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An investigation into the Mossbauer effect in Fe57 Woodrow, Janice Emily Jean 1964

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AN I N V E S T I G A T I O N  INTO T H E MOSSBAUEK E F F E C T  IN  Fe  by  JANICE B.  Sc.,  The  EMILY J E A N WOODHOW  University  of  British  Columbia,  A T H E S I S SUBMITTED I N P A R T I A L F U L F I L M E N T THE REQUIREMENTS FOR THE DEGREE OF MASTER OF in  the  SCIENCE  Department  of  PHYSICS  We a c c e p t to  THE  this the  thesis required  UNIVERSITY  as  conforming  standard  OF B R I T I S H COLUMEIA  April,  1964  i960  OF  In the  requirements  British  mission  for  for  purposes  I  an  reference  extensive  may b e  without  of  this  Department  that  and  by  the  study.  for  the  is  8,  Head  of  British  I  of  understood  at  the  thesis  fulfilment  University  shall  further  make  agree for  it  of  per-  scholarly or  t h a t , c o p y i n g or shall  of  freely  that  my D e p a r t m e n t  financial gain  Columbia,  partial  Library  this  permission*  Canada  in  degree  of  The U n i v e r s i t y Vancouver  advanced  It  thesis  thesis  copying of  granted  my w r i t t e n  this  agree  representatives.  cation  Date  for  Columbia,  available  his  presenting  not  be  by publi-  allowed  - i i ABSTRACT  In t h i s s t u d i e d as  thesis  the  Mossbauer e f f e c t  a f u n c t i o n of  the  l e n g t h of  into metallic and of  the  intensity  iron,  of  absorber  the  thickness.  a n d s h i f t s were Measurements  bauer e f f e c t , at  In each  the  case,  diffusion over  the  the  i r o n has of  the  been  C o ^  showed t h a t  the  w i t h i n the  r a n y e 156°K t o  l i n e shape,  l i n e w i d t h and t o  appearance first  and t h a t  increase  The o b s e r v e d t e m p e r a t u r e  corrections  made t o shift.  the  for  the  of  a n a p p r e c i a b l e MoG3=  t e n roinutes o f  diffusion  a d d i t i o n a l time i n t e n s i t y to  s h i f t of  the  is  The r e s u l t s  f o r both the  comparing the  required  source  and a b s o r b e r  The measurements the  i n t e r n a l magnetic  (the  W e i s s Law)  aade o v e r t h e  a n d a b s o r b e r , a0< 50°K,  For  of  the  field  at  absorber.  structure magnetic  *Q  measurements  at  absorber  tailed  saturation  (U20±20)°K  were  range  indicated  magnetization  made i n d i c a t i n g t h a t  curve  the  indicated that  the  internal  source  F o r ^6>^ 100°K,  the  Mossbauer l i n e d i s p l a y e d a h y p e r f i n e  the  dependent d i f f e r e n c e  than  i n the  that  internal  nuclei.  thicknesses  comparison of  .OlxlO^oe greater  source  minimum  the  temperature  was  that  d i f f e r e n c e s between the  These r e s u l t s  The l i n e p r o f i l e , w i d t h , four  theoretical  = 0°K f o r  a r i s i n g from the field  temperature  small temperature  l i n e w i d t h o c c u r r e d a t <*Q = 2h°K. magnetic  -  were  used.  f i e l d Jtf f o l l o w e d t h e  closely.  D  kev  How-  compression s h i f t  of ©  to  a b s o r b e d ih.h  measured s h i f t w i t h the  i n d i c a t e d a Debye t e m p e r a t u r e  time  ,1^5.  resonantly  an i s o m e r a n d a h y d r o s t a t i c  data before  *v78°K,  width,  r a d i a t i o n f o l l o w e d t h a t w h i c h was p r e d i c t e d b y t h e J o s e p h s o n e f f e c t . ever,  source  measured.  arises  .122,  time of  temperature  i n a hydrogen atmosphere  900°C  reduce  source  in metallic  -  .0002",  shift  and i n t e n s i t y were measured  .00035",  for  .00055", a n d .001". T h e d e -  these l i n e c h a r a c t e r i s t i c s  w i t h the  theoretical  values  - i i i -  required an extension of existing treatments, a discussion of which i s given in chapter five. In each case i t was observed that the Mossbauer line was accompanied by two small peaks, one on either side of the main line. The position of these peaks indicated that they were associated with a small or zero internal magnetic f i e l d at the site of some of the Fe  57 atoms.  -xiACKNOWLEDGMENT  The to  the  of  D r . 3.  this L.  University and  of  depth of  Of the  extends  her  many p e o p l e who a s s i s t e d  pletion to  writer  writer  work.  White,  the  i n the  In p a r t i c u l a r ,  Assistant  thesis  acknowledges  a large  the  have b e e n  efforts  and  progress  gratefulness and com-  gratitude  is  of P h y s i c s ,  owed  The  u n r e m i t t i n g a i d the  scope  attained.  involved in giving  special  of  Department  Tiie UBC C o m p u t e r D e p a r t m e n t and  debt  w i t h o u t whose  would never  many p e r s o n a  appreciation  preparation,  Professor,  B r i t i s h Columbia, this  sincerest  unselfish  assistance,  of: f o r numerous  test  results  programming Mr.  A. Fowler,  program which r e s u l t e d The received  v i a the  for  his  ingenious modifications to  i n much s a v i n g o f  N a t i o n a l Research  NRC s c h o l a r s h i p .  time  Council  and for  the  cost the  financial  assistance  -iv-  TABLE OF CONTESTS  ABSTRACT  i i  AC^OWLEDGMEHT  xi 1  INTRODUCTION CHAPTER I  General Features of the M o B S b a u e r E f f e c t and I t s Uses as an Experimental  Tool  1.0  Introduction  h  1.1  Recoiless Emission and Absorption  5  1.2  Internal F i e l d E f f e c t s  7  1.3  L a t t i c e Ejynamics  8  CHAPTER I I Theory of Line Width, S h i f t s and I n t e n s i t i e s 2.0 2.1  Introduction \ Transmission of the Six Line Spectrum of F e ^  10 12  2.2  The Josephson and Related E f f e c t s  19  2.3  Isomeric S h i f t  25  2.h  Hyperfine S p l i t t i n g  27  2.5  E l e c t r i c Quadrupole S p l i t t i n g  32  2.6  L o c a l i z e d Modes  33  CHAPTER IIIExperimental Apparatus and Method 3.0  Introduction  39  3.1  Detectors  39  a) Hal C r y s t a l  39  b ) Ar-CH^ Proportional Counter  *H  i) Sensitivity i i ) Construction of the Counter  ^1 ^2  -viii) iv) v) vi)  Filling  the  Counter  1+3  Gas F l o w System  kk  Difficulties  45  Characteristics  of  the  Counter  k6  3.2  The E l e c t r i c a l  3.3  The C o n t r o l System  50  3.it  The A b s o r b e r  51  a)  The A b s o r b e r Mount  51  b)  Armco I r o n A b s o r b e r s  51  3.5  Apparatus  kS>  The Source  52  a)  The Mount  52  b)  Sources  53  3.6  The M e a s u r e d Q u a n t i t y  5k  3.7  Geometric  55  3.8  Background C o r r e c t i o n  56  3.9  Selection  of  the  58  CHAPTER I V D i f f u s i o n  of  Co  Effects  5 7  on R(v)  Absorber  into  Natural  Iron  h.0  Introduction  60  h.l  T h e o r y of  6l  Diffusion  a)  General Diffusion  b)  Theory of  k.2  Apparatus  ^•3  • Procedure  h.k  Experimental  1*.5  D i s c u s s i o n of  the  Theory  Diffusion  6l of  C o ^  i n t o Fe  62 1  6k 65  Results Results  67 68  «viC3IAPTER V 5.0  Temperature  Independent  Effects  Introduction  71 5.1  A B r i e f D i s c u s s i o n o f the Relevant Theory  72  a) T h e o r e t i c a l Mossbauer I n t e n s i t y  72  b) L i n e S h i f t Mechanisms  72 72  i ) Isomer S h i f t i i ) Debye Tempo i-ature D i f f e r e n c e iii)  73  Impurity E f f e c t s  73  i v ) Mas.? D e f e c t v) H y d r o s t a t i c  Compression  c) L i n e Broadening Mechanisms i ) Magnetic F i e l d E f f e c t s  iii)  Source and Absorber T h i c k n e s s  i v ) Random S h i f t s 5.2  73 7h 7^ 75  i i ) L o c a l i z e d Modes .  72  •  75 75  E x p e r i m e n t a l Procedure  75  a) Source and Absorber  75  b) Procedure  76  c) D i s c u s s i o n of the Procedure  77  5.3  C a l c u l a t i o n s and C o r r e c t i o n s  78  5. 4  Results  78  5.5  D i s c u s s i o n of R e s u l t s  !  CHAPTER'VI Temperature  and C o n c l u s i o n s  79  Dependent E f f e c t s  6.0  Introduction  82  6.1  B r i e / D i s c u s s i o n of R e l e v a n t Theory  83  a) Josephson E f f e c t  83  -vil-  6.2  li)  Hydrostatic  Compression E f f e c t  83  c)  Temperature  Dependence of  f  83  d)  Temperature  Dependence o f  the  Experimental Analysis perature  6.3  of  the  D i f f e r e n c e s Between  Magnetic F i e l d  Effects Source  of  Small  Tem-  and A b s o r b e r  Line  Shifts  Qh  b)  Line  Widths  85  c)  Line  Intensities  86  Experimental Procedure  86'  a)  Source  and Absorber  86  b)  Source  Temperature  c)  Greater  Than A b s o r b e r  Source  87  Temperature Less  Than A b s o r b e r  Teiirperature d)  Small  88  Temperature  D i f f e r e n c e s Between  Source  and A b s o r b e r  88  C a l c u l a t i o n s and R e s u l t s a)  Large  Temperature  D i f f e r e n c e s Between  89 Source  and A b s o r b e r b)  Small  89  Temperature  D i f f e r e n c e s Between  and A b s o r b e r 6.5  8k  a)  Temperature  60b  8^  D i s c u s s i o n of  Source 91  Results the  and C o n c l u s i o n s  Spectra  93  a)  The Shape of  b)  The L i n e  Shift  9^  c)  The L i n e  Width  96  d)  The L i n e  Intensity  97  93  -viiiAPPENDIX A  The Computer Program  APPENDIX B  The P r e p a r a t i o n  APPENDIX C  Geometric Corrections  to  APPENDIX D  Statistical  the  APPENDIX E  T h e R o o t Moan S q u a r e D i f f u s i o n  BIBLIOGRAPHY  of  98  an E n r i c h e d Fe^?  Design of  R(v)  Absorber  100 102  Experiment Depth  105 10? 108  -ix> L I S T OF I L L U S T R A T I O N S To f o l l o w Figure  1-1  T h e Debye f u n c t i o n  Figure  1-2  Typical  Figure  II'-1  T h e gamma s p e c t r u m  Figure  II-2  The t h e o r e t i c a l  Josephson  Figure  II-3  The t h e o r e t i c a l  hydrostatic  Figure  I I T h e  6  Mossbauer a b s o r p t i o n o f Fe--'  9 13  1  effect  temperature dependence  magnetic  spectra  22  compression of the  shift  internal  field  29  Figure  III-l  Typical F e - ^ spectra detected by a K a l c r y s t a l  Figure  III-2  A cross-sectional  diagram of the  k2  Figure  III-3  The t o p t e r m i n a l  Figure  Ill-it  The gas f l o w system o f t h e p r o p o r t i o n a l c o u n t e r  Figure  III-5  The s p e c i f i c  of the proportional  gravity  correction  counter  curve  Figure  III-7  and the count  The e f f e c t portional  Figure  III-8  the atmospheric  pres-  rate  it 5  o f t h e b r e a k down p u l s e s  i n the  h6  of the apparatus  mine t h e c h a r a c t e r i s t i c  used to  deter-  of the p r o p o r t i o n a l  counter III-9  Figure  III-10 T y p i c a l  ^7  The gas g a i n o f t h e p r o p o r t i o n a l  III-ll  counter  ^8  s p e c t r a obtained by the use of the  proportional Figure  pro-  counter  The arrangement  Figure  kh  kk  The r e l a t i o n s h i p between sure  ^3  f o r the  flowmeter III-6  *tG  proportional  counter  Figure  23  counter  A b l o c k diagram of the e l e c t r i c a l  k8 apparatus  k$  page  (  -x-  Figure  111-12  A circuit  diagram  of t h e standard  Figure  III-13  A schematic  Figure  111-1*4-  The h i g h temperature  Figure  III-I5  The l o w t e m p e r a t u r e  Figure  III-16  The g e o m e t r i c  Figure  III-17  The t r a n s m i s s i o n  diagram  pulse  of the switching source source  generator  circuit  mount mount  53  arrangement o f t h e apparatus  III-18  Figure  IV-1  The d i f f u s i o n  Figure  IV-2  The thermocouple  Figure  IV-3  A record  Figure  IV-h  The r e s o n a n t  IV-5  t h r o u g h F e a n d A l o f ik.k k e v  X/n a s a f u n c t i o n o f a b s o r b e r  Figure  IV-6  59 6k  and oven h e a t e r  of the diffusion absorption  arrangement  temperature  line  shape  f o r n=2  after  65. 66  each  run  67 parameters  as a f u n c t i o n o f  d i f f u s i o n time  The Mossbauer the  thickness  apparatus  The Mossbauer l i n e the  55  56  Figure  Figure  50 52  radiation  diffusion  k$  line  rms p e n e t r a t i o n  67 parameters depth  as a f u n c t i o n o f  o f Co57  67  Figure  V - l  The a b s o r p t i o n  spectrum  for t' =  .0002"  78  Figure  V-2  The a b s o r p t i o n  spectrum  f o rt' =  .00035"  79  Figure  V-3  The a b s o r p t i o n  spectrum  for t'  Figure  V—*»-  The a b s o r p t i o n  spectrum  f o rt ' =  Figure  VI-1  The temperature dependence  Figure  VI-2  The h y p e r f i n e  Figure  VI-3  The measured  Figure  VI-k  The temperature  Figure  VI-5  A comparison  Figure  VI-6  The t e m p e r a t u r e dependence  of the l i n e width  Figure  VI-7  The temperature  of £ - h  Figure  VI-8  A comparison  structure Josephson  = .00055" .001"  o f T andf  shift of the line  o f f a n d -|hZ\  89 90  intensity  o f 81 a n d 82  dependence  8l 84  1  o f t h e Mossbauer l i n e  dependence  So  91 92 93 9J 96  -1INTRODUCTION  t This out  by t h e  sbauer found  author  Effect  on c e r t a i n  has  A.J.P. other  that  to  equal  that  to  facts  emit of  that  tion,  E^/c,  shift  due t o  of the  nuclear  there exists  lattices  is  the  recoil  because  pared with the ternal  the  Stated  s i m p l y , the  fluorescence,  gamma r a d i a t i o n  nuclear  transition,  up b y t h e  line.  that  crystal there  In g e n e r a l ,  i n Chapter  I,  interaction  termining  the magnitude  radiating  nuclei,  it  shifts  w i d t h of the  f i e l d s . Because  E  of the  carried  literature  i n the  2  is f  very  the  absorption  of t h i s ,  the  . The e f f e c t  nucleus  bonding of  (*) R e f e r e n c e s t o t h e s e b o o k s w i l l Moss I I , p . _ and B o y l e , P . .  under  be  and t h e  fields  atoms  be g i v e n a s  at  broadening special  spectrum  and s t a t i c  com-  and, the  been u s e f u l  the  line  measurable.  lattice  has  in  p o s i t i o n of multipole  in crystals,  Prauenfelder,  by  absorp-  no  s u f f i c i e n t l y small  Mossbauer E f f e c t  values  is  Doppler  enough t o  is  precisely  d u r i n g e m i s s i o n or there  the  crystal  characterized  Mossbauer a b s o r p t i o n line  an-  die t o  almost  s m a l l ; however,  large  and d i r e c t i o n o f t h e  chemical  is  a whole so t h a t  becomes  between t h e  h a v i n g an e n e r g y  of  by is  is  n u c l e i bound i n  n e g l i g i b l e thermal  is  of  Q  for  occuranoe  are  report  Mossbauer E f f e c t  and i t s  nucleus  as  in determining nuclear  i n v e s t i g a t i n g the  Summaries  a finite probability f,  Observable energy occur  '  resonance  and,  work  Mossbauer Effect.-*- The M o s -  studied.  r e c o i l momentum o f t h e  taken  given  of the  book, "The Mossbauer E f f e c t "  v  and a b s o r b  the  gamma r a y  conditions,  in  aspects  and t h e o r e t i c a l  C o n f e r e n c e ^ and i n " T h e M o s s b a u e r E f f e c t "  B o y l e and H . E . H a l l . name f o r  experimental  been e x t e n s i v e l y  Mossbauer E f f e c t  ;  the  describes  i n Hans P r a u e n f e l d e r ' s  the Second  fact  thesis  and  in  p._,  indethe  moments,  '  -2measuring s m a l l ,  Prior attempt  relativistic  to  velocity  intensity  and t h e  and J . R .  trum,  they  take  absorption  lines  in  are  a natural  features  of t h i s were  the  lattice  as  a f u n c t i o n of  the a  diffusion  double  to  of the  features  would  and t o  as  and o f a b s o r b e r  study  source  that  line  the  rather  whether  the  source  thickness source  and,  of  make a  The  of  thorough  use  of  certain features  diffusion  p r o f i l e of the  between t h e  spec-  composed  or not  a f u n c t i o n of source  between  Margu-  e m i s s i o n and  than  was t o  the line  Mossbauer  theoretically.  thickness,  differences  of S.  o b t a i n e d by the  determine  difference  of the  i n many c a s e s ,  therefore,  and o f a b s o r b e r  of the  First,  intensity  absorption  and  into  line  absorber,  line shifts  and a b o s r b e r  it  is  l i n e w i d t h and  possible that  and w i d t h o f t h e features  be r e c o g n i z a b l e  learning  how t o  to  expected  Second,  achieve on t h e  the  as  of the  such,  s t u d y has  the  intensity  current  and  spectrum by t h e o r y  rather  than  a practical  narrow  lines,  i.e.,  basis  of the  half  measured life  has  theory  Mossbauer spectrum  l i n e s h o u l d be p r e d i c t a b l e  perimental errors.  those  fact  Zeeman e f f e c t  intensity  e x p l a i n a l l the  features  of the  of  as the  source.  importance.  ficient  study  by t h e  temperature  of the  e x p l a i n the  theoretical  an o u t s t a n d i n g example,  lattice  line  of the  This  was t h a t  of thi3 thesis  f u n c t i o n of temperature  diffusion  to  the  calculation  work,  the  successful  measurements  this  l i n e c o u l d be u n d e r s t o o d  considered source  rigorous  Mossbauer a b s o r p t i o n  iron  no c o m p l e t e l y  experimental  prior to  is  split  The purpose of the  to  i n t o account  iron  thesis,  . The most  Ehrman.^ In t h e i r  in natural  investigation  theory  l i n e shape,  d i d not  lines.  fit  spectrum  which  single  w r i t i n g of t h i s  h a d b e e n made t o  Mossbauer  lies  the  shifts.  is  put  A l l the  any  attributed  forth  insuf-  adequately. so t h a t  a  to  new ex-  importance,  that  l i n e widths  equal  of the  radiating  of  nucleus.  The t h e s i s , Mossbauer E f f e c t the  .  i n Chapter I  Chapter  deals  Mossbauer v e l o c i t y spectrum  shifts  f o r the  case  of  Pe^7.  equipment and t e c h n i q u e s perimental V,  II  the  w o r k done  dependence  dependence  of the  line  or  "Mossbauer  line  shape,  Chapter  used.  width  III  Chapters  absorber  line".  i  fundamentals  theoretical width,  the  annealing of the  be a b b r e v i a t e d  to  line  experimental the  source;  Chapter V I , the  Throughout the  the of  i n t e n s i t y and  describes  thickness;  of  aspects  I V , V and V I d e s c r i b e  and s h i f t s .  spectrum" w i l l  w i t h the  with the  Chapter I V , the  on t h e  "Mossbauer v e l o c i t y trum"  -  deals  ex-  Chapter  temperature  thesis  "Mossbauer  the  phrase,  spec-  Chapter  I  GENERAL FEATURES O F T H E MOSSBAUER E F F E C T and I T S USES AS AN EXPERIMENTAL TOOL  Introduction  1.0  Nuclear served, nuclei or  for  two  reasons.  displace  no o v e r l a p  resonant  the  of  the  the  greatly  increase the  so t h a t  any  later  the  emitting  that  the  whole  lattice  ening  by t h e  sociated  thermal  chapter Mossbauer  been  Effect.  in their  contained  herein w i l l  cifically  to  II  energy  the  gamma  absorbing  so t h a t  does not  motions  little  take  of the  under c e r t a i n  exists  is  so  a finite  tightly (or  the  atom,  lattice  and t h e rays  probability  absorbing) by the  and t h e  lines  conditions  bound t o  characterized  place.  nuclei  e m i s s i o n and a b s o r p t i o n  emitting  whole  line  absorption  However,  of  and  ob-  the  f,  lattioe  nucleus  re-  mass  the  of  Doppler broadthat  of  the  Doppler broadening  of  such  have  rather than  the  energy  and w i d t h  as-  transition.  in Frauenfelder's  and H . E . H a l l  Chapter  of the  recoil  has  mass  emitting  thermal  is  c a n n o t u s u a l l y be  absorption  the  nucleus the  rays  of the  there  energy  by the  nuclear  out.  than  recoil  the  the  width of  n e g l i g i b l e and t h e  This  Boyle  due t o  chapter),  motion  both  with the  summarized  the  recoil  from the  spread  rather  o f gamma  and r e s o n a n t  absorbing)  rather than  is  ground of the  (or  ease,  Hence,  rays  is  in this  lattioe  In t h i s  whole  gamma  occurs  effective  overlapping  that  nucleus.  two  the  line  Doppler broadening  be g i v e n  coils.  First,  emission  Second,  (to  absorption  primarily to  Sinoe t h i s  book,  present  background theory  "The Mossbauer  article,  be g i v e n  experimental  designed  Effect"^  "The Mossbauer  in abbreviated work d e s c r i b e d  form. in this  a  back-  has  been  and by  Effect",^  A.J.  the  theory  Theory r e l a t i n g thesis  is  well  given  spein  -5-  1.1  Recoiless  E m i s s i o n and  A gamma r a y centered  at  ting #r  is  the  energy  atom a n d E  /anc^  D  identical  nucleus,  E„ =  since  the  E  +  0  width  nuclear t r a n s i t i o n ,  = k#R  is  range  found t h a t  x  occur  only  E ^/mc^ > P occur  For example,  temperature^  /\E = -  where  k is  the  mass  of the  emit-  e m i t t i n g atom  where  be a b s o r b e d b y a n  energy  for  is  is  2  0  transition.  E  where T i s  However, f o r  absorption could very  little  for  emitted  Fe^,  of atomic  natural  10 kev < E <  100  Q  be e x p e c t e d  (Boyle,p.445),  E ^/mc^ =  ev.  .004  Q  absorption  is  expected.  nucleus,  Doppler broad-  thermal motions  mass m = 5 7 ,  the  and  of the  emit-  broadening  is  y  1-3  constant.  In marked c o n t r a s t  to  the  gamma e m i t t e d  kev,  e m i s s i o n and  .Olev  Boltzmann  the  reso-  overlap of the  from a f r e e  p r o d u c e d by t h e  a gas  = 300°K  (k#/mc )£  In s u c h a c a s e t h e  n e g l i g i b l e resonant  gamma s p e c t r u m  at  the  of the  order to  recoil.  For example,  ening  pected  In  E ^ / 2 m c ^ < zV  so t h a t  D  when a gamma r a y  atoms.  if  nuclear  10~-'ev . a n d h e n c e  of the  r e c o i l energy  also  i n which resonant  spectra  m is  1-2  In a d d i t i o n ,  ting  the  g a m m a . r a y must h a v e  d e f i n e d by t h e  absorption  spectrum  E,  line  P=4.5  of the  this  a b s o r p t i o n would  is  an e n e r g y  1-1  absorbing nucleus w i l l  energy  n u c l e u s has  " r e c o i l temperature".  nant  it  by a f r e e  Eq — Eo*"/2mc^  may be c a l l e d t h e  the  emitted  energy  Eg —  where E  Absorption  from a f r e e  nucleus,  is  ex-  the  one  which i s  lattice.  In t h i s  Table  1-1,  there  emitted  from a nucleus  case  is  it  exists  found t h a t  a finite  that  is  subject  tightly to  probability f,  the  that  bound i n t h e  conditions the  recoil  crystal  listed  in  energy  given  i above M is the  is  replaced  the  mass  by E  of the  0  2  /2Mc  whole  Doppler broadening of  and a b s o r p t i o n i.e.,  the  the  Table  ray  state  lattice  ray  the  initial  of the  is  not  state  from the  is  source  <L |exp(i k - -X - ) -I ~  -  \<L \exp{-  i  then  -  f  X  2  A  2  )  L  the  fraction  pression, to  the  f  o f gamma is  called  lattice  f  r  rays the  is  the  Fe  5 7  0 -  2  the  resonant  and  emission  absorption,  under which a  energy  other  loss  words,  the  of a phonon.  fraction  of  to  the  lat-  emission  If  of  indicates  recoilless  gamma  rays  (Frauenfelder,  p.30)  where X i s t h e d i s t a n c e o f t h e . e m i t t i n g atom f r o m e q u i l i b r i u m position.  without  factor  is  1-^  recoil  is  In the  g i v e n by the  for  0  D  g i v e n by a s i m i l a r Debye  ex-  approximation  equation  f =  exp(-3E  for  /2mo /2k© ) 2  solid.  420°K.  D  f  1-5  ) ex-l  In c h a p t e r  =  Debye a p p r o x i m a t i o n  2  of the  1-1.  the  0  J  ©d  k©j)  in figure  a function of 9 0,  In  Debye-Waller f a c t o r .  Debye t e m p e r a t u r e  (f3^/Q) i s p l o t t e d  At  loss  exp(-2W)  0  as  energy  where  S  i>  absorbed  vibration this  mc  where0^  recoil  expression  1  The  recoil  conditions  emission  .  1  ±  without  g i v e n by the  =  f  2  obtained.  by t h e  lattice, is  the  unchanged.  accompanied  of the  lists  emitted is  m, t h e  producing nuclear  1-1  may be  M->>  (k#/Mc ) E  r a y s w o u l d be s o s m a l l t h a t  would o v e r l a p ,  When a gamma  gamma  D o p p l e r b r o a d e n i n g by  Since  s u c h gamma  Mossbauer E f f e c t  the  emitted  lattice.  Mossbauer E f f e c t .  measurable  tice,  spectra  and t h e  2  Using x =  VI,  f  has  A t © = 300°K,  reduces  » the  function  been c a l c u l a t e d f =  .78  for  Fe  to  1-6  for 5 7  .  1.0  0.9  0.8 0 (x) _ l x  -1  0.7  0.6  0.5  0.2  oA  1  0.6  1  0.8  1.0 x  FIGURE 1-1  * ©>/e  1.2  lA  ±  1.6  1.8  2.0  The value of the Debye function, (x), as a function of the r a t i o of the Debye temperature, (h^, to the ambient temperature, Q .  -7T H E CONDITIONS WHICH MUST B E S A T I S F I E D TO PRODUCE AN OBSERVABLE MOSSBAUER E F F E C T a)  The e x c i t e d  state  must d e c a y t o a g r o u n d s t a t e  b) The ambient t e m p e r a t u r e temperature c)  The c r o s s  a n d t h e Debye section,<f , Q  i n t h e same  nucleus.  must b e l o w c o m p a r e d w i t h b o t h t h e r e c o i l temperature.  o f t h e gamma t r a n s i t i o n a t  resonance,  must be  large. d)  The e x c i t e d  e)  T h e Debye t e m p e r a t u r e , © [ ) ,  nitude  s t a t e must h a v e a l a r g e  as o r l a r g e r  f)  The g r o u n d s t a t e  an  absorber.  g)  The e x c i t e d  a  p r o b a b i l i t y f o r gamma  of the c r y s t a l  must be o f t h e same m a g -  than the r e c o i l temperature, must b e a v a i l a b l e  decay.  ©r.  i n s u f f i c i e n t q u a n t i t y t o make  s t a t e must be a v a i l a b l e  i n s u f f i c i e n t q u a n t i t y t o make  source. Table  Transmission ing  1-1  through a resonance  t h e v e l o c i t y between t h e s o u r c e  absorber  c a n be v a r i e d b y v a r y - ,  and a b s o r b e r , (see s e c .  1.4). The r e s u l t -  i n g Mossbauer spectrum has a width g r e a t e r  than  or equal to twice  of t h e n u c l e a r t r a n s i t i o n . The t r a n s m i t t e d  line  intensities  dependent  as a r e s u l t  factor,  (in the expression  1.2  f  Internal  Field  width  dependence o f t h e D e b y e - W a l l e r  f o r the line  intensities,  Eqn. II-18).  feature  o f the Mossbauer E f f e c t  o f the Mossbauer a b s o r p t i o n spectrum  i s the f a c t  i s narrow -  n a r r o w e r t h a n t h e Zeeman s p l i t t i n g o f t h e n u c l e a r m a g n e t i c example, splitting x  temperature  Effects  An i m p o r t a n t line  of the temperature  are  the width  f o r F e ^ i n Fe, the l i n e of the f i r s t  excited  width  state  10"^ e v . a n d 1.90 x 10"^ e v . ^  Therefore,  the  i n many  cases,  substates.  For  i s 9 x 10"^ e v . w h e r e a s  and t h e ground s t a t e  that  t h e Zeeman  o f Fe57 a r e 1.07  the Mossbauer E f f e c t  has been  -8-j  used t o  observe  and a l s o tures  to  the  at  X  and a b s o r b e r ,  the  o n l y i f the  different.  the  extent,  by t h e  present  is  nuclear  from the  1.3  Lattice  arises  charge  3d  effect  Fe,  of the  is  the  lattice  the  maxima o f t h e  that the  the  gives  rise  Chapter  1.4  II  excited  a 3d m e t a l ,  quadrupole dens-  The s h i f t  n  iron)  Fe,  and t h e  is  and g r o u n d s t a t e s  c o n f i g u r a t i o n of the  ca3e o f  the  7  struc-  s electron  of the  ,  a difference  Mossbauer e f f e c t lattice  and, to  are  electrons  a  limited  only other  isomer s h i f t  element arises  s h o u l d be v e r y s m a l l .  shifts  vibrations  Mossbauer  and a b s o r b e r  between t h e of the  in several  or absorber  ways.  determine the  8  First,  occur.  between t h e source  essentially a  nuclei  f  A difference source  and  and a b s o r b e r Third,  an i m p u r i t y i n t h e  which  i n Debye  absorber  (the  Joseph-  the  lattice  l o c a l i z e d mode. T h i s  fact affects  l o c a l i z e d mode  and b r o a d e n i n g o f t h e M o s s b a u e r a b s o r p t i o n  a detailed  analysis  of these e f f e c t s  is  between  v e l o c i t y at  Mossbauer spectrum.  and e s t a b l i s h e s  both a s h i f t  contains  is  source  dynamics. Second, d i f f e r e n c e s  i n a v e r a g e mass  i n temperature  a l l produce  to  i n which the  Mossbauer a b s o r p t i o n spectrum  emitting nucleus  lattice  electric  and a b s o r b e r .  F o r Fe57 i  states,  chemical  source  electron  lattice  dynamics o f source  and a d i f f e r e n c e son E f f e c t )  of  i n the  c h e m i c a l bond a n d h e n c e ,  determined by the  temperature  nuclear  Dynamics  imbedded i n f l u e n c e t h e  primarily  radii  chemical environment.  The d y n a m i c s o f t h e are  from a d i f f e r e n c e  c o n f i g u r a t i o n i n the  like  differences  i s o m e r s h i f t s , and t h e  i n f l u e n c e d by the  Co w h i c h ,  only  the  due t o  n u c l e u s between t h e  j^'(o)|^is  (primarily,  Zeeraan s p l i t t i n g o f v a r i o u s  small shifts  The i s o m e r s h i f t  Jy(0)J  seen  measure  i n source  moments. ity  d i r e c t l y the  spectrum.  .  Experiments  The equipment n e c e s s a r y  to  perform a Mossbauer experiment  consists  of  a)  a source  tive  to  the  quantity as  a  of  gamma  source,  measured  function  radiation, is  is  i n which both  line  of Lorenzian shape,  shown  in figure  spectrum.  Its  f7 a n d ,  section  gamma  for  absorption then  spectrum one  arises  absorption  arises  from the  sorption should  lines.  is  This  depth  as  from the line.  be c e n t e r e d  the at  of  all  source v =  as  I-2b.  overlap  0.  the  spectra  is  of  consist  oalled  half  life  at  f,  lines,  absorption  dip  the  of the  called  the  Mossbauer  lines  with the  are  dip  nuclear cross and  spectrum  Mossbauer lines  with  in this  corresponding the  the  single  emission  Mossbauer  line  In  Mossbauer  arid t h e  emission  identical,  a  of the  in this  one  and a b s o r b e r  the  the  least  emission  of  The  absorber  absorption  is  case with  Lorenzian  Each  line, the  the  the  and a b s o r b e r .  o n l y one  spectrum  rela-  radiation.  through  source  displays  of which,  o f gamma  Debye-Waller f a c t o r ,  of several  The c e n t r e  overlap If  If,  velocity  rays  p r i m a r i l y by t h e  by t h e  in figure  gamma  between  transmission  absorption. consist  the  transmission  the  a detector  and a b s o r p t i o n  determined  the  spectra appear  the  of  velocity  emission  I-2a.  width  transition,  will  the  and c )  transmission  relative  case  as  an a b s o r b e r ,  controllable  the  of the  b)  Mossbauer  thesis, abline  to follow page 9  1.0 Normalized Transmission  0 Velocity (a)  1.0 Normalized Transmission  0 Velocity FIGURE 1-2  Typical Mossbauer spectra f o r (a) a single l i n e emission and absorption spectrum, and (b) a s i x l i n e emission and absorption spectrum.  -10Chapter  THEORY O F L I N E W I D T H ,  The p r i m e o b j e c t i v e thesis  was t o  sbauer  spectrum  with  the  embedded  correlate  theory  in natural  INTENSITIES  the  iron  various  could then  velocity  of F e  shift i  5 7  iron  shifts  and  iron  n  be d i s c u s s e d  the  been used f o r t h e  be a t t r i b u t e d  iron  source  properties  II-l,  intensity  and an  in this  to  the  rather than to  iron  chapter  this of the  absorber,  with  special  obtained with a source  absorber. etcetera,  This  of  particular  of Fe57  is:  absorber  or the  or p o s s i b l y to  to  the  1= +7/2 I =  +5/2  I =  +3/2  I =  +1/2  steel  case  lattioe  source. the  Co  5 7  5 7  or  Effects embedded i n  For referenoe  Mossbauer E f f e c t  pur-  are  given  -Co57  7v  0.136  0.0  10%  9<¥ x  10"  o = i.4 = 15  x  IO"  9  ;i8 1 0  ! i  om  <S^ = 420°K  0>2/3k =0.023 e v . R =0.002 e v . in natural iron: a -2.17$ Table Selected  mev.  0.0144 mev.  r =4.5 c7  Debye t e m p e r a t u r e : Mean l a t t i c e e n e r g y : Free r e c o i l energy: Fe  Fe57  c o u l d be d i s c u s s e d w i t h  a n unknown c a u s e .  pertinent  width:  of  Mos-  of  Cross s e c t i o n of resonance: Internal conversion ooeffioienti  Abundance  iron  in  below.  T h e d e l a y scheme  Line  width,  Mossbauer spectrum  F*57 Table  shape,  and a n a t u r a l  alloy lattice  natural  poses,  e x p e r i m e n t a l work d e s c r i b e d  e a s e t h a n would have been p o s s i b l e had a s t a i n l e s s  some o t h e r found  line  which w i l l  being given to  chosen so t h a t  greater  the  of the  obtained with a source  relevant  attention  the  SHIFTS and  Introduction  2.0  was  II  II-l  Characteristics  of  Fe  5 7  (Fe  5 7  )  (lO"7 ec) S  in  -11The M o s s b a u e r s p e c t r u m  of  the Fe57 radiation  a  natural  Fe  The f a c t s nuclei that  absorber  that  are  the  split  is  calculated  g r o u n d and e x c i t e d  Into t h e i r  the  souce has  a  lations.  The f i n a l  equation  computer  and t h e  culated  is  the  ratio  and a b s o r p t i o n less  steel  of  the  through  spectra  are  its  effect,  temperature. the  source  effect. ation  of the  increased  difference  of  the  steel  is  devoted  the  in width.  between  Josephson  calcu-  7090  I.D.M. Also  cal-  iron  inhere  emission  both the  from a  (where b o t h  stain-  emission  cause  size  herein,  of  in chemical  such been  the  and s i g n t o  the  Is  discussion  either  for  and t h e herein  and o f  s i m i l a r to  shifts  shifted  that  the  thermal as  of  from  is  This  the effect  expansion  a function  a v e r a g e mass of the  the  of  between  Josephson  source/absorber  combin-  calculated.  causes  environment  effect,  and p l o t t e d  shifts  have  to  One s u o h m e c h a n i s m  effeot  i n Debye t e m p e r a t u r e  magnitude  the  s o u r c e and a b s o r b e r .  A difference  This  for  and  from a n a t u r a l  absorber  calculated  considered  absorbing  i n the  intensity,, of r a d i a t i o n  of which are  and a b s o r b e r  absorber.  absorber  and  i n Chapter V .  Mossbauer spectrum  Another mechanism t h a t  the  iron  chapter  by w h i c h the  The p o s s i b l e  difference  appear  chapter.  Zeeman e f f e c t  was p r o g r a m m e d  radiation  a stainless  o f ^wo f a c t o r s ,  both  by the  of  the  emitting  both considered  calculation  split)to  of t h i s  through  unsplit).  of a temperature  composed  II-lS)  intensity  unperturbed p o s i t i o n or  effeot is  mechanisms  are  section  of the  substates  a natural  are  The remainder various  (Eqn.  of the  source through  and a b s o r p t i o n  magnetic  first  states  finite thickness,  results  source transmitted  i n the  transmitted  shifts  of the  isomer  shift,  be e x p e c t e d w i t h  is  of Mossbauer s p e c t r a  is  nuclei  and  i n the  discussed  Je57  ±  s  source  and an  given.  a in  estimate  -12Considerable ducing the this the  attention  field  magnitude the  is  of the Fe^7  n u c l e i i n the  plete  o v e r l a p p i n g of the  temperature  effect  is  width  size  of the  any d i f f e r e n c e  source  and a b s o r b e r  Second,  of the  field  pro-  field  the  magnetic  transmitted  determines  i n the  results  of  magnetic i n an  incom-  thereby  giving  field  is  spectrum w i l l  temperature also  be  dependent.  The of the  the  magnetic  e m i s s i o n and a b s o r p t i o n s p e c t r a ,  a broadened Mossbauer spectrum. so t h a t  the  s p l i t t i n g so t h a t  at  the  r a d i a t i o n . The i m p o r t a n c e  twofold. First,  field  that  then given to  hyperfine s p l i t t i n g of the  magnetic  dependent  is  electric  magnetic  expected  q u a d r u p o l e moment p r o d u c e s  field.  to  Since  Fe  is  an e f f e c t  approximately  be s m a l l b u t c o u l d l e a d t o  similar  a cubic  to  lattice,  this  a broadened t r a n s m i s s i o n  spectrum.  The  final  mechanism c o n s i d e r e d  In the  Mossbauer e f f e c t ,  in  source  the  the  spectrum  lattice of  quences  of the  process  peak  transmitted  spectrum.  concludes  to  souroe  First  spectrum at  this  l o c a l i z e d modes.  essentially  an i m p u r i t y  a l o c a l i z e d mode w h i c h  atoms.  i s the  The d e r i v a t i o n o f t h e  There are  existence the  of a  two  theory  conse-  one p h o n o n  l o c a l i z e d mode recoilless  affects  frequency  peak o f  p r e d i c t i n g the  the seoond  chapter.  The f o l l o w i n g Margulius  rise  of the  is  s h i f t i n g and b r o a d e n i n g o f t h e  T r a n s m i s s i o n of the  and S .  of the  transmitted  the  2.1  giving  l o c a l i z e d modes.  i n the  that  emitting nucleus  thereby  n o r m a l modes  and s e c o n d ,  effect  the  is  Six  line  theory  and J . R .  Spectrum of  arises  EhrmaxP.  Fe^?  from t h a t  However, the  g i v e n b $ J . G . Dash e t facts  that  the  source  al,^ has  -13a f i n i t e thickness and the emission and absorption spectra each consist of s i x l i n e s , are considered. An exact equation giving the transmission as a function of v e l o c i t y between the source and absorber i s derived.  Since Pe i s a ferromagnetic material, the Zeeman effect o r , the coupling between the nuclear magnetic moments of each energy l e v e l and the internal magnetic f i e l d H, acting i n each ferromagnetic domain removes the degeneracy of the nuclear magnetic substates producing a set of equally spaced spin sublevels of energies Enf = mg^ H s  n  - I f raf I  II-l  where m i s the magnetic quantum number, g i s the nuclear gyromagnetic r a t i o andyu  n  i s the nuclear magneton. By l e t t i n g nij refer to the  magnetic  quantum number of the 14.4 kev state of F e ? and m^ t o the ground state 5  and by employing the selection rule Am  = 0 , + 1, (since the 14.4 kev  gamma ray of Fe57 i s known to arise from a dominantly magnetic dipole t r a n s i t i o n ) the t r a n s i t i o n between the 14.4 kev sublevels t o the ground state levels results i n a gamma spectrum of s i x components as shown i n figure I I - l .  raj = +3/2 + 1/2  \  ^  ME \  > 3.2 x 10" ev. 7  -1/2 -3/2 \ 14.4 kev m  k  = -1/2  1.9 x 10* ev, 7  + 1/2  Fig. I I - l  -14The the  rules  relative  intensities  governing magnetic  For  I - j 1-1  of these  I  is  n  malized  the  probability that  transmission II-l,  the  for  lines  W  are:  J k  p.459) in  m j - > m j + l  I  n  mj-nrnj  I  n  I  n  relative one t o 1/4,  has  gamma r a y  transition  and t h e  1/12,  a natural  iron lattice,  greater  so that  than  the  4.5  full  six  C(I -m ) 2  =  will  lines  of absorption splitting  10~^ e v . ,  m)(I  ra-l)  a constant. occur  is  transition are  3C, 2 C ,  Fe  given  figure  in  C, 2C,  3C  probabilities  absorber  5 7  l i n e width of the  (Boyle,  For  kev s t a t e  l i n e e m i s s i o n and a b s o r p t i o n  northe  line probabilities. 14.4  The  g i v e n by  C,  transition  A dilute  of the  the  II-2  2  = 1/2C(I  f o r the  1/4.  g i v e n by  l/2C(I-m)(I-m-l)  and C i s  six  and  1/6  the x  of the  =  j-*k  Hence,  are  i.e.,  normalized r e l a t i v e  1/12,  a similar pattern  ation  Fe^  7  and t h e kev  14.4  spectra  ground  radi-  must  always  considered.  The absorber,  lines the  the  intensities  six  1/6,  is  and  of the  -1  p r o b a b i l i t i e s , WJk.  state  be  intensity  gamma r a y s  dipole radiation^,  mj-^mj  where  six  Mossbauer v e l o c i t y of a central  symetrioal  centre study  line  of  the  phrase  line profiles.  same s h i f t tinction  is  about  the  "Mossbauer  and a l l o w s  the  However,  be made b e t w e e n  the  primed s u b s c r i p t s  will  the  be u s e d t o  central  line  line" will spectrum.  greatest  and b r o a d e n i n g m e c h a n i s m s to  consists,  for  indentical  l i n e with a complicated  Mossbauer v e l o c i t y  experimentally  studying  strong  in velocity  f o l l o w i n g , the  spectrum  the  source denote  This  line  lines  are  Mossbauer  and a b s o r b e r the  Fig.  be u s e d t o  sensitivity  satellite as  (See  set  absorber.  of  in  satellite  I-2b).  denote is  source  the  the easiest  experiments  subject  to  l i n e . When a n y i n the  In  the dis-  following,  to  -15The as  intensity  o f a n y component  of this  spectrum  is  calculated  follows:  The t r a n s i t i o n  m  j->  a) t h e t r a n s i t i o n b)  m  k  *s p r o p o r t i o n a l t o :  probabilities Wj  the population Pj  and  k  of the sublevel  which the t r a n s i t i o n  originates.  Above t h e t e m p e r a t u r e  of liquid  levels  t o be e q u a l and t h e r e f o r e ,  c a n be assumed  following. composed  Therefore,  of lines  a n d c f • "the t o t a l 0  2  tr \  z  of cross  section  resonant  absorption  , _1_ , * 1+oc  =  the population of the s u b -  (88°K)  absorption  W «ji c5 «y k  section  1.48 x I O "  1 8  i s the energy  cm  at which the absorption  intensity J-»k  of s i x lines  transmitted  section  is  is  II-4  2  line  c o n s i d e r i n g both the resonance e m i s s i o n speotrum be c o m p o s e d  cross  i n the  where:  k  cross  are not considered  ;  C  and E ^ « j «  nitrogen  the hyperfine resonant  2I|+1 2I +1  at  of Lorenzian shape,  at a relative  k - * J , i s centered.  and a b s o r p t i o n  each  source/absorber  Then,  spectrum  to  l i n e having a width P , the speed  v , i s f o r the  transmission:  foo  MJ.VO- I a W . J fXlh J(E-Ej :E J  The summation  k  1  expf-naftT W « ..C^i ..o(k-k',j-j7]clE v) rM j V * ^ K  J k  2  J  f  J  i s : j 1,...6 k 1 , . . .6  where:  ~  f ' = p r o b a b i l i t y of absorption without recoil, nrnumber o f atoms/cm^ a = f r a c t i o n a l abundance Fe- ' I ^ r t o t a l emitted intensity t=absorption thickness 5  3  -16When E j ( l - v / c ) k  sorption  i.e., a t  — E^i y,  the centre  o f one o f t h e r e s o n a n t  ab-  dips,  1^3.10=  OO  dv  IfiWjk ir  i  exp  ^  -W »j«3 K  II-6  L f r ?  - O O  (where  y = 2 ( E - E i y )/p ,  x = naf'<5 t )  k  f  0  Whioh has t h e s o l u t i o n : It(j,k) =  where J  (iW «j»  0  conventional Hence,  W  e  Joaw^jtxjexpt-Wk'j  J k  1  z.  • x/2)  f o r m i n T a b l e s o f F u n c t i o n s , b y Eugene Jahnke and F r i t z  the t o t a l  and t h e s o u r c e  transmitted  intensity  Emele.  have s p l i t ground and 14.4  (considering resonant  in  1 0  kev  absorption  is: 3^ ( r o t a l , v ) I £ j =  k k  e  tj.  J (lW , ,x)exp(-W t »x)8(k-k ,J-j') 0  k  J  k  2  In p a r t i c u l a r , called  II-7  i s t h e B e s s e l F u n c t i o n a n d may b e f o u n d t a b u l a t e d  i f both the absorber  levels, only)  x) 2  k  I  the  i f the source  nonmagnetic s t a i n l e s s  II-8  2  i n t e n s i t y o f the Mossbauer l i n e ,  the Mossbauer i n t e n s i t y  However,  ,  J  (v = 0 ) ,  c a n be c a l c u l a t e d  from t h i s  and a b s o r b e r have u n s p l i t l i n e s as steel  source  and a b s o r b e r ,  henceforth equation.  i n the case  the t o t a l  of  intensity be-  comes :  I  t  (Total,v) =  The above e x p r e s s i o n s  I J (ix/2) e  Q  f o r the  exp(-x/2)  Intensity  II-8'  considered  o f t h e gamma r a y s p e c t r u m . When t h e e l e c t r o n i c the  transmitted  intensity  T(Total,v) =  only the resonant  absorption is also  part  considered,  becomes:  (l-f)I  e  exp(-yu't')+fl  t  (Total,v)  exp(-^t')  II-9  -ITwhereyu'  is  the  gamma r a y s . of the  In  order to  calculations,  intensities for  electronic  the  example  f o r the  illustrate is  useful  two c a s e s  moment t h a t J -*l,  it  absorption coefficient  a thin  T(Total,v)=  to  source  (l-f)I  Q  a simple example,  determine the  and a b s o r b e r a r e so t h a t  -/* tVe-^ ' ,  t  14.4  the  relative  o f a s p l i t and an u n s p l i t  and e x p ( x ) - » l + x ,  0  with  o f Fe f o r  equation II-9  f l g l j ^ . j .  W  results  transmitted  line,  used.  kev  assuming  For t h i s  simple  becomes?  J k  (l-W «j»x) k  x &(k-k\j-j')  for  the  split  line  11-10  and:  T* ( T o t a l , v ) = ( l - f ) l  - / i V  e  e"^*  fl (l-x)  11-11  e  2  for  the  unsplit  C  =  line.  T ( T o t a l , v aoo)  ^  This,  split  ^ - fx  *  2  for the  unsplit  the  case  line  7]  Jk,  v  unsplit  f _fx S "  Therefore,  defined  above  is:  -T(Total,v=o)  11-12  T(Total,v=oo)  f o r the  whereas  T h e Htfossbauer i n t e n s i t y a s  case b e c o m e s :  _ f x 28 ~ 2 144  line  _  k  Jk.k'j'  ratio  i n the  ease:  W« W. ,  Y  2  11-13  J  of the thin  k  J  central  source  ~  fx 2  line  11-14  (v=0)  and a b s o r b e r  f o r the  s p l i t case to  approximation f o r  Fe57 In p r a c t i c e , sources  this  ratio  and a b s o r b e r s  is  r e d u c e d by a)  and b)  non-resonant  the  saturation  effects  backgrounds.  in  l s  7/36.  non-thin  -18In the implicitly practice, both  assumed t h a t It  is  upon t h e  of  have  are  of the  source the  and a b s o r b e r  occur  spectra  the  In f a c t ,  lines  a l l decays  sorption the  not.  source  and a b s o r p t i o n f,  d i s c u s s i o n above  thickness  without  thickness.  energy  into  six  transmission through a resonant  -e  was  width of the  a L o r e n z i a n shape  split  T(Total,v)  transmitted  intensities,  n e g l i g i b l e whereas  Mossbauer l i n e  Assuming t h a t of width P ,  and t h a t  the  lines,  equation  11-10,  for  was in  depends  both  that  loss  absorber  it  a  emission fraction  e m i s s i o n and  ab-  representing  the'Mossbauer  line  becomes:  _ / U  J«  I  IV /  , .... ,  'If"?  (£-E ) t' A 0  OO  11-15  w h e r e 7"= f a n o ^ t and x + d x  and ^?(x)dx  is  from the  surface  A source  prepared  the  number o f  of the  source.  by  Fe57m  atoms  lying  between  x  electroplating  Cc37 onto fusing  dif-  the Co57  active  Fe and t h e n  atoms  5  into the  so  is  the  resents  the  2N  a Gaussian d i s t r i b u t i o n of  exp(-x /t ) 2  rras d i f f u s i o n  total  into equation  has  radio-  that:  P(X) = where t  lattioe,  11-15  depth of the  number o f r a d i o a c t i v e yields:  T(Total,v)-  C  V * '  x i O  2  II-16  radioactive  atoms.  atoms and Nt  S u b s t i t u t i o n of t h i s  repfunction  -19-  exp 11-17  Thus,  for  a given source  calculated the  as  and a b s o r b e r  a f u n c t i o n of the  absorber.  The  R(v)-  is  plotted  (see  Sec.  various  2.2  i-f  3-7).  The c a l c u l a t e d thicknesses  The J o s e p h s o n  and R e l a t e d  of the  system  the  energy,  by J o s e p h s o n ,  In which the  function for  lattice  of the  is  and  the  spectrum for  i n Chapter V .  Effects  a gamma r a y , is  gamma r a y  who a s s u m e d t h a t  p o t e n t i a l energy  the  Mossbauer v e l o c i t y  w i d t h s and Mossbauer i n t e n s i t i e s  given  vibrations  position only.  system has  produce the  line  are  When a n u c l e u s e m i t s  discussed  source  +yW  a f u n c t i o n of v to  of  v e l o c i t y between t h e  be  )  as  conservation  t r a n s m i s s i o n can  T(Total,v)  absorber  The e n e r g y  relative  the  ratio:  T(Total, =  thickness  It  it  loses  thereby  of the  This  crystal atoms  and s o ,  effect was a  i n the  follows therefore,  .  Q  affected  energy. the  mass ^itij= E / c  that  wa8  by  the  first  conservative  crystal  was  a  the Hamiltonian  form:  11-19 i  -20and t h a t the  H=C,  change  the  in 6  total  is  lattice  therefore  energy.  Upon e m i s s i o n o f t h e  gamma  ray,  given by:  &C = < 6H > = Sm /m i  =  The average that is  the  i  < Pf/2m >  i  1  h  increased  0  2  1  of the  i  nucleus emits  the  b y & £ when t h e  gamma r a y  t  h  nucleus  gamma r a y . is  order  a relative  of:  sumptions  shift  for the  &£/E  =  Q  b)  i  the  i  0  Therefore,  emitted, to  it  i  the  oonserve  since  is  the  assumed lattice  gamma r a y e n e r g y energy.  This  must  results  <Yj>  ,  J o s e p h s o n made t h e  following  a l l have  the  same mass  and t h e  k i n e t i c energy  is  d i s t r i b u t e d among t h e m .  lead to  is  h a l f the  the  forces  the  conclusion  total  coupling the  lattice atoms  energy  were  (i.e.,  harmonic).  that:  11-22  i  i where  U is  ative  shift  the  lattice  of the  H/E  Q  in  as-  <,Y -)/m = 1/2U i  energy  11-21  The k i n e t i c e n e r g y  assumptions  <Y >and  crystal:  he a s s u m e d t h a t  These  is  2  estimate  T h e atoms  equally  gamma r a y  -<Y >/m c  order to  about  a)  2  i  be c o r r e s p o n d i n g l y r e d u c e d b y & £ i n  In  _  E /m c <Y >  k i n e n t i c energy t  n  v i b r a t i o n a l energy  gamma r a y e n e r g y  = -1/2U  /  c  2  per u n i t mass.  Hence,  the  rel-  is:  11-23  •21 • T h e Debye t e m p e r a t u r e room t e m p e r a t u r e , ,  3k6 m  giving a  of  (6),  iron  U can  relative  is be  shift  approximated  420  K and t h e r e f o r e ,  by i t s  classical  at  value  of:  i  = -2.53 x 1 0 " x  = -7-6  When t h e in  approximately  another  gamma  relative  energy  Debye m o d e l  mass o f  is  °K  for  300°K  11-24  absorbed,  shift  it  will  gain  energy,  resulting  of  "?2  Eo  unit  ray  1 3  per  -1/2 uj;  Sf  From t h e  10~  1 5  the  11-23  of  the  lattice  is:- 1  1 Z ([o)hOdO o)hOdo M / e x p ( h 0 )" - 1  U =  crystal,  the  total  vibrational  energy  1  per  2  fZo r*> +  1/2/  Z(^)hOd  11-25  wherein be  the  seen t h a t  second  term  U and  U* a r e  t e m p e r a t u r e and t h e of the i.e.,  gamma the  ray  shift  is  mass  energy of  the  the  zero point  functions of  the  of  both the  of the  of the  crystal  atoms c o m p r i s i n g  considering position  the  energy  temperature,  the  lattice.  emission  Mossbauer  lattice.  It  can  the  Debye  The n e t  and a b s o r p t i o n  line,  is  g i v e n by  shift processes,  the  expression:  Eo = 1/2  9flk Me  Eo  f /e% ePVe 3  2  XQ  I  x dx. 3  exp(x)-l  -9N'k 2M c T  ,e\ 2  Wh  3  q \ J°  /  *2*^®DJ  exp(x)-l  "8"  11-26  -22Thus,  a s h i f t c a n be c a u s e d  by any o r a c o m b i n a t i o n o f these  a)  a temperature  b)  a Debye t e m p e r a t u r e  c)  a difference  By c o n s i d e r i n g  difference's)  difference  i n t h e average  4 0,  &©£j = 0, b u t d 6  with the Josephson E f f e c t ) ,  between s o u r c e  three  and a b s o r b e r  between s o u r c e  mass (a M) o f t h e s o u r c e N  (which a r e t h e c o n d i t i o n s  i t seems  factors:  and a b s o r b e r and a b s o r b e r  associated  that:  11-27  69  where  /  C . d© _ 9Nk , 6  n  j  M  and  i s the lattice  Equation the  source  6 / J  x^dx  s p e c i f i c heat  11-27  temperatures  of the absorber  u s e d were  Josephson s h i f t  of the  12  crystal  i s p l o t t e d as a f u n c t i o n o f t h e temperature  a n d Debye t e m p e r a t u r e  The t e m p e r a t u r e  11-28  explxT-1  0  of the source  and a b s o r b e r ,  remained constant  355°K a n d 4 2 0 ° K .  i s not very s e n s i t i v e  Hence,  it  i t was i m p l i c i t l y  c a n be s e e n t h a t  t o the precise  remained c o n s t a n t . crystal line  must b e t a k e n  shift.  constant  Empirically  I  however,  i n t o account  Thus t h e t e m p e r a t u r e  pressure  (AL)  assumed t h a t  Pig.II-2.  a t 297°K a n d t h e Debye  In t h e above c a l c u l a t i o n o f t h e t e m p e r a t u r e gamma r a y e n e r g y ,  in  of  value  the  of ® p .  dependence o f t h e  t h e volume o f t h e  crystal  the thermal expansion of the  when e x a m i n i n g t h e t e m p e r a t u r e  coefficient  dependent  o f t h e gamma r a y e n e r g y  at  has two c o n t r i b u t i o n s : ^  =  A (1L\  I  ,AL  )a  (J-l)  a  /blnVx  11-29  5 -  ® = 0  355°K  3  (xlo^-3) -1 -2  -3  -  o Hi O  0  50  _1_  100  150  200  250  300  I50—Wio  k$o—5oV  H  %  6>(°K) FIGURE II-2  The t h e o r e t i c a l J o s e p h s o n e f f e c t a s a f u n c t i o n o f t h e a b s o l u t e t e m p e r a t u r e o f t h e s o u r c e . T h e t w o c u r v e s shown a r e t h o s e o b t a i n e d f o r a n a b s o r b e r t e m p e r a t u r e o f 293°K, a n d D e b y e t e m p e r a t u r e s o f 355°K a n d 4 2 0 ° K .  ro to  -23where t h e of the  first  term  is  the  Josephson  thermal expansion which,  I (*L) (jLL.) E  0  a  *P  which  is  opposite  times  smaller.  {hM.)  e  in sign to  This  term  is  at  v  the  Josephson  temperature  of the  K  . The s e c o n d t e r m  -1 V  II-3.  (6V / j P )  e  The t o t a l  and a b s o r b e r dependent  is  shift  isothermal  caused  therefore  shift  dependent  i f the  effect  per  11-30  °K  and a p p r o x i m a t e l y because  of the  ten  temper-  instantaneous  compressibility  is  a function of 9 in  plotted  by a d i f f e r e n c e  the  as  i n temperatures  d i f f e r e n c e b e t w e e n t h e s e two  figure  between  source  temperature  terms. C o n s i d e r now t h e  which^©-Q as  +0-15 x l O - t f  =  second  is  295°K,  a t u r e dependence =  term and t h e  more c o m p l i c a t e d c a s e  ^ 0. U n d e r t h e s e c i r c u m s t a n c e s ,  i n which &6  equation  = 0, b u t  c a n be  11-26  in  written  follows:  ©0/® x3dx  3dx _i Eo  2Mc2  WJ°  e -l x  PCI  {qT)%  f  9Kk  l6Mc2  D  D  11-31 which,  when 6 = 0 ,  reduces  to  11-32  In  order to  the  the  estimate  the  a s s u m p t i o n i s made t h a t  maximum v a l u e the  source  is  of such a s h i f t essentially  for  Fe-^  7  in  Fe,  a  Co57 lattice a Debye t e m p e r a t u r e with 11-32  o f ©•Q=385°K and t h a t  a Debye t e m p e r a t u r e g i v e sA(£L) o E  = -3-2  of  ©  x 10"  D  =  420°K.  the  absorber  is  an i r o n  Using these values,  with lattice  equation  H-33  •155  .150  -  .145-'  (xlO-15) .l'tO  -  .135 50 FIGURE  II-3  100  150  The t e m p e r a t u r e crystal  200  dependent l i n e  p l o t t e d as  250  shift  a f u n c t i o n of  the  300  350  a r i s i n g from the absolute  400  thermal expansion of  temperature  of  the  crystal.  the  whioh c o r r e s p o n d s t o of  x  -9.6  10~  between t h e equation above,  cm/sec.  4  source  and a b s o r b e r  E  other  since  reasonable as  low as  ature in  x  than  the  source  expect  that  absorber,  first  term  same a s s u m p t i o n s  in  as  cm/see.  4  Henoe, these  1  of:  11-35  if  such that  to  of the Mossbauer l i n e  o n l y s m a l l amounts  Debye t e m p e r a t u r e  to,  tend  4  contains the  shift  two e f f e c t s  not the  exactly actual  of the  equal to  shift  and a b s o r b e r w i l l  of Co57,  jt  source  will  the  is not  be  Debye t e m p e r -  caused by a be e x p e c t e d  difference to  be much  1.3 x 10~* . I n t h e a b o v e d i s c u s s i o n , t h e a s s u m p t i o n was t h a t 4  lattice.  This  proximately 60°K^  atom o f t h e  lattices  assumption Is  was  independent of the  usually valid  for crystals  region w i l l  be c o n s i d e r e d  temperature  consider the  case  and a b s o r b e r  orystal  differ,  Debye t e m p e r a t u r e  of each  is  source  ature  and t h e  tains  Co57 a t o m s , mass  of the  (the  4  Finally,  average  Under t h e  distance  11-34  of source  Debye t e m p e r a t u r e  of the  10~  385°K b u t n e a r e r  of the  When © 4 0, t h e  the  1 4  -1.3 x l O "  =  Debye t e m p e r a t u r e  less the  to  implies that  absorber  of:  a n d g i v e a maximum n e t  AA  However,  10-  and  o  o r a v e l o c i t y o f 5.6 annul each  increases.  in a shift  x  1-9  =  velocity  be c o n s i d e r e d .  term r e s u l t s  l£_)  A (  v e l o c i t y between s o u r c e  The n e g a t i v e  must a l s o  11-31  this  a relative  the  of the  & M = M-M  average  absorber  1  mass  i n which the  of the  the  average  but  above  mass  will  Since the be l a r g e r  ap-  later).  of  i n which the  same.  source  temperature  the temper-  source than  conthe  by an amount:  11-36  -25and  hence:  6 (  Under the that  the  i£) -  assumption that a v e r a g e mass  the  of the  Co57  to  obtained. Fe was  through t h i s  However,  kept  absorber  accompanied  and t h e causes  a change  b)  nuclear  wave  (chemical) nuclear wave  than  1  0  (See  the  of  preparation  .01  ~  1  and  57  of:  so  that  of the  the  source,  the  maximum s h i f t  ratio  of  possible  6  n  also,  l o c a l i z e d modes  and h o s t  due t o  mass  -39  difference  lattice).  from the  charge w i t h i n  a)  nuclear  the  changes.  transition at  In  the  state to  the  ground  between  the  nuclear  interaction  the  nuclear levels  charge  radii  volume. (the  the.se  order  to  functions  see  the  two a t o m i c  nucleus,  are  systems  compared.  shift)  two s t a t e s  sensitive  isomeric  w h i c h have  state charge  interaction  to  shift,  but  are  appreciably with  are  nuclear  The  isomeric  of the  wave f u n c t i o n o v e r l a p s  and c )  i n the  isomeric  energy  nuclear  electronic  functions  functions  Fe^'  i n the  only i f  ferent,  a shift  is  11-38  by an e l e c t r o s t a t i c  electronic  observed  55-85*  is  atoms  Shift  The d e c a y is  x  r a d i a t i n g nucleus  Isomerio  2.3  2  c a n be n e g l e c t e d  between  atoms  source  is:  ' ' ^> 1  of the  1 4  i n the  less  mechanism  a(iL) =  which  to  n-37  a v e r a g e mass  A/iLx = -1-25 x 1 0 "  w o u l d be  -  Cj.de  is  dif-  the  external the  different  same electronic  -26Only t h e d i f f e r e n c e thus  i n t h e isomer  shifts  between  t h e two systems  w i l l be  measured.  Using derived  a simple model, Frauenfelder  the following  S  expression  |ftZe  =  (<Re>  2  2  pp.53-57)*  (Prauenfelder,  f o r the isomeric  shift:  - <R&> )[|y o^ - | ¥ ( 0 > f ] 2  -  n  (  4  0  2 i n w h i c h Ze i s t h e n u c l e a r radius  a n d |VJ^{o)J^  pected  that  these  have  equation,  i  **  s  t h e major  total  i e  <R>  electron  i t was a s s u m e d shift  <.R>  =  2  that  4ftjJ>(r) r  2  upon a s i m p l e m o d e l ,  i t gives  charge  density at the nucleus. from t h e s e l e c t r o n s  at the nucleus.  over  It i s exas o n l y  In d e r i v i n g  the nuclear  this  involved  volume and t h a t :  dr  II-41  charge d e n s i t y .  requirements  mean s q u a r e  t h e wave f u n c t i o n o f t h e e l e c t r o n s  remained constant  w h e r e f{r) i s t h e n u c l e a r the three  i s the root  contribution arises  a f i n i t e probability density  i n t h e isomer  cludes  charge,  listed  Equation 11-40 t h e r e f o r e , i n -  above.  the correct  Although the equation  magnitude  i s based  f o r t h e isomer  shift.  57 The measured combination  of the difference  ground s t a t e s the  nucleus  that  for Fe  o f the source 5 7  presence in  this  shift  between  and, the difference  ,  of the chemical lattice  isomeric  environment  therefore,  measures t h e  t h e charge r a d i i o f t h e e x c i t e d and  between  and a b s o r b e r  the charge radius  w i t h Fe  the total  electron  density at  n u c l e i . Measured values''"  o f t h e ground s t a t e  i s larger  o f t h e atoms a n d t h e r e f o r e ,  5  indicate  than  that  depends upon t h e  i n which t h e embedding and a b s o r b i n g n u c l e i a r e l o c a t e d . The o f an isomer  shift  i n the source/absorber  work w o u l d i m p l y a d i f f e r e n c e  between  combination  considered  the chemical structures of  -27the  source  trons as  and a b s o r b e r .  as a r e s u l t  happens  served,  The F e ^  of the r e c o i l  i n the case  7  atoms  i n the source  due t o t h e p r e o e d i n g  o f i o n i c compounds,  i s p r o b a b l y due t o t h e p r e s e n c e  Measurements  made b y L . R . W a l k e r e t a l  so that  greater  than  any e x p e c t e d  .003  than  that  isomer  f o r F e . Hence, shift  ^ indicate  that|\^(n)|  positive  f o r Co  2  2 -<Rg>  is  negative,  s i g n - and magnitude  shift  L.  a c c o r d i n g t o t h e r e s u l t s of  t h e absence  o f phase  p r e c i a b l y with temperature,  and h e n c e ,  i s not temperature shift  dependent.  less  for  a l l e m i s s i o n and a b s o r p t i o n  events.  and  instead,  i s not uniform,  sbauer  l i n e would  should  also  result  rather  be t e m p e r a t u r e  than  al.  s h o u l d n o t change a p -  The above  i n the given source/absorber  shift  t;t  i t c a n be assumed t h a t  isomeric  the isomeric  K. Walker  changes,] ^ ( 0 ) j  the  2.4  <Re>  lattice.  15 cm/sec,  In  meric  knowing t h a t  s h o u l d have  i f ob-  i n the source  2 is  elec-  121 k e v t r a n s i t i o n  such a s h i f t ,  of the 1  do not l o s e  If this  analysis  combination assumption  the  iso-  assumes  that  i s uniform is  invalid  a broadening o f the Mos-  a line shift.  Such a l i n e b r o a d e n i n g  independent.  Hyperfine S p l i t t i n g  The e f f e c t  of the i n t e r n a l magnetic  field  on t h e e n e r g y  levels  57 o f t h e Fe At t h i s  n u c l e u s was m e n t i o n e d b r i e f l y a t t h e b e g i n n i n g o f t h i s  point,  magnetic  field The  intimately Since known,  it  i s appropriate  and i t s Internal  related  the actual the f i e l d  to begin a detailed  discussion of  chapter. this  effects. or h y p e r f i n e magnetic  to the electronic  contributions  field  structure  of the electrons  a t t h e Fe n u c l e u s  cannot  o f a system  o f the magnetic to this  be a c c u r a t e l y  field  o f atoms  is  material. i s not w e l l  predicted.  It  is  -28expected ties  as  however, w e l l as  a)  that  the  the  field  the  field  arising  field  ized  (80kg t o 100kg)  into d)  the  at  upon t h e the  following quanti-  nucleus:  polarization  conduction  4s  3d e l e c t r o n s  field  the  field  from the  from the  by the  the  depend  of the  core  s  (-300kg)  ized  c)  will  l o o a l magnetic  electrons b)  the  from the  same way as  conduction  4s  which are  the  electrons  core  polar-  is  polar-  which are  admixed  3d b a n d (90kg)  overlapping  orbital  i n the  eleotrons  contribution  from any unquenched a n g u l a r  momen-  tum o f t h e 3d e l e c t r o n s (50kg)  The v a l u e s p.  120).  given  The measured  for  each  internal  cause  magnetic  is  that  field  at  estimated  for  9 = 300°K f o r  Fe Fe  (MossII, is  v  -333koe  which  a b o v e ) must able to  contribute  hypothesis  the  there  is  field the  on one  term  by q u a n t i t y  quantity  itself  although  each e l e c t r o n  of magnitude  is  upon t h e  electron  field  the  result .  is  s-like  sum o f  to  this  the  net  addition to magnetic  mechanism  (c)  above.  the  states  so  produced  The t o t a l must  be  that  fields  field  each q u a n t i t y  contributions  of  4s  contributing signs  field sign to  bands  d-wave  the  therefore  electrons  with the  opposite  addition  from  i n mind however,  of the  reason-  and 3d  the  field  listed  A  would c a n c e l  a magnetic  but  in  functions  hyperfine  kept  those  field.  mixing of  b a n d wave  contributes  observed  Hence,  large  It  field  magnetic  separately  than  conduction  hyperfine  quantities.  spin.  field  (in  w h i c h comes f r o m a m i x i n g o f t h e  a sum o f t h e  larger  mechanism  w h i c h comes f r o m a c o v a l e n t  atom w i t h  several  a fifth  p.122) f o r  The n e g a t i v e  produced sura o f  II,  contribution  a negative  another atom.  that  a negative  (Moss  positive  functions  is  indicates  that  of  many  one  each atom,  orders  depending the  magnetic  yielding a  small  A ferromagnetic contains  a number o f  magnetization  of  subject  s u c h as F e ,  domains w h i c h are  the  sample  is  the  In the  presence  the  atomic  d i p o l e moments  earth,  the  essentially in  that  upon the ternal  effective  nucleus at  has the  a single nucleus  HnoSS  the  magnetization 5 e x t / ? e f f *  c a n be  external  t  ;  i  s  w i t h the the  Figure  -  T i l e  l a t  field  field  is  e f f  at  the  Weiss l a w .  the  This  the  effective  effective  in-  4  2  in  the  in  the  the  Since Fe  is  strongly  domain. In f a c t ,  same t e m p e r a t u r e  temperature  =  dehas  i n f l u e n c e d by  however,  the  1  field  includes  sample u s e d .  appreciably  of  1  magnetization  quantity  nucleus  magnetization  two f o l l o w e s s e n t i a l l y  of  the  not  acting  by:  ter  negligible for  are  magnetization  magnetic  d i r e c t i o n of the  magnitude  of  it  dependence  dependence  is  corhas as  shown i n  II-4.  eracy of  the  nuclear  spin sub-levels the  the  that _ H  spontaneous  The e f f e c t  of  is  of  that  domain  the  net  moments  .  field, J-exf  a S 8 U m e d  p r e d i c t e d by the  i n the  Thus,  The  s u c h as  spontaneous  i n space.  represented  along the  f i e l d which i s i  direction  the  magnetic  Jext  +  The i n t e r n a l magnetic  found that that  atoms  domain the  of  related  Fe  i n each  domain and I L ^  .  the  r e g i o n so t h a t  ferromagnetic  t  of  dimensions,  magnetized.  field  of  a unit vector  x  a small external  saturation  i n which M i s  e  the  produce  Jeff  H  sum o f  aligned to  field  absence  of  macroscopic  spontaneously  vector  these domains.  of  of  transition  L  ±  of  the  internal  f i e l d was  spin orientation  energies  to  (mege -  =  14.4  kev +  a set  n  - 3 / 2 - * - l / 2 denoted by L  EQ +  produce  = mgyu H. T h e r e f o r e ,  &E  =  seen to  x  will  the  remove  the  degen-  equally  spaced  transition  energy  of  be:  «i g )/i H g  g  n  II-i*3 (.2433)u H n  been  to  f o l l o w page  29  0.7 0.6  0.5  O.k 0.3  0.2 0.1 oTI 0T2 FIGURE I I - 4  0T3  oTk  0T5  X  0.6  0.7"  The t e m p e r a t u r e dependence o f the f i e l d of a ferromagnetic material Weiss l a w .  0.8  6.9  1.0  i n t e r n a l magnetic as g i v e n b y t h e  -30where  the  subscripts  The e n e r g i e s this  table  for  it  absorption  refer  the  six  L  L  that  lines  depends  upon the  l  2  iQ  '1 2  — » - i 2  -1  -d  2  2  i 2  _ »  -1  ,  3  2 i  5  excited  transitions  evident  Li4  L  the  is  raj L  to  the  are  given  energy  or  internal  o  E  E  '  E  p o s i t i o n of magnetic  field  +  Q  i  E  o  "  . l U l ^ H  2 Table  II  -  emission  at  the  or  nucleus.  «°393MnH  +  ,0393/ H  2  From  n  -  2  below.  .llH3jU H  E  2  nucleus.  Energy  I  2  the  the  .2»»33MnH  f  o  of  i n Table II-2,  Transition  k  " i 2  —>  and ground s t a t e s  n  2 57  The t r a n s i t i o n e n e r g i e s of the s i x s p e c t r a l l i n e s of Fe i n terms of the i n t e r n a l magnetic f i e l d at the n u c l e u s .  However,  since  the  The p o s i t i o n of although this  the  field.  absorber  But  exact sorber  of  all  then  the  one m u s t  overlapping are  hyperfine  the  of  width of of  the  remember  indicates  structure  of  If  is  the  emission lines  identical.  that  the  the  lines  symmetric  independent  fields the  state that  of  the  at  predicts. the  the  source w i l l  internal  to E Q ,  field  do d e p e n d upon  nuclei  i n the  source  To c o m p l e t e l y  Mossbauer l i n e same s i x  internal are  the  with respect  and  Mossbauer l i n e would be o b s e r v e d  w i t h the  fields  are  Mossbauer spectrum  internal  lifetime  six  the  Mossbauer l i n e  should the  differ,  statement,  overlap  the  of  remaining l i n e s  be w i d e r than this  positions  fields  different,  i n the  the  d i f f e r from that  appreciate  arises  absorption  from  lines.  source  s p l i t t i n g of of  the  to  the An  and the  absorber  ab-  -31so t h a t  the emission spectrum  must  be D o p p l e r s h i f t e d  i n order  to  achieve  t resonance. line  However,  is considered  upon t h e e m i s s i o n  t h e magnitude so that  line.  of the s h i f t  the size  depends  of the necessary Doppler s h i f t  The Mossbauer  line  o f t h e e m i s s i o n and a b s o r p t i o n  shift  lines  thereby  p r o d u c i n g a broad Mossbauer Since  the  is small,  difference  the internal between  field  fields  depends  upon t h e t e m p e r a t u r e  absorber  thereby  producing  given  the  is calculated  f o r the  A ? 3  line  -1  2  =  so that  i f the  be u n r e s o l v e d ,  i s temperature  the nucleo  line.  dependent,  i n t h e s o u r c e and  between  t h e s o u r c e and  mechanism)  a temperature  Using the t r a n s i t i o n  o f t h e Mossbauer l i n e  caused  energies by t h i s  as a f u n c t i o n o f t h e temperature d i f f e r e n c e  source and a b s o r b e r .  energy  at  difference  l i n e width f o r t h e Mossbauer  mechanism  six lines w i l l  (by t h e above-mentioned  i n Table I I - 2 , the broadening  spectra  line.  absorber  dependent  these  at the nucleus  the i n t e r n a l  depends  i s produced by t h e o v e r l a p p i n g  o f a l l s i x components of the emission  upon w h i c h e m i s s i o n  I f A H = H - H ' , then the difference  between  in transition  is:  -  2  4  3 3 ^ H  11-44  2 6  The b r o a d e n i n g o f t h e l i n e culation  e'°K  148  293 295 296 293 294 295 294  |AEj^l jk• w  '®  rie  r  e  s  u  l t  of this  cal-  i s given i n Table II-3-  9°K  179 253 293 352 405 458  i s t h e sum  -145  -116 - 63 0 58 110 164  HxlC^oe  H'xlCoe  aHxlO^oe  3-39 3-38 3-35 3-33 3.31 3-27 3-21  3.33 3-33 3.33 3-33 3.33 3-33 3-33  -.06 -.05 -.02 0 .02 .06 .12  Table  II -  Increase  in L  .0178  cm/sec  .0115 .0046  cm/sec cm/sec  0 cm/sec .0046 era/sec .0138 cm/sec .0276 cm/sec  3  The i n c r e a s e i n l i n e w i d t h as a f u n c t i o n o f t e m p e r a t u r e d i f f e r e n c e between s o u r c e a n d a b s o r b e r . T h e v a l u e o f 3 . 4 2 x l O ^ o e was u s e d f o r t h e i n t e r n a l f i e l d at the Fe nucleus at © 0°K  -322.5  Electric  Quadrupole S p l i t t i n g  I n a manner s i m i l a r t o t h a t of  P e  the  57  interacts  with  I =  3/2 spin  state  of t h i s to  state  produce,  For  the e f f e c t i v e into  interacts  the e f f e c t i v e  gradients,  i  the e l e c t r i c  split  q u a d r u p o l e moment field  gradient  h y p e r f i n e s p l i t t i n g , two  i n the case  t h e two s u b - s t a t e s  moment  at the nucleus to  internal electric  o f the magnetic  a pure quadrupole i n t e r a c t i o n  field  internal field  four sub-states,  with  i n the absence  by which t h e n u c l e a r magnetic  of a x i a l l y symetric  sub-states.  electric  a r e s h i f t e d b y an e n e r g y :  .  E=  [ 3 m -1(1 +1)1 f l - %  e2qQ  2 where q -  spin  (l/e)(d  J I  J  o V/bz ),  t h e maximum p r i n c i p a l  d  of the state  11-45  2  4 1 ( 2 1 - 1 ) I-  (3/2 f o r the excited  q u a d r u p o l e moment, m i s t h e m a g n e t i c  state  field 57  o f Fe  gradient,  ),  Q i s the  I  i s the  electric  q u a n t u m number o f t h e m a g n e t i c s u b -  state  a n d if i s t h e a n t i - s h i e l d i n g f a c t o r .  field  gradients,  In the presence  t h e a b o v e e x p r e s s i o n must b e i n c r e a s e d  of non-symetric  by a  factor  I  (1+ l / 3 \ ) , ?  where  - I ? ) / 5 ^Z  ^=(4?  the is  asymmetry p a r a m e t e r  II-**  2  (Moss I I , p . l 6 9 ) .  When t h e q u a d r u p o l e  c o m b i n e d w i t h t h e Zeeman s p l i t t i n g o f t h e s u b l e v e l s  state,  the precise  quadrupole  energy  interaction,  shifts  f o r the i n d i v i d u a l  depend upon t h e o r i e n t a t i o n  relative  to the axis  o f symmetry f o r t h e e l e c t r i c  general,  a r e n o t t h e same  for a l l m states.  Interaction  of the I  m states  -  3/2  due t o t h e  o f the magnetic  field  The energy  gradient shift  axis  and, i n  then b e -  comes  t-  e  fe*  (3cos  2  0-1)  11-47  -33where  0 is  angle  depends upon t h e  the  the  electric  also  metrical  is  Co^  lattice,  metry the  i n the  7  of t h i s  Fe-*  the  observed Since  source  tric  erature  2.6  that It  of the  source  electric  dependent  source  a net  electric the  the  line  field  electric is  energies  the  the  could exist  on t h e  consisted  of  would  is  temperature  dependent. a single  shift  the  quadrupole  rather  In the  line,  in the  emis-  other  hand,  positions  the  than a s p l i t  dependent,  the  s p e c i a l case  the  at  quadrupole  lines  If,  presence  c u b i c a l sym-  of the  of the  would s p l i t .  interaction  s p l i t t i n g w o u l d be u n r e s o l v e d s o t h a t  be t e m p e r a t u r e  of the  of  p.l6o).  (Moss I I ,  that  gradient  presence  l i n e s . S u c h a s h i f t w o u l d be t e m p e r a t u r e dependence  The  magnitude  the  quadrupole  could destroy  quadrupole i n t e r a c t i o n  or absorber  axis.  quadrupole i n t e r a c t i o n  w o u l d be a b r o a d e n e d M o s s b a u e r l i n e  also  the  p o s s i b l e however,  Mossbauer l i n e  the  crystal  so that  no e l e c t r i c is  would s h i f t  the  crystal  temperature  case t h e r e f o r e ,  quadrupole i n t e r a c t i o n  sorption  is  the  absorber.  so t h a t  small,  electric  b r o a d e n i n g would the  of the  and t h e  a good a p p r o x i m a t i o n o f a c u b i c a l l y sym-  expected  In t h i s  so that  were  is  lattice  lattice  effect  the  is  Fe  field  h a v i n g c u b i c a l symmetry,  and h e n c e ,  i n the  In t h e  spectrum shifts  it  iron  nuclei.  7  interaction sion  zero  structure  Since natural  w o u l d be p r e s e n t of  magnetic  temperature  a crystal  gradient zero.  between t h e  quadrupole i n t e r a c t i o n  For field  angle  presence of the  line  i n which  o f an  elec-  resonant  d e p e n d e n t due t o  line.  ab-  the  temp-  interaction.  L o c a l i z e d Modes  In the  source  being discussed  in this  thesis,  the  emitting  nucleus  57 (Fe  )  is  a mass m the  0  an i m p u r i t y i n t h e which i s  coupling  host  source  d i f f e r e n t from the  constant  k  Q  of the  lattice  mass  and as  m of the  i m p u r i t y atoms  is  an i m p u r i t y ,  lattice  different  atoms.  it  has  Further,  from the  coupling  -34constant quency  k of the  atoms.  ~vV  which d i f f e r s  lattice  the  of the  tice.  If  the  mass d i f f e r e n c e  quencies  are  shifted to  (i.e.,  is  c  creases.  between  is  5  of the  121 k e v  .  is  localised kev  changing  «r  &' r a y  '  >  r  7.9 e v .  much l a r g e r  2  /  a  than  L  Q  r  the  of that  number o f atoms  positive,  the  the  of  expected  i n the phonon  latfre-  a n d a q u a s i - l o c a l i z e d mode  n o r m a l phonon s p e c t r u m  upon  i n the  of the  an  and  of the  establishment o f a 121  Debye  kev  of  is  impurity  of the ray.  of  frequencies  continuum).  increases  l o c a l i z e d mode  neighbours  emission  14.4  as am i n very  small,  atom.  kev  state  The r e c o i l  energy  11-49  that  of the  Wp, t h e  lattice  energy  by t h e  l o c a l i z e d mode c a n  of  i s  1  .023 ev.  emission  decay  easily  Debye c o n t i n u u m o f n o r m a l p h o n o n  l o c a l i z e d mode  6&mh Ttm w  mean  may be h i g h l y e x c i t e d  a phonon w i t h t h e  ~~ °  closest  lAi  8  Debye  phonon f r e q u e n c i e s  order  maximum f r e q u e n c y  Involved  i n the  " /  Since w <  of a  w i t h i n the  a n d Wj) d e p e n d s  mode t h e r e f o r e ,  lifetime  frequencies  Moreover,  4  is:  =  tfray.  total  the  w  by the  i n the  N is  in Table I I - l ,  preceeded  £  of the  the  D  w .  which i s  is  p r i m a r i l y of the  As s e e n  a shift  0  lower  T h e number o f atoms  consisting  of F e ^  D  atoms,  Am = m -m i s  established  wocWjy w h e r e w  The d i f f e r e n c e  lattice  impurity causes  a t o m s b y a n amount  frequency w  -  n  E i n s t e i n m o d e l , where  The  fre-  0  from the  121  a  m  from t h a t  approximation,  which  T h e i m p u r i t y atom t h e r e f o r e h a s  of:  v  the  lattice  of by  The  the ex-  frequencies.  8  11-50 D  -35Equation are  not  assumes  11-52 valid  lifetime  Is  (as  Is  the  much l e s s  Using a  that  w < < wp a n d  case  than t h i s  the  Debye-Waller factor  than  the  host  mass.  (Moss  f  =  e  i n Armco i r o n )  equation  In t h i s  II,  Increases case the  these  conditions  then the  actual  gives.  one d i m e n s i o n a l m o d e l o f a  that  expression  o f Fe  If  Am/m>> 1.  0  lattice,  i f the  it  has  i m p u r i t y mass  Debye-Waller factor  is  b e e n shown is  greater  g i v e n by  the  p.8l):  where  _ 2 w  W ©c  1  11-51  m(m +• &m)  In  order  mass o f  that the  the  Increase  i n f be a p p r e c i a b l e ,  i m p u r i t y be a p p r e c i a b l y  less  it  is  than that  necessary  of the  host  that  the  lattice.  57 Since the  this  is  not  the  so  Debye-Waller factor  i n the  case,  in this  case  For comparison purposes considered. but t h i s the  time,  highest  localized vicinity  In t h i s to  case a l l  o f Fe is  the  case  of the  of the  Q  iron  natural  lattice  lattice,  iron  i n w h i c h am< O w i l l  from the  frequency w  impurity. This  of the  b y an amount  frequency which separates  f r e q u e n c y mode o f  that  frequencies  higher frequencies  in a natural  lattice.  also  be  phonons a r e  of the  order  1/fa,  Debye c o n t i n u u m t o  can  except  form a  with amplitude l o c a l i z e d  l o c a l i z e d mode when e x c i t e d ,  shifted  in  the  decay  by  17 exchanging change  t w o o r more p h o n o n s w i t h t h e  of energy  mode a r i s e s atoms.  between t h e  from the  At a b s o l u t e  normal phonon f r e q u e n c i e s  anharmonic  zero the  Debye c o n t i n u u m .  forces  lifetime  coupling the  of the  mode i s  This  and t h e  interlocalized  i m p u r i t y and t h e given  by  lattice  1 7  2 A To"  w  o  8  *  i ( Q ) ifrow ^ov2 V  V  D  fk'm  3  ( o) ^ v  2  n-52  -36where  v is  atomic I  mass,  velocity  k'  y (l-y^)o y> 3  and k ' p  For t h i s  reduced  is  sound,  are  case,  by t h e  effect  it  presence  In both sbauer  of  of the  can  has of  be u s e d t o  of the  mode  is  the  slow high  should  necessary.  not  be  The  of  qualitatively  (i.e., this  m o d e l may be u s e d since  the  lattice  extent  is  of  the  of  F o r s m a l l mass  and t h e Hence,  0  oscillating  localized  a result mode w i l l  E  1  =  the  exert  recoilless  between  Mos-  of the  mode  #ray  spectrum).  few n e i g h b o u r s . well  occupation  D  of the  emission of the  121  be  of  a  is  used.  rest  This  is  the the  energy  equal to  the  of  model,  ground s t a t e  if r a y ,  zero  l o c a l i z e d mode  localized state kev  spectrum  establish  on t h i s  approximately  the  To  mode a n d t h e Based  single  broad  upon t h e  lattice  case  and has  is  broad  Mossbauer  effect  this  factor  Since  some  in this  in-  the  peak w i l l  of the  the  p.29).  considered,  E i n s t e i n model of a  diferences ^m, w  of the be  first  the  is  Debye-Wailer  mode. The p r e s e n c e  case,  is  0  interaction  (Moss I I ,  scanning  in a potential  p r o b a b i l i t y of the as  Q  also  interaction  the  is  the  m  =.  s u p e r i m p o s e d upon t h e  v>  in either  upon t h e  impurity |nw .  at  the  a good a p p r o x i m a t i o n  l i m i t e d to  anharmonic  l o c a l i z e d modes  resolution  effect,  as  the  and l ( x )  e m i s s i o n and a b s o r p t i o n p r o c e s s e s .  l o c a l i z e d mode c a n  phonon spectrum  lengths  impurity  detect  3hort  enough so t h a t  Gruneisen constant,  b e e n shown t h a t  cases  or m u l t i p l e phonon- Y r a y  the  attenuation  i n d i c a t e d by a peak c e n t e r e d  lifetime  is  At h i g h e r t e m p e r a t u r e s  ,  creased. is  the  w  D  small.  energy  of  the  approximately  (n+£>nw  0  H-53  -37E\ -  121 kev, which i s the r e c o i l energy of the 121 kev  ray. I f m'  0  represents the mass of the radiating nucleus i n i t s excited state (i.e., F e  5 7 m  ) , then a  m  ' o = o + _Eg m  (Ey = Ik.k kev)  0  Hence the frequency before the emission of the  11-54  ray i s given by:  = Wot "^ &m /m ) 1  o  Assuming that the  0  emission preceeds the phonon emission from the l o c a l -  ized mode, the energy of the excited state of the nucleus i s E? = <(n + | » W  where  <n+ |>  11-53*  0  i s the expectation value of n+£, and the energy of the  ground state i s given by equation 11-53- Hence the change i n l a t t i c e energy caused by the emission of the H ray i s : AE  X  = <n + |>fi(w  -w )  f 0  Q  u-56  =-<nt |> |nw fem /m 2 = -|E» /ra c (<n + |>-hv 0  0  Q  Since w  Q  0  0  =: w , n w i l l be small (of the order of 10) hence, D  large dispersions of n are possible. As a consequence of the fluctuations i n n, a broadening of the zero phonon spectrum A = |E;/m c 0  2  ( <n > - <n> )^ (hw ) 2  2  0  of the order 11-57  i s expected. Furthermore, the absorbing nucleus, also an impurity i n the  l a t t i c e , should be i n the ground state since i t i s not subjected to a r e c o i l from a preceding  decay as i s the case of the source atom. Hence,  the change i n the l a t t i c e energy of the absorber when a Z[  ray i s absorbed i s  = (£E;/m c )(£fhv )  11-58  2  0  0  Hence, there w i l l be a s h i f t i n the zero phonon peak of the order b ( & E ) = |E ;/m c < n > ^w  11-59  2  i  D  o  For both cases considered e a r l i e r , the magnitude of these  two  e f f e c t s depend upon the r a t i o of the l i f e t i m e of the Source t o the l i f e time of the l o c a l i z e d mode which, i n turn, depends upon I AmI. For the same  (ami  , the l i f e t i m e f o r am< 0 i s somewhat greater than that f o r  ara> 0. For the example of P^e  57  absorber, i t i s seen that m  0  i n Armeo iron source and an Armco iron  = 57» m = 56 and &m=l. Since Am  i s small  compared with m, the e f f e c t of the l o c a l i z e d mode on the p o s i t i o n  and  width of the Mossbauer l i n e i s expected to be very small. Further, the e f f e c t should decrease as temperature increases since the l i f e t i m e decreases as the temperature  increases.  In the discussion of the l o c a l i z e d modes, i t i s to be remembered that the introduction of the impurity may r e s u l t i n a change of the chemi c a l bond strength, thus g i v i n g r i s e to the isomer s h i f t discussed e a r l i e r .  -39Chapter  III  E X P E R I M E N T A L APPARATUS a n d METHOD  Introduction  3.0  The gathering chapter. the  source,  of the  absorber  experimental  a n d gamma r a y data  The method by w h i c h t h e  carriage  is  covered  so t h a t  i n Chapter  its  ried  and t h e  velocity  temperature  A proportional counter details  of this  relative  the  as  temperature source  well  as  to  of the  drift  by t h e  and t h e  termine  the  of this  chapter  gamma r a y s  presence  experimental  thickness  is  produce the  source  c o u l d be  of the  source  and a b s o r b e r the  the in  given  this but  Mossbauer  was m o u n t e d on a  detect  the  lathe  controlled. c o u l d be  var-  c o u l d be m o n i t o r e d .  gamma r a d i a t i o n . T h e  characteristics  of the  geometry.  of absorber  describes  through the  o f v e l o c i t y was d e t e r m i n e d . C o r r e c t i o n s t o necessitated  the  to  in  described  electroplated  source  to  are  used  of the  counter  chapter.  second part  transmission  the  was c o n s t r u c t e d  summarized i n t h i s  The  of this  of both the  construction  was  thesis  I V . The absorber  A r r a n g e m e n t s w e r e made s o t h a t  are  source  subsequent thermal treatment  effect,  for this  spectrometer  the  the  absorber  method by w h i c h as  a function  measured t r a n s m i s s i o n  background r a d i a t i o n , the Further,  counter  a n a n a l y s i s was made t o  w h i c h would g i v e t h e most  were  de-  significant  results.  3.1  Detectors  a) Nal C r y s t a l The  experimental  data  contained  in this  thesis  was  obtained with  -40the  u s e o f two t y p e s  lial  crystal  an Ar-CHjj  of d e t e c t o r s .  gamma r a y s p e c t r o m e t e r  (methane)  crystal means  f o r the f i n a l  u s e d was l g " i n d i a m e t e r  . 0 1 0 " Be w i n d o w t o a l l o w t h e h i g h e s t  e n e r g y gamma r a y s i  was u s e d ;  (those l e s s  and l / 8 " t h i c k ,  than 16 kev) e m i t t e d b y the s o u r c e . The  was m o u n t e d o n a n R . C . A . 6 3 ^ 2 p h o t o r a u l t i p l i e r t u b e #12-4-M»JJ b y  o f Dow C o m i n g  200 f l u i d  ( s i l i c o n grease with v i s c o s i t y 1,100,000  from entering the photoniultiplier tube. 1300 v o l t s .  paratus al  with this  the  Pulses  F i g . III-l  crystal.  i n this  T h e t u b e was o p e r a t e d a t  illustrates  electrical  coarse,  sorter,  live  time 10,  scaler,  discriminator bias The s p e c t r a steel  7  light  an EHT  radiation obtained  of 2.50  apparatus  of  .B/xsec;  0,  fine,  are i n -  o b t a i n e d w i t h one v o l t t o 7.00 were  a)  i n steps  of  across  . 5 0 . The s e t -  f o r the a m p l i f i e r ,  i n t e g r a t i o n time constant  dif-  of  .16  12 d b s . , b) of the n u c l e a r data  window 9 . 7 ^ , g a i n f 6 0 , dead time  shown i n F i g . I I I - l  absorber,  Each spectrum i n t h i s  8.00, c 16,  and c)  kick-  f o r the  5^sec. are those  a , 0 C l " shim s t e e l  obtained using  absorber,  a  settings.  considerably increases  .001"  and no a b s o r b e r .  f i g u r e was o b t a i n e d i n t h e same c o u n t i n g t i m e ,  a n d w i t h t h e same e l e c t r o n i c absorber  of F e ^  C a l i b r a t i o n pulses from the pulse generator  time constant  attenuation,  stainless  t y p i c a l spectra  f i g u r e . These peaks were  on the other  ferentiation ^sec;  to prevent  f r o m t h e p h o t o r a u l t i p l i e r were f e d t h r o u g h t h e a p -  h e l i p o t and h e l i p o t s e t t i n g s  tings  tape  cs)  shown i n F i g . 1 1 1 - 1 1 w i t h t h e p h o t o m u l t i p l i e r r e p l a c i n g t h e p r o p o r t i o n -  counter.  cluded  covered  transmission p o s s i b l e f o r low  a n d t h e a s s e m b l y was w r a p p e d w i t h b l a c k e l e c t r i c i a n s  of  measurements  p r o p o r t i o n a l c o u n t e r was u s e d .  The N a l c r y s t a l with a  F o r the p r e l i m i n a r y i n v e s t i g a t i o n s an  The u s e o f t h e s t a i n l e s s  T,  steel  t h e s e p a r a t i o n o f t h e 6 . 9 k e v a n d ik.k k e v  O Hi O  (9 FIGUKE I I I - l  T y p i c a l Fe  spectra  shim s t e e l  absorber,  o b t a i n e d by the and (c)  .001"  use of  stainless  a Hal crystal with steel  absorber.  (a)  No a b s o r b e r ,  (b)  C a l i b r a t i o n p u l s e s are  .001" also  shown.  -41gamma r a y p e a k s and i n c r e a s e s t h e a b s o r p t i o n of t h e 0 . 9 k e v gamioa r a y . The  component of t h e  absorption  is  the  element  17-19$ Cr) w h i c h has 6.9  k e v gamma r a y  stance kev  found to  the  gamma r a y a n d t h e r e f o r e ,  i)  Ar-CH^  6.9  the  of  were  not  steel  k e v gamma r a y used f o r the  Proportional  used  792cm"  1 4 . U k e v gamma r a y .  1  is  for  A second  the sub-  s t r o n g l y than the  grease mentioned above. 14.4  selective  stainless  k e v gamma r a y more  poor r e s o l u t i o n of  ray  i n the  its  i n s e n s i t i v i t y to  Ar-CH^ given  absence  of  either  in this  of  thesis.  Both of  as w e l l  as  detailed  l4.4  these  the  subkev  6.9  Investigations.  Counter  best  perimentally,  The b e s t  that  r e s o l u t i o n at  a factor  crystal. they  14.4  14.4  of  at  kev i s  kev i s  least  the  proportional counter  the  is  construction  33.6$  of  than the  10-20$ best  is  to  Ex-  expected is  the at  Moreover,  high energies.  counter  the  have  efficiencies  of N a l c r y s t a l s .  i m p o r t a n t when t h e  with  p.70).  proportional counter their  the  w i t h an N a l  (Moss I I ,  resolutions  of  results  c a n be e x p e c t e d  c a n b e made i n s e n s i t i v e t o  particularly  consequently  5»5$ w h e r e a s  and hence  c a n b e made h i g h e r t h a n t h o s e  k e v gamma  14.4  experimental  two g r e a t e r of  the  grease and  compile the  c a n be made v e r y l a r g e  energies  fact  at  for  prompted the  resolution that  Another advantage  low  latter  silicon  with proportional counters,  been obtained with the  Nal crystal  shifts  used to  a proportional counter the  the  Cr or the  small energy  proportional counter  crystal  fact  3^321  this  Sensitivity  The  use  f o r the  1  silicon  absorb  b)  responsible for  a linear absorption coefficient  absorb  however,  steel  chromium (type  and 3 9 6 c m "  gamma r a y was t h e  stances,  stainless  be  This  used  -42with  an  Pe57 source  with that kev an  of the  gamma r a y  i n which t h e r e e x i s t s  transmitted  of the  14.4  14.4  k e v gamma r a y  50# m o r e  counts  above t h e  Ideally, counter  the  produced a r a t i o total  14.4  s h o u l d be u s e d  (the  linear  pressure,  its  at  14.4  use.  required  a very  power as  the  linear coefficient  —2  —1  since the  cm  10" the  14.4  gas  counter  u s e d was  since  of the  x  10cm" ) 1  gas  calculated  line  (Moss  to  but the  Ar-CH^ mixture,,on to  get  be e n t i r e l y  the  lasted  only  1  expense  of  Ar-CIfy  mix-  other  hand,  Ar at  14.4  kev  A r (a  good  approximation  the  6.9  C o u n t e r was  approximately  Is  counter  f l o w . No a t t e m p t was made t o  (l600 lbs.)  of  a useful stopping  be 5 7 $ a n d f o r  contamination  of  proportional  the  s t o p p i n g power o f t h e to  120  II,pp.70-71)•  of Kr at  obtained with the  of absorption of  100$. To a v o i d gas  one c y l i n d e r  results  90# A r ) , t h e  run w i t h a continuous gas  gas  4.35  use  t h e r e were  a krypton f i l l e d  (8" diameter)  . Assuming the  k e v x - r a y s was  essentially to  large  The use  i.e.,  i n the  the  counting rate  absorption coefficient  kev i s  satisfactory  of the  1:1.5*  kev l i n e t h a n  k e v gamma r a y s ,  ture precluded  x  of  14.4  k r y p t o n and t h e  4.27  rate  compared  a r i s i n g from the  Pound f o u n d t h a t  for  atmosphere the  to  counting rate  kev c o u n t i n g r a t e ,  preceding t r a n s i t i o n .  Ar-Cfuj p r o p o r t i o n a l counter  the  a high  kev  for x-rays  designed  recover  five  the  months.  i i ) Construction of the Counter The a  23" length  to  make t h e  the  centre  .001" remove  i  counter  of aluminium p i p e , window o f t h e  of the  thick: the  body o f the  Fig.  III-2)  t h i c k and 8 "  an a r e a  a r e a was  The i n s i d e o f t h e  dirt,  1/8"  counter,  pipe. This  (see  counter  six  was c o n s t r u c t e d inner diameter.  inches  square  covered with a piece  of  was  from In  cut  remove t h e  from  mylar  was g r o u n d w i t h a d r u m s a n d e r  b u t n o a t t e m p t was made t o  order  to  remaining small  to Signal  Gas  f o l l o w page  42  out  out  0  ring  groove  Brass D u r a l , push fit, sealed w i t h R 313  / \  Al  23  (1/8")  Window, 6"x6" .001" Mylar Coated with A l Sealed with R313  r  .001  W  .005" N i Dural Al  Legs  Gas FIGURE I I I - 2  A cross-sectional  diagram  Ar-CHj(  flow proportional  continuous  of  the  In  body of  the  counter.  J O surface then  irregularities.  for  mylar.  the  w i n d o w , a l u m i n i u m was  T h e m y l a r was  secured  B o t h ends were the  further  cleaned with  a l c o h o l and d r i e d w i t h an i n f r a - r e d l a m p . To p r o v i d e a  surface the  T h e i n t e r i o r was  centre wire.  two t e r m i n a l s  f i t t e d w i t h an R. counter  to  the  the  shows  the  i d e n t i c a l except G . 560/U  plug  pre-araplifier.  flow tube  details for  of  the  providing  the  fact  the  the  for  top  one  was  connecting  . O G l " tungsten  these  seals.  ation  possible.  i n the  iii)  T h e two  Filling  removeable  Great  tungsten  wire  the  that  was  it  not  was  c a r e was in this  was  possible  feasible  to  to  have  cylinder  of  a n d b o t h were  not  iron  then f i l l e d  sidual  amounts  attained  were  to  that  nickel  two n i c k e l  fashioned to  insure  ends  inake t h i s  to  oper-  no k i n k s  remained  atmospheric  pressure  process.  counter  with the was  of  operate  a very large,  merely evacuate the  the  counter  taken  designed to  Instead  and the  plates  installation  mixture.  were  and s o l d e r i n g i t s  of  wire,  Counter  The c o u n t e r so  seals  piece  the  ( . 0 0 5 " ) . The t u n g s t e n w i r e was p o s i t i o n e d i n t h e c o u n t e r b y t h r e a d two c o v a r  short  These  wire  t h r o u g h the  a  terminal.  for  welded to  it  e n d o f w h i c h t h e r e was  of  and a t e r m i n a l  that  T h e c e n t r e w i r e was  inner side R313.  top  means  conducting  spot  ing  each  aluminium with  f i t t e d w i t h a gas  Fig.III-3  were  to  d e p o s i t e d upon the  acetone,  gas  reiooved,  air  had to  still  t h i n window. Hence,  counter  and r e f i l l  be p l a c e d w i t h i n a  evacuated mixture,  remained  at  the  essentially  i m m e d i a t e l y . However,  at  for  gas  Both  cylinders closed  Since  good r e s o l u t i o n  hours w i t h a gas  gas  larger  i n t a k e was  operation.  counter,  eight  with the  slightly  same t i m e .  counter  ready  i n the  after  the  it  rewas  f l o w of  to.follow  page  h3  R . G . U. #560  Soft Gas  Cu  solder  wire Removable  out  .005"  •  plate  K i wire  UK Kovar  Spot welded —  FIGURE III-3  A detailed Ar-CH^  >  <  001"  diagram of  propoi"tional  the  seal  W wire  top  counter.  terminal  of  the  -44-  .05  cu.ft.Ar-,  a resolution  of  a p p r o x i m a t e l y 20$ a t  14.4  kev  was  at-  tained.  iv)  Oas FDow System  The system providing the counter with a continuous gas i s shown schematically i n Fig- IU-4.  stream of  With reference t o t h i s d i a -  gram, the parts used i n t h i s system are given below.  DESCRIPTION  PARTS Counter gas  90$  V  g  and  V  3  Single stage regulator  V  4  and  V  5  V V  ?  and  V  g  Flowmeter  Ar;  10% CH  MFC Matheson Co. Ltd.  4  Low pressure "Pancake" Regulator #70-A-B Needle valve  Rollason  Imperial diamond valves 693-C  Imperial Brass  #201  {latheson Co. Ltd.  Flowmeter  Rubber hose clamp  Cenco  Table I I I - l  The flowmeter was  placed i n the system so that a) the actual  flowrate of the gas could be measured and kept the same when the gas bottles were changed and b) so that f l u c t u a t i o n s or I n s t a b i l i t i e s i n the gas flow could be detected. The meter was  constructed from a block  of l u c i t e . The gas flowed through a narrow tube so that the flow rate . was  indicated by a small metal sphere supported  of gas. The meter was  i n t h i s tube by a flow  calibrated from . 0 1 t o . 0 8 c u . f t . A r f o r dry a i r .  The correction curve f o r other gases i s shown i n F i g . I I I - 5 - The  flow  C o u n t e r gas ~(90# A r , 10$  Proportional Counter  Clfy)  c+ O  Diamond valve  V8  Hi O H  Flovmeter  JfFIGURE I I I - 4  A b l o c k diagram showing the  gas  f l o w system of  the  Ar-CH^  proportional  fl) counter.  1.5  o •ri U  I%  1.0  o Ms O H H  § 0.5  CJH4 0.5 INH1+  N  2  CO  2ffl»  H S 2  Ar  11.0 Gas  FIGURE III-5  (  (9 1.5  2.0  S p e c i f i c Gravity  The s p e c i f i c g r a v i t y c o r r e c t i o n curve f o r the flowmeter. The correction i s made by multiplying the indicated flow by the correction f a c t o r .  -45rate used i n the experiments was  v)  .05 cu.ft./hr f o r the Ar-CH^ mixture.  Difficulties Since both the r e s o l u t i o n (as noted i n the f i l l i n g procedure)  and-the gas gain of the proportional counter depends upon the gas p u r i t y * i t was v i t a l that the gas flow rate be kept constant. The required constant flow rate was obtained by including the needle valve  i n the  flow system b e c a u s e without i t the gas flow was e s s e n t i a l l y determined by  P ~P /Zf where P c  &  r e s p e c t i v e l y and I  f  c  and P  g  r e f e r t o the counter and atmospheric pressures  t o the impedance i n the gas flow of the flowmeter  appeared t o be a function of the flow rate through the meter). These conditions were s u f f i c i e n t t o produce p o s i t i v e feedback and fluctuations i n the pressure i n the counter. Vn acts as a high f i x e d impedance t o the gas flow so that the flow i s determined by P - P / I £ , which Is much c  a  v  smaller than P - P / l so that the flow i s dominated by the f i x e d Impedc  a  f  ance and the fluctuations are smaller. In f a c t under these conditions, no fluctuations could be detected by the flowmeter.  The gain of the counter was found t o be (as expected) strongly dependent, upon the pressure In the counter and h e n c e t h i s pressure had to be: kept constant during the operation of the counter. Since i t operates at atmospheric pressure, any fluctuations in; the room pressure affected the gain of the counter. Small fluctuations due t o the v e n t i l ating fan i n the room were minimized by leaving the door of the room open during the course of the measurements as w e l l as l e t t i n g the fan run at at a constant speed. However, a strong c o r r e l a t i o n was found between, the. value of theatmospheric pressure and the count r a t i in.the 14.4 kev channel. Therefore, changes i n atmospheric pressure w e r e monitered and the  to follow page 45  1021  1020 _ Atmospheric Pressure ( i n nib.)  1019  1100  1200  1300  I50T5  T5ot  X65o~  Time  _J  1100  FIGURE I I I - 6  _  J  1200  _  L  1300 Time  I  1400  _  J  1500  _ L .  1600  An I l l u s t r a t i o n of the r e l a t i o n s h i p between the atmospheric pressure and the count rate i n the 14.4 kev channel. The gamma rays were detected by means of the proportional counter.  -46r e s u l t s were corrected f o r t h i s count rate d r i f t .  Another source of d i f f i c u l t y with the counter vas the fact that very large pulses of unknown o r i g i n occasionally came through the counter, e f f e c t i v e l y turning i t o f f f o r periods of approximately 10 milliseconds. Toe frequency of these break down pulses vas found t o increase as the E.H.T. of the counter increased. These pulses were random and i t was found, as shown i n P i g . III-7, that when they occured s u f f i c i e n t l y often, they d i s t o r t e d the d i s t r i b u t i o n of count rates about the mean count rate away from the s t a t i s t i c a l l y expected d i s t r i b u t i o n t o sueh an extent that the highest E.H.T. at which i t was possible to operate the counter was 1800 v o l t s .  In addition t o being a function of the gas flow rate and the counter pressure, i t vas found that the gain of the counter depended upon the t o t a l count rate of the source. However, the change i n the count rate f o r the experiment r e s u l t i n g gain s h i f t was  vl)  (approximately 17$)  was small enough so that the  negligible.  Characteristies o f the Counter  Using the same equipment as used i n the experiment and the NSEC #2 source, the c h a r a c t e r i s t i c s of the proportional counter as shown in Table I I I - 2 were found. In order t o determine the gas gain In the counter, the gain of the amplifier and the preamplifier system was measured The equipment was arranged as shown i n P i g . H I - 8 a f o r t h i s measurement. The gain was determined f o r s i x values of  and averaged. With r e f e r -  ence to the f i g u r e , the gain of the amplifier i s given by:  -1*7UI-1 = W  V( /C Cl  1+  C ) 2  The average gain of the preamplifier-amplifier was 2 x 10 . The equipment was then arranged as i n Pig.III-8b t o determine the gas gain of the proportional counter a3 a function of the E.H.T.  In t h i s  case  1l?*  =  =  III-2  X  q/C  where q i s the charge c o l l e c t e d by the proportional dcounter. The charge q', produced i n the counter when the gamma ray of energy £ Is:  t  i s absorbed  '  q' =  Et/e  [  III-3  where e i s the stopping power of the gas mixture. The value of £ f o r Ar 19 i s 26.4 ev/ion p r . G  g  The gas gain therefore, i s given by:  = q/q'  III-4  In the c a l c u l a t i o n of Gg, i t was assumed that the e f f e c t of CH^ with 20 £ =29-4 ev/ion p r .  on G  g  would be within the experimental error and  hence was not included In the c a l c u l a t i o n s . The gas gain i s plotted as a function of the E.H.T. i n F i g . I I I - 9 . In order t o determine the resolution of the counter, the spectrum transmitted through the absorber was analyzed by means of the C.D.C. 100 channel kicksorter. The spectrum was plotted by an X-Y recorder and the background  correction made. The resolution as taken from t h i s spectrum  Pulse Generator  Cathode Eay Oscillascope  V  2pf  T "  T  C  65pf  2  Proportional Counter  'out V i n  Pre-amplifier  Amplifier  (a)  Vin  v  out •  Proportional Counter  _L —  65pf Pre-amplifier  Amplifier  C a t h o d e Ray Oscillascope (b)  FIGURE I I I - 8  A b l o c k diagram showing the arrangement o f t h e a p p a r a t u s t o d e t e r m i n e (a) the g a i n o f the p r e - a m p l i f i e r and a m p l i f i e r , and (b) t h e gas g a i n o f the p r o p o r t i o n a l c o u n t e r as a f u n c t i o n of the EHT.  -48was: Res =  m-5  where W| Is the width of the 14.4  kev peak at | height and L i s the  distance from the low energy l i m i t t o the 14.4 kev peak. The noise, the s i g n a l to noise r a t i o and the r i s e time of the pulses were a l l determined by displaying the pulses from the counter and amplifying system, into a calibrated o s c i l l o s c o p e .  E.H.T.  Oas Gain  Resolution  Noise  SlgnalAoise  Rise Ttoe(2Q~80jg)  1650 volts  28 ±3  18.9#  2.88 kev  5.00  3.0 //sec  1700  9  3* ±3  16.C#  2.47  kev  5.83  3.0  «  1750  n  45±3  17.0$  1.80  kev  8.00  3.0  «  1800  n  58±3  16.5#  1.44  kev  10.00  3.0  "  1850  n  74±6  16.7*  1.08  kev  13.30  2.5  »  1900  H  100 ± 6  17.7#  .85 kev  17.00  2.5  "  1950  n  142  18.35S  .60 kev  24.00  2.5  M  2000  n  164 ±9  19.92  .50 kev  28.80  2.5  ±9  ,  •  Table III-2 The C h a r a c t e r i s t i c s of the Proportional Counter as a Function of E.H.T. I The s o l i d angle into which the detected x-rays were emitted was determined by the siae of the counter window (36 square inches) and the distance of the source from the counter (27"). This s o l i d angle was calculated t o be .04-95 steradians. The spectrum obtainable with the prop o r t i o n a l counter i s shown i n P i g . I I I - 1 0 . . Again, the c a l i b r a t i o n pulses from the pulse generator have been included. The settings used f o r t h i s spectrum were: a) amplifier d i f f e r e n t i a t i o n : J8 sec; integration time constant: 3 . 2 sec; attenuation c  0 , 2 0 d b b) Multichannel (C.D.C.) P.H.A.  •3 volts/channel, bias 5 and c) E.H.T. 1850 v o l t s .  to follow page 48  FIGURE I I I - 1 0  A t y p i c a l Fe^7 included.  spectrum  o b t a i n e d w i t h the  proportional  counter.  Calibration pulses  a2"e  -49-  3.2  The E l e c t r i c a l Apparatus The e l e c t r i c a l apparatus used i n the experimental work i s shown  schematically i n P i g . III-l!. A l i s t of the equipment i s given i n Table III-3•  Name of Apparatus  Description  Regulated high voltage supply  Model RE-5001AWtL, S e r i a l #218 Northeast S c i e n t i f i c Corporation  Pre-araplifier and amplifier  Amplifier unit type 1430A.Serial #1367- Dynatron Radio Ltd.  Single channel kicksorter  Pulse height analyzer, s i n g l e channel Model 510.Serial #2633  \  h  U.B.C. NP Scaler, S e r i a l #7  Standard pulse generator  1/3 HP General E l e c t r i c Motor  Lathe drive motor j  UND Laboratories Inc.  Switch  Berkeley type, Made at U.B.C.  Scalers Multi-channel pulse height analyzer.  100 channel kicksorter. Computing Devices of Canada Ltd.  #2HBW1  Table III-3 A c i r c u i t diagram of the pulse generator used f o r c a l i b r a t i o n purposes i s given i n Fig.III-12. Also included i n t h i s figure are the AC and DC supply c i r c u i t s . The helipot used i n the DC supply was made by Helipot Corporation. The U.B.C. NP s c a l e r was modified so that i t produced an output s i g n a l s u f f i c i e n t t o drive the Berkeley s c a l e r s . In addition, prov i s i o n f o r a dead time of 15 microseconds was made i n t h i s s c a l e r f o r i t was found that with the dead times b u i l t into the s c a l e r ( i . e . , . 8 , . 2 , 5 and 50 microseconds),  i t was impossible t o set the Berkeley so that they would  Single  Pre  Aoplifier  Amplifier  Proportional  Kicksorter Atten.  Window  Pulse Gen.  Baselir e  '-Tt  Tdiff,  1  Supply  60  E.H.T.  FIGURE I I I - l l  Standard Pulse Generator  Counter  Lathe Drive Motor  Channel  cps  u svitc *  A block diagram of  the  electrical  apparatBS'.  Disc.  Dead tLme  11 Out  D. FIGURE  C . Supply  111-12  A circuit pulses.  Pulse diagram of  the  standard  Out  Generator  pulse generator used to  A. C. produce  the  Supply  calibration  -50all  count  in  unison.  The C o n t r o l  3»3  System  A S o u t h Bend P r e c i s i o n L a t h e with at  a b e d l e n g t h 3§'  small  constant  from r e a c h i n g t h e piece The  of  was u s e d t o  source  absorber  or absorber,  the  lathe  l e v e l e d and k e p t w e l l  the motion of the  lathe  further vibration.  carriage  #C1 644z)  relative  to  the  i n the  source  building  was m o u n t e d u p o n a  i n t u r n was m o u n t e d u p o n a 4  l a t h e was a c c u r a t e l y  and frefe o f  move t h e  v e l o c i t y . Toprevent v i b r a t i o n s present  lead which  sure that  (Model A, c a t a l o g u e  f  s t a n d o f cement  oiled  at  was c o n s t a n t  blocks.  a l l times to  in-  and r e p r o d u c e a b l e  An a d d i t i o n a l p r e c a u t i o n a g a i n s t  the  ef-  •i  fects  o f r o o m v i b r a t i o n s was t o  proof box.  enclose  T h i s box a l s o tended t o  equipment w i t h i n t h e box d u r i n g the The g e a r s  of the  lathe  on d i r e c t  and c l u t c h d r i v e .  used t o  drive the  lathe.  the  range  of speeds  lathe  course  of the a range  A 1/3  Using a r a t i o  lathe  carriage  was c o n t r o l l e d b y means  lathe  drive.  ,2cm/sec t o  moving  the  i n the  so t h a t they  correct  of scalers  pulley  also  belt to  reduce  the  controlled the  c o u n t e d o n l y when t h e the  correct  vibrations  lathe  lathe to  lathe  carriage  carriage  s p e e d . One o f  is  the  operation of  n u m b e r o f t r a n s m i t t e d gamma r a y s ,  60 c p s m a i n s f r e q u e n c y . T h e s w i t c h i n g c i r c u i t  of  .0025cm/sec. An  arranged so t h a t  d i r e c t i o n and a t  recorded the  m o t o r was  lathe  o f two m i c r o s w i t c h e s a t t a c h e d  2.3cm. T h e m i c r o s w i t c h e s  Berkeley sealers  o f 80 p o s s i b l e s p e e d s ,  and d i r e c t i o n o f motion o f the  b e d . T h e s e m i c r o s w i t c h e s were  travelled  pair  The l i m i t s  the  experiments.  hp G e n e r a l E l e c t r i c  0 - r i n g #1820-72 was u s e d i n s t e a d o f a p u l l e y  i n a sound  fluctuations of  o f motor p u l l e y t o  o b t a i n a b l e was  from the  and d e t e c t o r  reduce temperature  provided  40 e a c h  1/5,  the  the  was  each  the  other  shown i n P i g . I I I - 1 3 .  Lathe d i r .  control relay  FIGURE  111-13  A schematic diagram showing the switching c i r c u i t used to control the d i r e c t i o n of the lathe t r a v e l and the operation of the s c a l e r s .  The temperature all  times  b y means  of  of the  and a b s o r b e r were m o n i t o r e d  a copper-constantan  A H e w l e t t - P a c k a r d DC v o l t m e t e r  thermocouple attached  was u s e d t o  measure  the  to  at them.  thermocouple  voltage.  The Absorber  3.4  a)  The A b s o r b e r Mount  T h e a b s o r b e r was  fixed to  mount c o n s i s t i n g o f t w o s q u a r e s diameter  circle  cut  tween two £ " p i e c e s of  source  aluminium.  damping out keep t h e source  from i t s  of  at  several cided  that  a constant  I r o n w o u l d be t h e were  u s e d were  led  to  was  a  .0002" .001"  a temperature  fixed to  course  of the were  used as  s m a l l s h i f t s were t o most  satisfactory.  .00035"  shim s t e e l  the  the  f o r the shield  lathe  a  two  2"  bepieces  foil  by  tending  to  regardless  of  a l " diameter  the  brass  carriage.  of  that  foils  950°C  l i k e w i s e annealed but  were  before  de-  Arraco  Armco i r o n w h i c h were  i n an Ar  b e i n g u s e d . The  fragile,  rol-  measurements  annealed  i n a n Hg a t m o s p h e r e .  o b t a i n and were  effect  was  pure  Also used f o r c e r t a i n  T h e Armco i r o n  f o i j . was  to  investigated,  it  H a m i l t o n Watch C o . These f o i l s  shim s t e e l  difficult  However,  T h e two f p i l s  thicknesses.  foil.  absorbers.  be  f o r a p e r i o d o f one h o u r a t  were  was p l a c e d  p r e l i m i n a r y work on t h e M o s s b a u e r  atmosphere  Armco f o i l s  absorber  a support  of  o f w h i c h had a  and u n i f o r m t e m p e r a t u r e  obtained from the and  b y means  Absorbers  of materials  since  each  T h e a b s o r b e r mount was s c r e w e d t o  Armco I r o n  types  The t h i n  b o t h as  c y l i n d e r w h i c h i n t u r n was t i g h t l y  In the  carriage  o f s t y r o f o a m and t h e n clamped b e t e e n  i n d u c e d v i b r a t i o n and as  absorber  lathe  aluminium,  centre.  The s t y r o f o a m a c t e d  temperature.  b)  the  Since  t h e y were  .001" the  sprayed  with A c r y l i c p l a s t i c to give them added strength and protect them from corrosion. Unless otherwise stated, a l l absorption curves included in t h i s thesis were obtained using the two Armco iron f o i l s mounted t o gether to give an e f f e c t i v e thickness of .00055". The reason f o r t h i s choice of thickness i s given i n section 3.9.  Appendix B outlines the  method that could be used i n the preparation of an enriched Fe-* absorber 7  3.5  The Source a) The Mount  The source mount had t o serve two purposes; t o r i g i d l y clamp the source t o the lathe bed and t o enable the temperature of the source to be varied from 100°K t o 500°K. The mount shown i n Pig.III-l4 was used for temperatures of 300°K and higher. The pressure within the pot was reduced t o 15 microns when the temperature was increased t o prevent the Fe source from oxidizing and to provide s u f f i c i e n t temperature insulation 'so that a constant source temperature could be maintained. In order t o maintain the reduced pressure without pumping, i t was necessary t o r e place the .001" mylar window o r i g i n a l l y used with the .005" mylar window shown i n F i g . III-14 and the covar seals had t o be coated with R313* The source was heated with the heating unit from a soldering iron, placed i n the inner pot.  Preliminary investigations proved that the use of the above pot f i l l e d with l i q u i d nitrogen t o obtain the temperatures i n the region of 100°K was not an acceptable method of lowering the temperature of the source. The vibrations from the b o i l i n g nitrogen (and other low temperature mixtures) i n the pot were transmitted t o °the source and hence, broad  Glass  to  Metal  Seal  ZD  7/8"  • &  "Stainless  •  Steel  Hoke Valve JTo P i r a n i Gauge  O ~ O <  >  V jl/8"  Brass  •  Hoke Valve  Source  .005  1"  Mylar  Holder  3 § E To  4/  1  Vacuum  •  Pump  FIGUR3 I I I - l U  The  s o u r c e mount u s e d  temperature.  for  source temperatures  at  or  above  room  ened Mossbauer l i n e come t a i s that  to  difficulty  an o b s e r v a b l e b u t  the  inner pot  shown i n F i g . I I I - 1 5 >  low temperatures through the  were  stainless  7"  i n diameter,  of  f l o w of  the  b y means o f and a v a r i a c  l  microns  tubes.  liquid  vapour and hence,  the  temperature  a 600 watt h e a t i n g u n i t p l a c e d i n the to which the  was f o u n d t h a t  heat  keep the  i n s u l a t i o n , the  the  source  pressure  of  the  inner surface  of  the  l i q u i d n i t r o g e n from reaching the  was u s e d t o  connect  found necessary  of  the  the  outer  pot.  p o t was  source,  and the  surround the  t u b i n g w i t h 2 " of  reach temperatures  below 200°K.  A piece  the  output s t a i n l e s s  c o l d n i t r o g e n gas  from the  absorber temperature  b)  by D r . J . 1  of  at  of  the  heater  constant.  reduced to  around three  To 15  quarters  the v i b r a t i o n  source  mount.  source  It  was  styrofoam i n order  r u b b e r t u b i n g was  room  controlled reservoir  level  of the  prevent  holder to  equipment box and t h e r e b y  remained steady  utsed f o r t h e  B„ W a r r e n . T h i s ,  was c o - p l a t e d w i t h F e  square  tube of the  was  rate  to  connected  remove  the  insured that  the  temperature.  Sources  The s o u r c e  Co^  steel  The  polyethylene tubing  to  of  As t h e  temperature  In order to  nitrogen reservoir  source  liquid nitrogen  a n d a n a l u m i n i u m r a d i a t i o n s h i e l d was e r e c t e d  of  to  the  voltage  the  u s e d was  2" styrofoam.  h e a t i n g u n i t was a t t a c h e d . it  The  n i t r o g e n vapour  of of  with  below 300°K.  The l i q u i d n i t r o g e n r e s e r v o i r  deep and s u r r o u n d e d b y a l a y e r  1  increased s l i g h t l y to  improve the  temperatures  obtained by f o r c i n g the steel  To o v e r -  shown i n F i g . 1 1 1 = 1 4 was r e p l a c e d  f o r a l l source  l i q u i d nitrogen decreased, had t o be  incalculable extent.  copper b a r .  absorbent  p r e l i m i n a r y i n v e s t i g a t i o n s was  and a l l o t h e r v  i n the  ratio  sources, of  1:500  The p l a t i n g s o l u t i o n , F e S O ^ ,  c o t t o n wrapped about the  was C o - * ? ,  prepared  1 mc o f  onto a one-quarter  was a b s o r b e d i n a  platinum anode.  The C o ^ f / F e  inch  piece mixture  to  follow  page 53  it! o d . 2 Glass  to  <  Stainless  Steel  metal seals  • / s  k  «  (.Brass)  (3/V deep)  Hollow p a r t i a l l y f i l l e d w i t h Pb shot  F I G U R E 111-15  Brass  7/8"  1?  <-  £"  .1 3/V  T h e s o u r c e mount u s e d f o r room t e m p e r a t u r e .  source  temperatures  below  •  was d e p o s i t e d b y r u b b i n g t h e current  copper rod with t h i s 2  two s o u r c e s  of  Co57  E n g i n e e r i n g C o r p o r a t i o n . The f w i resr et NSEC #2, was 4 m c . T h e Co57  second, on  The  d e n s i t y u s e d was 50 m a / c m .  The and  anode a r r a n g e m e n t .  .005"  AITOCO i r o n .  The c o s t  of  w  a  ob om th u c l e 1a r mcS cai n ed n c et h e o f t tahi n e es d e , f rNSEC # 1e , Nwas  99.9$  s  p  u  r  e  a  n  d electroplated  up-  the  Co57 was $75«00 p e r mc a n d a c h a r g e o f $75-00  was made f o r t h e  fused onto the i r o n f i n e d t o an a r e a o f  electroplating.  this  con-  7  treatment consisted  d u r i n g which the  source  was u s e d f o r t h e  experimental  3.6  m a t e r i a l was  ( A r m c o ) b a c k i n g i n a n a r g o n a t m o s p h e r e a t 900°C f o r .5 s q u a r e c e n t i m e t r e s . T h e C o ^ o f NSEC #1 was d i f -  one h o u r . T h e a n n e a l i n g o f NSEC #2 Briefly,  The a c t i v e  is of  described in detail f i v e steps  was h e a t e d t o results  i n Chapter IV.  of ten minutes  each,  900°C i n a n A r a t m o s p h e r e . included  in this  NSEC #2  thesis.  The measured Q u a n t i t y  The gamma r a y of the  q u a n t i t y m e a s u r e d was t h e  of  t h r o u g h a n Pe a b s o r b e r  absorber  ditions.  With  relative  infinite  to  the  source  as  occurs  o n l y whereas  electronic  and Mossbauer ( r e c o l l l e s s )  transmission with r e l a t i v e  at  and a b s o r b e r ,  The measured t r a n s m i s s i o n i s  This  con-  electronic  velocities,  absorption occur.  kev  velocity  under various experimental  very small relative  velocities  14.4  a f u n c t i o n of the  v e l o c i t y between s o u r c e  absorption  spectrum.  transmission of the  both  change  in  produces the Mossbauer v e l o c i t y n o r m a l i z e d by f o r m i n g t h e  w h e r e v e l o c i t y v s i l k e i a / p e c ( r a t h e r t h a n ,v  ratio:  ) was u s e d * f o r e x p e r i m e n t a l  con-  -55venienee. The measured l i n e i n t e n s i t y therefore i s h - l - R ^ v = 0) and w i l l d i f f e r from the calculated l i n e i n t e n s i t y ( Equation 11-11). This d e f i n i t i o n of h also means that the measured Mossbauer l i n e width  A  ( i . e . , the width at h/2) w i l l d i f f e r from the calculated width, d. In the following, the quantities h and A w i l l be considered except when a comparison between theory and experiment i s t o be made at which time h and A w i l l be adjusted t o f i t data f o r v = oo rather than v = ,l4cm/sec.  3.7 Geometric E f f e c t s on R(v) In Chapter I I the function R(v) was defined as T ( t o t a l , v ) which by the substitution of Eqn.11-17 becomes:  III-6  for a s i x l i n e emission and absorption spectra. This equation i s complete in that i t accounts f o r the source thickness and the hyperfine s p l i t t i n g of the 14.4 kev t r a n s i t i o n of F e ^ but the influence of the small but f i n i t e s o l i d angle into which the gamma rays can radiate and s t i l l be detected by the gamma ray spectrometer, Is ignored. This s o l i d angle produces a f i n i t e width t o the f i r s t order Doppler s h i f t obtained with a given v e l o c i t y and also increases the e f f e c t i v e thickness of the absorber and proportional counter and hence, must be taken into account when i n t e r preting the experimental measurements (see P i g . I I I - 1 6 ) . The parameter which determines the magnitude of the e f f e c t s i s 0, the angle between the  t'cOSj0  .„...yo Source  --vW-  Absorber  Proportional Counter  o o  FIGURE  III-16  A schematic r e p r e s e n t a t i o n of the arrangement of the s o u r c e , a b s o r b e r , and P r o p o r t i o n a l c o u n t e r . The s c a l e h a s b e e n a d j u s t e d t o e m p h a s i z e t h e e f f e c t on t h e v e l o c i t y , v , t h e a b s o r b e r t h i c k n e s s , t ' , and the p r o p o r t i o n a l count e r t h i c k n e s s , x, o f the geometry. Q  :  VJ7  -56l i n e Joining the source and detector and the vector at r e l a t i v e v e l o c i t y . The e f f e c t of the experimental geometry on R(v) has been c a l culated i n Appendix C. The r e s u l t of t h i s consideration i s that Eqn.III-6 must be replaced  by:  R(v,*) = 1-f  III-7  i n which the factor .97 corrects the o r i g i n a l l y calculated R(v,0  = 0) t o  the experimental conditions under which Rg^v,^) i s measured. The correction f a c t o r 197  i s that required f o r a .001"  absorber. Those factors necessary  f o r oth^r absorber thicknesses are given i n Appendix C.  3.8  Background Correction Inspection of the spectrum of Fe^7  shows that the 14.4  (see f o r example, Pig.III-10)  kev peak i s superimposed upon a continuum of back-  ground r a d i a t i o n . In the c a l c u l a t i o n of R ^ v ) ,  t h i s background count rate  must be subtracted so that only the count rate of the 14.4 i s considered  i n the r e s u l t s . Therefore R ^ v )  Rm(v)  =  W(v) -N N(v =.14)-N  kev t r a n s i t i o n  i s calculated as: IH-8  b  b  where M i s the t o t a l count rate at v and Nj, i s the background count rate. ; For zero r e l a t i v e v e l o c i t y between the source and absorber, and for equal (dead time corrected) counting times, the Fe57 spectra transmitted through the Fe absorber and through a 1/16"  aluminium absorber  FIGURE 111-17  T h e t r a n s m i s s i o n o f ik.k k e v r a d i a t i o n t h r o u g h a .00055" Armco i r o n a n d a l / l 6 " A l a b s o r b e r . F r o m s u c h a s p e c t r u m the b a c k g r o u n d c o r r e c t i o n v a s made.  -57were measured thus determining the background rate N b  showed that the aluminium  Calculations  absorber would absorb a l l of the 14.4 kev r a -  d i a t i o n incident upon i t so that only background radiation of reduced i n t e n s i t y would be measured when t h i s absorber was used. The area of the background r a d i a t i o n within the window settings of the single channel kicksorter was measured f o r both spectra ( see diagram,Pig.Ill-17)• The count rate of the background radiation transmitted through the aluminium absorber was measured. Hence,  N  is  =• W A I . )  b  A  where  A1  of the  i s the count rate  III-Q  transmitted through  the aluminium  absorber.  and therefore  A  R^v),  N(v)N(v=  Fe^ Al) N  III-10  A a i  .l4)-A (N ) P e  A  A 1  A1  A l l experimental measurements involving ^ ( v ) vised i n the determination of r e s u l t s given i n t h i s thesis are corrected i n t h i s manner for the background unless otherwise noted. The accuracy of the above correction i s dependent upon the accuracy with which the dead time was measured. In the experimental work the dead time was (8 - l ) # so that the error i n the determination of N was approximately 1#.  Moreover, since  i s almost 1/3 N(v), the t o t a l  error i n the determination of the background correction R ^ v ) proximately  .3$.  b  was  ap-  -583.9  Selection of the Absorber As mentioned  i n seotion  3«3»  a number of absorbers of various  thicknesses were available f o r the experimental investigations. I t was necessary therefore, to determine which of these absorbers would give the most s i g n i f i c a n t r e s u l t s i n a given time T, that i s , maximize the expression X/e,  where X i s the difference i n the count rate transmis-  sion at v - 0 and v = .l4cm/sec and e i s the s t a t i s t i c a l error f o r the measured count rates. In Appendix D i t i s shown that  X/e~  where N  s  which X/e  f(N ) s  g  e  r  .(  1 - e  m-ii  I (/<m* ) 0  i s the source strength . For Fe57, the absorber thickness x at i s a maximum, depends upon the value of the Debye-Waller  factor  ,f and upon the f r a c t i o n of absorber atoms which are F&57.  The value of X/e  was measured f o r various absorber thicknesses  for both NSEC #1 and #2. Moreover, f o r the same absorber thicknesses, was calculated using Eqn. I I I - l l f o r f - .4, Absorber Thickness .0002" .00035" .00055" .001" .00155"  .6 and  NSEC #1  NSEC #2  f = .4  f _ .6  f = .7  25.78 45-70 57.30 57.77 49.44  25.78 39-00 54.83 55.98  25.78 35.60 41.41 41.70 39.15  25-78 33.63 35.80 31-96 23.81  25.78 34.02 37.60 34.46 26.21  —  X/e  Table III-4 The values of X/e  f o r various absorber thicknesses  as measured f o r NSEC #1&2 and calculated f o r f .6 and  .4,  .7.  Since the reason f o r t h i s procedure was t o determine the optimum t h i c k ness of the absorber, the values found i n the above table have been nor-  -59malized 30 that  X/Q.for x = . 0 0 0 2  u  i s the same f o r a l l c a l c u l a t i o n s .  Hence, the trend of X/e f o r a l l cases can be compared. The count rates used f o r the NSEC #1 data required four minutes t o accumulate whereas those f o r NSEC #2 required one minute. Corrections were made f o r the dead time of the apparatus i n each case but correction f o r background radiation was not made since only r e l a t i v e values of X/e were required. y  The data i n Table I I I - 4 i s given i n graphical form i n F i g . I I I - l i B . The dependence of X/e on absorber thickness f o r NSEC #2 i s seen from t h i s figure t o be the same, within experimental e r r o r . X/e attains a maximum value at x = 5-7 x 10~ inches f o r f r .7 and at x = (8.0 £ ^ J x l O " ^ 4  inches f o r the NSEC #2. The experimental r e s u l t s show that there i s a n e g l i g i b l e difference i n the s i g n i f i c a n c e attained with absorber t h i c k nesses of 5.5 x 10~ inches and 1.0 x 10~3 inches. The choice of ab4  sorber thickness of 5.5 x 10 that the accumulation  inches was therefore based upon the fact  of r e s u l t s was faster with the thinner absorber  and i t s closeness t o the t h e o r e t i c a l l y determined optimum thickness (assuming f = .7 f o r F e ) of 5.7 x 10"^ inches. 5 7  i  60  50  \  ko  HSEC  #1  HSEC  #2  30  t z  .6  c 8  10  12  Jk  Absorber Thickness (xlO* FIGURE  III-18  Ik  16  inches)  T h e r a t i o o f t h e d i f f e r e n c e o f t h e c o u n t r a t e o f t h e Ik.k k e v r a y s , ^ t o t h e s t a t i s t i c a l e r r o r i n t h e c o u n t r a t e , e, as a f u n c t i o n o f ' the absorber thickness.  o  i vo  -6oChapter  D I F F U S I O N OF C o  4,0  IV  INTO NATURAL IRON  5 7  Introduction  The u s u a l method o f p r e p a r i n g a r a d i o a c t i v e sbauer  experiment  terial  onto  active  material  i n t o the  notice  that  diffusion  the  to  the  diffused  which the  effect  parameters  procedure  (height  diffusion i n one  which the  fact  that  surrounding the  the  used  in this  procedure  first  as  magnitude  experiment  was d o n e  In a  the  and v a r i a t i o n s  series  could  be  the  min-  was e q u a l t o  the  cal-  of thickness  .00055").  possible to  measure  it  and s h i f t  atoms  had  Co57 d i f f u s e s  r e s u l t i n g from the  was  made  i) a s  the  Mossbauer  a f u n c t i o n of  the  diffused.  of these parameters  radiating  of  w i t h optimum p r o p e r t i e s .  radioactive  the  absorber  to  i n a source  on d i f f u s i o n  i n t o the  natural  depth i s  iron,  the  due lattice  Fe57  initial  o f a weak s o l u t i o n o f Co i n F e . shifts  radio-  results  nucleus lattice  d i f f u s e the  o r two so t h a t  iron  h , width  The dependence the  and t o  remarkably improved the  a natural  a controlled diffusion  depth to  to  than  ma-  2  The s o u r c e  Mos-  radioactive  measured Mossbauer e f f e c t  (17.92 f o r  In a d d i t i o n , t h i s line  rather  the  lattice  a  the  Pound and R e b k a ^ were  procedure  obtainable.  evaporate  for  i n a c o n t r o l l e d and c o n t i n u o u s l y m o n i t o r e d f a s h i o n t o  imum d e p t h a t culated  steps  or  metallic  lattice.  i n a s i m i l a r manner b u t  f i v e short  Such  electroplate  a non-radioactive  Mossbauer e f f e c t  made of  is  source  in line  changes  from the  deposition process  to  n o n - u n i f o r m Co < J  a uniform  S i n c e a c c u r a t e measurements  w i d t h s were  to  lattice  of small  be m e a s u r e d w i t h t h i s  line'  source,  -61the  diffusion  Mossbauer  process  was  continued u n t i l  and a  large  i n t e n s i t y were p r o d u c e d .  The v e l o c i t y spectrum iliary  a narrow l i n e  peaks  associated  on e i t h e r  with  side  a zero  or  showed t h e  of the  small  main  presence  line.  o f two s m a l l  These peaks  i n t e r n a l magnetic  field  at  appear the  auxto  be  sites  of  Pe57  some o f t h e  atoms. Before the cedure  are  discussed,  particular chapter  4.1  to  this  contains  Theory of  a)  the  and method u s e d  a b r i e f summary o f t h e  thesis)  of diffusion,  experimental  is  results  i n the  theory  given.  diffusion  pro-  (both g e n e r a l  and  The r e m a i n d e r o f  and a d i s c u s s i o n  the  thereof.  Diffusion  General D i f f u s i o n Theory  Diffusion Law  apparatus  of  one s o l i d  into  another  is  governed by  Flick's  2 1  dm = DA d c ( x )  dt  IV-1  dx  w h e r e dm I s  the  of d i f f u s i n g  mass  atoms  coefficient  at  (D) g i v e s  diffusing  across  area  depth x,  and D i s  the  at  rate  the  A, c(x)  is  diffusion  which d i f f u s i o n  the  concentration  coefficient.  takes  place.  It  This depends  21 strongly  upon t e m p e r a t u r e  D  where ature,  D  D  is  =  D  0  1  is  the  equation  exp(-Q'/ft©)  a c o n s t a n t i s  and Q  according to  the  the  approximate  IV-2  gas  constant,  binding  © is  energy  the  absolute  of the  crystal.  temper-  In o r d e r t o to  know t h e  ing the  calculate  coefficient  activity  the  D. This  rms d i f f u s i o n  coefficient  of a radioactive  solute  depth,  it  is  o a n be c a l c u l a t e d  before  and a f t e r  necessary by  its  measur-  diffusion  22 and t h e n s u b s t i t u t i n g t h e s e q u a n t i t i e s  r  = e (l z  -  £  v/z  i n t o the  equation,  )  IV-3 2  where  r  is  the  ratio  of counting r a t e s ,  coefficient  of the  equation  subjected  is  detected to  the  b)  The r a d i o a c t i v e  material  diffusing  Into  along the  Fe compounds a t  b y t h e m b y a more a c c u r a t e  atoms  at  c(x)  same  depth x  _  from which (see  (x ) the  e~  x  dx.  The  linear above  before  diffusion  diffuses  is  T  850°C,  infinitely  into a serai-infinite  cylinder's  cyl-  axis. was  f o r Co  agrees w e l l with that  calculated  method. as  above,  the  concentration  exp(-x /4Dt)  of  dif-  17-4  2  Dt'  the  equation  - "V 2 D t '  rms p e n e t r a t i o n ,  IV-5 is  Diffusion  derived.  of C o ^  When a M o s s b a u e r s o u r c e onto an Fe  thin.  i s g i v e n by  Appendix E)  b) Theory of the  o f Co  2  the  juls  and R i c h a r d s o n i n t h i s  assumptions  Q  Jv.  y u Dt and  <fo i s  d i r e c t i o n of the  D obtained by C a r t e r  Under t h e  is  assumptions:  deposit  of  giving  two  The r a d i o a c t i v e  The v a l u e  fused  radiation.  a)  inder  z  lattice,  the  is  Fe^  7 m  7  into  Fe  p r e p a r e d by the  electrodeposltion  n u c l e u s , when i t  emits  the  14.4  kev  -63ray,  i s located  a b s o r b i n g Fe  57  i n a Co l a t t i c e  r a t h e r than an Fe l a t t i c e  57 n u c l e u s . As the Co^' i s d i f f u s e d  the c o n c e n t r a t i o n o f Co  57  i n t o the Fe  d i m i n i s h e s so t h a t the Fe  of an Fe l a t t i c e r a t h e r than a Co  like  57  lattice  n u c l e i become  reduces the d i f f e r e n c e i n Debye temperatures between the s o u r c e and absorber l a t t i c e s  treatment  part  l a t t i c e and are s u b j e c t t o e s s e n t i a l l y  the same bonds as are the Fe n u c l e i . The d i f f u s i o n treatment  ates the l i n e s h i f t  the  therefore,  and c h e m i c a l bond e f f e c t s  which i n t u r n , reduces or e l i m i n -  of the Mossbauer spectrum. The f a c t t h a t the d i f f u s i o n  i n c r e a s e s the observed Mossbauer i n t e n s i t y i m p l i e s t h a t the 57  bonds on the Co or  atoms i n the e l e c t r o p l a t e d or evaporated d e p o s i t are weak  inhoraogeneous. The d i f f u s i o n p r o c e s s a l s o v a r i e s the apparent  of the Mossbauer In  line.  section 2.4  i t was  noted t h a t any d i f f e r e n c e i n the  at  the n u c l e i of the source and absorber would produce  At  o 3 ° ° K, the i n t e r n a l f i e l d  ference i n i n t e r n a l f i e l d s  i n a Co l a t t i c e  is 3.045  x lO^oe.  line. lattice (a d i f -  o f . 2 9 5 x l O ^ o e . ) . Using e q u a t i o n I I - 4 4 ,  i n c r e a s e i n l i n e width produced by t h i s d i f f e r e n c e i n i n t e r n a l i s c a l c u l a t e d t o be 3 - 2 7  As the C o ^  the Fe l a t t i c e , the d i f f e r e n c e i n the i n t e r n a l f i e l d s  7  the  magnetic  x 1 0 " ^ ev.(compare w i t h the expected  f o r t h i s source and absorber o f 9 . 0 x 10~%v).  d e c r e a s i n g the l i n e  fields  a broadened  57 a t an Fe^' nucleus embedded i n an Fe  i s 3 - 3 3 x lO^oe., whereas t h a t  fields  width  width  i s diffused  into  decreases, thereby  width.  A summary o f the c o n d i t i o n s d e s c r i b e d by o t h e r authors under «  which s o u r c e s f o r the Mossbauer experiments i n Table  IV-1.  have been d i f f u s e d , i s g i v e n  -64-  Source Preparation  Source  Lattice Material  Diffusion Temperature  Diffusion Time  Atmosphere  Ref  Electro.  Ga ?  ZnO  1000°C  6 0 min.  air  22  Electro.  Co57 Co57  S.S.  900°C  6 0 min.  hydrogen  23  Armco Iron  900-1000°C  6 0 rain.  hydrogen  24  Co57 Co57  S.S.  950°C  CoPd  1000°C  Electro. Evap. Electro.  6  1 2 0 min.  vacuum  = 16  vacuum  25  Table IV-1 4.2  Apparatus  The apparatus used i n the d i f f u s i o n of the Co^ source into 7  a . 0 0 5 " AxWo iron l a t t i c e i s shown i n Pigs. I V - 1 and I V - 2 . The apparatus t o hold the source i n the quartz tube had t o be designed i n a manner such that I t s heat capacity was small, thus requiring a very short time to change from room temperature t o the temperature used f o r the d i f f u s i o n and i t s heat conductance was low so that l i t t l e heat was l o s t down the supporting tubel Also, i t s strength was s u f f i c i e n t so that the source could be moved i n and out of the furnace without the r i s k of dropping i t i n the quartz tube. A combination of an iron ( . 0 1 0 " ) boat and t h i n walled s t a i n l e s s s t e e l tubing met these s p e c i f i c a t i o n s . The iron boat was spot welded t o the s t e e l tubing. The ceramic two-hole tube within the s t e e l tubing ensured that the thermocouple wires d i d not short e i t h e r t o each other or to the s t a i n l e s s s t e e l tube and i n addition, increased the strength of the boat handle. The hot Junction of the chromel-alumel thermocouple was attached t o the iron boat by laying a piece of . 0 0 1 " shim s t e e l over the Junction and spot welding t h i s overlay t o the 'boat*.  In order t o have a continous record of the temperature during the d i f f u s i o n , the thermocouple voltage was recorded on a Brown recorder.  : j #5 Stainless  L —  Steel  Rubber Stopper  \S\U\\\\US\\\SWGlasa  Tubing  Tubing t  Chroiael - A l u m e l Thermocouple Hole Ceramic  <N3pot W e l d Spot  Tube  Weld. .001  <-  —  .001" Iron  Boat  Shim S t e e l Overlay  3'Ocia Diffusion  <-— E l e c t r i c 20  Gas in  (a)  Boat  cm  Furnace Quartz  Tube  r  #5  Rubber Stopper Gas "Out  —>"  Diffusion Boat Handle  50.6 FIGURE I V - 1  (a) the  The b o a t i n w h i c h t h e d i f f u s i o n apparatus.  C o ? s o u r c e was d i f f u s e d 5  cm  i n t o an F e l a t t i c e ,  o  _J  O  > 00 P (b)  The arrangement  of  as **"  -65Tbe IK Spectrol helipot (#495086) was used t o reduce the thermocouple voltage (at 900°C, V=:90mv) t o a value that could be fed into the Brown recorder (Model 153X12V-X-30A1, S e r i a l #6471253) which has a f u l l scale d e f l e c t i o n of ICtav. The potentiometer was set such that the b o i l i n g water-ice temperature difference corresponded t o lmv on the Brown recorder. The arrangement of the apparatus i s shown schematically i n P i g . IV-lb. The flow of gas was indicated by a water bubbler. Factors determining the use of argon were a) i t i s safer t o use ( p a r t i c u l a r l y at high temperatures) than hydrogen, b) the apparatus needed f o r an argon atmosphere i s much simpler than that required f o r a vacuum or hydrogen and c) i t provides the necessary inert atmosphere. The argon used was 99-9975$  i  k  -  Ar, .00b8# 0 * .0013JSN and .0004$ Hg- The oven heater arrangement i s 2  2  shown i n F i g . IV-2b where the oven used was a Hoskins E l e c t r i c Furnace (FD 303A, s e r i a l #33553, 110V, 5A). To obtain 900°C inside the furnace, the vari6c s e t t i n g had t o be 82.8V. A fan was arranged t o blow on the ext e r i o r of the quartz tube at the gas e x i t end t o Increase the rate at which the d i f f u s i o n boat and source returned t o room temperature at the end of each run.  4.3  Procedure t  a) The c e n t r a l section of the quartz tube was heated t o a predetermined temperature. This took about t h i r t y minutes. b) The source was placed i n the boat at the gas e x i t end of the quartz tube. The flow of argon was increased t o a flow producing 240 bubbles per minute when the gas tubing was inserted  into the water.  The flow was maintained at t h i s rate f o r at least f i v e minutes before a heating cycle commenced t o remove any a i r from the tube.  to follow page 65  T o Brown  Recorder  W W W W W V ^  Spot W e l d e d ^ — Oven J u n c t i o n ^ -  ->sSpot ^/Ice  C h r o m e l - A l u m e l ^ I K  3 turn  Spectrol  Thermocouple  Thermocouple  Mains  Solar  (a)  Variac  Set at  Oven H e a t e r  FIGURE I V - 2  Arrangement  82.8v  Arrangement  (b)  Schematic d i a g r a m showing (a) t h e thermocouple a r r a n g e m e n t and (b) t h e o v e n h e a t e r a r r a n g e m e n t used i n the d i f f u s i n g of Co-* i n t o a Fe l a t t i c e . 7  Welded Junetioi  -66-  c) The flow rate was decreased to 8 5 hubbies per minute and the system was allowed f i v e minutes to reach thermal equilibrium. d) The source and assembly were quickly inserted into the furnace t o the p o s i t i o n indicated i n P i g . IV-lb, t o raise the temperature of the source t o 900°C. The temperature  of the source was recorded  as a function of time. e) The source and assembly were then quickly withdrawn from the furnace t o the cool end of the quartz tube and held there i n the Ar atmosphere u n t i l the source returned t o room temperature. f ) The source was removed and mounted i n the Mossbauer apparatus. The count rate of both the 6 . 9 kev and 14.4 kev gamma rays were measured by putting the spectrum  Into the C.D.C. kicksorter. The  dead time losses were taken into account i n determining the count rates. Two source t o counter distances were used f o r t h i s measurement on each cycle - 1 5 3 - 3 cm and 6 9 . 2 cm. g) The Mossbauer i n t e n s i t y and the shape of the resonant absorption spectrum was measured using the 0 . 0 0 0 5 5 " Armco Iron absorber. h) The procedure was repeated from step b.  Since the count rate due t o the 4 me source was high enough t o cause coding troubles i n the kicksorter, only every tenth pulse was put into the kicksorter. In order t o do t h i s , the output of the amplif i e r was fed into the UBC NP s c a l e r and the input t o the second unit of the sealer triggered the kicksorter (coincidence mode used). Pig.IV-3 Is a reproduction of a t y p i c a l temperature versus time trace made on the Brown recorder.  to follow page  FIGURE IV-3  66  A reproduction of tb.e temperature versus time record used f o r the second d i f f u s i o n run.  -67-  In the measurement of the Mossbauer intensity and the r e sonant absorption spectrum, correction was made f o r the n o n ^ l ^ ^ kev background count rate within the single channel pulse height analyzer s e t t i n g s . This correction was explained i n Chapter I I I . Diffusion runs 1,  2 , and 3 were done as a group, each run l a s t i n g approximately l | hours.  However, at the end of run 3> trouble with the electronics delayed the d i f f u s i o n runs 4 and 5 u n t i l two days l a t e r .  4.4  Experimental Results  The resonant absorption l i n e shape a f t e r each d i f f u s i o n run i s shown i n P i g . I V - 4 . Also shown i n t h i s figure i s the l i n e shape obtained from the electroplated source before the source was annealed. Prom these curves the following quantities were determined: the f r a c t i o n a l resonant absorption h, the t o t a l l i n e width at one h a l f maximum height A, and the s i z e and d i r e c t i o n of the l i n e s h i f t i.These three quantities are summarized i n Table I V - 2 . Also, these quantities are plotted as a function of the d i f f u s i o n time i n figure IV-5 and the rms depth of the active layer i n figure I V - 6 . The l i n e s h i f t was calculated by means of the equation:  (A+C) -  b  where:  (B'D)  A i s the f r a c t i o n a l resonant absorption at -Vj_ B  n  n  it  n  n  n  -Vg  C  «  "  «  »t  i>  n  . V-^  D  u  n  ti  n  it  Vg  n  to follow page 67  v (cm/sec)  v (cm/sec) -.o8-.o6  -.04  -.02  C  .04  .02  .0—0-  4>'5 ^ 0  ..98  .06  .08  -,08 - . 0 6 - . 0 4 - . 0 2  10  .02  .04  -o-  ..96  -.9* -.92  R » -.90  -.88 .86 n  .84  =0  v (cm/sec) - . 0 8 - . 0 6 -.04 - . 0 2 | 0 . 0 2 -1  r  v (cra/aec) .04  1  .98  0  •96 x  .06 1  .08 1—  -08 - . 0 6 - . 0 4 - . 0 2 t  0  .02  r  0^.0  /  © \  o  /  /  . .92  R(v) . .90  .04 —1  ©.  ©v  ©  . .88  ©  . .86 = 4  . .84  n = 3  FIGURE IV-4  The resonant absorption l i n e shape a f t e r d i f f u s i o n run.  R  IMS P e n e t r a t i o n D e p t h  O  ©  H  1  '  (xlO^'cm) ro  1  — T ~  Mossbauer  Intensity  ro  S- o  8H3 S  cr  ro  O  Line .  1  Shift :  :,.,,  L i n e Width  ^  i  i -  (mm/sec) ,1  (ram/^sec)  &  ^  FIGURE I V - 6  T h e Mossfcauer l i n e w i d t h ,  p e n e t r a t i o n depth of t h e  intensity  Co57.  and s h i f t  as  a function  of  the  rms  ;  -68-  The d i f f u s i o n c o e f f i c i e n t was calculated by the method given in section 4.1.  Prom the data of the d i f f u s i o n runs, r = 0690. from  which D was calculated to be 2 x lO"^ cm /sec. This value of D i s con1  2  s i s t e n t with the value of D obtained f o r the d i f f u s i o n of P e ^ PeO and CoO at 900°C. x  was 3.3  x  into  The rms d i f f u s i o n depth f o r the t o t a l treatment  10"^cm (see Appendix E ) .  Diffusion Time  Total Diffusion Time  6°C  h (mm/sec)  A (mm/sec)  10 min.  10 min.  898C-C  .122  .38  -.007  10 min.  20 min.  895°C  .147  .34  -.0024  10 min.  30 min.  890°C  .126  •37  .0045  10 min.  40 min.  900°C  .137  .38  .0013  10 min.  50 min.  910°C  .145  • 31  .0035  (mm/sec)  Table IV-2 Summary of Diffusion of Co into .00055° Armco Iron  4.5  Discussion of Results An inspection of Table IV-2 shows that an appreciable value of ^  . •  ••  -  h appears a f t e r very l i t t l e d i f f u s i o n at 900°C (less than ten minutes). Further, the increase of h a f t e r the i n i t i a l ten minutes of d i f f u s i o n i s only from .122 t o .145. Prom these results i t can be concluded that i f one desired to measure the i n i t i a t i o n and build-up of h or of the Mossbauer intensity as a function of d i f f u s i o n time, then d i f f u s i o n runs shorter than ten minutes must be used. Hence, i t would be necessary to construct a d i f f u s i o n apparatus which would reach a stable temperature f a s t e r than the  'boat' used f o r t h i s experiment.  -69Although the rather  irregular,  diffusion  change  there  run.  As n o t e d  nuclei  i n the  is  i n the  a trend  earlier,  width of the  for  the  experimental  width to  a difference  lines  decrease with  of the  internal  was  each  fields  at  57 the  Fe  Therefore, Co^  the  i n t o the  7  radiating more  decrease Fe  Fe-^  thus  was  and a b s o r b e r  lattice  an F e  larger  changed  lattice.  than the  indicating that  main  line  line.  obtained but  is  for  appear  to  two d i f f u s i o n of points  satellite  the  be a l m o s t  used t o  satellites  were  still  of these  which  comparable  is  emission  peaks  therefore  the  source  fectively similar  to  the  to  the  zero. that  better  spectra  position  emitted  In equation  to  are  of the  first  noted  the  the to  one x  (1.5  three  and  the  side  on t h e  obtained  ram/sec  .80  peaks  Mosof  the  spectra  diffusion  from the  increased  runs  last number  of  field  amplitude. to  obtained  .86 mm/sec.  at  the  1.0  these  The  mm/sec,  f r o m an u n s p l i t ,  The s m a l l  absorption  a small region within the  stainless  that  that  existed.  i n C h a p t e r V , show t h a t  presence  from non-magnetic  was  of  fields still  evident  i n such a r e g i o n would emit  it  at  cm/sec  .031  on e i t h e r  resolution  i n t e r n a l magnetic  11-40  one  spectra  between  the  field  c a n be s e e n  the  discussed  is  diffusion  Co l a t t i c e  with c o n s i d e r a b l y reduced  peaks  attributed  The n u c l e i  it  and f o r  l i n e and Fe a b s o r b e r ,  i n which the  of the  in internal  curves  from the  the  present  satellite  unshifted are  absent  the  line.  l i n e width  of F i g . I V - 4 ,  source  that  line width,  theoretical  However,  plot  position  characteristic  absorption  a broadened  internal  b y two s m a l l p e a k s ,  unannealed  runs.  effective  possibly a difference  accompanied  These  the  The f i n a l  From an e x a m i n a t i o n sbauer  produces  in l i n e width indicates  n u c l e i from t h a t  7  s i m i l a r to  10"^ev.)  source  Fe^  7  nuclei  an u n s p l i t  is  line,  steels.  isomeric  shift  ef-  o  wa3  -70given by the e q u a t i o n :  5  _ 2 Ze  2  [ <Re>  w h e r e t h e q u a n t i t y <Re> <R> i s i n d e p e n d e n t diffusion  runs  ?  -  - <R >  2  o  <Rg>  c a n be r e g a r d e d  a  w h e r e oc i s a c o n s t a n t . electron  a variation to  fusion. in  i n i does  However, t h i s  fact  that  were  separated  some s o r t place.  0  (°)| at X  change  shift,  57  .  Assuming  shift  This  i n I would  imply a v a r i a t i o n i n  As i n d i c a t e d  variation  i n TableI V - 2 ,  f r o m - . 0 0 7 mm/sec  i n S. was n o t a p e r m a n e n t  separated  process  during the  then  implies that ^ ( O ) ^increased  runs  that  rv-7  the nucleus.  by l | h o u r s . T h i s time  of relaxation  f o r Fe  11-40  \%(0)\'  |^(o)f  o f % noted between  t h e two r u n s were  -  2  the line  as t h e i s o m e r i c  indeed occur.  . 0 0 4 5 mm/sec  the variation  i s negative  Hence a v a r i a t i o n  density |  a maximum o f  j [ \%(0)\  o f t h e a n n e a l i n g and t h a t  i * &= - * [ | y ( ) t -  the  2  g  with the  change.  3 and 4 i s a t t r i b u t e d  b y two d a y s w e r e a s dependent  with a long  change  time  constant  The break to the  a l l other  inX  dif-  runs  indicates must h a v e  that taken  -71Chapter V  TEMPERATURE INDEPENDENT  Introduction  5.0  In t h i s pendent of  EFFECTS  the  ment  effects  chapter  that  c o u l d produce  Mossbauer l i n e .  of  the  thickness,  of  For  each  thickness  h,  l i n e width A ,  is  t'  the  all  a line  as  were u s e d ,  the  shift  then describes  Mossbauer l i n e  of  and the  g i v e n of  either  chapter  the  Four values  1  .001". the  This  parameters t .  a summary  temperature or  the  a line  .00035",  p a r a m e t e r s measured were the line  broadening  experimental  a f u n c t i o n of  .0002",  inde-  s h i f t S . The r e s u l t s  the  absorber  . 0 0 0 5 5 " and  line  are  measure-  intensity  summarized  in  Table V - l . Equation B.C.'s R(v),  IBM 1 6 2 0  the  ation  of  show t h a t  the  the  source t h i c k n e s s , t ,  the  case  in  the  II-18  as  This  equation,  theory  yielding  is  the  p u b l i s h e d b y M a r g u l i e s and Ehrman b u t  in-  indicated at  p r o g r a m w i t h some  numerical  into  results.  the  the  If  t  Mossbauer e f f e c t that  account  end of of  of  The r e s u l t s  i n which case  such an e x t e n t  the  lines  Fe57  of  M a r g u l i e s and Ehrman i s  then the  can be t a k e n  a theoretical  U.  for  negligible  thesis, to  of  n u m e r i c a l l y on t h e value  source thickness.  theory  effect  results  source  computer A.  not  in this  calculated  programmed and e v a l u a t e d  m u l t i p l e e m i s s i o n and a b s o r p t i o n  effect  s o u r c e does  was  computer.  was b a s e d u p o n t h e  c l u d e d the  and  II-18  this test  R(v  by the  chapter. results  - o°)  in metallic  iron  numerical  evalu-  the valid  o n l y when  Mossbauer e f f e c t is  not  i n the  in  the  n e g l i g i b l e as  was  source  affects  / S i . The M o s s b a u e r  r e n o r m a l i z a t i o n of A detailed has  the  been  the  effect  equation  d i s c u s s i o n of  the  i n c l u d e d i n Appendix  -72-  5oi  A B r i e f Discussion of the Relevant Theory  a)  Theoretical  Mossbauer  The c o n d i t i o n s Mossbauer e f f e c t  for  conditions,  the  state,  The l a t t e r factor,f, effect.  of  £  also  theoretical R(v),  the  b)  the  primary factors  depends,  These  -  from the  are  R(v)  are  and the  the  lifetime  Debye  extent,  all  of  into  g i v e n by the  Debye-Waller  the  upon the  taken  of  temperature.  i n determining the intensity, £ ,  1-1.  Mossbauer  source  and  consideration  equation  III-7.  In  in terms  is  V - l  Mechanisms  Shift  and the  <.Rg> ^  shift,  (see  i n the  absorber  <Re>«  Chapter  chemical nuclei.  The m a g n i t u d e  of  II,  section  environment  2.3)  (i.e.  can be bonds)  This  shift,  however,  the  isomer  shift  is  between  detected  be  may  caused  calculated  equation  (S/c)i -  ii)  energy,  the  the  summarized i n t a b l e  important  a lesser  factors  of  R(v)  Line Shift  source if  to  d e r i v a t i o n of  a small difference  only  but  intensity,£,  = 1  Isomer  the  moot  recoil  characteristics  were  free  An i s o m e r by  three  the  the  thickness.  £  i)  are  the  which i n t u r n determines  absorber the  two  which govern  a given nucleus  Among t h e s e excited  Intensity  ^Ze [<R > 3c L  Debye T e m p e r a t u r e  It  is  2  ?  2  -  <Rg> ][|y(0)| 2  2  -|V^(0)| ] 2  J L  11-40  J  Difference  possibJ.e  for  a difference  between  the  Debye  temper-'  I -73atures of the source and absorber to produce a l i n e s h i f t . (See Chapter I I , section 2.2). The size of t h i s s h i f t can be calculated from the equation f  2Mc^[@g ill)  ^9  11-31 P^I  <£j^ e ^ I J  lSMc^r^^J  Impurity Effects  For F e ^ i n Fe, the l o c a l i z e d mode s h i f t i s expected to be 7  small. (See Chapter I I , section 2 . 6 ) . The s h i f t i s  (&/c)li± - >-<n>hwo x ~f( l o c a l ! zed mode) moc^ ^{gamma decay)  V-2  where " ( l o c a l i z e d mode) i s the relaxation time f o r the l o c a l i z e d mode to come to thermal equilibrium with the l a t t i c e , and T(gamma decay) i s the  l i f e t i m e of the ih.h kev state i n F e . T ( l o c a l i z e d mode) < 1 0 " 5 7  l 2  appears to be a reasonable value f o r T ( l o c a l i z e d mode). iv)  Mass Defect The d i f f u s i n g of the radioactive impurity into the source  l a t t i c e changes the average mass of the source and thus i t s i n t e r n a l energy r e l a t i v e to that of the absorber. (See Chapter I I , section 2.2). This mass defect, through i t s influence on the Internal energy, produces a s h i f t of the Mossbauer spectrum of  ( /c) s  i v  =  *M \ 2M c*J ,  0  v) Hydrostatic  "  • fJ x (M ) C g  16  "-37  Compression  The isomeric s h i f t has been shown to be dependent upon the 13 pressure t o which the l a t t i c e i s subjected. (See Chapter I I , section2.2) J  That Is, the electron configuration i s dependent upon the pressure. The pressure c o e f f i c i e n t i s given by  1  1 dp  Sol U n V ]  [dP  J  +  BQ\ olnVJ f o T ^  = - ( 2 . 6 H 0 . 1 0 ) x 10- /kg/cm l8  £  v-3  2  The f i r s t term represents the e f f e c t of the changing e l e c t r o n density at the nucleus, and the second, the r e l a t i v i s t i c e f f e c t of changing the mean v i b r a t i o n a l energy. The second e f f e c t i s l e s s than 5$ of the t o t a l , measured c o e f f i c i e n t and hence, the volume dependence of the  i isomeric s h i f t accounts f o r the major part of the pressure c o e f f i c i e n t . Thus a pressure difference between the source and absorber r e s u l t s i n a s h i f t of - the Mossbauer spectrum of (6/e)v -  1  c  ^1 SOTO \ f o l n V  dlnV  I T F  blnV\ 6  c  ilnV  v-4  c) Line Broadening Mechanisms  i ) Magnetic F i e l d E f f e c t s  In Chapter I I , section 2.4 i t was seen that the magnetic Zeeman s p l i t t i n g of the excited and ground state energy l e v e l s of F e ^  7  i s determined by the magnetic f i e l d at the nucleus. A difference of the magnitude of the i n t e r n a l f i e l d at the source and absorber n u c l e i produces a broadened Mossbauer l i n e . A difference i n the i n t e r n a l magnetic f i e l d s could arise from the p a r t i a l magnetization of e i t h e r or both of the source and absorber by the annealing process, or, by the presence  57 of the Co  i n the source l a t t i c e .  -75-  i i ) Localized Modes The l o c a l i z e d modes, (see above and Chapter I I , section 2 . 6 ) , also give r i s e to a l i n e broadening as a r e s u l t of the uncertainty of the occupation number of the l o c a l i z e d mode. The magnitude of the broadening i s given by  A  =  ^  n  2  >  "  < n > 2  )^  h v  " T ( l o c a l i z e d modes)  OQC  iii)  Source  and  Absorber  T(gamma decay)  V-5  Thickness  The l i n e width of the Mossbauer spectrum should equal 2 T where V i s the l i n e width of the nuclear t r a n s i t i o n . (See Chapter I I , section 2 . 1 ) . However, only when the source and absorber thicknesses 15  can be approximated to z e r o i s t h i s the case.  The use of nonzero  source and absorber thicknesses produces a broadened l i n e . The quantity R(v) depends upon both thicknesses, and hence, the expected l i n e width i n the nonzero thickness case should be determinable from the theoretical  spectrum.  i v ) Random S h i f t s Random isomeric s h i f t s , ( s e e Chapter I I , section 2.3) caused 57 by a variable chemical environment of the Ccr  n u c l e i could produce a  broadening of the Mossbauer spectrum rather than a s h i f t . In addition, a difference i n the quadrupole s p l i t t i n g of the excited state of the source and absorber would also produce a broadening of the l i n e . 5,2  Experimental Procedure  a) Source and Absorber  The source used f o r the comparison between theory and  e x p e r i m e n t was Iii3EC  The p r e p a r a t i o n and t r e a t m e n t  was d i s c u s s e d i n C h a p t e r s iron  .00020" t h i c k ,  thick,  armco  a n d h. s h i m s t e e l  b)  iron  .00035*' t h i c k ,  3.  this  source  were u s e d ; armco  iron  armco  1.  .00055"  .001" thick.  Procedure  T h e s o u r c e v/as m o u n t e d i n t h e  1. holder  2.  I I I a n d IV. F o u r a b s o r b e r s  of  (see  h o l d e r was  Chapter III) b u t the attatched  to  the  outer  lathe  high temperature  p o t was r.ot from the  27"  source  evacuated.  proportional  The  source  counter  window. The E .  2. measurements intervals the  the  were b e g u n . The gas  for  The  3.  c o u n t e r was t u r n e d o n 1 2 flow of  an h o u r b e f o r e measurements  c o l l e c t i o n of  and t h e  H . T . of  . 0 0 0 2 0 " Fe  a continuous r e c o r d of  of the  pulser for  imposed upon the  to  330,  setting  the  attenuation 1.0  volts  source  -  across  of  of  the  1.00  time  12  and the  at  throughout  of  constant  lathe. t h e r m o c o u p l e s were  connected  and a b s o r b e r t e m p e r a t u r e the to  helipot,  the  ^ . 0 0 i n steps  output  spectra  o f 0 . 5 0 were  731.  superkicksorter. at  " B a s e l i n e " of  The a m p l i f i e r s e t t i n g s  - l.c^s,  15y-*s.  a  and  begun.  C . D . C . 100 c h a n n e l  corresponding to  i n t e g r a t i o n time  d b , The d i s c r i m i n a t o r b i a s dead time  a b s o r b e r mount  s i n g l e c h a n n e l k i c k s o r t e r were p l a c e d  2 . 0 0 and 3.50  "Channel Width"  differentiation  were begun and a l s o  lk .k k e v s p e c t r u m b y t h e  amplitudes of  and a  the  3«5 v o l t s  T h e window s e t t i n g s pulser  c o u n t e r was c h e c k e d  a b s o r b e r was m o u n t e d i n t h e  k. T h e c o p p e r - c o n s t a n t a n  With  before  data.  mount was a t t a t c h e d  5.  the  hours  o n t h e NP I I  used were;  constant  -  scaler  was  0.8yus,  -776. The count rate of the gamma rays transmitted through the Fe absorber and through a l / l 6 " A l absorber were measured f o r equal counting times, i . e . correction was made f o r the instrumental dead time. 7. The areas within the single channel k i c k s o r t e r settings of the background  radiation transmitted through the Fe absorber and  through the A l absorber were measured. 8. The gamma ray transmission through the Fe absorber within the above window settings was measured f o r 22 d i f f e r e n t p o s i t i v e and negative v e l o c i t i e s of the absorber r e l a t i v e t o the source. The measurements were made i n groups of 4 , separated by a measurement of the transmission a t v = 0 , and v =  0  0  ( i . e . ,lhk cm/sec). A record of the  time at which the i n d i v i d u a l measuring periods ended was kept. 9 . Steps 6, 1, and 8 were repeated f o r the other three absorbers. c)  Discussion of the Procedure The data used i n t h i s section of the thesis was c o l l e c t e d  i n one continuous run t o eliminate any possible d r i f t s i n the apparatus. Furthermore, the source was kept a t room temperature t o eliminate a Josephson s h i f t , and a t room pressure t o eliminate a hydrostatic compression s h i f t i n the Mossbauer spectrum. Steps 6 and 7 were included so that the background correction, explained i n Chapter III, could be made on the data.  The numerous measurements of the transmission at  v - 0 , and v -°° were made so that Rj^v) could be plotted as a function f  of time t o check f o r s h i f t s i n the proportional counter.  Also, the  continuous determination of R ^ v ) automatically corrected f o r the decay of the source strength.  -78Calculations  5.3  and  Corrections  The t r a n s m i t t e d d a t a and c o r r e c t e d f o r  count  the  The b a c k g r o u n d c o r r e c t i o n  \  %l( ) v  F o r each  absorber  order  calculate  to  Chapter the  III).  x  was  calculated  rate  as  from the  measured  o u t l i n e d i n Chapter  III.  is  A  |  F e  ^  111-15  N ( v - 0)  pressure  was p l o t t e d  drift  For these p a r t i c u l a r  d r i f t was  H(v),  background count  thickness, the  rate,  of  the  a function, of  proportional  measurements  n e g l i g i b l e and hence,  as  however,  no c o r r e c t i o n  for  time  in  counter.(see it  was  counter  found that  drift  was  necessary. Using the was  calculated  and p l o t t e d  and V - 4 . The l i n e  5.4  background corrected values  shift  was  nesses  are  calculated  w i d t h of  lines  shown i n f i g u r e s  maximum i n t e n s i t y , the  h,  measured  The measured  ness  a f u n c t i o n of  N(v),  v in figures  from equation  the  ratio  ^(v)  V - l , V-2,  V-3  IV-6.  Results  The m o s s b a u e r  These  as  of  results  show t h e  increased.  of  each  of  the  four absorber  thick-  V - l , V - 2 , V-3 a n d V - 4 . F r o m t h e s e f i g u r e s  the  line  obtained for  at  values  Mossbauer l i n e s were o b t a i n e d . ^ h was of  determined  h,Aand  expected increase  & are of  Also,  and c a l l e d the  the  the  width,A•  summarized i n T a b l e V - l .  h a n d A as  the  absorber  thick-  to follow page 78  Figure V - l The Mossbauer absorption spectrum f o r an absorber of  .0002".  thickness  -79-  Absorber Thickness  /\(cm/sec)  h  $ (rata/sec)  .0002"  .070  .285  -.002  .00035"  .113  .300  -.003  .00055"  .169  .310  -.002  .001"  .228  .320  -.007  Table Measured Mossbauer Spectrum parameters thickness.  5.5  Discussion  of  Results  and  V a3  a  function  of  absorber  Conclusions  The l i n e s h i f t s measured f o r the f i r s t  t h r e e absorbers  used  agree w i t h i n the e x p e r i m e n t a l e r r o r (i . 0 0 1 mm/see) but d i s a g r e e w i t h t h a t o b t a i n e d f o r the f o u r t h a b s o r b e r . T h i s d i f f e r e n c e i s a t t r i b u t a b l e to  the f a c t t h a t the absorber used  the absorbers used  so s m a l l t h a t i t was  was  difference  concluded t h a t the  an isomer s h i f t s i n c e a l l o t h e r s h i f t mechanisms would  y i e l d the same s h i f t f o r the two The  s h i n s t e e l while  i n the f i r s t t h r e e cases were A m c o i r o n , The  between these two m a t e r i a l s was l i n e s h i f t was  i n the f o u r t h case was  types of a b s o r b e r s .  t h e o r y by which a t h e o r e t i c a l v a l u e of R(v) was  calculated  based upon t h a t p u b l i s h e d by S. M a r g u l i e s and J. R. Ehrman-'. In t h e i r  work, the t r a n s m i s s i o n c a l c u l a t e d f o r a gauKsian i n t h e n o t a t i o n used  T(v)  --  in this  was,  thesis:  (1 - f ) e x p ( ^ t ) ( i - $ ( ^ t ) | ' 2  t  1  1 -  source d i s t r i b u t i o n  exp  trf" as 2KI (EfS)* + f /k c  \ Wy ft/I  v-6  Tr /u_ 2  7*0  to follow page 79  V-£  The Mossbauer absorption spectrum f o r an absorber thickness  of  .00035".  -8oIt  i s seen  source  that  part  the Doppler s h i f t ,  of the i n t e g r a t i o n  that the Mossbauer absorption a n e g l i g i b l e source  S = vE /c,  has been  0  rather  than the absorber  included i n the part  and a l s o ,  i n t h e source has been n e g l e c t e d ,  implying  thickness.  The above e q u a t i o n  was m o d i f i e d  and  a l s o tire s i x l i n e e m i s s i o n  iron  e  to include  the s o u r c e t h i c k n e s s  and a b s o r p t i o n s p e c t r a o f F e - ' ' ' i n m e t a l l i c  The t r a n s m i s s i o n was n o r m a l i z e d  to y i e l d  R ( v ) by d i v i d i n g  through  by what Marcjulies and Ehnnun i n d i c a t e d t o be t h e n o n r e s o n a n t a b s o r p t i o n p o r t i o n of e q u a t i o n  V-6, e x p ( £ / * t ) | l 2  - §  j  . This p r o c e s s y i e l d e d t h e  equation  -rexp(-^t) [l 2  At  -$(^t)J  v ~ot>, t h e i u t u g r a l of e q u a t i o n I I - 1 3 s h o u l d be reduced t o f  the nonresonant t e x w so t h a t .R(v  ~ 1. I n p r a c t i c e ,  however,  times eval-  u a t i o n by t h e program o u t l i n e d i n append!;: A y i e l d e d R(v = °°) - . 9 7 • T e s t i n g p r o v e d t h a t R(v - <») i n c r e a s e d t o .9957 when t was reduced b y a f a c t o r of 10 t o ^.y<10  ' CJ . From t h e s e i - e s u l t s i t was concluded u  that  a q u a t i o n 1T-18 would y i e l d v a l i d r e s u l t s o n l y when t h e source  thickness  i s n e g l i g i b l e a n d t h a t t h e Mossbauer a b s o r p t i o n i n t h e source  affects  the it  calculated value vac concluded that  o f R ( v ) when t to f u l l y  i s not n e g l i g i b l e .  Furthermore,  a l l o w f o r the Mossbauer a b s o r p t i o n  ±  n  source when t i s n o t n e g l i g i b l e e q u a t i o n I I - I 8 must be m o d i f i e d t o  B(v)  = T c ( v ) / T ( v = ~») c  V-7  the  to follow page 8o  1.00  R(v)  Figure V-3  The Mossbauer absorption spectrum f o r an absorber of  ii  .00055*'.  thickness  -81where  T (v) c  =  (1  -  f)  exp(^ut) [l 2  -$  (!-/*)]  1  x exp  x expl w«v T P A 2  B y means o f values the  of  this  the  equation,  line  experimentally  R(v)  intensity determined  +  f  fwjkfTJ  V-8  -  |ft  c o u l d be  evaluated  n u m e r i c a l l y and  and l i n e w i d t h determined values.  CLE  and compared  the with  to follow page 8 l  1.00  Figure V-4  The Mossbauer absorption spectrum f o r an absorber thickness of ,001'  -82-  Chapter VI  TEMPERATURE DEPENDENT EFFECTS  6.0  Introduction  A summary of the temperature dependence of the quantities discussed i n Chapter I I i s given at the beginning of t h i s chapter. The experimental work discussed i n t h i s chapter was divided Into two sections; one section considered large temperature differences, i . e . greater than 50°K, and the other, s n a i l temperature differences, i . e . l e s s than 50°K. This d i v i s i o n was made so that very small changes caused by small* temperature differences could be measured as w e l l as the general temperature dependence. The analysis of the data at small temperature differences required the use of s p e c i a l formulae and techniques, both of which are discussed herein. The experimental procedure by which the data was accumulated f o r both sections i s given i n d e t a i l and the results summarized i n table and graphic form.  As i n Chapter V,  the relationship between the t h e o r e t i c a l and experimental temperature dependence i s investigated. I t i s seen that the agreement between the experimental and t h e o r e t i c a l Josephson e f f e c t i s good. The measurements indicate a Debye temperature of  h20°K  f o r both the source and absorber.  I t i s also found that the minimum l i n e width does not occur when the source and absorber are at the same temperature, a p o s s i b i l i t y that was discussed i n Chapter V. The shape of the Mossbauer spectrum proved to be temperature dependent. This dependence i s shown and discussed near the end of the chapter. In the l a s t section of the chapter, a discussion of the possible causes of the s l i g h t v a r i a t i o n between theory and experiment i s given.  -83-  6.1  B r i e f Discussion of Relevant Theory a)  Josephson E f f e c t The Josephson e f f e c t was discussed i n Chapter I I section 2 . 2 .  The magnitude of the e f f e c t i s calculated from equation 1 1 - 2 7 .  b)  Hydrostatic Compression E f f e c t The d e t a i l s of t h i s e f f e c t were given i n Chapter I I section 2 . 2 .  The magnitude of t h i s e f f e c t i s calculated from equation 1 1 - 3 0 .  c)  Temperature Dependence of f The Mossbauer i n t e n s i t y i s determined, i n part, by the Debye-  Waller factor, f , which i s given, i n the Debye approximation to the l a t t i c e vibrations, by the expression  / e* - 1  L  1-5  J  lo Jo  As the temperature increases, f decreases so that the i n t e n s i t y , h, also should decrease. Table VI-1 shows the v a r i a t i o n of f as a function of temperature. For t h i s table, f was calculated using © (OK)  D  (°K)  f  80  • 910  300  .784  100  .900  350  .755.  150  .875  400  .730  200  .843  450  .701  250  .813  500  .675  Table V I - 1  f  =  420°K  -34The source  and absorber  in table in  VI-1, T  figure  d)  its of  H, at  H affects  difference  6.2  the is  the  i n Chapter II,  emitting or  as  the  the  + V2,  will  Using the  the  effective  values  a f u n c t i o n of  of  f  given  temperature  w i d t h of  Mossbauer l i n e  an i n c r e a s e  be  the  2.4,  is  This  internal  First,  f i e l d s of  the  Mossbauer l i n e broadened,  or  a decrease  of  the  its  i n the  magnetic  temperature  temperature  i n two w a y s .  the is  Field  as  dependent  dependence the  source  also  and  increases.  intensity will  decrease  source-temperature  will  intensity.  Line  C is  section  Weiss Law.  Between  Effects Source  Small  Temperature  and A b s o r b e r  An e q u a t i o n t h a t  c a n be u s e d t o  S = ?a,v(A +• B) (A + C)  (C +• D) (B +• I))  V g - v±, the  of  Shifts  count  -  A is  the  rate  at  count -v-^,  assumption contained i m p l i c i t l y as  affects  Magnetic  i n t e r n a l magnetic  Differences  -  the  Mossbauer spectrum  Experimental Analysis  w h e r e AV  also  absorbing nucleus  g i v e n by the  between the  either  a)  f  a n d T'»  T  Dependence of  increases, the  Secondly, so t h a t  of  and T have b e e n p l o t t e d as  noted  dependence  reduce  thicknesses,  Temperature  absorber  dependence  VI-1.  As field,  temperature  seen below,  this  detect  line  shifts  is  VI-1  rate  at  and D i s in this  approximate  tv^, the  B is  count  equation  is  the  count  rate  at  that  & « A .  rate  -vg.  formula y i e l d s a value  at  The In  for  fact, 8  2.0  I 1.0  loo  150  200  250  300  350  too  **50  500  ©'(OK)  550 (a)  0.2  O Hi  0.1  r 3 icfe—'  T50  '200  250  300  350  too  450  0(°K) FIGURE V I - 1  The variation of the apparent source and absorber thicknesses, Jand of temperature.  500  550  (b) T', as a function  S  9  -85that  is  i n e r r o r by a f a c t o r  of  A more a c c u r a t e  1.5.  formula  (Moss  II,p  is  c  _  J»k.(l/h -  .75) ( B * -  3N/T  (R  w h e r e R* and. R ' a r e t h e  count  equation  IV-6  is  it  equation  VI-1  does  It to the  evaluate  is O  velocity  figure  II-3.  d(S/c)/d(0),  b)  that  ,  the  there,  If  S is  determined  was  plotted  the  line  V it  was  dependent  width.  It  is  w i d t h can be  The l o c a t i o n be  of  used,  the  the  h and / \ ,  S as  source.  a function of  from the  same d e f i n i t i o n s  total as  the  of  of  whereas  a function  6/c,  0 and then  9,  of 0  where c © „  the  l i n e width  magnetice  g r e a t e r than  found at  determined  accuracy of  be g o o d .  better  that  A0  is  in  slope,  Thus,  field  the  that  the  the  theoretically  small  location  A 0 - 0,  then  function  requires of  w i t h good  be  a  of  temperature  determination  l\ c a n  seen  The t r u e p o s i t i o n  Assuming t h a t  i n section 6.2.a,  H . As  b y M e a s u r i n g A as  done w i t h a f e w p o i n t s spectrum.  depended  ^11 - 0 a t = 0.  minimum r e q u i r e s  and the  statistics  imum l i n e w i d t h i s  equation  of  expected that i f  minimum l i n e  the  of  measurement of  internal  the  rather than  The d i s a d v a n t a g e  a knowledge  as  noted  w i d t h w o u l d be  counting  -v.  Widths  minimum l i n e  the  and  temperature.  m e a s u r e d l i n e w i d t h was  differences  -tv  a function  Debye  the  of A 0.  at  by c a r e f u l  m e a s u r e d as  determines  temperature  the  rates  Debye t e m p e r a t u r e  light,  In Chapter upon the  R)  not.  of  Line  IV-6  necessitates  possible,  d  R)  +  A, B,  C,  the  that min-  statistics and D have  c a l c u l a t e d from  the  -86-  A  -  2|v-i| +  dv ( h / 2 dP  - A) +  a  where dv -  - v  requires  A, B,  that  Mossbauer an  extent that the  large  c)  i n the  In  points  However, by  6.3  For large two  sets  of  linear portion,  equations  on t h e  C -  VI-2  D. This  on t h e  rates,  equation  l i n e a r p o r t i o n of  spectrum w i l l  so t h a t  V I - 1 and V I - 2 i t  be  A and B,  the  shifted to  such  and C and D ,  will  equation VI-2  for  small A0,  Source  is  not  valid  h varies of  noted that A or  determine  h appears  S can be  a  calculated.  h f r o m a l i m i t e d number  i n d e p e n d e n t o f Z\ a n d £> i s  very l i t t l e  as  and hence  can be  impossible.  determined  h a g a i n s t aQ f o r . l a r g e a f l .  Procedure  and A b s o r b e r  source  the  NSEC $ 2 .  and  the  absorber,  used f o r  For large the  shown i n f i g u r e  was u s e d o n l y when t h e s y s t e m was  source  to  is  either  Mossbauer spectrum,  was  similar  count  C)  Intensities  Experimental  holder  the  a similar expression  The  the  and D he l o c a t e d  i n t e r p o l a t i o n from a p l o t  a)  a n d dRg =  t h a t m u s t b e known b e f o r e  Therefore, of  A - B,  -  &0.  Line  variable  C,  spectrum.  not b o t h be for  dR^ =  o J  dv ( h / 2 di^  and the  a l l temperature  temperature  source  source  absorber.  differences  was m o u n t e d i n t h e  III-14.  followed  dependent  Hie source temperature  for  measurements  between the  source  high temperature  source  h o l d e r shown i n f i g u r e  111-15  was l o w e r e d b e l o w 0 ° C . A  small temperature  differences  between  -87-  For It  t h e s e measurements,  was d e t e r m i n e d t h a t  obtained with this on  the  lathe  b)  1. holder  and the 2.  h o l d e r was  of  the  Greater  the  lathe  on the  c o u l d be  III.  to  i n the 15  high  temperature  microns  clamped i n i t s  h o l d e r and  the  carriage.  were  put i n t o  results  used.  Than A b s o r b e r Temperature  firmly  The t h e m o c o u p l e s  a b s o r b e r was  holder described i n Chapter  was m o u n t e d f i r m l y  T h e a b s o r b e r was  c o l d j u n c t i o n s were  .00055" Fe  a p p e n d i x D ) . T h e a b s o r b e r was m o u n t e d  p o t was e v a c u a t e d  clamped onto 3.  The  Temperature  outer  (see  b y means  The s o u r c e  the  maximum s i g n i f i c a n c e o f  absorber,  carriage  Source  the  only  connected  to  an i c e - w a t e r  h.  The s e t t i n g  apparatus  5.  The b a c k g r o u n d c o r r e c t i o n ,  the  Varian  recorder.  mixture.  were  the  same  as  used  in  Chapter V . as  o u t l i n e d i n Chapter V,  was  determined. 6. was  The t r a n s m i s s i o n of  measured as  a f u n c t i o n of  the  gamma r a y s  velocity for  through the  lk p o s i t i v e a n d  absorber negative  vel6cities. 7. source  temperature 8.  readings was  T h e h e a t i n g u n i t was  of  allowed f o r  measurements  5,  mv, the  were  i n t o the  inner pot  and  the  increased.  Steps 3.15  inserted  6,  and 7 were  5.25 source  begun.  mv,  repeated  and 8.3  temperature  mv.  for  In each  t o become  source case stable  thermocouple sufficient before  the  time  c)  Source Temperature Less  1. in  the  Steps  1 to  low temperature 2.  reservoir  4 of  section  6.3.b  were  repeated with  the  source  holder.  A 600 watt h e a t e r  and the  Than A b s o r b e r T e m p e r a t u r e  variac  was  was p l a c e d I n t h e  set  to  liquid  nitrogen  produce a thermocouple  reading of  rav.  -1.95  3.  T h e b a c k g r o u n d c o r r e c t i o n was d e t e r m i n e d a s  4.  The t r a n s m i s s i o n o f  was m e a s u r e d as  a f u n c t i o n of  the  the  gamma r a y s  i n Chapter V.  through the  v e l o c i t y f o r 14  absorber  p o s i t i v e and  negative  velocities. 5. readings  d)  of  to  2 . 0 5 mv,  m v  repeated  for  5 of  section  thermocouple  6 . 3 « b were  w e r e made a t  +.0123 cm/sec,  Steps  give  of  1.00  measurements  1 to  collecting  3 of  w e r e made a t  section  temperature  the  velocities  6 . 3 . C were of  of  it  mv,  variac  l . ? 2 mv,  as  d i f f e r e n c e s was o f  counts the  was f o u n d t h a t  registered  order of  and 0  The  . 0 1 2 mv,  used f o r the  short  each  +.1444cm/sec, cm/sec.  variac  a n d - . 8 5 mv.  each temperature.  was n e g l i g i b l e i n t h e  The number o f  repeated.  . 2 5 mv,  same v e l o c i t i e s  measurements,  of measurements time.  mv, 1 . 5 2  The  ± .0062 cm/sec, • .0046 cm/sec,  thermocouple readings  In a l l the any s e t  repeated.  mv. T h e gamma r a y t r a n s m i s s i o n was d e t e r m i n e d a t  The measurements  cm/sec,  to  1 to  T h e gamma r a y t r a n s m i s s i o n was d e t e r m i n e d a t  in  source  «  thermocouple readings  and 2.55  2. set  Steps  give  temperature.  was  3> afccl 4 w e r e  S m a l l Temperature D i f f e r e n c e s Between Source and A b s o r b e r  set  i.Ol85  2,  - 3 - 0 0 mv a n d - 3 . 8 9  1. was  Steps  The  step  1.  counter  p e r i o d of  per point f o r  drift, data  small  10 g r e a t e r t h a n t h a t  for  -8 9  large  6.4  temperature  differences.  Calculations  a)  Large  For corrected  and Results  Temperature  D i f f e r e n c e s Between  each  temperature,  source  transmission values  illustrated  ^(v)  The l i n e  width,A,  was  shifts,  line  shift,  Mossbauer spectrum, s p e c t r u m was are  ,  calculated  was  valid  the  taken  symmetrical at  as  that  the  for large  of  the  temperature  graphs  v e l o c i t y about which  -|h. The measured v a l u e s  as  directly  width of  was m e a s u r e d d i r e c t l y f r o m t h e  a n d was  using  determined  r e c o r d e d as  ^ h . S i n c e e q u a t i o n V I - 1 was n o t  and A b s o r b e r  a f u n c t i o n of v e l o c i t y  h,  spectrum at the  was  and p l o t t e d as  i n f i g u r e V I - 2 . The i n t e n s i t y ,  from these p l o t s .  Source  these  of  the  the  parameters  given i n table V I - 2 .  *6°K  h  A (cm/sec)  JL ( c i n / s e c )  179  .093  .060  .0130  100  .124  .036  .0075  50  .166  .031  .0035  0  .174  .031  -.0003  -76  .168  .037  -.0055  -108  .160  ,o4i  -.0065  -138  .153  .o4o  -.OO85  Table The  In Josephson  gross  order  effect,  VI-2  temperature dependence of the of the Mossbauer spectrum  to  compare  several  the  parameters  measured temperature  corrections  had to  shift with  b e made w i t h t h e  the  measured  to follow page 89  FIGURE V I - 2  The Mossbauer spectrum obtained with a source temperature of 126°C and an absorber temperature of 26°C. This spectrum i l l u s t r a t e s both the temperature s h i f t and the appearance of the hyperfine structure.  -90shift.  First,  measured. Second, for  it  This  it  was  noted that  residual  at  s h i f t was  was n o t i c e d t h a t t h e  a l l measurements.  This  a small residual  A Q =0  subtracted  absorber  variation  from the  temperature,  was  shift  measured  was  shift.  6', was n o t  compensated  constant  by adding a s h i f t  b'  given by  -2.24  S'=  x lo"  where  =  dependent  hydrostatic  This in  Q' - 2 9 7 ° K ,  hydrostatic  figure  and the  II-3.  to  in  VI-3  figure  Josephson  VI-3  measured  of  the  shift  calculated  line  shift.  shift  shift  lias b e e n p l o t t e d • The s o l i d  shift  the  compression  measured Josephson shift  A0c  compression  The v a l u e s  Josephson  Q' =  1 5  was  was  as  the  added t o  calculated  measured are  Third,  shift,  the  in this  figure  from equation  11-27  is  measured  from the the  the the  data  three  summarized i n t a b l e a f u n c t i o n of  temperature shift.  contained  corrections,  V I - 3 . The  source  temperature  theoretical  u s i n g (P)  D  =  420°K  and  297°K.  IxlO ' cm/sec  •  '  -4  o'xlO" * cm/sec 1  cm/sec  6hc^°'  U  cm/sec  lcxio- ' cm/sec 1  473  130  -3.0  -1.3  6.4  139.2  379  65  -3.0  -1.3  4.5  73-8  349  35  -3.0  2.3  41.6  299  -3  -3.0  -1.3  0.0  -1.3  219  -55  -3.0  1.3  -3.4  -56.7  186  -65  -3.0  2.0  -4.9  -68.9  156  -85  -3.0  2.0  -6.2  -86.0  Table  '  -1.3  VI-3  The t h r e e c o r r e c t i o n s a p p l i e d t o t h e measured t e m p e r a t u r e of the Mossbauer spectrum t o o b t a i n the measured Josephson  shift shift.  to follow page 90  i  • 0  i  I  I  50  100  150  I I 200 250  Q FIGURE VI-3  I I 300 350  I I 4oo 450  I 500  U 550  (°K)  The measured temperature dependent Josephson s h i f t plotted as a function of the source temperature. The s o l i d l i n e i s the t h e o r e t i c a l s h i f t calculated f o r a Debye temperature of 420°K.  -91The l i n e figure value  intensity  has b e e n p l o t t e d  VI-4. Included i n this o f /the  figure  Mossbauer i n t e n s i t y , £ ,  account only the  is  a f u n c t i o n o f hQ  variation  withA.9. This  temperature v a r i a t i o n  t i c a l p o i n t ' s were c a l c u l a t e d  the  as  of  from equation  f  and not  of  variation of  theoretical takes  H . The  on t h e  11-18  the  in  into  theore-  IBM 1 6 2 0  computer.  b)  Small Temperature  Differences  Between  Source  and A b s o r b e r  i  The l i n e to  above,  These  first,  calculated  variation section  in Q',  6.4.a.  tabulated  shifts  were c a l c u l a t e d  S, b y e q u a t i o n shifts and t h e  and s e c o n d l y ,  were c o r r e c t e d hydrostatic  The u n c o r r e c t e d  i n table  VI-1  from the  for  the  two e q u a t i o n s  8^ f r o m e q u a t i o n  residual shift,  compression  shift  and c o r r e c t e d v a l u e s  referred  as  of  the  outlined  8, a n d  IV-6.  in  S^are  VI-4.  0'°K  S (m) cm/sec  cm/sec  cm/sec  S?(c)  S( ) ra  1  2  cm/sec  335  298.5  .00214  •.00270  .00241  .00297  323  297.5  .00113  .00157  .00113  .00157  315  297  .00093  .00131  .00112  .00150  310.5  297.5  .00087  .00123  .00043  .00082  298.5  299.5  -.00009  .00030  -.00042  .00000  279  299.5  -.00278  -.00232  -.00226  -.00190  268.5  297.5  -.00330  -.00310  -.00290  -.00270  250  297  -.00490  -.00479  -.00386  -.00340  Table The u n c o r r e c t e d  VI-4  and c o r r e c t e d s m a l l t e m p e r a t u r e Mossbauer spectrum.  shifts  of  the  to follow page 9 1  A 9 - 0-9'(°K) FIGURE VI-4  The measured and t h e o r e t i c a l temperature v a r i a t i o n of the Mossbauer i n t e n s i t y plotted as a function of the temperature difference between the source and absorber.  I n s p e c t i o n of shows,  as  expected,  usually the k  A  greater  magnitudes as  of  the  that  values  the  than the the  of  two a r e  S, a n d S  different,  magnitude of & . is  also  w i t h the  This  x  two s h i f t s  given i n table  X  a f u n c t i o n o f i\Q i n f i g u r e V I - 5 . F r o m t h i s  d(&/ )A$  6,  of  c  and  8  were  x  found to  d(Si/c)d0 = 3 . 2 8 x 1 C *  magnitude  difference  obvious i n the  VI-4 of  between  p l o t of  f i g u r e the  8, a n d  slopes  be  /°K  VI-4  /°K  VI-5  1 5  and  d(S /c)d9 = 2.36 x 1 0 " 2  The  latter  slope  is  equal, within experimental  obtained from the  measurements  gross  shifts  temperature  equation VI-3 is ature to  shifts.  lie  slope  on the  versus  The s h i f t s , solid  in  table  S, z  line.  - 1 5  For(Q)  of  as  a f u n c t i o n of  temperature  shifts.  The e x p e r i m e n t a l p o i n t s  results  indicate  at  AQ  = 24°K.  that  Hence i f  the it  of  agreement  the  shows  for calculating  £  =  420°K the  that  temperseen  theoretical  VI-6  small temperature of  are  differences  equation V I - 2 are  v a r i a t i o n of  width l i e s  the  not  obtained  Mossbauer  and s m a l l  shown w i t h t h e i r  assumed t h a t t h e  tabulated  c a l c u l a t i o n s were  for both large  minimum l i n e is  slope  f i g u r e V I - 3 a^d are  h used i n these  f r o m f i g u r e V I - 2 . F i g u r e V I - 6 shows t h e widths  the  /°K  a n d a b s o r b e r b y means  V I - 5 . The v a l u e s  to  is  l i n e widths calculated f o r  source  This  use  added t o  250°K and Q = 3 5 0 ° K  x 10  to  error,  previous section  temperature.  were  theoretical  d(S/c)/d6 =2.34  between the  made i n t h e  s u f f i c i e n t l y accurate  between Q =  The  1 5  error at  residual line  line  temperature  bars.  The  = 0°K but broadening  to follow page 92  -.008  -.00b  -.004  -.002  0  h  D  (cm/sec)  .002  .004  |-  .006  ^  line shift  S  x  <J) l i n e s h i f t gg .008 -20  -to  20  0  Fo  "60"  A0<° > K  FIGURE V I - 5  A comparison of the temperature dependent l i n e s h i f t s , S and gg, calculated from equations V I - 1 and IV-6. LT  at  A 0 = O°K i s  All,  at  A 0 = 0°K,  which is at  the  A0=O°K  figure the  VI-6  the  field  .0025  is  the  t a k e n as  AG  cm/sec.  .0305  calculated  required to (See  table the  .01  x 10^  broadening in  Mossbauer spectrum c a l c u l a t e d  at  c a l c u l a t i o n the  = 24°K.  oe  minimum  The t h e o r e t i c a l  from line  points  bars.  Spectra  on the  Mossbauer spectrum.  figure  VI-2,  that  0°K  is  and C o n c l u s i o n s  = 100°K, l 8 Q ° K  indicates  forAH  difference,  I I - 3 ) . The s o l i d l i n e  f i g u r e without error  the  field  produce a l i n e  H. For this  cm/sec  Results  The Shape o f  At  resulting value  dependence of  D i s c u s s i o n of  a)  r e s i d u a l i n t e r n a l magnetic  broadening of  i n d i c a t e d on the  6.5  some  difference  of  temperature  w i d t h was are  due t o  and  This the  -138°K a h y p e r f i n e s t r u c t u r e  structure,  appeared  shown a t A£) = 100°K i n  Zeeman s p l i t t i n g o f  A0°K  the  nuclear  levels  h  /_\(cm/sec)  335.0  297.5  36.5  .17  .0335  323.0  297.5  25.5  .18  .0325  315.0  297.0  18.0  .18  .0320  310.5  297.5  13.0  .18  .033^  298.5  299.5  -1.0  .18  .0318  279.0  299.5  -20.5  .17  .0330  250.0  297.0  -47.0  .17  .03^0  Table The l i n e  w i d t h of  temperature  the  VI-5  M o s s b a u e r s p e c t r u m as  d i f f e r e n c e s between  source  a f u n c t i o n of and  absorber  small  to follow page 93  .07  .06  8  (cm/sec)  .05  .03  -  -120  FIGURE VI-6  -3o  -1)0  0  120  160"  * 0=0-©TO  The v a r i a t i o n of the Mossbauer l i n e width as a function of the temperature difference between the source and absorber. The s o l i d l i n e and the points without error bars represent, the t h e o r e t i c a l temperature dependence calculated from the known temperature dependence of E.  _ 49  f'l of  the source  and a b s o r b e r  detected at these between only  differences  but also that  i n the i n t e r n a l are present.  AH r  5  .06 x 1 0  oe,  5  temperature difference  of this  line  This  shift  t o t h e measured  .031  cm/sec.  i s comparable  in transition  x 10  the e f f e c t  probabilities,  thereby b)  overlap,the  shift  line  resonant  absorption  absorber  that,  a  large  shift  of  is cm/sec.  .0195  f o r t h e two  II-2).  Moreover,  Therefore,  are  at the  greater velocities  c o r r e s p o n d i n g t o t h e s e two should reach  a maximum  structure.  Shift  shifts  agree,  within experimental  p r e d i c t e d by the Josephson  implies  source  of the  and a b s o r b e r  I n t h e t e m p e r a t u r e r a n g e c o n s i d e r e d (Q = 156°K t o 9 The measured  large  of the n u c l e i  is greatest  (See t a b l e  lines  producing the observed doublet The L i n e  levels  that  W j ^ f o r t h e s e two t r a n s i t i o n s  which the emission and a b s o r p t i o n  transitions  consis-  l i n e w i d t h a t A 0 = 0°K o f  than those of the f o u r remaining t r a n s i t i o n s . at  oe, f o r each  5  a line  energy  -3/2 ~> - l / 2 a n d 3/2 - » l / 2 .  transmission  splitting is  of the source  11-44, t o p r o d u c e  the  produces  H * a t A(9 - 2k°K, t h e n  on the energy  calculated, from equation  transitions  no l o n g e r  -3/2-> - l / 2 a t A 0 - l 8 Q ° K . T h e d i f f e r e n c e i n  energies  The s h i f t  difference  at the n u c l e i of the  To i l l u s t r a t e  c a n have  the t r a n s i t i o n  transition  fields  o e , a n d -.12  temperature d i f f e r e n c e s .  This  t o be  temperature d i f f e r e n c e s ,  A s s u m i n g t h a t II =  x 10  .12  magnetic  i s , the  levels  a multiple l i n e .  at these large  and a b s o r b e r  the  That  t h e Zeeman s p l i t t i n g o f t h e n u c l e a r  tent with the f a c t  consider  is sufficiently different  temperature d i f f e r e n c e s .  a broadening,  above  nuclei  i n this  combination,  temperature  effect  f o r (P)  0  =  with the  4 2 0 ° K . The agreement,  range and f o r t h i s  t h e q u a d r u p o l e moment  error,  =478°K),  source and  and the isomer  shift  are  magnetic  f i e l d s between the  source  and a b s o r b e r  at A 0 = 0 K could  be  57 caused by the whereas  the  fact  that  absorber  experimentally that from the  the  source  lattice the  is  lattice  natural Fe.  internal field  c o n t r i b u t i o n s of  its  is  at  an a l l o y of  However,  it  Co'  and Fe  has b e e n  an Fe n u c l e u s a r i s e s  own e l e c t r o n s  and depends o n l y  found  primarily  slightly  16 upon the  magnetization of the  internal  field  whereas  it  at  300°K a t  changes  to  surrounding atoms.  the Fe n u c l e i  3.0^5  x lO^oe  F o r example,  i n an F e  lattice  i n a Co l a t t i c e .  is  Since  the  3.33 x  10 oe 5  the  57 concentration  of  Co  i n the  source  any b r o a d e n i n g caused by t h i s  u s e d was l e s s  mechanism i s  t h a n 2.CO x 1 0  negligible. Also,  if  , the  57 alloying the  of  Co  magnetic  and Fe  field  absorber n u c l e i . that  a t A@  that  at  the  left the  source  the  absorber n u c l e i , increases.  nuclei  source at  the  as  the Mossbauer l i n e impossible  to  nuclei  is  since  the  the  of  the  the  i n the  dependence of  f.  apparent  to  amount o f  This  A0>o°K a n d d e c r e a s e very  small.  (See  increase source  than that  greater  field  at  than  decreases  the  of  a r e s i d u a l magnetic  as  the  d i f f e r e n c e i n H between  annealing treatment. domains are  field  Moreover,  subjected  to  appear.  Unfortunately,  it  is  this broadening. dependent b r o a d e n i n g mechanism the w i d t h of  thickness,  the  observed l i n e  through the so as  to  is  temperature  increase A  f o r A0 < 0 ° K . H o w e v e r , t h e v a r i a t i o n i n 7" i s  figure VI-l)  the  indicates  of  mechanism would o p e r a t e A  is  then  a s m a l l amount o f b r o a d e n i n g o f  can be e x p e c t e d the  source  magnetic  magnetic  f i e l d s so t h a t  c o u l d be o p e r a t i n g t o change  the  the  An a d d i t i o n a l t e m p e r a t u r e that  at  presence of  s h o u l d be l e s s  field,  t h a t A H = 0 f o r tsQ>0°K  field  a result  estimate  i n t e r n a l magnetic  A p o s s i b l e cause  boarders  nonuniform magnetic  the  fact  magnetic  and a b s o r b e r  i n the  source  However, the  - 0°K the  temperature the  at  affected  for  to follow page 9 5  L  I  -120 -100  L _  -.03  _ J L  -80  j -.02  L _  J  - 6 0 -40  L  -20  I  J  _ !  _ 1  _ L  0  20  40  oO  oO  L__^£JMJ  -.01  0 AH(xl05oe)  J  . 0 1 •»  I  1  ;  100 120  ii  i .02  FIGURE VI-7 The difference between the t h e o r e t i c a l and measured Mossbauer i n t e n s i t y plotted as a function of the temperature d i f f e r e n c e between t h e s o u r c e a n d a b s o r b e r a n d the d i f f erence between the i n t e r n a l magnetic f i e l d s .  -96temperature independent or, at the most, t h e i r temperature  variation  i s within the experimental e r r o r . Hence any v a r i a t i o n of these two quantities, such as that found by S . DeBenedetti et al^was not observed  28 Figure II-4 showing S/c  with t h i s source and absorber combination. as a function of 0 f o r  ( P )  0  =  > 4 2 0  dependence of S/c on <Q above 0  O  K  and(P)  2 0 0 ° K ,  0  -  i l l u s t r a t e s the weak  355°K  so that the agreement between  experimental and t h e o r e t i c a l l i n e s h i f t s merely means that any v a r i ation caused by a temperature dependence of ®  0  i s small compared with  the experimental e r r o r . This weak dependence of <Q) on 0 also means D  that an e r r o r of ( H )  D  =  4 2 0 ° K  ± 2 0 ° K  must be associated with the assignment of  to the source and absorber. In order to reduce t h i s error,  and to detect a difference i n Debye temperature between the source and absorber, measurements would have to be conducted i n the region below 0 r l 5 O ° K .  o The residual s h i f t found f o r £ . 0 = 0  K  does not arise from  a difference i n Debye temperature between the source and absorber. Since the source was an a l l o y of Co ? and Fe, i t i s expected that 5  ©t> < © D  w i l  i c h would produce a positive s h i f t whereas the observed  residual s h i f t i s negative. I t i s concluded, therefore, that the residual s h i f t i s e s s e n t i a l l y an isomer s h i f t . c) The Line Width The observed temperature dependence of the l i n e width i s predicted, within experimental error, by the temperature dependence of the i n t e r n a l magnetic f i e l d and i t s e f f e c t upon the Zeeman s p l i t t i n g s of the nuclear l e v e l s of the source and absorber n u c l e i . This r e s u l t , however, i s based upon the observation that the minimum l i n e width occurs at AQ - 2k°K and not at A 0 = O ° K . The difference i n the i n t e r n a l  to follow page 96  Figure VI-8  The temperature dependence of the Debye-Waller factor, f , and the approximate area beneath the Mossbauer spectrum, ihA.  -97-  d)  The L i n e  Intensity  The t e m p e r a t u r e d e c r e a s e as  Q increases.  a t u r e dependent, to  explains  the  demonstrated p l o t t e d as has  the  dependence However,  temperature  temperature  i n figure VI-4.  a definite  h should  and t h e r e f o r e  dependence of  of  h.  f  A are  interesting  to  is  note  at  a p p r o x i m a t e l y A 0 = 2k°K supporting the  temperature  dependence  temperature It  is  area under the as  alone  is  insufficient  fact  is  clearly  This  T h e d i f f e r e n c e "between h a n d £  Q. F r o m t h i s  that  the  of  h,  in this  of A.H. T h i s  f i g u r e V I - 7 has  curve  of  f  that the  appears  figure curve  of  is  agreement  seen same  to  6.1.d.  a minimum  figure VI-6,  be t h e  The  resultant  and H . product,  |hA, which i s  should follow  In f i g u r e V I - 8 ^,-hkand f it  of  the  therefore  dependence  do have t h e  these l i m i t s the  curve  similar to  Mossbauer spectrum,  figure  two q u a n t i t i e s  corresponding values  h a s "been  c o n c l u s i o n t h a t A H =r 0 a t A Q = 2k°K.  expected  f.  temper-  r e l a t i o n s h i p was d i s c u s s e d i n s e c t i o n  It  dependence  that  r e l a t i o n s h i p between A H and h . The p r o b a b l e  mechanism p r o d u c i n g t h i s  the  predicts  a f u n c t i o n o f kQ i n f i g u r e V I - 7 . T h e A Q a x i s  illustrates  of  f  sinceJg  dependence  a l s o been l a b e l e d w i t h the  thereby  of  that,  are  between  temperature  b r e a k s i down.  the  same  p l o t t e d as  the  approximate  temperature functions  2 0 0 ° K a n d kOQ°K,  these  dependence b u t beyond  of  -98Appendix  A  T H E COMPUTER PROGRAM  In order line  with those  to  compare  This  at  UBC C o m p u t i n g C e n t r e  Toronto's  e v a l u a t i o n was  the  evaluation  correct the  of  steps,  3x10* ev.  The t i m e  53 m i n u t e s  arrangements  per  difficulty  the  that  inherent of  and the  r e q u i r e d to  p o i n t u s i n g the each  of  the  eval-  IBM 1 6 2 0  University  four  of  .97  tests  were  step  size  step size, -3xl0"?ev  The number o f thicknesses the  e v a l u a t i o n of  was t h a t  the  R(v=°o)  further  the  a  to values  points  was  120  that  hence  3°,  U n i v e r s i t y of the  and  lxlO"^ev,  program u s i n g these  Toronto's  points would  R(v= °°) s h o u l d be 1. R ( v = <*>),  than I. Since t h i s  a series  than  the  correct,  value  was l e s s  u s e d were  program t o  however,  step size,  v u s e d f o r v =00,  rule  IBM 7 0 9 0 .  .970125 r a t h e r  i n the  for  integration,  absorber  complete  p r o g r a m u s e d was remained,  values  IBM 1 6 2 0 .  send the  on the  of  evaluate  were  replacing  the  be  formulated using Simpson's  values  limits  i n which these values  Before  had to  Mossbauer  The d i f f i c u l t y i n p r o d u c i n g  The f i n a l  IBM 1 6 2 0 of  IBM 7 0 9 0 a t  integral.  where t h e  a b o v e p r o g r a m was  value  and the  the  o n l y 20 m i n u t e s If  be  II-18  the  done n u m e r i c a l l y u s i n g t h e  p r o g r a m was  w e r e made t o  Computing Centre  to  equation  in determining suitable  601,  required for  require  be  the  integration.  number o f  were  of  program l a y  limits  was  to  parameters of  Computing C e n t r e . Originally,  for  theoretical  obtained experimentally  uated. the  the  limits  of  check  varied.  of  a l l d i v i s i o n s by c o n s t a n t s ,  error  runs were  indicating that  made o n t h e  calculated  program,  was  integration,  In each  it  The  case, the  was  changed  by m u l t i p l i c a t i o n s ,  the the  however,  error  the  thought  or  made o n  by  lay by  and by  the elsewhere.  -99replacing Eight,  Simpson's rule by a Gaussian formula f o r  four-point Gaussian integrations  integration did  the  former program but  Some f i n a l A-l. Dr.  i n d i c a t e d above.  results  This  L.  V  White,  program y i e l d e d the  obtained with this  Department  E(v,ti)  used over  i n an average time  Bothithe o r i g i n a l and f i n a l B.  were  of  the  of  are  the  range  of  same r e s u l t s  2-|- m i n u t e s  program are  programs  integration.  per  point.  summarized i n T a b l e  on f i l e  i n the  care  of  P h y s i c s , UBC.  R(v,t )  Rfv/fc^)  R(v,tJ)  • 9060  .8614  .8099  .7393  .9^00  .9171  .8892  .8481  .04  .9690  .9662  .9627  .8572  .11  .9723  .9720  -9717  .9710  «.9726  .9729  .9723  .9726  2  cm/sec 0  if  Table A - l . . Test results and tl, r e f e r  of to  the the  numerical e v a l u a t i o n of absorber t h i c k n e s s e s of . 0 0 0 5 5 " and . 0 0 1 " .  as  »  1  t  R(v). t i , t , t3 .0002", .00035", 2  -100Appendix B 57 THE PREPARATION OF AH ENRICHED Fe ABSORBER  As i n v e s t i g a t i o n was made i n t o the p o s s i b i l i t y of making 57 57 an enriched Fe absorber. Enriched F e " i s supplied i n the compound FegO^ so that the following procedure was devised to prepare the metallic absorber from the oxide. 1. Reduce the f e r r i c oxide t o Fe by placing the oxide i n a hydrogen atmosphere at 900°C f o r one hour. 2. Add HgSO^ (dilute) t o the Fe to prepare F e g f S Q i ^ containing 10 mg of Fe/ml. 27 3. Prepare the following e l e c t r o p l a t i n g solution 2.5 ml Fe (S0l!)3 2  85.O ml (HH^JgCgOi, (saturated) 1.0 ml 3M HgSO^ 4. Use the following p l a t i n g c h a r a c t e r i s t i c s current O.85 amps voltage 8 - 1 0 volts time 2.25 hours I n i t i a l pH k f i n a l pH less than 7 i n i t i a l solution colour yellowish green time f o r colourless s o l u t i o n 0.75 hours 5. Use Pt f o r the anode and Cu f o r the cathode. 6. Maintain the p l a t i n g s o l u t i o n temperature at 70°C 7. Offset the tendancy of the p l a t i n g s o l u t i o n to become basic by dropwise additions of 3M HgSOij. The above procedure was used t o produce several discs of natural Fe electroplated on Cu. The l a y e r of Fe was shiny unevenly d i s t r i b u t e d i n every case.  but was  This uneven d i s t r i b u t i o n made  -101the preparation of the enriched absorber impractical since a uniform absorber thickness i s necessary f o r the i n t e r p r e t a t i o n of the e x p e r i mental r e s u l t s .  Also, the thickness of the absorber had to be known  but no method could be devised whereby the e l e c t r o p l a t e d l a y e r could be measured except by destroying the absorber.  t  D  -102Appendix C GEOMETRIC CORRECTIONS TO R(v}, To determine the e f f e c t the experimental geometry had on R(v), the two  assumptions  1. the- source thickness i s zero and 2. the emission and absorption spectra each consist of a single line were made.  Under these conditions equation II-18 becomes  IQ JO  C-l  Jo In the above equation = 2trsin<})  C-2  ijVD) ~ (1 - e  J2> =  r  VQ/C  )  where/<p i s the l i n e a r absorption c o e f f i c i e n t f o r Ar C-3 C-4  o- (E,v ,(J^) = f o b r / 4 / [ E - E o ( l + ^ c o s ) J + (- /4 2  2  m  2  0  57 and n i s the number of Pe When (pr  c-5  . o atoms/cm .  0,^(0) = _rv(0),77 W>)  cos0 =  = 7^(0),  1 and  -<3m(E,v )n  R(v .0) = 1 - f  0  o  2*ffB - E o ( l + /3)J 4- r A " 2  When 0 used.  m  i s small, the substitution  2  C-6  costf) = 1 r<^/2 can be  For a proportional counter window of 6" width, and a source to  counter distance of 15",  <t> = .2rad. m  and the use of the expansion of  cos(j) introduces an error of .1% i n the value of cos0.Hence, when<J)  m  i s small  -103-  -r\&) = 2K® - g?/6)  C-2'  71(0)= 7 J ( 0 ) [ l + » A 2 # ]  C-3'  where the f a c t that yu x = 0 . 8 5 f o r the Ar-CH^ proportional counter p  has been used, and exp(-4n/cos0) = exp(-c£n)(l - , ^ )  C-7  2  Hence Jd0^(0)7J(^)exp(-c^n/co&0) = 2 i t 7 ^ O ) e x p ( - c 4 n ) J ^ ( 0  -<jfi/6)  x (1 + ..l»2tfp)(l - . 7 ^ ) C-8 Also  exp(-  $a/cos0) = eatp(-<&(0 = O)ii)[l - <^{0 = 0 ) 0 3 ^ / 2 ]  C-U*  i n the d e r i v a t i o n of which the approximation (E - E o ( l *• J3)) - 2 p 57  has been made. C ( 0 = o)ni has a maximum value at E = E Q of h ( f o r Fe  ,  m  a .001" t h i c k natural Fe absorber, and f = .7)» and an average value of 1 over the range of integration, hence  ^°  [E -£o(I + 8 ^f "TV /  °' ' n  d0e f  f "7J(0)e  0  |^L£_£  _d"r  cos  (0 - .  +  1 ,  [E-£oO^P  - ,y^w,,  • #  160(0- 1$*)  ^  -  ^  .  ^ }  -104Thus  i 1  2<t MoUt/3)] + r ^ f % < 0 -«v'/4)(i - . 2 % f ) J C-10  - f f l - Affd£exp(-c^n)  .  where A i s the geometrical correction f a c t o r . Values of A corresponding to the four absorber thicknesses used i n the experimental work are given below.  Absorber Thickness  i  Correction Factor - A  .0010  .97  .00055  -98  .00035"  •99  .0002*  .996  -105-  Appendlx D STATISTICAL DESIGN OF THE EXPERIMENT  Since a v a r i e t y of absorber thicknesses vere a v a i l a b l e f o r the experimental work, (.0002", .00035",  and .001"), the following  analysis was made t o determine which thickness would give the most s i g n i f i c a n t r e s u l t s i n a given counting time T.  The assumption made  i n t h i s analysis i s that the emission and absorption spectra consist of single lines. Assuming that  cos0  = 1, the change i n count rate from that at  v - oo, t o that at v = 0, i s given by  = N(vr <*>) - N(v = 0)  >  = N e - ^ - N^fe-'**  + U - f U'**  s  = where N  g  V  D-l  >  ( i -  i s the count rate with no absorber, and/^ and/* are the m  e  l i n e a r absorption c o e f f i c i e n t s of the Mossbauer and electronic absorption respectively.  Also  (-  dE  D-2  I (Eo-E) + r 2  =  e  ^  2  I  where I ^ x / 2 ) - ^ ( 1 / ^ / 2 ) 0  m  0 y y n j  2  A  x/2)  i s the Bessel function of the f i r s t  order . 1 0  The s t a t i s t i c a l error i n the count rate i s  e  = (M e->**)* Q  Then, f o r the given counting time T, the absorber thickness x, f o r  D-3  -io6which the most s i g n i f i c a n t r e s u l t s are obtained i s that value at which the r a t i o X/e has i t s maximum value. X/e = f ( H T ) * e " ^ 5  B  so that f o r a fixed N  s  / a  How I {^x/2))  ( l - e'  D»4  1UmX/2  0  and T, whenX/e i s a maximum  dCVe)/dx = 0 s/l9  /2[l  - e"^ I ^x/2)]  +  2  0  - W2e" * /  /  l X / 2  ^ * *Jo(r*K/2) lh  Ma  /  I ( a x/2) 1  )  D-5  m  where the relationship I o ' ( z ) = I i ( z ) has been used.  Using z  =ju x/2 m  e  the condition determining the value of x at which X/e i s a maximum i s that  e  I ( z ) = /* /^um+/ie) -tMia/^m 0  e  + yue> e" I].(*) z  D-6  Since the experimental measurements were done with Zeeman s p l i t spectra rather than with the single l i n e spectra that the above theory assumes, a difference between the experimentally determined value of x at whichX/e i s a maximum and the t h e o r e t i c a l value i s not unexpected.  -107Appendix E THE ROOT MEAN SQUARE DIFFUSION DEPTH  Using the assumptions given i n Chapter IV, the number of diffused atoms between x and x +dx i s  c(x)=  Qexp^-x /4Dt) 2  E-l 2  Therefore, the mean square penetration depth x i s  ~T x  fx Oexp_(-x ADt )dx 2  2  ^ TX  _ Joo-\f-h-Dt  ~  2Dt  E-2  texp(-x /4Dt)dx g  Hence, the root mean square d i f f u s i o n depth i s  = (act)  5  E-3  -108Bibliography 1.  R.<L. Z.  2.  Mossbauer,  Physik,  4.  B.  M. J .  "The Mossbauer E f f e c t " ,  Compton,  and A . H . Schoen, 196l",  I r ^ l ,  W. A . B e n j a m i n ,  Inc.,  Sept.  Boyle,  13-16,  and H . e .  "Reports  Wiley,  Hall,  Society,  "proceedings  Mossbauer E f f e c t ,  New Y o r k ,  i n Physics",  London, 1962,  of  the  Saclay,  1962.  The Mossbauer e f f e c t ,  on P r o g r e s s  Physical  (eds.),  on the  Strickland,  The I n s t i t u t e V o l . 25,  of  A.  C ,  Physics  pp 4 4 1 - 5 2 5 .  S . M a r g u l i e s , a n d J . R. Ehmnan, T r a n s w i n s i o n a n d l i n e b r o a d e n i n g o f r e s o n a n c e r a d i a t i o n i n c i d e n t o n a reson.'.oice a b s o r b e r , N u c l e a r I n s t r . ,  12, 6.  (ed.),  France,  and the  in  1962.  Conference  A. J .  von Gammastrahlung  (1958)  Second I n t e r n a t i o n a l  (ed.),  5.  Kernresonanzfluoreszenz 124  H. Frauenfelder, New Y o r k ,  3.  151,  131  (I96la)  J . G . B a s h , R. D . T a y l o r , D . E . N a g l e , P . P . C r a i g , a n d W. M . 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