UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

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

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1982_A1 K54.pdf [ 6.6MB ]
Metadata
JSON: 831-1.0085513.json
JSON-LD: 831-1.0085513-ld.json
RDF/XML (Pretty): 831-1.0085513-rdf.xml
RDF/JSON: 831-1.0085513-rdf.json
Turtle: 831-1.0085513-turtle.txt
N-Triples: 831-1.0085513-rdf-ntriples.txt
Original Record: 831-1.0085513-source.json
Full Text
831-1.0085513-fulltext.txt
Citation
831-1.0085513.ris

Full Text

MUONIUM AND POSITRONIUM AS MICROPROBES OF SURFACES AND SOLIDS by ROBERT FRANCIS XAVIER KIEFL B.Sc. C a r l e t o n U n i v e r s i t y , 1976 M.Sc. The U n i v e r s i t y of B r i t i s h C olumbia, 1978 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES P h y s i c s Department We ac c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA January 1982 ^ R o b e r t F r a n c i s X a v i e r K i e f l I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e head o f my department 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 . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f 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 C o l u m b i a 2075 Wesbrook P l a c e V a n c ouver, Canada V6T 1W5 i i ABSTRACT The p r o p e r t i e s of muoniumU'e') and p o s i t r o n i u m ( e + e ~ ) are a l t e r e d s i g n i f i c a n t l y i n the presence of m a t t e r . The study of t h e s e e x o t i c H - l i k e atoms p r o v i d e s a unique p e r s p e c t i v e on atomic i n t e r a c t i o n s w i t h atoms, s u r f a c e s , and s o l i d s . T h i s theme i s e x p l o r e d i n a v a r i e t y of h o s t s . The c r o s s s e c t i o n f o r s p i n 1 p o s i t r o n i u m t o be c o n v e r t e d t o s p i n 0 p o s i t r o n i u m d u r i n g c o l l i s i o n s w i t h 0 2 m o l e c u l e s has been measured from 120 °K t o 630 °K i n an S i 0 2 powder moderator u s i n g a p o s i t r o n l i f e t i m e t e c h n i q u e . The r e s u l t s i n d i c a t e t h a t p o s i t r o n i u m does not t h e r m a l i z e i n the powder below 450 °K. 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 i n c r e a s e s s l i g h t l y w i t h t e mperature above 450 °K. A t h e o r y f o r s p i n c o n v e r s i o n of p o s i t r o n i u m by a s p i n 1 m o l e c u l e i s d e v e l o p e d and used t o i n t e r p r e t the d a t a . Muon S p i n R o t a t i o n measurements, i n S i 0 2 , A l 2 0 3 , and MgO powders a t low temperature i n an atmosphere of He i n d i c a t e t h a t muonium emerges from the s u r f a c e s r e g a r d l e s s of the ambient temperature of the powder. The muonium s p i n r e l a x a t i o n r a t e i n A l 2 0 3 i n a He(or Ne) atmosphere i s found t o be a l i n e a r f u n c t i o n of the f r a c t i o n of s u r f a c e a r e a not c o v e r e d by adsorbed He(or Ne). The 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 o f f a dsorbed He and Ne atoms have been measured t o be 11.0±0.2 A 2 and 8.9±0.2 A 2, r e s p e c t i v e l y . The f i r s t o b s e r v a t i o n s of muonium i n the condensed phases of A r , K r , and Xe are p r e s e n t e d . The d a t a i n d i c a t e t h a t t h e r e i s a h i g h p r o b a b i l i t y of muonium f o r m a t i o n i n a l l c a s e s . The s p i n r e l a x a t i o n r a t e of muonium i n s o l i d Xe i s t e n ti m e s t h a t i n the l i q u i d , where the random l o c a l f i e l d s from the n u c l e a r moments of 1 3 9 X e and 1 4 ^ e a r e averaged by a d d i t i o n a l t r a n s l a t i o n a l m o t i o n . i v TABLE OF CONTENTS INTRODUCTION 1 I . POSITRONS AND POSITRONIUM 3 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 i o n 4 2. Ps A n n i h i l a t i o n 5 3. E x p e r i m e n t a l Techniques 6 1. L i f e t i m e Technique 6 2. A n g u l a r C o r r e l a t i o n 7 3. Doppler Broadening 8 4. Quenching of o-Ps i n M a t t e r 8 I I . POSITRONIUM IN S i 0 2 POWDER 10 1 . Ps Formation 10 2. Ps T h e r m a l i z a t i o n i n S i 0 2 Powder 12 3. E f f e c t of 0 2 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 Below 0.l9eV .. 14 3. Oxygen H y p e r f i n e T r a n s i t i o n s Below 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. Quenching of o-Ps i n S i 0 2 Powder 17 1 . S p e c i a l Case X.B t « 1 ( A d i a b a t i c A p p r o x i m a t i o n ) 19 2. S p e c i a l Case k B t >> 1 ( S t r o n g C o l l i s i o n A p p r o x i m a t i o n ) 19 5. E f f e c t of 0 2 on the Quenching of o-Ps i n S i 0 2 Powder 21 I I I . TEMPERATURE DEPENDENCE OF CONVERSION QUENCHING OF ' o-Ps BY 0 2 IN S i 0 2 POWDER 24 1 . E x p e r i m e n t a l 25 2. Procedure and R e s u l t s 28 3. D i s c u s s i o n 31 1 . T h e r m a l i z a t i o n 34 2. Anomolous S p i n Exchange i n o-Ps + 0 2 S c a t t e r i n g 35 4. Summary and C o n c l u s i o n s 37 V IV. MUONS, MUONIUM AND ^ + SR 38 1. Source of P o l a r i z e d Muons 39 2. Muon Decay 41 3. Muon S p i n R o t a t i o n 42 1. Free Muons i n a T r a n s v e r s e Magnetic F i e l d 44 2. Free Muonium i n a T r a n s v e r s e Magnetic F i e l d ... 45 4. Mu S p i n R e l a x a t i o n 50 1. Random L o c a l M a g n e t i c F i e l d s (RLMF) 51 2. Random A n i s o t r o p i c D i s t o r t i o n 52 3. Chemical R e a c t i o n 53 4. S p i n Exchange 53 5. The »»*SR Spectrum i n a T r a n s v e r s e F i e l d 54 V. MUONIUM IN INSULATING POWDERS 55 1 . Mu Format i o n 55 2. Mu T h e r m a l i z a t i o n 58 3. Mu Bound S t a t e s on Oxide S u r f a c e s 60 4. Mechanisms f o r Mu S p i n R e l a x a t i o n i n a Powder .... 61 1. N u c l e a r Magnetic Moments 62 2. Paramagnetic I m p u r i t i e s 63 3. M o t i o n a l Narrowing 66 4. Random A n i s o t r o p i c D i s t o r t i o n 66 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 in. a Powder 67 1. S p e c i a l Case X 6 t << 1 ( A d i a b a t i c A p p r o x i m a t i o n ) ._. 67 2. S p e c i a l Case x.5 t » 1 ( S t r o n g C o l l i s i o n A p p r o x i m a t i o n ) 68 6. E f f e c t of Adsorbed I n e r t Gas on the S p i n R e l a x a t i o n 69 V I . LOW TEMPERATURE STUDY OF MUONIUM IN A l 2 0 3 , S i 0 2 AND MgO POWDERS 7 0 1. Mu i n the V o i d s of Oxide Powders a t 6°K 71 1. E x p e r i m e n t a l D e t a i l s 71 2. E l e c t r o n i c s 73 3. A n a l y s i s and R e s u l t s 75 4. D i s c u s s i o n 78 1. .Mu i n S i 0 2 Powder a t 6°K i n a He Atmosphere 79 2. Mu i n MgO Powder a t 6°K 81 3. Mu i n A 1 2 0 3 Powder (5°K - 20°K) 82 5. Summary and C o n c l u s i o n 83 2. S p i n R e l a x a t i o n of Mu i n A 1 2 0 3 Powder w i t h Adsorbed He/Ne 84 1. E x p e r i m e n t a l D e t a i l s 84 2. E l e c t r o n i c s 88 3. Procedure 88 4. A n a l y s i s and R e s u l t s 89 5. D i s c u s s i o n 90 1. A d s o r p t i o n I s o t h e r m s of He on A l 2 0 3 90 2. Mu Sp i n R e l a x a t i o n i n A l 2 0 3 Powder W i t h Adsorbed He 92 3. A d s o r p t i o n Isotherms of Ne on A l 2 0 3 95 4. Mu S p i n R e l a x a t i o n i n A 1 2 0 3 Powder W i t h Adsorbed Ne 95 6. S t a t u s of the ATTD Model 96 7. C o n c l u s i o n 97 v i V I I . MUONIUM IN THE CONDENSED PHASES OF A r , Kr AND Xe .. 98 1 . E x p e r i m e n t a l 99 2. Data A n a l y s i s and 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 and S o l i d Ar 103 2. Mu i n L i q u i d and S o l i d Xe 106 3. Mu i n L i q u i d and S o l i d Kr 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 IN A POWDER 112 APPENDIX I I . PS SCATTERING OFF 2 ELECTRON ATOM (MOLECULE) 118 1. T o t a l E l e c t r o n S p i n S t a t e s 119 2. S p a c i a l S t a t e s 120 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 of T o t a l E l e c t r o n S p i n 121 4. P h y s i c a l S t a t e s of Ps s p i n (IIz,) 122 5. The T m a t r i x i n | k l m l l i s s i > R e p r e s e n t a t i o n 123 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=1 126 7. The T o t a l C r o s s S e c t i o n 127 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 128 9. E v a l u a t i o n of T s M j 1 ; j ' l') i n the L i m i t kR « 1 130 APPENDIX I I I . DIRECT THERMALIZATION OF MUONIUM IN THE VOIDS OF OXIDE POWDERS 133 APPENDIX IV. ADSORPTION OF ATOMS ON A SURFACE 137 1. Van Der Waals Two D i m e n s i o n a l Gas 137 2. T i g h t B i n d i n g Model 139 3. S i n g l e Atom A d s o r p t i o n 140 4. Mean S u r f a c e D w e l l Time 141 BIBLIOGRAPHY 142 v i i LIST OF FIGURES CHAPTER I 1. The Energy Spectrum For o-Ps A n n i h i l a t i o n 5 2. Decay Scheme For 2 2 N a 6 CHAPTER I I I 1. Apparatus For Measuring o-Ps L i f e t i m e In A Powder 26 2. E l e c t r o n i c s For Measuring o-Ps L i f e t i m e 27 3. 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 2 Powder 29 4. Decay Rate of o-Ps V e r s u s 0 2 C o n c e n t r a t i o n 30 5. C o n v e r s i o n Rate C o n s t a n t Of Ps V e r s u s Temperature 31 CHAPTER IV 1. Muon Decay Parameters . . 42 CHAPTER VI 1. The v + SR Apparatus "Beaver" 71 2. The Target C r y o s t a t Assembly 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 2 76 5. Temperature Dependence of the Mu S p i n R e l a x a t i o n Rate i n A l 2 0 3 Powder 79 6. The v + SR Apparatus " E a g l e " 85 7. The Target V e s s e l And C r y o s t a t Used To Study Mu Sp i n R e l a x a t i o n V e r s u s He/Ne Coverage 86 8. The Gas H a n d l i n g System 87 9. Mu S p i n R e l a x a t i o n Rate In A l 2 0 3 V e r s u s Adsorbed He 90 10. Mu S p i n R e l a x a t i o n Rate In A l 2 0 3 V e r s u s Adsorbed Ne 91 CHAPTER V I I 1. The Target V e s s e l For Condensed Noble Gases 99 2. Two Freqency P r e c e s s i o n Of Mu In S o l i d Ar 102 3. Mu P r e c e s s i o n In L i q u i d And S o l i d Ar 104 4. Mu P r e c e s s i o n In S o l i d Kr 107 APPENDIX I I I 1. Imagined C r o s s S e c t i o n Of The Mu Powder G r a i n P o t e n t i a l 133 v i i i LIST OF TABLES CHAPTER I I I 1. 0 2 C o n v e r s i o n Rate Constant V e r s u s Temperature .... 32 CHAPTER VI 1. »i*SR R e s u l t s i n B u l k and Powdered Oxides 78 2. P r o p e r t i e s Of Oxide Powders 82 CHAPTER V I I 1. (i*SR r e s u l t s i n condensed A r , K r and Xe 100 2. Mu F r a c t i o n In Gas Phase A r , Kr And Xe 101 APPENDIX I I 1. The M a t r i x < k l m l l r s s z 2. The M a t r i x < k l m l l x s s i klmSS z Sfi> k l m l I x s s z > 124 1 25 i x ACKNOWLEDGEMENT I f e e l v e r y f o r t u n a t e t o have had the o p p o r t u n i t y t o work (and a t tim e s laugh) w i t h the numerous peop l e who d i r e c t l y and I n d i r e c t l y h e l p e d me d u r i n g the c o u r s e of t h i s r e s e a r c h . My s u p e r v i s o r s Dr. John B. Warren and Dr. J e s s e H. Brewer have been i n v a l u a b l e s o u r c e s of a d v i c e and encouragement w h i l e never i n f r i n g i n g upon my s c i e n t i f i c freedom. I o f f e r them my s i n c e r e s t t h a n k s . Much of the work c o n t a i n e d i n t h i s t h e s i s r e q u i r e d the e x p e r t i s e <. of v e r y c a p a b l e p e o p l e i n the e x e c u t i o n s t a g e of the e x p e r i m e n t s . For t h i s I thank Dr. C h r i s J . Oram, Dr. Glen M. M a r s h a l l , C a r l W. Clawson, Dave P. Spencer, Dale R. Harshman, and C.A. F r y . The e n t i r e MSR group has always been accomodating and h e l p f u l so I take p l e a s u r e i n t h a n k i n g them. Dr. Dave M. Garner d e s e r v e s s p e c i a l thanks f o r h i s a s s i s t a n c e i n the computing a s p e c t of the ex p e r i m e n t s (even a t 3:00 AM i n the mo r n i n g ) . I am ve r y g r a t e f u l t o Dr. B i r g e r B e r g e r s e n f o r s e v e r a l h e l p f u l d i s c u s s i o n s on the q u e s t i o n of Ps t h e r m a l i z a t i o n i n a powder and , t o Dr. John B e r l i n s k y f o r d i s c u s s i o n s on the t h e o r y of s p i n exchange. I would a l s o l i k e t o thank a l l the s t a f f a t UBC, e s p e c i a l l y A l Morgan, A l B i s h o p , C h r i s S t e v e n s , Doug Maas, f o r t h e i r 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 f o r d r a f t i n g many of the f i g u r e s i n t h i s t h e s i s . My w i f e Robin s t i l l l o v e s me a f t e r a l l t h o s e owl s h i f t s . Her u n d e r s t a n d i n g support and l o v e have always been a source of comf o r t and s t r e n g t h and I d e a r l y thank her f o r them. F i n a l l y I thank R i c h a r d and Theresa K i e f l who have gave me l o v e , encouragement and the o p p o r t u n i t y t o do my b e s t . I am X f o r e v e r g r a t e f u l and I d e d i c a t e t h i s t h e s i s t o you b o t h . 1 INTRODUCTION Pure l e p t o n i c atoms such as muonium (Mu) (»*e~) and p o s i t r o n i u m (Ps) (e*e~) p l a y an i m p o r t a n t r o l e i n p a r t i c l e • p h y s i c s . T h e i r p r o p e r t i e s a re f r e e from s t r o n g i n t e r a c t i o n e f f e c t s so they p r o v i d e an i d e a l t e s t i n g ground f o r e l e c t r o m a g n e t i c and weak i n t e r a c t i o n t h e o r i e s . Thus the p a r t i c l e p h y s i c i s t i s i n t e r e s t e d i n the p r o p e r t i e s of f r e e atoms, such a s : 1. The a n n i h i l a t i o n r a t e and decay mode of 1 3S and 1 1S s t a t e 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+) c o n v e r s i o n p r o b a b i l i t y (weak i n t e r a c t i o n ) . 3. The h y p e r f i n e s p l i t t i n g (1 3 S - 1 1S) and the Lamb s h i f t ( 2 P x . -2Sj£ s p l i t t i n g ) of both Mu and 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 ) . In c o n t r a s t , i t i s the d e v i a t i o n s from f r e e atom be h a v i o u r i n the presence of ma t t e r t h a t a r e of i n t e r e s t t o the muon or p o s i t r o n s c i e n t i s t , concerned w i t h t h e ' c h e m i c a l or p h y s i c a l p r o p e r t i e s of the probe and/or h o s t . The Ps or Mu atom may f u n c t i o n as a microprobe of i t s environment and can p r o v i d e a unique p e r s p e c t i v e on atom-molecule, a t o m - s u r f a c e , or a t o m - s o l i d i n t e r a c t i o n s . Very few t e c h n i q u e s a r e s e n s i t i v e t o the a c t i o n of i n d i v i d u a l atoms on a time s c a l e as s m a l l as 10" 9 s. The fundamental decay p r o p e r t i e s of the e + and n* i n m a t t e r , w e l l known from p a r t i c l e p h y s i c s , make such o b s e r v a t i o n s p o s s i b l e . T h i s t h e s i s i s a c o l l e c t i o n of f o u r e x p e r i m e n t s concerned w i t h the b e h a v i o u r of these H - l i k e atoms i n m a t t e r : 2 1. 1 3S — > 1 1S s p i n c o n v e r s i o n of Ps o f f 0 2 m o l e c u l e s i n an S i 0 2 powder moderator between 120-630 °K. 2. Mu p r o d u c t i o n i n the v o i d s of S i 0 2 , A l 2 0 3 , and MgO powders at 6°K. 3. Mu i n t e r a c t i o n w i t h A l 2 0 3 powder s u r f a c e s w i t h adsorbed He or Ne between 7-30°K. 4. The f i r s t o b s e r v a t i o n s of Mu i n the condensed phases of A r , K r , and Xe. The f i r s t t h r e e c h a p t e r s a r e devoted t o Ps. The f i r s t c h a p t e r p r o v i d e s t h e n e c e s s a r y background i n f o r m a t i o n on Ps and the methods of s t u d y . The second c h a p t e r i s concerned p r i m a r i l y w i t h the c h a r a c t e r i s t i c s of Ps decay i n S i 0 2 powder, w i t h and w i t h o u t paramagnetic 0 2 gas. The reader i s r e f e r r e d t o Appendix I I 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 of Ps i n c i d e n t o f f a s p i n 1 atom or m o l e c u l e . The f i r s t experiment i s p r e s e n t e d i n Chapter I I I . The l a s t f o u r c h a p t e r s may be c l a s s i f i e d as the Mu p a r t of the t h e s i s . Chapter IV p r o v i d e s background i n f o r m a t i o n on Mu and the 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 of Mu and the »**SR spectrum i n o x i d e powders a r e of p r i m a r y concern i n Chapter V. The second and t h i r d e x p e r i m e n t s a r e p r e s e n t e d i n Chapter VI whereas the l a s t experiment i s the s u b j e c t of Chapter Chapter V I I . 3 CHAPTER I : POSITRONS AND POSITRONIUM In t h e b e g i n n i n g t h e r e was D i r a c (1930), who p o s t u l a t e d t h a t v a c a n c i e s i n a f i l l e d sea of n e g a t i v e energy e l e c t r o n s t a t e s would m a n i f e s t themselves p h y s i c a l l y as p o s i t i v e l y c harged p a r t i c l e s or a n t i - e l e c t r o n s . Anderson (1933) was the f i r s t t o observe p o s i t r o n s i n c l o u d chamber photographs of cosmic r a y showers. The p r o d u c t i o n of p o s i t r o n - e l e c t r o n p a i r s from h i g h energy gamma r a y s was observed s h o r t l y a f t e r w a r d s ( B l a c k e t t 1933). These e x p e r i m e n t a l r e s u l t s s p a r k e d a l a r g e e f f o r t t o d e v e l o p a t h e o r y f o r p o s i t r o n s i n m a t t e r . P i r e n n e (1946) was one of the f i r s t t o pe r f o r m c a l c u l a t i o n s on the energy l e v e l s of p o s i t r o n i u m (e*e~) whose e x i s t e n c e was f i r s t p o s t u l a t e d by M a h o r o v i c i i (1934). Wheeler (1946) and Ore and P o w e l l (1949a) c a l c u l a t e d a n n i h i l i a t i o n r a t e s from s=0 and s=1 ground s t a t e s . Meanwhile e x p e r i m e n t a l s t u d i e s on Ps were j u s t •becoming p o s s i b l e as p o s i t r o n s o u r c e s such as 2 2 N a and 6 < 1Cu became a v a i l a b l e . The work of Deutsch (1951) f i r m l y e s t a b l i s h e d the e x i s t e n c e of p o s i t r o n i u m . - D e s p i t e the g r e a t amount of e x p e r i m e n t a t i o n s i n c e those e a r l y days i t was not u n t i l 1974 t h a t the f i r s t e x c i t e d s t a t e of Ps was observed ( C a n t e r 1975), made p o s s i b l e by the development of monoenergetic beams of low energy p o s i t r o n s (Canter 1972). In t h i s c h a p t e r , the decay p r o p e r t i e s of f r e e p o s i t r o n s and Ps i n m a t t e r and the e x p e r i m e n t a l t e c h n i q u e s i n p o s i t r o n a n n i h i l a t i o n a r e revi e w e d . 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 i o n S i n c e p o s i t r o n s a r e a n t i - e l e c t r o n s , the p o s i t r o n - e l e c t r o n s t a t e i s u n s t a b l e t o a n n i h i l a t i o n i n t o gamma ra y s through 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 . The charge c o n j u g a t i o n o p e r a t o r t r a n s f o r m s every p a r t i c l e i n t o i t s a n t i p a r t i c l e . For a fermio n a n t i f e r m i o n s t a t e such as e*e~, the charge c o n j u g a t i o n or C-p a r i t y i s where Jt i s the r e l a t i v e o r b i t a l a n g u l a r momentum and s i s the t o t a l s p i n ( W i l l i a m s 1971). Gammas or photons a r e s e l f c o n j u g a t e w i t h C - p a r i t y -1 by v i r t u e of the t r a n s f o r m a t i o n p r o p e r t i e s of the e l e c t r o m a g n e t i c v e c t o r p o t e n t i a l under the C o p e r a t i o n . S i n c e 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 s c o n s e r v e C - p a r i t y , t he a n n i h i l a t i o n of Ps i n t o n gammas must be such t h a t i+s+n i s even. I f the a n n i h i l a t i o n o c c u r s from an ^ =0 s t a t e ( i . e . : t he ground s t a t e of* P s ) , t h i s r e s t r i c t s t he a n n i h i l a t i o n from s=0 and s=1 s t a t e s ( i . e . : 1 1S and 1 3S atomic s t a t e s of Ps) t o an even and odd number of gammas r e s p e c t i v e l y . A n n i h i l a t i o n i n t o a s i n g l e gamma cannot conserve both momentum and energy so the s=1 s t a t e must decay i n t o on odd numbers of photons g r e a t e r than 1. The a n n i h i l a t i o n r a t e i n t o n gammas i s p r o p o r t i o n a l t o c n where o (=1/137) i s the f i n e s t r u c t u r e c o n s t a n t , so t h a t the s=0 and S=1 s t a t e s decay p r i m a r i l y i n t o two and t h r e e gammas r e s p e c t i v e l y . 5 1*2 Ps A n n i h i l a t i o n When p o s i t r o n s a r e i n j e c t e d i n t o matter they may c a p t u r e an e l e c t r o n t o form para-Ps ( w r i t t e n p-Ps or P s ( 1 1 S ) ) or o r t h o - P s ( w r i t t e n o-Ps or P S ( 1 3 S ) ) , i n a s t a t i s t i c a l r a t i o of one t o th r e e . . As s t a t e d above, the p-Ps decays i n two 511 KeV gammas whereas o-Ps decays i n t h r e e gammas w i t h a c o n t i n u o u s energy spectrum (see F i g u r e 1.1). The mean decay r a t e ( 1 / l i f e t i m e ) i n ENERGY (mc2) F i g u r e 1*1 The energy spectrum f o r o-Ps a n n i h i l a t i o n (Ore 1949a). vacuum f o r p-Ps and o-Ps have been measured t o be 799±11 x 10 7 s " 1 ( T h e r i o t 1967) and 0.7056±0.0007 x 10 7 s " 1 ( G i d l e y 1978) r e s p e c t i v e l y . 6 I»3 E x p e r i m e n t a l Techniques The study of p o s i t r o n s i n ma t t e r i s based on d e t e c t i o n of the a n n i h i l a t i o n quanta. The r e l e v a n t o b s e r v a b l e s a re the mean a n n i h i l a t i o n r a t e , the a n g l e between the gammas, the energy, p o l a r i z a t i o n and number of gammas. The most commonly used means of 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 a r e the l i f e t i m e , a n g u l a r c o r r e l a t i o n and d o p p l e r b r o a d e n i n g t e c h n i q u e s (West 1973). 1-3*1 L i f e t i m e Technique 2 2 N a s o u r c e s a re o f t e n used f o r measuring p o s i t r o n l i f e t i m e s because the decay p o s i t r o n i s f o l l o w e d i n most decays by the e m i s s i o n of a n u c l e a r gamma of energy 1274 KeV w i t h i n 10" ^ s (see F i g u r e 1*2). The time d e l a y between the n u c l e a r gamma and the a n n i h i l a t i o n quanta can be measured w i t h F i g u r e 1*2 Decay scheme f o r 2 2 N a . s c i n t i l l a t i o n d e t e c t o r s . S m a l l p l a s t i c s c i n t i l l a t o r s ( 2cm l o n g x 2cm d i a m e t e r ) p r o v i d e e x c e l l e n t t i m i n g r e s o l u t i o n (2.5 x 1 0 " 1 0 s) but s u f f e r 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 are e s s e n t i a l when s t u d y i n g s h o r t l i f e t i m e s i n s o l i d s and l i q u i d s . The r e l a t i v e l y l o n g l i f e t i m e of o-Ps i n gases and powders may be s t u d i e d u s i n g l a r g e r Nal d e t e c t o r s which have much b e t t e r energy r e s o l u t i o n and e f f i c i e n c y , a l t h o u g h the t i m i n g r e s o l u t i o n (4 t o 5 x 10' 9 s) i s not as good as may be a c h i e v e d w i t h p l a s t i c s c i n t i l l a t o r s . I•3•2 A n g u l a r C o r r e l a t i o n The a n g l e between two photons from e + e~ (s=0) a n n i h i l a t i o n i s g i v e n as & = p j _ / m 0 G p4.^< maC (west, 1973) E q u a t i o n 1-1 where p x i s the p a i r momentum component p e r p e n d i c u l a r t o d i r e c t i o n of e m i s s i o n and m0 i s 'the e l e c t r o n res't mass. The a n g u l a r d i s t r i b u t i o n between the a n n i h i l a t i o n quanta can be measured u s i n g a l o n g s l i t a n g u l a r c o r r e l a t i o n a p p a r a t u s which measures the c o i n c i d e n c e c o u n t i n g r a t e between two d e t e c t o r s as a f u n c t i o n of a n g l e d e f i n e d by d e t e c t o r 1, the p o s i t r o n s o u r c e , and d e t e c t o r 2. A t y p i c a l a n g u l a r r e s o l u t i o n i s 0.5 mrad. The decay of mean t h e r m a l i z e d p-Ps c o n t r i b u t e s a narrow component t o the a n g u l a r d i s t r i b u t i o n of the a n n i h i l a t i o n quanta s i n c e the p a i r momentum (of o r d e r (kT2m)"^ ) i s s m a l l i n comparison t o a n n i h i l a t i o n s 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 . 8 I'3*3 Doppler Broadening I n f o r m a t i o n on the p a i r momentum d i s t r i b u t i o n can 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 or G e - L i gamma d e t e c t o r t o measure the Doppler b r o a d e n i n g of the a n n i h i l a t i o n l i n e a t 511 KeV. In f i r s t o r d e r the s h i f t i n energy AE = h v ~ m . c z ~ p„c/2 ( H o t z ^ 1 9 6 8 ) E q u a t i o n l m 2 where hi/ i s the energy of the d e t e c t e d gamma and p„ i s the component of p a i r momentum a l o n g the d i r e c t i o n of e m i s s i o n . The Doppler broadening t e c h n i q u e a n a l y z e s a l l momentum c h a n n e l s s i m u l t a n e o u s l y and i s t h e r e f o r e much f a s t e r and does not r e q u i r e h i g h e + s t o p p i n g d e n s i t i e s nor s t r o n g s o u r c e s as does the a n g u l a r c o r r e l a t i o n t e c h n i q u e . However the r e s o l u t i o n of 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 t o around 1 KeV a t 511 KeV. 1-4 Quenching of o-Ps i n M a t t e r The p r o p e r t i e s of o-Ps a r e s i g n i f i c a n t l y a l t e r e d i n the pres e n c e of m a t t e r . The decay r a t e of o-Ps i n matter can be ex p r e s s e d X=X.0+X.^ where X 0 i s the f r e e decay r a t e and X.^  i s the quenching r a t e a s s o c i a t e d w i t h the medium, due p r i m a r i l y t o s p i n c o n v e r s i o n and/or p i c k o f f . S p i n c o n v e r s i o n i s when o-Ps i s c o n v e r t e d t o p-Ps v i a a s p i n exchange i n t e r a c t i o n w i t h paramagnetic s p e c i e s such as H, NO, and 0 2 . T h i s i s co v e r e d i n more d e t a i l i n Appendix I I . In s h o r t , c o l l i s i o n s i n v o l v i n g o-Ps and a paramagnetic m o l e c u l e do not conse r v e the z component of s p i n of the e l e c t r o n on Ps and thus o-Ps—>p-Ps c o n v e r s i o n • i s p o s s i b l e . P i c k o f f quenching o c c u r s when the p o s i t r o n i n o-Ps 9 a n n i h i l a t e s w i t h a v a l e n c e e l e c t r o n from the h o s t . These two quenching p r o c e s s e s a re e a s i l y d i s t i n g u i s h a b l e u s i n g a n g u l a r c o r r e l a t i o n or -doppler b r o a d e n i n g . S p i n c o n v e r s i o n t o p-Ps r e s u l t s i n a narrow p a i r momentum d i s t r i b u t i o n d e t e r m i n e d by the t h e r m a l motion of the p-Ps atom. In c o n t r a s t , p i c k o f f a n n i h i l a t i o n w i t h a v a l e n c e e l e c t r o n from the host r e s u l t s i n a r e l a t i v e l y broad p a i r momentum d i s t r i b u t i o n , d e termined p r i m a r i l y by the h i g h momentum of the v a l e n c e e l e c t r o n . For example the mechanism f o r quenching of o-Ps by 0 2 i n s i l i c a g e l (Chuang I974)and s i l i c a powder ( K i e f l 1978)moderators has been d e t e r m i n e d t o be s p i n c o n v e r s i o n as opposed p i c k o f f a n n i h i l a t i o n r e s u l t i n g from a c h e m i c a l r e a c t i o n . 10 CHAPTER I I : POSITRONIUM IN S i 0 2 POWDER 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 o x i d e powders e x h i b i t t h r e e components (Brandt 1968). A v e r y f a s t (< 1 ns) component i s due t o p-Ps and f r e e e + d e c a y ( w h i l e s l o w i n g down or t h e r m a l i z e d ) . A 'second 2-3 ns component i s a t t r i b u t e d t o p i c k o f f a n n i h i l a t i o n of o-Ps w i t h i n the powder g r a i n s and the l o n g e s t 1 i v e d component (~140 ns) i s thought due t o o-Ps i n the v o i d r e g i o n s of the powder. In t h i s c h a p t e r the f o r m a t i o n , t h e r m a l i z a t i o n , and quenching of o-Ps i n S i 0 2 powder w i t h and w i t h o u t 0 2 are d i s c u s s e d . T h i s i s r e l e v e n t t o e x p e r i m e n t a l r e s u l t s i n Chapter I I I on the c o n v e r s i o n quenching of o-Ps w i t h 0 2 i n an S i 0 2 moderator. 11•1 Ps F o r m a t i o n There a r e a t l e a s t t h r e e models t h a t may be used t o e x p l a i n Ps f o r m a t i o n i n o x i d e powders the Ore gap model, the spur model, and the s u r f a c e f o r m a t i o n model. These models are not v e r y u s e f u l i n making q u a n t i t a t i v e p r e d i c t i o n s of the Ps f r a c t i o n , e s p e c i a l l y i n such a complex medium as a powder. The purpose of s t a t i n g them here 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 u n d e r s t a n d i n g of the v a r i o u s p r o c e s s e s which may l e a d t o Ps format i o n . A c c o r d i n g t o the Ore gap model (Ore 1949b), f i r s t proposed f o r gases and and then extended t o m o l e c u l a r s o l i d s ( W a l l a c e 1960), 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 charge exchange w i t h 11 atom (or m o l e c u l e ) A E g u a t i o n 11•1 i n the energy r e g i o n ( E J 0 K ) -E B) < E < Eex (termed the Ore gap) where E, o n i s the i o n i z a t i o n energy of atom, E e x i s the f i r s t e x c i t a t i o n energy, and E g (6.8eV) i s the b i n d i n g energy of Ps. Below E; o n - E B , Ps f o r m a t i o n i s e n e r g e t i c a l l y f o r b i d d e n and above E e x e x c i t a t i o n and i o n i z a t i o n a r e thought t o dominate the dE/dx. In the spur model (Mogensen 1974) a t h e r m a l i z e d e* i s a t t r a c t e d t o , and 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 i t s own r a d i a t i o n t r a c k (composed of s m a l l branches c a l l e d s p u r s ) . T h i s model has been p a r t i c u l a r l y s u c c e s s f u l i n e x p l a i n i n g e x p e r i m e n t a l r e s u l t s i n l i q u i d s . In the s u r f a c e f o r m a t i o n model p o s i t r o n s c a p t u r e an e l e c t r o n a t the s u r f a c e where i t may be e n e r g e t i c a l l y f a v o u r a b l e . Such s u r f a c e f o r m a t i o n has been o b s e r v e d w i t h low energy p o s i t r o n s i n c i d e n t on m e t a l or o x i d e c o a t e d m etal s u r f a c e s (Canter 1974). P r e v i o u s t o these r e s u l t s Brandt had proposed on the b a s i s of 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 2 , MgO and A l 2 0 3 powders t h a t o-Ps i s formed w i t h i n the powder g r a i n s and then d i f f u s e s t o the s u r f a c e where i t i s e j e c t e d i n t o the v o i d s , presumably because of a n e g a t i v e work f u n c t i o n a t the s u r f a c e (Brandt 1968). The e v i d e n c e p r e s e n t e d f o r t h i s ambient te m p e r a t u r e t h e r m a l d i f f u s i o n (ATTD) model i s t h a t the 2-3 ns component ( a t t r i b u t e d t o o-Ps i n s i d e the powder g r a i n s ) i n c r e a s e s a t the expense of the 140 ns component ( a t t r i b u t e d t o o-Ps i n the v o i d s ) as the t emperature i s lowered or the p a r t i c l e 12 s i z e i n c r e a s e s . However the l i f e t i m e s p e c t r a may a l s o be c o n s i s t e n t w i t h s u r f a c e f o r m a t i o n so t h a t the r o l e of s u r f a c e s i n Ps f o r m a t i o n i n o x i d e powders remains u n c l e a r . II-2 Ps T h e r m a l i z a t i o n i n S i 0 2 Powder Whether i t formed a t the s u r f a c e or i n the b u l k the o-Ps appears t o be e j e c t e d from the o x i d e s u r f a c e w i t h a k i n e t i c energy of o r d e r 1eV ( F o r d 1976). For example the k i n e t i c energy of o-Ps e j e c t e d from MgO powder has been measured t o be 0.25 ± 0.10 eV ( C u r r y 1971). The Ps atom i s v e r y l i g h t , so t h a t the mean energy l o s s per c o l l i s i o n w i t h the o x i d e s u r f a c e i s much s m a l l e r than f o r a h e a v i e r atom such as Mu. In a p u r e l y c l a s s i c a l e s t i m a t e , i n which the Ps i m p a r t s momentum t o i n d i v i d u a l s u r f a c e atoms, F o r d (1976) e s t i m a t e s the t h e r m a l i z a t i o n time i n l i g h t l y packed S i 0 2 powder t o be -140 ns. A 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 of the problem employing the one d i m e n s i o n a l D e v o n s h i r e quantum t h e o r y of g a s - s u r f a c e i n t e r a c t i o n i s d e v e l o p e d i n Appendix I . We f i n d t h a t i n S i 0 2 powder (35 A r a d i u s , d e n s i t y of 0.056 gem" 3) at 121°K the c a l c u l a t i o n y i e l d s ~30 ns f o r Ps of 1eV t o reach 0.0l25eV (145°K) w i t h most of t h a t time (25 ns) spent below 0.03eV. T h i s c a l c u l a t i o n uses the Debye temperature of b u l k S i 0 2 (470°K) (Zemansky 1968) and a p u r e l y r e p u l s i v e Morse p o t e n t i a l a t the s u r f a c e w i t h range parameter a=0.5 A" 1. The r e s u l t i s not s e n s i t i v e t o the p o t e n t i a l parameters and agrees r o u g h l y w i t h the p u r e l y c l a s s i c a l e s t i m a t e of F o r d (1976). I t s h o u l d be p o i n t e d out t h a t i t i s assumed i n t h e s e c a l c u l a t i o n s t h a t the Ps atom samples the e n t i r e s u r f a c e a r e a 13 w i t h e q u a l p r o b a b i l i t y . I t does not take i n t o account the aggre g a t e s t r u c t u r e of the powder which may have d r a s t i c e f f e c t s on the mean f r e e p a t h and t h e r m a l i z a t i o n t i m e . Thus the above r e s u l t must be c o n s i d e r e d a lower l i m i t on the t h e r m a l i z a t i o n t i m e . In f a c t the e x p e r i m e n t a l r e s u l t s p r e s e n t e d i n Chapter I I I i n d i c a t e t h a t the Ps has a t h e r m a l i z a t i o n time much l a r g e r than t h i s i n S i 0 2 powder below 450 K. II»3 E f f e c t of 0 2 on Ps T h e r m a l i z a t i o n 0 2 i n t r o d u c e d i n t o the v o i d r e g i o n s of a powder s h o u l d tend t o d e c r e a s e the t h e r m a l i z a t i o n t i m e . Low energy o-Ps (< 5eV) may l o s e energy through c o l l i s i o n s w i t h 0 2 v i a e l a s t i c s c a t t e r i n g , h y p e r f i n e t r a n s i t i o n s , r o t a t i o n a l e x c i t a t i o n and v i b r a t i o n a l e x c i t a t i o n . However, we f i n d t h a t below ~0.03 eV the major c o n t r i b u t i o n i s from e l a s t i c s c a t t e r i n g which most l i k e l y i s i n s u f f i c i e n t t o cause t h e r m a l i z a t i o n a t low 0 2 d e n s i t i e s ( l e s s than 10" 1 9 cm' 3). I I '3-1 V i b r a t i o n a l E x c i t a t i o n The energy l o s s r a t e of Ps due t o v i b r a t i o n a l e x c i t a t i o n can be w r i t t e n where n i s the 0 2 c o n c e n t r a t i o n , v i s the Ps v e l o c i t y , cvv\ i s the 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 from v i b r a t i o n a l s t a t e v t o v', AEVV> = E/ - Ey, i s the energy d i f f e r e n c e between s t a t e s and Equat i o n 11 • 2 P 1 V i s the p r o b a b i l i t y t h a t 14 v i b r a t i o n a l s t a t e v i s o c c u p i e d . The energy of v i b r a t i o n a l s t a t e v i s a p p r o x i m a t e l y Ev - fly (v +~ E q u a t i o n 11 • 3 where A y = 1580 cm" 1 f o r 0 2 m o l e c u l e s ( L e v i n e 1975). At room temp e r a t u r e P 0 -~ 1 and E q u a t i o n I I «2 can be s i m p l i f i e d t o 4? 1 L ~ HIT <£y« AE 0 V< E q u a t i o n I I -4 I f i t i s assumed c0\ ~ 10" 1 6 cm 2 and cB)>> =0 f o r v >1, then Ps of energy 1eV i n 0 2 gas a t a d e n s i t y of 1 0 1 9 cm" 3 w i l l l o s e energy a t a r a t e of 4.8 KeV/ns. However, below the t h r e s h o l d energy ( A E 0 I =.l9eV) t h i s c h a n n e l i s c l o s e d . I I ' 3 - 2 R o t a t i o n a l E x c i t a t i o n Below 0.19eV Only odd r o t a t i o n a l s t a t e s of 1 6 0 2 ( 3Eg) a r e a l l o w e d as a consequence of the s p a t i a l symmetry of the e l e c t r o n s and p e r m u t a t i o n symmetry of the 1 6 0 n u c l e i (Tinkham 1964). In f i r s t o r d e r ( i g n o r i n g the e f f e c t of 0 2 s p i n ) the r o t a t i o n a l e n e r g i e s a r e ( L e v i n e 1975): c m E q u a t i o n I I -5 The r a t e of energy l o s s can be w r i t t e n i | ) r o f = n i r l > Pj<*jj'AEjj' E q u a t i o n I I - 6 where r j = ( 2 j f i ; e / ^ C 2 J i s the p r o b a b i l i t y t h a t the 0 2 m o l e c u l e i s i n r o t a t i o n a l s t a t e j (v - 0 ) , i s the s c a t t e r i n g c r o s s s e c t i o n f o r j --> j ' t r a n s i t i o n s and AEj-j' 15 i s the energy l o s s ( g a i n ) per t r a n s i t i o n . A rough e s t i m a t e on d E / d t ) r o f f o r Ps of energy 0.1eV i n 0 2 a t 300°K can be made by assuming P 7 = 1 ( s i n c e j = 7 i s the most l i k e l y o c c u p i e d s t a t e a t 300 °K) and c79 = 10" 1 6 cm 2 ( w i t h a l l o t h e r <fj-> =0).The e s t i m a t e y i e l d s dE/dt -~ 0.6eV/ns. The t h r e s h o l d 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 1 3 ) i s ^0.014eV.-However, i t i s s e v e r e l y i n h i b i t e d a t e n e r g i e s below ~0.03eV because of the s wave n a t u r e of low energy Ps s c a t t e r i n g c o u p l e d w i t h t o t a l a n g u l a r momentum c o n s e r v a t i o n (see S e c t i o n A l l . 9 ) 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 0.03eV Even when 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 , the Ps may l o s e energy i n e l a s t i c a l l y t h r ough h y p e r f i n e t r a n s i t i o n s between the s t a t e s of t o t a l a n g u l a r momentum J = j + s where s i s the 0 2 s p i n and j i s the r o t a t i o n a l a n g u l a r momentum of the m o l e c u l e . The 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 the same j i s 2.5 x lO"*eV and o n l y weakly dependent on j (Townes 1955). The r a t e of energy l o s s can be w r i t t e n In the a p p r o x i m a t i o n where t o t a l e l e c t r o n s p i n i s c o n s e r v e d , J J ' t r a n s i t i o n s a t low energy must o r i g i n a t e from s p i n exchange(or s z changing, c o l l i s i o n s ) , s i n c e j , and s a r e c o n s e r v e d (see Appendix I I ) . A v e r y crude e s t i m a t e of the energy l o s s r a t e due t o J J ' t r a n s i t i o n s can be made by assuming the e n e r g i e s and c r o s s s e c t i o n s a re independent of j and t h a t « , =er. =er-„ (the s p i n exchange c r o s s s e c t i o n f o r Ps s c a t t e r i n g o f f 0 2 ) . Then i t Equat i o n 11•7 16 f o l l o w s : dE\ ^ n i r [ / - J _ J t /^VP E q u a t i o n I I - B S u b s t i t u t i n g n = I0 1 9cm". 3, <re)C = 4 x 10" 1 9 cm 2 (the s p i n exchange c r o s s - s e c t i o n d e f i n e d as (28/7)*^ (See E q u a t i o n A l l . 2 0 ) where i s the measured 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 ( K l o b u c h a r 1980)), T = 300°K, and AE = 2.5 x l0"*eV y i e l d s d E / d t ) h y ^ ~ 1 0 " 7 eV/ns. Thus i t can be n e g l e c t e d i n comparison t o the r a t e of energy l o s s due t o s u r f a c e c o l l i s i o n s . I I - 3 - 4 E l a s t i c S c a t t e r i n g The energy l o s s r a t e due t o e l a s t i c s wave s c a t t e r i n g a t i n c i d e n t energy E can be w r i t t e n (Mobley 196.7) 4^1 = n t r 2 E n o / M E q u a t i o n I L 9 where i s the e l a s t i c 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 2Em/M i s the mean energy l o s s per c o l l i s i o n f o r Ps of mass n s c a t t e r i n g o f f 0 2 of mass M. There i s c o n s i d e r a b l e u n c e r t a i n t y i n the v a l u e chosen f o r s i n c e the s c a t t e r i n g a t low e n e r g i e s i n v o l v e s o n l y one p a r t i a l wave (see Appenix I I ) . Thus the e l a s t i c c r o s s s e c t i o n can v a r y tremendously depending upon o n l y the s-wave e l e c t r o n s p i n q u a r t e t and d o u b l e t phase s h i f t s ( s e e Appendix I I ) . For example 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 i s o n l y 1 0 " 1 9 cm 2 -- about f o u r o r d e r s of magnitude s m a l l e r the p h y i c a l c r o s s s e c t i o n . However f o r the purpose of i l l u s t r a t i o n we chose ee\ =10" 1 6 cm 2 and n = l 0 1 9 cm" 3. One then f i n d s dE/dt) f i| ^ 1.5 x 10" 5 eV/ns which i s the l a r g e s t c o n t r i b u t i o n t o 17 the energy l o s s r a t e due t o 0 2 but c o n s i d e r a b l y s m a l l e r than the energy l o s s r a t e due t o the S i 0 2 powder. I I • 3 • 5 C o n c l u s i o n The presence of 0 2 gas d e c r e a s e s the time r e q u i r e d t o reach ~ 0.03eV due t o i n e l a s t i c s c a t t e r i n g p r o c e s s e s , but has l i k e l y has l i t t l e e f f e c t a t lower e n e r g i e s where s wave e l a s t i c s c a t t e r i n g dominates ( p r o v i d e d the e l a s t i c c r o s s s e c t i o n i s not much g r e a t e r than 10" 1 6 c m 2 ) . S i n c e the " t h e r m a l i z a t i o n " time i n S i 0 2 powder a t 121°K i s p r i m a r i l y due t o s c a t t e r i n g below 0.03eV i t i s u n l i k e l y t h a t the presence of 0 2 has a s u b s t a n t i a l e f f e c t on t h i s t i m e . I I ' 4 Quenching of o-Ps i n S i 0 2 Powder C o n s i d e r an ensemble of o-Ps i n a powder a t temperature T w i t h s u r f a c e a r e a A and f r e e volume V F=(V - V s a) i (j ) . In the most g e n e r a l c a s e , t h e r e e x i s t b oth bound ( b i n d i n g energy B) and f r e e s t a t e s . L e t X F and \ 3 be the a n n i h i l a t i o n r a t e s f o r o-Ps i n the bound and f r e e s t a t e s . These can be w r i t t e n Ac = V. ^ + X E q u a t i o n 11•10 where i s the c o l l i s i o n f r e q u e n c y w i t h the s u r f a c e , Pq i s the p r o b a b i l i t y f o r p i c k o f f a n n i h i l a t i o n per c o l l i s i o n , kp i s the p i c k o f f r a t e from the bound s t a t e , and X.„ i s the f r e e a n n i h i l a t i o n r a t e (7.056 r s " 1 ) . The bound s t a t e quenching r a t e , X.p , can be e s t i m a t e d from the l i f e t i m e of o-Ps adsorbed on 18 s u r f a c e s of s i l i c a g e l . The observed l i f e t i m e of 30 ns (Chuang I 9 7 3 ) a t 300°K c o r r e s p o n d s t o a quenching r a t e of ~ 26 */S~ 1. P a can be e s t i m a t e d from the o-Ps quenching i n Ar gas (due t o p i c k o f f ) as > a ~ * E q u a t i o n 11•11 where i s the c r o s s s e c t i o n f o r p i c k o f f a n n i h i l a t i o n (1.3 x 10~ 5 A 2 ) ( C e l i t a n s 1964) and e i s t h e a r e a of an Ar atom over which the e l e c t r o n d e n s i t y i s a p p r e c i a b l e (assumed t o be 10 A 2 ) . In c o m p l e t e l y d i s p e r s e d S i 0 2 powder a t 300°K (35 A r a d i u s , d e n s i t y = 0.056 g cm" 3) where the e n t i r e s u r f a c e area i s e q u a l l y a c c e s s a b l e t o the Ps, v<, = 6 x 1 0 ~ 1 1 S ( a c c o r d i n g t o E q u a t i o n A I . 2 4 ) . T h i s y i e l d s v<_ P„ of o r d e r 0.8 (/S" ' but must be c o n s i d e r e d an upper l i m i t on s i n c e c l u m p i n g of the powder g r a i n s i n t o a g g r e g a t e s may i n c r e a s e the mean f r e e p a t h and decrease vc . T h i s p o s s i b i l i t y w i l l be d i s c u s s e d f u r t h e r i n Chapter I I I . The form of the o-Ps l i f e t i m e spectrum depends on how x p t compares w i t h u n i t y , where t i s the mean s u r f a c e d w e l l t i m e , e v a l u a t e d i n Appendix IV. I f the s u r f a c e bound Ps behaves as a 2 d i m e n s i o n a l gas then t can be w r i t t e n : p ^, E q u a t i o n I I •12 where A i s the t h e r m a l wavelength of the o-Ps, v- i s the mean o-Ps v e l o c i t y , and P t i s s t i c k i n g p r o b a b i l i t y on the s u r f a c e . The two l i m i t i n g c a s e s X^t >> 1 and X.pt << 1 can e a s i l y be e v a l u a t e d , whereas the i n t e r m e d i a t e c a s e X^t ~ 1 i s more c o m p l i c a t e d and w i l l i n g e n e r a l not y i e l d a decay spectrum which i s t r a c t a b l e . 19 I I • 4• 1 S p e c i a l Case X B t << 1 ( A d i a b a t i c A p p r o x i m a t i o n ) T h i s i s the o n l y p o s s i b i l i t y c o n s i d e r e d i n the l i t e r a t u r e ( F o r d 1976). I t i m p l i e s t h a t the Ps adsorbs and desorbs many ti m e s b e f o r e decay, l e a d i n g t o a s i n g l e e x p o n e n t i a l decay spectrum. The decay r a t e kfi i s then an average of X.^  and X.p w e i g h t e d by the f r a c t i o n of time spent i n each s t a t e . Xft = A B + 6 ~ ° 0 - V E q u a t i o n 11-13 where a i s the f r a c t i o n of time spent i n the bound s t a t e . In th e r m a l e q u i l i b r i u m o can be e x p r e s s e d ( a c c o r d i n g t o E q u a t i o n A I V.15): j + e ~ F C A / K T \ / F / / M E q u a t i o n 11-14 I I - 4 - 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 A p p r o x i m a t i o n ) T h i s case has not been c o n s i d e r e d p r e v i o u s l y , but the en s u i n g r a t e e q u a t i o n s a re w e l l known from the t r a p p i n g model f o r p o s i t r o n s i n me t a l s (Brandt 1967). The assumption \&t » 1 i m p l i e s t h e r e i s no d e s o r b i n g from the s u r f a c e b e f o r e decay and t h i s l e a d s t o a decay spectrum which i s a sum or d i f f e r e n c e of two e x p o n e n t i a l s . T h i s can be seen as f o l l o w s . D e f i n e n F ( t ) and n B ( t ) t o be the number of o-Ps atoms i n the f r e e and bound s t a t e s r e s p e c t i v e l y . The t r a p p i n g r a t e on the s u r f a c e can be w r i t t e n v, P, , where v. i s the c o l l i s i o n r a t e w i t h l* ."C Cr the s u r f a c e and Pt i s the p r o b a b i l i t y f o r t r a p p i n g (or 20 a d s o r p t i o n ) per c o l l i s i o n . I t i s assumed t h a t the c o u p l i n g between a l i g h t atom such as Ps and the phonons a t the s u r f a c e i s weak so t h a t Pj. << 1. T h i s c o r r e s p o n d s t o t r a p p i n g which i s t r a n s i t i o n r a t e r a t h e r than d i f f u s i o n r a t e l i m i t e d . n f and n^ then s a t i s f y the f o l l o w i n g r a t e e q u a t i o n s . n F = - ) l F n F - n F E q u a t i o n 11-15 = ' ^ f c ^ B +" E q u a t i o n 11-16 The i n i t i a l c o n d i t i o n s n F ( t = 0 ) = n 0 and n f r(t=0) = 0 l e a d t o s o l u t i o n s E q u a t i o n 11-17 r\bfr) = ^ ^ [e- - e J AF-ZW + RV, E q u a t i o n 11-18 The decay spectrum w i l l then be of the form ^ F - A p t V . F i / I p ' ^ + ^ f * E q u a t i o n 11-19 T h i s i s the f a m i l i a r two component s o l u t i o n d e r i v e d by Brandt (1967) f o r p o s i t r o n t r a p p i n g i n d e f e c t s , except t h a t i n t h i s case X.& > \f + v^P± l e a d s 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 r a t h e r than a sum. In h i g h l y d i s p e r s e d powders where Ph « |X K - X.F | 21 E q u a t i o n I I ' 2 0 A l t e r n a t i v e l y i n h i g h l y compacted powders such as s i l i c a g e l where vc » |x.p - X F| n(" t ) ~ n o < 2 E q u a t i o n 11-21 The assumption k^t >> 1 and the consequences d i s c u s s e d above agree q u a l i t a t i v e l y w i t h e x p e r i m e n t s on s i l i c a g e l and S i 0 2 powder. In l i g h t S i 0 2 powder, th e quenching r a t e of o-Ps i s a l i n e a r f u n c t i o n of ( G i d l e y 1976) as g i v e n i n E q u a t i o n 11-20, whereas i n s i l i c a g e l the quenching r a t e of o-Ps i s independent of pore s i z e (Chuang 1972) or e q u i v a l e n t l y , i / , , as suggested by E q u a t i o n 11-21. I I - 5 E f f e c t of 0 2 on the Quenching of o-Ps i n S i 0 2 Powder 0 2 m o l e c u l e s i n the gas phase a r e p a r a m a g n e t i c , p o s s e s s i n g two u n p a i r e d e l e c t r o n s . The c o n v e r s i o n quenching of o-Ps w i t h 0 2 can be w r i t t e n -P< + o. E q u a t i o n 1 1 - 2 2 S i n c e p-Ps has a l i f e t i m e of o n l y 0.125 ns, s p i n c o n v e r s i o n q uenching i s e a s i l y o b s e r v a b l e i n the l i f e t i m e spectrum. At low energy, the o-Ps — > p-Ps c o n v e r s i o n c r o s s s e c t i o n can be w r i t t e n (see Appendix I I ) 22 6^ £- I T s i n 2 7 V E q u a t i o n 11-23 where 6a i s the s c a t t e r i n g phase s h i f t f o r t o t a l e l e c t r o n s p i n S and o r b i t a l a n g u l a r momentum 0, independent of the r o t a t i o n a l s t a t e of the m o l e c u l e . The most r e c e n t room temperature v a l u e of <yc i s 1.0 ± 0.1 x 10" 1 9 cm 2 ( K l o b u c h a r 1980). The p i c k o f f c r o s s s e c t i o n w i t h 0 2 i s of o r d e r 1 0 " 2 1 cm 2 ( e s t i m a t e d from quenching i n pure N 2 gas, C e l i t a n s 1964) and can thus be n e g l e c t e d i n comparison w i t h 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 . When 0 2 gas i s a d m i t t e d i n t o S i 0 2 powder, a c e r t a i n f r a c t i o n w i l l adsorb onto the s u r f a c e . The a d s o r p t i o n of 0 2 on S i 0 2 s u r f a c e s has 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 energy i s r o u g h l y 1100°K e s t i m a t e d from a BET (B r u n a u e r - E m m e t t - T e l l e r ) p l o t (Brunauer 1938) of the a d s o r p t i o n d a t a . ESR data of H atoms on the S i 0 2 s u r f a c e i n the presence of adsorbed 0 2 i n d i c a t e t h a t the 0 2 remains paramagnetic on such a s u r f a c e a t tempe r a t u r e s a t l e a s t as low as 100°K ( S u r i n 1973). F u r t h e r m o r e , 0 2 adsorbed onto the pores of s i l i c a g e l a t 300°K has been shown t o be an e f f e c t i v e c o n v e r s i o n quenching agent of o-Ps (Chuang 1974). Thus when 0 2 gas i s a d m i t t e d i n t o the v o i d s of S i 0 2 , the f r e e and bound a n n i h i l a t i o n r a t e s of o-Ps must be m o d i f i e d from 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 c o n v e r s i o n c r o s s s e c t i o n of f r e e o-Ps w i t h adsorbed 0 2 ( s u r f a c e d e n s i t y n& ), <r* i s the 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 f r e e o-Ps w i t h f r e e 0 2 (gas d e n s i t y n g ) , v 3 i s the v e l o c i t y of f r e e o-Ps, v 5 i s the 2 d i m e n s i o n a l v e l o c i t y of adsorbed o-Ps, <tj~ i s the 2 d i m e n s i o n a l 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 bound o-Ps w i t h bound 0 2 (~<fJ!.z ) t v° z i s the v e l o c i t y of f r e e 0 2 and i s the 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 f r e e 0 2 w i t h bound o-Ps. When n i s much l e s s than the monolayer c o v e r a g e , n 5 ~ Hg/ e E q u a t i o n 11-26 where b = 1100°K i s the 0 2 b i n d i n g energy on S i 0 2 (see E q u a t i o n A I V - 9 ) . E q u a t i o n s 11-13, 11-20 and 11-21 s t i l l h o l d as l i m i t i n g c a s e s . For example, i n h i g h l y d i s p e r s e d powders where X p t >> 1 and v^P^«\kB - X F | E q u a t i o n 11-20 y i e l d s -[ \>, C P« f H + 6l n6) + n3u3 t A.] t E q u a t i o n 11-27 24 CHAPTER I I I : TEMPERATURE DEPENDENCE OF CONVERSION QUENCHING OF o-Ps BY 0 2 IN S i 0 2 POWDER Co n v e r s i o n quenching of o-Ps by paramagnetic 0 2 i n gas moderators a t 300°K has 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 . [See f o r example (K l o b u c h a r 1980)]. At 300°K the s p i n exchange c r o s s s e c t i o n f o r o-Ps + 0 2 ( d e f i n e d i n Appendix I I as 27/8 tim e s 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 ) i s 2 x 10 3 t i m e s s m a l l e r than the s p i n exchange c r o s s s e c t i o n s f o r Mu + 0 2 (Fleming 1981a) and H + 0 2 (Gordon 1981). The o-Ps s p i n exchange c r o s s s e c t i o n w i t h 0 2 ( 4 x 1 0 " 1 9 cm 2) i s c o n s i d e r e d t o be be anomolously low i n comparison w i t h the p h y s i c a l c r o s s s e c t i o n , b e i n g r o u g h l y 1000 times s m a l l e r . In f a c t , some, e a r l y a u t h o r s ( C e l i t a n s I964)were c o n v i n c e d t h a t quenching was not due t o s p i n exchange because i t was so s m a l l . However, a n g u l a r c o r r e l a t i o n measurements (Chuang 1974) and d o p p l e r b r o a d e n i n g measurements ( K i e f l 1978) d i s c u s s e d i n S e c t i o n I».4 p r o v i d e c l e a r e v i d e n c e t h a t the quenching by 0 2 i n the gas phase i s dominated by s p i n c o n v e r s i o n . The temperature dependence of the s p i n c o n v e r s i o n r a t e may h e l p improve our u n d e r s t a n d i n g of t h i s v e r y i n t e r e s t i n g i s o t o p e e f f e c t . In t h i s c h a p t e r a p o s i t r o n l i f e t i m e experiment i s d e s c r i b e d i n which the o-Ps + 0 2 --> p-Ps + 0 2 c o n v e r s i o n r a t e has been measured from 121°K t o 630°K u s i n g an S i 0 2 powder moderator. The r e s u l t s a r e p e r t i n e n t t o both c o n v e r s i o n quenching of o-Ps w i t h 0 2 and t o the b e h a v i o u r of o-Ps i n S i 0 2 powder. 25 111*1 E x p e r i m e n t a l A 3 nCi 2 2 N a p o s i t r o n s o u r c e , p r e p a r e d from a NaCl s o l u t i o n , was d r i e d and sandwiched between 1 am n i c k e l f o i l . The source was embedded i n S i 0 2 powder (mean p a r t i c l e r a d i u s 35A and d e n s i t y 0.056 g e m - 3 ) , s e a l e d w i t h a copper o - r i n g i n a welded s t a i n l e s s s t e e l vacuum chamber ( F i g u r e 111 -1 ) and outgassed at 10" 5 t o r r f o r a p e r i o d of 12 ho u r s . T h i s removes most of the adsorbed H 20 from the s u r f a c e ( C a b o t ) . P r o v i s i o n s were made t o i n p u t e x t r a d r y grade 0 2 gas (99.65% 0 2 , 0.03% A r , 0.05% N 2, 2 ppm C 0 2 , 20 ppm hydrocarbons) v i a a gas h a n d l i n g system c o n s t r u c t e d from 1/4 i n c h s t a i n l e s s s t e e l t u b i n g and s t a i n l e s s s t e e l b e l l o w s v a l v e s . The 0 2 p r e s s u r e w i t h i n the vacuum chamber was measured w i t h a Matheson a b s o l u t e p r e s s u r e gauge a c c u r a t e t o ±5 t o r r . Two copper c o n s t a n t a n t h e r m o c o u p l e s , i n s e r t e d i n s t a i n l e s s s t e e l w e l l s (see F i g u r e 111 - 1 ), were used t o monitor the tempe r a t u r e and i t s u n i f o r m i t y (±2°K over the chamber volume). The temperature was c o n t r o l l e d t o w i t h i n ±2°K over the range of study (121°K t o 630°K). The tar g e t ' chamber was c o o l e d below 300°K by c i r c u l a t i n g c o l d N 2 gas around the v e s s e l h e l d i n a st y r o f o a m c r y o s t a t , whereas h i g h e r t e m p e r a t u r e s were a c h i e v e d w i t h h e a t i n g t a p e . The p o s i t r o n l i f e t i m e measurements were made a t TRIUMF u s i n g the *+SR data a c q u i s i t i o n system (see S e c t i o n V I * 1 * 2 ) . The time d e l a y between the n u c l e a r gamma (1274 KeV) from 2 2 N a (see S e c t i o n I•3•1) and the subsequent p o s i t r o n a n n i h i l a t i o n r a d i a t i o n was measured w i t h two 4 i n c h diameter by 4 i n c h l o n g Nal d e t e c t o r s u s i n g a s t a n d a r d fast.-slow c o i n c i d e n c e c i r c u i t 26 stainless steel thermocouple wells copper-constantan thermocouples welded stainless steel chamber To 0 2 supply, vacuum gauge and vacuum pump copper o-ring i0 2 powder 22 Na positron source Scale cm F i g u r e 111*1 Apparatus f o r measuring o-Ps l i f e t i m e i n S i 0 2 powder i n an 0 2 atmosphere. (see F i g u r e 111*2). T i m i n g i n f o r m a t i o n was o b t a i n e d from the anode output v i a c o n s t a n t f r a c t i o n d i s c r i m i n a t i o n , whereas energy ( p u l s e h e i g h t ) i n f o r m a t i o n was d e r i v e d from the dynode o u t p u t by p a s s i n g the p u l s e t h rough a s p e c t r o s c o p y a m p l i f i e r and s i n g l e c h a n n e l a n a l y z e r . The energy r e s o l u t i o n of these 27 Ortec 473A const, frac di sc. dynocje P o w d e r Nal #1 Ortec 473A const. frac| disc. anode o-Ps decay 22 Na decay followed by 28 MeV y EG&G GPIOO/N p i leup gata EG&G GP100/l|l p ileup gate Ortec 471 spect. amp. | Ortec 471 spect. amp. start stop TDC 100 time d i g i t i z e r Ortec 455 S.C .A. CAMAC MBD-11 PDP-11/40 computer F i g u r e 111•2 E l e c t r o n i c s and dat a a c q u i s i t i o n system f o r measuring 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 event 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 ray and a s i n g l e 28 s t o p p u l s e i n the energy window 400 t o 450 KeV, both w i t h i n a 2 »s g a t e . The time i n t e r v a l between the s t a r t and s t o p p u l s e s was d i g i t i z e d w i t h an EG&G TDC100 c l o c k . The o v e r a l l t i m i n g r e s o l u t i o n was 5 ns FWHM as determined w i t h a 6 0 C o source (which produces two v i r t u a l l y s i m u l t a n e o u s r r a y s a t 1170 and 1330 KeV). The energy window f o r a good s t o p was chosen below the 511 KeV photopeak (due t o 2r a n n i h i l a t i o n ) i n o r d e r t o i n c r e a s e the s e n s i t i v i t y t o o-Ps decay which has a c o n t i n u o u s a n n i h i l a t i o n spectrum by v i r t u e of i t s 3r decay (see F i g u r e 1*1). However, due t o the poor energy r e s o l u t i o n of these d e t e c t o r s and the Gompton s c a t t e r i n g of 511 keV gammas, the s t o p d e t e c t o r was s t i l l s e n s i t i v e t o some 2 gamma a n n i h i l a t i o n s . I l l * 2 P r o c e d u r e and R e s u l t s L i f e t i m e s p e c t r a c o n s i s t i n g of 100,000 e v e n t s were c o l l e c t e d f o r a t l e a s t f i v e 0 2 p r e s s u r e s a t each t e m p e r a t u r e . F i g u r e s 111*3 (a) & (b) show the e f f e c t of 0 2 on the l i f e t i m e spectrum of o-Ps i n S i 0 2 powder. The prompt a n n i h i l a t i o n i s due t o f r e e e + , p-Ps and o-Ps w i t h i n the powder g r a i n s , whereas the l o n g l i v e d component i s due t o o-Ps i n the v o i d r e g i o n s of the powder. Good f i t s were o b t a i n e d assuming a s i n g l e e x p o n e n t i a l decay r a t e over the f i t t i n g range 30-500 ns, where N 0 i s the n o r m a l i z a t i o n , X i s the o-Ps decay r a t e , and Bg i s a time independent background. F i g u r e 111*4 shows two sample p l o t s of X. as a f u n c t i o n of 0 2 c o n c e n t r a t i o n ( i n the gas phase) d e t e r m i n e d from Equat i o n 111*1 29 100000 10000 t-1000 b 100 200 300 400 TIME IN NSEC (5 NSEC/BIN) 600 ib) CO o i _ ) 100000 f 10000 b-1000 b-100 100 200 300 400 TIME IN NSEC (5 NSEC/BIN) 600 F i g u r e I I I - 3 (a) P o s i t r o n l i f e t i m e spectrum i n evacuated S i 0 2 powder a t 295 °K. (b) Same w i t h 0 2 gas d e n s i t y of I 0 , 9 c i r r 3 . the p r e s s u r e . Good f i t s were o b t a i n e d a t a l l temperatues assuming a l i n e a r dependence. The s l o p e of each l i n e g i v e s the t o t a l c o n v e r s i o n r a t e c o n s t a n t k 0 d e f i n e d from 30 X ~ \t V\g i XCftyd) E q u a t i o n I I I - 2 T h i s r a t e c o n s t a n t k c i s p l o t t e d i n F i g u r e I I I - 5 as a f u n c t i o n 0 2 CONCENTRATION (CM"3 X 10,e ) F i g u r e I I I - 4 Decay r a t e of o-Ps v e r s u s 0 2 c o n c e n t r a t i o n (iv, ) a t 295°K and 632°K. of t e m p e r a t u r e . The data p o i n t s a r e g i v e n i n Tab l e 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 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 temperature i n an S i 0 2 powder moderator.. Note t h e r e i s no o b s e r v a b l e dependence on powder d e n s i t y . 111•3 D i s c u s s i o n I t i s c l e a r from F i g u r e 111*5 t h a t the r a t e c o n s t a n t , kc , i s independent of temperature below 450°K. T h i s may seem s u r p r i s i n g i f one e x p e c t s the adsorbed 0 2 t o p l a y a r o l e i n the quenching s i n c e t h e r e i s a l a r g e v a r i a t i o n i n adsorbed 0 2 i n the range 121 t o 450°K. For example, a t a gas d e n s i t y n^ = 1 0 1 9 cm" 3, the d e n s i t y of adsorbed 0 2 , ( a c c o r d i n g t o E q u a t i o n 11*26) v a r i e s from 2 x 1 0 1 1 cm" 2 a t 450°K t o 2.5 x 10 1• cm" 2 a t 121°K. I f one assumes t h a t a o-Ps adsorbed on the s u r f a c e behaves as a 32 Ta b l e 111*1. 0 2 C o n v e r s i o n Rate Constant V e r s u s Temperature Temperature C o n v e r s i o n Rate Powder D e n s i t y Constant (°K)t2 (I0" i zcm 3s"') (g/cc) 645 1.68*0.04 0.056 636 1.75*0.05 0.161 530 1.34*0.04 0.056 422 1.22*0.05 0.056 290 1.26*0.05 0.056 290 1.23*0.04 0.161 183 1.18± 0.04 0.056 121 1.29*0.05 0.056 116 1.22*0.05 0.161 2 d i m e n s i o n a l gas atom w i t h s u r f a c e v e l o c i t y v s = (rrkT/2m) and t h a t the 2 d i m e n s i o n a l c o n v e r s i o n c r o s s s e c t i o n s c a l e s w i t h the 3 d i m e n s i o n a l c o n v e r s i o n c r o s s s e c t i o n (<J£s ~ * ) , then the above 0 2 s u r f a c e c o n c e n t r a t i o n s c o r r e s p o n d t o X.& ( « 5 s v 5 n 3 ) of o r d e r 10 2 «s" 1 and 10 5 u s - 1 a t 450°K and 121°K r e s p e c t i v e l y . The observ e d quenching r a t e a t n 9 = 1 0 1 9 c m - 3 i s o n l y 13 >»s~1 and independent of t e m p e r a t u r e . However, the da t a a r e t o t a l l y c o n s i s t e n t w i t h the " s t r o n g c o l l i s i o n " model assumptions \9t >> 1 and v^P^ « |X.p - X.P | , d e s c r i b e d i n S e c t i o n 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 decay r a t e E q u a t i o n 111*3 Note t h a t i n t h i s l i m i t i n g case X i s l i n e a r l y dependent on n^ a t low s u r f a c e coverage and independent of X^(the bound s t a t e 33 a n n i h i l a t i o n r a t e ) , as o b s e r v e d . The e s s e n t i a l p o i n t i s t h a t X.& i s s u f f i c i e n t l y l a r g e t h a t once t h e o-Ps i s adsorbed on the s u r f a c e i t i s l o s t from the Ps ensemble, independent of p r e c i s e l y how l a r g e X B i s . The t h i r d term i n E q u a t i o n I I I * 3, i / c (P +Pj. ) , due t o quenching by the S i 0 2 a l o n e , i s much l e s s than 1 us" ' , s i n c e the. decay r a t e w i t h no 0 2 i s c l o s e t o the f r e e decay r a t e at a l l t e m p e r a t u r e s (see f o r example F i g u r e I I I . 4 ) . The f i r s t term (*' v n ) i s the c o n v e r s i o n r a t e i n the gas phase, whereas the second term («•/ v, n, ) i s the c o n v e r s i o n r a t e due t o f r e e o-Ps c o l l i d i n g w i t h adsorbed 0 2 . The t o t a l c o n v e r s i o n r a t e c o n s t a n t , k c, (the term i n square b r a c k e t s ) was observed t o be independent of powder d e n s i t y f o r the two d e n s i t i e s s t u d i e d (0.056 gem" 3 and 0.161 gem" 3). S i n c e v0 d i f f e r s i n these two powders by a f a c t o r of a p p r o x i m a t e l y 2.5 (see Appendix I ) , t h i s i m p l i e s t h a t the c o n v e r s i o n r a t e of unbound o-Ps by gas phase 0 2 i s much l a r g e r than the c o n v e r s i o n r a t e of unbound o-Ps by adsorbed 0 2 ( i . e . : v- n, « c' va n. ) . I f one s e t s cc ^ then a t 121°K t h i s r e d uces t o A » n 3 / n 3 - 2 - 5 x 10 cm E q u a t i o n I I I - 4 where d=v3 /v^ i s the mean f r e e p a t h between s u r f a c e c o l l i s i o n s . W h i l e not i n a c c o r d w i t h d = 0.6 x 10" 5 cm, c a l c u l a t e d from E q u a t i o n AI*24 i n Appendix I f o r the h i g h e r d e n s i t y powder, the above l i m i t i s s t i l l q u i t e r e a s o n a b l e s i n c e E q u a t i o n AI*24 i s based on the assumption of s p h e r i c a l p a r t i c l e s e v e n l y d i s t r i b u t e d i n space, and t h e r e f o r e r e p r e s e n t s o n l y a lower l i m i t on d. 34 111•3•1 T h e r m a l i z a t i o n There a r e a t l e a s t two f a c t o r s which i n d i c a t e t h a t the o-Ps i s not t h e r m a l i z e d below 450°K. 1. The r a t e c o n s t a n t k c ~ v g i s independent of T below 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 change below 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 tfg3 v a r i e s as 1 / ( T ) Z below 450°K. There 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 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 of o-Ps by H atoms (Hara 1 9 7 5 ) i n d i c a t e t h a t the 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 energy below t h e r m a l e n e r g i e s . T h i s would g i v e r i s e ^ t o a c o n v e r s i o n r a t e c o n s t a n t p r o p o r t i o n a l t o ( T ) 2 i f the o-Ps i s t h e r m a l i z e d . 2. The r a t e c o n s t a n t g^ v 3 a t 300°K (1.2 ± 0.1 x 1 0 " 1 2 c m 3 s " 1 ) i s s u b s t a n t i a l l y h i g h e r than the r a t e c o n s t a n t of 0.8 ± 0.1 x 10" 1 2 cm 3s" 1 measured a t 300°K i n an Ar moderator. I t i s worth p o i n t i n g out t h a t the s p i n exchange c r o s s s e c t i o n of Mu+0 2 i s the same i n Ar gas and powder moderators ( M a r s h a l l 1978). In thes e e x p e r i m e n t s the Mu i s known t o be t h e r m a l i z e d (see S e c t i o n V * 2 ) . A p o s s i b l e e x p l a n a t i o n of why the o-Ps may not t h e r m a l i z e as i n d i c a t e d i n S e c t i o n 11*2 has t o do w i t h the assumption t h a t the e n t i r e s u r f a c e a r e a i s e q u a l l y a c c e s s i b l e t o the Ps. A c c o r d i n g t o the manufacturer ( C a b o t ) , the p r i m a r y p a r t i c l e s ( 35A r a d i u s spheres) a r e f u s e d i r r e v e r s i b l y i n t o l a r g e a g g r e g r a t e s t r u c t u r e s w i t h d i m e n s i o n s as l a r g e as 20000A. These a g g r e g r a t e s t r u c t u r e s a re m e c h a n i c a l l y e n t a n g l e d i n t o a g g l o m e r a t e s , which support the l a r g e f r e e volume a s s o c i a t e d w i t h the powder. A v e r y l i g h t atom such as Ps has a th e r m a l wavelength of 60A a t 450°K, which i s of the same o r d e r as the s p a c i n g of the p r i m a r y p a r t i c l e s i n the a g g r e g r a t e s . I t i s c o n c e i v a b l e t h a t a t lower t e m p e r a t u r e s the Ps s c a t t e r s p r i m a r i l y o f f the s u r f a c e s of the a g g r e g r a t e s , t h u s i n c r e a s i n g the mean f r e e p a t h and thus the t h e r m a l i z a t i o n time by a l a r g e f a c t o r 35 (.~102 or more). T h i s l a r g e i n c r e a s e i n the mean f r e e p ath c o u l d a l s o h e l p e x p l a i n the independence of the r a t e c o n s t a n t on powder d e n s i t y . One i m p l i c a t i o n of t h i s h y p o t h e s i s i s t h a t the o bserved quenching r a t e i n e vacuated S i 0 2 i s not p u r e l y due t o p i c k o f f a n n i h i l a t i o n as p r e v i o u s l y suggested ( F o r d 1976), s i n c e d e c r e a s i n g by a f a c t o r of 100 i m p l i e s t h i s r a t e i s o n l y of o r d e r 0.008 u s - ' (see S e c t i o n 11*4). The o b s e r v e d quenching r a t e would then c o r r e s p o n d p r i m a r i l y t o the t r a p p i n g r a t e on the s u r f a c e (see E q u a t i o n 11*20). T h i s might be t e s t e d a t low t e m p e r a t u r e s by d e p o s i t i n g a f i l m of He on the s u r f a c e . T h i s would most l i k e l y e l i m i n a t e the p o s s i b i l i t y of t r a p p i n g s i n c e Ps i s not l i k e l y bound t o such a s u r f a c e . Such a medium might be i d e a l f o r d e t e r m i n i n g the vacuum decay r a t e of o-Ps, a s u b j e c t of g r e a t i n t e r e s t s i n c e i t p r o v i d e s a t e s t of quantum e l e c t r o d y n a m i c s . 111*3*2 Anomolous S p i n Exchange i n o-Ps + 0 2 S c a t t e r i n g At low energy, the s p i n exchange c r o s s s e c t i o n of o-Ps o f f a 2 e l e c t r o n m o l e c u l e can be w r i t t e n (see Appendix I I ) s h i f t s f o r e l a s t i c s wave s c a t t e r i n g o f f the i s o t r o p i c p a r t of the molecule-atom i n t e r a c t i o n and independent of the m o l e c u l a r r o t a t i o n a l s t a t e . The s m a l l s p i n exchange c r o s s s e c t i o n f o r Ps + 0 2 compared w i t h the p h y s i c a l c r o s s s e c t i o n and those of Mu + 0 2 and H + 0 2 a t 300°K can be e x p l a i n e d q u a l i t i v e l y as f o l l o w s . In where are the s p i n q u a r t e t and s p i n d o u b l e t phase Equat i o n 111*5 36 the case of Mu + 0 2 and H + 0 2 a t 300°K t h e r e are many p a r t i a l waves which c o n t r i b u t e t o both e l a s t i c and i n e l a s t i c s c a t t e r i n g i n v o l v i n g r o t a t i o n a l e x c i t a t i o n . In the case of Ps + 0 2 a t 300°K, t h e s e i n e l a s t i c c h a n n e l s are c l o s e d a l o n g w i t h any non-s-wave s c a t t e r i n g . Thus o n l y i n the case of Ps + 0 2 s c a t t e r i n g does the s p i n exchange c r o s s s e c t i o n depend on o n l y two phase, s h i f t s . T h i s may l e a d t o a s m a l l s p i n exchange when 60 -6 a nir. T h i s i s somewhat analogous t o the Ramsauer Townsend e f f e c t i n low energy p o s i t r o n ( e l e c t r o n ) - a t o m s c a t t e r i n g , where a c a n c e l l a t i o n i n the s wave c o n t r i b u t i o n l e a d s t o a v e r y s m a l l e l e a s t i c c r o s s s e c t i o n . The i n c r e a s e i n the c o n v e r s i o n r a t e c o n s t a n t above 450°K may p a r t l y be due t o the i n c r e a s e i n the mean t h e r m a l v e l o c i t y . There i s a l s o i n d i c a t i o n t h a t i n c r e a s e s w i t h t e m p e r a t u r e . The room temperature v a l u e of ( 1-0 ± 0.1 x 10" 1 9 c m 2 ) , o b t a i n e d u s i n g an Ar moderator ( K l o b u c h a r 1980), i s s l i g h t l y lower than at 540°K (1.3 ± 0.1 x 1 0 " 1 9 cm 2) and ^ a t 630°K (1.5 ± 0.1 x 1 0 " 1 9 c m 2 ) , o b t a i n e d from the p r e s e n t d a t a u s i n g an S i 0 2 powder moderator. T h i s i n c r e a s e i n cL c o u l d be the r e s u l t of a s m a l l p-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 t e m p e r a t u r e s , or p o s s i b l y a weakening of the i n t e r f e r e n c e between q u a r t e t and d o u b l e t s-wave s c a t t e r i n g . I l l - 4 Summary and C o n c l u s i o n s 1. 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 2 i n S i 0 2 powder has been measured over the temperature range from 121°K t o *630°K. Below 450°K, the c o n v e r s i o n r a t e was obser v e d t o be independent of T. There a re i n d i c a t i o n s t h a t the o-Ps does not t h e r m a l i z e a t lower t e m p e r a t u r e s , p o s s i b l y because of l a r g e mean f r e e path r e s u l t i n g from c l u m p i n g of the powder g r a i n s . The c o n v e r s i o n c r o s s s e c t i o n measured at 530°K and 630°K i s s l i g h t l y h i g h e r than p r e v i o u s l y measured a t 300°K i n gas moder a t o r s . 2. The anamolously low s p i n exchange c r o s s s e c t i o n of o-Ps i n 0 2 i s e x p l a i n a b l e i n terms of the s-wave n a t u r e of low energy Ps s c a t t e r i n g caused by the i s o t r o p i c p a r t of the Ps - 0 2 i n t e r a c t i o n . 38 CHAPTER IV : MUONS, MUONIUM AND „ +SR U n l i k e the p o s i t r o n , the e x i s t e n c e of the muon was not p r e d i c t e d t h e o r e t i c a l l y . I t was d i s c o v e r e d i n cosmic ray ex p e r i m e n t s (Anderson 1937, S t r e e t 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 t o e x p l a i n the n u c l e a r f o r c e . Muons can 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 a mass of 103 MeV/c 2, r o u g h l y 200 times t h a t of an e l e c t r o n . They a r e s p i n 1/2 p a r t i c l e s , and come i n both p o s i t i v e and n e g a t i v e c h a r g e s . As i n the case of e l e c t r o n s , they do not p a r t i c i p a t e i n s t r o n g i n t e r a c t i o n s . The magnetic moment of the muon i s v e r y c l o s e t o efi/m* c, as p r e d i c t e d by the D i r a c e q u a t i o n . Muon decay was one of the f i r s t e x p e r i m e n t s t o show t h a t p a r i t y 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 (Garwin 1957, Friedman 1957). Not u n e x p e c t e d l y , the p o s i t i v e muon may c a p t u r e an e l e c t r o n t o form the H - l i k e atom c a l l e d muonium or Mu. The f i r s t o b s e r v a t i o n of Mu was made by Hughes e t a l . (Hughes 1960). The p r o p e r t i e s of muons and Mu atoms a r e of tremendous importance i n p h y s i c s s i n c e they p r o v i d e an almost i d e a l t e s t i n g ground f o r e l e c t r o m a g n e t i c and weak i n t e r a c t i o n t h e o r i e s (Hughes 1977). The advent of the "meson f a c t o r y " i n the 1970's has r e v o l u t i o n i z e d muon p h y s i c s by p r o v i d i n g i n t e n s e • beams of p o l a r i z e d muons f o r e x p e r i m e n t a l s t u d y . A p a r t from i t s fundamental r o l e i n p a r t i c l e p h y s i c s , the muon has become a u s e f u l probe i n n u c l e a r p h y s i c s , s o l i d s t a t e p h y s i c s and p h y s i c a l c h e m i s t r y . N e g a t i v e muons can be used t o probe n u c l e a r s t r u c t u r e because t h e i r atomic o r b i t s o v e r l a p the n u c l e u s . P o s i t i v e muons have been employed p r i m a r i l y as magnetic probes 39 i n s o l i d s t a t e p h y s i c s . The Mu atom (*/*e~) i s of s p e c i a l i n t e r e s t i n p h y s i c a l c h e m i s t r y because i t can be c o n s i d e r e d a l i g h t i s o t o p e of H, the muon h a v i n g 1/9 the p r o t o n mass. A d e s c r i p t i o n of the Muon S p i n R o t a t i o n ( j i * S R ) t e c h n i q u e would be i n c o m p l e t e w i t h o u t f i r s t d i s c u s s i n g the source of p o l a r i z e d muons and the p r o p e r t i e s of muon decay. These are. b r i e f l y p r e s e n t e d i n the f i r s t two s e c t i o n s of t h i s c h a p t e r . The ti*SR t e c h n i q u e i s then e x p l a i n e d w i t h emphasis on the t r a n s v e r s e f i e l d t e c h n i q u e . The 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 f o r Mu are then i n t r o d u c e d . F i n a l l y , the form of the »« + SR spectrum i s d e r i v e d . IV«1 Source of P o l a r i z e d Muons I n t e n s e beams of medium energy p r o t o n s (~100 »»A a t 500 MeV), i n c i d e n t on a s u i t a b l e p r o d u c t i o n t a r g e t , a r e c u r r e n t l y b e i n g used a t meson f a c i l i t i e s such as TRIUMF, SIN and LAMF as a source of v mesons — the " n u c l e a r g l u e " p a r t i c l e s . The most common source of p o l a r i z e d muons i s from weak decay of n's which have a f r e e l i f e t i m e of 26 ns. The m a s s l e s s n e u t r i n o obeys a two component Weyl e q u a t i o n and i s thus an h e l i c i t y e i g e n s t a t e . C o n s e r v a t i o n of energy, Equat i o n IV-1 3-f \v,y = Iplol y> 3.5 i %? -- \?Y i E q u a t i o n IV»2 40 t o t a l a n g u l a r momentum, and l i n e a r momentum r e q u i r e t h a t the »-i s an h e l i c i t y e i g e n s t a t e and monoenergetic a t 4.2 MeV i n the r e s t frame of the tr* . A secondary b e a m l i n e , c o n s i s t i n g of l a r g e d i p o l e m a g n e t s ( f o r momentum s e l e c t i o n ) and qu a d r u p o l e magnets ( f o r f o c u s s i n g ) , i s used t o t r a n s m i t charged p a r t i c l e s from the p r o d u c t i o n t a r g e t t o the e x p e r i m e n t a l a r e a . The muon h e l i c i t y remains v i r t u a l l y unchanged d u r i n g passage t h r o u g h t h i s beamline s i n c e the c y c l o t r o n f r e q u e n c y f o r a muon i n a magnetic f i e l d of s t r e n g t h B ( q B / d i ^ c ) ) and the p r e c e s s i o n frequency (qq* B/( 2m^c)) a r e almost the same ( g ^ - 2 ) . The f i r s t s t o p p i n g muon c h a n n e l s were d e s i g n e d t o c o l l e c t backward d e c a y i n g muons from p i o n s i n f l i g h t . These t y p e s of c h a n n e l s r e s u l t i n a r e l a t i v e l y h i g h energy muon beam, ~50 MeV, w i t h p o l a r i z a t i o n ~0.8. R e c e n t l y , i t was d i s c o v e r e d ( P i f e r 1976) t h a t a f l u x of u* can be o b t a i n e d from v* decay on or near the s u r f a c e of the p r o d u c t i o n t a r g e t . The t e c h n i q u e of p r o d u c i n g h i g h l y p o l a r i z e d i n t e n s e f l u x e s of th 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 and e x p l o i t e d a t TRIUMF (Oram 1981). The muons have energy 4.2 MeV and a r e almost c o m p l e t e l y p o l a r i z e d because the p i o n s a re a t r e s t i n the l a b frame. S u r f a c e muons are p a r t i c u l a r l y u s e f u l i n p*SR exp e r i m e n t s because of t h e i r low energy and h i g h p o l a r i z a t i o n . 41 IV*2 Muon Decay Muons decay 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 and two n e u t r i n o s w i t h a l i f e t i m e of 2199.4 ns (Wu 1966). —> e " -r- V4. +• V^f E q u a t i o n IV-3 The decay p r o p e r t i e s of 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 -c u r r e n t i n t e r a c t i o n of the f o l l o w i n g form Hx = 9A [ %^C\-^s)%][^A^)%] + herni+m c o n j u g a l E q u a t i o n IV-4 ( W i l l i a m s 1971), where g^ i s a c o n s t a n t , the \ and the ^ are the f i e l d o p e r a t o r s f o r l e p t o n i , a r e the D i r a c m a t r i c e s , and A i s a summation i n d e x . Each term i n square b r a c k e t s has a v e c t o r - a x i a l v e c t o r (V-A) form. The product terms i n v o l v i n g v e c t o r 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 , and thus connect s t a t e s of o p p o s i t e p a r i t y . Non c o n s e r v a t i o n of p a r i t y , s i m u l t a n e o u s l y d i s c o v e r e d by Wu (1957) i n t h e be t a decay of n u c l e i and by o t h e r groups (Garwin 1957, Friedman 1957) i n the ver y muon decay p r o c e s s now b e i n g d i s c u s s e d , l e a d s t o an asymmetric muon decay, depending on the p s e u d o - s c a l a r <o-/'>'ne , where <&"> i s the muon p o l a r i z a t i o n v e c t o r and ne i s the 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 , the energy a n g u l a r d i s t r i b u t i o n of the e + can be w r i t t e n M - c(uu)[ I t DM <^> '^Isi)] E q u a t i o n IV'5 where u = E/E^ 4 X i s the p o s i t r o n energy i n u n i t s of E ) > v i v = mA /2. 42 F i g u r e IV*1 shows the decay parameter C(w) and D(w). Note t h a t 0.5 - 0 . 4 r • i i . | 1 1 i • / • ,/DM i i i , i • i i 0.2 0.4 0.6 0.8 1.0 OJ = E / E max F i g u r e IV-1 Muon decay parameters C(o) and D(o) the asymmetry changes s i g n w i t h energy and t h a t the d i s t r i b u t i o n of p o s i t r o n e n e r g i e s , C ( u ) , i s weighted towards E m 4 x . The average of C(u)*D(u) i s t h e o r e t i c a l l y 1/3 and has been measured t o be 0.324 ± 0.004 ( C r o n i n 1968)'. IV*3 Muon S p i n R o t a t i o n The t e c h n i q u e s of muon s p i n r o t a t i o n (y*SR ) i n v o l v e measuring the decay r a t e of the muon i n a p a r t i c u l a r d i r e c t i o n as a f u n c t i o n of time a f t e r the muon a r r i v a l i n the t a r g e t . In a t y p i c a l n*SR ex p e r i m e n t , the muon a r r i v a l i s s i g n a l l e d by one or more s c i n t i l l a t i o n c o u n t e r s and i t s decay by the passage of a h i g h energy p o s i t r o n t h r o u g h a p o s i t r o n t e l e s c o p e , c o n s i s t i n g of two or t h r e e s c i n t i l l a t i o n 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 as a range f i l t e r . The h i s t o g r a m of time d e l a y s between these 43 e v e n t s ( t h e »/ + SR spectrum) has the f o l l o w i n g form ( f o r n* decay) Nn'tt) NJt) jj„ en,) C Aft fiMD ' DM<2^)>-aMj + B 3 o JI' iir E q u a t i o n IV«6 where N ^ t ) = N e i s the t o t a l number of muons i n the ensemble a t time t , r* i s the »* l i f e t i m e , e ( o ) i s the e f f i c i e n c y of the p o s i t r o n t e l e s c o p e f o r d e t e c t i n g a p o s i t r o n of energy u, n' i s the s o l i d a n g l e subtended by the p o s i t r o n t e l e s c o p e , Bg i s a time independent background, and <e/'(t)> i s the muon p o l a r i z a t i o n v e c t o r ( w h i c h , i n g e n e r a l , i s time dependent). C a r r y i n g out the i n t e g r a t i o n y i e l d s a spectrum of the form Nj.lt) = N0C~f/t/1l 1 t A<^(-t)>'fr] + % E q u a t i o n IV-7 where m i s the d i r e c t i o n of the p o s i t r o n t e l e s c o p e , N 0 i s the n o r m a l i z a t i o n , and A 0 i s the maximum p o s s i b l e asymmetry. In a t y p i c a l t>*SR a p p a r a t u s , A 0 ~ 0.3 and N o/N~0.03. I t i s c l e a r from E q u a t i o n IV«7 t h a t the »» + SR spectrum a l l o w s the e x p e r i m e n t e r t o measure the time e v o l u t i o n of the muon p o l a r i z a t i o n v e c t o r <<r^(t)> — both i t s magnitude and d i r e c t i o n . In t r a n s v e r s e f i e l d n *SR , a magnetic f i e l d i s a p p l i e d p e r p e n d i c u l a r t o the i n i t i a l muon p o l a r i z a t i o n <Z/i(0)> , and the p o s i t r o n t e l e s c o p e d i r e c t i o n m. As might be e x p e c t e d , the muon p o l a r i z a t i o n p r e c e s s e s about the a p p l i e d f i e l d d i r e c t i o n , g i v i n g r i s e t o o s c i l l a t i o n s i n the >i + SR spectrum. The time e v o l u t i o n of <c> / 1(t)> f o r f r e e muons and Mu i n a t r a n s v e r s e f i e l d w i l l now be d e r i v e d . 44 IV'3'1 Free Muons i n a T r a n s v e r s e Magnetic F i e l d The s p i n H a m i l t o n i a n f o r an i s o l a t e d muon i n a magnetic f i e l d B a l o n g the 2 d i r e c t i o n i s 2 2. E q u a t i o n IV«8 where = g^eB/^m^c, ~ 2, m^  i s the muon mass, and a* are P a u l i s p i n m a t r i c e s . The energy e i g e n s t a t e s a r e then |«i> = =1> and \t2> = 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 . I f the muons a r e i n i t i a l l y i n a pure s t a t e X [ |£,> +- J E q u a t i o n IV« 9 then 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 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 -2- 2 E q u a t i o n IV-11 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. E q u a t i o n IV«12 45 The time e v o l u t i o n of muon p o l a r i z a t i o n v e c t o r i s most e a s i l y e v a l u a t e d i n terms of the e x p e c t a t i o n v a l u e of e^* = «f + irf , t w i c e the muon s p i n r a i s i n g o p e r a t o r . I t i s c l e a r <c*> = Re<tf^ + >? <<fy> = Im<a/'*> and > = 0. In m a t r i x n o t a t i o n , o 2. O E q u a t i o n IV-13 I t i m m e d i a t e l y f o l l o w s from TrE iu)t = e E q u a t i o n IV*14 t h a t < 6 ^ (*")> - s i n W A + Equat i o n IV*15 Thus the p o l a r i z a t i o n v e c t o r r o t a t e s about the a p p l i e d f i e l d d i r e c t i o n a t a s i n g l e f r e q u e n c y , uM, as e x p e c ted c l a s s i c a l l y . IV*3*2 Free Muonium i n a T r a n s v e r s e Magnetic F i e l d The s p i n H a m i l t o n i a n f o r an i s o l a t e d Mu atom i n a magnetic f i e l d a l o n g the 2 d i r e c t i o n can be w r i t t e n 2" 3* -J- "four-**"' + "K aic« 6* E q u a t i o n IV*16 Z Z. The f i r s t term i s the h y p e r f i n e c o u p l i n g between the muon and e l e c t r o n . In the ground s t a t e , u0/2v = 4463.302 MHz (Casperson 1975). The l a s t two terms a r e muon and e l e c t r o n Zeeman i n t e r a c t i o n s w i t h the a p p l i e d f i e l d . In the I tr'*1 ee > b a s i s , ** ** 1 3£- Z-where o and p r e f e r t o ±1 r e s p e c t i v e l y , 46 w i t h c o r r e s p o n d i n g e i g e n v a l u e s - - l/J_ +- UJo J -z - ^ 0 _ t w . 1 ^ E q u a t i o n IV'17 E q u a t i o n IV'18 where E q u a t i o n IV-19 UJ 0 E q u a t i o n IV-20 E q u a t i o n IV«21 where B 0=1585G and These a r e the 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 H atom(see f o r example Brewer 1976). C o n s i d e r a muon i n i t i a l l y p o l a r i z e d a l o n g the x d i r e c t i o n , which 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 Mu s t a t e i s a m i x t u r e of the two s t a t e s 47 = [ l6,> + co$q> l e t > - 5in k , > J J-E q u a t i o n 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 can then be w r i t t e n E q u a t i o n IV'23 In m a t r i x n o t a t i o n , 1 i eOi>&> 1 C O S f 7 V o V - $\v\q> H o o s mjp "•i _L V o cosq> i E q u a t i o n IV-24 The time e v o l v e d d e n s i t y m a t r i x . i - n t l £ j > 48 j_ cob £, co$_q e 7 o - 5 m f e H o 4 T 7 H E q u a t i o n IV-25 where ho,- = e. - c . J < J As i n the case of the f r e e muon, i t i s s u f f i c i e n t t o e v a l u a t e the e x p e c t a t i o n v a l u e of tf/14 , t w i c e the muon s p i n r a i s i n g o p e r a t o r . I t i s easy t o show 6«] = < 6 ( l ^ " l 6 J > o o O a na>scf> o O o o -xcoscf* o 1 COS(D O O O E q u a t i o n IV-26 I t f o l l o w s i m m e d i a t e l y from 49 <6^y - T r l P f r ) ^ + J 4- S l ^ <^  f c o s ^ e- J E q u a t i o n IV-27 t h a t ^ 2--E q u a t i o n IV-28 In low f i e l d s x << 1 or B << 1585 G, the p e r i o d of the u,„ and o 3 4 f r e q u e n c i e s i s of o r d e r 0.225 ns and i s u s u a l l y not o b s e r v a b l e w i t h a t y p i c a l v*SR a p p a r a t u s , where t i m i n g r e s o l u t i o n i s ~1 ns. Thus h a l f the muon p o l a r i z a t i o n i n Mu i s not o b s e r v e d , and the r e m a i n i n g two terms i n <e^> can be e x p r e s s e d (i + x 1 ) * J E q u a t i o n IV-29 — — — J E q u a t i o n IV-30 In v e r y low f i e l d s , < 10G, where n-r^  <<; 1 ( T^ , =2.2 «s i s the 50 muon l i f e t i m e ) the s p l i t t i n g i s n e g l i g i b l e and the e x p r e s s i o n s i m p l i f i e s t o (<5xy= -L_ C0t> LO-~t E q u a t i o n IV-31 where u_=B«1.40 MHz-G" 1 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 are stopped i n matter as bare muons or Mu, the f r e e H a m i l t o n i a n i s p e r t u r b e d by the s u r r o u n d i n g medium i n v a r i o u s ways which l e a d t o decay of the muon p o l a r i z a t i o n <c*> . Much of the i n t e r e s t i n »/*SR i s f o c u s s e d on the s p i n r e l a x a t i o n of muons (Mu), 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) + host s t a t e . So f a r as s p i n r e l a x a t i o n i s concerned the bare muon i s a pure magnetic probe, s e n s i t i v e t o the l o c a l magnetic 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 c o u p l e d t o the e l e c t r o n , whose magnetic moment i s 200 tim e s l a r g e r . I t s response t o the l o c a l magnetic environment a r e i n d i r e c t but i n g e n e r a l 103 times f a s t e r than the bare muon (due t o the f a s t e r p r e c e s s i o n f r e q u e n c y ) . Muons i n Mu atoms a r e a l s o s e n s i t i v e t o e l e c t r i c f i e l d s s i n c e i n g e n e r a l these w i l l a l t e r the muon e l e c t r o n c o u p l i n g and thus the p r e c e s s i o n f r e q u e n c i e s of Mu. The i n f o r m a t i o n o b t a i n e d from n +SR of Mu atoms i s s i m i l a r t o t h a t o b t a i n e d from ESR of H atoms. There a re f o u r b a s i c mechanisms by which <<r / H(t)> of Mu decays i n a h o s t . 1. Random l o c a l magnetic 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 Exchange 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 Magnetic F i e l d s (RLMF) ' C o n s i d e r a Mu atom a t p o s i t i o n i n a hos t c o n t a i n i n g magnetic moments a t p o s i t i o n s f 1 f f 2 .. . f v . The f r e e s p i n H a m i l t o n i a n f o r Mu must be m o d i f i e d t o i n c l u d e the d i p o l e - d i p o l e c o u p l i n g between the Mu and the host moments. The p e r t u r b i n g i n t e r a c t i o n can be w r i t t e n E q u a t i o n IV-32 where E q u a t i o n IV«33 i s the magnetic f i e l d a t the Mu s i t e due t o the moment a t r,-(Abragam 1961). Thus the e f f e c t i v e f i e l d a t the s i t e of Mu atom B e f p - Bappl . 'etJ ^ B>d/p E q u a t i o n IV-34 where ABd- = p,-In a l a t t i c e where the host moments a r e f i x e d and u n p o l a r i z e d , A B ^ i s randomly d i s t r i b u t e d about z e r o , g i v i n g r i s e t o a b r o a d e n i n g i n the Mu f r e q u e n c i e s , or ( e q u i v a l e n t l y ) a decay of the muon p o l a r i z a t i o n a m p l i t u d e . A s p e c i f i c example of t h i s w i l l be g i v e n i n S e c t i o n V«4. I f AB d l^ i s f l u c u a t i n g q u i c k l y due t o motion of the Mu or the host moments, t h i s tends t o average the p e r t u r b i n g f i e l d so the Mu atom sees the average f i e l d . T h i s phenomenon i s commonly r e f e r r e d t o as m o t i o n a l h a r r o w i n g . 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 C o n s i d e r a Mu atom l o c a l i z e d a t a s i t e f i n a l a t t i c e . The ground s t a t e w a v e f u n c t i o n may be p e r t u r b e d by the l a t t i c e , a l t e r i n g the h y p e r f i n e i n t e r a c t i o n between the muon and 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 can be e x p r e s s e d ^ = n a t i o n IV. 35 where A i s a 4 x 4 t e n s o r . In the case of an i s o t r o p i c p e r t u r b a t i o n ( i . e . : A = ftAooI), the energy e i g e n v a l u e s and e i g e n s t a t e s a re i d e n t i c a l t o tho s e o b t a i n e d i n the f r e e Mu c a s e , except t h a t the h y p e r f i n e s p l i t t i n g i s m o d i f i e d t o " h ( u 0 + A u 0 ) • I n the g e n e r a l case the 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 and <<r^(t)> has s i x f r e q u e n c i e s c o r r e s p o n d i n g t o a l l p o s s i b l e u.j, even i n z e r o a p p l i e d f i e l d . Three of the s e (of o r d e r u 0 ) a r e not n o r m a l l y observed because of t h e i r h i g h f r e q u e n c y . A l l of them may depend on the o r i e n t a t i o n of the l a t t i c e w i t h r e s p e c t t o the muon p o l a r i z a t i o n . Zero f i e l d o s c i l l a t i o n s of Mu have r e c e n t l y been observ e d i n s i n g l e c r y s t a l S i 0 2 below 50°K (Brewer 1981). In f u s e d S i 0 2 , the o r i e n t a t i o n of the l a t t i c e w i t h r e s p e c t t o the i n i t i a l muon p o l a r i z a t i o n i s random the a n i s o t r o p i c d i s t o r t i o n c a uses s p i n r e l a x a t i o n . As i n the case of random l o c a l magnetic f i e l d s , r a p i d motion of the Mu atom l e a d s t o an a v e r a g i n g of the d i s t o r t i o n over many s i t e s , and a subsequent weakening of the r e l a x a t i o n . 53 IV«4'3 Chemical R e a c t i o n A Mu atom may r e a c t c h e m i c a l l y w i t h a m o l e c u l e t o form a Mu compound. The magnetic environment of the muon changes d r a m a t i c a l l y a t the i n s t a n t of f o r m a t i o n ( e x c ept i f the Mu compound i s a r a d i c a l i n which the muon e l e c t r o n c o u p l i n g i s almost the same as f o r Mu), so t h a t the muon s p i n v e c t o r q u i c k l y f a l l s out of phase w i t h the r e m a i n i n g members of the Mu ensemble. In the gas phase the Mu r e l a x a t i o n r a t e can be w r i t t e n t r e ^ v n , where i s the c r o s s s e c t i o n f o r the r e a c t i o n , v i s the mean t h e r m a l v e l o c i t y of Mu and n i s the c o n c e n t r a t i o n of r e a c t a n t . IV'4«4 S p i n Exchange As i n the case of Ps, the z component of the Mu e l e c t r o n s p i n i s not a co n s e r v e d q u a n t i t y i n c o l l i s i o n s w i t h paramagnetic m o l e c u l e s such as NO (S=1/2) or 0 2 (s=1). T h i s l e a d s t o h y p e r f i n e t r a n s i t i o n s of Mu which r e s u l t i n a l o s s of coherence of muon s p i n s . The r e l a x a t i o n r a t e f o r Mu due t o s p i n exchange can be w r i t t e n \- f Jex• t m / 2 E q u a t i o n IV-36 where <*ev i s the s p i n exchange c r o s s s e c t i o n d e f i n e d i n Appendix I I and f i s a c o n s t a n t depending on the s p i n of the paramagnetic m o l e c u l e (f = 3/4 f o r NO and 8/9 f o r 0 2 , Fle m i n g 1981a). 54 IV-5 The »i + SR Spectrum i n a T r a n s v e r s e F i e l d I t i s c l e a r t h a t the p r e c e s s i o n a m p l i t u d e s hj*+ and A^ M f o r muons and Mu i n ma t t e r have some time dependence 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 the time dependence be r e p r e s e n t e d by the f u n c t i o n s R ^ + ( t ) and R ^ f t ) w i t h R^ + (0) = R r t l 4(0) = 1. Then the n*SR spectrum i n a moderate t r a n s v e r s e f i e l d f o r a p o s i t r o n t e l e s c o p e i n the x d i r e c t i o n can be w r i t t e n , u s i n g E q u a t i o n s IV-7 and IV-29, E q u a t i o n IV-37 where E q u a t i o n IV-38 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 i n the p l a n e of p r e c e s s i o n d i f f e r s o n l y by a phase f a c t o r . In v e r y weak f i e l d s (< 10G) one may use E q u a t i o n IV-31 i n s t e a d of E q u a t i o n IV-29 so t h a t z E q u a t i o n IV-39 55 CHAPTER V : MUONIUM IN INSULATING POWDERS Mu p r e c e s s i o n has p r e v i o u s l y been observed i n s e v e r a l o x i d e powders such as S i 0 2 , A l 2 0 3 and MgO ( M a r s h a l l 1978, K i e f l 1979). As i n the case of Ps, a f r a c t i o n of the Mu emerges i n t o the v o i d r e g i o n s of the powder. T h i s has been v e r i f i e d through the i n t r o d u c t i o n of paramagnetic 0 2 gas, which r a p i d l y r e l a x e s the Mu i n the v o i d s t h r o u g h s p i n exchange a t a r a t e c o n s i s t e n t w i t h t h a t measured i n an Ar or N 2 gas moderator. As a p r e l u d e t o the e x p e r i m e n t a l r e s u l t s on Mu i n these o x i d e s a t low tem p e r a t u r e , t h i s c h a p t e r d e a l s w i t h the f o r m a t i o n , t h e r m a l i z a t i o n , and s p i n r e l a x a t i o n of Mu i n these h i g h l y d i s p e r s e d media. The form of the r e l a x a t i o n f u n c t i o n R ^ M ( t ) w i l l be d e r i v e d under 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 Format i o n The 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 p o s i t i v e muons i s a c o m p l i c a t e d p r o c e s s , e s p e c i a l l y i n condensed matter or a powder. As i n the case of Ps, e p i t h e r m a l , s p u r and s u r f a c e p r o c e s s e s can r e s u l t i n Mu f o r m a t i o n . However, r e c e n t l y i t has 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 the a p p l i c a t i o n of an e l e c t r i c f i e l d i n bu l k S i 0 2 and s e v e r a l hydrocarbon l i q u i d s ( I t o 1981). T h i s i s i n c o n t r a s t t o the case of Ps, where such a f i e l d i n h i b i t s c o m b i n a t i o n of e* and spur e~ ( I t o 1979). These r e s u l t s weaken the spur model h y p o t h e s i s f o r Mu f o r m a t i o n , a t l e a s t i n th e s e c a s e s . The problem of e p i t h e r m a l Mu f o r m a t i o n can be f o r m u l a t e d as 56 f o l l o w s . D e f i n e f * ( E t ) t o be the f r a c t i o n of muon ensemble i n charge s t a t e i at energy E and time t . I g n o r i n g double charge exchange [ v a l i d below 10KeV f o r p r o t o n s (Tawara 1973)] the ft- 's obey the f o l l o w i n g c o u p l e d i n t e g r o - d i f f e r e n t i a l e q u a t i o n s 5f = - f, (E, + )Tt 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 ' 2+ 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') i s the t r a n s i t i o n r a t e between a muon i n an i n i t i a l s t a t e of energy E, charge i and a f i n a l s t a t e of energy E' and charge j . I t i s g e n e r a l l y a c c e p t e d t h a t most of the charge exchange o c c u r s i n the 2 KeV t o 20 eV r e g i o n , where the muon has a v e l o c i t y comparable w i t h the v a l e n c e e l e c t r o n s i n most atoms (F l e m i n g 1981b). At t h e r m a l e n e r g i e s , charge exchange may be e n e r g e t i c a l l y f o r b i d d e n . S i n c e the t h e r m a l i z a t i o n time i n condensed m a t t e r i s e s t i m a t e d t o be of or d e r 1 0 ~ 1 1 S (Brewer 1975), a l l the charge exchange o c c u r s much too f a s t t o be o b s e r v a b l e v i a the u*SR t e c h n i q u e ( t i m i n g r e s o l u t i o n 10' 9 s ) . The o b s e r v a b l e q u a n t i t i e s i n a i>*SR experiment a re the Mu 57 f r a c t i o n F M i ) ( t ) and the d i a m a g n e t i c f r a c t i o n F^<-(t) where o E q u a t i o n V-4 f > m => 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 t o d i s t i n g u i s h from a bare muon w i t h the x +SR t e c h n i q u e , s i n c e the time e v o l u t i o n of the muon p o l a r i z a t i o n i s v i r t u a l l y the same i n both c a s e s and thus c o n t r i b u t e s t o the d i a m a g n e t i c f r a c t i o n . In p r i n c i p l e , i f the t r a n s i t i o n r a t e s t;j(EE') a re known, one can s o l v e e q u a t i o n s V-1, V-2 and V«3 and p r e d i c t F M and F / t 1 +. However, i n g e n e r a l , they a r e not e a s i l y measurable or c a l c u l a b l e . T h i s i s why Mu f o r m a t i o n i n condensed matter has remained somewhat of a mystery. Mu p r e c e s s i o n has a l s o been obser v e d i n b u l k samples of MgO and S i 0 2 . L o n g i t u d i n a l f i e l d measurements i n d i c a t e t h a t i t i s a l s o p r e s e n t i n b u l k A l 2 0 3 ( M i n a i c h e v 1970). The Mu f o r m a t i o n p r o b a b i l i t y i s not p o s i t i v e l y c o r r e l a t e d t o the degree of a g g r e g a t i o n of the o x i d e , s u g g e s t i n g t h a t Mu f o r m a t i o n i s a bu l k r a t h e r than a s u r f a c e phenomenon. 58 V'2 Mu T h e r m a l i z a t i o n Below the low e s t e l e c t r o n e x c i t a t i o n energy of a non-m e t a l l i c medium (or of Mu, whichever i s s m a l l e r ) , Mu must l o s e 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 v i b r a t i o n s (phonons). Once t h e r m a l i z e d , i t d i f f u s e s t h rough the l a t t i c e a t a r a t e which i s , i n g e n e r a l , s t r o n g l y t e m p e r a t u r e dependent. At v e r y low t e m p e r a t u r e s , the Mu atom may become t r a p p e d i n the l a t t i c e f o r t i m e s g r e a t e r than the muon l i f e t i m e . T r a p p i n g of H atoms i n bulk i n s u l a t o r s has been observed i n many ESR e x p e r i m e n t s (Foner 1960, W e i l 1981). The p r o c e s s by which Mu e v e n t u a l l y reaches the i n t e r s t i c e s of o x i d e powders i s not c l e a r l y u n d e r s t o o d . In the t h e r m a l d i f f u s i o n model, Mu t h e r m a l i z e s w i t h i n the g r a i n and then proceeds t o d i f f u s e t o the s u r f a c e where i t i s e j e c t e d i n t o the v o i d s , presumably because of a n e g a t i v e work f u n c t i o n a t the s u r f a c e . (Such a work f u n c t i o n might a r i s e from the l a t t i c e d i s t o r t i o n induced by an i n t e r s t i t i a l Mu atom). T h i s model has been used t o e x p l a i n j#*SR r e s u l t s from S i 0 2 powder (70A and 140A mean d i a m e t e r ) at 300°K. ( M a r s h a l l 1978). There i s no i n d i c a t i o n t h a t the Mu r e e n t e r s the powder g r a i n s , s u g g e s t i n g t h a t the work f u n c t i o n a t the s u r f a c e i s much g r e a t e r than 300°K. However, the assumption t h a t a l l the Mu i n i t i a l l y t h e r m a l i z e s w i t h i n the g r a i n s may not be v a l i d a t a l l temp e r a t u r e s and under a l l c o n d i t i o n s . A l a r g e work f u n c t i o n a t the s u r f a c e , the s m a l l p a r t i c l e s i z e and the presence of a b u f f e r gas i n the i n t e r s t i c e s may f a v o u r d i r e c t t h e r m a l i z a t i o n i n the v o i d s . The m o t i v a t i o n f o r i n t r o d u c i n g t h i s model i s p r o v i d e d i n Chapter VI i n l i g h t of the e x p e r i m e n t a l d a t a on S i 0 2 , A 1 2 0 3 and MgO powders 59 i n a He atmosphere a t 6 °K. I t s f e a s i b i l i t y i s q u a l i t a t i v e y demonstrated i n Appendix I I I . Once the Mu reaches the v o i d s w i t h an energy l e s s than the work f u n c t i o n a t the s u r f a c e , i t w i l l t h e r m a l i z e v i a phonon e x c i t a t i o n d u r i n g c o l l i s i o n s w i t h the s u r f a c e or by s c a t t e r i n g o f f a b u f f e r gas, i f p r e s e n t . The t h e r m a l i z a t i o n of an atom i n a powder i s t r e a t e d i n Appendix I , u s i n g the 1-dimensional D e v o n s h i r e t h e o r y f o r gas-s u r f a c e i n t e r a c t i o n s . In t h i s a p p r o x i m a t i o n , the time r e q u i r e d f o r 1eV Mu (8600°K) t o r e a c h 35°K i n A l 2 0 3 a t 7°K (SA = 220 m 2/g, P =0.56 g-cm" 3, e 0 = 880°K and Morse s u r f a c e p o t e n t i a l parameters a =0.5 A' 1, D=0) i s 40 ns, w i t h most of the time b e i n g spent below 300°.K. The presence of a s m a l l amount of monatomic b u f f e r gas i n the i n t e r s t i c e s reduces the t h e r m a l i z a t i o n time s u b s t a n t i a l l y . In the case of s-wave s c a t t e r i n g , the time r e q u i r e d t o go from energy E; t o E.^  i s r o u g h l y (Mobley 1 966). n i s the number d e n s i t y of the b u f f e r gas. The time r e q u i r e d t o go from 8600°K t o 35°K i s o n l y !,,32 ns i n 1 t o r r of He. We may c o n c l u d e t h a t the i n f o r m a t i o n o b t a i n e d from n*SB. e x p e r i m e n t s on such powders 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. E q u a t i o n V«6 where m i s the Mu mass M i s the mass of the b u f f e r atom «y i s the s wave c r o s s s e c t i o n (~10A 2) 60 V-3 Mu Bound S t a t e s on Oxide S u r f a c e s H atoms have been s t a b i l i z e d on s i l i c a and alumina s u r f a c e s below 120°K (Golubev 1965). The a c t i v a t i o n energy on the s u r f a c e ranges from 500°K t o 1500°K ( i n u n i t s of k ) . T h i s r e p r e s e n t s 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 r e s u l t s , i t i s p o s s i b l e t o e s t i m a t e the b i n d i n g energy of Mu t o a s i m i l a r s u r f a c e . The s u r f a c e - a t o m i n t e r a c t i o n i s assumed t o be r e p r e s e n t e d by a Morse P o t e n t i a l •1Q.X. -<a z_> -2qz: - < a - v V (20 = D l e - 2 e ) E q u a t i o n V-7 The b i n d i n g energy of the deepest bound s t a t e can be w r i t t e n (Morse, l o c . c i t . ) ° 2.' n ru 2 M M , 1 U J J J E q u a t i o n V-8 where d = (2m,, D)"/fia and € n 0 i s the c o r r e s p o n d i n g b i n d i n g energy of the H atom. For example, i f CQ = 1000°K and a =0.5A, then to" = 850°K. T h i s s e r v e s t o i l l u s t r a t e the i s o t o p e dependence one may expect between Mu and H bound on such s u r f a c e s . I t i s of some i n t e r e s t t o c o n s i d e r the f r a c t i o n of time a Mu atom w i l l spend i n a bound s t a t e on the powder s u r f a c e a t 300 °K, g i v e n t h a t the b i n d i n g energy i s of o r d e r 850 °K. T h i s 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 thermodynamics g i v e n i n Appendix IV. One f i n d s , f o r example i n 70 A S i 0 2 a t a powder d e n s i t y of 0.04 g cm 3, t h a t the Mu spends o n l y 7.7 % of i t s time on the s u r f a c e . I t i s c l e a r t h a t t o t r a p Mu on the s u r f a c e of such powder f o r any a p p r e c i a b l e amount of t i m e , one 61 must experiment a t t e m p e r a t u r e s much l e s s than the b i n d i n g energy. V«4 Mechanisms f o r Mu S p i n R e l a x a t i o n i n a Powder The mechanisms f o r Mu s p i n r e l a x a t i o n i n a powder ( d i s c u s s e d i n g e n e r a l i n S e c t i o n IV«4) depends on whether the Mu i s bound t o the s u r f a c e or c o l l i d i n g f r e e l y w i t h i t . S p i n r e l a x a t i o n of Mu adsorbed on an u n r e a c t i v e s u r f a c e has t h r e e b a s i c o r i g i n s : 1. D i p o l e - d i p o l e i n t e r a c t i o n w i t h n u c l e a r moments or d i s t a n t paramagnetic i m p u r i t i e s . T h i s i s p a r t i c u l a r l y e f f e c t i v e when the Mu i s s t a t i c on the 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 of the Mu atom due t o the ato m - s u r f a c e i n t e r a c t i o n . A g a i n , t h i s i s most e f f e c t i v e when the Mu i s s t a t i o n a r y on the s u r f a c e . 3. S p i n exchange 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 the s u r f a c e . T h i s , on the o t h e r hand, i s most e f f e c t i v e when the Mu i s mo b i l e on the s u r f a c e . I f the Mu i s c o l l i d i n g f r e e l y w i t h the s u r f a c e , the e f f e c t s of the f i r s t two a r e d i m i n i s h e d c o n s i d e r a b l y due t o m o t i o n a l n a r r o w i n g . The s i t u a t i o n i s analogous t o s p i n r e l a x a t i o n i n a gas s i n c e the powder g r a i n s i n t h i s case a c t as l a r g e gas m o l e c u l e s . Thus s p i n exchange i s ex p e c t e d t o be dominant i n the case of desorbed Mu. 62 V«4«1 N u c l e a r Magnetic Moments I f Mu i s adsorbed on a s u r f a c e which p o s s e s s e s n u c l e a r magnetic moments the d i p o l e - d i p o l e c o u p l i n g between the Mu and the moments l e a d s t o a b r o a d e n i n g i n the p r e c e s s i o n frequency d i s t r i b u t i o n or e q u i v a l e n t l y , s p i n 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 of the 2 component of magnetic f i e l d produced a t the Mu s i t e by a moment at p o s i t i o n f and z component of s p i n I z can be w r i t t e n A ^ ' > = < I x l & J I , > = Y x I ~ ( l - 3 c o * V ) — •-- E q u a t i o n 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 the a n g l e between f and the 2 a x i s and rz = ge/4m/1.c. The s h i f t i n the Mu p r e c e s s i o n f r e q u e n c y about the 2 a x i s can then be w r i t t e n i n f i r s t o r d e r where ~iH - I T T • 1-7 M H * 6 H E q u a t i o n V- 1 0 p r o v i d e d t h a t A B ^ << Ba^ |,'e<|. I f t h i s c o n d i t i o n i s not s a t i s f i e d , p r o p e r account must be taken of A B ^ and AB . I f one assumes t h a t on a s u r f a c e o n l y the n e a r e s t neighbour c o n t r i b u t e s t o A B ^ , then the mean squared f r e q u e n c y s h i f t i s -3, r & E q u a t i o n V-1 1 T h i s i s the second moment due t o d i p o l a r b r o a d e n i n g by u n l i k e s p i n s (Abragam 1961). In a powder, e may be averaged t o o b t a i n dur = J J - r a t i ) V R x Mu lb 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 example, i n the case of Mu adsorbed on A l 2 0 3 a t a d i s t a n c e of 1.5A from an 2 7 A 1 n u c l e u s , 2[(AB^!/ ) 2 ] = 3.9G. T h i s agrees r e a s o n a b l y w e l l w i t h the ESR l i n e w i d t h (4.27 G) of H atoms s t a b i l i z e d on an A l 2 0 3 s u r f a c e (Golubev 1965). For comparison, the ESR l i n e w i d t h on S i 0 2 i s o n l y 0.87 G, and i s p r o b a b l y due t o a random a n i s o t r o p i c d i s t o r t i o n , s i n c e o n l y 2% of n a t u r a l S i has n u c l e a r moments. Thus the r e l a x a t i o n r a t e f o r Mu atoms f r o z e n on an A l 2 0 3 s u r f a c e i s e s t i m a t e d a t 2vy^y&^J =19 ?s - 1 V'4'2 Paramagnetic I m p u r i t i e s I f the Mu i s f r o z e n on the s u r f a c e of a l a t t i c e , the d i p o l e - d i p o l e i n t e r a c t i o n between Mu and an i m p u r i t y a t p o s i t i o n f- l e a d s t o a s h i f t i n the p r e c e s s i o n f r e q u e n c y . 4U>''~ l i ^ l ( 1 - 3 ^ ) 1 1 E q u a t i o n v.,4 hi I f t h e r e a re N i m p u r i t i e s , then the t o t a l s h i f t u = £ Au' . In a s t a t i s t i c a l model (Anderson 1951, Abragam 1961), the frequency d i s t r i b u t i o n x(u)du i s p r o p o r t i o n a l t o the volume of phase space, such t h a t u < E Au < o + du. -XM - S £ £(V ^ w) it dfj E q u a t i o n V'15 64 where E q u a t i o n V'16 and where V i s the h a l f space a s s o c i a t e d w i t h the s o l i d . F o l l o w i n g (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 w i t h the s o l i d . where B' = 11^j r-^ v (1 - 3 c o s 2 6 ) / r 3 = , _ T r " * W , | f l a _ T r j J X j J 9 \T3 a r t / v Equat i o n V'17 where n = d e n s i t y of i m p u r i t i e s . D e f i n e 65 \ = JUL* i i i 5 TV I \uj n E q u a t i o n V*18 S u b s t i t u t i n g E q u a t i o n V»17 i n t o E q u a t i o n V«16 y i e l d s ~ _l E q u a t i o n V-19 For example, a 2% F e ~ 3 i m p u r i t y i n A l 2 0 3 r e l a x e s Mu on a s u r f a c e a t a r a t e X ~ 400>.s- 1 I f the Mu i s moving on the s u r f a c e , or c o l l i d i n g f r e e l y w i t h the s u r f a c e , then the Mu may undergo s p i n exchange w i t h paramagnetic i m p u r i t i e s . The r e l a x a t i o n r a t e i n a f r e e Mu s t a t e can be w r i t t e n ) = f m V c Sex E q u a t i o n V-20 2 where i s the c o l l i s i o n f r e q u e n c y w i t h the s u r f a c e , m i s the c o n c e n t r a t i o n - o f paramagnetic i m p u r i t y on the s u r f a c e , ecx i s the s p i n exchange c r o s s s e c t i o n , and f a c o n s t a n t of o r d e r u n i t y , depending 'on the s p i n of the i m p u r i t y (see S e c t i o n I V ' 4 ' 4 ) . For example, i n 70 A S i 0 2 powder a t a d e n s i t y of 0.04 g cm' 3 a t 6 °K w i t h ce> =9.2 A 2 and m=4 x 10" 3A- 2, the r e l a x a t i o n r a t e of Mu s h o u l d be of o r d e r 120 «s" 1. 66 V«4'3 M o t i o n a l Narrowing In the p r e c e d i n g s e c t i o n s , i t has been assumed t h a t t h e r e a r e no f l u c t u a t i o n s i n the random l o c a l magnetic f i e l d e x p e r i e n c e d by the Mu atom. However, a v e r y l i g h t atom such as 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 , and t h i s may l e a d t o a m o t i o n a l n a r r o w i n g of the p r e c e s s i o n frequency d i s t r i b u t i o n , o r , e q u i v a l e n t l y , a r e d u c t i o n i n s p i n r e l a x a t i o n r a t e . I f the l o c a l magnetic f i e l d i n the r e f e r e n c e frame of the Mu atom i s f l u c t u a t i n g randomly, c h a r a c t e r i z e d by some r e l a x a t i o n time , and i f ( A o 2 ) i r << 1 (the l i m i t of f a s t f l u c t u a t i o n s ) , then the obs e r v e d r e l a x a t i o n w i l l be reduced t o } ~. ^ ? c E q u a t i o n V-21 V'4'4 Random A n i s o t r o p i c D i s t o r t i o n I t i s d i f f i c u l t t o p r e d i c t what e f f e c t the s u r f a c e w i l l have on the h y p e r f i n e i n t e r a c t i o n of Mu. I f the ESR l i n e w i d t h of H on S i 0 2 i s due t o t h i s e f f e c t , then one might assume t h a t the r e l a x a t i o n of Mu on an S i 0 2 s u r f a c e t o be of o r d e r i r r M j < 0.87G = 4.2 us*'. Note t h a t the r e l a x a t i o n r a t e of Mu t r a p p e d i n f u s e d S i 0 2 (3.9 u s ' 1 ) (Brewer 1981) i s thought t o be due t o t h i s ef f e c t . 67 V«5 The R e l a x a t i o n F u n c t i o n R ^ M ( t ) f o r Mu i n a Powder The d e r i v a t i o n of the form of the Mu r e l a x a t i o n f u n c t i o n R y j y ( t ) i n a powder i s c o m p l e t e l y analogous t o the d e r i v a t i o n of the l i f e t i m e spectrum f o r o-Ps i n a powder (see S e c t i o n 11 • 4 ) . C o n s i d e r an ensemble of Mu atoms i n a powder a t temperature T, 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 energy B. D e f i n e X F and X.5 t o be the s p i n r e l a x a t i o n r a t e s f o r Mu i n the f r e e and adsorbed s t a t e s . The form of R M l ( ( t ) (which d e s c r i b e s how the muon s p i n i n Mu r e l a x e s w i t h time) depends on how \et compares w i t h u n i t y , where, as b e f o r e , t i s the mean s u r f a c e d w e l l time f o r Mu on the s u r f a c e (see Appendix I V ) . V«5«1 S p e c i a l Case 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 ) As i n the case of Ps, 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 where the Mu ad s o r b s and desorbs many t i m e s w h i l e s t i l l p o l a r i z e d . The p o l a r i z a t i o n a m p l i t u d e decays a c c o r d i n g t o a s i n g l e e x p o n e n t i a l decay r a t e — the average over f r e e and bound s t a t e s , weighted by the f r a c t i o n of time spent i n each s t a t e . Thus E q u a t i o n V«22 w i t h \ft •= <* )l B +" ^--"°0 E q u a t i o n V-23 where a i s the f r a c t i o n of time spent i n the bound s t a t e (see E q u a t i o n A I V - 1 5 ) . I t i s assumed t h a t the Mu atom hops many times 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 , so t h a t the d e s o r b i n g - a d s o r b i n g p r o c e s s does not s i g n i f i c a n t l y c o n t r i b u t e t o any m o t i o n a l n a r r o w i n g . Combining e q u a t i o n s AIV«15 and V-23 y i e l d s 68 r BAT As + 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.g t >> 1 ( S t r o n g C o l l i s i o n A p p r o x i m a t i o n ) The d e r i v a t i o n i s i d e n t i c a l t o t h a t c o n t a i n e d i n S e c t i o n II*4»2. The r e s u l t i n g Mu r e l a x a t i o n f u n c t i o n i s a sum or d i f f e r e n c e of e x p o n e n t i a l s . ,. ^ . E q u a t i o n V»26 The l i m i t i n g f-orms a r e as 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 of Adsorbed I n e r t Gas on the Spin R e l a x a t i o n I f an i n e r t gas such as He or Ne i s a d m i t t e d i n t o the v e s s e l , a f r a c t i o n of the gas ad s o r b s on the bare s u r f a c e , thus d e c r e a s i n g the c o l l i s i o n f r e q u e n c y of the Mu w i t h the bare s u r f a c e a c c o r d i n g t o = V t {P) [ I ~ ^0,1 E q u a t i o n V-29 Mu where n, i s the d e n s i t y of adsorbed atoms and efl i s the e l a s t i c c r o s s s e c t i o n f o r Mu s c a t t e r i n g o f f an adsorbed atom. A l s o the a v a i l a b l e s u r f a c e area f o r a d s o r p t i o n d e c r e a s e s t o f\ ir\) - flto') I I " 4 " ^'7 E q u a t i o n V-30 E q u a t i o n V'23 must then be r e w r i t t e n w i t h Vf ^ E q u a t i o n V'31 B/kT v a l i d when X.gt << 1 and « 1. S i m i l a r l y , E q u a t i o n s V'27 and V'28 must be r e w r i t t e n E q u a t i o n V-32 v a l i d when x ^ t >> 1, v 0(n,)P f << \\B - kF\ and . , - Apt E q u a t i o n V'33 v a l i d when kgt » 1, ^ ( n , ) ^ >> | x, - X p| . 70 CHAPTER VI : LOW TEMPERATURE STUDY OF MUONIUM IN A l 2 0 3 , S i 0 2 AND MgO POWDERS The s i m p l i c i t y of the H atom make i t an i d e a l o b j e c t of fundamental a t o m - s u r f a c e i n t e r a c t i o n s t u d i e s . In a d d i t i o n , the c h e m i s t r y and p h y s i c s of H atoms on s o l i d s u r f a c e s may have i n d u s t r i a l a p p l i c a t i o n s , e s p e c i a l l y i n the f i e l d of c a t a l y s i s . S i n c e Mu can be c o n s i d e r e d a l i g h t i s o t o p e of H, h a v i n g 1/9 the mass of H, the i n t e r e s t i n H n a t u r a l l y extends t o Mu. In the pas t t h i s i n t e r e s t has been c o n f i n e d t o s t u d i e s i n the gas, l i q u i d , and s o l i d phases. In t h i s c h a p t e r the f i r s t e x p e r i m e n t a l study of Mu i n t e r a c t i n g w i t h s o l i d s u r f a c e s i s p r e s e n t e d . The p r e s e n t study c o n s i s t s of two e x p e r i m e n t s i n which muons have been i n j e c t e d i n t o h i g h s p e c i f i c s u r f a c e area o x i d e powders. In the f i r s t e x p e r i m e n t , the f r a c t i o n of muons which emerge i n t o the v o i d s as Mu a t an ambient temperature of 6°K was measured. T h i s o b v i o u s l y has i m p o r t a n t consequences i n r e g a r d t o the f e a s i b i l i t y of s t u d y i n g Mu s u r f a c e phonomena. I t a l s o r e p r e s e n t s a t e s t of the t h e r m a l d i f f u s i o n model ( S e c t i o n V»2), s i n c e , a t low te m p e r a t u r e , the d i f f u s i o n l e n g t h b e f o r e decay i s ex p e c t e d t o be s m a l l i n comparison w i t h the p a r t i c l e s i z e . For example, i n b u l k f u s e d q u a r t z , the Mu i s b e l i e v e d t o be s t a b i l i z e d a t a s i n g l e s i t e below 50°K (Brewer 1981). The second experiment was undertaken t o examine the i n t e r a c t i o n of Mu w i t h r - A l 2 0 3 ( K n o z i n g e r 1978) s u r f a c e s w i t h v a r y i n g amounts of adsorbed He and Ne. 71 VI•1 Mu i n the V o i d s of Oxide Powders a t 6°K VI • 1 • 1 E x p e r i m e n t a l D e t a i l s The n*SR a p p a r a t u s i s show i n F i g u r e V I • 1 . A beam of s p i n B 2 COUNTER B I COUNTER 0 . 0 0 5 " MYLAR F l COUNTER F2 COUNTER 10 cm 20 CARBON DEGRADER D COUNTER VACUUM TARGET CRY0STAT ASSEMBLY CARBON DEGRADER F i g u r e VI--1 The v*SR a p p a r a t u s "Beaver". Note t h a t the p o s i t r o n t e l e s c o p e s a re a l o n g the beam d i r e c t i o n . p o l a r i z e d p o s i t i v e " s u r f a c e muons" of momentum 28 MeV/c from the M13 ir-x c h a n n e l a t TRIUMF (//e* r a t i o ~ l ) was c o l l i m a t e d to. a 3/4 i n c h d i ameter s p o t , and passed through a t h i n (0.010 i n c h ) d e f i n i n g c o u n t e r (D), b e f o r e e n t e r i n g a He gas f l o w c r y o s t a t . The dE/dx• f o r 28 MeV/c muons i s r o u g h l y 6 t i m e s t h a t f o r 28 MeV/c p o s i t r o n s , so t h a t s u r f a c e muons can e a s i l y be d i s c r i m i n a t e d from beam p o s i t r o n s . The i n c i d e n t muon r a t e was t y p i c a l l y 10,000 n*/s. The t a r g e t c r y o s t a t assembly i s shown i n F i g u r e V I • 2 . The M y l a r windows (35 mg/cm 2), the t h i n c o u n t e r (29.5 mg/cm 2), h e l i u m gas (16 mg/cm2) and a l u m i n i z e d M y l a r 72 TARGET SUPPORT STYROFOAM VESSE GLASS WOOL .005" MYLAR COLD He GAS 29MeV/c JU.V" .0005 ALUMINIZED MYLAR LHe VAPOURIZER Ge THERMOMETER POWDER TARGET LHe SUPPLY o • i _ 5 cm F i g u r e VI•2 The t a r g e t - c r y o s t a t assembly used t o study Mu i n o x i d e powders. windows (3.5 mg/cm2) sum t o a t o t a l of 84 mg/cm2. S i n c e the range of a 28 MeV/c s u r f a c e muon i s o n l y 140 mg/cm2 (of Carbon), 56 mg/cm2 of . t a r g e t m a t e r i a l a r e s u f f i c i e n t t o s t o p a l l the muons. P o s i t r o n s from muon decay were d e t e c t e d i n two t e l e s c o p e s p o s i t i o n e d upstream and downstream of the t a r g e t . Helmholz c o i l s were used t o a p p l y a magnetic f i e l d p e r p e n d i c u l a r t o the muon p o l a r i z a t i o n d i r e c t i o n . Temperature measurements were made w i t h a C r y o C a l CR2500H germanium r e s i s t o r w i t h an a b s o l u t e a c c u r a c y of b e t t e r than 30 m°K a t a l l t e m p e r a t u r e s . A ve r y c a r e f u l human temperature c o n t r o l l e r m a i n t a i n e d the temperature t o w i t h i n ±250 m°K. A l l 73 measurements were performed w i t h the powders h e l d i n a He atmosphere at a p r e s s u r e of 760 t o r r . VI •1•2 E l e c t r o n i c s F i g u r e VI«3 shows a schematic of the e l e c t r o n i c s , d e s i g n e d t o measure the time i n t e r v a l between a s i n g l e muon e n t e r i n g the t a r g e t and i t s decay. At low i n c i d e n t muon r a t e s (<< 1/T^ ) t h i s t a s k i s s t r a i g h t f o r w a r d , but a t h i g h e r r a t e s p r e c a u t i o n s must be taken t o m i n i m i z e d i s t o r t i o n e f f e c t s due t o p i l e u p (more than one muon i n the t a r g e t a t the same t i m e ) . A "good" event s a t i s f i e s the f o l l o w i n g c o n d i t i o n s . 1. A muon a r r i v e s a t t=0 w i t h the microprogrammable branch d r i v e r (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 the time i n t e r v a l -P < t < 0 (P i s the p i l e u p gate l e n g t h ; -~l0»/s). T h i s c o n d i t i o n s t a r t s the time d i g i t i z e r . 2. A p o s i t r o n event d e f i n e d by B1-B2 or F1«F2 o c c u r s at t = r (T < P) w i t h no second muon h a v i n g e n t e r e d the t a r g e t i n the time i n t e r v a l 0 < t < T. T h i s s t o p s the c l o c k . 3. No second muon or second e l e c t r o n i s d e t e c t e d i n the time i n t e r v a l r < t < P. A h i s t o g r a m of t h e s e time d e l a y s i s termed a f*SR spectrum. The >» + SR d a t a a c q u i s i t i o n system c o n s i s t s of a PDP-11/40 computer and a CAMAC i n t e r f a c e , d r i v e n by a microprogrammable branch d r i v e r (MBD-11). The dead time a s s o c i a t e d w i t h p r o c e s s i n g an event i s o n l y 20ns. T h i s i s made p o s s i b l e by u s i n g the r e l a t i v e l y f a s t MBD t o p r o c e s s an event and increment the a p p r o p r i a t e h i s t o g r a m b i n i n the PDP-11/40 memory. 74 F i g u r e VI•3 * +SR e l e c t r o n i c s t h a t are used t o measure the time i n t e r v a l between 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 and R e s u l t s n*SR s p e c t r a were taken i n a t r a n s v e r s e magnetic f i e l d of 8G i n o r d e r t o e v a l u a t e the Mu p r e c e s s i o n a m p l i t u d e and i t s r e l a x a t i o n r a t e . The da t a were f i t t e d t o the f u n c t i o n a l form d e r i v e d i n S e c t i o n IV«5 Equat i o n VI•1 w i t h S H M ( t ]= fh^M R ^ f t ) cost^t + 1 E q u a t i o n VI•2 where y and 0^  a r e the i n i t i a l phases f o r f r e e muon and Mu p r e c e s s i o n (determined by the o r i e n t a t i o n of the t e l e s c o p e r e l a t i v e t o the muon p o l a r i z a t i o n v e c t o r a t t = 0 ) . The muon r e l a x a t i o n r a t e s were v e r y s m a l l f o r S i 0 2 and MgO (-0.05 y s " 1 ) , so the f i t s were i n s e n s i t i v e t o whether R / 1 ( t ) was g a u s s i a n ( e ~ 5 + ) or e x p o n e n t i a l (e"*^" ). In the case of A l 2 0 3 , which has a h i g h c o n c e n t r a t i o n of n u c l e a r moments, b e t t e r f i t s were o b t a i n e d w i t h a g a u s s i a n form w i t h p *~ 0.2 <<s"' . In t h e o r y , one e x p e c t s the r e l a x a t i o n f u n c t i o n f o r s t a t i c muons i n t e r a c t i n g w i t h o n l y n e a r e s t neighbor d i p o l e s t o be g a u s s i a n (Pake 1948). In r e a l i t y , t he presence of paramagnetic i m p u r i t i e s and muon m o b i l i t y c o m p l i c a t e the s i t u a t i o n . However, s i n c e t h i s study f o c u s s e s on Mu i n th e s e t a r g e t s , i t s u f f i c e s t o say t h a t the Mu p r e c e s s i o n parameters A ^ M and a r e o n l y weakly c o r r e l a t e d w i t h the f r e e muon r e l a x a t i o n f u n c t i o n R ( t ) , and a g a u s s i a n form y i e l d e d good f i t s i n a l l c a s e s . 76 The g e n e r a l form of 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 the v o i d s of a powder i s too c o m p l i c a t e d t o use f o r f i t t i n g . However, the l i m i t i n g forms c o n s i s t of a s i n g l e e x p o n e n t i a l or a d i f f e r e n c e of e x p o n e n t i a l s ( S e c t i o n V«5). I f a f r a c t i o n of the Mu remains i n s i d e the powder, then an a d d i t i o n a l component must be added. In the p r e s e n t e x p e r i m e n t , good f i t s were o b t a i n e d w i t h a s i n g l e e x p o n e n t i a l r e l a x a t i o n f u n c t i o n except i n the case of 140A S i 0 2 powder (see F i g u r e I V ' 4 ) . In t h i s 0 .15 r-0 10 -0 05 -0 00 --0 05 --0 10 I--0 15 -i 1 1 1 r A A A A A rt "i r _L 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 ) ( d e f i n e d i n E q u a t i o n VI.2 ) f o r 140A S i 0 2 powder i n a He atmosphere a t 6°K. p a r t i c u l a r c a s e , a sum of e x p o n e n t i a l s [ R ^ ( t ) = a exp(-X.,t) + ( 1-a) e x p ( - X . 2 t ) ] gave a s u b s t a n t i a l improvement i n the per degree of freedom (315 f o r 301 degrees of freedom, compared w i t h 401 f o r 303 degrees of freedom). 77 The Mu f r a c t i o n s were e v a l u a t e d from the f i t t e d parameters AMn bY n o r m a l i z i n g t o the f r e e muon a m p l i t u d e i n A l . I t was assumed t h a t a l l the muons i n j e c t e d i n t o A l p r e c e s s a t the f r e e muon Larmor frequency B*13.55 KHz G"1 (B i n Gau s s ) . The f r e e muon p r e c e s s i o n background due t o muons s t o p p i n g i n p a r t s of the t a r g e t v e s s e l o t h e r than the powder i t s e l f was dete r m i n e d w i t h an a n t i f e r r o m a g n e t i c F e 2 0 3 powder t a r g e t . Muons i n F e 2 0 3 e x p e r i e n c e v e r y l a r g e l o c a l magnetic f i e l d s and thus do not c o n t r i b u t e t o the f r e e muon Larmor f r e q u e n c y . The Mu f r a c t i o n s were e v a l u a t e d as FM„ = (swp\c)_ E q u a t i o n VI•3 The v a l u e s f o r A^ i n A l and F e 2 0 3 were o b t a i n e d from c o a r s e l y packed d a t a taken a t 45G, f i t t e d t o E q u a t i o n s VI•1 and VI•2 w i t h S r t M ( t ) = 0 . T a b l e VI «-l g i v e s the Mu r e l a x a t i o n r a t e s and f r a c t i o n s measured a t 6°K i n 760 t o r r of He. For co m p a r i s o n , the low temperature r e s u l t s f o r b u l k o x i d e s and the room temperature r e s u l t s f o r both b u l k and powdered samples a r e i n c l u d e d . The temperature dependence of the Mu r e l a x a t i o n r a t e i n the A l 2 0 3 powder i s shown i n F i g u r e V I - 5 . 78 TABLE VI • 1 . »/ + SR r e s u l t s i n b u l k and powdered o x i d e s . T a r g e t Temperature Mu F r a c t i o n Mu R e l a x a t i o n (K) (%) R a t e r s " 1 ) S i O i b u l k f u s e d 6 SiO-z b u l k f u s e d 295 SiOt-^powder (70 A) 6 SiOi powder (70 A) 295 SiOa powder (140 A°) 6 SiOi powder (140 A) 295 A l j O a b u l k f u s e d 6 AlzO-» b u l k f u s e d 295 A l t O * powder (75 A) 6 Al-jO, powder (75 A) 295 MgO s i n g l e XL 6 MgO s i n g l e XL 295 MgO powder (300 A) 6 MgO powder (300 A) 295 a The (1) and (2) r e f e r t o the two components r e s o l v e d i n the f i t . b Rough e s t i m a t e u s i n g Mu asymmetries o n l y . c E s i t m a t e based on the m i s s i n g f r a c t i o n presuming i t t o be Mu J ( K i e f l 1979|) e (Spencer 1981) f ( M a r s h a l l 1 9 7 8 ) ^ (Brewer 1 9 8 1 ) 79* 3*'* 79 t 3 b , 3 3.3 i 0.5 0.2010.05 4 9 t 3 0.46t0.03 61 ± 3 d 0.18± 0.03 35 t 5 0 ) * ' b 3 5 * 5 (2) 4.1* 0.7 0.16*0.05 (1) (2) *iSt 20 0.I8i0.03 >80c' >80'^ >20 >20 29 t 3 0.35t0.05 35 t 14^ 11.3 £ 4.4 35 t 10 \ 3 5 1 1 0 b l 6.3*1.4 2..0 t 0.5 12 i 3 0.22i.0.03 15 t 3 1 .9 ± 0.5 79 o LU CO LU (X or cr x <X ox 6.00 5.00 h 4.00 3.00 2.00 h 1.00 h 0.00 6 8 10 12 TEMPERRTURE (K) 14 16 F i g u r e VI •5 The temperature dependence of the Mu s p i n r e l a x a t i o n r a t e i n A 1 2 0 3 powder i n a He atmosphere. The arr o w s i n d i c a t e p o i n t s t h a t a r e o f f s c a l e . VI - I .4. D i s c u s s i o n VI • 1•4 • 1 Mu i n S i 0 2 Powder a t 6°K i n a He Atmosphere I t i s worth w h i l e t o p o i n t out the v a r y i n g c o n d i t i o n s under which the S i 0 2 powder t a r g e t s i n Ta b l e VI.1 were s t u d i e d . . The room t e m p e r a t u r e 70A and 140A t a r g e t s were evacuated t o 10" 6 t o r r , which e f f e c t i v e l y removes adsorbed water (see 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 gas 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 . Thus i t had adsorbed water p l u s whatever 0 2 and N 2 was adsorbed d u r i n g c o o l i n g (4 x 10" 3 0 2 80 atoms per A 2, 1.6 x 10" 2 N 2 per A 2, assuming t h a t a l l the a i r i n the t a r g e t v e s s e l was a d s o r b e d ) . The 140A t a r g e t a t 6°K was ev a c u a t e d t o 10~ 1 t o r r a t 270°K b e f o r e f u r t h e r c o o l i n g and thus had adsorbed water but l i t t l e 0 2 or N 2. A l s o , the c o l d runs were made i n a dense atmosphere of He (760 t o r r of He at 6°K c o r r e s p o n d s t o a d e n s i t y of ~0.01 g/cm 3). In r e g a r d t o the r e s u l t s t aken i n a He atmosphere t h e r e i s no Mu f o r m a t i o n i n He gas (F l e m i n g 1981), so t h a t the observed Mu o r i g i n a t e s from the powders. A l t h o u g h t h e r e i s i n s u f f i c i e n t i n f o r m a t i o n t o c o n c l u s i v e l y s t a t e t h a t the s u r f a c e s p l a y no r o l e i n Mu f o r m a t i o n , the l a r g e Mu f r a c t i o n i n the bu l k o x i d e s ( s e e T a b l e VI.1) i s i n d i c a t i o n t h a t Mu f o r m a t i o n i s a bu l k phenomenon. In some c a s e s the observed Mu f r a c t i o n i n the powder i s l e s s than i n the b u l k . T h i s may p a r t l y be e x p l a i n e d i n terms of a Mu component t r a p p e d i n s i d e the powder g r a i n s where the r e l a x a t i o n i s ve r y f a s t . Another p o s s i b i l i t y i s t h a t a p o r t i o n of the Mu r e a c t s e p i t h e r m a l l y w i t h s u r f a c e groups such as (OH)" t o form a d i a m a g n e t i c s u r f a c e s t a t e such as (OMu)". In b o t h the 70A powder and 140A powder, a s l o w l y r e l a x i n g Mu component was ob s e r v e d (0.46 ± 0.03 (is" 1 and 0.157 ± 0.05 c s " ' , r e s p e c t i v e l y ) . T h i s component cannot be due t o Mu i n s i d e the powder g r a i n s , s i n c e Mu i s s t a t i c i n f u s e d S i 0 2 below 50°K and has a r e l a x a t i o n r a t e of 3.3 ± 0.5 K S ~ 1 , a t t r i b u t a b l e t o random a n i s o t r o p i c d i s t o r t i o n ( S e c t i o n IV«4«2). T h e r e f o r e , the s l o w l y r e l a x i n g component i n the S i 0 2 powder must be due t o Mu which has t h e r m a l i z e d i n the v o i d s . T h i s i s i n c o n t r a d i c t i o n t o the ambient temperature t h e r m a l d i f f u s i o n model ( S e c t i o n V-2) s i n c e , a c c o r d i n g t o t h i s model, no Mu would escape the powder 81 g r a i n s at low temperature where t h e r e i s l i t t l e or no d i f f u s i o n . In f a c t , a second f a s t r e l a x i n g component (see F i g u r e VI-4) i n the 140A powder (4.1 ± 0.7 c s " 1 ) i s most l i k e l y due t o Mu which i s t r a p p e d i n s i d e the powder g r a i n s . The reason t h a t such a component was not obse r v e d i n the 70A powder i s p r o b a b l y due t o the s m a l l e r p a r t i c l e s i z e . A l s o the presence of 0 2 on the s u r f a c e c o u l d r a p i d l y r e l a x Mu w i t h i n 15A of the s u r f a c e ( e s s e n t i a l l y the e n t i r e powder g r a i n ) . Such 0 2 was not p r e s e n t i n the 140A powder. I t may seem s u r p r i s i n g a t f i r s t t h a t 0 2 on the s u r f a c e s does not have a more pronounced e f f e c t on the r e l a x a t i o n of Mu i n the e x t r a g r a n u l a r r e g i o n . A c c o r d i n g t o E q u a t i o n V-20 one would expect t h a t such a c o n c e n t r a t i o n of 0 2 on the s u r f a c e would l e a d t o a Mu s p i n r e l a x a t i o n r a t e of r o u g h l y 120 **s ~ 1 . However, i t i s w e l l known t h a t He r e a d i l y a d s o r b s on such s u r f a c e s a t t h i s temperature (6°K) (See f o r examples Dash 1975). I t i s l i k e l y t h a t a He f i l m s h i e l d s the Mu from d e p o l a r i z i n g e f f e c t s of the s u r f a c e . More e v i d e n c e f o r t h i s w i l l be g i v e n s h o r t l y . VI'1-4-2 Mu i n MgO Powder a t 6°K T h i s sample was s t u d i e d under the same c o n d i t i o n s as the 70A S i 0 2 , i n an open-top non-evacuated v e s s e l , and thus had H 20, 0 2 and N 2 on the s u r f a c e s . A s i n g l e l o n g l i v e d component was r e s o l v e d whose a m p l i t u d e a g r e e s w e l l w i t h the room temperature v a l u e . However, the r e l a x a t i o n r a t e was s u b s t a n t i a l l y l e s s at 6°K (0.22 ± 0.03 «s*') compared w i t h t h a t a t 300°K (1.9 ± 0.5 x s - 1 ) . C o n s i d e r i n g the 82 low p u r i t y of the MgO powder (see T a b l e V I - 2 ) , i t i s u n l i k e l y t h a t t h i s component i s due t o Mu i n s i d e the powder g r a i n s . As i n the case of the S i 0 2 , i t i s a t t r i b u t e d t o Mu i n the e x t r a g r a n u l a r r e g i o n s h i e l d e d from the s u r f a c e by a He f i l m . VI-1-4-3 Mu i n A l 2 0 3 Powder (5°K - 20°K) T h i s t a r g e t a l s o had adsorbed H 20, 0 2 and N 2 (3 x 10"" 0 2 per A 2 and 1.2 x 10~ 3 N 2 per A 2 assuming t h a t a l l the a i r was adsorbed on the powder s u r f a c e s ) . I n a d d i t i o n , the sample c o n t a i n e d an 1.8% paramagnetic F e * 3 i m p u r i t y . ( s e e Table VI.2) Ta b l e V I - 2 . P r o p e r t i e s of o x i d e powders. Powder S i O i Cabot EH5 D e n s i t y (g/cc) 0.04 S i 0 2 0.04 Cabot M5 Al zO> 0.56 Dav i son SMR-7-7563 MgO Matheson Coleman B e l l MX 65-05 0.12 S u r f a c e Area (mVg) 400 200 225 not a v a i l a b l e P a r i c l e S i z e ( d i a . ) (A) I m p u r i t i e s 70 Na(20-40 ppm) P (<300 ppm) A l l o t h e r element l e s s than 30 ppm (see Cabot) 140 same as above 75 F e ( 1 . 8 % ) f r o m (Fe^O,) SOy (.2%) S i 0 2 (.08%) Na„0 (.03%) C l * (<.01%) 300 Na (.5%) C1(.01%) Ca (.05%) Ba(.005%) SO* (.02%) K (.005%) NH„(OH) (.02%)Sr(.005%) In (.01%) Heavy M e t a l s ( . 0 0 3 % ) Mn(5ppm) 83 The observed l o n g - l i v e d component (0.35 ± 0.05 »/S~1 ) must not be due t o Mu i n the powder g r a i n s or adsorbed d i r e c t l y on the s u r f a c e , s i n c e the F e " 3 i m p u r i t y would r e l a x such Mu a t a r a t e of r o u g h l y 400 i i S " ' (see S e c t i o n V«4'2). As i n the case of the 70A S i 0 2 , the adsorbed 0 2 s h o u l d r e l a x e x t r a g r a n u l a r Mu c o l l i d i n g d i r e c t l y w i t h the A l 2 0 3 s u r f a c e . A g a i n , the s m a l l r e l a x a t i o n i s a t t r i b u t e d t o the presence of a He f i l m on the o x i d e s u r f a c e . C o n v i n c i n g 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 F i g u r e V I • 5 , which d i s p l a y s the temperature dependence of X i n A1 2 0 3 . The s h a r p i n c r e a s e i n X. above 12.5 ± 0.5°K i s a t t r i b u t e d t o v a c a n c i e s i n the f i l m as the f i r s t monolayer be g i n s 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 of t h i s phenomenon i s the s u b j e c t of the f o l l o w i n g e x p e r i m e n t . VI•1•5 Summary and C o n c l u s i o n A s l o w l y r e l a x i n g Mu component ( X ~ 0 . 2 vs~ 1 ) has been obser v e d i n A l 2 0 3 , S i 0 2 , and MgO powders i n a He atmosphere 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 the powder g r a i n s c o l l i d i n g f r e e l y w i t h He c o a t e d o x i d e s u r f a c e s . In the 140 A S i 0 2 powder an a d d i t i o n a l f a s t r e l a x i n g component was a l s o r e s o l v e d and a t t r i b u t e d t o Mu t r a p p e d i n s i d e the powder g r a i n s . The d r a m a t i c i n c r e a s e i n Mu s p i n r e l a x a t i o n r a t e i n A l 2 0 3 above 12.5 °K i s thought t o be due t o e v a p o r i z a t i o n of the He f i l m . In c o n c l u s i o n , i t appears t h a t a s i z e a b l e f r a c t i o n of the Mu atoms emerge i n t o the v o i d r e g i o n s of t h e s e o x i d e powders r e g a r d l e s s of temperature and t h a t the presence of a He f i l m on the o x i d e s u r f a c e s i n h i b i t s Mu s p i n r e l a x a t i o n . 84 VI-2 S p i n R e l a x a t i o n of Mu i n A l 2 0 3 Powder w i t h Adsorbed He/Ne In t h i s s e c t i o n f u r t h e r e v i d e n c e i s p r e s e n t e d which s u b s t a n t i a t e s the above c o n c l u s i o n a t l e a s t i n the case of A 1 2 0 3 powder. The Mu s p i n r e l a x a t i o n r a t e has been measured as a f u n c t i o n of adsorbed gas per u n i t s u r f a c e a r e a at c o n s t a n t t e m p e r a t u r e . Both He and Ne gases were used as an a d s o r b a t e . Isotherms . were measured a t 7.3°K and 10.4°K f o r He and 28.7°K and 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 , l i n e a r l y d e c r e a s i n g f u n c t i o n of adsorbed gas below monolayer c o m p l e t i o n and i s v i r t u a l l y independent of the amount of adsorbed gas a t h i g h e r c o v e r a g e s . VI «2-1 E x p e r i m e n t a l D e t a i l s The TRIUMF/Lawrence B e r k e l y L a b o r a t o r y (LBL) s u r f a c e muon a p p a r a t u s , " E a g l e " (see F i g u r e VI«6), was used i n t h i s e x p e r i m e n t . The main d i f f e r e n c e between t h i s a p p a r a t u s and t h a t d e s c r i b e d i n S e c t i o n VI•1 i s the a d d i t i o n a l l e f t and r i g h t p o s i t r o n t e l e s c o p e s , which a r e i n g e n e r a l more s u i t e d f o r t r a n s v e r s e f i e l d »*SR s i n c e they a re l e s s s e n s i t i v e t o s c a t t e r i n g of beam p o s i t r o n s . The TRIUMF M9 v~n c h a n n e l equipped w i t h a 3m l o n g Wien f i l t e r v e l o c i t y s e l e c t o r , or "DC s e p a r a t o r " , s e t a t E = 3.9 kV/cm x, B 48 G y, was used t o o b t a i n a c l e a n beam ( e + c o n t a m i n a t i o n u n d e t e c t a b l e ) of 28 MeV/c " s u r f a c e muons". An added e f f e c t of the s e p a r a t o r was t o r o t a t e the muon p o l a r i z a t i o n by a s m a l l v e r t i c a l a n g l e (9°) r e l a t i v e t o the beam d i r e c t i o n . Muons e n t e r i n g the 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 F I counter Hucite observation window carbon degrader R 2 counter RI counter acuum cryostat M diameter collimator B 2 counter 0 0 3 mylar window 29 MeV/c / i + F i g u r e V I • 6 The M + S R a p p a r a t u s " E a g l e " . Note the f o u r p o s i t r o n t e l e s c o p e s . 3/4 i n c h d i a m e t e r spot b e f o r e d e t e c t i o n by a t h i n (0.010 i n c h ) s c i n t i l l a t o r . T y p i c a l i n c i d e n t muon r a t e s were 30,000/s. The He l e a k t e s t e d t a r g e t v e s s e l (see F i g u r e V I ' 7 ) was c o n s t r u c t e d from s t a i n l e s s s t e e l w i t h two 1 i n c h d i ameter 0.001 i n c h t h i c k s t a i n l e s s s t e e l windows. The A 1 2 0 3 powder sample ( w e i g h i n g 10.5g) was baked a t 500°C 86 G LAS S WOOL ft* stopping region •005" MYLAR •21. M e V i i + c . 0 0 0 5 ALUMINIZED MYLAR STYROFOAM SPACER COLD He GAS He (Ne) ATMOSPHERE Ge THERMOMETER O.OOl" stainless steel WINDOW A l 2 0 3 POWDER TARGET (I0.5g) cm F i g u r e VI«7 The powder t a r g e t v e s s e l and c r y o s t a t used t o study 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 the t a r g e t v e s s e l i s i s o l a t e d from the He atmosphere of the c r y o s t a t . f o r 24 hours i n a i r and c o o l e d i n a d e s s i c a t o r b e f o r e b e i n g p l a c e d i n t h e . t a r g e t v e s s e l . Adsorbed water on r - A l 2 0 3 forms h y d r o x y l groups w i t h h eated above 100°C, so t h a t the s u r f a c e was l i k e l y t e r m i n a t e d by a l a y e r of h y d r o x y l groups ( K n o z i n g e r 197 6)-. The v e s s e l was then s o l d e r e d t o the gas h a n d l i n g system and ev a c u a t e d t o 10" 5 t o r r f o r a p e r i o d of 24 hours immediately p r i o r t o the expe r i m e n t . A C r y o C a l 2500L germanium r e s i s t o r w i t h an a b s o l u t e a c c u r a c y of b e t t e r than ± 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 was used t o mo n i t o r the temp e r a t u r e and c o n t r o l i t t o w i t h i n ± 60 m°K. 87 The a l l - m e t a l ( s t a i n l e s s s t e e l and copper) gas h a n d l i n g a p p a r a t u s ( F i g u r e VI«8) was composed p r i m a r i l y from 1/4 i n c h VI G A S = S INLET TO PUMPING STATION STANDARD VOLUME CRYOSTAT INSIDE WALL-TARGET V E S S E L -F i g u r e VI«8 The gas 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 of He/Ne on r - A l 2 0 3 powder. t u b i n g , Swagelock f i t t i n g s and Nupro b e l l o w s v a l v e s . The vapour p r e s s u r e measurements i n the t a r g e t v e s s e l were made w i t h a W a l l a c e and T i e r n a n p r e s s u r e gauge (0-800 t o r r ) ( G 2 ) , a c c u r a t e t o ±0.5 t o r r . The p r e s s u r e d i f f e r e n c e measurements were made w i t h a Matheson 6301 s t a i n l e s s s t e e l a b s o l u t e p r e s s u r e gauge (0-760 t o r r ) ( G l ) , a c c u r a t e t o ± 2 t o r r . The system was evac u a t e d t o 10" a t o r r and He l e a k t e s t e d p r i o r t o the e x p e r i m e n t . High p u r i t y grade "He (99.995%) and p u r i f i e d grade 2 0 N e (99.99%) were used as a d s o r b a t e s . 88 VI«2«2 E l e c t r o n i c s The e l e c t r o n i c s a r e as d e s c r i b e d i n S e c t i o n VI•1 w i t h a minor m o d i f i c a t i o n t o a l l o w a c q u i s i t i o n of f o u r h i s t o g r a m s i n s t e a d of the two used i n the f i r s t e x p e r i m e n t . VI•2•3 Procedure The e x p e r i m e n t a l procedure c o n s i s t e d of a d m i t t i n g a c o n t r o l l e d amount of a d s o r b a t e i n t o the t a r g e t v e s s e l h e l d at c o n s t a n t t e m p e r a t u r e , r e c o r d i n g the vapour p r e s s u r e , and c o l l e c t i n g a »i + SR spectrum. The p r e c i s e s t e p s were as f o l l o w s ( w i t h the system i n i t i a l l y under vacuum and a l l v a l v e s c l o s e d except V4, which was open f o r the e n t i r e e x p e r i m e n t ) . 1. V1 was opened t o p r e s s u r i z e the s t a n d a r d volume (1368 ± 20 c c ) bounded by V1, V2, and V3. 2. The p r e s s u r e on gauge G1 was r e c o r d e d . 3. V2 was opened and c l o s e d t o admit a s m a l l amount of gas ( 80 cc a t STP) i n t o the t a r g e t v e s s e l volume bounded by 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 the amount of gas a d m i t t e d i n t o the t a r g e t volume c o u l d be c a l c u l a t e d . 5. The p r e s s u r e and temperature i n the t a r g e t v e s s e l were s t a b i l i z e d over a 30 minute p e r i o d and the vapour p r e s s u r e r e c o r d e d (G2). 6. An ti*SR spectrum was' taken ( 4 m i l l i o n e v e n t s ) over a p e r i o d of about one or two h o u r s . 7. The p r e s s u r e i n the t a r g e t v e s s e l , G2, was r e c o r d e d a g a i n . The d i f f e r e n c e from b e f o r e and a f t e r the run was t y p i c a l l y 1 or 2 t o r r , i n d i c a t i n g t h a t the system was v e r y c l o s e t o t h e r m a l e q u i l i b r i u m . 8. Steps 3 through 7 were r e p e a t e d u n t i l more than a monolayer of gas was adsorbed. A f t e r w a r d s the powder was removed and r e p l a c e d w i t h a p i e c e of aluminum of volume m /p^ulk ' w ^ e r e m * s the m a s s of powder 89 (10.5 g) and p^lk * s t h e d e n s i t y of b u l k A l 2 0 3 (3.7 g/cm 3). Steps 1 through 5 were r e p e a t e d . The p r e s s u r e i n the t a r g e t v e s s e l was found t o be a l i n e a r f u n c t i o n of the amount of gas a d m i t t e d f o r each temperature s t u d i e d . The amount of gas adsorbed on the A l 2 0 3 s u r f a c e s ( i n u n i t s of cc a t STP) a t a g i v e n t e mperature and vapour p r e s s u r e was d e t e r m i n e d from the d i f f e r e n c e between the amount of gas a d m i t t e d w i t h and w i t h o u t powder. The s p e c i f i c s u r f a c e a r e a (225m 2/g as s p e c i f i e d by Da v i s o n C h e m i c a l s ) and t o t a l mass of the powder (10.5 g) were then used t o determine the number of adsorbed atoms per u n i t s u r f a c e a r e a . VI'2-4 A n a l y s i s and R e s u l t s The »« + SR data a n a l y s i s was done e x a c t l y as d e s c r i b e d i n S e c t i o n V I•1•3. Good f i t s were o b t a i n e d w i t h a s i n g l e e x p o n e n t i a l r e l a x a t i o n f u n c t i o n f o r the Mu p r e c e s s i o n s i g n a l . The r e s u l t s from the l e f t and r i g h t t e l e s c o p e s were averaged as a l a s t s t e p . The 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 or the number of adsorbed "He atoms per u n i t a r e a a t 7.3°K and 10.4°K i s shown i n F i g u r e V I - 9 . For compa r i s o n , the vapour p r e s s u r e i n the v e s s e l a t each coverage i s a l s o p l o t t e d . S i m i l a r r e s u l t s f o r 2 0Ne adsorbed on A l 2 0 3 a r e shown i n F i g u r e VI-10. 90 (_> UJ CO a. or GC X ex UJ or 12 10 8 6 4 2 0 1 1 1 $ 10 4°K 1 1 1 1 1 I _ } 7-3-K • • 10 4° K • — 1 f 7-3°K 1 1 1 * • V ' V . 1 . 1 i i i i H500 H400 LU cn 3 0 0 | LU DC CL 200 or Z) o a 6 8 10 12 14 ATOMS/(NANOMETER)2 16 18 20 iOO 0 F i g u r e 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 2 0 3 powder v e r s u s adsorbed He a t 7.3°K and 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 . VI•2•5 D i s c u s s i o n VI-2«5«1 A d s o r p t i o n Isotherms of He on A 1 2 0 3 Vapour p r e s s u r e i s o t h e r m s a r e a common means of d e t e r m i n i n g the amount of gas r e q u i r e d t o complete a monolayer. The p o i n t on. the i s o t h e r m a t which the d e n s i t y of adsorbed atoms becomes a l i n e a r f u n c t i o n of vapour p r e s s u r e i s a rough i n d i c a t i o n of the monolayer d e n s i t y (Brunauer 1938). T h i s i s r e f e r r e d t o as the p o i n t B method. 91 LU CO or g cc X (X _ J LU or 12 10 8 6 4 2 0 1 1 1 { 30-3° K 1 1 1 I I 1 - j 28-7° K \ -— \ — i • V 30-3° K f " 28 7° K \ * * 1 1 1 1 ' .y A° » a A ' 1 1 1 1 1 0 6 8 10 12 14 ATOMS/(NANOMETER)4 300 200 100 0 or CO CO LU or Q_ or Z> o 16 18 20 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 i n A l 2 0 3 powder v e r s u s adsorbed Ne a t 30.3°K and 28.7°K. A p p l y i n g t h i s method t o the 7.3°K and 10.4°K "He a d s o r p t i o n i s o t h e r m s (see F i g u r e VI-9) y i e l d s a monolayer d e n s i t y n^(7.3°K) = 0.125 ± 0.01A" 2 and n w(10.4°K) = 0.10 ± 0.01A" 2. The e f f e c t i v e h a r d sphere c r o s s s e c t i o n assuming a c l o s e packed 2 d i m e n s i o n a l a r r a y , = — , i s then 8.6 ± 0.5A 2 a t 10.land 7.3 ± 0.5A 2 a t 7.3°K. For comparison the hard sphere c r o s s s e c t i o n of "He on g r a p h i t e a t 4.2°K, o b t a i n e d from the monolayer d e n s i t y , 0.123 92 A 2 (Dash 1975), i s 7.9 A 2, whereas f o r l i q u i d He a t 4.2 °K, assuming a s i m p l e c u b i c p a c k i n g f a c t o r i s irp _ 2 / 3 / 4 ) = 10.4 A 2. A s t o n (1955) has compared the p r o p e r t i e s of adsorbed He t o those of b u l k He under an e f f e c t i v e p r e s s u r e . The s l i g h t temperature dependence of n m i n d i c a t e s t h e r e may be some t h e r m a l e x p a n s i o n of the f i l m between t h e s e two t e m p e r a t u r e s , a l t h o u g h the non-u n i f o r m i t y of these s u r f a c e s a l o n g w i t h the rough n a t u r e of the p o i n t B method cannot be r u l e d out as a cause. VI'2'5-2 Mu S p i n R e l a x a t i o n i n A l 2 0 3 Powder With Adsorbed He The »i + SR data taken on A l 2 0 3 w i t h adsorbed He i s a l s o shown i n F i g u r e VI«9. Below monolayer c o m p l e t i o n , the Mu s p i n r e l a x a t i o n r a t e f i t s w e l l t o a l i n e a r f u n c t i o n of adsorbed He, independent of t e m p e r a t u r e . Above monolayer c o m p l e t i o n , the r e l a x a t i o n r a t e r a p i d l y l e v e l s o f f a t a c o n s t a n t v a l u e X.0 ^ 0.54 ± 0.05 as- 1 . The i n t e r p r e t a t i o n i s q u i t e s i m p l e . The Mu r e l a x a t i o n r a t e i s p r o p o r t i o n a l t o the f r a c t i o n of exposed s u r f a c e a r e a : E q u a t i o n VI•4 where n, i s the d e n s i t y of adsorbed atoms i n the f i r s t l a y e r o n l y , i s the t o t a l e l a s t i c c r o s s s e c t i o n f o r Mu s c a t t e r i n g o f f an adsorbed He atom, k i s a c o n s t a n t ( t o be d i s c u s s e d s h o r t l y ) and X.0 i s a c o n s t a n t r e l a x a t i o n r a t e u n r e l a t e d t o the M u - A l 2 0 3 s u r f a c e i n t e r a c t i o n . There a r e a t l e a s t t h r e e f a c t o r s which c o n t r i b u t e t o X.0-93 1. In a t r a n s v e r s e f i e l d of 8 G, the s p l i t t i n g (n) of the 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"1 . The e f f e c t of f i t t i n g t h ese two f r e q u e n c i e s t o a s i n g l e component y i e l d s an apparent r e l a x a t i o n X ~ n. 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 s l i g h t l y magnetic s t a i n l e s s s t e e l c a u s i n g 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 used i n the experiment was 99.995%, so t h a t i m p u r i t i e s may have caused a s m a l l r e l a x a t i o n . At low coverage, n, i s s i m p l y the t o t a l amount of adsorbed gas ( t h i s e x p l a i n s the l i n e a r b e h a v i o u r of X.(n) below monolayer c o m p l e t i o n ) whereas a t h i g h e r c o v e r a g e s , n, --> n m , the monolayer d e n s i t y , and thus X.(n) — > X.0. There was no marked temperature dependence i n X.(n), and t h e r e f o r e the combined data a t 7.3 and 10.4°K were f i t t e d t o a l i n e a r f u n c t i o n a t low cov e r a g e , y i e l d i n g k = 32.9 ± 0.3 u s - ' and a** = 11.0 ± 0.2 A 2. Not s u r p r i s i n g l y , the ha r d sphere c r o s s s e c t i o n of h e l i u m , «$Z a t 7.3 and a t 10.4°K (7.3 ± 0.5A 2 and 8.6 ± 0.5A 2 r e s p e c t i v e l y ) d e t e r m i n e d from the a d s o r p t i o n i s o t h e r m s i s s l i g h t l y d i f f e r e n t than . T h i s c o u l d e a s i l y be e x p l a i n e d c l a s s i c a l l y as due the f i n i t e s i z e of the Mu atom which a l s o c o n t r i b u t e s t o the He-Mu s c a t t e r i n g c r o s s s e c t i o n . Perhaps more c o r r e c t l y t h i s d i f f e r e n c e can be e x p l a i n e d by the f a c t t h a t c^l i s d e t e r m i n e d by the by the He-He-surface i n t e r a c t i o n , whereas i s d e t e r m i n e d by the He-Mu-surface i n t e r a c t i o n . The l i n e a r dependence of X(n) a t low coverage 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 A T f o r w H i c h "2-E q u a t i o n VI•5 f o r w^ ' ) ^ y ^ o l T R . t f ^ j v i J z Equat i o n VI•6 C o n s i d e r i n g the s t r o n g b i n d i n g of H t o the bare s u r f a c e s ( S e c t i o n V«3) and the l a r g e v a l u e of X.g e x p e c t e d because of the F e + 3 i m p u r i t y ( 400 «s' 1 f o r s t a t i c Mu, see S e c t i o n V«4«2) l i m i t i n g case 2 must be f a v o u r e d . I t i s d i f f i c u l t t o e x t r a c t any more i n f o r m a t i o n from the the p r e s e n t r e s u l t s , such as a v a l u e f o r P^. , w i t h o u t making 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 t o k from s p i n exchange w i t h the F e * 3 on the s u r f a c e i s not known. F u r t h e r e x p e r i m e n t s on a paramagnetic f r e e sample would e l i m i n a t e t h i s unknown. Furthermore o n l y an upper l i m i t on the c o l l i s i o n f r e q u e n c y w i t h the s u r f a c e i s k n o w n ) s i n c e Mu a t low t e m p e r a t u r e s may s c a t t e r o f f the aggregate s t r u c t u r e s r a t h e r than o f f the p r i m a r y p a r t i c l e s j u s t as i n the case of Ps (see S e c t i o n I I I . 3 . 1 ) . T h i s c o u l d cause a s i g n i f i c a n t d e c r e a s e i n vc . D e s p i t e these u n c e r t a i n t i e s i t i s i n t e r e s t i n g t o e s t i m a t e P^  under the assumptions t h a t the e n t i r e s u r f a c e a rea i s e q u a l l y a c c e s s i b l e t o the Mu, i n which case v£ (0) a t 7.3°K i s 4.3 x 10" y s " 1 , and 95 t h a t the s p i n exchange r a t e i s n e g l i g a b l e . One then f i n d s the t r a p p i n g p r o b a b i l i t y P# t o be 0.00074. In the f u t u r e i t s h o u l d be p o s s i b l e t o measure P i n w e l l c h a r a c t e r i z e d samples where vc i s known. VI«2'5'3 A d s o r p t i o n Isotherms of Ne on A l 2 0 3 A p p l y i n g the p o i n t B method g i v e s an e f f e c t i v e h a rd sphere c r o s s s e c t i o n f o r adsorbed Ne, <s§l ~ 7 * 9 1 1«° ^ 2 a t 30.3°K and 8.6 ± 1.0 A 2 a t 28.7°K. For comparison, on g r a p h i t e near 20°K i s 7.37A 2 (Huff 1975), whereas a l a y e r of l i q u i d neon at 27.1°K c o r r e s p o n d s t o ajfe = 7.23 A 2. The agreement between a l l t h e s e methods i s i n d i c a t i o n t h a t Ne atoms a r e much l i k e hard s p h e r e s . VI'2«5'4 Mu S p i n R e l a x a t i o n i n A l 2 0 3 Powder With Adsorbed Ne The shape of the r e l a x a t i o n r a t e v e r s u s d e n s i t y of adsorbed Ne atoms was i d e n t i c a l t o t h a t observed f o r He. As i n the case of He, no marked temperature dependence was ob s e r v e d between 30.3°K and 28.7°K. F i t t i n g the r e g i o n below 0.11 atoms A" 2 to E q u a t i o n VI»2 w i t h X 0 = 0.85 ± 0.08 xS" 1 g i v e s k = 31.4 ± .3 „s" 1 and <f^u6 = 8.9 ± 0.2A 2. The v a l u e of a ^ e o b t a i n e d from the f i t i s i n f a i r agreement w i t h e^fe o b t a i n e d from the a d s o r p t i o n i s o t h e r m s , and i s s u b s t a n t i a l l y lower than cjj^. T h i s might be due t o the s t r o n g e r a t t r a c t i o n the Mu f e e l s toward the Ne, which c o u l d enhance the p r o b a b i l i t y of t r a p p i n g and thus decrease the the e l a s t i c c r o s s s e c t i o n . 96 S u r p r i s i n g l y , the v a l u e o b t a i n e d f o r k i s c l o s e t o t h a t observed a t the lower t e m p e r a t u r e s (7.3°K and 10.4°K) i n the case of He. U n f o r t u n a t e l y , s i n c e the temperature dependence f o r the s p i n r e l a x a t i o n r a t e and the t r a p p i n g r a t e i n E q u a t i o n IV'4 are both unknown, t h i s o b s e r v a t i o n cannot be used t o c o n c l u s i v e l y d i s t i n g u i s h one from the o t h e r . The measureably l a r g e r v a l u e of X.0 i n the case of Ne i s p r o b a b l y due t o the h i g h e r i m p u r i t y l e v e l i n the Ne as opposed t o the He. VI-2-6 S t a t u s of the ATTD Model A c c o r d i n g t o the ambient temperature t h e r m a l d i f f u s i o n (ATTD) model the Mu t h e r m a l i z e s a t the ambient temperature i n s i d e the g r a i n s , d i f f u s e s t o the o x i d e s u r f a c e , and i s e j e c t e d i n t o t h e v o i d s because of a n e g a t i v e work f u n c t i o n . However, s i n c e Mu i s s t a t i c i n bulk o x i d e s such as S i 0 2 below 50 °K (Brewer 1981),' the ATTD model f a i l s t o e x p l a i n the c o p i o u s amount of Mu i n the v o i d s of these powders a t low t e m p e r a t u r e s . A s i m p l e model t h a t e x p l a i n s why Mu might 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 g i v e n i n Appendix I I I 97 VI•2•7 C o n c l u s i o n I t has been shown t h a t Mu emerges from 75A A l 2 0 3 powder g r a i n s below 30°K i n low d e n s i t y He and Ne atmospheres (10-760 t o r r ) . 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 dependent on the amount of exposed s u r f a c e a r e a . T h i s s u b s t a n t i a t e s the c o n c l u s i o n s reached i n the p r e v i o u s experiment ( S e c t i o n V I . 1 . 5 ) . The e f f e c t i v e c r o s s s e c t i o n f o r e l a s t i c s c a t t e r i n g of Mu o f f adsorbed He (7.3 t o 10.4°K) and Ne (28.7 t o 30.3°K) have been measured t o be 11.1 ± 0.1A 2 and 8.9 ± 0.2A 2 r e s p e c t i v e l y . A new t e c h n i q u e f o r s t u d y i n g the p r o p e r t i e s of the s e o x i d e s u r f a c e s and adsorbed atoms on them has been demonstrated. 98 CHAPTER V I I : MUONIUM IN THE CONDENSED PHASES OF A r , Kr AND Xe M o t i v a t i o n f o r s t u d y i n g Mu i n the condensed phases of A r , K r , and Xe i s easy t o f i n d . F i r s t l y , when a muon s t o p s i n matter i t l e a v e s a hot 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 has been c o n s i d e r a b l e debate i n r e c e n t y e a r s on the e f f e c t of t h i s t r a c k on Mu f o r m a t i o n and r e l a x a t i o n ( P e r c i v a l 1981, Walker 1981). S i n c e the p r o p e r t i e s of the t r a c k a re phase dependent i t i s of i n t e r e s t t o study the e f f e c t of phase (gas, l i q u i d and 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 . In t h i s c h a p t e r the f i r s t o b s e r v a t i o n s of Mu i n the condensed phases of A r , Kr and Xe are r e p o r t e d . These measurements, a l o n g w i t h the e x i s t i n g gas phase d a t a ( F l e m i n g 1981b) r e p r e s e n t the f i r s t complete M'SR study of an element i n a l l t h r e e phases. S e c o n d l y , H atoms have been s t a b i l i z e d and s t u d i e d i n s o l i d A r , K r , and Xe u s i n g ESR. The ESR measurements a r e s e n s i t i v e t o l a t t i c e - i n d u c e d p e r t u r b a t i o n s of the h y p e r f i n e c o n t a c t i n t e r a c t i o n s as w e l l as t o the presence of n u c l e a r moments. In p r i n c i p l e , i s o t o p i c dependence of these e f f e c t s can be s t u d i e d u s i n g n*SR . F i n a l l y , t h e r e have been s e v e r a l s t u d i e s of c h e m i c a l r e a c t i o n s of H atoms w i t h i m p u r i t i e s such as C « H 1 0 d e p o s i t e d i n s o i i d Xe (Kinugawa 1 978, I w a s a k i 1978). In the p a s t »i + SR has been used v e r y s u c c e s s f u l l y 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 atom c h e m i s t r y i n gas and l i q u i d phases. I f t h e r e i s a l a r g e Mu f r a c t i o n i n the s o l i d phase of the s e elements i s o t o p i c e f f e c t s of the above mentioned H atom r e a c t i o n s i n the s o l i d phase c o u l d be s t u d i e d . 99 VI1-1 E x p e r i m e n t a l The muon beam has 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 has been shown i n F i g u r e V I • 5 . The t a r g e t v e s s e l ( F i g u r e V I I • 1 ) was c o n s t r u c t e d from copper and s t a i n l e s s copper - constantan thermocouple heater wires gas Inlet condensed target 29 MeV/c 0 0 5 " mylar window scale cms. 0 I 5 copper target vessel heater heater wires cold finger from cryostat F i g u r e V I I • 1 The t a r g e t v e s s e l used t o condense noble gases. s t e e l except f o r the mylar window. The v e s s e l was mounted on a r o t a t a b l e L N 2 c r y o s t a t 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 of the gas c o u l d be observed p r i o r t o f a c i n g i t towards the beam. The t a r g e t gases ( u l t r a h i g h p u r i t y Xe, u l t r a h i g h p u r i t y K r , and r e s e a r c h grade Ar) were a d m i t t e d v i a the gas i n l e t tube ( F i g u r e V I I - 1 ) . The v e s s e l was equipped w i t h two h e a t e r s , one on 100 the gas i n l e t t o p r e v e n t b l o c k a g e s , and a second mounted between the copper v e s s e l and the c o l d f i n g e r of the c r y o s t a t f o r temperature c o n t r o l . Two copper c o n s t a n t a n thermocouples were used t o mon i t o r the temperature ( c o n t r o l l e d t o ±1°K). The >/ + SR 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. V I I * 2 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 taken i n a t r a n s v e r s e f i e l d of 65G i n or d e r t o determine the f r e e muon f r a c t i o n . The data 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 VI•1 w i t h ^ ( t ) g i v e n by E q u a t i o n VI*2 and S ^ y ( t ) =0 The parameter of i n t e r e s t , A^  , d i d not depend on whether R ^ ( t ) was g a u s s i a n or e x p o n e n t i a l . The f r e e muon f r a c t i o n s i n T a b l e V I I • 1 were d e t e r m i n e d from HflCn) - f\^CYc^03) E q u a t i o n VI 1*1 P*SR s p e c t r a were a l s o taken i n a low t r a n s v e r s e f i e l d 8 G i n o r d e r t o determine the Mu f r a c t i o n . The da t a were f i t t o E q u a t i o n V* 1 w i t h S^ ( t ) and S ^ ( t ) g i v e n by E q u a t i o n V*2 and w i t h R ^ ( t ) = e" p . In a l l cases except l i q u i d Ar and s o l i d K r , good f i t s were o b t a i n e d w i t h a s i n g l e component Mu r e l a x a t i o n f u n c t i o n ( R ^ * ^ e'^ ) . There were seven f r e e p a r a m e t e r s , N 0, , AMK ' UM m ' *Mn ' a n c ^ B3 * T ^ e P a r a m e t e r s *y t * a n < 3 yu were h e l d f i x e d a t the v a l u e s o b t a i n e d from the h i g h f i e l d d a t a . In the cas e s of l i q u i d Ar and s o l i d K r , c o n s i d e r a b l y b e t t e r f i t s were o b t a i n e d w i t h a two component Mu r e l a x a t i o n f u n c t i o n [(R*^ l f(t) oe-A>-t + (. _ 0 ) e - ^ ] . The improvement i n the x 2 was s i g n i f i c a n t : 101 T a b l e V I I «1. *i + SR r e s u l t s i n condensed Ar,Kr,and Xe. Target Free Muon Hu Hissing Mu Relaxation Mu Hyperfine Substance Temperature Fr a c t i o n 3 Fraction (s) 1 3 Fraction Rate(s) S p l i t t i n g 0 K % % % usee - 1 MHz Liquid Ar 85 1.6 + 1.0 (1) 48 + 6 3 + 29 (1) 0.65 + 0.12 (2) 49 + 28 (2) 19.0 + 11.0 Sol id Ar 77 0.8 + 0.2 91 + 9 8 + 9 0.15 + 0.03 4463.8 ± 6.0 Liquid Xe 162 3.3 + 0.8 43 + 9 54 + 10 2.07 + 0.21 Solid Xe 150 5.0 + 3.3 79 + 25 16 + 28 19.0 + 2.5 Liquid Kr 120 6.5 + 0.1 57 + 10 36 + 10 3.6 + 0.9 Solid Kr 90 1 + 1.8 (1) 71 + 7 0 + 10 (1) 6.68 + 0.20 4462.9 ± 3-7 (2) 29 + 7 (2) 0.89 + 0.04 These fractions were determined from data at ~70 G. These fractions 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 The vacuum hyperfine s p l i t t i n g is 4463.302 MHz. from 164 (145 degrees of freedom) t o 126 (143 degrees of freedom) f o r s o l i d Kr and from 169 (145 degrees of freedom) t o 154 (143 d e g r e e s . o f freedom) f o r l i q u i d A r . The Mu f r a c t i o n s i n T a b l e V I I • 1 were d e t e r m i n e d from E q u a t i o n VI«3. For comparison the r e s u l t s i n the gas phase are shown i n T a b l e VII«2. In the case of s o l i d Xe, the r e l a x a t i o n r a t e of Mu was so f a s t t h a t a r e l i a b l e measure of the Mu asymmetry and r e l a x a t i o n r a t e was not p o s s i b l e i n low f i e l d . I t was n e c e s s a r y t o use data taken a t 70G(where the p r e c e s s i o n s i g n a l i s more c o m p l i c a t e d ) to d e t e r m i n e the Mu f r a c t i o n and r e l a x a t i o n r a t e . The d a t a were f i t t o E q u a t i o n VI•1 w i t h 1 02 Tab l e V I I . 2 . Mu F r a c t i o n i n Gas Phase A r , K r , and X e ? b b Tar g e t P r e s s u r e Mu F r a c t i o n Free Muon F r a c t i o n (atmospheres) % % Ar 2.5 75 24 Kr 0.9 100 3 Xe 0.65 100 3 « from ( 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 M J 2~ . ? -/It ' i J E q u a t i o n V I I - 2 T h i s e x p r e s s i o n i s v a l i d a t i n t e r m e d i a t e t r a n s v e r s e f i e l d s 65G (see S e c t i o n I V ' 5 ) . The parameters A^, p, N 0, and «^ were f i x e d a t the v a l u e s o b t a i n e d from the c o a r s e l y binned d a t a . The parameters ^ ( n o t t o be c o n f u s e d w i t h + and *M„) and n are f u n c t i o n s of the a p p l i e d f i e l d (B) and the h y p e r f i n e frequency ( u 0 ) (see S e c t i o n I V ' 3 ' 2 ) . In t h i s f i t , u 0 was s e t t o the vacuum v a l u e , 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., 0Mi< and Bj. The r e l a x a t i o n r a t e of Mu i n s o l i d Ar and Kr was s u f f i c i e n t l y s m a l l so t h a t the two normal Mu f r e q u e n c i e s c o u l d be r e s o l v e d a t an i n t e r m e d i a t e f i e l d of 66G (See S e c t i o n I V . 3 . 2 ) . The d i f f e r e n c e between the s e f r e q u e n c i e s i s a f u n c t i o n of B and u 0 ( E q u a t i o n IV.31) and thus p r o v i d e s a measurement of o 0 . The two freq u e n c y p r e c e s s i o n of Mu i n s o l i d Ar a t 66 G i s shown i n F i g u r e V I I - 2 . The h y p e r f i n e s p l i t t i n g d e t ermined i n 103 0.50 0.25 5 0 0 0 -0.25 -0.50 0.00 0.05 0.10 0.15 0.20 0.25 0.30 TIME IN MSEC (2 NSEC/BIN) F i g u r e V I I • 2 Two f r e q e n c y Mu p r e c e s s i o n s i g n a l i n s o l i d Ar a t 77°K i n h i g h f i e l d ( 66 G). t h i s way i s a l s o g i v e n i n Ta b l e V I I « 1 . V I I • 3 D i s c u s s i o n 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 both l i q u i d and s o l i d Ar i s e x t r e m e l y s m a l l (<2%). T h i s i s a s t r o n g i n d i c a t i o n t h a t most of the muons form Mu, s i n c e t h e r e a r e no s t r o n g l o c a l f i e l d s t o r e l a x f r e e muon p o l a r i z a t i o n . In f a c t , t h e r e i s no source of even weak l o c a l magnetic f i e l d s , s i n c e t h e r e a r e no s i g n i f i c a n t amounts of n a t u r a l l y o c c u r i n g i s o t o p e s of Ar w i t h n u c l e a r 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 s i g n a l (F^ = 91 ± 9%) 1 04 was observed w i t h a s m a l l r e l a x a t i o n r a t e (X, = 0.15 ± 0.03 »/S~ 1). The Mu p r e c e s s i o n s i g n a l s i n s o l i d and l i q u i d argon a r e shown i n F i g u r e V I I ' 3 . In l i q u i d A r , t h e r e i s a s l o w l y r e l a x i n g component 0.25 | 1 1 1 1 1 1 1 1 0.15 -0.05 --0.05 --o.i5 y -0.25 -0.0 .0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 TIME IN uSEC (10 NSEC/BIN) .0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 TIME IN uSEC (10 NSEC/BIN) F i g u r e V I I • 3 (a) S i n g l e f r e q u e n c y Mu p r e c e s s i o n i n l i q u i d Ar a t 85°K a t low f i e l d ( 8 G); (b) In s o l i d Ar a t 77°K. (X. = 0.65 ± 0.17 «s"') which a c c o u n t s f o r o n l y 48 ± 6% of the muon p o l a r i z a t i o n . The r e l a x a t i o n i s l i k e l y due t o a r e a c t i v e i m p u r i t y p r e s e n t at a ppm l e v e l . In a d d i t i o n , t h e r e i s 105 i n d i c a t i o n of a v e r y f a s t - r e l a x i n g component (X. = 19 ± 11 D S " 1 ) , a c c o u n t i n g f o r the r e m a i n i n g muon p o l a r i z a t i o n and o b s e r v a b l e o n l y w i t h i n the f i r s t 100 ns. The l o n g - l i v e d Mu s i g n a l d emonstrates t h a t the i n e r t l i q u i d or i t s i m p u r i t i e s are not r e s p o n s i b l e f o r t h i s r e l a x a t i o n . I t i s p o s s i b l e t h a t the f a s t r e l a x i n g component i s due t o Mu i n t e r a c t i n g w i t h 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 of 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 atoms. T h i s w i l l be d i s c u s s e d f u r t h e r i n S e c t i o n VII«4. E l e c t r o n s p i n resonance e x p e r i m e n t s on H atoms i n a s o l i d Ar l a t t i c e 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 s i t e s (Foner 1960). The major component has a h y p e r f i n e s p l i t t i n g s h i f t of 0.46% r e l a t i v e t o the vacuum v a l u e . At t e m p e r a t u r e s above 39°K, the H atoms become m o b i l e . In the p r e s e n t e x p e r i m e n t , we have measured the mean h y p e r f i n e s p l i t t i n g of Mu i n s o l i d argon a t 77°K t o be 4463.8 ± 6MHz, which g i v e s a s h i f t of 0.01 ± 0.13%. A l t h o u g h t h i s does not agree w i t h the H atom r e s u l t s , the two measurements are not d i r e c t l y comparable because i n the ESR experiment the H atoms a r e t r a p p e d at f i x e d l a t t i c e s i t e s , whereas i n the p r e s e n t y*SR experiment the Mu atoms are p r o b a b l y d i f f u s i n g t h r ough the l a t t i c e so t h a t the h y p e r f i n e p e r t u r b a t i o n Is*-%.-e* (see S e c t i o n IV-4-2) i s averaged over many s i t e s . 106 VII»3«2 Mu i n L i q u i d and S o l i d Xe As i n the case of A r , the f r e e muon f r a c t i o n i n l i q u i d and s o l i d Xe i s s m a l l (3.3 ± 8% and 5 ± 3%, r e s p e c t i v e l y ) . I n s o l i d Xe, the Mu component a c c o u n t s f o r 79 ± 25% of the muon ensemble. The l a r g e e r r o r i s due t o the f a s t r e l a x i n g n a t u r e of the s i g n a l . In l i q u i d Xe, the Mu component ac c o u n t s f o r o n l y 43 ± 9% of the muons. I t i s p o s s i b l e t h a t t h e r e i s a l s o a f a s t r e l a x i n g Mu component as i n the case of l i q u i d A r , but i t i s not r e s o l v a b l e . The Mu r e l a x a t i o n r a t e i n c r e a s e s s h a r p l y from 2.1 ± 0.2 us " 1 i n the l i q u i d t o 19.0 ± 2.5 ^ s " 1 i n the s o l i d . The n u c l e a r d i p o l e moments of 1 2 9 X e and 1 3 1 X e , which comprise 26.44% and 21.18% of the n a t u r a l l y o c c u r r i n g Xe, are l i k e l y r e s p o n s i b l e f o r the f a s t r e l a x a t i o n i n the s o l i d . M o t i o n a l n a r r o w i n g (see S e c t i o n V«4«4), an e f f e c t g r e a t l y enhanced i n the l i q u i d , might cause such a d i s c o n t i n u i t y between phases. The l a r g e r r e l a x a t i o n r a t e i n l i q u i d Xe compared w i t h l i q u i d Ar i s l i k e l y due t o the h i g h e r i m p u r i t y c o n t e n t i n the Xe. The ESR spectrum 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 the Xe l a t t i c e a t 4.2°K i s multicomponent w i t h an o v e r a l l s pread of 98.2 G. Each f r e q u e n c y c o r r e s p o n d s t o a p a r t i c u l a r magnetic environment, d e t e r m i n e d p r i m a r i l y by the i s o t o p i c c o m p o s i t i o n s of i t s n e a r e s t n e i g h b o u r s . Thus, the mean l o c a l f i e l d 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 of o r d e r 25 G. T h i s c o r r e s p o n d s t o a Mu r e l a x a t i o n r a t e of o r d e r 2n«1.4 M HzG'^SG = 220 *»s"1. The observed Mu r e l a x a t i o n r a t e at 150°K i s o n l y 19.0 ± 2.5 «s" '. Thus, i t i s l i k e l y t h a t the Mu i s d i f f u s i n g r a p i d l y a t t h i s t e mperature or p o s s i b l y t r a p p e d i n 1 07 d e f e c t s where the l o c a l f i e l d s a r e much s m a l l e r . I t s h o u l d be noted t h a t H atoms are mo b i l e i n s o l i d Xe a t te m p e r a t u r e s above 20°K (Kinugawa 1978). VII«3«3 Mu i n L i q u i d and S o l i d Kr A g a i n , the f r e e muon f r a c t i o n s i n s o l i d and l i q u i d Kr are s m a l l (1.4 ± 1.8% and 6.5 ± 0.1%, r e s p e c t i v e l y ) . As i n the case in I iquii Kr of l i q u i d Xe, the Mu component^accounts f o r o n l y 57 ± 10% of the muon ensemble, whereas the Mu f r a c t i o n i n s o l i d K r , 99 ± 10%, i s c o n s i s t e n t w i t h 100% Mu f o r m a t i o n w i t h no m i s s i n g f r a c t i o n . The Mu r e l a x a t i o n i n l i q u i d Kr i s a g a i n l i k e l y due t o i m p u r i t i e s , s i n c e m o t i o n a l n a r r o w i n g i s ex p e c t e d t o quench any r e l a x a t i o n due t o 8 3 K r , which a c c o u n t s f o r 11.48% of the n a t u r a l K r . Perhaps the most i n t e r e s t i n g **SR spectrum i n t h i s experiment i s f o r s o l i d K r . As mentioned i n S e c t i o n V I I - 2 , a c o n s i d e r a b l y b e t t e r f i t t o the d a t a was o b t a i n e d u s i n g a two component Mu r e l a x a t i o n f u n c t i o n (see F i g u r e VII«4). The sum of thes e components a c c o u n t s f o r 99 ± 10% of the muon ensemble. One i n t e r e s t i n g i n t e r p r e t a t i o n i s t h a t t h e r e e x i s t t r a p p i n g s i t e s due t o v a c a n c i e s or d e f e c t s . The s i t u a t i o n i s analogous t o the t r a p p i n g of Mu on a s u r f a c e d e s c r i b e d i n S e c t i o n V«5 w i t h X.3t >> 1. In the p r e s e n t case X.g c o r r e s p o n d s t o the r e l a x a t i o n r a t e of t r a p p e d Mu. T h i s i s e x p e c t e d t o be s m a l l , s i n c e the RLMF ( S e c t i o n IV«5«1) due t o 8 3 K r (11.48% of n a t u r a l Kr p o s s e s s i n g a magnetic moment of -0.969 nv ) f a l l s o f f r a p i d l y as 1 / r 3 . The r e l a x a t i o n r a t e of i n t e r s t i t i a l Mu, b e i n g much c l o s e r t o the n u c l e a r moments, i s ex p e c t e d t o be l a r g e r . As i n V«5, we d e f i n e 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) F i g u r e V I I ' 4 The Mu p r e c e s s i o n s i g n a l s o l i d Kr at 90°K i n a magnetic f i e l d of 10.7G v as the t r a p p i n g r a t e . Under th e s e c o n d i t i o n s , one would expect a r e l a x a t i o n f u n c t i o n E q u a t i o n V I I • 3 ( i . e . : a sum of e x p o n e n t i a l s as o b s e r v e d ) . U s i n g the f i t t e d a m p l i t u d e s and r e l a x a t i o n r a t e s y i e l d s v = 1.7±0.4 e s " 1 , X F = 5.0±0.4 ^ s " 1 and X.& = 0.89±0.04 u s ' 1 . One may a l s o make a rough e s t i m a t e on the hopping r a t e between i n t e r s t i t i a l s i t e s by comparing X F w i t h the c a l c u l a t e d s t a t i c 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 t r a p p i n g s i t e s i n a f a c e c e n t e r e d c u b i c l a t t i c e such as s o l i d Kr have t e t r a h e d r a l or o c t a h e d r a l symmetry. We w i l l assume t h a t the Mu atoms hop between o c t a h e d r a l s i t e s which have a l a r g e r Mu-nucleus 109 s e p a r a t i o n (1.418 A) (Foner 1960). The s t a t i c r e l a x a t i o n r a t e of Mu a t one of these s i t e s may be c a l c u l a t e d from e q u a t i o n V'13, assuming one of the e i g h t n e a r e s t n e i g h b o u r s i s 8 3 K r ( n u c l e a r moment=-0.969 n u c l e a r magnetons). The r e s u l t y i e l d s X=15 */S~ 1 which i s a f a c t o r of t h r e e l a r g e r than the observed r e l a x a t i o n r a t e . T h i s i n d i c a t e s t h a t the hopping i s r a t h e r slow-- 45 ns~ 1 a c c o r d i n g t o e q u a t i o n V'21 (assuming T i s the mean time between h o p s ) . A l s o the Mu atoms hop o n l y 26 times b e f o r e t r a p p i n g . T h i s model c o u l d be t e s t e d by s t u d y i n g the e f f e c t of a n n e a l i n g and temperature on the »/ + SR spectrum. V I I - 3 - 4 M i s s i n g F r a c t i o n s I t i s c l e a r from these r e s u l t s t h a t i n s o l i d A r , Kr and Xe a l l the muon p o l a r i z a t i o n i s a c c o u n t e d f o r , whereas i n the l i q u i d s r o u g h l y h a l f the muon p o l a r i z a t i o n i s m i s s i n g a f t e r 100 ns. R e c e n t l y , such r e s u l t s have been e x p l a i n e d q u a l i t a t i v e l y (Walker 1981)in terms of an expanding t r a c k model. A c c o r d i n g t o t h i s model, the Mu i s formed e p i t h e r m a l l y and b e g i n s t o d i f f u s e randomly from some p o i n t beyond the end of the t r a c k . At some time T, the c o n c e n t r a t i o n of t r a c k s p e c i e s , a l s o d i f f u s i n g , o v e r l a p s w i t h the Mu f o r a s h o r t p e r i o d of t i m e , d u r i n g which the s p i n exchange r e l a x a t i o n r a t e i s l a r g e . Thus, one would expect a f r a c t i o n of the muon p o l a r i z a t i o n t o be l o s t w i t h i n time T. In c o l d l i q u i d s , such as l i q u i d Ar a t 85°K, the d i f f u s i o n r a t e s may be s u f f i c i e n t l y s m a l l so t h a t the r e l a x a t i o n of the f a s t component i s b a r e l y o b s e r v a b l e . In s l i g h t l y warmer l i q u i d s , such as l i q u i d Xe a t 150°K and l i q u i d Kr a t 120°K, the d i f f u s i o n r a t e s a re too l a r g e , and -T too s m a l l , f o r the f a s t 110 r e l a x i n g component t o be obse r v e d . I t i s argued t h a t i n the s o l i d phase the d i f f u s i o n r a t e s a re too s m a l l and T too l a r g e (>> 2.2 us) f o r the f a s t component t o be observed. V I I ' 4 C o n c l u s i o n s As i n the gas phase, most of the muons i n the l i q u i d and s o l i d A r , Kr and Xe form 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 the case of Ar gas, where a 25% muon f r a c t i o n i s observed (Fleming 1981). No such f r e e muon f r a c t i o n was observed i n e i t h e r l i q u i d or s o l i d A r . The elements of A r , Kr and Xe a r e i d e a l s u b s t a n c e s t o study M u - l a t t i c e s t a t e s because of t h e i r l a r g e (^100%) Mu f o r m a t i o n p r o b a b i l i t y and t h e i r s i m p l e monatomic s t r u c t u r e . I t may be p o s s i b l e t o make more d i r e c t comparisons w i t h e x i s t i n g ESR data of t r a p p e d H atoms a t lower t e m p e r a t u r e s where perhaps the Mu a l s o becomes t r a p p e d . There i s an i n d i c a t i o n t h a t the Mu i n t e r a c t s w i t h i t s own r a d i a t i o n t r a c k i n the l i q u i d e l e m e n t s , l e a d i n g t o a f a s t r e l a x i n g Mu component. 111 CONCLUDING REMARKS I t has been demonstrated i n the p r e c e d i n g c h a p t e r s how Ps and Mu s t u d i e s can p r o v i d e a unique p e r s p e c t i v e on atom-m o l e c u l e , 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 . In the f u t u r e , i t i s p o s s i b l e t h a t Mu and Ps c o u l d p r o v i d e a t e s t i n g ground f o r fundamental t h e o r i e s d e a l i n g w i t h atom-s u r f a c e i n t e r a c t i o n . Measurements of such q u a n t i t i e s as s i n g l e atom b i n d i n g e n e r g i e s , s u r f a c e d i f f u s i o n r a t e s and a d s o r p t i o n 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 re a l m of p o s s i b i l i t y . U n t i l now, the c h e m i s t r y of Mu on s u r f a c e s and i n s o l i d s has been n e g l e c t e d . I t i s hoped t h a t t h i s s tudy w i l l s t i m u l a t e some i n t e r e s t i n t h i s d i r e c t i o n . 1 1 2 APPENDIX I THERMALIZATION OF GAS ATOMS IN A POWDER The mean energy l o s s per c o l l i s i o n f o r a t h e r m a l beam of atoms can be w r i t t e n E q u a t i o n Al•1 where T^ = the temperature of the beam. t3- = the i n i t i a l energy of the gas atom. c| = the f i n a l energy of the gas atom. P( «?,€*) = the p r o b a b i l i t y per c o l l i s i o n f o r a t r a n s i t i o n e1* --> c| . P ( € ^ , t | ) can be approximated i n a o n e - d i m e n s i o n a l t h e o r y f o r atom s u r f a c e s c a t t e r i n g , f i r s t d e v e l o p e d by D e v o n s h i r e (1937) and r e c e n t l y reviewed by Goodman (1971). The e f f e c t of u s i n g a o n e - d i m e n s i o n a l c a l c u l a t i o n i n c r e a s e s the t r a n s i t i o n r a t e by a s m a l l f a c t o r (2 or 3) i n ca s e s where the t h r e e - d i m e n s i o n a l c a l c u l a t i o n has 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 atom p o t e n t i a l i s r e p r e s e n t e d by a f u n c t i o n E q u a t i o n A l • 2 where z and Z are the d i s p l a c e m e n t s of the gas atom and s u r f a c e atom r e s p e c t i v e l y . V ( z ) i s o f t e n chosen t o be a Morse p o t e n t i a l )/(*)= Vie. -2c- ) . E q u a t i o n A l •3 - 2 A Z - a z \ E q u a t i o n A l • 4 D and a are the depth and range of the p o t e n t i a l , r e s p e c t i v e l y . 1 1 3 The u n p e r t u r b e d gas atom s t a t e s a r e s o l u t i o n s t o L n 0 x y ^ ' J E q u a t i o n A l - 5 The t r a n s i t i o n r a t e from €^ — > ef i s g i v e n as E q u a t i o n A l • 6 where |s ; > i s the i n i t i a l s t a t e of the s o l i d a t temperature T s w i t h energy €*. The sum i s over a l l f i n a l s t a t e s of the s o l i d , |s^>. The m a t r i x element i n v o l v i n g the s o l i d i s most e a s i l y e v a l u a t e d by e x p r e s s i n g Z i n terms of phonon• c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s ( A s h c r o f t 1976). l)A/ ^~ J ^ W ^ ^ ' E q u a t i o n AI-7 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 t o normal mode q, a^ c r e a t e s one of the same, i s the phonon f r e q u e n c y , M i s the mass of the s o l i d atoms, and N i s the number of atoms i n the s o l i d . E q u a t i o n A l • 8 where 1 14 \ - e E q u a t i o n A l • 9 ~ . 1  ^ ^ W ^ A T 5 ; — & ~~ i Equat i o n 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 of modes, E q u a t i o n A l • 1 1 where u p i s the Debye f r e q u e n c y and n i s the volume of the s o l i d , i n t o E q u a t i o n AI-8 y i e l d s L - H - S - = ^ [ to ( n w + 0 f ^ - « f f +• Equat i o n A l • 1 2 E q u a t i o n A l • 1 3 where the g s u p e r s c r i p t has been dropped. Thus E q u a t i o n A l • 1 4 The p r o b a b i l i t y per c o l l i s i o n f o r s c a t t e r i n g i n t o an energy 1 15 i n t e r v a l cU^ i s Equat i o n A l • 1 5 where PC^^ = /JQQ \ 2 i s the d e n s i t y of f i n a l s t a t e s i n one x. d i m e n s i o n w i t h n o r m a l i z a t i o n L and D = no c o l l i s i o n f r e q u e n c y . i s the 6 er i A J i s e Equat i o n A l • 1 6 For a Morse p o t e n t i a l , + Equat i o n Al•17 1 16 ere / i ' - ( 2 W 6 f ) ' / 2 / ^ d = (2wD),/l/tifl tiW - 6+-6:1 Equat i o n Al•18 S u b s t i t u t i n g E q u a t i o n AI-18 i n t o E q u a t i o n A l • 1 , . Eq u a t i o n A l • 1 9 By the p r i n c i p l e of m i c r o s c o p i c r e v e r s i b i l i t y i < 6 f i v ( z ) U ; > f = - \<6C) vU))^>r E q u a t i o n Al•20 P f + tus) - e P U + f c w , 0 E q u a t i o n Al•21 CD Wt> -^/fcT 5 fc /JL - / \ Eq u a t i o n Al•22 Thus, i n t h e r m a l e q u i l i b r i u m , t h e r e i s no mean energy t r a n s f e r . The r a t e of energy l o s s can be w r i t t e n cU - A H V 6 E q u a t i o n A l -23 where va i s the c o l l i s i o n f r e q u e n c y w i t h the s u r f a c e , can be 1 17 e s t i m a t e d as V P E q u a t i o n Al-24 where N i s the number of p a r t i c l e s per u n i t mass, V F i s the f r e e volume per u n i t mass, R i s the r a d i u s of the powder g r a i n s and v i s the mean v e l o c i t y of the atom. E q u a t i o n A l '25 where SA i s the s p e c i f i c s u r f a c e a r e a of the powder, p i 0) l (j i s the d e n s i t y of the s o l i d , and p i s the d e n s i t y of the powder. The time r e q u i r e d f o r an atom of mean energy Ei t o reach 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 AI-23 Equat i o n A l • 2 6 where AE i s g i v e n by E q u a t i o n s AI'22 and AI'18. 1 18 APPENDIX I I PS SCATTERING OFF 2 ELECTRON ATOM (MOLECULE) In t h i s a ppendix, a t h e o r y f o r low energy s c a t t e r i n g of o-Ps o f f a t w o - e l e c t r o n atom i s d e v e l o p e d . I t i s assumed t h a t the s p i n (s) of the atom and t h u s t h e s p a c i a l symmetry of the e l e c t r o n s i n the atom i s c o n s e r v e d d u r i n g c o l l i s i o n s . When s=0 (e. g . , He) t h e r e i s no s p i n c o n v e r s i o n , but when s=1 (e . g . , H e * ( 2 3 S ) ) , o-Ps may be c o n v e r t e d t o p-Ps v i a s p i n exchange. The e f f e c t of a l a r g e magnetic f i e l d on the observed quenching r a t e i s c o n s i d e r e d . F i n a l l y , the r e s u l t i s g e n e r a l i z e d t o a 2 e l e c t r o n m o l e c u l e . The p a r t i c l e 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 of o n l y the s p a c i a l c o o r d i n a t e s . Thus, e f f e c t s due t o the s p i n - s p i n and s p i n - o r b i t c o u p l i n g a r e n e g l e c t e d . The s p i n of each p a r t i c l e and the t o t a l e l e c t r o n s p i n a r e t h e r e f o r e c o n s e r v e d q u a n t i t i e s . However, the d i r e c t i o n of the e l e c t r o n s p i n a s s o c i a t e d w i t h the o-Ps i s not n e c e s s a r i l y c o n s e r v e d d u r i n g c o l l i s i o n s , because of the exchange degeneracy a s s o c i a t e d w i t h the e l e c t r o n s ( i . e . , the e l e c t r o n s may i n t e r c h a n g e ) . T h i s i s the b a s i c p r i n c i p l e behind s p i n exchange. 119 A 11•1 T o t a l E l e c t r o n S p i n S t a t e s S t a t e s of t o t a l e l e c t r o n s p i n |SS 2p> (p the z component of p o s i t r o n s p i n ) span an i r r e d u c i b l e sub-space Cs^n w i t h r e s p e c t t o 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 . B a s i s v e c t o r s f o r C*p,'„r g e n e r a t e the i r r e d u c i b l e r e p r e s e n t a t i o n r of the p e r m u t a t i o n group S 3. The b a s i s v e c t o r f o r r 3 / 2 t r a n s f o r m s a c c o r d i n g t o the one d i m e n s i o n a l , t o t a l l y symmetric i r r e d u c i b l e r e p r e s e n t a t i o n A, of S 3 [ u s i n g n o t a t i o n of Tinkham ( 1 9 6 4 ) ] , whereas the b a s i s v e c t o r s f o r r , / z t r a n s f o r m a c c o r d i n g t o the two d i m e n s i o n a l i r r e d u c i b l e r e p r e s e n t a t i o n E of S 3. For example, E q u a t i o n A l I • 1 E q u a t i o n A l I • 2 E q u a t i o n A l I • 3 where t(- i s the row index of the i r r e d u c i b l e r e p r e s e n t a t i o n 1 20 A 11 •> 2 S p a c i a l S t a t e s The space of a s y m p t o t i c s t a t e s f o r Ps (momentum k and o r b i t a l a n g u l a r momentum lm) i n c i d e n t on a two e l e c t r o n atom ( s p i n s) a r e spanned by the t h r e e s t a t e s where £ > , f f 1 = J*Lk|r,t.r r l] r/fri+Cf 1 V%,-rj) and * l 2 5 ( f , f 2 ) i s the two e l e c t r o n w a v e f u n c t i o n , symmetric f o r s=0 and a n t i s y m m e t r i c f o r s=1. There a r e two i r r e d u c i b l e subspaces a s s o c i a t e d w i t h each v a l u e of s. For s=1, one has A 2 symmetry ( t o t a l l y a n t i s y m m e t r i c ) and one has E symmetry. For s=0, t h e r e i s one w i t h A, symmetry and one w i t h E symmetry. For example, E q u a t i o n A l I • 4 1 Urn E f, s.i> . - k \- t f VrW& ft 1 + < f T ( r , r f ) flf, f, ~) ] E q u a t i o n A l I • 5 121 E q u a t i o n A l I • 6 A I I ' 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 of T o t a l E l e c t r o n S p i n These s t a t e s a re found i n the d i r e c t p r o d u c t space c. Cf x <L.S^ ;„ an& must have A 2 symmetry. The v a l u e of S u n i q u e l y d e t e r m i n e s the i r r e d u c i b l e r e p r e s e n t a t i o n r ^ " " ' a s s o c i a t e d w i t h the s p a c i a l wave f u n c t i o n . For S=3/2, r'P"1 = A, and thus r ^ 4 " = A 2. For S=1/2, r 5 r m = E and thus r * P v r = E. m m S ' i 5*'s--| f>= ||!jcrV\ / l,S=l>]5 - - | S . f,f> Equat i o n A l I • 7 I Vcjem S=£ S z 5 * i p> = J ^ f U i m E £ ^ l > I S - I S t p > E q u a t i o n A l I • 8 Ittm 5 = i S z *-o J ^ f - U i ^ E f , s«o>IS-lS. f*f> E q u a t i o n A l I • 9 S i n c e the H a m i l t o n i a n and thus the T m a t r i x a r e i n v a r i a n t under the group o p e r a t i o n s of S 3, the m a t r i x elements of T v a n i s h between s t a t e s b e l o n g i n g t o d i f f e r e n t i r r e d u c i b l e r e p r e s e n t a t i o n s of S 3 or t o d i f f e r e n t rows of the same 1 22 i r r e d u c i b l e r e p r e s e n t a t i o n of S 3 / a c c o r d i n g t o a g e n e r a l m a t r i x element theorem (Tinkham 1964). S i n c e we have imposed the c o n d i t i o n t h a t s i s c o n s e r v e d , the T m a t r i x i s d i a g o n a l i n the above b a s i s . The d i a g o n a l elements are T ^ S = < YJlm E S, s IT | k i m E f, s> E q u a t i o n A l l • 1 0 E q u a t i o n A l I • 1 1 A I I - 4 P h y s i c a l S t a t e s of Ps s p i n ( I I Z ) C o n s i d e r the Ps s p i n s t a t e s : E q u a t i o n A l I • 1 2 These s t a t e s a l s o have A 2 symmetry, but a r e not w e l l d e f i n e d s t a t e s of t o t a l e l e c t r o n s p i n i n g e n e r a l . Thus, the T m a t r i x i s not d i a g o n a l i n t h i s r e p r e s e n t a t i o n . 123 A II»5 The T m a t r i x i n | k l m l l ^ s s z > R e p r e s e n t a t i o n 'Define T » S < ( I I , « . ; J , I . , S . ' ) « <\cjlm I I . s s J T I k J J ^ r ' l l . S S ; > - - S s E q u a t i o n A l I • 1 3 The m a t r i x < k l m l I z s s z |klmSS^sp> and the T m a t r i x i n the | k l m l l z s s z > r e p r e s e n t a t i o n ( i n terms of T^ ,* ) a r e g i v e n i n the f o l l o w i n g t a b l e s . T a b l e A I I . 1 . The M a t r i x < k l m l I z s s z |klmSS rs*> s s f 1 1 i i 1 i / I 1 / D 0 0 0 3/z % \ % 3/2 & '/z % Yz Vz 'A -\ -l/z -% ~>/z '/z K Yi •Vz / 1 1 1 J / 0 1 I Vis / -1 1 1 '43 0 o 1 1 1 1 1 0 1 o 1 0 ; -1 I o 0 D I D 4 i 1 1 -1 % -s 1 D 1 '1 I -I 1 -1 \ 0 0 1 -1 '4 1 1 o D -l l O o o % 1 -1 0 o -1 0 0 0 o 'Vfz -Vfz 1 25 T a b l e A l l . 2 . The M a t r i x <klml I t s s z | T | klml" !^ s's'^ 1 I s' K i r r s sx 1 1 i o 1 1 1 0 i 1 O 1 1 o / 0 -/ o / o -/ O /' o - / O 1 o -1 o / 1 / 1 1 1 1 / 1 1 1 o o o 0 1 1 i 1 0 o 0 - / -1 -1 - / 0 o o o i / 1 1 q 1 1 b d d i -J 1 1 d d o o / 1 d b -d 1 1 ' o -d b 1 o / 0 d b d / -1 / 0 b d d o o / 0 d b -d i 1 / -1 d -d c i 0 1 -1 d b - d i -1 / -1 Q o o ' -/ d -d b 1 1 0 o e. 1 o o o e 1 -1 o o e o o 0 o e. 3 ^ d = ~~ 'kjj £ ~ ' let 3 126 A II'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 The s c a t t e r i n g a m p l i t u d e f o r a g i v e n t r a n s i t i o n I'l^'s's^. — > I I 2 s s 2 . i s g i v e n as k K CO E q u a t i o n A l I • 1 4 E q u a t i o n A l I • 1 5 The t o t a l c r o s s s e c t i o n f o r such a t r a n s i t i o n Equat i o n A l I • 1 6 The o-Ps-->p-Ps 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 b t a i n e d by a v e r a g i n g over i n i t i a l s t a t e s of o-Ps w i t h s=1 and summing over f i n a l s t a t e s of p-Ps 3 i - s ^ s . = -IT "^Jj-t+n " >ki E q u a t i o n AII-17 27 k* 3 s S i n c e the s c a t t e r i n g m a t r i x i s u n i t a r y Tk^ may be p a r a m e t e r i z e d i n terms of a s i n g l e r e a l a n g l e , the s c a t t e r i n g phase s h i f t f o r s c a t t e r i n g i n a s t a t e of Ps o r b i t a l a n g u l a r momentum/, t o t a l e l e c t r o n s p i n S, and atomic s p i n s. -, • r 5 * E q u a t i o n A l I • 1 8 1 27 Thus the c o n v e r s i o n c r o s s s e c t i o n can be r e w r i t t e n g> . r <~7h 'I r>A / 7 -2. 1-1 L-r 27 k J-co E q u a t i o n A l l • 1 9 The s p i n exchange c r o s s s e c t i o n i s d e f i n e d as k r JL=d ~ HI <4 E q u a t i o n A l l - 2 0 6 In a l a r g e magnetic f i e l d , where (10) and (00) h y p e r f i n e s t a t e s of Ps are mixed^, the r e l e v a n t c o n v e r s i o n r a t e i s from Iz= ±1 L, = 0 . 27 k J.=o J ' E q u a t i o n A l I • 2 1 A I I - 7 The T o t a l C r o s s S e c t i o n The t o t a l c r o s s s e c t i o n f o r s=1 128 = J - o r J : (2i+\)L 2 | T ^ ' | Z + JT^'j J = ±JE 12. (2U))l 2 sin* 5, 4-5/" J 3 k*-E q u a t i o n A l I • 2 2 The t o t a l c r o s s c r o s s f o r s=0 i s s i m p l y E q u a t i o n A l l - 2 3 whereas the c o n v e r s i o n c r o s s . s e c t i o n i s z e r o . A I I • 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 In the case of Ps of o r b i t a l a n g u l a r momentum 1, s c a t t e r i n g o f f a m o l e c u l e w i t h r o t a t i o n a l a n g u l a r momentum j , i n v a r i a n c e 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 the t o t a l a n g u l a r momentum, J = j + 1, be c o n s e r v e d . T h i s a l l o w s f o r t h e p o s s i b i l i t y of i n e l a s t i c c o l l i s i o n s i n v o l v i n g r o t a t i o n a l e x c i t a t i o n . The 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 from r o t a t i o n a l s t a t e j --> j ' ( A r t h u r s 1960) i n a t o t a l e l e c t r o n s p i n s t a t e S ( m o l e c u l a r s p i n s) can be w r i t t e n E q u a t i o n A l I • 2 4 J5» where T ( j ' l ' j l ) i s the T m a t r i x element'between i n i t i a l s t a t e 129 j l and f i n a l s t a t e j ' l ' . The 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 from i n i t i a l s t a t e j l l z s s — ? j ' l ' I ^ s s ' ^ can be w r i t t e n ^ % a ^ j ) J = ^ j ' l x ! T r f j U ' r , r , , s . , - , j / i ^ O r E q u a t i o n A l I • 2 5 The c o r r e s p o n d i n g 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 ^ ' ' J ' ^ f f c - ^ ^ ^ ' ^ , ) ' T O ' ^ - T W ' ) E q u a t i o n A l I • 2 6 As w i l l be shown i n the f o l l o w i n g s e c t i o n J S S * Cj'l'7ji) - ^ ( e 6 - I ) E q u a t i o n A I L 2 7 i n the low energy l i m i t kR << 1, where R i s the range of the p o t e n t i a l . In t h i s l i m i t , L0 ; J 7 \ J J 2 7 ^ o . / E q u a t i o n A l l . 2 8 Thus the c o n v e r s i o n c r o s s s e c t i o n averaged over r o t a t i o n a l s t a t e s of t h e m o l e c u l e i s s i m p l y ^-^ v * J E q u a t i o n A l l - 2 9 Note t h a t the phase s h i f t s a r e independent of the r o t a t i o n a l s t a t e of the m o l e c u l e . 130 SSs , , A II«9 E v a l u a t i o n of T ( j 1 ; j 1) i n the L i m i t kR << 1 We assume t h a t the s c a t t e r i n g s t a t e s of t o t a l e l e c t r o n s p i n S and m o l e c u l a r s p i n s a t low energy e v o l v e 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 +$>J9"+rt + V S * E q u a t i o n A I L 3 0 where B 0 i s the r o t a t i o n a l c o n s t a n t f o r the mo l e c u l e and V ^ 3 i s an e f f e c t i v e l o c a l i n t e r a c t i o n between the Ps and the molecule which depends on the t o t a l e l e c t r o n s p i n S and m o l e c u l a r s p i n s. Now c o n s i d e r m a t r i x elements of V S s * S d r J a I> < k J - J * j j e l k j j - ^ » > < k j j . ^ l r w > J l i t / V€\ r w )< P w Ik ' j ' j > iX>< ic j j . ' ^ V-' / *'JT, J '> E q u a t i o n A l l • 3 1 where f i s t h e v e c t o r from the Ps cm t o m o l e c u l a r CM and u i s Si the m o l e c u l a r a x i s u n i t v e c t o r . V ( f , u ) can then be expanded as Vsrra)« ^ <f(r) ftr.a) • ^ ^)Yjf)Y»1*>\ E q u a t i o n A l I • 3 2 where we have assumed t o be a r e a l f i n i t e f u n c t i o n of f and r- o , i n v a r i a n t under u — > -o. U s i n g E q u a t i o n A l I • 3 3 131 E q u a t i o n A l l • 3 4 E q u a t i o n A l I • 3 5 I t f o l l o w s : 1- J U i j * j ''2. Equat i o n A l l - 3 6 At 300°K, k f o r Ps i s 0.113A" 1, and a t 600°K i s s t i l l o n l y 0.163A" 1. I f we assume t h a t the p o t e n t i a l i s n e g l i g a b l e f o r R > 3A, then the i n t e g r a l over r i s s m a l l f o r a l l v a l u e s of 1 L 1', except 1=L=1'=0. Thus the m a t r i x element of V 5 s can be ap p r o x i m a t e d t o 132 ^ ho S k K' A ) dr^ J J 7 ^ 0 E q u a t i o n A l l -37 T h i s i m p l i e s t h a t the s c a t t e r i n g i s dominated by the s wave e l a s t i c component i n v o l v i n g o n l y the i s o t r o p i c p a r t of the Ps mo l e c u l e 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 element i n the T m a t r i x can t h e r e f o r e be e x p r e s s e d u ' JJ E q u a t i o n A l l - 3 8 where the phase s h i f t s 6^* are independent of j . T h i s i s p r e c i s e l y the c o n c l u s i o n reached by A r t h u r s and Da l g a r n o c o n c e r n i n g low energy s c a t t e r i n g of e l e c t r o n s by a r i g i d r o t o r ( A r t h u r s 1960). They found t h a t s c a t t e r i n g o f f a non s p h e r i c a l p o t e n t i a l such as y[r>£> ) ~ f CD v < 3a* -5J£3 - o-rjl r>34„ E q u a t i o n A l l ' 3 9 was c o m p l e t e l y e l a s t i c s-wave f o r e l e c t r o n e n e r g i e s below 0.01 Rydberg, which c o r r e s p o n d s t o a wave v e c t o r k l e s s than 0.187 A. I t i s i m m e d i a t e l y c l e a r from the C l e b s h Gordon c o e f f i c i e n t C C j L j 1 j T M j ^ ) i n E q u a t i o n A l l - 3 6 t h a t j z i s a l s o c o n s e r v e d i n t h i s low energy l i m i t . 133 APPENDIX I I I DIRECT THERMALIZATION OF MUONIUM IN THE VOIDS OF OXIDE POWDERS The f a i l u r e of the ATTD model (see S e c t i o n VI.2.6) n e c e s s i t a t e s an a l t e r n a t i v e e x p l a n a t i o n f o r how Mu emerges i n t o the v o i d r e g i o n s a t low te m p e r a t u r e . In t h i s a ppendix, the p o s s i b i l i t y t h a t Mu t h e r m a l i z e s d i r e c t l y i n the v o i d s i s examined. A c r o s s s e c t i o n of the Mu-powder p o t e n t i a l can be imagined as i n F i g u r e A l 1 1 • 1 . Given t h a t t h e r e i s a c o l l i s i o n of a CM I 70A DIA. ~3A > 0) F i g u r e A I I I . 1 . Imagined c r o s s s e c t i o n of the Mu-powder g r a i n p o t e n t i a l . Jo Mu atom (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 p o s s i b l e outcomes. 1. The atom i s s c a t t e r e d e l a s t i c a l l y o f f the s u r f a c e , l o s i n g a f r a c t i o n of i t s energy. S i n c e the th e r m a l wavelength f o r a 100°K Mu atom i s much l e s s than the atomic 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 atoms. Thus the mean energy l o s s i s 3mE/M, where m i s the Mu mass and M i s the 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 i m p o r t a n t near t h e r m a l e n e r g i e s . 3. The atom e n t e r s the g r a i n w i t h k i n e t i c energy E - V 0, 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 the g r a i n . In t h i s c a s e , the Mu might become t r a p p e d i n the 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 the 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 the g r a i n w i t h k i n e t i c energy E - V 0, l o s e s energy v i a phonons and i s e x p e l l e d from the s u r f a c e b e f o r e t h e r m a l i z a t i o n . T h i s can be thought of as 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 . For E < V 0, outcomes 1 and 2 are the o n l y ones a l l o w e d e n e r g e t i c a l l y . I t may be assumed t h a t f o r Mu e n e r g i e s g r e a t e r than some t h r e s h o l d energy E ^ (dependent on the type of powder and the g r a i n r a d i u s , R ) , outcome 4 dominates. Thus t h e r e i s , i n e f f e c t , an energy window V 0 < E < E ^ where t h e r m a l i z a t i o n w i t h i n the powder i s most l i k e l y . The p r o b a b i l i t y f o r a p a r t i c l e of energy, E, t o be t r a n s m i t t e d i n t o the powder g r a i n can be e s t i m a t e d from the t r a n s m i s s i o n p r o b a b i l i t y f o r square p o t e n t i a l b a r r i e r of h e i g h t V 0 < E. In one dimens i o n ( L e i g h t o n 1959), Equat i o n Al11•1 T h e r e f o r e , a s u f f i c i e n t 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 d i r e c t l y i n the v o i d s i s f o r ( E + ^ - V 0 ) / V 0 << 1, s i n c e ^ ( E ) i s then s m a l l over the e n t i r e energy window V 0< E <E^ . E t h can be e s t i m a t e d by assuming t h a t a Mu atom which e n t e r s a g r a i n l o s e s energy through e l a s t i c c o l l i s i o n s w i t h i n d i v i d u a l atoms u n t i l i t reaches a c r i t i c a l k i n e t i c energy, E f r-V 0 , a t which p o i n t i t becomes t r a p p e d . E ^ can be approximated a s : 135 t h E q u a t i o n A l I I - 2 where l 2 i s the mean squared d i s t a n c e between c o l l i s i o n s and R 2 / l 2 i s the mean number of c o l l i s i o n s r e q u i r e d f o r a Mu atom t o s c a t t e r randomly the r a d i u s of the g r a i n s (R) ( R e i f 1965),. l o s i n g an average 2m(E - V 0)/M per c o l l i s i o n . I 2 s h o u l d be on o r d e r of the mean atomic s p a c i n g squared ( 1 0 A 2 ) . S i n c e Mu becomes t r a p p e d i n S i 0 2 below 50°K, a v a l u e f o r E^ - V 0 i s somewhat a r b i t r a r i l y chosen t o be 100°K. The above parameters, i n s e r t e d i n t o E q u a t i o n AIII«2, y i e l d E ^ - V 0 ~ 0.033eV. In o t h e r words i f a Mu atom i s t o d e p o s i t the l a s t of i t s k i n e t i c energy w i t h i n a powder g r a i n and s t o p , then i t s k i n e t i c energy i n s i d e the g r a i n must not exceed about 0.033 eV. The work f u n c t i o n a t the s u r f a c e i s not known, a l t h o u g h the b a r r i e r h e i g h t must be much g r e a t e r than 300°K, s i n c e the Mu does not r e e n t e r the S i 0 2 powder a t room t e m p e r a t u r e . In ord e r t o i l l u s t r a t e the f e a s i b i l i t y of t h i s model, V 0 i s chosen t o be 2 eV, which y i e l d s 2U< 0.13 over the e n t i r e energy window ( V 0 , E ^ ). Another i m p o r t a n t f a c t o r which may i n f l u e n c e the p r o b a b i l i t y f o r d i r e c t t h e r m a l i z a t i o n i n the v o i d s i s the number of c o l l i s i o n s a Mu atom makes w i t h the s u r f a c e i n the s e n s i t i v e energy window ( V 0 f E ^ ) . In an evacuated powder sample, t h i s may be e s t i m a t e d as n c = (E//, ~ VO)/^ e-^A > where AE i s the energy l o s s per s u r f a c e c o l l i s i o n . U s i n g AE = 3mE^/M, r\c = 1.5 i n the S i 0 2 powder. The presence of a b u f f e r gas such as He w i l l tend t o reduce n^ due t o the a d d i t i o n a l energy l o s s between s u r f a c e 1 36 c o l l i s i o n s . I f 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 atoms 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 , then n c ~ ( E ^ - V 0 ) / A E . In t h i s e q u a t i o n A E = U 6 J E q u a t i o n A I I I - 3 i s the mean energy l o s s t h r o u g h e l a s t i c c o l l i s i o n s w i t h the b u f f e r gas (of 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 ) a f t e r t r a v e l l i n g a d i s t a n c e d (the mean f r e e p a t h between s u r f a c e c o l l i s i o n s ) . I n s e r t i n g = 10A 2, n = 1 0 2 1 cm" 3 (an i d e a l gas a t 760 t o r r and 7°K), = MH<, = 3720 MeV/c 2 and d = 1800A (mean d i s t a n c e between s u r f a c e c o l l i s i o n s i n S i 0 2 powders R = 35A p =0.04 gem" 3) i n t o E q u a t i o n AIII«3. y i e l d s AEe =0.68 eV and n,= 0.04. A c c o r d i n g t o t h i s model, the p r o b a b i l i t y t h a t a Mu atom t h e r m a l i z e s d i r e c t l y i n the v o i d s of o x i d e powders i s determined by p r o c e s s e s o c c u r r i n g below the e l e c t r o n e x c i t a t i o n t h r e s h o l d (E < '6eV). I t i s most p r o b a b l e when the p a r t i c l e s i z e i s s m a l l and the work f u n c t i o n f o r Mu a t the o x i d e s u r f a c e i s l a r g e . The presence of a b u f f e r gas such as He may enhance t h i s p r o b a b i l i t y p r o v i d e d t h a t the mean f r e e p a t h between gas c o l l i s i o n s i s l e s s than the mean f r e e p a t h between s u r f a c e c o l l i s i o n s . 1 37 APPENDIX IV ADSORPTION OF ATOMS ON A SURFACE C o n s i d e r gas atoms c o n t a i n e d w i t h i n a volume, V, i n c o n t a c t w i t h an a d s o r p t i v e s u r f a c e of a r e a A. L e t € 0 be the b i n d i n g energy t o the s u r f a c e , and N s be the number of atoms on the s u r f a c e , N^ the number of atoms i n the gas. D e f i n e ns = N 5/A and n<j = N 3/V. The a d s o r p t i o n i s o t h e r m n 5 v e r s u s n^ a t c o n s t a n t T w i l l , i n g e n e r a l , depend on the atom m o b i l i t y on the s u r f a c e . Two i d e a l s i t u a t i o n s , r e v i e w e d i n more d e t a i l by (Dash 1975) are c o n s i d e r e d h e r e . 1. A d s o r p t i o n of a Van der Waals gas atom of a r e a e on a smooth s u r f a c e . 2. A d s o r p t i o n of t i g h t l y bound atoms on a s u r f a c e w i t h \./e a d s o r p t i o n s i t e s per u n i t a r e a . A IV«1 Van Der Waals Two D i m e n s i o n a l Gas The t o t a l energy of N s atoms on the s u r f a c e i s where t 0 i s the b i n d i n g energy t o the s u r f a c e , p/ i s the momentum of atom i , and U ( r , .. r^) -is the t o t a l energy of i n t e r a c t i o n between atoms on the s u r f a c e . In the case of hard E q u a t i o n AIV-1 d i s c s , 6 7~Jn E q u a t i o n AIV'2 The p a r t i t i o n f u n c t i o n , Z f o r the adsorbed atoms 138 6 V M ™ E q u a t i o n AIV«3 where A= (h 2/2rrmkT) i s the th e r m a l de B r o g l i e w a v e l e n g t h , dr i s an element of phase space, and Q i s g i v e n by: <5 ) e d r'-' ^ E q u a t i o n AIV-4 At low d e n s i t i e s , where N s«/A << 1, E q u a t i o n AIV-5 The f r e e energy, F, f o r the adsorbed atoms i s then E q u a t i o n AIV-6 The c h e m i c a l p o t e n t i a l 2 V E q u a t i o n AIV'7 The c h e m i c a l p o t e n t i a l f o r the atoms i n the gas phase (assuming an i d e a l gas) i s g i v e n by a* = kT In T n«/l3J J E q u a t i o n AIV-8 139 In e q u i l i b r i u m , ^ = ^ , and thus n & = n ^ A ^ e E q u a t i o n AIV.9 At v e r y low d e n s i t i e s , ns i s l i n e a r i n n^, s i n c e n^tye « 1. Near monolayer .co>mpletion as er\s tends t o 1 and Q ( E q u a t i o n A I I I ' 1 ) tends t o z e r o , so t h a t n s i s bounded by 1 /'« (we have i g n o r e d the geo m e t r i c p a c k i n g f a c t o r from the 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 , the f r e e energy f o r t h e N 3 atoms adsorbed on 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 i s (Dash 1975) F = -kT In [ UUUU'&)l'N.l)c % | n [ i - r W ] E q u a t i o n AIV-10 where n^c i s the f r a c t i o n a l c o v e r a g e . The c h e m i c a l p o t e n t i i a l f o r the adsorbed 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 the c h e m i c a l p o t e n t i a l of gas phase atoms ( g i v e n by E q u a t i o n AIV-8) may be equated w i t h t h a t of the s u r f a c e atoms, y i e l d i n g : * A — r i j A + & E q u a t i o n AIV-12 At low co v e r a g e , x\,o « 1, na i s l i n e a r i n n 3 1 40 3 6* /k-T £ E q u a t i o n AIV«13 whereas f o r l a r g e n^ , n*. = \/e. Note t h a t a t a low cov e r a g e , both Van Der Waals model and t i g h t b i n d i n g y i e l d a l i n e a r r e l a t i o n s h i p between n s and n^ w i t h the p r o p o r t i o n a l i t y c o n s t a n t d i f f e r i n g by a f a c t o r A 2/* (-1 f o r He a t 7°K). A IV-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 (Ps or Mu) i s t h e r m a l i z e d i n a system w i t h s u r f a c e a r e a A and t o t a l f r e e volume V F, i t i s d e s i r a b l e t o know the f r a c t i o n of time spent-on the s u r f a c e , averaged over times much l a r g e r than the d w e l l time on the s u r f a c e or the mean time between s t i c k i n g s . T h i s f r a c t i o n , c, may be o b t a i n e d from the 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 and 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 gas bound t o the s u r f a c e by t0 . 141 E q u a t i o n AIV-16 f o r t i g h t l y bound atoms. A IV'4 Mean S u r f a c e D w e l l Time I t i s a l s o of i n t e r e s t t o e s t i m a t e the mean time spent on the s u r f a c e per s t i c k i n g t o the s u r f a c e b e f o r e d e s o r p t i o n o c c u r s (Crampton 1980). The d e n s i t y of atoms on the s u r f a c e can a l s o be w r i t t e n ls ij J E q u a t i o n AIV- 17 where \y i s the mean th e r m a l v e l o c i t y and i s the p r o b a b i l i t y t h a t an atom which s t r i k e s the s u r f a c e w i l l a dsorb. The s t i c k i n g of an atom t o a s u r f a c e i s an i n e l a s t i c p r o c e s s i n which the i n c i d e n t momentum of the atom and i t s b i n d i n g energy a re t r a n s f e r r e d t o t o the l a t t i c e v i a phonon i n t e r a c t i o n . In the case of a low d e n s i t y 2 d i m e n s i o n a l gas, the above e x p r e s s i o n may be equated t o E q u a t i o n AIV«8 y i e l d i n g E q u a t i o n AIV-18 142 BIBLIOGRAPHY Abragam, A., 1961, N u c l e a r Magnetism, O x f o r d P r e s s , London, 126. Anderson, C. D., 1933, Phys. Rev. 43, 491. Anderson, C. D. and S. H. Neddermeyer, 1937, Phys. Rev. 5J_, 884. Anderson, P. W., 1951, C. R. Acad. S c i . 82, 342. A s h c r o f t , N. W. and N. D. Mermin, 1976, S o l i d S t a t e P h y s i c s , H o l t R i n e h a r t and Winston, 781. A s t o n , J . G., S. V.' R. M a s t r a n g e l o and R. J . T y k o d i , 1955, J . Chem. Phys. 23, 1633. A r t h u r s , A. M. and A. D a l g a r n o , 1960, P r o c . R. Soc. London A256, 540. B l a c k e t t , P. M. and G. P. O c c h i a n l i n i , 1933, P r o c . Roy. Soc. A139, 699. B r a n d t , W, 1967, P o s i t r o n a n n i h i l a t i o n , Academic P r e s s , , 180, . B r a n d t , W. and R. P a u l i n , 1968, Phys. Rev. 2J_, 193. Brewer, J . H., 1 9 8 1 , H y p e r f i n e I n t e r a c t i o n s 8, 375. Brewer, J . H., K. M. Crowe, F. N. Gygax and A. Shenck, 1975, Muon P h y s i c s , V o l I I , e d i t e d by V. W. Hughes and C. S. Wu, Academic P r e s s , New York, 3. Brunauer, S., P. H. Emmett and E. T e l l e r , 1938, J . Am. Chem. Soc 60, 309. Cabot C o r p o r a t i o n , T e c h n i c a l R e p o r t , C a b - O - S i l P r o p e r t i e s and  Funct i o n s , a v a i l a b l e from Cabot C o r p o r a t i o n , 125 High S t r e e t , B o s t o n , MA 02110, U. S. A. C a n t e r , K. F., A. P. M i l l s and S. Berko, 1975, Phys. Rev. L e t t . 34, 177. C a n t e r , K. F., A. P. M i l l s J r . and S. Berko, 1974, Phys. Rev. L e t t . 33, 7. C a n t e r , K. F., P. G. Coleman, T. C. G r i f f i t h and G. R. Heyland, 1972, J . Phys. B5, L167. Casp e r s o n , D.E., T.W. Crane, V.W. Hughes, P.A. Souder, R.D. Stambaugh, P.A. Thompson, H. O r t h , G. zu P u t l i z , H.F. Kasp a r , H.W. R e i s t , A.B. D e n i s o n , 1975, Phys. L e t t s . 59B, 397. 143 Chuang, S.Y. and S.J. T Q Q , 1973, Can. J . Phys. 5J_, 823. Chuang, S. Y. and S. J . Tao, 1974, A p p l . Phys. 3, 1 9 9 . Crampton, S.B., 1980, J o u r n a l De Physique C7, 249. C r o n i n , J . W. , 1968, P r o c . I n t . Conf. High Energy P h y s i c s , 14th, 289. C u r r y , S. M. and A. L. Shawlow, 1971, Phys. L e t t . 37A, 5. Dash, J . G., 1975, F i l m s On S o l i d S u r f a c e s , Academic P r e s s , New York, . De u t s c h , M. , 1951, Phys. Rev 8 J 3 , 866. D e v o n s h i r e , A. F., 1938, P r o c . Roy. Soc. A158, 269. D i r a c , P. A. M., 1930, P r o c . Roy. Soc. A126, 360. F l e m i n g , D. G., D. M. Garner and R. J . M i k u l a , 1981a, H y p e r f i n e I n t e r a c t i o n s 8, 337. F l e m i n g , D. G., R. J . M i k u l a and D. M. Garner, 1981b, H y p e r f i n e I n t e r a c t i o n s 8, 307. Foner, S. N., E. L. Cochran, V. A. Bowers and C. K. J e n , 1960, J . Chem. Phys. 3_2, 963. F o r d , G. W., L. M. Sander and T. A. W i t t e n , 1976, Phys. Rev. L e t t . 36, 1269. Friedman, J . and V . T e l e g d i , 1957, Phys. Rev. _1_05, 1 681. Gar n e r , D. M., 1978, Ph. D. T h e s i s , U n i v e r s i t y of B r i t i s h C o l u m b i a , C h e m i s t r y Department. Garwin, R.L., L.M.Lederman, and W.Weinrich, Phys. Rev., 105 14 15 G i d l e y , D. W., A. R i c h , P. W. Z i t e w i t z and D. A. L. P a u l , 1978, Phys. Rev. L e t t . _40, 737. Golubev, V.B., 1965, R u s s i a n J o u r n a l of P h y s i c a l C h e m i s t r y 2S_, 1 395. Goodman, F., 1971, S u r f . S c i . 24, 667. Goodman, F., 1972, S u r f . S c i . 3_0, 1. Gordon, E. B., B. I . Ivanov, A. P. Perminov, A. N. Ponomarev, V. L. T a l roze and S. G. K h i d i r o v , 1973, JETP L e t t e r s 17, 395. 1 44 Hara, S.and P.A. F r a s e r , 1975, J.Phys.B Atom. Molec. Phys. 8, L472. H o t z , H. P., J . M. M a t h i e s e n and J . P. H u r l e y , 1968, Phys. Rev. 170, 351. H u f f , G.B. and J.G. Dash, 1976, J . Low Temp. Phys. 2_4, 155. Hughes, V.W., D.W. McColm, K. Z i o c k , and R. P r e p o s t , 1960, Phys. Rev. L e t t . 5, 63. Hughes, V.W. and T . K i n o s h i t a , 1977, Muon P h y s i c s , V o l . I , e d i t e d by V.W. Hughes and C.S. Wu, Academic P r e s s , New York, 11. I t o , Y., B. W. Ng, Y. C. Jean and D. C. Walker, 1981, H y p e r f i n e I n t e r a c t i o n s 8, 355. I t o , Y. and Y. Tabata, 1979, P r o c . 5th I n t e r n . Conf. P o s i t r o n A n n i h i l . , 325. I w a s a k i , M., K. Toriyama, H, Muto and K. Nunome, 1978, Phys. L e t t . 56, 494. K i e f l , R.F., 1978, M.Sc. T h e s i s , U n i v e r s i t y of B r i t i s h Columbia, Department of P h y s i c s , . K i e f l , R. F., J . B. Warren, G. M. M a r s h a l l , C. J . Oram, J . H. Brewer, D. J . Judd and L. D. S p i r e s , 1979, H y p e r f i n e I n t e r a c t i o n s 6>, 185. Kinugawa, K., T. M i y a z a k i and H. Hase, 1978, J . Phys. Chem. 82, 1697. K l o b u c h a r , R. L. and P. J . K a r o l , 1980, J . Phys. Chem 84, 483. K n o z i n g e r , H., 1976, Adv. C a t a l y s i s 25, 184. K n o z i n g e r , H. and P. Ratnasamy, 1978, C a t a l . Rev. - S c i . Eng. _T7 , 31 . L e i g h t o n , R.B., 1959, P r i n c i p l e s of Modern P h y s i c s , M c g r a w - H i l l , New York, 149, . L e v i n e , I . N., 1957, M o l e c u l a r S p e c t r o s c o p y , John W i l e y & Sons, New York, 160. M a r s h a l l , G. M., J . B. Warren, D. M. Garner, G. S. C l a r k , J . H. Brewer and D. G. F l e m i n g , 1978, Phys. L e t t . 65A, 351. Manaichev, E.V.,G.G. Myasishcheva,Yu V. Obukhov,V.S. Roganov,G.I. S a v e l ev and V.G. F i r s o v , 1970, JETP 3J_, 849. 1 45 Mobley, R.M., J . Amato, V.W. Hughes, J.E. Rot h b e r g , and P.A. Thompson, 1966, J . Chem. Phys. £7, 3074. Mogensen, 0. E., 1974, J . Chem Phys. 60, 998. M o h o r o v i c i i , S., 1934, A s t r o n . Nachr., 94. Oram, C.J., J.B. Warren, G.M. M a r s h a l l , and J . Doornbos, 1981, N u c l . I n s t r u m . Meths. 179, 95, Ore, A. and J . L. P o w e l l , 1949a, Phys. Rev. 75, 1696. Ore, A., 1949b, U n i v . i Berger Arbok N a t u r v i t e n s k a p . Rekke 9, . Pake, G. E., 1948, J . Chem. Phys. J_6, 327. P e r c i v a l , P.W., 1981, H y p e r f i n e I n t e r a c t i o n s 8, 315. P i f e r , A.E., T. Bowan, and K.R. K e n d a l l , 1976, N u c l . I n s t r . Methods K35, 39. P i r e n n e , J . , 1946, A r c h . S c i . Phys. et Nat. 28, 273. R e i f , F., 1965, Fundamentals of S t a t i s t i c a l and Thermal P h y s i c s , M c G r a w - H i l l , New York, 16, . Spencer, D.P. and J.H. Brewer, 1981, P r i v a t e Communication, . S t r e e t , C. and E. Stevenson, 1937, Phys. Rev. 5±, 1005. S u r i n , S.A.,G.M. Zhidomirov,B.N. Shelimov and V.B. K a z a n s k i i , 1970, T e o r e t i c h e s k a y a i E k s p e r i m e n t a l naya Khimiya 6, 353. Tawara, H. and A. Russek, 1973, Rev. Mod. Phys. 45, 178. Temansky, M. W., 1968, Heat and Thermodynamics, 5th e d i t i o n , M c G r a w - H i l l , New York, 318. T h e r i o t , E. D. J r . , R. H. B e e r s , V. W. Hughes, 1967, P h y s i c s R e p o r t s 5, 215. Tinkham, M., 1964, Group Theory and Quantum Mechanics, McGraw-H i l l , New York, 253. Townes, C H . and A.L. Schawlow, 1955, Microwave S p e c t r o s c o p y , McGraw H i l l , New York, 183, . W e i l , J . A., 1981, H y p e r f i n e I n t e r a c t i o n s 8, 371. Walker, D.C., 1981, H y p e r f i n e I n t e r a c t i o n s 8, 329. W a l l a c e , P.R., 1960, S o l i d S t . Phys. J_0, 1. 1 46 West, R. N., 1973, Advances i n P h y s i c s 22, 263. Wheeler, J . A., 1946, Ann. N. Y. Acad. S c i . 46, 221. W i l l i a m s , W. S. C., 1971, An I n t r o d u c t i o n t o Elementary  P a r t i c l e s , 2nd E d i t i o n , Academic P r e s s , New York, 166 ; 349. Wu, C. S., E. Ambler, R. W. Hayward, ' D. D. Hoppes and R. R. Hudson, 1957, Phys. Rev. J_05, 1413. Zemansky, M.W., 1968, Heat and Thermodynamics, McGraw H i l l , New York, 318, . PUBLICATIONS R.F. K i e f l , J . B . Warren, G.M. Marshall , C . J . Oram, and C.W. Clawson, 1981, Muonium in the Condensed Phases of Ar , Kr, and Xe, J . Chem. Phys. 74, 308. R.F. K i e f l , J . B . Warren, C . J . Oram, G.M. Marshal l , J . H . Brewer, D. Harshman, and C.W. Clawson, 1981, Surface Interaction of Muonium in Oxide Powders at Low Temperature, Submitted for publication in J a n . , 1982. R.F. K i e f l , J . B . Warren, G.M. Marshall , C . J . Oram, J . H . Brewer, D.J . Judd, and L.D. Spires, 1979, Muonium and Positronium in Oxide Powders, Hyperfine Interactions 6, 185. R.F. K i e f l , 1981, Thermalization of Muonium in Oxide Powders at Low Temperature, Hyperfine Interactions 8, 359. G.M. Marshal 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. Fry, 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 in 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 Graff , 1981, Zero-Field MSR in an Insulator Spin Glass, Hyperfine Interactions 8, 757. 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0085513/manifest

Comment

Related Items