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Nuclear magnetic resonance in single crystals of tin and aluminum. Jones, Edward Peter 1962

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NUCLEAR MAGNETIC RESONANCE I N SINGLE CRYSTALS OF T I N AND ALUMINUM by EDWARD PETER JONES B . A . S c . , U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1958 M.Sc, U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1959  A T H E S I S SUBMITTED I N P A R T I A L F U L F I L M E N T OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n t h e Department of PHYSICS  We a c c e p t t h i s  t h e s i s as conforming t o the  required standard  THE UNIVERSITY OF B R I T I S H COLUMBIA November, 1962  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study.  I further agree that permission  for extensive copying of this thesis for scholarly purposes may  be  granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  Department The University of British Columbia, Vancouver 8, Canada. Date  i  1  . vnCS  - i i -  ABSTRACT  Nuclear magnetic resonance  studies  i n single  c r y s t a l s o f a l u m i n u m a n d t i n h a v e b e e n done a t l i q u i d temperatures* a function field  The K n i g h t s h i f t  i n a constant  f o rd i f f e r e n t values of f i e l d Knight s h i f t  the f i r s t time. also studied in  i n t i n has been s t u d i e d  of c r y s t a l orientation  anisotropic  and temperature.  i n t i n was o b s e r v e d  t h e magnetic f i e l d .  been c a l c u l a t e d  The  directly for  t o depend on t h e c r y s t a l The s e c o n d  as  magnetic  The l i n e w i d t h o f t h e t i n r e s o n a n c e  and found  helium  was  orientation  moment o f t h e l i n e h a s  i n terms o f d i p o l e - d i p o l e  i n t e r a c t i o n s and  i n d i r e c t exchange i n t e r a c t i o n s between n u c l e i o f d i f f e r e n t m a g n e t i c moments a n d c o m p a r e d w i t h t h e e x p e r i m e n t a l r e s u l t s . The external  field  K n i g h t s h i f t was s t u d i e d  f o r b o t h t i n and aluminum i n a s e a r c h f o r  de H a a s - v a n A l p h e n t h e s e was f o u n d . determined  as a f u n c t i o n of  type o s c i l l a t i o n s .  No i n d i c a t i o n o f  An u p p e r l i m i t f o r t h i s e f f e c t was  f o r each sample.  -vi-  ACKNOWLEDGMENT  I w i s h t o e x p r e s s my s i n c e r e g r a t i t u d e t o D r . Myer B l o o m f o r t h e v a l u a b l e i n s t r u c t i o n a n d i n s p i r a t i o n he p r o v i d e d t h r o u g h o u t  my c a r e e r a s a g r a d u a t e  s t u d e n t , and  f o r h i s s u p p o r t i n o b t a i n i n g f i n a n c i a l a s s i s t a n c e f o r my studies. I w i s h a l s o t o e x p r e s s my g r a t i t u d e t o D r . D. L I . W i l l i a m s f o r h i s h e l p i n t h e m e a s u r e m e n t s r e p o r t e d i n t h i s w o r k a n d f o r many u s e f u l p a r t i c u l a r l y a s r e g a r d s work i n s i n g l e  suggestions  crystals.  My t h a n k s a r e d u e t o D r . R o g e r Howard who i n t r o d u c e d me t o t h e I B M 1 6 2 0 c o m p u t e r a n d who c a r r i e d o u t t h e computation  d e s c r i b e d i nChapter  4.  I w i s h t o acknowledge t h e f i n a n c i a l  support  provided by a f e l l o w s h i p given by t h e I n t e r n a t i o n a l  Nickel  Company o f C a n a d a a n d r e s e a r c h f u n d s made a v a i l a b l e b y t h e N a t i o n a l Research  Council.  - i i i -  TABLE OF CONTENTS CHAPTER  Page  1  INTRODUCTION  1  2  THEORY OF THE KNIGHT SHIFT  4  3  EXPERIMENTAL APPARATUS AND TECHNIQUES  4  THE KNIGHT SHIFT AND LINE WIDTH OF THE TIN SINGLE CRYSTAL  5  17  27  THE POSSIBILITIES OF DE HAAS-VAN ALPHEN TYPE OSCILLATIONS IN THE KNIGHT SHIFT  45  APPENDIX A  THE SECOND MOMENT OF THE RESONANCE LINE DUE TO MISALIGNMENT OF THE CRYSTAL SLICES  B  52  THE ANISOTROPY OF THE NUCLEAR MAGNETIC RESONANCE IN WHITE TIN:  E. P. JONES  AND D. L I . WILLIAMS C  CIRCUIT DIAGRAMS OF PARTS OF THE SPECTROMETER  REFERENCES  55 57  62  -iv-  L I S T OF  ILLUSTRATIONS  FIGURE  Page  1  B l o c k Diagram of the Spectrometer  2  S c h e m a t i c D i a g r a m o f t h e Low  18  Temperature  System  21  3  D e r i v a t i v e o f t h e Sn R e s o n a n c e  26  4  D e r i v a t i v e of the A l Resonance  26  5  The  Knight  Shift  i n T i n as a F u n c t i o n  of  the C r y s t a l O r i e n t a t i o n i n the Magnetic F i e l d 6  P l o t s of  £2LI1  28  and  d  (  X  '  d  The  Z X 1  "  )  for  d  L o r e n t z i a n and 7  +  Gaussian  L i n e Shapes  L i n e W i d t h o f t h e T i n Resonance as  31 a  F u n c t i o n of the C r y s t a l O r i e n t a t i o n i n the Magnetic F i e l d A 8  P l o t of A2  9  The  Knight  33  2  a s a F u n c t i o n o f kt? *  Shift  41  i n T i n as a F u n c t i o n  of  Temperature 10  The  Knight  Shift  43 i n T i n as a F u n c t i o n  of  Magnetic F i e l d 11  The  Knight  Shift  43 i n Aluminum as a  of Magnetic F i e l d  Function 50  - V -  12  The K n i g h t  Shift  i n T i n as a F u n c t i o n  of  Magnetic F i e l d  51  Cl  The P o u n d - K n i g h t - W a t k i n s O s c i l l a t o r  C2  M o d i f i e d T e k t r o n i x 162 Waveform Generator  C3  Phase S h i f t e r  '  59  and H o r i z o n t a l A m p l i f i e r f o r  t h e T e k t r o n i x 360 O s c i l l o s c o p e C4  58  Phase S e n s i t i v e Detector  60 61  CHAPTER I  INTRODUCTION  The  change i n t h e n u c l e a r magnetic r e s o n a n c e  quency o f a n u c l e u s in  i n a m e t a l f r o m t h a t o f t h e same  fre-  nucleus  a n o n - m e t a l l i c s t a t e was f i r s t e x p l a i n e d i n t e r m s o f a n  electron-nucleus  i n t e r a c t i o n b y Townes, H e r r i n g a n d K n i g h t . 1  T h i s change i n resonance frequency,  the Knight  s h i f t , has  been s t u d i e d e x p e r i m e n t a l l y and t h e o r e t i c a l l y q u i t e s i v e l y f o r many m e t a l s a n d a l l o y s . t h i s work a r e summarized i n a r e v i e w a r i g o r o u s t h e o r e t i c a l treatment  exten-  The r e s u l t s o f much o f a r t i c l e by K n i g h t ^ and  of the Knight  shift i s  g i v e n by Abragam . 3  Because of t h e s m a l l s k i n depth of almost a l l metals,  n u c l e a r m a g n e t i c r e s o n a n c e e x p e r i m e n t s have been  a l m o s t e n t i r e l y c o n f i n e d t o s t u d i e s i n m e t a l p o w d e r s whose i n d i v i d u a l p a r t i c l e s i z e s are l e s s than the s k i n depth of the metal. shift  I n any m e t a l w i t h c u b i c symmetry, t h e K n i g h t  i s independent of t h e o r i e n t a t i o n of t h e metal  with respect  t o t h e e x t e r n a l magnetic f i e l d .  crystal  However, i f  t h e symmetry o f t h e c r y s t a l i s l e s s t h a n c u b i c , t h i s i s n o t true.  The n u c l e a r m a g n e t i c r e s o n a n c e s i g n a l w i l l  be  -2-  broadened when observed i n a powder whose p a r t i c l e s are randomly oriented i n the magnetic f i e l d or shifted when observed i n a single c r y s t a l . The purpose of t h i s work has been to study the nuclear magnetic resonance i n metal single c r y s t a l s , i n p a r t i c u l a r , 31  white t i n and aluminum.  The advantages of using a single  c r y s t a l to study the nuclear magnetic resonance of t i n ares (a)  the anisotropy of the Knight s h i f t due to the tetragonal  symmetry of the t i n c r y s t a l can be studied d i r e c t l y j  (b) the  l i n e shape of the resonance can be studied without the large broadening caused by the anisotropic Knight shifty and (c) the measurements of the resonance frequency can be made more accurately because the resonance l i n e i s narrower. 11Q  117  The Sn  and Sn  isotopes were studied i n a  white t i n single c r y s t a l as a function of c r y s t a l orientation in the magnetic f i e l d , as a function of the magnetic f i e l d i t s e l f , and as a function of temperature i n the l i q u i d helium range.  The two parameters which describe the Knight s h i f t i n  a metal with tetragonal symmetry were determined.  A slight  dependence of the Knight s h i f t on temperature and external magnetic f i e l d was observed.  The l i n e width of the t i n  resonance was studied and found to have contributions from both the dipole-dipole interactions of nearest neighbours and from i n d i r e c t exchange interactions which couple two nuclei of d i f f e r e n t magnetic moments by way of the conduction electrons.  -3-  A search f o r a v a r i a t i o n of the Knight s h i f t  caused  by o s c i l l a t i o n s i n t h e diamagnetic s u s c e p t i b i l i t y as the external f i e l d  i s varied  (de H a a s - v a n A l p h e n e f f e c t )  made i n s i n g l e c r y s t a l s o f b o t h t i n a n d a l u m i n u m . l i m i t t o t h e e f f e c t has been s e t f o r b o t h m e t a l s .  was  An u p p e r  CHAPTER 2  THEORY OF THE KNIGHT SHIFT  The K n i g h t  shift  i n m e t a l s w i t h t e t r a g o n a l symmetry  can be e x p l a i n e d i n terms o f t h e h y p e r f i n e i n t e r a c t i o n  which  i n v o l v e s e s s e n t i a l l y t h r e e terms i n t h e s p i n H a m i l t o n i a n and which couples  the electron spins t o the nuclear spins.  c o u p l i n g e n e r g y i s s m a l l compared  t o t h e atomic energy  t i n g s s o t h a t p e r t u r b a t i o n t h e o r y c a n be used.  The split-  Further, the  e l e c t r o n s a r e assumed t o b e n o n - i n t e r a c t i n g , i . e . , t h e f r e e e l e c t r o n model i s used. The H a m i l t o n i a n  f o r t h e magnetic i n t e r a c t i o n of t h e  electrons with the nucleus  4  can be w r i t t e n as  (1) *U - 2 & h I  $L S & ( r )  Y  where  £ i s t h e Bohr  magneton  y i st h e nuclear gyromagnetic r a t i o X i s the n u c l e a r  spin  # i s the e l e c t r o n o r b i t a l J3 i s the e l e c t r o n  quantum number  spin  X i s the r a d i u s v e c t o r from the nuclear the e l e c t r o n  spin  spin to  -5-  The  first  term of t h i s H a m i l t o n i a n  bution t o the electron-nucleus because the o r b i t a l angular (bismuth  i s a notable  will  g i v e a l m o s t no  i n t e r a c t i o n i n most m e t a l s  momentum i s q u e n c h e d o r n e a r l y  exception).  In other words, the  n e t i c c o n t r i b u t i o n i s u s u a l l y s m a l l compared t o paramagnetic c o n t r i b u t i o n . be d e t e c t e d , the Knight varied.  however, by  shift  a t low  lower  The  diamagnetic  the observation  be d i s c u s s e d  c o n t r i b u t i o n might  of o s c i l l a t i o n s  i n Chapter 5.  The  shift . 5  For  and  shift  standard  Knight  only shift  i s o t r o p i c p a r t of  d e r i v a t i o n of the Knight  given i n d e t a i l f o l l o w i n g mainly 5  tetragonal  the  i n c r y s t a l s w i t h t e t r a g o n a l symmetry.  The  Slichter .  couples  no  gives r i s e t o the Knight  c r y s t a l s of c u b i c symmetry o r t o the  Knight  give  c r y s t a l s of  l a s t term i n the Hamiltonian  is  and  account f o r the a n i s o t r o p i c part of the  s - e l e c t r o n s t o the nucleus in  in  For  t h a n c u b i c symmetry, t h e second  c o n t r i b u t i o n t o the Knight  shift.  diamag-  the  t h i r d terms, the d i p o l a r terms, of the Hamiltonian  symmetry, t h e y  so  t e m p e r a t u r e s as the magnetic f i e l d  This point w i l l  c r y s t a l s w i t h no  contri-  This i s being  work r e p r e s e n t s  the f i r s t  the treatment  given  done f o r c o m p l e t e n e s s . direct observation  t r o p i c c o n t r i b u t i o n s t o the Knight d e r i v a t i o n the type  shift  is by  Since  of the  now  this  aniso-  s h i f t , by g i v i n g t h i s  of i n f o r m a t i o n o b t a i n a b l e i n s u c h  an  e x p e r i m e n t s h o u l d be more c o m p r e h e n s i b l e . For  simplicity, first  consider  the i n t e r a c t i o n s  -sinvolving only the last  term,  o r t h e c o n t a c t term,  i nthe  Hamiltonian  - i f E e y h I • S cf C r )  U  C  (2)  1  Because t h e e l e c t r o n s and n u c l e i a r e o n l y w e a k l y t h e c o m p l e t e wave f u n c t i o n product  be  can be approximated  by t h e  o f t h e many p a r t i c l e e l e c t r o n i c a n d n u c l e a r wave  functions,  The  V  interacting,  V  V  and  e  antisymmetrized  [  n  e l e c t r o n i c wave f u n c t i o n i s c o n s i d e r e d t o  6  (3)  where  r  ^  1 e  s  (  fN  - u  the s p i n term,  f e  e*~*  r  1 ) P  ^kS  ( l )  U 'k's «'< > 2  r  (N)  , the Bloch function including  i n the permutation.  c o n t r i b u t i o n of the electron-nuclear i n t e r -  a c t i o n s f r o m a l l e l e c t r o n s t o one n u c l e u s i s  (4)  •••• V fkc*Ds ^  a n d p i s an i n d e x t o i n d i c a t e an o d d o r e v e n  number o f i n t e r c h a n g e s The  -  -7-  Since the operator there  S  cf ( r ) i n v o l v e s  a r e no c o n t r i b u t i o n s f r o m terms i n w h i c h e l e c t r o n s  exchanged.  Thus u s i n g e q u a t i o n  (3), we c a n r e w r i t e  H r ^ ^ i j - f j[V"  (5)  *  If  o n l y one e l e c t r o n ,  the electrons  external  k  s  (1) ... ] d T d T 1  (4)  V ...  2  a r e q u a n t i z e d along the z - d i r e c t i o n by the  state f i e l d H , the only c o n t r i b u t i o n to equation Q  comes f r o m S ^ .  % j  (6)  ^  =  t h e sum i s o v e r  all  ^  I M ° ) !  ^3  and s ,  2  and f ( k , s )  »B  F  s t a t e d e s c r i b e d b y t h e wave v e c t o r The f a c t o r  m  s  i s just  & >  + £ or -  of the e l e c t r o n i c  the nucleus.  For a given k, equation  a  given  - f  1  ? *  I  z  J  k  wave f u n c t i o n e v a l u a t e d  2 & < i )  (6) c a n b e w r i t t e n  f(fe,i)  x  s is  \ and u ( o ) i s t h e  ,k X~~  >  k_ a n d s p i n c o o r d i n a t e  s p a t i a l part  (7)  S  i s the Fermi  d i s t r i b u t i o n function g i v i n g the p r o b a b i l i t y that  occupied.  (5)  T h e i n t e g r a l t h e n becomes  z  where  [ V  J(  are  +  f(k,-i)J  2 $ < - J )  | u (o)| k  at  -8-  The f a c t o r i n square b r a c k e t s i s the average c o n t r i b u t i o n of the  s t a t e t o the z component of m a g n e t i z a t i o n of the sample.  If t h i s f a c t o r i s c a l l e d the e l e c t r o n s  u,  z k  , the t o t a l z m a g n e t i z a t i o n of  i n a sample of a u n i t volume i s fi  z  "  ( 8 )  k"  ^~k  The t o t a l s p i n s u s c e p t i b i l i t y f o r a u n i t volume can be  defined  as  O)  and  i£ "  *  H s  o  the s p i n s u s c e p t i b i l i t y f o r one v a l u e of  (10)  x  H  k  k  °  which means t h a t  <io»)  X  s  \x  A  s k  The t o t a l e f f e c t i v e i n t e r a c t i o n f o r the j t h n u c l e a r s p i n i s then  -9-  In order ing  t h a t t h i s sum may b e e v a l u a t e d ,  q u a n t i t y must b e c o n s i d e r e d .  > ) d E ^ dA A  d e f i n e d a s t h e number o f a l l o w e d J k - v a l u e s c y l i n d r i c a l volume o f  Js: s p a c e h a v i n g  i s  lying within a small  a cross-section  dA anci l y i n g b e t w e e n t h e e n e r g y s u r f a c e s E ^ a n d E The c o o r d i n a t e s  the f o l l o w -  k  o f t h e s u r f a c e a r e d e n o t e d b y A.  number o f s t a t e s dN b e t w e e n E ^ a n d E ^ + d E  k  + dE^ . The  i s given  Slimming t h e c o n t r i b u t i o n s o v e r t h e w h o l e c o n s t a n t  area  total by  energy  surface.  (12)  dN - dEj^  g(E  k  E, k  P  S  P(E  \  const  )  d  f o r a u n i t volume.  l a t e r , N(Ep) c a n be s i m i l a r l y d e f i n e d s t a t e s per u n i t energy i n t e r v a l atomic volume V . evaluated  (13)  i n terms of these  For convenience  as t h e d e n s i t y o f  a t t h e F e r m i s u r f a c e f o r one  T h e sum i n e q u a t i o n  Q  an  \  ) i s a d e n s i t y of s t a t e s per u n i t energy i n t e r v a l at  the energy s u r f a c e E^  by  , A) dA  ( 1 1 ) c a n now b e  q u a n t i t i e s i f t h e sum i s r e p l a c e d  integral  £L | u ( o ) | k  Z  X  S  •s  -  K j  (o)| X 2  S  g<  E k  >  A )  d  %  d  A  -10-  d e p e n d s on t h e F e r m i f u n c t i o n s and  on t h e d i f f e r e n c e  s t a t e , one w i t h  i n e n e r g y between two s p i n s  spin parallel  any s t a t e s  Jk  i n the  t o t h e magnetic f i e l d  other with spin a n t i p a r a l l e l . for  f ( k , £) a n d f ( k ,-£)  Clearly, ^C  k  and t h e  w i l l b e t h e same  S  h a v i n g t h e same v a l u e o f  .  I t i s there-  s f o r e assumed t h a t DC the  energy  can be w r i t t e n as a f u n c t i o n o n l y of  E^.  xt  d4)  - X  <\ >  s  k  Equation  (15)  2.  ( 1 3 ) c a n now be w r i t t e n  |u  ( o ) l  k  2  X  - flu,.(o)l X  S  2  x  The a v e r a g e v a l u e o f constant energy i s defined  <E  S  g(E  <(lu  k  (o)l  )  E  ,A) dA d E  f c  k  as I  (16)  )  \ i2 |u, ( o ) \ o v e r a s u r f a c e o f  u 2  k  k  (o)l  2 g(E  k  ,A) dA  k  ,A) dA  k  g(E,  \  W  ,A) dA  I k  2  (o)l g(E  ?< k E  >  -11-  so  that  <i7) sujoii x;- k)v° K * V !  )12  X  (  Because t h e e l e c t r o n s p i n s a r e e i t h e r p a i r e d o f f or t h e e l e c t r o n s p i n e n e r g y s t a t e s a r e u n o c c u p i e d , f o r a l l Ek near  (not w i t h i n a region  o f t h e order o f kT) t h e Fermi  3XL (  f a c e Ep, t h e c o n t r i b u t i o n s o f susceptibility Fermi surface.  will  outside  as)  S  E  k  ) t o the t o t a l  sur-  spin  come o n l y f r o m e n e r g y s t a t e s n e a r t h e <^l k ( ° ) \ ^  I f i t i s assumed t h a t  varies slowly with  not  E^ , t h i s  u  2  E  i n t u r n c a n be t a k e n  the integral  s U  k  <Kwi\[x <\)?^)^ B  (o)i X^2  U s i n g e q u a t i o n ( 1 0 ' ) , we c a n w r i t e  Xk  X - £k X k s  s  X  s  j  or  g ( E ,A) dEk dA — k  (Ek > e ( E  >> A  k  ^  d  A  -12-  s  (19)  Combining equations  ( 1 1 ) , ( 1 8 ) , a n d ( 1 9 ) , we g e t t h e i n t e r -  a c t i o n w i t h t h e j t h n u c l e a r s p i n t o be  (20,  x  •  i„  [f  0*  <°>i > %  xK  2  s  T h i s i n t e r a c t i o n r e p r e s e n t s an e x t r a m a g n e t i c f i e l d the a p p l i e d magnetic f i e l d shift.  The K n i g h t  HQ w h i c h g i v e s r i s e t o t h e K n i g h t  s h i f t c a n t h e n be w r i t t e n  AH H_  (21)  If  added t o  t,  i s d e f i n e d as t h e r a t i o l«A<»>\ \  where  I  \u (o)\ A  :  2 i s the p r o b a b i l i t y d e n s i t y of e l e c t r o n s i n  a f r e e atom, t h e n t h e e x p r e s s i o n f o r t h e K n i g h t  s h i f t c a n be  w r i t t e n i n t e r m s o f t h e f r e e atom h y p e r f i n e c o u p l i n g a ( s ) - J f L -yn  (22)  Equation  (21) then  becomes  £\u (o)\ A  constant.  -13-  £S  (23)  In order  =  a  (  s  )  ! ^ P  t o c o n f o r m w i t h more u s u a l n o t a t i o n , X . 2  r e w r i t t e n as  X  p  M,  where  "X  p  i s the P a u l i  p e r u n i t mass a n d M i s t h e a t o m i c Equation in  M  s  n  a  s  been  susceptibility  mass.  (23) i s t h e e x p r e s s i o n f o r t h e K n i g h t  a crystal with at least  c u b i c symmetry.  t e t r a g o n a l symmetry as does w h i t e represents the i s o t r o p i c  shift  I f the c r y s t a l  t i n , then equation  part of the Knight s h i f t .  (23) As  was  mentioned e a r l i e r , the d i p o l e - d i p o l e i n t e r a c t i o n s i n the Hamiltonian of equation  ( 1 ) w i l l c o n t r i b u t e an a n i s o t r o p i c  7 term t o the expression f o r the Knight s h i f t . Consider  the Hamiltonian J  < > 24  If  °V4anis - -  t h e a n g l e between H  equation  (25)  J.1  Q  3 r (S • j ; )  L 3 r  r  5  and t h e r a d i u s v e c t o r £  i s ©c ,  ( 2 4 ) becomes  ^ a n i s " " ^  m  I  ( 1  "  3  c  o  s  2  ^  )  r  "  3  where t h e + o r - s i g n i s d e t e r m i n e d by t h e e l e c t r o n s p i n being p a r a l l e l or a n t i p a r a l l e l t o the magnetic f i e l d  H . Q  has  -14-  T o d e t e r m i n e t h e e n e r g y d i f f e r e n c e due t o t h i s a nucleus of spin J  for  equation  Hamiltonian  ( A i j => 1 ) , we must i n t e g r a t e  ( 2 5 ) o v e r t h e e l e c t r o n i c wave f u n c t i o n s i n a way  s i m i l a r t o t h a t done f o r t h e i s o t r o p i c c a s e . volume, o n l y t h e  2 Q H  N(E )  Q  For a unit  e l e c t r o n s w i t h unbalanced  F  spin  near t h e Fermi s u r f a c e w i l l c o n t r i b u t e t o t h e i n t e r a c t i o n a r i s i n g from ^  a n  i *  I f  v  S  o  d i f f e r e n c e due t o ^ H a n i g  (26)  A W  a  n  i  "  s  i  s  t  in  V  e  the  Y  e  a  t  o  m  i  volume, t h e energy  c  is  7* * $ ' 2 $ H V N ( E ) 0  x  where  n  0  (3 c o s *  F  2  k  < I f  * ek  - 1 ) |r 1-3  *  dv) /E  F  i s t h e e l e c t r o n i c wave f u n c t i o n o f an e l e c t r o n  k  t h state.  I f we i n t r o d u c e  an a v e r a g e wave f u n c t i o n  so that represents  i n space o f the conduction then equation Knight (27)  e l e c t r o n s near the Fermi  density surface,  (26) g i v e s f o r t h e a n i s o t r o p i c p a r t o f t h e  shift A  H  anis o H  To e v a l u a t e angles  t h e average e l e c t r o n  ©  - 26  2  V  N(E ) F  °  1 V * ( 3 c o s * . - 1) J 2  t h i s i n t e g r a l , we l e t t h e f i e l d and  (p a n d t h e r a d i u s v e c t o r  _r  |r\" ^dxdydz 3  have t h e p o l a r have p o l a r  -15-  angles  and  $  w i t h r e s p e c t t o t h e x , y, z  coordinate  2  system.  ( 1 - 3 c o s ex. ) c a n t h e n b e e x p r e s s e d i n  The t e r m  t e r m s o f t h e s e a n g l e s b y a w e l l known a d d i t i o n t h e o r e m i n terms o f t e s s e r a l harmonics ©  and  i>  i n the angles  ©  a?  and  and  respectively 2  (28)  ^ ( 3 cos =c 2  1) - £  (-l) P m  m 2  (cos  6 ) P "" (cos  0)  m  2  m=-2  x  We assume t h a t  V  e  im($  - £> )  can be r e p r e s e n t e d by a m i x t u r e o f  f u n c t i o n s of the form  4^  *  (W.  )  + W  p-wave  and  IT2  -~—  ( V (s  +  - V  ).  T h e s e wave f u n c t i o n s a r e r e a l a n d t h e r e f o r e  r e p r e s e n t quenched  p  orbitals.  F o r a x i a l symmetry, t h e  e l e c t r o n d e n s i t y c a n be w r i t t e n as  YV*  (29)  - g(r)  [ A ( X  2  + y ) + Gz ] 2  2  = r  2  g(r)  +  (C  -  A)  x. c o s © j 2  where  g ( r ) i s a r a d i a l d i s t r i b u t i o n f u n c t i o n whose f o r m i s  unimportant and  (30)  f o r this calculation.  ( 2 8 ) into equation  A  H  a  n H  l  o  S  - £  S u b s t i t u t i n g equations ( 2 9 )  ( 2 7 ) , we g e t  2  V  Q  N(Ej.) q ( 3 c o s  2  0-1)  -16-  where  ©  i s t h e angle between t h e z a x i s and t h e a p p l i e d  magnetic f i e l d ,  and q i s g i v e n by  ^ ( 3  cos ©  - 1) l r \ " V  2  3  dV  bo  (C - A)  q i s the e l e c t r i c f i e l d  f  r g(r) dr  g r a d i e n t a t t h e n u c l e u s caused by  e l e c t r o n s near t h e F e r m i c s u r f a c e . In  o r d e r t o e x p l o r e t h e o r i e n t a t i o n dependence o f  the Knight s h i f t w i t h respect t o t h e magnetic f i e l d , rewrite this  e x p r e s s i o n as A H  (31)  we c a n  J  „  §=== o  -  constant  • (3 c o s  z  © - 1)  H  The  t o t a l Knight s h i f t  including the anisotropic part i s  then g i v e n by equations  ( 2 3 ) a n d ( 3 1 ) a n d can.be w r i t t e n  (32)  -  - ~ o H  K + $ K J (3 c o s  2  © - 1)  8  -17-  CHAPTER 3  EXPERIMENTAL APPARATUS AND TECHNIQUE  The  nuclear magnetic resonance spectrometer  t h i s work i s e s s e n t i a l l y o f s t a n d a r d schematic  design.  diagram of t h e spectrometer.  used i n  Figure 1 i s a  The c i r c u i t  diagrams  a r e a l l g i v e n i n A p p e n d i x C. S i n c e t h e o b j e c t o f t h e e x p e r i m e n t s has been t o measure r e s o n a n c e f r e q u e n c i e s and l i n e shapes, a Pound-KnightWatkins  9  marginal  o s c i l l a t o r was c h o s e n f o r t h i s w o r k :  o s c i l l a t o r was v e r y s l i g h t l y m o d i f i e d f r o m t h e o r i g i n a l Instead of a 6J6 tube,  a 396A was u s e d a s t h e o s c i l l a t o r  The design. tube.  A l s o t h e r f a m p l i f i e r was f e d f r o m t h e c a t h o d e s o f t h e 396A i n s t e a d o f from t h e g r i d t o which t h e sample c o i l attached.  The c a p a c i t a n c e f o r t h e r e s o n a n t  v i d e d by a V a r i c a p  was  c i r c u i t was p r o -  (variable capacity diode).  The  frequency  o f t h e o s c i l l a t o r was t h e n s w e p t b y v a r y i n g t h e v o l t a g e across the diode.  The v o l t a g e s o u r c e  f o r the diode  p r o v i d e d by t h e sawtooth waveform from a m o d i f i e d waveform g e n e r a t o r  1 0  '  1 1  .  was  Tektronix  The m o d i f i e d w a v e f o r m g e n e r a t o r c a n  p r o v i d e a n e g a t i v e g o i n g s a w t o o t h v o l t a g e f r o m 100 t o 0 v o l t s w i t h p e r i o d s which vary from s e v e r a l seconds t o s e v e r a l  hours.  Power  Au<Aio  A mptif tar  Osci HaW  Ho ri70f\+al Arripli-Ficr  CRO  PUce S k . f W  P. K.  W  £e.r\e.i"Vwe  Occi I l a t e r  Director  Co jrvffcr ar\d" Prir+'er  142  ^tco rder  I M 00  I  P. K . w .  F i g u r e 1.  B l o c k Diagram of the Spectrometer  -19-  Any  v o l t a g e b e t w e e n 10 a n d 100 v o l t s c a n b e u s e d a s t h e i n i t i a l  v o l t a g e o f t h e s a w t o o t h and t h e s a w t o o t h rundown c a n be s t o p p e d at any t i m e w i t h t h e v o l t a g e r e t u r n i n g t o i t s i n i t i a l Using a Varicap instead of a v a r i a b l e  a i r condenser  set value.  eliminates  n o i s e f r o m a d r i v i n g m o t o r a n d p r o v i d e s a more u n i f o r m f r e q u e n c y sweep, e s p e c i a l l y a t t h e s l o w e r  rates.  Following the marginal o s c i l l a t o r t w i n t e e narrow  band audio; a m p l i f i e r .  15 c p s w e r e u s e d network  i n t h i s work:  i s a m o d e l 216  Two b a n d w i d t h s  a 23% bandwidth  a n d a 1.3% ( a b o u t .2 c p s ) n e t w o r k .  at  (about 4 c p s )  The l a t t e r  w i d t h c o r r e s p o n d s t o a time c o n s t a n t of about  White  band-  5 seconds and  t h u s was u s e d o n l y f o r c o m p a r a t i v e l y s l o w f r e q u e n c y s w e e p s . The  phase s e n s i t i v e d e t e c t o r used  i s i n principle  12 t h e same a s S i n i s t e r ' s The  , a l t h o u g h d i f f e r e n t tubes were employed.  s i g n a l r e c o r d e r was a V a r i a n r e c o r d e r , m o d e l G11A. The  frequency of the o s c i l l a t o r  as i t swept  t h e s i g n a l was m o n i t o r e d b y a H e w l e t t - P a c k a r d  through  electronic  c o u n t e r , m o d e l 524C, w i t h t h e a p p r o p r i a t e p l u g - i n u n i t . f r e q u e n c y m e a s u r e d was i n a l l c a s e s t h e a v e r a g e .1 s e c o n d .  frequency  The over  The f r e q u e n c y r e a d i n g was r e c o r d e d b y a H e w l e t t -  P a c k a r d d i g i t a l r e c o r d e r , m o d e l 516B. turn activated  an i n d i c a t o r  The d i g i t a l r e c o r d e r i n  pen on t h e s i g n a l r e c o r d e r each  time the e l e c t r o n i c  c o u n t e r measured t h e f r e q u e n c y .  method o f m e a s u r i n g  and r e c o r d i n g f r e q u e n c y gave f a s t , a c c u r a t e ,  and f r e q u e n t m o n i t o r i n g o f t h e o s c i l l a t o r  i  This  frequency as i t  -20-  passed through t h e n u c l e a r magnetic resonance  signal.  The magnet u s e d t h r o u g h o u t t h i s w o r k was a V a r i a n r o t a t i n g magnet w i t h t w e l v e i n c h p o l e f a c e s a n d a 2 1/4 i n c h g a p a n d was c a p a b l e gauss.  of g i v i n g a magnetic  f i e l d o f 11.4 k i l o -  M o d u l a t i o n o f t h e m a g n e t i c f i e l d was a c c o m p l i s h e d  u s i n g t w o m o d u l a t i o n c o i l s e a c h wound w i t h 60 t u r n s o f No. 18 copper w i r e on b a k e l i t e f o r m s w h i c h were mounted around t h e pole caps. oscillator  The m o d u l a t i o n c o i l s were s u p p l i e d by an a u d i o t h r o u g h a 20 w a t t power  amplifier.  A s c h e m a t i c diagram o f t h e low t e m p e r a t u r e i s g i v e n i n F i g u r e 2.  system  The h e l i u m c r y o s t a t i s a s t a n d a r d  d o u b l e g l a s s dewar s y s t e m .  Temperatures  down t o 1.15°K w e r e  o b t a i n e d b y pumping on t h e l i q u i d h e l i u m w i t h a 3 i n c h pump. ing  T h e t e m p e r a t u r e was c o n t r o l l e d b y c o n t r o l l i n g t h e pump-  s p e e d w i t h a n e e d l e v a l v e i n p a r a l l e l w i t h a l£ i n c h  vacuum v a l v e , a n d was m e a s u r e d b y o b s e r v i n g t h e v a p o u r s u r e o f t h e h e l i u m w i t h a m e r c u r y manometer one c o n t a i n i n g d i - b u t y l p t h a l a t e .  i nparallel  preswith  The d e w a r s a n d dewar c a p  were mounted s o t h a t t h e y c o u l d s l i d e the  Kinney  i n and o u t from between  magnet p o l e f a c e s . The s a m p l e was m o u n t e d on t h e e n d o f a 3/8 i n c h  t h i n w a l l s t a i n l e s s s t e e l tube which formed tor  the outer  of t h e c o a x i a l l i n e c o n n e c t i n g t h e sample c o i l  marginal o s c i l l a t o r .  The c e n t r e c o n d u c t o r  conduc-  tothe  of the c o a x i a l  l i n e was a No. 32 c o p p e r w i r e h e l d i n p l a c e w i t h t e f l o n s p a c e r s .  Kmney Pump  He Cy ltr\der  Needle Valv/e  To Return Line  He Trorvs-fe «•  lo is| pWorv JacUt (9)  Oil Trap  6  H  3  X  Oil Bottler-  Fore. M  F i g u r e 2.  Schematic. Diagram of the Low Temperature System  Oil  -22-  T h r e e s e m i - c i r c u l a x r a d i a t i o n s h i e l d s made f r o m s h e e t m e t a l c o p p e r w e r e mounted on t h e s t a i n l e s s s t e e l c o a x i a l The  line,.,  t i n s i n g l e c r y s t a l was made f r o m " E x t r a P u r e "  t i n , V S - 1 5 1 , o b t a i n e d f r o m t h e V u l c a n D e t i n n i n g Company.  The  p u r i t y q u o t e d was 9 9 . 9 9 9 + % t i n w i t h i m p u r i t i e s o f . 0 0 0 8 % l e a d and  .000018% i r o n .  T h e s i n g l e c r y s t a l was g r o w n b y p o u r i n g  molten t i n i n t o a § i n c h by \ i n c h by 4 i n c h g r a p h i t e mold, then s l o w l y  ( o n e i n c h e v e r y 14 m i n u t e s ) w i t h d r a w i n g t h e m o l d  from t h e furnace. orientation,  I n o r d e r t o grow a c r y s t a l w i t h t h e d e s i r e d  that i swith the  IpOl]  direction approximately  p e r p e n d i c u l a r t o t h e c r y s t a l a x i s , a seed c r y s t a l w h i c h had the  d e s i r e d o r i e n t a t i o n was u s e d .  T h e s e e d was j o i n e d t o  the  m o l t e n t i n j u s t o u t s i d e t h e f u r n a c e mouth b y d r a w i n g o u t  a s m a l l s t r e a m o f t i n f r o m t h e main m e l t w i t h a g l a s s r o d . The  i n t e r f a c e o f t h e s e e d a n d m o l t e n t i n was " p u d d l e d " o r  stirred until  t h e seed began t o m e l t back f r o m t h e i n t e r f a c e .  The m o l d was t h e n w i t h d r a w n f r o m t h e f u r n a c e a s d e s c r i b e d above.  T h e c a s t t i n s l a b t h u s o b t a i n e d was e t c h e d  lytically  i na fairly  electro-  d i l u t e HC1 s o l u t i o n , a n d i f no g r a i n  b o u n d a r i e s w e r e a p p a r e n t , a n X - r a y p h o t o g r a p h was t a k e n back r e f l e c t i o n photograph) tion. one  t o determine t h e c r y s t a l  (Laue  orienta-  T h i s method w i l l g i v e t h e o r i e n t a t i o n t o w i t h i n  about  degree. The  manner.  seeds were p r e p a r e d and a n a l y s e d i n a s i m i l a r  Instead o f drawing t h e molten t i n out t o j o i n a seed  -23-  as d e s c r i b e d  above,  first  t h i s p o i n t t o s o l i d i f y does so t o form a  t i p of  i t was drawn o u t t o a f i n e p o i n t .  c r y s t a l which i s then propagated it  The single  along the whole specimen  i s withdrawn from the furnace.  By u s i n g t h i s seed  as  placed  p r o p e r l y i n t h e m o l d , a new s e e d w i t h a n o r i e n t a t i o n c l o s e r t h e d e s i r e d o r i e n t a t i o n c a n be g r o w n . until all  a suitable  subsequent  seed  crystal  for  q u i t e near  saw i n t o s l i c e s  were e t c h e d  the desired o r i e n t a t i o n c o n t a i n i n g the  about  then glued together  t h e y were about  w i t h Q dope.  c r y s t a l was made u p o f  axis  H C l a n d HNO3 They were  laminated  a n d was a b o u t  ^ i n c h by  were a l i g n e d p r o b a b l y t o w i t h i n  one d e g r e e a n d p o s s i b l y b e t t e r . seem t h e n t o be o f  The e r r o r o f a l i g n m e n t  t h e same o r d e r o f m a g n i t u d e  A l t h o u g h no s u b g r a i n b o u n d a r i e s w e r e o b s e r v e d , have been p r e s e n t  and c o u l d g i v e  one-half  The e x p e r i m e n t s  degree.  to misalignments of o f m i s a l i g n m e n t was  in  photograph.  t h e y may w e l l  a misalignment of  perhaps  performed were m o s t l y  l e s s t h a n a few d e g r e e s , observed.  would  as t h e e r r o r  determining the c r y s t a l o r i e n t a t i o n from the X - r a y  evidence  was  inch.  The c r y s t a l s l i c e s  sensitive  for  These  .3 mm t h i c k .  The r e s u l t i n g  thirty slices  [oOl]  1 mm t h i c k .  i n a mixture of concentrated  a few m i n u t e s u n t i l  \ i n c h b y 3/8  continued  i s o b t a i n e d which i s then used  i t was s l i c e d a l o n g p l a n e s  with a jeweller's slices  is  crystals.  Once a c r y s t a l obtained,  This process  to  not and no  -24-  The  a l u m i n u m c r y s t a l was o b t a i n e d c o m m e r c i a l l y f r o m  M e t a l s Research L t d . , Cambridge.  I t was a c y l i n d r i c a l  speci-  men \ i n c h d i a m e t e r b y 3/4 i n c h l o n g a n d was made f r o m 99.9999% pure For crystals,  aluminum. e x p e r i m e n t s w i t h b o t h t h e aluminum and t i n  t h e s a m p l e c o i l was wound d i r e c t l y o n t h e c r y s t a l  rather than having the c o i l of  some i n s u l a t o r  and c r y s t a l  s e p a r a t e d by a l a y e r  l i k e e l e c t r i c a l tape or mylar.  In order to  have t h e o s c i l l a t o r o s c i l l a t e a t t h e c o r r e c t f r e q u e n c y w i t h minimum c a p a c i t y , a f o u r s t r a n d 4^ t u r n c o i l wound f r o m No. 28 w i r e was u s e d on t h e t i n c r y s t a l  and a f i f t y  turn c o i l of  No. 28 w i r e was wound a r o u n d t h e a l u m i n u m c r y s t a l . the  coil  d i r e c t l y on t h e a l u m i n u m c r y s t a l  to-noise r a t i o  c o n s i d e r a b l y , presumably  improved  because  f a c t o r o f t h e c o i l was i n c r e a s e d a n d b e c a u s e of  t h e c o i l was r e d u c e d , a l l o w i n g t h e c o i l  number o f t u r n s .  Winding the signal-  the f i l l i n g  the inductance  t o have a g r e a t e r  U s i n g No. 32 w i r e f u r t h e r r e d u c e d t h e i n d u c -  t a n c e , b u t w i t h more t h a n a b o u t 40 t u r n s i n t h e c o i l ,  the low  Q of the inductance prevented the o s c i l l a t o r from o s c i l l a t i n g . Because  the magnetic f i e l d  d r i f t s somewhat d u r i n g  an e x p e r i m e n t , t h e p r o c e d u r e f o l l o w e d was t o m e a s u r e t h e magnetic f i e l d just  f r e q u e n t l y w i t h a probe c o n t a i n i n g D 0  o u t s i d e t h e dewars.  2  mounted  T h e d e u t e r o n r e s o n a n c e was o b s e r v e d  d i r e c t l y o n an o s c i l l o s c o p e t h r o u g h a w i d e b a n d a u d i o amplifier  w h i l e a t t h e same t i m e t h e o s c i l l a t o r f r e q u e n c y was  -25-  measured w i t h t h e H e w l e t t P a c k a r d f r e q u e n c y meter. measurements o f t h e a n i s o t r o p i c K n i g h t s h i f t , f i e l d was  m e a s u r e d b e f o r e and a f t e r  For  the  the magnetic  each s e t of r e a d i n g s  taken at a p a r t i c u l a r c r y s t a l o r i e n t a t i o n .  The s e c o n d  field  m e a s u r e m e n t o f t e n a g r e e d w i t h t h e f i r s t t o w i t h i n one p a r t i n 6 0 0 , 0 0 0 a n d n e v e r d i f f e r e d b y more t h a n f i v e p a r t s i n 10  for  the  m e a s u r e m e n t o f t h e t i n r e s o n a n c e t o be c o n s i d e r e d v a l i d .  For  t h e m e a s u r e m e n t s d e s c r i b e d i n C h a p t e r 5, somewhat more  c a r e was  taken i n measuring the f i e l d .  The  b e f o r e and a f t e r e v e r y m e t a l r e s o n a n c e . a n d  f i e l d was  measured  d i d not d i f f e r  by  more t h a n two p a r t s i n 6 0 0 , 0 0 0 f o r t h e m e a s u r e m e n t t o be considered  valid. Almost a l l of the resonances were r e c o r d e d u s i n g  5 o r 10 s e c o n d t i m e c o n s t a n t s i n t h e p h a s e s e n s i t i v e  detector.  N e a r l y a l l o f t h e t i n r e s o n a n c e s w e r e o b s e r v e d u s i n g a modul a t i o n of l e s s than o n e - t h i r d gauss.  F o r t h e aluminum  r e s o n a n c e s w h e r e t h e r e seemed t o be some p r o b l e m s o f  satura-  t i o n a n d w h e r e o n l y t h e r e s o n a n c e f r e q u e n c y was w a n t e d , m o d u l a t i o n u s e d was  i n c r e a s e d f r o m l e s s t h a n two g a u s s  about f o u r gauss f o r t h e l a s t s e t of measurements. and 4 show t y p i c a l t i n a n d a l u m i n u m s i g n a l s a t 1.15°  the to  Figures K.  3  F i g u r e 4.  D e r i v a t i v e o f t h e A l Resonance  F i g u r e 3.  D e r i v a t i v e o f t h e Sn R e s o n a n c e  -27-  CHAPTER 4  THE KNIGHT SHIFT AND  L I N E WIDTH OF ;  THE T I N SINGLE CRYSTAL  Measurements of t h e n u c l e a r m a g n e t i c resonance 117  119  Sn  a n d Sn  i n Chapter tion  of  i n the s i n g l e c r y s t a l of white t i n described  3 w e r e made a s a f u n c t i o n o f t h e c r y s t a l o r i e n t a -  i n the applied magnetic f i e l d ,  a p p l i e d magnetic f i e l d , the l i q u i d helium range. results.  and as a f u n c t i o n o f t e m p e r a t u r e i n Figure  5 s u m m a r i z e s some o f t h e s e  The d i f f e r e n t c u r v e s show t h e v a r i a t i o n o f t h e  Knight s h i f t  as t h e e x t e r n a l f i e l d  a n g l e o f 180° fields  as a f u n c t i o n of t h e  at temperatures  o f 10.1 k i l o g a u s s  i s rotated  through  an  o f 4.2°K a n d 1.15°K a n d a t  a n d 6.13 k i l o g a u s s .  The  Knight  s h i f t was f o u n d  t o be e q u a l w i t h i n e x p e r i m e n t a l e r r o r s f o r  both isotopes.  Measurements  of the Knight s h i f t  as a  f u n c t i o n o f c r y s t a l o r i e n t a t i o n i n two d i f f e r e n t p l a n e s o f 13  rotation  ( a p p e n d i x B) a s s u r e d t h a t  ment o f t h e n u c l e u s showed t e t r a g o n a l  the e l e c t r o n i c symmetry.  environ-  A l l of the  m e a s u r e m e n t s r e c o r d e d i n t h i s c h a p t e r w e r e made i n t h e c r y s t a l plane containing v a l u e s o f K and KJJ magnetic  fields.  the  [00l]  axis.  Table 1  f o r d i f f e r e n t temperatures  and  gives applied  -28-  Fig.  5.  The Knight S h i f t i n T i n as a F u n c t i o n of the C r y s t a l O r i e n t a t i o n i n the Magnetic F i e l d ( O r i e n t a t i o n measured from the [001] a x i s )  -29-  TABLE 1  Temperature  Field  K x 10  1.15° K  10.1 k g  71.9 ± .1  5.4 ± .1  4.2° K  10.1 k g  71.6 + .1  5.4 ± .1  70.7 ± .2  5.4 ± .2  1.15° K  6.13 k g  KJ  4  x 10  4  As h a s b e e n n o t e d b y B l o e m b e r g e n a n d R o w l a n d , K° i s p o s i t i v e i n d i c a t i n g t h a t q o f e q u a t i o n (30) i s p o s i t i v e . F u r t h e r m o r e , K{ Because  (  i s q u i t e l a r g e , a l m o s t t e n p e r c e n t o f K.  the hyperfine  i n t e r a c t i o n f o r p-wave f u n c t i o n s i s  l e s s t h a n f o r s-wave f u n c t i o n s a n d b e c a u s e  only the aniso-  t r o p i c p a r t o f t h e p-wave f u n c t i o n i n t e r a c t i o n c o n t r i b u t e s t o K', , t h e l a r g e v a l u e o f K' i n d i c a t e s a s u b s t a n t i a l p-wave u  f u n c t i o n c o m p o n e n t i n t h e e l e c t r o n i c wave f u n c t i o n . I n m e t a l s whose s p e c i m e n  s i z e i s l a r g e compared t o  t h e i r s k i n d e p t h , t h e power a b s o r b e d b y t h e s a m p l e i s p r o p o r t i o n a l t o X'  , t h e r e a l and i m a g i n a r y p a r t s o f t h e n u c l e a r  s p i n s u s c e p t i b i l i t y , r a t h e r t h a n t o X " a l o n e a s w o u l d be t h e case f o r non-metals  o r f o r m e t a l s whose p a r t i c l e s i z e i s  s m a l l compared t o t h e i r s k i n d e p t h . Seymour*  4  Chapman, R h o d e s , a n d  have d e t e r m i n e d t h a t t h i s e f f e c t would d e c r e a s e t h e  -30-  measured z e r o of the d e r i v a t i v e of t h e observed a b s o r p t i o n s i g n a l by about  .3 o f t h e l i n e w i d t h a s d e t e r m i n e d f r o m  maxima o f t h e d e r i v a t i v e .  T h i s would  decrease K by  the  about  -4 .2 x 10 would  o r l e s s t h a n o n e - h a l f p e r c e n t o f t h e v a l u e o f K.  a l s o d e c r e a s e t h e v a l u e o f K[  i t s value.  The  (  by, a b o u t one p e r c e n t o f  c o r r e c t i o n s t o K and K' (  s m a l l e r because,  as n o t e d by K a r i m o v  It  may  w e l l be even 15  and S h c h e g o l e v  , for  r e s o n a n t c i r c u i t s o f l o w Q a s i s t h e c a s e when t h e c o i l o f  the  circuit  X'  to  i s wound on a m e t a l s p e c i m e n ,  t h e o b s e r v e d a b s o r p t i o n may  the c o n t r i b u t i o n of  be q u i t e s m a l l .  That  this  c o u l d be t r u e i n o u r c a s e i s s u b s t a n t i a t e d b y t h e l a r g e of  s y m m e t r y shown b y t h e o b s e r v e d a b s o r p t i o n c u r v e .  shows t h e a s y m m e t r y e x p e c t e d i n t h e d e r i v a t i v e o f and G a u s s i a n l i n e s h a p e s .  The r a t i o  Figure 6  Lorentzian  of the amplitudes of the  e x t r e m a o f t h e d e r i v a t i v e o f t h e L o r e n t z i a n c u r v e i s .39 of  t h e G a u s s i a n c u r v e i s .55, b u t t h e r a t i o  d e r i v a t i v e was the  about  .7.  observed resonance  T h i s means e i t h e r  the l i n e shape of line  s h a p e o r t h a t t h e d i s p e r s i o n mode d o e s n o t c o n t r i b u t e w i t h t h e a b s o r p t i o n mode t o t h e o b s e r v e d r e s o n a n c e . (  and  of the observed  i s "squarer" than a Gaussian  c o r r e c t i o n s t o K and K'  degree  equally Since the  c o m p u t e d on t h e b a s i s o f e q u a l c o n -  t r i b u t i o n s f r o m b o t h t h e a b s o r p t i o n a n d d i s p e r s i o n modes a r e j u s t b a r e l y l a r g e r t h a n t h e e r r o r s i n t h e m e a s u r e d K and  ,  when t h e a d d i t i o n a l p o i n t o f u n e q u a l c o n t r i b u t i o n s i s c o n s i d e r e d i t seems u n n e c e s s a r y t o make any c o r r e c t i o n s  to  -31-  Lorerctzian Lwe Skape.  d»  1  \  1  Gaussian  dV  dv  i  Lir\e SKape  /-  — —  \  ^  i  Figure  i  6.  i  -32-  the observed  results.  F i g u r e 7 shows t h e l i n e w i d t h  o f t h e t i n specimen  as d e t e r m i n e d b y t h e e x t r e m a i n t h e d e r i v a t i v e s o f a b s o r p t i o n lines.  The l i n e w i d t h  cannot be e x p l a i n e d  d i p o l e c o u p l i n g between n e a r e s t moment,  AoJ  f  2  f  o  r  i n terms o f d i p o l e -  neighbours alone.  The s e c o n d  d i p o l e - d i p o l e coupling f o r a substance  16 w i t h two s p i n s i s g i v e n b y V a n V l e c k  (33)  AGJj  2  "  ( A 6 J  2 *  , , ^ K I + 1 ) S kk  I  2  )  i  i  +  (A0J  2  I  )  I  S  where 3 ) - -j Y II 4 I 4  (^60  2 AT  T T  T  4  (1 - 3 c o s  2  O  j k  )  2  r ° " Jk 6  and (1-3  cos  2  ©jk)  Jk  are t h e c o n t r i b u t i o n s t o a nucleus I a n d u n l i k e n u c l e i l a b e l l e d S. ©..=»() o n l y t h e t w o n e a r e s t JK  from l i k e n u c l e i l a b e l l e d  I n t h e case o f t i n , f o r  neighbours give a s i g n i f i c a n t  c o n t r i b u t i o n t o t h e s e c o n d moment.  When t h e n a t u r a l a b u n d -  ances o f t h e two t i n i s o t o p e s a r e c o n s i d e r e d ,  this  formula  2  -33-  F i g u r e 7*  The L i n e Width of the T i n Resonance as a F u n c t i o n df the C r y s t a l O r i e n t a t i o n i n the Magnetic F i e l d ( O r i e n t a t i o n measured from the [OOl] a x i s )  -34-  g i v e s a s e c o n d moment o f .15 ( k c / s )  a t fl = 0. T a k i n g t h e jk s e c o n d moment o f t h e o b s e r v e d s i g n a l t o b e a p p r o x i m a t e l y e q u a l to  2  t h e square of t h e s e p a r a t i o n o f t h e extrema of t h e d e r i v a 2  t i v e o f t h e r e s o n a n c e s i g n a l , we g e t a r e s u l t o f 2 ( k c / s ) , more t h a n t e n t i m e s t h e t h e o r e t i c a l v a l u e . is  I f the l i n e  shape  i n f a c t G a u s s i a n , t h e e x p e r i m e n t a l s e c o n d moment i s  1.6  (kc/s) . I f t h e c a l c u l a t e d s e c o n d moment d u e t o d i p o l e - d i p o l e 2  i n t e r a c t i o n s i s s u b t r a c t e d f r o m t h e o b s e r v e d s e c o n d moment, 2 t h e r e r e m a i n s a n i s o t r o p i c s e c o n d moment o f .6 ( k c / s ) .  One  p o s s i b l e o r i g i n o f t h i s s e c o n d moment i s b r o a d e n i n g d u e t o m i s a l i g n m e n t o f t h e s l i c e s w h i c h make u p t h e c r y s t a l (although t h i s b r o a d e n i n g w i l l n o t be i s o t r o p i c ) . calculation (34)  A\>[  =»  2  - (  }  K,{ 7 H , Q  ) 12 c o s  2  0 cos * sin  2  0 sin **  + — sin  4  0  2  and  e x p r e s s i n g t h e degree of misalignment.  4  The c o n t r i b u t i o n t o  i s b e l i e v e d t o be t h e e x p l a n a t i o n f o r p r e v i o u s l y .  sin ot-  oC i s a parameter  s e c o n d moment w i l l b e g r e a t e s t f o r 0 «• 4 5 ° .  broad l i n e s  2  0 i s t h e a n g l e between t h e [ p O l ]  c r y s t a l a x i s and t h e magnetic f i e l d ,  the  A simple  ( s e e A p p e n d i x A ) shows t h i s s e c o n d moment t o b e  ( V - U )  where  specimen  I n o r d e r t o e x p l a i n t h e measured  Misalignment reported second  -35-  moment o f .6 ( k c / s ) , c * . must b e s e t a t 1 8 ° , a n high figure.  Further, a study  of magnetic f i e l d  improbably  of t h e l i n e width as a f u n c t i o n  f o r 0 =» 45° r e v e a l e d no c h a n g e i n t h e l i n e  w i d t h when t h e f i e l d went f r o m 10.1 k i l o g a u s s t o 6.13 k i l o g a u s s . The  u n c e r t a i n t y i n t h e m e a s u r e m e n t o f t h e l i n e w i d t h was a b o u t  .1 k c / s .  Assuming t h e change i n l i n e w i d t h over  magnetic f i e l d  t o b e t h e maximum v a l u e w i t h i n t h i s u n c e r t a i n t y ,  we f i n d t h e c o r r e s p o n d i n g mean a m i s a l i g n m e n t If slices  c h a n g e i n t h e s e c o n d moment w o u l d  o f <=*. =• 4°, s t i l l  a somewhat h i g h  i t i s assumed t h a t m i s a l i g n m e n t  (or misalignment  large i s o t r o p i c broadening  i n d i r e c t exchange i n t e r a c t i o n s  t h e r e must b e some  fairly from  18 '  between t h e two t i n  isotopes which contributes t o the l i n e width.  and  does n o t  mechanism s u c h as t h a t a r i s i n g 17  suggested  value.  of the c r y s t a l  due t o s u b g r a i n b o u n d a r i e s )  c o n t r i b u t e t o the l i n e w i d t h , then  Shchegolev  t h i s region of  K a r i m o v and  t h a t t h i s mechanism i s p r e s e n t  a t t r i b u t e d a s e c o n d moment o f a b o u t .5 ( k c / s )  i nt i n  to i t .  I n d i r e c t exchange i n t e r a c t i o n s i n metals  result  f r o m a c o u p l i n g b e t w e e n t h e m a g n e t i c moments o f t w o n u c l e i b y way If  of t h e i r conduction  e l e c t r o n s and t h e h y p e r f i n e  o n l y t h e c o n t a c t term o f t h e s p i n Hamiltonian  (1) i s c o n s i d e r e d ,  this coupling i s purely s c a l a r  interaction.  of equation 1 9  .  The  exchange i n t e r a c t i o n energy between u n l i k e n u c l e a r s p i n s i n a metal  i s of the form  17  -36-  (35)  °H  -  £  £  A  I  ± i  . I.  ±  JL 6 Van V l e c k  shows t h a t t h e s e c o n d moment o f t h e a b s o r p t i o n  line  o f t h e n u c l e i w i t h one m a g n e t i c moment due t o t h e e x c h a n g e i n t e r a c t i o n w i t h t h e n u c l e i o f a n o t h e r m a g n e t i c moment i s (36)  h  2  ( AV)  2  -  i 1(1 + 1) f 3  A,,  1  .  2  xj  where f i s t h e f r a c t i o n o f t h e l a t t i c e s i t e s o c c u p i e d by t h e n u c l e i w i t h t h e d i f f e r e n t m a g n e t i c moment.  He a l s o shows i n  t h e same p a p e r t h a t e x c h a n g e i n t e r a c t i o n s b e t w e e n n u c l e i o f l i k e m a g n e t i c moments do n o t c o n t r i b u t e of  the resonance  t o t h e s e c o n d moment  line.  Several  a s s u m p t i o n s a r e made t o c a l c u l a t e A ^ j . I t •*\ 2-1.2 i s assumed t h a t t h e e n e r g y E i s g i v e n by E = . , , where — 2m* m* i s t h e e f f e c t i v e mass o f t h e e l e c t r o n s w i t h wave number k. K  k  The number vectors  o f o r b i t a l s t a t e s Z ( k ) d k i n t h e s p a c e o f wave Vd3k i s assumed t o be _ , t h e number f o r f r e e 3  (2TT)  electrons.  Also the excited states  extend from the Fermi l e v e l , is,  there  Finally,  3  E^,  Ep = ^ ^F 2m* 2  2  to infinity,  a r e no e n e r g y g a p s j u s t a b o v e t h e F e r m i  that  surface.  i t i s assumed t h a t t h e m a i n c o n t r i b u t i o n t o t h e  e x c h a n g e i n t e r a c t i o n comes f r o m v a l u e s Ik  }  a r e assumed t o  - k/l  2  i s very  s m a l l , and t h a t  of |jkj ~  such |k»|  that jkpl,  -37-  being  t h e wave v e c t o r o f a f i l l e d  unfilled state.  Under t h e s e  s t a t e a n d .k'  o f an  assumptions, the s c a l a r coupling  c o e f f i c i e n t becomes  (37)  '  A..-  :  :  2 ^ ( 2 ^ + l ) ( 2 I j + 1)  1 J  (—  2 ^ ^ , 0 0 8 ( 2 ^ % , )  \  \ i j j R  I  F  3  F  - s i n 2 k R .< F  where  i s the atomic  volume  m* i s t h e e f f e c t i v e mass o f t h e e l e c t r o n i n v o l v e d i n t h e exchange §  interaction  i s the r a t i o of the hyperfine i n t e r a c t i o n i n c r y s t a l t o t h a t i n a f r e e atom  11  i s the observed atomic hyperfine s t r u c t u r e splitting  R  t i  i s t h e s e p a r a t i o n o f t h e two n u c l e a r 1^  spins  and I j  k p i s t h e wave v e c t o r o f a n e l e c t r o n a t t h e F e r m i surface It i sd i f f i c u l t g i v e n by t h i s formula  t o ascribe a value  to A j for t i n i  b e c a u s e t i n does n o t have even a n e a r l y  s p h e r i c a l Fermi surface.  However, a s s u m i n g t h a t t h i s  formula  f o r A ^ j i s n o t e n t i r e l y i n v a l i d , we c a n make a n o r d e r o f m a g n i t u d e e s t i m a t i o n o f t h e s e c o n d moment due t o e x c h a n g e  3  -38-  broadening.  I t i s assumed t h a t m = m .  T h i s i s not t r u e f o r  all  e l e c t r o n s a s i s shown b y t h e many p e r i o d s o f t h e de H a a s -  van  Alphen o s c i l l a t i o n s  OQ  in tin '  v a r y i n g b e t w e e n .05 a n d 1.0. m^ m e a s u r e m e n t s . —— i s f o u n d m wave v e c t o r k  F  pp  rn •  which give values of  From e l e c t r o n i c s p e c i f i c  t o b e 1.2.  d e p e n d s on w h i c h p a r t o f t h e F e r m i  surface i s approximated  u s i n g t h e measured v a l u e s of t h e e l e c t r o n i c s p e c i f i c Within the l i m i t s of these approximations, J  =1,  heat  Likewise the value of the  b e i n g c o n s i d e r e d , b u t some a v e r a g e v a l u e c a n be  assume  —  heat.  i t i s reasonable  a n d t o assume some a v e r a g e v a l u e o f  V  a  to  measured  i n the d i f f e r e n t f i n e s t r u c t u r e l e v e l s of the ground s t a t e of 21  t h e f r e e atom o f t i n . In order t o perform perform  t h e sum o v e r  the f i r s t  t h e summation o f e q u a t i o n n-1 s h e l l s o f n e a r e s t  ( 3 6 ) , we  neighbours,  a n d r e p l a c e t h e sum b y a n i n t e g r a l f o r t h e r e s t .  (38) sin' (2k R j) - 4k R j cos(2k Rij) sin(2k Rij) i  F  i  F  ±  F  F  dV  2-  Rin  dV  -39-  where A r e p r e s e n t s t h e c o n s t a n t terms i n e q u a t i o n dV = 47TRJLJ volume.  . . (39)  *ij>  d i  2  The  ^ Z. j=n  a n d  ^  *-  *  s  h e  n  u  m  D  er  (37),  o f atoms p e r  unit  t h r e e i n t e g r a l terms g i v e  . 2 , p. .c ,2 f c o s a A. . = 4 T T ^ ( 2 k ) A ) — + 2  p  C  1 J  sin a — — + sin b 2  5  5  3 a  5 a  j _ _2  x  15b  where  ... £  s i b  = -  \  _  16 5b "  2  4  dx  and  b = 2a = 4 k  For  large b  ( i n our case b  F  8 0 ) , s i b c a n be e x p a n d e d i n  a s e r i e s , and  t h e sum  (40)  2 - 4 T r ? A ( 2 k ) 5 {\ c o s ^ a (. 3 a  £ A j-n  1 J  2  Rin  c a n be  2  written  5  F  4 cos b 3 b  3  + s i n f a + 16 c o s b 5a 5b 5  4sin b  3  5  +  4  0  (J^} W J 6  b  -40-  U s i n g a n IBM 1 6 2 0 c o m p u t e r , e q u a t i o n e v a l u a t e d f o r k p b e t w e e n .5 x 10® cm" i n t e r v a l s o f .01 b y p e r f o r m i n g (84 n e a r e s t n e i g h b o u r s )  -1  a n d 2.5 x 1 0  at  1  shells  a n d r e p l a c i n g t h e sum b y a n i n t e g r a l F i g u r e 8 shows t h e r e s u l t «r A,  2  J-J A t h e sum i s v e r y s e n s i t i v e t o k p .  t h i s calculation with a plot of  23  be s e e n ,  cm  8  t h e sum o v e r f i f t e e n  from t h e s i x t e e n t h s h e l l outwards. of  ( 3 8 ) was  vs  kp.  As c a n  2  The v a l u e o f k  F  o b t a i n e d from t h e e l e c t r o n i c s p e c i f i c heat of t i n i s 8  —1  1.5 x 10  cm" , w h i c h  the curve.  happens t o be near  Thus a s m a l l e r r o r  i n kp, although i t w i l l  a considerable error i n  , will A  T  another  value might.  a maximum p o i n t o f  give less error  F u r t h e r , t h i s v a l u e o f kp w i l l  from e l e c t r o n i c s p e c i f i c heat measurements. = 1.5 x 1 0  8  cm" ,  part of equation  2.  1  i j  A  - 1,75 x 1 0  IF  J  give the  coefficient unless the  c o r r e c t v a l u e o f kp i s c o n s i d e r a b l y l a r g e r than t h a t  P  than  2  upper l i m i t t o t h e exchange broadening  k  cause  6 2  expected  For ,  The  integral  ( 3 8 ) c o n t r i b u t e s o n l y a b o u t one p e r c e n t t o  t h i s sum, a n d t h e t e r m s n e g l e c t e d , t h o s e o f t h e o r d e r 4TT9 ( 2 k p ) for  5  ^°(^) ] 6  a  r  e  d  o  w  n  b  y  a  factor of 10 . 8  In fact,  t h i s v a l u e o f k p , t h e most s i g n i f i c a n t c o n t r i b u t i o n comes  from o n l y the f i r s t  two n e a r e s t n e i g h b o u r s i n t h e f i r s t s h e l l . 10 —54 Assuming V t o b e 10 c / s , we g e t A =» 2.66 x 10 and u s i n g e q u a t i o n ( 3 6 ) , we g e t ( A D ) - .57 ( k c / s ) . This e x c e l l e n t a g r e e m e n t b e t w e e n t h e o r y a n d e x p e r i m e n t c a n be no a  2  2  -41-  •4  6  8  1.0  1.2  k  1.4 F  x  16 IO'  8  F i g u r e 8.  1.8  2.0  2.2  2.4  •42  more t h a n f o r t u i t o u s c o n s i d e r i n g t h e a p p r o x i m a t i o n s i n v o l v e d i n d e r i v i n g t h e f o r m u l a f o r A^^, l e t a l o n e c o n s i d e r i n g t h e approximations i n t h e value of the constants such as V  and ^ .  a  However, t h e agreement i s e n c o u r a g i n g i n s o f a r a s i t g i v e s some j u s t i f i c a t i o n f o r c h o o s i n g k j , i n n o n - c u b i c m e t a l s  from  measurements o f t h e e l e c t r o n i c s p e c i f i c heat and i n d i c a t e s that the approximation of a s p h e r i c a l Fermi surface i n the d e r i v a t i o n o f t h e e x p r e s s i o n f o r A-|j may n o t b e p a r t i c u l a r l y restrictive.  B e c a u s e t h i s r e s u l t c a n be c o n s i d e r e d o n l y an  o r d e r o f m a g n i t u d e c a l c u l a t i o n , no a t t e m p t t o s e p a r a t e t h e 2 quantities  "X  p  and  J  h a s b e e n made .  F i g u r e s 5 a n d 9 show t h e t e m p e r a t u r e d e p e n d e n c e o f the S n  1 1  ^ and S n  1  1  9  resonance.  The measured change o f  i  n  H  g o i n g f r o m 4.2°K t o 1.15°K was o n l y - . 0 0 2 % o r c o n s i d e r a b l y l e s s t h a n 1% o f t h e t o t a l K n i g h t s h i f t . model h o l d s ,  i s expected t o v a r y as V 2  t h e a t o m i c volume . measurements  I f a free Q  electron  , where V  The a c c u r a c y o f t h e t e m p e r a t u r e  0  i s  dependence  i s t o o poor over t h i s range o f temperature t o  enable any statement about temperature v a r i a t i o n o f t h e K n i g h t s h i f t other than i t i n c r e a s e s s l i g h t l y as t h e temperature i s decreased.  F i g u r e 5 shows t h a t o n l y K a n d n o t K*  v  i s tempera-  t u r e dependent w i t h i n t h e a c c u r a c y o f t h e experiment. 117 T h e m a g n e t i c f i e l d d e p e n d e n c e o f t h e Sn Sn  1 1 9  r e s o n a n c e s shown i n F i g u r e s 5 a n d 10 i s f a i r l y  and small  over t h e range o f f i e l d s c o n s i d e r e d , but i sw e l l o u t s i d e  -43-  .770 H  /o  760  F i g u r e 9.  F i g u r e 10.  The K n i g h t S h i f t i n T i n a s a f u n c t i o n of Temperature  The K n i g h t S h i f t i n T i n a s a F u n c t i o n of Magnetic F i e l d ( M a g n e t i c f i e l d measured i n t e r m s o f t h e n o n - m e t a l Sn resonance)  -44-  experimental e r r o r .  No simple e x p l a n a t i o n of the f i e l d  dependence seems t o e x i s t s i n c e X  i s b e l i e v e d t o be  n  field  X"'  independent.  The f i e l d dependence can be e x p l a i n e d perhaps  i n terms of p o l a r i z a t i o n of the i o n c o r e s , or i n terms of some diamagnetic f i e l d  dependence.  -45-  CHAPTER 5  THE  POSSIBILITIES  OF DE HAAS-VAN ALPHEN TYPE  OSCILLATIONS I N THE  As m e n t i o n e d  KNIGHT SHIFT  i n C h a p t e r s 1 and 2, i f t h e  orbital  a n g u l a r momentum o f t h e e l e c t r o n s i n a m e t a l i s n o t c o m p l e t e l y quenched,  t h e r e e x i s t s t h e p o s s i b i l i t y o f an  oscillatory  b e h a v i o u r of t h e K n i g h t s h i f t caused by o s c i l l a t i o n s i n t h e d i a m a g n e t i c s u s c e p t i b i l i t y as t h e e x t e r n a l f i e l d T h i s e f f e c t w o u l d be s i m i l a r 22 23 effect the  *  i s varied.  t o t h e de H a a s - v a n A l p h e n  24 '  w h i c h h a s y i e l d e d s o much i n f o r m a t i o n  F e r m i s u r f a c e o f many m e t a l s .  The  about  idea f o r this  experi-  ment f i r s t a r o s e i n d i s c u s s i o n s i n t h i s l a b o r a t o r y b e t w e e n D r . R. B a r r i e a n d D r . M. B l o o m who  mentioned  i t t o T.  The p r o b l e m o f c a l c u l a t i n g t h e d i a m a g n e t i c f i e l d  P.  Das.  at the  n u c l e u s i s r a t h e r f o r m i d a b l e , a l t h o u g h a t t e m p t s a t i t have 25 b e e n made b y Das authors  2  6  '  2  Knight s h i f t The  7  ,  2  8  and Sondheimer ,  2  9  3  0  and e s t i m a t e s o f t h e change i n t h e  h a v e b e e n made. de H a a s - v a n A l p h e n e f f e c t i s a l o w t e m p e r a t u r e ,  high magnetic f i e l d done a t 1.2°K  ,  f o l l o w e d by o t h e r  phenomenon.  T y p i c a l l y , experiments are  i n m a g n e t i c f i e l d s b e t w e e n 15 and 100  kilogauss.  -46-  Although the temperatures required are r e a d i l y  accessible,  t h e s e m a g n e t i c f i e l d s a r e somewhat h i g h f o r t y p i c a l magnetic resonance s t u d i e s .  nuclear  To o b t a i n f i e l d s o f 100  kilogauss,  22 Shoenberg  and o t h e r s h a v e u s e d p u l s e d m a g n e t s whose r a t e o f  g change of f i e l d  i s something  l i k e 5 x 10  gauss/sec,  u n s u i t a b l e f o r n u c l e a r magnetic r e s o n a n c e work.  The  d e s c r i b e d h e r e was  10  at  performed i n a f i e l d  of about  quite work kilogauss  1.15°K. The m a g n e t i c f i e l d  a t t h e n u c l e u s due  to orbital  motion of e l e c t r o n s at the bottom of a non-degenerate  band i s  26  g i v e n by Y a f e t  t o be  (41)  AH  -  (4TT -  D)  • Xd  • H  ;X(j i s t h e d i a m a g n e t i c s u s c e p t i b i l i t y t e n s o r , D i s  where the  d  d e m a g n e t i z i n g c o e f f i c i e n t t e n s o r , and H i s t h e  magnetic f i e l d .  The  tensor  "Xd  will  applied  have a terra w h i c h i s  i n d e p e n d e n t o f H and a t e r m w h i c h o s c i l l a t e s a s H i s v a r i e d . T h e s e two t e r m s w i l l be s u p e r i m p o s e d on t h e p a r a m a g n e t i c 27 term which g i v e s r i s e t o the Knight s h i f t .  Stephen  c a l c u l a t e d t h e o s c i l l a t o r y p a r t of t h e s h i e l d i n g <Td  •»  4TT  , a n d h a s f o u n d i t t o be  has  factor,  -47-  (42)<T (osc) - -  ^okT V ^ H  1 2 N 7 r  d  0  ^ ^ ^ ^ ^ ^ J m ^  , £ n=l  \v*  ^ sinh  H  f™ ™ 2  »o  H  N where  ^  i s the density of electrons involved i n t h i s interaction  m* i s t h e e f f e c t i v e mass o f t h e e l e c t r o n | i i s t h e " e f f e c t i v e " Bohr magneton Q  ^  i s the Fermi  Q  (m* r e p l a c e s  m)  energy  I ( n ) i s an i n t e g r a l t o be e v a l u a t e d k i s Boltzmann's  numerically  constant  T i s temperature As c a n b e s e e n f r o m e q u a t i o n p e r i o d i c a l l y i n ^.  Further,  ( 4 2 ) , c5",j(osc) v a r i e s  s i n c e m* d e p e n d s o n t h e o r i e n t a -  t i o n of the Fermi surface, that i s the c r y s t a l o r i e n t a t i o n in  the magnetic f i e l d ,  will  be a n i s o t r o p i c .  10 k i l o g a u s s , w h e r e  the s i n /  n i T  T  0  )  f a c t o r means <5~ ( o s c )  For comparatively ^°  d  low f i e l d s such as  500, t h i s a n i s o t r o p y  would  ^o completely randomly  mask a n y e f f e c t i n a powder whose p a r t i c l e s a r e  o r i e n t e d i n the magnetic For  l i q u i d helium  f i e l d s of the order  field. o f 10 k i l o g a u s s a n d f o r  temperatures, the amplitude of the f i r s t  term  -43-  of e q u a t i o n  (42) i s .2  < >  [^ <  43  o s c  d  >]max  ^  12NTT —  -  r  .  _  w h e r e S t e p h e n g i v e s 1 ( 1 ) a v a l u e o f a b o u t . 1. i n s u i t a b l e values f o r t h e parameters,  *  I f we s u b s t i t u t e  we f i n d t h a t — — — i s H  o f t h e o r d e r o f IO*" f o r b o t h a l u m i n u m a n d t i n .  This i s less  6  t h a n one p e r c e n t o f t h e K n i g h t s h i f t limit  and would be an upper  t o t h e e f f e c t s i n c e D h a s b e e n assumed t o b e z e r o .  period of the o s c i l l a t i o n s  i s g i v e n b y A ^ L - ^ P - ^ = 2w .  The Using  t h e e x p e r i m e n t a l v a l u e f o r t h e p e r i o d s o f aluminum and t i n oo  g i v e n b y Shoenberg"" , we f i n d t h a t a t 10 k i l o g a u s s a l u m i n u m h a s a p e r i o d o f a b o u t 30 g a u s s a n d t i n h a s one b e t w e e n 20 g a u s s a n d 60 g a u s s ,  depending  on t h e c r y s t a l o r i e n t a t i o n  i n t h e magnetic  field. Because t h e t h e o r y f o r these diamagnetic tions i n the Knight s h i f t  oscilla-  i s a t b e s t o n l y an o r d e r o f  m a g n i t u d e t h e o r y , i t was f e l t w o r t h w h i l e  t o s e a r c h f o r them  i n s p i t e of t h e i r small p r e d i c t e d amplitude. were a l l done a t 1 . 5 ° K a t f i e l d s near method o f r e c o r d i n g t h e d a t a t o e n s u r e  The m e a s u r e m e n t s  10 k i l o g a u s s .  The  greatest possible  a c c u r a c y has a l r e a d y been d e s c r i b e d i n Chapter  3.  were r e c o r d e d a t i n t e r v a l s o f a few gauss over  a range of  magnetic  field  Signals  c o r r e s p o n d i n g t o a t l e a s t one p e r i o d .  The  -49-  r e s u l t s of the experiments are shown i n F i g u r e s 11 and 12. W i t h i n the a c c u r a c y of the experiment, n e i t h e r the aluminum nor t i n c r y s t a l gave any i n d i c a t i o n of these diamagnetic oscillations.  An upper  Knight s h i f t was  l i m i t t o t h e i r c o n t r i b u t i o n t o the  found t o be about  . 0 0 2 % f o r the aluminum  c r y s t a l and . 0 0 1 % f o r the t i n c r y s t a l . S i m i l a r experiments were attempted on a bismuth s i n g l e c r y s t a l which s h o u l d have a much g r e a t e r amplitude of cT^(osc) than e i t h e r t i n or aluminum. from bismuth was  However, no  signal  observed at l i q u i d helium temperatures,  26 p r o b a b l y because of the long r e l a x a t i o n time of bismuth  .  -50-  AH / H *> 0  .150 .149 148 •WL  o o o  °°  o  o  o  o  .o  °  o 8  o  o  8  o  8  S  8  8  ° -  o  8  •  8  o  .146.  o o  ° 0  ,145.  .144.  6.578 6.SM) 6.582 6J84 6.5S6 6558 6.595 6.592 6.594 6.596 6.598  (l^uW •  F i g u r e 11.  -  t^k  The Knight S h i f t i n Aluminum as a F u n c t i o n of Magnetic F i e l d (Magnetic f i e l d measured i n terms o f t h e deuteron resonance)  -51-  H k •772 o  .17 \ 770h  o  o o  «  «  °  0  °  o  0  o  0  0  a  e  0  0  9  o  769 .768 .767 .766 .765 -  15 "30  IS.9I  15.92  F i g u r e 12.  IS.93  15.94  J 1 15.95 15.96  1  1  1  1  1  1  L_V^  15.97 IS.96 15.99 16.00 16.01 16.0P 16.03  The K n i g h t S h i f t i n T i n a s a F u n c t i o n o f Magnetic F i e l d (Magnetic f i e l d measured i n terms o f t h e n o n - m e t a l Sn r e s o n a n c e )  -52-  APPENDIX A  THE SECOND MOMENT OF THE RESONANCE L I N E DUE TO MISALIGNMENT OF THE CRYSTAL S L I C E S  The r e s o n a n c e f r e q u e n c y  i n a single crystal  t e t r a g o n a l symmetry measured f r o m i t s a v e r a g e ^ (l+K)  value  i s  Q  V - I A  A ~d\  where  with  equation  =  Kf \)  .  Q  (3 c o s  2  This equation  ©  - 1)  f o l l o w s from  ( 3 2 ) . We assume t h a t t h e s l i c e s a r e m i s a l i g n e d  such t h a t © v a r i e s between © a measure o f t h e m i s a l i g n m e n t  0  - oC a n d ©  Q  + oC , w h e r e »C i s  of t h e c r y s t a l s l i c e s .  To  c a l c u l a t e t h e s e c o n d moment o f t h e a b s o r p t i o n s i g n a l d u e t o misalignment,  we c o n s i d e r t h e f o l l o w i n g q u a n t i t i e s .  average value of V  i s defined by  V where  f(.v) dy  frequency  The  -  "V f ( V ) dV  i s the p r o b a b i l i t y that the resonance  i s between  V  and  "J + dV  .  I f we assume an  -53-  isotropic distribution and  ©  0  Thus  between t h e two l i m i t s  ©  Q  -oc  + ©< , t h e n  f (u)du -  where  in ©  -  7—  b-a  u « c o s ©, b = c o s ( ©  4 ^ dv  —  b-a  0  civ -  f ( v ) dv  - o c ) , and a » cos(© + o c ) . 0  "v" c a n b e w r i t t e n b ^  _ 2  C  ( 3 u - 1)  J a  b - a  2  (b'  Similarly  ~y2~  d  u  + b a + a' - 1)  5  4  c a n be w r i t t e n b  V»  . (-^ii.)  2  C  Ou  - 6u  4  2  + !)  d  u  a  3  a  j |  ( b  4  +  b  3  a +  a  2 2 b  +  2(b  b  2  +  a  4  }  _  + ba + a ) + l j 2  The s e c o n d moment due t o m i s a l i g n m e n t o f t h e c r y s t a l is  slices  -54-  (V  - V  )'  v  . P S i'2  2  ^ ( b  4  +  5  b =» cos(© -ot)  If  v  2  - (U)  and  11  2  2  )  I  a ) 4  a - cos(© + <*),  )12 cos  2  (b a + b a 3  5  2  2  then  © c o s * sin © sin <* + 2  i  2  sin  4  2  © sin oC 4  + ba ) 3  PHYSICS  Volume 1, number 3  THE  ANISOTROPY  LETTERS-  OF T H E NUCLEAR MAGNETIC IN W H I T E T I N *  1 May 1962  RESONANCE  E . P . JONES * * and D. L L E W E L Y N W I L L I A M S * * * Department of Physics, University of British Columbia, Vancouver 8, B.C., Canada Received 17 April 1962  The nuclear magnetic resonance signal in white tin has been studied in some detail in the powder ) , and it has been established that the observed broad line is a consequence of the anisotropy of the Knight shift in tin together with the effect of nuclear spin exchange between different i s o topes 4). Bloembergen and Rowland from observations on thallium, have suggested that the exchange interaction need not be isotropic. 1 - 3  In the hope of clarifying these ideas, a direct study of the anisotropy of both the Knight shift and the line width has been c a r r i e d out in a single c r y s tal specimen of white tin. The specimen was constructed in the form of a multiple layer sandwich of 0.1 mm thick oriented in tin layers separated by 0.05 m m layers of M y l a r ; the whole cemented together with a silicone resin spray. The tin layers were formed by etching down 1 m m thick tin slices cut from a single crystal. The signals were observed with a Pound-Knight spectrometer, and in view of the comparatively weak tin signal, measurements were taken at the lowest available temperature (~ 1 . 1 5 ° K ) . A steady magnetic field of 10 kilogauss was produced by a Varian 12 inch rotatable magnet and was monitored by the deuteron resonance in D 2 O . Rotations were * Research supported by the National Research Council of Canada. ** Holder of International Nickel Company of Canada Research Fellowship. *** National Research Council of Canada Postdoctoral Fellow.  Fig. 1. The (110) plane containing the [001] and the [111] directions, and the plane perpendicular to it. The magnetic field is rotated in either of these planes. performed in the mutually perpendicular planes shown in fig. 1. The observed anisotropy in the Knight shift and in the line width determined by the maxima in the derivative of the S n ^ " resonance are shown in figs. 2 and 3. The anisotropy was checked for 1 8 0 ° symmetry by two points at 2 2 5 ° and 3 1 5 ° . The Knight shift is closely expressible in terms 109  Volume 1, number 3  PHYSICS  1 May 1962  LETTERS 60-\  o (jTo) plane * plane perpendicular to (110) K + K'„ (3 cos'& -/)  o (l Io) plane * plane perpendicular  to (llo)  SO  76 _^4 0 74 ^3.0  %  ^1*  I  70  'I [III]  J  [00\]  .68 0°  45'  90'  135'  180°  0°  11  of the e x p e c t e d r e l a t i o n f o r t e t r a g o n a l  symmetry  45'  90'  135'  180°  Fig. 3. The Snll9 line width as a function of crystal orientation in the magnetic field.  Fig. 2. The Sn ^ Knight shift as a function of crystal orientation in the magnetic field. 6)  any c a s e ,  such a m i x i n g would i n c r e a s e o u r value  o f A" b y o n l y o n e p e r c e n t a n d w o u l d h a v e a n e g l i g i ^  = K + \ K{  (3 c o s 6  b l e effect on J£j[.  - 1) ,  2  One c a n r e a d i l y understand e r r o r s i n an analysis w h e r e S i s the a n g l e b e t w e e n t h e m a g n e t i c f i e l d a n d  of the p o w d e r l i n e s h a p e s i n c e a w e i g h t i n g  the (001) a x i s .  c o r r e s p o n d i n g to the a n i s o t r o p y o f the l i n e w i d t h  O u r r e s u l t s a r e c o m p a r e d with  s h o u l d be u s e d .  o t h e r v a l u e s i n t a b l e 1.  factor  T h i s factor probably explains most  of the d i s c r e p a n c y b e t w e e n the r e s u l t s q u o t e d i n  Table 1. Knight shift constants. K x  Ref. ) Ref. ) Ref. 3) Present experiment 1  2  10  4  70.9 + 0.7 75.7 71.3 + 0.2  K\  t a b l e 1.  x io  4  asymme-  W o r k i s c o n t i n u i n g o n a d e t a i l e d i n v e s t i g a t i o n of  2.3 6.6 ± 0.6 5.4 + 0.2  t h e s e p o i n t s t o g e t h e r w i t h a study of the S n  1  1  7  res-  onance.  T h e a n i s o t r o p y of t h e l i n e w i d t h d i f f e r s i n b o t h f o r m a n d m a g n i t u d e f r o m that e x p e c t e d f r o m d i p o l a r broadening alone.  T h e o b s e r v e d l i n e - s h a p e showed s o m e  t r y w h i c h a p p e a r e d to b e a f u n c t i o n of o r i e n t a t i o n .  T h e mean value i s i n a g r e e -  m e n t w i t h that of K a r i m o v a n d S h c h e g o l e v f o r the  W e w o u l d l i k e to t h a n k D r . E . T e g h t s o o n i a n a n d M r . A . L . C a u s e y f o r p r o v i d i n g the t i n s i n g l e  crys-  tal and D r . M y e r B l o o m f o r i l l u m i n a t i n g d i s c u s sions.  s e c o n d m o m e n t (1.2 ( k c / s ) ) a n d i f t h e a d d i t i o n a l 2  b r o a d e n i n g i s a t t r i b u t e d t o the e x c h a n g e  interaction,  this interaction h a s a l a r g e anisotropic component. A d e t a i l e d e x p r e s s i o n f o r t h i s i s a v a i l a b l e 5) a n d c a l c u l a t i o n s o n the i m p l i c a t i o n s of the o b s e r v e d anisotropy are in progress. In t h e s e e x p e r i m e n t s the s a m p l e s i z e i s m u c h g r e a t e r than the e l e c t r o m a g n e t i c  skin-depth.  Theo-  r y > i n d i c a t e s that t h e l i n e - s h a p e s h o u l d then be a n 7  e q u a l m i x t u r e of a b s o r p t i o n a n d d i s p e r s i o n m o d e s . No obvious contribution f r o m the d i s p e r s i o n mode was o b s e r v e d , vations  3  i n agreement with previous o b s e r -  >8) that t h e r e s e e m s to be a n effective  skin-depth f o r nuclear magnetic resonance whi ch i s g r e a t e r t h a n the e l e c t r o m a g n e t i c  110  s k i n - d e p t h . In  References  1) B.R.McGarvey and H.S.Gutowsky, J. Chem. Phys. 21 (1953) 2114. 2) N. Bloembergen and T.J.Rowland, Acta Met. 1 (1953) 731. 3) Yu.S.Karimov and I. F.Shchegolev, J. Exptl. Theoret. Phys. (USSR) 40 (1961) 1289; translation: Soviet Phys. JETP 13 (1961) 899. 4) M.A.Ruderman and C.Kittel, Phys. Rev. 96 (1954) 99. 5) N.Bloembergen and T.J.Rowland, Phys. Rev. 97 (1955) 1679. 6) A.Abragam, The principles of nuclear magnetism (Oxford University Press, 1961), p. 205. 7) A.C.Chapman, P.Rhodes and E. F.W.Seymour, Proc. Phys. Soc. (London) B 70 (1957) 345. 8) A.G.Redfield, Phys. Rev. 101 (1956) 67.  -57-  APPENDIX C  This appendix i n c l u d e s c i r c u i t commerical items  of the spectrometer.  on t h e a p p a r a t u s a r e The and  diagrams of the  One  o r two  non-  comments  appropriate.  heaters  of the Pound-Knight-Watkins  oscillators  o f t h e W h i t e a m p l i f i e r s a r e s u p p l i e d by a 6 v o l t  storage  battery i n p a r a l l e l with a Heathkit Battery Eliminator. A dc s o u r c e  f o r t h e h e a t e r s was  found necessary  especially  f o r t h e W h i t e a m p l i f i e r s t o e l i m i n a t e 60 c / s i n t e r f e r e n c e . The  P o u n d - K n i g h t - W a t k i n s o s c i l l a t o r u s e d t o measure  the d e u t e r o n resonance employed a V a r i c a p  i n t h e same way  shown i n F i g u r e C l , e x c e p t t h a t a 90 v o l t b a t t e r y and Helipot provided Tektronix  t h e v o l t a g e sweep i n s t e a d  162  waveform  The  initial  T e k t r o n i x 162  terminated F i g u r e C2  v o l t a g e o f t h e sweep o f t h e  modified  the s a w t o o t h run-down i s begun or or P u l s e  shows t h e m o d i f i c a t i o n s o f t h e w a v e f o r m  t h e T e k t r o n i x 162  modified  i s set w i t h the c o n t r o l  w i t h t h e s w i t c h m a r k e d G a t e Out  i n dotted squares.  a 100  generator.  waveform g e n e r a t o r  m a r k e d V e r n i e r , and  of the  as  The  Out. generator  r e s t o f t h e c i r c u i t c a n be f o u n d i n  waveform g e n e r a t o r  manual.  K  +/oo to Oy .OO/  6  A6  F i g u r e C2.  2  Z.4 3.2 45 x I0&n  6.5  8  10  /4  M o d i f i e d T e k t r o n i x 162 Waveform Generator  /O  S<xv*toofA OuTpuT  225  V  250  v  Horizontal Amplifier f o r t h e T e k t r o n i x 360 Oscilloscope  Phase S h i f t e r  Figure  C3.  225V  F i g u r e C4.  Phase S e n s i t i v e Detector  -62-  REFERENCES  1  C . H . T o w n e s , C . H e r r i n g , W. D . K n i g h t : 852  Phys.  Rev. 77  (1950)  2  W. D . K n i g h t :  Solid State Physics  3  A . Abragam:  The P r i n c i p l e s o f N u c l e a r Magnetism, C h . I V  4  A . Abragam:  The P r i n c i p l e s o f N u c l e a r Magnetism,  2 93  (1956)  p . 172  5  C.  P. S l i c h t e r :  Nuclear Magnetic Resonance, t o be  published L. P a u l i n g , E . B. W i l s o n : Mechanics,  p 232  7  N . Bloembergen,  8  A . Abragam:  9  D. G . W a t k i n s :  1  0  R. Blume:  1  1  E . Sawatzky:  1  2  N. A. Shuster:  1  3  E . P. Jones,  1  4  A . C . Chapman, P . R h o d e s , Soc.  1  5  6  T. J . Rowland:  Ph.D. Thesis,  R S I 32 743  RSI 22 254  University of B.C.  Physics  (1961)  L e t t e r s JL 1 0 9  E . F . W. S e y m o u r :  Proc.  (1962)  Phys.  (195*)  Yu. S. Karimov, I . F . Shchegolev: JETP 13 899 Phys.  (1952)  (1951)  D. L I . W i l l i a m s :  J . H . Van V l e c k :  Magnetism, p 205  (1961)  Ph.D. Thesis,  B LXX 345  (1953)  Harvard U n i v e r s i t y  J E T P (USSR) 4 0 1 2 8 9  (1961),  (1961)  R e v . 74 1168  17  M . A . Ruderman,  C. K i t t e l :  1  N . Bloembergen,  T . J . Rowland:  8  A c t a M e t . 1. 7 3 1  The P r i n c i p l e s o f N u c l e a r  transl: 1  I n t r o d u c t i o n t o Quantum  Phys.  (1948)  R e v . 96 99  Phys.  (1954)  Rev. 97 1679  (1955)  -63-  Abragain:  The P r i n c i p l e s of Nuclear Magnetism, p 207  V, Gold, M. J, P r i e s t l y :  P h i l . Mag. 5 1089 (.1960)  Tolansky, G. 0. Forester: Shoenberg: G. Chambers: B. Pippard:  Progress i n Low Temperature Physics II, p 226 Can. J. Phys. 34 1395 (1956) Reports on Progress i n Physics 22 176 (1960)  P. Das, E. H. Sondheimer: Yafet:  P h i l . Mag, 32 315 (1941)  P h i l . Mag, 5 529 (1960)  J. Phys. Chem. Solids 21 99 (1961)  J. Stephen: I. Kaplan:  Phys. Rev, 123 126 (1961) J. Phys. Chem, Solids 23 826 (1962)  J. Stephen: Proc. Phys. Soc. 79 987 (1962) E. Hebborn, M. J. Stephen:  Proc, Phys. Soc. 80 991 (1962)  L, Sagalyn, J . A. Hofmanni  B u l l . APS, Ser, II 7 226 (1962)  Phys. Rev, 127 68 (1962)  

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