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Muonium and positronium as microprobes of surfaces and solids Kiefl, Robert Francis 1982

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MUONIUM AND POSITRONIUM AS MICROPROBES OF SURFACES AND SOLIDS  by  ROBERT FRANCIS XAVIER K I E F L B.Sc.  Carleton  U n i v e r s i t y , 1976  M.Sc. The U n i v e r s i t y o f B r i t i s h  Columbia,  1978  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS  FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY  in THE FACULTY OF GRADUATE STUDIES Physics  We a c c e p t t h i s  Department  t h e s i s as conforming  to the required  standard  THE UNIVERSITY OF B R I T I S H COLUMBIA January  ^Robert  1982  Francis Xavier  Kiefl  In p r e s e n t i n g  this  thesis i n partial  f u l f i l m e n t of the  r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e of B r i t i s h Columbia, I agree that it  freely  the L i b r a r y  a v a i l a b l e f o r r e f e r e n c e and s t u d y .  agree that p e r m i s s i o n for  University  f o r extensive  s c h o l a r l y p u r p o s e s may  for  financial  shall  of  K  h lj ^ I CS  The U n i v e r s i t y o f B r i t i s h 2075 W e s b r o o k P l a c e V a n c o u v e r , Canada V6T 1W5  thesis  Columbia  my  It is thesis  n o t be a l l o w e d w i t h o u t my  permission.  Department  further  be g r a n t e d by t h e h e a d o f  copying or p u b l i c a t i o n of t h i s  gain  I  make  copying of t h i s  d e p a r t m e n t o r by h i s o r h e r r e p r e s e n t a t i v e s . understood that  shall  written  i i  ABSTRACT  The  properties  altered  of m u o n i u m U ' e ' ) and  significantly  positronium(e e~)  i n t h e p r e s e n c e of m a t t e r .  these  exotic  atomic  i n t e r a c t i o n s w i t h a t o m s , s u r f a c e s , and  is explored The spin  H-like  atoms  i n a v a r i e t y of  provides  a  a  lifetime  positron  °K t o 630  p o s i t r o n i u m does not spin  conversion  interpret  by  the  a  i n an  technique.  thermalize  t e m p e r a t u r e a b o v e 450 positronium  °K  cross  spin  Si0  A 1  2  The  powders  temperature  muonium emerges f r o m t h e  2  of t h e Ne).  fraction The  and  8.9±0.2 A , 2  The  on  theme  converted  molecules  has  been  powder m o d e r a t o r  using  2  indicate  increases  theory  for  molecule  to  °K.  The  slightly  spin  is  that  with  conversion  developed  and  of  used  i n an  2  Al 0 , 2  a t m o s p h e r e o f He  surfaces  t h e p o w d e r . The  Si0 ,  regardless  of  3  to  MgO  indicate that the  ambient  muonium s p i n r e l a x a t i o n r a t e i n  a t m o s p h e r e i s f o u n d t o be a l i n e a r  of s u r f a c e a r e a  and  not c o v e r e d  by  function  adsorbed  He(or  c r o s s s e c t i o n s f o r muonium t o s c a t t e r e l a s t i c a l l y  a d s o r b e d He and  of  i n a H e ( o r Ne)  3  of  data.  a t low  temperature  t o be  results  Muon S p i n R o t a t i o n measurements, i n  Al 0  s o l i d s . This  i n t h e powder b e l o w 450  section  °K.  study  hosts.  0 positronium during c o l l i s i o n s with 0 120  The  unique p e r s p e c t i v e  cross section for spin 1 positronium  measured from  are  +  Ne  be  11.0±0.2  A  2  respectively.  first  of A r , K r , and  atoms h a v e been m e a s u r e d t o  off  observations Xe  a high p r o b a b i l i t y  o f muonium i n t h e c o n d e n s e d  are presented.  The  data  of muonium f o r m a t i o n  phases  i n d i c a t e that there  in a l l cases.  The  is  spin  relaxation liquid, of  1 3 9  Xe  motion.  r a t e o f muonium i n s o l i d Xe i s t e n t i m e s t h a t  where t h e random l o c a l and  1  4  ^e  are  fields  from t h e  nuclear  a v e r a g e d by a d d i t i o n a l  i n the moments  translational  iv  TABLE OF CONTENTS  INTRODUCTION  1  I . POSITRONS AND POSITRONIUM 1. C o n s e r v a t i o n o f C h a r g e C o n j u g a t i o n P a r i t y Annihilation 2. P s A n n i h i l a t i o n 3. E x p e r i m e n t a l T e c h n i q u e s 1. L i f e t i m e T e c h n i q u e 2. A n g u l a r C o r r e l a t i o n 3. D o p p l e r B r o a d e n i n g 4. Q u e n c h i n g o f o-Ps i n M a t t e r II.  i n e*e'  3 4 5 6 6 7 8 8  POSITRONIUM IN S i 0 POWDER 10 1 . Ps Formation 10 2. P s T h e r m a l i z a t i o n i n S i 0 Powder 12 3. E f f e c t o f 0 on Ps T h e r m a l i z a t i o n 13 1. V i b r a t i o n a l E x c i t a t i o n 13 2. R o t a t i o n a l E x c i t a t i o n B e l o w 0 . l 9 e V .. 14 3. Oxygen H y p e r f i n e T r a n s i t i o n s B e l o w 0.03eV .... 15 4. E l a s t i c S c a t t e r i n g 16 5. C o n c l u s i o n 17 4. Q u e n c h i n g o f o-Ps i n S i 0 Powder 17 1 . S p e c i a l Case X. t « 1 ( A d i a b a t i c Approximation) 19 2. S p e c i a l C a s e k t >> 1 ( S t r o n g C o l l i s i o n Approximation) 19 5. E f f e c t o f 0 on t h e Q u e n c h i n g o f o-Ps i n S i 0 Powder 21 2  2  2  2  B  B  2  2  I I I . TEMPERATURE DEPENDENCE OF CONVERSION QUENCHING OF ' o-Ps BY 0 I N S i 0 POWDER 1 . Experimental 2. P r o c e d u r e a n d R e s u l t s 3. D i s c u s s i o n 1 . Thermalization 2. A n o m o l o u s S p i n E x c h a n g e i n o-Ps + 0 Scattering 2  2  2  4. Summary a n d C o n c l u s i o n s  24 25 28 31 34 35 37  V  I V . MUONS, MUONIUM AND ^ SR 1 . S o u r c e o f P o l a r i z e d Muons 2. Muon Decay 3. Muon S p i n R o t a t i o n 1. F r e e Muons i n a T r a n s v e r s e M a g n e t i c F i e l d 2. F r e e Muonium i n a T r a n s v e r s e M a g n e t i c F i e l d 4. Mu S p i n R e l a x a t i o n 1. Random L o c a l M a g n e t i c F i e l d s (RLMF) 2. Random A n i s o t r o p i c D i s t o r t i o n 3. C h e m i c a l R e a c t i o n 4. S p i n E x c h a n g e 5. The »»*SR S p e c t r u m i n a T r a n s v e r s e F i e l d +  38 39 41 42 44 ... 45 50 51 52 53 53 54  V. MUONIUM I N INSULATING POWDERS 1 . Mu F o r m a t i o n 2. Mu T h e r m a l i z a t i o n 3. Mu Bound S t a t e s on O x i d e S u r f a c e s 4. M e c h a n i s m s f o r Mu S p i n R e l a x a t i o n i n a Powder .... 1. N u c l e a r M a g n e t i c Moments 2. P a r a m a g n e t i c I m p u r i t i e s 3. M o t i o n a l N a r r o w i n g 4. Random A n i s o t r o p i c D i s t o r t i o n 5. The R e l a x a t i o n F u n c t i o n R ^ y ( t ) f o r Mu i n . a Powder 1. S p e c i a l C a s e X t << 1 ( A d i a b a t i c A p p r o x i m a t i o n ) ._. 2. S p e c i a l C a s e x. t » 1 ( S t r o n g C o l l i s i o n Approximation) 6. E f f e c t o f A d s o r b e d I n e r t Gas on t h e S p i n Relaxation 6  5  V I . LOW TEMPERATURE STUDY OF MUONIUM I N A l 0 , S i 0 AND MgO POWDERS 1. Mu i n t h e V o i d s o f O x i d e P o w d e r s a t 6°K 1. E x p e r i m e n t a l D e t a i l s 2. E l e c t r o n i c s 3. A n a l y s i s a n d R e s u l t s 4. D i s c u s s i o n 1. .Mu i n S i 0 Powder a t 6°K i n a He A t m o s p h e r e 2. Mu i n MgO Powder a t 6°K 3. Mu i n A 1 0 Powder (5°K - 20°K) 5. Summary a n d C o n c l u s i o n 2. S p i n R e l a x a t i o n o f Mu i n A 1 0 Powder w i t h A d s o r b e d He/Ne 1. E x p e r i m e n t a l D e t a i l s 2. E l e c t r o n i c s 3. P r o c e d u r e 4. A n a l y s i s a n d R e s u l t s 5. D i s c u s s i o n 1. A d s o r p t i o n I s o t h e r m s o f He on A l 0 2. Mu S p i n R e l a x a t i o n i n A l 0 Powder W i t h A d s o r b e d He 3. A d s o r p t i o n I s o t h e r m s o f Ne on A l 0 4. Mu S p i n R e l a x a t i o n i n A 1 0 Powder W i t h A d s o r b e d Ne 6. S t a t u s o f t h e ATTD M o d e l 7. C o n c l u s i o n 2  3  2  2  2  3  2  3  2  2  67 68 69 70 71 71 73 75 78 79 81 82 83  2  3  84 84 88 88 89 90 90  2  3  92 95  3  3  55 55 58 60 61 62 63 66 66 67  95 96 97  vi  V I I . MUONIUM I N THE CONDENSED PHASES OF A r , K r AND Xe .. 98 1 . Experimental 99 2. D a t a A n a l y s i s a n d R e s u l t s 100 3. D i s c u s s i o n 103 1. Mu i n L i q u i d a n d S o l i d A r 103 2. Mu i n L i q u i d a n d S o l i d Xe 106 3. Mu i n L i q u i d a n d S o l i d K r 107 4. M i s s i n g F r a c t i o n s 109 4. C o n c l u s i o n s 110 CONCLUDING REMARKS  111  APPENDIX I . THERMALIZATION OF GAS ATOMS I N A POWDER  112  APPENDIX I I . PS SCATTERING OFF 2 ELECTRON ATOM (MOLECULE) 1. T o t a l E l e c t r o n S p i n S t a t e s 2. S p a c i a l S t a t e s 3. P h y s i c a l A s y m p t o t i c S t a t e s o f T o t a l E l e c t r o n S p i n 4. P h y s i c a l S t a t e s o f Ps s p i n (IIz,) 5. The T m a t r i x i n | k l m l l s s > R e p r e s e n t a t i o n 6. The S p i n C o n v e r s i o n C r o s s S e c t i o n f o r s=1 7. The T o t a l C r o s s S e c t i o n 8. G e n e r a l i z a t i o n t o a 2 E l e c t r o n M o l e c u l e 9. E v a l u a t i o n o f T M j 1 ; j ' l') i n t h e L i m i t kR « 1  118 119 120 121 122 123 126 127 128 130  APPENDIX I I I . DIRECT THERMALIZATION OF MUONIUM I N THE VOIDS OF OXIDE POWDERS  133  APPENDIX I V . ADSORPTION OF ATOMS ON A SURFACE 1. Van D e r W a a l s Two D i m e n s i o n a l Gas 2. T i g h t B i n d i n g M o d e l 3. S i n g l e Atom A d s o r p t i o n 4. Mean S u r f a c e D w e l l Time  137 137 139 140 141  BIBLIOGRAPHY  142  i  i  s  vi i  L I S T OF FIGURES  CHAPTER I 1. The E n e r g y S p e c t r u m F o r 2. D e c a y Scheme F o r N a  o-Ps A n n i h i l a t i o n  5 6  2 2  CHAPTER I I I 1. A p p a r a t u s F o r M e a s u r i n g 2. 3. 4. 5.  o-Ps L i f e t i m e I n A Powder  E l e c t r o n i c s For Measuring o-Ps L i f e t i m e P o s i t r o n L i f e t i m e S p e c t r a i n S i 0 Powder D e c a y R a t e o f o-Ps V e r s u s 0 C o n c e n t r a t i o n C o n v e r s i o n R a t e C o n s t a n t Of Ps V e r s u s T e m p e r a t u r e 2  2  CHAPTER I V 1. Muon Decay P a r a m e t e r s  26 27 29 30 31  . . 42  CHAPTER V I 1. The v + SR A p p a r a t u s " B e a v e r " 71 2. The T a r g e t C r y o s t a t A s s e m b l y 72 3. j/ SR E l e c t r o n i c s , 74 4. The Mu P r e c e s s i o n S i g n a l F o r 140A S i 0 76 5. T e m p e r a t u r e Dependence o f t h e Mu S p i n R e l a x a t i o n Rate i n A l 0 Powder 79 6. The v + SR A p p a r a t u s " E a g l e " 85 7. The T a r g e t V e s s e l And C r y o s t a t Used To S t u d y Mu S p i n R e l a x a t i o n V e r s u s He/Ne C o v e r a g e 86 8. The Gas H a n d l i n g S y s t e m 87 9. Mu S p i n R e l a x a t i o n R a t e I n A l 0 V e r s u s A d s o r b e d He 90 10. Mu S p i n R e l a x a t i o n R a t e I n A l 0 V e r s u s A d s o r b e d Ne 91 +  2  2  3  2  3  2  3  CHAPTER V I I 1. The T a r g e t V e s s e l F o r C o n d e n s e d N o b l e G a s e s 2. Two F r e q e n c y P r e c e s s i o n Of Mu I n S o l i d A r 3. Mu P r e c e s s i o n I n L i q u i d And S o l i d A r 4. Mu P r e c e s s i o n I n S o l i d K r  99 102 104 107  APPENDIX I I I 1. I m a g i n e d C r o s s S e c t i o n Of The Mu Powder Potential  133  Grain  vi i i  L I S T OF TABLES  CHAPTER I I I 1. 0 C o n v e r s i o n R a t e C o n s t a n t V e r s u s T e m p e r a t u r e CHAPTER V I 1. »i*SR R e s u l t s i n B u l k a n d P o w d e r e d O x i d e s 2. P r o p e r t i e s Of O x i d e P o w d e r s 2  .... 32 78 82  CHAPTER V I I 1. (i*SR r e s u l t s i n c o n d e n s e d A r , K r a n d Xe 2. Mu F r a c t i o n I n Gas Phase A r , K r And Xe  100 101  APPENDIX I I 1. The M a t r i x 2. The M a t r i x  124 1 25  <klmll ss <klmll ss r  z  x  i  k l m S S Sfi> klmlI ss > z  x  z  ix  ACKNOWLEDGEMENT  I f e e l v e r y f o r t u n a t e t o have h a d t h e o p p o r t u n i t y (and  at  h e l p e d me d u r i n g t h e c o u r s e  of  this  research.  s u p e r v i s o r s D r . J o h n B. W a r r e n a n d D r . J e s s e H. B r e w e r invaluable infringing  sources  of  advice  upon my s c i e n t i f i c  and  encouragement  freedom.  I offer  t h a n k s . Much o f t h e work c o n t a i n e d i n t h i s e x p e r t i s e <. o f  Marshall,  thesis  Carl  W.  I t h a n k D r . C h r i s J . Oram,  so  deserves  computing  aspect  I  special of  thanks  required  the experiments  the  Glen  M.  accomodating  i n the  ( e v e n a t 3:00 AM i n t h e  t o Dr. Birger Bergersen for several  and , t o D r . John B e r l i n s k y  of s p i n e x c h a n g e .  Dr.  f o r h i s assistance  h e l p f u l d i s c u s s i o n s on t h e q u e s t i o n o f P s  UBC,  sincerest  t a k e p l e a s u r e i n t h a n k i n g them. D r . Dave M.  m o r n i n g ) . I am v e r y g r a t e f u l  powder  never  C l a w s o n , Dave P. S p e n c e r , D a l e R. Harshman,  and C.A. F r y . The e n t i r e MSR g r o u p h a s a l w a y s been helpful  have been  while  them my  My  very capable people i n t h e e x e c u t i o n stage of t h e  experiments. For t h i s  Garner  work  t i m e s l a u g h ) w i t h t h e numerous p e o p l e who d i r e c t l y a n d  Indirectly  and  to  I would a l s o l i k e  thermalization  in a  f o r d i s c u s s i o n s on t h e t h e o r y t o thank a l l t h e  staff  at  e s p e c i a l l y A l M o r g a n , A l B i s h o p , C h r i s S t e v e n s , Doug Maas,  for their  t e c h n i c a l a s s i s t a n c e and Jean H o l t  of t h e f i g u r e s My  wife  inthis Robin  for drafting  many  thesis. still  l o v e s me a f t e r a l l t h o s e o w l s h i f t s .  Her u n d e r s t a n d i n g s u p p o r t a n d l o v e have a l w a y s been a s o u r c e  of  c o m f o r t a n d s t r e n g t h a n d I d e a r l y t h a n k h e r f o r them. Finally  I thank R i c h a r d and Theresa K i e f l  l o v e , encouragement and t h e o p p o r t u n i t y t o  do  who have gave me my  best.  I  am  X  f o r e v e r g r a t e f u l and  I dedicate this  thesis  t o you  both.  1  INTRODUCTION  P u r e l e p t o n i c atoms s u c h positronium  (Ps)  •physics. Their effects  (e*e~)  properties  so  they  as  muonium  play  an  are  free  provide  an  physicist  i s interested  important role from  and  in particle  strong  ideal  e l e c t r o m a g n e t i c a n d weak i n t e r a c t i o n  (»*e~)  (Mu)  interaction  testing  ground  for  t h e o r i e s . Thus t h e p a r t i c l e  i n the p r o p e r t i e s of  free  atoms,  such  as:  1.  The a n n i h i l a t i o n r a t e a n d d e c a y mode o f 1 S a n d 1 S of Ps ( e l e c t r o m a g n e t i c i n t e r a c t i o n ) .  2.  Mu(M e") — > Mu(»»"e ) interact ion).  3.  The  3  +  +  conversion  1  probability  states (weak  h y p e r f i n e s p l i t t i n g (1 S - 1 S ) and t h e Lamb s h i f t ( 2 P x . 2Sj£ splitting) o f b o t h Mu and Ps (electromagnetic interaction). In  3  contrast,  1  i t i s t h e d e v i a t i o n s f r o m f r e e atom b e h a v i o u r  i n t h e presence of matter t h a t a r e of i n t e r e s t positron  scientist,  concerned  p r o p e r t i e s of t h e probe and/or function  as  a  with host.  i n t e r a c t i o n s . Very i n d i v i d u a l atoms on fundamental  The  Ps  decay  time  scale  properties  muon  or  or  Mu  atom  may  and can p r o v i d e a  atom-surface, or atom-solid  few t e c h n i q u e s a r e s e n s i t i v e a  the  t h e ' chemical or p h y s i c a l  microprobe of i t s environment  u n i q u e p e r s p e c t i v e on a t o m - m o l e c u l e ,  to  as  of the e  small +  t o t h e a c t i o n of as  10"  9  s.  The  a n d n* i n m a t t e r , w e l l  known f r o m p a r t i c l e p h y s i c s , make s u c h o b s e r v a t i o n s p o s s i b l e . This thesis  i s a collection  of four  experiments  w i t h t h e b e h a v i o u r o f t h e s e H - l i k e atoms i n m a t t e r :  concerned  2  1.  1 S —> 1 S spin conversion S i 0 powder m o d e r a t o r b e t w e e n 3  1  2  o f Ps o f f 0 120-630 °K.  2  molecules  i n t h e v o i d s o f S i 0 , A l 0 , a n d MgO  i n an  2.  Mu p r o d u c t i o n a t 6°K.  3.  Mu interaction with A l 0 o r Ne between 7-30°K.  4.  The f i r s t o b s e r v a t i o n s o f Mu i n t h e c o n d e n s e d p h a s e s o f A r , K r , and Xe.  2  2  The chapter  first  3  three chapters  provides the necessary  the  without II  characteristics  paramagnetic 0  2  are  devoted  1  atom  Ps.  The  i s concerned  o f Ps decay i n S i 0  g a s . The r e a d e r  or molecule.  to  He  first  b a c k g r o u n d i n f o r m a t i o n on P s a n d  The f i r s t  2  primarily  p o w d e r , w i t h and  i s referred  f o r a t h e o r y of s p i n exchange s c a t t e r i n g  spin  powders  3  powder s u r f a c e s w i t h a d s o r b e d  t h e methods o f s t u d y . The s e c o n d c h a p t e r with  2  to  Appendix  of Ps i n c i d e n t o f f a  experiment  i s presented i n  Chapter I I I . The  last  f o u r c h a p t e r s may be c l a s s i f i e d  the t h e s i s . Chapter  a s t h e Mu p a r t  of  I V p r o v i d e s b a c k g r o u n d i n f o r m a t i o n on Mu and  t h e Muon S p i n R o t a t i o n (»/*SR ) t e c h n i q u e . The c h a r a c t e r i s t i c s o f Mu a n d t h e »**SR s p e c t r u m i n o x i d e p o w d e r s a r e o f p r i m a r y in  Chapter  V. The s e c o n d a n d t h i r d  Chapter  VI w h e r e a s t h e l a s t  Chapter  VII.  experiments  experiment  concern  are presented i n  i s t h e s u b j e c t of  Chapter  3  CHAPTER I : POSITRONS AND POSITRONIUM  In that  the beginning  vacancies  in  would  manifest  states charged first  t h e r e was  particles  t o observe  cosmic  ray  a  filled  or  effort  showers.  (1946)  was  in  rays  of  physically  cloud  chamber  was  observed  the  first  Na  and  6 < 1  Meanwhile as  Cu  available.  became  positron  firmly  e s t a b l i s h e d the existence  great  amount  not  until  (Canter  experimentation  1974 t h a t t h e f i r s t  Ps  in  a large  matter.  existence  Pirenne  was  first  ( 1 9 4 6 ) a n d Ore and and  s=1  s t u d i e s on Ps were  just  such  work o f D e u t s c h  as (1951)  positronium. - Despite  s i n c e those  the  e a r l y d a y s i t was  e x c i t e d s t a t e o f Ps  was  observed  1 9 7 5 ) , made p o s s i b l e by t h e d e v e l o p m e n t o f m o n o e n e r g e t i c  beams o f l o w e n e r g y p o s i t r o n s ( C a n t e r In  sparked  sources  The of  afterwards  r a t e s f r o m s=0  experimental  possible  of  of  c a l c u l a t i o n s on t h e  Wheeler  (1949a) c a l c u l a t e d a n n i h i l i a t i o n  •becoming 2 2  Mahorovicii  in  t o perform  (1934).  states.  ( 1 9 3 3 ) was t h e  shortly  for positrons  postulated  ground  positively  photographs  results  ( e * e ~ ) whose  Powell  as  Anderson  energy l e v e l s of p o s i t r o n i u m by  postulated  of n e g a t i v e energy e l e c t r o n  experimental  a theory  one  who  The p r o d u c t i o n o f p o s i t r o n - e l e c t r o n p a i r s  These  to develop  (1930),  anti-electrons.  positrons  1933).  sea  themselves  f r o m h i g h e n e r g y gamma (Blackett  Dirac  1972).  t h i s c h a p t e r , t h e d e c a y p r o p e r t i e s o f f r e e p o s i t r o n s and matter  and  the  a n n i h i l a t i o n are reviewed.  experimental  techniques  in positron  4  1-1  C o n s e r v a t i o n of Charge C o n j u g a t i o n P a r i t y  i n e e~ +  A n n i h i l a t ion Since positrons state  are anti-electrons,  i s unstable  electromagnetic  annihilation  interaction.  transforms  every  antifermion  state  parity  to  The  p a r t i c l e into  where  are  self  transformation potential  gammas  properties  under  interactions must  C  such  that  and  3  respectively.  structure primarily  or  Gammas  or  of t h e  electromagnetic Since  C-  angular  -1 by v i r t u e  vector  electromagnetic of  t h e ground s t a t e  Ps  into  of* Ps),  f r o m s=0 a n d s=1 s t a t e s  n  this  (i.e.:  1 S 1  o f P s ) t o an e v e n a n d odd number o f gammas into a single  of photons g r e a t e r than  constant,  1971).  the annihilation  (i.e.:  Annihilation  gammas i s p r o p o r t i o n a l  For a fermion  relative orbital  b o t h momentum a n d e n e r g y s o t h e s=1 s t a t e numbers  operator  i+s+n i s even. I f t h e a n n i h i l a t i o n  the annihilation  1 S atomic states  through  conjugation  (Williams  the  rays  conjugation  charge  operation.  conserve C-parity, be  gamma  with C-parity of  the  o c c u r s f r o m an ^=0 s t a t e restricts  spin  conjugate  positron-electron  its antiparticle.  Jt i s t h e  momentum a n d s i s t h e t o t a l photons  into  charge  such a s e*e~, t h e  is  the  to so  c  n  that  gamma  cannot  must d e c a y  1. The a n n i h i l a t i o n  where  o  the  s=0  i n t o two a n d t h r e e gammas  (=1/137) and  S=1  respectively.  conserve  i n t o on odd rate  into n  i s the states  fine decay  5  1*2 P s A n n i h i l a t i o n When p o s i t r o n s electron (written  t o form p a r a - P s o-Ps  or  t h r e e . . As s t a t e d whereas spectrum  are injected  o-Ps  (written  PS(1 S)), 3  above,  decays  i n t o m a t t e r t h e y may c a p t u r e an  in  p-Ps o r P s ( 1 S ) a statistical  t h e p-Ps decays in  ) or  1  i n two  ortho-Ps  r a t i o o f one t o 511  KeV  gammas  t h r e e gammas w i t h a c o n t i n u o u s e n e r g y  ( s e e F i g u r e 1 . 1 ) . The mean d e c a y  rate  (1/lifetime) in  ENERGY (mc ) 2  F i g u r e 1*1 (Ore 1 9 4 9 a ) .  The e n e r g y  spectrum  f o r o-Ps  annihilation  vacuum f o r p - P s a n d o-Ps h a v e been m e a s u r e d t o be 799±11 s"  1  (Theriot  respect ively.  1967)  and  0.7056±0.0007 x 1 0  7  s"  1  (Gidley  x  10  7  1978)  6  I»3  Experimental Techniques The  the  s t u d y o f p o s i t r o n s i n m a t t e r i s b a s e d on  annihilation  annihilation  of  t h e gammas,  the  energy,  a n d number o f gammas. The most commonly u s e d means  c a r r y i n g out i n v e s t i g a t i o n s of p o s i t r o n s  angular  of  q u a n t a . The r e l e v a n t o b s e r v a b l e s a r e t h e mean  r a t e , t h e a n g l e between  polarization  detection  correlation  and  doppler  are the  broadening  lifetime,  t e c h n i q u e s (West  1973).  1-3*1  Lifetime  Technique  Na  sources  lifetimes  because  2 2  are often  used  for  t h e decay p o s i t r o n  i s followed  by t h e e m i s s i o n o f a n u c l e a r gamma o f 10" ^ s gamma  (see Figure and  the annihilation  F i g u r e 1*2  scintillation x (2.5  2cm x  1 * 2 ) . The  10"  quanta  2 2  can  1 0  s)  but  suffer  1274  be  KeV  decays within  the nuclear  measured  with  Na .  detectors. Small p l a s t i c provide  energy  positron  i n most  t i m e d e l a y between  D e c a y scheme f o r  diameter)  measuring  scintillators  excellent  timing  ( 2cm l o n g resolution  from poor e f f i c i e n c y and energy  7  r e s o l u t i o n . They a r e e s s e n t i a l when s t u d y i n g solids  and  liquids.  The  g a s e s a n d p o w d e r s may be which  have  although  much  relatively studied  better  larger  resolution  t h e t i m i n g r e s o l u t i o n (4 t o 5 x 1 0 '  a s may be a c h i e v e d  with p l a s t i c  in  l o n g l i f e t i m e o f o-Ps i n  using  energy  short l i f e t i m e s  Nal and  detectors efficiency,  s) i s not as  9  good  scintillators.  I•3•2 A n g u l a r C o r r e l a t i o n The  angle  b e t w e e n two p h o t o n s f r o m e e~ +  (s=0) a n n i h i l a t i o n  i s given as & where  p  x  direction angular measured  =  pj_/m G 0  i s the of  using  and  momentum and  between  a long s l i t  measures t h e c o i n c i d e n c e a  a  pair  emission  distribution  p4.^< m C  f u n c t i o n of angle  m  0  the  1973) E q u a t i o n  1-1  component  perpendicular  to  i s 'the e l e c t r o n res't mass. The annihilation  angular  counting  (west,  can  c o r r e l a t i o n apparatus  1, t h e p o s i t r o n  r e s o l u t i o n i s 0.5  be  which  r a t e b e t w e e n two d e t e c t o r s  d e f i n e d by d e t e c t o r  d e t e c t o r 2. A t y p i c a l a n g u l a r  quanta  as  source,  mrad.  The  d e c a y o f mean t h e r m a l i z e d p - P s c o n t r i b u t e s a n a r r o w component t o the  angular  distribution  p a i r momentum ( o f o r d e r annihilations  of t h e a n n i h i l a t i o n quanta since the (kT2m)"^ ) i s s m a l l  in  comparison  i n v o l v i n g h i g h momentum v a l e n c e e l e c t r o n s .  to  8  I'3*3 D o p p l e r  Broadening  Information  on  t h e p a i r momentum d i s t r i b u t i o n c a n a l s o be  o b t a i n e d by u s i n g a h i g h r e s o l u t i o n Ge o r G e - L i to  measure  511  KeV. I n f i r s t  order the s h i f t  where hi/ i s t h e e n e r g y  Doppler  detector  t h e Doppler broadening of t h e a n n i h i l a t i o n  AE = h v ~ m . c  component  gamma  i n energy  ~ p„c/2  z  of  ( H o t z  the detected  ^  1 9 6 8 )  gamma  o f p a i r momentum a l o n g t h e d i r e c t i o n  broadening  technique  line at  analyzes  E q u a t i o n  and  l m 2  p„ i s t h e  o f e m i s s i o n . The  a l l momentum  channels  s i m u l t a n e o u s l y a n d i s t h e r e f o r e much f a s t e r a n d d o e s n o t r e q u i r e high  e  stopping  +  angular c o r r e l a t i o n  densities  The  t o around  of  matter.  The  +  0  quenching  1 KeV a t 511 KeV.  0  decay  altered  converted  and/or  i s t h e f r e e decay  to  pickoff.  p-Ps v i a a  paramagnetic  species  more d e t a i l  i n Appendix  paramagnetic  Spin spin  r a t e a n d X.^ i s t h e  conversion exchange  to spin  i s when o-Ps i s interaction  with  s u c h a s H, NO, a n d 0 . T h i s i s c o v e r e d i n 2  I I . In short, c o l l i s i o n s  involving  o-Ps  m o l e c u l e do n o t c o n s e r v e t h e z component o f  s p i n o f t h e e l e c t r o n on P s a n d t h u s Pickoff  i n the  r a t e o f o-Ps i n m a t t e r c a n be  r a t e a s s o c i a t e d w i t h t h e medium, due p r i m a r i l y  conversion  possible.  as does t h e  o f o-Ps i n M a t t e r  e x p r e s s e d X=X. X.^ where X  a  sources  p r o p e r t i e s o f o-Ps a r e s i g n i f i c a n t l y  presence  and  strong  t e c h n i q u e . However t h e r e s o l u t i o n o f p r e s e n t  day d e t e c t o r s i s l i m i t e d  1-4 Q u e n c h i n g  nor  quenching  occurs  o-Ps—>p-Ps  conversion • i s  when t h e p o s i t r o n  i n o-Ps  9  annihilates  with a valence electron  quenching  processes  correlation results  or  are  -doppler  motion  of  the  p-Ps  relatively  broad  pair  of  example t h e mechanism f o r q u e n c h i n g  determined resulting  silica  t o be s p i n  These  t o p-Ps  powder  contrast,  valence  o f o-Ps by 0  (Kiefl  reaction.  pickoff  2  in a  determined  electron.  For  in silica gel  1978)moderators  c o n v e r s i o n as opposed p i c k o f f  from a c h e m i c a l  by t h e  the host r e s u l t s  distribution, the  two  using angular  conversion  In  from  momentum  by t h e h i g h momentum  I974)and  Spin  atom.  with a valence electron  (Chuang  host.  momentum d i s t r i b u t i o n d e t e r m i n e d  annihilation  primarily  the  distinguishable  broadening.  i n a narrow p a i r  thermal  easily  from  h a s been  annihilation  10  CHAPTER I I : POSITRONIUM I N S i 0  Positron components  lifetime  (Brandt  t o p-Ps a n d f r e e e  spectra i n oxide  1968). A v e r y decay(while  +  'second 2-3 n s component o-Ps  within  the  POWDER  2  fast  powders  exhibit  three  (< 1 n s ) component  i s due  s l o w i n g down o r t h e r m a l i z e d ) .  i s attributed  to pickoff  a n n i h i l a t i o n of  powder g r a i n s a n d t h e l o n g e s t 1 i v e d  (~140 n s ) i s t h o u g h t  due t o o-Ps i n  the  void  A  component  regions  of  the  powder. In  this  chapter  q u e n c h i n g o f o-Ps discussed. III  This  in  the  Si0  2  formation,  powder  with  t h e r m a l i z a t i o n , and and  i s relevent to experimental  on t h e c o n v e r s i o n q u e n c h i n g o f  o-Ps  with  without results 0  2  0  2  are  i n Chapter  in  an  Si0  2  moderator.  11•1 Ps  Formation  There a r e a t l e a s t Ps  formation  in  t h r e e m o d e l s t h a t may be u s e d t o e x p l a i n  o x i d e powders  t h e Ore g a p m o d e l , t h e s p u r  model, and t h e s u r f a c e f o r m a t i o n model. very  useful  fraction, purpose  in  making  especially of s t a t i n g  understanding  quantitative  These  models  predictions  i n s u c h a c o m p l e x medium a s a  of  are not the  powder.  Ps The  them h e r e i s s i m p l y t o p r o v i d e a q u a l i t a t i v e  of t h e v a r i o u s p r o c e s s e s  which  may  lead  to  Ps  format i o n . According  t o t h e O r e gap m o d e l  f o r gases and and then  extended  to  (Ore 1949b), f i r s t  proposed  molecular  (Wallace  solids  1 9 6 0 ) , Ps f o r m a t i o n o c c u r s e p i t h e r m a l l y v i a c h a r g e e x c h a n g e w i t h  11  atom  in  (or molecule) A  the  energy region  where E,  i s the  on  excitation Below E; above E  - E ,  Ps  B  and  < E < E  B  Eg  ex  11•1  ( t e r m e d t h e Ore  gap)  e n e r g y o f atom, E  is  e x  the  first  ( 6 . 8 e V ) i s t h e b i n d i n g e n e r g y of  formation  e x c i t a t i o n and  e x  -E )  J 0 K )  ionization  energy,  on  (E  Eguation  is  energetically  forbidden  Ps. and  i o n i z a t i o n are thought t o dominate  the  dE/dx. In  the  attracted i t s own  spur  t o , and  model  This  track  the  electron  has  energy  Al 0 2  on 3  then  of  been  results  surface  model  where  formation  incident  on  p o w d e r s t h a t o-Ps diffuses  to the  has  to  e*  is  diffusion to  i n t h e v o i d s ) as  the  called  successful  in  capture  an  be  been or  these  energetically  observed oxide  with  coated  had  spectra  in S i 0 ,  MgO  work  2  powder  function for this  (ATTD) model i s t h a t inside 140  temperature  metal  Brandt  evidence presented  o-Ps  low  results  s u r f a c e where i t i s e j e c t e d  i n c r e a s e s a t the expense of the o-Ps  may  i s formed w i t h i n the  1 9 6 8 ) . The  (attributed  branches  positrons  it  metal  1974). P r e v i o u s  temperature thermal component  small  t h e b a s i s of p o s i t r o n l i f e t i m e  (Brandt  thermalized  liquids.  v o i d s , presumably because of a n e g a t i v e surface  a  particularly  in  formation  Such s u r f a c e  (Canter  proposed  and  the  positrons  surfaces  and  surface  at  favourable.  (composed  model  explaining experimental In  1974)  e v e n t u a l l y combines w i t h , a f r e e e l e c t r o n from  radiation  spurs).  (Mogensen  the  ns component  the  powder  grains into  the  at  the  ambient 2-3  grains)  (attributed  i s lowered or the  ns  to  particle  12  size  increases.  consistent i n Ps  with  formation  II-2  However  the  surface in oxide  Ps T h e r m a l i z a t i o n  formation  to  be  in Si0  ejected  energy of o r d e r of  o-Ps  0.10  eV  mean  1eV  energy  1 9 7 1 ) . The loss  s m a l l e r than f o r classical  a  1976). For  which  powder  is  developed  the Ford  (35 A r a d i u s ,  calculation  ~30  ns  for  0.056 Ps  of  1eV  temperature  (Zemansky surface  1968) with  and  a purely  range  parameter  a=0.5  It  should  calculations  be  t h a t the  estimate  of F o r d  pointed  out  at  in  121°K  to reach  of b u l k  A" .  agrees  the  Si0  Si0  2  the  0.0l25eV This  (470°K)  2  potential The  1  s e n s i t i v e t o t h e p o t e n t i a l p a r a m e t e r s and the p u r e l y c l a s s i c a l  ns.  gas-surface  s p e n t b e l o w 0.03eV.  r e p u l s i v e Morse  to the  -140  f i n d that 3  calculation  Debye  momentum  of  gem" )  (25 n s )  the  purely  of t h e p r o b l e m e m p l o y i n g  (145°K) w i t h most o f t h a t t i m e uses  the  estimates  theory  ±  i s much  a  powder t o be  2  0.25  that  In  imparts  A p p e n d i x I . We of  energy  surface  Mu.  (1976)  quantum  density  yields  Ps  so  o-Ps  kinetic  kinetic  light,  as  packed S i 0  in  the  been m e a s u r e d t o be  such  s l i g h t l y more r i g o r o u s t r e a t m e n t  interaction  bulk  w i t h the oxide  atom  Devonshire  surfaces  surface with a  Ps atom i s v e r y  time i n l i g h t l y  dimensional  i n the  example the  powder has  atoms,  be  unclear.  the oxide  collision  in  also  so t h a t t h e r o l e of  s u r f a c e or  heavier  surface  thermalization  one  per  estimate,  individual  A  (Ford  may  Powder  2  from  e j e c t e d f r o m MgO (Curry  spectra  powders remains  Whether i t formed a t the appears  lifetime  at  result roughly  the  i s not with  (1976).  that  Ps atom s a m p l e s t h e  it  i s assumed i n  entire  surface  these area  13  with  equal  probability.  aggregate structure on  t h e mean f r e e  result  must  indicate this  2  II»3 E f f e c t  2  the experimental r e s u l t s  of 0  into the void  1 9  is  from  can  be  thermalization  i n Chapter I I I  t i m e much l a r g e r  than  r e g i o n s o f a powder s h o u l d  rotational that  elastic  2  v i a elastic  excitation below  tend  scattering,  and v i b r a t i o n a l  ~0.03  scattering  t o cause t h e r m a l i z a t i o n  eV  the  major  w h i c h most l i k e l y i s  a t low 0  2  densities  (less  cm' ). 3  I I '3-1 V i b r a t i o n a l The  presented  above  t i m e . Low e n e r g y o-Ps (< 5eV) may  However, we f i n d  contribution  10"  on t h e  the  on Ps T h e r m a l i z a t i o n  2  transitions,  insufficient  Thus  effects  powder b e l o w 450 K.  energy through c o l l i s i o n s w i t h 0  excitation.  i n t o account the  time.  considered a lower l i m i t  introduced  hyperfine  than  take  o f t h e powder w h i c h may have d r a s t i c  to decrease the thermalization lose  not  t h a t t h e Ps h a s a t h e r m a l i z a t i o n  in Si0  0  does  p a t h and t h e r m a l i z a t i o n  be  time. In fact  It  energy  Excitation  loss  rate  o f Ps due t o v i b r a t i o n a l  excitation  written  Equat ion where n i s t h e 0 the AE > VV  P 1  V  cross section = E / - Ey,  2  11 • 2  c o n c e n t r a t i o n , v i s t h e Ps v e l o c i t y ,  c \  is  for scattering  v to  v',  from v i b r a t i o n a l  i s the energy d i f f e r e n c e is  state  between s t a t e s the  vv  and  probability  that  14  s t a t e v i s o c c u p i e d . The e n e r g y  vibrational  of v i b r a t i o n a l s t a t e  v i s approximately Ev where  A  - fly ( =  y  temperature  1580 P  0  cm"  1eV i n 0  at  rate  0  I  2  of  cm  1 6  permutation  of  AE <  Equation  0V  of 1 0  cm"  1 9  of  spatial  symmetry o f t h e  1 6  16  0  ( Eg) are allowed 3  2  symmetry  0  nuclei  the e f f e c t of 0  energy energy  2  of  the  (Tinkham  as  electrons  1964).  In  a and  first  spin) the r o t a t i o n a l energies  1975):  The r a t e o f e n e r g y  i|)  l o s s c a n be  r o f  =  rj=(2jfi;e molecule  n i  rl>  Pj<*jj'AEjj'  the  scattering cross section  I I -5  Equation I I - 6 i s the p r o b a b i l i t y  2  the 0  Equation  m  written  / ^C J  that  2  lose  0.19eV  c  where  will  3  i s closed.  E x c i t a t i o n Below  the  I I -4  B)>  4.8 K e V / n s . However, b e l o w t h e t h r e s h o l d  (ignoring  (Levine  A t room  a n d c > =0 f o r v >1, t h e n Ps o f  2  odd r o t a t i o n a l s t a t e s  consequence  are  10"  ( L e v i n e 1975).  11 • 3  I I «2 c a n be s i m p l i f i e d t o  H I T <£ y «  =.l9eV) t h i s c h a n n e l  Only  molecules  2  gas a t a d e n s i t y  II'3-2 Rotational  order  for 0  1  ~ ~  0  energy  ( A E  L  i t i s assumed c \  a  Equation  -~ 1 and E q u a t i o n  4? 1 If  +~  v  i s i n r o t a t i o n a l state f o r j -->  j (v  - 0),  is  j ' t r a n s i t i o n s and AEj-j'  15  i s the energy l o s s  (gain)  pert r a n s i t i o n .  A r o u g h e s t i m a t e on d E / d t ) f  f o r P s o f e n e r g y 0.1eV i n  r o  at  300°K c a n be made by a s s u m i n g P  likely other  <fj-> =0).The  estimate  yields  = 10"  79  dE/dt  inhibited  at  1 6  -~  f o r pure r o t a t i o n a l e x c i t a t i o n ( A E  However, i t i s s e v e r e l y  cm  (with a l l  2  0.6eV/ns. 1 3  energies  total  a n g u l a r momentum c o n s e r v a t i o n  11 * 3•3 Oxygen H y p e r f i n e T r a n s i t i o n s  below  Even lose energy the  ~0.03eV  and  All.9)  0.03eV  inelastically  through hyperfine  transitions  between  a n g u l a r momentum J = j + s where s i s t h e 0  2  j i s t h e r o t a t i o n a l a n g u l a r momentum o f t h e m o l e c u l e .  s p l i t t i n g between J s t a t e s c o r r e s p o n d i n g t o t h e  2.5  coupled  r o t a t i o n a l e x c i t a t i o n i s p r o h i b i t e d , t h e P s may  states of t o t a l  spin The  when  (see Section  Below  The  ) i s ^0.014eV.-  b e c a u s e o f t h e s wave n a t u r e o f l o w e n e r g y P s s c a t t e r i n g with  2  = 1 ( s i n c e j = 7 i s t h e most  s t a t e a t 300 °K) a n d c  occupied  threshold  7  0  x lO"*eV and o n l y  same  j is  w e a k l y d e p e n d e n t on j (Townes 1 9 5 5 ) . The  r a t e o f e n e r g y l o s s c a n be w r i t t e n  Equat ion I n t h e a p p r o x i m a t i o n where t o t a l  electron  spin  t r a n s i t i o n s a t l o w e n e r g y must o r i g i n a t e f r o m s  z  changing,  collisions),  since  j ,  i s conserved,J J ' spin  exchange(or  and s a r e c o n s e r v e d (see  Appendix I I ) . A very crude e s t i m a t e of the energy l o s s rate to  J J ' transitions  cross  sections  spin  exchange  can  be  section  due  made by a s s u m i n g t h e e n e r g i e s a n d  a r e independent of j and cross  11•7  that  « , =er.  =er-„  (the  f o r P s s c a t t e r i n g o f f 0 ) . Then i t 2  16  follows:  dE\  J t  ^  /^VP  Substituting exchange  19  is  dE/dt) ^ h y  <r  3  cross-section  (Klobuchar  J  n = I0 cm". ,  where  the  nir[/-  the  =  e)C  defined  as  measured  4  II-3-4 E l a s t i c  spin AE  =  ntr 2  is  (See E q u a t i o n  =  2.5  the  off  the  only cross  s-wave  Appendix only  0  to elastic  s  wave  chose  cm  2  Equation  for  wave  Ps  of  i s considerable  (see  Appenix  IL9  section mass  four  1 6  e  x 10"  5  cm  2  and  cross  o r d e r s of magnitude  s e c t i o n . However f o r t h e p u r p o s e n=l0  1 9  cm" . 3  of One  n  energies  I I ) . Thus t h e  and d o u b l e t p h a s e  the spin conversion  and  uncertainty  t h e s c a t t e r i n g a t low  spin quartet  about  at  ( M o b l e y 196.7)  There  since  partial  --  e \ =10"  d E / d t ) | ^ 1.5 fi  one  scattering  s e c t i o n c a n v a r y t r e m e n d o u s l y d e p e n d i n g upon  phyical cross we  for  electron  1 9  yields  i n comparison to  s wave s c a t t e r i n g c r o s s  o f mass M.  2  I I ) . For example  10"  section  l0"*eV  Eno/M  elastic  i n the v a l u e chosen  elastic  All.20)  cross x  spin  collisions.  2Em/M i s t h e mean e n e r g y l o s s p e r c o l l i s i o n  involves  (the  2  Scattering  i n c i d e n t e n e r g y E c a n be w r i t t e n  scattering  cm  1 9  conversion  to surface  The e n e r g y l o s s r a t e due  where  10"  e V / n s . Thus i t c a n be n e g l e c t e d  7  r a t e o f e n e r g y l o s s due  4^1  x  (28/7)*^  1 9 8 0 ) ) , T = 300°K, and  ~10"  _  Equation II-B  only  shifts(see section  is  smaller  the  illustration then  finds  eV/ns w h i c h i s t h e l a r g e s t c o n t r i b u t i o n  to  17  the  energy l o s s  r a t e due t o 0  energy l o s s r a t e  II•3•5  effect  scattering  at  dominates  much g r e a t e r  than  it  i s u n l i k e l y that  powder a t 121°K  powder.  scattering  p r o c e s s e s , but has  energies  where  s  wave  cross section  i s p r i m a r i l y due t o s c a t t e r i n g 2  b e l o w 0.03eV  has a s u b s t a n t i a l  an  2  effect  Powder  e n s e m b l e o f o-Ps i n a powder a t t e m p e r a t u r e T  s u r f a c e a r e a A and f r e e volume V = ( V - V ) j ) . In F  free  i s not time i n  2  Q u e n c h i n g o f o-Ps i n S i 0  general  elastic  cm ). Since the " t h e r m a l i z a t i o n "  the presence of 0  likely  time.  Consider  the  to reach  2  (provided the e l a s t i c 1 6  2  with  gas d e c r e a s e s t h e t i m e r e q u i r e d  lower  10"  Si0  II'4  2  due t o i n e l a s t i c  little  on t h i s  than the  Conclusion  0.03eV  has  smaller  due t o t h e S i 0  The p r e s e n c e o f 0 ~  but c o n s i d e r a b l y  2  case,  states.  there  Let X  F  bound a n d f r e e  Ac  exist  and \  V. ^  b o t h bound  T h e s e c a n be  the  i(  (binding  most  e n e r g y B) and  be t h e a n n i h i l a t i o n r a t e s  3  states.  =  sa  f o r o-Ps i n  written  + X Equation  where  i s the c o l l i s i o n  probability pickoff  rate  annihilation  for  pickoff  from rate  frequency with  a n n i h i l a t i o n per c o l l i s i o n ,  the  bound  (7.056 s " ) .  X.p , c a n be e s t i m a t e d f r o m  the surface,  1  r  the  state,  is  the  kp i s the free  The bound s t a t e q u e n c h i n g  rate,  of  X.„ i s  q  the  lifetime  and  P  11•10  o-Ps  adsorbed  on  18  surfaces  of s i l i c a  I973)at P  of  30 ns (Chuang  300°K c o r r e s p o n d s t o a q u e n c h i n g r a t e o f  can  a  g e l . The o b s e r v e d l i f e t i m e  be  estimated  f r o m t h e o-Ps q u e n c h i n g  ~  26  */S~ . 1  i n A r g a s (due t o  p i c k o f f ) as >a where (1.3  is x 10~  5  ~  the  *  E q u a t i o n 11•11  cross  section  A ) ( C e l i t a n s 1964) a n d e  2  In completely  density  dispersed  i s appreciable  Si0  3  t o t h e P s , v<, = 6 x  AI.24).  This  considered  yields  v<_ P„  an u p p e r l i m i t  10~ of  The  will  1 1  S  surface area (according  order  0.8  i s equally  to  (/S" ' b u t  Equation must  on s i n c e c l u m p i n g o f t h e powder  be d i s c u s s e d  f o r m o f t h e o-Ps l i f e t i m e  grains  and d e c r e a s e v . c  f u r t h e r i n Chapter I I I . s p e c t r u m d e p e n d s on  how  p  evaluated  i n A p p e n d i x I V . I f t h e s u r f a c e bound Ps b e h a v e s a s a 2  p where A Ps two  limiting  evaluated, complicated is  ^,  t  cases whereas  and w i l l  tractable.  time,  t c a n be w r i t t e n :  i s the thermal  v e l o c i t y , and P  t i s t h e mean s u r f a c e d w e l l  x t  with  gas then  where  be  compares  dimensional  unity,  o f an A r atom  (assumed t o be 10  i n t o a g g r e g a t e s may i n c r e a s e t h e mean f r e e p a t h This p o s s i b i l i t y  annihilation  powder a t 300°K (35 A r a d i u s ,  2  = 0.056 g cm" ) where t h e e n t i r e  accessable  pickoff  i s the area  2  over which the e l e c t r o n d e n s i t y A ).  for  Equation  I I •12  w a v e l e n g t h o f t h e o - P s , v- i s t h e mean  i s sticking X ^ t >> the  1  probability and  intermediate  i n general  on t h e s u r f a c e . The  X. t << 1 p  not y i e l d  case  o-  can  easily  X^t ~ 1  be  i s more  a decay spectrum  which  19  I I • 4• 1 S p e c i a l Case X This (Ford  i s the only  1976).  It  times before spectrum.  possibility  implies  decay,  The  t << 1 ( A d i a b a t i c  B  leading  decay  rate  w e i g h t e d by t h e f r a c t i o n Xft  =  that  A  considered  a  +  literature  single  exponential  p  i n each  state.  6 ~ ° 0 -V  E q u a t i o n 11-13  where a i s t h e f r a c t i o n o f t i m e s p e n t i n t h e b o u n d thermal  equilibrium  o  decay  i s t h e n an a v e r a g e o f X.^ a n d X.  fi  of time spent  B  i n the  t h e Ps a d s o r b s a n d d e s o r b s many  to k  Approximation)  c a n be e x p r e s s e d  (according  state.  In  t o Equation  AIV.15):  j  e~  +  F C A / K T  \/F//M  I I - 4 - 2 S p e c i a l Case X. t >> 1 ( S t r o n g &  This ensuing  case has rate  this two  i s no d e s o r b i n g  exponentials.  This  n ( t ) and  n  F  B  c a n be w r i t t e n  previously,  surface  and  P  t  but t h e  known f r o m t h e t r a p p i n g  1 9 6 7 ) . The  assumption  from t h e s u r f a c e  model  \t &  »  1  b e f o r e decay and difference  of  ( t ) t o be t h e number o f o-Ps atoms i n r e s p e c t i v e l y . The t r a p p i n g  v, P, , where v. l*  the  Approximation)  c a n be seen a s f o l l o w s .  f r e e a n d bound s t a t e s  surface  considered  are well (Brandt  Collision  t o a d e c a y s p e c t r u m w h i c h i s a sum o r  Define the  i n metals  there  leads  been  equations  for positrons implies  not  E q u a t i o n 11-14  ."C  r a t e on t h e  i s the c o l l i s i o n  rate  with  Cr  i s the p r o b a b i l i t y  f o r trapping  (or  20  adsorption)  per  collision.  It  is  assumed  that  the coupling  between a l i g h t atom s u c h a s Ps a n d t h e phonons a t is  weak  s o t h a t Pj. <<  transition  corresponds to trapping  r a t e r a t h e r than d i f f u s i o n  then s a t i s f y n  the f o l l o w i n g rate =  F  =  The  1. T h i s  initial  -)lF F  -  n  conditions  rate limited.  n  surface which i s and  f  n^  equations. n  '^fc^B  the  F  Equation  +"  11-15  Equation  n (t=0) = n F  and  0  11-16  n (t=0) = 0 lead t o fr  solutions  Equation  r\ fr) =  ^  b  ^  [e-  -e  J  AF-ZW + RV, The  decay spectrum w i l l  i s the f a m i l i a r  (1967)  for positron  Equation  11-18  Equation  11-19  t h e n be o f t h e f o r m  ^F-AptV.Fi This  11-17  /Ip'^  +^f*  two component s o l u t i o n trapping  c a s e X. > \f + v^P± l e a d s &  derived  by  Brandt  i n d e f e c t s , except that  in this  t o a d i f f e r e n c e of e x p o n e n t i a l s  t h a n a sum. In h i g h l y d i s p e r s e d  p o w d e r s where  P  h  «  |X  K  - X. | F  rather  21  Equation Alternatively  i n h i g h l y compacted  where v  |x. -  »  c  The  2  a  ~  n <2  k^t  powder. I n l i g h t  >>  is  Si0  function  independent  Effect 0  and  of pore  of 0  2  the  consequences  with experiments  on  (Gidley  size  1976)  1972)  or  two u n p a i r e d  gel  as  g e l the quenching  (Chuang  11-21  and  r a t e o f o-Ps  is  given r a t e of  in o-Ps  equivalently,  i/, ,  11-21.  on t h e Q u e n c h i n g  o f o-Ps  in Si0  2  Powder  m o l e c u l e s i n t h e gas p h a s e a r e p a r a m a g n e t i c ,  2  gel  discussed  silica  powder, t h e q u e n c h i n g  2  of  as s u g g e s t e d by E q u a t i o n  II-5  1  11-20, w h e r e a s i n s i l i c a  Equation  silica  Equation  o  assumption  linear  as  F  above a g r e e q u a l i t a t i v e l y Si0  such  X |  p  n("t)  powders  II'20  e l e c t r o n s . The  conversion quenching  possessing  o f o-Ps  with  0  2  c a n be w r i t t e n  -P< S i n c e p-Ps  has a l i f e t i m e  quenching  is easily  energy,  t h e o-Ps  —>  written  (see Appendix  of  only  observable p-Ps II)  + 0.125  o.  Equation ns,  i n the l i f e t i m e  conversion  cross  spin  11-22  conversion  s p e c t r u m . At section  can  low be  22  6^  £27  where S  6  I T s i n V  E q u a t i o n 11-23  i s t h e s c a t t e r i n g phase s h i f t  a  and o r b i t a l  a n g u l a r momentum  for total  electron  spin  0, i n d e p e n d e n t o f t h e r o t a t i o n a l  s t a t e o f t h e m o l e c u l e . The most r e c e n t room t e m p e r a t u r e v a l u e o f <y  c  i s 1.0 ± 0.1 x 10"  section  with 0  in pure N  1 9  2  ( K l o b u c h a r 1 9 8 0 ) . The p i c k o f f  i s of o r d e r 1 0 "  2  gas, C e l i t a n s  2  cm  cm  2 1  1964) a n d  can  comparison with spin conversion c r o s s When  0  fraction will Si0  i s admitted  roughly plot  be  neglected in  section. Si0  2  powder,  (Brunauer  0  2  2  estimated  0  2  on  surface  a BET ( B r u n a u e r - E m m e t t - T e l l e r )  i n the presence of adsorbed 0  remains paramagnetic  the  from  1938) o f t h e a d s o r p t i o n d a t a . ESR d a t a o f H atoms  l e a s t a s l o w a s 100°K ( S u r i n pores  of s i l i c a  2  indicate  Thus when 0  2  1973).  Furthermore,  0  2  adsorbed  g e l a t 300°K h a s been shown t o be an  gas i s a d m i t t e d i n t o t h e v o i d s  a n d bound a n n i h i l a t i o n  that  on s u c h a s u r f a c e a t t e m p e r a t u r e s a t  e f f e c t i v e c o n v e r s i o n q u e n c h i n g a g e n t o f o-Ps (Chuang  free  a certain  a d s o r b o n t o t h e s u r f a c e . The a d s o r p t i o n o f  1100°K  on t h e S i 0  onto  into  thus  quenching  s u r f a c e s h a s been s t u d i e d p r e v i o u s l y . The b i n d i n g e n e r g y i s  2  the  gas  2  ( e s t i m a t e d from  2  cross  of  1974). Si0 , 2  the  r a t e s o f o-Ps must be m o d i f i e d f r o m  t h o s e g i v e n i n E q u a t i o n 11-10 t o E q u a t i o n 11-24  E q u a t i o n 11-25  23  where  i s the conversion  adsorbed  0  section  (surface  2  cross  d e n s i t y n ), <r*  of  free  free 0  o-Ps,  v  free  g  the 2 dimensional  a d s o r b e d o - P s , <tj~ i s t h e 2 d i m e n s i o n a l  conversion  f o r bound o-Ps w i t h bound 0  z  0  2  and  i s the conversion  bound o - P s . When n  n where  5  cases. and  cross  b = 1100°K i s t h e 0  i s the  3  v e l o c i t y of  cross  for  free  B  -X  e  of f r e e  2  Equation  b i n d i n g e n e r g y on S i 0  1 1 - 1 3 , 11-20 a n d 11-21 s t i l l  F  section  0  2  2  with  | Equation  11-26  (see Equation  h o l d as  limiting  F o r e x a m p l e , i n h i g h l y d i s p e r s e d p o w d e r s where X  v^P^«\k  cross  i s the v e l o c i t y  section  with  i s much l e s s t h a n t h e m o n o l a y e r c o v e r a g e ,  ~ Hg/  AIV-9). Equations  (~<fJ!. ) t v ° z  2  o-Ps  (gas d e n s i t y n ) , v  2  is  5  of  i s the conversion  &  f o r f r e e o-Ps w i t h  velocity  section  p  t >> 1  11-20 y i e l d s  -[ \>, C P« f H + 6l n ) + 6  nu3 t A.] t 3  Equation  11-27  24  CHAPTER I I I : TEMPERATURE DEPENDENCE OF CONVERSION QUENCHING o-Ps  Conversion moderators  BY 0  IN S i 0  2  POWDER  q u e n c h i n g o f o-Ps  at  f o r o-Ps + 0  spin conversion  by  paramagnetic  1980)].  2  2  1 9  cross section) i s 2 x 10  cm )  i s considered  2  times  3  smaller than the  (Fleming  2  1981a) a n d H +  to  be  be  smaller.  convinced  In f a c t ,  some, e a r l y a u t h o r s  0  2  low i n  roughly  1000  ( C e l i t a n s I964)were  t h a t q u e n c h i n g was n o t due t o s p i n e x c h a n g e b e c a u s e i t  was s o s m a l l . However, a n g u l a r 1974)  with  anomolously  comparison with the p h y s i c a l cross s e c t i o n , being times  cross  ( d e f i n e d i n A p p e n d i x I I a s 27/8 t i m e s t h e  ( G o r d o n 1 9 8 1 ) . The o - P s s p i n e x c h a n g e c r o s s s e c t i o n  ( 4x10"  i n gas  2  A t 300°K t h e s p i n e x c h a n g e  s p i n e x c h a n g e c r o s s s e c t i o n s f o r Mu + 0 0  0  300°K h a s been w e l l s t u d i e d e x p e r i m e n t a l l y . [ S e e  f o r example (Klobuchar section  2  OF  and d o p p l e r  in  Section  in  the  b r o a d e n i n g measurements ( K i e f l  I».4 p r o v i d e  gas  c o r r e l a t i o n measurements  phase  (Chuang  1978) d i s c u s s e d  c l e a r e v i d e n c e t h a t t h e q u e n c h i n g by 0  is  dominated  temperature  dependence  of  improve our  understanding  the of  by  spin  spin  this  conversion.  conversion very  The  r a t e may  interesting  2  help  isotope  effect. In t h i s chapter in  which  the  measured from  a positron lifetime  o-Ps + 0  2  --> p-Ps + 0  2  2  conversion  121°K t o 630°K u s i n g an S i 0  r e s u l t s a r e p e r t i n e n t t o both conversion 0  experiment  and t o t h e behaviour  o f o-Ps i n S i 0  2  2  i s described r a t e h a s been  powder m o d e r a t o r . The  q u e n c h i n g o f o-Ps powder.  with  25  111*1  Experimental  A  3  nCi  2 2  Na  positron  source,  prepared  from  a  s o l u t i o n , was d r i e d a n d s a n d w i c h e d b e t w e e n 1 am n i c k e l source  was embedded i n S i 0  density  0.056 g e m ) , - 3  stainless 10"  for a  p e r i o d o f 12 h o u r s .  to  2  input  extra  d r y grade  steel  from  1/4  inch  2  gas  stainless  s t e e l b e l l o w s v a l v e s . The 0  gauge a c c u r a t e  pressure  2  t o ±5 t o r r .  wells  thermocouples, i n s e r t e d i n  (see Figure  111 - 1 ),  were  t e m p e r a t u r e a n d i t s u n i f o r m i t y (±2°K o v e r t e m p e r a t u r e was c o n t r o l l e d (121°K t o 630°K). The  300°K  by  styrofoam  circulating  The  cold  target' N  Section  to  2  stainless  monitor the  t h e chamber  chamber  was  volume).  t h e range of cooled  below  gas around t h e v e s s e l h e l d i n a temperatures  were  achieved  tape.  positron  lifetime  measurements  u s i n g t h e *+SR d a t a a c q u i s i t i o n delay  used  t o w i t h i n ±2°K o v e r  c r y o s t a t , whereas h i g h e r  with heating  system  were  made a t TRIUMF  ( s e e S e c t i o n V I * 1 * 2 ) . The  between t h e n u c l e a r gamma (1274 KeV) f r o m I•3•1)  and  t h e subsequent  positron  r a d i a t i o n was m e a s u r e d w i t h two 4 i n c h d i a m e t e r Nal  0  2  system c o n s t r u c t e d  Two c o p p e r c o n s t a n t a n  time  at  t h e vacuum chamber was m e a s u r e d w i t h a M a t h e s o n a b s o l u t e  pressure  study  outgassed  T h i s removes most o f t h e  2  t u b i n g and s t a i n l e s s  within  The  welded  0 , 0.03% A r , 0.05% N , 2 ppm C 0 , 20 ppm h y d r o c a r b o n s )  v i a a gas h a n d l i n g steel  in a  (Cabot).  2  P r o v i s i o n s were made  f o i l . The  r a d i u s 35A a n d  ( F i g u r e 111 -1 ) a n d  a d s o r b e d H 0 from t h e s u r f a c e  (99.65%  (mean p a r t i c l e  s e a l e d w i t h a copper o - r i n g  s t e e l vacuum chamber  torr  5  powder  2  NaCl  detectors  using  a  standard  by 4  2 2  Na  (see  annihilation inch  fast.-slow c o i n c i d e n c e  long  circuit  26  To 0 supply, vacuum gauge vacuum pump 2  and  stainless steel thermocouple wells  copper o-ring  i0  copper-constantan  2  powder  thermocouples  22  welded stainless steel chamber  Scale  Na positron source  cm  Figure 111*1 Apparatus f o r m e a s u r i n g o-Ps l i f e t i m e i n Si0 powder i n an 0 a t m o s p h e r e . 2  2  (see F i g u r e  111*2). Timing  anode  output  energy  (pulse height)  v i a constant  o u t p u t by p a s s i n g single  information  channel  fraction  was  analyzer.  from  through a spectroscopy The  from t h e  discrimination,  i n f o r m a t i o n was d e r i v e d  the pulse  obtained  energy  whereas  t h e dynode a m p l i f i e r and  resolution  of  these  27  dynocje  P  o w d e r  o-Ps decay  Nal #1  22 Na decay followed by 28 MeV y  Ortec 473A Ortec 473A const, frac const. frac| disc. di sc. anode  EG&G GPIOO/N EG&G GP100/l|l Ortec 471 | Ortec 471 p i leup gata p ileup gate spect. amp. spect. amp.  Ortec 455 S.C.A.  start  stop  TDC 100 time digitizer  CAMAC  PDP-11/40 MBD-11  computer  F i g u r e 111•2 E l e c t r o n i c s and data a c q u i s i t i o n m e a s u r i n g o-Ps l i f e t i m e s .  d e t e c t o r s was 14% FWHM a t 1274 KeV. A good e v e n t  system f o r  c o n s i s t e d of a  s i n g l e s t a r t p u l s e c o n s i s t e n t w i t h a 1274 KeV r r a y a n d a s i n g l e  28  stop  pulse  i n t h e e n e r g y window 400  »s g a t e . The digitized  t i m e i n t e r v a l between  with  an  was  5  resolution  EG&G ns  1330  KeV).  t h e 511  The  KeV  increase  annihilation 1*1).  (due  determined with a  2r  t o o-Ps  to  the  Ill*2  still  P r o c e d u r e and  Lifetime collected  for  at  its  poor  free e , +  long l i v e d powder. decay  least  in Si0  five 0  2  source  1170  chosen in  decay  and below  order  (see  resolution  of 511  2  of  to  Figure of t h e s e  keV gammas, t h e s t o p  within  100,000  events  were  p r e s s u r e s at each temperature.  p o w d e r . The  component i s due fits  Co  t o some 2 gamma a n n i h i l a t i o n s .  consisting  p-Ps and o-Ps  Good  3r  energy  (a) & (b) show t h e e f f e c t  s p e c t r u m o f o-Ps  timing  Results  spectra  F i g u r e s 111*3  to  sensitive  s t o p was  was  decay w h i c h has a c o n t i n u o u s  d e t e c t o r s and t h e Gompton s c a t t e r i n g d e t e c t o r was  6 0  at  annihilation)  s p e c t r u m by v i r t u e o f due  overall  simultaneous r rays  to  sensitivity  However,  The  e n e r g y window f o r a good  photopeak  the  clock.  as  ( w h i c h p r o d u c e s two v i r t u a l l y  both w i t h i n a 2  t h e s t a r t and s t o p p u l s e s  TDC100 FWHM  t o 450 KeV,  of 0  2  on  the  lifetime is  due  t h e powder g r a i n s , whereas  the  t o o-Ps  prompt a n n i h i l a t i o n  i n the v o i d  regions  were o b t a i n e d a s s u m i n g a s i n g l e  of  the  exponential  rate Equat ion  over the f i t t i n g  r a n g e 30-500 n s , where N  X is  decay  the  o-Ps  background.  Figure  f u n c t i o n of 0  2  rate,  111*4  concentration  and  shows  Bg two  0  is  i s the a  sample  ( i n t h e gas p h a s e )  111*1  normalization,  time plots  independent of X. a s a  determined  from  29  100000  10000 t-  1000 b  100  200  300  400  600  TIME IN NSEC (5 NSEC/BIN)  100000 f  10000 bib)  CO o i_)  1000 b-  100  100  200  300  600  400  TIME IN NSEC (5 NSEC/BIN) Figure III-3 (a) P o s i t r o n l i f e t i m e spectrum i n evacuated Si0 powder a t 295 °K. ( b ) Same w i t h 0 gas d e n s i t y of I0 cirr . 2  , 9  the  2  3  pressure.  Good  assuming a l i n e a r total  conversion  fits  were  obtained  d e p e n d e n c e . The s l o p e rate constant  k  0  at  a l l temperatues  of each l i n e  defined  from  gives  the  30  X ~ \t V\  g  This  rate constant  0  k  2  c  i  XCftyd)  Equation  i s p l o t t e d i nFigure  CONCENTRATION  I I I - 5 as  (CM" X 10 3  ,e  points a r egiven  function  )  Figure III-4 Decay r a t e o f o-Ps v e r s u s (iv, ) a t 295°K a n d 632°K.  o f t e m p e r a t u r e . The d a t a  a  III-2  0  2  concentration  i n Table I I I • 1 .  31  U J C O  m z: CM  O  1 .0  0  90  180 270 360 450 540 630 720 TEMPERATURE (K)  F i g u r e 111*5 Conversion rate constant versus temperature in an S i 0 powder m o d e r a t o r . . N o t e t h e r e i s no o b s e r v a b l e d e p e n d e n c e on powder d e n s i t y . 2  111•3 It is  Discussion i s clear  independent  surprising quenching range  f r o m F i g u r e 111*5 t h a t t h e r a t e c o n s t a n t ,  k  of  seem  temperature  below  i f one e x p e c t s t h e a d s o r b e d 0  450°K. 2  since there i s a large v a r i a t i o n  121  to  450°K.  F o r example,  This  may  t o play a role i n adsorbed 0  3  2  If  from  2 x 10  1 1  cm"  2  a t 450°K t o 2.5 x 1 0 • c m " 1  2  ,  i n the 2  i n the  a t a g a s d e n s i t y n^ = 1 0  cm" , t h e d e n s i t y of adsorbed 0 , ( a c c o r d i n g t o E q u a t i o n varies  c  1 9  11*26)  a t 121°K.  one assumes t h a t a o-Ps a d s o r b e d on t h e s u r f a c e b e h a v e s  as a  32  Table  111*1. 0  2  Conversion  Temperature  Conversion Rate Constant (I0" cm s"')  (°K)t2  iz  645 636 530 422 290 290 183 121 116  Powder D e n s i t y (g/cc)  3  1.68*0.04 1.75*0.05 1.34*0.04 1.22*0.05 1.26*0.05 1.23*0.04 1.18± 0.04 1.29*0.05 1.22*0.05  2 dimensional that  Rate Constant Versus Temperature  g a s atom w i t h s u r f a c e v e l o c i t y v  the 2 dimensional  3 dimensional above  0  order  10  2  2  conversion  conversion  surface «s"  0.056 0.161 0.056 0.056 0.056 0.161 0.056 0.056 0.161  1  5  cross section  us  = (rrkT/2m)  and  cross section scales with the (<J£ ~  * ),  s  concentrations correspond  and 1 0  s  t o X. ( « &  then v 5 s  ) of  n 5  the  3  a t 450°K a n d 121°K r e s p e c t i v e l y . The  - 1  observed quenching rate a t n  9  = 10  1 9  cm  - 3  i s only  13  >»s~ a n d 1  independent of temperature. However,  the  data are t o t a l l y c o n s i s t e n t with the "strong  c o l l i s i o n " model a s s u m p t i o n s \ t 9  described decay  in  Section  >>  1 a n d v^P^ «  |X.p  - X.  |  ,  11*5, which l e a d t o a s i n g l e e x p o n e n t i a l  rate  Equation Note t h a t i n t h i s low  P  limiting  surface coverage  and  case X i s l i n e a r l y independent  of  111*3  d e p e n d e n t on n^ a t  X^(the  bound  state  33  annihilation is  r a t e ) , a s o b s e r v e d . The e s s e n t i a l p o i n t  sufficiently  surface  l a r g e t h a t once t h e  i t is  lost  from  p r e c i s e l y how l a r g e X The  third  term  q u e n c h i n g by t h e S i 0 decay  rate  with  temperatures (*' v n )  in  no  is  0  2  i s close  the conversion  0.161  Figure rate  of  i/ (P +Pj. ) ,  v  approximately  differs  0  2.5  first  to  conversion  term  free  rate  o-Ps  constant,  was o b s e r v e d t o be i n d e p e n d e n t (0.056 gem"  i n t h e s e two p o w d e r s by a  (see Appendix I ) , t h i s  than  conversion c'  The  r a t e due  r a t e o f unbound o-Ps by g a s p h a s e 0  v- n , «  to  1 us" ' , s i n c e t h e .  III.4).  conversion the  due  c  f o r t h e two d e n s i t i e s s t u d i e d  3  of  i n t h e gas phase, whereas t h e  2  gem" ). Since  the  t o the f r e e decay rate a t a l l  w i t h a d s o r b e d 0 . The t o t a l  o f powder d e n s i t y  on  independent  i s much l e s s t h a n  ( t h e term i n square b r a c k e t s )  c  I I I * 3,  («•/ v, n, ) i s t h e c o n v e r s i o n  colliding  i s adsorbed  ensemble,  Equation  alone,  2  Ps  &  i s .  (see f o r example  second term  k ,  B  the  o-Ps  i s t h a t X.  c  a  ^  and  factor  implies that the i s much  2  larger  r a t e o f unbound o-Ps by a d s o r b e d 0  v n. ) . I f one s e t s c  3  then  at  2  (i.e.:  121°K  this  Equation  III-4  reduces t o  A  where  d=v /v^ 3  n  »  3  / n  on  distributed limit  2-5  x  cm  10  w i t h d = 0.6  x  10"  5  cm,  AI*24 i n Appendix I f o r t h e h i g h e r  above l i m i t based  -  i s t h e mean f r e e p a t h b e t w e e n s u r f a c e  While not i n accord Equation  3  on d.  isstill the  quite  reasonable since  assumption  i n space, and  of  therefore  spherical  collisions.  calculated  density  powder, t h e  Equation  AI*24  particles  represents  from  only  a  is  evenly lower  34  111•3•1  Thermalization  There a r e a t l e a s t i s not t h e r m a l i z e d 1.  two f a c t o r s w h i c h i n d i c a t e t h a t t h e o-Ps  b e l o w 450°K.  The r a t e c o n s t a n t k ~ v i s independent of T b e l o w 450°K. T h i s i s e a s i l y e x p l a i n e d i f v does not c h a n g e b e l o w 450°K. The a l t e r n a t i v e e x p l a n a t i o n i s t h a t tfg v a r i e s a s 1 / ( T ) b e l o w 450°K. T h e r e i s no t h e o r e t i c a l j u s t i f i c a t i o n f o r t h i s . C a l c u l a t i o n s of t h e s p i n c o n v e r s i o n c r o s s s e c t i o n o f o-Ps by H atoms (Hara 1 9 7 5 ) i n d i c a t e t h a t t h e c o n v e r s i o n c r o s s s e c t i o n i s o n l y weakly dependent on e n e r g y below thermal energies. This would give rise ^ t o a conversion rate constant p r o p o r t i o n a l t o ( T ) i f t h e o-Ps i s t h e r m a l i z e d . c  g  Z  3  2  2.  The r a t e c o n s t a n t g^ v a t 300°K (1.2 ± 0.1 x 1 0 " cm s" ) i s substantially higher than the rate c o n s t a n t o f 0.8 ± 0.1 x 10" cm s" measured a t 300°K i n an A r m o d e r a t o r . I t i s w o r t h p o i n t i n g o u t t h a t t h e s p i n e x c h a n g e c r o s s s e c t i o n o f Mu+0 is the same i n A r g a s a n d powder m o d e r a t o r s ( M a r s h a l l 1 9 7 8 ) . I n t h e s e e x p e r i m e n t s t h e Mu i s known t o be thermalized (see Section V*2). 1 2  3  3  1  1 2  3  1  2  A as  possible  explanation  indicated i n Section  the  entire  According ( 35A  surface  11*2 h a s t o do w i t h t h e a s s u m p t i o n  area  is  t o the manufacturer radius  aggregrate  spheres)  accessible  (Cabot),  are  primary  powder.  A  very  the  mechanically large  free  entangled volume  of  conceivable off free  the  primary  that a t lower  particles  and  thus  associated  in  order  thermal as  the aggregrates.  the I ti s  t e m p e r a t u r e s t h e Ps s c a t t e r s p r i m a r i l y  t h e s u r f a c e s of t h e a g g r e g r a t e s , path  into  l i g h t atom s u c h a s P s h a s a  w a v e l e n g t h o f 60A a t 450°K, w h i c h i s o f t h e same spacing  particles  These  agglomerates, which support  irreversibly  the Ps.  s t r u c t u r e s w i t h d i m e n s i o n s a s l a r g e a s 20000A. are  fused  the  to  large  structures  the  equally  that  into  aggregrate  with  o f why t h e o-Ps may n o t t h e r m a l i z e  thus  increasing  the  mean  t h e t h e r m a l i z a t i o n t i m e by a l a r g e f a c t o r  35  (.~10  or more). T h i s  large  also  help  the  2  explain  powder d e n s i t y . observed pickoff  One  rate  a n n i h i l a t i o n as by  o r d e r 0.008 u s '  (see  Equation  100  t e m p e r a t u r e s by  depositing  w o u l d most l i k e l y is  not  ideal of  likely  interest  rate constant  purely  (Ford  1976),  this  might  rate  of He  on  possibility  it  due  the of  to  since of rate  on  the  t e s t e d at surface.  trapping  low This  since  Ps  S u c h a medium m i g h t  be  vacuum d e c a y r a t e of o-Ps,  since  the  i s only  rate  be  on  that  i s not  trapping  bound t o s u c h a s u r f a c e .  f o r d e t e r m i n i n g the great  the  could  observed quenching  the This  a film  eliminate  implies  1 1 * 4 ) . The  11*20).  2  suggested  would then correspond p r i m a r i l y to surface  the  this hypothesis i s  previously  Section  mean f r e e p a t h  of  in evacuated S i 0  a f a c t o r of  (see  -  i n the  independence  i m p l i c a t i o n of  quenching  decreasing  increase  provides  a  test  a  subject  of  quantum  electrodynamics.  111*3*2 A n o m o l o u s S p i n At  low  energy, the  E x c h a n g e i n o-Ps  s p i n exchange c r o s s  a 2 e l e c t r o n m o l e c u l e can  are  where shifts the  for elastic  0  2  and  s t a t e . The  compared w i t h H + 0  2  at  be  the  written  (see  spin quartet  the  i n t e r a c t i o n and small  be  s e c t i o n of o-Ps  and  explained  off  II)  Equat i o n  111*5  spin doublet  phase  isotropic  independent of  s p i n exchange c r o s s  physical cross  300°K can  Scattering  2  Appendix  s wave s c a t t e r i n g o f f t h e  molecule-atom  rotational  + 0  s e c t i o n and qualitively  the  section  part  molecular f o r Ps  t h o s e of Mu as  of  follows.  +  + 0  2  In  36  the  case  of Mu  + 0  and  2  H + 0  a t 300°K t h e r e a r e many  2  waves w h i c h c o n t r i b u t e t o b o t h e l a s t i c involving  rotational  300°K, t h e s e  excitation.  inelastic  the  shifts. This  spin  energy  eleastic  i n the c a s e of  o f Ps + 0 w i t h any  +  0  positron  only  i n the  0  Townsend  (electron)-atom  non-s-  two  phase,  -6  nir.  a  a  effect  scattering,  s wave c o n t r i b u t i o n l e a d s t o  at  2  scattering  2  l e a d t o a s m a l l s p i n e x c h a n g e when 6  increase  partly  be  due  in  the conversion  to the  increase  There i s a l s o i n d i c a t i o n  that  room  of  temperature  value  u s i n g an A r m o d e r a t o r  1 9  Ps  scattering  in  where  very  a  small  cross section.  The  at  case  e x c h a n g e c r o s s s e c t i o n d e p e n d on  T h i s may  cancellation  10"  the  i s somewhat a n a l o g o u s t o t h e Ramsauer  low  may  inelastic  channels are c l o s e d along  wave s c a t t e r i n g . Thus o n l y does  In  and  partial  540°K (1.3 cm ), 2  ± 0.1  obtained  moderator. This  1  i n cL  scattering.  x  10"  1 9  cm ), 2  1980), i s s l i g h t l y  cm ) 2  and  c o u l d be  wave c o n t r i b u t i o n e x p e c t e d a t h i g h e r w e a k e n i n g of t h e  ± 0.1  from the p r e s e n t  increase  velocity.  increases with temperature. ( -0  1 9  a b o v e 450°K  i n t h e mean t h e r m a l  (Klobuchar x 10"  rate constant  ^ data the  obtained  lower  a t 630°K (1.5 u s i n g an result  Si0  The  than  ± 0.1 2  x  powder  of a s m a l l  p-  t e m p e r a t u r e s , or p o s s i b l y a  i n t e r f e r e n c e between q u a r t e t  and  doublet  s-wave  Ill-4  1.  Summary a n d C o n c l u s i o n s  The c o n v e r s i o n r a t e c o n s t a n t w i t h 0  in Si0  2  has been m e a s u r e d o v e r t h e t e m p e r a t u r e 121°K  t o *630°K.  was o b s e r v e d t o be indications  that  Below  range  from  450°K, t h e c o n v e r s i o n r a t e  independent the  powder  2  of  T.  There  are  o-Ps d o e s n o t t h e r m a l i z e a t  lower temperatures, p o s s i b l y  because  free  c l u m p i n g o f t h e powder  path  resulting  from  g r a i n s . The c o n v e r s i o n c r o s s 530°K  of l a r g e  section  a n d 630°K i s s l i g h t l y  measured  h i g h e r than  mean  at  previously  m e a s u r e d a t 300°K i n g a s m o d e r a t o r s . 2.  The a n a m o l o u s l y o-Ps  in  0  2  low s p i n exchange  i s explainable  p a r t o f t h e Ps - 0  of  i n t e r m s o f t h e s-wave  n a t u r e o f l o w e n e r g y Ps s c a t t e r i n g isotropic  cross section  2  caused  interaction.  by  the  38  CHAPTER I V : MUONS, MUONIUM AND  Unlike predicted  the  positron,  theoretically.  experiments  +  the It  existence  was  o f t h e muon was n o t  discovered  in  cosmic  ray  1937,  Street  1937) i n a s e a r c h  f o r the  p i o n , a p a r t i c l e p r e d i c t e d by  Yukawa  to  nuclear  f o r c e . Muons c a n be c o n s i d e r e d  heavy e l e c t r o n s , h a v i n g  103 1/2 As  (Anderson  „ SR  MeV/c , roughly  the  a n d come i n b o t h p o s i t i v e a n d n e g a t i v e  i n t h e c a s e o f e l e c t r o n s , t h e y do n o t p a r t i c i p a t e  interactions.  The  experiments  to  i s v i o l a t e d i n weak i n t e r a c t i o n s ( G a r w i n  electron  in  unexpectedly,  the  strong  close to  equation.  Muon d e c a y was one o f t h e f i r s t  1957). Not  charges.  m a g n e t i c moment o f t h e muon i s v e r y  efi/m* c, a s p r e d i c t e d by t h e D i r a c  parity  a mass o f  200 t i m e s t h a t o f an e l e c t r o n . They a r e s p i n  2  particles,  explain  positive  muon  show  that  1957, F r i e d m a n  may  capture  t o f o r m t h e H - l i k e atom c a l l e d muonium o r Mu. The  observation  of  an first  Mu was made by Hughes e t a l . (Hughes 1 9 6 0 ) . The  p r o p e r t i e s o f muons a n d Mu atoms a r e o f t r e m e n d o u s i m p o r t a n c e i n physics  s i n c e they provide  electromagnetic The  polarized  of muon  muons  the  "meson  probe  testing  physics for  by  factory"  experimental  in  nuclear  physics,  in  the  1977). 1970's h a s  Apart  t h e muon  has  state  N e g a t i v e muons c a n be u s e d t o p r o b e  structure  their  atomic  orbits  overlap  P o s i t i v e muons h a v e been e m p l o y e d p r i m a r i l y a s  of  from i t s become  physics  physical chemistry. because  for  i n t e n s e • beams  study.  solid  ground  (Hughes  providing  fundamental r o l e i n p a r t i c l e p h y s i c s , useful  ideal  a n d weak i n t e r a c t i o n t h e o r i e s  advent  revolutionized  an a l m o s t  a  and  nuclear  the nucleus.  magnetic  probes  39  in  solid  interest light  state  physics.  t h e muon h a v i n g  description  w o u l d be polarized briefly  Mu  atom  (*/*e~)  i n p h y s i c a l c h e m i s t r y because i t can  i s o t o p e o f H, A  The  without  muons  the  ti*SR t e c h n i q u e  be  and  first  i n the f i r s t  two  a  mass. ) technique  (ji*SR  discussing  properties  special  considered  the p r o t o n  of t h e Muon S p i n R o t a t i o n  incomplete  presented  1/9  i s of  the  source  of  o f muon d e c a y . T h e s e are.  s e c t i o n s of t h i s c h a p t e r .  The  i s t h e n e x p l a i n e d w i t h e m p h a s i s on t h e t r a n s v e r s e  field  t e c h n i q u e . The  then  introduced.  b a s i c t y p e s of s p i n r e l a x a t i o n  Finally,  the  form  of  for  t h e »« SR  Mu  spectrum  +  are is  derived.  IV«1  S o u r c e o f P o l a r i z e d Muons I n t e n s e beams of medium e n e r g y  MeV),  incident  on  of v mesons —  common s o u r c e  (~100  »»A  a s u i t a b l e production t a r g e t , are  b e i n g u s e d a t meson f a c i l i t i e s source  protons  s u c h a s TRIUMF, SIN and  the " n u c l e a r  glue"  particles.  at  currently LAMF a s a The  o f p o l a r i z e d muons i s f r o m weak d e c a y o f  most  n's  Equat i o n which  have a f r e e l i f e t i m e of 26 n s . The  a two  component W e y l  and  is  3-f  \v,y  3.5  i  thus  an  %?  massless  500  IV-1  n e u t r i n o obeys  equation  = Iplol --  helicity  \?Y  i  y> Equation  e i g e n s t a t e . C o n s e r v a t i o n of  IV»2  energy,  40  t o t a l a n g u l a r momentum, a n d l i n e a r momentum r e q u i r e t h a t t h e »is rest  an  e i g e n s t a t e and monoenergetic  a t 4.2 MeV i n t h e  f r a m e o f t h e tr* . A s e c o n d a r y b e a m l i n e , c o n s i s t i n g o f  dipole (for  helicity  magnets(for  focussing),  production  momentum  s e l e c t i o n ) and quadrupole  i s used t o t r a n s m i t c h a r g e d p a r t i c l e s  target  remains v i r t u a l l y  to  since the cyclotron  are  almost  the  ( g ^ - 2 ) . The f i r s t  were d e s i g n e d t o c o l l e c t flight.  These  types  channels  discovered  f r o m v* d e c a y The  (Pifer  result  of  pions  in a relatively  production  producing highly polarized  TRIUMF (Oram 1 9 8 1 ) . The muons h a v e e n e r g y  frame.  polarized  because  target.  i n t e n s e f l u x e s of  low energy and h i g h  at  4.2 MeV a n d a r e a l m o s t  the pions are at rest  S u r f a c e muons a r e p a r t i c u l a r l y  because of t h e i r  high  Recently, i t  t h e s e s u r f a c e muons was s u b s e q u e n t l y d e v e l o p e d a n d e x p l o i t e d  completely  in  1976) t h a t a f l u x o f u* c a n be o b t a i n e d  on o r n e a r t h e s u r f a c e o f t h e  technique  of  (qq* B/( 2m^c))  b a c k w a r d d e c a y i n g muons f r o m  of  field  s t o p p i n g muon c h a n n e l s  e n e r g y muon beam, ~ 5 0 MeV, w i t h p o l a r i z a t i o n ~ 0 . 8 . was  beamline  f r e q u e n c y f o r a muon i n a m a g n e t i c  same  the  helicity  through t h i s  s t r e n g t h B ( q B / d i ^ c ) ) and t h e p r e c e s s i o n f r e q u e n c y  magnets  from  t h e e x p e r i m e n t a l a r e a . The muon  unchanged d u r i n g passage  large  useful  i n the l a b  i n p*SR e x p e r i m e n t s  polarization.  41  I V * 2 Muon D e c a y Muons d e c a y v i a weak i n t e r a c t i o n  i n t o an e l e c t r o n  a n d two  n e u t r i n o s w i t h a l i f e t i m e o f 2199.4 n s (Wu 1 9 6 6 ) .  —> The  decay  current Hx  e "  +• V^f  -r- V4.  Equation IV-3  p r o p e r t i e s o f muons a r e d e s c r i b e d w e l l by a c u r r e n t -  i n t e r a c t i o n of t h e f o l l o w i n g form 9A [ %^C\-^s)%][^A^)%]  =  + herni+m  conjugal E q u a t i o n IV-4  (Williams  1 9 7 1 ) , where g^ i s a c o n s t a n t ,  the  field  operators  i s a  summation  A  vector - a x i a l vector  f o r lepton i , index.  vector  and the  Each  term i n square b r a c k e t s terms  discovered  n u c l e i a n d by o t h e r muon  asymmetric <o- '>'n , e  Wu  (Garwin  (1957)  process  now  muon  decay,  depending  <&">  being  M  -  c(uu)[ I t  +  decay o f the  discussed,  leads  on  pseudo-scalar  the  i s t h e muon p o l a r i z a t i o n  d i s t r i b u t i o n of the e  parity,  1957) i n  t o an  v e c t o r and n i s e  t h e p o s i t r o n momentum d i r e c t i o n . More s p e c i f i c a l l y , angular  has a  and thus  of  i n t h e beta  1957, F r i e d m a n  decay  where  /  groups  p a r i t y . Non c o n s e r v a t i o n by  are  involving  and a x i a l v e c t o r components a r e p s e u d o - s c a l a r s ,  simultaneously  ^  a r e the D i r a c m a t r i c e s , and  (V-A) f o r m . The p r o d u c t  connect s t a t e s of opposite  very  the \  t h e energy  c a n be w r i t t e n  DM <^> '^Isi)] Equation IV'5  where  u = E/E^  4 X  i s the p o s i t r o n energy  i n units of E  )>viv  = m /2. A  42  F i g u r e IV*1 shows t h e d e c a y p a r a m e t e r C(w) a n d D ( w ) .  i  r •  i  .  |  Note  that  1 i  1  •  / 0.5  ,/DM •  i  -0.4  i  0.2  i  ,  OJ =  F i g u r e IV-1  i  E/E  asymmetry changes  of  positron  C ( u ) , i s weighted  be 0.324 ± 0.004 ( C r o n i n  1/3 a n d h a s been  more  of  muon  of time a f t e r  spin  rotation  m 4 x  . The  measured  scintillation  counters  The  ) involve  particular  direction  i n the target.  In a  i s s i g n a l l e d by one o r  a n d i t s d e c a y by t h e p a s s a g e o f a  h i g h energy p o s i t r o n through a p o s i t r o n two o r t h r e e s c i n t i l l a t i o n  (y*SR  t h e muon a r r i v a l  n*SR e x p e r i m e n t , t h e muon a r r i v a l  a range f i l t e r .  E  1968)'.  m e a s u r i n g t h e d e c a y r a t e o f t h e muon i n a  typical  towards  Rotation  techniques  as a f u n c t i o n  1.0  max  average of C(u)*D(u) i s t h e o r e t i c a l l y  The  i  s i g n w i t h e n e r g y and t h a t t h e d i s t r i b u t i o n  energies,  I V * 3 Muon S p i n  i 0.8  Muon d e c a y p a r a m e t e r s C ( o ) a n d D ( o )  the  to  •  0.6  0.4  t e l e s c o p e , c o n s i s t i n g of  c o u n t e r s w i t h some a b s o r b e r a c t i n g a s  histogram  of  time  delays  between  these  43  events  ( t h e »/ S R s p e c t r u m )  has t h e f o l l o w i n g form  +  Nn'tt)  jj„ en,) C Aft  NJt)  fiMD  ( f o r n* d e c a y )  DM<2^)>-aMj + B  '  JI' iir  o  Equation where  N^t)  =  3  N  e  is  the  total  IV«6  number o f muons i n t h e  e n s e m b l e a t t i m e t , r* i s t h e »* l i f e t i m e , e ( o ) i s t h e e f f i c i e n c y of  the positron  telescope  n' i s t h e s o l i d a  time  Carrying  vector  is  <<r^(t)> —  and A  A<^(-t)>'fr]  0  time  +%  u,  Bg i s  the  muon  dependent).  E q u a t i o n IV-7  telescope,  i s t h e maximum p o s s i b l e  N  0  i s the  asymmetry.  In  o  +  evolution  of  both i t s magnitude transverse  field  to the i n i t i a l  telescope  the  muon  a from  the experimenter to polarization  vector  and d i r e c t i o n .  n*SR  ,  a magnetic  field  i s applied  muon p o l a r i z a t i o n <Z (0)>  , and t h e  /i  d i r e c t i o n m. As m i g h t  p r e c e s s e s about t h e a p p l i e d  to oscillations  derived.  is  t h e »» S R s p e c t r u m a l l o w s  IV«7 t h a t  polarization  /1  is  /  general,  0  perpendicular  <c> (t)>  <e '(t)>  the d i r e c t i o n of t h e p o s i t r o n  measure t h e time  rise  in  and  telescope,  t>*SR a p p a r a t u s , A ~ 0.3 a n d N / N ~ 0 . 0 3 . I t i s c l e a r  Equation  positron  1 t  f/t/1  0  normalization,  In  (which,  N C~ l  =  m  typical  background,  of energy  out t h e i n t e g r a t i o n y i e l d s a spectrum of t h e form  Nj.lt) where  a positron  a n g l e s u b t e n d e d by t h e p o s i t r o n  independent  polarization  fordetecting  be e x p e c t e d , field  the  muon  direction, giving  i n t h e >i S R s p e c t r u m . The t i m e e v o l u t i o n o f +  f o r f r e e muons a n d Mu i n a t r a n s v e r s e  field  will  now be  44  IV'3'1 F r e e Muons i n a T r a n s v e r s e The s p i n H a m i l t o n i a n field  f o r an i s o l a t e d  where  muon  in  a  magnetic  =  Equation  2. g^eB/^m^c,  ~  2, m^  s p i n m a t r i c e s . The e n e r g y  =1> a n d \t If  >  IV«8  i s t h e muon mass, and a* a r e  eigenstates  are  then  |«i>  =  w i t h e i g e n v a l u e s ±huf/2 r e s p e c t i v e l y .  =  2  t h e muons a r e i n i t i a l l y  X then  Field  B along the 2 d i r e c t i o n i s  2  Pauli  Magnetic  [  i n a pure  |£,> +-  state  J  Equation  IV«9  t h e c o r r e s p o n d i n g d e n s i t y m a t r i x a t t=0 i s  2. E q u a t i o n IV'10 In m a t r i x n o t a t i o n J-  2-  2-  i-  E q u a t i o n IV-11  2  2-  The d e n s i t y m a t r i x a t a l a t e r  -  time t i s g i v e n as  e* 4-  J-  e  2. 2.  Equation  IV«12  45  The easily +  evolution  evaluated  ,  irf  time  of  muon  i n t e r m s of  t w i c e the  the  muon s p i n  Re<tf^ > <<fy> = Im<a '*> +  expectation value  >  =  0.  In  matrix  follows  = <c*>  «f =  notation,  O  immediately  e^*  2.  o  It  of  i s most  r a i s i n g operator. It i s clear  and  /  ?  polarization vector  Equation  IV-13  Equation  IV*14  Equat ion  IV*15  applied  field  from  TrE =  e  iu)t  that  <  Thus t h e  (*")>  6^  sin  -  p o l a r i z a t i o n vector  d i r e c t i o n at  a single  W  +  A  rotates  about  f r e q u e n c y , u , as M  the  expected  I V * 3 * 2 F r e e Muonium i n a T r a n s v e r s e M a g n e t i c The field  spin Hamiltonian  a l o n g the  f o r an  2 d i r e c t i o n can  2" 3*  i s o l a t e d Mu  be  first  1975).  In  The  interactions  the last with  magnetic  c  hyperfine coupling  ground s t a t e , two  terms  the  p r e f e r t o ±1  Equation  Z.  u /2v  are  applied  0  field.  respectively,  between  the  and  I tr'* e 1  1  3£-  and  (Casperson  electron  In t h e  IV*16  muon  = 4463.302 MHz  muon  ** **  where o and  atom i n a  -J- "four-**"' + "K ai « 6*  term i s the  electron.  Field  written  Z  The  classically.  e  Z-  Zeeman >  basis,  46  E q u a t i o n IV'17 with corresponding  eigenvalues J z  -  -  +-  l/J_  UJo  _  - ^ 0  t w .  1  ^  E q u a t i o n IV'18 where E q u a t i o n IV-19  UJ  0  Equation  IV-20  Equation  IV«21  where B =1585G a n d 0  These a r e t h e f a m i l i a r  B r e i t - R a b i energy e i g e n v a l u e s f o r the  atom(see f o r example Brewer Consider  1976).  a muon i n i t i a l l y  p o l a r i z e d along the x d i r e c t i o n ,  w h i c h c a p t u r e s an u n p o l a r i z e d e l e c t r o n a t t = 0 . The r e s u l t i n g state  i s a mixture  H  o f t h e two s t a t e s  Mu  47  =  [ l6,> +  co$q> l e > - 5 i n  k,>J  t  JEquation  IV-22  The c o r r e s p o n d i n g d e n s i t y m a t r i x a t t=0 c a n t h e n be w r i t t e n Equation  IV'23  Equation  IV-24  In m a t r i x n o t a t i o n ,  1  i eOi>&>  1  o  The  COS  f  o  7 V s mjp  "•i o  time evolved d e n s i t y m a t r i x  V  - $\v\q> H o  _L  V  cosq>  i  .i-  n  t  l£j>  48  j_  cob  -5mf e  o  £,  H  co$_q e 7  o  7 4  H  T  Equation where ho,J As  = e. - c . < J  in  evaluate  the  =  of  the  free  muon, i t i s s u f f i c i e n t t o  o f tf/  14  ,  twice  the  muon  spin  Equation  IV-26  I t i s e a s y t o show  <6 l^"l6 > (  o na>scf>  o 1 COS(D  It  case  the expectation value  raising operator.  6«]  IV-25  J  o  o -xcoscf* O  f o l l o w s immediately from  O  a  O  o  o O  O  49  <6^y  -  T r l  Pfr)^ J +  S l ^  4-  <^  cos  f  ^  e-  J  Equation  IV-27  Equation  IV-28  that  ^  2--  In and  o  3 4  low f i e l d s x <<  1585 G, t h e p e r i o d o f t h e u,„  f r e q u e n c i e s i s o f o r d e r 0.225  observable resolution not  1 o r B <<  with  a  typical  i s ~1 n s . Thus h a l f  observed,  and  the  v*SR  ns  and  is  apparatus,  usually where  t h e muon p o l a r i z a t i o n  remaining  two  terms  in  in <e^>  not  timing Mu  is  c a n be  expressed  (i + x ) * 1  — — — In v e r y  low f i e l d s ,  J  J  < 10G, where n-r^ <<; 1 ( T^,  Equation  IV-29  Equation  IV-30  =2.2  «s  is  the  50  muon  lifetime)  the  splitting  i s n e g l i g i b l e and t h e e x p r e s s i o n  simplifies to  (<5y= x  -L_ C0t> LO-~t  where u_=B«1.40 MHz-G"  1  E q u a t i o n IV-31  from e q u a t i o n IV.21.  IV«4 Mu S p i n R e l a x a t i o n When muons a r e s t o p p e d i n m a t t e r a s b a r e muons o r free  Hamiltonian  i s perturbed  v a r i o u s ways w h i c h l e a d t o d e c a y Much o f t h e i n t e r e s t  by  the  Mu,  the  s u r r o u n d i n g medium i n  o f t h e muon p o l a r i z a t i o n  i n »/*SR i s f o c u s s e d on t h e s p i n  <c*>  .  relaxation  o f muons ( M u ) , s i n c e t h i s y i e l d s  i n f o r m a t i o n on muon (Mu) + h o s t  state.  So f a r a s s p i n r e l a x a t i o n  i s c o n c e r n e d t h e b a r e muon i s a  pure  magnetic  probe,  sensitive  to  the  e n v i r o n m e n t . The muon i n a Mu atom i s s t r o n g l y electron,  whose  magnetic  moment  response t o the l o c a l magnetic general  103  precession electric  is  200  environment  magnetic  coupled  t o the  times  larger. I t s  are indirect  but i n  t i m e s f a s t e r t h a n t h e b a r e muon (due t o t h e f a s t e r  f r e q u e n c y ) . Muons i n Mu atoms a r e a l s o fields  since  in  general  these  e l e c t r o n c o u p l i n g and t h u s t h e p r e c e s s i o n  will  +  sensitive alter  to  t h e muon  f r e q u e n c i e s o f Mu. The  i n f o r m a t i o n o b t a i n e d f r o m n S R o f Mu atoms i s obtained  local  similar  to  that  f r o m ESR o f H a t o m s . T h e r e a r e f o u r b a s i c mechanisms by  w h i c h <<r (t)> o f Mu d e c a y s /H  i n a host.  1. Random l o c a l m a g n e t i c f i e l d s . 2. Random a n i s o t r o p i c d i s t o r t i o n . 3. S p i n E x c h a n g e r e a c t i o n s . 4..'Chemical r e a c t i o n s .  51  IV«4«1 Random L o c a l M a g n e t i c F i e l d s 'Consider  a  Mu  atom  at  position  m a g n e t i c moments a t p o s i t i o n s Hamiltonian coupling  f  1  f  f  f o r Mu must be m o d i f i e d between  interaction  the  Mu  (RLMF) i n a host .. . f .  2  v  The  containing free  spin  to include the d i p o l e - d i p o l e  a n d t h e h o s t moments. The  perturbing  c a n be w r i t t e n  Equation  IV-32  Equation  IV«33  where  is  the magnetic  (Abragam  a t t h e Mu  site  due t o  1 9 6 1 ) . Thus t h e e f f e c t i v e f i e l d B fp e  where AB -  -  Bappl.'etJ  ^  B>d/p  the  host  In  a  unpolarized,  lattice AB^  where  i s randomly  t o a broadening  this will to  at  o f Mu  r,-  atom  Equation  moments  distributed  i n t h e Mu  be g i v e n  IV-34  frequencies,  motion  of  This  harrowing.  are  about  fixed  zero,  and  giving  or ( e q u i v a l e n t l y ) a  A specific  example  of  i n S e c t i o n V « 4 . I f AB ^ i s f l u c u a t i n g q u i c k l y dl  the  average the p e r t u r b i n g field.  moment  at the s i t e  d e c a y o f t h e muon p o l a r i z a t i o n a m p l i t u d e .  due  the  = p,-  d  rise  field  Mu  field  phenomenon  is  o r t h e h o s t moments, t h i s s o t h e Mu commonly  atom  sees  referred  tends t o  the  average  t o as  motional  52  IV«4-2 Random A n i s o t r o p i c D i s t o r t i o n Consider  a Mu atom l o c a l i z e d a t a s i t e  ground s t a t e wavefunction altering  the  may  hyperfine  be  f  in a lattice.  perturbed  interaction  by  between  e l e c t r o n . T h i s p e r t u r b i n g i n t e r a c t i o n c a n be ^  lattice,  the  muon  n a t i o n  4  x  4  perturbation  (i.e.:  tensor. A  =  eigenstates are i d e n t i c a l  In  the  ftAooI), t o those  case  the  of  an  I V . 35  isotropic  energy e i g e n v a l u e s and  obtained  that the hyperfine s p l i t t i n g  and  expressed  =  where A i s a  except  the  The  i n t h e f r e e Mu  i s modified  case,  to "h(u Au )• +  0  I n  0  t h e g e n e r a l c a s e t h e p e r t u r b a t i o n i s a n i s o t r o p i c a n d <<r^(t)> h a s six  frequencies corresponding  applied  field.  observed  because of t h e i r h i g h  on  Three  the orientation  polarization. observed fused  Zero  of  of t h e field  i n single crystal  Si0 , 2  t o a l l p o s s i b l e u.j, e v e n  these  (of order  frequency.  lattice  with  in  u ) a r e not normally 0  A l l o f them may depend respect  to  the  Si0  2  below  50°K  of the l a t t i c e  i s random t h e  (Brewer  1981).  muon p o l a r i z a t i o n  anisotropic  causes  s p i n r e l a x a t i o n . As i n t h e c a s e o f random l o c a l  fields,  r a p i d m o t i o n o f t h e Mu atom l e a d s t o an a v e r a g i n g many s i t e s , a n d a s u b s e q u e n t  In  with respect t o the  initial  relaxat ion.  muon  o s c i l l a t i o n s o f Mu have r e c e n t l y been  the o r i e n t a t i o n  d i s t o r t i o n over  zero  distortion  weakening  magnetic of t h e of  the  53  IV«4'3 C h e m i c a l  Reaction  A Mu atom may r e a c t compound.  The  dramatically compound  chemically  magnetic  radical  of  out  of  phase  of  formation  the  muon  (except  changes  i f the  Mu  i n w h i c h t h e muon e l e c t r o n c o u p l i n g i s  a l m o s t t h e same a s f o r M u ) , s o t h a t falls  a m o l e c u l e t o f o r m a Mu  environment  at the instant  is a  with  with  the  t h e muon s p i n v e c t o r  remaining  members  of  quickly the  Mu  e n s e m b l e . I n t h e g a s p h a s e t h e Mu r e l a x a t i o n r a t e c a n be w r i t t e n t r ^ v n , where  i s the cross  e  mean  thermal  velocity  of  section Mu  and  f o r the r e a c t i o n , v i s the n  i s the concentration  of  reactant.  IV'4«4 S p i n As spin  Exchange  i n t h e c a s e o f P s , t h e z component o f  i s not a conserved q u a n t i t y  molecules  such  as  NO  (S=1/2)  i n c o l l i s i o n s with or  0  t r a n s i t i o n s o f Mu w h i c h r e s u l t  of  muon  s p i n s . The r e l a x a t i o n r a t e  can  be w r i t t e n  where  <*  ev  Appendix I I  f Jex•  i s the and  f  paramagnetic molecule 1981a).  t m  spin  (s=1).  electron  paramagnetic leads t o  i n a l o s s of  coherence  f o r Mu due t o s p i n  /2 exchange  Mu  This  2  hyperfine  \-  the  exchange  Equation cross  section  defined  IV-36 in  i s a c o n s t a n t d e p e n d i n g on t h e s p i n o f t h e ( f = 3/4 f o r NO a n d 8/9  for 0 , 2  Fleming  54  The »i SR S p e c t r u m i n a T r a n s v e r s e  IV-5  Field  +  It muons  i s clear and  Mu  that the precession amplitudes in  matter  have  some  time  hj*+ a n d A ^  dependence  M  for  due t o  i n t e r a c t i o n w i t h h o s t . L e t t h e t i m e d e p e n d e n c e be r e p r e s e n t e d by t h e f u n c t i o n s R ^ ( t ) a n d R ^ f t ) w i t h R^ (0) = R +  the  n*SR s p e c t r u m  telescope IV-7  and  +  rtl4  (0) =  i n a moderate t r a n s v e r s e f i e l d  i n the x d i r e c t i o n can  be  written,  1.  Then  fora positron  using  Equations  IV-29,  Equation  IV-37  Equation  IV-38  where  The  spectrum  f o r a t e l e s c o p e i n any d i r e c t i o n  precession d i f f e r s (<  o n l y by a p h a s e f a c t o r .  10G) one may u s e E q u a t i o n  i n t h e p l a n e of  In very  weak  IV-31 i n s t e a d o f E q u a t i o n  fields IV-29 so  that  z  Equation  IV-39  55  CHAPTER V : MUONIUM I N INSULATING POWDERS  Mu p r e c e s s i o n h a s p r e v i o u s l y been o b s e r v e d powders such a s S i 0 , A l 0 2  As  i n t h e case  2  introduction  a n d MgO ( M a r s h a l l 1978, K i e f l  3  of Ps, a f r a c t i o n  r e g i o n s o f t h e powder. of  This  a  oxides  at  formation,  prelude low  2  to  temperature,  this  h i g h l y d i s p e r s e d m e d i a . The  form  will  gas, which r a p i d l y  the experimental  and  M  verified  through  the  relaxes the with  gas moderator.  2  thermalization,  R^ (t)  been  s p i n exchange a t a r a t e c o n s i s t e n t  t h a t m e a s u r e d i n an A r o r N As  1979).  o f t h e Mu emerges i n t o t h e v o i d  has  paramagnetic 0  Mu i n t h e v o i d s t h r o u g h  i n s e v e r a l oxide  r e s u l t s on Mu i n t h e s e  chapter  deals  with  the  s p i n r e l a x a t i o n o f Mu i n t h e s e of  the  relaxation  function  be d e r i v e d u n d e r v a r i o u s e x p e r i m e n t a l c o n d i t i o n s .  V«1 Mu F o r m a t i o n The muons  t h e r m a l i z a t i o n and n e u t r a l i z a t i o n  of e n e r g e t i c  i s a complicated process, e s p e c i a l l y  o r a p o w d e r . As i n t h e c a s e processes  can  result  positive  i n condensed  of Ps, epithermal,spur  and  matter surface  i n Mu f o r m a t i o n . However, r e c e n t l y i t h a s  been shown t h a t Mu f o r m a t i o n  i s u n a f f e c t e d by t h e a p p l i c a t i o n o f  an e l e c t r i c  2  field  i n bulk S i 0  (Ito  1981). T h i s  field  i n h i b i t s combination  results least  weaken  i n these The  and s e v e r a l  i s i n c o n t r a s t t o t h e case  the  hydrocarbon  o f P s , where s u c h a  o f e* a n d s p u r e~ ( I t o 1 9 7 9 ) .  s p u r model h y p o t h e s i s  liquids  These  f o r Mu f o r m a t i o n , a t  cases.  p r o b l e m o f e p i t h e r m a l Mu f o r m a t i o n c a n be f o r m u l a t e d a s  56  f o l l o w s . D e f i n e f * ( E t ) t o be t h e f r a c t i o n o f muon charge  state  i  exchange [ v a l i d  a t energy  5f  2+  =  E and time t . I g n o r i n g double  b e l o w 10KeV f o r p r o t o n s  obey t h e f o l l o w i n g c o u p l e d  ensemble  charge  (Tawara 1 9 7 3 ) ] t h e  integro-differential  in  f - 's t  equations  - f , (E, + ) T t t„ ( E E') + t j E E') J J E' +Trti,(E'E)f^W+ U f ' e l f . t ' f J j E '  E q u a t i o n V«1  3£  E q u a t i o n V«2  E q u a t i o n V-3 where  t,:(EE')  initial E'  s t a t e of energy  and  charge muon  i s the  charge  has  a  may be e n e r g e t i c a l l y condensed matter a l l the  observable  between  i and a f i n a l  i s g e n e r a l l y accepted  a muon i n an  s t a t e of  energy  t h a t most o f t h e where  the  comparable w i t h the valence e l e c t r o n s i n  1981b). At t h e r m a l e n e r g i e s , charge  exchange  f o r b i d d e n . S i n c e t h e t h e r m a l i z a t i o n time i n  i s estimated to charge  exchange  o b s e r v a b l e v i a t h e u*SR t e c h n i q u e The  rate  i n t h e 2 KeV t o 20 eV r e g i o n ,  velocity  most atoms ( F l e m i n g  1975),  E, c h a r g e  j . It  exchange o c c u r s  transition  quantities  in a  be  of occurs  order much  10~ S 1 1  t o o f a s t t o be  (timing resolution i>*SR  (Brewer  experiment  10'  9  s).  a r e t h e Mu  57  fraction F  M i )  (t) and t h e d i a m a g n e t i c  fraction  F^<-(t) where  o  f>m  Equation  V-4  => T t f , ( E t ) + f . , ( E t ) J E q u a t i o n V«5  The  Mu" i o n i s d i f f i c u l t  the  x SR  technique,  +  polarization  to distinguish  since  i svirtually  the  the  c o n t r i b u t e s t o the diamagnetic In one F  /t1+  principle,  from a  time  same  evolution  i n both  However,  c a l c u l a b l e . This  in  i f the t r a n s i t i o n  general,  they  Si0 . 2  and  thus  and  predict  F  M  and  i n condensed  matter  has  mystery.  Longitudinal  also present  i n bulk A l 0  probability  i s not  aggregation  cases  t h e muon  r a t e s t ; j ( E E ' ) a r e known,  Mu p r e c e s s i o n h a s a l s o been o b s e r v e d and  of  with  a r e not e a s i l y measurable or  i s why Mu f o r m a t i o n  r e m a i n e d somewhat o f a  muon  fraction.  c a n s o l v e e q u a t i o n s V - 1 , V-2 a n d V«3 .  bare  2  i n b u l k s a m p l e s o f MgO  f i e l d measurements i n d i c a t e t h a t i t i s 3  (Minaichev  positively  The  Mu  formation  to  the  degree of  t h a t Mu f o r m a t i o n  i s a bulk  correlated  of the oxide, suggesting  r a t h e r t h a n a s u r f a c e phenomenon.  1970).  58  V'2 Mu  Thermalization  Below the l o w e s t metallic  medium  electron  excitation  ( o r o f Mu, w h i c h e v e r  energy t h r o u g h e x c i t a t i o n of l a t t i c e thermalized, in  strongly  bulk  greater  than  the  Weil  powders  d i f f u s i o n model,  voids,  is Mu  not  clearly  thermalizes  (Such  a  work  t h e Mu  assumption  function  for  (Foner  interstices  the  grain  work  A  size  interstices  may  valid  large and  at  work the  function  2  powder  at  the  model  has  (70A a n d  140A  suggesting that  thermalizes  a l l temperatures function of  t h e work  t h a n 300°K. However, t h e  and  at the surface, a  buffer  favour d i r e c t t h e r m a l i z a t i o n  for introducing  i n t o the  1 9 7 8 ) . T h e r e i s no i n d i c a t i o n  initially  presence  then  might a r i s e from the l a t t i c e  i s much g r e a t e r Mu  and  where i t i s e j e c t e d  t h e powder g r a i n s ,  a l l the  n o t be  conditions.  in l i g h t  reenters  that  may  motivation  i n the l a t t i c e  understood. In the thermal  j#*SR r e s u l t s f r o m S i 0  at the surface  particle  low  reaches the  within  mean d i a m e t e r ) a t 300°K. ( M a r s h a l l  grains  very  i n d u c e d by an i n t e r s t i t i a l Mu a t o m ) . T h i s  been u s e d t o e x p l a i n  function  At  i n many ESR e x p e r i m e n t s  presumably because of a n e g a t i v e  distortion  that  Once  muon l i f e t i m e . T r a p p i n g o f H atoms i n  t o d i f f u s e t o the surface  surface.  lose  1981).  oxide  proceeds  non-  (phonons).  dependent.  The p r o c e s s by w h i c h Mu e v e n t u a l l y of  a  at a rate which i s ,  become t r a p p e d  i n s u l a t o r s h a s been o b s e r v e d  1960,  vibrations  temperature  t e m p e r a t u r e s , t h e Mu atom may  of  i s s m a l l e r ) , Mu must  i t d i f f u s e s through the l a t t i c e  general,  times  energy  2  2  under a l l the small  gas  in  i n the voids.  t h i s model i s p r o v i d e d  o f t h e e x p e r i m e n t a l d a t a on S i 0 , A 1 0  w i t h i n the  3  i n Chapter and MgO  the The VI  powders  59  in  a  He  atmosphere  demonstrated  at  6 °K. I t s f e a s i b i l i t y  i s qualitativey  i n Appendix I I I .  Once t h e Mu r e a c h e s t h e v o i d s w i t h an e n e r g y work  function  at  the  surface,  i tw i l l  by  scattering  a buffer gas, i f present. The  thermalization  of  an  atom i n a powder i s t r e a t e d i n  Appendix  I , using the 1-dimensional Devonshire theory  surface  interactions.  for  the  t h e r m a l i z e v i a phonon  e x c i t a t i o n during c o l l i s i o n s with the surface or off  l e s s than  1eV Mu  m /g,  P  2  parameters  In t h i s a p p r o x i m a t i o n , the time  (8600°K) t o r e a c h 35°K i n A l 0 2  =0.56  f o r gas-  g-cm" , 3  e  0  3  at  7°K  required  (SA  =  220  = 880°K a n d M o r s e s u r f a c e p o t e n t i a l  a =0.5 A ' , D=0) i s 40 n s , 1  with  most  of  the  time  b e i n g s p e n t b e l o w 300°.K. The  presence  the  interstices  In  the  o f a s m a l l amount o f m o n a t o m i c b u f f e r g a s i n  reduces the t h e r m a l i z a t i o n  c a s e o f s-wave s c a t t e r i n g ,  e n e r g y E; t o E.^ i s r o u g h l y ( M o b l e y  time  substantially.  t h e time r e q u i r e d  t o go from  1 966).  E q u a t i o n V«6 where m M «y n The torr  is is is is  the the the the  Mu mass mass o f t h e b u f f e r atom s wave c r o s s s e c t i o n (~10A ) number d e n s i t y o f t h e b u f f e r g a s . 2  t i m e r e q u i r e d t o go f r o m 8600°K t o 35°K i s o n l y ,,32 ns !  in 1  o f He. We  may  conclude  that  t h e i n f o r m a t i o n o b t a i n e d f r o m n*SB.  e x p e r i m e n t s on s u c h p o w d e r s w i l l  p e r t a i n t o t h e r m a l i z e d muonium.  60  V-3 Mu Bound S t a t e s on O x i d e  Surfaces  H atoms have been s t a b i l i z e d on s i l i c a b e l o w 120°K ( G o l u b e v r a n g e s from  1965).  and alumina  surfaces  The a c t i v a t i o n e n e r g y on t h e s u r f a c e  500°K t o 1500°K ( i n u n i t s o f k ) . T h i s  represents  a  rough e s t i m a t e of the b i n d i n g energy t o the s u r f a c e . From  the  H  atom  results,  b i n d i n g e n e r g y o f Mu t o interaction  a  similar  •1Q.X.  = Dl e  V(20 binding  (Morse,  surface.  The  surface-atom  i s assumed t o be r e p r e s e n t e d by a M o r s e -2qz:  The  i t i s p o s s i b l e t o estimate the  energy  of  2  Potential  -<a - < a z_> z - v  e  )  the deepest  Equation  V-7  bound s t a t e c a n be w r i t t e n  loc.c i t . )  °  2.'  2n  ru M  M  ,  U J J  1  where d = (2m,, D ) " / f i a a n d €  n 0  Equation  J  i s the corresponding  V-8  b i n d i n g energy  of t h e H atom. F o r e x a m p l e , i f CQ = 1000°K a n d a =0.5A, t h e n =  850°K.  may e x p e c t It  This  serves t o i l l u s t r a t e  t h e i s o t o p e d e p e n d e n c e one  b e t w e e n Mu a n d H bound on s u c h  i s o f some i n t e r e s t will  spend  surfaces.  t o c o n s i d e r the f r a c t i o n of  Mu  atom  300  °K, g i v e n t h a t t h e b i n d i n g e n e r g y i s o f o r d e r 850  in  time  A p p e n d i x I V . One f i n d s ,  °K.  This  thermodynamics  f o r e x a m p l e i n 70 A S i 0  3  time  on  the  surface.  I t i s clear  at a  2  powder d e n s i t y o f 0.04 g c m , t h a t t h e Mu s p e n d s o n l y 7.7 its  a  i n a bound s t a t e on t h e powder s u r f a c e a t  f r a c t i o n may be e s t i m a t e d by a p p l y i n g some s i m p l e given  to"  %  of  t h a t t o t r a p Mu on t h e  s u r f a c e o f s u c h powder f o r any a p p r e c i a b l e amount o f  time,  one  61  must  experiment  at  temperatures  much  less  than the b i n d i n g  energy.  V«4 M e c h a n i s m s f o r Mu S p i n The  mechanisms  f o r Mu  (discussed  i n general  is  to  bound  the  Relaxation spin  i n Section  surface  or  r e l a x a t i o n o f Mu a d s o r b e d on an basic  i n a Powder  relaxation  in  a  powder  IV«4) d e p e n d s on w h e t h e r t h e Mu colliding  freely with  unreactive  surface  i t . Spin  has  three  origins:  1.  Dipole-dipole i n t e r a c t i o n w i t h n u c l e a r moments o r d i s t a n t paramagnetic i m p u r i t i e s . This i s particularly effective when t h e Mu i s s t a t i c on t h e s u r f a c e .  2.  Random a n i s o t r o p i c d i s t o r t i o n o f t h e Mu atom due t o t h e atom-surface i n t e r a c t i o n . Again, this i s most e f f e c t i v e when t h e Mu i s s t a t i o n a r y on t h e s u r f a c e .  3.  Spin e x c h a n g e w i t h u n p a i r e d f r e e e l e c t r o n s on t h e s u r f a c e . T h i s , on t h e o t h e r h a n d , i s most e f f e c t i v e when t h e Mu i s m o b i l e on t h e s u r f a c e .  If  the  the  first  Mu i s c o l l i d i n g  narrowing. gas  two The  are  f r e e l y with the surface,  diminished  situation  s i n c e t h e powder g r a i n s  molecules.  considerably  t h e e f f e c t s of  due  to  motional  i s analogous t o spin r e l a x a t i o n i n a in  this  case  act  as  large  gas  Thus s p i n e x c h a n g e i s e x p e c t e d t o be d o m i n a n t i n t h e  case of desorbed  Mu.  62  V«4«1 N u c l e a r M a g n e t i c I f Mu magnetic  i s adsorbed moments  Moments on  a  surface  or  equivalently,  in  spin  nuclear  by a moment a t p o s i t i o n  the  precession  frequency  r e l a x a t i o n . The e x p e c t a t i o n  v a l u e o f t h e 2 component o f m a g n e t i c site  possesses  t h e d i p o l e - d i p o l e c o u p l i n g b e t w e e n t h e Mu a n d  t h e moments l e a d s t o a b r o a d e n i n g distribution  which  field  produced  at  the  f a n d z component o f s p i n I  z  Mu  c a n be  written  A^'>=  <Ixl  &JI,>=YxI~(l-3co*V) — •--  Equation  V-9  where B ^ i s g i v e n i n E q u a t i o n IV«33, © i s t h e a n g l e b e t w e e n f and the  2  axis  r  and  = ge/4m .c. The s h i f t  z  /1  i n t h e Mu p r e c e s s i o n  f r e q u e n c y a b o u t t h e 2 a x i s c a n t h e n be w r i t t e n  where  ~i  -  H  1-7  ITT•  in first  6  MH*  E q u a t i o n V- 1 0  H  provided  that  satisfied,  AB^  proper  one assumes  that  <<  B^|,' |. a  account on  contributes to A B ^  a  e<  If  this  surface  only  the  , t h e n t h e mean s q u a r e d  This  is  spins  (Abragam 1 9 6 1 ) .  condition  must be t a k e n o f A B ^  -3, the  order  r  &  a n d AB  nearest  dur = J J - r a t i ) V lb  R  not . If  neighbour  frequency s h i f t i s  E q u a t i o n V-1 1  s e c o n d moment due t o d i p o l a r b r o a d e n i n g I n a p o w d e r , e may  is  be a v e r a g e d  by  unlike  to obtain  xMu E q u a t i o n V» 12  63  or  15  —/ r  E q u a t i o n V-13  For  e x a m p l e , i n t h e c a s e o f Mu a d s o r b e d on A l 0  of  1.5A  2  f r o m an  2 7  A 1 n u c l e u s , 2[(AB^!/ ) ] 2  reasonably well with  the  stabilized  2  the  on  an A l 0  linewidth  surface  3  ESR l i n e w i d t h on S i 0  ESR  2  a random a n i s o t r o p i c d i s t o r t i o n , nuclear an  Al 0 2  surface  If  f-  leads  li^l  H  2% o f n a t u r a l  atoms  S i has  f o r Mu atoms f r o z e n on  2vy^y&^J  ?s - 1  =19  on  the  surface  of  a  lattice,  i n t e r a c t i o n b e t w e e n Mu a n d an i m p u r i t y  to a shift  4U>''~  of  agrees  Impurities  t h e Mu i s f r o z e n  dipole-dipole  = 3.9G. T h i s  1965). F o r comparison,  since only  i s estimated at  V'4'2 P a r a m a g n e t i c  distance  0.87 G, a n d i s p r o b a b l y due t o  moments. Thus t h e r e l a x a t i o n r a t e 3  a  ( 4 . 2 7 G)  (Golubev  i s only  at  3  i n the precession  the  at position  frequency.  (1-3^)11  E q u a t i o n v.,4 hi  If  there  a r e N i m p u r i t i e s , t h e n t h e t o t a l s h i f t u = £ Au' . I n a  statistical distribution  model (Anderson x(u)du  space, such that  -XM -  1951, Abragam  i s proportional  S£  1961),  to  the  the  frequency  volume of phase  u < E Au < o + d u .  £(V ^ w )  it  df  j  E q u a t i o n V'15  64  where  Equation and where V Following  is  the  half  space  associated  where B' = 11^j r-^ v (1 -  , _  the  solid.  (Abragam 1961)  where n' i s 2tr s t e r a d i a n s a s s o c i a t e d  =  with  V'16  Tr"*W,|fl  9  \T3  3cos 6)/r 2  a  _  with the s o l i d .  3  T r j J X j J  a r t /  v  Equat ion where n = d e n s i t y o f i m p u r i t i e s .  Define  V'17  65  iii  \ = JUL*  5  TV  I \uj  n  V*18  Equation S u b s t i t u t i n g E q u a t i o n V»17  ~ For  If with  the  Mu  be  where  impurity  3  in Al 0 2  m o v i n g on t h e s u r f a c e ,  t h e n t h e Mu may  i m p u r i t i e s . The  undergo  =  f 2  m V  spin  surface  freely  exchange  with  i n a f r e e Mu  state  Sex  c  -of  Equation  paramagnetic  s p i n exchange c r o s s d e p e n d i n g 'on  a t 6 °K w i t h  the  r a t e o f Mu  c  e>  impurity  section, spin  I V ' 4 ' 4 ) . F o r e x a m p l e , i n 70 A S i 0 3  Mu on a  or c o l l i d i n g  relaxation rate  i s the c o l l i s i o n frequency with  unity,  relaxes  written  concentration  g cm'  3  V-19  1  is  the s u r f a c e ,  )  the  Equation  X ~ 400>.s-  paramagnetic can  yields  _l  e x a m p l e , a 2% F e ~  at a rate  i n t o E q u a t i o n V«16  =9.2  A  s h o u l d be o f o r d e r  2  2  and  f  of  the  powder  a n d m=4 120  the surface,  m  is  the  e  is  on t h e s u r f a c e , a  constant impurity  V-20  of  cx  order  (see S e c t i o n  at a density  of  0.04  x 10" A- , the r e l a x a t i o n 3  «s" . 1  2  66  V«4'3 M o t i o n a l  are  Narrowing  In  the  no  fluctuations  experienced  preceding sections,  by  in  the  i t h a s been assumed t h a t  random  local  m o t i o n a l narrowing of the p r e c e s s i o n  magnetic  and  a reduction  field  fluctuating  in  the  i n spin  observed r e l a x a t i o n  will  } ~. ^  ?  is difficult  have on t h e h y p e r f i n e H on S i 0  2  of  2  (3.9  ef f e c t .  of  us' ) 1  a  the  I f the l o c a l Mu  by some r e l a x a t i o n  of f a s t  atom  is  time  ,  f l u c t u a t i o n s ) , then  the  E q u a t i o n V-21  c  Distortion  to  predict  what e f f e c t  the surface  will  i n t e r a c t i o n o f Mu. I f t h e ESR l i n e w i d t h o f  Mu on an S i 0  4.2 u s * ' . N o t e t h a t Si0  frame  rate.  i s due t o t h i s e f f e c t , t h e n one m i g h t assume t h a t  relaxation  to  be r e d u c e d t o  V'4'4 Random A n i s o t r o p i c It  relaxation  reference  << 1 ( t h e l i m i t  i  a n d t h i s may l e a d  frequency d i s t r i b u t i o n , o r ,  randomly, c h a r a c t e r i z e d  i f (Ao ) r 2  field  t h e Mu atom. However, a v e r y l i g h t atom s u c h a s  Mu may be e x t r e m e l y m o b i l e on a s u r f a c e ,  equivalently,  magnetic  there  2  s u r f a c e t o be o f o r d e r i r r  the relaxation (Brewer  rate  1981) i s  o f Mu t r a p p e d  thought  Mj<  the  0.87G =  in  fused  t o be due t o t h i s  67  V«5 The R e l a x a t i o n F u n c t i o n R ^ ( t ) f o r Mu i n a Powder M  The d e r i v a t i o n o f t h e f o r m o f t h e Ryj (t)  Mu  relaxation  i n a powder i s c o m p l e t e l y a n a l o g o u s  y  the l i f e t i m e Consider  spectrum  an  f o r o-Ps i n a powder  t o t h e d e r i v a t i o n of (see Section  F  a n d X. t o be t h e s p i n 5  form  of R  B.  T,  Define  r a t e s f o r Mu i n t h e f r e e and  states.  muon s p i n  i n Mu r e l a x e s w i t h t i m e ) d e p e n d s on how  unity,  The  relaxation  adsorbed  with  11 • 4 ) .  e n s e m b l e o f Mu atoms i n a powder a t t e m p e r a t u r e  where t h e r e e x i s t s a bound s t a t e w i t h b i n d i n g e n e r g y X  function  M l (  ( t ) ( w h i c h d e s c r i b e s how t h e \t e  compares  w h e r e , a s b e f o r e , t i s t h e mean s u r f a c e d w e l l  time  f o r Mu on t h e s u r f a c e ( s e e A p p e n d i x I V ) .  V«5«1 S p e c i a l C a s e X. t << &  1 (Adiabatic  Approximation)  As i n t h e c a s e o f P s , t h i s c o r r e s p o n d s t o a s i t u a t i o n t h e Mu a d s o r b s a n d d e s o r b s many t i m e s w h i l e s t i l l p o l a r i z a t i o n amplitude decays decay  rate  —  by t h e f r a c t i o n  the average  p o l a r i z e d . The  according to a single  exponential  o v e r f r e e a n d bound s t a t e s ,  of time spent  i n each  state.  where  weighted  Thus E q u a t i o n V«22  with \ft •= <* )l B  +"  where a i s t h e f r a c t i o n Equation AIV-15).  ^--"°0  E q u a t i o n V-23  of time spent  i n the  bound  state  (see  I t i s assumed t h a t t h e Mu atom h o p s many t i m e s  on t h e s u r f a c e b e f o r e d e s o r b i n g , s o t h a t t h e d e s o r b i n g - a d s o r b i n g process  does  not  significantly  contribute  to  any  n a r r o w i n g . C o m b i n i n g e q u a t i o n s AIV«15 a n d V-23 y i e l d s  motional  68  r  As  BAT  +  1  B7*T  E q u a t i o n V«24 In  the l i m i t of  << 1 t h i s  reduces t o E q u a t i o n V>25  V-5'2  S p e c i a l Case X. t >> 1 ( S t r o n g C o l l i s i o n  The d e r i v a t i o n II*4»2.  Approximation)  g  The  i s identical  resulting  Mu  to that  relaxation  contained function  d i f f e r e n c e of exponentials.  in is a  Section sum o r ,.  ^ .  E q u a t i o n V»26 The l i m i t i n g  f-orms a r e a s b e f o r e E q u a t i o n V«27  and  - Apt  J.  .  i  E q u a t i o n V« 28  69  V«6 E f f e c t  o f A d s o r b e d I n e r t Gas on t h e S p i n R e l a x a t i o n  I f an i n e r t g a s s u c h a s He vessel,  a fraction  or  Ne  of t h e gas a d s o r b s  decreasing the c o l l i s i o n frequency  i s admitted on t h e b a r e  of  the  Mu  into  the  s u r f a c e , thus  with  the  bare  surface according to V t {P) [ I ~  =  ^0,1  E q u a t i o n V-29 Mu  where  n, i s t h e d e n s i t y o f a d s o r b e d  cross section available  atoms and e  f o r Mu s c a t t e r i n g o f f an a d s o r b e d  s u r f a c e area  i s the e l a s t i c  fl  atom.  Also  the  f o r adsorption decreases t o  f\ ir\) - flto') I  4 " ^'7  I"  Equation  E q u a t i o n V'23 must t h e n be r e w r i t t e n  V-30  with  Vf  ^  E q u a t i o n V'31  B/kT  valid  when  X.gt << 1 a n d  «  1. S i m i l a r l y ,  Equations  V'27 a n d V'28 must be r e w r i t t e n  Equation valid  when x ^ t >>  and  1, v ( n , ) P 0  f  <<  \\  B  -  V-32  k\ F  . ,  - Apt E q u a t i o n V'33  valid  when k t g  »  1, ^ ( n , ) ^  >>  | x,  - X |. p  70  CHAPTER VI : LOW TEMPERATURE STUDY OF MUONIUM I N A l 0 , S i 0 2  2  AND  object  of  3  MgO POWDERS  The  s i m p l i c i t y o f t h e H atom make i t an  fundamental chemistry  atom-surface  and p h y s i c s  industrial  interaction  o f H atoms  applications,  of  past  this  liquid, study  H,  the  s t u d i e s . In a d d i t i o n , the solid  a light  interest  o f Mu i n t e r a c t i n g w i t h s o l i d study  consists  muons have been i n j e c t e d i n t o h i g h powders.  In  the  first  have  of c a t a l y s i s . 1/9  the  i n H n a t u r a l l y e x t e n d s t o Mu. I n t h e  i n t e r e s t h a s been c o n f i n e d  present  may  i s o t o p e o f H, h a v i n g  to  studies  and s o l i d phases. In t h i s chapter  The  surfaces  e s p e c i a l l y i n the f i e l d  S i n c e Mu c a n be c o n s i d e r e d mass  on  ideal  the f i r s t  surfaces of  in  the gas,  experimental  i s presented.  two e x p e r i m e n t s i n w h i c h  specific  surface  area  oxide  e x p e r i m e n t , t h e f r a c t i o n o f muons w h i c h  emerge i n t o t h e v o i d s a s Mu a t an a m b i e n t t e m p e r a t u r e o f 6°K was measured. T h i s o b v i o u s l y has important the  feasibility  represents  of  studying  Mu  a t e s t of the thermal  consequences i n regard t o  surface  d i f f u s i o n model  s i n c e , a t low t e m p e r a t u r e , t h e d i f f u s i o n expected  to  example, i n stabilized  be s m a l l bulk  at  a  phonomena.  single  quartz, site  the  length before  below  Mu  Mu  with  r Al 0 -  2  3  (Knozinger  o f a d s o r b e d He a n d Ne.  decay  50°K ( B r e w e r  is  s i z e . For  i s believed  s e c o n d e x p e r i m e n t was u n d e r t a k e n t o e x a m i n e t h e  also  (Section V»2),  i n comparison with the p a r t i c l e  fused  It  to  be  1 9 8 1 ) . The  interaction  1978) s u r f a c e s w i t h v a r y i n g  of  amounts  71  VI•1  Mu i n t h e V o i d s o f O x i d e P o w d e r s a t 6°K  VI • 1 • 1 E x p e r i m e n t a l The  n*SR  Details  a p p a r a t u s i s show i n F i g u r e  B2  VI•1.  A beam o f s p i n  COUNTER CARBON DEGRADER  B I COUNTER  D COUNTER 0 . 0 0 5 " MYLAR  Fl  VACUUM TARGET CRY0STAT ASSEMBLY  COUNTER CARBON DEGRADER  F2 COUNTER  cm 20  10  F i g u r e VI--1 The v*SR a p p a r a t u s " B e a v e r " . N o t e that p o s i t r o n t e l e s c o p e s a r e a l o n g t h e beam d i r e c t i o n .  polarized p o s i t i v e "surface  muons" o f momentum  M13 ir-x c h a n n e l a t TRIUMF ( / / e * r a t i o ~ 3/4  inch  diameter spot,  d e f i n i n g counter The  dE/dx•  MeV/c  f o r 28  positrons,  discriminated typically Figure (29.5  from  MeV/c so  positrons.  V I • 2 . The M y l a r windows 2  helium  a He  surface  10,000 n*/s. The t a r g e t  mg/cm ),  l ) was c o l l i m a t e d  gas  gas  muons  flow  can  to.  (0.010  a  inch)  cryostat.  muons i s r o u g h l y 6 t i m e s t h a t  that  beam  28 MeV/c f r o m t h e  and passed t h r o u g h a t h i n  (D), before entering  the  f o r 28  easily  be  The i n c i d e n t muon r a t e was  cryostat  assembly  (35 mg/cm ), 2  (16 mg/cm ) 2  and  the  i s shown thin  aluminized  in  counter Mylar  72  COLD He GAS  TARGET SUPPORT STYROFOAM VESSE GLASS WOOL .005" MYLAR 29MeV/c  Ge THERMOMETER  JU.V" POWDER TARGET  .0005 ALUMINIZED  LHe SUPPLY  MYLAR LHe VAPOURIZER  5  o •  i _  cm  Figure VI•2 The t a r g e t - c r y o s t a t a s s e m b l y u s e d t o s t u d y Mu i n o x i d e p o w d e r s .  windows ( 3 . 5 mg/cm ) sum t o a t o t a l 2  of  r a n g e o f a 28 MeV/c s u r f a c e muon i s o n l y 56  mg/cm  of .target  2  material  84  mg/cm .  Since  2  140 mg/cm  2  the  (of Carbon),  are s u f f i c i e n t t o stop a l l the  muons. P o s i t r o n s f r o m muon d e c a y were d e t e c t e d positioned were  i n two  upstream and downstream o f t h e t a r g e t . Helmholz c o i l s  used  t o apply a magnetic  polarization  field  p e r p e n d i c u l a r t o t h e muon  direction.  T e m p e r a t u r e m e a s u r e m e n t s were made w i t h a germanium m°K  at  resistor  w i t h an a b s o l u t e a c c u r a c y  a l l temperatures.  controller  telescopes  maintained  A  very  careful  t h e temperature  CryoCal  CR2500H  o f b e t t e r t h a n 30 human  temperature  t o w i t h i n ±250 m°K. A l l  73  m e a s u r e m e n t s were p e r f o r m e d atmosphere at a pressure  VI •1•2  and  i t s d e c a y . At  is straightforward,  muon  in  satisfies  the  1.  held  in  a  He  torr.  e l e c t r o n i c s , designed  t i m e i n t e r v a l b e t w e e n a s i n g l e muon e n t e r i n g low but  taken to minimize d i s t o r t i o n one  powders  VI«3 shows a s c h e m a t i c o f t h e  t o measure the  task  760  the  Electronics  Figure  target  of  with  the  target  following  i n c i d e n t muon r a t e s at higher  the  1/T^ ) t h i s  rates precautions  e f f e c t s due at  (<<  to  same  pileup  the  must  (more  be  than  t i m e ) . A "good" e v e n t  conditions.  A muon a r r i v e s a t t=0 with the microprogrammable branch driver (MBD) not busy and w i t h no n* h a v i n g e n t e r e d the t a r g e t i n t h e t i m e i n t e r v a l -P < t < 0 (P i s the pileup gate l e n g t h ; -~l0»/s). T h i s condition s t a r t s the time digitizer.  2. A p o s i t r o n e v e n t d e f i n e d by B1-B2 o r F1«F2 o c c u r s a t t = r (T < P) w i t h no s e c o n d muon h a v i n g e n t e r e d t h e t a r g e t i n t h e t i m e i n t e r v a l 0 < t < T. T h i s s t o p s t h e c l o c k . 3. No s e c o n d muon o r s e c o n d e l e c t r o n i n t e r v a l r < t < P. A h i s t o g r a m of The  >» SR +  c o m p u t e r and  event  relatively appropriate  data  i s t e r m e d a f*SR  is  (MBD-11). The only  f a s t MBD  in  the  time  spectrum.  a c q u i s i t i o n s y s t e m c o n s i s t s of a PDP-11/40  a CAMAC i n t e r f a c e , d r i v e n  branch d r i v e r an  these time delays  i s detected  20ns. to  histogram bin  by  a  microprogrammable  dead time a s s o c i a t e d  with  This  is  made p o s s i b l e by  process  an  event  i n the  and  PDP-11/40 memory.  processing using  increment  the the  74  Figure VI•3 * S R e l e c t r o n i c s t h a t a r e u s e d t o measure t h e t i m e i n t e r v a l b e t w e e n an i n c i d e n t muon and i t s decay p o s i t r o n (Garner 1981). +  75  VI •1•3 A n a l y s i s a n d R e s u l t s n*SR  spectra  were t a k e n  i n a transverse magnetic  8G i n o r d e r  to evaluate  t h e Mu  relaxation  rate.  data  derived  The  precession  were f i t t e d  f i e l d of  amplitude  and i t s  t o t h e f u n c t i o n a l form  i n S e c t i o n IV«5  Equat ion VI•1 with  S ( t ] = fh^M R ^ f t ) cost^t  +  HM  1 where y  a n d 0^  precession relative  t o t h e muon rates  so t h e f i t s  were  5 +  )  a high  by  to  In r e a l i t y , mobility  of  at  whether  nuclear  R (t)  parameters  for static  the  paramagnetic  situation.  , which has  better  fits  were  muons  interacting (Pake  impurities  and  However, s i n c e t h i s  1948). muon study  t a r g e t s , i t s u f f i c e s t o say that the A^  M  and  w i t h t h e f r e e muon r e l a x a t i o n f u n c t i o n f o r m y i e l d e d good f i t s  3  gaussian  f o r m w i t h p *~ 0.2 <<s"' . I n t h e o r y , one  the presence of  on Mu i n t h e s e  muon 1  was  / 1  moments,  Mu  (-0.05 y s " ) ,  2  of  and  t = 0 ) . The  a n d MgO  2  VI•2  the telescope  n e i g h b o r d i p o l e s t o be g a u s s i a n  complicate  precession  vector  muon  (e"*^" ) . I n t h e c a s e o f A l 0  the relaxation function nearest  orientation  small f o rS i 0  insensitive  with a gaussian  only  focusses  the  were v e r y  concentration  expects  phases f o r f r e e  polarization  or exponential  obtained  with  are the i n i t i a l  (determined  relaxation  (e~  Equation  i n a l l cases.  Mu  a r e only weakly c o r r e l a t e d R (t),  and  a  gaussian  76  The  general  form o f t h e r e l a x a t i o n  function  i n t h e v o i d s o f a powder i s t o o c o m p l i c a t e d However, t h e l i m i t i n g difference  of  forms c o n s i s t  exponentials  be  added.  In  the present  i n t h e c a s e o f 140A S i 0  powder  0 10 -  i  0 05 -  I f a f r a c t i o n of the  good f i t s were  "i  must  obtained  function  except  (see Figure  1 1r  1  V«5).  experiment,  relaxation  0 .15 r-  fitting.  an a d d i t i o n a l component  with a single exponential 2  t o use f o r  f o r Mu  of a s i n g l e e x p o n e n t i a l or a  (Section  Mu r e m a i n s i n s i d e t h e p o w d e r , t h e n  R^(t)  IV'4).  In  this  r  A A A A A rt  0 00 -0 05 -0 10 I_L -0 15 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2  TIME IN uSEC (16 NSEC/BIN)  F i g u r e VI•4 The Mu p r e c e s s i o n s i g n a l S (t)(defined i n Equation V I . 2 ) f o r 140A S i 0 powder i n a He a t m o s p h e r e a t 6°K. 2  particular  case,  a exp(-X.,t) + ( 1-a) the  per  a  sum  of  exponentials  e x p ( - X . t ) ] gave a s u b s t a n t i a l 2  degree of freedom  [R^(t)=  improvement  in  ( 3 1 5 f o r 301 d e g r e e s o f f r e e d o m ,  c o m p a r e d w i t h 401 f o r 303 d e g r e e s o f f r e e d o m ) .  77  A  Mn  The  Mu f r a c t i o n s were e v a l u a t e d  bY  normalizing  to  the  f r e e muon a m p l i t u d e  assumed t h a t a l l t h e muons i n j e c t e d muon  Larmor  frequency  from t h e f i t t e d  B*13.55  i n A l . I t was  into A l precess  KHz G"  parameters  at the  (B i n G a u s s ) .  1  free  The f r e e  muon p r e c e s s i o n b a c k g r o u n d due t o muons s t o p p i n g i n p a r t s o f t h e target v e s s e l other an  than  antiferromagnetic  experience  very  contribute  to  t h e powder i t s e l f was Fe 0 2  powder  3  determined  target.  l a r g e l o c a l magnetic  fields  Muons and  t h e f r e e muon L a r m o r f r e q u e n c y .  with  in  thus  The Mu  do  Fe 0 2  3  not  fractions  were e v a l u a t e d a s  (swp\c)_  F„ = M  Equation The  values  packed  f o r A^ i n A l a n d F e 0 2  data  taken  at  3  were  45G, f i t t e d  obtained  from  t o Equations  VI•3  coarsely  VI•1 and VI•2  with Srt (t)=0. M  T a b l e V I «-l g i v e s t h e measured  at  temperature results  6°K  results  f o r both  temperature  in  Mu  760  relaxation  torr  and  the  fractions  room  t h e low  temperature  a n d p o w d e r e d s a m p l e s a r e i n c l u d e d . The  d e p e n d e n c e o f t h e Mu r e l a x a t i o n  powder i s shown i n F i g u r e  and  o f He. F o r c o m p a r i s o n ,  f o r bulk oxides bulk  rates  VI-5.  rate  in  the  Al 0 2  3  78  TABLE V I • 1 . »/ SR r e s u l t s +  Target  Temperature (K)  S i O i bulk fused SiO-z b u l k f u s e d SiOt-^powder (70 A) SiOi powder (70 A) S i O a powder (140 A°) SiOi (140  powder A)  6 295 6 295 6  i n b u l k and powdered o x i d e s .  Mu F r a c t i o n (%)  Mu R e l a x a t i o n Raters" ) 1  7 9 * 3*'* b , 79 t 3 3  3.3 i 0.5 0.2010.05  49t 3  0.46t0.03  61 ± 3  0.18± 0.03  d  35 t 5 0 ) * 3 5 * 5 (2)  '  b  4 . 1 * 0.7 (1) 0.16*0.05 (2)  295  *iSt 20  0.I8i0.03  AljOa bulk fused Al O-» b u l k f u s e d A l t O * powder (75 A) A l - j O , powder (75 A)  6 295  >80 ' >80'^  >20 >20  29 t 3  0.35t0.05  295  35 t 14^  11.3 £ 4.4  MgO MgO MgO (300 MgO (300  6 295  35 t 10 \ 35110  6.3*1.4 2..0 t 0.5  6  12 i 3  0.22i.0.03  295  15 t 3  1 .9 ± 0.5  z  s i n g l e XL s i n g l e XL powder A) powder A)  6  c  b l  a The ( 1 ) a n d (2) r e f e r t o t h e two c o m p o n e n t s resolved i n the f i t . b Rough e s t i m a t e u s i n g Mu a s y m m e t r i e s o n l y . c E s i t m a t e b a s e d on t h e m i s s i n g f r a c t i o n p r e s u m i n g i t t o be Mu J ( K i e f l 1979|) e ( S p e n c e r 1981) f (Marshall 1978) ^ (Brewer 1 9 8 1 )  79  6.00  5.00 h  o  LU  4.00  CO  LU 3.00 (X  or  cr x  2.00 h  <X  ox 1.00 h  0.00  6  8  10  12  14  16  TEMPERRTURE (K) Figure VI •5 The t e m p e r a t u r e d e p e n d e n c e o f t h e Mu s p i n r e l a x a t i o n r a t e i n A 1 0 powder i n a He a t m o s p h e r e . The arrows i n d i c a t e points that a r eo f f s c a l e . 2  VI - I .4.  Discussion  VI • 1•4 • 1 Mu i n S i 0 It which  3  2  Powder a t 6°K i n a He A t m o s p h e r e  i s worth while theSi0  t o point  powder t a r g e t s  2  out thevarying  i n T a b l e V I . 1 were s t u d i e d .  room t e m p e r a t u r e 70A a n d 140A t a r g e t s torr,  which e f f e c t i v e l y  conditions  were  evacuated  removes a d s o r b e d w a t e r  under . The  t o 10"  6  ( s e e C a b o t ) . The  70A t a r g e t a t 6°K was a d m i t t e d i n t o a He g a s f l o w c r y o s t a t  i n an  open t o p v e s s e l w i t h o u t e v a c u a t i o n .  water  p l u s whatever  0  2  and N  2  Thus i t h a d a d s o r b e d  was a d s o r b e d d u r i n g  cooling  (4 x 1 0 " 0 3  2  80  atoms p e r A ,  1.6 x 1 0 "  2  the  target  vessel  evacuated t o had  10~  1  2  was  N  per A , assuming t h a t a l l the a i r i n 2  2  adsorbed).  t o r r a t 270°K b e f o r e  adsorbed water but l i t t l e  made i n a d e n s e  atmosphere  corresponds to a density In  regard  The 140A t a r g e t a t 6°K was  0  f u r t h e r c o o l i n g and t h u s  or N . A l s o ,  2  t h e c o l d r u n s were  2  of  He  of ~0.01  (760  torr  at  6°K  3  t o t h e r e s u l t s t a k e n i n a He a t m o s p h e r e t h e r e i s i n He g a s ( F l e m i n g  Mu  from t h e powders. A l t h o u g h t h e r e  originates  1981), so t h a t  to c o n c l u s i v e l y state that  i n Mu f o r m a t i o n , Table  He  g/cm ).  no Mu f o r m a t i o n  information  of  VI.1)  indication  that  Mu  observed  i s insufficient  the surfaces  t h e l a r g e Mu f r a c t i o n i n is  the  the  play  bulk  formation  no r o l e  oxides(see is a  bulk  phenomenon. I n some c a s e s t h e o b s e r v e d Mu f r a c t i o n i n t h e powder i s l e s s than i n the bulk. of  a  Mu  relaxation of  component i s very  T h i s may p a r t l y be e x p l a i n e d  trapped  to form a diamagnetic  surface  with  surface  i s that  component  was  observed  cs"', r e s p e c t i v e l y ) . This the and  r e l a x i n g component  which  has t h e r m a l i z e d  0.5  i n the S i 0  i n the voids.  2  according  to  relaxing  a n d 0.157 ± 0.05  i n fused  Si0  2  Mu  This  inside  below  50°K  K S ~ , attributable 1  IV«4«2). T h e r e f o r e ,  powder must be due  ambient temperature thermal d i f f u s i o n  since,  1  slowly  component c a n n o t be due t o  distortion(Section  slowly  the  ( 0 . 4 6 ± 0.03 (is"  h a s a r e l a x a t i o n r a t e o f 3.3 ± anisotropic  portion  s t a t e s u c h a s (OMu)".  powder g r a i n s , s i n c e Mu i s s t a t i c  random  a  g r o u p s s u c h a s (OH)"  I n b o t h t h e 70A powder a n d 140A powder, a Mu  terms  i n s i d e t h e powder g r a i n s where t h e  f a s t . Another p o s s i b i l i t y  t h e Mu r e a c t s e p i t h e r m a l l y  in  to the  to  Mu  i s i n contradiction to model  (Section  V-2)  t h i s m o d e l , no Mu w o u l d e s c a p e t h e powder  81  g r a i n s a t low t e m p e r a t u r e where t h e r e In f a c t , a second the  140A powder  is little  f a s t r e l a x i n g component (4.1 ± 0.7  o r no  (see F i g u r e  c s " ) i s most l i k e l y  VI-4)  due t o Mu  1  i s trapped  i n s i d e t h e powder g r a i n s .  component  was n o t o b s e r v e d i n t h e 70A powder i s p r o b a b l y  the  smaller p a r t i c l e  surface  could  (essentially in  size.  rapidly  Also  relax  the Mu  presence  within  that  of  15A  t h e e n t i r e powder g r a i n ) . Such 0  may  seem  surprising  0  which  such  a  due t o on  2  the  of the s u r f a c e  was  2  at f i r s t that 0  d o e s n o t have a more p r o n o u n c e d  not  present  the  extragranular  would  lead  to  a  on t h e  surfaces  region.  According  is  at t h i s likely  to Equation  of  0  on  2  known  that  temperature that  He  (6°K)  a He f i l m  readily  of  Mu  V-20  one  the  Mu s p i n r e l a x a t i o n r a t e o f r o u g h l y  However, i t i s w e l l surfaces  2  e f f e c t on t h e r e l a x a t i o n  would expect t h a t such a c o n c e n t r a t i o n  It  reason  in  t h e 140A powder. It  in  The  diffusion.  adsorbs  surface 120 * * s ~ . 1  on  (See f o r e x a m p l e s D a s h  s h i e l d s t h e Mu  e f f e c t s o f t h e s u r f a c e . More e v i d e n c e f o r  such 1975).  from d e p o l a r i z i n g  this  will  be  given  shortly.  V I ' 1 - 4 - 2 Mu This  i n MgO  sample  Powder a t 6°K was  s t u d i e d under  70A S i 0 ,  i n an o p e n - t o p  0  on t h e s u r f a c e s .  2  2  and N  2  A s i n g l e long agrees  well  relaxation  with  non-evacuated  t h e same c o n d i t i o n s a s t h e v e s s e l , a n d t h u s had  l i v e d component was r e s o l v e d whose the  room  temperature  value.  r a t e was s u b s t a n t i a l l y l e s s a t 6°K  c o m p a r e d w i t h t h a t a t 300°K ( 1 . 9 ± 0.5  xs  - 1  ).  H 0, 2  amplitude  However, t h e  (0.22 ± 0.03 Considering  «s*') the  82  low  purity  of  that  t h i s component  i s due t o Mu i n s i d e t h e powder g r a i n s .  the  case  Si0 ,  of  extragranular  t h e MgO powder  the  shielded  V I - 1 - 4 - 3 Mu i n A l 0 2  This per  A  2  target  on  the  Powder  3  also  a n d 1.2 x 1 0 ~  adsorbed  i t i s attributed  2  region  N  3  (see Table V I - 2 ) ,  per A  2  Powder  Density (g/cc)  2  assuming that  surfaces).  c o n t a i n e d an 1.8% p a r a m a g n e t i c F e *  Table VI-2. Properties  Mu  3  In  and N all  addition,  (mVg)  Impurities  Paricle Size(dia.) (A)  Si0 C a b o t M5  0.04  200  140  same a s above  Al O> 0.56 D a v i son SMR-7-7563  225  75  Fe(1.8%)from SOy (.2%) Si0 (.08%) Na„0 (.03%) Cl* (<.01%)  0.12  t h e sample  of oxide powders.  Surface Area  N a ( 2 0 - 4 0 ppm) P (<300 ppm) A l l other element l e s s t h a n 30 ppm ( s e e C a b o t )  (Fe^O,)  2  not available  300  2  a i r was  impurity.(see Table VI.2)  70  MgO Matheson Coleman Bell MX 65-05  the  film.  the  400  z  in  (3 x 10"" 0  2  0.04 SiOi C a b o t EH5  2  As i n  (5°K - 20°K)  2  powder  to  f r o m t h e s u r f a c e by a He  had a d s o r b e d H 0 , 0 2  i t i s unlikely  Na (.5%) C1(.01%) Ca (.05%) Ba(.005%) SO* (.02%) K (.005%) NH„(OH) ( . 0 2 % ) S r ( . 0 0 5 % ) In (.01%) Heavy M e t a l s ( . 0 0 3 % ) Mn(5ppm)  83  The o b s e r v e d l o n g - l i v e d component not  be  the  due  surface,  rate  t o Mu  0  2  with  should  2  the  Al 0  relax  directly  relaxation  i s a t t r i b u t e d t o t h e p r e s e n c e of a  Mu  at  on a  2  3  extragranular  surface. Again, He  Mu  the small  film  on  the  surface. Convincing  Figure  VI•5,  e v i d e n c e f o r t h i s He f i l m h y p o t h e s i s  i s shown i n  which d i s p l a y s the temperature dependence of X i n  A1 0 . The s h a r p i n c r e a s e to  must  400 i i S " ' ( s e e S e c t i o n V « 4 ' 2 ) . As i n t h e c a s e o f  colliding  2  )  1  i m p u r i t y would r e l a x such  3  70A S i 0 , t h e a d s o r b e d  oxide  »/S~  i n t h e powder g r a i n s o r a d s o r b e d d i r e c t l y  since the F e "  of r o u g h l y  the  ( 0 . 3 5 ± 0.05  i n X. a b o v e 12.5 ± 0.5°K i s a t t r i b u t e d  3  vacancies  in  the  film  as  the  first  monolayer begins t o  e v a p o r a t e . A more d e t a i l e d e x a m i n a t i o n o f t h i s phenomenon i s t h e subject  of the f o l l o w i n g experiment.  V I • 1 • 5 Summary and A slowly observed  in  Conclusion  relaxing Al 0 , 2  Mu  component  S i 0 , a n d MgO  3  2  oxide  an  relaxing  additional  attributed increase  fast  t o Mu t r a p p e d in  Mu  surfaces.  I n t h e 140 A  component  atoms  regardless the  oxide  emerge  has  Si0  colliding 2  i n s i d e t h e powder g r a i n s . The 2  3  the void regions  powder  dramatic  film.  of these oxide  of  i n h i b i t s Mu s p i n r e l a x a t i o n .  the  powders  o f t e m p e r a t u r e a n d t h a t t h e p r e s e n c e o f a He f i l m surfaces  and  a b o v e 12.5 °K i s  i t appears that a s i z e a b l e f r a c t i o n into  been  was a l s o r e s o l v e d  t h o u g h t t o be due t o e v a p o r i z a t i o n o f t h e He  Mu  1  t h e powder g r a i n s  spin relaxation rate i n A l 0  In c o n c l u s i o n ,  vs~ )  p o w d e r s i n a He a t m o s p h e r e a t  6 °K. I t i s a t t r i b u t e d t o Mu o u t s i d e f r e e l y w i t h He c o a t e d  (X~0.2  on  84  VI-2  S p i n R e l a x a t i o n o f Mu i n A l 0 2  In  this  section  further  Powder w i t h A d s o r b e d He/Ne  3  evidence  i s presented  s u b s t a n t i a t e s t h e above c o n c l u s i o n a t l e a s t p o w d e r . The Mu s p i n r e l a x a t i o n function  of  adsorbed  gas  rate  per  has  unit  t e m p e r a t u r e . B o t h He a n d Ne g a s e s were I s o t h e r m s . were and  measured  i n t h e case of A 1 0 2  been  measured  surface area used  as  and  higher  a  an  adsorbate.  a t 7.3°K a n d 10.4°K f o r He and 28.7°K  f u n c t i o n of a d s o r b e d gas below  is virtually  as  3  at constant  30.3°K f o r Ne. The s p i n r e l a x a t i o n r a t e i s a s t e e p ,  decreasing  which  monolayer  linearly completion  i n d e p e n d e n t o f t h e amount o f a d s o r b e d g a s a t  coverages.  VI «2-1 E x p e r i m e n t a l The  Details  TRIUMF/Lawrence B e r k e l y L a b o r a t o r y (see F i g u r e  (LBL)  "Eagle"  experiment.  The main d i f f e r e n c e b e t w e e n t h i s a p p a r a t u s and t h a t  i n S e c t i o n VI•1  positron  telescopes,  transverse  field  scattering The filter  TRIUMF  B  »*SR  are  since  v~n  additional in  they  general are  in  this  left  and  more  suited for  less  channel  right  sensitive  to  e q u i p p e d w i t h a 3m l o n g Wien  s e l e c t o r , o r "DC s e p a r a t o r " , G y,  undetectable)  effect  polarization  M9  48  contamination added  which  the  used  o f beam p o s i t r o n s .  velocity  kV/cm x,  is  was  muon  apparatus,  described  VI«6),  surface  of  the  was of  used 28  set  at  E  =  3.9  t o o b t a i n a c l e a n beam ( e MeV/c  separator  by a s m a l l v e r t i c a l a n g l e  was  "surface to  rotate  (9°) r e l a t i v e  muons". the  +  An muon  t o t h e beam  d i r e c t i o n . Muons e n t e r i n g t h e t a r g e t r e g i o n were c o l l i m a t e d t o a  85  r  FI  counter Hucite observation window  carbon degrader  R2 counter RI counter  acuum  cryostat  M diameter collimator B2  counter  0 0 3 mylar window  29 MeV/c / i  Figure VI•6 The M positron telescopes.  3/4  +  SR  +  apparatus "Eagle".  inch diameter spot before d e t e c t i o n  scintillator. The  He  constructed inch thick  Typical leak  by a t h i n  i n c i d e n t muon r a t e s were  tested  target  vessel  from s t a i n l e s s s t e e l w i t h  Note  the  (0.010  four  inch)  30,000/s.  (see F i g u r e  VI'7)  two 1 i n c h d i a m e t e r  was  0.001  s t a i n l e s s s t e e l windows.  The A 1 0 2  3  powder s a m p l e ( w e i g h i n g  10.5g) was b a k e d a t 500°C  86  COLD He G A S  He (Ne) ATMOSPHERE  G LAS S WOOL ft* stopping region • 0 0 5 " MYLAR •21. M e V i i c  Ge  THERMOMETER  +  O.OOl" stainless steel WINDOW  .0005 ALUMINIZED  Al 0 2  MYLAR  POWDER  3  TARGET (I0.5g)  STYROFOAM SPACER cm  Figure VI«7 The powder t a r g e t v e s s e l a n d c r y o s t a t u s e d t o s t u d y Mu r e l a x a t i o n v e r s u s He/Ne c o v e r a g e . Note t h e target vessel i s isolated f r o m t h e He a t m o s p h e r e o f t h e cryostat.  for  24 h o u r s i n a i r and c o o l e d  placed  in  hydroxyl likely  the. target  evacuated  to  dessicator  100°C, s o t h a t  by a l a y e r o f h y d r o x y l  10"  5  before 2  groups w i t h h e a t e d above  terminated  a  being  v e s s e l . A d s o r b e d w a t e r on r - A l 0  . The v e s s e l was t h e n s o l d e r e d  prior  in  torr  groups  t o t h e gas  for a period  3  forms  the surface (Knozinger  handling  197 6)-  system  o f 24 h o u r s  was  and  immediately  t o the experiment. A  CryoCal  2500L  germanium  resistor  with  an  absolute  a c c u r a c y of b e t t e r  t h a n ± 30 m°K a t a l l t e m p e r a t u r e s s t u d i e d  used  t h e t e m p e r a t u r e a n d c o n t r o l i t t o w i t h i n ± 60  m°K.  to  monitor  was  87  The a l l - m e t a l apparatus  (stainless  steel  and  ( F i g u r e VI«8) was composed  copper)  primarily  gas from  handling 1/4  inch  VI G  A  S  = S  INLET  STANDARD VOLUME  TO PUMPING STATION  CRYOSTAT INSIDE WALL-  TARGET VESSEL-  Figure VI«8 The g a s h a n d l i n g system used t o d e p o s i t c o n t r o l l e d amounts o f He/Ne on r - A l 0 powder. 2  tubing,  Swagelock f i t t i n g s  a n d N u p r o b e l l o w s v a l v e s . The v a p o u r  p r e s s u r e measurements i n t h e t a r g e t W a l l a c e a n d T i e r n a n p r e s s u r e gauge ±0.5  torr)(Gl), a  vessel  were  made  with  a  (0-800 t o r r ) ( G 2 ) , a c c u r a t e t o  t o r r . The p r e s s u r e d i f f e r e n c e m e a s u r e m e n t s were made w i t h a  Matheson  10"  3  6301  stainless  a b s o l u t e p r e s s u r e gauge ( 0 - 7 6 0  a c c u r a t e t o ± 2 t o r r . The s y s t e m was  t o r r a n d He l e a k  purity  steel  tested  prior  to  the  g r a d e "He ( 9 9 . 9 9 5 % ) a n d p u r i f i e d g r a d e  used as a d s o r b a t e s .  evacuated experiment.  2 0  Ne  to High  ( 9 9 . 9 9 % ) were  88  VI«2«2 E l e c t r o n i c s The  electronics  minor m o d i f i c a t i o n  are  to  as  allow  described  in Section  acquisition  i n s t e a d of t h e two u s e d i n t h e f i r s t  of  VI•1 w i t h a  four  histograms  experiment.  VI•2•3 Procedure The  experimental  procedure  consisted  of  c o n t r o l l e d amount o f a d s o r b a t e i n t o t h e t a r g e t constant  temperature,  recording  the  c o l l e c t i n g a »i SR s p e c t r u m . The p r e c i s e (with  the  system  initially  vessel  vapour  steps  +  admitting held  at  pressure,  were  as  a  and  follows  u n d e r vacuum and a l l v a l v e s  closed  e x c e p t V4, w h i c h was open f o r t h e e n t i r e e x p e r i m e n t ) . 1.  V1 was o p e n e d t o p r e s s u r i z e t h e s t a n d a r d c c ) bounded by V 1 , V2, and V 3 .  2.  The p r e s s u r e  3.  V2 was opened ( 80 c c a t STP) V2.  4.  The p r e s s u r e on G1 was r e c o r d e d a g a i n so t h a t t h e amount o f gas a d m i t t e d i n t o t h e t a r g e t v o l u m e c o u l d be c a l c u l a t e d .  5.  The pressure and temperature i n t h e t a r g e t v e s s e l were s t a b i l i z e d o v e r a 30 m i n u t e p e r i o d and t h e v a p o u r pressure recorded (G2).  6.  An i*SR spectrum was' t a k e n ( 4 p e r i o d o f a b o u t one o r two h o u r s .  7.  The p r e s s u r e i n t h e t a r g e t v e s s e l , G2, was r e c o r d e d again. The d i f f e r e n c e f r o m b e f o r e a n d a f t e r t h e r u n was t y p i c a l l y 1 o r 2 t o r r , i n d i c a t i n g t h a t t h e s y s t e m was v e r y close to thermal e q u i l i b r i u m .  8.  S t e p s 3 t h r o u g h 7 were r e p e a t e d u n t i l more t h a n a m o n o l a y e r o f gas was a d s o r b e d .  on gauge G1 was  volume  a n d c l o s e d t o a d m i t a s m a l l amount o f gas i n t o the t a r g e t v e s s e l volume bounded by  t  aluminum  of  20  recorded.  million  events) over a  A f t e r w a r d s t h e powder was removed and r e p l a c e d of  (1368 ±  v o l u m e m /p^ulk ' ^ w  e  r  e  m  *  s  the  m  a  s  with a s  piece  o f powder  89  (10.5  g) and p ^ l k * through  s  t  h  density  Steps  1  vessel  was f o u n d t o be a l i n e a r  admitted  for  5  e  each  a d s o r b e d on t h e A l 0 2  temperature  and  difference  were  Chemicals)  function  The Section  »« SR  of t h e The  g/cm ). 3  i n the target  amount  of  gas  amount  of  gas  pressure  was  determined  surface and  area  (225m /g 2  as  the  and w i t h o u t  specified  t o t a l mass o f t h e powder adsorbed  given  from  t h e amount o f g a s a d m i t t e d w i t h  and  by  ( 1 0 . 5 g) were  atoms  per  unit  Results  data  +  VI•1•3.  exponential  analysis  Good  fits  relaxation  a last  was done e x a c t l y a s d e s c r i b e d were  function  The r e s u l t s f r o m t h e l e f t  obtained  a  single  f o r t h e Mu p r e c e s s i o n  signal.  and r i g h t t e l e s c o p e s  with  in  were a v e r a g e d  as  step.  The adsorbed  at  (3.7  3  area.  VI'2-4 A n a l y s i s  Figure  2  studied.  t h e n u s e d t o d e t e r m i n e t h e number o f surface  Al 0  s u r f a c e s ( i n u n i t s o f c c a t STP) a t a  3  p o w d e r . The s p e c i f i c Davison  bulk  r e p e a t e d . The p r e s s u r e  temperature  vapour  between  of  Mu s p i n  r e l a x a t i o n r a t e as a f u n c t i o n  "He atoms p e r u n i t a r e a a t 7.3°K and 10.4°K i s shown i n  V I - 9 . For comparison, the vapour p r e s s u r e  each  o r t h e number o f  coverage  a d s o r b e d on A l 0 2  3  is  also  a r e shown  plotted. i n Figure  in  the  vessel  Similar results for VI-10.  2 0  Ne  90  12  1  1  1  1  1  1  1  1  I  $ 10 4°K  10  _  } 7-3-K  •  • 10  (_>  UJ CO  8  —  K  H400  1  a.  or  4° •  H500  f  LU  7-3°K  cn  300  6  | LU  DC GC  X  4  200  ex UJ  or  iOO  2  • V ' V 0  1  1  1 *  6  . 1 . 1  8  10  i  i  i  i  12  14  16  18  CL  or  Z)  oa  0  20  ATOMS/(NANOMETER)  2  Figure VI•9 The Mu s p i n r e l a x a t i o n r a t e i n A l 0 powder versus a d s o r b e d He a t 7.3°K a n d 10.4°K. The d o t s r e p r e s e n t the vapour p r e s s u r e a t each c o v e r a g e . 2  VI•2•5  3  Discussion  VI-2«5«1 A d s o r p t i o n I s o t h e r m s o f He on A 1 0 2  Vapour p r e s s u r e  3  i s o t h e r m s a r e a common means o f d e t e r m i n i n g  t h e amount o f g a s r e q u i r e d t o c o m p l e t e a m o n o l a y e r . The p o i n t on. the isotherm a t which t h e d e n s i t y of adsorbed linear  f u n c t i o n of vapour p r e s s u r e  monolayer  density  p o i n t B method.  atoms  becomes  i s a rough i n d i c a t i o n  (Brunauer 1938). T h i s i s r e f e r r e d  a  of t h e  t o as the  91  12  1  1  1  1  1  1  I  I  1  { 30-3° K  -  10 LU CO  8  -  \  j 28-7° K  \  —  i  6  or  •  30-3° K  V  —  or 300  g 4  cc  f  X (X _J  LU  \  "  *  200  28 7° K *  2  or  100 A°  »  1  1  1  12  14  16  .y  0  1  1  1 ' ' 1  1 6  0  8  10  a  CO CO LU  or Q_  or  Z> o  A  1  18  0  20  ATOMS/(NANOMETER)  4  F i g u r e VI•10 The Mu s p i n r e l a x a t i o n r a t e v e r s u s a d s o r b e d Ne a t 30.3°K a n d 28.7°K.  Applying isotherms  (see Figure  at  2  sphere cross  array,  =  —  VI-9) y i e l d s a monolayer d e n s i t y  at  3  powder  adsorption n^(7.3°K)  a n d n (10.4°K) = 0.10 ± 0 . 0 1 A " . The e f f e c t i v e 2  w  s e c t i o n assuming a c l o s e , i s t h e n 8.6 ± 0.5A  2  packed 2 dimensional  a t 1 0 . l a n d 7.3 ± 0.5A  7.3°K. F o r c o m p a r i s o n t h e h a r d s p h e r e c r o s s  graphite  2  t h i s method t o t h e 7.3°K a n d 10.4°K "He  = 0.125 ± 0.01A" hard  i nA l 0  4.2°K, o b t a i n e d  from t h e monolayer  section  2  o f "He on  density,  0.123  92  A  2  (Dash  1975),  is  7.9  A ,  w h e r e a s f o r l i q u i d He  2  assuming a simple c u b i c packing Aston of  ( 1 9 5 5 ) has  bulk  He  of  between t h e s e  film  i n d i c a t e s t h e r e may  u n i f o r m i t y of these  in  relaxation  taken  of  on A l 0 2  Below  rate f i t s  relaxation ± 0.05  data VI«9.  independent  r u l e d out as a  2  +  He  slight  some  3  3  well to a linear  temperature.  rate rapidly  Above  2  to  expansion  although  the  non-  of  the  cause.  Powder W i t h  Adsorbed  He  He  i s a l s o shown  completion,  the  f u n c t i o n of  adsorbed  He,  completion,  the  v a l u e X. ^  0.54  monolayer  l e v e l s off at a constant  Mu  spin  0  as- . 1  The  interpretation  i s p r o p o r t i o n a l to the  i s q u i t e s i m p l e . The fraction  Mu  relaxation  of exposed s u r f a c e  where  n,  only,  is  the  d e n s i t y of a d s o r b e d  i s the t o t a l an  adsorbed  s h o r t l y ) and Mu-Al 0 2  3  X.  0  elastic  He  atom,  i s a constant  surface  which c o n t r i b u t e to  atoms i n t h e  cross section for k  is  a constant  relaxation  0  first  VI•4 layer  Mu  scattering  ( t o be  discussed  rate unrelated  i n t e r a c t i o n . There are at l e a s t X. -  rate  area:  Equation  off  those  temperature  thermal  w i t h adsorbed  monolayer  A .  3  temperatures,  Spin R e l a x a t i o n in A l 0  »i SR  Figure  two  be  °K,  / / 4 ) = 10.4  s u r f a c e s a l o n g w i t h the rough nature  p o i n t B method c a n n o t be  The  _ 2  u n d e r an e f f e c t i v e p r e s s u r e . The m  V I ' 2 ' 5 - 2 Mu  i s irp  c o m p a r e d t h e p r o p e r t i e s of a d s o r b e d  dependence of n the  factor  a t 4.2  three  to  the  factors  93  1.  I n a t r a n s v e r s e f i e l d o f 8 G, t h e s p l i t t i n g ( n ) o f t h e Mu p r e c e s s i o n f r e q u e n c i e s i s 0.178 »is" . The e f f e c t o f f i t t i n g these two f r e q u e n c i e s to a single component y i e l d s an a p p a r e n t r e l a x a t i o n X ~ n. 1  2.  The t a r g e t v e s s e l was c o n t r u c t e d from slightly magnetic s t a i n l e s s s t e e l causing a s m a l l f i e l d inhomogenity over the e f f e c t i v e t a r g e t volume.  3.  The He u s e d i n the experiment i m p u r i t i e s may h a v e c a u s e d a s m a l l At  gas  l o w c o v e r a g e , n, i s s i m p l y  was 9 9 . 9 9 5 % , relaxation.  the t o t a l  ( t h i s e x p l a i n s the l i n e a r behaviour  completion) monolayer  whereas density,  at and  higher thus  so  amount o f  adsorbed  o f X.(n) b e l o w m o n o l a y e r  coverages, X.(n) — >  n,  -->  n , m  7.3  and  10.4°K  coverage, y i e l d i n g  «$Z  were  fitted  0  the hard  at  7.3  10.4°K  slightly  at  determined  different  than  correctly is  2  sphere c r o s s s e c t i o n of (7.3  from  c l a s s i c a l l y a s due t h e f i n i t e contributes  .  ±  the  2  and  helium,  8.6  ± 0.5A  isotherms  2  is  c o u l d e a s i l y be e x p l a i n e d  s i z e of the  Mu  atom  cross section.  t h i s d i f f e r e n c e c a n be e x p l a i n e d  which  also  P e r h a p s more  by t h e f a c t  that  by t h e by t h e H e - H e - s u r f a c e i n t e r a c t i o n ,  i s d e t e r m i n e d by t h e H e - M u - s u r f a c e The  0.5A  adsorption  This  t o t h e He-Mu s c a t t e r i n g  determined  f u n c t i o n a t low  -  surprisingly,  respectively)  a linear  data  k = 32.9 ± 0.3 u s ' a n d a** = 11.0 ± 0.2 A .  Not  and  to  the  X. . T h e r e was no m a r k e d  t e m p e r a t u r e d e p e n d e n c e i n X.(n), a n d t h e r e f o r e t h e c o m b i n e d at  that  c^l  whereas  interaction.  l i n e a r d e p e n d e n c e o f X ( n ) a t low c o v e r a g e i s c o n s i s t e n t  w i t h two l i m i t i n g c a s e s d i s c u s s e d  i n S e c t i o n V'6. They a r e :  94  g for  AT  wHich  "2-  Equation  for  w^')^  y^olTR.  t  f ^ j v i  VI•5  J  z Equat i o n VI•6 Considering the (Section Fe  + 3  strong  binding  of  H  to  impurity  ( 400  «s'  is  difficult  the p r e s e n t r e s u l t s ,  1  for  static  Mu,  see  of the  Section  t o e x t r a c t any more i n f o r m a t i o n  such as a  value  for  u n j u s t i f i e d a s s u m p t i o n s . The c o n t r i b t i o n the  Fe*  on t h e s u r f a c e  3  on a p a r a m a g n e t i c  free  V«4«2)  o n l y an u p p e r  the s u r f a c e  i s known since  the  particles  aggregate  without  t o k from s p i n  making exchange  on t h e c o l l i s i o n  Mu a t low  structures  eliminate  this  frequency with  temperatures  rather  than  unknown.  off  may  scatter  the  primary  j u s t a s i n t h e c a s e o f Ps ( s e e S e c t i o n I I I . 3 . 1 ) .  cause  uncertainties assumptions t o t h e Mu,  would  limit  )  P^. ,  from t h e  i s n o t known. F u r t h e r e x p e r i m e n t s  sample  Furthermore  could  surfaces  c a s e 2 must be f a v o u r e d .  It  off  bare  V«3) a n d t h e l a r g e v a l u e o f X.g e x p e c t e d b e c a u s e  limiting  with  the  a i t that  significant is  decrease  interesting  the e n t i r e  to  in  v . c  estimate  Despite  these  P^ u n d e r  surface area i s e q u a l l y  i n w h i c h c a s e v (0) a t 7.3°K i s 4.3 x 10" £  This  the  accessible y s " , and 1  95  that  the  spin  exchange r a t e  trapping probability  P# t o be 0.00074. I n t h e f u t u r e  be p o s s i b l e t o measure P is  i s n e g l i g a b l e . One t h e n f i n d s t h e i t should  s a m p l e s where v  in well characterized  c  known.  VI«2'5'3 A d s o r p t i o n I s o t h e r m s o f Ne on A l 0 2  Applying  t h e p o i n t B method g i v e s an e f f e c t i v e  cross section 8.6  ±  1.0  f o r a d s o r b e d Ne, <s§l ~ * 7  A  20°K i s 7 . 3 7 A 27.1°K  3  2  9  1  1  «° ^  a t 28.7°K. F o r c o m p a r i s o n ,  2  a  t  sphere  30.3°K  and  on g r a p h i t e  (Huff 1975), whereas a l a y e r of  corresponds  2  hard  liquid  near  neon  at  t o ajfe = 7.23 A . The a g r e e m e n t b e t w e e n a l l 2  t h e s e methods i s i n d i c a t i o n  t h a t Ne atoms  are  much  like  hard  spheres.  VI'2«5'4 Mu S p i n R e l a x a t i o n  in Al 0 2  The s h a p e o f t h e r e l a x a t i o n Ne  atoms  was i d e n t i c a l  3  Powder W i t h A d s o r b e d  Ne  rate versus d e n s i t y of adsorbed  t o t h a t o b s e r v e d f o r He. As i n t h e c a s e  of He, no m a r k e d t e m p e r a t u r e  dependence  30.3°K  t h e r e g i o n b e l o w 0.11 atoms A "  and  28.7°K.  E q u a t i o n VI»2 w i t h X „s"  1  a n d <f^ = 8.9 u 6  Fitting 0  e  obtained  substantially  gives k  =  between  31.4  o b t a i n e d from the f i t i s i n f a i r  to .3  from  the  adsorption  l o w e r t h a n cjj^. T h i s m i g h t  agreement  isotherms,  be due t o t h e  and i s stronger  attraction  the  probability  of t r a p p i n g and thus decrease t h e t h e e l a s t i c  section.  ±  2  2  e  e^f  1  observed  ± 0.2A .  The v a l u e o f a ^ with  = 0.85 ± 0.08 x S "  was  Mu f e e l s t o w a r d t h e Ne, w h i c h c o u l d e n h a n c e t h e cross  96  Surprisingly, observed case  the  a t the lower  value  obtained  temperatures  f o r k i s close t o that  (7.3°K  and  10.4°K)  i n the  o f He. U n f o r t u n a t e l y , s i n c e t h e t e m p e r a t u r e d e p e n d e n c e f o r  the s p i n r e l a x a t i o n r a t e and t h e t r a p p i n g r a t e i n E q u a t i o n are  both  unknown,  this  observation  cannot  be  IV'4  used  to  c o n c l u s i v e l y d i s t i n g u i s h one f r o m t h e o t h e r . The  m e a s u r e a b l y l a r g e r v a l u e o f X. i n t h e 0  probably  due  t o the higher  impurity level  case  of  Ne  is  i n t h e Ne a s o p p o s e d  t o t h e He.  V I - 2 - 6 S t a t u s o f t h e ATTD M o d e l According (ATTD)  model  to the  the Mu  ambient  temperature  thermalizes  at  inside the g r a i n s , d i f f u s e s t o the oxide i n t o t h e v o i d s because of a since  Mu  is  static  in  negative bulk  model  oxides  (Brewer  1981),' t h e ATTD  fails  amount  o f Mu i n t h e v o i d s o f t h e s e  thermal  t h e ambient  diffusion temperature  s u r f a c e , and i s e j e c t e d  work  function.  such as S i 0 to  explain  powders a t low  2  However,  b e l o w 50 °K the  copious  temperatures.  A s i m p l e m o d e l t h a t e x p l a i n s why Mu m i g h t t h e r m a l i z e d i r e c t l y i n the v o i d s  i s given  i n Appendix I I I  97  VI•2•7  Conclusion  I t h a s been shown t h a t Mu emerges grains torr).  below  of  conclusions The  exposed reached  surface  He  (7.3  surfaces  This  Al 0 2  powder  3  (10-760  d e p e n d e n t on  the  substantiates  the  experiment  section for elastic  (Section  VI.1.5).  scattering  of  Mu  t o 10.4°K) a n d Ne ( 2 8 . 7 t o 30.3°K) have  been m e a s u r e d t o be 11.1 ± 0.1A A new t e c h n i q u e  area.  i n the previous  effective cross  adsorbed  75A  30°K i n l o w d e n s i t y He and Ne a t m o s p h e r e s  The Mu s p i n r e l a x a t i o n r a t e i s l i n e a r l y  amount  off  from  2  f o r studying  a n d 8.9 ± 0.2A  2  respectively.  t h e p r o p e r t i e s of these  a n d a d s o r b e d atoms on them h a s been d e m o n s t r a t e d .  oxide  98  CHAPTER V I I : MUONIUM I N THE CONDENSED PHASES OF A r , K r AND Xe  Motivation  f o r s t u d y i n g Mu i n t h e c o n d e n s e d p h a s e s o f A r ,  K r , a n d Xe i s e a s y t o f i n d . it  when a muon s t o p s  i n matter  l e a v e s a h o t t r a c k of e x c i t e d and i o n i z e d s p e c i e s . There  been  considerable  track  on Mu f o r m a t i o n  1981). is  Firstly,  d e b a t e i n r e c e n t y e a r s on t h e e f f e c t and  relaxation  (Percival  of t h i s  1981,  Walker  S i n c e t h e p r o p e r t i e s of t h e t r a c k a r e phase dependent i t  of i n t e r e s t  t o study the e f f e c t  of  s o l i d ) on Mu f o r m a t i o n and r e l a x a t i o n . observations  phase  (gas, l i q u i d  In t h i s chapter  (Fleming  an e l e m e n t  and  the f i r s t  o f Mu i n t h e c o n d e n s e d p h a s e s o f A r , K r a n d Xe a r e  r e p o r t e d . These measurements, a l o n g w i t h t h e e x i s t i n g data  has  1981b) r e p r e s e n t  i n a l l three  Secondly,  the f i r s t  gas  phase  complete M'SR study of  phases.  H atoms have been s t a b i l i z e d a n d s t u d i e d i n s o l i d  A r , K r , a n d Xe u s i n g ESR. The ESR m e a s u r e m e n t s a r e s e n s i t i v e lattice-induced  perturbations  of  i n t e r a c t i o n s as w e l l as t o the presence principle, using  isotopic  hyperfine  of n u c l e a r  contact  moments.  In  d e p e n d e n c e o f t h e s e e f f e c t s c a n be s t u d i e d  n*SR . Finally,  reactions  there  have  used  very  atom c h e m i s t r y fraction  been  several  o f H atoms w i t h i m p u r i t i e s s u c h  s o i i d Xe ( K i n u g a w a 1 978, been  the  to  in  Iwasaki  successfully  1978). In  studies  of  as C « H  deposited i n  the  1 0  past  chemical  »i SR +  has  i n e x p l o r i n g i s o t o p i c e f f e c t s of H  i n gas a n d l i q u i d p h a s e s . I f t h e r e  i s a large  t h e s o l i d phase of t h e s e elements i s o t o p i c  Mu  effects  of t h e a b o v e m e n t i o n e d H atom r e a c t i o n s i n t h e s o l i d p h a s e c o u l d be  studied.  99  VI1-1  Experimental  The  muon beam h a s been d e s c r i b e d i n S e c t i o n V I • 1 • 1 . The LBL  s u r f a c e muon a p p a r a t u s h a s been shown i n F i g u r e V I • 5 . The t a r g e t vessel  ( F i g u r e V I I • 1 ) was c o n s t r u c t e d f r o m c o p p e r a n d s t a i n l e s s copper - constantan thermocouple heater  gas  wires  Inlet copper target vessel heater  condensed target  29 MeV/c  heater wires 0 0 5 " mylar window scale cms. 0 I  5  Figure VII•1 gases.  s t e e l except rotatable LN of  the  gas  The t a r g e t v e s s e l u s e d  to  condense  noble  f o r t h e m y l a r window. The v e s s e l was mounted 2  cryostat could  be  beam. The t a r g e t g a s e s Kr,  cold finger from cryostat  on  a  such t h a t c o n d e n s a t i o n and s o l i d i f i c a t i o n observed p r i o r  to facing  i t towards the  ( u l t r a h i g h p u r i t y Xe, u l t r a  high  purity  a n d r e s e a r c h g r a d e A r ) were a d m i t t e d v i a t h e g a s i n l e t  tube  ( F i g u r e V I I - 1 ) . The v e s s e l was e q u i p p e d w i t h two h e a t e r s , one on  100  the gas i n l e t  t o prevent  blockages,  a n d a s e c o n d mounted b e t w e e n  the copper v e s s e l and  the  cold  temperature  Two  copper constantan  control.  used t o monitor  the temperature  finger  of  (controlled  the  t h e r m o c o u p l e s were  t o ±1°K).  The >/ S R e l e c t r o n i c s have been d e s c r i b e d  i n S e c t i o n V I * 1*3.  +  VII*2  cryostat for  Data A n a l y s i s and R e s u l t s  n*SR s p e c t r a were t a k e n order  to  determine  the  i n a transverse f i e l d  free  muon  fraction.  of  65G  The  data  in were  c o a r s e l y binned  and f i t t e d t o E q u a t i o n  Equation  a n d S ^ ( t ) =0 The p a r a m e t e r o f i n t e r e s t , A^ , d i d  VI*2  f r e e muon f r a c t i o n s  i n Table  HflCn)  in  Equation with  to  U  M  m  the  *Mn '  a n c  ^ 3 * ^ B  T  e  of  obtained -A>-t  oe  +  P  Mu f r a c t i o n . The d a t a  a  r  except  by  V*2  )  e  - ^ ] .  e  t  e  component  r  s  *y t *  a n <  3  field  relaxation  The i m p r o v e m e n t i n t h e x  Mu  relaxation N ,  ,  0  yu were h e l d data.  function 2  and  Ar and s o l i d K r ,  In the  Kr, considerably better f i t s  w i t h a two component Mu 0  m  8 G  were f i t t o  Equation  liquid  from t h e h i g h  l i q u i d Ar and s o l i d  (. _  a  VI 1*1  i n a low t r a n s v e r s e f i e l d  ( R ^ * ^ e'^ ) . T h e r e were s e v e n f r e e p a r a m e t e r s , '  The  from  Equation  were o b t a i n e d w i t h a s i n g l e  fixed at the values obtained cases  were d e t e r m i n e d  3  . In a l l cases  p  exponential.  - f\^CYc^0 )  determine  R ^ ( t ) = e"  function MK '  VII•1  or  V* 1 w i t h S^ ( t ) a n d S ^ ( t ) g i v e n  good f i t s  A  gaussian  s p e c t r a were a l s o t a k e n  order  by  y  n o t d e p e n d on w h e t h e r R ^ ( t ) was  P*SR  VI•1 w i t h ^ ( t ) given  was  were  [(R*^ (t) lf  significant:  101  T a b l e V I I «1.  Target Substance  *i S R r e s u l t s +  Temperature K  L i q u i d Ar  77  L i q u i d Xe  Hu Fraction ( s ) %  3  1.6  85  Sol i d Ar  Free Muon Fraction % 1.0  +  i n condensed Ar,Kr,and  (1) 48  +  6  (2) 49  +  28  91  +  0.8  +  0.2  162  3.3  +  0.8  43  +  S o l i d Xe  150  5.0  +  3.3  79  +  L i q u i d Kr  120  6.5  +  0.1  57  90  1  +  1.8  S o l i d Kr  Hissing Fraction %  13  Xe.  Mu Relaxation Rate(s) usee  Mu Hyperfine Splitting MHz 0  -1  3 + 29  (1) 0.65  +  0.12  (2) 19.0  +  11.0 0.03 0.21  8  +  9  0.15  +  9  54  +  10  2.07  +  25  16  +  28  19.0  +  2.5  +  10  36  +  10  3.6  +  0.9  (1) 71  +  7  0  +  10  (1) 6.68  +  0.20  (2) 29  +  7  (2) 0.89  +  0.04  9  These f r a c t i o n s were determined from data at ~70  4463.8 ± 6.0  4462.9 ± 3-7  G.  These f r a c t i o n s were determined from data at ~8 G except for s o l i d Xe where the f r a c t i o n was  from  ~70 G data. c  from  The vacuum hyperfine s p l i t t i n g i s 4463.302 MHz.  164  (145  degrees  freedom) f o r s o l i d 154  Kr and  (143 d e g r e e s . o f  of from  In fast that rate  was  the  169  126  (143 d e g r e e s  (145 d e g r e e s  f r o m E q u a t i o n VI«3.  of Mu  of  freedom)  to  fractions  in  For  comparison  i n t h e gas p h a s e a r e shown i n T a b l e VII«2. case of s o l i d  a reliable not p o s s i b l e  Xe,  the r e l a x a t i o n  m e a s u r e o f t h e Mu i n low  field.  t a k e n a t 70G(where t h e p r e c e s s i o n determine  to  f r e e d o m ) f o r l i q u i d A r . The  T a b l e V I I • 1 were d e t e r m i n e d the r e s u l t s  freedom)  t h e Mu  t o E q u a t i o n VI•1  f r a c t i o n and with  rate  o f Mu  asymmetry and  I t was  signal  relaxation  was  so  relaxation  n e c e s s a r y t o use  data  i s more c o m p l i c a t e d ) t o rate.  The  d a t a were f i t  1 02  Table  V I I . 2 . Mu F r a c t i o n i n Gas P h a s e A r , K r , and X e ?  Target  Pressure (atmospheres)  Ar Kr Xe  Mu F r a c t i o n %  2.5 0.9 0.65  b  b  F r e e Muon F r a c t i o n %  75 100 100  24 3 3  « f r o m ( F l e m i n g 1981b) fc> e r r o r e s t i m a t e i s 5%  S  M u  / t } = (LMU { cos'f cos[(u)--Jl)t +" f 2~  . '  This  expression  i at intermediate  obtained  (u ) 0  of  relaxation  sufficiently be r e s o l v e d IV.3.2).  small at  an  In t h i s  The in  rate  of  two  *M„)  0  Mu  in  65G  «^  were  data.  and  n  The are  frequency  was s e t t o t h e vacuum  0  Mi<  was  so t h a t t h e two n o r m a l Mu f r e q u e n c i e s  could  field  IV.31) and thus  VII-2.  precession  solid  and Bj. Kr  frequency  Figure  and  0  and  f i t , u  The d i f f e r e n c e b e t w e e n t h e s e 0  shown  N ,  VII-2  fields  (B) a n d t h e h y p e r f i n e  intermediate  of B and u ( E q u a t i o n 0  p,  l e a v i n g f o u r f r e e p a r a m e t e r s , A ^ , X.,  The  o .  transverse  with +  the a p p l i e d f i e l d  (see S e c t i o n IV'3'2).  value,  Equation  from t h e c o a r s e l y binned  p a r a m e t e r s ^ ( n o t t o be c o n f u s e d functions  -/It  J  i s valid  at the values  J ?  ( s e e S e c t i o n I V ' 5 ) . The p a r a m e t e r s A^, fixed  M  of  Ar  and  66G  (See  frequencies  provides o f Mu  a  Section  i s a function  measurement  of  i n s o l i d A r a t 66 G i s  The h y p e r f i n e s p l i t t i n g d e t e r m i n e d i n  103  0.50  0.25  5  0  0  0  -0.25  -0.50 0.00  0.05  0.10  0.15  TIME IN S E C M  0.20  0.25  0.30  (2 NSEC/BIN)  Figure VII•2 Two f r e q e n c y Mu p r e c e s s i o n A r a t 77°K i n h i g h f i e l d ( 66 G ) .  t h i s way  VII•3  i s a l s o given  i n Table  in  solid  VII « 1 .  Discussion  VII«3-1  Mu  i n L i q u i d and S o l i d Ar  The f r e e muon component i n b o t h extremely  small  (<2%). T h i s  liquid  i s a strong  t h e muons f o r m Mu, s i n c e t h e r e a r e no relax  signal  free  muon  polarization.  e v e n weak l o c a l m a g n e t i c amounts  of  naturally  fields, occuring  and  solid  indication  strong  In f a c t ,  local  there  is  t h a t most o f fields  i s no s o u r c e  s i n c e t h e r e a r e no isotopes  Ar  to of  significant  of  Ar  with  signal  (F^  = 91  nuclear  moments. In s o l i d  A r , a l a r g e Mu p r e c e s s i o n  ±  9%)  1 04  was The  observed with a small r e l a x a t i o n rate Mu p r e c e s s i o n  Figure VII'3. 0.25  |  0.15  -  0.05  -  -0.05  -  -o.i5  y  -0.25  0.0  signals in solid  In l i q u i d  Ar, there  (X, = 0.15 ± 0.03 »/S~ ). 1  and l i q u i d  a r g o n a r e shown i n  i s a s l o w l y r e l a x i n g component  1  1  1  1  1  1  1  .0.2  0.4  0.6  0.8  1.0  1.2  1.4  1.6  1.4  1.6  1  TIME IN uSEC (10 NSEC/BIN)  .0.0  0.2  0.4  0.6  0.8  1.0  1.2  TIME IN uSEC (10 NSEC/BIN)  Figure VII•3 (a) S i n g l e frequency Mu precession in liquid A r a t 85°K a t l o w f i e l d ( 8 G ) ; (b) I n s o l i d Ar a t 77°K.  (X. = 0.65 ± 0.17 «s"') w h i c h a c c o u n t s f o r o n l y 48 ± 6% o f t h e muon polarization. impurity  The  present  relaxation  is  at  level.  a  ppm  likely In  due  to  addition,  a  reactive there  is  105  indication  of a very  accounting only  for  within  demonstrates responsible relaxing  f a s t - r e l a x i n g component  the  remaining  the  first  100  that  the  inert  (X. = 19 ±  muon p o l a r i z a t i o n a n d ns.  The  liquid  long-lived  i s due t o Mu  DS" ), 1  observable Mu  signal  or i t s i m p u r i t i e s are not  for this relaxation. I t i s possible  component  11  that  i n t e r a c t i n g with  the  fast  the r a d i a t i o n  t r a c k c o n s i s t i n g o f f r e e e l e c t r o n s , i o n s and e x c i t e d a t o m s . will  be d i s c u s s e d Electron  Ar  lattice  further i n Section  VII«4.  s p i n r e s o n a n c e e x p e r i m e n t s on H atoms i n  a t 4.2°K i n d i c a t e a t l e a s t two t r a p p i n g  1 9 6 0 ) . The m a j o r component h a s a h y p e r f i n e 0.46%  relative  t o t h e vacuum v a l u e .  t h e H atoms become m o b i l e . measured  the  mean  this  does  splitting  not agree w i t h  measurements a r e not d i r e c t l y experiment  the  H  atoms  whereas i n t h e present diffusing Is*-%.-e*  are  sites  (Foner  s h i f t of  experiment, o f Mu  have  ±  0.13%.  t h e H atom r e s u l t s ,  t h e two  comparable trapped  so t h a t  we  i n s o l i d argon a t o f 0.01  because  at f i x e d  in  the hyperfine  I V - 4 - 2 ) i s a v e r a g e d o v e r many  the  lattice  y*SR e x p e r i m e n t t h e Mu a t o m s a r e  through the l a t t i c e  (see S e c t i o n  solid  splitting  77°K t o be 4463.8 ± 6MHz, w h i c h g i v e s a s h i f t Although  a  A t t e m p e r a t u r e s a b o v e 39°K,  In the present  hyperfine  This  ESR  sites, probably  perturbation sites.  106  VII»3«2 Mu i n L i q u i d a n d S o l i d Xe As solid  i n t h e c a s e o f A r , t h e f r e e muon f r a c t i o n i n l i q u i d Xe i s s m a l l  ( 3 . 3 ± 8% a n d 5 ± 3%, r e s p e c t i v e l y ) . I n s o l i d  Xe,  t h e Mu component a c c o u n t s f o r 79 ± 2 5 % o f t h e muon  The  large  signal. of  error  i s due  In l i q u i d  and  to  the  fast  relaxing  ensemble.  nature  of  the  X e , t h e Mu component a c c o u n t s f o r o n l y 43 ± 9%  t h e muons. I t i s p o s s i b l e  Mu component a s i n  the  that  case  there i s also  of  liquid  Ar,  a fast but  relaxing  i t i s not  resolvable. The us"  Mu  relaxation  dipole  moments  increases sharply 1  of  1 2 9  X e and  Xe  1 3 1  21.18% o f t h e n a t u r a l l y o c c u r r i n g the  fast  relaxation  Section  in  in liquid  higher  the  solid.  The  Motional  narrowing  (see  enhanced i n t h e l i q u i d ,  might  liquid  relaxation  A r i s l i k e l y due  t o the  c o n t e n t i n t h e Xe.  ESR s p e c t r u m f o r H atoms t r a p p e d a t i n t e r s t i t i a l s i t e s  i n t h e Xe l a t t i c e  a t 4.2°K i s m u l t i c o m p o n e n t  spread  G.  of  magnetic  98.2  of  with  determined  i t s nearest  primarily  by  e x p e r i e n c e d by an i n t e r s t i t i a l H atom i s o f  This  corresponds  to  a  Mu  relaxation  rate  1  diffusing  ±  2.5  «s" '.  the  isotopic  order  of  M H z G ' ^ S G = 220 *»s" . The o b s e r v e d Mu r e l a x a t i o n 19.0  overall  n e i g h b o u r s . T h u s , t h e mean  field  only  an  Each frequency corresponds t o a p a r t i c u l a r  environment,  compositions  is  nuclear  responsible f o r  b e t w e e n p h a s e s . The l a r g e r  Xe c o m p a r e d w i t h  impurity  The  , w h i c h c o m p r i s e 26.44% a n d  Xe, a r e l i k e l y  V « 4 « 4 ) , an e f f e c t g r e a t l y  cause such a d i s c o n t i n u i t y rate  f r o m 2.1 ± 0.2  t o 19.0 ± 2.5 ^ s " i n t h e s o l i d .  i n the l i q u i d  1  rate  order rate  Thus, i t i s l i k e l y t h a t  r a p i d l y a t t h i s temperature  or  possibly  at  local 25  G.  2n«1.4 150°K  t h e Mu i s  trapped  in  1 07  defects  where  the  local  f i e l d s a r e much s m a l l e r .  n o t e d t h a t H atoms a r e m o b i l e 20°K ( K i n u g a w a  I t should  i n s o l i d Xe a t t e m p e r a t u r e s  be  above  1978).  VII«3«3 Mu i n L i q u i d a n d S o l i d K r Again, small of  the  f r e e muon f r a c t i o n s i n s o l i d a n d l i q u i d  Kr a r e  (1.4 ± 1.8% a n d 6.5 ± 0.1%, r e s p e c t i v e l y ) . As i n t h e in I iquii K r  l i q u i d X e , t h e Mu c o m p o n e n t ^ a c c o u n t s f o r o n l y  muon e n s e m b l e , w h e r e a s t h e Mu f r a c t i o n consistent with The  Mu  100% Mu f o r m a t i o n  relaxation  in  liquid  since motional  narrowing  relaxation  due  ,  8 3  Kr  57 ± 10% of t h e K r , 99 ± 10%, i s  w i t h no m i s s i n g  impurities,  to  in solid  Kr  fraction.  i s again  likely  i s expected t o  which  accounts  case  due t o  quench  any  f o r 11.48% o f t h e  natural Kr. Perhaps experiment  the is  considerably  most  interesting  for solid  better  **SR  spectrum  K r . As m e n t i o n e d  f i t t o the data  component Mu r e l a x a t i o n f u n c t i o n  was  in  this  i n Section VII-2, a  obtained  using  a  two  ( s e e F i g u r e V I I « 4 ) . The sum  of  t h e s e c o m p o n e n t s a c c o u n t s f o r 99 ± 10% o f t h e muon e n s e m b l e . One interesting due  to vacancies  trapping X. t 3  interpretation  >>  of  Mu  (Section  that there  o r d e f e c t s . The s i t u a t i o n on  a  1. I n t h e p r e s e n t  r a t e of trapped  is  Mu. T h i s  IV«5«1) due t o  surface  exist  i s analogous  described g  8 3  Kr  the  relaxation  (11.48% of n a t u r a l Kr p o s s e s s i n g off  relaxation  Mu, b e i n g  nuclear  to the  i s e x p e c t e d t o be s m a l l , s i n c e t h e RLMF  v  of  sites  i n S e c t i o n V«5 w i t h  c a s e X. c o r r e s p o n d s t o  m a g n e t i c moment o f -0.969 n ) f a l l s rate  trapping  interstitial  rapidly  as  a  1 / r . The 3  much c l o s e r t o t h e  moments, i s e x p e c t e d t o be l a r g e r . As i n V«5, we  define  108  0.20  -0.20  0.0  0.2  0.4  0.6  0.8  1.0  1.2  1.4  1.6  TIME IN uSEC (10 NSEC/BIN)  Figure VII'4 The Mu p r e c e s s i o n i n a m a g n e t i c f i e l d o f 10.7G  v  as  the  trapping  expect a r e l a x a t i o n  rate.  s i g n a l s o l i d K r a t 90°K  Under t h e s e c o n d i t i o n s ,  one w o u l d  function  Equation (i.e.:  a sum o f e x p o n e n t i a l s  amplitudes X  and  as  relaxation  observed). rates  yields  the  fitted  v = 1.7±0.4 e s " , 1  = 5.0±0.4 ^ s " a n d X. = 0.89±0.04 u s ' . 1  F  Using  VII•3  1  &  One may a l s o make a r o u g h between static face  interstitial  on  s i t e s by c o m p a r i n g X  r e l a x a t i o n r a t e . The i n t e r s t i t i a l  centered cubic  octahedral between  estimate  lattice  symmetry.  octahedral  We sites  which  assume have  with  hopping  rate  the c a l c u l a t e d  trapping  such as s o l i d will  F  the  sites  in a  K r have t e t r a h e d r a l o r that a  the  Mu atoms hop  larger  Mu-nucleus  109  separation Mu  at  (1.418 A) ( F o n e r  1 9 6 0 ) . The s t a t i c  one o f t h e s e s i t e s may be c a l c u l a t e d f r o m e q u a t i o n  a s s u m i n g one o f t h e e i g h t n e a r e s t moment=-0.969  nuclear  neighbours  l a r g e r than the  rate.  the hopping  This  according  i n d i c a t e s that  t o equation  T h i s model c o u l d annealing  ns.  by  relaxation 45 ns~  1  studying  trapping.  the  effect  roughly  Recently,  Fractions  i s accounted  i n s o l i d A r , K r and Xe f o r , whereas  h a l f t h e muon p o l a r i z a t i o n i s m i s s i n g  such r e s u l t s  1981)in  have  been  explained  i n the a f t e r 100  qualitatively  t e r m s o f an e x p a n d i n g t r a c k m o d e l . A c c o r d i n g  t h i s m o d e l , t h e Mu i s f o r m e d e p i t h e r m a l l y a n d b e g i n s t o randomly  from  some  the  with  of  track  a  fraction  time T. I n diffusion the  liquids, diffusion  cold  species,  also  Mu f o r a s h o r t p e r i o d o f t i m e ,  s p i n exchange r e l a x a t i o n r a t e  expect  of  the  of  is  diffuse  large.  diffusing, during  Thus,  liquids,  such  as  liquid small  Ar  at  which  one  t h e muon p o l a r i z a t i o n t o be l o s t  r a t e s may be s u f f i c i e n t l y  would within  85°K,  In s l i g h t l y  warmer  s u c h a s l i q u i d Xe a t 150°K a n d l i q u i d K r a t 120°K, are  too  the  so t h a t t h e r e l a x a t i o n  f a s t component i s b a r e l y o b s e r v a b l e .  rates  to  p o i n t b e y o n d t h e end o f t h e t r a c k . A t some  t i m e T, t h e c o n c e n t r a t i o n overlaps  of  +  t h e muon p o l a r i z a t i o n  (Walker  1  i s t h e mean t i m e between  i s c l e a r from these r e s u l t s t h a t  liquids  (nuclear  slow--  26 t i m e s b e f o r e  tested  Kr  a n d t e m p e r a t u r e on t h e »/ SR s p e c t r u m .  VII-3-4 Missing It  be  8 3  observed  i s rather  V'21 ( a s s u m i n g T  h o p s ) . A l s o t h e Mu atoms hop o n l y  is  V'13,  The r e s u l t y i e l d s X=15 * / S ~  magnetons).  which i s a f a c t o r of three  all  r e l a x a t i o n r a t e of  the  l a r g e , a n d -T t o o s m a l l , f o r t h e f a s t  110  relaxing solid  component t o be o b s e r v e d . I t  phase  the  diffusion  i s argued  that  i n the  r a t e s a r e t o o s m a l l and T t o o l a r g e  (>> 2.2 us) f o r t h e f a s t component t o be o b s e r v e d .  VII'4 Conclusions As i n t h e g a s p h a s e , most o f t h e muons i n  the  liquid  and  solid  A r , K r a n d Xe f o r m Mu. The o n l y r e a l d i s c r e p e n c y i s i n t h e  case  of  A r g a s , where a 2 5 % muon f r a c t i o n  i s observed  1 9 8 1 ) . No s u c h f r e e muon f r a c t i o n was o b s e r v e d or  in either  liquid  solid Ar. The e l e m e n t s  o f A r , K r a n d Xe a r e i d e a l  Mu-lattice  states  probability  and t h e i r  possible of  (Fleming  because simple  t o make more d i r e c t  of t h e i r  large  monatomic  substances t o study (^100%) Mu f o r m a t i o n  structure.  comparisons  with e x i s t i n g  t r a p p e d H atoms a t l o w e r t e m p e r a t u r e s where  a l s o becomes There radiation relaxing  It  perhaps  may  be  ESR d a t a the  Mu  trapped. i s an i n d i c a t i o n  track  in  the  Mu c o m p o n e n t .  that  liquid  t h e Mu i n t e r a c t s  elements,  leading  with to  i t s own a  fast  111  CONCLUDING REMARKS  It and  Mu  has  studies  molecule, In  been d e m o n s t r a t e d i n t h e p r e c e d i n g can  provide  a  on  t h e f u t u r e , i t i s p o s s i b l e t h a t Mu a n d P s c o u l d  interaction.  atom b i n d i n g e n e r g i e s ,  theories  atom-  Until  now,  the  been n e g l e c t e d .  some i n t e r e s t  dealing  surface d i f f u s i o n  chemistry  rates  atom-  direction.  and  adsorption  of p o s s i b i l i t y .  o f Mu on s u r f a c e s a n d i n s o l i d s  I t i s hoped t h a t t h i s s t u d y  in this  with  provide  Measurements of such q u a n t i t i e s as s i n g l e  p r o b a b i l i t i e s a r e a l l w i t h i n the realm  has  perspective  how Ps  a t o m - s u r f a c e and a t o m - s o l i d i n t e r a c t i o n s .  a t e s t i n g ground f o r fundamental surface  unique  chapters  will  stimulate  11 2  APPENDIX I THERMALIZATION OF GAS ATOMS I N A POWDER  The  mean  energy  loss per c o l l i s i o n  f o r a t h e r m a l beam o f  atoms c a n be w r i t t e n  Equation Al•1 where  T^ = t- = c| = P( «?,€*) = 3  t h e t e m p e r a t u r e o f t h e beam. t h e i n i t i a l e n e r g y o f t h e g a s atom. t h e f i n a l e n e r g y o f t h e g a s atom. the probability per collision t r a n s i t i o n e* --> c| .  for  a  1  P(€^,t|)  can  be  approximated  atom s u r f a c e s c a t t e r i n g , and  recently  first  factor  calculation  developed  r e v i e w e d by Goodman  one-dimensional c a l c u l a t i o n small  i n a one-dimensional theory f o r  (2  or  3)  by  Devonshire  (1937)  ( 1 9 7 1 ) . The e f f e c t o f u s i n g a  increases the t r a n s i t i o n rate  by  a  i n c a s e s where t h e t h r e e - d i m e n s i o n a l  h a s been done.  In t h e D e v o n s h i r e t h e o r y , t h e s u r f a c e r e p r e s e n t e d by a  atom  potential  is  function Equation Al•2  where  z a n d Z a r e t h e d i s p l a c e m e n t s o f t h e g a s atom a n d s u r f a c e  atom r e s p e c t i v e l y . )/(*)=  Vie.  V ( z ) i s o f t e n c h o s e n t o be a M o r s e -2c-  )  potential  . E q u a t i o n A l •3  -2AZ  -az \  Equation Al•4 D and a a r e t h e depth and range  of the p o t e n t i a l ,  respectively.  11 3  The u n p e r t u r b e d L n  x  y  0  gas atom s t a t e s a r e s o l u t i o n s t o Equation Al-5  ^ ' J  The t r a n s i t i o n  r a t e from  €^ — >  ef i s g i v e n a s  Equation where  |s > ;  i s the i n i t i a l  w i t h energy  €*. The sum  |s^>.  matrix  The  i s over a l l f i n a l  element  e v a l u a t e d by e x p r e s s i n g a n n i h i l a t i o n operators  l)A/  ^~  J^W^  s t a t e of the s o l i d  Z  involving in  terms  the s o l i d  one  of  mass o f t h e s o l i d  the  i s most  T  s  solid, easily  phonon• c r e a t i o n  and  Equation  AI-7  ( A s h c r o f t 1976).  ^  '  where a ^ a n n i h i l a t e s a phonon c o r r e s p o n d i n g creates  at temperature  s t a t e s of  of  Al•6  t h e same,  t o n o r m a l mode q, a^  i s t h e phonon f r e q u e n c y , M i s t h e  a t o m s , and N i s t h e number  of  atoms  in  the  solid.  Equation where  Al•8  1 14  \ -  e Equation  ~  .  Al•9  1  ^  &  ^W^AT  ; —  5  ~~ i  Equat ion A l • 1 0 S u b s t i t u t i n g a t h r e e d i m e n s i o n a l Debye d e n s i t y o f modes, Equation Al • 1 1 where  u  solid,  i n t o Equation AI-8  L-H-S-  p  is  the  =  Debye  ^  [  frequency  and n i s t h e volume o f t h e  yields  to ( n  w  + 0 f ^ - « f f  +•  Equat ion Al•12  Equation where t h e g s u p e r s c r i p t  has  been  Al•13  dropped.  Thus  Equation Al• 1 4 The  probability  per  collision  for  scattering  i n t o an e n e r g y  1 15  i n t e r v a l cU^ i s  Equat i o n A l • 1 5 where  PC^^ =  dimension  /JQQ \  with  2  i s t h e d e n s i t y of f i n a l  normalization  L  and  D =  s t a t e s i n one x. i s the  no collision  frequency.  6  eriAJi s e  Equat i o n Al•16 For a Morse  potential,  +  Equat i o n Al•17  1 16  ere  / i '-  (2W6 )'  /  /^  2  f  d = (2wD) /tifl ,/l  tiW -  6+-6:1  Equat i o n S u b s t i t u t i n g E q u a t i o n AI-18  By  into Equation A l • 1 ,  the p r i n c i p l e o f m i c r o s c o p i c  i < 6  f  i v ( z ) U ; > f = -  f + tus)  P  CD  -  W  t>  .  Equation  Al•19  Equation  Al•20  Equation  Al•21  reversibility  \<6 )  vU))^>r  C  e  P U + f c w , 0  -^/fcT  fc  5  /JL - / \  Equation Thus, i n t h e r m a l e q u i l i b r i u m , The  t h e r e i s no mean e n e r g y  r a t e o f energy l o s s can - AH V  cU  Al•18  Al•22  transfer.  be w r i t t e n  6  E q u a t i o n A l -23 where v  a  i s the c o l l i s i o n  frequency  w i t h the s u r f a c e ,  c a n be  1 17  estimated as VP  E q u a t i o n Al-24  where N i s t h e number o f p a r t i c l e s p e r u n i t mass, V  F  i s the free  v o l u m e p e r u n i t mass, R i s t h e r a d i u s o f t h e powder g r a i n s a n d v i s t h e mean v e l o c i t y  of t h e atom.  E q u a t i o n A l '25 where SA i s t h e s p e c i f i c d e n s i t y of the s o l i d , The t i m e  s u r f a c e a r e a of t h e powder, p ) j i s t h e  and p  i0  l(  i s t h e d e n s i t y o f t h e powder.  r e q u i r e d f o r an atom o f mean e n e r g y E  i  t o r e a c h E^  i s o b t a i n e d by i n t e g r a t i n g E q u a t i o n A I - 2 3  Equat i o n Al•26 where AE i s g i v e n by E q u a t i o n s A I ' 2 2 a n d A I ' 1 8 .  1 18  APPENDIX I I PS SCATTERING OFF 2 ELECTRON ATOM  In Ps  t h i s appendix, a theory  o f f a two-electron  spin  (s) of  electrons (e.g.,  f o r l o w e n e r g y s c a t t e r i n g o f o-  atom i s d e v e l o p e d . I t i s assumed t h a t  t h e atom  and  thus  there  i s no  the  t h e s p a c i a l symmetry o f t h e  i n t h e atom i s c o n s e r v e d d u r i n g  He)  (MOLECULE)  collisions.  spin conversion,  When  b u t when s=1  s=0  (e.g.,  H e * ( 2 S ) ) , o-Ps may be c o n v e r t e d t o p - P s v i a s p i n e x c h a n g e .  The  effect  rate  3  of a l a r g e magnetic f i e l d  is considered.  Finally,  electron  molecule.  The  particle  only  the spacial coordinates. spin-orbit coupling  and  the t o t a l  However,  the  i s generalized  electron  are neglected.  The s p i n o f e a c h  spin are therefore  t h e d i r e c t i o n of the e l e c t r o n  exchange degeneracy a s s o c i a t e d may  exchange.  to  a  2  T h u s , e f f e c t s due t o t h e s p i n - s p i n  i s not n e c e s s a r i l y conserved during  electrons spin  result  i n t e r a c t i o n s a r e assumed t o be a f u n c t i o n o f  and  o-Ps  the  on t h e o b s e r v e d q u e n c h i n g  interchange).  This  with  conserved  quantities.  spin associated collisions, the electrons  i s the basic  particle  with the  because  of  ( i . e . , the  p r i n c i p l e behind  119  A 11•1 T o t a l E l e c t r o n S p i n States positron  of t o t a l  spin)  States  electron spin  |SS p> (p t h e z component 2  s p a n an i r r e d u c i b l e  s u b - s p a c e Cs^n  to p e r m u t a t i o n of the e l e c t r o n s p i n c o o r d i n a t e s . for  C*p,'„  r  generate  permutation group according  The  3  A, o f S  t(  basis  [using  3  the basis vectors  dimensional  where  S .  irreducible  t o t h e one d i m e n s i o n a l ,  representation whereas  the  irreducible  is  the  row  for r  vector totally  transform  representation  index  for  r  3  /  2  symmetric of  respect  Basis  representation  notation , / z  with  Tinkham  according  of  vectors r  of the  transforms irreducible (1964)], t o t h e two  E of S . F o r example, 3  of t h e i r r e d u c i b l e  Equation  AlI • 1  Equation  AlI•2  Equation  AlI•3  representation  1 20  A 11 • > 2 Spacial  States  The s p a c e o f a s y m p t o t i c orbital (spin  angular  momentum  states lm)  f o r Ps  (momentum  k  i n c i d e n t on a two e l e c t r o n  s ) a r e s p a n n e d by t h e t h r e e  and atom  states  where  £ > , f 1 = J*Lk|r,t.r l] r/fri+Cf 1 and  *  l  2  5  (f,f )  i s t h e two e l e c t r o n w a v e f u n c t i o n ,  2  s=0 a n d a n t i s y m m e t r i c There each  value  V%,-rj)  r  f  are of  antisymmetric)  symmetric  for  f o r s=1.  two i r r e d u c i b l e s.  For  s=1,  and  one  has  subspaces one  has  associated with A  symmetry  2  (totally  E s y m m e t r y . F o r s=0, t h e r e i s one  w i t h A, symmetry a n d one w i t h E s y m m e t r y . F o r e x a m p l e ,  Equation AlI•4  1 U r n E f, s.i>  . -k  \- t f  V W& r  f 1 t  + <fT(r,r ) f  flf, f, ~) ]  Equation  AlI•5  121  Equation  A I I ' 3P h y s i c a l Asymptotic  S t a t e s of T o t a l E l e c t r o n Spin  These s t a t e s a r e found <L.^;„  &  an  S  determines  must  have  A  AlI•6  i n the d i r e c t product symmetry.  2  The  value  s p a c e c.  x  Cf  of S uniquely  the i r r e d u c i b l e representation r^""' associated  with  t h e s p a c i a l wave f u n c t i o n . For  S=3/2, r'P" = A, a n d t h u s 1  E and thus  r*P  mmS'i  5*'s--|  r ^ " = A . F o r S=1/2, r r 4  5  2  m  =  = E.  v r  f>=  ||!jcrV\/l,S=l>]5--|S.  f,f> Equat ion A l I • 7  I Vcjem S=£ S  z  5  * i p> = J ^ f U i m E £ ^ l > I S - I S  p>  t  Equation  Ittm 5 = i S  z  *-o  J ^ f - U i ^ E f , s « o > I S - l S . f*f>  Equation Since the Hamiltonian the  group  between  and thus  operations states  representations  of  the T matrix are invariant  S , 3  belonging of  AlI•8  S  3  or  under  the matrix elements of T vanish to  to  AlI•9  different  different  rows  irreducible of  the  same  1 22  irreducible element  representation  theorem  condition  that  (Tinkham  S  =  < YJlm  states  of t o t a l  not  diagonal  Since  also  we  have  matrix  imposed t h e  i s diagonal  in  the  elements are E f, s>  Equation  All•10  Equation  AlI•11  Equation  AlI•12  Z  have  electron  in this  to a general  o f Ps s p i n ( I I )  C o n s i d e r t h e Ps s p i n  states  according  1964).  E S, s IT | k i m  A II-4 Physical States  These  3 /  s i s conserved, the T matrix  a b o v e b a s i s . The d i a g o n a l T ^  of S  states:  A  2  spin  symmetry, but a r e not w e l l i n general.  representation.  defined  Thus, t h e T m a t r i x  is  123  A II»5 The T m a t r i x  i n |klmll^ss > Representation z  'Define T» (II,«.;J I. .')« S  ,  <\cjlm  ,  <  S  I I . s s J T I  k J J ^ r ' l l . S ; > S  --Ss  Equation The  matrix  |klmll ss > z  following  z  <klmlI ss z  z  |klmSS^sp>  and  the  T  r e p r e s e n t a t i o n ( i n t e r m s o f T^,* ) a r e  tables.  AlI•13  matrix  i n the  given  i n the  T a b l e A I I . 1 . The  Matrix  1  s  s  /z  3  1  i  /  1  1  1  /  0  1  I  /  -1  1  1  o  1  1  1  0  0 1  1 o  1  0  -1  I  o  0  D  I  D  i  1  1  -1  1  D  1  '1  I  -I  1  -1  0  0  1  -1  1  1  o  D  l  O  o  o  1  -1  0  o  0  0  0  o  1  ;  z  i  i  1  %  % \  |klmSS s*>  z  r  I  /  3/  2  1  &  D  /  '/z  %  -l/z  -\  'A  f  <klmlI ss  0  0  Yz  Vz  -%  ~>/z  Yi  K  '/z  0  •Vz  J Vis  '43  4  %  -s \  '4 -l  % -1  'Vfz -Vfz  1 25  Table A l l . 2 .  I  The M a t r i x  1 1 /  s'  K i  r  i  /  r  i  -J  o  o  1  1  s  s  1 1  1 1  1  1  '  1  /  /  0  /  -1  /  0  o  o  /  0  i  1  /  -1  i  0  1  -1  i  -1  /  -1  o  o  '  -/  1  0  o  o  o  o  1 -1  o  o  o  o  /  1  1  1  1  o  -/  i  1 o  /  1  /  1  1  -/  1  O 1  0 o 0  i /'  1 o  -/  /  1  1  -/  -1  -1  O  1  1 o  O  1  o -1 o  1 o -/  o  o  0  0 o o o  d  d  b  d d  b  -d  -d  b  d  b  d  d b  d d d  b  -d  -d  c  d  d  b  -d Q  d  -d  b  e. e e  e.  0 o  3 d =  1 0  | T | k l m l " ! ^ s's'^  z  q  o  o  1  0  o  i  t  x  1  1  1  <klml I s s  ~~ 'kjj  3  £  ~  ^ ' let  126  A I I ' 6 The S p i n C o n v e r s i o n  Cross  The s c a t t e r i n g a m p l i t u d e  S e c t i o n f o r s=1  f o r a given t r a n s i t i o n  I'l^'s's^.  — >II ss2. i s given as 2  kK CO  The t o t a l c r o s s s e c t i o n  f o r such  Equation  AlI•14  Equation  AlI•15  a transition  Equat ion A l I • 1 6 The  o-Ps-->p-Ps  averaging final  over  Since  initial  cross  section  is  obtained  s t a t e s o f o-Ps w i t h s=1 and summing  by over  s t a t e s o f p-Ps 3  =  conversion  i-s^s. -IT  27  k  *  "^Jj-t+n  the s c a t t e r i n g matrix  " >ki  Equation  3s i s unitary T^ k  may be  AII-17  parameterized  i n terms of a s i n g l e r e a l a n g l e , t h e s c a t t e r i n g phase s h i f t scattering electron  in  a  s t a t e o f Ps o r b i t a l  s p i n S, and a t o m i c -, • r 5 *  spin  a n g u l a r momentum/,  for total  s.  Equation  AlI•18  1 27  Thus t h e c o n v e r s i o n g>  cross  s e c t i o n c a n be r e w r i t t e n . r <~7h 'I r>A / 7 -2.  1-1 L-r 27 k J-co  Equation A l l • 1 9  The s p i n e x c h a n g e c r o s s  k ~  r  HI 6  as  <4  Equation field,  where ( 1 0 ) and ( 0 0 ) h y p e r f i n e  Ps a r e mixed^, t h e r e l e v a n t L,  i s defined  JL=d  In a l a r g e magnetic of  section  conversion  rate  All-20 states  i s from I= z  ±1  =0.  27  k  J.=o  J  ' Equation  A I I - 7 The T o t a l C r o s s The t o t a l  cross  Section  section  f o r s=1  AlI•21  128  J :  J-or  =  = ±JE 3 k*-  The  (2i+\)L  12. (2U))l  t o t a l cross cross  whereas t h e c o n v e r s i o n  2  |T^'|  +  Z  2 s i n * 5,  JT^'j  4-5/"  off  J  t h e c a s e of Ps o f o r b i t a l  +  inelastic  1,  be  conserved.  collisions  angular  This  spin  (Arthurs  All-23  momentum 1, s c a t t e r i n g  momentum  involving  1960) i n a t o t a l  j , invariance  angular  momentum, J  a l l o w s f o r t h e p o s s i b i l i t y of rotational  average c r o s s s e c t i o n f o r s c a t t e r i n g j'  Equation  Molecule  a molecule with r o t a t i o n a l angular  j  AlI•22  cross .section i s zero.  under r o t a t i o n s r e q u i r e s o n l y t h a t t h e t o t a l =  Equation f o r s=0 i s s i m p l y  A II•8 Generalization to a 2 Electron In  J  excitation.  from r o t a t i o n a l  electron spin state S  The  s t a t e j --> (molecular  s ) c a n be w r i t t e n  J5» where T (j'l'jl)  i s the T matrix  Equation  AlI•24  element'between i n i t i a l  state  129  jl  and  final  scattering  state  j ' l ' . The  from i n i t i a l  state  average  jll ss  — ?  z  written  ^  x  cross  j'l'I^ss'^  ^j)  % a  section can  ! T fjU'r r, s. -,j/i^Or r  corresponding  ,  ,  spin conversion  ^ ' ' J ' ^ f f c -  As w i l l be J  S  S  7  in  the  low  potential.  L  0  ;  J  -  Thus t h e  ^  ,  (e  In t h i s  that  o cross  section ,  )  '  T  O'^-T  W ' )  Equation  AlI•26  Equation  AIL27  section I)  1, where R i s t h e r a n g e o f  the  the phase s h i f t s molecule.  .  section  /  Equation  averaged  over  All.28  rotational  i s simply  v  s t a t e of t h e  kR <<  ^  s t a t e s of t h e m o l e c u l e  Note  -  6  ' ^  AlI•25  limit,  conversion  ^-^  limit  cross  ^  following  ^  energy  \ J J 27  7  ^  shown i n t h e  * Cj'l' ji)  be  J= ^ j ' l  Equation The  for  *  J  are  Equation independent of the  All-29  rotational  130  A II«9 E v a l u a t i o n  SSs  of T  We assume t h a t S and molecular  ,,  ( j 1 ; j 1) i n t h e L i m i t kR << 1  t h e s c a t t e r i n g s t a t e s of t o t a l  s p i n s a t low energy e v o l v e  electron  spin  i n time a c c o r d i n g t o  some e f f e c t i v e H a m i l t o n i a n H - - i S where B an  0  +$>J9"+rt  + V  S  *  Equation  i s the r o t a t i o n a l constant  effective  local  c o n s i d e r matrix elements of V  * S d r J a I>  and V ^  3  is  i n t e r a c t i o n between t h e Ps and t h e m o l e c u l e  w h i c h d e p e n d s on t h e t o t a l e l e c t r o n Now  f o r the molecule  AIL30  s p i n S and m o l e c u l a r  s p i n s.  S s  <kJ-J*jjelkjj-^»><kjj.^lrw>  J lit  / V \ r w )< P w Ik 'j' j > iX>< ic j j . ' ^ V-' / *'JT, J '> €  Equation  All•31  where f i s t h e v e c t o r f r o m t h e P s cm t o m o l e c u l a r CM a n d u i s Si the molecular a x i s u n i t v e c t o r . V ( f , u ) c a n t h e n be e x p a n d e d a s  V rra)« s  ^  <f(r)ftr.a)•  ^  ^)Yjf)Y»1*>\ Equation  where r-o,  we h a v e assumed i n v a r i a n t under u — >  t o be a r e a l  finite  function  AlI•32  of f and  -o. Using  Equation  AlI•33  131  It  Equation  All•34  Equation  AlI•35  follows:  1- J U i j* j  ''2.  Equat ion A l l - 3 6 At  300°K, k f o r Ps i s 0 . 1 1 3 A " , 1  0.163A" . 1  at  600°K  is  still  I f we assume t h a t t h e p o t e n t i a l i s n e g l i g a b l e  3A, t h e n t h e i n t e g r a l o v e r except  and  1=L=1'=0.  approximated to  Thus  r i s small the  matrix  f o r a l l values element  of  of V  5 s  only  for R > 1 L 1', can  be  132  ^  ho  S ' kK  JJ  A  ) dr^  7^  E q u a t i o n A l l -37  0  This implies that the scattering involving  i s dominated  by  only the i s o t r o p i c  the  elastic  component  molecule  i n t e r a c t i o n . The c o r r e s p o n d i n g m a t r i x e l e m e n t  s  wave  p a r t o f t h e Ps i n the  T  m a t r i x c a n t h e r e f o r e be e x p r e s s e d  '  u  JJ  Equation  where t h e p h a s e s h i f t s 6^* a r e i n d e p e n d e n t This  i s precisely  the  rigid  rotor  of j .  c o n c l u s i o n reached  D a l g a r n o c o n c e r n i n g low e n e r g y  scattering  of  by A r t h u r s and  electrons  ( A r t h u r s 1 9 6 0 ) . They f o u n d t h a t s c a t t e r i n g  spherical potential  y[r>£> ) ~  f  Rydberg, It CCjLj this  1  CD  v<  elastic  3a*  r>34„  which corresponds  T  M  t o a wave v e c t o r k l e s s t h a n 0.187 A. from t h e C l e b s h Gordon  j ^ ) i n Equation A l l - 3 6 that  low energy  limit.  All'39  s-wave f o r e l e c t r o n e n e r g i e s b e l o w 0.01  i s immediately c l e a r j  a  o f f a non  Equation completely  by  such as  -5J£3 - o-rjl  was  All-38  j  z  coefficient  i s also conserved i n  133  APPENDIX  I I I DIRECT THERMALIZATION OF MUONIUM I N THE VOIDS OF OXIDE POWDERS  The  failure  necessitates  t h e ATTD  model  an a l t e r n a t i v e e x p l a n a t i o n  the v o i d r e g i o n s possibility  of  that  at  low  Mu  temperature.  thermalizes  (see Section  VI.2.6)  f o r how Mu emerges In  this  directly  appendix,  i n the voids  into the is  examined. A cross as  s e c t i o n o f t h e Mu-powder p o t e n t i a l c a n be  i n Figure Al11•1.  Given that there  is a collision  imagined  of a  70A DIA. ~3A  CM I >  0)  Figure AIII.1. Imagined powder g r a i n p o t e n t i a l .  J  o  Mu atom  cross  section  of  (E < 6 eV) w i t h a powder g r a i n t h e r e a r e f o u r  t h e Mu-  possible  outcomes. 1.  The atom i sscattered e l a s t i c a l l y o f f the surface, losing a f r a c t i o n of i t s energy. Since t h e thermal wavelength f o r a 100°K Mu atom i s much l e s s t h a n t h e a t o m i c d i m e n s i o n s , momentum i s t r a n s f e r r e d p r i m a r i l y t o i n d i v i d u a l a t o m s . Thus t h e mean e n e r g y l o s s i s 3mE/M, where m i s t h e Mu mass a n d M i s t h e s u r f a c e atom mass.  134  2.  The atom may be c a p t u r e d i n a s u r f a c e s t a t e v i a phonon i n t e r a c t i o n . T h i s i s o n l y important near thermal e n e r g i e s .  3.  The atom e n t e r s t h e g r a i n w i t h k i n e t i c e n e r g y E - V , l o s e s energy v i a phonons, and t h e r m a l i z e s w i t h i n t h e g r a i n . I n this case, t h e Mu m i g h t become t r a p p e d i n t h e l a t t i c e a t v e r y low t e m p e r a t u r e s i f t h e d i f f u s i o n r a t e i s s m a l l .  4.  The atom e n t e r s t h e g r a i n w i t h k i n e t i c e n e r g y E - V , l o s e s energy v i a phonons and i s e x p e l l e d from t h e s u r f a c e before thermalization. T h i s c a n be t h o u g h t o f a s a g e n e r a l i z a t i o n of i n e l a s t i c s c a t t e r i n g .  0  0  For  E < V ,  energetically. than and  o u t c o m e s 1 and 2  0  It  may  are  the  only  allowed  be assumed t h a t f o r Mu e n e r g i e s  some t h r e s h o l d e n e r g y E ^  ( d e p e n d e n t on t h e t y p e  greater  of  powder  t h e g r a i n r a d i u s , R ) , outcome 4 d o m i n a t e s . Thus t h e r e  effect, within  an  energy  t h e powder  The  window  i s most  probability  transmitted  into  <  0  E < E^  a  particle  powder  where t h e r m a l i z a t i o n  grain  of  energy,  E,  c a n be e s t i m a t e d  f o r square p o t e n t i a l  V  (Leighton  < E. I n one d i m e n s i o n  i s , in  likely.  for  the  V  transmission probability 0  ones  barrier  to  be  from the  of  height  1959),  Equat ion Al11•1 Therefore,  a sufficient  in the voids s m a l l over E  t h  i s for (E ^ - V )/V +  as:  at  0  <<  1, 0  can  be  estimated  i n d i v i d u a l atoms u n t i l 0  0  t h e e n t i r e e n e r g y window V <  enters a grain loses  V ,  c o n d i t i o n f o r Mu t o t h e r m a l i z e  which  point  energy  by  since  is  then  E <E^ .  assuming  through  ^(E)  directly  t h a t a Mu atom w h i c h  elastic  i t reaches a c r i t i c a l i t becomes t r a p p e d .  collisions  kinetic  energy,  with E fr  E ^ c a n be a p p r o x i m a t e d  135  Equation AlII-2  t h  where l R /l 2  i s t h e mean s q u a r e d  2  i s t h e mean number  2  scatter  randomly  the  l o s i n g an a v e r a g e order  of  the  other  of c o l l i s i o n s  required  radius  of  the  mean  atomic  spacing  below  2  chosen  50°K,  a  (R) ( R e i f  1965),.  should  2  be  2  value  for  E^ -  V  0  is  t o be 100°K. The a b o v e p a r a m e t e r s , -  V  ~  0  0.033eV.  In  i f a Mu atom i s t o d e p o s i t t h e l a s t o f i t s k i n e t i c  i n s i d e t h e g r a i n must n o t e x c e e d a b o u t The work f u n c t i o n a t t h e s u r f a c e height  must  be  does not r e e n t e r the S i 0 to  on  ( 1 0 A ) . S i n c e Mu  e n e r g y w i t h i n a powder g r a i n and s t o p , t h e n i t s k i n e t i c  barrier  and  f o r a Mu atom t o  I  squared  i n t o E q u a t i o n AIII«2, y i e l d E ^  words  collisions  grains  0  arbitrarily  inserted  between  2m(E - V ) / M p e r c o l l i s i o n .  becomes t r a p p e d i n S i 0 somewhat  distance  illustrate  2  2 eV, w h i c h y i e l d s  2U<  0.033 eV. i s n o t known, a l t h o u g h t h e  much g r e a t e r  t h a n 300°K, s i n c e t h e Mu  powder a t room t e m p e r a t u r e .  the f e a s i b i l i t y 0.13  of t h i s model, V over  energy  0  the  entire  which  may  In  order  i s chosen  t o be  energy  window  ( V , E ^ ). 0  Another probability  important for direct  factor  thermalization  window  (V  be e s t i m a t e d a s n  0 f  = (E//, ~ O ) / ^ - ^ A > where AE  l o s s per surface c o l l i s i o n . Si0  2  powder.  i n the  sensitive  E ^ ) . I n an e v a c u a t e d powder s a m p l e , V  c  e  U s i n g AE  the  i n t h e v o i d s i s t h e number  o f c o l l i s i o n s a Mu atom makes w i t h t h e s u r f a c e energy  influence  is  the  this  may  energy  = 3mE^/M, r\ = 1.5 i n t h e c  The p r e s e n c e o f a b u f f e r gas s u c h a s He w i l l  t o r e d u c e n^ due t o t h e a d d i t i o n a l e n e r g y l o s s  between  tend  surface  1 36  collisions. atoms (E^ A  is  If  the  mean  d i s t a n c e between c o l l i s i o n s w i t h gas  i s l e s s than t h a t between s u r f a c e c o l l i s i o n s , - V )/AE  . In t h i s  0  E  U  =  the  J  energy  n  c  ~  equation  6  mean  then  loss  Equation  through e l a s t i c  AIII-3  c o l l i s i o n s with the  b u f f e r g a s ( o f d e n s i t y n, s wave s c a t t e r i n g c r o s s s e c t i o n  and  mass  path  ) after  between (an  powders  R  =  distance 35A  free  = 10A , n = 1 0 2  = M , = 3720 M e V / c  2 1  2  H<  surface  collisions  gem" ) i n t o E q u a t i o n 3  t o t h i s model, the p r o b a b i l i t y  thermalizes d i r e c t l y processes  i n t h e v o i d s of oxide  in  cm"  3  and d Si0  2  AIII«3. y i e l d s  t h e work  that  when t h e p a r t i c l e  f u n c t i o n f o r Mu a t t h e o x i d e  t h a t t h e mean f r e e p a t h  t h a n t h e mean f r e e p a t h  size  surface  atom  collisions.  threshold is  small  i s l a r g e . The probability  between gas c o l l i s i o n s  between s u r f a c e  Mu  powders i s d e t e r m i n e d  p r e s e n c e o f a b u f f e r g a s s u c h a s He may e n h a n c e t h i s provided  a  o c c u r r i n g below t h e e l e c t r o n e x c i t a t i o n  (E < ' 6 e V ) . I t i s most p r o b a b l e and  ( t h e mean  Inserting  between  p =0.04  d  =0.68 eV a n d n,= 0.04. According  by  distance  i d e a l g a s a t 760 t o r r a n d 7 ° K ) , (mean  e  a  surface c o l l i s i o n s ) .  = 1800A  AE  travelling  is  less  1 37  APPENDIX I V  ADSORPTION OF ATOMS ON A SURFACE  Consider with  an  gas atoms c o n t a i n e d  adsorptive  surface  e n e r g y t o t h e s u r f a c e , and N  of  w i t h i n a v o l u m e , V, i n c o n t a c t area  be t h e  s  A. L e t €  number  0  of  be t h e b i n d i n g atoms  s u r f a c e , N^ t h e number o f atoms i n t h e g a s . D e f i n e n<j  =  N /V.  will, Two  The a d s o r p t i o n  3  isotherm  n  5  versus  i n g e n e r a l , d e p e n d on t h e atom m o b i l i t y  ideal  situations,  considered  i n more d e t a i l  s  the  = N /A  and  5  n^ a t c o n s t a n t on  the  Adsorption of a smooth s u r f a c e .  2.  Adsorption adsorption  Van  der  Waals  o f t i g h t l y bound atoms on s i t e s per u n i t area.  Van Der W a a l s Two D i m e n s i o n a l  The t o t a l  surface.  by (Dash 1975) a r e  energy of N  s  g a s atom o f a r e a a  surface  e on a  with  t  is  0  the  binding  atoms on t h e s u r f a c e i s  energy  momentum o f atom i , and U ( r , .. interaction  between  atoms  \./e  Gas  Equation where  T  here.  1.  A IV«1  reviewed  n  on  to  r^)  the  -is the  AIV-1  s u r f a c e , p/ i s t h e total  energy  on t h e s u r f a c e . I n t h e c a s e o f  of hard  discs,  7~Jn  6 The p a r t i t i o n  function, Z  for the adsorbed  Equation atoms  AIV'2  138  V M ™  6  where A= (h /2rrmkT) 2  i s t h e t h e r m a l de B r o g l i e  an e l e m e n t o f p h a s e s p a c e , <5  )  e  d  Equation  AIV«3  wavelength,  dr i s  and Q i s g i v e n by: r  '-' ^  Equation  AIV-4  At l o w d e n s i t i e s , where N « / A << 1, s  The  The  free energy,  chemical  F, f o r t h e a d s o r b e d  Equation  AIV-5  Equation  AIV-6  Equation  AIV'7  atoms i s t h e n  potential  2 V  The an  chemical p o t e n t i a l  f o r t h e atoms i n t h e g a s p h a s e  (assuming  i d e a l g a s ) i s g i v e n by  a* = kT In T n«/lJ 3  J  Equation  AIV-8  139  In e q u i l i b r i u m , n =  ^  = ^ , and thus  n ^ A ^  &  At v e r y low d e n s i t i e s , n monolayer tends  .co>mpletion  i slinear  s  a s er\ t e n d s s  t o z e r o , so t h a t n  geometric  Equation  e  1. Near  t o 1 and Q (Equation A I I I ' 1 )  i s bounded by 1 /'« (we have i g n o r e d t h e  s  packing factor  i n n ^ , s i n c e n^tye «  AIV.9  from t h e s t a r t ) .  A IV.2 T i g h t B i n d i n g Model In t h i s c a s e , t h e f r e e energy  for theN  a s u r f a c e w i t h N = k/a t i g h t b i n d i n g s i t e s  F = -kT In [  %  |  n  [ i -  atoms a d s o r b e d  3  i s ( D a s h 1975)  UUUU'&) 'N.l)c l  r W ] Equation  where for  n^c  i s the f r a c t i o n a l coverage.  the adsorbed ~  The c h e m i c a l  AIV-10  potentiial  atoms i s  " ^°  *  ^ £ I - Ylsd]  E q u a t i o n AIV-1 1  In t h e r m a l e q u i l i b r i u m t h e c h e m i c a l p o t e n t i a l o f gas phase ( g i v e n by E q u a t i o n A I V - 8 ) s u r f a c e atoms, *  on  may  be  equated  with  that  atoms  of the  yielding: A  At low c o v e r a g e ,  — rij A x\,o «  + &  Equation  1, n  a  i slinear  in n 3  AIV-12  1 40  6* /k-T  3  £  E q u a t i o n AIV«13  w h e r e a s f o r l a r g e n^ , n*. = \/e. Note tight  that  a t a low c o v e r a g e ,  binding yield a linear  b o t h Van D e r W a a l s m o d e l a n d  r e l a t i o n s h i p between n  the p r o p o r t i o n a l i t y constant d i f f e r i n g  by a f a c t o r  a n d n^ w i t h  s  A /* 2  (-1  for  He a t 7 ° K ) .  A I V - 3 S i n g l e Atom A d s o r p t i o n I f a s i n g l e atom  ( P s o r Mu) i s t h e r m a l i z e d i n a s y s t e m w i t h  s u r f a c e a r e a A and t o t a l the  fraction  much l a r g e r between  f r e e volume V , F  of time spent-on  i t i s d e s i r a b l e t o know  the s u r f a c e , averaged over  times  t h a n t h e d w e l l t i m e on t h e s u r f a c e o r t h e mean  time  stickings.  This  fraction,  c, may be o b t a i n e d f r o m t h e  above low d e n s i t y a p p r o x i m a t i o n s f o r n  s  a n d n^ .  (A = n.  +  U4 Yr Y\* P E q u a t i o n AIV»14  1 /)/\  E q u a t i o n AIV-15  f o r a 2 - d i m e n s i o n a l g a s bound t o t h e s u r f a c e by t . 0  141  Equation for t i g h t l y  bound a t o m s .  A I V ' 4 Mean S u r f a c e D w e l l It the  Time  i s a l s o o f i n t e r e s t t o e s t i m a t e t h e mean t i m e  surface  occurs  AIV-16  per  (Crampton  sticking 1980).  to  the  surface  spent  on  before desorption  The d e n s i t y o f atoms on t h e s u r f a c e c a n  a l s o be w r i t t e n ij  ls  J  E q u a t i o n AIV- 17  where \y i s t h e mean t h e r m a l v e l o c i t y a n d t h a t an atom w h i c h of  an  atom  to  i n c i d e n t momentum transferred case  to  probability  s t r i k e s the surface w i l l  adsorb.  The s t i c k i n g  a s u r f a c e i s an i n e l a s t i c  process  i n which the  of to  the  the  atom  lattice  and  i t s binding  energy  v i a phonon i n t e r a c t i o n .  of a low d e n s i t y 2 d i m e n s i o n a l g a s ,  may be e q u a t e d  i s the  the  above  are  In the  expression  t o E q u a t i o n AIV«8 y i e l d i n g  Equation  AIV-18  142  BIBLIOGRAPHY  Abragam, A., 1 9 6 1 , N u c l e a r M a g n e t i s m ,  O x f o r d P r e s s , L o n d o n , 126.  Anderson,  C. D.,  1933, P h y s . Rev. 4 3 , 4 9 1 .  Anderson,  C. D. a n d S. H. N e d d e r m e y e r , 1937, P h y s . R e v . 5J_, 884.  Anderson,  P. W.,  1951, C. R. A c a d .  S c i . 8 2 , 342.  Ashcroft, N. W. a n d N. D. M e r m i n , 1976, H o l t R i n e h a r t and W i n s t o n , 781. Aston, J . G., S. V.' R. M a s t r a n g e l o J . Chem. P h y s . 2 3 , 1633. A r t h u r s , A. M. a n d A. D a l g a r n o , 540. Blackett, P. M. a n d A139, 6 9 9 . B r a n d t , W,  State Physics,  R. 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K i e f l , 1981, Thermalization of Muonium in Oxide Powders at Low Temperature, Hyperfine Interactions 8, 359. G.M. M a r s h a l l , J . B . Warren, C . J . Oram, and R.F. K i e f l , 1981, A Search for Muonium to Antimuonium Conversion, To be published in Phys. Rev. D. C . J . Oram, C . A . F r y , J . B . Warren, R.F. K i e f l , and J . H . Brewer, 1981, Observation of the 2S State of Muonium in Vacuum, To be published i n J . Phys. B. Y . J . Uemura, C.Y. Huang, C.W. Clawson, J . H . Brewer, R.F K i e f l , and A.M. de G r a f f , 1981, Z e r o - F i e l d MSR in an Insulator Spin G l a s s , Hyperfine Interactions 8, 757.  

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