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The fermi surface of copper by positron annihilation Petijevich, Peter 1966

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THE FERMI  SURFACE OF COPPER BY POSITRON ANNIHILATION by PETER  B.A.Sc.,  A THESIS  PETIJEVICH  U n i v e r s i t y of B r i t i s h  Columbia,  1963  SUBMITTED IN PARTIAL FULFILMENT OF  THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of PHYSICS  We a c c e p t t h i s required  t h e s i s as c o n f o r m i n g t o t h e  standard  THE UNIVERSITY  OF B R I T I S H COLUMBIA  September,  1966  In presenting  t h i s thesis i n p a r t i a l f u l f i l m e n t of  the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. mission f o r extensive  I f u r t h e r agree that per-  copying of t h i s t h e s i s f o r s c h o l a r l y  purposes may be granted by the Head o f my Department or by h i s representatives,,  I t i s understood that copying or p u b l i -  c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n p e r m i s s i o n .  Department o f  FH/SICS  The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada. D  a  t  e  -Sep/ I91G  ABSTRACT  A study o f th© Form I s u r f a c e o f copper a t room t c m p e r s w r e has been made by moans o f e p o s i t r o n a n n i h i l a t i o n t e c h n i q u e . A p o s i t r o n a c t i v a copper s i n g l e c r y s t a l was p l a c e d midway between two ••point" s c i n t i l l a t i o n c o u n t s r s o p e r a t e d In tie;© c o i n c i d e n c e .  Th® c o -  i n c i d e n c e count r a t e was measured f o r v a r i o u s c r y s t a l o r i e n t a t i o n s and t h e count r a t e I n t e r p r e t e d as a measure o f t h a d l e m e t e r o f t h o Fermi  Tho e x p e r i m e n t y i e l d s a Forrol s u r f s c o t h a t except f o r protrusions  In t h o  {111^  subtend an a n g l e o f about 20° a t £  w  Is S p h e r i c s !  surface.  In k - i p a c o  d i r e c t i o n s which aro estimated 0.  r e s u l t s a r e c o n s i s t e n t w i t h those obtained  to  W i t h i n exportirtental e r r o r tho by o t h o r methods near 0° K.  ACKNOWLEDGEMENTS  The a u t h o r w i s h e s helpful  discussions  a l s o due t o D r .  to Dr.  G.  Jones f o r  D. L l . W i l l i a m s  for  this  h i s many  his  work.  helpful  and s u g g e s t i o n s .  The a u t h o r w i s h e s Risebrough of  the  d i f f r a c t i o n study of  ment o f  Civil  in t h i s  work.  to thank Dr.  E.  Department o f M e t a l l u r g y  Thanks are  the c r y s t a l  used  T e g h t s o o n i a n and M r . N. for  the c r y s t a l  S.  rotation  H, de J o n g o f  assembly.  Fraser  the  work.  the e n g i n e e r ' s  t o thank Mr. A.  R.  assistance with  in the p r e s e n t  E n g i n e e r i n g f o r making a v a i l a b l e  the design of  their  a l s o extended to Professor  The a u t h o r a l s o w i s h e s with  gratitude  a d v i c e and k i n d s u p e r v i s i o n t h r o u g h o u t t h e d u r a t i o n o f Thanks are  x-ray  to express his  the  transit  for his  Departused  assistance  TABLE OF  CONTENTS  Page Chapter  I  INTRODUCTION  1  Chapter  II  MOTION  7  OF ELECTRONS IN METALS  A.  Introduction  7  B.  The O n e - E l e c t r o n A p p r o x i m a t i o n  8  C.  The F r e e - E l e c t r o n  D.  The C r y s t a l L a t t i c e  12  E.  The R e c i p r o c a l  17  F.  Motion  10  Model  Lattice  o f an E l e c t r o n  1.  Perturbation  2.  Effect of Electron  Theory  in a Crystal L a t t i c e f o r Weak P e r i o d i c P o t e n t i a l s  20  C o r r e l a t i o n s on t h e F e r m i 22  Surface 3.  Justification  k.  Energy  of the One-Electron Approximation  22  Band C a l c u l a t i o n s by u s e o f t h e O n e - E l e c t r o n 23  Model 5.  18  The F e r m i  S u r f a c e o f Copper:  Theory  and E x p e r i m e n t  24  A N N I H I L A T I O N OF POSITRONS  27  A.  Introduction  27  B.  Free A n n i h i l a t i o n o f P o s i t r o n s  27  C.  P o s i t r o n A n n i h i l a t i o n from  29  Chapter  D.  III  1.  Bound E l e c t r o n - P o s i t r o n  2.  Positronium  3.  Positronium  a Bound S t a t e Systems  29 30  Annihilation  A n g u l a r C o r r e l a t i o n o f Two-Photon A n n i h i l a t i o n o f Positrons  30 3i  Page E.  Angular C o r r e l a t i o n Geometries  33  1.  33  Introduction  2(a).  Wide S l i t  Geometry  2(b).  Determination of  33  Fermi  Surfaces  by W i d e  Slit  Geometry  35  3(a).  Point  37  3(b).  Determination of  Geometry Fermi  Surfaces  by u s e o f  Point  Geometry h. F.  Point  A n n i h i l a t i o n of  G.  Positrons  in E l e c t r o n  Gases  ^0  of  Electron-Electron  Interactions  kO  2.  Effect  of  Electron-Positron  Interactions  k]  3.  Effect of Electron-Electron Interactions Comparison w i t h  Positrons  in Real  Introduction  2.  Effect  of  the P e r i o d i c  3.  Effect  of  Core A n n i h i l a t i o n  of  and  Electron-Positron k\  Experiment  1.  Lifetimes IV  kO  Effect  A n n i h i l a t i o n of  H.  Geometry  1.  h.  Chapter  Col l i n e a r  38  k2 Metals  kk  Positrons  Crystal  Lattice  in M e t a l s  EXPERIMENTAL ARRANGEMENT Crystal  kk  kk h$ k$ hi  A.  Metal  B.  Orientation of  C.  Spatial  Stability  D.  Crystal  Holder  52  E.  Holder  Support  53  F.  Gamma C o u n t e r s  55  G.  Electronics  57  H.  Stability  of  and P o s i t r o n  Potential  Source  Crystal of  N o t c h and P i n A s s e m b l y  Electronics  kS 50 52  57  Page Chapter  V  RESULTS AND CONCLUSIONS  61  A.  Introduction  61  B.  Experiment  62  C.  Results  63  D.  Interpretation 1.  Effects  2.  Resolution Point  3.  of  of  the Results  Finite  Source  Function for  65 and D e t e c t o r  Finite  Size  Detectors  and  Crystal  Resolution Finite  66  Function  for  Finite  Detectors  and  Crystal  E.  Interpretation  F.  Accuracy  of  65  67 the Results  Attainable with  the Method  69 70  1.  R e s o l u t i o n and C o u n t i n g R a t e  70  2.  Stability  71  G.  Discussion  72  H.  Conclusions  73  SOLUTION OF ABEL'S INTEGRAL EQUATION  75  B.  EXPECTED ANGULAR CORRELATION CURVE WIDTH  77  C.  EFFECT OF CORE ANNIHILATION  80  APPENDICES  .A..  Bibl iography  8l  viii  LIST OF TABLES  Table  Page Comparison of Theoretical Dimensions  and Experimental Fermi Surface 26  L I S T OF FIGURES  Fi gure  Page  1.  Body-centered  Cubic  Structure  ]h  2.  Face-centered  Cubic  Structure  15  3.  Hexagonal  Close-packed  h.  Brillouin  Zone f o r F . C . C .  5.  Copper  6.  Two-Photon A n n i h i l a t i o n  32  7.  Wide-slit  Geometry  32  8.  Wide-slit  Geometry  9.  Region  Fermi  Surface  Structure Lattice  Details  19 25  Results  Sampled by W i d e - s l i t  16  35 Geometry  35  10.  Point  11.  Region  12.  Col l i n e a r P o i n t  13.  Annihilation  14.  P o s i t r o n A n n i h i l a t i o n Rates  h3  15.  Experimental  48  16.  N o t c h and P i n A r r a n g e m e n t  51  17.  Crystal  Holder  Sh  18.  Remote C o n t r o l  5h  19.  Preamplifier  20.  Coincidence  21.  Experimental  22.  Fermi  Geometry  38  S a m p l e d by P o i n t  Geometry  Geometry  38  in E l e c t r o n  Gases  Arrangement  and S h a p e r  Circuit  Circuit  Surface  hi  58 59  Results Neck  38  Details  Sh 65  CHAPTER I  INTRODUCTION  One o f  the  fundamental  to f i n d a s o l u t i o n of  the  f i n e d by a c r y s t a l l i n e  problems  Schrodinger equation for  solid.  possible considerable effort  Since a d i r e c t has been d i r e c t e d  m o d e l s w h i c h a p p r o x i m a t e t h e many-body  system.  i s t h e n p o s s i b l e t o make p r e d i c t i o n s a b o u t system t h a t  can be c o m p a r e d w i t h  desirable  t o be a b l e  that  c a n be c o m p a r e d w i t h  they  m o d e l s and a l s o  in the t h e o r y o f  s o l u t i o n of  this  system  problem  is  deis  not  toward the p r o d u c t i o n of By t h e u s e o f  the p r o p e r t i e s of  those derived  solid state  t h e many-body  these models the  from experiment.  t o make r e l i a b l e m e a s u r e m e n t s o f  because  the  the p r e d i c t i o n s o f  crystalline It  is  these p r o p e r t i e s  the various  it  thus so  theoretical  t h e d a t a may be u s e d t o s u g g e s t newer and  better  models.  A special o c c u r s when t h e  case of  crystalline  this solid  problem that is a m e t a l .  p r o b l e m c a n be c o n s i d e r e d t o h a v e s t a r t e d w i t h  is p a r t i c u l a r l y  interesting  Work on t h i s a s p e c t o f the  free-electron  the  approximation  2  of  Sommerfeld  do  not  (1928)  in which  interact with  each  i t was assumed  other  or with  Despite  its oversimplification of  usually  gives  quantitative effect the  of  the  is  perform, large  model. would  but  body  be f u r t h e r  tron-electron lem  has  on  Most  the  culty  (Reitz,  1955;  hundred the  band  advent  number  of  of  limited  1958).  to metals  of  are  than  bands  on  of  that to  the  in metals  cores  are  difficult  accord  with  Sommerfeld  experiment  aspect (Pines,  of  elec-  the  of  probPines,  1955;  are s t i l l  because  of electron-electron  of  (1928),  the e f f e c t of  at present  however,  the  aspect  ion  with  this  Bohm a n d P i n e s  II),  this  the simple  include on  satisfactory  Hartree  agreement  theory  include  in b e t t e r  does  progress  of  to  solid  lattice.  the  that  calculations  that  possible  the e f f e c t s  and S e i t z  for metals computing  calculations  However,  in order  the e f f e c t of  data  (Chapter  o f Wigner  calculations high-speed  situation,  the studies  the  the crystal  necessary  accurate  of  the  based d i f f i -  interaction  1958).  t h e work  extensive  experiment.  energy  including  Callaway,  Since  of  physical  of  Considerable  p a r t i c l e model  adequately  of  t o assume  studies  of  t h e o r e t i c a l work  results  i t were  the early  calculations  independent of  if  ion cores  usually  experimental  interactions. from  that  to  reasonable  improved  resulted  1963).  is  so  leads  of available it  is  the electrons  However,  Inclusion  the problem  it usually  However,  it  an e x t e n s i o n  (1930).  and Fock  complicates  results.  The e x t e n s i v e  essentially  the  the actual  be o b t a i n e d ,  ion cores.  (1928),  greatly  qualitative  results  the  problem  Bloch  to  good  that  with  a  have  been  machines  that  agree  few e x c e p t i o n s ,  the a l k a l i  group  or  (1933)  on  sodium,  performed.  h a s made well good  In  possible  well recent an  w i t h one another accord  the a l k a l i n e  over  years  increasing and  with  with experiment  earth  group  a  is  (Callaway,  3  An e x c e p t i o n t h a t agreement  is of particular  interest  is copper.  o f r e c e n t e x t e n s i v e band c a l c u l a t i o n s w i t h e x p e r i m e n t  ( S e g a l 1, 1962; B u r d i c k ,  1963).  The e x p e r i m e n t a l  situation  Fermi  surface which describes  absolute zero, niques  and Webb,  These  I960).  part  (the " b e l l y " )  directions. copper tions  t h i s model  t h e Fermi  1962;  Burdick,  The e x p e r i m e n t a l on f a i r l y  pure metal  II,  the surface  factory  study o f metals  possibility  methods one g e n e r a l l y Fermi  surface.  p r e d i c t e d by r e c e n t  low t e m p e r a t u r e s path  (Harrison  For example,  band  ( ~4  that are  central fill}  surface of calcula-  boundary.  the temperature Finally,  I960).  complicates the  low temperature  phase  a l s o e l i m i n a t e s the  dependence o f t h e " s h a r p -  states  interest  This  impossible a s a t i s -  i t i s t o be n o t e d t h a t  examines o n l y the e l e c t r o n  I t w o u l d be o f c o n s i d e r a b l e  K) d u e t o t h e  and Webb,  i t makes  The r e s t r i c t i o n t o l o w t e m p e r a t u r e s  surface  resonance,  along the  t h i s model f o r t h e F e r m i  The r e q u i r e m e n t o f l o w t e m p e r a t u r e s  o f examining adequately  n e s s " o f t h e Fermi  results  ("necks")  s u c h as s o d i u m a n d p o t a s s i u m i n w h i c h  transitions occur.  tech-  t e c h n i q u e s m e n t i o n e d above a r e a l l l i m i t e d t o u s e  restrictive.  study o f a l l o y s .  at  1963).  specimens at very  is rather  level  f o r c o p p e r p r o p o s e d by P i p p a r d  protrusions  r e q u i r e m e n t o f l o n g e l e c t r o n i c mean f r e e requirement  cyclotron  surface consists of a spherical  in Chapter  i s i n good a c c o r d w i t h (Segall,  surface  together with eight  As d i s c u s s e d  For example, the  i n c l u d e t h e methods o f anomalous  A l l f i v e methods g i v e  in a c c o r d w i t h t h e model o f t h e F e r m i In  thoroughly  f i v e major e x p e r i m e n t a l  magnetoresistance, magnetoacoustic e f f e c t ,  and t h e de H a a s - v a n A l p h e n e f f e c t .  (1957).  1962).  rather  the highest occupied e l e c t r o n energy  h a s been s t u d i e d by a t l e a s t  (Harrison  skin effect,  ( S e g a 11,  i s good  is also  u n i q u e s i n c e t h e e l e c t r o n i c p r o p e r t i e s o f c o p p e r h a v e been more i n v e s t i g a t e d t h a n t h o s e o f any o t h e r m e t a l  Here t h e  that  in these  r e s i d e near t h e  t o be a b l e  t o probe the  k  entire  valence  band.  A technique volves  a study of  the metal  small  does not  suffer  there  is a chance  from s t r i c t  that  c the speed o f  energy of lation will  Lee-Whiting 1 ized)  before  essentially  if  (1955)  indicate  of  annihilation  the  is  photons  radians)  photon p a i r ,  sample  that  reasonably  the p o s i t r o n  the  a  nd  by De B e n e d e t t i many e l e m e n t s  et  that  al.  is  The  related  to by  momentum  i s a b o u t 0.51  rest  The  Mev  (annihi-  and s c a t t e r i n g by t h e  small.  sample  Since c a l c u l a t i o n s  is e s s e n t i a l l y  at  rest  the  gamma-ray  (therma-  the p a i r  pairs emitted  o f M o n t g o m e r y and B e r i n g e r the  is  in p o s i t r o n  The e a r l y  (1950).  two gamma-rays  Since then a n g u l a r  work  (19^2) e s t a b l i s h e d emitted  The f i r s t d e t a i l e d a n g u l a r d i s t r i b u t i o n was  and compounds h a v e been made.  correlation  The s u b j e c t  in obtained  studies  has been  t h e w o r k on F e r m i  surfaces  of  the  of  reviewed  (i960).  Most o f  by  electron.  correlation of  positron annihilation.  In  result.  f r o m 180°.  i n s o l i d s has been o b s e r v e d by many w o r k e r s .  (193*0  with  m the e l e c t r o n i c  c e n t r e - o f - m a s s momentum o f  t i m e c o i n c i d e n c e and n e a r c o l l i n e a r i t y o f  by W a l l a c e  ( ~1 0  gamma-ray a t t e n t u a t i o n  the sample  The a n g u l a r  Klemperer  antiparallel  it annihilates,  that  antiparallel  l i g h t , and a t h e a n g u l a r d e v i a t i o n  so that  be n e g l i g i b l e  Upon c o l l i s i o n  i s t h e component o f p a i r  the photons emanating from the  radiation)  method  the a n n i h i l a t i n g e l e c t r o n - p o s i t r o n p a i r  t h e s i m p l e r e l a t i o n p^ = mce w h e r e p^ p e r p e n d i c u l a r to the n e a r l y  In t h i s  in-  t h e p a i r may a n n i h i l a t e e a c h o t h e r .  antipara11 elism  t h e c e n t r e - o f - m a s s momentum o f  mass,  in m e t a l s .  i s bombarded w i t h p o s i t r o n s .  e v e r y s u c h a n n i h i l a t i o n two n e a r l y deviation  from the above l i m i t a t i o n s  the a n n i h i l a t i o n o f p o s i t r o n s  t o be s t u d i e d  an e l e c t r o n , nearly  that  by t h e p o s i t r o n a n n i h i l a t i o n  5  t e c h n i q u e has u s e d t h e " w i d e - s l i t " m e t h o d the  f r e e - e l e c t r o n model  a slice it  through the  it  Fermi  is easy sphere  has been p o s s i b l e t o o b t a i n  within  a metal  at  temperatures  the m e l t i n g p o i n t f i c a t i o n of  (Gustafson  Fermi-Dirac  effects  t o show t h a t  (Chapter  et  i n f o r m a t i o n about the d e t a i l e d  1965).  the wide  study of  Fermi  making  effects  this  thesis  technique which o f f e r s  are  (col l i n e a r p o i n t  i n t h e new  The p r i n c i p l e s o f are d i s c u s s e d  1955)  to  beyond  direct  veri-  resolution  and  h i g h e r momentum which are  dis-  for yielding quantisurfaces.  (Colombino et  volume  al.,  1963;  in Fuji-  improved r e s o l u t i o n compared  a g a i n o b s c u r e d by c o r e  describes advantages geometry")  to  annihilation  in Chapter  a development o f  the  for  Fermi  the  it  than appears  a r i s i n g from the presence o f role  technique  i t d i f f i c u l t t o use the method f o r  g e o m e t r y " methods because  important  Fermi  samples  an  accurate  topology.  s u r f a c e more q u a n t i t a t i v e l y  or "point  less  results  effects  surface  W i t h t h i s method Fermi  shapes of  A l t h o u g h t h i s method p e r m i t s  The w o r k o f geometry"  been u s e f u l  has been u s e d by a few w o r k e r s  lattice  lattice)  of  electrons  g e o m e t r y " method w h i c h samples a c y l i n d r i c a l  s l i t method, the  and o t h e r  the  due t o f i n i t e  the c r y s t a l  t h e m e t h o d has n o t  this  providing a rather  ( c o r e a n n i h i l a t i o n and o t h e r  tative  wara,  K (Stump,  However,  in C h a p t e r  momentum s p a c e  By t h e u s e o f  III).  1963)>  al.,  cussed  A "point  t h i s w i d e - s l i t method  ranging from  a r i s i n g from the presence o f III,  On t h e b a s i s  III).  t h e momentum d i s t r i b u t i o n o f  statistics.  other complicating factors  (Chapter  study of  "point surfaces.  is p o s s i b l e to examine p o s s i b l e by t h e  the  "wide-slit"  c o r e a n n i h i l a t i o n and h i g h e r momentum the  crystal  lattice  play  a  relatively  technique.  t h i s new " c o l l i n e a r p o i n t  II1 of  the present work.  geometry" This chapter  technique is  followed  6  by C h a p t e r detail.  IV  in which the e x p e r i m e n t a l  Finally,  i n C h a p t e r V some e x p e r i m e n t a l  application of this room t e m p e r a t u r e surface  new m e t h o d t o a s t u d y o f t h e  are presented.  f o r copper which  obtained at  very  that  These  is d i s c u s s e d  Fermi  surface of  copper  used t o c o n s t r u c t  the copper  by t h e more p r e c i s e  Fermi  study.  a  at  Fermi  surface  conventional  The d i s c u s s i o n c l o s e s w i t h a s t a t e m e n t  h a v e b e e n drawn f r o m t h e p r e s e n t  i n some  r e s u l t s o b t a i n e d f r o m an  r e s u l t s are  i s t h e n compared w i t h  low t e m p e r a t u r e s  niques mentioned above. clusions  arrangement  of  techthe  con-  7  CHAPTER I I  MOTION OF ELECTRONS IN METALS  A.  Introduction  If so t h a t  one c o n s i d e r s a c r y s t a l l i n e  the motion of  the  relatively  can  t o a good a p p r o x i m a t i o n ( Z i m a n ,  for  the  system  Here  it  NZ  composed o f N a t o m s e a c h w i t h A, i s d e n o t e d by  27i"1i  I960)  write  low  temperatures  c a n be n e g l e c t e d ,  the  Schrodinger  one  equation  MZ  -e  constant  t e r p r e t a t i o n of  the  the  1 T -  lattice  j  is the e l e c t r o n i c  E1P  is p e r f e c t  z electrons.  and t h a t o f n u c l e u s  meaning;  Planck's  massive n u c l e i  r/ifVi "i^^TFTFT  has been assumed t h a t  the usual  fairly  as N  r-i*JfcV-  s o l i d at  the c r y s t a l  The p o s i t i o n v e c t o r o f  by f^- .  The  in the above H a m i l t o n i a n  for  rest  the system.  is well-known  is  electron  remaining symbols  c h a r g e , m the e l e c t r o n i c  and E t h e e n e r g y e i g e n v a l u e  terms  and t h a t  have mass,  The  in-  (Ziman,1960).  8  The f i r s t  term r e p r e s e n t s  the p o t e n t i a l the  energy of  the  the e l e c t r o n s  t h i r d term the p o t e n t i a l  action.  If  energy  more t h a n o n e t y p e o f  is e a s i l y  (2-1)  k i n e t i c energy  g e n e r a l i z e d and  of  the e l e c t r o n s ,  in the n u c l e a r  of  atom  leads  the  second  Coulomb f i e l d ,  the e l e c t r o n - e l e c t r o n is present  the  and  Coulomb  inter-  Hamiltonian of  t o an e x p r e s s i o n t h a t  is only  equation slightly  more c o m p l i c a t e d .  B.  The O n e - E l e c t r o n A p p r o x i m a t i o n  The a b o v e e q u a t i o n damental solid  equations  in the theory o f  one has a v e r y  possible  to s o l v e  the  for  the  less (the  s y s t e m t h e n becomes  ^(7, •••*,.)  dition term  ^(>p  replaced  one-electron  However,  5  in f a c t since  interacting particles  tractable  i f one t r e a t s  (Anderson,  one o f in a  it  by  is q u i t e  the e l e c t r o n s The wave  point  ?i i s -e  and v o l u m e e l e m e n t  4  as  improbsta-  function  (2-2) functions  (including spin).  If,  the  Hartree  in a d -  interaction  equation  with  ;  if  it  l^-Oj-)! d£  the  Vfovjft  In t h e s e e q u a t i o n s  interpretation  fun-  1963)  i t s a v e r a g e v a l u e one o b t a i n s  eigenvalue  the  typical  Nevertheless  one-electron approximation).  one-electron  e  i = l,2,...Nz.  simple j at  r  is  to the o n e - e l e c t r o n a p p r o x i m a t i o n , the e l e c t r o n - e l e c t r o n  is  where  a  and  Schrodinger equation d i r e c t l y .  independent  where the  general  solids.  l a r g e number o f  lem c a n be made more o r tistically  is q u i t e  is  the average  is noted t h a t »  t  n  e  potential J  ^  since  interaction  t e r m has a  the charge d e n s i t y o f  electron  energy a s s o c i a t e d w i t h e l e c t r o n  i  Then t h e  c a n be  quantity  i n t e r p r e t e d as t h e p o t e n t i a l  of e l e c t r o n j  (Raimes,  In o r d e r calculates  (i^ )  to solve  the  antisymmetry functions;  this  requirement  the  which are  i  coordinates.  does not obey  a self-consistent  (spatial with  the P a u l i  t h e p r o d u c t f u n c t i o n g i v e n by However,  it  by f o r m i n g a s u i t a b l e given  required.  the  antisymmetric equation  is p o s s i b l e to s a t i s f y  linear  }  combination of  the  product  yJi*»J If  identical)  for  two e l e c t r o n s  the determinant  one has vanishes,  = in  accord  exclusion principle.  set of one-electron equations  Fock e q u a t i o n s ;  ^  by;  Using the determinanta1 f u n c t i o n better  cloud  is o b t a i n e d .  However,  condition.  and s p i n c o o r d i n a t e s  result  in  t h e f u n c t i o n IP must be  Slater determinant,  i s a n t i s y m m e t r i c as  in t u r n used  fermions  xs*) which  in the charge  H a r t r e e e q u a t i o n s one assumes a s e t o f  and t h e n c e a new s e t o f y  Since e l e c t r o n s are  (2-2)  i  196l).  H a r t r e e e q u a t i o n s and s o o n , u n t i l  in the e l e c t r o n  energy o f e l e c t r o n  it  is p o s s i b l e to c o n s t r u c t  (Anderson,  1963). ^  These a r e ^  the  a Hartree-  10  where  <r d e n o t e s  Hartree-Fock method  equations,  (Messiah,  1957), nuclear However, its  use  1 9 6 4 ) , and s o l i d s t a t e  theory of  (Anderson,  t h e method w i l l  (Hartree,  1963).  be l i m i t e d  to  metals.  electrons  i n d e p e n d e n t l y o f each o t h e r as  In t h e  Hartree is not  taken  into account.  In  general,  correlations will  reduce  (Raimes,  to  196l).  the  ignore a l l Thus t h e  its  one-electron  1955).  the p r o p e r t i e s of a f r e e - e l e c t r o n  the  their  include only  H a r t r e e method f r e q u e n t l y y i e l d s  (Reitz,  spin  by t h e o n e - e l e c t r o n method  it  In p a r t i c u l a r , gas.  This  it  better  been  C.  Model  one  replaces  the  charge d i s t r i b u t i o n of  the  ion cores  to  useful  gas c a n be r e g a r d e d a s a c r u d e model o f a m e t a l .  If  re-  used  c a n be u s e d  is p a r t i c u l a r l y  is  some o f  since a free-electron  The F r e e - E l e c t r o n  time  method.  l i m i t a t i o n s , t h e H a r t r e e m e t h o d has o f t e n  in the t h e o r y o f m e t a l s  electrons  system because  spend l e s s o f  than to  of  method,however,  Coulomb c o r r e l a t i o n s and  correlations  than does the u n m o d i f i e d H a r t r e e - F o c k  Despite  Hartree-Fock  in making the e l e c t r o n s of metals  solid  Coulomb c o r r e l a t i o n  the energy o f a many-electron  In a t r e a t m e n t  found b e t t e r  In t h e  in the  i n t r o d u c e a c o r r e l a t i o n between  inclusion of  c o r r e l a t i o n s are e f f e c t i v e near each o t h e r .  Thus t h e  i s assumed  and H a r t r e e - F o c k m e t h o d s t h e  e x c l u s i o n p r i n c i p l e does  like spin.  derive  problems  in a t o m i c  (Coulomb c o r r e l a t i o n ) .  the e l e c t r o n s  sults  importance  general  to a v o i d each o t h e r  n o t move  them  is o f c o n s i d e r a b l e  c a s e o f a more  the  in a  will  often  a special  known a s  Coulomb r e p u l s i o n , t h e e l e c t r o n s  tend  of  they are a c t u a l l y that  (Brown,  Although these equations are  Due t o t h e i r m u t u a l will  the P a u l i  1962)  label.  in what f o l l o w s , t h e d i s c u s s i o n o f in the  model.  the s p i n  by a  11  uniform distribution of positive charge, and the charge distribution of the valence electrons by a uniform distribution of negative charge so that the net charge is zero, the Hartree equation simplifies to (2-4)  in,  For a cube of side L, containing N electrons it is seen that a solution of (2-4) is ^  (2-5)  |£  (2-6)  provided that 6 (  *  ) =  Application of the usual periodic boundary conditions (Ziman, I960; Raimes, 1961): ^(x+L.y.z) =^(x,y+L,z) = t=27r(  n]  ,z+L)  then yields  T+n j+n k) 2  3  (2-7)  where the n- are positive or negative integers or zero and T, j*, £ are unit t  vectors along the cube edges. From (2-7) it is seen that the integers n,. representing "orbital" states (Raimes, 196l) define a lattice in k-space.  Since each cube of side  2 7T will contain one such orbital state, the number of orbital states per L unit volume of k-space is L Thus in a volume element dk* of T<-space there 3 , ^F*are L .dk orbital states. Generalizing slightly, it is seen that for a metal 8 TT' 3  of volume v there are 2vdk electron states (spin degeneracy included) in a volume element dk of k-space. It is evident from (2-6) that the surfaces of constant energy are spheres.  Thus, as a consequence of Fermi-Dirac statistics, the occupied  12  region of  k-space a t  dense s p h e r e .  the a b s o l u t e zero of  Denoting the  and o b s e r v i n g t h a t  radius of  there are  following condition  temperature w i l l  this  sphere  N electron states  (the  within  be a u n i f o r m l y  Fermi the  sphere)  Fermi  by  sphere  the  is o b t a i n e d ;  ( 2 v )4?rk^=N (8T')3 or,  k = (37TN) p  ( V )  and t h e  Fermi  energy  is  2-m I  2-*i  D.  F  The  Crystal  Lattice  In a c r y s t a l l i n e known a s  the c r y s t a l  p o s s i b l e to generate of a u n i t  cell.  V I  s o l i d the atomic n u c l e i  lattice.  Due t o t h e p e r i o d i c i t y o f  the e n t i r e  (Ziman,  set of If  1964)  lattice  the u n i t  l e l e p i p e d c o n t a i n i n g one a t o m , t h e c r y s t a l lattice.  On t h e o t h e r  q u i r e d the c r y s t a l  hand,  lattice  is  if a unit  the various  easily  seen t h a t  if  cells,  e a c h c o n t a i n i n g one a t o m  a^,  and  are  c a n be c h o s e n  lattice  lattice  ]  vectors  ]  + 1 ? 2  array  it  the  cell  lattice),  t o be a p a r a l -  s a i d t o be a  Bravais is  re-  l a t t i c e with a basis  since  m u s t be s p e c i f i e d .  any  lattice  through a vector of  2  is  concept  i s c o n s i d e r e d t o be made up o f  (Bravais  \a  is  array  w i t h more t h a n one a t o m  in the u n i t  by a t r a n s l a t i o n  T =  where a ^ ,  atoms  the c r y s t a l  f r o m any o t h e r  cell  cell  this  p o i n t s by u s e o f  s a i d t o be a B r a v a i s  the p o s i t i o n s of  reached  form a p e r i o d i c  the  unit  p o i n t can  be  form  + 1 ^  d e f i n e d by t h e e d g e s o f  It  (2-8)  the u n i t  cell,  and  is  13  lj,  1^» a n d 1^ a r e i n t e g e r s .  Equation  l a t t i c e made up o f u n i t  cells  l a t t i c e with  However,  reside  a basis).  at equivalent  containing  sites within  Most m e t a l s c r y s t a l l i z e are  the body-centered cubic,  packed  structures,  spectively. For  (2-8) i s a l s o  valid  for a crystal  more t h a n o n e a t o m e a c h  (Bravais  i n t h i s c a s e t h e t w o a t o m s s o l i n k e d must t h e i r respective  unit  cells.  i n one o f t h r e e s t r u c t u r e s .  the face-centered  cubic  d i a g r a m s o f w h i c h a r e shown  These  and t h e hexagonal  in Figures  structures.  e x a m p l e , l i t h i u m , sodium, and p o t a s s i u m a r e b o d y - c e n t e r e d c u b i c  close-packed  and g o l d  structure  are face-centered cubic. are beryllium,  close-  1, 2, a n d 3» r e -  N e a r l y a l l o f t h e common m e t a l s have o n e o f t h e s e  copper, s i l v e r  three  whereas  Examples o f t h e hexagonal  magnesium and z i n c .  F i g u r e 1: Body-Centered.Cubic  Structure  F i g u r e 2: Face-Centered Cubic S t r u c t u r e  16  k^^^  ;  —j  H  •  >i  y/ai a,  F i g u r e 3: Hexagonal C l o s e - P a c k e d  Structure  17  E.  The R e c i p r o c a l  In known a s  the  the  by means o f  Lattice  theory  reciprocal the  of metals lattice.  reciprocal  b*^ = 2K ^  x a ^,  a^-  The u n i t (Kittel,  1956)  satisfying  1  2  integers  cell,  zone,  Bragg  a general  c a n be s e e n  that  2  to  This  introduce a lattice  lattice  c a n be  defined  mj&  +  a^ , D^-2/ra*^  or  a y a*  2  x a^ and m^ , m^,  and  x a*^  zero.  in the  shortest  (2-9)  3  x a"  reciprocal  lattice  non-zero r e c i p r o c a l  is  then  lattice  obtained vectors  condition  (i< + CT = k  w h e r e T< i s  1964)  + m b*  ]  negative  by f i n d i n g t h e  the  m K  l y a ^  or  convenient  vector  b^ = 2 7 r x  x a*  may be p o s i t i v e o r  is  (Ziman,  lattice  G*=  where  it  vector  2k-Ci -  in the  - G so t h a t  (2-10)  2  reciprocal each  space.  zone boundary  From t h i s  equation  i s normal  to a  it  recip-  G  rocal  lattice  first  Brillouin In  primitive  vector  at  zone o r  particular,  its midpoint. the  reduced  for  The c e l l  to  formed  is  known a s  the  one may t a k e  the  zone.  a face-centered  translation vectors  thus  cubic  lattice  be  3, = a(t + j) 2 a \ = a t f + ft) 2 3 = a(f* + ft) 2 5  where a  is  the  length of  a c u b e e d g e and T ,  j , and ft a r e  unit  vectors  along  18  the  cube e d g e s .  From t h e a b o v e e q u a t i o n s G = 27t  Thus 2* a  the  shortest  (m -rr^+m^) T  (+?+J+K) and t h e n e x t  2.7[(+2k).  The  midpoints) in  Figure  F.  reciprocal shortest  intersection  defines  the  of  first  then e a s i l y  (n^+n^-m^j  lattice  are  + (-rr^+ir^+m^) K j  vectors  are  the s i x v e c t o r s  the planes Brillouin  normal  obtains  vectors  2 r ( + 2 7 ) ; _27r(+2j) ; a a  to these  z o n e shown as  the e i g h t  vectors  the  (at  truncated  their octahedron  k.  M o t i o n o f an E l e c t r o n If  the  first  in a C r y s t a l  two t e r m s  t h e a s s u m p t i o n made t h a t  the  attached  so t h a t  moving  +  1  non-zero  one  to in a  their  nuclei  lattice  of  Lattice  in the  ion cores  Hamiltonian of are  closed  (2-1)  shells  are  retained  of electrons  rigidly  one has a s y s t e m o f n o n - i n t e r a c t i n g  ion c o r e s ,  equation  (2-1)  reduces  electrons  to  [-i" *w>Ji"fT  .,o  v  where V ( r )  is  the p e r i o d i c p o t e n t i a l  simple Bravais  lattice  a*^ r e s p e c t i v e l y ; an e l e c t r o n  It  is well  to  the  with  in the  lattice  known(Heine,  (2  due t o t h e  dimensions  w h e r e N^ , N  will  lattice.  N a 2  and N^ a r e  2 >  =  I960;  Tinkham,  +  Now c o n s i d e r  a  and N^a^ a l o n g a*j , a* ,  2 >  2  integers.  The p o t e n t i a l  have t h e p e r i o d i c i t y o f  V(r")  and  the  energy  and of  lattice; (2-12)  T) 1964)  that  s o l u t i o n s of  (2-11)  subject  i !<• r Uj»(r) the  conditions has  usual  are  the  Bloch  the p e r i o d i c i t y of  the  lattice;  periodic  (2-12)  boundary  5 5  Nj  £  U £ (r)  conditions'^^)  1^T» (r+N_a* ) g i v e s k*-a = 2 r n , , k  functions  =  fLa, = rub*.  N  1  +  rub*-  N  2  = u^  = 2rri N  +  = e (r+f)  = ^(r+N^a*^)  Toa*  2rn„, N  ^(i*)  J2,tT_  3  u  .  where  Application  = ^fc^^  whence 3  j*(?)  2^ 2)  =  of  F i g u r e k: B r i l l o u i n Zone f o r Face-Centered Cubic L a t t i c e  where  > r^,  that,  just  n^ are  and  as  reason  J  J  By rocal the  first  (2-10). the  =  e  k  \Z i s o f t e n  vectors  restricted  scheme  translating  zone.  that  be vdiL  the vector  ^]  '  s  a  ^  s  o  to t h e f i r s t  the energy  °f  the f i r s t  that  de-  (reduced R. recip-  surround  defines  in a s i m i l a r manner.  The e x t e n d e d z o n e scheme with  £ (k)  this  condition  by  zone and t h e n e x t p o l y h e d r o n  £ (k)  For  function of  given  valued  spectrum  = 1  form.  symmetrically  the energy  c o m p a r i s o n o f an e n e r g y  e  B r i l l o u i n zone through  Higher zones a r e d e f i n e d  k*.  itS-f gives  Bloch  u s e o f t h i s e x t e n d e d z o n e scheme f u n c t i o n o f fc f o r a l l  states  B r i l l o u i n zone  of these polyhedra are again  second B r i l l o u i n zone.  seen  £ is not u n i q u e l y  i s a many-valued  G, o n e c a n f o r m p o l y h e d r a  The v o l u m e b e t w e e n  is  orbital  8T->  (2-8) a n d (2-9)  regions of the f i r s t  The f a c e s  k it  -fff-F C  e  In t h i s  lattice  it follows  e x p r e s s i o n for  there w i l l  Use o f  i(E + u) • r  since ^(^)  zone scheme).  Thus  ^  the vector  case,  dk o f k - s p a c e .  s i n c e a.-b*. = 27r£- . termined  From t h i s  in the f r e e e l e c t r o n  in a volume element  1  integers.  c a n be w r i t t e n  By  as a s i n g l e -  then p e r m i t s  easy  the p a r a b o l i c free e l e c t r o n  spec-  S°{k) = f i k .  trum  2  2  2m 1.  Perturbation  If small,  Theory  the p e r i o d i c p o t e n t i a l  standard perturbation  the e l e c t r o n  f o r Weak P e r i o d i c  theory  V("r) o f t h e c r y s t a l (Anderson,  £ (k) = fi k  However, a  Fourier  lattice  1963; Ziman,  i s assumed  1964) y i e l d s  for  energy  €($ - a\£) +<lc|vtf)| where  Potentials  since  lv(?) 1 E ^ l  is the energy of the unperturbed  V (?)  series  E > + £ K E  free electron  has t h e p e r i o d i c i t y o f t h e l a t t i c e  ( Z i m a n , 1964)  2  state  | k> .  i t may be w r i t t e n  as  21  < = ^ V_ e  V(r) so t h a t  t h e m a t r i x e l e m e n t <k*  If  condition  this  is s a t i s f i e d ,  and t h e e x p r e s s i o n f o r  general where  equation  not  follows  be s p h e r i c a l .  is nonzero o n l y  reduces  + v  that  need n o t  be s p h e r i c a l ,  (2-13)  surfaces  the  Fermi  level  at  in c o n t r a s t  From t h e p e r t u r b a t i o n e x p a n s i o n  € = € (£)  surface  if  a degeneracy  is e q u i v a l e n t  other words, a Brillouin (Ziman,  a first in o r d e r  to  the  zone boundary.  it  =€f  free  tem-  electron  c a n be s e e n t h a t  occurs.  B r a g g c o n d i t i o n "k expansion  However,  This  near  ik•r (?)  -  e  u (P) R  is not  ignore a l l  t o o b t a i n an e x p r e s s i o n f o r  degeneracy  con-  (2-10).  it  of  by T< = (k*+(j) a  +  near  possible  the  i G•r ~ ~)  coefficients ^Q  In  is s t i l l  in a s e r i e s  a ^ e  the energy.  of  v a l i d when k l i e s  a zone boundary  ik-r ft  the zone boundary d e f i n e d  approximation, to  = (k* + "5)  wave f u n c t i o n = e  the  2  (2-13)  near  t o expand the e l e c t r o n  Tlf When k* l i e s  in  zero of  to the p e r f e c t l y  (2—13)  $<) = €  the p e r t u r b a t i o n  1964)  will  € (£)  the a b s o l u t e  2 dition  0=  2, 2 € = fi k . 2m  f o r wh i c h  method f a i l s  is s i m p l y C U | V(r) )  2  the energy  In p a r t i c u l a r ,  i f £ - £ + G = 0.  to  /vJ  n  £p i s t h e h i g h e s t o c c u p i e d e n e r g y  perature, case  it  |  the m a t r i x element  the energy  S ( i c ) = €(£)  From t h i s  | V(r)  iG - r  form (2-14)  it  is p o s s i b l e ,  e x c e p t a ^ and  The e n e r g y  is  as  a^ ^ +  then g i v e n  by  £(K) = V + 1/2 (€°(K)+ £(£+£) + 1/2 0  where  the minus  sign  refers  to s t a t e s  " i n s i d e " the zone boundary  (  Ik* - Gl^G*/ ) IGI  2  22  and where the p l u s s i g n (  Ik*• Gl >IG1 ) . IGJ 2  refers  to s t a t e s  From ( 2 - 1 5 ) i t  is again  will  be n o n - s p h e r i c a l .  ally  possess d i s c o n t i n u i t i e s at  of  (1963), or  Effect of  2.  seen  that  In a d d i t i o n , t h e e n e r g y  Ziman  Electron  considerable  i n t e r a c t i o n on t h e  C o r r e l a t i o n s on t h e  interest  Fermi  possess a " s h a r p " differ  in shape  Fermi  (but  not  Fermi  surface  Fermi  surface  field  (Cornwel1,  3.  for  interacting electrons  surface  discussion Kittel  neglects  electron-electron of  1964)  effects,  electron-electron  (Luttinger,  I960) a l t h o u g h Fermi  i s b a s e d upon  I960;  in a c r y s t a l in general  does  it  will  s u r f a c e of a system  Such an a n a l y s i s  has t h e same s y m m e t r y a s  i n t e r a c t i n g through a Hartree-Fock  of  a l s o shows  that  the  self-consistent  of  the One-electron A p p r o x i m a t i o n  the o n e - e l e c t r o n a p p r o x i m a t i o n f o r a system  i s p r o v i d e d by t h e  theory  (Raimes,  Bohm-Pines  theory of plasma  1957; 1 9 6 l ) a m e t a l  is  a r e embedded.  This  theory  a s s o c i a t e d w i t h a quantum o f e n e r g y  shows t h a t (plasmon)  hw  of  oscil-  r e g a r d e d as a plasma  . composed o f a u n i f o r m d i s t r i b u t i o n o f p o s i t i v e b a c k g r o u n d c h a r g e electrons  gener-  1964).  Justification  In t h i s  Fermi  from the  interacting electrons  A j u s t i f i c a t i o n of  lations.  (Luttinger,  (Cornwel1,  for electrons  surface  Surface  Such c o n s i d e r a t i o n s  i n symmetry)  non-interacting electrons. the  largely  Fermi  a system of ,i n t e r a c t i n g e l e c t r o n s  surface  will  A detailed  to examine the e f f e c t s  surface.  C o r n w e l 1 , 1964) show t h a t  s p e c t r u m £= t (k")  Fermi  (1964).  the o n e - e l e c t r o n a p p r o x i m a t i o n which is of  the  i n many t e x t b o o k s , f o r e x a m p l e ,  Since the concept of a w e l l - d e f i n e d  it  boundary  in general  the zone b o u n d a r i e s .  t h e s e d i s c o n t i n u i t i e s may be f o u n d  (1955), Anderson  " o u t s i d e " the zone  in which  p l a s m a o s c i l l a t i o n s may > f. and t h a t  plasmons  be in  the  23  a metal  at o r d i n a r y  temperatures  c o n s e q u e n t l y may be o f t e n part  of  the e f f e c t i v e  will  ignored.  n o r m a l l y be  in t h e i r  The t h e o r y a l s o shows  Coulomb i n t e r a c t i o n  ground s t a t e that  is a s s o c i a t e d w i t h  the  and  long-range  the plasmons.  o The  r e m a i n i n g Coulomb i n t e r a c t i o n has an e f f e c t i v e  short  that  often  oscillations  band c a l c u l a t i o n s a r e  the Hartree-Fock  equations  tions are d i f f i c u l t various  approaches  Since  for  the  first  calculation  a hundred energy of  these  calculations  culation  for  proaches  (Slater,  data  tions  1958).  o f energy  and S e i t z  same m e t a l . ten  For example,  times  by u s e o f  trivalent,  over Many  cal-  various  is f a i r l y  satisfactory  (1963)  the  (Callaway,  On t h e  1958).  few d e t a i l e d c a l c u l a t i o n s  For c o p p e r ,  (Slater,  the agreement w i t h  however,  1963).  at  least  Agreement of  ap-  is there 1958).  quadrivalent  with experiment  is good.  s a t i s f a c t o r y agreement Relatively  betw een  few c a l c u l a t i o n s  or pentavalent  metals.  is  often  available  for  other the  a dozen  the  f o r w h i c h d e t a i l e d band c a l c u l a t i o n s e x i s t ,  for  the  band c a l c u l a t i o n s w i t h e x p e r i m e n t  valent  (Callaway,  (1933), well  h a v e been p u b l i s h e d .  and B u r d i c k  periment  calcula-  and a p p r o x i m a t i o n s h a v e been e m p l o y e d .  For the a l k a l i metals  h a v e been p e r f o r m e d  for beryllium  numerical  As these  S e g a l 1 (1962)  only  Model  1963).  h a v e been p u b l i s h e d .  metals  so  w o r k on p l a s m a  the One-electron  for metals alone  hand, w i t h the e x c e p t i o n of c o p p e r , metals  is  1956).  (Callaway,  s o d i u m has been made more t h a n  not s a t i s f a c t o r y .  A which  i n p r i n c i p l e b a s e d on t h e  h a v e been f o r t h e  The a g r e e m e n t  experimental  (1955,  s o d i u m by W i g n e r  band c a l c u l a t i o n s  "'I  The e x p e r i m e n t a l  Band C a l c u l a t i o n s by u s e o f  Energy s o l u t i o n of  t o o may be n e g l e c t e d .  has been d i s c u s s e d by P i n e s  Energy  4.  it  range o f  noble  calcula-  recent work For the  by  few d i -  i t appears  that  t h e o r y and e x -  have been  However,  the  performed calculations  2k  f o r aluminum appear 1958). the  t o be  in q u a l i t a t i v e  For the t r a n s i t i o n elements  results 5.  t e n d t o be r a t h e r  The  Fermi  the c a l c u l a t i o n s are  qualitative  Surface of  The e n e r g y  accord with experiment  Copper;  (Callaway,  T h e o r y and  band c a l c u l a t i o n s o f  c o p p e r show q u a n t i t a t i v e  agreement  has  calculated  the " n e c k s " which  as w e l l  as  the  radius of  the average " b e l l y . "  very d i f f i c u l t  Experiment  and B u r d i c k  with experiment.  radius of  the  and  1958).  S e g a l 1 (1962)  for  (Callaway,  lie  For example,  in the  Fermi  {ill}  surface. 8  (1963)  For  Segal 1  directions the  neck  -1  0 . 2 8 t 0 . 0 3 x 10 cm w h i c h compares w e l l 8 -1 w i t h t h e e x p e r i m e n t a l v a l u e o f 0 . 2 6 x 10 cm ( J o s e p h and T h o r s e n , 1 9 6 4 ) . F o r 8 -1 a v e r a g e " b e l l y " r a d i u s he o b t a i n s 1.33 - 0.01 x 10 cm w h i c h i s i n good 8 -1 radius  Segal 1 o b t a i n s a v a l u e o f  accord with work of  the experimental  Morse  (i960).  pared with experiment Figure  5-  All  value of about  The d e t a i l e d in T a b l e  I.  r a d i u s k^. = 1 . 3 6 5 x 10 cm  refers  to the neck  the hexagonal  refers of  to the  the hexagonal  d i r e c t i o n of changed. (1962).  o b t a i n e d from  B u r d i c k and S e g a l 1  The s y m b o l s u s e d in terms o f  in T a b l e  I are  .  In  the  table  t h e number  Further  and k ^ ^  comin  Fermi  (1)  under  (2)  refer  sound waves used  to the d i f f e r e n t  The numbers values  (l)  in the paper  r  center  by Bohm and  k  center to  (k)  o b t a i n e d when  in the m a g n e t o a c o u s t i c measurements  d e t a i l s may be f o u n d  k  under  r a d i u s as measured a l o n g a l i n e p a s s i n g t h r o u g h t h e  k^g  the  defined  r a d i u s as measured a l o n g a l i n e p a s s i n g t h r o u g h the  B r i l l o u i n z o n e f a c e and o n e c o r n e r .  the  are  the f r e e - e l e c t r o n  z o n e f a c e and t h e m i d p o i n t o f one e d g e , w h e r e a s  neck  a p p e a r i n g under  x 10 cm  -1  sphere  of  r e s u l t s of  dimensions are expressed 8  1.32  is  Easterling  the  25  F i g u r e 5: Copper Fermi< S u r f a c e D e t a i l s  TABLE I COMPARISON  OF THEORETICAL AND EXPERIMENTAL FERMI SURFACE FOR COPPER  (AFTER BOHM AND EASTERLING, 1962)  FERMI SURFACE DIMENSION  BOHMEASTERLING (a)  (1). (2)  '100  ( 0  '110  (2) (3) (4)  .  "  (2) /k *100' 110  ;ioo M 10  V  ( 0  "  (2) /k 'lOO 110  DIMENSIONS  1.036 1.100 0.957 0.956 0.959 0.956 0.852 0.815 0.284 0.195 0.191 1.11  t 0.021  t t t i t t t t  0.060 0.01 1 0.011 0.032 0.01 1 0.009 0.021 0.042 t 0.01 1 t 0.01 1  SEGALL ( c ) 1 .04 f 0.015 0.94 ± 0.015 0.87 0.015 0.76 * 0.015 0.29 0.02 0.14 0.02  SEGALL 1.02 t 0.94 t 0.87 t 0.81 t 0.24 t 0.21 t  1.10  1 .09  (d) 0.015 0.015 0.015 0.015 0.02 0.02  ROAF (b) 1 .076 0.943  0.200 1.14 BURDICK ( e ) 1.05 ± 0.02 0.97 t 0.02 0.84 t 0.02 0.77 t 0 . 0 1 5 0.31 t 0.04 0.15 t 0.04 0.17 t 0.02 1.09  7  (a) M a g n e t o a c o u s t i c e f f e c t (b) D i m e n s i o n s d e d u c e d by R o a f (1962) f r o m S h o e n b e r g ' s d e H a a s - v a n A l p h e n d a t a (1962), P i p p a r d ' s (1957), a n d M o r t o n ' s (i960) a n o m a l o u s s k i n e f f e c t measurements. (c) Band t h e o r y c a l c u l a t i o n ( C h o d o r o w p o t e n t i a l ) (d) Band t h e o r y c a l c u l a t i o n ( - ^ - d e p e n d e n t p o t e n t i a l ) (e) Band t h e o r y c a l c u l a t i o n ( C h o d o r o w p o t e n t i a l )  27  CHAPTER I I I  A N N I H I L A T I O N OF POSITRONS  A.  Introduction  In been done of  the  in the  years considerable  and e x p e r i m e n t a l  by p o s i t r o n  lection  rules  are  to take  satisfied  likely  and  up t h e  relevant  chapter  to the  outlines  study of  a n n i h i l a t i o n may p r o c e e d  by  l i n e a r momentum, a n g u l a r momentum, e n e r g y  one-photon a n n i h i l a t i o n able  This  work  has some  Fermi  Positrons  Free p o s i t r o n - e l e c t r o n l o n g as  results  and e x p e r i m e n t a l  annihilation.  Free A n n i h i l a t i o n o f  p h o t o n s , as  theoretical  f i e l d of positron a n n i h i l a t i o n .  theoretical  surfaces  B.  recent  can o n l y o c c u r  recoil  in a t y p i c a l  ( J a u c h and R o h r l i c h ,  momentum.  s o l i d over  1955).  1,  2,  3 or  and o t h e r  In t h i s  95% o f  is  se-  respect,  i n t h e p r e s e n c e o f an e x t e r n a l The o n e - p h o t o n p r o c e s s  more  rather  the p o s i t r o n s a n n i h i l a t e  by  system untwo  28  o r more  photons. For p o s i t r o n - e l e c t r o n p a i r s ,  tions,  the p r o b a b i l i t y  that  for annihilation  fine  structure  processes dicate  is  (1954)  the  n>3  are  ignored.  r a t i o of  the  spin-averaged  processes  i s ^%-  agreement  f o r a l u m i n i u m , v i z . ^/rjg  sion of  section  with a free electron  this  In f a c t  372  was  first  detailed  cross  for  the  than  be  Thus, occurs  the  incurred  for  the  if  result  in-  two-photon  1956). of  This  Basson  the a n n i h i l a t i o n o f predominantly  by t h e  posiemis-  1954).  two-photon a n n i h i l a t i o n o f a f r e e obtained  is  considerations  sections  by D i r a c  (1930)  ( i m p l y i n g no Coulomb d i s t o r t i o n ) .  result  considera-  smaller  ( B e r k o and H e r e f o r d ,  in a metal  for  error will  the e x p e r i m e n t a l  (Graham and S t e w a r t ,  The c r o s s  wave" c a l c u l a t i o n  with  little  = 406 ± 5 0 .  conduction electrons  two p h o t o n s  Thus  =  3t  is  rule  by a f a c t o r o f o r d e r < * w h e r e w  2.  in reasonable  trons with  limit,  i n t o n photons  from s e l e c t i o n  i n t o n+1 p h o t o n s  and n  and t h r e e - p h o t o n ratio  for annihilation  constant  f o r which  that  aside  spin-averaged  cross  positron  by means o f a  In t h e  "plane  non-relativistic  s e c t i o n per e l e c t r o n  reduces  to  2 where r of  0  =  mc  is  the c l a s s i c a l  t h e p o s i t r o n and e l e c t r o n .  relative  velocity  per unit  time of a n n i h i l a t i o n )  ent of  velocity,  R where N i s  the e l e c t r o n  =  r a d i u s and v t h e  Although t h i s  v approaches  being given  electron  zero,  cross  section diverges  the a n n i h i l a t i o n  of a positron  relative  rate  i n an e l e c t r o n  velocity as  the  (i.e.probabi1ity gas  is  independ-  by  Nvoj" = *r  density.  0  cN  (3-0  Because o f the success o f the e l e c t r o n electrons trons  in m e t a l s ,  i t m i g h t be e x p e c t e d t h a t  gas model o f t h e c o n d u c t i o n  t h e mean l i f e t i m e s o f  posi-  i n m e t a l s w o u l d be a p p r o x i m a t e l y t h o s e e x p e c t e d o n t h e b a s i s o f  y i e l d i n g values  of  duction electron  (3-1)  -9  J_ ~ 10 s e c a n d i n v e r s e l y p r o p o r t i o n a l t o t h e c o n R density. The o b s e r v e d l i f e t i m e s , h o w e v e r , a r e s u r p r i s i n g l y T =  -10 constant  (f~  2 x 10  from one metal bly arises  to another  (Wallace,  from the n e g l e c t  deriving  (3-0  (Section  H) .  C.  sec) d e s p i t e  and w i l l  the large I960).  be e x a m i n e d  Bound E l e c t r o n - P o s i t r o n  in e l e c t r o n  Much o f t h i s  of the p o s i t r o n - e l e c t r o n  density  disagreement  interaction  after  State  Systems  t o t h e h y d r o g e n atom was s u g g e s t e d by M o h o r o v i c i c  the experimental  discovery of the positron  a n d O c c h i a l i n i , 1933). tronium  (Ruark,  e e e e)  In a d d i t i o n t o t h i s  1945) s e v e r a l  were c a l c u l a t e d  three-electron  systems  in  below  The p o s s i b l e e x i s t e n c e o f a bound p o s i t r o n - e l e c t r o n s y s t e m analogous  proba-  involved  in the d i s c u s s i o n o f l i f e t i m e s  P o s i t r o n A n n i h i l a t i o n f r o m a Bound  1.  variation  other  two-electron  bound s t a t e s  +  e)  (1934)  shortly  1932;  Blackett  s y s t e m known a s p o s i  " p b l y e l e c t r o n " systems  to possess s t a b l e  being stable  (Anderson,  7  (e  (e e e , e e e , and  (Wheeler,  by 0.20 e v ( H y l l e r a a s ,  1946), t h e  1947) a n d t h e  f o u r - e l e c t r o n s y s t e m by 0.11 e v ( H y l l e r a a s a n d O r e , 1947).  However,  three  both because o f  the  and f o u r - e l e c t r o n  small  p r o b a b i l i t y of their  up by c o l l i s i o n s three  systems a r e u n l i k e l y  (Deutsch,  and f o u r - e l e c t r o n  b e i n g f o r m e d , and b e c a u s e . o f  1953).  systems  The l i f e t i m e a g a i n s t  is estimated  In a d d i t i o n t o t h e s e p o l y e l e c t r o n bound s t a t e s  t o be o b s e r v e d  their  these  ready  break  annihilation of  these  t o be ~ 1 0 " ^ s e c .  systems, dynamically  stable  o f p o s i t r o n s w i t h molecules o r ions a r e a l s o p o s s i b l e .  For  30  example,  calculations  by a b o u t 0 . 2 3 ev ev  (Simons,  indicate  (Neamtan  1953).  et  l i t t l e experimental  and L e e ,  1964).  2.  al.,  A g a i n , as  systems,  that  a l s o composed o f  in the  case o f  a resemblance  the  three  and  stable 4.65  four-electron  "compounds"  gy  those of  levels  will  be h a l f  the p o s i t r o n i u m  1.06 A .  that  of  the  Deutsch  hydrogen atom,  h y d r o g e n and  i o n i z a t i o n energy  A discussion of  f o r example,  3.  to the hydrogen  a p a i r of o p p o s i t e l y charged p a r t i c l e s . half  in,  s h o u l d be  and p o s i t r o n c h l o r i d e by a b o u t  w o r k has been done on t h e s e  mass o f p o s i t r o n i u m i s  is  1962)  H  (Green  . P o s i t r o n ium  Positronium bears  Thus  p o s i t i o n hydride e  the  fine  its  i s 6 . 8 ev  Bohr  atom  in t h a t  Since  the  it  is  reduced  the p o s i t r o n i u m e n e r radius  twice  as  large.  and t h e p o s i t r o n Bohr  s t r u c t u r e o f p o s i t r o n i u m may be  radius  found  (1953).  Positronium Annihilation  P o s i t r o n i u m decays  by two and t h r e e  quantum a n n i h i l a t i o n ; the  'S  Q  •>  (para-)  state  decaying  by t w o - p h o t o n a n n i h i l a t i o n and t h e  S^  (ortho-)  state  3 by t h r e e  photons,  selection  rules.  = 1.25 -7 10  The n  lifetime  of  the  S^  state  b e i n g f o r b i d d e n by  f o r two-photon a n n i h i l a t i o n  s e c and t h a t  for  is  given  the  by  t h r e e - p h o t o n a n n i h i l a t i o n b y ^ = 1.4  x  3 n  times is  x 10  two-photon decay  sec. become  formed  should  Thus f o r  annihilation  T^l.25 x 10  i n any e x c i t e d  radiate  optically  1 0  state to  the  f r o m the ground s t a t e  s e c and (other  (n = 1)  the  T = 1.4 x 10^ s e c . If the p o s i t r o n i u m 1 3 than S s t a t e s o r the 2 S^ s t a t e ) it q  ground s t a t e  before  annihilating  (Deutsch,  1953).  Since  life-  p o s i t r o n i u m f o r m a t i o n does  not  seem l i k e l y  in  metals  (Wa11 a c e  1960;  on l i f e t i m e s  by K u g e l  in the present and d e c a y  Kanazawa et  work.  in s o l i d s ,  Angular  are  (Heitler,  part  emitted  rather  liquids,  extensive  and g a s e s  reference  conserved  the c e n t e r frame,  to the  formation. r a t i o n of  for  laboratory  The c a l c u l a t i o n light  If velocity  of  the  relative  light  of  off  by t h e  between  )  it  (1964).  two p h o t o n s  in m o t i o n  of  ener  frame  relative  the  to  the  is very small  center  the  transaber-  1964).  compared t o  a simple  compared  be  of  Lorentz  the  expression  from a n t i p a r a 1 1 el ism.  each photon w i l l  de-  two p h o t o n s c a n  used to c a l c u l a t e  is very small  so t h a t  (each o f  center-of-mass  is p o s s i b l e to d e r i v e  the p a i r  two p h o t o n s  v  (i960).  Positrons  ( B e c k e r and S a u t e r ,  velocity  furthe  formation  by W a l l a c e  system from the  is s i m i l a r to that optics  work  t h e p h o t o n d i r e c t i o n s may  two-photon  the photon d i r e c t i o n s  case the k i n e t i c energy of  is  recent  positronium  f r a m e by means o f an a p p r o p r i a t e  electron  ( v/c^lO  the  from col l i n e a r i t y  in r e l a t i v i s t i c  the d e v i a t i o n  carried  of  reviewed  in the  o f mass f r a m e  the angle  The d e p a r t u r e  if  the  n o t be d i s c u s s e d  h a s been d i s c u s s e d by G r e e n and L e e  f o u n d by t r a n s f e r r i n g t h e c o l l i n e a r frame  subject  has been  in o p p o s i t e d i r e c t i o n s  If  f r o m 180°.  mass  f o r example,  1966, p o s i t r o n i u m w i l l  and momentum a r e  1954).  laboratory  1965; s e e a l s o ,  C o r r e l a t i o n o f Two-Photon A n n i h i l a t i o n o f  Energy gy mc )  al.,  al.,  The  Positronium chemistry  D.  et  to the  In  this  energy  h a v e an e n e r g y  very  2 nearly it  equal  is seen  given  t o mc .  Thus e a c h  from F i g u r e  6 that  p h o t o n has momentum mc, and s i n c e  the  transverse  is  small  component o f p a i r momentum  is  by  (3-2) This  is  the  correlation  fundamental studies.  angle-momentum  relation  used  in two-photon  angular  32  33  E.  Angular Correlation  Geometries  I n t r o d u c t ion  1.  There are e s s e n t i a l l y  t h r e e ways by w h i c h t h e  Fermi  been s t u d i e d by u s e o f a n g u l a r c o r r e l a t i o n o f a n n i h i l a t i o n angular below  c o r r e l a t i o n e x p e r i m e n t s have u s e d t h e " w i d e  (Stewart,  "point"  geometry  (Colombino et  described  in t h i s  in which  the angle  (a)  2  does not v a r y .  are  B e r k o and P l a s k e t t ,  7, t h e  al.,  1963,  w o r k has  has  radiation. geometry  Most  described  been done by u s e o f a  Fujiwara  1964'  t h e s i s u t i l i z e s a t h i r d method  Wide S l i t  There  ment s i n c e  a l t h o u g h some r e c e n t  1957)  slit"  surface  1965).  ("collinear  The w o r k  point  These methods a r e o u t l i n e d  geometry")  below.  Geometry  several  variations  but  1958)  it  are  m o u n t e d b e h i n d two s l i t s  counted  that  the wide  is s u f f i c i e n t  the methods a r e e q u i v a l e n t .  gamma-ray p a i r s  of  In  s l i t method  to consider  (Stewart,1957;  Stewart's  t h i s arrangement,  shown  i n t i m e c o i n c i d e n c e by means o f  c a n be t a k e n  to  lie  in the h o r i z o n t a l  arrange-  in  detectors plane.  The r a d i o a c t i v e  sample c o n t a i n i n g a n n i h i l a t i n g e l e c t r o n - p o s i t r o n p a i r s  allowed  in a l i n e p e r p e n d i c u l a r t o the p l a n e o f ,  these  t o move  two s l i t s .  c o i n c i d e n c e at (Stewart,  Further  1957)  By m o v i n g t h e  various for  experimental  source v e r t i c a l l y ,  a n g l e s may be o b s e r v e d .  the d i s t a n c e s  shown  =  100  in.  X  =  1.5  in.  h  =  0.050 i n .  details  photons e m i t t e d  Some t y p i c a l  in the f i g u r e  D  can be f o u n d  and midway  in S t e w a r t ' s  are  paper.  Figure  is  between, in  time  dimensions  From F i g u r e  7 a n d by (3-2)  p  =  z  where z i s t h e v e r t i c a l  i t i s seen  displacement.  now t h i s a p p r o a c h  model  i t i s seen  slice  t h r o u g h t h e Fermi  of  course,  that  approximately,  mc9 = 2mcz/D  (3-3)  Thus  component o f p a i r momentum p^ i s d i r e c t l y If  that,  i n t h i s method t h e t r a n s v e r s e  proportional  is considered  to the displacement z.  in terms o f t h e f r e e - e l e c t r o n  f o r each s e t t i n g z o f t h e d i s p l a c e m e n t , one samples a v o l u m e a s shown  i s d e t e r m i n e d by t h e f i n i t e  in Figure  The s l i c e  9.  thickness,  resolution of the experimental  arrange-  ment.  Following the analysis incidence count  rate w i l l  o f Stewart  c*  P  j3(p-) dp  i s o t r o p i c so that  for  =  P  2 r  +  P  2 z  =  2  P  x  +  ^(p)  Py  = J>(p) , t h e c o u n t  jfip)  <*  z  2 where p  _  OO i s t h e d e n s i t y o f momenta o f t h e a n n i h i l a t i n g p a i r s .  n(p )  thec o -  dp  x  — oo  assumed  that  be g i v e n by  n( J where ^ ( p )  i t i s seen  (1957)  2  +  2K  Pf  d _ = Pf  rate  27T  If ^ ( p ) is  becomes  C (p)  pdp  2 P • Thus, d i f f e r e n t i a t i o n o f t h i s z  expression  n(p^) g i v e s  f(p  x  ) <X -  P. so t h a t  by (3-3)  it follows  '  d  P*  that  P ( p ) <* -1 z  For t h e i d e a l  dn ( p , )  1  dn(z). dz  free-electron  case  (3-4)  in which  i t i s assumed  that-  36  (a)  The momentum o f the  is zero  thermalized p o s i t r o n s are  electron (b)  the p o s i t r o n  (i.e.  neglected  t h e momenta  compared t o  of  the  momenta.  The e l e c t r o n momentum d i s t r i b u t i o n  is  i s o t r o p i c and  t o be o f u n i f o r m d e n s i t y j> up t o t h e maximum, o r  taken  Fermi  momentum p and z e r o f o r p > p . one  r  F  has  n(p„)  j°  Pdp  or  n which  (pj"(pj  i s t h e e q u a t i o n o f an  inverted  As m i g h t be e x p e c t e d , f r o m an e x a c t viation very  is  inverted  small.  nearly  and b e y o n d ,  verted  parabola.  to the  are  shown  Fermi  p  "However,  it  in the  8.  In  case of  the statement But  larger  this  counting rate curves the a l k a l i  that  " t a i l s " are results  for  that  their  They  estimate  s h o u l d be n o t e d t h a t  core a n n i h i l a t i o n c a l c u l a t i o n s .  is  there  in are  the  metals  through the  deexhibit  alkaline  s o d i u m and c o p p e r the angle  attribute  have e s t i m a t e d  deviate  s u p e r i m p o s e d on t h e  f i g u r e ©> d e n o t e s  B e r k o and P l a s k e t t  metals  the a l k a l i  as o n e p a s s e s  to a n n i h i l a t i o n of p o s i t r o n s w i t h  (core a n n i h i l a t i o n ) .  c o p p e r and f i n d  the experimental  For c o m p a r i s o n , the in F i g u r e  (3-5)  parabola.  behaviour.  increasingly  momentum p .  the copper curve cores  confirms  free electron  earths  1957)  parabola.  This  ->*)  most o f  the e l e c t r o n s  the  reasonable  (Stewart,  corresponding the  tail  of  of  the  ion  core c o n t r i b u t i o n agreement  in-  with  for  experiment.  d i f f i c u l t i e s associated with  These d i f f i c u l t i e s w i l l  be d i s c u s s e d  these  below.  37  2  (b)  Determination of  If the  z axis  a particular  in the wide  correlation  1958).  due t o a n o n s p h e r i c a l statements  samples a s l i c e  a metal  it  is  the  (Majumdar,  1965)  Fermi  it  the  source of  region.  occur  at  the  been  al.  (1965)  to study  the  Fermi  methods.  Metals  t h i s method Berko,  1962;  Plaskett,  3  as  these  so t h a t  obtained  (Berko,  that  average difficult  the  the  radii  should Also,  higher  shape  al., 1962),  that  1965),  of  (Stewart et  in several  by t h e m o r e p r e c i s e  al., been  cases  h a v e been e x a m i n e d  beryllium  h o l m i u m , e r b i u m , and y t t r i u m  in  con-  conventional  (Stewart et  a l u m i n u m and c o p p e r  (Berko  (Williams et  by al., and  al.,1966).  Geometry  t h i s m e t h o d two " p o i n t " d e t e c t o r s  arrangement  method  l i m i t a t i o n s , t h e m e t h o d has  single crystals et  it  the  of  the observed " b r e a k s "  a number o f m e t a l s ;  (Donaghy  magnesium  as w e l l  Point  In slit  1962),  of  results  form o f  i n c l u d e sodium  1958)  (a)  surface  the  in the  Despite  basis  momentum.  t h e e l e c t r o n wave f u n c t i o n may m o d i f y  surface.  firming qualitatively  Fermi  being  quanti-  curve which  suggests  c u r v e no l o n g e r c o r r e s p o n d t o  used  on t h e  a v e r a g i n g makes  correlation Fermi  t o make  i s an  the  the  as  that  curve s u f f i c i e n t l y  of  interpreted  giving a result  c o r r e s p o n d i n g to the  angular  (Berko  that  correlation  with  orientation  fact  correlation  1962)  the  the  in the a n g u l a r  is  is a l i g n e d  shape o f  surface  the a n g u l a r angular  crystal  difficult  Fermi  In a d d i t i o n , t h i s  angles  has  difficulty  l i t h i u m by Donaghy e t  momentum c o m p o n e n t s o f  the  Geometry  crystal  is u s u a l l y  topology of  sharp " b r e a k "  work w i t h  found that  anisotropy  surface  single  choice of  t h r o u g h momentum s p a c e ,  to observe  by W i d e S l i t  Although this  A major  a considerable  recent  d i r e c t i o n of  s l i t method,  about  such measurements.  over  Surfaces  c u r v e may d e p e n d upon t h e  and P l a s k e t t ,  tative  Fermi  d i s c u s s e d above.  With t h i s  are  used  arrangement  in p l a c e o f the  the  wide  r e g i o n o f momen-  turn s p a c e  that  i s sampled  m i n e d by t h e d e t e c t o r  i s an  geometry.  i n f i n i t e c y l i n d e r whose d i a m e t e r The a r r a n g e m e n t  Using a n o t a t i o n s i m i l a r to that slit  case,  it  i s seen t h a t  used  for counting rate at  is  illustrated  is  in F i g u r e  in the d i s c u s s i o n o f p  one e x p e c t s  deter-  the  (Figure  —  The s o l u t i o n o f  this  integral  (Appendix  I)  wide  11)  (  equation  10.  3  "  6  )  is  oo rz If,  as f o r  the wide  slit  c a s e , ^(p)  s u r f a c e and z e r o o u t s i d e , given  it  i s t a k e n t o be c o n s t a n t  is seen  from  (3-6)  that  i n s i d e the  the c o u n t i n g r a t e  z  2 so t h a t 3  a plot of n (b)  \ p*  - P*  (3-9)  2 vs.  p^  is  Determination of  1inear.  Fermi  S u r f a c e s by Use o f  Point  Geometry  The u s e o f p o i n t g e o m e t r y w o u l d be e x p e c t e d t o y i e l d a n g u l a r relation  curves  geometry  c o n t a i n i n g more s t r u c t u r e  since  method  (Fujiwara,  t h a n do t h o s e o f  the  1965).  Fermi  surface  Although the  B e r k o and P l a s k e t t  difficult.  region  results definitely  (1958),  wide  i n momentum s p a c e .  o f c o p p e r has been made by  momentum c o m p o n e n t s a g a i n makes q u a n t i t a t i v e surface  cor-  than those c h a r a c t e r i z i n g the  the method samples a s m a l l e r  A preliminary study of  Fermi  is  by n(p )«  slit  Fermi  this  show more  detail  the c o p i o u s presence of  higher  interpretation  in terms o f  the  39  I F i g u r e 10; P o i n t Geometry  >Figure. 11:, Region Sampled by P o i n t Geometry  i  F i g u r e 12: C o l l i n e a r P o i n t Geometry  40  Col l i n e a r  4.  Point  Geometry  In t h i s method t h e r a d i o a c t i v e m e t a l "point"  detectors  remain c o l l i n e a r and the c o i n c i d e n c e count  as a f u n c t i o n o f c r y s t a l e l e c t r o n model space.  This  the c r y s t a l  the n  Fermi  o( p^  .  orientation  t h i s method a l s o  volume e l e m e n t  crystal  surface. If  it  (Figure  about  so t h a t  i s , from  i s assumed t h a t  (3-9),  a l l other  rate  obtained  v o l u m e o f momentum  t h e o r i g i n o f momentum s p a c e a s  o r i e n t a t i o n t o be i n t e r p r e t a b l e  That  a n d t h e two  On t h e b a s i s o f t h e f r e e -  one would e x p e c t  if  variations  in c o u n t -  in terms o f r a d i i  (and h e n c e p^) i s z e r o ,  8  c o n t r i b u t i o n s to the counting rate  c o r e a n n i h i l a t i o n s a r e i s o t r o p i c and i f of  12).  samples a c y l i n d r i c a l  rotates  orientation varies  ing r a t e w i t h  single crystal  h i g h e r momentum e f f e c t s  it  i s assumed t h a t  is roughly  to then  from  the net e f f e c t  i s o t r o p i c , then  variations  in t h e c o i n c i d e n c e c o u n t i n g r a t e a r e a d i r e c t measure o f f l u c t u a t i o n s o f t h e Fermi  F.  surface  radius.  Annihilation of Positrons  in E l e c t r o n  When p o s i t r o n s a n n i h i l a t e  Gases  in a dense gas o f i n t e r a c t i n g  t h e momentum d i s t r i b u t i o n o f t h e p h o t o n p a i r s  should d i f f e r  electrons  from the s i m p l e  F e r m i - D i r a c momentum d i s t r i b u t i o n c h a r a c t e r i s t i c o f n o n - i n t e r a c t i n g and shown a s c u r v e a o f F i g u r e electron-electron Kahana, of  these  pairs  Several  attempts  from the e f f e c t  (Daniel  of  a n d Vosko,1960;  h a v e been made t o c a l c u l a t e  the e f f e c t  1965)-  Effect of Electron-electron  If  arises  c o r r e l a t i o n s on t h e momentum d i s t r i b u t i o n o f t h e gamma-ray  (Hatano e t a l . ,  1.  This deviation  and e l e c t r o n - p o s i t r o n c o r r e l a t i o n s  I960; 1963). various  13-  electrons  Interactions  t h e e l e c t r o n - p o s i t r o n c o r r e l a t i o n s a r e i g n o r e d and o n l y t h e  electron-electron t r i b u t i o n of  correlations  (Hatano et  al.,  t h e momentum d i s t r i b u t i o n o f f o r an e l e c t r o n i c d e n s i t y shown  in curve  the p o s i t r o n acts h a v e been  Daniel  1965)-  such a system o f  Figure  ideal  It  13.  of  Electron-Positron  positron-electron pair  In t h i s  is  calculation  exception of  and V o s k o h a v e  in a sea o f  the  inter-  calculated  interacting electrons, in sodium they  t o be n o t e d t h a t  a Bethe-Goldstone  interacting electrons,  two-body c o r r e l a t i o n s  h a v e been a c c u r a t e l y  force,  and  obtain  the  in t h i s  case  correlations  equation for Kahana  between  accounted the  the n e g l e c t  of  d e n s i t i e s and n o t  electron densities  metals.  Effect of  Hatano et  found  in real  Electron-electron  (i960;1963)  c of  screening of  for  and E l e c t r o n - p o s i t r o n  Hatano e t  relatively  i n c l u d e both  For  the  relative  Their  result  i s shown  momentum d i s t r i b u t i o n o f  al.  be low  Interactions  al.  to a p p r o x i m a t e l y  the  attractive  c o r r e l a t i o n s may the  13.  the  with  the  Bij1-Dingle-Jastrow  h a v e been a b l e  Figure  However,  By s t a r t i n g w i t h a wave f u n c t i o n o f  and e l e c t r o n - p o s i t r o n c o r r e l a t i o n s . 13.  for.  these other  j u s t i f i e d only for high electron  a  t h e p o s i t r o n and  no o t h e r c o r r e l a t i o n s a r e c o n s i d e r e d .  (1965) h a v e s u g g e s t e d t h a t  3•  the  Interactions  the c o r r e l a t i o n s a s s o c i a t e d w i t h  electron-positron  obtain  of  t h e p h o t o n p a i r momentum d i s t r i b u t i o n g i v e n by c u r v e  annihilating electron  Figure  to that  probe since e l e c t r o n - p o s i t r o n  By an a p p r o x i m a t e s o l u t i o n o f  obtained  identical  corresponding to that  b of  a s an  be  dis-  ignored.  Effect  2.  i n t o c o n s i d e r a t i o n , t h e momentum  t h e gamma-ray p a i r s w i l l  acting electrons  results  taken  type,  electron-electron in curve  d of  the photon p a i r s  they  42  w(p) w(0) in w h i c h  =  1 + 0.13 (p/p. )  the c o e f f i c i e n t s a r e  be c o m p a r e d w i t h K a h a n a ' s to  -  2  )  +  k  ...  independent of e l e c t r o n d e n s i t y .  result(for  an e l e c t r o n  =  1  +  0.262(p/p, )  This  is  density corresponding  to roughly  stantial  t h e two  4.  between  Comparison w i t h  theoretical  curves  experimental al.  for  t o be  sodium than are  (1961)  with curve  However,  Figure  those of  13).  sodium appears shown  Donaghy e t  c u s s e d below)  al.  is  on t h e b a s i s o f a n g u l a r make a d e f i n i t e  this  is  is a sub-  to  t o assume  that  give  some i n s i g h t  the  into  the  r a t h e r d i f f i c u l t t o do b e c a u s e  agreement w i t h  results It  is  rather  Stewart's experimental  Daniel  satisfactorily  On t h e o t h e r  13.  13 and t h u s  hand, that  and V o s k o ,  by t h e  are  to note however,  for core a n n i h i l a t i o n  d i f f i c u l t to estimate  the v a r i o u s  it  models.  Hatano  results  s i m p l e S o m m e r f e l d model  i n good a c c o r d w i t h  c o r r e l a t i o n work a l o n e  c h o i c e between  Stewart's  of  results  r e c e n t w o r k by Donaghy e t  important  involve a correction  which  there  i n s o d i u m c a n be c o n s i d e r e d  i t m i g h t be r e a s o n a b l e  Kahana o r  as  to give  in F i g u r e  is seen t h a t  F o r e x a m p l e , a l t h o u g h t h e c o m p u t a t i o n s by  in b e t t e r  are e x p l a i n e d j u s t  (curve a,  gas  in F i g u r e  uncertainties.  appear  It  s o d i u m c o u l d be u s e d t o c h o o s e t h e " b e s t " o f  shown  annihilation process.  k  Experiment  r o u g h l y a p p r o x i m a t e an e l e c t r o n studies of  )  f  results.  Since the c o n d u c t i o n e l e c t r o n s  experimental  0.233(p/p f  F e r m i momentum.  difference  +  2  f  w h e r e p^. = fik^. i s t h e u s u a l  of  f f  sodium)  w(p) w(0)  et  0.02(p/p  f  that (to  (Carbotte,  a 1.(1965)  Kahana's the be  results dis-  1966).  is d i f f i c u l t at  Thus  present  to  i.5o  p/p  F i g u r e 13s Momentum D i s t r i b u t i o n o f A n n i h i l a t i o n ' Photons Emanating From an I n t e r a c t i n g E l e c t r o n Gas ( A f t e r Hatano e t a l . , 1965)  o ©  Kanazawa e t a l . B e l l and Jorgenson Kahana <a  -p  a  h  i.  6  F i g u r e ll(.: P o s i t r o n A n n i h i l a t i o n Rates ( A f t e r Kanazawa e t a l . , 1965)  G.  A n n i h i l a t i o n of  Positrons  Metals  I n t r o d u c t ion  1.  A quantitative the a n g u l a r difficult is  in Real  correlation  (Wallace,  the presence  of  Carbotte,  electrons.  Both o f  relation  values  radiation  1966).  the  An e a r l y  greater  made by De B e n e d e t t i  et  yield  from m e t a l s  One o f  Coulomb r e p u l s i o n ,  ume v  (the  the  lattice is  Crystal  (1950).  exceedingly  sources of  difficulty  i n b o t h t h e p o s i t r o n and of  the  crystal  elec-  lattice. core  a c o n t r i b u t i o n to the a n g u l a r  Lattice  cor-  from a n n i h i l a -  Potential  the e f f e c t  of  the c r y s t a l  lattice  They c o n s i d e r e d a s i m p l e model  t h e p o s i t r o n wave f u n c t i o n  " e x c l u d e d volume")  upon  alone.  to consider  al.  due t o  crystal  than those which would a r i s e  Periodic  attempt  the  from the a n n i h i l a t i o n of p o s i t r o n s w i t h  these e f f e c t s of &  Effect of  of  due t o t h e p e r i o d i c p o t e n t i a l  tion with free conduction electrons  2.  the e f f e c t  h i g h e r momentum c o m p o n e n t s  complication arises  at  of  of a n n i h i l a t i o n  I960;  t r o n wave f u n c t i o n s Another  treatment  around the nucdeus.  was  in w h i c h ,  is e x c l u d e d from a Outside  this  vol-  excluded  e volume  t h e p o s i t r o n wave f u n c t i o n  wave f u n c t i o n positron duces  to  tive  regions  v /v which  t o be a c o n s t a n t  of  is the  The  s u r r o u n d i n g the " e x c l u d e d volume"  f o r example,  intensity  taken  is c o n s i d e r e d a s i m p l e p l a n e wave.  h i g h e r momentum c o m p o n e n t s  expected,  is  these  from the  into  electron  r e s t r i c t i o n of regions  the  thereby  intro-  t h e p o s i t r o n wave f u n c t i o n as w o u l d  Heisenberg u n c e r t a i n t y  h i g h e r momentum c o m p o n e n t s  r a t i o of  and t h e  t h e e x c l u d e d volume  is  to the  principle. related  The  to the  be  rela-  quantity  volume o f a u n i t  cell.  e Typical 1956;  values  of  v /v g  range  L a n g and De B e n e d e t t i ,  f r o m 0.05 1957).  i n b e r y l l i u m t o 0.29  The r e s u l t s  however  in barium  generally  (Lang,  disagree  45  with experiment probably  (Lang,  1 9 5 6 ; B e r k o and P l a s k e t t ,  1958),  3.  interaction  E f f e c t of  Core  (Wallace,  and P l a s k e t t  (1958)  from the  elec-  e l e c t r o n wave f u n c t i o n s  Berko  I960).  Annihilations  By u s e o f u n c o r r e l a t e d  one-particle  have c a l c u l a t e d  the a n g u l a r  d i s t r i b u t i o n of photons  s u l t i n g from the a n n i h i l a t i o n o f p o s i t r o n s w i t h  core e l e c t r o n s .  l a t i o n s w e r e done f o r c o p p e r and a l u m i n u m and a r e with experiment.  However,  to the shape o f  assess  the v a l i d i t y  (1966)  indicate  importance  al.  H.  that  the method.  Also,  it  The  good  is q u i t e  by R o c k m o r e and  by  (1965)  to  Carbotte considerable  Berko-Plaskett  Stewart  sensi-  is d i f f i c u l t  calculations  The  calcu-  agreement  i n t e r a c t i o n may be o f  calculations.  re-  method  and by  has  Terrell  (1965).  If (3-1),  of  Positrons  typical  in  values  lifetimes  agreement  with  addition,  the observed  of  Metals  of  the e l e c t r o n  the o r d e r of  the e x p e r i m e n t a l l y lifetimes  are  (Wallace,  i960)  electron  density.  Thus a n n i h i l a t i o n  annihilation Neither  in which  can the  triplet  whereas  the e f f e c t s  lifetimes  be  predicts in metals the  interpreted  of metals  obtained.  lifetimes  essentially  (3-1)  of  density  10"* s e c a r e  observed  density,  since  so t h a t  recent  the e l e c t r o n - p o s i t r o n  to other metals  Lifetimes  into  t h e p o s i t r o n wave f u n c t i o n of  in f a i r l y  t h e c o m p u t e d momentum d i s t r i b u t i o n  in core a n n i h i l a t i o n  been a p p l i e d et  disagreement  b e i n g due t o c o r e a n n i h i l a t i o n and t o c o n t r i b u t i o n s  tron positron  tive  the  of  ~2  This  cannot  substituted is  x lo'^  independent a direct  are  of  in  sec.  dependence  c a n be  in terms o f p o s i t r o n i u m  p o s i t r o n i u m w o u l d have a l o n g l i f e t i m e o f a b o u t  In  electron on  be r e g a r d e d a s  Coulomb f o r c e s  dis-  the  free  neglected. formation  10^ s e c  which  is not observed. would  lead  tronium  On t h e o t h e r h a n d , r a p i d  t o a l i f e t i m e o f ^5  (Wallace,  observed  I960).  lifetimes of  rate  dicate the  that  positron  rate  positron  is also  and Kanazawa  lifetime  et a l .  shown  work o f Stewart  in Figure  \k.  time c a l c u l a t i o n s botte  (1966)  process.  (1965).  is very s e n s i t i v e  Kahana  curve that  (1961).  f o r electrons Inclusion  is also  The  predicts  (1963),  to correlation  effects, that  the  r e c e n t w o r k o f Kahana(l963),  positron (Figure  between t h e  l i f e t i m e s that 14).  H i s work  are also  i s i n rough agreement w i t h t h e ex-  Kanazawa  et a l .  (1965) h a v e  o f z e r o momentum, t h e i r  of effects  of the c r y s t a l  some i n s i g h t  into the core  calculated  results  lattice  very d i f f i c u l t although preliminary  has a l r e a d y p r o v i d e d  than the  These s t u d i e s i n -  i n t o a c c o u n t t h e two-body c o r r e l a t i o n s  g i v e s an a n g u l a r c o r r e l a t i o n  rate  larger  posi-  on t h e a n n i h i l a t i o n  (1956),  by F e r r e l 1  a r e a t t h e same p l a c e .  takes  annihilation  of singlet  considerably  correlations  in o r d e r o f magnitude agreement w i t h experiment  the  times that  conversion  sec.  and t h e a n n i h i l a t i n g e l e c t r o n ,  perimental  to singlet  b e i n g d i r e c t l y d e p e n d e n t on t h e p r o b a b i l i t y  and e l e c t r o n  which a c c u r a t e l y  this  investigated  (1965),  the p o s i t r o n  annihilation  x lo'"  sec, four  of electron-positron  i n m e t a l s h a v e been  C a r b o t t e and Kahana  However,  ~2  The e f f e c t s  x 10^  triplet  being  into  w o r k by  lifeCar-  annihilation  CHAPTER  EXPERIMENTAL  The s c h e m a t i c d i a g r a m ment f o r order  the " c o l l i n e a r p o i n t  t o be d e t e c t e d ,  c o p p e r had t o p a s s mators If  within  time  a d i s t a n c e o f about  was m e a s u r e d .  the  10  T h i s made  the p o s i t r o n a c t i v i t y  it  (T^  12 f e e t  a c o i n c i d e n c e was  resolution of  required for  the a p p a r a t u s ,  of  the  the  o n l y photon p a i r s  for  the  in C h a p t e r  III.  crystal.  scaler. of  crystal counts  the decay  and t h u s o b t a i n a m e a s u r e o f  2  colli-  coincidence  r e c o r d i n g a p r e d e t e r m i n e d number o f  = 12.9 hr)  lead  r e c o r d e d by t h e  correct  In  of  from the copper  For each o r i e n t a t i o n o f  p o s s i b l e to a c c u r a t e l y  s p a c e a n i s o t r o p y as d i s c u s s e d  single crystal  by t h e two c o u n t e r s w i t h i n  usee,  arrange-  in the p r e s e n t work.  inch diameter aperture  momentum w e r e c o u n t e d .  interval  15 shows t h e e x p e r i m e n t a l  g e o m e t r y " method used  through the 0.25  r e s o l v i n g time of  zero transverse the  in F i g u r e  a gamma-ray p a i r was d e t e c t e d  In t h i s w a y ,  ARRANGEMENT  photons from the p o s i t r o n - a c t i v e  t h a t were p l a c e d at  circuit  IV  of  momentum  PREAMP AND SHAPER  tfaltTl)  . DETECTOR ."SYSTEM  LEAD COLLIMATOR  [113 t  Cu SINGLE CRYSTAL  AXIS OP ROTATION  COINCIDENCE UNIT  SCALER  '(in)  LEAD COLLIMATOR  K a l ( T l ) DETECTOR SYSTEM FREAMP AND SHAPER F i g u r e l£:  Experimental  Arrangement•(Schematic)  A.  Metal  Crystal  and P o s i t r o n  The c o p p e r c r y s t a l in d i a m e t e r  by 3.84  was c h o s e n a s  mm h i g h ,  the metal  (Chapter  ll)  Copper  2.  in t h i s  with  £l 1 l j  the  is s t a b l e at o r d i n a r y in the  crystal  The p r e s e n t  o r sample  Also,  is well  known  to a s c e r t a i n  and  the  used  neutron  readily  became an a d e q u a t e  all  A conventional  rendered  Another  copper  source.  since  provided a  it  whereas  the  nearly  conventional  ( B e r k o and  Plaskett,  gamma r a d i a t i o n  a n n i h i l a t i o n r a d i a t i o n thus  radiations.  that  positron  t o some p o s i t r o n s o u r c e s ,  is n e a r l y  so  i r r a d i a t i o n the  p o s i t r o n s throughout the c r y s t a l  from o t h e r  is  i m p u r i t i e s need n o t be c o n s i d e r e d .  " s h i n e " p o s i t r o n s on t h e sampl>e s u r f a c e  n a t i n g from the c r y s t a l  o f economy.  temperatures  a p p r o a c h was a d v a n t a g e o u s  in c o n t r a s t  noninterference  easy  reasons:  p o s i t r o n s o u r c e was  s i n c e by t h e r m a l  itself  uniform d i s t r i b u t i o n of source geometries  relatively  several  form o f very pure s i n g l e c r y s t a l s  The u s e o f a c o n v e n t i o n a l unnecessary  that  it  copper  Copper  t h e new p o s i t r o n a n n i h i l a t i o n t e c h n i q u e  sample environment  1958).  surface of  its axis.  mm  experiment.  obtained  3.  e x p e r i m e n t was c y l i n d r i c a l , 3.56 d i r e c t i o n along  Fermi  thus making  usefulness of in t h i s  used  t o be e x a m i n e d by t h i s m e t h o d f o r  The t o p o l o g y o f  1.  Source  ema-  ensuring  important c o n s i d e r a t i o n  l o n g - l i v e d source of  comparable  is  strength  w o u l d h a v e been p r o h i b i t i v e l y e x p e n s i v e .  The m a i n d i s a d v a n t a g e convenience  associated with  coincidence counting rates  the  of  this  type of p o s i t r o n source  short h a l f - l i f e .  Due t o t h e  is the  in-  relatively  r e s u l t i n g from use o f " p o i n t " geometry  it  was  low  necessary  to have a  relatively  A l t h o u g h the c o i n c i d e n c e ( "1  count per  second)  counting  this  several tion,  the  it  the c r y s t a l  ideal  crystal  crystal  escape.  Taking these  seen  dimensions given et  al.  will B.  (1957)  escape  above  and E v a n s  from the  Orientation  of  it  (1955),  fjnil  relative  then p l a c e d brought fined  by t h e  consideration  On t h e o t h e r  irradiation with  long-  less  the  to h e i g h t small  than  its  results  percent  of  in  it  order  thermal to  choose crystal  chosen  extent,  as p o s s i b l e .  by u s e o f  resolu-  in the  s a m p l e was  so t h a t  five  of  hand,  the  i t was d e s i r a b l e  into account,  to as  With  the  from  Price  the  positrons  Crystal  used  in the p r e s e n t  d i r e c t i o n along the o t h e r  to a f i d u c i a l  its axis.  fill}  {"ill}  w o r k was a c y l i n d e r  By means o f x - r a y  directions of  l i n e on t h e c r y s t a l  in a " n o t c h and p i n " arrangement  successive  The u s e o f a  the p o s i t r o n s produced  to e s t i m a t e ,  that  to  sample.  (Laue b a c k - r e f l e c t i o n ) termined  upon  w o u l d be a s  is easy  it d i f f i c u l t  t h a n a c e r t a i n minimum s i z e  roughly equal  The c o p p e r c r y s t a l sessing a  few o f  detectors,  adequate  view of angular  small.  In a d d i t i o n ,  considerations  from the " p o i n t "  t i m e , making  From t h e p o i n t o f  available.  be a c y l i n d e r w i t h d i a m e t e r  mCJ)'  difficulty.  larger  enough so t h a t  (»»50  b e g i n n i n g o f a r u n was  good s t a t i s t i c s .  positron activity  currently  large  the  was d e t e r m i n e d a f t e r  t o have the c r y s t a l  fluxes  positron activity  s h o u l d be v a n i s h i n g l y  have s u f f i c i e n t  neutron the  this  c o n f l i c t i n g requirements.  is necessary that  rate at  points with  s o u r c e would have o b v i a t e d  The s i z e o f  initial  rate f a l l s with  e s t a b l i s h many e x p e r i m e n t a l lived  high  directions  two " p o i n t " d e t e c t o r s .  of  the  T h u s , as  diffraction  the c r y s t a l  base.  pos-  were d e -  The c r y s t a l  (Figure  16)  crystal  into the d i r e c t i o n  c a n be s e e n  which,  upon  was  from the  rotation, de-  discussion  ALUMINUM SUPPORTING PLATE  F i g u r e 16: Notch and P i n Assembly  52  in Chapters  II  in the { i l l } C.  Spatial  and  III,  the necks o f  directions) Stability  the  Fermi  c a n be e x a m i n e d by t h i s  o f N o t c h and P i n  ment.  c a r e was  Stability  and c a r e f u l l y  taken  to maximize the  measurements  spatial  that motion of  t h e c e n t r e o f mass o f  less  than 0.10  inches a l o n g the d i r e c t i o n  less  than 0.004  will  results  tronic  D.  setting  indicated  i t s mean p o s i t i o n was of  the detectors  mechanical  and  instability  in Chapter  engendered  V,  this  As  uncertainty  by c o u n t i n g s t a t i s t i c s o r  is elec-  Holder  In t h e adhesive  first  to a small  run t h e c o p p e r c r y s t a l  cylindrical  a l u m i n u m was u s e d f o r  absorption  cross  s e c t i o n and  a l u m i n u m has a v e r y crystal  residual  dial  drift.  Crystal  purity  from  con-  arrange-  in the c o i n c i d e n c e c o u n t - r a t e .  presented  compared t o the u n c e r t a i n t y  the  the v e r n i e r  (vertical)  This  of<~0.2 percent  small  s t a b i l i t y of  The m e a s u r e m e n t s  the c r y s t a l  c a u s e s an u n c e r t a i n t y be s e e n f r o m t h e  t h e n o t c h and p i n a s s e m b l y  position.  inches h o r i z o n t a l l y .  occur  arrangement.  w e r e made by v a r y i n g  o b s e r v i n g the c r y s t a l  (which f o r copper  Assembly  In t h e d e s i g n and c o n s t r u c t i o n o f siderable  surface  h o l d e r made o f  s h o r t h a l f-1 i f e  p i n on t h e h o l d e r  (Livingstone,  damage s o t h a t upon a r r i v a l  high p u r i t y aluminum.  in a d d i t i o n , the a c t i v i t y (2.30 m i n . ) .  1963)  induced  in  High  irradiated  The c y l i n d e r composed o f which  16).  f i r s t experimental  r u n , but  the epoxy s u f f e r e d c o n s i d e r a b l e  the c r y s t a l  epoxy  neutron  i n t o a tube c o n t a i n i n g a notch  (Figure  T h i s m e t h o d was u s e d f o r t h e to e x p e c t a t i o n  by means o f  t h e h o l d e r s i n c e a l u m i n u m has a s m a l l  and h o l d e r was d e s i g n e d t o f i t  engaged a s m a l l  was a t t a c h e d  was  found separated  contrary radiation  from the  holder.  53  T h i s made use o f  it  necessary  to  remount t h e c o p p e r c r y s t a l  second  run t h e c r y s t a l  was  force  i n t o t h e end o f a t h i n - w a l l e d a l u m i n u m h o l d e r a s T h i s h o l d e r had a b o u t t h e as d i d t h e f i r s t  reduce s c a t t e r i n g o f there  is  (0.033  E.  same g r o s s d i m e n s i o n s  except  that  inches)  16.  The c r y s t a l  to a v e r n i e r  dial  plate.  heavy p l a t e  This  cantilevered  17.  ( 4 . 7 2 mm d i a m e t e r by 3 9 - 4 mm region to  With t h i s  t h e gamma r a d i a t i o n by t h e  at  holder f i t s  further  arrangement  thin  covering  (Armaco  DV4)  that  holder support  i s shown  i t s other end.  rods w h i c h ,  these  residual  mechanical  i n s t a b i l i t y of  t u b e was  attached  b o l t e d to a heavy aluminum was  for c l a r i t y ,  h a v e been o m i t t e d f r o m  s t a b i l i z i n g e l e m e n t s made the c r y s t a l  it p o s s i b l e to  the  reduce  the  t o t h e n e g l i g i b l e amount m e n -  s t a b i l i t y d i s c u s s i o n above.  To t h e end o f  the  vernier  w h i c h c o u l d accommodate a s m a l l attached  vernier  radioactive  was s e c u r e l y  This  The s u p p o r t i n g t u b e was s t a b i l i z e d by means  The u s e o f  wheel  holder.  in  contains  i n t u r n was b o l t e d t o an a l u m i n u m beam t h a t  figure.  in the  the  i n t o the s u p p o r t i n g tube which  t h e p i n on t h e c r y s t a l  o f a s y s t e m o f d i s c s and  the  in F i g u r e  Support  engages  trol  irradiation)  o f aluminum.  a notch that  pulley  (before  indicated  radiation.  A s i m p l i f i e d schematic diagram of Figure  fitted  i t had a h o l l o w c e n t r a l  the a n n i h i l a t i o n  l i t t l e attenuation of  Holder  tioned  by  f r e s h e p o x y b e f o r e a r u n c o u l d be made. In t h e  long)  on a new c y l i n d e r  dial  to a long,  s h a f t was a t t a c h e d  diameter  cord.  s u p p o r t e d rod  a simple pulley  By use o f  t h e c o r d and  (Figure  18)  another  i t was p o s s i b l e t o e a s i l y  s e t t i n g from a d i s t a n c e o f about ten f e e t  copper c r y s t a l .  wheel  The v e r n i e r  dial  was  from  con-  the  conveniently  5k  COPPER CRYSTAL  \  \  \  \  \  ^ A L U M I N U M HOUSING  N  N  \^,~-LOCATING PIN  F i g u r e ; 1 7 : C r y s t a l ~ aridir-Holder  Cu CRYSTAL AND HOLDER \  ..SUPPORTING TUBE11  PULLEY ~^+y-"- _ 7  _  >  VERNIER DIAL z_ PULLEY  CORD  -ANGULAR CONTROL ROD  TELESCOPE  F i g u r e 18: Remote C o n t r o l  55 read from t h i s this  arrangement  (less  F.  distance  than  Gamma  the  by means o f  radiation  5 mrem p e r 8  hours).  Counters  m o u n t e d on RCA inch  6342  in d i a m e t e r  chosen because a small  consisted of  p h o t o m u l t i p l i e r tubes. by two  inches  i t was e c o n o m i c a l  transit  incidence  time spread  resolution time.  long.  These c r y s t a l s were  6342  The  and r e a d i l y  ( ^ 4 nsec),  two Harshaw N A l ( T l )  thus  On t h e o t h e r  available.  measured e f f i c i e n c y o f t h i s would p r o b a b l y inches  in  the  individual  require a crystal  cylinders  634-2  The  f a c i l i t a t i n g the  hand,  the d e t e c t o r  short  at  least  four  a s s e m b l i e s were e n c l o s e d  means o f a " C o - n e t i c N e t i c " In a d d i t i o n e a c h  lead f o i l  for  bolted to a large 0.25  inches  was t w e l v e  alloy  shield,  inches  the  double  in d i a m e t e r  in l i g h t - t i g h t  (manufactured  a l u m i n u m h o u s i n g was w r a p p e d w i t h  radiation s h i e l d i n g purposes.  (6x6x3  in d i a m e t e r  and  cylindrical  in) and 3  s h i e l d e d by  by P e r f e c t i o n about  lead block c o n t a i n i n g a c y l i n d r i c a l inches  long.  inches  0.2  E a c h d e t e c t o r was  Mica  rigidly aperture  The s o u r c e t o d e t e c t o r  distance  feet.  The c h o i c e o f  source to detector  s i z e was b a s e d on s e v e r a l cussed .  were  For example,  d e t e c t o r s was a b o u t 0 . 4 5 , t o  has  co-  crystals  aluminum h o u s i n g s , each p h o t o m u l t i p l i e r tube b e i n g m a g n e t i c a l l y  of  also  was  height.  The d e t e c t o r  Co.)  crystals  p h o t o m u l t i p l i e r tube  c h o s e n as a c o m p r o m i s e b e t w e e n economy and e f f i c i e n c y .  four  With  h a z a r d t o t h e o p e r a t o r was f o u n d t o be s m a l l  The " p o i n t " d e t e c t o r s  one  a t e l e s c o p e m o u n t e d on a t r i p o d .  considerations.  distance  and c o l l i m a t o r  These w i l l  aperture  how be b r i e f l y  dis-  56  The Table  most  important  c o n s i d e r a t i o n was t h a t o f r e s o l u t i o n .  I i t c a n be s e e n t h a t t h e d i a m e t e r  2.r %]0  mc.  t  r a t e change at crystal (Chapter  Hence,  (about  in order  half  V) w i t h  full  to observe a reasonable  that expected  will  width  a t h a l f maximum o f ^]/2^  counting  and p o i n t  detectors)  d i s t a n c e between s u c c e s s i v e arrangement.  Since  check  incidence rate a l s o very  small.  The  i s amply s a t i s f i e d  (Chapter  For D  the c o l l i m a t o r aperture I  /2  o  u  r e c  '.  This  i t i s seen t h a t t h e c o s m i c r a y be n e g l i g i b l e .  V) showed t h a t t h e t o t a l  diameter  then  gives  (Chapter  Indeed,  background co-  the r a d i o a c t i v e copper c r y s t a l  12 f e e t a c h o i c e o f d  absent)  = 0.25 inches f o r V) a r e s o l u t i o n f u n c t i o n  as r e q u i r e d .  v  w  a  s  compromise between c o i n c i d e n c e  a  0  t o t h e Fermi  the r e s o l u t i o n function width  surface topology.  For example,  t o W, ..~ 2 r  f r o m W. ,„~ kr 1/2  the c o i n c i d e n c e  l '  s i n c e t h i s was t h e v e r t i c a l  count r a t e should  =  c h o i c e o f ^-\/2^  r a t e and s e n s i t i v i t y  r e c  12 f e e t f o r t h e s o u r c e t o  =  convenient  ( i . e . the rate with  h a l f w i d t h o f W, ,?kr  S  f l o o r s o f t h e tower w h i c h housed t h e e x p e r i m e n t a l  to the coincidence  an e x p e r i m e n t a l  0  t h e c o n d i t i o n 2D>c^'> where f i s t h e c o i n c i d e n c e  r e s o l v i n g time,  contribution  '  r  below.  work, a c h o i c e o f D  d e t e c t o r d i s t a n c e was p a r t i c u l a r l y  of  coincidence  f o r a point source  be f u r t h e r d i s c u s s e d  In t h e p r e s e n t  was  surface "necks" i s  o r i e n t a t i o n s associated with the necks, a r e s o l u t i o n f u n c t i o n  choice o f  circuit  o f t h e Fermi  From  o  1/2  would  count  reduction reduce  o  c o u n t r a t e by a f a c t o r ^ 1 6 ( f r o m a b o u t 60 c o u n t s p e r m i n u t e  (maximum) t o a b o u t k c o u n t s p e r m i n u t e with the s p e c i f i c  (maximum).)  a c t i v i t y o f the source  These problems a r e f u r t h e r d i s c u s s e d  Problems a s s o c i a t e d  must a l s o be t a k e n  i n C h a p t e r V.  into consideration.  57  G.  E l e c t r o n i cs The s c h e m a t i c  used  in the e x p e r i m e n t .  gamma-ray  g i v i n g as o u t p u t  p u l s e s were fed  the  After  a positive  coincidence  are  a Schmitt-type,  of  i n t o the the  coincidence  coincidence  the present about  1 ma t h r o u g h  that  pulses  from the p u l s e  Stability  of  the  time set  Corp.  (this  Such  at  nsec.  ~10  scaler.  firm  is  no  and s h a p i n g c i r c u i t  19 and 20  signal,  when o f  shown  current  pulse  line,  sufficient  control tunnel (-  at  was  diode.  This  10 v o l t s ) arrived  varies  adjusted  In t h i s  20. the  input  amplitude, The  output  was  which  circuit  adjust-  threshold  to give a standing current  from  giving  a 25 n s e c o u t p u t p u l s e  in F i g u r e  control  and  respectively.  c l i p p e d by a s h o r t e d d e l a y  shaped t o g i v e  circuit  shapers  10 n s e c )  small  level.  current  enough  to  o c c u r r e d o n l y when two 25 n s e c  simultaneously  the c o i n c i d e n c e  circuit  (ie.  within  the  re-  input.  Electronics  Since the small  in F i g u r e s  This  then  an o u t p u t p u l s e  s o l v i n g time of  fairly  25 n s e c w i d e .  resolving  the p r e a m p l i f i e r  sensitivity  work t h i s  ensure  H.  discriminated,  zero-crossover discriminator c i r c u i t .  f r o m t h e d i s c r i m i n a t o r was  For  is a m p l i f i e d ,  and p r e a m p 1 i f i c a t i o n , t h e n e g a t i v e  a bipolar zero-crossing signal.  ment o f  from a  r e c o r d e d by a E 11 OA d e c a d e  shown  t h e p h o t o m u l t i p l i e r c o l l e c t o r was  went  electronics  pulse a r i s i n g  p u l s e about  by O x f o r d E n g i n e e r i n g  diagrams of  circuit  differentiation  triggered  crystals  the  existence). The c i r c u i t  of  the N a I ( T l )  were then  The E 110A was m a n u f a c t u r e d in  15 i n d i c a t e s  into a coincidence unit with  resulting coincidences  longer  Figure  A photomultiplier current  i m p i n g i n g on one o f  and s h a p e d ,  The  diagram of  count-rate  (about s i x percent)  variation it  is  expected  important  from the p r e s e n t that  drifts  in the  method  is  coincidence  F i g u r e 1 9 * preamp and Shaper C i r c u i t u\  TEST INPUT  1N752  +10  SHORTED STUB (T>IL FEET OF 100JICOAXIAL CABLE)  V  10K COINCIDENCE SENSITIVITY  3.3K  3.3K  2N1195  2.7K  0.1 fit  OUTPUT 3 o  •H  o (D O  c  ©  •H o C  INPUTS  •rt!  o1  o  (5  -AAAAy—-  68A  2N797  o CM | ©  TD-3  L V _  10 TURNS ON FOT . ill CORE  60  counting large  drifts  caution the  r a t e be m i n i m i z e d . in the  tower  which housed the area  course of  ambient  the  the experimental  drifts  in the  count  apparatus.  set-up  curves  d i d not o c c u r .  One s u c h  pre-  temperature  done by o p e n i n g  of doors  from the main  building  temperature  was m o n i t o r e d  throughout  t h e maximum f l u c t u a t i o n s  V)  that  air  t 0.8°C.  (Chapter  to ensure  in t h e ambient  T h i s was e a s i l y  The a m b i e n t  being about  were taken  r a t e d i d not o c c u r .  fluctuations  two e x p e r i m e n t s ,  temperature  precautions  so f r e e c i r c u l a t i o n o f  t h e t o w e r was p o s s i b l e .  the  of  coincidence  i n v o l v e d m i n i m i z i n g the  in the experimental to  Several  During both  were r e p e a t e d  To w i t h i n  the  runs,  from the certain  to ensure  statistical  that  mean points large  uncertainty,  22 t h e p o i n t s w e r e f o u n d t o be r e p r o d u c i b l e . that  the  coincidence  over  a 2k h o u r  c o u n t i n g r a t e was  period.  Tests  stable  with  a Na  to b e t t e r  than  source  t 0.6  indicated percent  61  CHAPTER' V  RESULTS AND CONCLUSIONS  A.  I n t r o d u c t i on  The e x p e r i m e n t a l IV.  The c r y s t a l  [ill]  a x i s was  s u p p o r t was the a x i s o f  t i o n s would s u c c e s s i v e l y 111j  d i r e c t i o n s of  standard x-ray  the  the  rate to  Fermi  i n c l i n e d at  at  the  the  expect peaks  c o i n c i d e n c e count  experimental  the d e t e c t o r s .  direcThese  x-ray  the  by  facilities.  coincidence  c o r r e s p o n d i n g to the " n e c k s " in the  r a t e at  Fermi  the  been d e t e r m i n e d  Metallurgy  directions.  {ill}  t h r e e - f o l d symmetry about a  d a t a on t h e  Chapter  upon r o t a t i o n , { i n }  one would e x p e c t  orientations  15 o f  as s h o w n , s o t h a t  had p r e v i o u s l y  III,  as a c o n s e q u e n c e o f  available  an a n g l e ,  Department o f  in Chapter  crystal  in F i g u r e  in the d i r e c t i o n of  s u r f a c e , which f o r copper occur  in the  shown  r o t a t i o n and s o t h a t  point  techniques  r i s e at  i s as  copper c r y s t a l  As p o i n t e d o u t count  arrangement  {'llj  intervals  surface of  of  copper  of  Also,  a x i s one w o u l d 120°.  From  (Table  I)  the  one  would expect  for a point crystal  13 p e r c e n t on t h e b a s i s o f does not B.  the assumptions of  i n c l u d e h i g h e r momentum e f f e c t s  settings of  two  Chapter  IV.  of  This  about estimate  s u c h as c o r e a n n i h i l a t i o n .  runs d e s c r i b e d b e l o w ,  than  0.3  Mev o f e n e r g y  nsec output pulse  a b o u t 0 . 3 Mev so t h a t i n one o f  positron activity,  o r a b o u t 50 h o u r s .  the  (12.9 h o u r s ) .  (^0.1  dence  rate,  percent  true  the  rate since  This  rate  absent)  the  of  the c r y s t a l  and d e t e c t o r s The e r r o r  This  misalignment affects  percent.  run was  the  true  about  four  random  coincidence  coincidence  the h a l f - l i f e o f rate  runs where  it  50 mCi  half-lives  rate).  No  the s o u r c e  (i.e.  the  rate  was m e a s u r e d and f o u n d t o be  faces  true  was with  about coinci-  c o n t r i b u t e d about  0.4  the copper c r y s t a l  and  was v e r t i c a l  c o u l d be a c h i e v e d  arrangement. residual  t o a 25  rate.  the " p o i n t " d e t e c t o r  in t h i s  correspond  circuit.  The a x i s d e f i n e d by t h e c e n t e r o f mass o f the c e n t e r s  gave r i s e  i s n e g l i g i b l e compared t o the  t h e end o f  coincidence  voltage  run b e i n g a b o u t  of  threshold  gamma-ray e x p e n d i n g more  crystals  The b a c k g r o u n d c o i n c i d e n c e  except near  to the  each  the b e g i n n i n g o f each  percent  this  copper c r y s t a l  two c o u n t s p e r h o u r .  This  F o r 50 mCi o f p o s i t r o n a c t i v i t y  be n e g l i g i b l e  radioactive  shaping  the d u r a t i o n of each  a t t e m p t was made t o m e a s u r e so s h o r t  the N a l ( T l )  from the a s s o c i a t e d  The s o u r c e s t r e n g t h a t  rate w i l l  the d i s c r i m i n a t o r  the p u l s e s h a p e r s were about 0 . 6 v o l t s .  t o a gamma-ray e n e r g y o f  0.1  an e f f e c t  E x p e r iment  For the  of  and p o i n t d e t e c t o r s  so t h a t  alignment  of  by means o f a s i m p l e plumb bob  a l i g n m e n t was e s t i m a t e d the c o i n c i d e n c e count  t o be l e s s r a t e by  than  less  1 mm  than  63  The h i g h p u r i t y o b t a i n e d from Metals neutron Ltd. x-ray  fill]  techniques  neutron  C.  the  at  the  Chalk River the  f a c i l i t i e s of crystal  Department o f M e t a l l u r g y  of  Canada  standard  facilities  before  the  Results  results  second  are  t h e c o l l i n e a r p o i n t g e o m e t r y m e t h o d two  shown  in F i g u r e 2 1 .  The  results  giving first  in o r d e r  to  reduce  run a s m a l l e r a n g u l a r  statistics.  for  results of the second  1.25 p e r c e n t run.  differed  In  the f i r s t  In t h i s  from that  the e r r o r  results  the  s e c o n d run a r e  run m o s t p o i n t s  relative  first  the e x p e c t e d  corrected  120°  the  If  All  for  periodicity.  f o r decay o f  coincidences.  curve  the s o u r c e  the  as e x p e c t e d .  first  The c u r v e s ^\/2  r e a d i n g s were t a k e n a t  o n e as s u me s a u n i f o r m l y d e n s e  h i g h e r momentum e f f e c t s  c a n be n e g l e c t e d ,  shown a b o v e s h o u l d be a m e a s u r e o f  ~ ^.9 room Fermi  in  the  21.  crystal  It  in the is seen  hours)  holder  counting that  the At  evidence  21 have  and f o r  the  and  about s i x p e r c e n t .  in F i g u r e  In thus  two o v e r  run can be s e e n some shown  necessary  a b o u t 64-00 c o u n t s  The p e a k  of  one,  better  in F i g u r e  o r i e n t a t i o n of  run by a b o u t 6 0 ° .  it  However,  by a f a c t o r o f  c o n s i s t e n t w i t h an e f f e c t  hand s i d e o f  ).  a l s o shown  represent  s e c o n d run t h e  presented are  left  (by ~1 /  improvement  the  r u n , an e x p l o r a t o r y  s t a t i s t i c s , making  s t a t i s t i c s , an  of  r u n s w e r e made.  r a n g e was c o v e r e d , m a k i n g p o s s i b l e  r a t e c u r v e was a l s o s h i f t e d by a b o u t 6 0 ° ,  of  thermal  A t o m i c Energy  x-ray  was  i r r a d i a t i o n s were p e r f o r m e d .  to p a i r points  the  in the p r e s e n t work  w e r e d e t e r m i n e d by  most p o i n t s were t a k e n w i t h o n l y 2.5 p e r c e n t  the  used  L t d . o f C a m b r i d g e , E n g l a n d and was  d i r e c t i o n s of  To t e s t The  Research  i r r a d i a t e d at The  copper s i n g l e c r y s t a l  been  background  temperature. v o l u m e and assumes  then the peaks  the diameter o f  the  in the  Fermi  that  curves  volume  in  the  i?9  65  directions.  {ill}  The w i d t h o f t h e s e p e a k s s h o u l d be a m e a s u r e o f  diameter of the "necks" that occur cussion  below  indicates  consistent with  D.  that  this  in the simple  £l 1 1^  directions.  The  the  dis-  i n t e r p r e t a t i o n o f the r e s u l t s  t h e known d i m e n s i o n s o f t h e c o p p e r  Fermi  is  surface.  I n t e r p r e t a t i o n of the R e s u l t s  1.  E f f e c t s of F i n i t e  With  be s u b s t a n t i a l l y  finite this  crystal  effect,  approximated eight  and  Size  reduced  from the expected  detector size.  by a s p h e r e  indicated  t h e change  in c o i n c i -  t o t h e p r e s e n c e o f the " n e c k s " o f the Fermi s u r f a c e  ("the  s u r f a c e as g i v e n  belly") of  are t r u n c a t e d cones. in F i g u r e 22.  be a p p r o x i m a t e l y a c c o u n t e d  13 p e r c e n t b e c a u s e o f  The  radius p The  finite  Q  in Table upon w h i c h  Fermi  I c a n be  d e t e c t o r and  S u r f a c e Neck  a r e mounted  crystal  or "density"  Details  of  roughly  r e l e v a n t dimensions o f such  f o r by u s e o f a r e s o l u t i o n  F i g u r e 22;  the  In o r d e r t o e s t i m a t e t h e e x p e c t e d s i z e  the e x p e r i m e n t a l Fermi  "necks" which  cone a r e  Detector  the present experimental c o n f i g u r a t i o n  d e n c e c o u n t i n g r a t e due will  S o u r c e and  size  a  can  function.  66  By  t h e use o f t h i s d e n s i t y f u n c t i o n t h e "Mass" o r e f f e c t i v e  appropriate  r e g i o n s o f momentum s p a c e c a n be f o u n d  the expected  coincidence counting  This a cylinder  and an e s t i m a t e made o f  r a t e change.  i s done by p e r f o r m i n g  an e l e m e n t a r y  volume  c o n t a i n i n g a n e c k a t each end and then  that obtained  from a s i m i l a r  t h e r e a r e no n e c k s .  integration  Letting  volume o f  over  comparing the r e s u l t  performed  the effective  integration  in a d i r e c t i o n  volume o f a c y l i n d e r  in which  in a direction  i n w h i c h t h e r e a r e no n e c k s be 2m^ a n d t h a t o f a p a i r o f n e c k s be 2m^, seen t h a t a measure o f t h e r e l a t i v e ratio m m  2  and  i s given  is  by t h e  l 1  =  |r  w I i 2  rate variation  i t  where  m  m. n  count  with  iff  = Jr  o• P ( r ) dz  dr  P(r)  dz  da  and  ^b+Aa+oc  0  de  dt  +  I  2T|f j(r)[h  +(b-fH*n«Q  (5-0 dr  _P(r) i s t h e r e s o l u t i o n f u n c t i o n .  2.  Resolution Function f o r F i n i t e  For t h e case crystal  source  of circular  the resolution  Detectors  and P o i n t  detectors of diameter  Crystal  d and a " p o i n t "  function i s  P(r)  1  -  Dr d  r  <  d D  (5-2) /(r) where D i s t h e s o u r c e was tion  obtained  r > f  = 0  to detector distance.  as a g e n e r a l i z a t i o n o f a n u m e r i c a l  a p p e a r s t o be q u i t e d i f f i c u l t .  This expression  f o r j°(r)  e x a m p l e ; an a n a l y t i c d e r i v a -  For the present  experimental  arrangement  67  o n e has d D  1.70 x 10  =  p  -3  o  b h  .  If  o n e now c h o o s e s  =  -3 5 . 1 0 mc x 10  =  1.00 mc x 10  =  (Chapter  I)  J  iu  p  -  P  0 . 7 0 x 10  =  Q  , mc  (5-3)  k  equations  (5-1)  average b e l l y necks are effect  and  (5-2)  radius  y i e l d m^  (Chapter  I).  It  t a k e n t o be c y l i n d e r s o f  drops to  = 0.12. is  Here p^  i s t a k e n t o be  i n t e r e s t i n g to note that  radius  b instead of  - 0.087, a substantial  truncated  the  if  the  cones,  the  change,  m, 3.  Resolution Since the  whose a x i s was detectors,  it  Function for  copper c r y s t a l  i n c l i n e d at  is d i f f i c u l t to give function.  f u n c t i o n would appear  t o be t h e  fir) £ i s an e f f e c t i v e  function  r a t i o of  (r)  <V e  Finite  the  a simple a n a l y t i c  «  cylinder  l i n e d e f i n e d by expression for  f i r s t choice of  e  V 2  -  the  the  resolution  (5-4)  0 /  (as  " s e e n " by a d e t e c t o r ) .  r e s o l u t i o n f u n c t i o n u s e d by L a n g  correlation  Crystal  following;  results  (wide s l i t  geometry)  s l i t w i d t h was 0 . 5 .  one a s s u m e s an e f f e c t i v e — r_2 2  70.5° w i t h  A reasonable  source width to d e t e c t o r  arrangement  f  angular  and  i n t h e p r e s e n t w o r k was a  source dimension  is s i m i l a r to the  rection of  Detectors  used  an a n g l e o f  resolution or density  where  Finite  source s i z e of  If, £  (1956)  This for  in which for  the  k mm one  cor-  the  present finds  68  Using this obtains  r e s o l u t i o n f u n c t i o n and t h e p a r a m e t e r s  = 0.082 which  i s somewhat  s m a l l e r than  one  (5-3)  the p r e v i o u s  value  0.12  o b t a i n e d assuming n e g l i g i b l e source s i z e .  I n c l u d i n g , now, f o r v a r i a t i o n s  b p e r m i t t e d by t h e p r e s e n t  data  uncertainty  ™2  in  is 0.004.  on o t h e r p a r a m e t e r s  repeating  low-temperature  (Table  The d e p e n d e n c e o f  s u c h as  T.  the  resultant  the c a l c u l a t e d  the value chosen f o r  t h e c a l c u l a t i o n f o r «< =  I),  <tf  is  values  The c a l c u l a t i o n y i e l d s m_  'Z m  0.004.  shape o f  Thus t h e p r e s e n t m e t h o d s h o u l d be q u i t e the  Fermi  It  s h o u l d be n o t e d t h a t  -r f u n c t i o n s . j*(r)  and 0 . 0 5 0  correlation the second Figure are  e  respectively  in F i g u r e 2 1 .  2  2  is 32°  and  ^(r)  Despite  this  -[lllj  qf e4  I).  Fermi  surface with  to the  and 4 5 °  respectively  difficulty, the  it  Fermi  This angle  i s d e f i n e d by t h e  the  are  values  consistent with  detailed  is seen surface  consistent  of  the  fit  results  0.082 shown  angular  however;  to the d a t a  that  reso-  of  the c o r r e s p o n d i n g  ( A p p e n d i x B)  a better  d i r e c t i o n s , s u b t e n d i n g an a n g l e o f  Table  0.072  l  For example the  give  h a l f maximum o f  r e s o l u t i o n f u n c t i o n may g i v e  21.  results  resolution functions. 2 -_r  width at  c o n s i s t e n t w i t h a model o f  the  the experimental  both of which are  The f u l l  curves  sensitive  =  surface.  w i t h a c o n s i d e r a b l e number o f lution  of  i l l u s t r a t e d by  3  +  in  so  shown  the e x p e r i m e n t a l  that in  results  in w h i c h " n e c k s " o c c u r  "X ~ 20° a t  (circular)  k*  region of  B r i l l o u i n z o n e b o u n d a r y : ~\ = 2  = 0  (see  contact  of  tan/.kj.j  where  in  the k^  km  is  the neck  radius at  zone boundary  from k  Thus,  taking  t h e z o n e b o u n d a r y and k =  ()(  is the d i s t a n c e o f  this  0. i n t o c o n s i d e r a t i o n the u n c e r t a i n t i e s  in the  available  Fermi  surface  dimensions  to expect  an e f f e c t  the  volume  Fermi  effects  of  (Table  about  it  21,  five  and  in the  to eight  resolution  percent  Although t h i s  is evident  that  f u n c t i o n one  in the p r e s e n t  i s assumed t o be u n i f o r m l y d e n s e and  c a n be n e g l e c t e d .  in F i g u r e  I)  if  higher  is consistent with  a more a c c u r a t e  work  led  if  momentum  the  knowledge o f  is  results  the  shown  resolution  of  function  is e s s e n t i a l  believed  that  if  the method  a more c o n v e n i e n t  a good e s t i m a t e  of  the  is to b e ^ q u a n t i t a t i v e  choice of  resolution  crystal  value.  orientation  f u n c t i o n , thus o b v i a t i n g  It  should  this  is permit  source  of  difficulty. E.  Interpretation  of  The a b o v e  the  Results  interpretation  due t o h i g h e r momentum e f f e c t s However, Fermi  the  lattice  vector  However,  if  hexagonal appears  that  the  losses  curve  associated with  be m e r e l y a l o n g the  t o assume t h a t  distant  this  the  results of  difficulty by a  the  since  sampling  Since  contribution will  applies  t o a zone  in c o p p e r the  from the c e n t e r  of  non-  the zone,  be f a i r l y  annihilation  the  cylinder.  small  it and  contribution.  on c o r e  annihilation  given  by B e r k o  the core  annihilation  c o n t r i b u t i o n c a n be made.  i s made by a p p r o x i m a t i n g t h e c o r e a n n i h i l a t i o n a n g u l a r by a p a r a b o l a  the  reciprocal  perpendicular  occur.  core  of  a similar consideration  is not a p p r o x i m a t e l y rate w i l l  contributions.  translated axis of  complications  the c o n t a c t  not cause  t o a zone boundary,  relatively  (1958) an e s t i m a t e  This estimate  usually  essentially  axis  neglected  ion core a n n i h i l a t i o n  be m a s k e d by t h e much l a r g e r  By u s e o f  lation  are  results  zone f a c e w i l l  in the c o u n t i n g  zone f a c e s  probably  will  normal  cylinder  reasonable  Plaskett  lies  directions  face small  will  w i t h a hexagonal  momentum s t a t e s  For o t h e r  and  the  h i g h e r momentum c o m p o n e n t s  surface  affected  of  and c o n s i d e r i n g t h i s  parabola  and  corre-  to correspond  to a  70  Fermi  surface with  t r i b u t i o n of  P  the " i o n  sphere a s s o c i a t e d with ( A p p e n d i x C) percent  core  or  sphere" with  This  is not  Accuracy  attainable  the present work.  (1966) i n d i c a t e t h a t  is perhaps of  with  t o p i c are 1.  discussed  counting  rate.  is necessary  to  reduce  a factor  '  v J  to  half-width of  in d e t e c t o r  the  of  8.2  present  t o be j u s t i f i e d by  In a d d i t i o n ,  recent  the  calcula-  c o r e a n n i h i l a t i o n c a l c u l a t i o n s may  be  believed.  surface  in o r d e r  that  accuracy  i t may be c o m p a r e d w i t h  determination.  Various  l6.  in the p r e s e n t method  improve the the  aspects  of  is the  low  the d e t e c t o r s o l i d angle  This estimate  by a f a c t o r o f  changed,  the from  two, w i t h  s o l i d angle  reduces  by a f a c t o r o f  the c o i n c i d e n c e  is f o r a " p o i n t " c r y s t a l .  the  s o u r c e s t r e n g t h used ~ 50 mCi  r e s o l u t i o n by a f a c t o r o f  r e s o l u t i o n f u n c t i o n by a f a c t o r o f  t h e d e c r e a s e w o u l d be e v e n more s e v e r e .  of  estimated  the expected e f f e c t  to c o n s i d e r the u l t i m a t e  limiting factor  In o r d e r  reduction  creased  c a n be r o u g h t l y  Fermi  Method  interest  Fermi  (concentric)  the  this  below.  important  reduce the  lution  it  the  con-  R e s o l u t i o n and C o u n t i n g R a t e An  to  the  the present method,  o t h e r methods used f o r  of  c a l c u l a t i o n does not appear  Attainable with  It  that  inconsistent with  c o n s i d e r a b l y more d i f f i c u l t t h a n e a r l i e r  F.  By p r o p e r l y w e i g h i n g t h e  core a n n i h i l a t i o n should lower  resolution of  by C a r b o t t e  .  the c o n d u c t i o n e l e c t r o n s ,  A more d e t a i l e d  statistics  mc x 10  Fermi  to about 6 p e r c e n t .  results.  tions  that  1*  = Q  Thus,  rest of  in o r d e r t o  two  (i.e.  two)  four.  it This  c o u n t i n g r a t e by For a f i n i t e improve the  the experimental  reso-  arrangement  i n t h e p r e s e n t w o r k w o u l d need t o be  to about a C u r i e .  crystal  Problems a s s o c i a t e d w i t h  the  s u c h s o u r c e s w o u l d become much more s e v e r e t h a n t h o s e e n c o u n t e r e d  so  un-  inhandling far.  71  However,  a more  activity. River,  important  With p r e s e n t l y  in the p r e s e n t  ficiency.  This  to  f l u x e s at AECL,  involve  an  Chalk  o f much more  increased detection  r a t e c o u l d be  from t h i s d i s c u s s i o n t h a t  as a s o u r c e )  ( l e s s t h a n 40 f e e t )  crystals.  individual detector efficiency of  the c o i n c i d e n c e count  is thus evident  crystal  improvements  and by u s i n g l a r g e r N a l ( T l )  improve the  so t h a t  priate  neutron  thermal  than  ef-  c o u l d be a c c o m p l i s h e d by l o w e r i n g t h e d i s c r i m i n a t i o n  the pulse shapers  possible  It  available  specific  crystal.  Other p o s s i b l e  "2  from c o n s i d e r a t i o n s o f  i t w o u l d be d i f f i c u l t t o o b t a i n a p o s i t r o n a c t i v i t y  a Curie  of  limitation arises  cannot  be  may t h e n  0 . 4 5 by a  resolution  distances are  4.  (using the <-»3  t o be u s e d .  be  factor  i m p r o v e d by a f a c t o r  i m p r o v e d by more t h a n a f a c t o r  source to d e t e c t o r  long-lived source, of  the  It  levels  if  present  reasonable  An a p p r o -  c o u r s e , w o u l d be p r o h i b i t i v e l y e x p e n s i v e ,  at  present.  2.  Stabi1ity  Another  important  s t a b i l i t y of  the  mechanically  so t h a t  be r e d u c e d t o  limiting factor  arrangement.  less  percent.  i n c i d e n c e c o u n t i n g r a t e due t o t h e reduced below ^ 0 . 1 mentation.  Thus,  percent  more t h a n a f e w t e n s o f uncertainties  ~1  However,  greatly  f o r c o i n c i d e n c e count  Allowing,  the u n c e r t a i n t y  thousands per  to set  increased  rates  in the  in the  complexity of 1 to  the p r a c t i c a l  c o u l d be o b t a i n e d  point.  stable could co-  the e l e c t r o n c i s cannot  at m o s t , a few hours p e r p o i n t ,  percent  electrical  in the c o i n c i d e n c e c o u n t i n g r a t e  i n s t a b i l i t y of  without  c o u n t i n g s t a t i s t i c s would appear attainable.  and  The s y s t e m c o u l d be made s u f f i c i e n t l y  the u n c e r t a i n t y  t h a n 0.1  is the mechanical  10 s e c " ^  l i m i t on t h e total  instrurange, precision  counts of  leading to  be  no  statistical  G.  Di s c u s s i o n By d e c r e a s i n g  factor  of  f o u r and  the  s o l i d angle  increasing  u s i n g t h e maximum a v a i l a b l e improve the  r e s o l u t i o n of  the d e t e c t o r  thermal  of  d u c e d t o r-0.1  by u s e o f  convenient the  choice of  resolution  and o t h e r with  function.  with which,  be s p h e r i c a l If  careful  such  incidence  counting  electrons  of  (Table  percent  ~1  (Shoenberg,  resolution  ion cores  assumptions  in c e r t a i n  metals.  It  might  associated with  periodic crystal  potential  associated with  or  Fermi  composed o f  the e f f e c t  momentum e f f e c t s  effects  various  of  annihilation  competitive  crystal  found  orientations,  of  counting If  the  positrons  r a t e due t o  such a s e p a r a t i o n  surface  diameters  such t r a c t a b l e a l l o y i n g on c o r e  localization of  the  Alphen  o f p o t a s s i u m was  t o an  metals  if  to  Brillouin  investigate the p o s i t r o n  zone  are cowith anproves accuracy certain  annihilation  of  higher in  i t may be p o s s i b l e t o e x a m i n e h i g h e r  p r o x i m i t y of  of  effects  t h e de Haas van  a l s o be p o s s i b l e t o the  re-  1965).  valence electrons.  can be made a b o u t  the c o n s t i t u e n t  surface  to  while  c o u l d be  t e c h n i q u e w o u l d be  from the c o i n c i d e n c e  alloys  two  i t may be p o s s i b l e t o s e p a r a t e  i t may p e r m i t m e a s u r e m e n t s  percent  of  a good e s t i m a t e  core  for  and  In a d d i t i o n , a more  r a t e c o n t r i b u t i o n due t o a n n i h i l a t i o n o f  n i h i l a t i o n of positrons with  of  Fermi  for  drift  should permit  perhaps  by a  s h o u l d be p o s s i b l e  procedures.  the present  the  it  by a f a c t o r  absence of  except  measurements,  improved  the  I)  for example,  to ^ 0 . 1  made w i t h  possible,  in the  h i g h e r momentum e f f e c t s ,  the o t h e r methods  effect  recycling  a detector  by a f a c t o r ^ 2  The e l e c t r o n i c  orientation  Thus,  fluxes  arrangement  »*» 1 p e r c e n t .  crystal  efficiency  neutron  the present  maintaining statistics percent  associated with  boundaries.  the  momentum  73  H.  Conclus ions By u s e o f  a new p o s i t r o n a n n i h i l a t i o n t e c h n i q u e  " c o l l i n e a r p o i n t geometry" a n i s o t r o p i c at sistent with with  t h e . p i c t u r e of  at  faces  done by o t h e r w o r k e r s  to  indicate  with  better  t r i b u t i o n of the  Fermi  of  statistics  and  surface  over  methods,  the  possible  to apply  considerable of  Fermi  r e s u l t s of  the p r e s e n t work are  first  B r i l l o u i n z o n e s u b t e n d s an a n g l e  the  h a v i n g " n e c k s " whose  zero.  in t h i s work are not o f  and  of  work  sufficient  precision  c o n t r i b u t i o n from core a n n i h i l a t i o n s .  statistics  treatment  be r e q u i r e d b e f o r e  further  However,  of  the  con-  details  of  ascertained.  r e s o l u t i o n can be  in the p r e s e n t work temperature  surface of  contact  i n d i c a t e d by t h e more a c c u r a t e  r e s o l u t i o n , a more d e t a i l e d  it  range than  various metals.  l i t t l e w o r k has  improved  substantially  s h o u l d be p o s s i b l e t o s t u d y ,  the method t o a s y s t e m a t i c since  con-  the  t o p o l o g y c a n be  interest  be  surface  absolute  a much l a r g e r  c o p p e r was f o u n d t o  Fermi  core a n n i h i l a t i o n s w i l l  those employed  detail,  The  k*-space as  near  the e x t e n t  surface of  the  results obtained  Since the over  of  the o r i g i n of  The  Fermi  room t e m p e r a t u r e .  the hexagonal  a b o u t 20°  the  employing  is the case f o r  In a d d i t i o n study of  it  alloys.  been done on t h e  in  other  s h o u l d be This Fermi  is  of  surface  alloys..  In a d d i t i o n t o Fermi  surfaces,  some v a l u e lution  its  the method o f  in the  study of  and s t a t i s t i c s  p r o m i s e as a t o o l col 1 inear point  core e l e c t r o n  for  the  of  g e o m e t r y may p r o v e t o be o f  annihilation.  i t may be p o s s i b l e t o s e p a r a t e  c o i n c i d e n c e c o u n t i n g r a t e due t o c o r e e l e c t r o n s  investigation  With  improved  reso-  the c o n t r i b u t i o n to  from the c o n t r i b u t i o n  the  arising  from the test  of  conduction electrons. core a n n i h i l a t i o n In c l o s i n g  Fermi fact  surfaces that  col l i n e a r Berko  the  has  it  T h i s w o u l d make p o s s i b l e an  calculations.  should perhaps  be n o t e d t h a t  very  been done by means o f p o i n t g e o m e t r y .  preliminary  p o i n t geometry  and P l a s k e t t  experimental  results of  Fujiwara  (1965)  show much more s t r u c t u r e  l i t t l e w o r k on In v i e w o f  the  o b t a i n e d by u s e o f  non-  t h a n do t h o s e o b t a i n e d  (1958)  u s i n g the wide  s l i t method, the n o n c o l 1 i n e a r  g e o m e t r y m e t h o d may p r o v e  t o be a u s e f u l  complement t o the  col 1inear  by  point  point  geometry method.  Finally,  it  germanium d e t e c t o r s energy  c a n be n o t e d t h a t  (Dearnaley  the advent  and N o r t h r o p ,  r e s o l u t i o n : may make p o s s i b l e f u r t h e r  geometry method. together with  1966)  the  the usual  angular  col 1 inear  z  It  = + p .  Here  Q  is seen  components  that (  in the  s e l e c t i o n , one c o u l d a c h i e v e  to the  case,  = o<P  e x t e n d i n g f r o m p^ P  possessing excellent  improvements  Fermi  surface  0  = |Sp  <X and ^ a r e p o s i t i v e  Q  instead of constants  the method h o l d s p r o m i s e f o r  p>p  )  since  o( and/?  a system the  b e made v e r y  one c o u l d s a m p l e a c y l i n d e r  t o p^  point  discrimination  topology since  momentum s p a c e s a m p l e d by t h e p r o p o s e d m e t h o d c o u l d for  lithium drifted  F o r e x a m p l e , by c o m b i n i n g some D o p p l e r s h i f t  v e r y much more s e n s i t i v i t y  Thus,  of  Q  is  the study of  region  the  -  of  small.  i n momentum  f r o m p^ =  and p  with  p^  space  to  F e r m i momentum.  h i g h e r momentum  c o u l d be c h o s e n t o be g r e a t e r  than  unity.  75  APPENDIX A  SOLUTION OF A B E L ' S  The s o l u t i o n o f A b e l ' s S (x)  is given  =  (Bocher, u(x)  =  (x - vT  s  EQUATION  equation  0< <* < 1 ,  f (a)  =  0  by  in  d dx  7T<*  7T It  integral  "(y)dy, 1929)  INTEGRAL  i s d e s i r e d to f i n d the  | J  A.  S(t)dt t)-*  (A-l)  (x.-  s o l u t i o n of the  related  integral  equation  Ok  f(x)  f q(t)dt  =  2 Let r(  w  =  -  -x )  x  =  Z  2  , r  x  v  =  (w)  =  -  t -f(x)  and dv -  r  =  -2tdt  and  r  q(t)dt  G(v)dv  (w - vR  (t* X ) t V  G(v)  =  q(t)  =  - 2t A p p l i c a t i o n of G(w)  d_f dw  =  dy  dx  dx  dw  (A-l)  q(  r=y)  -  2 iT^v"  whe re  a.  (A-2)  gives  =-(  d ( r(v)dv T dw } (w - v)  =  consider  -] 2x  dx  =  -  _L d 2TA d x  r  VI (v)dv (w - v ) *  s i nee  7.6  Substitution  f o r v and  G(w)  From  (A-2)  one  =  has  w  yields  /C-f(t)  - _ i d 2 7Tx dx G(w)  =  q  NT^W  J[ ( t  =  -2 ^ Combining t h i s  l a s t equation  with  l  -1 *  A  dx  r  (A-3)  T  q(x)  -2x (A-3)  then g i v e s  •a  g(x)  -2t dt] x )t  t f ( t ) dt (t^ x*)*  the d e s i r e d  result  APPENDIX B  EXPECTED ANGULAR CORRELATION CURVE WIDTH  Certain first  outlined.  about  the  figure  geometric  relations  Consider the  axis.  For  the  rotation,  for  the  calculation will  t h r o u g h an a n g l e ^  two p o s i t i o n s  , of  be  a v e c t o r "p*  (1 and 2) one ^has f r o m t h e  adjacent  that  P,  =  •p*  =  2  (P«  sinf ,  (p„  p*2 c a n be o b t a i n e d Uc?  2  Solving  after -  correlation  1.  P,  for  In o r d e r  ,  cosf  Q  p  distance  ) •P  2  J  1  +  curve  surface  cylinders  of  is  ,  p  cos^ )  Q  f r o m p o i n t p*^ t o t h e the  constant"*  line  defined  in the  distance  2 c o s \j/ the  cos  shape  equation  d^  = |<*p*2 - P*j J  <j>  ( cos  approximations are  n  e  sin ^  a, =  1*  in Chapter  between  the  finds +  2  the  made;  b  +  V.  ( i n momentum s p a c e )  the d i s t a n c e  o  assumed t o be a p p r o x i m a t e d  h and d i a m e t e r  distance  2 <j>  (hence the h a l f w i d t h o f  " n e c k s " are  height  by  (B-2)  same m e a n i n g as  An e f f e c t i v e  sin^  B-l)  ^  =0  several  The F e r m i  (  W  )  siny  Q  the p e r p e n d i c u l a r  to c a l c u l a t e  distance  p  determination of  h have the 2.  ,  0  s i n y cos^  The p e r p e n d i c u l a r  angular  needed  dj_  tips of  _h. 2  Here  is used.  by b and  This  t h e v e c t o r s "p*j  cos  y  78'  and p »  The q u a n t i t y  2  +  h  2  where p  p  Q  i s assumed t o be g i v e n  =  Q  p^  i s t h e m a g n i t u d e o f t h e F e r m i momentum.  F  C o r e a n n i h i l a t i o n and o t h e r  3.  by p  h i g h e r momentum e f f e c t s  are  neglected. W i t h t h e above s i m p l i f i c a t i o n s , c u l a t i o n o f the "mass" or e f f e c t i v e  the problem reduces  c o l u m e o f a c y l i n d e r o f b a s e a^u -  height  h if  i t s density  varies  t o be i n d e p e n d e n t o f h. p^ - Py p l a n e , origin.  The ( c i r c u l a r )  i t s center  =  r =  r  =  5  d^_ c o s ©  Y  =  d  d '  ±  cose  .  This density  axis  i s assumed  r e s t s on t h e  a d i s t a n c e dj^ f r o m t h e  is  + Y aj/  -  d±  | +V  -  dj_  2  sin©  6>©= s i n  (ft-ejjf-  i  2  sine  )  d ( _ a )  6<®= s i n ~  d^?a^  The e f f e c t i v e  m  (f)  volume o f t h e c y l i n d e r  =  ®  2K  I  £{f) where R  =  equation  rr^  Mr)  r  is  dr  da  i s g i v e n by  = siiiL  R  m(o)  i s the c o i n c i d e n c e count  (5-1)  ( i . e . a "neck")  A  and t h e a n g u l a r c o r r e l a t i o n c u r v e  (  r  and  2  base o f t h e c y l i n d e r  l y i n g on t h e p^  2  r  ^ ( r ) <s< e " ^  The e q u a t i o n o f t h e c i r c l e  r  Here  as  to the c a l -  r a t e a n i s o t r o p y e x p e c t e d a t j> = Q  ).  A choice of g  =  J_  2  gives  a curve with  h e i g h t m„  = 0.082 and a  l o n e o b t a i n s 0.050 and m  full 45  w i d t h a t h a l f maximum o f a b o u t 32 . respectively.  It would thus appear  For g that  =  better  J_  h  agreement w i t h  the ex-  79 perimental  c u r v e s i s o b t a i n e d i f one c h o o s e s t h e r e s o l u t i o n f u n c t i o n -_r e ( s e e a l s o C h a p t e r V): s i n c e one then o b t a i n s = 0.062 v  |» ( r )  =  3  m, and  Wj  ~  40°,  in f a i r  accord w i t h the experimental  curves,  80  APPENDIX C  EFFECT OF CORE A N N I H I L A T I O N  If  it  i s assumed t h a t  by a s o r t o f " i o n c o r e F e r m i electrons  by a  geometry,  for  z  =  (concentric) the  t h e c o r e e l e c t r o n s can be r o u g h l y d e s c r i b e d  sphere" with Fermi  r a t i o of  radius p  sphere of  radius p  =  the c o i n c i d e n c e c o u n t i n g rate  h.  =  /  ~  N core  h  2  f  ?  r  r  d  r  c  I 2 Kb r  2  w e r e £ i s t h e momentum s p a c e d e n s i t y  (Chapter  V),  m  2  to m  2  1  f  m, Il Il  + +  *1  .  i_  "  K  (assumed c o n s t a n t )  and m^ t h e c o n t r i b u t i o n o f  f o r p o i n t geometry the  m,  the  2  contributions  =  m jj m^  m  at  for  the c o r e e l e c t r o n s .  ion c o r e e l e c t r o n s , anisotropy  +  m^  3  mc s o t h a t  reduced  2  p a p e r one c a n e s t i m a t e  1  ]_ + m.  =  h  2  = 1.15  and  l  0.75  l  Thus c o r e a n n i h i l a t i o n w o u l d be e x p e c t e d t o a f f e c t by a b o u t 2 5 p e r c e n t .  seen  V *  m  anisotropy  is  is  h P,  h  18 x 1 0 "  it  2  From B e r k o and P l a s k e t t ' s  ^  slit  t h e n e c k mass c o n t r i b u t i o n  coincidence counting rate  m1  p £  f o r wide  dr  L e t t i n g m^ be t h e be 11 y ' m a s s " c o n t r i b u t i o n , m  from  one h a s ,  0 N cond  that  and t h e c o n d u c t i o n  p  the c o i n c i d e n c e count  rate  Bl BL I OGRAPHY  ANDERSON,  C. 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