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Nuclear magnetic resonance in single crystals of tin and cadmium. Sharma, Surendra Nath 1967

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The U n i v e r s i t y o f B r i t i s h FACULTY OF GRADUATE  Columbia  STUDIES  PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE.OF DOCTOR OF PHILOSOPHY  of  SURENDRA NATH SHARMA M. Sc. U n i v e r s i t y of B r i t i s h Columbia, 1962  FRIDAY, APRIL 28th, 1.9.67.,.. a t 3:30 P.M. IN ROOM 301, PHYSICS  (HENNLNGS.) BUILDING  COMMITTEE IN CHARGE Chairman.:  S..D. Cavers  M. Bloom L.W. Reeves C.F. Schwerdtfeger E x t e r n a l Examiner:  E. Teghtsoonian B.G. T u r r e l l D.L1-. W i l l i a m s R.G. Barnes  Department of P h y s i c s Iowa S t a t e Uni.ver.sity.. .  Ames, U . S . A . '  Research S u p e r v i s o r : D..L1. W i l l i a m s  NUCLEAR MAGNETIC RESONANCE,IN SINGLE CRYSTALS OF TIN AND  CADMIUM  ABSTRACT A s y s t e m a t i c study of the K n i g h t s h i f t parameters has been c a r r i e d out on s i n g l e c r y s t a l specimens of t i n ..and .cadmium over a range from 1 K to j u s t below the . m e l t i n g - p o i n t s . The l i n e widths observed i n cadmium are.,.approximately h a l f the widths quoted by o t h e r workers e n a b l i n g more p r e c i s e measurements to be made. These measurements show the a n i s o t r o p y of the K n i g h t s h i f t - i n cadmium..changes s i g n between 4 K- and 77. .K Pronounced ..changes i n the K n i g h t s h i f t parameters are observed upon a l l o y i n g cadmium w i t h mercury. The temperature...dependence of the iso.tropic K n i g h t s h i f t i n t i n can be e x p l a i n e d by the volume e f f e c t s and.the.phonon c o n t r i b u t i o n s when the.tempe r a t u r e exceeds the Debye ...temperature. Comparing the temperature dependence data w i t h p r e s s u r e dependence r e s u l t s of o t h e r w o r k e r s . t h e , . e x p l i c i t temperaturedependence i s extracted.. I t appears t h a t the v a r i a t i o n i n the a n i s o t r o p y of the K n i g h t s h i f t above the Debye temperature i s l a r g e l y caused by a change i n l a t t i c e parameters, s p e c i f i c a l l y a change in,-c/a r a t i o . Measurements, of the l i n e w i d t h ^ a ^ h e l i u m temperature i n i s o t o p i c a l l y .pure Sn and the. same i s o t o p e i n n a t u r a l .tin 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 are r e p o r t e d . Second moments f o r d i f f e r e n t c r y s t a l o r i e n t a t i o n s have .been .computed A n a l y s i s of the. i s o t o p i c a l l y .pure t i n data l e a d s an e v a l u a t i o n of r e l a t i v e c o n t r i b u t i o n s • f r o m the v a r i o u s s h e l l s , to the p s e u d o - d i p o l a r i n t e r a c t i o n For the i s o t o p i c a l l y pure t i n , the measurements at the h e l i u m and room temperatures enable an e v a l u a t i o n of the s p i n l a t t i c e r e l a x a t i o n time T^. T^T i s c o n s t a n t over the temperature range and c l o s e l y isotropic.  GRADUATE STUDIES  i e l d o f Study:  Seminar  N u c l e a r Magnetic  i n NMR  -D.Ll. Williams  Quantum Theory of S o l i d s - R . B a r r i e S t a t i s t i c a l Mechanics Advanced Magnetism  -R; B a r r i e -M.  Bloom  Low Temperature P h y s i c s - J . B . Brown Electronic  Instrumentation -F.K.Bowers  Resonance  PUBLICATION  "The Temperature Dependence o f the K n i g h t S h i f t i n T i n and Cadmium" S.N. Sharma and D . L l . -Williams, P r o c e e d i n g s of the I n t e r n a t i o n a l Conference on Magnetic Resonance and R e l a x a t i o n ( L j u b l j a n a , September 1966.) ( I n p r e s s ) .  NUCLEAR  MAGNETIC  CRYSTALS  OF  RESONANCE  T I N AND  IN  SINGLE  CADMIUM  by  SURENDRA  M.Sc.,  A  N A T H ' SHARMA  The U n i v e r s i t y  THESIS  SUBMITTED  IN PARTIAL  REQUIREMENTS . DOCTOR  in  of British  FOR  OF  the  THE  Columbia,  FULFILMENT DEGREE  OF  PHILOSOPHY  Department  >  PHYSICS  accept  required  THE  this  thesis  OF  March,  @  as conforming  to the  standard  UNIVERSITY  THE  OF  Of  We  1962  BRITISH  COLUMBIA  1967  Surendra Nath Sharma  1967  In  presenting  for  an  that  advanced  thesis  or  by  publication 'my  of  i t freely  that  of  may  be  representatives. thesis  British  of  Columbia  the  granted It  by  requirements  Columbia,  the  Head  shall  of  of  this  my  that not  agree  and  copying  i s understood gain  I  f o r reference  for extensive  for financial  f  of  British  available  permission  ft(L^jn,6<^>  Canada  fulfilment  University  permission.  of  8,  the  purposes  this  written  University  Vancouver  his  at  in partial  make  agree  for scholarly  Department The  shall  I further  Department  without  thesis  degree  ths Library  study,  or  this  be  copying allowed  - i i -  ABSTRACT  A  systematic  study  of  the  Knight  shift  parameters  has  been  carried o  out to  on  single  just  are  crystal  below  the  approximately  precise  specimens  melting half  measurements  t i n and  points.  the  to  of  be  The  cadmium  line  widths  quoted  made.  These  over  widths  by  other  a  range  observed  in  workers,  measurements  from  1  cadmium  enabling  show  that  the  more aniso-  o tropy  of  the  Pronounced ing  be  changes  cadmium The  Knight  with  temperature dependence explicit  above  in  by  the  the  Debye  parameters,  the  with  effects  and  the  temperature. dependence  dependence the  a  the  change  line  are  reported.  crystal  orientations  and  room  the  to  to the  an  time,  T]_.  isotropic.  isotope  been  and  77  upon  enable T^T  an  shift  K.  alloy-  i n t i n can  contributions  of  the  when  the  temperature  other  anisotropy  workers  the  helium  of  a  the  Knight  shift  change  in  lattice  temperature  in  isotopic-  in natural  computed.  t i n as  moments Analysis  relative  a  function  of  for  the  different  of  the  isotopically  contributions,  from  interaction.  t i n , the  measurements  evaluation  i s constant  by  of  ratio.  Second  evaluation  pure  phonon  caused  at  pseudo-dipolar  isotopically  temperatures  relaxation closely  leads  have  Knight  results  i n c/a  width  orientation  For  K  observed  Comparing  i n the  i s largely  crystal  shells,  are  4  extracted.  variation  temperature  of  i s  same  various  between  parameters  isotropic  the  t i n data  sign  the  Debye  specifically  S n ^ ^  shift  and  pure  pure  changes  of  pressure  that  Measurements ally  Knight  dependence  temperature appears  the  volume  exceeds data  i n cadmium  o  mercury.  temperature  explained  It  shift  K  over  of the  the  at  the  helium  spin-lattice  temperature  range  and  the  -iii-  TABLE OF CONTENTS CHAPTER  Page  1  INTRODUCTION  1  2  EQUIPMENT  4  3  REVIEW OF THE THEORY OF THE KNIGHT SHIFT 3.1  The Knight S h i f t  11  3.2  O r b i t a l Paramagnetism  13  3.3  Core P o l a r i z a t i o n  15  3.4  Anisotropy  16  3.5  Temperature Dependence o f the Knight s h i f t  18  4  EXPERIMENTAL RESULTS AND piSCUSSION  20  4.1  Tin  ,  20  4.2  Cadmium  24  4.3  Discussion  31  5  6  THEORY OF LINE-WIDTH AND LINE SHAPES  45  5.1  D i p o l a r Broadening  46  5.2  I n d i r e c t Exchange I n t e r a c t i o n  48  5.3  P s e u d o - d i p o l a r Broadening  52  EXPERIMENTAL RESULTS AND DISCUSSION  SUGGESTIONS FOR FURTHER EXPERIMENTS  56 69  APPENDIX A  B REFERENCES  Second Moment i n I s o t o p i c a l l y Pure T i n as a F u n c t i o n o f Orientation  70  D e t e r m i n a t i o n o f the C o r r e c t Resonance Frequency i n T i n  72 75  -IV-  LIST  OF  ILLUSTRATIONS  FIGURE  Page  1  Block  Diagram  2  Schematic  3  The  4  Anisotropy  o f the Spectrometer  o f Low  Isotropic  Temperature  KnightShift  i n the Knight  5  System  8  i n T i n as Shift  a  Function  i n T i n as  a  o f Temperature  Function  of  Temperature 5  The K n i g h t Function  6  The  22 Shift  i n T i n , f o r Various  of Crystal  Isotropic  Temperatures,  as  a  Orientation  Knight  Shift  23  i n Cadmium  as  a  Function  of  Temperature 7  Anisotropy  26 i n the Knight  Shift  i n Cadmium  as  a  Function  of  Temperature 8  9  27  The K n i g h t  Shift  a  of Crystal  Function  The K n i g h t Helium  i n Cadmium,  Shift  as  10  D e r i v a t i v e s o f Cadmium  11  Explicit  Temperature  Tin  Function  12  a  Explicit Cadmium  a  Temperatures,  as  a  28  at Liquid  Function  Resonance  Dependence  Nitrogen  of Crystal  and  Orientation  Dependence  Function  Signals  30  of the Knight  Shift  i n 34  of  the Knight  Shift  i n  of Temperature  38  13  Anisotropy  of the Knight  Shift  vs. c/a Ratio  for T i n  14  Anisotropy  of the Knight  Shift  vs. c/a Ratio  f o r Cadmium  15  The of  Line-Width Crystal  29  o f Temperature  Temperature as  f o r Various  Orientation  i n Cadmium,  Temperatures,  as  21  o f the Isotopically  Pure  T i n as  a  41  Function  .  43  1  Orientation  57  16  The L i n e  Width  o f the Isotopically  17  The  Width  o f Sn  18  Plots  19  Plot  of  20  Plot  o f B^j and  Pure  T i n i n the Basal  Plane  58  119 Line of  i n Natural  T i n i n the Basal  2 ~ ( l - 3 c o s 2Q ^ Q ,, - )\2 ^ r-6 7^ v s . 9 f o r V a r i o u s s  r  '  J~(1-3  i j  W  ;  i j  r  .2 s2 -6 cos 9..) rT v s . <$> ( i n - t h e B a s a l 2  n  2  J ^ j vs. r^j  6  Plane  Values Plane)  of  59  60 61 66  -v-  ACKNOWLEDGEMENT  I his  wish  constant I  am  preting I Dr. some the  R.  help  am  o f my  grateful  Howard  o f my  data  The Canada  sincere  to Prof.  gratitude  Myer  t o D r . B. G. M.  pure  i n the form  Bloom  this  L I .Williams f o r  work.  f o r h i svaluable  Turrellfor  help  i n inter-  helpful discussions,  f o r helping  a n d t o D r . H.  t i nsingle  assistance  i s highly  financial  throughout  Agrawal  on t h e computer  cheerful  t o D r . D.  results.  and Krishna  isotopically  matters  my  and encouragement  indebted  some  The cal  to express  to  i n the computation  E. Schone  f o r the loan  of of  crystal.  of the Physics  Workshops  staff  i n techni-  appreciated  assistance o f summer  of the National research  grants  Research  Council  i s gratefully  of  acknowledged  CHAPTER  I  INTRODUCTION When  an  moment  and  the  nuclei  now  an  radio  it  This  also  seen  The  applied  energy  nance  the  the a  given  resonance  a  is  field  to  the  with  the  Larmor  resonance  nucleus  each  subjected  perpendicular  is  possessing to  to  this  different  is  defined  in  the  nucleus  can  magnetic  field.  spectral  frequency  are  The  varying  'line'  d.c.  parameters  of  In  terms  system  be  as  consists  The  and  shape  physical  the  however,  field  because  neighbouring  the  used  that  absorbs  applied  of  If  such  practice,  the  frequency  i s observed.  field, —  field  e l e c t r o n s and  experiment radio  magnetic  magnetic  the  from  then  a  frequency  phenomenon.  frequency  and  d.c.  frequency,  surrounding  fields,  a  Larmor  from  absorption with  frequency  nuclei,  field  local  local  I,  the  magnetic  'local'  and  rate  of  around  Larmor  this  momentum  is applied  by a  measure  identical  corresponds  is  feels  nuclei. the  precess  frequency  field  of  angular  r . f . field  energy. the  assembly  vector a  probe  of  of  to  observing  around of  sum  the  this  the  reso-  line  and  interest.  20 While nce  investigating  frequency  resonance magnetic usually  field.  which  then  the  measured shift  It  between  field  can  the  induces felt  i n metals  particular  This  positive,  magnetic  able  any  frequency  interaction  is  of  NMR  by  be  a  Knight a  explained and  nucleus.  by  the  found  metal  non-metallic  polarization  the  in  frequency  nucleus  the  i n f o r m a t i o n about  isotope  in a i n  W.D.  was  shifted  compound  i s known  as  taking  into  conduction  in  the  conduction  Thus  the  Knight  e l e c t r o n s near  same  Knight  electrons.  the  the from  the  account  of  conduction  that  can  the  Fermi  the external  shift  the  spins  provide surface  i  limited  the to  past  the  powdered  experiments samples  with  on  the  Knight  particle  size  shift less  measurements than  the  is  applied  metals.  In  and  hyperfine  The  electron  shift  resona-  were  electro-  valuin  magnetic r.f.  skin  depth,  field  into  For  cubic  a  in order  the  crystal  the  with  respect  to  metals  with  crystal  symmetry  such  metals  result  the  Any  can  observed  result  be  Knight  substantial  lower  broad  orientation  in considerably  shift  external  in a  crystallites. clearly  achieve  penetration  of  the  sample.  tation  of  to  i s  independent  magnetic, than  line  field.  cubic.  of  This  crystal  i s not  Experiments  on  orien-  true  the  powders  due  to  the  random  orientation  dependence  of  the  Knight  shift  i f single narrower  crystal  lines  specimens  hence  are  and  used.  increasing  the  for  of  the  line  This  shape  would  accuracy  of  the  i t  might  measurements. However, appear (i)  from  In  nuclei to  a  experiments  reading  single  the  that  i n a  i n the  the  rest poor  signal  Since  noise  which  i n the ratio  f o r the  are  fraction  only  of  few  are  as  the  which  microns  sitting  necessitates  a  easy  as  reasons. total  i s very  nuclei  i s a  not  following  phenomenon  crystal which  crystals  the  resonance  region,  nuclei  to  paragraph  specimen.  depth  the  in single  experiment  i n the  powder  skin  of  last  crystal  participating  ted  a  NMR  number  small  happen  only,  idle.  as  This  sensitive  compared  to  take  of  be  situa-  part  while  results  set  of  in  equip-  ment. (ii) This the  The  observed  i s also data.  previous At  resonance  caused  The  by  f i r s t  experiments the  skin  reason have  commencement  frequency depth i s a  been of  i s not  effects major  limited  this  work  and  the  true  complicates  hurdle  and  to  powder  two  the  resonance  only  the  frequency.  analysis  because  of  of  this  specimens.  experiments  )  had  been  (33 ) recently studied The  performed  on m e t a l  single  crystals  but  Schone \  'has  since  also  cadmium. object  parameters  of  the  i n t i n and  present cadmium  work from  was  (i) to  helium  study  the  temperatures  to  Knight just  shift below  the  -3-  melting of  point,  crystal  dence  of  Benedek  and  been  The  present  ssure has  We.  at  Knight  shift  Kushida,  the  attributed  to  results  dependence  been  ( i i ) to  orientation  the  has  and  derived have  over  various and  line  temperatures.  in  effects  been  combined  Knight  shift,  the  investigated  widths  i t s a n i s o t r o p y , has  volume  the  the  variation  have  of  study  entire the  the  Knight  and  an  with and  The  temperature  been  as  studied.  shift  other  the  as  shapes  intrinsic  temperature  line-width  and  with  a  function depenFollowing  temperature  temperature  workers'  intrinsic  dependence.  results,  on  temperature  pre-  dependence  range. a  function  of  crystal  orienta-  119 tion  in  isotopically  natural  t i n .  luation  of  An  the  analysis  enabled  an  of  Analysis  of  relative the  Sn  and  the  of  line  shapes  isotopically  pure  of  the  the  at  liquid  spin-lattice  helium  same  t i n data  pseudo-dipolar contributions  line-widths  evaluation  temperature.  pure  and  relaxation  has  from room  isotope  in  enabled  various  the  shells.  temperatures  time,  T^,  at  room  has  eva-  CHAPTER-II EQUIPMENT The  apparatus  standard  in  design.  cription  of  each  used  for  the  Figure  unit  used  1  e x p e r i m e n t a l work  shows  the  is given  is conventional  schematic  diagram.  A  and  brief,  des-  below.  (3,9) A A  PKW  6922  type  vacuum  cathodes  of  oscillating  tube  the  were  tie  the  oscillator  the  amplifier value  (varicap) mum  the  frequency  stable ilOOmv  a  be  and  was  varied  i n the  in  level  6J6  i n the  of  p¥  low  a a  voltage point  wide  and  range.  voltage  determined several  with  quite  low  thus  various  at  The  noise.  The  the  osci-  i n order  time  variable  was  and  to  same  diode  the  the  oscillator worked  Experiments  from  mini-  reso-  was  very  gain  C2  roughly  than  to  capacitor  capacitor  kept  greater  hours)  section.  capacitor  small  C±  C2.  by  constructed.  back  the  sensitive  always  was  feeds  A  C2  was  oscillator  mylar  circuit.  over  changes  which  resonant  Since  10^  minor  amplifier,  over  practically (2  .with  operating  with  experiments.  r f  .1  single  used  was  the  a  in parallel  i n frequency peak-peak  i n place  through  at  could  2-8pf  during  nance  coupled  PC116  used  oscillator  llator,  C^of  was  detector-  very  well  at  1.15°K  to  o 450  K  were  performed  and  found  that  the  mentioned  high  r f  levels,  The  ab.ove up  to  IV  frequency  of  the  l i t u d e (0-100  volts)  p-p,  levels  equipment used  was  f o r the  oscillator  linear  r f  sawtooth  was  had  to  be  adequately  high  swept  available  used.  sensitive  temperature by  applying  from  a  It  was  even  for  experiments. a  variable  amp-  modified Tektronix  (32 ) wave 1  form  m.sec. The  generator to  audio  tee  narrow  tor  an  Several  several  band  Aluminum 5%  type  162  hours  output  of  which  generate  linear  sweeps  of  from  duration. the  amplifier  marginal  model  single, c r y s t a l  bandwidth  could  White  oscillator  216A. was  used  To as  twin-tees with  test the  was  the  fed  a  performance  specimen  centre  into  at  room  frequencies  White of  the  twindetec-  temperature.  from  15  to  400c  Power  Ampli-  fier  Atten-  Audio  Oscillator  Horizontal  uator Phase  PKW ting  Amp.  fCRO  and  Oscilla  Narrow  Detectoi  Shifter  Band  Phase  Amplifier  tive  SensiDetector  Modified Tektronix  Counter  162.  and  Recorder  Printer  Magnet  Field  Monitoring  Oscillator  — (  CRO)-  Battery Helipot  and  -6-  were  tried  mum  f o ra  the  same.  findings  a n d i t was twin-tee This  found  that  o f 20 c/s,  the signal  keeping  other  rather  peculiar  behavior  and the Naval  Research  Lab. Report  Following  the narrow  band  to noise  ratio  variables  (S/N)  o f t h e PKW  i s not inaagreement  amplifier  was  maxi-  detector  with  the  Watkin's  (15)  i s t h e phase  sensitive  detector  The s i g n a l  recorder  which  (35 ) in  principle  used  of Schuster's  a V a r i a n r e c o r d e r , model  The  frequency  Packard  models  digital  activated  measurements  524C  were  made  r e c o r d e r s models  516B a n d 5622A  measured  frequency  was  external  rotation  before  supplied  pole  c a n be  from  field  faces  so that  the actual  an average  magnetic  12" diameter table  Hewlett-Packard  The f r e q u e n c y  pen on the s i g n a l  graduated  with  and 5245L.  indicator  with  .  G11A.  an  The  of  as that  was  counters,  The  i s t h e same  and a  field  a reservoir  over  which  a  by a  2 - l / 4 " gap.  was  was  Varian  print.  magnet  i s mounted  on a  i n the plane  on a t least  water  a  second.  t o thesample  cooled by  i n turn  i t made  rotatable  switched  The c o o l i n g  i n turn  o f .1  Hewlett-  which  time  The magnet  relative  The magnet  by  respectively,  period  supplied  started.  recorded  recorder every  orientation  determined. experiment  was  was  electronic  24  hours  f o r t h e magnet  the running  was  t a p water.  6 This  gave  a  Field turns pole  by a  pump.  the liquid  over  pair  current  supplied  was  used  audio  a  period  of coils,  forms,  was  was  i n 10  on b a k e l i t e  system  cryostat  achieved  Kinney  over  1  by a Hewlett-Packard  glass  temperature  pressure  achieved  wire  low temperature  inch  o f about  The m o d u l a t i o n  driven  doubledewar  three  was  18 c o p p e r  faces.  The  stability  modulation  o f No.  amplifier  est  field  mounted  hours.  wound, w i t h  60  around  the  by a Williamson  oscillator,  i s shown  each  of several  i n figure  2.  An  to achieve  low temperatures.  1.15°K  by pumping  on t h e l i q u i d  helium  bath  was  using  determined  type  power  model202D.  used  The temperature  magnetic  ordinary The  helium  by measuring  t h e o i l manometer.  low-  with the  a  -7-  The in  sample  diameter.  copper  wire  The on  was  mounted  The c e n t r a l held  field  the oscilloscope  line  conducting wire  i n place  magnetic  on a c o a x i a l  with  teflon  measurements  directly.  The  of stainless  of the coaxial  steel  line  3/8  was  inch  a No.  36  spacers. were  made  by  displaying  sample  used  was  the proton  glycerine  which  was  signal situa-  (10) ted  just  outside  the  field  phonics, run  was  tubing 32  was  assembled  a n d was  about  one  wire  was  foot  glycerine  (11-60) Mc/s. T h u s  a  90 v o l t s  with  the varicap.  thus  a very  ncy  was  llent  for  a  tal  with  symmetry hexagonal  such  mounted  probes a  This  were  cases  probe  voltage  enabled  extremely  the crystals axis  i n 10  used  were  with  o f symmetry  and had a  end o f the  glass  tube  5  coaxial  mm.  out-  the frequency f o r each  No.  116 was  used  used  fine  control  high  measured  axis  i nthe from  i n series  resonance  The c i r c u i t  signal before  to the  i n shape.  used.  o f the frequency and  The p r o t o n meter.  range  field  was  2pf capacitor  No.  on one e n d  A  respect  perpendicular  copper  obtained  parallel  t o the specimen  of  was  cylindrical  symmetry  made  connector  available  micro-  f o r the varicap  , and very was  monitor  experimental  diameter  covered  a varicap  frequency  field  orientation  of tetragonal  perpendicular axis  3  was  of the field.  6  i n a  which  with  100K H e l i p o t .  the magnetic field  made  i n parallel  The v a r i a b l e and a  I t h a d a BNC  permanently  suitable  stability,  particular Ajll  was  on t h e H e w l e t t - P a c k a r d  frequency  each  t o the box and on the other  a c c u r a t e measurement  read  In,all  conductor.  on  2-8pf,  dry cell  until  outside  from  circuit.  t h e magnet  to  box, to avoid  inch  diameter.  resonance  brass  in>length and h a l f  side  capacitor,  heavy  used  t h e b o x was  sample  line  circuit  with  hooked  Four  on  T h e  used  a  small  permanently  as the inner  directly  dewar.  shock-mounted  The c o a x i a l  line  A  nitrogen  i n a  mounted  complete.  copper  which  the liquid  freque-  had  to noise and a f t e r  exce-  ratio. each  run  t i n  crys-  crystal. The n a t u r a l  to the specimen  and t h e cadmium  t o the specimen  axis,  crystal  axis  were  axis  of  with obtained  -8-  F i g u r e 2.  Schematic o f Low Temperature  System.  -9-  commercially 99.9997o. etched  from  Each  before  Metals  crystal making  Research  was  one  Ltd.,, Cambridge  inch  i n length  and had a  purity  of  and 3/8" i n diameter  a n d was  measurements.  The•isotopically  pure  t i n crystal  was  grown  i n the form  of a  hollow  o cylinder.  The a x i s  specimen basal  axis.  plane  field  Two  symmetry  s u i t a b l e ^ m o u n t s were  and with  the tetragonal  made  made  axis  an angle  t o make  o f symmetry  o f 30  with the  the measurements i n the plane  i nthe  of the  rotation. The  from  of tetragonal  Cd-Hg  Semi  alloy  Elements  diameter.  crystal,  with  Inc. Saxonburg,  The h e x a g o n a l  axis  12  atomic  U.S.A.  o f symmetry  percent  I t was  .5  mercury inch  was  long  was  perpendicular  No.  40 was  obtained  and.4  to the  mm.  i n  specimen  axis. In the  specimen.  ween For  a l l the experiments  the c o i l higher  between  layer  trials erature  coil  would  insulation  experiments,  The c o i l  five  s t i l l  optimum  was  wound  oscillate  between  mylar  was  and l i q u i d  or s i xlayers  a t room  f o r the c o i l  used  as  around  insulator  nitrogen  bet-  experiments.  o f mylar  were  used  only  ,one  layers t o work signal  o f mylar at high  the Q  o f the  so that  to  For high  I t was  temp-  high  experimenta-  the lowest  the  experiment.  found  two o r t h r e e  of the circuit  obtained with  as p o s s i b l e  a t the temperature  After  be wound.  temperatures.  t o n o i s e was  as c l o s e l y  could  made  I t was  f o r the o s c i l l a t o r  of the coil.  frequency  with  and specimen.  nitrogen  frequency  resonance  temperature  the c o i l  i n liquid  resonance  f o r the oscillator that  five  d i d not oscillate  of desired  thick  used  specimen.  t o put the specimen  observed  coil.  as  wire  f o rhelium  experiments  and t o know-the a  o f .001 i n c h  and the  oscillator  necessary  enough  temperature  o f mylar  operate  layer  and the specimen  the c o i l  The  lly  One  a copper  oscillator  Q  -10-  The ght  used  f o r the c o i l  of the crystal  copper is  wire  bomb  very  the  and t o keep.it  designed  important.  plane  rotation,  results  served  as a  to hold  o f the  support  mounted  in-the  As  already  described  In  case  and s i g n a l  used and a  Time  i n the phase cadmium  topically  tudes  were  less  pure  experiments  correct  than  t i n very  time used.  no  worked  problem  up  couple  cylindrical  o f the crystal does  not  l i e i n  o f degrees, The•copper  enabled  i t will  bomb n o t  t h e sample  t o be  .5  could  s e c , 2.5  even  modulation  was  seconds  well  be  with  20  displayed  s e c . a n d 5.0  f o r isotopically  of the line  t o 30  very  cycle  f o r any o f the specimens  In a l l the experiments  one f i f t h  constants  by a  but also  detector  detector  small  A  The mounting  and anisotropy.  t i n the signal  constants of  sensitive  position.  the wei-  orientation.  t h e PKW  pure  respectively.  modulation  shift  t o support  o f the crystal  and i s o f feven  t o n o i s e was  of isotopically  oscilloscope.  o f symmetry  Knight  enough  i n the desired  f o r the crystal  rigidly  twin-tee  1  not strong  the crystal.  I f the axis  o f field  givewrong only  was  was  width used.  pure  seconds  used.  For high  and higher  on the were  t i n , natural t i n  at helium was  used.  temperatures However,  f o r iso-  temperature  modulation  ampli-  CHAPTER REVIEW In>the shift and  shift  detailed  i s treated.  shift  the nuclei  conduction  the  e l e c t r o n s c a n be  plus  some  system  ion  + * ^\  terms  treatment  The  electrons.  of the  i s given  Knight  by  dependence  Abragam  (1)  of the  stationary  t o be  Q  feel  i s the Hamiltonian  field  H  n_ b  I  z  +  energies  The motion  Q  c a n be  N  .,(i,  ^  electron  appro-  o f the  written as,  j)  3.1.1  electrons The  of  to i o n cores  i  of the i o n cores.  o f the conduction  due  of  approximation,  The H a m i l t o n i a n  Q ^  f o r the conduction  i s the Hamiltonian  points.  the potential  N  )+ ^ Y ^ H  Z  a r e t h e Zeeman  lattice  interaction  an assembly  the adiabatic  of electron.repulsion.  (L +2S^  consider  u n c o r r e l a t e d and the free  electrons  _,• •  H  of the hyperfine  L e t us  Under  of external magnetic  n  i n terms  i o n cores.  assumed  a l l value  +  cores,  of the theory  the temperature  explained  t r e a t e d as  used.  ' n-  = ^  where  be  in,presence p  M  SHIFT  •  has been  c a n be  over  section  .  e l e c t r o n s and N  i o n cores  may  review  and comprehensive  and conduction  the  ximation  brief  KNIGHT  Shift:  Knight  between  THEORY•OF'THE  a  In the last  •The K n i g h t The  n  A  S l i c h t e r ^ ^ .  3.1  THE  following sections  i s given.  Knight  OF  I I I  i n the f i e l d  third  electrons  and  of  fourth  and the n u c l e i  and  (i, j) ^) It  i s the interaction represents  energy ^| 1  ( j )  Hamiltonian  the hyperfine. i n t e r a c t i o n  o f the j t hnucleus  =  between  (  j  )  z  energy  (  i  Y * g > I ^  )  +  L  (  i  and  and  j t h nucleus.  i s of interest  here.  The  c a n be w r i t t e n a s ,  1 -Y* H I 2SY-(r:3)iq>i; J 5 Z T ^ g ( l S . T R F ) i=l 3 J 0  i t he l e c t r o n  )  (  j  <-2_  i=T  r^i  )  +  i  i  -  ( s ( V r \ i ) ' )' r ^ i 3.1.2  1  The  electron  with It  j .  The  operators prime  has already  last  on  been  are labelled the 3 r d term  evaluated  with means  superscript that  at the o r i g i n  i t does  i and nuclear not contain  and the value  included  operators singularity. i n the  term. The  expectation  value  of  with.respect  to.t h e e l e c t r o n  wave  function  gives rons, are  the Knight  ^  i s normalized  simply  having most not  shift.  Bloch  over  contribute.  primed  term  i s zero  metals  with  symmetry  will  be  known  treated  as  Knight may  ^ <  j  where  earths  with  where  cubic  U^Cr) i s a  the orbital  are exception), symmetry,  angular  electfunctions  function  momentum i n  the second  term  does  the contribution of the  1  lower  than  contribute  cubic  term  I f only  i s the only  contact  term  to the Knight  and i s responsible  i n one o f t h e f o l l o w i n g  - - W ^ ' m ^  electron gives  (rare  The e l e c t r o n wave  1  i l c , r  Since  a l l the conduction  sections.  term,  shift  f o r anisotropy.  The l a s t  term  contributing to the  i s considered,  i nthe  i n  This  (3.1.2)  isotropic  the expectation  value  of  be w r i t t e n a s ,  >  )  lattice.  b u t i t does  the contact  shift.  volume.  /  For metals (28)  i s over  l/ (k",f)=UR(r)e  as  i s quenched,  t h e sum  the crystal  functions  same.periodicity  metals  Since  +  If? * I< y  i s the eigenvalue being  rise  found  to Pauli  J  (0)f  £  )  P  of S ^  at the site  and|4^[is  the probability  o f j t hnucleus.  susceptibility.  -. - ? j L X<H ,P  I t c a n be  3.1.3  The  shown  sum  of the  appearing  i t h  in. (3.1.3)  that  !  r  where at  Pp  i s the probability  the site  ^(J))= The  - m  Pauli  of the nucleus.  o  I  (  j  o f an e l e c t r o n , with Thus  -8jy«H ^I  )  0  z  susceptibility*X  (5f»)- -W&i»  F  ( j ) 2  VN(E )P F  =p N(E ). F  -^7*H l(i)v P 6  energy,  being  p  F  Therefore  F  - -y*H ij (n|2'vXpP ) b  The.first Zeeman the  term  tional  The  surface.  magnetic  AH=-MH V*pPF 0  3.1.4  p  i n t h e above  energy.  Fermi  :l>  second  expression term  I t c a n now  field and  AH,  found  we g e t  2  p  Fermi  c a n be  recognized  as  i s the contribution of only be  seen  due t o t h e s e  that  the nucleus  electrons, which  j  the nuclear s electrons at  feels  an  addi-  c a n be w r i t t e n  as  -13-  This  i s known  3.2  as  Orbital In  been  the previous  quenched  i n general. moment  section  I t was  and hence  magnetic  Knight  shift  ( K ^  s  o  ).  Paramagnetism.  outlined.  true  isotropic  the theory  assumed  does  there  that  not contribute  For metals contributes  of the isotropic the orbital  Knight  angular  t o the paramagnetism.  with,partially f i l l e d to the paramagnetic  shift  has  momentum i s This  non s bands,  i snot  the  susceptibility.  orbital  This  i s  (22) because,  as Kubo  reduced metals  by a  with  fraction  has pointed  of the total  bands,  e  * o  where  S  0  A  =  2 frr+V(r)+m  c a n be  c a n be an  shown  and i n  appreciable  b<y c o n s i d e r i n g  the  +2/3S-H  A  2  z  be w r i t t e n  one  This  f^"(P- "^'P)  +  2 - ^ A 2mc  J- 2mc  A  paramagnetism  temperature  2  2  .fl.^-S-Cp.A+A.p)  A" c a n b e  orbital  paramagnetism i s  * l  +  z  and  i s the degeneracy  q  paramagnetism.  + V(r) +  =  the spin  ,  2  =  T  where  o  degenerate  Hamiltonian  of  T/T ,  factor  out, i n metals  +2f2»S-H  a s 2A=Hxf,  where  divA=*0 a n d c u r l A = H ~ .  Using  this  value  gets  3^ - i t r ^ CW-2S) = The  pH«(L+2S)  magnetic  lity  by  where  F  * =  moment 2  3.2.1 of the metal  energy  ^F b y M=-(———)  and the s u s c e p t i b i -  H-*0  - ( ^ 2 ~ ) ^ c o n s t ,  i s the free  i s given  and i s given  by  F=Nf - k T i : log(l+exp((^-Ei)/kT)) and  t h e summation taken  electrons  and  over  3.2.2  a l l the states.  i s the-thermodynamic  N . i s • t h e • t o t a l number  potential.  ""7=%  a  t  0°K  only.  of  -14-  is  an eigenvalue  denoted F  by  cp (Sj  - N £  where  +  )  and eqn.  <p  -kT ^  using  (3.2.2)  l o g (1-texp ( (  diagonal  i n vector  expanded  i n the following  )/kT)  first  term  tion  band  will  n o t be  moment  k~£^' ^  i n which ^  which  we  T r ^ S - l ^ ) )  -  Ir^(St)  last  summation  to  term  over  k* s p a c e  each  approximation,  M M ^ l ^ f ^  j y b w  +  contributes  present  and  a s ^ i s  Tr(<^(3^)) c a n  be  ^  TO,*-/  to the diamagnetic  Since  we  here.  are looking The  and so o n l y  second occurs  m  0  m  susceptibility  f o r paramagnetic  term  requires  3.2.3  2  a  o f the conduccontributions, i t  permanent  f o r ferromagnetic  metals,  magnetic a  case  are not interested i n .  The  over  o f <H ,  way  =  considered  i s diagonal  i n t i g h t "binding  2  electrons.  t o be  function  )  nj The  as a  (SO)  the representation  Tr(q>t(&|))  c a n be w r i t t e n  below.  T r (  (Af ) = Now  of  value  gives  the paramagnetic  a l l possible  and a  summation  -» o f k.  energy over  values  susceptibility. i t i s changed  a l l the values  Substituting  To e v a l u a t e  into  an  of E ( £ ) which n  the thermodynamic  the  integration correspond  relation for X  gives 3.2.4  where  9t> (E)=f(E). /  absence  of H  f ( E ) i s the Fermi  has been  substituted.  function The  and f o r E  f a c t o r — — 87T  per  unit  •Xpp^o  c r y s t a l volume  consists  Dara  per spin  of three  state  i n k  m  the value  i s the density  of  i n the states  3  space.  terms  <a) 3.2.5  This surplus  i s the well  electron  spins  known with  2 \=  p> N ( E p ) a s  before.  Pauli-spin their  susceptibility  magnetic  moments  and a r i s e s  parallel  t o H.  because  of  -15-  r  *)so  n n '  * *  J  n  3  n  1  i s the contribution  the  queening  appropriate small  of spin-orbit  o f the orbital angular  as compared  angular  momentum  t o . ^  ( n E | 2 i | n ' R ) (n'R|  L|nE))  coupling.  coupling  This  momentum,by  and.symmetry.  and i t i s not an  mixing  3.2.6  l i f t s  i n other  The c o n t r i b u t i o n  important  slightly  states  of  i sn e g l i g i b l y  term.  (c) 3. 2. 7  This  arises  i t a l  angular  orbital  matrix  elements  metals  have  and  mainly  Core  which  This levels  s electrons metals  significant  Q  3.3  between  Transition  "X has  n'.  into  3.1 w h e r e  surface.have  no c o u p l i n g  This  ect  i s not strictly  electron  electrons.get  electron  of  partially  the orb-  with L  quenched  has  matrix  and d i f f e r  are only  as they  the contribution  Fermi  This  state  by  i n mag-  contributions  f i l l e d  band.  Most  band  and f o r themX©is  have  a  partially  from of the negli-  f i l l e d  d  band  value.  i t i s assumed  core  there  i n the conduction  are exceptions  states,  approximation  t h e same v a l u e  i n t h e same  i s outlined,  core  ground  binding  means t h a t  shift  the  have  excited  Polarization:  In•section  trons.  an occupied  In the tight  the states  number  o f the uh-occupied  operator,  momentum.  between  quantum  gible.  of mixing  momentum  angular  elements netic  because  spins  c a n be e x p l a i n e d has spin \ because  o n n=2,  1=0  closed  with  true.  through  polarized  that  the unpaired  the closed  the exchange  and contribute  as  o f the magnetic electrons.  s type  shell  The u n p a i r e d  qualitatively  shell  of the contact  term  electrons  electrons  electron  t o the magnetic  field  There  H . 0  As  a  elec-  interact result,  field  Suppose  at the  or core  spins  interaction.  follows.  to the Knight  at  with  the  thenuelei.  the conduction  L e t us c o n s i d e r  a r e two e l e c t r o n s  i t s  i n this  s  effshell,  -16-  one  with  will  be  s p i n f denoted pulled  forces.  a  b i t  In.other  words  site  which  i t would  ;  ^'  renormalization  tra  This  field  due  to  upon  the  degree  depend  field  magnetic  the  experienced field  and  o  r  e  otherwise  due  netic  wave  ^  electrons.  the  other  82^  and  the  conduction  to  the  S2K  with-spin  electron  functions  pushed  are  in  S2I  The as  distorted  a  in  electron  result such  a  of  exchange  fashion  2  "I^2&|£^  ( I ^ S ' f ^  and  outside  2 that  S2t  by  by  a  m  P l  have  w i l l  of  core of  x  e  >  been  also  the  i H  w  for  true  paramagnetic  is  in  zero  at  presence  be  of  (|vy  functions,  the  The  p o l a r i z a t i o n of nuclei  the  for  polarization.  the  longer  o  but  be  wave  n  (0)1 I £>l  extent  the  in  the  -  |V+>  1  2  )•  Thus,  net  mag-  seen  will  o*  sees  of  field  this  polarized  (0)|  nucleus  conduction  always  2  the.nuclear the  a  electrons.  d i r e c t i o n of  This  the  ex-  applied  character.  (13) Cohen in  L i  a t e s  et  and  Na.  317o  and  be  The fine  They  have  used  smaller 2P  increase  in  higher  electrons as  self  comparison for  are  they  a  the  contribution  consistent  the  hyperfine  between atomic  also  have  out  L i  and  to  wave  core  polarization  functions  and  calcul-  coupling  constant  for  Na  that  contribution  shows  the  L i  and  Na  number.  polarized,  node  field  due  at  the  but  do  not  nuclear  contribute  to  the  hyper-  site.  Anisotropy: So  which to  worked  The  coupling  .3.4  have  5.57o  respectively. should  a l  do  the  that  f a r , we not  axis  the  have  depend of  been upon  symmetry  primed  term,  applied  magnetic  shift.  The  nuclear  dipole  in  nucleus  in  metal.  cause  the  the  the of  not  field of  dis-cussing  H  Q  this  The  crystal.  considered  so  the If  field  only  term  external  we  far,  contributes  orientation  dipolar  contributions  d i r e c t i o n of  the  and  the  to  the  left  back  depends the  dependent  of  go  to  the  magnetic to  upon  eqn.  unpaired  is  the  electron  unconsidered  thus  shift  field  relative  (3.1.2)  the  anisotropy shift  Knight  we  find  d i r e c t i o n of  of  the  energy spins far  is  the  Knight of  the  outside H  a n  -^  s o  ,  the  where  H  aniso= Let  then  -Y*g|U ^  the angle  a  n  i  s  T h e s i g n to  H  Q  wave  .  l  - S r . J ^ j ) the radius  2  designates  function  ^  rT]) ' r  ^  3.4.1  r" a n d H  vector  0  be  denoted  b y cC,  becomes  r "  whether  The e x p e c t a t i o n  shift.  )  iV^a-teoB *)  =  o  (  between  eqn. (3.4.1)  H  (S  B l o e m b e r g e n ^  the electron  value-of  a n d sum o v e r  3.4.2  3  spin,  eqn.(3.4.2)  with  a l l t h e IT s t a t e s  has worked  this  q  - 1)  i s anti-parallel respect  gives  or  parallel  to the electron  the anisotropic  out f o raxially  symmetric  Knight  case.  The  result i s ,  H  where  q q  in The  2  F  c o s ^-  o f symmetry  total = A H  I  S  ^ H  a  n  i  i s given  s  3  l ^ d V ^  wave  chosen  between  shift  +  O  electron  has been  i s the angle Knight  1) r "  3.4.3  of the anisotropy  to the conduction  (3.4.3)  ( 3cos 9 2  F  i s t h e measure  axis  AH  0  ^Jty*(3  »  p  F  related the  = (1 JT-H N(E )  aniso  3.4.4  F  i n charge  functions  t o be.the  the magnetic  distribution  (3.4.4).  z axis.  f i e l d  H  Q  and is. :  In the  The angle  discussion appearing  a n d t h e symmetry  axis.  by  o  2 o  r  M  = K  i  s  +/3/UN(E ) F  o  q  ( 3 cos 0 2  p  -  1)  H 1 = For  i  s  + j K '  o  any a r b i t r a r y  AH For  K  =  quency  white  in  the case The  factor.  t i n , of  +  when  shift  1)  the f i e l d  H_,_ s i n  ,  3.4.5  i s  9 than  the crystal  The. r e l a t i o n s h i p  AH_4_ a n d , t h e r e f o r e , t h e r e s o n a n c e  i s oriented  reverses  when  with q  p  9=0.  This  i s negative,  i s seen which  fre-  i n case  i s seen  thallium.  expression I f g  -  , AH|| i s g r e a t e r  F  i s higher  of  Knight  q  (3 c o s 9  angle,  H,, c o s 9  positive  2  f o rthe isotropic  i s anisotropic, as a r e s u l t  i tw o u l d  Knight also  o f an anisotropic  shift  produce  contains  the Lande  an anisotropy  Pauli.spin  i n  paramagnetism,  1  g  the g can  -18-  be  expressed 8(9)  which  as =  2  approximates  distinguish 3.5  i s o  between  i  8  s  i  v  gl  these  2  two  Dependence  e  sin 9)^  the formof  Temperature  K  +  (g|| c o s 9  n  b  eqn.  and hence  i t i s not possible  to  contributions. of the Knight  Shift:  y  ~ 3  Hso  w h e r e JL i s t h e v o l u m e  over  spin  susceptibility.  In the free  dent  to a very  electron  (3.4.5)  good  which  ^(£,o)  electron  approximation.  at the nuclear  site  i s normalized  Pp  model  "Xp  and  i s the  i t i s temperature  indepen-  i s the probability  of finding  c o n s i d e r e d t o be  temperature  and i s a l s o  Pauli  the  independent. Soon  after  the discovery,  experiments  were  undertaken  t o measure t h e  (27) Knight  shift  as a  that  the Knight  with  temperature In  o  solely  •  from ^  0 T  The  V  following  dependence.  made  ^>T  'V  of the equation  o f the Knight  )  model  that  v  o f the thermal  ^ T  measurements  ,SlnK  P  (315L.) • &  expansion  constant.  U s e was  ^ l n V .  side  the contribution  made  Preliminary  o f temperature  thermal  o f the assumption  >.  .  Cs, L i and Cu by using  expansion.  hand  dependence  tocthe  the temperature  V  the left  erature  .-.  /SlnK  function  and K u s h i d a ^  keeping  "  a weak  i n Na, Rb,  thermal _  P  was  o f temperature  attributed  the validity  ,-SlnK  is  was  shift  while  examine  where  shift  1958 Benedek  of.the•Knight Kgm/cm  function  They  experiments  revealed  and the v a r i a t i o n o f the  on volume  sample. dependence  p r e s s u r e s up were  the temperature  ,.  i n a  • 4 t o 10  position  dependence  o f the thermodynamic  to  comes  relation 3.5.1  i s e x p e r i m e n t a l l y m e a s u r e d , ("^^[y ) T ^ T T ^ p  expansion  -  a n d ( ^ ^ ) i s the intrinsic  temp-  shift..  V <V > ) l n  V was  L  1  T  proposed  The c h a r a c t e r i s t i c  period  V to explain of lattice  the intrinsic vibration  temperature  i s o f the order  -19-  of  -12  10  between  sec.  while  -6  -9  10  vibration second be  P  an  time  P ( The  in  the  F (  P  second  first  right  of  o )  T  + W  term  V  and )  0  C  /  electron  can  local be  a  i t can  pressibility  and  C  i s  phonons  the  metal. is  not  averages  the  What  v  comes  lattice  Then  i t  the  should  _ 0  V~  zero  Using  in  the  volume  adjust  occupied  themselves Thus  equilibrium  Pj, i s  atomic  volume  below  ^  0  and of the  PF(t)=Pp(\£) zero  point  value  of  only  i n  the  dilation  2 =  (3U, i.where  p  i s the  energy  density  associated  longitudinal  phonons  to  adiabatic  with  the  com-  longi-  of  the  of  specific  heat  comes  assumption,  that  one  from  the  the  specific  longitudinal  heat  of  the  phonons  crude  the  longitudinal  third  of  the  total  specific  heat  phonons,  2  .  the  vibrations.  -> P  ^lnK  a  known.  the  from  the  ^  fraction  exactly  internal  and  varies  assumption  strain.  as  because  dilation.  0  i s the  the  taken,  to  showni.thatl(AV/V )  contribution  Making C  U  thermal  2  0  be  spins  follow  this  changes  around  average  —"  operator,  cannot  functions  of  0  (19)  tudinal  state  2KV/V ) 4  } 0  just  electron  be•neglected.  wave  otherwise. Pp(t)^Pp(V )  0  of  Under  rapid  expanded  time  expansion  (4V/V ) i s  ((V(t)-V ) /Vo)=  can  i t produces  this  V  spins  i s expected.  (3.5.1A)  The  then  i n the  approximation  Q  and  electron  time  temperature.  vibrates,  function  V  i n X^>  with  relaxation  the  eqn.  lattice.  temperature -  Thus  change  instantaneously to  )  f c  F  the  no  lattice  varying  at  Q  hence  i s changing  the  atom  spin-lattice  seconds.  on  which  F  almost  V  and  term  As by  -10  the  _ 1  F  ,Fv  , 9P v IT  - . -  -20CHAPTER EXPERIMENTAL In  the experiments  parameters  was  carried  reported out as  a  IV  RESULTS  here  a  AND DISCUSSION :  systematic  function  study  of the Knight  o f temperature  i n both  shift  t i nand c a d o  mium  single  below  crystals.  the melting Temperatures  77°K,  195°K,  specimen  temperature  lled  and 450°K.  contact  covered  was  was  with  liquid  pressure. was  temperatures used.  were  from  carried  For experiments  the temperature  achieve  o i l bath  to within  4.1  380°K  and above, To  range  the experiments  at the atmospheric  thermocouple. temperature  a t which  i n thermal  respectively  temperature  1.1  K  to  just  points.  300°K,  was  The  The  helium  o u t were  a t 4.2°K  and  liquid  and  higher  than  temperature  by  a  room  77°K  nitrogen  For the experiments measured  1.15°K,  4.2°K, the  baths  at dryi c e  copper-constantan temperature  of the bath  could  a  be  high contro-  1°C.  T i n the  made  by  first  direct  Jones  measurement  and W i l l i a m s  (42)  .  of the Knight A  sliced  shift  sandwich  parameters  single  i n t i n was  crystal  was  used  and (3)  the  measurements  have  made  were  experiments  The  present  and  the temperature  The  results  shift the  For the  to helium  using  powdered  measurements  a  were  range  was  f o r the isotropic  a r e shown  results  limited  as  a  function  o f the above  the single  frequency  on  sample a  extended knight  to just  shift  only.  Barnes  and the c r o s s e d  single  crystal, below  coil  without  and  i n figures  3  Borsa  technique.  slicing i t ,  the melting point  and the anisotropy  o f temperature  mentioned  crystal  at which  made  temperatures  of t i n .  i n the knight  and 4  together  with  experiments.  experiments  the derivative  the observed  o f the observed  resonance signal  frequency, i . e .  i s zero,  i s not  (12) the  true  magnetic  resonance, frequency f i e l d  line-width.  the correct  The  because  of mixing  o f modes  resonance  frequency  i s higher  experimental results  have  been  corrected  . by  For constant a  fraction  f o r this  of the  (Appendix  B).  Jones  and  <j> B a r n e s • Present  Will-iams  and  Borsa  work  Temperature Figure  3.  The  Isotropic  °K  Knight  Shift  i n T i n as  a  Function  of  Temperature.  Figure  4.  Anisotropy  i n the  Knight  Shift  i nT i n  as  a  Function  of  Temperature.  -23-  0 Helium D Liquid ©Dry  -45  0  Figure  5.  The as  between  Knight a  Shift  Function  of  N2  Temperature  Temperature  » Room  Temperarure  +380°  K  M 5 0 °  K  45 Angle  Ice  Temperature  ;  90 H  Q  and  135  (001)  i n T i n , for Various  Temperatures,  Crystal' OrientationAC  10  KGOMJS.  -24-  Plots respect 5  to  to  of  the  the  indicate  used  (001) the  for these  4.2  Knight  shift  axis  at  accuracy  as  the  and  experiments  a  function  different  the  was  of  the  field  temperatures  completeness  of  the  orientation  a r e • shown data.  with  in  The  figure  reference  S ^ C ^ .  Cadmium: Measurements  was  the  on  Cd  firsttomeasure  have  the  been  Knight  made  shift.  by He  quite also  a  few  workers.  estimated  the  Masuda^  '  contribution  of  (34) s  and  p-type  tropic  Knight  point.  in  the  first  functions.  shift,  Barnes  a l l the  ured  wave  and  above  Knight  using  Borsa  Styles  a  (3)  in a  Seymour  measured  super-regenerative detector, also  experiments shift  and  made  measurements  powdered  sliced  specimens  sandwich  of  were  single  only  until  these  used.  crystal  the  the  iso-  melting  parameters (33)  but  Schone  meas-  of  cadmium,for  the  time. For  a l l the  recorded  varied  the  was  line  from:0ii5  to  previous experiments, from  quite 0.7  l.OKc/s  to  narrow.as  Kc/s.  1.5Kc/s.  compared  This  including  increased  Schone's,  However,  to  the  the  i n the  above  the  line-width  present  experiments,  accuracy  of  the  experiments  and  varied  measurements.  . 113 Experiments crystal tin  without  were  slicing  experiments. Figures  temperature. comparison. field  The  and  Figure  present  i n Cd  without  7  8  of  the the  shows  line-width  in sign  Cd  in a  similar  f o r cadmium  Knight  shift  natural  almost  and  to  plot  the have  the  (0001)  have  4.2°K been  are  and  the  77°K.  plotted  on  as  also  shift  a  a  an  the CdC^.  function  of  there  function  for  of  temperatures.  anisotropy In  as was  plotted  as  for various  that  single  experiments  anisotropy  Knight  axis  revealed  between  correction,  of  cadmium  i n a l l respects  resonance  previous experiments the  experiments  on  were  reference  relative  changes  out  i t and  show  Results  orientation The  shift  6  carried  figure extended  i n the 9  the scale  Knight results, just  to  -25-  emphasize workers  this.  and  stantially results It  This  result  i t i s not narrower  clear  lines  a r e .more  reliable.  has  found  been  orientation  of  the  The  that  i s connected  this  reason  since  this  cases  are  f i e l d  i s parallel  dicular  Cd-Hg  governs shown  to  Hg  variation  c/a  of  helium  the  10.  helium  . 04047«.  the  systems  of  that  results In  felt  changes  means  the  line  i t i s  signal  presently  other  the  of  view that  in  of  shape  but  other the  the  themixture  understood  with of  symmetric  orientation  being  the  the  modes  i t seems  electromagnetic field.  with  likely  of  Two  when  sub-  present  magnetoresistance  i s almost  to  i n cadmium known  temperatures  Both  natural are  considerably  made  to  to  noise  of  fact  study  variation Cd-Hg  cadmium  be  has  has  of  single a  c/a  1.9023.  calculated  extreme  the  field  Cd  magnetic perpen-  the  Knight  crystal  with  ratio, The  shift  at  12  room  temperature  for helium  temperature  but  Cd-Hg.  at  shift  the  higher  at  than at  liquid  was,,unfortunately, the  was  same  f o r cadmium  at  a  Cd-Hg  can  these-measurements  ratio  the of  Pure  Knight  cadmium  also  signal  for  the  corresponding values make  a  1.8857 w h e r e a s  i s not  than  between  undertaken.  with  kc/s  work  anisotropy of  The  cide  six  this  This  i s not  relationship  ratio  value  9.447» h i g h e r  the  in  field.  (0001),  was  equal  py  discrepancy exists.  observed  penetration  i n hexagonal  percent  At  with  figure  the  temperature,  the  the  for this  to  the  the  crystal:  explore  parameters atomic  i n  why  with  (0001) .  alloy To  that  the  in contradiction  observed  magnetic  changes.  i s  that  the  line  width  whereas  the  line  width, i n n a t u r a l  only  helium  found  to  temperature, cadmium about  and  170°K.  nitrogen about  .371*.002%,  and  the  these An  2:1.  varied  anisotro-  values  coin-  attempt  temperature  temperature  cadmium  be  This  was  varied between  was  but  the  because  from  four  to  0.5-0.7kc/s.  F i g u r e 6. The I s o t r o p i c Knight  Shift  i n Cd as a F u n c t i o n of Temperature.  -28-  jQ  Figure  i  3£  8.  60  The as  Knight a  ,  9Q  .  Angle  between  Shift  i n Cadimium,  Function  of Crystal  H  0  12,0  and  ,  150  •  180  (0001)  for Various  Orientation*/.'  Temperatures, to  /< S o w s .  -29-  350  Liquid  Nitrogen  345  340!  1  Cgu> 3  Q Liquid 335>  Helium  a CO  XI 60 •i-l  330  "30  Figure  9.  The  '  Knight  Helium  50~  1  90  120 Ho  Angle  between  Shift  i n Cadmium,  Temperatures,  as  a  and  '  "T50  '  180  (0001)  at Liquid  Nitrogen and  Function of Crystal  Orientation.  Observed s i g n a l when H  Q  i s p a r a l l e l to (0001)  -31-  Thus as  the experiment  the accurate  but  i t showed  at  liquid  measurement  nitrogen  of  un-ambiguously  temperature  the Knight  that  there  shift  was  and  not  a  success  anisotropy  was  no  change  are obtained  at  1.2°K  i s  so f a r  concerned  i n the sign  of the  anisotropy.  4.3  Discussion:  (i)  Comparison  (a)  T i n The  noise  ratio  workers than  most  with  accurate  data.  results  i s greatest.  i t i s seen  that  other  that  When  comparing  the anisotropy  p r e v i o u s l y measured.  This  where  the present  observed  i s even  at  true  the signal  results  this  with  previous  temperature  f o r the single  to  i s higher  crystal  (42) work The  of  Jones  present  crystal (001) in  and W i l l i a m s  value  mounting  axis.  0.086±0.001  t o ensure  This  the previous For  of  where  that  indicates  an  anisotropy  i s believed  the plane  that  a  of  o f 0.08Q.7 w a s  t o be the  misalignment  observed.  o  a  result  field  o f more  rotation  o f about  careful  contained  t e n degrees  the  occurred  work.  temperatures  higher  than  helium,  the only  other  results  available  (3) are is  t h e powder some  measurements  disagreement  and  b i l i t y  o f powder  endent  line-width-which  source  of error.  nated  by  higher  more  and  Borsa  that  powder  the case  temperatures  where  time  I t i s seen  indicates  analysis  not  relaxation  this  .  assumes and the  the degree an  i t i s t o be  there  of  orientation  possibly line  that  this  width  indep-  i s the  becomes  expected  relia-  that  domi-  this  valid.  Cadmium For  and  becomes  The  i s clearly  the s p i n - l a t t i c e  approximation . (b)  i t i s felt  results.  At  of Barnes  cadmium  single  a  crystal  considerable form.  amount  A.comparison  of  data  i s shown  i s available i n figures  both 6  and  i n powder 7 and  i t i s  -32-  seen  that  atures cate  large  the  zero  present this  anisotropy  work  alone  are  nearly  to  have  should  one  the  In  be  half  changed  noted  those  present  particular sign  that  quoted  results.  liquid  whereas  the by  at  helium  other  temper-  workers  line-widths observed  others  Possibly  and  the  in  i t is felt  broad  lines  indithe  that observed  33) '  are  these  evidence  seen  It  vindicates  others  observe  i s  anisotropy.  (3 by  discrepancies exist.  of  due  to  oscillations  oscillations  their  in  presence  the  but  i n  the  specimen  probably  Knight  used  the  shift.  d i d not  other  An  reveal  specimens  attempt  any  were  to  significant  of  higher  purity. (ii)  Analysis As  shift  already  may  and  the  may  be  be  temperature  stated  separated  other  due  separated  In mined  of  both  to by  in  using  t i n and  In  may  the  be  two  the  cadmium of  absence  of  any  the  expression the  one  dependence  due  to  dependence.  thermal These  of  the  Knight  expansion  contributions  (3.5.1)  thermal  expansion  w o r k e r s >  The data  temperature  temperature  other  obtained.  3.5,  contributions,  explicit  from'measurements  dependence  section  into  an  dependence:  analysis available  2  5 )  contribution  and  the  can  explicit upon  to  explicit  out  the  deter-  temperature  i s concentrated sort  be  this. temperature  ^lnK dependence,^  ), B a r n e s  cally  by  putting ^ ^ )  where  no  zone  model  is a  T  =  boundaries  negative  and  Borsa  - ~ .  This  are  made  attempt  is correct  present.  contribution  an  to  The  the  to  under  result  Knight  work  the  of  i t out  free  volume  theoreti-  electron effect  model  under  this  shift.  Tin Recently pressure the This  Matzkanin  dependence  explicit result  volume i s  of  and the  Scott  (25)  have  Knight  shift  contribution.  They  in contradiction  to  at  carried room  found  Barnes  and  out  experiments  temperature  that  which  ;&nK , 31 nV C ^ ^ - l ( T>lnV T dT  Borsa's  x  on  the  determine  ) =+0.6110.04. 'pP  theoretical  speculation  -33-  and  i s an•indication  predicting to  reliable  the details If  for  reference  the free  results  electron  approximation  f o r experimental quantities  results  f o r (^5^ L  (^j" ^ Y (^"^ o l n v i. oT  ) P  n  (37),  together with  the intrinsic  temperature  range.  a r e combined  constant.  temperature  In evaluating  ,The r e s u l t s  Various earlier  which  possible  i n sections  a r e shown  dependence  (*?^ ^ )  are  of  sensitive  3.1  t o 3.3.  facts  expansion  c a n be  and  Scott's  results  from  obtained over a l l assumed  (3^ ) nK  that  V  olnV 11.  to the Knight  T h e  Matzkanin  i thas been  n  i n figure  contributions  with  the thermal  2i T is  i s not capable  o f t h e model.  the present  the term  the  that  that  shift  have  the results  been  outlined  of the Knight  (43) shift  measurements  contact range is  contribution  and that  of temperature,  t h e dominant  w i l l  i n superconducting  be n e g l e c t e d . K = ! * j a  where  p  p  Thus  are explicable  the Korringa relation  are very  term.  t i n  good  indications  i n the following  The c o n t a c t  contribution  i n terms  i s satisfied  that  over  the contact  treatment  shift  i s  F  t h e symbols  have  their  usual  meanings.  Since  'XpWCgN(Ep), ••: •  77  F  after  N(Ep). of  this  appears  K  c o n t a i n s t o o many  t o be  variables  formidable as they  i . e . g,  a l l c a n be  a  Pp a n d  function  temperature. a r e two p o s s i b l e - s o u r c e s  influence  function Let  us  t r y to analyse i . e . we  the Fermi  spectrum  o f the lattice  f o r the electronic  Kushida^) of  F  simplification  The problem  There the  wide  contributions  K=| " g - n . P N ( E ) Even  a  907»  contribution  a l l other  ,to t h e K n i g h t  of  vibration  spectrum  temperature  and that  dependence;  o f Fermi  distribution  states. the accuracy  assumeXp  distribution  o f an i n t r i n s i c  t o be  function  o f t h e model  temperature  independent  i s negligible.  i s c o n s i d e r e d as a modulation  due t o Benedek.and and that  The e f f e c t  o f the atomic  volume  the  effect  o f t h e phonon  and as a  result  T  -34-  Figure  11.  Explicit a  Temperature  Function  of  Dependence  Temperature.  of  the  Knight  Shift  i n Tin  as  -35-  the  probability  term  of  being  is  the  mean  It  has  already  2  found  In  V=V  with  the  the  case  ^ C  i s the  evaluating  of  •3T  ;  constant  Benedek  Kushida's  the  temperature,  last  is  F  ^ ( V / V  of  time  i t s equilibrium  a t  a  Fermi and  value  (^~) V . Q  (BU) )  0  and  •  2 V  U  =  V  0  i s the  internal  energy  density  phonons.  is very  K  / v y  v  heat  =  v  small  v V  due  size  o  to  and  of  °  longitudinal this  term  Phonons.  l e t us  For  C=C /3  put  v  =  V  v  o  than  J l i s the  result,  is•a  purpose  of  then  V  C  higher and  the  (* V 2  c  the  special  Debye  atomic  which  result  of  the  model  dependence  as  the  specific  plotted  explicit  case  of  volume.  the  a  V  temperature—77-  is valid  as  a  function  temperature  of  i s that  It  i s thus  generalized  k  = w h e r e seen  f o r temperatures  should  ^expansion ^ l n K  be  dependence  noted  and Y i s ,  the 1  that  v  Gruneisen ^  P  (  k  is  that  higher  theory  F  should have  heat.  as  temperature  flc It  contains  than  outlined  the  in  chapter.  The ture  from  £>C  temperatures  and  probability  Vo  Boltzmann's  Debye  f  ^;(v/v )  F  nucleus  instantaneous  ih.  P  "^F  6P  an  the  ^PF  1 v  i s the  w  at  compressibility  t i n (^^~)  2P  oC-  For  2  approximate  N  P  volume  _J  o  i  =  site  the  0  specific  the  .?)lnK  of  longitudinal  *  T  , where  2  with  that  V  adiabatic  /2ilnK_ where  shown  interacting  nuclear  1^7=  i s the  associated  the  )  deviation  been  Xv/Vof  where  —  at  t P F  F  electron  (  square  _ P  Fermi  •—  electron  i  a  does  in  not  figure  obey  H. 3 the T  well  the  as  I t can law  at  the be low  same  tempera-  specific seen  heat  that  the  temperatures.  P v c  - -——  =  constant.  cC  where ofis Thus  the  coefficient  of  linear  -36-  and  i  *  "2 K V / V O )  2?F The  left  hand  tin  with  the  p  ?(V/V )  V  =  V  l  above  value  of  (^  )  =  Matzkanin  n  x  K  'V  O  the  V=V  0  Z  , ^ ^ ?)T  of  — F  _ X " oC  side  _ i 2  F  P  n K  expression  from  i s e v a l u a t e d by  figure  11.  For  1  0  .  K  and  -SK(V)  Scott  have  plotted  (  L  S  v  )  —  0  vs(  ).  =  3 +  4  our  model,  since  0  i t i s  constant  seen  2  2  that  dependence from  felt  that  this  on  single  the  i s of  ^ ( v / V  value good  the  the can  i n  ^PF  F  P  v = V o  pressure  derivative  Knight  Z v  =  V  O  obtained  the  However  obtained  that  _  with  shift.  The, f a c t  i  __^JI___  —  dependence  better  crystal.  )  agreement  pressure be  of  G  the  requires  from  the  value  the  types  tion  due  does  a  not  phonon  higher  are  function for  contribution  erature at  experiments,  t o Xp as  while-it other  of  the  equal  of  low  exists  temperature. temperature  which  contribution,  The  but  i s  ture of  4  shows  Barnes  results  range  nitrogen  accurate  dependence ^  p  F  P  3(V/V  there  model I t  by  the  data.  F  Z>  It  i s  experiments  obtained  from  0  i s  l i t t l e  seems  to  contribu-i .  work  i s possible to  phonon  the  for  that low  T>^9Q an-  temp-  contribution  :  of  slope  second  temperatures.  Figure  present  the  i n magnitude  dominated  Cadmium:  points  determination of  results.  i s comparable  at  pre-  T T K — - > — ;  that  experiments the  1  of  the  from  pressure  values  indicates  from  obtained  very  2  two  results  5  i _  =  2K(V ) 2)(v/Vo)  a  >  their  KocPp,  An  ssure  0  From  o  v  0  i Thus  n  ^(v/V )v=vo  Q  In  » 9  X. f o r  3.810.2 '  Kiso(Vo)  2K(V )  T  combining  but  at and  and are  for  dry dry  a  ice  K £  s  o  Borsa i n  the  fair  vs are  temperature also  shown  agreement  helium  with  temperature.  temperature.  ice temperatures.  No  plot  f o r cadmium.  t h e r e ,with their The  Barnes  errors  results present  measurements I f  the  and  were  The  over  Borsa's  involved.  a l l the  results made  experimental  show  between  T h e  temperaa  change  liquid  measurements  i n  -37-  this as  range  shown  a r e combined  by the dotted  Kushida Knight  their ture  indicate  (^ "ST l  n  ) V  K  v  constant  over  has  been  worked  the  variation  lations  n  i n this  (  2  a l  1  n  c  a l l the temperature  of ^ - and f  a r e from  the Knight results and  out.  V  reference  T>  range,  be n o t e d  T h a s a hump  low  More  and r e l i a b l e  For  explicitly  T )  a  a t room  tempera,  P  ^ l n c  and  aT  C ? ) ^ l n a cT l  temperature  i n figure  n  as  K  dependence  12 t o g e t h e r  o f c£n a n d oC u s e d "  i n these  with  calcu-  the Griineisen  constant,  a r e from  are certainly  some  inaccurate.  i n t h e same  i s required  what  temperature  to establish  region.as  the existence  of a  cadmium  the c o m p r e s s i b i l i t y i s not independent  i n ^ a p p e a r i n g  i n (3.5.3)  c a n n o t be  of  temperature^^  neglected  at high  temperatures,  o  c o n t r i b u t i o n a t 300 K holds  •  maximum.  el  relation  with  i  the.term  The  \  n  of the  V temperature  and  data  look  a r e combined  T  ( ^  K  The values  It  .  n  T  f o r  ^ T  l  I t i s shown  ( 1 1 ) . The r e s u l t s  (JLiU^)  will  dependence  )  t  (^ ) ? l n a cT  ( 2 4 ) a n d f o r y,  t h a t ^ v s  plot  i s largely  f o r (  the explicit  reference should  shift  Treating.  1°K t o 300°K.  (-+U  on the pressure  P  c a n be worked  o u t from  )  3 T  vs temperature  so  region.  i n cadmium  )  K  K±  the  I f the present  f o r (? X n c l  our data,  ^ ^ e x p e r i m e n t s  that  dependent.  results then  line  and Rimai's  shift  temperature  with  i s about  f o rcadmium^"^  from  10% o f the t o t a l 150°K  t o 450°K  effect.  .  A  The  Gruneisen  calculation  of  ^PF n f o r T=300°K 9n=172°K^ ^> from present r e s u l t s g i v e s a value ^ ( V / v ) ^ / , e q u a l t o 48 w h i c h seems r a t h e r l a r g e . Unfortunately — - ° ^ ^ ' — could not ^ W v ) 2  S  2  e  v  v  0  2  K  2  0  be  evaluated  and  from  the figure  Pressure  the pressure  obtained  dependence  then  a  comparison  that  o f Muto  f o r it.from  i n a  single  i s possible.  et a l  dependence  results  the present  results  crystal  can give  Perhaps  a more  i s necessary  to evaluate  because  more  of a  V  o  large  c a n n o t be  compared.  reliable.results  sophisticated technique the lattice  scatter  vibration  and like effects.  Figure  12.  Explicit in  Temperature  Cadmium  as  a  Dependence  Function  of  of the 1  Knight  Temperature.  Shift  -39-  (iii)  A n i s o t r o p y and Temperature  (a)  T i n No  variation  temperatures. and  i n anisotropy i s observed  Since  no measurements  dry i c e temperatures,  temperature . range. and at  dependence:  the results T°R and v  coefficient the  Q  volume  Above  <j£is  of linear  some  said  made  parameter  and should  so  I t i s found•that been  at constant  nitrogen  V(t)  temperature,  linearly  i s t h e volume  n o t be c o n f u s e d  from  i n this  i tvaries  oC= - 7 . 3 i 0 . 2 .  used  nitrogen  i t s variation  , where  e  have  liquid  9JJ=195°K^^,  (v(t) / V )  an  and l i q u i d  between  regarding  temperature,  expansion  experiments,  been  helium  K ± =  as  expansion.  of thermal  dependence  c a n be  t h e Debye  c a n be e x p r e s s e d  a t 0°K.  coefficient  nothing  have  between  with the  The v a l u e s o f  referecne  (37).  f o rt i n give  a  The  value  cC=-4. 2 5 1 1 . 8 .  One in  of the possible  the Pauli  that  X  from  Xp-  with  a  anisotropy  The a  metals if  should  be n o t e d ,  i n X  function i s a  mechanism  been with  c/a 4  ratio  as a  1-  mentioned symmetry I t seems  have  while  gives  lower  profound  should  have  n  i  s  contribution increases  Since  i t i s tempting  they  to K niso  cubic.  of H  In other  to conclude  and hence effect  a  p  on K  .  a  j [  ^  n  s  o  i  s  o  .  S  to  K  a  n  £  i s non-zero  a  n  ^  s  o  o  .  I t has  only f o r  contributes  Any change  i n volume  Experiments  s  o n Mg  the  interaction  eqn. 3.4.1.  Q  H  comes  i n opposite  of the electron  The changes  n  go  the contribution  constant.  on t h e symmetry on q  n  a  words  that  i fc / a remains  effects  H  to %  i n magnitude  i s the dipole-dipole  a  e t a l show  to speculate that  anisotropy and i s not related  by the Hamiltonian  than  l i t t l e  major  decreases.  o  the anisotropy  of A l e x e n d r o v ^  AX= 3j|-X]_  the contribution  appreciably  unaltered  a  rise  reasonable  i n the lattice  K  that  that  o f temperature,  that  tribution  The measurements indication  diamagnetic  which  t o t h e a n i s o t r o p y c a n be  however,  and i s described  not change should  .  i s an  i n temperature  distance  already  This  I t should  directions,  at  susceptibility  i s positive.  rise  contributions  of  H  a  only  n  ^  s  o  i n thec/a charge  which  dis-  keep c / a  single  crystal  -40-  can  provide valuable  1.623 7 ^ Barnes ent  which  and  while  remains  Borsa  c/a. r a t i o bur  on  than  variation  /^aniso\  _  powdered  specimen  pure  on  Cd,  Cd-Hg  283°K  do  alloy  i n anisotropy. i n K  a  i  n  ^ ( c / a )  The  first  term  on  c/a  ratio  as  function  erature  from  the  dependence  ted  contribution  and  (  s  can  o  T^anisos  P  a  regard. to  of  single  not  for  A  be  ;  of  be  7T  K  of  the  that  (  to  written \  ,  plot  )^  at  second  term  i n c/a  which  of  to  in  a  differ-  anisotropy  to'be. d e s c r i b e d  later,.show  conclusive  yet.  and  the  second As  due  term  already  to  the  change  is explocit mentioned  negligible.  in  temp-  Since  the expec3K ( — ^ ^  enable  the  l  s  o  ( ~  —)  because  ys  using  are  will  c/a  the  large  results  errors  involved.  T  on  a  enable  ratio  single  crystal  at  c/a=l  the  structure  the  evaluation  f o r t i n i S i shown  will  give  of  the  reliable  explicit  temperature  from  Lee  and  values  in figure  straight, line.  is responsible  White's^*^  temp-  f o r the  variation  dependence. Raynor^  2 3  -*  It  The and  for o^and  seems in K  values  below The  13.  a  It  that  n  i  of  273°K  behaviour  s  o  is  seen  above and  c/a  9  D  that  used  have  been  below  9  cubic  stru-  D  present.  is tetragonal For  of  Scdtt's  evalu-  a  ratio  450°K  understood  the  has  K niso>  explicit  is  means,  which  change  contribution  s h o u l d be  1  out  cture.  of  c/a  constant c/a.  9 D = 1 9 5 ° K ( ^ ,. i t i s a  worked  Tin  experiments  as  >>T  temperature  calculate  i n Kaniso  l i t t l e  from. 2 7 3 °  to  i  dependence  is  not  any  equal  / 9 ^ a n i s o \ :'•  P  p r e s s u r e dependence  ^  change  there  However  crystal,  i t i s not  represents the  Kaniso the  Thus  /9(c/a) T  right  from  used  f o r T»9D,  that  ratio  ^3 ^ c i n i s o  3^  erature  c/a  —-) a r e known, t h e e v a l u a t i o n o f (— r—) would P : d(c/a) T the e x p l i c i t temperature dependence. Matzkanin.and  is felt  data  a  alloy,  not: i n d i c a t e  ff(c/a) It  has  473°K.  Cd-Mg  ( n/ ° "^T ation of could  Mg  1  change  -dT  i n this  constant  of  measurements  remarkable The  the that  .  a  information  with  c/a  anisotropy i s becoming  =0.5456  and  i s zero. more  can  be  I f c/a  nearly  cubic  compared of  with  t i n increases,  and  as  a  result  by the  any  -42-  anisotropy Figure the  13  should shows  expected  (b)  zero.  this  i n  i s  Tne c/a  does  !  fact  true  and  increase  the  with  variation  temperature.  of  K  a  n  i  s  is  o  in  direction,  anisotropy  as  compared  q  is also  at  that  towards  Cadmium.  The  F  go  helium  to  i n the  t i n .  q  It  temperature.  and  nitrogen  ion  of  this  the  goes  This  charge  requires  a  shift  i s positive  F  decreased.  Knight  to  above  zero  implies  Cd  i s more  77°K.  at  some  that  distribution detailed  of  at  As  the  some  of  and  temperature  the  dependent  temperature  temperature  i s spherically  calculation  temperature  becomes  negative  between  symmetrical. electron  decreases,  An  wave  helium explanat-  functions  for  cadmium. The  variation  in K  a  n  i sented  as  K  a  n  i  s  =  o  i  s  i s linear  o  ( V ( t ) / V » ) •• .  The  By  extrapolation  around  At  450°K  ratio  is also  crystal  distribution. Cd-Hg  alloy To  c/a=  decreased.  structure  single  further  14  i t i s  T h i s  shows  a  the  temperature.  The.results  specimen  with  parameters  a  drastically. with  Experiments  wi,th  that  were  1.0  that  the  can  be.repre-  qp  i s  found  to  i s zero  for a  c/a  vs  which  be ratio  i s decreased  c/a  temperature  the  a  charge  lowering of  ratio  i n c/a  cadmium  temperature  with  c/.a.  a  plot  relationship  helium  coincide  and  has  spherical  anisotropy.  crystal.  systems  meters  as  that  towards  i n hexagonal  parameters  172°K,  D  ofpCfor  found  and  means  meters  the  value  1.894  i s heading  Figure  Oj), 9 =  ot  cC=+l3.3t0.2. 1.868.  above  of  Mg.  Cd  by  ratio  at  the  Knight  crystal  was  the  addition  temperatures  the  values  about  and  d i d hot  experimental error.  Cd,  shift  done  that  Barnes They  and  single  revealed  helium  pure  c/a  i n Cd-Hg  have  undertaken  a t o m i c "L  i n the  study  At  of  between  of  of  para-  at Hg  the  changes para-  170°K.  Borsa  on  observe Cd-Mg  Cd-Mg any  and  alloy  change  Cd-Hg  powder in  have  these the  -44-  sarae  crystal  affect the  the  number  components  should 1.0  affect  atomic  %  12.0  atomic  take  the  it  structure.  the  %  the  case,  because  of  Since  alloy  as  has  compared  broad  thermal  the  K  plot  s  o  vs  , Thus  c/a  a  v a l e n c e .so  and  the  Cd.  no  charge  in  the  distribution  seen  that In  the  the  lattice  ratio,  Perhaps  a  the  does  each  types of  alloy  of  the  could  and  previous  single  not  nucleus.  impurities  be  results  crystal.  smaller anisotropy i s effect  between  f o r Cd=1.8857)  two  not  parameters  surrounding  light  a n i s o t r o p y i n Cd-Hg  alloying  re-distribution  t o -1. 8 7 1 6 ^ ^ ( c / a  i t i s  c/a  that  charge  change  directions.  lower  to  same  atom  ratio  larger  lines  c/a  i  c/a  a  expect  per  the  to  a n i s o t r o p y vs  n  the  1.9023.  the  a  of  in opposite  the  no  to  the  However  ratio  Cd-Mg  in.this  symmetry  changes  Hg  i s natural  Since  electrons  i s expected.  Mg  c/a  of  A l l have  expected,  observed  encountered.  expansion ratio  at  for  Cd.  data helium  f o r Cd-Hg  single  temperature  crystal  could  not  be  i s  available,  compared  with  -45-  CHAPTER THEORY The field  resonance  H is  W=  Q  ^H  Q  jected  to  a  occurs  at  <«>„ a s  in  the  limit  rf  level  nity  of  and  an  depends  of  w  the  r f  field  i s swept. the  case,  i n an  gyromagnetic  r f  goes  SHAPES  external ratio.  homogeneous  I f  the  Hi  perpendicular to  The  nucleus  level  absorbs  to  drop  in a  a  very  sharp  through  a  local spread  and  nuclei  has  a  interaction. the  In  magnetic  magnetic i n the finite  the  line-width  the  fields,  at  mechanisms  with  field  Larmbr width. i t is  calculated  other  embedded  crystal  neighbours,  However  are  a  are  relative,orientation  several  the  exceed  the  local  upon  has  causes  causing  ensemble.  produce  tions  nucleus  i s the  frequency  in this  a  LINE  PKW  magnetic  nucleus Ho,  sub-  resonance  energy  box.  only  The  minimum  is  at  plot  i n the  of  v i c i -  uOo.  on  spin  Y  L I N E - W I D T H AND  of  frequency  H|—>0,  In.practice made  where  Q  the  <*>,  THE  frequency  variable  vs  OF  V  has  each and  that  width.  positions  of  relative  and  the  the This  that  the  experiments  of  a  magnitude  nuclei.  different  absqrpjtion by  line the  and  line-widths  leads  conclusion  the  This  discussion  chapter  each  posiThis  function  dipole-dipole  experimental to  which  sites. is a  ..a  fashion  Since  orientations  i s caused  are  in,an orderly  the  values.at  for broadening.  i n t i n so  arranged site,  broadening  found  are  and  other's  different  frequency  responsible  studies  nuclei  different  The  dipolar  i n ;bulk matter  is  is limited  generally that  there  limited.to to  the  spin  -L c a s e .  Since  r  due  to  quence life  t h e r e - i s no  this, interaction  i s absent.  of  The  time  a of  finite a  T  1 #  state.  AE.Ti»ft or  quadrupole  A&>»1/T,  From  moment  present  Another  spin-lattice uncertainty  type  in a ;  of  relaxation principle  spin Isystem,  broadening time  T-^  is a  broadening conse-  characterises  the  -46-  and  shorter  case  of  T-^  t i n T^=35  i temperature The  5.1  i s , the  is  relevant  The  Hamiltonian H  (  Hjj  is; the  H„ d  and  to  the  i t s contribution  contributions  H=H -Hl j w h e r e  and  1°K  contribute  to  line-width. the  In  line-width  the  at  this  w i l l  now-~,be  discussed  i n  detail.  broadening.  total  Z  at  i t w i l l  negligible.  Dipolar  is  ms.  more  i s  2  the  perturbing  -  of  £  £_  j=l  k=l  a  system  main  having  Hamiltonian  Hamiltonian ( i C i k .  N  spins  r  r  i  ) ( I  k  a  representing  responsible  3 ( i  in  k  . f  for i  k  the  line  )  magnetic  f i e l d  Zeeman  H  Q  energy  broadening.  )  1  where  F j  sider  only  can  now  is  k  two  be  H =  radius  spins  written  i L ^ b  d  where  the  A=  1  "  I]^ l2 z  is  the.polar  l l e l  to  H  (1  z  standard  text  The  the  total  tude  of  It  is  as  the  r f  thus  vs  + l[  angle  on  the  expression  for  I^)  Ij  k  and  subscript  familiar  T .  For  k  from  r.  simplicity,  The  dipolar  l e t us  con-  Hamiltonian  form:  0)  (1  -  3  cos 9) 2  describing  the  and  have  F  each  data  %  is  //  is  x"is  that  the  in  absorption  and  frequency in  cos  absorbed  f i e l d  parameters  j  o r i e n t a t i o n of a  similar  r,  form  the  which  z  axis  can  be  being  para-  found  in  a  NMR^.  energy  seen  joining the  more  3  experimental  absorption  omit  a  -  C,-D,.E  Q t  vector  and  in  i  k  (A+B+C+D+E+F)  B=TJ(IIT2 9  r ^  varied  -X'(*j)-*1T^ ( k T Z )  1  ^  ^  is  expressed  frequency.  per  unit  the  imaginary  a  good and  absorption  %"(36)  NMR  Under  time  is  index,  for  such  2  ^(E -E .-h<0) n  n  the  > where r f H^  }  rate of  Hi  of  no is  of  shape  the of and  power the  energy  saturation the  susceptibility  information  line-width,  s  |(nlH")|  fixed  a l l the as  the  of  assumption  2H^X^  of a  terms  the  P=  part  contains  line  in  ampli-  ' tt  X=(7-iX). absorbed  descriptive  intensity.  The  Z=  ^2.exp(-E /kT)  and n,n' a r e t h e quantum  n  numbers  referring  t o the  spin.  2  n  Let  f ( W ) - H,|(n|^ |n)| n,n' *•  g(E -E .^o)  x  n  n  7."=2|^-f(«).  then  kTZ Since to  7]' a n d f(">) a r e r e l a t e d ,  determine  moments.  X"  and the l i n e  T h e n t h moment  a  theoretical  shape.  Use  i s defined  computation  i s made  o f f (<0)  o f the-method  enables  of  one  second  as  •4> If  n=2,  to  retain  Van  d  quantity  t h e terms  Vleck^  H  It  t h e above  3  =  8  ^.  A  i s called  and B  the second  i n the dipolar  moment.  Hamiltonian  I t i s only as d i s c u s s e d  by  Thus  ^ l ^ Z l ( A j k + B j k ) r j l  c a n be  necessary  shown  5.1.1  that  •Tr-cm;,/^*  <*">= -  ^  = f Y^ !(I+DN" ' ^ b ^ 2  F  b ^ = ( ( l - 3  sum  over  restricted sum  over  sum  <^>II = pfa  H  T  T  I z  I ' and H  II 1 1' H^ j.Hjj  z  spin  I.  i s t o be  However,  and f o r second  occupied eqn.  summation  sites  (5.1.3)  which  moment  equal  under  two  the, total T ' T  convenient  +H^  A  species  times  the  to  of spins,  Hamiltonian  denoted  i s given  by primed  and un-  by  ) energies  interactions  between  ^represents  use a  T T '  1  are the dipolar  study,  to F  fraction  5.1.4:  a r e t h e Zeeman  interaction  the  calculations  o f t h e two  spin  systems  1  dipolar  over  i t i s more  i s just  reduces  taken  p j k  (HZ+HP+ (H^ +Hj H  by  sites  contains  T T  The  occupied  Thus F  below,  1  ) r ^ )•  only  KI+1)  letters  =  where  over  t h e sample  primed  k  crystal  a l l sites. 2  If  sites  a l l N  k  ^ i i  cos 9 j  of the lattice  to  5.1.3  1  2 where  5 . 1 . 2  ,u ~  T r ( /  unlike  o f the like  nucei.  the magnetisation  nuclei  I  1  ,  I I ' a n d H^ : is the  I f the resonance of spin  I and I  system  of species only.  The  I i s  -48-  truncation because should  o f the perturbation  i t sf l i p - f l o p be  dropped.  contribution  2 I ,  (  i s the fraction  around  5.2  a  spin  I.  Indirect It  widths many  The  exceed  the  same  sites  by  using  i t hspin  in  L  species  gives  an  and  additional  5.1.5  k  sites  second  occupied  moment  mentioned  that  dipolar  by  i s equal  the primed  to  ^ '  a r e found  w  spin  ' ^ ) n  ^  +  system  A  U  ^|'i"''"  interacts  t o be  electrons.  a nuclear  spin  nuclear  o f spins  a  substantial  field  f o r this  v i a these  with  by  observed  and temperature  by  and gets spins  an  behaviour. An  electron  occupying  time  passing  effects.  through  Moving  different  which  indep-  The.nuclear  polarized.  interaction  line-  amount f o r  and s p i n - l a t t i c e r e l a x a t i o n  are responsible  with  pairs  the experimentally  line-width  anisotropies  of this  functions.  second a  pair part  indirect  order  L  i  spins  under  i  the radius  further  lattice  depends  on  their  omitted  under  of this  the  coupling  3  *1  conduc-  c a n be  esti-  and R j .  Hamiltonian  The  between  i s  ri(S'f"i) i , i - ) r  5.2.1  I  i t hspin  the assumption  sites  interaction  consideration  from  involves  theory.  i  vector  size  at the lattice  c +2(jy fil .(-|  •  has been  and  perturbation  and the e l e c t r o n  r-P^, i s  form  interaction  of the nuclear-electron  87T - r , =T -*83YiIi-S«r )  =  spin-spin  The  of nuclear  1  where  t h e Zeeman H a m i l t o n i a n  Interaction.  together  wave  dependent  I-S  o f the primed  attention  orientations.  Consider  the  total  electrons  coupling  with  special  2  line-widths  interacts  electron  spin  jr>  f  out shift  Computation  mated  n o t commute  deserves  moment  > '  been  The  electron  thus  mutual  tion  The  are coupled  lattice  presence  the calculated  ruling  the  does  of the lattice  has already  conduction  spins  I , + 1  Exchange  elements.  endent  H  The  B  t o the second  <^V£T?I* F'  term  II' H^  Hamiltonian  that  to the electron. L  i s quenched.  A  The  term  similar  -49-  Hamiltonian electron.  t  involving  S7T  w ^ . o O c r  i  H  +  1  in  yk'  s  of  k  s  K  S  only.  Then  spin  and  H^g  through  the  same  s-type  !  +  f ^ * ^ - s S ( r \ )  Y  the  theory  s'  system  due  to  the  presence  of  H^g  is  given  as  by  second  t  1  E(k)  -E(k') ^ ^  1  ^  s*  is  is  the  cross  Writing E  j  coupling  jth  I s  B '| H ^ s )  over  as  of  the  Hj  ith  1  !  where-C- ^  and  !  1  Yl  ^j^Ven : 4  states  function  spin  explicitly  = C Z _ 5 _ I . . _ I k s  H j \ % . • )+<  excited  Bloch  excited, state and  ( ^ s |  a l l tjie  product  terms  intermediate  A  and  the  ( ^ ' B . ' \ H l < f e 8 ) C J f g s l - H i s l g f ')  summation  in,the  between  electron-nuclear  ^ S L C E W - E C k ' ) ) "  s  coupling  _,_ 87T„  .  +(  ^  A  the  i  1  i  energy  T  • k'  s'.  an  spin  j  H  perturbation  :  The  for  consider  g  A2)_ k  us  and  change  order  written  =|ii.. p*y i .sX(r )  s  = The  be  Let  electrons  ^  can  causes  spin we  k'  and a  s-| H  and  spin  s  X  the, two  ^  i t back  of  to  H  spin  operator.  transition  .j b r i n g s  ^  )  ]  ..5.2.3  orientations  The  the  the  ^  interest  electron  ground  to  lies an  state.  have  _ L±± E(k)-E(k')  J  •  ^  . I.+ J  complex  conji  5.2.3  (2) The of  an  luate ally i j H  e  extra the  total  >^ cs ks'  i s what  H^-j  in  energy  statfes }'•%',s- has  zero  be  nuclear of  to  be  found  as  the  Hamiltonian,  spin  i  and  first given  j , the  performed.  '/ S f t r j ) / l ^ s ) ( — — - — —  We  defined  otherwise.  as  the  Then  (5.1.1).  To  a l l  evai n i t i -  have  . Lex • I .+comple  g  restriction  by  therefore  S g ( r )| l f r , , . •) v — — :  conjugate  j  p r o b a b i l i t y , where the  order;contribution  summation'•. o v e r  E (k, s ) - E (k ' , s ' )  unoccupied be  the  will  because  ii» occupied fcToccunied  ,p(k,s)  pied,  &E  (.^,s  p  X  Let  term  occupied  -c  p  energy  from  p(k~,s)=l the  i f state  summation  can  k, s 1  is  easily  occube  ij removed.  H  variation  with  e x  must the  be  averaged  temperature  over of  the  an  ensemble  electrons'.  to  take  This  into  can  be  account  i t s  done  replacing  by  -50-  p(k,s)  by f ( k , s ) ,  ij  -  ex"  H  ^  C  k  "  Thus _  E(K,s)-E(k' ,s')  * j  ,  1  s'  1  '  + complex  jC  =C  function.  ( V p s ' [ s S ( r . ) | ^ ) ( U ^ ^ i  I  the Fermi  l  conjugate  (s'| S|s) ( s l s f s ^ - I j  r  x  5  >  2  t  4  k' s  1  •  '  E  The  Fermi  zero the  levels  i  g  above  delta-functions  (  R  >  s  of spin  a l l the energy E^i  X V ^ ^ f C ^ s X l - f ^ . s  'J C r ^ j ^ X ^ l S C r )  _E  (  .)  s  ~  up and s p i n  states  E^  the Fermi  vary  k  -.,  level  slowly  +  down, d i s t r i b u t i o n s level  are unoccupied.  with  ) )  ;  up t o t h e F e r m i  g  1  energy  which  complex  coincide.  makes  At absolute  are occupied  The m a t r i x  conju  anda l l  elements  t h e dependence  ofthe  of  on 6X  the  electron  spin  of  the electron  in  a  form  energy  spin  (  It  i s seen j  ing  '1  field  {  has a  1  i j  i s known  simplify  1  as a r e s u l t scalar  as t h e exchange  the computation  spherical  Jij  =  The  ex  energy  9 r > 2 * £ exchange  where primed  surfaces,  * H V  0  o f summation  constant  over  c  o  m  P  l  e  x  conjugate)  s and s' f o r S=l/2.  I  interaction  alone  ^  *  J  k  and into  Hamiltonian  and i s independent  2kpR  assumptions  T  electron  cos2kpR  ( —  Hamiltonian with  the exchange  total  written  » +  an e f f e c t i v e  )  \  The  be  absent:  of J various  ^f i^h variety  may  form  .4  J  "  and eqn. (5.2.4)  the energy  J  2  H  were  approximation  ^ k > ( V k l ^ ' j > H t f r > * ( f t (1 • f ( E '  i - appears  that  = A J  H* ex  to  t o a good  ECE)-E(E')  the factor  J  Thus  c o o r d i n a t e c a n be o m i t t e d  C-  where  small.  as i f the magnetic  ij ex  H  very  '  W  mixed  mass  +  *  J  In order  t o b e made. m*  one gets  Assumf o r J „  sin2kpR  ~u  two m a g n e t i c  has been  -  have  of spin.  ) ingredients i s  j k ' I j * V  separated  5.2.5  into  pairs  o f unprimed  and  pair.  f o r t h e system  i s n o w H=  R -ffi^ z  +H  e x  where  H  e  x  i s  given  by  (a)  Exchange The  ,  (5.2.5)  second  T %  r  (  H  s  ^  interaction  d .  '  ex»/Sc)  I f no primed  moment  (ii)  However does  x  the presence  Exchange  interaction  Neglecting  the cross  Tr(H *  In E  e  2  k  . b  terms, M*))  Q  T r ( ^  4  does  ex  n o t commute  increases presence second  moment  resulting tre  moment  H  e x  ..ahd  then  the cross  \s  there  term i n  a net contribution  moment i s  j[ ,  + ^ I ( I ' +1) F ' 1  k  3.2.6 moment:  the fourth  moment  i s given  by  2  has only  i s more h e a v i l y  moment,  changes  unaffected  i s narrower,  of the line  *  n o t change t h e  ,  i ft h e primed  a  small  a r e s u l t (AW ) i s i n c r e a s e d  which  the line  while  has a  weighted  1  even  t o an extent  coupling  remains  line  k  does  variety  ( H i , M- ) a n d c a n n o t b e d r o p p e d . u x  moment  o f exchange  l  interaction  a n c  )  with ,  the fourth  second  and the. fourth  ,(H ,  x  i  the computation o f the fourth  av  as  + H  Q  ^ J  = YL - n J ^ j l i ' I j  i s present  2  Yj-.F'  e x  (5.1.4).  - f t 1 ' ( I ' + 1 ) F ' ) £ bI j  2  ,  by eqn.  As a consequence  and the t o t a l  H^ll'Cl'+DftVj.  then H  o f t h e exchange  ingredient  n o t commute w i t h moment  present  given  i fthe primed  % (J-Vjft 1(1+1) F + 4 Y i  (b)  were  and i t i s s t i l l  the second  & J  I i s  2  ingredient  Thus  second  to  H  system  * Tr(Mx)  commute.  g  +  of spin  moment:  2  (i)  H  moment  and the second  shape  the fourth  sharper effect  depends  peak  As a r e s u l t i t  upon  J.  i n such  moment  k e p p i n g <AW> c o n s t a n t .  This  that the  i s increased.  and enhanced  as compared  Thus t h e  a way  wings.  on a n y o f t h e moments.  by t h e wings  i s absent,  The cen-  The  fourth  t o the second i s known  The  as  moment  exchange  (41) narrowing. when  According  J»half  |w-w^~j w h i c h  t o Anderson-Weiss  line-width, keeps  the resonance  t h e moments  finite.  model curve  , which  has been  i s Lorentzian  with  verified,  a cut o f f at  If  more  increased,  unlike  ctrons  neglected. i  The  as the f o u r t h  between  like  nuclei  the coupling  section,  written  between  two n u c l e i  the change  i n evergy  Hamiltonian  between  below  =  i l  H  +  narrows  roughly  the line  while  i  i  i  i  r . ( S - r - ) ^  v i a the conduction e l e was spins  convenience: 3  )  r i  i 2  H  Thus  an e l e c t r o n and nuclear  f o r the reader's  = ! ^ g p y I « s £ ( r ) + 2 p r I . (S-3 i  moment'.  moment i s  due t o n o n s e l e c t r o n s  2  %-s  the second  broadens i t .  interaction  and j are again  then  broadening.  considering  i n the last  i s present,  as w e l l  interaction  nucei  Pseudo-dipolar While  of nuclei  discussed,  the exchange  between  5.3  one s p e c i e s  as already  speaking, that  than  5.3.1a  Similarly  H  I j S  The  H  IS  =  j l  H  +  j 2  H  coupling  =  (  =  H  where  H  1  +  energy  j l  H  + H  between  > +•<  by  H  ,  2  i 2  spin  +  j 2  H  i and j i s given  t h e sum  >  of contact  change  i n the energy  second  order  <  o f the system  perturbation  s'1 l H  +  H  terms  theory  2 ^ k s ) (  jH  1  +  H  and H  as: 2  c a n be  ^ k • 8 ')  l  a  ~f^  *2  separated  3.  three  g  E(k")-E(K')  s  (  ^  '  ^  H  ^  s  K  ^  N  ^ E(K)-E(Ic')  =£j  " f j 7  where  parts 1  ( y k . s ' | H a  into  (^•B'|Hi|4icB)(yE'|H ^ .s')  x ~  1  (2) Eks  2  ^  s  ) ( ^  k  s | H  EOc)-E(E')  = a^ +a  2  + a-$.  2  l ^ , s ' )  2  t h e sum  of dipolar  due t o t h e p r e s e n c e  12) EKS  by  .  H-j. r e p r e s e n t s  The given  i l  5.3.2b  terms  o f H^g i s  -53-  The result  term  (5.3.2a)  indicated  Hamiltonian. interest can  be  where  taken Hp  has  =  The last  smaller B*£j  T .  The  total  unprimed  <A^>-^-I 4  Since  t j  =  and  C  B  i k  adding  larger  than  <  l  k  1  +  B  i j  , - * J  l  > ^  k  l  has  by  to the  that  a  2  Hamiltonian,  5  -  treated  contribution.  Hamiltonian  w i l l  3  '  3  i n the The  be  con-  s t i l l  form  wave as  and  functions  i s a  compli-  and v a r i e s  the dipolar  h  Hamiltonian.  f o r the resonance  f i "  ,  r c k' 2  ,  2  k  Yjft ZI 2  of  5.3.4  r. .  a  v  t h e same  e  l  k  ,  R  . J  3  dimensions,  we  can write  B*  as  follows  j  l  as  3  -3  i  electron  constant  a l l the interactions,  2 3 (B^.+gg'/S r ^ , ) b  r  interaction  t h e same  , ) Y pure  Y  x  * i  b  2  l  k  . .  'tin the second  <1+B  can also  r  ) bj  moment i s , 5.3.5  2  i  j  j  be w r i t t e n  • 2 (A' cos4d?+D') (1-2 c o s 2 8 ) + E ' 1  <Ati>=  i t i s found  of  (6)  r.. ) b...  isotopicailly  expression  i s the term  to the line-width,  i n the dipolar  +i;I'(I +l) 3  J  + ( l - f f l  <^>=|l(I+l)  to the x  * i j  the pseudo-dipolar  including  £ B * X j  2  p  b  Hpgg^Q.^p  term  f o r s-type  (5.3.3)  i s given  B ^ j and g ^ ^  term  the pseudo-dipolar  moment,  ,+  H  The  neglected.  ( B . . +g  i k  a  of the contact  i s quadratic  (1+1) *  ,=fi.J  For  This  by  .  C  and  section,  - ^ i ' V t f j - i j ) - i l )  where B*.=  term  o f the last  r ^ .  species  a  the technique  I t i s zero  second  adding  section.  form  represents  f o r large  3  r  by  o f a2, which  a^, which  function.  calculated  i n the last  computation  i s much  and c a n be  be  considered  e  account  the  been  out a  contribution  tribution,  could  carries  ( l i ' I j  section,  cated  a. i  using  into  ^  £ d .  that  I f one  here,  has already  i n the form 2 cos ®  (8)  (Appendix A ) .  2 s i n 8+F'(3  2 cos ®  -  2 1)  5.3.6  -54-  In  the theoretical  assumed single  that  the resonance  crystal  signal,  yl' a . r e r e a l  observed equal  resonance  f(tO),  fractional  i s a  and  resonance  calculation has  of  ui  0  t o be  error  A .  error  A  us  i n Wo.  The  J(  In a  constant  i s lower shape  than  c  to )  forward.  In the case  reason  This  case  the  6  1  magnetic  field  o f the line-width  signal  i s known  line  shapes, has  exactly  moment  u  then  some  t o be  will  be  where  affected  which  <A« > -<^ V  (  2  (i&^-.A,)  to  uncertainty  Experimental  mentally it.  an  ) f ( 6 0 ) d w +  &  .  result  second  moment.  i n errors  of  and  = 2bM (^ - A )+( A - A ) 1  1  2  2  I f <*»^>is  2  computed  f o r two  known  ^ t h e n  ? 2bM (^ - A ) . 1  1  2  2  as  In  by  5.3.7  i s the required  corrections  to the  =1  1  M2  approxi-  be i n  =M +2AbM + 2  the  added  a n d may  0  f o r  i s  w h e r e Jf(«o)d<o  w - w  observed  external  r  f ( w ) d w  metal  i s the  l/3rd  7. ( w )  of  r f s u s c e p t i b i l i t y and b  by  second  the  been  %' and  not give  i n this  +2A/(  i s that  of the line-width  may  f a r , i t has  where  For unknown  fraction  moment  f ( W ) d ^  e  known.  of the observed  2  w-  so  i.e. f ( w ) = X " ( ^ ) + b  the experimental  second  The  o f the complex  *fc .  i.e. a  how  r  =  of  frequency.  J(W-WO+A)  <Ab?-> =  parts  moment,  exactly  0  i s not true.  i s straight  examine  u is  o f modes  frequency  made  resonance  Let  mixture  I f the line  observed by  this  contribution  of the second  feequency  imaginary  mixture.  mation  calculation  ;  i s negligible. i n Uo c a n b e  computation  the theory  shown  that  left  hand  side  i s known  and  the %  error  due  estimated.  o f the second  the second  i t ' s derivative  I t c a n be  The  moment  i s recorded  moment:  i s calculated and the second  f o r f^-^n) moment  while  calculated  experifrom  -55-  I J(}>-Vo) f' 3  ^ f ^ . v o ) The in  above  expression  arbitrary  V  •/•  3  units.  U  (M- v ) d ( f - ^ o ) 0  f ' (v-y,,)d(y-Vo)  o  )  contains For  the  discrete  f' (U-VB)  derivative frequency  f'(^-Vo) values  which  can  be  expressed  -56CHAPTER RESULTS Isotopically  AND  VI  DISCUSSION  Pure T i n .  Figures  15  a n d 16  show  the line-widths  a t t h e maximum  slope  ($1))  n  m. s l . at  helium  applied  temperature  magnetic  frequency signal  was  luation fixed,  J  field.  )X&^-0  mi  s  i ,  observed  .  increased  away  from  Thus  i t i s concluded  erimental  lower  with  spin-lattice  frequency. was  that  amplitude. Keeping  increased  the signal  could  after  seen,  which  than  on  the  the  the eva-  orientation line-width.  i n the wings,  J^IO kc/s.  o f f  J, first  enabled  i t disappeared  the cut o f f frequency  to the  and the cut  limit This  respect  the crystal  b y more  be  with  i n the centre  to s e t an experimental  resonance  o f resonance  orientation  i s Lorentzian  modulation  10 k c / s  into This  the noise. sets  an  exp-  on J .  measurements  amplitude  of crystal  small  amplitude  limit  Line-width  combined  a  modulation  the centre  line  In order  with  the modulation  function  The  of the correct  With  lation  as a  smaller  the helium relaxation  were  than  a  also  sixth  temperature time  a t room  made  a t room  temperature  of the line-width. line-widths  leads  temperature,  as a  with  The data  a  modu-  obtained  to the evaluation function  of  of the  crystal  orientation,  Natural T i n . 119 Figure Measurements  17  shows  were  t h e 1ine-widths,.  also  made  i n Sn  isotope,  10° o f f t h e (001) a x i s .  i n the basal  The s i g n a l s  plane.  recorded 2  have  been  used  i n the computation  o f the second  moments.  The  factor  ,>  J appearing The  i n the second  results  shown  a r e shown  i n table  1.  moment  e x p r e s s i o n has been  i n figures  18 a n d 1 9 .  The  calculated  second  moment  f o r 21  bj i J  shells.  results  are  -57-  (001)  Plane  J  i  I  I  -45  i  i  i  0.0  15.  The  1—'.  Rotation  1  45 Angle  Figure  of  Line  Resonance  between  Width as  a  of  1  1  . 90 H  0  the  and  Function  of  135  (001)  Isotopically Crystal  Pure  1  Sn  Orientation.  -58-  — T h e o r e t i c a l  I  i  0  5  10  i 15 Angle  Figure  16.  The  line  Width  • 20 between  of  the  Experimental Line  +  Calculated  q  and  Width  ^  from ^ ( l + B j j )  i 30  25 H  Check  6  1 35  -J 40  2  b-y  ——-I 45  (100)  Isotopically  Pure  Sn  119  i n the  Basal  Plane.  -59-  Angle  Figure  17.  The  Line  Width  between  of  S n  i  i  y  H  0  in  and  (100)  Natural  Tin  in  the  Basal  Plane.  -62-  Table The Direction  of  1.  Second H  Moments  Second  Q  10° o f ( 0 0 1 )  It  should  Ttyis  be  i s because  modulation the in  emphasized  amplitude  slope  3.9+  0.4  (100)  3.7+  0.4  (110)  1.2+0.1  that  the wings  these  a r e lower  c a n n o t be  i s too large,  of the absorption  an exaggerated  Moment  limits  observed  with  the signal  curve.  Instead,  (Kc/s)  2  o f the second a  moments.  low modulation.  recorded  I f the  i s not proportional to  the line  i s broadened  resulting  secondmoment.  Discussion:  It the  has been  cut o f f frequency  exchange  keeps  the line  J»( S*->)  m  g  ^  .  i s Lorentzian  I t appears  that  model  to this  model  the line  i s Lorentzian with  a  was  finite,  and the second  t o examine  made  whether  by using  moment  the line  parameters  are ( ^  shown  (001),  A',  D',  (100),  E' and E'.  of the line-widths i n figures  )m. s i . °  ( 1 1 0 ) a n d 0=75  15  The v a l u e s  as a  a n d 16 b y  The e x c e l l e n t  (5.3.6).  applied.  exchange  line-  narrowed  The e x p e r i m e n t a l  a  fur-  line-  o ,3?  =25  thus  function of ® the solid  agreement  extreme  i s proportional to the  o along  c a n be  that  c u t o f f | ( ^ - i->t, )| ~ J w h i c h  i s really  the expression  and  are observing  the Anderson-Weiss  In order  culation  we  to which  check  widths  i n the centre  lines  t h e moments  width.  that  narrowed  According  ther  observed  between  were  used  to evaluate  obtained  were  used  a n d <#>_.. T h e  line  together  i n the  theoretical  with  the  the calresults  experimental  t h e two e s t a b l i s h e s t h a t  we a r e  -63-  indeed Weiss  observing model  extreme  ( Abragam  exchange  pp.  narrowing.  According  to the Anderson-  intensity.  For a Lorentzian  107)  <S^> =  where  £  (2%)  i s the half  =J3  C ^ )  m  J(S«)  s  f f l  l  <S^> where  C  .'. K  K  l  B  the second  ( 1 +  =| I  The  b  a l l the  next  B  (1+1) -ft  5 1 d+Bij) J  contains  line  , i.  or alternately = C  at half  and  -  Theoretically (5.3.6)  width  step  by  moment  pure  t i ni s given  by  , f b - j  i : j  i n isotopically  J  ( f f v i )  m  #  8  l  I  >  ( i )  Y$  2  i j =  ( ^ ) m . s l .  ( i i )  constants. was  to evaluate  the relative  pseudo-dipolar  contributions,  2 .(l+B^j  ) , from  variables  various  which  independent  suggests  eqns.  can  be  described  B^j  f o r the f i r s t  values  from  mation  eqn.  K ( 1 + B The  U  )  2  shells.  E q n . 5.3.6  that  we  out o f system by o n l y  four  three  shells  the remaining  2  ! +K(l+B  experimental  i  can n o t form ( i i ) .  )  I b 2  2  line-widths  a  four  s e t o f more  i t was  are equal  only  the second  are different  shells  2  Since  variables,  ( i i ) c a n be w r i t t e n  Z b I  contains  each  than  moment,  assumed  from  independent  that  eqn.  linearly ( i i ) ,  the values  other  to a constant.  four  and that  Under  this  of  the approxi-  as  2 i  2  « ( l + B  along  i  3  )  (001),  ^ ^ ( l + B ^ 3  2  (100),  & rest  2  =( 8 » )  2  (110) and ©=54.7356°,  m  s  l  ( i i i )  <3?=25°  2 with  appropriate  independent from  eqns.  the various  values  of  ^  b  ^  were  S o l u t i o n o f these shells.  The r e s u l t s  (1+Bil) 0.33  relative 2  '  gave  obtained  a  set of four  the relative  a r e summarized  linearly contributions  i n table  2.  c o n t r i b u t i o n s from  (1+Bi2) 0.83  t o form  equations  Table The  used  2  (1+Bi3) 1.19  2  various (1+Bi 0.89  )  2  shells.  2.  -64-  These ®'s  and  ment  c£,'s.  with  the  This  is  An that  values  third  A  in  table  that ent  the  values.  first was of  which  time  also Bij  on  of  j .  shells  A  to  us  J^j  the  assume  by  in  the  estimate  figures  evaluate  are  for  computation 15  the  two  shells  a l l equal.  disagreement  i . e . J^IO  are  in excellent  The  results  ~  made  of  the  much  and  thus  those  assuming from  obtained  experimental  the  gave  results.  -3  Bij=Bij  of  agree-  c o n t r i b u t i o n s by  different  .  for a l l  evaluated.  are  kc/s.  the  16  the  may  j^jfor  line-widths  with  dependence  be  the  been  c o n t r i b u t i o n s and  J  of  relative  radial  calculating  and  c o n t r i b u t i o n s have  relative  value  the  first  onwards  the  rough  to  i n marked  experimental Let  these  made  for  were  limit  3.  in  experimental  inclusive,  lower  used  plotted  values  line-widths  later  results  the  shell,  using  The  attempt  the  were  r^.  The  results  B^j  is  of  can are  the  case  of  set  by  tabulated  same  contributions of simpler  be  the  the  form  as  differ-  spherical  <n k£ 2  Fermi  surface  priate m*=.5m  to  t i n we  and  e  where  E =  F  kp,  as  10  shall 0.370  p  8 k =1.336  kp  is given  by  use  expression  this  Rydberg  Ep=  (Phys.  Rev.  cm  .  The  ( 1 + B  n  )  (1+B  2  1.80  nance to  find  kp  of  a  from  504(1966)).  theory the  more  following  This  approvalues:  gives  in  i  2  )  figure  this  Value  of  20.  3.  2  (1+B  4.56  i  3  )  ( l + B ^  2  6.59  2  4.89  Tin. the  computation  frequency  i t .  to 149,  place  r a d i a l -.dependencer.of~\JwjV•''©btaifled-ffdm  Table  In  In  -1  w e l l ' a s ^ t h a t ' > o f "guy-; i s , p l o t t e d  Natural  .  The  has  fraction  to of  of be  the  experimental  c o r r e c t e d by  the  second  adding  line-width to  be  a  moment,  fraction  added  is  of  the  observed  the  determined  reso-  line-width by,  the  -65-  fraction skin  of mixture  depth  ratio  o f %'.  i s anomalous, o r normal  of the derivative  and  Lorentzian  lines  .66  and  We  mental  .535. second  whereas  a  a  Gaussian  peaks  and  i s t h e same  f o r the normal  deduce  line  has a  was  of the derivative.  as a  i s neither The  ivative  a  by  anomalous  but a  completely  true  resonance  correction  case.  A  of  f/  good  (5^/3)  correction  f o r the normal  (£ /4) w a s u  of  since  Curve  and  the  even  ratio  between  the line  shape  the  at helium  and  considered  using  i s .53  to the observed  case  experi-  line-width,  ( i i ) that  anomalous  i ti s  fitting  ( i ) that  orientation  f o r Gaussian  depth  not obtained  implies  i s related  .39  skin  peak-to-peak  nor completely  frequency  and  the  peak-to-peak  (<5v) i s t h e  4.  f i t was  This  of crystal  normal  i s .55  2(£v) , where  whether  The  i s not Gaussian  The e x p e r i m e n t a l  function  depending  the anomalous  moment  f o r a l l orientations.  not changing  eratures.  the line  second  tried  depth  where&for  that  to 1  temperature.  skin  are approximately  function  the  depth  may  from ^  at helium  respectively  moments  Lorentzian  is  'b' c a n v a r y  skin temp-  zero  der-  f o rthe  reasonable  i n this  analysis. In  order  ermination as  of  of  factors.  difference (<5y/o)„  error  arising  absolute  between  The  from  a  of the error  moments  Since  were  are-shown  frequency  possible  to the probable  error  gives  of  10° o f f  of  H  errors  2bMj&  incorrect  (WS")  and  i s linear  due  together  i n the experiment.  to a  with  which  would  The e r r o r s  i n  <Sv >, =2bK  seem  t o be t h e  quoted  i n  4.  <S&>i  2bM  r  (.«*)  error  percentage  Error  l  (001)  3. 7 3 3  3. 7 1 1  .022  .066  (100)  3.612  3.553  .059  .177  27c 5%  1.100  0.968  .032  .096  9%  " (110) .  A,  frequency  the  det-  (SV&)  i n M>„  x  <S*>2  Q  4  (f»/2>o)  Table Direction  term  i n an  using  the error  i n table  error  involved  calculated  the-correction  t h e two v a l u e s  results  maximum  a r e due  the size  Vo, t h e s e c o n d  correction  the  1  to estimate  table  It  is clear  incorrect weighted ly  to  from  resonance by  the  cause  moment  ^ ( F + f F where ej).  In  mental  one  term  these  of  the  m  s  l  technique  which  in  go  J  moments  and are  parameters  F'  to  the  (1+B). the  could  not  seeing  the the  error second  signal  to  the  caused  by  moment  is heavily  in  the  using  wings  expression  an  is  like-  (5.3.6),  the  as  J  k  to  (5.3.6).  evaluate  expression  limits  only  ( £ ^ )  J,  independent  as  be  the  rise  written  However,  lower  As  gave  i n eqn  three  in  -l)+.01917ZUi -<5y >expt.  2  along  back  be  cos ®  i t is possible  moments  can  t i n can  97„.  of  this.  like  limit  i s around  than  +G'(3  >  constant  second  second  error  upper  inability  for natural  is a  the  frequency  more  principle  values,  that  the  ) J ( ? v )  1  G'  cross  use  4  wings,  much  Making second  table  B -and y  \  *-  H J ^ i  directions.  (5.3.5)  already and  estimated.  s  m  and  using  The  them  results  for  using  Having  deduce  stated,  known  s  the  the  experi-  found  these  value  present  give  a  and  the  of  the  experimental  reasonable  even  ®  values  for  negative  value  2 for  5~JjQr' •  inorder second would  to  It  is  see  the  moments. be  felt  that  signal  One  of  the  the  cross  sufficient  accuracy  and  give  should  Karimov  term in  and  compared  obtained  =2.5±0.lkc/s.  Spin-lattice  =  s/3TT(^)  because  of  m  T-,.  g  ^  .  line  This  However  on  results  time,  the is  in  use  get  It  for  radial  measurements the  be  averaging  device  values  the  moment  direction.  dependence on.the  calculated  calculation would  that  can of  of  This  emphasized  parameters  be  determined  both  line-width dipolar  with  J in  and  B.,  white  moments.  They  T^.  line-width is given  true  second  (001)  should  signal  accurate  the  a l l the  the  with  some  more  from  moments  made  relaxation  Lorentzian  and  derections  second  the  to  i s measured  Schegolev  and  a  wings  information  tin. powder J  the  9  have  contribution.  the  useful  may  best  0 = c o s ~ * " (1/3) , w h e r e  eliminate  For  in  one  i f there  presence  of  T^  i s no  by  the  expression  c o n t r i b u t i o n to  c o n t r i b u t i o n s , the  the  line-width  line-width is  des-  -68-  c r i b e d - by  a  T  -1 T where From  T  i s  2  1  ble  the  at  case  the  of  out.  given  and  temperature  line-width  which  isotopically at  contribution  worked  a  contribution  i s the  T°K  can  data  helium  t o (£v^ at  due  of  pure  higher  to  and  T^, room  helium  T-^  t i n the  and  helium  Results  includes  worked  out.  It  contribution. i s equal  to  and  at  Direction  i s Lorentzian  temperatures.  helium  room  T^  temperature  at are  (£jj) i n  °K  294  2.96  1  following  has  been  below.  0.9  294  the  temperature  the  kc/s.  (001)  results  apprecia-  5.  1.42  (001) .  away  combining  summarized  1  _Lto  well  i s no  Thus  room  (001)  (001)  until  There  temperature.  temperatures,  Temperature  _Lto  these  contribution.  line  Table  Combining  be  spin-lattice  (Sv/-S» )  centre  line-width  l  spin-spin  i s the In  T  • +  =J37T  where  from  -1  T.2  these, T i  below:  -1 =  2  as  2  2.38  values  f o r T^  at  -  room  temperature,  o 294  K,  have  been  obtained. o  _L  TjT  along  (001)  =  36-2  m.  sec.  K  T]T  J_ to  (001)  =  37-2  m.  sec.  °K.  It  i s known  that  T-jT a t  is  constant  over  the  1°K  =  35  temperature  m  and  sec.°K.  range  and  Present closely  results  indicate  isotropic.  that  T-|T  -69-  Suggestions a.  Recently experiments  room  temperature  limitation data  are  the  on  been  powder  available  deriving  only  most  shift  obvious on  further  pressure  done  on  at  room  t i n and  accuracy  regarding  crystals  to  the  t i n and  just  below  as  i s  the  Knight  powder well  the  shift  at  specimens.  as  limiting  model  t r y are  of  of  cadmium  temperature  experiments  single  experiments  dependence  technique.in  general conclusions  The Knight  of  have  for  the  the  fact  The  that  possibility  proposed  by  pressure  dependence  cadmium  at  the  Benedek  of  &  Kushida.  of  the  various  constant  temp-  point.  The  results  the  contri-  o eratures thus  ranging  o b t a i n e d would  bution  of  in  ratio  c/a  The results the as  volume  would  a  function  the  give  due  to  change or  variation  then  the  ments  accurate The  exactly  shift  constant  these  experiments  temperature  entire  ( i ) the  i t i s a  anisotropy  lies  can  variation An  of  temperature.  One  of  with  i n this  contribution  anisotropy  ( i i ) i t is explicitly effect  fact  Knight  out.  that  shift  a l l the  will  i n Cd  the  change  of  volume  should  the  a  °  over p  be  effects able  temperature  two. to  present  shift  temperature  between  the  Knight  If both  the  of  77°K dotted  ambiguity.  to  i s  found  that  experi-  available temp,  &.195°K line  purely  phenomena  these  entire  on  dependent  i t i s  information  by  the  operation.  advantage  over  i s shown  resolve  a  i s due  The  conclusive  variation  range  of  temperature  sorted  hence  expected  forward  of  with  of  expression-1-  range  i n the  the  straight  the  the  combined  effect  with  the  us  be  dependable  of  of  give  variation  ratio,  a  combined  dependence  be  the  ( i i ) the  temperature.  will  also  (i) regarding  and  temperature  others  Measurements  from  Knight at  melting  information  calculation  (iii)  known.  the  the  The  i n c/a  and  to  explicit  contributions  over  dependable  anisotropy  would  of  to  range.  whether  phenomenon the  of  over  K  obtained  results  determine  4,  give  whole-temperature  anisotropy  b.  on  77  effects  results  The  be  from  in  would  range. i s  not  figure  -70APPENDIX SECOND MOMENT I N I S O T O P I C A L L Y The-second  moment  i n isotopically  ^ - ^ ( I + l W ' ^ C g V r ^ + B i j ) P = where  s£  %  a constant  having  2  i  :  j  FUNCTION  OF'ORIENTATION  t i n c a n be w r i t t e n a s , -l)  and  J?jis- the d i s t a n c e between i s the angle  sites  between  i nunits  o f (  )  2  (PzCcosQ^))  the dimensions  p non^equivalent  teraction  pure  (3cos 0  2  ( d + b i P / j i i  the  8^j:  T I N AS A  N  k X  -Tuis  PURE  A  (p=2 i n . S n ) ,  o f frequency,  N=total  i and j t h spin  r ^ j and H .  b^jis  0  Vti/a  ) .  CD  2  number  i runs  1  of spins  i n reduced a measure  the index  units,  over  i nthe crystal  J^^v^^a,  i . e .  o f thepseudo-dipolar i n -  Eqn. ( i ) c a n be r e - w r i t t e n  as  follows:  ^-••^v-I^^jCjC-Ba^cosO^)) 2  where  ^ We  {l+b ) J>~l refer  theapplied  fixed  with  (001)  respectively.  Using  P  \  2  t]  respect  theaddition  Hoand  to thecrystal. The three  theorem  (cos9 .)=|r^Y  r"ijt o a coordinate  X ,Y ,Z C  axes  C  a r e chosen  C  a r emutually  for the spherical  (Q^.,^.)  i  2  field  Y  ?  m  (  system  along  (X ,Y ,Z ) C  C  (100),  C  (010) a n d  perpendicular.  harmonics  ® , $ )  •m=-z Substituting  <^>4^3>  (iii)  2  in  crystal  t h e XZ p l a n e  oC j = i  (3^ji,  obtain  2^ i=I  The  i n ( i i ) we  ®,*)YJ J.-1'  has r e f l e c t i o n means  that  symmetry  f o r atom  Q i j = 9 i j a n d <^j=.-^"j i . ,  n  .(8,©  m=-2m'=-2 i n t h e XZ a n d YZ p l a n e s .  j , there  Reflection  i s another  atom  i n t h e YZ p l a n e  Reflection  j ' which has means  that  for  d v )  • 71-  each  atom  j there, i s another  <P ,ii ~ 1?-'nS •  Since  C  an  atom  j  1  ' which  1  A.\f " C i j " ' , It  i sclear  terms  £  mm  A l l  e  i  (  "  m  terms  n  '  )  <  ?  operations  hold,  f o r each  and atom  j there i s  and^,,p  type  +-e- < i  m +  77 +  j i n eqn.  for a  ( i v ) ,f o r t h e above  fixed  ° ) $j-jl  ^  e  <  1  (  symmetry,  involves  m,m'  m 4 t n  '*>T(  1 e  (  r a + m  '>^jfe" fa ™') 1  J )  4  .  ( i v ) may b e r e - w r i t t e n ,  keeping  only  terms  i nwhich  (m+m ) 1  even.  . ^ ^ f ^  / X I means  The  Writing  r  J  i  p  rafc that only  (v) e x p l i c i t l y  -fih 4JT-'  2  ^ > =  — P  (~) (A(P  +D where  A=  B  i j  =  c  i j =  Dij  p  2 0 (  (  p  2 l (  =  j  =  Fij= Fo!r  (  p  2  1  2  e  e  i j )  p  2 2 ( 6 ^ )  2 0 (  e  2  i j »  2  tetragonal  ) » 2 . ^  *>W' m+m'= e v e n  2  cos4$  terms  + BP  ( P l ( 0 ) )  2  2  A..,,etc.  cos  ( 2 ^ )  2  symmetry  i  C  1  ) « « « - -  ,  > 4 i ' ' p  a r e kept.  9  jr«)p  1  9  C  r  t  ,  '  )  *  (  _,(e)  we g e t  c o s ( 2 ^ . )  2  ) )  j  1  c o s ( 4 ^ j )  i j ) )  ( 9 |  . < » i  2  + E  ^ p f c j  (P22(6ij)) ( P  (»))  2  L  2  2 2  (P 2(®))  ^ 2 2 ( 9 ^ ) 5  A i -  i  has^L j = ° ^j " , S i j - Q i j *  j  (m+m") = o d d a r e z e r o  eqn.  f  E  ' which  1  symmetry  t h e sum o v e r  i j  t  with  Now is  h  j  has  of the following  , =  both  9^=0^,1,  that  atom  B=C=0.  2  d  (»)  + F  P  2  2 2  (8>cos(2§)+C(P ^(®))  (P 0(®)) 2  2  2  )  ,  cos(2$)  v  )  -72-  APPENDIX DETERMINATION  OF  THE  CORRECT  B  RESONANCE  FREQUENCY IN ;  TIN  (29) Bloembergen much  larger  determined bility.  has  than by  T h e  the r . f . skin  both  the real  observed  ssion  shown•that  i n case  depth,  and  conductor,  the observable  imaginary  line-shape  of a  parts  function  with  absorption  of the nuclear  can then  be  dimensions line-shape i s  r f  suscepti-  represented by  the  expre-  /  ; /  • f ( w ) = a X ( « ) + *(*>) where 'a'  and %  no  are the real  i s the-mixing . The  resonance  us  of  until  by  , i|M  of exact  well  does  t o have  away  o f it and  a  of the-r.f.  i s the derivative  due  from  to V  susceptibility  correct  and  i s  t i n , i t has been  the centre  x " i n eqn.  the line  and  resonance shift  para-  essential.  established This  the observed  that  fact  the line  immediately  function  c a n be  the•expression K  L*-,.^) )-  where A=.(l/T ) , 2  M^)  ( f«o )=<<>-<«>o, b = a " ^  and K  (2) contains  a l l the constants  of the  expression. Our gives be  main  us  the line  written  where  oC  interest  1  lies  width,  i n the evaluation  and  w«the  resonance  of correct frequency.  values  o f L,  Expression  which  (2) c a n  as  s a r e the parameters  ( A , w«, b  a n d K)  t o be  was  to the  assymmetric  of the Knight  frequency.  (1) a n d  I f there  correspond  to the correct  knowledge  frequency  of f(w).  would  makes  not correspond  resonance pure  2  =  parts  o f the derivative  contribution  isotopically  the forms  represented y  In order  the knowledge  Lorentzian  The  of the derivative  In.case  signal  from'. ."X.', ' t h e z e r o  f r e q u e n c y .  gives  imaginary  parameter.  frequency.  the zero  meters  and  experimentally observed  contribution  now  (1)  determined.  Values  of  i s  -73-  are val  measured,  -tt  around  i n arbitrary  the observed  units,  zero  of  as the  a  function  ofu)^ i n t h e  derivative.  Since  frequency  there  will  inter-  be  some  2 scatter,  the  minimized.  information To  achieve  1 for  n  a l lof .  Now  m  considered.  P  i s f e d i n the  this  1  the  form  derivatives  the variations  i n  eC  n  from  decimal  i s a  The  computer.  <A^))  with  to<4i  must  vanish  so  guessed  values  are  respect  i s  the  i n i t i a l l y  Then  ^m"*^  meters.  (y^-fCe^,  -Vgfc  i This  y ^ v s . ^ a n d  set of  place  simultaneous  Newton-Raphson  Iteration of  ^ m  was  method  carried  the desired  equations  was  out  parameters  for corrections  extended  until  'SeCp  there  i n kc/s.  to  solve  was  no  ^  cC n  these change  to  the  equations on  the  para-  by  third  that  PKW  Oscillating  Detector  -75-  REFERENCES  1.  Abragam,  A.;  2.  Alexandrov  3.  B a r n e s , R.G.  Principles  of Nuclear  e t a l ; J.E..T.P. and Borsa,  12,  25  Magnetism,  Chem.  of Solids  T. ; J . P h y s „  Chem.  Solids,  5.  Bloembergen,  N.  and Rowland,  T.J.; Acta  6.  Bloembergen,  N.  and Rowland,  T , J . ;Phys.  7.  Bloembergen,  N.;  8.  Bloom,  9.  Busk,  R.S.;  10.  Buss,  L.  11.  Chang,  12.  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