r e s u l t s a t f i n i t e temperatures.  i n f i n i t e temperature,  the s h i f t v a n i s h e s , and a L o r e n t z i a n shape  investilimit", However,  appears  124  when the c u t - o f f r a d i u s t h a t o f Abragam  shrinks  ( r e f . ( 1 8 ) , page 126).  each paramagnetic i m p u r i t y the  to z e r o .  l i m i t i n g r e s u l t i s the same as  I n the " s h o r t  life-time correlation  limit",  has a time-independent magnetic moment e q u a l t o  average magnetic moment p e r spin.-  dipoles  This  a r e thus d i r e c t l y a p p l i c a b l e  The r e s u l t s o b t a i n e d f o r the c l a s s i c a l  to t h i s case.  When the sample i s i n the  form o f a powder, the side-bands a r e smeared out i n t o a shape s i m i l a r t o t h a t due  t o the a n i s o t r o p i c K n i g h t - s h i f t  o f a p o l y c r y s t a l l i n e sample.  I f the  unperturbed l i n e i s s u f f i c i e n t l y narrow, these s i d e - b a n d s s h o u l d be and  t h e i r i n t e n s i t i e s and p o s i t i o n s  concentration  An low  S = 1/2  case.  At  numerically.  T = 4.2°K  s l i g h t l y w i t h the a n t i c i p a t e d ,  on the i m p u r t i t y  example i n which the b r o a d e n i n g - i s due t o  As e x p e c t e d , the g e n e r a l f e a t u r e s  t h a t i n the  CH^  information  and v a l u e o f the i n d u c e d magnetic moment.  temperatures was a l s o i n v e s t i g a t e d  S = 1.  only  give  resolvable  4.2°K .  2  impurities at  I n t h i s example, we have  o f t h e b r o a d e n i n g a r e t h e same as the l i n e shape f u n c t i o n  c r y s t a l f i e l d D-parameter f o r 0  as the average magnetic moment v a r i e s o n l y  D-parameter a t  0  varies 2  in  s l i g h t l y with the  A t lower temperatures, the e f f e c t o f the D-parameter  would be more pronounced.  125  APPENDIX  We s h a l l now show t h a t when T ,  and to  x~  where  l  z  o  - - P 1 - p  p  and  c  n  I > 3 ,  n = pN ,  E q s . [2.16],  - T ^ t  T  f o r some f i x e d  [2.18b] and [2.19] l e a d t o  '  -pN-l/2  _  ( 1  pN-1/2  N i=l  [  1  +  _ J ^ x p  f  (  Y  i  .  j  t  )  ]  j  = /2irN  i n t h e l i m i t when  N —>  oo .  To t h i s end we f i r s t  write  f ( r . ,t) = 1 - e ( r . , t ) . 3  Eq.  [2.19],  [A2.1]  3  can then be r e w r i t t e n as  pN + 1 =  N I j=l  Nz  z [ l - e(r\,t)] n  1 + z  - z e(rj,t) 0  z  Q  N  Q  Q  + z  e(r\,t)  I  ^r—7-—  rr-v— 1 + z  Q  Q j  =  [A2.2]  2  1  1  +  ^  _  v  (  7  .  >  t  )  From Eq.[A2.2], we o b t a i n a f t e r some s t r a i g h t f o r w a r d a l g e b r a i c  manipulations,  126  where  "  z  Q  = x  x  Q  =  + n  Q  ,  °  ,  (  1  [A2.3]  [A2.4]  N  ( r t)  E  Substituting Eqs. [A2.1], [A2.4] and [A2.5] into Eq. [2.18b], we have  G  „  M  ( Z O ) =  j  .  £ j-1  <*o + ? > N  p-N+1  Now,  —* <o x  ?>  +  X  N  P  N  o  ,  (x  o  +  (1 + x )  Q  P^N  1  x  + ?  ( 2 X  )  N  Q  [  1  " ^ j . t ) ]  ^  +  [ A 2  .  6 ]  j  —  o< o + C > x  N  >  t)] )  (1 + g ) ( r . , t ) [ 2 + 2x + E; - s ( r . , t ) - 2x e(r~ , t ) - ^ e C ^ . t ) ] N  j=r  2  Q  (x  w 2.  [  ? N  N  1 +  V  i  Q  1  1 - e(r. ,t) J  1  +  i 1 - e(F ,t) J £ = j = l I 1 + (x + ) [ 1 - s ( r N  and  =  f I  N  i  .PN±JL_  £  p  0  n  (1 + x ) \1 + (x + c ) [ l - e ( r . , t ) ] 2  Q  G  N  We have  g"(z ) = N { -P- 0  1  + P(N) ) ,  [A2.7]  127  where  1  P(N) = N  (*o  +  ?)  V  N  2o  x  N (1 + c ) ( r . t ) [ 2 + 2 x  1  N  £  >  j=l  N  From Eqs. [2.13],  N  > "  = R(N)S(N)x  R(N) = [1 +  and  S(N) =  j-  (2)  P(N) = 0  following  ,  ^  P^  _  N  p  N r n' 1  N  _  1  | I  Q  eC^.t) - 2x^(7.,t)-  + C ) [ 1 - t(J.,t)]  }  N  {  1  +  ( x  o  ?N  +  ) f ( Y  j'  t }  W  ^ ( r . ,t)]  2  }  ^(XQ) ,  n Q  p  -  have  -nN-1 - ] [ l+  (1 -  X  [A2.8]  P  )  C  J  N  N  ,  IN  [A2.9]  C i f C r .t) \ .  J  [1 + x  to prove t h a t ,  inequality  P  1  I  (3)  (x  2  x  where  + cN  c  = ( o + ?  0  Q  (1 + x ) { 1 +  [A2.1] and [A2.3], we  C~Vz )  B e f o r e we p r o c e e d  4>  +  R(N) = 1  Q  - x e(r ,t)](l + x 0  j  i n the l i m i t and  (4)  N —>  Q  +  oo ,  S(N) = 1 ,  ? N  [A2.10]  )  (1)  we s h a l l  £ ^ = 0, e s t a b l i s h the  :  -Jl If is  - T <_ t <_ T  sufficiently  large,  f o r some  T ,  co(r,q)  r  (£ ^ 3) ,  and  N  then N I | e ( r . , t ) | <_ M l o g N , j=l . 3  [A2.ll]  128  where  M  i s a constant  independent o f  To prove i n e q u a l i t y  N.  [ A 2 . l l ] , we  according  to the f o l l o w i n g scheme :  From Eqs.  [2.10] and  [A2.1], we  first  i > j  implies that  have  iio(r  | (r  I  ,t)| <  e  l a b e l the l a t t i c e  ,q )  t  I  p(q )|1 - e  .  ioo ( r . ,q . ) t Since  I1 - e  3  \l ^[icoCT k=l *  I  3  q)t] 3  l I irr|co(r.,q.)t| k  ]  3  k  =i  | (T U  q.)t|  j 5  < e  |u(r <  [wCr  aT  ,q  a  T  /  )t[ e  r  l  r .J  where  a max. { rr. ^| a ) ( 7 . , q . ) | „ ==max.  we have  |slr..t>|<I  {  }  aT/r^" 1  M  pCq)2|e J  R  5  1  =  ,q ) t | J  \  J  r^ >  sites  r.. .  129  aT/r* where  D =  aTe  N  N and  < j=l  We in  a unit  family  now  which  a  I  [A2.12] r.  volume  d v  I  T  r  1  r  N  J=l  t h e maximum d i s t a n c e and  s p h e r i c a l regions  <  =  denote by  has  of concentric  S  N  J  shall  cell  D  }  a  contains centred  =  1  f  between  sites.  at the o r i g i n  »--'»  Let  two S a  points be  a  :  k  a  such  that (a)  r  - r  N  <_ d  N  d+1  and  Now,  l,-..,k-l)  ,  a  (b)  number o f s i t e s  (c)  N = N  as  (a =  fc  contained  in  define  r(a)  S  is  N  ,  .  For  each  a,  we  k —>  oo ,  we  may  such  assume t h a t  that  —>  fa)  v  _  ™  .  k Comparing  this  l i m i t with  = v  ,  we  have  k  &  r  N  =  k  r  (  k  )  [A2.13]  130  Let  e  Since  a = l,...,k - 1  lim e, .. = lim [r(k) - r(k - 1)] = lim (r„  k-^  k"*  k-1  0  the value of e  Let  = r(a + 1) - r(a)  a  \  00  .  - r„  \-l  ) < d , ~  is bounded .  a  e = max  {e }  From [A2.13], there exists an integer m > 0 , such that  |r  - rfa)| <_y  N  whenever  a >_ m  [A2.14]  a We can always choose an integer  1  N  N  «  I  Q =  =  ~+ l .  I  r  I j = N _ l r.  a = 2  a  1+  a > p + 1 , we have  N  ya  j=N J  1  a=p+l j=N^_j+l r  j  Now, for each  such that  k  3=1  where  p  _1_ +1 r j  N <  3  a-1  -  a  l  N  V l  3  N  rr  . a-1  p> m  and  r(p) >_ 4e  131  N  a -  N a  - l  [ r ( a - 1) - e]  4irf —  v  r.(a)  2  r(a-l)  4TT f  r dr  f  V  r  < Q +  < Q +  a=p+l  4lTf  ^  r(a)  ^  Q  n  + MIL!  {  . 16TTf  r dr  r(p)  ( r - 2e)  2  l o g  dr r  ( r ( k ) )  ^  T  < Q + - ^3v 7- log N +  <_  64irf  l°g  N  ( r - 2e)  r ( k )  r(k)  16irf  r dr  r(a-l)  r(p)-2e  <  ( r - 2e)'  have  N 3-1  ;  2 ,  r(a) r(a-l)  T h e r e f o r e , we  A  r ar [ r ( a - 1) - e ]  f°  16lTf  r  _  l o g  I.  [ r ( p )  VI,  | log ^ |  _  2 e  ] }  16lTf  + — —  s u f f i c i e n t l y large  |  N  / \  |log[r(p) -  n  r  2e]  [A2.15]  132  From Eqs. [A2.12] and Eq.  [A2.ll] with  We  (1)  lim r  [A2.15], we o b t a i n the i n e q u a l i t y r e p r e s e n t e d by  M =  D  .  can now prove the f o u r  limits.  = 0  Denoting  y =  max j <• | l + z  Q  - z e(r\ ,t)| 0  from Eq. [A'2.5], we have  UI N  f o r some  Since  K  lim  l o  i a T ^ { l  -TT  "T7)N  i  f  ^  {  l*ol  +  1  + |z | +  N  = 0 ,  (2)  The p r o o f o f  (3)  lim  R(N) = 1  we have  l|m P(N) = 0  .  From Eq. [A2.9], we have  0  ,t)|}  yM l o g N }  [A2.16]  1  which i s independent o f  ^  + 1 * ^  lim r  N .  = 0 .  i s s i m i l a r to t h a t i n (1) .  [A2.17]  133  U - pk  f l o g R(N)= N \ l o g [1 + (1 - p)e ~I.  Expanding the f i r s t  (1 - p ) r  >,  ] - p l o g [1 +  - l o g [1 +  J  P  51 P  two l o g a r i t h m i c f u n c t i o n s i n t o power s e r i e s , and  s u b t r a c t i n g one from the o t h e r , we get  l o g R(N) = N ( l -  jJ  p) 4 2  (-if  | - log [ 1 +  N k  +  1  —  »] .  Then k-1  oo  | l o g R(N)| < N ( l - P )  2  | g  2  |  I (-D k=l  k  (  1  ~  P  - N(1  Now,  p)2u  -k >,  k+1  < na - P) ig {i 2  k-1 h  )  1  . |  i  o  , g  -  ( 1  1  j,  1  +  Ni1rnrj- r i T ^ r } +  i  + ( 1  +  "  P H P  " " tA2  we have  implying  that  l i m l o g R(N) = 0  and  l^m R(N) = 1  ^N  p  fig - } i^u  2  P  1 1  [A2.19]  181  j  134  (4)  lim S(N) = 1 .  Expanding the p r o d u c t  i n Eq. [A2.10], we  S(N) = i + I <-c r  have  I  N  £=1  j6 +-..+Jl =£ [1 + x  1  jn  J  1  N  - x e(r  Q  0  j S  t)]  1  TT  (1 + x  Q  +  ? ) N  £.=0,1  e(r ,t) N  " - x e(r ,t)]  %  £  [1 + x  Using  V = max •( >j | l+ x  Q  0  N  (1 + x  N  :  N  Q  - x e ( rf. t ) | | l + x 0  J  +  Q  ?  Q  +  ) ,  ? ) N  the above  expression  N  l e a d s to  |S(N) "  1|  I <-?/{  =I  I  }  £.=0,1  <  IV*|C I*{  I  N  ~~ £.=1  | (r t)|\...*| ? ,t)| } l N  E  I £ + . • -+£ =£ 1  N  £.=0,1  x  1  -  N I £=1  N [V|C | I N  |s(r  j=l  K(N)[K(N) - 1] K(N) - 1  J  t).|]  s (  J  A  N  J  N where  K(N) = v|?„|  I  | e (Ft)|  .1=1  .  3  From Eqs. [ A 2 . l l ] and [A2.16], we have 2  lim  Therefore,  l i m |s(N) - l l = 0  lim ^ j|  = VKM  K(N)  l o  =0  N )  and  l i m S(N) = 1  F i n a l l y , from Eqs. [ A 2 . 3 ] ,  [A2.4] and [A2.17], we have  lim  z  ^xo  From E q s . [ 1 6 ] ,  [A2.6],  [A2.8],  o  = x  l i m c_ = l i m ^ n J^OD  [A2  ]_ - p  [A2.17]-[A2.29],  -0N-1 x  = -r—2—  o  Q  we have  N  o  iSl^ + x f(r 1  J  / 2TTN  x  P-PN-1/2  J  P  j  o  ,t)]  0  ( 1  +  1 x  o>  pN-1/2 J  2  j=l  = lim /2W  [ 1  +  ^ 1  f  (  F  - p  f  t  J  -  )  ]  136  CHAPTER 3  EFFECT OF THE EXCHANGE AND MAGNETIC DIPOLE-DIPOLE INTERACTIONS ON THE  SPECIFIC HEAT OF PARAMAGNETIC  1.  SALTS AT VERY LOW  TEMPERATURES  INTRODUCTION.  The method o f a d i a b a t i c d e m a g n e t i z a t i o n o f paramagnetic proposed o r i g i n a l l y by Debye (53) and Giauque (54) and f i r s t a p p l i e d by de Haas, Wiersma and Kramers  salts  successfully  (55) has made i t p o s s i b l e to c a r r y  out  experiments i n the m i l l i k e l l v i n range o i temperatures.  the  temperature dependence o f the s p e c i f i c heat o f paramagnetic s a l t s a t  this  temperature range has become i m p o r t a n t f o r a t l e a s t  it  determines the e f f i c i e n c y  of  temperature reached by d e m a g n e t i z a t i o n .  is  still  negligible. magnetic  reason,  two r e a s o n s .  First,  o f the s a l t as a c o o l a n t , and the lower l i m i t Second, the s p e c i f i c heat  one o f the p r i m a r y thermometric parameters  For  1.  For this  :  temperatures below 1 K e l v i n ,  itself  i n the m i l l i k e l v i n range.  the l a t t i c e s p e c i f i c heat i s  The c o n t r i b u t i o n t o the s p e c i f i c heat i n the absence o f e x t e r n a l  field  i s then m a i n l y due t o the f o l l o w i n g e f f e c t s  The i n t e r a c t i o n  :  o f the i n d i v i d u a l  paramagnetic  i o n s w i t h the c r y s t a l  o f the i n d i v i d u a l  paramagnetic  ions with t h e i r  field ; 2.  The i n t e r a c t i o n  nuclear  magnetic d i p o l e and e l e c t r i c quadrupole moments ( " h y p e r f i n e i n t e r a c t i o n " ) ;  1 3 7  3.  The exchange i n t e r a c t i o n o f p a r a m a g n e t i c i o n s ;  4.  The magnetic d i p o l e - d i p o l e i n t e r a c t i o n o f p a r a m a g n e t i c i o n s .  ( We d i s r e g a r d some o t h e r much s m a l l e r e f f e c t s ) .  I n many s a l t s , t h e p a r a m a g n e t i c i o n s have a d e g e n e r a t e ground  state  when t h e y a r e f r e e and t h e s p l i t t i n g due t o t h e c r y s t a l f i e l d i s much l a r g e r than t h e t h e r m a l energy o f t h e i o n s a t  T = 1  Kelvin.  The e f f e c t o f t h e  c r y s t a l f i e l d on t h e s p e c i f i c h e a t i s then i n d i r e c t and o n l y g i v e s r i s e t o an a n i s o t r o p i c g - t e n s o r .  We s h a l l r e s t r i c t our c a l c u l a t i o n s o n l y t o such  cases.  I t has been a c c e p t e d from b o t h t h e o r e t i c a l c o n s i d e r a t i o n s and e x p e r i m e n t a l e v i d e n c e (56) t h a t , f o r s a l t s i n w h i c h t h e i n t e r a c t i o n s ( i n c l u d i n g exchange, magnetic d i p o l e - d i p o l e and h y p e r f i n e ) a r e s m a l l compared t o  kT  where  i n v e s t i g a t i o n and  k  T  i s the l o w e r l i m i t o f t h e t e m p e r a t u r e under  i s t h e Boltzmann c o n s t a n t , t h e s p e c i f i c h e a t  C  can  v  be r e p r e s e n t e d by t h e e x p r e s s i o n :  constant  [ 3 . 1 ]  w h i c h i s o b t a i n e d by r e t a i n i n g t h e f i r s t term i n t h e 'high t e m p e r a t u r e ' expansion s e r i e s of the s p e c i f i c heat,  [ 3 . 2 ]  138  Here,  as  i s well-known  A  N  n  =  (57,  58),  coefficient  of  i s t h e number o f p a r a m a g n e t i c  Hamiltonian which interactions,  includes  and  J  N  ions  in  <H >  i n the volume  the exchange,  i s the e f f e c t i v e  magnetic  spin  H /(2J + 1 ) ; n  = Trace  n  V  ,  N  and  H  dipole-dipole,  q u a n t u m number o f  the  i s the and  hyperfine  paramagnetic  ions.  One to  would  the energy  expect  that  r e p r e s e n t e d by  H  at  temperatures  T  ,  the behaviour of  such C  that  kT  will  v  is  deviate  comparable from  -2 the  T  law.  paramagnetic deviations  we  terms shall  salts  (59).  theoretical two  Recent  at m i l l i k e l v i n To  point  experiments  see what  of view,  on  temperatures  sort  we  the s p e c i f i c  calculate  i n t h e e x p r e s s i o n r e p r e s e n t e d by disregard  i n fact  of deviations  shall  o f a number show  c a n be  [3.2].  expected  In our  interaction,  of  considerable  the c o e f f i c i e n t s  Eq.  c o m p l e t e l y the h y p e r f i n e  heat  from  of the  the first  calculations,  since  eventually  we  s h a l l a p p l y our t h e o r e t i c a l r e s u l t s o n l y t o the c a s e o f c e r i u m magnesium n i t r a t e i n which the h y p e r f i n e i n t e r a c t i o n of the cerium ions i s absent. For that  ions with hyperfine  interactions,  i t has  the c o n t r i b u t i o n s  to the s p e c i f i c  heat  interaction  and  that  due  been p o i n t e d C  t o t h e exchange and  v  due  out by  to the  magnetic  Daniels  (60)  hyperfine  dipole-dipole  interactions  _3 are we  additive shall  up  to the term  assume t h a t  T  i n Eq.  the paramagnetic  [3.2].  ions  q u a n t u m s t a t e w h o s e a n g u l a r momentum i s  J  In  the f o l l o w i n g  are a l l a l i k e .  They  are  and  calculations,  are i n the  assumed  to form  same a  139  Bravais lattice with an axial symmetry and the interaction consists of exchange and magnetic dipole-dipole interactions, both of which are anisotropic.  The coefficients  X  2  » A  3  s  and  X^  have been calculated by Van  Vleck (61) and Joseph and Van Vleck (62) for the less general case of isotropic magnetic g-factor, isotropic nearest neighbour exchange interaction, and a simple cubic lattice.  Opechowski (63) obtained expressions for  with the most general anisotropic interactions.  A  2  Daniels (60) ignored exchange  interaction and obtained general expression up to  X^  for a pure magnetic  dipolar interaction, but he gave numerical values only for the  X^  coefficient.  Marquard (64), in a recent paper which we saw only after completing our calculations, gave general formulae for the coefficients  X^ , X^ ,  and  X^ . However, when evaluating them numerically (for the case of GdCl ), he 3  assumed magnetic isotropy.  _3 After obtaining a general expression for  C  y  to the term  T  in  the next section, we shall specialize the expression to the case of cerium magnesium nitrate (CMN)  in Section 3.  A l l the general expressions given below  have already been published by us (65) together with the numerical computations of the coefficients  Aj_  ( i =2,  3).  However, these numerical computations  have been based on the values of the lattice constants of CMN which, according to the latest crystallographic data, are not quite correct.  In this thesis the  numerical computations have been redone using the new more accurate lattice constants (66).  It should be added that, very recently but quite indepen- .  dently, D.J. O'Keeffe (67) has calculated  X  2  ,  A  3  and  X^  for magnetic  140  dlpole-dipole  i n t e r a c t i o n without  taking  i n t o account exchange  and has computed them n u m e r i c a l l y f o r the case o f CMN. chosen, because paramagnetic  (59, 68) :  (1)  The s a l t  i t i s used most e x t e n s i v e l y ,  s a l t s , as a thermometer i n the m i l l i k e l v i n range,  amount o f s p e c i f i c heat d a t a have been c o l l e c t e d down to about (3)  deviation  and  (4)  magnetic  o f the s p e c i f i c h e a t from the  the c o n t r i b u t i o n dipole-dipole  CMN i s  among (2)  i n t e r a c t i o n rather  l a t t i c e sums o f t h e rhombohedral  are  calculated  law has been o b s e r v e d ,  than exchange i n t e r a c t i o n  a s s i g n e d a v a l u e which corresponds  from (60, 6 8 ) .  l a t t i c e , which i s the l a t t i c e o f CMN,  n u m e r i c a l l y by means o f a d i g i t a l computer. (59) down t o about  large  0.003K,  to the s p e c i f i c heat comes p r e d o m i n a n t l y  The  w i t h experiment  T  interaction  0.004K  The r e s u l t s  i f the exchange c o n s t a n t i s  to a f e r r o m a g n e t i c i n t e r a c t i o n .  At  lower temperatures, h i g h e r o r d e r terms i n the expansion would have t o be taken i n t o  account.  agree  141  2.  GENERAL THEORY  In the absence o f e x t e r n a l magnetic f i e l d s , system o f  N  the H a m i l t o n i a n o f a  i n t e r a c t i n g paramagnetic i o n s i s g i v e n by the f o l l o w i n g  expression : H =  I  H  [3.3a]  H.. '= VV.J'J. + D..  r and  D. . = —  Here the terms  V.. J . - J . ij x j  paramagnetic i o n s  i  and  \  y,-y.  a r e the i s o t r o p i c j  w i t h exchange  are t h e i r magnetic d i p o l a r i n t e r a c t i o n .  ,  [3.3b]  3(y*. . r . .) (y".-r. .) > ±—±3 J — i — I .  [3.3c]  3  exchange  i n t e r a c t i o n between  constant J\  and  a  n  d  t  '  i e  t  e  r  m  s  y"j_ a r e r e s p e c t i v e l y  the t o t a l a n g u l a r momentum and the magnetic moment a s s o c i a t e d w i t h i o n i , and  7^  i s the v e c t o r j o i n i n g i o n i t o i o n j .  From the p a r t i t i o n  function  Z = tre-«  where T  k  Z  o f the system w i t h  / k T  i s the Boltzmann c o n s t a n t , the s p e c i f i c h e a t  can be c a l c u l a t e d by means o f the f o l l o w i n g r e l a t i o n  C  v  a t temperature  (61) :  142  c  Formally  Z  _v  i-a_  =  T  2 9 i o ^  can be expanded i n powers o f  Z =  I  (trl)  T ^  ,  (-l) ^  that i s ,  (^) <H >  n  n  n  ,  [3.5]  n=0  where  <H  > =  — — trl  N and  I  i s the i d e n t i t y  I f we vanishes  operator with  sbstitute  identically  [3.4]  Eq.  (which  we  t r l=  [3.5],  to Eq.  shall  (2J +  I  Our  task i s then  The the  Nk  k  to evaluate  <H  I  Hk  2  2  >  using  show l a t e r ) , we  ^-.<»6. -2.<H!» -3 N  1)  that  get  ....-;  +  <H>  [ 3  .  6 ]  3  3 <H >  and  the fact  m a g n e t i c moment a n d t h e g - t e n s o r  .  o f an i o n i a r e r e l a t e d  by  equation,  y  where  g  ia  =  3  i s t h e Bohr magneton.  E i«Y 8  Y  Using  J  Y  Eq.  '  fa  Y  =  [3.7], we  x  »-y»'  z )  express  >  Eq.  [ 3  '  7 ]  [3.3b] i n  143  a form which i s more c o n v e n i e n t f o r l a t e r c a l c u l a t i o n s  H  ij  £  =  K  ijaB ia j@ J  >  J  -..  ....  [3.8a]  .2 a  n  d  K  ijaB  =  V  ijag  3  +  3  l  (  g  " ~T~  isa je& g  IJ  (Here we n o t e t h a t  K .^  respect  In terms of the  to  8  iaa jpB ija ijp 8  r  r  i s symmetrical with respect  ±  a, g ) .  I i j  K .^ ±  's,  <W>  to  ,  )  [ 3  i ,j  <tf > 2  ,  '  8 b ]  and w i t h  and  <H >  can  3  be r e w r i t t e n as :  [3.10]  c,'e'  <  H  >  i " > j " J>y  =  i j j , a  a  .  K  iJ«3 i'j'a'g' i"j"a"g" K  K  t r ( J  ia jB i'a' j'g' i"a" J  J  J  J  g V,'g"  J „ „)/(2J + 1 ) . [3. N  j  g  From the i d e n t i t y  t  it  r  J  ia  =  0  f  o  r  a  1  1  i s obvious t h a t f o r  i  i  and  a ,  ^ j , < ..> H  = 0 ,  implying  <H> = 0 .  Similarly  144  2  we can see t h a t o n l y one k i n d o f term kinds  identities  t  and  3 H. .  o f terms,  r  (  2  to  <H >  3  and H..H..H. . ,  c o n t r i b u t e t o <H > .  ^  J  (  J+  1  )  (  2  J+  1  )  N  +  and two  Using the  *ag  1  t r ( J . J.„J. ) = \ J ( j + 1 ) ( 2 J + 1 ) i a i g iY 6 J  where  contributes  g i v e n by Rushbrooke and Wood (58) :  i a V  J  H.. ij  e' agY  N + 1  y  e „ agY  ,  i s d e f i n e d by  1  i f (agY) i s an even p e r m u t a t i o n o f  -1  'agY  i f (agY) i s an odd p e r m u t a t i o n o f  0  (xyz) , (xyz) ,  otherwise ;  we g e t  <H > = ± J 2  2  ( J  +  I  1 ) ^  K ijag 2  i>J a,(  <H > - I J 3  9  3  ( j 4" i )  "  y  y  [3.12]  '  K  i>j>k ,g,Y  1  K J  a  a  — J (J + l ) 36 2  2  I  I  i>j i  ( I  a  K.  J^^kiYa  B  K..  ) + 3  - 3( y K.. a  Eqs.  [3.6],  [3.12],  1  K..  J  2  a,g,Y  l j a a  J  a  1  ) y (K..  FL  a  g,Y  j  B  and [3.13] then g i v e a g e n e r a l  Y  3  a  3  K.. l  j  3  Y  K..  V  0  l  j  Y  a  ) I 2  [3.13]  /  expression  for  C  y  up t o  145  the term  T  _3  We l a t t i c e with  f o r a general  s h a l l now  We  a l s o choose a c o o r d i n a t e system which  the d i r e c t i o n s o f the p r i n c i p a l axes o f the c r y s t a l .  a l l i o n s , we  Introducing  assume t h a t the paramagnetic i o n s form a B r a v a i s  a x i a l symmetry.  coincides with for  lattice.  Then  have  8  ia3  8  1  < S  8  2  ag a ' 8  8  x  '  8  3  8  i |  the a b b r e v i a t i o n s ,  >. . iia  =  1 g r. . r . . °a i i a  1 2 Q. . = g r. . ii a r. . a i i a ij  and  J  We  obtain  2  K. .  . =  (  V r  Substituting this expression but  for  2  8 + V. . ) 6 2  -  3 ij  6  a  and  <H >  - — r  we  P. . P.  3 l j a ljf i j  i n t o Eqs. [3.12] and  straight-forward a l g e b r a i c manipulation,  <H >  0  i j a3  [3.13] and a f t e r a l o n g  finally  o b t a i n the  i n terms o f e x p l i c i t l a t t i c e sums :  expressions  146  z. 2  <H > 2  = |  eV  (J + i )  ^  i"I >>i-1 rTT.  + 9(g  |e j 2  (J + i ) (  2  2  2  2  2  ) I  2  s  3z.  v.  I  -4)  g  ^ i.  2  2  4  2  (1 -  i>j r  ) + ^ J (J + l ) 2  2  [3.14]  2  i>3  r. .  2  <« > = \ ^ +  ^ {  3  i  - f + ^ i ^ ± > j  . ij r  ± > j  r  ij  6 (gl -g ) 2  2  I  +  f j> <j  +  D'{  T  Z  6 ?  <• , i>i>k J  J  T T  r..  .  ~ 4 7  [  i  '-r^r r  E -  4 i + g;  r. . ki 2 7  + ecgi -  1  8  r  r t>3>k  '  r . .r., r; . i j jk k i  jk k±  2.  J  8|>-T  9  ^ k' ki ?  ) (  g  Q k"\l  ^ ) %  +  \ . f ]  9(?  r j k  V. .V., . 3z - i L J ^ (1 - - ^ i ) + 3 . £ r  L k i  ki  )  j  ' V V ^ k l ' V ' V V  2  2,2  3  + <<£ -  4  +  2 7  ± > 2 > k  ^  V }  i>i  j  ^  i j  r  - I  r..  +  +  2  2  1  i>i  (gi - g )]  +  .  V. , 3Z .. d - ^ ) 2  +  £ g 2  [ 2  V  • V  V  )  V  1  "  147  3.  THE  CASE OF CERIUM MAGNESIUM NITRATE  In  specializing  magnesium n i t r a t e , assumption,  CMN,  the Eqs.  [3.14] and  (Ce Mg (N0 ) •24 2  3  3  1 2  [3.15] to the case o f c e r i u m  H 0), 2  we  s h a l l make the  which i n t h i s case i s p r o b a b l y j u s t i f i e d ,  simplifying  t h a t the exchange  i n t e r a c t i o n i s p r i m a r i l y i m p o r t a n t o n l y between n e a r e s t n e i g h b o u r s , t h a t i s , we  s h a l l put  otherwise.  V.. = V The  when i o n s i and  cerium i o n s i n CMN  i are nearest neighbours  form a rhombohedral  and  V..  = 0  l a t t i c e w i t h the l o n g  d i a g o n a l of each rhombohedron b e i n g p a r a l l e l to the t r i g o n a l a x i s o f the crystal.  The  rhombohedral  c e l l whose dimensions  are  unit c e l l  can be i n s c r i b e d i n t o a nexagonal  unit  (66)  a = 11.004 A  o c = 17.296 A  and  The of  distance  d  between the n e a r e s t n e i g h b o u r s , i . e . the l a t t i c e c o n s t a n t  the rhombonedral  lattice,  i s then  = 0.780a  I f we  measure the magnitude and components of v e c t o r  and use the f o l l o w i n g a b b r e v i a t i o n s i n Eqs.  [3.14] and  r„  t a k i n g a as  [3.15] :  unit  148  -g J(J + 1) x  T = a k 3  w  n  Va  J  2  2  H  a 9a  3  and  2  Y =  we get  <H > 2  -2 c 2  j  Nk  (5 + Y ^ ) S  - 6(1 - Y ) ( 2 - Y ) S 2  1  4  - 12(1 - Y ) 2  d  <W >  T  3  Nk  J  6J(JT1T  /  Y  2  ^  S  7  T  -jC- 7 +  [  2  6 r  2  +  Y  " Y )d 3  6 Y  18w  2  ;  2  [3.16]  3 2 -^-(1-Y)]w 2  Z  - ^ ) w d  2  d  +  2  2  2  3a  +  +  +  (1 ~ — ) w d  3  + 9(1 - Y ) S  2  2  -  3w  +  3  2  )S  5  + 3(1 -  6 Y  )S  6  + 9[S  y  - S  + Y\S  1 Q  8  +  Y (S 2  - 3S ) 1 2  8  "  3S  11>  + Y (S 6  9  +  - S  1 3  )]  149  + w[(-  + Y )S . +  4  IF  6(1 - l )S k  LLF  15  3w (1 - Y ) [4 + — ( 1 ' - — ) 4d d 2  2  3  where  z  i s the z-component of  d.  2  +  9(S + 2Y S + Y \ ) ]  + — ( 1 p 3  The l a t t i c e sums  2  16  8  1?  - '—')•] } p  ,  [3.17]  2  (k =  18)  and t h e i r numerical v a l u e s , which were c a l c u l a t e d by means of an IBM 7044 computer, are l i s t e d below :  S 1  S 2  = J JH  = 39.334 x; 1  J  = I z . / r ? . = 14.911 J ' 2  l j  1 J  S q  = I ?./  S  = J l / r . = 64.785  z  = 7.303  r  9  J  S  S  5  6  =  =  J = 710.540 3 3 3 j ,k r. . r., r. . xj j k kx , . L  ^  7  C  j ,k .rf .rf. r f . xj j k kx 3  =  2  4  8  -  6  2  9  150  .ik ki- -».ikyki  (x  S, =  t  3,  3  =  )  5  (x.. x, . + y ..y,. . ) z . , z, .  i  ^  \  j,k  j,k  y  > ;  z jk k i . 2  i.i .ik  ( x  x  +  yjiyik^^k^ki i-  ij  + y . .y  xj j k  =  y  +  (x. . x  )z  M  5  2  36.867  r ? . r ? . r, . xj j k kx 5  y  (  = 128.899  x  k i i j x  +  y  ki ii y  5  y.,y .) jk^ki ki ij 1  '.ik ki J  /  ;  2 z, . z. .  j k kx x j = 3.901  r?.r?.r. . xj j k kx  2  )  jk k i  'li'ik  n , ^2 z . . z .. z, . ^1 -Ik k x =  y  jk k i  ) (x.. x, . +  + y . .y  M  i - l -Ik  i k k i  y  r-  'lrjik jkkx 5 5 i5r c c r r r ij  „  I j,k  i-  r ^ r~" r  j,k  '13 =  = n.711  = 93.999  j ," k  12  i  3  V  £  k  r . r ^ rf. xj j k kx  j,k  =  k  2  ( x . .x...  S  i  i  L>  10  11 ** "*  k  3 5 5 r r ij jk k i  r  z  s  = 187.138  5  r . . r ., r. .. x j j k . kx  j,k  s  +  Z  z  = -7.324  )  =  70.634  151  z  'S,c =  = 43.585  M k  r  5 a  =  s  16  2  j  £  (  k ki r  *,ik*ki  ,.>. f ^ k  r  +  - , k ki y  y  )  1  =  4  5  >  7  2  3  5 5 jk ki r  (x.. x, . + y.. y. . ) z . . z. .  S  -  I  l-^-^  , ~ z.. z, 2  ia=  ; U  (j) J  Where  kk  -1  l  kk l  • 7.587  jkk i  J  s  -1  r ? , r. .  (j) k  5  2  H T T - «•<>««  k r  r. . jk k i  t h e symbol  £  means  that  summation  i s taken only  over  those values  of  (J) j  such  that  i o n j i s a nearest  The  sums  S-^  to be a t t h e o r i g i n .  (k =  where with in for  d d  each  (m = 1,  m  = d,  m  sum  which  and S  k  1, ...,18)  A l l other  r  2, n  3)  m  neighbour of i o n i .  ions  have  = r ^ d j+ n d 2  are the basis are integers.  i s characterized  are evaluated  2  their  + n d 3  given  the i o n i by  ,  o f the rhombohedral  T h e number  by an i n t e g e r  taking  positions  3  vectors  by  of terms  such  that  taken  lattice  into  a l l those  account  terms  152  -n  are by  taken i n t o  account.  (  k  )  l  n  m  l n  An a c c u r a c y  C  k  (m = 1, 2, 3)  )  s u f f i c i e n t f o r our p u r p o s e s was  obtained  taking  and  n  ( k )  =40  for  1 < k < 4 ,  n  ( k )  =4  for  5 < k < 13 ,  = 20  f o r 14 < k < 18 .  n ^  The Daniels  sums  S-^ , S  2  ,  and  S3  have.already  ( 6 0 ) , a n d much more a c c u r a t e l y b y P e v e r l e y  been  computed b e f o r e b y  and M e i j e r  (69).  Peverley o  and  M e i j e r have used  the old values  of l a t t i c e  constants  ( a = 10.92 A ,  o c = 17.22 A ) . with  those  of Peverley  constants used  recent  f o r Sj , S  ,  and  the  o l dvalues  of lattice  of  S-L , S  and  a  and  2  ,  S3  of S^  ,  2  and  i f we a l s o  S3  agree  almost  use the o l d values  exactly  of lattice  I n t h e above n u m e r i c a l c o m p u t a t i o n s , we h a v e o o a = 11.004 A a n d c = 17.296 A , a n d t h e agree w i t h i n  constants  (60, 6 5 ) .  are not sensitive  1%  with  This  to small  those  shows change  obtained  that  using  the values  i n the values o f  c .  We now T = 1.187 x 1 0 [3.17].  65).  values  2  f o r S-^ , S  and M e i j e r  (see reference  t h e most  results  Our r e s u l t s  _ 3  substitute K  Neglecting  ,  the values  J = 1/2  and t h e v a l u e s o f  terms c o n t a i n i n g  S  k  , into  Y ( n > 2), n  g^ = 1.84  we  ,  Eqs. [3.6], obtain  g  /(  = 0.1 ,  [ 3 . 1 6 ] , and  the n u m e r i c a l  153  e x p r e s s i o n f o r the s p e c i f i c heat  of CMN  i n CGS  units :  [3.18]  where  t> = 6.533 + 10.720y  b  and  T  3  i s expressed  to  b  experimentally s m a l l e r than  b  3  ,  2  <  0.1/1.84 ,  3%  discrepency to  S  1 8  we  put  values of  b  Y = 0  2  for  b  2  [3.18],  and  i n these  b  can be n e g l e c t e d .  2  + 1.115w  g i v e n by  70), gj  (  Since, seems to  be  equations.  g i v e n by  3  equation  O'Keeffe  and much l a r g e r f o r  b  3  .  (67). The  [3.18] f o r The  w = 0  discrepency i s  o r i g i n of  this  i s not easy to f i n d because i n O'Keeffe's c a l c u l a t i o n s the sums  d e f i n e d above do not  We  2  the c o n t r i b u t i o n o f terms c o n t a i n i n g  as g i v e n by Eq.  do not e x a c t l y agree w i t h those of about  ,  - (32.457 + 37.070Y )w + 1.266(1 - Y ) w  2  (from paramagnetic resonance data) (60, 0.1,  The  2  in millikelvin.  Y = g^/gji  and  2  + 1.409w  2  = -25.366 - 95.466Y  'If Y  + 0.703w(l - y )  2  2  s h a l l next  e x p r e s s i o n of Eq.  [3.18]  occur  S^  explicitly.  determine the v a l u e of ( i n which we  put  w  by  comparing our  Y = 0 ), i . e . ,  theoretical  3  ,  154  C  Nk  b  and  b  with  b  v  ^  =  b  ^I  +  T  I  value  data  U  N  I  °  T  1.409w  ^  F  + 1.115w  2  o f M e s s e t a l . ( 5 9 ) . We heat  (4 + n)mK  n = 0, 1,  then  fitting  [  3  -  1  9  J  2  of the s p e c i f i c  Eq. [ 3 . 1 9 ] , a n d we by thus  N  = -25.366 - 32.457w + 1.266w  3  T =  obtained  (  = 6.533 + 0.703w +  2  the experimental  experimental  into  2  a t each  ,  3  first  substitute the  temperature  16  s o l v e Eq. [3.19] f o r w.  Eq. [3.19] a t v a r i o u s  The v a l u e s  temperatures  of  w  are plotted  versus  -4 the or  temperature  curves  and  the i n t e r v a l  represented  Due t o e i t h e r  5.  b  are and  and w h i c h  2  T  from  4 mK  w = -0.122  w = -0.438  and into  and  a r e marked  gives b  3  by  to  the value 20 mK.  w = -0.438  i n F i g . 34 b y  A  of the  T  of  term w  varies  T h e maximum respectively.  Eq. [ 3 . 1 9 ] , we o b t a i n t h e two  i n F i g . 34 t h e two c u r v e s  w = -0.249  coefficients Table  w  of  the effects  points o r both,  by t h e p o i n t s marked  we g i v e  w = -0.249 of  of  w = -0.122  addition,  choice  in  over  minimum v a l u e s  By p u t t i n g  In  i n F i g . 33.  the uncertainty of the experimental  considerably and  T  which  and  the smallest value  , respectively.  correspond  x ,  of  f o rthe four d i f f e r e n t  • and o  b  to  respectively.  2  .  values  The v a l u e s of  w  w = 0 The ofthe  are listed  0.4-  0.2  0  -1  I  I  l_  10  '  <  I  I  15  I  L  20  T (m K ) Fig.  33.  Values of  -w  o b t a i n e d by  et a l . (59) at v a r i o u s  f i t t i n g Eq.  temperatures.  [3.19] to e x p e r i m e n t a l data of Mess  .005b  .002  .001 F i g . 34.  100 S p e c i f i c heat o f CMN as f u n c t i o n o f - temperature. The s o l i d curve r e p r e s e n t s e x p e r i m e n t a l d a t a of mess e t a l . (59). C a l c u l a t e d v a l u e s are i n d i c a t e d by P  A ^ O O  -0.438,  ' *' ' respectively. X  a  n  d  0 >  c o r r e  sponding  to  w = 0, -0.1222, -0.249,  and  T a b l e 5.  w  b  0  6.533  -25.366  -0.122  6.468  -21.381  -0.249  6.445  -17.223  -0.438  6.495  -11.003  Values of (T  b  2  and  i s i n unit of  b  3  mK)  2  for single crystal  CMN.  158  S i n c e we term the In  do n o t know, o f c o u r s e ,  becomes l a r g e r  than  w  the effect  the uncertainty of experimental  q u e s t i o n open as t o what v a l u e any c a s e ,  where  i s certainly  of  w  should  p o i n t s , we  be regarded  negative, which would  of the  indicate  T  -4  leave  as t h e b e s t . a  ferromagnetic  interaction.  It w = -0.417  may b e p o i n t e d o u t t h a t  i n F i g . 34. c o r r e s p o n d i n g  to  i s g i v e n by t h e formula  =  Nk  which  the curve  i s not very  much  6.495 _ 11.003 T  2  different  v  Nk  =  T  from  6.4 2  3  the formula  12 T  _ , _ „ T > 6.5 mK -  3  g i v e n b y A b r a h a m a n d E c k s t e i n (71) i n t h e c o n c l u d i n g p a r a g r a p h  of a recent  paper.  data  In that paper,  the authors  h o w e v e r do n o t seem t o b e d i f f e r e n t  6.5  mK.  used from  their those  own e x p e r i m e n t a l o f Mess  which  e t a l . (59) a b o v e  159  APPENDIX  The rather vague statement on p.152  "An accuracy s u f f i c i e n t for  our purposes etc. ..." i s not meant to imply that the numerical values of the l a t t i c e sums  S , 1  S.,  as given on pages 149-151 are correct to  0  lo  the t h i r d decimal d i g i t .  It only implies the author's b e l i e f that these  values when substituted into the f i n a l formulae for the s p e c i f i c heat w i l l give values whose uncertainty i s less than that of the experimental values. Dr. J.R.H. Dempster has pointed out to the author that i n the case of some of the l a t t i c e sums one can make more precise statements about the uncertainty of these numerical values by considering the rounding-off errors i n some d e t a i l .  Let us f i r s t consider  .'"-Since all^jterms are p o s i t i v e , the  contribution due to terms outside a sphere with radius  h  can be  approximated  by an i n t e g r a l  T  -  h " V  dr  —  h r  4ir 3Vh  3  9.22 3  where  a = 1  lattice.  and  V  i s the volume of a unit c e l l of the rhombohedral  The accuracy of the computer i s  7  digits.  If the summation i s  done i n which terms are arranged i n descending order of their magnitudes,  160  then a l l terms in  with value less than  S  ±  x 10  - 39.334 x 10  7  7  » 4 x 10  are automatically disregarded by the computer. outside a sphere with radius  h,  ±- = 4 x h  IO"  6  This implies that terms  such that  6  ,  6  were discarded, resulting in an underestimate of  9.22 x / 4 x 10  6  . 2 x 10  by about  2  (In the actual summation, not a l l terms are in descending order, and the error may be less).  Since the sphere with radius  h  is contained inside  the region of summation defined by  -40 _< n-^, n , 113 <_ 40 2  (page 151),  we have not introduced any additional error (which would be the case i f n^"^  had been taken too small).  39.33 < S  ±  We thus have finally  - 39.35  161  S i m i l a r a n a l y s i s can g i v e c o r r e c t e d v a l u e s As t o t h e double sums outside  S^,  S^,  of  errors introduced  S , S3,  and  S^ .  by n e g l e c t i n g  terms  2  t h e r e g i o n , which i s d e f i n e d by  -4 <^ i > 2 » 3 — n  n  >  n  may be more i m p o r t a n t than the r o u n d i n g - o f f can be e s t i m a t e d  In p r i n c i p l e , both e r r o r s  by e v a l u a t i n g a double i n t e g r a l o f the form  16ir V  o v e r the domain  errors.  D :  2  f(r  l S  r )dr dr 2  x  2  2  r , r  2  >_ h ;  |Tj - r~1 2  >_ 1 .  Due t o the d i f f i c u l t y  i n e v a l u a t i n g the i n t e g r a l , we have note., a t t e m p t e d ^ e s t i m a t i n g the e r r o r s i n these double sums.  The e r r o r s may be one o r two o r d e r s  than t h a t o f  L a t t i c e sums  S^ .  p r e c i s i o n , which has  15  S^g  d i g i t s of accuracy.  o f magnitude l a r g e r  were e v a l u a t e d  using  double  R o u n d i n g - o f f e r r o r s here a r e  n e g l i g i b l e as compared t o e r r o r s a r i s i n g from n e g l e c t i n g terms o u t s i d e the r e g i o n d e f i n e d by  -20 <^ n-^, n , n ^ £_ 20 2  We have r e a s o n t o b e l i e v e t h a t the e r r o r s a r e about a few t e n t h o f S  14'  S  18  *  1% f o r  162  -The rounding-off errors in the lattice sums of Chapter II (Eqs.  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