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The interactions of muonium with silica surfaces Harshman, Dale Richard 1986

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THE INTERACTIONS OF MUONIUM WITH SILICA  SURFACES  by DALE RICHARD HARSHMAN B.Sc,  P a c i f i c Lutheran U n i v e r s i t y ,  1978  M . S c , Western Washington U n i v e r s i t y ,  1980  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  in THE FACULTY OF GRADUATE STUDIES Department of P h y s i c s  We accept t h i s  t h e s i s as conforming  to the r e q u i r e d  standard  THE UNIVERSITY OF BRITISH COLUMBIA January 1986  © Dale R i c h a r d Harshman, 1986  In  presenting  degree  this  at the  thesis  in partial  University of  British  fulfilment  of  Columbia,  I agree  freely available for reference and study. copying  of  department  this or  publication of  thesis by  for scholarly  his  this thesis  or  her  requirements  for  may  representatives.  It  be is  Department The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3  advanced  that the Library shall make it  I further agree that permission  purposes  an  granted  for extensive  by the head  understood  that  for financial gain shall not be allowed without  permission.  DE-6(3/81)  the  of  my  copying  or  my written  ABSTRACT  The  behavior  powders has Results silica  been s t u d i e d u s i n g  the techniques  i n d i c a t e d i f f u s i o n and  groups.  t r a p p i n g behavior  Specifically,  been shown to i n h i b i t  of the muonium atoms on the c o n c e n t r a t i o n of  the presence of the s u r f a c e h y d r o x y l  the motion of muonium on the s i l i c a  These s t u d i e s have a l s o p r o v i d e d  information regarding  has  by the l o c a l  the  surface groups  the o r i g i n of  S p e c i f i c a l l y , a random a n i s o t r o p i c d i s t o r t i o n of the muonium induced  2  surface.  r e l a x a t i o n of the muon s p i n p o l a r i z a t i o n f o r muonium on the s i l i c a  interaction,  Si0  of muon s p i n r o t a t i o n (uSR).  s u r f a c e , which i s s t r o n g l y i n f l u e n c e d by  hydroxyl has  of muonium on the s u r f a c e of f i n e (35 A mean r a d i u s )  the  surface.  hyperfine  s u r f a c e environment of the muonium atom,  been shown to be a p r i n c i p a l c o n t r i b u t o r to the r e l a x a t i o n of the muon  ensemble s p i n p o l a r i z a t i o n , whereas the random l o c a l magnetic f i e l d s the n e i g h b o r i n g this result, hydroxyl  protons  induced  been a t t r i b u t e d to an a s s o c i a t e d  by the n e i g h b o r i n g  hydroxyls.  A new  a d s o r p t i o n isotherm  s t u d i e s were a l s o performed, w i t h ^He  also  the  at 6  i n c r e a s i n g s u r f a c e coverage, s u g g e s t i n g  r e s u l t s with  regard  surface i s s i g n i f i c a n t .  to the o r i g i n s of muonium  K,  fractional  i n d i c a t e that the muonium  charge exchange c r o s s s e c t i o n at the s i l i c a i m p l i c a t i o n of these  surface  data.  These r e s u l t s c l e a r l y  p r o b a b i l i t y decreases w i t h  From  spin relaxation  which show the muonium asymmetry to be s t r o n g l y i n f l u e n c e d by s u r f a c e coverage.  to  hyperfine  f o r the case of random a n i s o t r o p i c h y p e r f i n e d i s t o r t i o n s , has  been developed to e x p l a i n the Gas  were found to p l a y only a minor r o l e .  the observed s t r o n g dependence of the r e l a x a t i o n on the  c o n c e n t r a t i o n has  distortion, theory,  hydroxyl  due  formation that  the  The formation  - i i i (i.e.,  s u r f a c e or bulk formation) i s as y e t u n c l e a r , however, s i n c e the  p r e c i s e r o l e played by the adsorbed  helium atoms i s not known.  These i n v e s t i g a t i o n s have a l s o been extended where the f i r s t  to platinum loaded  s u r f a c e r e a c t i o n of muonium has been observed;  silica,  the r e a c t i o n  r a t e of muonium w i t h the s u r f a c e of oxygen-covered p l a t i n u m m i c r o c r y s t a l s was found  to be 3.5 ± 0.15 [is~^ •  - ivTABLE OF CONTENTS  ABSTRACT  i i  LIST OF TABLES  ix  LIST OF FIGURES  x  ACKNOWLEDGEMENTS  xiii  CHAPTER I . INTRODUCTION A.  B.  C.  D.  1  Muons and Muonium  3  1.  Muon C h a r a c t e r i s t i c s  4  2.  T h e r m a l i z a t i o n of P o s i t i v e Muons i n Matter  6  3.  Muonium Formation and C h a r a c t e r i s t i c s  8  Time E v o l u t i o n of the Muonium S p i n S t a t e  12  1.  Muonium i n Vacuum  12  2.  I n t e r a c t i o n s with  the Environment  15  The I n t e r a c t i o n s of Muonium w i t h S i l i c a  17  1.  Muonium i n Bulk S i l i c a  18  2.  Muonium on S i l i c a  19  3.  Muonium Formation i n F i n e S i l i c a Powders  20  4.  Extragranular  Muonium P r o d u c t i o n  25  The I n t e r a c t i o n s of Hydrogen and Deuterium w i t h S i l i c a  27  1.  28  Surfaces  Hydrogen D i f f u s i o n i n Bulk S i l i c a  - v 2.  Hydrogen and Deuterium on S i l i c a S u r f a c e s  28  CHAPTER I I . EXPERIMENTAL TECHNIQUE A.  B.  C.  D.  31  A c c e l e r a t o r s and Beamlines  31  1.  The TRIUMF C y c l o t r o n F a c i l i t y  31  2.  Muon P r o d u c t i o n and T r a n s p o r t  33  The uSR / MSR Technique 1.  Zero and L o n g i t u d i n a l F i e l d  2.  Transverse  Experimental  Field  (ZF and LF)  (TF)  42 42  Apparatus and Data A q u i s i t i o n  44  1.  The uSR Spectrometer  44  2.  E l e c t r o n i c s and L o g i c  48  3.  Targets  52  4.  Cryogenics  58  5.  Gas Handling  60  Data A n a l y s i s  63  1.  Transverse  Field  Spectra  2.  Zero and L o n g i t u d i n a l F i e l d  CHAPTER I I I . THEORY OF MUONIUM RELAXATION A.  41  Spin Relaxation Functions  63 Spectra  64  65 65  - vi-  B.  1.  Random L o c a l Magnetic  2.  Random A n i s o t r o p i c H y p e r f i n e D i s t o r t i o n s (RAHD)  71  3.  Chemical  83  4.  Spin Exchange (SE)  84  5.  Superhyperfine  85  Reactions  Fields  (RLMF)  66  (CH)  I n t e r a c t i o n s (SHF)  Dynamical R e l a x a t i o n F u n c t i o n s  88  1.  Gaussian-Markovian  89  2.  Strong C o l l i s i o n - M a r k o v i a n P r o c e s s  3.  D i f f u s i o n i n the Presence  Process  91  of Traps  102  CHAPTER IV. EXPERIMENTAL RESULTS AND INTERPRETATIONS A.  B.  C.  Muonium on S i l i c a  106  Surfaces  107  1.  Transverse F i e l d Results  108  2.  Zero and L o n g i t u d i n a l F i e l d R e s u l t s  118  Muonium on the S u r f a c e of Helium  Coated  Silica  130  1.  R e l a x a t i o n Rate Versus ^He Coverage  130  2.  Muonium Asymmetry Versus ^He Coverage  132  Muonium On the S u r f a c e of Supported  Platinum C a t a l y s t s  1.  Unloaded S i l i c a  Support  2.  P l a t i n u m Loaded S i l i c a :  3.  P l a t i n u m Loaded S i l i c a : 0.1% and 1.0%  134 134  0.001% and 0.01%  137 139  - viiCHAPTER V. CONCLUSIONS AND FUTURE DIRECTIONS A.  B.  142  Summary of R e s u l t s  142  1.  D i f f u s i o n and T r a p p i n g  142  2.  R e l a x a t i o n Mechanisms  143  3.  Muonium Formation  146  4.  C a t a l y t i c Chemistry  Probability  147  Future D i r e c t i o n s  148  1.  Theoretical  149  2.  Experimental  150  APPENDIX I . THE TIME EVOLUTION OF THE | i SPIN POLARIZATION IN MUONIUM FOR A GENERALLY ANISOTROPIC HYPERFINE INTERACTION +  A.  B.  C.  153  Observables - C r y s t a l and D e t e c t o r Frames  153  1.  154  Spin Relaxation Functions  The Spin H a m i l t o n i a n f o r I s o l a t e d  Muonium  161  1.  E v a l u a t i o n of the H y p e r f i n e Term  161  2.  E v a l u a t i o n of the Zeeman Term  167  Isolated  Muonium i n Zero F i e l d  168  1.  R e l a x a t i o n Due to a C y l i n d r i c a l D i s t o r t i o n  170  2.  R e l a x a t i o n Due to a P l a n a r D i s t o r t i o n  173  - viii 3.  D.  Cylindrical  and P l a n a r D i s t o r t i o n s Combined  I s o l a t e d Muonium i n an E x t e r n a l Magnetic  Field  177  180  1.  Longitudinal F i e l d Relaxation Function  184  2.  Transverse F i e l d Relaxation Function  185  APPENDIX I I . ULTRA-LOW ENERGY MUON PRODUCTION  (nSOL)  187  1.  C u r r e n t Status of Slow P o s i t r o n P r o d u c t i o n  188  2.  Band Gap E m i s s i o n of e  from I o n i c C r y s t a l S u r f a c e s  190  3.  Comparison of e  WRT  Band Gap E m i s s i o n  193  4.  C a l c u l a t i o n s f o r P o s i t i v e Muon E m i s s i o n Y i e l d  195  5.  P r o t o t y p e Apparatus  199  6.  Measurements  199  7.  Backgrounds  203  +  and \i  +  +  APPENDIX I I I . COLLISION FREQUENCY OF THERMAL MUONIUM A.  205  Derivation  205  1.  Low D e n s i t y L i m i t  205  2.  High D e n s i t y L i m i t  206  APPENDIX IV. TABULATED TRANSVERSE FIELD DATA  207  REFERENCES  210  - ix LIST OF TABLES  CHAPTER I . INTRODUCTION 1.  P r o p e r t i e s of Muons ( u + . u )  5  2.  P r o p e r t i e s of Muonium (Mu)  10  3.  Muonium F r a c t i o n s F „ and T r a n s v e r s e F i e l d R e l a x a t i o n  -  M  " - U  M u  Rates \ j _  l u  f o r Bulk and Powdered S i l i c a  21  CHAPTER I I . EXPERIMENTAL TECHNIQUE 1.  Beam Parameters f o r the M20 Secondary Channel (a) Backward Decay Muons a t 75° (M20-A) and 37.5° (M20-B) (b) Simultaneous Decay Muons on M20  37  2.  Surface Muons a t 75° (M20-A) and 37.5° (M20-B)  39  3.  P h y s i c a l P r o p e r t i e s of the S i 0  54  4.  Experimental Target C h a r a c t e r i s t i c s (a) S i 0 T a r g e t s (b) P l a t i n u m Loaded S i 0 T a r g e t s  v  Powder  2  59  2  2  CHAPTER IV. EXPERIMENTAL RESULTS AND INTERPRETATIONS 1.  F i t R e s u l t s of T h r e e - S t a t e Model f o r Mu on S i 0 (a) Sample S i 0 ( l ) prepared a t 110 °C (b) Sample S i 0 ( 3 ) Prepared a t 600 °C  2  Surfaces  .... 115  2  2  APPENDIX I V . TABULATED TRANSVERSE FIELD DATA 1.  S i 0 ( l ) Prepared  2.  S i 0 ( 3 ) Prepared a t 600 °C; \  3.  S i 0 ( 2 ) Prepared a t 110 °C;  2  at 110 °C;  2  2  v  M  u  s  Temperature  207  vs Temperature  20b  vs Temperature  209  - x LIST OF FIGURES CHAPTER I . INTRODUCTION 1.  Muon Decay Parameters  7  CHAPTER I I . EXPERIMENTAL TECHNIQUE 1.  TRIUMF C y c l o t r o n F a c i l i t y  32  2.  M20  35  3.  Schematic R e p r e s e n t a t i o n s of u.SR Techniques (a) Zero and L o n g i t u d i n a l F i e l d  Secondary Channel  (b) T r a n s v e r s e  43  Field  4.  The  " E a g l e " \iSR Spectrometer  5.  Data A c q u i s i t i o n E l e c t r o n i c s  50  6.  L o g i c L e v e l Diagram f o r a "Good" Event  53  7.  Thermogravimetric  56  8.  Gas  Handling  Plot  45  for Cab-0-Sil  System  61  CHAPTER I I I . THEORY OF MUONIUM RELAXATION 1.  S t a t i c L o n g i t u d i n a l F i e l d Gaussian  2.  S t a t i c L o n g i t u d i n a l F i e l d L o r e n t z i a n Kubo-Toyabe  72  3.  S t a t i c Zero F i e l d Random A n i s o t r o p i c H y p e r f i n e Interaction Relaxation Function (Lorentzian)  77  S t a t i c T r a n s v e r s e F i e l d Random A n i s o t r o p i c H y p e r f i n e Interaction Relaxation Function (Lorentzian)  79  4.  5.  Kubo-Toyabe  70  S t a t i c Zero F i e l d Random A n i s o t r o p i c H y p e r f i n e Interaction Relaxation Function (Modified Lorentzian)  6.  Zero and L o n g i t u d i n a l F i e l d Data f o r Bulk Fused S i 0  7.  Superhyperfine  I n t e r a c t i o n Diagram  2  ...  80  ....  82 87  - xi8.  9.  10.  11.  12.  13.  14.  Dynamic T r a n s v e r s e F i e l d Gaussian (Gaussian-Markovian) Dynamic Zero F i e l d Gaussian (Strong-Collision)  Kubo-Tomita 90  Kubo-Toyabe 94  Dynamic Zero F i e l d L o r e n t z i a n Kubo-Toyabe (Strong C o l l i s i o n )  96  Dynamic Zero F i e l d L o r e n t z i a n Random A n i s o t r o p i c Hyperfine I n t e r a c t i o n Relaxation function (Strong C o l l i s i o n )  98  Dynamic Zero F i e l d L o r e n t z i a n Random A n i s o t r o p i c Hyperfine Interaction Relaxation Function (Strong C o l l i s i o n ) - C y l i n d r i c a l Component  100  Dynamic Zero F i e l d L o r e n t z i a n Random A n i s o t r o p i c Hyperfine I n t e r a c t i o n Relaxation Function ( S t r o n g C o l l i s i o n ) - P l a n a r Component  101  Dynamic Zero F i e l d M o d i f i e d L o r e n t z i a n Random A n i s o t r o p i c Hyperfine I n t e r a c t i o n Relaxation Function (Strong C o l l i s i o n ) - C y l i n d r i c a l Component 103  CHAPTER IV. EXPERIMENTAL RESULTS AND INTERPRETATIONS 1.  2.  3.  T r a n s v e r s e F i e l d Mu R e l a x a t i o n Rate vs Temperature f o r Muonium on S i l i c a S u r f a c e s Prepared a t 110 °C and a t 600 °C  109  Hyperfine-Structure Interval v vs Temperature f o r Muonium on S i l i c a S u r f a c e s Prepared a t 110 °C  112  0 0  S u r f a c e Hop Rate v vs Temperature f o r Muonium on S i l i c a S u r f a c e s Prepared a t 110 C and 600 "C  117  Zero and L o n g i t u d i n a l F i e l d S p e c t r a f o r Muonium on the S i l i c a S u r f a c e (110 C p r e p a r a t i o n ) a t 7.0 ± 0.2 K  119  0  U  4.  U  5.  6.  7.  Zero and L o n g i t u d i n a l F i e l d S p e c t r a f o r Muonium on the S i l i c a S u r f a c e (600 °C p r e p a r a t i o n ) a t 3.6 ± 0.2 K  123  Zero and L o n g i t u d i n a l F i e l d S p e c t r a f o r Muonium on the S i l i c a S u r f a c e (600 "C p r e p a r a t i o n ) a t 16.0 ± 0.1 K  126  Zero and L o n g i t u d i n a l F i e l d S p e c t r a f o r Muonium on the S i l i c a S u r f a c e (110 °C p r e p a r a t i o n ) a t 25 ± 0.5 K  127  - xii8.  9.  10.  11.  12.  13.  Zero and L o n g i t u d i n a l F i e l d S p e c t r a f o r Muonium on the S i l i c a S u r f a c e (600 °C p r e p a r a t i o n ) a t 30 ± 0.5 K  129  T r a n s v e r s e F i e l d Mu R e l a x a t i o n Rate vs ^He Coverage f o r S i l i c a Prepared a t 110 "C and a t 600 "C  131  T r a n s v e r s e F i e l d Mu Asymmetry vs ^He Coverage f o r S i l i c a Prepared a t 110 °C  133  T r a n s v e r s e F i e l d Mu R e l a x a t i o n Rate vs Temperature f o r unreduced and H-reduced, 0.0% P t Loaded S i l i c a  135  T r a n s v e r s e F i e l d Mu R e l a x a t i o n Rate vs Temperature f o r 0.001% and 0.01% P t Loaded S i l i c a  138  T r a n s v e r s e F i e l d Mu R e l a x a t i o n Rate vs Temperature f o r 0.1% and 1.0% Pt Loaded S i l i c a  140  APPENDIX I I . ULTRA-LOW ENERGY MUON PRODUCTION (uSOL) 1.  Target O r i e n t a t i o n WRT  I n c i d e n t \x beam  2.  S c a t t e r i n g Chamber and DQQ  +  Spectrometer  197 200  - xili  -  ACKNOWLEDGEMENTS  I t i s a p l e a s u r e to acknowledge the h e l p and support of my s u p e r v i s o r , J.H.  Brewer, who  has p r o v i d e d me  i n s i g h t s i n t o the p h y s i c a l w o r l d . J.B. Warren, D.LI. W i l l i a m s and my  Ph.D. I am  D.M.  committee and  Stewart,  esthetic  g r a t i t u d e to  P h y s i c s department, the members of  staff.  p a r t i c u l a r l y g r a t e f u l to D.J. Arseneau,  Garner, R. K e i t e l , R.F.  and  I wish a l s o to express my  the UBC  the TRIUMF  w i t h many new  research  K i e f l , R.F.  K.M.  Marzke, M.  Crowe, D.G.  Senba, D.P.  Fleming,  Spencer,  J.  Y.J. Uemura and J . Worden f o r t h e i r a s s i s t a n c e w i t h some of the  experiments  and  f o r e n l i g h t e n i n g d i s c u s s i o n s , and e s p e c i a l l y to R.E.  f o r h i s h e l p and guidance I would a l s o l i k e  to thank my  u n d e r s t a n d i n g nature was spent w r i t i n g  i n the development of the RAHD r e l a x a t i o n  Turner theory.  wonderful w i f e , Sandra, who's •  f r e q u e n t l y put to the t e s t by my many l a t e n i g h t s  t h i s work.  Above a l l ,  I wish  to express my  s i n c e r e s t g r a t i t u d e to my  R i c h a r d and LaVonne Harshman, f o r t h e i r  support and  to them t h a t t h i s work i s l o v i n g l y d e d i c a t e d .  parents,  encouragement, and i t i s  - 1 CHAPTER I —  INTRODUCTION  The work presented  i n t h i s d i s s e r t a t i o n concerns the i n t e r a c t i o n s of  p o s i t i v e muons ( u ) and muonium atoms (p. e~, Mu) w i t h the s u r f a c e s of f i n e l y +  +  divided s i l i c a  powders (35 A mean r a d i u s ) .  first  i n v e s t i g a t i o n of the d i f f u s i o n and t r a p p i n g b e h a v i o r ,  detailed  T h i s r e s e a r c h r e p r e s e n t s the and the  r e l a x a t i o n mechanisms, f o r muonium on s u r f a c e s . P o s i t i v e muons and muonium atoms have proven to be i d e a l  microscopic  probes of magnetic systems as w e l l as i s o t o p i c probes of proton/hydrogen d i f f u s i o n mechanisms and chemical  reactions [1-3].  present work a r i s e s because of these  The m o t i v a t i o n f o r the  f e a t u r e s , and the d e s i r e to extend  the  s t u d i e s of muons and muonium to i n t e r a c t i o n s w i t h a s u r f a c e environment. T h i s work develops  a b a s i c q u a l i t a t i v e understanding  of the behavior of  muonium on s u r f a c e s and c o u l d c o n c e i v a b l y l e a d to the study  of s u r f a c e  magnetism and the e x t e n s i v e use of muonium as an i s o t o p i c probe of hydrogen catalysis.  Experimental  methods such as NMR,  ESR, LEED, e t c . , which are  w i d e l y used i n the study of adatom a d s o r p t i o n , g e n e r a l l y r e q u i r e a  fairly  h i g h d e n s i t y of atoms, which has obvious  to the  s t a t i s t i c a l mechanics of a d s o r p t i o n . r o t a t i o n ) techniques of  indeed, of  In c o n t r a s t , the uSR (muon s p i n  [1-3] employed i n the present work r e q u i r e o b s e r v a t i o n  one muon ( o r Mu atom) a t a time.  a l l o w s no p o s s i b i l i t y  r a m i f i c a t i o n s w i t h regard  f o r any study  T h i s f e a t u r e of the e x p e r i m e n t a l  of u - u , u -Mu or Mu-Mu i n t e r a c t i o n s ;  at p r e s e n t l y a c h i e v a b l e stopped  +  +  +  muon d e n s i t i e s ,  two muons o r muonium atoms must be an extremely Specifically,  the present work has p r o v i d e d  d i f f u s i o n and t r a p p i n g behavior  method  rare  the mutual encounter occurrence.  i n f o r m a t i o n concerning the  of muonium on the s i l i c a  s u r f a c e , as w e l l as  - 2 the e f f e c t  of the l o c a l s u r f a c e environment on the h y p e r f i n e i n t e r a c t i o n of  the muonium atom.  In the case of the l a t t e r , a theory has been  developed  d e s c r i b i n g the time e v o l u t i o n of the \i s p i n p o l a r i z a t i o n i n muonium f o r a +  g e n e r a l l y a n i s o t r o p i c h y p e r f i n e i n t e r a c t i o n , which can adequately e x p l a i n the muonium r e l a x a t i o n d a t a . was  a l s o found  The  behavior of muonium on the s i l i c a  to e x h i b i t a s t r o n g dependence on the c o n c e n t r a t i o n of  surface hydroxyl  groups.  A g r e a t d e a l of i n t e r e s t has been generated of  hydrogen atoms w i t h c a t a l y t i c  surfaces.  present study i s t y p i c a l of those used Because muonium can be it  surface  thought  silica  interaction  powder used  i n the  as support m a t e r i a l s f o r c a t a l y s t s .  of as a l i g h t  i s i d e a l l y s u i t e d f o r t h i s type of study.  p r o v i d e d the f i r s t  The  c o n c e r n i n g the  chemical i s o t o p e of hydrogen, In f a c t ,  the p r e s e n t work has  study of muonium on p l a t i n u m loaded s i l i c a  surfaces, i n  which the r e a c t i o n r a t e of muonium w i t h the oxygen-coated s u r f a c e s of platinum m i c r o c r y s t a l s was  measured.  T h i s d i s s e r t a t i o n i s o r g a n i z e d i n t o f i v e c h a p t e r s and The  the  present c h a p t e r , Chapter  two  I, i s p r i m a r i l y i n t r o d u c t o r y .  appendices.  I t provides  background i n f o r m a t i o n r e g a r d i n g the p r o p e r t i e s of muons and muonium, the effect  of the l o c a l environment on the time e v o l u t i o n of the n  p o l a r i z a t i o n f o r both charge e x p e r i m e n t a l and  states, a brief  Chapter  t h e o r e t i c a l \iSR s t u d i e s which are p e r t i n e n t to the p r e s e n t  muon p r o d u c t i o n and  acquisition,  experiments.  I I , the d i s c u s s i o n focuses on the s p e c i f i c  techniques employed i n the present i n v e s t i g a t i o n . of  spin  s y n o p s i s of p r e v i o u s  work and a d i s c u s s i o n of r e l e v a n t hydrogen atom In  +  I n c l u d e d are d e s c r i p t i o n s  t r a n s p o r t , the |iSR apparatus,  t a r g e t p r e p a r a t i o n and  experimental  e l e c t r o n i c s and  data  the methods of data a n a l y s i s employed.  - 3 Chapter I I I p r o v i d e s a g e n e r a l  t h e o r e t i c a l d i s c u s s i o n c o n c e r n i n g the  time e v o l u t i o n of the p. s p i n p o l a r i z a t i o n f o r the f i v e known muonium s p i n +  " r e l a x a t i o n " mechanisms, and the a s s o c i a t e d particular  spin relaxation functions.  importance are the r e l a x a t i o n f u n c t i o n s ,  a p p l i e d magnetic f i e l d , the muonium h y p e r f i n e  Of  f o r both zero and  which a r i s e from a random a n i s o t r o p i c d i s t o r t i o n o f  i n t e r a c t i o n (see Appendix I ) .  These f u n c t i o n s a r e  used i n the a n a l y s i s and i n t e r p r e t a t i o n of some of the d a t a . The  e x p e r i m e n t a l r e s u l t s a r e presented and d i s c u s s e d  These r e s u l t s i n d i c a t e d i f f u s i o n and t r a p p i n g silica  surface  hyperfine  i n Chapter IV.  behavior of muonium on the  and suggest a random a n i s o t r o p i c d i s t o r t i o n of the muonium  i n t e r a c t i o n as a p r i n c i p a l c o n t r i b u t o r  the u- s p i n on the s i l i c a +  to the d e p o l a r i z a t i o n of  surface.  F i n a l l y , Chapter V p r o v i d e s a b r i e f summary of the s u b j e c t along w i t h a d i s c u s s i o n of p o s s i b l e Appendix I c o n t a i n s the u  +  to date,  future d i r e c t i o n s .  the d e t a i l e d d e r i v a t i o n s  s p i n p o l a r i z a t i o n i n muonium s u b j e c t  of the time e v o l u t i o n of  to a g e n e r a l l y  hyperfine  i n t e r a c t i o n , along w i t h the a s s o c i a t e d  described  i n Chapter I I I .  anisotropic  spin relaxation  functions  L a s t l y , Appendix I I o u t l i n e s an experiment which i s designed to study the  i n t e r a c t i o n s of muons and muonium atoms w i t h "macroscopic" s u r f a c e s , and  draws h e a v i l y on knowledge a l r e a d y  gained i n the study of p o s i t r o n s  ( e ) and +  positronium ( e e ~ , P s ) . +  I.A  Muons and Muonium Some of the c h a r a c t e r i s t i c s o f muons and muonium a r e d i s c u s s e d  following  few pages.  i n the  - 4 I.A.I  Muon C h a r a c t e r i s t i c s The muon  i n 1937.  was f i r s t  observed  [4,5] as a component of cosmic r a y s  Muons a r e u n s t a b l e l e p t o n s , having a r e s t mass of about 105.7  MeV/c , and apart from t h e i r f i n i t e 2  l i f e t i m e can i n n e a r l y every r e s p e c t be  c o n s i d e r e d heavy e l e c t r o n s ( o r p o s i t r o n s ) .  Some of the p r o p e r t i e s of muons  are g i v e n i n Table 1.1. The most common source of muons i s from the decay of charged (it ,it~).  Pions  +  nonconserving it+  ( s p i n = 0) decay v i a weak i n t e r a c t i o n i n the p a r i t y  processes [6]  •> u . + + v  7 t - > L i + v  and  (I«l)  w i t h a f r e e mean l i f e t i m e of 26.030(23) nanoseconds ( n s ) . of the p i o n , the decay i s s p a t i a l l y being emitted  I n the r e s t  i n opposite d i r e c t i o n s .  I n the case o f 7t decay the muon and +  (i.e.,  s p i n a n t i p a r a l l e l to momentum),  decay they are both emitted w i t h p o s i t i v e h e l i c i t y  -  spin p a r a l l e l  to momentum).  frame  i s o t r o p i c w i t h the muon and n e u t r i n o  n e u t r i n o both have n e g a t i v e h e l i c i t y whereas f o r n  pions  Since n e u t r i n o s possess  zero ( o r near  r e s t mass, the momentum of the emitted muon i n the r e s t  (i.e., zero)  frame i s 29.8 MeV/c,  which t r a n s l a t e s i n t o a k i n e t i c energy of 4.1 MeV. L i k e the p i o n , the muon a l s o decays v i a the weak i n t e r a c t i o n , to the p a r i t y v i o l a t i n g u . + -> e  +  + v  e + nv  according  r e a c t i o n s [6] and  w i t h a f r e e mean l i f e t i m e of T  u - > e + v + v ^ e ^ i = 2.19695(6) \is [ 7 ] .  (I«2) I n c o n t r a s t to p i o n  decay, muon decay i s s p a t i a l l y a n i s o t r o p i c i n the c e n t e r of mass frame; the muon p r o v i d e s a p r e f e r r e d d i r e c t i o n ( i t s s p i n o r i e n t a t i o n ) as a r e f e r e n c e .  T a b l e 1.1 Property  P r o p e r t i e s of Muons  (n ,fi-) +  Value  (symbol)  Charge  e  H+,u  Spin  s  1/2  Rest Mass  \  105.6596 MeV/c  Mean Free  Lifetime  -  = ±1.60225 x 1 0 "  2.19695(6)  2  1 9  Coulombs  m (a) = 206.76859(29) 0.7570 m^ 0.1126123(6) m (a) fi  \is (b)  g-Factor  -2[l.001165895(27)] ( c )  Magnetogyric R a t i o  2  K e  8.5165 x 1 0 s G 2% x 13.5544 kHz/G 4  V  1  28.0272(2) x 1 0 " 0.00484 n 3.1833417(39) u.  Magnetic Moment  1 8  1  MeV/G  e  h  Compton Wavelength  de B r o g l i e Wavelength  1.86758 fm  v  Mi  h(2nm kT)" [i  1/2  (a) D.E. Casperson, et a l . , Phys. Rev. L e t t . 38.. (b) K.L. G i o v a n e t t i , (c) J.M.  2.99 A 25.29 A  =  9  5  6  (300 K) (4.2 K)  (1977).  et a l . , Phys. Rev. D 29_, 343 (1984).  B a i l e y , et a l . , Phys. L e t t . 55B, 420 (1975).  (a)  - 6 The  maximum momentum of the decay e l e c t r o n i s g i v e n m c u.  = [p c e  2  2  2  + m c ] e 2  4  1 / 2  + p c e  ;  p  m a X  e  T h i s maximum occurs when both the n e u t r i n o the same d i r e c t i o n , o p p o s i t e the s p i n s of the n e u t r i n o  =  52.827 MeV/c  to that of the decay e l e c t r o n .  (negative  For t h i s  case  h e l i c i t y ) and the a n t i n e u t r i n o ( p o s i t i v e to balance the s p i n of the  I n weak i n t e r a c t i o n s , the momentum of the e ( e ~ ) tends s t r o n g l y to  +  be  (1.3)  and a n t i n e u t r i n o are emitted i n  h e l i c i t y ) c a n c e l , l e a v i n g the p o s i t r o n ( e l e c t r o n ) u (u~).  by the r e l a t i o n  +  ( a n t i ) p a r a l l e l t o i t s s p i n , so t h a t the high  along(opposite) Since  energy e ( e ~ ) tends to e x i t +  the | i ( u ~ ) s p i n . +  the t o p i c of t h i s d i s s e r t a t i o n concerns o n l y p o s i t i v e muons, the  discussions  henceforth  w i l l be c o n s t r a i n e d  accordingly.  mass of the p o s i t r o n i n comparison w i t h the mass m  By n e g l e c t i n g the  of the muon, the  p r o b a b i l i t y per u n i t time dW(e,9) f o r the e m i s s i o n of a p o s i t r o n of energy E i n the e l e m e n t a l s o l i d angle dco a t an angle 9 w i t h r e s p e c t  to the muon s p i n  d i r e c t i o n can be expressed as [6] dW(e,9) = ^ where u  +  e =  ^/^  [e (3-2e)][l + P 2  =  ensemble.  ^^ y ° m  2  a n (  L  m a x  *  p  r  cos(9)]  dedu  (1.4)  e p r e s e n t s the degree of p o l a r i z a t i o n of the  E q u a t i o n 1.4 i s w r i t t e n i n terms of an i s o t r o p i c average  energy spectrum C ( e ) = e ( 3 - 2 e ) and an asymmetry f a c t o r D ( e ) = P ( 2 e - l ) / ( 3 - 2 e ) , 2  both of which a r e shown i n F i g u r e  1.1 f o r a muon ensemble p o l a r i z a t i o n of  P=l.  I.A.2  Thermalization of Positive Muons in Matter The  slowing  down o f a u"" i n matter i n v o l v e s 1  s e v e r a l stages of energy  - 7  1.0  I  I  0.8 0.6 0.4 0.2 / D (  0.0 -.2  £  )  -  -.4  I  0.0  0.2  I I  I  0.4  0.6  0.8  1.0  e = E / E max F i g u r e 1.1 P o s i t r o n energy spectrum from muon decay (upper curve) and energy dependence o f the asymmetry f a c t o r f o r 100% p o l a r i z e d (P=l) muons (lower c u r v e ) . The p o s i t r o n energy i s g i v e n i n u n i t s of the maximum p o s s i b l e e m i s s i o n energy E_„„ = 52.827 MeV.  - 8 l o s s mechanisms [ 1 ] . l o s e energy  A h i g h energy  \i  +  i n t e r a c t i n g w i t h matter  by s c a t t e r i n g w i t h e l e c t r o n s .  will  When the | i v e l o c i t y  first  approaches  +  t h a t of the v a l e n c e e l e c t r o n s of the t a r g e t atoms ( c o r r e s p o n d i n g to a | i kinetic  energy  through  i o n i z a t i o n , i n accordance  keV,  energy  of 2-3  loss s t i l l  keV),  the energy  l o s s per u n i t  time occurs  w i t h the Bethe e q u a t i o n  occurs through  [8].  primarily Below ~2  c o l l i s i o n s w i t h e l e c t r o n s , except i n  t h i s case the Bethe e q u a t i o n does not h o l d s i n c e the e l e c t r o n s now a degenerate  gas.  In t h i s energy  +  behave as  r e g i o n , a muon can a l s o capture and  e l e c t r o n s i n i t s i n t e r a c t i o n s w i t h the t a r g e t medium, forming  lose  short-lived  n e u t r a l h y d r o g e n - l i k e muonium ( f i e ~ ) atoms; i n many cases, the n e u t r a l +  muonium atom i s the f a v o r e d charge  s t a t e as the \i  +  threshold for this capture/loss cycle. down through  The  v e l o c i t y drops  below the  f i n a l muonium atom then  slows  subsequent n o n - i o n i z i n g c o l l i s i o n s w i t h atoms and/or  molecules. The  e f f e c t of the slowing down process on the u.  +  been g i v e n e x t e n s i v e c o n s i d e r a t i o n by many authors n e g l i g i b l e i n s o l i d s , where the charge the h y p e r f i n e p e r i o d .  +  found  s p i n w i t h i t s environment.  comparable to the h y p e r f i n e p e r i o d  has  to be  exchange c y c l e s are much s h o r t e r than  d e p o l a r i z a t i o n can indeed occur s i n c e the charge  In gases,  study  however,  exchange c y c l e s may  be  [11,12].  Muonium Formation and Characteristics The  [13],  [9,10], and  T h i s i s , of course, good news i f one wishes to  the i n t e r a c t i o n of the \i  I.A.3  spin polarization  d e t a i l s of muonium (Mu)  but i t was  f o r m a t i o n was  not u n t i l 1960  obtained  [14].  f o r m a t i o n were f i r s t that d i r e c t  discussed i n  e x p e r i m e n t a l evidence  1952 of i t s  Some of the p r o p e r t i e s of muonium are g i v e n i n  - 9 Table 1.2.  The reduced mass of the e l e c t r o n i n muonium i s about 0.996 t h a t  f o r hydrogen, making the Bohr r a d i i and i o n i z a t i o n p o t e n t i a l s of muonium and hydrogen e s s e n t i a l l y the same. like a light  Consequently,  i s o t o p e of hydrogen [1,15,16],  0.1131 the r e s t mass m^ o f hydrogen.  muonium behaves c h e m i c a l l y  having a r e s t mass  equal t o  U n l i k e hydrogen, however, muonium i s a  p u r e l y l e p t o n i c system whose p r o p e r t i e s a r e c a l c u l a b l e to extreme p r e c i s i o n e n t i r e l y from  first  system to be used  principles.  As a r e s u l t , the muonium atom i s an i d e a l  f o r t e s t s of quantum e l e c t r o d y n a m i c s , and has been w i d e l y  employed as such. A completely g e n e r a l H a m i l t o n i a n f o r the h y p e r f i n e i n t e r a c t i o n between the \i  +  and e~ s p i n s , i n the presence  o f a magnetic f i e l d B and a l l o w i n g f o r  e f f e c t s due to an a n i s o t r o p i c environment, can be w r i t t e n  K. = (h/2Tt)( S - y ^ ) • B Mu ^'e ~op u ~op ~  +  e  Y  where y  = £  2n:(Y )  ;  a  n  a  e  Y^  =  2TC(  Y  e the e l e c t r o n and the muon, S ~op  (h/2n) W : (s ) » ~op ~op  (1.5)  e  v  ;  ) a r e the r e s p e c t i v e magnetogyric u and S a r e the c o r r e s p o n d i n g ~op  s p i n o p e r a t o r s and W i s a second  r  ratios of  dimensionless  rank t e n s o r r e p r e s e n t i n g the c o n t a c t  h y p e r f i n e i n t e r a c t i o n , which has been e x p l i c i t l y g e n e r a l i z e d here  to i n c l u d e  the p o s s i b i l i t y of an a n i s o t r o p i c Mu atom, as might be imposed by a s o l i d medium. second  I n vacuum, of c o u r s e , W reduces rank  to a c o n s t a n t m u l t i p l y i n g  the u n i t  tensor.  For i s o t r o p i c muonium ( i . e . , having a s p h e r i c a l l y symmetric h y p e r f i n e interaction),  the e i g e n v a l u e s of the s p i n H a m i l t o n i a n  ( E q u a t i o n 1.5) a r e  g i v e n i n terms of t h e i r r e s p e c t i v e w e a k - f i e l d quantum numbers (F,mp) by the B r e i t - R a b i formula  [17], namely  - 10 T a b l e 1.2 Property  P r o p e r t i e s of Muonium (Mu) (symbol)  Value  Rest Mass  m  Mu  Reduced  m  Mu  Mass  Bohr Radius Ground  ^ o^Mu a  S t a t e Energy  Magnetogyric R a t i o  H y p e r f i n e Frequency  de B r o g l i e  Wavelength  ( Y  v  R O T  Mu  oo  ,Mu  )M  0.1131  m  R  0.9952  m  e  = 207.8 ni e = 0.9956 m£  0.5315 A  = 1.0044 ( a )  -13.54 eV  = 0.9956 ( R J  triplet;  = 8.8 x 1 0  Q  H  H  U  6  s  -  1  G"  1  « 2it x 1.4 MHz/G theo. = 4463.3185(6.5) MHz (a) expt. = 4463.30235(52) MHz (a) 2.979 A (300 K) = 2.967 \ J (300 K)  (a) D.E. Casperson, et a l . , Phys. Rev. L e t t . 38_, 956 (1977).  . . = - 7- E ~ (g v?) |B|m_ ± i E ( l + 2m X + _ rl . l - i 4 o u ^o '~' F 2 o*F F-(j ± >*  E  KO  X ) 2  J  1 / 2  ;  F  (1.6) ] BI X - ( g H* + g  |#  e  and  where E  i s the zero f i e l d  Q  = 1, 0, -1  o  r  (F=l) and s i n g l e t  h y p e r f i n e energy s p l i t t i n g  (F=0) s t a t e s , g  and g are the r e s p e c t i v e g - f a c t o r s and (i,  e  e u ii and u r a r e the e l e c t r o n and muon Bohr magnetons. 0  between the t r i p l e t  0  Denoting v  00  0  = E /h to o  be the h y p e r f i n e - s t r u c t u r e i n t e r v a l (~4463.3 MHz [ 1 8 ] ) , e v a l u a t i o n o f E q u a t i o n 1.6 f o r the four e x i s t i n g s p i n c o u p l i n g s t a t e s then g i v e s  v  i  1 „ 1 TT i T h 1 4  =  E  1  1  v  3 0  =  +  =  „  ,  v  -  00  ;  1  E„ = -r v - v h 3 4 00 -  T-  v ;  0  1 = T- E 2 h  r  = 2  1 „ v. = 7- E.  4  1 4  ^ r l 2 ^ 2-.1/2 + I 7+ v.l 00 4 00 +  v  T  v  L  1 = - 7- v  4 4  4  00  -  J  rl L  2  ^  7- v 4 o o  (1.7)  2-11/2  + v. +  J  w i t h the d e f i n i t i o n V  I  ^  ^  ^  D  = i C l v . l ± \\\)  T  Choosing the a x i s of q u a n t i z a t i o n to be along energy e i g e n s t a t e s  (1.8)  the magnetic f i e l d , the  |j> of i s o t r o p i c muonium can be r e p r e s e n t e d  the i n d i v i d u a l s p i n e i g e n f u n c t i o n s  i n terms of  |m ,m > as LI e  1  |1> =  |+,+>  ;  |2> = s|+,-> + c|-,+>  |3> =  |-,->  ;  |4> = c|+,-> - s|-,+>  (1.9)  where the amplitudes s ( s i n e ) and c ( c o s i n e ) a r e d e f i n e d as Ti - [1 /2 1  s =  x = (g  e  x -|l/2 , TTPy\ (1 + x ) |B| |B| - g = = a  2  ^  1  0  /  2  n  d  1 . . x -,1/2 - L 9 i 10 J /2 (1 + x ) r  c  =  2  1  /  2  (I.10)  S p e c i f i c F i e l d Parameter  0  where one has the n o r m a l i z a t i o n c o n d i t i o n s the h y p e r f i n e f i e l d .  1 +  2  + c  Note that i n zero f i e l d ,  2  = 1, and B (» 1585 G) i s ' o  s = c = 1//2 and v i 2  = l  v  0 0  .  - 12 I.B  Time E v o l u t i o n The  -  of the Muonium S p i n  four h y p e r f i n e  states  are,  the muons a r r i v e w i t h a p r e f e r r e d  State  i n general,  u n e q u a l l y populated  p o l a r i z a t i o n ( d i r e c t e d o p p o s i t e to  d i r e c t i o n of e m i s s i o n from p i o n decay), w h i l e the normally unpolarized. initial u state  |a >  =  0  |+,+>  and  longitudinal f i e l d , the  Thus, choosing the  the  where the  t h i s same d i r e c t i o n such t h a t of the H a m i l t o n i a n , but  the  h a l f of  triplet"  w h i l e the  and  Q  g i v e n by  p  l  =  1 2  state the  0  initial  ;  p  2  =  f  s  |b >  |a > 0  =  0  =  |+,->  =  Q  |+,->.  In  |+,+>  =  = s|2>  |1>  i s an  + c|4>  remaining h a l f i s formed i n the  a along  along eigenstate  i s not.  the muonium ensemble i s formed i n the  2  5  p  With  "polarized  "mixed"  four h y p e r f i n e  3  °  =  ;  P  4  =  f  c  state  states  are  2  Coulomb i n t e r a c t i o n which governs the  i n t e r a c t i o n between the \x  +  the  on  s p i n and  the  ( I  the  order of  -  U )  e l e c t r o n capture p r o c e s s  s p i n p o l a r i z a t i o n i n s o l i d s , the  r i s e to phase o s c i l l a t i o n s i n the  s t a t e s at f r e q u e n c i e s  I.B.I  state  r e l a t i v e p o p u l a t i o n s of the  a n e g l i g i b l e e f f e c t on  give  |b >  the  the p r o b a b i l i t i e s  Although the has  |a >  the  the  ensemble forms i n  o r i e n t a t i o n energy i s q u a n t i z e d  state  these d e s i g n a t i o n s ,  state  are  a x i s along  e x t e r n a l magnetic f i e l d B i s d i r e c t e d  i n i t i a l muon p o l a r i z a t i o n , the  |b >,  spin quantization  other h a l f i n the  the  captured e l e c t r o n s  s p i n p o l a r i z a t i o n d i r e c t i o n , h a l f of the Mu  +  since  s p i n of the superposition  hyperfine  e l e c t r o n i n muonium does |b > Q  of the  the h y p e r f i n e - s t r u c t u r e  hyperfine  interval V Q. Q  Muonium i n Vacuum Consider the  muonium.  time e v o l u t i o n  In l o n g i t u d i n a l f i e l d ,  of the muon s p i n p o l a r i z a t i o n i n f r e e the  polarized  triplet  eigenstate  |a > Q  is a  - 13 s t a t i o n a r y s t a t e , w h i l e the mixed s t a t e eigenstates. the  Since the s t a t e  time dependence o f s t a t e  |b > i s a s u p e r p o s i t i o n n  |a > i s s t a t i o n a r y , one must only Q  o f two determine  |b >. R e c a l l i n g E q u a t i o n 1.7, and d e f i n i n g Q  oa . = 2nv . and u>. . = 2n(v. .) = 2it(v. - v.)» one then f i n d s [1] J 3 i j i j i 3 |b(t)>  = e~ 2 {[ i u  In zero f i e l d ,  t  2 s  2 + c  exp(iu) t)]|+,-> 2 4  the s t a t e  (|  where s = c = 1//2 and  o s c i l l a t e s w i t h a frequency and  + sc[ l-exp( i o ^ t ) ] |-,+> } (1.12)  | |  to  0Q  - oo  = 2 T C V , the s t a t e 0 0  between the i n i t i a l h y p e r f i n e  state  |b(t)>  |+> >n -  |-,+>||, i n which the muon s p i n d i r e c t i o n i s r e v e r s e d .  s p i n p o l a r i z a t i o n o f the muons i n the s t a t e  |b(t)>  i s given  The  by the r e l a t i o n  |b(t)>z, where 0-3 i s the muon P a u l i s p i n m a t r i x f o r  p£l|(x,t)z = <b(t) | projections  00  along the q u a n t i z a t i o n  a x i s ( z - a x i s ) and z i s the  corresponding unit vector.  By combining t h i s w i t h the 100% p o l a r i z a t i o n o f  the muons i n the s t a t i o n a r y  state  |a >, the time dependence of the t o t a l Q  muon ensemble i n l o n g i t u d i n a l f i e l d 1  i  i s given 1  P)f(x,t) = \ [1 + P£„(x,t)]z = J + J  x  ^  +  by [1]  c o s  ( o/. ) w  [  t  2^—]  (1.13)  1 + x In t r a n s v e r s e applied |b > Q  field  perpendicular  t o the i n i t i a l muon p o l a r i z a t i o n , the s t a t e s  a r e not e i g e n s t a t e s  initial  state vectors  longitudinal f i e l d  ( T F ) , where the e x t e r n a l magnetic f i e l d B i s  and n e i t h e r one i s s t a t i o n a r y .  |a > and |b > can be w r i t t e n  basis  Q  Q  |m^,m>|| . e  |a > and Q  I n t h i s case, t h e  i n terms o f the  By expanding these s t a t e s i n terms o f  the i s o t r o p i c muonium energy e i g e n s t a t e s  |j>, g i v e n  i n E q u a t i o n 1.9,  time dependence f o r each o f these two s t a t e s i s found to be [1]  the  |a (t)> = i [  e " " ! ! ^ + (s + c ) e " 2 | 2 > 1  0  1 1  i u  c^e'^l^]  + e" 3 |3> - (s iu  t  t  (1.14) |b (t)> = i [ - e " " ! ^ ^ + (s - c ) e 1  _ i a )  0  + e"  ia3  2 |2> t  3 | > + ( s + c)e" '» |4>] t  lu  t:  3  Because the magnetic f i e l d  ( B ) i s o r i e n t 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 z  muon p o l a r i z a t i o n , and s i n c e the muonium e l e c t r o n i s o n l y i n t e r a c t i n g the muon s p i n , a l l o f the motion of the \i c o n f i n e d to the x-y p l a n e .  with  s p i n i n the muonium s t a t e i s  +  T h i s being the case, the time e v o l u t i o n o f the  muon p o l a r i z a t i o n f o r the e n t i r e muonium ensemble i s g i v e n by the complex quantity P ^ ( x , t ) - l[< (t)|(o-£ + io-£)|a(t)> + <b(t)|(o-f + ia£)|b(t)>]  (1.15)  a  where  and  a r e the x and y P a u l i s p i n o p e r a t o r s .  the \i p o l a r i z a t i o n along the i n i t i a l +  r e p r e s e n t s the \i  +  Here the r e a l p a r t i s  x d i r e c t i o n and the imaginary  part  p o l a r i z a t i o n along the y d i r e c t i o n , p e r p e n d i c u l a r t o both  x and z ( i . e . , x x y = z ) .  Substituting  the e x p r e s s i o n s f o r the s t a t e  v e c t o r s g i v e n i n E q u a t i o n 1.14 i n t o E q u a t i o n 1.15 then g i v e s the r e s u l t  c o s ( - ^ t) [ c o s ( - | ^ + o)t  = exp(iw_t)  6 = (c - s ) = 2  2  X 2  /  (1 + x Most o f the experiments low f i e l d  limit  (x «  2  and  Q = |<OJ  2 3  - w  - i6sin(-22. + Q ) t ] ) = - ^ ( l  + x ) 2  1  /  (1.16)  2  - l]  J  r e p o r t e d i n t h i s d i s s e r t a t i o n were performed  1).  In this l i m i t i n g  case, the r e a l p a r t o f  i n the  - 15 E q u a t i o n 1.16  s i m p l i f i e s to g i v e  Re{p^(t)} « j where u_ =  [1]  cos(u_t) [cos(fit) + C O S ( O D as d e f i n e d  2TCV_  -  q o  i n E q u a t i o n 1.8.  + Q)t]  (1.17)  Since the  frequency  0  i n g e n e r a l too h i g h to be observed e x p e r i m e n t a l l y , except i n h i g h fields,  E q u a t i o n 1.17  describes  a s i g n a l w i t h h a l f of the  is  ( C O Q + Q)  initial  transverse | i spin +  p o l a r i z a t i o n amplitude (asymmetry) which o s c i l l a t e s a t the Larmor frequency modulated at a beat frequency equal to Q.  A more e l a b o r a t e  developed elsewhere [19,20] f o r cases where the muonium e l e c t r o n  formalism i s interacts  with i t s environment.  I.B.2  Interactions with the Environment The  result  i n t e r a c t i o n of the u-  in a depolarization  I n s t r u c t i v e at t h i s p o i n t relaxation.  The  or a r e l a x a t i o n of the u to d e f i n e  i n some cases  s p i n ensemble.  It is  what i s meant by d e p o l a r i z a t i o n  versus  +  encompasses a l l v a r i e t i e s of  i n t e r a c t i o n s i n which the  i n p r i n c i p l e be  echo t e c h n i q u e s ) ;  s p i n w i t h i t s environment may  term " d e p o l a r i z a t i o n "  dynamics, i n c l u d i n g ensemble c o u l d  +  phase coherence of  spin  the  recovered at some l a t e r time ( e . g . , by  whereas " r e l a x a t i o n " a p p l i e s  spin spin  to those i n t e r a c t i o n s which  r e s u l t i n a s t r i c t l y i r r e v e r s i b l e l o s s of ensemble p o l a r i z a t i o n , such as the  case of a d i f f u s i n g magnetic probe.  "relaxation" i s applied applied  herein  In the the \x  +  i n the  i n a somewhat g e n e r i c  fashion  and  however, the  term  w i l l generally  be  same manner.  case of a bare u ,  s p i n w i t h the  Conventionally,  in  +  s p i n r e l a x a t i o n occurs v i a the  l o c a l magnetic f i e l d d i s t r i b u t i o n .  muonium, however, the \i  +  i s strongly  coupled to the  i n t e r a c t i o n of  In the  case of  e l e c t r o n so t h a t  i n weak  - 16 magnetic f i e l d s triplet  the f r a c t i o n (50%)  ( F = l , m^  = +1)  strong  the order of the e l e c t r o n ' s  a l s o s e n s i t i v e to e l e c t r i c  hyperfine  field  gradients  generalized  interaction.  i n E q u a t i o n 1.5.  are  +  than a bare  u . +  spin polarization i s  +  thereby induce  I t was  may  anisotropics  f o r t h i s reason that W  s p i n i n a Mu  atom.  The  or  f i v e known mechanisms  are:  Random L o c a l Magnetic F i e l d s  (depolarization)  (2) (3) (4)  Random A n i s o t r o p i c H y p e r f i n e D i s t o r t i o n s Chemical R e a c t i o n s Spin Exchange  (depolarization) (relaxation, for (relaxation)  (5)  Superhyperfine Interactions  (depolarization)  static  of  (non-diffusing)  " d e p o l a r i z a t i o n " a p p l i e s only probe (Mu  atom).  corresponding spin r e l a x a t i o n functions, Chapter I I I .  Of  interactions.  associated  These f u n c t i o n s  Owing to i t s r e l a t i v e l y mobile i n the  stopping  a  These mechanisms, along w i t h  the  are d i s c u s s e d  in  i n more d e t a i l  w i t h random a n i s o t r o p i c are d e r i v e d  the  static  hyperfine  i n d e t a i l i n Appendix I .  l i g h t mass, the muon (or Mu  medium.  TF)  i n the case of  p a r t i c u l a r importance to the present study are  relaxation functions  atom) may  T h i s motion or hopping may  result  be in  e f f e c t i v e r e l a x a t i o n r a t e which d i f f e r s i n magnitude i n comparison to s t a t i c value.  T h i s d i f f e r e n c e comes about because the e f f e c t s of  i n t e r a c t i o n ( s ) governing the  and  +  induce d e p o l a r i z a t i o n  (1)  Here the d e s i g n a t i o n  was  the  cause d e p o l a r i z a t i o n f o r both bare u.  f o u r other mechanisms t h a t can  r e l a x a t i o n of the u-  i s thus  Thus, i n a d d i t i o n to i n t e r a c t i o n s w i t h  l o c a l magnetic f i e l d s , which can there  the u  and  spin-one  or other mechanisms that  the muonium e l e c t r o n w a v e f u n c t i o n and  i n t o the muonium h y p e r f i n e  Mu,  coupling,  polarized  like a polarized  times more s e n s i t i v e to l o c a l magnetic f i e l d s  Because of the r a t h e r  distort  of muonium that forms i n the  s t a t e behaves m a g n e t i c a l l y  o b j e c t w i t h a magnetic moment on about 103  -  time e v o l u t i o n of the \i  +  very an the  the  s p i n p o l a r i z a t i o n are  averaged by the  the motion, hence the  s p e c i f i c i n t e r a c t i o n ( s ) and  term "motional a v e r a g i n g " .  the  time s c a l e s  produce an e f f e c t i v e r e l a x a t i o n r a t e that has ("motional narrowing") or an comparison to the  increased  s t a t i c value.  s p i n exchange are not  involved,  to chemical r e a c t i o n s  a f f e c t e d by m o t i o n a l a v e r a g i n g , but random a n i s o t r o p i c  and  indeed  and  defining %  f l u c t u a t i o n s as fast  «  distortions  affected. presence  <AOJ > 2  Assuming a Gaussian d i s t r i b u t i o n of random  to be  c  sensed by  f l u c t u a t i o n s becomes  X  hyperfine  +  of s t a t i c n u c l e a r d i p o l e s .  the  c o r r e l a t i o n time of the  the L I , the +  local  field  s p i n r e l a x a t i o n r a t e i n the  limit  of  [21]  x  (1.18)  C  Ll  or  relaxations  t r a d i t i o n a l example i s a L I hopping s t o c h a s t i c a l l y i n the  The  can  magnitude ("motional broadening") i n  a r i s i n g from random d i p o l a r f i e l d s ,  fields,  the motion  e i t h e r a reduced magnitude  R e l a x a t i o n s due  s u p e r h y p e r f i n e i n t e r a c t i o n s are  Depending upon  2  <Aco > i s the second moment of the frequency d i s t r i b u t i o n f o r the  where  random l o c a l f i e l d The  [22].  e f f e c t of hopping on  from random a n i s o t r o p i c i s not  as  I.C  The  case of the  Interactions  Much of studies  hyperfine  straightforward  t h i s i n the  the  shape of the  relaxation functions  d i s t o r t i o n s or s u p e r h y p e r f i n e  to determine.  interactions  However, a d e t a i l e d d i s c u s s i o n  of  former i s g i v e n i n Chapter I I I .  o f Muonium w i t h  Silica  the work presented i n t h i s d i s s e r t a t i o n stems from  i n v o l v i n g both muonium i n bulk s i l i c a and  b r i e f summary of  arising  these s t u d i e s  i s therefore  on s i l i c a  earlier  surfaces.  presented here, along w i t h  A  - 18  -  d i s c u s s i o n s on those p o i n t s of p a r t i c u l a r  I.C.I  Muonium i n B u l k  r e l e v a n c e to the present work.  Silica  E x t e n s i v e s t u d i e s have been made on muonium i n bulk quartz  [23-27],  where most of the phenomena of i n t e r e s t a r i s e from a n i s o t r o p i c of  the muonium h y p e r f i n e i n t e r a c t i o n .  distortions  Zero f i e l d measurements of muonium i n  s i n g l e c r y s t a l quartz have r e v e a l e d three f r e q u e n c i e s a t low temperatures 77 K ) .  These f r e q u e n c i e s , which obey the sum  c o n s t a n t but have amplitudes  rule v  With t h i s p i c t u r e ,  the three observed  t r a n s i t i o n s between three l e v e l s , and  At h i g h e r temperatures  +  v  1 2  23»  T h i s r e s u l t i s c o n s i s t e n t w i t h an  s p i n H a m i l t o n i a n i n which the h y p e r f i n e t e n s o r has  to  = v  r  e  m  a  *  t h a t v a r y , as the c r y s t a l i s r o t a t e d about  i n i t i a l muon s p i n p o l a r i z a t i o n .  symmetry.  1 3  (<  n  the  effective  t h r e e p r i n c i p a l axes of  f r e q u e n c i e s then  correspond  as such are l a b e l l e d a c c o r d i n g l y .  (near room temperature),  the muonium h y p e r f i n e  i n t e r a c t i o n has an a n i s o t r o p y which i s symmetric about the c - a x i s of the c r y s t a l due  to m o t i o n a l a v e r a g i n g .  In t h i s  ( h i g h temperature) case,  h y p e r f i n e t e n s o r W can be broken down i n t o an i s o t r o p i c p a r t <W> term 6W  £  S  £  and  = 6u  c  S£ as the p r o j e c t i o n s of the e  respectively, H In  h f  a s s o c i a t e d w i t h a d i s t o r t i o n along the c - a x i s .  the a x i a l l y  = (h/2ii) W:(S ' » ~op  zero f i e l d ,  6  v  the  = O)Q and 0  By  denoting  and u"" s p i n s a l o n g the c - a x i s , 1  symmetric c o n t a c t h y p e r f i n e H a m i l t o n i a n becomes  )  S^ ~op'  =  (h/2it){<W>(s ^~op  e  1  •~op' sM  + 6W ( S c^ c  e  S^)} C  J  '  (1.19)  w i t h the c - a x i s o r i e n t e d p e r p e n d i c u l a r to the i n i t i a l muon  p o l a r i z a t i o n , an o s c i l l a t i o n of 0.412(4) MHz c-axis oriented p a r a l l e l  i s observed;  however, w i t h  to the i n i t i a l muon p o l a r i z a t i o n , the  the  oscillation  a  - 19 d i s a p p e a r s , as p r e d i c t e d In fused and  quartz,  the  the d e p o l a r i z a t i o n  owing to the hyperfine  zero f i e l d  of the  1.19. hyperfine  o s c i l l a t i o n s are  random magnitude, symmetry and to the  of the \i  +  o r i e n t a t i o n of the muonium  i n i t i a l muon s p i n .  of random a n i s o t r o p i c h y p e r f i n e  time e v o l u t i o n  suppressed  s p i n i s enhanced v i a ensemble dephasing,  d i s t o r t i o n with respect  discussion the  the  by E q u a t i o n  -  d i s t o r t i o n s and  A more g e n e r a l their effect  on  s p i n p o l a r i z a t i o n f o r s t a t i c muonium i s g i v e n  i s Appendix I .  I.C.2  Muonium on S i l i c a I t has  Si0 ,  can  2  low  long  been known that  be used i n the  temperatures  powder g r a i n s ,  [29] .  0  2  I t i s thought that  Mu  evacuated S i 0  extragranular  2  due  gas moderator at one powder a c t s  finally  first  the  escapes i n t o  reported  powder [30], where the  r a t e was  verified  by  the  introduction  atmosphere [31] , thus demonstrating t h a t  2  the muonium formed f i n d s i t s way  f o r a l l three o x i d e s .  Of  a l l the  linearly  2  i n an  the  Si0  investigations 2  3  and  MgO,  to the  oxides t e s t e d ,  Si0  of  was  2  0  2  argon 2  [32-34]  show that  extragranular 2  0  with  paramagnetic  with r e s u l t s o b t a i n e d w i t h 0  Later  for fine  emergence of  the  i n f i n e oxide powders, namely S i 0 , A 1 0  c e r t a i n f r a c t i o n of  i n 1978  observed to i n c r e a s e  l i k e a v e r y coarse moderator gas.  c o n c e r n i n g Mu  and  to s p i n exchange i n t e r a c t i o n s w i t h the  m o l e c u l e s , i n a manner c o n s i s t e n t  region  surface  r e g i o n was  spin depolarization  concentration,  even a t  p o s i t r o n i u m i s formed i n  analogous phenomena f o r muonium was  muonium i n t o the The  p r o d u c t i o n of p o s i t r o n i u m i n vacuum [28],  and  grains.  (35 A mean r a d i u s )  gas.  f i n e i n s u l a t i n g powders, such as MgO  d i f f u s e s r a p i d l y to the  v o i d between the The  Surfaces  found  to  a  - 20  -  have the h i g h e s t f o r m a t i o n p r o b a b i l i t y f o r Mu, a v a i l a b l e i n the s m a l l e s t g r a i n s i z e ) ; produce the h i g h e s t y i e l d  and  ( p o s s i b l y because i t was  the 35 A S i O j powder was  of e x t r a g r a n u l a r muonium (>97%  r e g a r d l e s s of the ambient temperature of the powder. the 35 A s i l i c a  powder the obvious  i n t e r a c t i o n of \i  I.C.3  +  candidate  of Mu  found  formed [ 3 0 ] ) ,  This l a s t  p o i n t made  f o r f u r t h e r s t u d i e s of  and muonium with s u r f a c e s , the s u b j e c t of t h i s  the  thesis.  Muonium Formation i n Fine S i l i c a Powders The muonium f r a c t i o n s f o r bulk fused q u a r t z as w e l l as f o r 35 A and  A mean r a d i u s s i l i c a  powders are g i v e n i n T a b l e 1.3.  As  i n the case  p o s i t r o n i u m f o r m a t i o n , muonium f o r m a t i o n i n f i n e oxide powders may thermal,  e p i t h e r m a l , spur and/or s u r f a c e p r o c e s s e s .  Because the  draw a simple analogy  atoms.  First  70  of  involve atomic  b i n d i n g energy of p o s i t r o n i u m i s about h a l f t h a t of muonium, i t i s to  to  difficult  between the f o r m a t i o n p r o b a b i l i t i e s f o r the  two  l e t us ask whether muonium f o r m a t i o n i n f i n e oxide powders i s  a b u l k or a s u r f a c e phenomenon; s u r f a c e f o r m a t i o n of p o s i t r o n i u m has, f o r i n s t a n c e , been observed metal-oxide  f o r low energy p o s i t r o n s i n c i d e n t on metal  surfaces [35].  one would expect surface area.  the Mu  I f muonium f o r m a t i o n i s indeed  surface related,  f r a c t i o n to i n c r e a s e w i t h i n c r e a s i n g s p e c i f i c  From the v a l u e s g i v e n i n T a b l e 1.3,  appear to be p a r t i c u l a r l y dramatic, f o r m a t i o n of muonium i n s i l i c a The  t h i s e f f e c t does not  i f i t e x i s t s at a l l , suggesting  powders takes p l a c e p r i m a r i l y i n the  p o s s i b i l i t y of some charge exchange o c u r r i n g a t the s i l i c a  not however r u l e d The  and  t h a t the bulk.  surface i s  out.  next q u e s t i o n i s whether Mu  e p i t h e r m a l or spur p r o c e s s e s .  f o r m a t i o n occurs v i a thermal,  In the spur model [36], muonium  formation  - 21 T a b l e 1.3  Muonium F r a c t i o n s ( F ) and T r a n s v e r s e F i e l d R e l a x a t i o n Rates (\ ) f o r Bulk and Powdered S i l i c a . M u  Mu  Sample  T (K)  Bulk fused  Si0  2  Si0  2  powder (70  A)  2  powder (35  A)  M u  (%)  X™  (us" ) 1  6  79 +  3  3.3  295  79 +  3  0.20 + 0.05  (a)  6  bulk 35 + s u r f a c e 35 +  5 5  + 0.7 4.1 0.16 + 0.05  (b) (b)  45 + 20  0.18 + 0.03  (c)  6  49 +  3  0.46 + 0.03  (b)  295  61 +  3  0.18 + 0.03  (b)  295  Si0  F  (a) J.H. Brewer, H y p e r f i n e I n t e r a c t i o n s 8^, 375  + 0.5  (1981).  (b) R.F. K i e f l , 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 Columbia (c)  G.M.  M a r s h a l l , e t a l . , Phys. L e t t . 65A, 351  (a)  (1982).  (1978).  The measurements on the 70 A mean diameter powder were performed atmosphere.  i n a helium  - 22  -  comes about when a t h e r m a l i z e d L I combines w i t h an e l e c t r o n from  the  +  r a d i a t i o n track that i t i t s e l f  produced  while stopping.  I t has been shown,  f o r p o s i t r o n i u m formed v i a a spur mechanism, t h a t the a p p l i c a t i o n of an electric  field  inhibits  S i m i l a r experiments  the combination  of e  +  w i t h the spur e  -  [37].  c o n c e r n i n g muonium f o r m a t i o n have shown the Mu  p r o b a b i l i t y i n bulk S i 0  to be independent  2  to 60 kV/cm [38], s u g g e s t i n g t h a t Mu governed by a spur mechanism.  of a p p l i e d e l e c t r i c  f o r m a t i o n i n bulk S i 0  However, the analogous  s i l i c a powders have not as yet been  2  formation  f i e l d s of  i s probably  experiments  up not  using  fine  performed.  For the case of e p i t h e r m a l ( o r hot atom) f o r m a t i o n , the L I undergoes a +  s e r i e s of charge-exchange p r o c e s s e s as i t slows Recent r e s u l t s  [39] on the f o r m a t i o n of muonium and  l i q u i d s , where the spur model i s most popular processes p l a y a s i g n i f i c a n t Finally,  microseconds  t h a t \i •*• Mu +  One  of s p u r s . 2  independent  3  show  (T < 10 K ) , s h o r t e n i n g to picoseconds  T h i s p r o c e s s , however, does not seem l i k e l y  near  for s i l i c a  the muonium f r a c t i o n i s observed  to be  i n bulk fused q u a r t z .  l a s t p o i n t can be made by drawing a t t e n t i o n to the f a c t  exists a s t a t i s t i c a l l y  t h a t there  s i g n i f i c a n t d i s c r e p a n c y i n the muonium f o r m a t i o n  p r o b a b i l i t y between f i n e s i l i c a 1.3,  that epithermal  on a thermal b a s i s over times as long as  powders s i n c e , from T a b l e 1.3, temperature  [40], suggest  [41] on muonium f o r m a t i o n i n A 1 0  a t low temperatures  room temperature.  "muonated" r a d i c a l s i n  r o l e , even i n the presence  r e c e n t experiments  c l e a r evidence  down, as d i s c u s s e d e a r l i e r .  the muonium f r a c t i o n s F  powder and  bulk fused S i 0  addition, F  M  2  powders and  bulk fused q u a r t z .  From T a b l e  measured a t 295 K f o r the 35 A r a d i u s S i 0 Mu ^ w  o  are 61 ± 3% and 79 ± 3%,  f o r bulk fused S i 0  2  i s found  respectively.  to be independent  of  In temperature  - 23 whereas f o r the 35 A powder F., decreases Mu may be e x p l a i n e d by the f a c t  to 49 ± 3% a t 6 K.  t h a t i n powders the Mu atoms have the  p o s s i b i l i t y of i n t e r a c t i n g with the g r a i n s u r f a c e s .  There a r e two p o s s i b l e  mechanisms a s s o c i a t e d w i t h the s u r f a c e t h a t might account of  the muonium f r a c t i o n ;  field  causes  c o v a l e n t bonding,  f o r the r e d u c t i o n  which i n zero and l o n g i t u d i n a l  no d e p o l a r i z a t i o n of the \i s p i n but which i n t r a n s v e r s e +  removes muons from at  These r e s u l t s  field  the muonium ensemble, or i o n i z a t i o n of the muonium atom  the s u r f a c e , which has the same e f f e c t . First  silica  c o n s i d e r the p o s s i b i l i t y of c o v a l e n t bonding.  s u r f a c e s a r e covered w i t h h y d r o x y l groups [42,43]  chemically inert  f o r muonium of thermal  energies.  G e n e r a l l y , the and a r e l i k e l y  I t may be e n e r g e t i c a l l y  p o s s i b l e f o r a s t o p p i n g \i to exchange w i t h a h y d r o x y l proton; +  because  this  type of process r e q u i r e s non-thermal e n e r g i e s , however, one would not expect it  to be temperature  dependent, making i t i n c o n s i s t e n t w i t h o b s e r v a t i o n s .  Now c o n s i d e r the p o s s i b i l i t y of i o n i z a t i o n a t the g r a i n s u r f a c e s . Recent p o s i t r o n experiments implanted  [44] show t h a t when e  into ionic crystals  +  of keV e n e r g i e s a r e  they a r e r e e m i t t e d i s o t r o p i c a l l y from the  s o l i d s w i t h a continuum of e n e r g i e s having a maximum approximately the band gap energy  of the s o l i d .  equal to  T h i s phenomenon has f u r t h e r been shown t o  be a s s o c i a t e d w i t h p o s i t r o n i u m d i f f u s i n g  to the s u r f a c e and subsequently  dissociating. In of  1972 i t was p o s t u l a t e d t h a t Ps c o u l d be f i e l d - i o n i z e d  l e a v i n g a s u r f a c e [45] .  the anomalously energy  account f o r  l a r g e e m i s s i o n e n e r g i e s or the c o r r e l a t i o n w i t h the band gap  of the s o l i d .  Auger-emitted  T h i s , however, does not adequately  i n the process  An a l t e r n a t e e x p l a n a t i o n [44] i s t h a t the p o s i t r o n i s  when the Ps e l e c t r o n f a l l s  i n t o an a c c e p t o r s t a t e a t the  - 24 s u r f a c e of the c r y s t a l . It  i s q u i t e p o s s i b l e t h a t the same mechanism(s) governing  e  emission  +  may be i n v o l v e d i n the i n t e r a c t i o n s of muonium w i t h i o n i c s u r f a c e s such as fine s i l i c a  powders.  A d e t a i l e d d i s c u s s i o n of t h i s p a r t i c u l a r phenomenon i s  g i v e n i n Appendix I I , and thus no e l a b o r a t e e x p l a n a t i o n s w i l l be g i v e n h e r e . S u f f i c e i t to say t h a t w i t h the model j u s t d e s c r i b e d the maximum energy of the emitted  corresponding  to the Mu e l e c t r o n recombining  with a hole at  the bottom of the v a l e n c e band, can be w r i t t e n as = (E g  max where E  + AE ) - $ v  v  - R  M u  y  oo  i s the band gap energy,  g  + $  (1.20)  e  -  AE  i s the width of the v a l e n c e band, R V  the b i n d i n g energy  Q  of muonium i n vacuum, §?_ i s the e l e c t r o n a f f i n i t y Mu  bottom of the c o n d u c t i o n band and $ surface.  is  °o  a t the  i s the muonium work f u n c t i o n a t the  I n analogy w i t h p o s i t r o n i u m s t u d i e s , the maximum k i n e t i c  energy  f o r Mu e m i s s i o n i s the n e g a t i v e of i t s work f u n c t i o n , which i s g i v e n by S  = (E^ - R j  MU  U  where E ^  U  2  (*f  +  *J) of muonium a t the s u r f a c e and  i s the L I  +  A n e g a t i v e work f u n c t i o n has been p o s t u l a t e d f o r muonium on  s u r f a c e s [30,33,34].  conservative estimate f o r $ with R  (1.21)  i s the b i n d i n g energy  work f u n c t i o n . Si0  +  For S i 0 , E 2  Mu  = 10.7 eV [46] and one may assume a  o f 0 ± 1 eV.  S u b s t i t u t i n g these v a l u e s , a l o n g e and r a t h e r c o n s e r v a t i v e e s t i m a t e s f o r AE and $ , i n t o E q u a t i o n 1/  GO  1.20, one can conclude  —  t h a t Mu i o n i z a t i o n a t the s u r f a c e of f i n e  powders i s e n e r g e t i c a l l y f e a s i b l e .  silica  However, s i n c e one does not expect  l o n g - l i v e d h o l e s i n the v a l e n c e band, E q u a t i o n 1.20 i s an o v e r e s t i m a t e . Assuming t h i s model to be c o r r e c t , i t can be e a s i l y argued process would indeed be temperature  that  dependent simply because a t lower  this  - 25 temperatures a Mu s u r f a c e , thereby  atom w i l l  -  spend a l a r g e r f r a c t i o n of i t s l i f e  enhancing the p r o b a b i l i t y of e n c o u n t e r i n g  on  the  a hole.  More  e x t e n s i v e measurements of the muonium f o r m a t i o n p r o b a b i l i t y have been made as p a r t of t h i s d i s s e r t a t i o n , f o r S i 0 Si0  2  I.C.4  powders.  2  powders as w e l l as f o r helium  T h i s i s d i s c u s s e d i n Chapter  coated  IV.  Extragranular Muonium Production Two  models concerning  the p r o d u c t i o n of Mu  i n f i n e oxide powders have been put f o r t h ; (TD) model [30]  and  t h e r m a l i z a t i o n (DT)  another  up i n the e x t r a g r a n u l a r The  termed the thermal  which w i l l be r e f e r r e d to as the  model [33,34].  bulk phenomenon, but present  one  i n the e x t r a g r a n u l a r  Both models assume Mu  differing  diffusion  direct  formation  e x p l a n a t i o n s of how  region  the Mu  to be  atoms  end  region.  TD model i s an a d a p t a t i o n of a model o r i g i n a l l y a p p l i e d to  positronium  diffusion  the powder g r a i n s and  [47], which assumes t h a t the Mu then d i f f u s e  atoms t h e r m a l i z e i n  to the s u r f a c e where they may  be e j e c t e d  from the s u r f a c e v i a a n e g a t i v e work f u n c t i o n mechanism.  As mentioned  e a r l i e r , Mu  [26] .  is static  i n bulk fused S i 0  assumes t h a t the s i l i c a and  a  below about 50 K  If  one  g r a i n s are of the same s t r u c t u r e as bulk fused  that the g r a i n i n which the u  then the reduced  2  +  d i f f u s i o n expected  stops remains at the ambient at low  Si0  2  temperature,  temperatures appears to c a s t  doubt on the TD model s i n c e i t would p r e d i c t a temperature dependence i n the probability  f o r the p r o d u c t i o n of e x t r a g r a n u l a r Mu,  e x i s t i n g data.  However, l o c a l h e a t i n g of the g r a i n s due  d e p o s i t e d by the s t o p p i n g d i f f u s i o n and  i n c o n t r a d i c t i o n with  may  p l a y an important  subsequent e j e c t i o n of the Mu  to the energy  r o l e i n the  bulk  atoms from the oxide g r a i n s .  Calculations  [48]  of t h i s e f f e c t estimate  f o r muons s t o p p i n g  i n a 35 A r a d i u s S i 0  2  an energy d e p o s i t i o n of 0.3 powder g r a i n ; assuming a  keV  uniform  temperature d i s t r i b u t i o n w i t h i n the g r a i n , t h i s t r a n s l a t e s i n t o an average temperature i n c r e a s e of ~300 crude and  Bear i n mind that these  thus o n l y i n d i c a t e an order  i s known to d i f f u s e very Thus the muonium has  of magnitude.  r a p i d l y (at least  At  c a l c u l a t i o n s are t h i s temperature,  i n c r y s t a l l i n e quartz  Mu  [26]),  a h i g h p r o b a b i l i t y of a r r i v i n g at the s u r f a c e i n a  s h o r t e r p e r i o d of time. average energy of E^= assist  K.  Furthermore, t h i s temperature corresponds to  (3/2)kT « 0.04  i n the e j e c t i o n of Mu  eV f o r the muonium atom, which  from g r a i n s u r f a c e s .  an  may  I f the muonium work  Mu function $  i s indeed  negative  at the s i l i c a  s u r f a c e , the muonium atoms  w i l l escape the powder g r a i n s w i t h k i n e t i c energy E, +  |$^ |. u  the powder g r a i n s ( e x t r a g r a n u l a r r e g i o n ) , a muonium atom w i l l o u t s i d e s i n c e i t would r e q u i r e only a few condition E^ «  |$ | Mu  to be met.  The  DT model was  Thus the TD model can  l a r g e (<S>  eV)  n e g a t i v e Mu  f r a c t i o n to be  w i t h experiments, but  s u b j e c t o n l y to  the  of  region.  thermalization  T h i s model p r e d i c t s the  temperature independent, i n b e t t e r agreement  i t is difficult  n e g a t i v e work f u n c t i o n .  the  work f u n c t i o n at the s u r f a c e of  the p o s s i b i l i t y of d i r e c t  atoms i n the e x t r a g r a n u l a r  e x t r a g r a n u l a r Mu  explain  T h i s model p o s t u l a t e s the e x i s t e n c e of a  the powder g r a i n s , which p r o v i d e s of the Mu  indeed  o r i g i n a l l y proposed to circumvent the q u e s t i o n  ~ -2  remain  hypothesis.  temperature dependent d i f f u s i o n . Mu relatively  likely  e l a s t i c c o l l i s i o n s f o r the  e x i s t i n g data on e x t r a g r a n u l a r muonium p r o d u c t i o n , v a l i d i t y of the g r a i n h e a t i n g  Once o u t s i d e  to e x p l a i n the o r i g i n of such a l a r g e  - 27 R e c a l l the e x p r e s s i o n f o r the muonium work f u n c t i o n $ Mu E q u a t i o n 1.21.  The n e g a t i v i t y o f  Mu  given i n  i s of course i n f l u e n c e d  by many  f a c t o r s , however there a r e two phenomena which a r e o f p a r t i c u l a r i n t e r e s t . One i n v o l v e s  the d i s t o r t i o n o f the muonium h y p e r f i n e  other i n v o l v e s what i s termed the " p h o t o e l e c t r i c consider  the e f f e c t of the former.  i n t e r a c t i o n and the  size effect".  I f the muonium h y p e r f i n e  d i s t o r t e d by v i r t u e of being on the s i l i c a g r a i n s u r f a c e s , i s o t r o p i c part  of the h y p e r f i n e  the vacuum v a l u e ) ,  would decrease a c c o r d i n g l y  interaction i s  so that the  i n t e r a c t i o n i s reduced ( i . e . , v  the atomic b i n d i n g  L e t us f i r s t  Q 0  l e s s than Mu  energy of muonium on the s u r f a c e  with respect  E^  to the vacuum v a l u e R , thereby 00  Mu enhancing the n e g a t i v i t y of $ photoelectric  size effect.  photoelectric  yield  .  Now  consider  the l a t t e r case o f  Both the e l e c t r o n work f u n c t i o n and  f o r small  (< 50 A r a d i u s )  Ag p a r t i c l e s were s t u d i e d  with  r e s u l t s i n d i c a t i n g an decrease of a few p e r c e n t i n the e l e c t r o n work function  along w i t h a c o r r e s p o n d i n g i n c r e a s e  f a c t o r of 1 0 sizes the  2  over the macroscopic s u r f a c e  i n the p h o t o e l e c t r i c  v a l u e f o r the s m a l l e s t  yield  by a  particle  [ 4 9 ] . Depending upon the o r i g i n of the i n t e r a c t i o n , t h i s decrease i n  e l e c t r o n work f u n c t i o n may a c t to enhance the n e g a t i v i t y of the muonium Mu  work f u n c t i o n $  f o r the same m a t e r i a l .  Thus t h i s e f f e c t may a l s o a s s i s t  in  increasing  the p r o b a b i l i t y of e x t r a g r a n u l a r  no  conclusion  can be drawn a t t h i s time.  I.D  muonium p r o d u c t i o n ; however,  The Interactions of Hydrogen and Deuterium with S i l i c a Although muons a r e c o n s i d e r e d to be heavy e l e c t r o n s  ( o r p o s i t r o n s ) , the  b e h a v i o r of p o s i t i v e muons and muonium i n matter i s more r e m i n i s c e n t of  - 28  -  protons and hydrogen than of p o s i t r o n s and hydrogen and deuterium brief  with s i l i c a  positronium.  The  i n t e r a c t i o n s of  s u r f a c e s has been e x t e n s i v e l y s t u d i e d ; a  s y n o p s i s of what i s p r e s e n t l y known about the behavior of  hydrogen and deuterium  i n bulk s i l i c a  ( f u s e d and  silica  s u r f a c e s w i l l be presented  I.D.I  Hydrogen D i f f u s i o n i n Bulk S i l i c a Results  [50]  s i n g l e c r y s t a l ) and  obtained f o r hydrogen i n s i n g l e c r y s t a l q u a r t z a t low  temperatures,  axes.  L i k e the o b s e r v a t i o n s made f o r muonium i n quartz  i n d i c a t e h y p e r f i n e a n i s o t r o p i e s along three [23-27],  f r e q u e n c i e s f o r hydrogen are assumed to a r i s e from  between three l e v e l s . suggests  T h i s correspondence  (ESR)  involving  principle the  hydrogen  temperatures.  the d i f f u s i o n of protons i n s i n g l e  crystal  q u a r t z have shown t h a t the r e c o v e r y of a gamma p u l s e - i n d u c e d frequency at  306 K f o l l o w s a t ~ ^  50  seconds), i n d i c a t i v e of one-dimensional  data  [52]  dependence over an extended  / 2  c o n s t a n t of about 5 x 1 0 d i f f u s i o n i s suspected temperature  I.D.2  diffusion  - 6  cm /s. 2  of about 0.25  [51].  A n a l y s i s of  eV and a d i f f u s i o n  As mentioned e a r l i e r ,  one-dimensional  f o r muonium i n s i n g l e c r y s t a l q u a r t z a t  this  [23-27].  e f f e c t s of i o n i z i n g r a d i a t i o n (gamma-rays) on the s u r f a c e  p r o p e r t i e s of s i l i c a - g e l have been e x t e n s i v e l y i n v e s t i g a t e d u s i n g  this  optical  Hydrogen and Deuterium on S i l i c a Surfaces The  shift  p e r i o d of time (up to  i n d i c a t e s that protons d i f f u s e p r e f e r e n t i a l l y along the  a x i s ( c - a x i s ) w i t h an a c t i v a t i o n energy  (<  transitions  between muonium and  t h a t they occupy the same s i t e at low  Experiments  on  here.  120 K)  observed  both  ESR  [53,54]. it  S i l i c a - g e l has a r a t h e r d i f f e r e n t  i s comprised  of l a r g e porous  composed of non-porous S i 0 size.  s t r u c t u r e than powdered  silica;  p a r t i c l e s , whereas the powdered m a t e r i a l i s  g r a i n s , which are i n g e n e r a l much s m a l l e r i n  2  In these s t u d i e s , r a d i a t i o n induced d i s s o c i a t i o n of the s u r f a c e  h y d r o x y l (OH) hydrogen  groups was  atoms which can be s t a b i l i z e d on the s i l i c a - g e l  temperatures. hydrogen  observed along w i t h the subsequent  ESR  As  the temperature  observed  raised  surface at  low  from 123 K to 153 K the  s i g n a l i n t e n s i t y decreased, c o r r e s p o n d i n g to a r e d u c t i o n i n the  s t a b l e H atom p o p u l a t i o n . reactive.  was  f o r m a t i o n of  The adsorbed H atoms were a l s o found  to be  highly  In p a r t i c u l a r , chemical r e a c t i o n s w i t h oxygen and e t h y l e n e were i n the temperature  the H-ethylene  range from 123 K to 153 K, which suggests  r e a c t i o n i n v o l v e s the f o r m a t i o n of an e t h y l  that  radical.  Measurements of the. s p i n - l a t t i c e r e l a x a t i o n time and l i n e width i n the presence of oxygen i n d i c a t e t h a t the average and an oxygen molecule i s about 10 A.  The  s e p a r a t i o n between adsorbed  authors suggest  that the  atoms are l o c a t e d i n deep " m i c r o s l i t s " where the oxygen molecules penetrate.  Weak ESR  sidebands, p o s s i b l y due  between u n p a i r e d e l e c t r o n s and  cannot  the h y d r o x y l p r o t o n s , were a l s o observed  and  In p a r t i c u l a r ,  at 200-300 °C f o r 8 hours p r i o r to i r r a d i a t i o n were found  to e x h i b i t a s i g n a l w i t h a g-value i d e n t i c a l  to that of the d i p h e n y l p i c r y l  h y d r a z y l r a d i c a l , and a s t r u c t u r e a p p a r e n t l y due  to the h y p e r f i n e  i n t e r a c t i o n between the odd e l e c t r o n of the r a d i c a l and protons.  T h i s was  OH  by deuterium.  groups  hydrogen  to h y p e r f i n e i n t e r a c t i o n s  found to be dependent upon pretreatment of the s i l i c a - g e l . samples degassed  H  v e r i f i e d by r e p l a c i n g the hydrogen  p r i o r to i r r a d i a t i o n was  The  the h y d r o x y l  atoms of the s u r f a c e  s i g n a l o b t a i n e d f o r samples degassed  at 500  °C  i n d i c a t i v e of an enhancement i n the b u l k f o r m a t i o n  - 30 of  F - c e n t e r s due In  1975,  to the c a p t u r e of e l e c t r o n s by oxygen l a t t i c e v a c a n c i e s .  measurements [55] were made of the h y p e r f i n e i n t e r v a l v  both hydrogen and deuterium temperature.  -  adsorbed  for  on the s u r f a c e of fused q u a r t z a t room  Results indicate reductions i n v  hydrogen and deuterium,  Q 0  respectively.  0 0  of 0.12%  and 0.13%  for  In a d d i t i o n to the r e d u c t i o n i n the  i s o t r o p i c h y p e r f i n e i n t e r a c t i o n , an a n i s o t r o p i c h y p e r f i n e i n t e r a c t i o n ( d i s t o r t i o n ) was by < 0.4% electric  from field  a l s o i n t r o d u c e d , producing h y p e r f i n e s p l i t t i n g s  the vacuum v a l u e s ; t h i s a n i s o t r o p y has been a t t r i b u t e d a t the s u r f a c e .  h y p e r f i n e i n t e r a c t i o n due d e t a i l elsewhere The  differing  The  to an  p e r t u r b a t i o n of the hydrogen atom  to an e l e c t r i c  field  has been d i s c u s s e d i n some  [56,57].  i n t e r a c t i o n s of gas-phase deuterium  been s t u d i e d [58] w i t h r e s u l t s  atoms w i t h s i l i c a  showing evidence  s u r f a c e s have  f o r a chemical r e a c t i o n of D  atoms w i t h w i t h these s u r f a c e s , s i g n a l e d by the f o r m a t i o n of SiO-D bonds. Both Cab-O-Sil  and  porous Vycor  g l a s s (amorphous) s u r f a c e s were s t u d i e d .  For C a b - O - S i l , the f o r m a t i o n of SiO-D groups was corresponding  decrease  i n SiO-H groups,  f a v o r i n g l i b e r a t i o n of the l i g h t e r however, no The  s i g n i f i c a n t decrease  Cab-O-Sil  s u r f a c e used  same s u r f a c e used The  accompanied by a  s u g g e s t i n g an exchange r e a c t i o n  isotope.  In the case of Vycor  i n the SiO-H group p o p u l a t i o n was  i n the ESR  s t u d i e s of deuterium  observed.  on s i l i c a was  the  i n the present work on muonium.  t e x t up to t h i s p o i n t has been a g e n e r a l i n t r o d u c t i o n to muons and  muonium atoms and  t h e i r c h a r a c t e r i s t i c s , and has  p r o v i d e d a review of the  r e s u l t s of p r e v i o u s s t u d i e s i n v o l v i n g muonium and hydrogen i n bulk and on s i l i c a detail,  glass,  surfaces.  The  present work w i l l now  silica  be c o n s i d e r e d i n more  b e g i n n i n g w i t h a d i s c u s s i o n of the e x p e r i m e n t a l  technique.  - 31  CHAPTER I I —  II.A  -  EXPERIMENTAL TECHNIQUE  Accelerators and Beamllnes At the present time, there are three "meson f a c t o r i e s " i n e x i s t e n c e :  (1) Los Alamos Meson P h y s i c s F a c i l i t y  (LAMPF), (2) S c h w e i z e r i s c h e s  fur  Nuklearforschung  (SIN) and  All  three f a c i l i t i e s  c u r r e n t l y support r a t h e r l a r g e s c a l e \xSR r e s e a r c h  programs.  (3) T r i - U n i v e r s i t y Meson F a c i l i t y  Institut  In a d d i t i o n , CERN, Brookhaven, JINR, G a t c h i n a ( L e n i n g r a d ) and  BOOM f a c i l i t y  a t KEK  a l s o support ongoing  \iSR r e s e a r c h .  d e s c r i b e d i n t h i s d i s s e r t a t i o n were conducted channels of the TRIUMF c y c l o t r o n  II.A.1 The  on the M9  experiments  and M20  secondary  facility.  present l a y o u t of the TRIUMF f a c i l i t y  i s shown i n F i g u r e I I . 1 .  [1-6] i s a s e c t o r f o c u s s e d H~  a c c e l e r a t i n g protons to e n e r g i e s ranging from 180 c u r r e n t s of 170 uA a t 520 MeV  [7].  The  b u r s t every 43  to 520 MeV  -  "macroscopic"  ( n o r m a l l y ) of a 5 ns  ns. efficiency  i o n s through a t h i n carbon " s t r i p p e r " f o i l  the two e l e c t r o n s and  e x t r a c t e d p r o t o n energy The  at maximum  p r o t o n beam has a 100%  The p r o t o n beam i s e x t r a c t e d w i t h e s s e n t i a l l y 100% p a s s i n g the H  The  c y c l o t r o n capable of  duty c y c l e and a m i c r o s c o p i c time s t r u c t u r e c o n s i s t i n g  foils.  The  the  The TRIUMF Cyclotron F a c i l i t y  TRIUMF a c c e l e r a t o r  off  (TRIUMF).  thus  by  stripping  e f f e c t i v e l y r e v e r s i n g the charge of the i o n s .  i s s e l e c t e d by the r a d i a l p o s i t i o n . o f  the  stripper  r e s u l t i n g protons then swerve out of the machine through  available extraction ports.  A more d e t a i l e d d i s c u s s i o n of the  The  three  recombination  magnet and beam o p t i c s a s s o c i a t e d w i t h the e x t r a c t i o n system i s g i v e n  REMOTE HANDLING FACILITY  CHEMISTRY ANNEX  PROTON HALL EXTENSION  42 MeV' ISOTOPE PRODUCTION CYCLOTRON  BLIB(P) MESON HALL  ,Ml3(n/u) (Ml1(n)  M9(n/u) M20(g) INTERIM -RADIOISOTOPE LABORATORY NEUTRON /-ACTIVATION I ANALYSIS  SERVICE ^ ANNEX EXTENSION ION SOURCE 3  M15(ij)  X  BATHO BIOMEDICAL LABORATORY  - 33 elsewhere [ 8 ] . During  high i n t e n s i t y  ( u n p o l a r i z e d ) o p e r a t i o n , a p r o t o n beam i s  e x t r a c t e d down Beam L i n e 4 and then  t r a n s m i t t e d a t c u r r e n t s as low as 1 nA  down Beam L i n e 4B (maximum 1 LIA) or 4A (maximum 10 LIA) . channels 520  are u t i l i z e d  MeV.  f o r nucleon  experiments a t e n e r g i e s  I n a d d i t i o n , a 20 to 30 LIA p r o t o n  Both of these between 180 and  beam can be e x t r a c t e d down Beam  L i n e 2C f o r i s o t o p e r e s e a r c h and p r o d u c t i o n .  F i n a l l y , a 130 to 140 LLA, 500  MeV p r o t o n beam i s e x t r a c t e d down Beam L i n e IA and passed through two p i o n production  t a r g e t s (1A-T1 and 1A-T2) and u l t i m a t e l y dumped i n t o a molten  l e a d t a r g e t a t the T.N.F. (Thermal Neutron F a c i l i t y ) . ( p o l a r i z e d ) beam o p e r a t i o n i s u s u a l l y shared Typically,  the p i o n p r o d u c t i o n  Low i n t e n s i t y  between 4B, 4A and IB.  t a r g e t a t 1A-T1 i s a 10 mm  t h i c k (long)  water c o o l e d p y r o l i t i c g r a p h i t e s t r i p and the one a t 1A-T2 i s a 100 mm t h i c k ( l o n g ) water c o o l e d b e r y l l i u m s t r i p . v i a nuclear r e a c t i o n s . along Beam L i n e IA. at 1A-T1: M13  Pions  are produced a t these  S i x secondary channels  Three channels  are c u r r e n t l y o p e r a t i o n a l  simultaneously  e x t r a c t it-mesons or muons  [ 9 ] , M i l and a newly commissioned channel,  p o s i t i v e muons i n the momentum range 21 - 29.8 MeV/c. channel  p r o h i b i t s pion transport.  e x t r a c t it-mesons or muons a t 1A-T2: M8, i s d e d i c a t e d  M20 [14] i s used p r i n c i p a l l y f o r  II.A.2  M15, t h a t e x t r a c t s  The l e n g t h of the M15  Three secondary channels  M9 [10] p r i m a r i l y ( i n recent y e a r s )  targets  simultaneously  to TC cancer  therapy,  to the TPC (Time P r o j e c t i o n Chamber) and  LISR  experiments.  Muon Production and Transport  To begin  t h i s d i s c u s s i o n o f muon p r o d u c t i o n and t r a n s p o r t , we focus our  a t t e n t i o n on the M20 secondary channel  [14] .  The M20 channel  (shown i n  - 34 F i g u r e II.2) i s mainly  a decay muon channel which views the 1A-T2 t a r g e t a t  55° w i t h r e s p e c t to the primary  proton beam d i r e c t i o n .  i n j e c t i o n system i n c o r p o r a t i n g two quadrupole d o u b l e t s separated by a 42.5° bending  momentum of the p a r t i c l e s emitted focuses them a t the s l i t s  ten  quadrupole decay s e c t i o n which i s designed  (SL1).  t h i s system s e l e c t s the  The i n j e c t i o n system I s f o l l o w e d by a  muons produced by pions t h a t decay i n f l i g h t  two  of 12 msr.  from the p i o n p r o d u c t i o n t a r g e t a t 1A-T2  and  doublet  (Q1&2, Q3&4)  magnet ( B l ) , which has an acceptance  In a d d i t i o n to p r o v i d i n g the p a r t i c l e c o l l e c t i o n ,  particles  I t c o n s i s t s of a  to c o l l e c t and t r a n s p o r t  along i t s l e n g t h .  The  t h a t emerge from the decay s e c t i o n are c o l l e c t e d by a quadrupole  (Q7 & Q8) and focused  through  a second bending  magnet (B2) which has  e x i t p o r t s , one (M20-A) a t 75° and the other (M20-B) a t 37.5° to the  secondary  beam d i r e c t i o n b e f o r e the B2 bender.  crossed-field velocity  s e p a r a t o r i s i n c o r p o r a t e d i n t o M20-B and used to  reduce the p o s i t r o n c o n t a m i n a t i o n "spin  A Wien f i l t e r or  o f the beam and may a l s o be used as a  rotator". Muons can be t r a n s p o r t e d through M20 i n any one of three o p e r a t i o n a l  modes; " c o n v e n t i o n a l " , " c l o u d " , or " s u r f a c e / s u b s u r f a c e " . muons ( L I or LI ) a r e produced by pions decaying +  -  i n flight  Conventional along  the l e n g t h  of  the decay s e c t i o n between B l and B2.  is  s p a t i a l l y i s o t r o p i c and the r e s u l t i n g muons have a momentum of 29.8  MeV/c. lab to  I n i t s r e s t frame, the p i o n decay  The decay muons born i n the d i r e c t i o n of the p i o n momentum i n the  frame a r e termed "forward  muons" and those born i n a d i r e c t i o n  the p i o n momentum are c a l l e d  kinematics,  "backward muons".  From  opposite  relativistic  the l a b frame momenta a r e t y p i c a l l y ~140 MeV/c and ~86 MeV/c,  respectively.  The decay s e c t i o n of M20 i s designed  to t r a n s m i t o n l y  those  F i g u r e II.2  The M20  secondary channel showing both l e g s (A and B ) .  muons having (i.e.,  a s m a l l angular d i v e r g e n c e  backward and forward muons).  from the i n i t i a l  T h i s f e a t u r e not o n l y narrows the two  a v a i l a b l e c o n v e n t i o n a l momenta, but a l s o g i v e s r i s e polarization. through  The M20 channel  p i o n momentum  can be tuned  to a h i g h (85%)  to t r a n s p o r t backward muons  e i t h e r M20-A o r M20-B w i t h low p o s i t r o n c o n t a m i n a t i o n .  a "simultaneous"  In addition,  tune i s a v a i l a b l e which s i m u l t a n e o u s l y d e l i v e r s  "low -  contamination"  backward and forward  respectively.  Because c o n v e n t i o n a l muons are produced from an extended  source  (pions decaying  i n flight  beam spot s i z e a t the f i n a l  decay muons to M20-A and M20-B,  along the l e n g t h of the decay s e c t i o n ) the  focus i s g e n e r a l l y r a t h e r l a r g e .  beam parameters [11] f o r M20 o p e r a t i n g i n backward mode and  The measured simultaneous  mode are g i v e n i n T a b l e s I I . 1 ( a ) and I I . 1 ( b ) , r e s p e c t i v e l y . Cloud muons (p. or u~) a r e produced by pions decaying +  the p r o d u c t i o n t a r g e t a t 1A-T2 and B l . forward muons a r e p r e s e n t .  i n flight  between  In t h i s mode, both backward and  However, because the i n j e c t i o n system does not  d i s c r i m i n a t e on the angular d i v e r g e n c e  of decay muons as much as does the  decay s e c t i o n between B l and B2, the beam p o l a r i z a t i o n i s r e l a t i v e l y low (50-60%).  At p r e s e n t ,  there a r e no c a l c u l a t e d or measured beam parameter  v a l u e s f o r c l o u d muons on the r e c e n t l y r e b u i l t v e r s i o n o f M20; but on M9 the beam p o l a r i z a t i o n f o r c l o u d muons a t 77 MeV/c i s ~30%. S u r f a c e muons [12] ( o n l y u ) a r e produced from T C t h a t decay a t r e s t on +  the s u r f a c e of the p i o n p r o d u c t i o n t a r g e t .  +  Muons produced i n t h i s manner  have s e v e r a l advantages over c l o u d or c o n v e n t i o n a l muons.  Unlike cloud  muons, f o r example, s u r f a c e muons i n c l u d e o n l y the f o r w a r d l y - d e c a y i n g component.  T h i s f e a t u r e , coupled w i t h the acceptance  of the i n j e c t i o n  system and the k i n e m a t i c s o f T t decay, g i v e s r i s e to two important +  - 37 Table II.1(a)  Backward Decay Muons a t 75° (M20-A) and 37.5° (M20-B)  Beam Parameters  M20-A  M20--B  Total Flux  2.5 x 1 0 / s e c 5.9 x 1 0 / s e c 6  5  Central Luminosity  H H~ +  2.9 x 1 0 V s e c / c m  C e n t r a l Momentum (P)  86.4 MeV/c  Momentum Spread (AP/P)  9.6%  E l e c t r o n Contamination  0.3% 1.3%  n Polarization  85%  Beam Spot (fwhm)  7.2 cm 9.5 cm  X  y Divergence  63 mr 70 mr  X  y  Table I I . 1 ( b )  2  Simultaneous Decay Muons on M20 M20--B  Beam Parameters  M20-A  Total Flux  1.6 x 1 0 / s e c 3.6 x 1 0 / s e c  1.6 X 1 0 / s e c 4.6 X 1 0 / s e c  6  6  5  Central Luminosity  n  +  1.6 x  lO'Vsec/cm  C e n t r a l Momentum (P)  85.5 MeV/c  Momentum Spread (AP/P)  6.7%  5  2  3.0 X IO */ sec/cm 1  A l l r a t e s a r e f o r a 100 Liamp p r o t o n beam i n c i d e n t on a 10 cm Be t a r g e t  2  - 38 c h a r a c t e r i s t i c s of s u r f a c e u :  (1) s u r f a c e u  +  i n a d i r e c t i o n opposite  a r e a t l e a s t 99.9% p o l a r i z e d  +  to the beam momentum and (2) the s u r f a c e n  beam  +  momentum d i s t r i b u t i o n has a sharp "edge" a t a momentum of 29.8 MeV/c, which t r a n s l a t e s i n t o a k i n e t i c energy of 4.1 MeV. d e f i n e d energy, s u r f a c e \i r a t h e r s m a l l range  Because of t h e i r low and w e l l  have a h i g h (~140 mg/cm ) s t o p p i n g  +  spread.  Another advantage of s u r f a c e \x a r i s e s because the u +  directly  from the p i o n p r o d u c t i o n  s m a l l source source. final  f o r surface u  +  thereby  target.  producing  stopping  d e n s i t y as w e l l . +  Owing to  of the s u r f a c e muon beam, s u r f a c e The s m a l l beam spot and h i g h  make i t p o s s i b l e to stop muons i n s m a l l  In p a r t i c u l a r ,  the work d e s c r i b e d i n t h i s  d i s s e r t a t i o n i n v o l v e s the use of low d e n s i t y S i 0 not have been c a r r i e d out without  2  powder t a r g e t s and c o u l d  a high i n t e n s i t y s u r f a c e u  measured beam parameters f o r M20 o p e r a t i n g Table  t a r g e t i s imaged a t the  a s m a l l (~2 cm diameter) beam s p o t .  d e n s i t y of s u r f a c e | i  and/or low d e n s i t y t a r g e t s .  originate  This feature provides a rather  the p r o d u c t i o n  the low energy and monochromatic nature muons have a h i g h s t o p p i n g  +  i n comparison to the extended c o n v e n t i o n a l muon  I n the case of s u r f a c e  focus  d e n s i t y and a  2  beam.  +  The  i n s u r f a c e mode are g i v e n i n  II.2. By  t u n i n g the channel to lower momenta, i t i s p o s s i b l e to c o l l e c t and  transport target. muons.  which are produced by T C decaying  i n s i d e the p i o n  +  These muons a r e , f o r l a c k of a b e t t e r term, c a l l e d Consider  the c r o s s s e c t i o n i n the p i o n s t o p p i n g  a width Ay and l o c a t e d i n s i d e the p r o d u c t i o n emitting surface.  The decay u~*" emitted  production  subsurface  distribution  having  t a r g e t a d i s t a n c e y from the  from TC stopped i n t h i s r e g i o n , must  t r a v e r s e the d i s t a n c e y through the p r o d u c t i o n  +  target before  r e a c h i n g the  T a b l e II.2  S u r f a c e Muons a t 75° (M20-A) and 37.5° (M20-B)  Beam Parameters  M20-A  Total Flux  2.7 x 1 0 / s e c  Central Luminosity  1.6 x 10^/sec/cm  C e n t r a l Momentum (P)  29.4  Momentum Spread (AP/P)  6.4%  7.1%  E l e c t r o n Contamination ( e / u )  40/1  40/1  Polarization  100%  100%  Divergence  1.5 x 1 0 / s e c  6  +  Beam Spot (fwhm)  M20-B  +  x  6  2  MeV/c  y  4.5 cm 4.3 cm  x  ———  y  All  r a t e s are f o r a 100 uamp p r o t o n beam i n c i d e n t on a 10 cm Be t a r g e t  - 40 emitting being  Thus the L I a r e degraded by the p r o d u c t i o n  +  target  +  c o l l e c t e d and t r a n s p o r t e d down the beamline.  decay u  R  surface.  The range R  Q  before  of the  i s approximated by [13]  7/2 = k P ' o o o  where P  Q  i s the i n i t i a l \i  a constant target).  (II.1) +  momentum ( i n t h i s case, 29.8 MeV/c) and k  that depends upon the s t o p p i n g medium ( i . e . ,  The  utility  Q  multiplied  of subsurface  P < 29.8 MeV/c.  muons i s e a s i l y understood  S i m i l a r to L I i n the p r o d u c t i o n +  t r a n s p o r t e d beam i s g i v e n approximately  LI rate i s +  by the a p p r o p r i a t e decay  the s t o p p i n g d i s t r i b u t i o n of the t r a n s p o r t e d s u b s u r f a c e  is  the p r o d u c t i o n  I t can then be e a s i l y argued that the subsurface  p r o p o r t i o n a l to the range R  Q  u  factor.  by now c o n s i d e r i n g +  beam of momentum  t a r g e t , the range R o f the  by [13]  7/2 R = kP '  (II.2)  where the constant k depends on the sample i n which the beam i s stopped, (i.e.,  k » 140mg/cm  2  (29.8 M e V / c )  - 7 / 2  , f o r surface u ) . +  For a given  spread  i n momentum and t a k i n g range s t r a g g l i n g i n t o c o n s i d e r a t i o n ( t y p i c a l l y ~10% of  the range [ 1 4 ] ) , an estimate  g i v e n by AR « [ ( 0 . 1 ) = k[(0.1)  2  + (3.5 A P / P ) ]  1 / 2  2  + (3.5 A P / P ) ]  1 / 2  2  2  From t h i s i t i s obvious limited  of the t o t a l  y i e l d s a dramatic  decrease  i n AR.  spread AR i s then  R P  7 / 2  that d e c r e a s i n g  r e d u c t i o n i n the s t o p p i n g  stopping  the momentum b i t e r e s u l t s i n o n l y a  spread AR, w h i l e d e c r e a s i n g  the momentum  Thus the use of s u b s u r f a c e muons makes i t  - 41 p o s s i b l e to tune f o r a d e s i r e d s t o p p i n g  range spread  AR.  An attempt i s now being made to produce a low energy (0 to ~10 keV) j i beam by u t i l i z i n g  the knowledge gained  i n low energy p o s i t r o n p r o d u c t i o n  r e s e a r c h and drawing the a p p r o p r i a t e a n a l o g i e s . show t h a t when e reemitted  +  of keV e n e r g i e s a r e implanted  isotropically  a maximum approximately regarding  Recent measurements [15] i n t o L i F and NaF, they a r e  from the s o l i d s w i t h a continuum of e n e r g i e s equal  having  to the a l k a l i h a l i d e band gap. D e t a i l s  the p h y s i c s i n v o l v e d and a d e s c r i p t i o n o f the p r o t o t y p e  apparatus  to t e s t f o r the analogous a"" phenomenon a r e g i v e n i n Appendix I I . 1  II.B  The iiSR / MSR Technique The  experiments r e p o r t e d i n t h i s d i s s e r t a t i o n were performed u s i n g t h e  conventional  t i m e - d i f f e r e n t i a l |iSR technique  measurement one r e c o r d s  the a r r i v a l  time t  [16-18] . o f the u  +  decay a t time t , and then c o n s t r u c t s a time histogram e  d e f i n e d by  At = ( t  g  - t^).  T h i s technique  unambiguously a s s o c i a t e a g i v e n e Normally, t h i s requirement to be present  +  I n t h i s type of and i t s subsequent f o r the i n t e r v a l s  r e q u i r e s the a b i l i t y to  w i t h the muon from which i t was e m i t t e d .  i s s a t i s f i e d by a l l o w i n g only one muon a t a time  i n the sample.  I n g e n e r a l , the t i m e - d i f f e r e n t i a l uSR spectrum observed  i n a direction  d e f i n e d by the u n i t v e c t o r n w i t h r e s p e c t to the beam momentum can be expressed as -tlx N(t) = N e o where N  Q  ^ f l + A P L  Mu  i s a normalization constant, x  and Aj^ a r e the i n i t i a l u  u u  ( t ) ' n + A,, p\, ( t ) . n l + B Mu u  (II.4)  J  i s the mean muon l i f e t i m e , A^  i n s t r u m e n t a l asymmetries f o r the \x and muonium +  +  - 42 s i g n a l s , r e s p e c t i v e l y , P^(t) represents u  s i g n a l and  +  M u  (t)  *  s  t  n  e  corresponding  q u a n t i t y f o r the muonium s t a t e and  B i s a time independent background.  the constant  II.B.l  p  the muon s p i n p o l a r i z a t i o n f o r the  Zero and L o n g i t u d i n a l F i e l d (ZF and L F )  In zero f i e l d ,  one a c q u i r e s  information regarding  of the | i s p i n p o l a r i z a t i o n by o b s e r v i n g  the time dependence  and comparing the LISR s p e c t r a a t  +  angles 9=0° and 9=180° w i t h r e s p e c t to the i n i t i a l muon s p i n d i r e c t i o n . schematic r e p r e s e n t a t i o n of t h i s i s g i v e n i n F i g u r e 11.3(a).  A  I n these  d i r e c t i o n s , the observed uSR s p e c t r a a r e  9 -  0°; N°(t)  = N°  e  _  t  /  ^ [ l + A°  O  zz  Ll  L  (t) + A° Mu  G  M u  ( t ) ] + B°  ZZ  J  (II.5) 9 = 180°; N ° ( t ) = N 1 8  1  8  0  e  _  t  /  ^  [ l- A  O  1 8  Ll  V  B  (t) - A ^ G ^ t ) ] + zz Mu Z Z  1 8 0  J  where the time e v o l u t i o n of the muon s p i n f o r the \x and Mu s i g n a l s a r e +  Mu  Li  represented  by the two zero  field  relaxation functions, G j ^ t )  a n <  *  G  ^ ^» t  z z  respectively.  II.B.2  Transverse  F i e l d (TF)  I n weak t r a n s v e r s e f i e l d s  (B^ «  B ) , the e v o l u t i o n of the muon Q  s p i n p o l a r i z a t i o n i n muonium can be t r e a t e d u s i n g p e r t u r b a t i o n theory [ 1 6 ] . I n t h i s approximation, opposite  muonium |a(t)> = |1> precesses  to t h a t of a f r e e L I i n the same f i e l d , w i t h a Larmor  u> = - 103OJ^, w h i l e Mu  triplet +  i n a sense frequency  f o r Mu i n the mixed s t a t e |b(t)> = s|2> + c|4> the u  s p i n p o l a r i z a t i o n o s c i l l a t e s a t a high frequency  +  which i s on the order of  - 43 F i g u r e 11.3(a)  F i g u r e I I . 3 Schematic r e p r e s e n t a t i o n s of the |iSR techniques; F i g u r e 11.3(a) shows the zero and l o n g i t u d i n a l f i e l d c o n f i g u r a t i o n and F i g u r e 11.3(b) shows the c o r r e s p o n d i n g diagram f o r t r a n s v e r s e magnetic f i e l d . F i g u r e s taken from Y . J . Uemura, Ph.D. T h e s i s , U n i v e r s i t y of Tokyo, 1981.  - 44 the h y p e r f i n e frequency (DQ'Q ;  -  However, s i n c e the experimental  timing  r e s o l u t i o n i s t y p i c a l l y about 2 ns, the h y p e r f i n e o s c i l l a t i o n i s g e n e r a l l y not o b s e r v a b l e , making t h i s h a l f of the muonium ensemble appear to be completely d e p o l a r i z e d .  The  transverse f i e l d  r e p r e s e n t e d i n F i g u r e 11.3(b). t r a n s i t i o n frequencies v  a n c 1 2  In v e r y weak f i e l d s  ^ 23 v  p o s i t r o n s p e c t r a take the simple N (t) = N  +  where  A  e  * [l + A  Mu xx^>  and  G  c&jj^  C  0  S  a  r  e  PP  a  r  o  x  i  m  a  t  +  s i g n a l and G^(t:)  II.C  4j]  +  and  ) (  I  I  >  6  )  the  r e l a x a t i o n f u n c t i o n f o r the  f u n c t i o n f o r muonium.  and Data A c q u i s i t i o n  the course of these experiments,  d e s c r i b e the d i f f e r e n t  both the experimental  apparatus  S i n c e i t i s not p o s s i b l e to  stages of development here, o n l y the present s t a t e of  affairs  i s d i s c u s s e d i n any  II.C.l  The |iSR  detail.  Spectrometer  ("Eagle") spectrometer,  shown i n F i g u r e I I . 4 , was  take f u l l advantage of the p r o p e r t i e s of s u r f a c e muons. important  thus the |iSR  t e l e s c o p e w i t h r e s p e c t to the i n i t i a l muon  data a c q u i s i t i o n system have e v o l v e d .  The uSR  equal and  B  i s the c o r r e s p o n d i n g  E x p e r i m e n t a l Apparatus During  ly  the  form  s p i n d i r e c t i o n , G^ ( t ) i s the t r a n s v e r s e f i e l d xx +  e  (B^ < 10 G),  are the muon and muonium phase a n g l e s , d e f i n e d by  o r i e n t a t i o n of the p a r t i c u l a r  \x  t  G^ ( t ) cosfw t - $ xx u  u  K u  geometry i s s c h e m a t i c a l l y  One  d e s i g n c o n s t r a i n t s a r i s e s because of the r e l a t i v e l y  designed  to  of the most s h o r t range  - 45 -  Figure II.4  The  "Eagle"  LISR  spectrometer.  - 46 (~140  mg/cm ) and l a r g e m u l t i p l e s c a t t e r i n g of s u r f a c e \i , +  2  the requirement end,  of m i n i m i z i n g the amount of mass i n the beam path.  the spectrometer i s evacuated  counter a r r a y and isolated  which d i c t a t e s  sample s i t u a t e d  To  this  to a p r e s s u r e of ~5 microns w i t h both the inside.  The  spectrometer vacuum i s  from the beam l i n e and c y c l o t r o n vacuum by a 76 Lim (0.003") mylar  window through which the \i  +  e n t e r s the  spectrometer.  A f t e r e n t e r i n g the spectrometer, the p o s i t i v e muons pass through a v a r i a b l e c o l l i m a t o r and are d e t e c t e d by a 0.305 mm scintillator  ("D"  counter) b e f o r e f i n a l l y  Four p o s i t r o n t e l e s c o p e s , each comprised scintillators assembly,  (0.012") t h i c k  s t o p p i n g i n the t a r g e t of two 6.35  mm  (0.25") p l a s t i c  (B1-B2, F1-F2, R1-R2, L1-L2), are arranged around  p e r p e n d i c u l a r (B and F t e l e s c o p e s ) and p a r a l l e l  t e l e s c o p e s ) to the beam d i r e c t i o n .  The  assembly.  the  target  (R and L  B t e l e s c o p e (up stream) i s p r o v i d e d  w i t h a 5 cm hole i n i t s c e n t e r to pass the \i  +  beam.  The F t e l e s c o p e (down  stream) i s a l s o p r o v i d e d w i t h a 5 cm h o l e , p r i m a r i l y i n t e n d e d to pass beam p o s i t r o n s thereby r e d u c i n g p o s s i b l e backgrounds due  to beam c o n t a m i n a t i o n . 2  The  p o s i t r o n t e l e s c o p e a r r a y subtends a t o t a l  steradians.  The  two  solid  angle of about -j(4n)  counters c o m p r i s i n g each of the f o u r  t e l e s c o p e s are separated by a 2.5  cm  positron  t h i c k g r a p h i t e moderator.  T h i s has  e f f e c t of i n c r e a s i n g the p o s i t r o n asymmetry by c u t t i n g o f f the low end of the  o f the M i c h e l spectrum  energy  [16] and h e l p s prevent s c a t t e r e d beam  (29.8  MeV/c) p o s i t r o n s from f i r i n g one of the p o s i t r o n t e l e s c o p e s .  light  produced  The  i n the counters ( s c i n t i l l a t o r s ) i s t r a n s m i t t e d through  bottom of the vacuum chamber v i a UVT and a m p l i f i e d  by RCA  the  L u c i t e l i g h t guides and  the  then d e t e c t e d  8575 p h o t o m u l t i p l i e r s .  The v a r i a b l e c o l l i m a t o r (2.5 cm t h i c k b r a s s ) , immediately  upstream  of  - 47 the D-counter, 8.0,  p r o v i d e s f o u r e a s i l y s e l e c t a b l e c o l l i m a t o r diameters (5.2,  10.8 and 18.0 mm) and serves to d e f i n e the incoming \i  +  beam. The  c o l l i m a t o r i s p o s i t i o n e d between the B l and B2 counters such that  decay  p o s i t r o n s from L I stopped i n the c o l l i m a t o r have o n l y a s m a l l p r o b a b i l i t y of +  f i r i n g both counters of the B - t e l e s c o p e , which would r e s u l t The  i n a bad event.  present spectrometer has four p a i r s of c o i l s which p r o v i d e  magnetic  fields  Helmholtz  coils,  i n three orthogonal d i r e c t i o n s .  A p a i r of water-cooled  having a mean diameter of 56 cm and a B/I f a c t o r of 4.63  G/A, p r o v i d e s a magnetic  field  i n the " t r a n s v e r s e - v e r t i c a l " d i r e c t i o n ( i . e ,  t r a n s v e r s e to the i n c i d e n t muon momentum and v e r t i c a l i n the l a b frame). I n principle,  these c o i l s  can produce  6.5 kG, however because  of the s m a l l  t u r n i n g r a d i u s of s u r f a c e muons i n a magnetic  field  spectrometer i t s e l f ,  to f i e l d s below about  the apparatus i s l i m i t e d  A p a i r of a i r - c o o l e d c o i l s , field  not i n Helmholtz  i n the " l o n g i t u d i n a l " d i r e c t i o n ( i . e . ,  momentum) from 0-12 G.  When connected  f a c t o r of about  With  1 G/A.  and the geometry of the 500 G.  configuration, provide a  along the i n c i d e n t muon  i n series,  the c o i l s have a B/I  t h i s arrangement, one can study the  l o n g i t u d i n a l p o l a r i z a t i o n of the L I s p i n as a f u n c t i o n of time by o b s e r v i n g +  the decay The  spectrum  i n the F and B t e l e s c o p e s .  remaining two p a i r s of c o i l s a r e a i r - c o o l e d and p r o v i d e s m a l l  (~1 G) bucking f i e l d s i n the " t r a n s v e r s e - h o r i z o n t a l " ( i . e . ,  transverse to  the i n c i d e n t muon momentum and h o r i z o n t a l i n the l a b frame) and l o n g i t u d i n a l directions.  These bucking c o i l s a r e used  to a c h i e v e zero f i e l d  plus or  minus ~100 mG; however, they do not a u t o m a t i c a l l y compensate f o r time dependent d r i f t s which may i n t r o d u c e s m a l l f i e l d several  hours.  f l u c t u a t i o n s (~0.2 G) over  The  l i m i t a t i o n s of the E a g l e spectrometer,  namely the i n a b i l i t y  r e s e a r c h i n a h i g h (> 500 G) f i e l d  or a s t a b l e zero f i e l d  be overcome w i t h the commissioning  of a new apparatus  c u r r e n t l y under c o n s t r u c t i o n . c o o l e d Helmholtz directions; The  coils  T h i s apparatus  environment,  has three p a i r s of water-  t h a t produce magnetic f i e l d s i n three o r t h o g o n a l  longitudinal,  t r a n s v e r s e - v e r t i c a l and t r a n s v e r s e - h o r i z o n t a l .  the other two can produce maximum f i e l d s of o n l y ~ 100 G. o r i e n t a t i o n o f the h i g h f i e l d +  f i e l d Helmholtz  (< 6 kG) w h i l e The l o n g i t u d i n a l  c o i l s makes p o s s i b l e r e s e a r c h i n h i g h  enter the spectrometer  t r a j e c t o r i e s , except  along the f i e l d  l i n e s and thus  f o r f o c u s i n g e f f e c t s , remain u n a f f e c t e d .  p a i r s can of course be operated  independently  fields, their  The two low to produce  f i e l d s of up t o ~ 100 G i n the two t r a n s v e r s e d i r e c t i o n s ; however, they be u t i l i z e d  p r i m a r i l y as bucking  t h r e e Helmholtz probes  A feedback  will  system c o u p l i n g a l l  to produce a s t a b l e , time independent  environment w i t h a s t a b i l i t y Hall  coils.  p a i r s a l o n g w i t h two s t r a t e g i c a l l y p l a c e d 3-dimensional  w i l l be used  will  dubbed "Omni" which i s  l o n g i t u d i n a l c o i l s a r e capable of producing h i g h f i e l d s  s i n c e the | i  to do  zero  Hall  field  of ~10 * G, l i m i t e d by the s e n s i t i v i t y of the -1  probes.  II.C.2  E l e c t r o n i c s and Logic  The  t i m e - d i f f e r e n t i a l data a c q u i s i t i o n e l e c t r o n i c s has e v o l v e d d u r i n g  the course of the present study, p r i m a r i l y due to the I n t r o d u c t i o n of the LeCroy  4204 TDC.  From a uSR p o i n t of view, the 4204 possesses s e v e r a l  attractive features. memory and a nominal  Two of the more important  a t t r i b u t e s are i t s buffer  time r e s o l u t i o n of 156.25 ps.  With the i n c o r p o r a t i o n  of a b u f f e r memory, the 4204 TDC e f f e c t i v e l y combines a l l the f u n c t i o n s  - 49 performed by the TRIUMF B080 1 GHz TDC and EG&G C212 p a t t e r n u n i t , as employed i n the p r e v i o u s event  p r o c e s s i n g dead The  system [ 1 9 ] .  T h i s f e a t u r e g r e a t l y reduces the  time.  s i g n a l s from the p h o t o m u l t i p l i e r s a r e t r a n s m i t t e d along ~30 m of  c o a x i a l c a b l e b e f o r e being d i s c r i m i n a t e d and routed shown i n F i g u r e I I . 5 . the D-counter thereby  through the NIM  A | i e n t e r i n g the spectrometer +  generating  a p u l s e a t a time t  first  logic  passes through  which both  starts  r*  the TDC and a l s o t r i g g e r s a p i l e u p gate the subsequent decay e  +  event.  a p o s i t r o n p r e f e r e n t i a l l y along detected  that d e f i n e s the time window T f o r  At a l a t e r  time t  +  decays, e m i t t i n g  I f the decay e  is  +  by one o f the f o u r p o s i t r o n t e l e s c o p e s , w i t h i n the p r e s e l e c t e d time which stops the TDC.  If a  i s not d e t e c t e d w i t h i n the time T, the TDC i s a u t o m a t i c a l l y r e s e t .  Constant signals.  f r a c t i o n d i s c r i m i n a t o r s (CFD) were used on the c r i t i c a l  The d i s c r i m i n a t e d p u l s e s from the counters  p o s i t r o n t e l e s c o p e s are routed two-fold  +  i t s spin direction.  window T ( t y p i c a l l y 10 [is), a p u l s e i s generated decay e  the \i  e  i n t o f o u r separate  comprising  the f o u r  coincidence u n i t s .  c o i n c i d e n c e s (e+E^ = E.^l'E.^2, e t c . ) ensure that accepted  correspond  to decay p o s i t r o n s that pass through both  timing  The  events  t e l e s c o p e counters and  the carbon degrader s e p a r a t i n g them.  The t h i r d c o i n c i d e n c e , shown i n F i g u r e  II.5,  study.  was not employed i n the present  coincidence u n i t s are l o g i c a l l y the TDC s t o p .  "OR-ed", w i t h  Simultaneous w i t h s t o p p i n g  four t e l e s c o p e s a r e a l s o routed converter u n i t s ,  thereby  The outputs  o f the f o u r  the r e s u l t i n g  p u l s e s e r v i n g as  the TDC, p u l s e s from each of the  to s e t i d e n t i f i c a t i o n b i t s i n NIM-ECL  i d e n t i f y i n g which e  +  t e l e s c o p e was f i r e d .  are w r i t t e n Into the PDP-11 memory v i a a CAMAC computer-logic  The data  interface  which i s s e r v i c e d by a Bi-Ra Microprogrammed Branch D r i v e r (MBD-11).  The  - 50 -  TARGET  !x"i •  ! •  ; J  [D]  IT"!  !B!  \ £ F D V \ CFD / I \ / \ / '  IDISO  D©6  V  —  I •  bisd  ^STOP MUST C ARRIVE BEFORE ITS B U S Y T  I  START  FOR ADDITIONAL POSITRON COUNTER CIRCUITS  )  F i g u r e I I . 5 Data a c q u i s i t i o n e l e c t r o n i c s . p o s i t r o n t e l e s c o p e i s shown.  Only the c i r c u i t  f o r one  - 51 MBD  reads the memory b u f f e r of the 4204 TDC,  the t e l e s c o p e that  generated  the event, and  increment  the histogram b i n c o r r e s p o n d i n g to the measured time  (t„ - t..). failure, The is  then performs  identifies  the n e c e s s a r y f u n c t i o n s r e q u i r e d  To guard a g a i n s t p o s s i b l e l o s s of data due  the data are p e r i o d i c a l l y updated of having o n l y one \x  requirement  +  on an RL02 d i s k . p r e s e n t i n the sample a t a time  D-counter  and  lifetimes).  latched  into  EG&G O r t e c ) .  p u l s e i n the  +  f o r a p r e s e t time T ( n o r m a l l y set to 4-8  muon  I f a p o s i t r o n i s d e t e c t e d d u r i n g the data gate p e r i o d T,  "good" event i s logged and however "bad"  100/N  on an i n c i d e n t u  p i l e u p gate ( d a t a gate) i s t r i g g e r e d  interval  to computer  f u l f i l l e d w i t h the use of the p i l e u p gate (model: GP  The  the a p p r o p r i a t e h i s t o g r a m updated.  events t h a t , i f l e f t  the s p e c t r a .  The  unsuppressed,  There  [i  +  a are  would i n t r o d u c e d i s t o r t i o n s  two most important p r o c e s s e s that produce  events are e a r l y second  events ( u - u - e ) , where the second  u  +  "bad"  arrives  d u r i n g the p e r i o d T but b e f o r e t , and  l a t e second u  where the second p. a r r i v e s a f t e r  I n e a r l i e r v e r s i o n s of the data  e  +  aquisition electronics  [19]  t . e  events  +  (|j,-e-^),  the u - u - e events were r e j e c t e d i n l o g i c  by  v e t o i n g m u l t i p l e c l o c k s t o p s , thereby c a u s i n g the c l o c k to time out. u-e-u  events on the other hand, which must be r e j e c t e d a f t e r  stopped, were r e j e c t e d  u n i t when the second \x  then read and  c l e a r e d by the MBD.  4204 TDC,  the u-u-e  earlier version.  detected.  the new  rejected  The C212  system  has  now  and TDC  incorporating  were the  i n the same manner as i n the  u-e-p. events, however, are r e j e c t e d  which r e j e c t s m u l t i p l e h i t e v e n t s . into  was  With  events are s t i l l  The  +  The  the c l o c k  i n software by s e t t i n g a fake p a t t e r n i n the  o b s o l e t e C212  to  i n the TDC  itself  Thus the i n c o r p o r a t i o n of the 4204  TDC  the data a q u i s i t i o n e l e c t r o n i c s has made i t p o s s i b l e to process a l l  - 52 "bad"  -  event r e j e c t i o n i n hardware thereby g r e a t l y r e d u c i n g the event  p r o c e s s i n g dead The  time.  r e j e c t i o n of the Li-Li-e events p l a c e s a r e s t r i c t i o n on the maximum  r a t e at which one  can take data u s i n g the t i m e - d i f f e r e n t i a l  technique.  S i n c e the time s t r u c t u r e of the TRIUMF c y c l o t r o n has a p e r i o d which i s much s m a l l e r than the muon l i f e t i m e , one  of 43  can assume t h a t  ns,  the  i n c i d e n t muons a r r i v e w i t h a time d i s t r i b u t i o n c l o s e l y d e s c r i b e d by P o i s s o n statistics.  From t h i s assumption,  the "good" event r a t e R  and d e n o t i n g the i n c i d e n t r a t e | i  i s g i v e n by  by RQ  +  [19]  O  R  = R  g  For R  Q  o  exp(-2R T ) ^ o ;  = (2T)  - 1  v  , the "good" event r a t e R  g  i s maximized.  With a  (II.7) '  typical  gate width of 8 LIS, the maximum "good" event r a t e occurs f o r a L I stop r a t e +  R  Q  of about  e /s +  62 kHz.  T h i s t r a n s l a t e s i n t o a p o s i t r o n event r a t e of 2k-3k  per p o s i t r o n t e l e s c o p e .  The  l o g i c l e v e l diagram  f o r a "good" event i s  shown i n F i g u r e I I . 6 , and a more d e t a i l e d d i s c u s s i o n of bad e f f e c t on LISR s p e c t r a can be found elsewhere  II.C.3  events and  their  [19].  Targets  The  SiO^ powder used  high s p e c i f i c  i n these experiments  s u r f a c e a r e a (390 ± 40 m /g 2  was  chosen  because  [20]) and h i g h y i e l d  e x t r a g r a n u l a r muonium p r e v i o u s l y observed at 300 K  of i t s  (61 ± 3%)  [21] and at 6 K  of  [22,23].  Some of the p h y s i c a l c h a r a c t e r i s t i c s of t h i s powder are g i v e n i n T a b l e I I . 3 . The  s u r f a c e s of these powders n o r m a l l y have ~4.5  groups  per nm , 2  c o r r e s p o n d i n g to about  chemisorbed  hydroxyl  (OH)  h a l f of the s u r f a c e S i atoms being  a s s o c i a t e d w i t h a s u r f a c e h y d r o x y l [20] .  When evacuated  a t room  temperature  -  53 -  Good Event B —|,  T-  P e Gate  1  Routing Bits Reject  [J  Dead Time Computer Busy /i.Stop  Gated e  y  ^j-  F i g u r e I I . 6 L o g i c l e v e l diagram f o r a "good" event.  T a b l e I I . 3 P h y s i c a l C h a r a c t e r i s t i c s of the SiO~ Powder Property  Value  Supplier  Cabot C o r p o r a t i o n , 125 High S t r e e t , Boston, MA., 02110 (U.S.A.)  S e r i a l Number  EH-5  Density  0.033 gram/cm  (unpacked)  3  S p e c i f i c Surface Area  390 +/- 40 m /gram  Mean G r a i n S i z e  35 Angstrom  Major  Na P Other  Impurities  Hydroxyl  Concentration  2  (mean r a d i u s )  20-40 ppm < 300 ppm < 30 ppm  average 3.5-4.5 groups per nm maximum ( c a l c . ) 7.8 groups per nm  The above v a l u e s a r e taken from r e f e r e n c e [ 2 0 ] .  2  2  - 55 (in  a vacuum equal to 1 0  - 2  T o r r ) or heated  undergo " r e v e r s i b l e d e h y d r a t i o n " .  above 110°C, the powder s u r f a c e s  In t h i s p r o c e s s , the s u r f a c e h y d r o x y l s  combine to form H 0 which, when r e l e a s e d , l e a v e s behind 2  groups ( S i - O - S i ) . powder begins  additional siloxane  Above about 800 °C t h i s h y d r o l y s i s i s completed  to s i n t e r .  and the  The term r e v e r s i b l e d e h y d r a t i o n means t h a t the  powder s u r f a c e s can be r e s t o r e d to t h e i r o r i g i n a l s t a t e by e i t h e r to  a i r or immersion i n water; w i t h the t a r g e t geometry used  study, t h i s r e s t o r a t i o n process ambient atmospheric  moisture.  exposure  i n the present  takes about 24 hours i n a i r , s u b j e c t to the I t i s t h e r e f o r e p o s s i b l e to vary the s u r f a c e  d e n s i t y of h y d r o x y l s , and indeed  study the r e a c t i o n s of v a r i o u s  w i t h the s u r f a c e h y d r o x y l groups [24-26] .  The thermogravimetric  curve (measured a t one atmosphere), f o r the S i 0  2  powder used  molecules analysis  i n the p r e s e n t  study, i s shown i n F i g u r e I I . 7 . Four from  t a r g e t s were prepared w i t h the S i 0  the manufacturer's  w i t h the same S i 0  2  specifications.  2  6  [27] .  onto  The platinum loaded samples were  Briefly,  the S i 0  2  the l o a d i n g procedure  powder s u r f a c e s .  T h i s molecule 2  produce s u r f a c e P t atoms.  move about on the s i l i c a  described  involves f i r s t physisorbing  hydrogen atmosphere a t 500 °C, v i a the r e a c t i o n H P t C l to  prepared  powder, but i n t h i s case hydrogen-reduced, with f o u r of  a t A r i z o n a S t a t e U n i v e r s i t y , a c c o r d i n g to procedures  elsewhere H PtCl  powder e s s e n t i a l l y u n a l t e r e d  F i v e other t a r g e t s were  these having a non-zero p l a t i n u m l o a d i n g . prepared  2  i s then reduced 6  + 2H  2  -> Pt° + 6HC1,  Because of the h i g h temperatures,  the Pt atoms  s u r f a c e and e v e n t u a l l y begin a g g r e g a t i n g .  l e v e l s of p l a t i n u m l o a d i n g were chosen f o r the c a t a l y s t 0.001%, 0.01%, 0.1% and 1.0% by weight.  in a  samples;  Five  0.0%,  A l l samples were c h a r a c t e r i z e d by  well-known gas a d s o r p t i o n techniques a t S t a n f o r d U n i v e r s i t y .  The s p e c i f i c  - 56  -  TEMPERATURE  (°C)  F i g u r e II.7 Thermogravimetric a n a l y s i s of C a b - O - S i l . Above 110 °C, the weight l o s s i s caused by a-gradual l o s s of water as the h y d r o x y l s undergo condensation. F i g u r e taken from r e f e r e n c e [ 2 0 ] .  - 57 s u r f a c e a r e a of the S i 0  support, which had been hydrogen-reduced  2  d u r i n g the  sample p r e p a r a t i o n , was measured u s i n g the B.E.T. a d s o r p t i o n i s o t h e r m t e c h n i q u e [28] ( i n t h i s case, N  a t 77 K) and found to be 320 ± 20 m /g. 2  2  T h i s i s somewhat s m a l l e r than the manufacturer's s p e c i f i c a t i o n of 390 ± 40 m /g f o r the unreduced  Si0  2  2  powder.  P l a t i n u m d i s p e r s i o n s (# of P t atoms a t  s u r f a c e / t o t a l # of Pt atoms I n sample) were measured i n both the 0.1% and 1.0%  P t loaded samples  by hydrogen  c h e m i s o r p t i o n [ 2 9 ] , and were found to be  1.0 ± 0.02 and 0.39 ± 0.02, r e s p e c t i v e l y .  The percentage of the t o t a l  s u r f a c e area of the loaded c a t a l y s t s which i s a t t r i b u t a b l e to the Pt atoms is  then 0.08% f o r the 0.1% sample and 0.34% f o r the 1.0% sample. The  t a r g e t s were prepared by compressing  the S i 0  2  powder i n t o  stainless  s t e e l vacuum v e s s e l s , onto which a 25 Lim or 50 Lim s t a i n l e s s window was then TIG  (Tungsten I n e r t Gas) welded  steel  or e l e c t r o n beam welded.  t a r g e t s were used both f o r c l e a n l i n e s s and because  prepare some of the samples  by a l i q u i d  The t a r g e t s were evacuated  arising was  - 6  s t e e l tubing, using  from the t a r g e t  assembly  t r a p , to reduce the p o s s i b i l i t y o f c o n t a m i n a t i o n  from backstreaming.  measured to be 1 0  prior  through a 110 cm l e n g t h of  The pumping system was i s o l a t e d  nitrogen cold  of the need to  i n vacuum f o r a p e r i o d of 10-12 hours  0.635 cm (0.25") outer diameter (0.4 cm I.D.) s t a i n l e s s a d i f f u s i o n pump.  stainless  by baking i n vacuum a t h i g h temperatures. A l l  heat treatments were performed to the experiments.  Welded  The p r e s s u r e a t the i n p u t of the d i f f u s i o n pump  T o r r , whereas the u l t i m a t e p r e s s u r e a t the t a r g e t  ( a f t e r baking a t T > 110 °C) was measured to be 10*" - 1 0 ~ T o r r .  Although  "low-magnetic"  existed  5  6  s t e e l s ( t y p e s 316-L and 321) were used, there s t i l l  some remnant m a g n e t i z a t i o n , which was found to i n t r o d u c e a s m a l l r e l a x a t i o n of the muonium s p i n due to induced f i e l d  inhomogeneity.  Specific  details  - 58 r e g a r d i n g the t a r g e t s used i n t h i s work are g i v e n i n T a b l e I I . 4 .  A l l but  one of the t a r g e t s used i n the present study had 25 |im TIG welded  windows.  The  e x c e p t i o n was  t a r g e t S i 0 ( 3 ) , which had a 50 |im e l e c t r o n beam welded 2  window.  II.C.4 The  Cryogenics evacuated samples  gas-flow cryostat low-temperature  were i n s e r t e d  (model: 10DT  environment,  through the top of a J a n i s  ^He  Super-VariTemp) which p r o v i d e s a u n i f o r m  v a r i a b l e from 1.8  to 300 K.  The J a n i s  cryostat  i s mounted through the top of the E a g l e spectrometer w i t h the c r y o s t a t extending down between the f o u r p o s i t r o n t e l e s c o p e s . in  the beam path the t a i l  To minimize  tail  the mass  outer vacuum s h i e l d i s removed, making the  c r y o s t a t i n s u l a t i n g vacuum contiguous w i t h the E a g l e vacuum chamber. muons e n t e r the c r y o s t a t by f i r s t a l u m i n i z e d mylar heat s h i e l d and  p a s s i n g through a 76.2 then a 0.127  mm  The  urn (0.0003")  (0.005") mylar window  s e p a r a t i n g the helium gas thermal bath of the "sample space" from the i n s u l a t i n g vacuum. The  temperature  and the ^He i n t o a PID  i s r e g u l a t e d by a d j u s t i n g the v a p o r i z e r heater c u r r e n t  flow r a t e through a needle v a l v e . [30] temperature  The heater i s i n c o r p o r a t e d  feedback system along w i t h the thermometer  (mounted on the o u t s i d e of the t a r g e t v e s s e l ) which monitors the sample temperature.  For temperatures  platinum r e s i s t o r was germanium r e s i s t o r was  i n the range 300 K - 75 K,  used, and f o r the range 100 K - 1.8 employed.  Some q u e s t i o n may  a calibrated K, a c a l i b r a t e d  a r i s e as to whether the  thermometers measure the " t r u e " temperature of the sample ( i . e . , sample a t thermal e q u i l i b r i u m w i t h the helium thermal b a t h ) .  i s the  This  was  - 59 T a b l e 11.4(a)  Si0  2  Targets  Targets  Characteristic  Si0 (l) (Vol. = 2  Si0 (2) (Vol. =  *  4.48 + 0.05 cm ) 3  2  Si0 (3) (Vol. =  4.48 + 0.05 cm ) 3  2  Si0 (4) (Vol. =  6.63 + 0.05 cm ) 3  2  6.08 + 0.05 cm )  T a b l e 11.4(b)  3  Mass of Powder Packing Density Surface Area  1.50 + 0.05 g 0.335 + 0.015 g/cm 585 + 79.5 m  Mass of Powder Packing Density S u r f a c e Area  0.50 + 0.05 g 0.112 + 0.012 g/ cm 195 + 39.5 m  3  Mass of Powder Packing Density S u r f a c e Area  0.72 + 0.05 g 0.109 + 0.008 g/cm 281 + 48.3 m  3  Mass of Powder Packing Density Surface Area  1.76 + 0.05 g 0.289 + 0.011 g/cm 686 + 89.8 m  3  2  2  2  3  2  P l a t i n u m Loaded S i O , T a r g e t s  Targets  Pt(l) (Vol.  Value  Characteristic  (0.0% loading) = 4.38 ± 0.05 cm ) 3  Value  Mass of Powder Packing Density Surface Area  1.50 + 0.08 0.61 + 0.03 480 + 25.6  g g/cm m  3  2  Pt(2) (Vol.  (0.001% l o a d i n g ) = 4.38 ± 0.05 cm )  Mass of Powder Packing Density S u r f a c e Area  1.50 + 0.08 0.61 0.03 480 + 25.6  g g/cm m2  Pt(3) (Vol.  (0.01% loading) = 4.38 ± 0.05 cm )  Mass of Powder Packing Density Surface Area  1.50 + 0.08 0.61 + 0.03 480 + 25.6  g g/cm 2  m  Mass of Powder Packing Density Surface Area  1.50 + 0.08 0.61 + 0.03 480 + 25.6  g g/ cm m  Mass of Powder Packing Density S u r f a c e Area  1.50 + 0.08 0.61 + 0.03 480 + 25.6  g g/cm m2  Pt(4) (Vol.  Pt(5) (Vol.  3  3  (0.1% loading) = 4.38 ± 0.05 cm ) 3  (1.0% loading) = 4.38 ± 0.05 cm ) 3  3  3  3  2  3  - 60 t e s t e d e x p e r i m e n t a l l y by f i r s t increasing The  s t e p s , and  data were found  s t e p p i n g through  to be q u i t e i n s e n s i t i v e indicating  to the order i n which the  t h a t the sample was  chemisorbing altering  h a n d l i n g system used f o r p h y s i s o r b i n g o r  c o n t r o l l e d amounts of d i f f e r e n t gases onto the sample s u r f a c e . the s u r f a c e c h a r a c t e r i s t i c s i n t h i s f a s h i o n , and  a s s o c i a t e d change i n the muonium b e h a v i o r ,  observing  one gains f u r t h e r i n s i g h t  the r e l a x a t i o n and  d i f f u s i o n behavior of muonium on s u r f a c e s .  present  i s d e p o s i t e d on the s i l i c a  study, ^He  surfaces.  apparatus  i s q u i t e t y p i c a l f o r the purpose a t hand and  volume  (three d i f f e r e n t  ± 0.16  (1331  ± 28  cm , 3  cm ) 3  36.5  ± 0.6  cm  and  3  valve  ± 0.5  cm ), 3  model:  of the r e a d i n g and  taken i n t o account  gauge was  observed.  and Vg was During  F.S./A°C.  3  gas  ensure t h a t any  handling  c o n s i s t s of a  doser  experiments,  The  V  s  p r e s s u r e i n the  220BHS-2A1-B-1000)  Torr.  The  latter  T h i s gauge i s  source of e r r o r  no "zero d r i f t "  dead volume i n the output  measured to be 35 ± 2 cm ,  gas  the  i s a l s o temperature compensated  i n the data a n a l y s i s , and  The  In  the  into  a standard volume  on the output.  of measuring p r e s s u r e s In the range 0-100  w i t h an a s s o c i a t e d e r r o r of 0.01% not  28.2  by a B a r a t r o n gauge (MKS  a c c u r a t e to w i t h i n 0.15%  The  doser volumes were used f o r these  and a metering  system i s monitored capable  indeed i n  Gas Handling  F i g u r e II.8 shows the gas  41.9  order.  e q u i l i b r i u m w i t h the helium bath of the c r y o s t a t .  III.0.5  By  the temperature range In  then r e p e a t i n g these measurements i n r e v e r s e  temperature p o i n t s were taken, thermal  -  not i n c l u d i n g  section after  was  i n the valves  the t a r g e t s .  d e p o s i t i o n , the sample temperature was  kept  low enough to  gas atoms r e a c h i n g the s u r f a c e would be adsorbed; f o r ^He  on  - 61 -  PRESSURE RELIEF VALVE  GAS INPUTr=^V\  Figure I I . 8  Gas h a n d l i n g  system.  w £  BYPASS VALVE  Si0  the temperature  2 >  was t y p i c a l l y kept below 10 K.  To estimate  the amount  of gas r e q u i r e d f o r f r a c t i o n a l or complete monolayer coverage, one c a l c u l a t e s the change i n p r e s s u r e A P i n the doser volume, a c c o r d i n g to the simple  equation  ^rz— RT  AP = f x  (II.8)  d  where f i s the f r a c t i o n of the s u r f a c e area to be covered, A i s the s u r f a c e a r e a o f the sample, R i s the gas c o n s t a n t (1.036 x 1 0 * -  is  9  t o r r cm  the area covered by an a d s o r p t i o n atom or molecule X (~10 ^-  helium on s i l i c a ) , volume  5  3  K  - 1  ),  a  K  cm , f o r 2  i s the doser volume of the system and T i s the doser  temperature.  F o r p h y s i s o r p t i o n of atoms and m o l e c u l e s , procedure  the f o l l o w i n g gas h a n d l i n g  was used:  (1) With V^, V , V^, V and Vg c l o s e d , open V system u s i n g the turbo pump. 3  5  (2) Once the system i s evacuated, (3) Open  2  and evacuate the  close V . 2  to p r e s s u r i z e doser volume to d e s i r e d l e v e l .  (4) Once.the d e s i r e d p r e s s u r e i s . a t t a i n e d , c l o s e on the B a r a t r o n gauge. (5) Open  or V  5  (depending  on requirements  and note  p l a c e d on flow  pressure  rate).  (6) Open V and wait f o r the system to come to e q u i l i b r i u m , (may take 30 m i n u t e s ) . 6  (7) Note p r e s s u r e on B a r a t r o n gauge. (8) Take d a t a . (9) Note p r e s s u r e once a g a i n to ensure equilibrium.  t h a t the system was near  (10) Repeat t h i s procedure as many times as necessary d e s i r e d s u r f a c e coverage.  to a c h i e v e the  Because one  i s n o r m a l l y d e a l i n g w i t h s m a l l q u a n t i t i e s of gas  r a t e s , the  temperature of the  and  sample i s only b r i e f l y perturbed  small  by  flow  this  procedure; such temperature f l u c t u a t i o n s were n o r m a l l y u n o b s e r v a b l e .  II.D  Data Analysis The  fits  MINUIT [31]  minimization  to the data and  package was  to generate the  used to p r o v i d e  least-squares  s t a t i s t i c a l e r r o r s on the  function  parameters.  II.D.l  Transverse Field Spectra  For  the  transverse  field  w i t h the f u n c t i o n g i v e n  data,  the raw  i n Equation II.6.  f i e l d muonium r e l a x a t i o n f u n c t i o n was G^(t)  = ex [-(\  s p e c t r a were f i t s e p a r a t e l y , In these f i t s ,  the  assumed to be of the  transverse  form  + \ )t]  M u  P  (II.9)  Q  Mu where \  q  («  ) Is the r e l a x a t i o n r a t e due  determined by a measurement at low monolayers of He 4  on  centers  on  the oxide s u r f a c e by  of i n t e r e s t  to the  present Mu  the r e l a x a t i o n r a t e \^ were t r e a t e d weighted  temperature (~6  the g r a i n s u r f a c e s ;  alumina powders have shown that Mu  .  just  fitted  K)  inhomogeneity and with  from the  such a helium f i l m . the  was  several  e a r l i e r experiments [22,23] on  i s protected  study are  The  to f i e l d  fine  depolarization The  two  parameters  i n i t i a l muonium asymmetry A^  asymmetries f o r each p o s i t r o n  and  telescope  i n d e p e n d e n t l y , whereas the r e l a x a t i o n r a t e s were combined i n a  average.  -  II.D.2  64  -  Zero and Longitudinal Field Spectra  For  the  zero and  longitudinal field  data,  the F and B s p e c t r a were  first  t r e a t e d s e p a r a t e l y by removing the r e s p e c t i v e backgrounds u s i n g  "t<0"  time b i n s .  After this,  spectrum d e f i n e d A S Y  ,  r»v>  they were combined to form an  asymmetry  by -  B°I  - rAt> -  [N (t> - B ] + B  Bp  [N ( t ) - B ]  i n which the muon l i f e t i m e i s a u t o m a t i c a l l y d i v i d e d out. spectrum was  the  then f i t u s i n g E q u a t i o n  1 1 . 1 0 with  The r e s u l t i n g  the a p p r o p r i a t e  relaxation  f u n c t i o n assumed. In p r a c t i c e , the angle,  with  respect  two  to the  telescopes  do not  i n general  t a r g e t , or e f f i c i e n c i e s .  have the same s o l i d  These d i f f e r e n c e s are F B  parameterized i n terms of a r e l a t i v e e f f i c i e n c y parameter a a s s o c i a t e d c o r r e c t i o n i s g i v e n by the e q u a t i o n  Q  = N /N . Q  q  + a )ASY - ( 1 - a ) o o ( 1 + a ) - ( 1 - a )ASY  The  (1  A S Y  The  (corr)  =  Q  r e s u l t i n g asymmetry spectrum Is then r e f e r r e d to as a  asymmetry".  (H-H)  Q  "corrected  - 65  -  CHAPTER III — THEORY OF MUONIUM RELAXATION  To  o b t a i n a c l e a r i n t e r p r e t a t i o n of  important  be  1  be  experimental r e s u l t s , i t i s  to understand the d i f f e r e n t r e l a x a t i o n mechanisms through which  the a"" s p i n may can  the  lost  depolarized.  For muonium, the  through i n t e r a c t i o n s w i t h the  ensemble s p i n p o l a r i z a t i o n  environment by means of the  five  r e l a x a t i o n mechanisms: (1)  Random L o c a l Magnetic F i e l d s  (2) (3) (4)  Random A n i s o t r o p i c H y p e r f i n e Chemical R e a c t i o n s S p i n Exchange  (5)  Superhyperfine  Distortions  Interactions  I n t h i s c h a p t e r , these mechanisms, along w i t h the functions, functions  are  discussed.  The  ensemble i n i t i a l l y  Ill.A  i n the  these  Since the mixed s t a t e w i l l be  polarized  |b > Q  spin  relaxation  i s normally  r e s t r i c t e d to the h a l f of  triplet  state  relaxation  the muonium  |a >. Q  Spin Relaxation Functions For a muonium atom, the  is  e f f e c t of d i f f u s i o n on  i s also considered.  unobservable, t h i s d i s c u s s i o n  associated  defined  by  the  expectation  v a l u e of the L I s p i n p o l a r i z a t i o n +  equation  <S^ (t)> = Tr{S^ ( t ) [ i ~op ~op 44 •>op ^op l  L  +  L  v  (S^ (0) ~op  6  J  = i-Trfs^J ( t ) p (0)} + Tr{(s^(t) ~op  H  u where IT' and  U  e  are  s  1  p  • )]U ~in' "s  "-~op  s  I  and  respectively,  + (S^CO) • £^ )]u  associated  s p i n dynamics of the muon, P ( 0 )  density  w i t h the  e  m a t r i x f o r the  i s the  pj£ = 2 » S ^ n  e~  initial g  environment and  (III.l)  (0) p (0)) .  ~op  the u n i t o p e r a t o r s f o r the yr n  (0)} '  n  s|M  ~in  J  spins,  spin density  Is the i s the  initial  operator spin  incoming muon s p i n  polarization vector.  66 -  I n g e n e r a l , the f i r s t  term i n E q u a t i o n  III.l  i s zero.  reduces  to  zero because the t r a c e over any s p i n v e c t o r o p e r a t o r  T h i s makes  it  p o s s i b l e to express the e x p e c t a t i o n v a l u e , < S ^ ( t ) > , i n terms of a second p  rank s p i n - s p i n a u t o c o r r e l a t i o n tensor § ( t ) , d e f i n e d as  g(t) =  Tr{s^ '°  (t)  (0) ~ °  P  P  = Tr{exp[iHt(2*/h)]  p  (0)}  -  (III.2)  s£ (0) e x p [ - i H t ( 2 * / h ) ]  s£ (0)  p  where H i s the s p i n H a m i l t o n i a n  of the system.  p  P g  (0)}  With t h i s , E q u a t i o n  III.l  can be w r i t t e n as <S^ (t)> - g ( t ) • s|J ~op ~ i n  (III.3) '  e  where the time e v o l u t i o n of the u  +  s p i n p o l a r i z a t i o n i s completely  determined by the motion tensor § ( t ) . i s d e f i n e d as the r e l a x a t i o n t e n s o r . tensor  includes o s c i l l a t o r y  are omitted  III.A.l  In transverse  terms c o r r e s p o n d i n g  i n the d e f i n i t i o n of the t r a n s v e r s e  Random L o c a l M a g n e t i c F i e l d s  In the context and  I n zero and l o n g i t u d i n a l f i e l d , | ( t ) field,  however,  this  to Larmor p r e c e s s i o n which field  r e l a x a t i o n tensor.  (RLMF)  of the s p i n - s p i n i n t e r a c t i o n between a magnetic probe  some weak d i p o l a r f i e l d  distribution,  the s p i n Hamiltonians  muonium and f o r p o s i t i v e muons a r e m a t h e m a t i c a l l y  for triplet  e q u i v a l e n t , except that  the former has a magnetic moment which i s ~103 times g r e a t e r than the latter. The triplet  general  spin Hamiltonian  f o r a muonium atom, i n the p o l a r i z e d  s t a t e , i n t e r a c t i n g with N nuclear  H = f j l =  (H^ + jfl) + H  Z  s p i n s i s g i v e n by the e q u a t i o n (III.4)  - 67 where  r e p r e s e n t s the d i p o l a r i n t e r a c t i o n s between the muonium atom and  the N n e i g h b o r i n g n u c l e i ,  r e p r e s e n t s the quadrupolar  n u c l e a r s p i n s (due f o r i n s t a n c e to e l e c t r i c presence dipolar of  of the muonium atom) and H  g r a d i e n t s induced  a r e the Zeeman terms.  defines  triplet  by the  Note t h a t any  i n t e r a c t i o n s among the n u c l e a r s p i n s have been n e g l e c t e d .  the d e s i r e to t r e a t  and y  field  i n t e r a c t i o n s of the  Because  muonium the same as a p o s i t i v e muon, one  to be the s p i n o p e r a t o r of the magnetic probe ( L I or Mu, e t c . ) +  (= s  defining J  ) as the c o r r e s p o n d i n g  2ICY  magnetogyric r a t i o .  s  i  as the s p i n o p e r a t o r of the j  corresponding  magnetogyric r a t i o ,  By f u r t h e r  tti n u c l e u s , and y  (  T  *— 2icy ) as the  =  T  the Zeeman term i s w r i t t e n as  N H  Z  = (h/2u) y ( S ~ ' S  T (h/2*) Y ( ' ) ^ J ^ O D ~ J=l terms can be w r i t t e n  • B)J -  ~OP  In a d d i t i o n , the d i p o l a r H and  d  j  = (h/2it)  2  j J  ~  ( Y.)(r.)~ ^ s 'j'* j '  (III-5)  B  t  V  [(S • J ) - 3(n.. S ) ( n . . J )] -~op ~op^ j ~op j ~op '  3  j  Y  1  j  L1  v  ; v  (III.6)  J  the quadrupole terms a r e H<j = (h/2*) ^ 1  [3(n. • j j ) p  2  - J ( J + 1)]  (III.7)  where n^ i s the u n i t v e c t o r i n the d i r e c t i o n from the muonium s p i n to the j "* 1  1  nucleus  o p e r a t o r jjj The  l o c a t e d a t a d i s t a n c e r_., J ( J + 1) i s the e i g e n v a l u e and OJ^ r e p r e s e n t s the s t r e n g t h of the quadrupolar  zero and low f i e l d  (external f i e l d  «  local field)  of the  interaction.  spin relaxation  f u n c t i o n s f o r a magnetic probe i n t e r a c t i n g w i t h a random l o c a l magnetic f i e l d were f i r s t  d i s c u s s e d i n 1967 [ 1 , 2 ] .  I n t h i s f o r m u l a t i o n , the  quadrupole i n t e r a c t i o n s are assumed to be n e g l i g i b l e and the d i p o l a r i n t e r a c t i o n s , which a r e i n g e n e r a l d e s c r i b e d by a q u a n t a l l o c a l magnetic field  o p e r a t o r , are approximated by a s t a t i c  (continuous)  effective  local  - 68 dipole the  f i e l d H.  With these assumptions, the d i p o l a r  i n t e r a c t i o n term takes  simple form N 7  H. = (h/2*) Y (S J  • H)  's^~op  The e f f e c t i v e f i e l d the magnitude  (III.8)  ;  d i s t r i b u t i o n i s further  of each component being d i s t r i b u t e d a c c o r d i n g to a continuous  Gaussian d i s t r i b u t i o n f u n c t i o n  of width A/y  G  and  ) =  Y ^ exp[- - i - ^ i /2TI A 2A  f (|H|) = [ ^ - ] /2TC A G  3  exp[-  Y  ]  ;  i = x, y, z  zz  2A  the zero f i e l d  =J f +  ( t )  t  1  (III.10)  2  - A * ) 2  2  distribution [3].  spin relaxation  a system of s t a t i c l o c a l d i p o l a r 8  (III.9)  ] [ 4 * |H| ]  s  where A i s the second moment of the f i e l d distribution,  , g i v e n by s  Y  f (H  assumed to be i s o t r o p i c , w i t h  e x  function  Assuming  this  of a magnetic probe i n  f i e l d s i s found to be [1,2]  P("  k  (III.11)  which i s the f a m i l i a r s t a t i c G a u s s i a n Kubo-Toyabe f u n c t i o n .  In E q u a t i o n  I I I . 1 1 , The 1/3 component corresponds to the component of the l o c a l f i e l d directed  p a r a l l e l to the i n i t i a l u  +  spin p o l a r i z a t i o n ( i . e . ,  the beam  d i r e c t i o n , z - a x i s ) , w h i l e the damped o s c i l l a t i o n of the 2/3 component from the x and y components ( i . e . , polarization)  of the random l o c a l  normal to the i n c i d e n t  muons  field.  The a p p l i c a t i o n of a l o n g i t u d i n a l  field B  (directed  along the z - a x i s )  can be used to e f f e c t i v e l y "decouple" the magnetic probe s p i n from the static local fields, dipolar  interaction.  arises  thereby quenching the d e p o l a r i z i n g For small l o n g i t u d i n a l  fields,  e f f e c t s of the  one o b t a i n s the  e x p r e s s i o n [1,2] 2 8  zz  L  ( t , U  )  =  2~ * " L  1  L  ^" 2  6 X P  A  fc  )  c o s  (  t 0  L ^ t  (III.12)  W  2A^ 1 2 2 + — j - J d t exp(A T ) sin(o) x) uT o fc  L  Li  where, OJ = y B .  T h i s f u n c t i o n i s shown f o r v a r i o u s v a l u e s o f OJ  T  Li  III.l.  S LI  i n Figure  LI  P o s i t i v e muons and the techniques  of uSR are i d e a l l y  suited f o r  s t u d y i n g r e l a x a t i o n f u n c t i o n s i n any ( o r zero) e x t e r n a l magnetic f i e l d , and thus  provided  the f i r s t  experimental  function given i n Equation In the l i m i t  observation  III.11.  of "randomly ordered" moments ( i . e . ,  i n the l a t t i c e ) ,  the l o c a l magnetic f i e l d  approximates a L o r e n t z i a n d i s t r i b u t i o n d i s t r i b u t i o n (HWHM = a/y^),  f  ( i)  L  H  =  IT  L  [4] of the Kubo-Toyabe  3 2  a  distributed  d i s t r i b u t i o n a t the u  [5-7].  The L o r e n t z i a n  +  randomly  site  field  can be w r i t t e n as  2 2^ + Y H, 's i  i = x, y, z  5  (III.13)  and 3 f (|H|) = - ^ § [ - 2  ~  L  *  For  l  the case o f s t a t i c  takes 8  zz  j-*"] t ^ | H | ]  Y  (III.14)  2  2  + a  |H| J  g  local fields,  the zero f i e l d  spin relaxation function  the form [8] (  t  )  =  J  +  f t  1  "  a t  )ex (-at)  As i n the case of the Gaussian I I I . 11,  (III.15)  P  Kubo-Toyabe f u n c t i o n , g i v e n i n E q u a t i o n  the s t a t i c L o r e n t z i a n f u n c t i o n g ^ C t )  time dependent 1/3 component.  e x n  ibits  the c h a r a c t e r i s t i c  Furthermore, an a p p l i e d l o n g i t u d i n a l  field  Figure I I I . l Static longitudinal f i e l d spin relaxation function for a G a u s s i a n random l o c a l f i e l d , p l o t t e d f o r v a r i o u s v a l u e s of av/A.  >  will effectively  magnetic probe and  "decouple"  the s p i n - s p i n i n t e r a c t i o n between the  the random f i e l d s a c c o r d i n g to the e q u a t i o n  [8]  (III.16) - a[l + where j  U  n  and  j  J. 1  ) ] /dx L o 2  [ j ( o ) x ) exp(-aT)] 0  L  denote S p h e r i c a l B e s s e l f u n c t i o n s , and u  = y B . s Li  JJ  This  f u n c t i o n i s shown i n F i g u r e I I I . 2 f o r s e l e c t e d v a l u e s of oo . Lt  The  exact quantum mechanical  s o l u t i o n f o r the time dependence of  the  s p i n p o l a r i z a t i o n , assuming the H a m i l t o n i a n of E q u a t i o n I I I . 4 , has a l s o been investigated are found  [9-11].  The  to d e v i a t e from  zero f i e l d  2/A).  i n the long time t a i l ,  limit, has  involving  the p. and +  appear as coherent  r e c e n t l y been observed  1 9  F The  can be understood  obtained  solutions allow  as  by  spin-flip dilute  T h i s type of r e l a x a t i o n f u n c t i o n  e x p e r i m e n t a l l y f o r \i  +  +  intuitively  n e i g h b o r i n g n u c l e i which, i n the  oscillations.  w i t h r e s u l t s s u g g e s t i n g the u two  (t)  T h i s d e v i a t i o n , which m a n i f e s t s i t s e l f  n o t i n g t h a t the exact quantum mechanical transitions  z z  the Kubo-Toyabe f u n c t i o n , g i v e n i n E q u a t i o n  I I I . 1 1 , a t long times ( t » extra o s c i l l a t i o n s  relaxation functions 8  to be l o c a l i z e d  in alkali  fluorides  along the <110>  [12],  a x i s , between  nuclei. corresponding  transverse f i e l d  i n t e r a c t i n g w i t h l o c a l magnetic f i e l d s ,  f u n c t i o n , f o r a magnetic probe i s d i s c u s s e d i n both the s t a t i c  and  dynamic l i m i t s i n s e c t i o n I I I . B .  III.A.2  Random A n i s o t r o p i c H y p e r f i n e D i s t o r t i o n s (RAHD)  In condensed media, the muonium h y p e r f i n e c o u p l i n g may  be p e r t u r b e d  to the e l e c t r o s t a t i c i n t e r a c t i o n between the muonium e l e c t r o n and  the  due  - 72  -  Figure III.2 S t a t i c longitudinal f i e l d spin relaxation function for a L o r e n t z i a n random l o c a l f i e l d , p l o t t e d f o r v a r i o u s v a l u e s of u^/a.  - 73 lattice.  As d i s c u s s e d  -  i n s e c t i o n I.C.I,  a c l a s s i c example of  i n t e r a c t i o n i s the case of muonium i n bulk q u a r t z . the ground s t a t e w a v e f u n c t i o n i s t r a n s m i t t e d d i p o l e - d i p o l e c o u p l i n g of the u  and  +  e  t o t a l Hamiltonian  due  i n c l u d e d i n the W  tensor of the h y p e r f i n e  H = (h/2Tc)(y If  S e ~op e  to the u  spins.  -  The  The  this  r e s u l t i n g change i n v i a the magnetic  +  c o n t r i b u t i o n to  to the h y p e r f i n e - l a t t i c e i n t e r a c t i o n i s i m p l i c i t l y  - y  LI  ~op  ;  1 • B ~  +  term of the  Hamiltonian  (h/2Ti) W : (S » ^~op  S* ) ~op  &  (III.17)  1  J  the d i s t o r t i o n of the h y p e r f i n e c o u p l i n g i s i s o t r o p i c , a s h i f t  hyperfine-structure interval v shift  the  will  0 0  i n the energy e i g e n v a l u e s .  The  occur,  along w i t h a  zero f i e l d  corresponding  eigenfunctions  system w i l l however remain f u n c t i o n a l l y u n a l t e r e d  i n the  f o r the  from the vacuum h y p e r f i n e  s t a t e s so that no a d d i t i o n a l time dependence of the L I s p i n p o l a r i z a t i o n i s +  induced.  In g e n e r a l , however, the d i s t o r t i o n may  components, and  i n t h i s case one  have some a n i s o t r o p i c  observes dramatic  e f f e c t s even i n zero  field. The  time e v o l u t i o n of the \i  +  a n i s o t r o p i c muonium h y p e r f i n e Appendix I .  The  spin polarization for a generally  i n t e r a c t i o n i s discussed  approach that i s taken i n v o l v e s expanding the  tensor W i n terms of s p h e r i c a l harmonics and coefficients u)  L m  to parameterize  contact  have r e f l e c t i o n symmetry, the antisymmetric zero.  u s i n g the  the d i s t o r t i o n .  t e n s o r W i n v o l v e s only d i p o l e - d i p o l e and  identically  i n some d e t a i l i n hyperfine  expansion  Because the  hyperfine  i n t e r a c t i o n s , both of which  p a r t of the expansion i s  Although i n t e r a c t i o n s of t h i s  type may  produce d i f f e r e n t  d i s t o r t i o n s i n the e l e c t r o n w a v e f u n c t i o n of each muonium atom, they do lead  hot  to a t r u e ( i r r e v e r s i b l e ) r e l a x a t i o n of the muon s p i n v e c t o r f o r the  i n d i v i d u a l muonium atoms.  For an ensemble of muonium atoms, however,  - 74 d e p o l a r i z a t i o n can occur v i a ensemble dephasing, p r o v i d e d that t h e r e i s a random d i s t r i b u t i o n i n the d i s t o r t i o n s o f i n d i v i d u a l muonium atoms i n the enemble.  To d e s c r i b e the ensemble r e l a x a t i o n , an approximation o f T r { p ( 0 ) } g  is  adopted, where each o f the w^'s *  to  some d i s t r i b u t i o n f u n c t i o n ^ n j ^ m ^ " Tr{p (0)} s  - H  / dco  f  2m  2 m  s  assumed to be d i s t r i b u t e d a c c o r d i n g w  *  t h i s approximation, one has  t n  (co J  (111.18)  2  —OO  The motion  tensor f o r the ensemble i s then approximated by CO  - II  g(t)  s  m  / dw, _o> 2  m  2  f_ (OJ ) Tr{s^ ( t ) 2m ~op ~op 0  m  l  }  J  (III.19)  where the t r a c e over the muon s p i n o p e r a t o r s i s i n c l u d e d  i n the i n t e g r a l .  The  spin r e l a x a t i o n functions associated with a s p e c i f i c d i s t o r t i o n  are  then c a l c u l a t e d by a v e r a g i n g over the a p p r o p r i a t e oo Of p a r t i c u l a r  field  interest  2m  distributions.  to the p r e s e n t study a r e the zero and t r a n s v e r s e  s p i n r e l a x a t i o n f u n c t i o n s f o r a randomly o r i e n t e d system  such as i n  the case of muonium i n b u l k fused quartz o r on the s u r f a c e of f i n e powders.  symmetry  silica  To c a l c u l a t e these f u n c t i o n s , one must a l s o average over a l l  possible orientations.  C o n s i d e r the combination o f a c y l i n d r i c a l  distortion  coupled w i t h a p l a n a r d i s t o r t i o n , which a r e parameterized by the f r e q u e n c i e s M O) Q 2  and u 2 2 »  respectively.  Assuming each o f these f r e q u e n c i e s t o be  d i s t r i b u t e d according to a L o r e n t z i a n or L o r e n t z i a n - l i k e d i s t r i b u t i o n with zero average, as d i s c u s s e d i n Appendix I , one has M  f ^ o , ^ )  =  -  ( 20J U  f +  (  — ^ H -  °20J  M  L(u ) 22  M  2  M +  2  - 2 (2n)  }  (111.20)  ^ 2 2 ^ •!  M where O^Q and a distributions.  2  2  r e p r e s e n t the r e s p e c t i v e widths  (HWHM) o f the f r e q u e n c y  I g n o r i n g the mixed s t a t e component o f the muonium ensemble,  - 75 the p o l a r i z e d t r i p l e t muonium s t a t i c r e l a x a t i o n f u n c t i o n i n zero f i e l d i s found t o be rh,,. 8  zz  (  t  )  1  1  r i  " 6 t +  This result  1 ^  r 1  M  ~ 2 ^  1  J °22^  _  e  X  M  \  P(" 2 ^  E X  t"  p  (111.21)  ^°22  +  3  /  2  7  1 0  2O>]  can be b e t t e r understood by c o n s i d e r i n g the two cases  p u r e l y c y l i n d r i c a l d i s t o r t i o n and a p u r e l y p l a n a r d i s t o r t i o n , If  one n e g l e c t s  III.21  *£  of a  separately.  the p l a n a r component o f the h y p e r f i n e d i s t o r t i o n ,  Equation  becomes  ( t  Notice  '°20>  J  =  T  +  e x p  t" T  /  2  7  ° o^  T  < ' > III 22  2  that as t •» « , t h i s f u n c t i o n tends to 1/6, ( o r 1/3 o f the i n i t i a l  polarization  of the t r i p l e t muonium ensemble).  The time independent 1/6  component of the t o t a l ensemble s p i n p o l a r i z a t i o n arises  because there e x i s t s  a non-trivial  (residual polarization)  zero frequency.  T h i s can be  understood i n t u i t i v e l y by drawing an analogy w i t h random d i p o l a r f i e l d s and n o t i n g t h a t f o r a random h y p e r f i n e i n t e r a c t i o n , a x i s w i l l be d i r e c t e d along  the z - a x i s ( i . e . ,  the c y l i n d r i c a l d i s t o r t i o n  along  the i n i t i a l muon s p i n  p o l a r i z a t i o n ) 1/3 of the time on average. If  on the other hand one n e g l e c t s  hyperfine d i s t o r t i o n , Equation  g  rh._ zz  ( t ; C T  M > 22 )  1 6 C  +  11.21 becomes  1 M • > " 2 °22^  r i  =  1  e  , 1 11 1 M v 3 t " 4 22^ 1  a  the c y l i n d r i c a l component of the  r 1 M \ X  e x p  p  ^ 2  °22^  1 M v ^"4 22^ 0  N o t i c e here that as t -> », t h i s f u n c t i o n approaches z e r o . simply  r e f l e c t s the f a c t  \ J.J.1 • ^ J  t  This  that, unlike a c y l i n d r i c a l d i s t o r t i o n ,  d i s t o r t i o n generates no n o n - t r i v i a l  zero  frequencies.  result a planar  )  - 76 From c o n s i d e r i n g these  two  -  l i m i t i n g cases,  i t i s obvious  p l a n a r component of the d i s t o r t i o n i s r e s p o n s i b l e f o r d r i v i n g  that  the  the f u n c t i o n  rh g  z z  (t)  i n Equation  III.21  f u n c t i o n , along w i t h  to zero at long times.  the two  l i m i t i n g cases  In an e x t e r n a l magnetic f i e l d B,  A t y p i c a l example of  this  Is p l o t t e d i n F i g u r e I I I . 3 .  the" problem of c a l c u l a t i n g  the  r e l a x a t i o n f u n c t i o n s f o r a random a n i s o t r o p i c h y p e r f i n e i n t e r a c t i o n becomes somewhat more d i f f i c u l t , a r e , however, a few for  simple  example, t r i p l e t  and x «  1).  Hamiltonian  e s p e c i a l l y f o r a randomly o r i e n t e d system. l i m i t i n g cases  muonium i n the l i m i t  In t h i s l i m i t ,  t h a t can be t r e a t e d . of "high f i e l d s "  Consider,  (i.e.,  co^  »  o"  the i s o t r o p i c f r e q u e n c i e s dominate so t h a t  the  These c a l c u l a t i o n s are g i v e n i n Appendix I f o r the case  a randomly o r i e n t e d system. that, i n this l i m i t ,  R e s u l t s f o r the l o n g i t u d i n a l f i e l d  a longitudinally applied f i e l d w i l l  the random a n i s o t r o p i c h y p e r f i n e i n t e r a c t i o n .  case show  completely  decouple  In the t r a n s v e r s e f i e l d  = [2n)~  l  j'dB o  sinB  case,  fdQ o  (III.24)  3 2 M 3 _____ x {exp[- -g- s i n B o c o s ( 9 ) t ] exp[- j /2/3 a 2 2  For e a r l y times time b e h a v i o r g_Ct)  - \  ( t -> 0 ) , one  and  2 |3cos B -  o  2Q  + -2-  4 )t]  111]}  to o b t a i n the  approximate the r e l a x a t i o n f u n c t i o n with the  ex [-( % P  can expand the i n t e g r a n d  2 Q  short  expression (111.25)  2  TC/6  The  of  o b t a i n s ( o m i t t i n g the Larmor p r e c e s s i o n p a r t ) the r e l a x a t i o n f u n c t i o n g!\t)  exact  2m  can be approximated by i t s d i a g o n a l elements alone ( s e c u l a r  approximation).  one  There  s o l u t i o n of E q u a t i o n  the expansion  approximation  I I I . 2 4 , c a l c u l a t e d n u m e r i c a l l y , as w e l l as  given i n Equation  III.25  are p l o t t e d f o r  1.0  —.2  H 0.  '  r  <  r  1.  t  r  2.  3.  (JMS)  F i g u r e I I I . 3 S t a t i c zero f i e l d RAHD r e l a x a t i o n f u n c t i o n s assuming L o r e n t z i a n frequency d i s t r i b u t i o n s w i t h zero averages. The c y l i n d r i c a l component of the d i s t o r t i o n i s r e p r e s e n t e d by the long-dashed curve (^20 10 u s ) , whereas the p l a n a r d i s t o r t i o n component (022 10 u s ) i s r e p r e s e n t e d by the short-dashed c u r v e . The s o l i d l i n e i s the combined r e l a x a t i o n f u n c t i o n f o r equal c y l i n d r i c a l and p l a n a r components (o"2o 2 2 10 u s ) . A l l t h r e e curves have been n o r m a l i z e d to equal 1 at t=0. - 1  =  =  - 1  =  - 1  0  =  -  78 -  comparison as a f u n c t i o n of time i n F i g u r e I I I . 4 . times,  these  The  two f u n c t i o n s a r e v i r t u a l l y  Notice  that a t e a r l y  i n d i s t i n g u i s h a b l e i n shape.  assumption o f a L o r e n t z i a n ( o r L o r e n t z i a n - l i k e ) d i s t r i b u t i o n may  not i n g e n e r a l be a p p r o p r i a t e , an i n f i n i t e second moment.  simply  because a L o r e n t z i a n d i s t r i b u t i o n has  A more a p p r o p r i a t e  approximation may be made by  assuming a m o d i f i e d L o r e n t z i a n , w i t h a Gaussian damping. assuming t h i s frequency  d i s t r i b u t i o n w i l l of course  The a f f e c t of  be r e f l e c t e d  i n the  shape of the c a l c u l a t e d r e l a x a t i o n f u n c t i o n , and can most e a s i l y be understood by c o n s i d e r i n g the simple hyperfine d i s t o r t i o n . f(co  2 Q  )  cylindrical  I n t h i s case one d e f i n e s the d i s t r i b u t i o n  [n e ^ e r f c a ) ) "  =  example of a p u r e l y  1  [  °  ] exp^u^/a^jx ]  20  2  u  20  20  +  CT  where X i s a damping parameter ( t y p i c a l l y l e s s than one), complimentary e r r o r f u n c t i o n .  (111.26)  2  2  Using  and e r f c ( X )  t h i s d e f i n i t i o n , the s t a t i c zero  isa field  r e l a x a t i o n f u n c t i o n i s found to be  g^(t,\;a Z Z  Z  9 n  )  U  = i +  (6 e r f c ( X ) ) "  1  {e ° + a  =  /2/3 O" Q. 2  function i s v i r t u a l l y  20  t  erfc(X + a  t h i s , one should  2 Q  t/(2X))}  Note that f o r s m a l l v a l u e s  of X ,  this  i n d i s t i n g u i s h a b l e from the f u n c t i o n d e r i v e d u s i n g a  Lorentzian d i s t r i b u t i o n (Figure I I I . 3 ) .  i n c r e a s e d , the i n i t i a l  (III.27)  U  T h i s f u n c t i o n has been c a l c u l a t e d and i s p l o t t e d i n  F i g u r e I I I . 5 f o r X = 0 . 0 and 0 . 1 .  standard  e r f c ( x - a' t / ( 2 X ) )  0  Z  + e where O^Q  2  6  decay begins  However, as X i s  to mimic a Gaussian shape.  Because of  be able to put an upper l i m i t on X f o r data e x h i b i t i n g an  exponential-like i n i t i a l  decay shape.  - 79 -  1.0  0.  1.  2.  3.  (/xs)  t  Figure III.4 S t a t i c t r a n s v e r s e f i e l d RAHD r e l a x a t i o n f u n c t i o n . The s o l i d curve i s the exact s o l u t i o n of E q u a t i o n 11.24, c a l c u l a t e d n u m e r i c a l l y , and the dashed curve i s the expansion approximation of E q u a t i o n 11.25. Both f u n c t i o n s have been e v a l u a t e d f o r O^Q °"22 ^ us and have been nomalized to equal 1 a t t=0. =  =  - 1  0.0  0.2  0.4  0.6  t  (/us)  0.8  1.0  F i g u r e I I I . 5 S t a t i c zero f i e l d RAHD s p i n r e l a x a t i o n f u n c t i o n f o r a pure c y l i n d r i c a l d i s t o r t i o n , assuming the form of E q u a t i o n I I I . 2 7 . The f u n c t i o n i s p l o t t e d f o r two v a l u e s of the damping parameter \, 0.0 and 0.1, f o r the case of c?20 10 j i s . The curves have been n o r m a l i z e d to equal 1 a t t=0. =  - 1  - 81 An i n t e r e s t i n g p o i n t can now be made by comparing the i n i t i a l s l o p e s of the zero f i e l d  r e l a x a t i o n f u n c t i o n g i v e n i n E q u a t i o n III.21 and the h i g h  transverse f i e l d  f u n c t i o n of E q u a t i o n I I I . 2 4 .  By d e f i n i n g m ^  the i n i t i a l s l o p e s of the zero and t r a n s v e r s e f i e l d one  _ [1 + /3  z f  -  ( a /a ) ] = — ~ — — —  relaxation functions,  M  Thus, one f i n d s t h a t m ^ f a s t e r i n zero f i e l d understood field,  >  •  /3  indicating  (III.28)  t h a t the d e p o l a r i z a t i o n r a t e i s  than i n t r a n s v e r s e f i e l d .  r e s u l t can be  a l l t h r e e components (x,y,z) of the h y p e r f i n e d i s t o r t i o n c o n t r i b u t e  the r e l a x a t i o n of the u  +  s p i n , whereas i n h i g h t r a n s v e r s e f i e l d ,  For Mu i n bulk s i l i c a ,  i s o t o p i c ) are r e l a t i v e l y  (RAHD) [13]; i n t e r a c t i o n s w i t h  insignificant.  Because of these two t e s t case f o r the  RAHD s p i n r e l a x a t i o n f u n c t i o n s developed  zero and l o n g i t u d i n a l f i e l d  quartz a t 7.0 ± 0.1 K a r e p l o t t e d zero f i e l d  nuclei  I n an amorphous environment  f e a t u r e s , Mu i n bulk fused s i l i c a p r o v i d e s an e x c e l l e n t  The  Si  I t i s a l s o known t h a t muonium  i n b u l k q u a r t z below about 50 K [ 1 3 ] .  zero and t r a n s v e r s e f i e l d  2 9  the h y p e r f i n e d i s t o r t i o n s a r e d i s t r i b u t e d  randomly both i n o r i e n t a t i o n and magnitude. static  ( z - a x i s ) components.  the \& s p i n p o l a r i z a t i o n r e l a x e s v i a random  anisotropic hyperfine distortions  such as bulk fused s i l i c a ,  one i s  to the H a m i l t o n i a n and e f f e c t i v e l y  i g n o r e a l l but the i s o t r o p i c and c y l i n d r i c a l  (4.6%,  T h i s important  by c o n s i d e r i n g the problem i n terms o f d i m e n s i o n a l i t y ; i n zero  a b l e to make a s e c u l a r approximation  is  to be  can d e f i n e the r a t i o • —  to  and m ^  here.  s p e c t r a f o r muonium i n bulk f u s e d  i n Figure III.6.  The curve through t h e  data i s a f i t o f E q u a t i o n 11.10 to the d a t a , assuming the s t a t i c  RAHD f u n c t i o n of E q u a t i o n I I I . 2 1 .  The f i t gave a Chi-square  of 86.2 f o r 53  - 82 -  !_ __  d  d  — 6  00  o o  •  CD o o  •  OJ o o  •  o  O  •  o  F i g u r e I I I . 6 Zero and l o n g i t u d i n a l f i e l d data f o r muonium i n bulk f u s e d q u a r t z a t 7±1 K. The l i n e through the zero f i e l d data ( c i r c l e s ) i s a f i t of the RAHD f u n c t i o n to the d a t a . The l o n g i t u d i n a l f i e l d data shown a r e f o r 0.5 G ( t r i a n g l e s ) , 1.0 G (diamonds) and 2.0 G ( s q u a r e s ) .  - 83 degrees  of freedom and  the f i t t e d  d i s t o r t i o n parameters were 5.7 respectively.  The  in  found  the f i t and  triplet  -  r e s u l t s f o r the c y l i n d r i c a l and  (+1.5/-1.2) u s  - 1  and 6.2  (+0.54/-0.52)  muonium asymmetry parameter was  to be 0.118  (+0.0030/-0.0030).  the p l a n a r  allowed  us  - 1  ,  to vary  R e c a l l t h a t the muonium  asymmetry r e f l e c t s the f r a c t i o n of muonium formed i n the sample; the  result  o b t a i n e d here i s c o n s i s t e n t w i t h the c o r r e s p o n d i n g v a l u e o b t a i n e d i n low transverse  field.  As has been d i s c u s s e d , a random h y p e r f i n e i n t e r a c t i o n can be c o m p l e t e l y decoupled  i n longitudinal field  the zero f i e l d 7 K d a t a , t h i s order of a few Gauss.  for  »  0*2 «  F  m  r o m  almost the f i t of  translates into a longitudinal f i e l d  on  the  As can be seen i n F i g u r e I I I . 6 , the d e c o u p l i n g  behavior i s c o n s i s t e n t w i t h  this. M  F i n a l l y , by s u b s t i t u t i n g zero f i e l d  f i t i n t o the approximation  g i v e n i n E q u a t i o n I I I . 2 5 , one \  M u  of 2.9  the v a l u e s f o r O^Q and  + 0.5  us  - 1  ,  obtained i n the  f o r the t r a n s v e r s e f i e l d f u n c t i o n  obtains a transverse f i e l d  which i s c o n s i s t e n t w i t h e x i s t i n g  relaxation rate transverse f i e l d  data [ 1 3 ] . Although  a L o r e n t z i a n d i s t r i b u t i o n of f r e q u e n c i e s does reproduce  b a s i c f e a t u r e s of the quartz d a t a , i t i s probably not the most assumption.  model t h a t can be used  should  realistic  Perhaps u s i n g the type of d i s t r i b u t i o n d e f i n e d i n E q u a t i o n  III.26 would be b e t t e r , but there i s at present no obvious or  the  to help d e c i d e what the c o r r e c t  p h y s i c a l argument distribution  be.  Ill.A.3 The  Chemical Reactions coherence  (CH)  of the s p i n p o l a r i z a t i o n of the u  +  i n the muonium  state  can ( i n t r a n s v e r s e f i e l d ) be l o s t w i t h molecules which  through c o l l i s i o n s of the muonium atoms  r e s u l t i n c h e m i c a l r e a c t i o n s forming diamagnetic  products [14].  With t h i s mechanism, the muon s p i n p o l a r i z a t i o n must  exponentially.  I n the gas phase,  the t r a n s v e r s e f i e l d  relaxation  decay  function  i s g i v e n by the e x p r e s s i o n = e x p [ - xf  g^(t)  exp[-(n  t] =  v a  c h  ) t]  (III.29)  where r\ i s the number of i n t e r a c t i n g molecules per u n i t volume, v i s the mean r e l a t i v e v e l o c i t y between the muonium atom and the molecules and a 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 . In  transverse f i e l d ,  a u  +  p r e c e s s e s about 103 time slower than  triplet  muonium i n the same f i e l d ,  such that i f a chemical r e a c t i o n produces a u  a diamagnetic environment,  the u  muonium ensemble.  +  I n zero and l o n g i t u d i n a l f i e l d ,  z - a x i s r e g a r d l e s s of whether the u  diamagnetic  Ill.A.4 In as  in  +  +  however, such  reactions  s p i n remains p o l a r i z e d along  i s i n the muonium s t a t e or i n a  state.  Spin Exchange (SE) c o l l i s i o n s w i t h paramagnetic m o l e c u l e s , h y p e r f i n e t r a n s i t i o n s  |m^,m> = |+,+> e  •*• | +  can take p l a c e .  reactions i n transverse f i e l d , to  +  i s e f f e c t i v e l y removed from the p r e c e s s i n g  produce no o b s e r v a b l e r e l a x a t i o n s i n c e the u the  is  such  As i n the case o f c h e m i c a l  the decay of the muon s p i n p o l a r i z a t i o n due  s p i n exchange i s found to vary e x p o n e n t i a l l y w i t h time so t h a t , i n  transverse f i e l d , g^(t)  the r e l a x a t i o n f u n c t i o n i s w r i t t e n  = exp[- X  S e L  t ] = e x p [ - ( f T, v a j t ]  (III.30)  - 85 where TJ i s a g a i n t h e number d e n s i t y of i n t e r a c t i n g m o l e c u l e s , v i s t h e mean r e l a t i v e v e l o c i t y between the muonium atom and the m o l e c u l e s , a  i s the se  s p i n exchange c r o s s s e c t i o n and f i s a f a c t o r which depends on b o t h t h e s p i n of t h e paramagnetic m o l e c u l e s and the o r i e n t a t i o n o f t h e e x t e r n a l magnetic f i e l d with respect  t o the i n i t i a l muon s p i n p o l a r i z a t i o n .  I n l o n g i t u d i n a l f i e l d , where t h e q u a n t i z a t i o n a x i s i s a l o n g the i n i t i a l muon s p i n p o l a r i z a t i o n , s p i n exchange causes h y p e r f i n e p r o b a b i l i t y of s c 2  2  = (1 + x ) 2  - 1  .  t r a n s i t i o n s with a  Thus i n terms of t h e s p e c i f i c  field  parameter x, t h e s p i n exchange r e l a x a t i o n f u n c t i o n f o r muonium i n a longitudinal field  i s w r i t t e n as [14,15]  g _ _ ( t , x ) = exp[- \ J Transverse f i e l d  e  t ] = exp[-(| - v a  studies  the f a c t o r f t o e q u a l 8/9 f o r 0 studies  - 1  t]  (III.31)  [16] o f t h e temperature dependence o f t h e s p i n  exchange r e a c t i o n o f muonium w i t h 0  field  )(l + x ) 2  g e  2  2  (S=l)  and NO (S=l/2) have shown t h a t  and 3/4 f o r NO, whereas i n l o n g i t u d i n a l  [15] the f a c t o r f was found t o e q u a l 64/27 and 2 f o r 0 and 2  NO, r e s p e c t i v e l y .  Ill.A.5 The  Superhyperfine Interactions (SHF) time e v o l u t i o n o f t h e muon s p i n p o l a r i z a t i o n i n t h e muonium s t a t e  can a l s o be i n f l u e n c e d by t h e s u p e r h y p e r f i n e i n t e r a c t i o n s between t h e u n p a i r e d e l e c t r o n of t h e muonium atom and n e i g b o r i n g non-zero magnetic moments. a coordinate  To d e s c r i b e  nuclei with  t h i s i n t e r a c t i o n , one f i r s t  defines  system w i t h t h e muon l o c a t e d a t t h e o r i g i n , t h e e l e c t r o n  p o s i t i o n e d a t r a d i u s s, t h e i n t e r a c t i n g n u c l e u s l o c a t e d a t r a d i u s ^ ( d i r e c t e d a l o n g t h e z ' - a x i s ) , and t h e d i s t a n c e between t h e e l e c t r o n and t h e  - 86 nucleus  d e f i n e d by the v e c t o r r which o r i g i n a t e s from the e l e c t r o n .  schematic  diagram of t h i s i s shown i n F i g u r e I I I . 7 .  d e s i g n a t i o n s , the H a m i l t o n i a n  = (a - b) ( S ~op  S h f  • J ) + 3b ~op'  e  v  these  f o r the s u p e r h y p e r f i n e (SHF)  between the e l e c t r o n and a nucleus of s p i n J ~op H  With  interaction  can be w r i t t e n as 1  [171 '  ( S , J ,) z z '  (III.32)  e  ;  A  g where S ,  and J , are the r e s p e c t i v e z'-components of the e l e c t r o n and  z  n u c l e a r s p i n s and "contact"-like  the s u p e r h y p e r f i n e parameters (a & b ) , c o r r e s p o n d i n g  term and  the to a  d i p o l e - d i p o l e term, r e s p e c t i v e l y , are d e f i n e d as ,2  and  (III.33) , b  If  e  2 ^ e  =  8  ^o  8  *  J>  f  ,3+  ^ J  d  r  J  ,2  I  ?(S)  2 cos ( t ) - In  r3  I [  3  ^  r Here g  and  are the g - f a c t o r and Bohr magneton of the n e i g h b o r i n g  n u c l e u s , r e s p e c t i v e l y , and i  Is the angle d e f i n e d by the v e c t o r s r and  Take as an example an i s o t r o p i c w i t h a s i n g l e nucleus one  has  b=0  and  z=z'  of s p i n J (>1), such  R.  s u p e r h y p e r f i n e i n t e r a c t i o n of muonium i n zero magnetic f i e l d .  t h a t the t o t a l H a m i l t o n i a n  In t h i s  case  can be w r i t t e n i n the  form : (S ) + a(s . J ) ' a ~op ~ O p ' ^~Op ~0p f o u r e i g e n v a l u e s of t h i s H a m i l t o n i a n a r e  H = (H  h f  + H  s h f  ) - W  (III.34)  e  e  v  The X  X  1  = \ 4  " " \  2,4  where o)  (h/2*)  00  u  uu  + § J L  [(h/2ir>  0 0  ;  + a] ± \  X  = \  (h/2«)  {[(h/2x)  U ( J 0  U o ( J u u  - f ]  - f  ( j '+ l )  (III.35)  L  2  + j ( j+  l)a } 2  1 / 2  i s the h y p e r f i n e - s t r u c t u r e i n t e r v a l of the p e r t u r b e d muonium atom,  Figure III.7 Diagram r e p r e s e n t i n g from r e f e r e n c e [ 1 7 ] .  the s u p e r h y p e r f i n e  interaction.  Taken  - 88 and  the c o r r e s p o n d i n g  t r a n s i t i o n frequencies  d e f i n e d as u>^s= ( 2 i t / h ) ( \ ^ X  3  « (l/4)(h/2Tc)a)Qg, the  order  Frequencies  (JJQQ.  observable  i n zero  -  field  Since  between the e i g e n s t a t e s  to a f i r s t  t r a n s i t i o n f r e q u e n c i e s oo^, of order u due  0 0  approximation, u i 2  are g e n e r a l l y not  to t i m i n g l i m i t a t i o n s .  and u>  t  f r e q u e n c i e s , which simply  Bearing  large value  r e l a t i o n f o r the  field  zero  of J , one  are  « \  2  of  t h i s i n mind,  terms i n v o l v i n g  i m p l i e s i g n o r i n g the s i n g l e t  Assuming a r e l a t i v e l y  \^  experimentally  an a p p r o x i m a t i o n can be made by i g n o r i n g the o s c i l l a t o r y these  23  are  then o b t a i n s an  state. approximate  r e l a x a t i o n f u n c t i o n of the o b s e r v a b l e  muonium  ensemble, namely g  f z z  (t)  - \  {1 + c o s ( o ) t ) + c o s ( u t ) } 12  (III.36)  2 3  Comparison to muonium i n vacuum shows t h a t the g  shf zz  time-independent p a r t  ( t ) ( r e s i d u a l p o l a r i z a t i o n ) i s reduced from 1/2 The  contact  term of the s u p e r h y p e r f i n e  i n t e r a c t i n g nucleus of t h i s ,  to be w i t h i n about one  one would expect a s u p e r h y p e r f i n e  than a simple  to  1/6.  interaction requires  the  Bohr r a d i u s of the muon. i n t e r a c t i o n to be much  d i p o l e - d i p o l e i n t e r a c t i o n , and  of  Because  stronger  t h e r e f o r e more d i f f i c u l t  to  decouple i n l o n g i t u d i n a l f i e l d .  III.B  Dynamical Relaxation Functions Up  to now  in a solid  the d i s c u s s i o n s on s p i n d e p o l a r i z a t i o n f o r a magnetic probe  have assumed the magnetic probe to be  environment.  Owing to i t s r e l a t i v e l y  s t a t i c with respect  l i g h t mass, however, the muon ( o r  muonium atom) may  be very mobile i n the s t o p p i n g medium.  hopping may  the  s t a t i c case.  alter  to i t s  T h i s motion or  shape of the r e l a x a t i o n f u n c t i o n i n comparison to  T h i s phenomenon comes about because the e f f e c t s of  the  the  "  - 89 i n t e r a c t i o n ( s ) which govern the time e v o l u t i o n of the u in  the s o l i d  the  +  etc.),  hence  the case of a magnetic  s t a t i c nuclear dipoles,  the  Process  Gaussian-Markovian  In  of  are averaged by the motion of the probe ( u , Mu,  polarization  term " m o t i o n a l a v e r a g i n g " .  III.B.l  of  spin  +  local  field  < H^t)  where T  c  H (0) ±  = 1/v  probe.  >  .2 -A_ e x p ( - t / T )  =  (III.37)  C  i s the c o r r e l a t i o n time of the f i e l d  ( t , v ) = exp{-A /v 2  x x  of a l a r g e number of  has a l s o been a p p l i e d  i n t e r a c t i n g with a Gaussian l o c a l f i e l d  t r a n s v e r s e e x t e r n a l magnetic  which i s the f a m i l i a r  2  field,  yielding  field.  to the case of a i n a strong  the a n a l y t i c  external  result  [ e x p ( - t v ) - 1 + tv]}  formula of Kubo and Tomita  t h i s e x p r e s s i o n behaves  Assuming  induces a g r a d u a l change i n the l o c a l  The Gaussian-Markovian assumption  G  fluctuation.  that the f l u c t u a t i o n of the  i s determined by the c u m u l a t i v e e f f e c t  probe  i s assumed to  i s c h a r a c t e r i z e d by the e q u a t i o n  random p r o c e s s e s , each of which  magnetic  field  p r o c e s s , where the c o r r e l a t i o n of the  a G a u s s i a n random p r o c e s s a u t o m a t i c a l l y i m p l i e s local field  In the o r i g i n a l r e s e a r c h  the m o d u l a t i o n of the l o c a l  f o l l o w a "Gaussian-Markovian" field  s t o c h a s t i c a l l y i n the presence  the motion induces a m o d u l a t i o n or f l u c t u a t i o n of  as sensed by the magnetic  Kubo and Toyabe [1,2]  fluctuating  probe hopping  (III.38) [18].  I t i s obvious  p r o p e r l y i n the slow m o d u l a t i o n l i m i t  (v/A «  that 1),  where the r e l a x a t i o n f u n c t i o n e x h i b i t s a G a u s s i a n shape, as w e l l as i n the f a s t modulation l i m i t decay.  (v/A »  1 ) , where the shape  This function i s plotted  resembles an e x p o n e n t i a l  f o r v a r i o u s v a l u e s of A/v  i n Figure  III.8.  F i g u r e I I I . 8 Dynamic Kubo-Tomita high t r a n s v e r s e f i e l d s p i n r e l a x a t i o n f u n c t i o n (Gaussian-Markovian process) p l o t t e d f o r v a r i o u s v a l u e s of A/v.  - 91 The assumption of a Gaussian-Markovian process may  not, however, be a  good one, p a r t i c u l a r l y f o r the case of d i f f u s i o n i n the presence of t r a p s . T h i s i s because a magnetic probe jumping frequency v would  likely  from s i t e to s i t e w i t h a hopping  sense a sudden change i n the l o c a l  d i s t r i b u t i o n and not an a d i a b a t i c or g r a d u a l one.  field  T h i s type of b e h a v i o r i s  i d e a l i z e d by the "Strong C o l l i s i o n Model" [ 1 9 ] .  III.B.2  S t r o n g C o l l i s i o n Model  In  the s t r o n g c o l l i s i o n model, the l o c a l f i e l d  sensed by the magnetic  probe i s assumed to change a b r u p t l y upon c o l l i s i o n , w i t h the l o c a l  field  d i s t r i b u t i o n b e f o r e and a f t e r t h i s c o l l i s i o n being completely u n c o r r e l a t e d . In  t h i s approximation, the time e v o l u t i o n of the dynamical  f u n c t i o n G..(t,v) i s c o n s t r u c t e d from an i n f i n i t e xi  relaxation  s e r i e s of d i s c r e t e  static  r e l a x a t i o n f u n c t i o n s a c c o r d i n g to the e q u a t i o n  G (t,v) = i ±  where g ^ n  I n=o  g^Ct.v)  (111.39)  ( t , v ) i s the r e l a x a t i o n f u n c t i o n f o r the magnetic probes that jump  times i n time t .  The f i r s t  term (n=0)  i n the s e r i e s of E q u a t i o n III.39 i s  e a s i l y understood to be  g  i i  >  (  t  '  v  )  =  e x  P(~  v t  )  S i i ^ )  (III.40)  where e x p ( - v t ) i s the p r o b a b i l i t y t h a t the magnetic time t , and S j ^ C O the  probe does not hop i n  the s t a t i c r e l a x a t i o n f u n c t i o n .  1 S  The f o l l o w i n g term i n  s e r i e s d e s c r i b e s the process i n which the magnetic probe hops a t time t^  (0 < t^ < t) and i s expressed as OO  g ^ ( t , v ) = v / dt  L  e~  v  (  t  " l t  )  g  i i  (t-t ) e~ 1  v t  l  g.^^)  (111.41)  By i n t r o d u c i n g the L a p l a c e  transforms  CO  CO  f ( s ) = / dt e ~  S t  ± 1  8 i l  (t)  and  the exact s o l u t i o n i n the frequency i  I  F^CB.-V)-  v  ffif  n  v)  n=o  Laplace  static  G  XX  G (t)  S t  (III.42)  ±±  domain becomes  (111.43)  f  relaxation function G ^ ( t , v ) , for a  r e l a x a t i o n f u n c t i o n 8^^(t)»  o  n  e  must c a l c u l a t e the i n v e r s e  t r a n s f o r m of E q u a t i o n I I I . 4 3 .  The  Q  = / dt e "  f (s+v) • , _** i i  To o b t a i n the time domain dynamical specific  F.^s)  time domain dynamical  transverse f i e l d  relaxation function  ( t , v ) , f o r the case of a magnetic probe i n t e r a c t i n g w i t h a Gaussian  random l o c a l f i e l d , c o l l i s i o n formula  has been n u m e r i c a l l y c a l c u l a t e d  [20] u s i n g a s t r o n g  s i m i l a r to t h a t of E q u a t i o n I I I . 4 3 .  This c a l c u l a t i o n  was  performed  u s i n g the " K o r r e k t u r - V e r f a h r e n " ( I t e r a t i o n Procedure)  [21,22].  Comparison of the r e l a x a t i o n f u n c t i o n s o b t a i n e d i n t h i s manner  w i t h the Gaussian-Markovian  approximation  method  of the Kubo-Tomita formalism  g i v e n i n E q u a t i o n I I I . 3 8 , r e v e a l s t h a t the two  cases are n e a r l y i d e n t i c a l  except  t h a t the s t r o n g c o l l i s i o n f u n c t i o n e x h i b i t s a s l i g h t l y  rate.  T h i s d i s c r e p a n c y i s p a r t i c u l a r l y n o t i c a b l e i n the l i m i t of slow  hopping  (v/A  or v/a «  respectively).  1, f o r a Gaussian  slower  decay  or L o r e n t z i a n d i s t r i b u t i o n ,  However, the d i f f e r e n c e between the r e l a x a t i o n f u n c t i o n  o b t a i n e d u s i n g the s t r o n g c o l l i s i o n approximation a Gaussian-Markovian of  [18],  and  t h a t o b t a i n e d assuming  process i s so s m a l l t h a t the simple a n a l y t i c  E q u a t i o n III.38 i s g e n e r a l l y p r e f e r r e d f o r data  expression  analysis. Q  The  dynamical  zero f i e l d  spin relaxation function G ( t , v ) z z  f o r the  case  of  a magnetic  probe i n t e r a c t i n g w i t h a Gaussian random l o c a l f i e l d  been c a l c u l a t e d III.43.  has  also  [4] u s i n g the s t r o n g c o l l i s i o n model g i v e n i n E q u a t i o n  This function i s plotted  i n F i g u r e I I I . 9 f o r v a r i o u s v a l u e s of A/v.  Comparison of F i g u r e I I I . 9 w i t h the Gaussian-Markovian  curves of Kubo-Toyabe  [1,2] i n d i c a t e s t h a t , as i n the case of the t r a n s v e r s e f i e l d  function  Q  G ^ C t . v ) , the zero f i e l d  curves generated w i t h the s t r o n g c o l l i s i o n model  decay a t a s l i g h t l y slower r a t e than those based on the  Gaussian-Markovian  approximation, p a r t i c u l a r l y i n the l i m i t of slow hopping (v/A « The modulation of the l o c a l f i e l d the  has a marked e f f e c t on the shape of  long time t a i l of the r e l a x a t i o n f u n c t i o n as w e l l .  d i s c u s s e d , the zero f i e l d  As has a l r e a d y been  s t a t i c r e l a x a t i o n f u n c t i o n s f o r both Gaussian and  L o r e n t z i a n random l o c a l f i e l d asymmetry at long times.  d i s t r i b u t i o n s , e x h i b i t a 1/3  r e c o v e r y of the  For slow modulations of the l o c a l f i e l d ,  r e c o v e r y i s suppressed, and f o r Gaussian random l o c a l f i e l d , asymptotic  1).  this  f o l l o w s the  form  G ( t , v ) <* j exp(- j z z  where the f a c t o r of 2/3  vt)  3/A  (III.44)  i n the exponent can be understood i n t u i t i v e l y  n o t i n g t h a t , on average, 1/3 p r e s e r v e d f o r each hop.  ; for t »  by  of the l o n g i t u d i n a l ( z - a x i s ) p o l a r i z a t i o n i s  In the l i m i t of f a s t f l u c t u a t i o n s (v/A »  1 ) , the  Gaussian l i n e shape begins to mimic an e x p o n e n t i a l , due to m o t i o n a l narrowing, such t h a t G ( t , v ) « exp(-2A t/v) 2  z z  which  (III.45)  tends to zero at long times. A p p l i c a t i o n of the s t r o n g c o l l i s i o n model of E q u a t i o n III.43 to the  problem of a magnetic  probe i n t e r a c t i n g w i t h a L o r e n t z i a n l o c a l  field  - 94 -  Figure III.9 Dynamic zero f i e l d Gaussian Kubo-Toyabe s p i n r e l a x a t i o n f u n c t i o n ( S t r o n g C o l l i s i o n P r o c e s s ) p l o t t e d f o r v a r i o u s v a l u e s of A / v .  y i e l d s the c o r r e s p o n d i n g dynamical  zero f i e l d  relaxation function G  z z  T h i s f u n c t i o n i s shown i n F i g u r e III.10 f o r v a r i o u s v a l u e s of a/v. f e a t u r e to make note of i s t h a t , u n l i k e i t s Gaussian function G (t,v) z z  l i m i t of slow hopping,  and  for t »  3/A,  of the hop  one  G (t,v) z z  This, l i k e  frequency.  the Gaussian  fluctuations,  (v/a »  1 ) , the  to zero a t long times; however, i t has of v, as e x p l a i n e d above.  technique has been shown to be a powerful  s t u d y i n g the d i f f u s i o n and  F i g u r e s I I I . 9 and  to  (III.46)  case, tends  w i t h a random l o c a l f i e l d  fact, In the  * exp(-4at/3)  Thus, the zero f i e l d  field  In  form  the p e c u l i a r f e a t u r e of being independent  of  effect.  o b t a i n s an e q u a t i o n s i m i l a r  E q u a t i o n I I I . 4 4 , whereas i n the l i m i t of f a s t r e l a x a t i o n f u n c t i o n takes the  A major  c o u n t e r p a r t , the  does not e x h i b i t a m o t i o n a l narrowing  the i n i t i a l decay r a t e i s q u i t e independent  (t,v).  tool for  t r a p p i n g behavior of a magnetic probe i n t e r a c t i n g (Gaussian or L o r e n t z i a n ) d i s t r i b u t i o n .  Comparison  I I I . 8 a l s o r e v e a l s the advantage p r o v i d e d by the zero  technique, as opposed to the t r a n s v e r s e f i e l d  ( p r e c e s s i o n ) method, f o r  such s t u d i e s . A d d i t i o n a l d i s c r i m i n a t i o n between s t a t i c and dynamic systems can be o b t a i n e d u s i n g l o n g i t u d i n a l f i e l d , still  where the r e l a x a t i o n f u n c t i o n would  e x h i b i t an e x p o n e n t i a l decay a t long times, even i n r e l a t i v e l y  magnetic f i e l d s .  T h i s behavior can be understood  muonium i n the i n t e r m e d i a t e and h i g h f i e l d s t r o n g c o l l i s i o n assumption field  also  by c o n s i d e r i n g the case of  limits,  of E q u a t i o n I I I . 4 3 .  i n the context of the  The  static  r e l a x a t i o n f u n c t i o n can be o b t a i n e d by combining  AI.110 w i t h the d e f i n i t i o n s of E q u a t i o n AI.21.  high  longitudinal  Equations AI.109  For l o n g i t u d i n a l f i e l d s  and of  F i g u r e I I I . 10 Dynamic zero f i e l d L o r e n t z i a n Kubo-Toyabe s p i n r e l a x a t i o n f u n c t i o n ( S t r o n g C o l l i s i o n P r o c e s s ) p l o t t e d f o r v a r i o u s v a l u e s of a/v.  s t r e n g t h (0 < X < Tt/2), the observable  intermediate  " r e l a x a t i o n f u n c t i o n " ( o m i t t i n g the modulating g ( t , X ) « j ( l + cos X) 2  z z  ;  2  field  Q  1 / 2  ]  (III.47)  parameter, d e f i n e d i n E q u a t i o n  From the s t r o n g c o l l i s i o n model, one then w r i t e s i n frequency  F  (s+v) = Z  time p a r t ) i s w r i t t e n  X = arcsin[l/(l+x )  where x (= | B | / B ) i s the s p e c i f i c I.10.  static longitudinal  1  +  C  °  2  x  ,  (111.48)  v(l-cos x)  2(s+v) -  Z  s  space  Z  and by t a k i n g the i n v e r s e L a p l a c e  transform of E q u a t i o n  I I I . 4 8 , one o b t a i n s  the r e l a x a t i o n f u n c t i o n G  (t,v,X) = y ( l + c o s X) e x p [ - J ( l - c o s X ) t ] _j  ZZ  (III.49)  «_  l i m i t (X ->• 0, x -v <»), i s the  N o t i c e t h a t only f o r the extreme h i g h f i e l d r e l a x a t i o n completely  decoupled  f o r muonium, u n l e s s v = 0.  T h i s argument  can be extended to any r e l a x a t i o n mechanism, as long as the "high  field"  ( s e c u l a r approximation) l i m i t a p p l i e s .  0.01.  In the present work, x «  Since the strong c o l l i s i o n model can be a p p l i e d to any  static  r e l a x a t i o n f u n c t i o n , the dynamical s p i n r e l a x a t i o n f u n c t i o n s f o r the case of a random h y p e r f i n e i n t e r a c t i o n can a l s o be obtained Taking  the L a p l a c e  Equation .rh, f  zz  III.21  ( s )  6^  =  1 2  +  + i[s  +  Substituting this  a  M  2 J 2  l  -i-l  "  (t,v). ZZ  r  l  M if  °2 )L  6<2  S  +  2  + 3/273  .1  r e l a x a t i o n f u n c t i o n of  1-2  a  o )T - 1 ( 1 l  2Q  transform  M  J 22^  expression into Equation  c a l c u l a t e the i n v e r s e L a p l a c e function G  zero f i e l d  III.43.  gives  lr  N  transform of the s t a t i c  using Equation  (11.50) a^)[s  +  3/271  c^)]"  2  I I I . 4 3 , one can n u m e r i c a l l y  to o b t a i n the time domain r e l a x a t i o n  T h i s f u n c t i o n i s shown i n F i g u r e III.11  for selected  -  1.0 -4 0.8  -  0.6  -  0.4  -  1  -  98  L  IT  +i  0.  1. t  2.  3.  (yUs)  F i g u r e I I I . 1 1 Dynamic zero f i e l d L o r e n t z i a n RAHD s p i n r e l a x a t i o n f u n c t i o n (Strong C o l l i s i o n P r o c e s s ) p l o t t e d f o r s e l e c t e d v a l u e s of the hop r a t e v, where O^Q = u s . The f u n c t i o n i s normalized to equal 1 a t t = 0 . =  i  U  - 1  - 99 values  of v, where both O^Q  M 22  a  A N A  a  r  e  e  c  l  u  a  l  t o  ^  u  us  - x  i . As  i n the case  of a magnetic probe i n t e r a c t i n g w i t h a L o r e n t z i a n random l o c a l d i s t r i b u t i o n ( F i g u r e I I I . 1 0 ) , the i n i t i a l i n F i g u r e III.11  i s completely  decay r a t e of the f u n c t i o n shown  u n a f f e c t e d by the hop  which i s somewhat c o u n t e r - i n t u i t i v e , a r i s e s d i r e c t l y L o r e n t z i a n or L o r e n t z i a n - l i k e frequency  field  r a t e v.  This feature,  from the assumption of  distributions.  T h i s behavior  is  a l s o independent of the d i m e n s i o n a l i t y of the d i s t r i b u t i o n s i n c e i n the case of a L o r e n t z i a n random l o c a l  field  the d i s t r i b u t i o n i s  whereas f o r random h y p e r f i n e d i s t o r t i o n s one  has  three-dimensional,  both a one-dimensional  c y l i n d r i c a l component d i s t r i b u t i o n p l u s a two-dimensional p l a n a r distribution.  Thus the behavior  shown i n F i g u r e III.11  component  i m p l i e s that f o r  L o r e n t z i a n and L o r e n t z i a n - l i k e d i s t r i b u t i o n s ( o f a l l dimensions), at  the  e a r l y times i s independent of the motion of the magnetic probe.  particularly totally  i n s t r u c t i v e to c o n s i d e r  the two  l i m i t i n g cases  c y l i n d r i c a l or t o t a l l y p l a n a r d i s t o r t i o n .  shape  It i s  of e i t h e r a  In F i g u r e III.12  the  r e l a x a t i o n f u n c t i o n generated by assuming o n l y a c y l i n d r i c a l d i s t o r t i o n of M the muonium h y p e r f i n e  interaction (i.e.,  = 0) i s p l o t t e d .  The  effect  motion on t h i s component of the r e l a x a t i o n f u n c t i o n i s to suppress the time t a i l ,  even f o r s m a l l hop  long time t a i l °20  =  ^  *  S  tends to z e r o .  s t l o w n  *  n  r a t e s ; f o r s u f f i c i e n t l y high hop The  f i g u r e III.13.  rates,  case of a p u r e l y p l a n a r d i s t o r t i o n Since i n the s t a t i c l i m i t  of  long this (i.e.,  this function  already  tends to zero, the e f f e c t of hopping i s not v e r y n o t i c e a b l e at long  times.  Instead,  evident  at e a r l y times, where i t serves  the e f f e c t of hopping on the r e l a x a t i o n f u n c t i o n i s more to reduce the depth of the minimum.  T h i s same procedure can be a p p l i e d to c a l c u l a t e the dynamical rh transverse f i e l d r e l a x a t i o n function G (t,v). Taking the L a p l a c e  transform  - 100 -  1.0  H  0.  •  L  1.  2.  t  3.  (yU-S)  F i g u r e I I I . 12 Dynamic zero f i e l d L o r e n t z i a n RAHD s p i n r e l a x a t i o n f u n c t i o n f o r a pure c y l i n d r i c a l d i s t o r t i o n ( S t r o n g C o l l i s i o n P r o c e s s ) p l o t t e d f o r s e l e c t e d v a l u e s of the hop r a t e v, where a = 10 u s . The curves have been normalized to equal one at t=0. - 1  20  - 101  1.0  H  0.8  H  1  L  0.  1.  2.  3.  (/xs)  t  F i g u r e I I I . 13 Dynamic zero f i e l d L o r e n t z i a n RAHD s p i n r e l a x a t i o n f u n c t i o n f o r a pure p l a n a r d i s t o r t i o n ( S t r o n g C o l l i s i o n P r o c e s s ) p l o t t e d f o r s e l e c t e d v a l u e s of the hop r a t e v, where 0^2 10 u s . The curves have been normalized to equal one a t t=0. =  - 1  - 102 of the approximation i n E q u a t i o n  £<•> • T  i -  [  +  1 °2o  -  I I I . 2 5 , one  obtains  <&]r  l  +  ("LSI)  TC/O  Substituting  t h i s i n t o Equation  inverse Laplace  transform  independent of the hop The  III.43  and  numerically  y i e l d s a time domain f u n c t i o n which i s  frequency  These f u n c t i o n s have been found  dependence at e a r l y times,  L o r e n t z i a n and L o r e n t z i a n - l i k e d i s t r i b u t i o n s . same d i s t o r t i o n symmetries, but which has a f i n i t e Equation  The  chooses a d i f f e r e n t  strong c o l l i s i o n  assumes the  frequency  distribution  r a t e v.  Notice  i s shown i n F i g u r e III.14  that f o r s m a l l hop  v dependence at e a r l y times, becomes more apparent. argument that can  dynamical f u n c t i o n d e r i v e d  L o r e n t z i a n " d i s t r i b u t i o n of E q u a t i o n  c a l c u l a t e d n u m e r i c a l l y and  III.26  by has  f o r various values  been of  r a t e s , t h i s f u n c t i o n e x h i b i t s no  but as v i s i n c r e a s e d , m o t i o n a l  narrowing  As mentioned, however, t h e r e i s no obvious p h y s i c a l  enable one  to decide which d i s t r i b u t i o n i s most s u i t a b l e .  D i f f u s i o n i n the Presence of T r a p s  Thermally and  I f i n s t e a d one  second moment, such as the d i s t r i b u t i o n d e f i n e d i n  assuming the "modified  III.B.3  owing to the assumption of  I I I . 2 6 , one would expect the r e s u l t i n g f u n c t i o n s to e v e n t u a l l y  m o t i o n a l l y narrow.  the hop  distortion  to L o r e n t z i a n ( o r L o r e n t z i a n - l i k e )  d i s t r i b u t i o n s of dimension l e s s than t h r e e . e x h i b i t no m o t i o n a l  completely  to random a n i s o t r o p i c  i n t e r a c t i o n s have thus f a r been c a l c u l a t e d assuming the  parameters to be d i s t r i b u t e d a c c o r d i n g  to  the  v.  dynamical r e l a x a t i o n f u n c t i o n s corresponding  hyperfine  calculating  a c t i v a t e d d i f f u s i o n and  i m p u r i t i e s has  t r a p p i n g of p o s i t i v e muons at  been observed by many authors  [23,24].  Two  defects  t h e o r i e s have  - 103 -  1.0  0.  1. t  2.  3.  (ylc-S)  F i g u r e III.14 Dynamic zero f i e l d m o d i f i e d L o r e n t z i a n RAHD s p i n r e l a x a t i o n f u n c t i o n f o r a pure c y l i n d r i c a l d i s t o r t i o n ( S t r o n g C o l l i s i o n P r o c e s s ) p l o t t e d f o r s e l e c t e d v a l u e s of the hop r a t e v, where a Q - 10 u s and the parameter X = 0.1. The f u n c t i o n s are normalized to equal one a t t=0. - 1  2  - 104 been proposed to e x p l a i n p o l a r i z a t i o n , one McMullen and be  -  the e f f e c t of such phenomena on the u  suggested by Kehr et a l . [20]  Zaremba [25]  and  Petzinger  and  [26,27] .  another put  applicable  to any  In the case of  latter  some of the  r e l a x a t i o n mechanism, whereas the  the  theory has  transverse  can  the  c o l l i s i o n process i s assumed, making t h i s formalism l a t t e r case only  to muons or t r i p l e t muonium i n t e r a c t i n g w i t h a l o c a l d i p o l a r The  f o r t h by  Both of these t h e o r i e s  extended to a muon i n the muonium s t a t e as w e l l .  former theory, a s t r o n g  spin  +  been used i n the  field  data.  frequency d i s t r i b u t i o n due  applies  field.  present work i n the a n a l y s i s  of  I t assumes a Gaussian a p p r o x i m a t i o n f o r  to d i p o l a r f i e l d s and  expresses the  spin  r e l a x a t i o n f u n c t i o n i n terms of time dependent s i t e o c c u p a t i o n p r o b a b i l i t i e s and  autocorrelation  functions.  The  i n c l u s i o n of the  o c c u p a t i o n p r o b a b i l i t i e s a l l o w s f o r the p o s s i b i l i t y i n thermal e q u i l i b r i u m w i t h r e s p e c t  relaxation function for a multi-state TI  G™(t)  = e x p [ - r ( t ) ] = exp[-  Here the  sum  theory, the  system can  site  that the muons are  to t h e i r s i t e occupancy.  second-order time dependent p e r t u r b a t i o n  M  time dependent  not  Using  transverse  be w r i t t e n as  field  spin  [26,27]  t t / dt'/ dt" N ( f ' H ^ C f - t " ) ] o o 1  9  £ a\ i=l  extends over n s t a t e s ;  ±  2  i s the  second moment of the  (III.52)  frequency  th d i s t r i b u t i o n f o r the i o c c u p a t i o n of the autocorrelation The  i  state, s t a t e and  i s the $^(t)  ^  s  time dependent p r o b a b i l i t y f o r t n e  corresponding  the  site  function.  evaluation  of E q u a t i o n III.52 can be  facilitated  d i m e n s i o n l e s s l i n e w i d t h parameter a [28] , d e f i n e d  by i n t r o d u c i n g  a  as  CO  a = / dt exp(-t/T ) [ d T ( t ) / d t ] o ^  (III.53)  - 105 where x  i s the mean muon l i f e t i m e .  assumed so t h a t the f u n c t i o n s 't'^^Ct)  For s i m p l i c i t y , a  r  e  s t o c h a s t i c hopping i s  g i v e n by an e x p o n e n t i a l of the form  e x p ( - t / x ^ ) , where x^ i s the mean d w e l l time i n the i * " * Equations a =  III.52 and I I I . 5 3 , one o b t a i n s the simple  I a\ £ { N ( t ) } 1=1 ±  [\ u  + \ )~ i  where £{N^(t)} i s the L a p l a c e p r o b a b i l i t y , w i t h the i m p l i c i t  1  state.  By combining  result (111.54)  l  t r a n s f o r m of the  state occupation  t r a n s f o r m v a r i a b l e s = 1/x...  F o r the case  of an e x p o n e n t i a l r e l a x a t i o n , T ( t ) = a t / x ^ , and f o r a Gaussian 1  2  relaxation,  2  F(t) = j at l ^ x  m  Thus, the problem of c a l c u l a t i n g  f o r a m u l t i - s t a t e system has been reduced  the r e l a x a t i o n f u n c t i o n  to d e t e r m i n i n g  which a r e i n g e n e r a l the s o l u t i o n s to a s p e c i f i e d  the £{N^(t)},  set of rate equations.  - 106  -  CHAPTER IV — EXPERIMENTAL RESULTS AND INTERPRETATIONS  Previous  to the present  the g r a i n s of f i n e oxide present  study,  surfaces.  and  work i t was  shown [1-3]  that muonium escapes  powders, i n c l u d i n g the s i l i c a  r e s i d e s i n the e x t r a g r a n u l a r  T h i s phenomenon was  powders used i n the  r e g i o n and  on the g r a i n  f u r t h e r shown to be t o t a l l y independent of  the ambient temperature of the powder g r a i n s (see s e c t i o n I . C ) . the l a r g e s p e c i f i c  s u r f a c e area (390  e x t r a g r a n u l a r muonium, p r o v i d e d chosen f o r the present d e s o r p t i o n and Zero, in  study  ± 40 m /g  [4]) and  2  the 35 A s i l i c a  by  the h i g h y i e l d  of the i n t e r a c t i o n s ( i . e . ,  the present  work to study  transverse f i e l d the behavior  surface  techniques  of muonium on s i l i c a  transverse  (< 10 G) data were taken to i n v e s t i g a t e the  surface hydroxyl hypothesis protons  concentrations.  diffusion,  have been used  In the i n i t i a l  dependence of the t r a n s v e r s e f i e l d  surfaces  of t h i s work ( s e c t i o n I V . A . l ) , temperature  muonium r e l a x a t i o n r a t e f o r s e v e r a l These s t u d i e s were prompted by  t h a t a d i p o l e - d i p o l e i n t e r a c t i o n between muonium and  the the  might be a p r i n c i p a l c o n t r i b u t o r to the r e l a x a t i o n of the u  p o l a r i z a t i o n f o r muonium on the s i l i c a be c o r r e c t , a t h r e e - s t a t e model was d i f f u s i o n and  surface.  ( s e c t i o n IV.A.2), u s i n g  +  spin  Assuming t h i s h y p o t h e s i s  s u r f a c e and  includes  to  the the  A second s e t of experiments were then performed  zero and  longitudinal field  techniques,  i n f o r m a t i o n on the shape of the r e l a x a t i o n as w e l l as the behavior.  hydroxyl  a l s o developed, which d e s c r i b e s  t r a p p i n g of muonium on the s i l i c a  p o s s i b i l i t y of d e s o r p t i o n .  was  surfaces.  ( s e c t i o n IV.A). field  stages  uSR  of  powder, t h i s m a t e r i a l  s p i n r e l a x a t i o n mechanisms) of muonium w i t h  l o n g i t u d i n a l and  Because of  T h i s i n f o r m a t i o n was  to o b t a i n  decoupling  used to d i s c r i m i n a t e between d i f f e r e n t  - 107 r e l a x a t i o n mechanisms, and  prompted the development of a new  t h e o r y , i n v o l v i n g random h y p e r f i n e some of the d a t a . transverse  A third  field,  anisotropies,  the  surface  d i f f u s i o n and  investigate  the  catalysts.  These experiments p r o v i d e d the  first  behavior  of  data were taken  to  surfaces  of p l a t i n u m  observation  o r i g i n f o r one  of the  surface  muonium on  to c o n s t r u c t  the  silica  chemical  also the  i n d i v i d u a l l y subtle;  an unbroken c h a i n  of l o g i c l e a d i n g  surface  i n bulk q u a r t z , may  total  such as  the  to random a n i s o t r o p i c  silica  (due  In t h i s s e c t i o n , data are  surface  as w e l l as  silica  d i s t o r t i o n s of to  a l s o a r i s e from other r e l a x a t i o n mechanisms mainly to the  p r o t o n s ) or perhaps s p i n exchange i n t e r a c t i o n s ( w i t h any  information  the  However, s i g n i f i c a n t c o n t r i b u t i o n s  random l o c a l magnetic f i e l d s  f o r t h to e x t r a c t  operable f o r  s p i n p o l a r i z a t i o n f o r muonium on  interaction.  s p i n r e l a x a t i o n may  impurities).  to some c l e a r  Surfaces  e x p e r i e n c e r e l a x a t i o n due  the muonium h y p e r f i n e  however, they  surface.  Muonium on S i l i c a As  the  loaded  s i t e s f o r muonium on  d e d u c t i o n s c o n c e r n i n g which s p i n r e l a x a t i o n mechanism(s) are  the  of the  ( i n t h i s case p l a t i n u m ) , and  r e s u l t s of these experiments are  a l l o w one  IV.A  surface  surface.  The do  trapping  field  i n t e r a c t i o n s of muonium w i t h the  r e a c t i o n of muonium w i t h s u r f a c e s  silica  which i s used to i n t e r p r e t  e f f e c t of f r a c t i o n a l  F i n a l l y ( s e c t i o n IV.C), t r a n s v e r s e  suggested a p o s s i b l e  relaxation  set of experiments ( s e c t i o n IV.B), a g a i n i n  were done to study the  coverages of helium on muonium.  -  presented and  surface  hydroxyl  paramagnetic  arguments are  put  c o n c e r n i n g the motion of the muonium atoms the  o r i g i n of  the  relaxation interaction.  on  - 108 -  IV.A.l  Transverse Field Results Mu  The of  transverse f i e l d  i n v e r s e temperature  muonium r e l a x a t i o n r a t e \  i s shown as a f u n c t i o n  f o r two sample p r e p a r a t i o n s i n F i g u r e IV.1; the  circles  are the data obtained f o r sample S i 0 ( l ) prepared  squares  r e p r e s e n t the date taken w i t h sample S i 0 ( 3 ) prepared  Let  2  a t 110 °C and the  2  us f i r s t  c o n s i d e r the 110 °C d a t a .  i n t e r p r e t e d as f o l l o w s :  Qualitatively,  a t 600 °C.  these data a r e  The p l a t e a u below about 8 K i s due to muonium  " l o c a l i z e d " i n a host a d s o r p t i o n s i t e (by which i s meant a v e r y common s h a l l o w p o t e n t i a l w e l l ) , and the peak which occurs a t about 25 K i s taken to be due to t r a p p i n g a t l e s s common d e p o l a r i z a t i o n c e n t e r s ( t r a p s i t e s ) . Mu the low temperature  p l a t e a u to the minimum a t about 16 K, \  because of m o t i o n a l narrowing host s i t e s .  due to hopping  From  (T) decreases  of the muonium atom between  Between the minimum and the 25 K peak, the hopping  becomes  s u f f i c i e n t l y r a p i d f o r the muonium atom to reach the t r a p s i t e s b e f o r e i t Mu decays. is  As the temperature  i s i n c r e a s e d beyond the peak temperature,  seen to decrease m o n o t o n i c a l l y .  T h i s decrease  \  (T)  i s a t t r i b u t e d to  d e t r a p p i n g and e v e n t u a l d e s o r p t i o n of the muonium atom from  the g r a i n  surfaces. If  one p i c t u r e s the s i l i c a  powder t a r g e t as a uniform d i s t r i b u t i o n of  s p h e r i c a l p a r t i c l e s of r a d i u s R and mass d e n s i t y p , packed to an o v e r a l l Q  mass packing d e n s i t y p, the maximum c o l l i s i o n frequency  of the Mu atoms w i t h  the g r a i n s u r f a c e s i s e a s i l y shown to be (see Appendix I I I )  where N i s the number of s p h e r i c a l p a r t i c l e s i n the sample, V ^ u i s the f r e e  - 109 -  .00  .05  .10 1/T  .15  .20  .25  (K"')  F i g u r e IV.1 T r a n s v e r s e f i e l d muonium r e l a x a t i o n r a t e as a f u n c t i o n o f i n v e r s e temperature f o r muonium on the s u r f a c e s of f i n e s i l i c a powders (mean g r a i n r a d i u s 35 A) with samples prepared a t 110 °C ( f i l l e d c i r c l e s ) and a t 600 °C (open s q u a r e s ) . The l i n e s shown are f i t s of the t h r e e - s t a t e model d e s c r i b e d i n the t e x t .  volume of the sample (V - V muonium atom.  -  *  the mean thermal v e l o c i t y of  a n c  - 110  In the case of S i 0 , p 2  v  Q  1 S  « 2.2  g/cm .  From E q u a t i o n  3  Mu might expect \^ (T) to e x h i b i t some dependence on the packing which would presumably be more e v i d e n t at h i g h e r  the  IV. 1,  density  temperatures.  one  p,  In order  to  Mu test  f o r a p o s s i b l e packing  d e n s i t y dependence of \  (T), transverse  measurements were made u s i n g sample S i 0 ( 2 ) , which has  a mass  2  d e n s i t y of about 1/3  t h a t of sample S i 0 ( l ) , 2  of the these r e s u l t s w i t h r e v e a l s no  those  obtained  prepared  at 110  f o r the higher  s i g n i f i c a n t d i f f e r e n c e s below ~65  K,  packing °C.  packing  l e a d i n g one  Comparison density  to conclude t h a t  the r e l a x a t i o n r a t e i s l a r g e l y independent of the t a r g e t packing the  temperature and  packing  the i d e a that at low  ranges s t u d i e d .  t a b u l a t e d i n Appendix  density i n  This result i s consistent  temperatures, the muonium atoms are  p r i m a r i l y to motion on  field  the s u r f a c e s of the s i l i c a  with  constrained  grains.  These data  are  IV. Mu  Now  consider  the temperature dependence of \  prepared  at 600  hydroxyl  c o n c e n t r a t i o n i s reduced.  d i f f u s i o n and prepared  °C, a l s o shown i n F i g u r e IV.1,  t r a p p i n g behavior  at 110  °C;  These data  as o r i g i n a l l y  f o r sample  f o r which the  Si0 (3) 2  surface  i n d i c a t e the same g e n e r a l observed f o r sample  however, there are some important  differences.  Si0 (l) 2  In  p a r t i c u l a r , one  observes that the r e d u c t i o n i n the c o n c e n t r a t i o n of s u r f a c e Mu h y d r o x y l groups i s accompanied by a decrease i n \ . Moreover, t h i s e f f e c t seems to be more e v i d e n t at lower (< 30 K) temperatures. In a d d i t i o n to the Mu observed g e n e r a l seen to s h i f t This s h i f t  reduction i n \  to h i g h e r  , the p o s i t i o n of the " t r a p p i n g peak" i s  temperatures w i t h reduced h y d r o x y l  concentration.  i n p o s i t i o n can f o r i n s t a n c e be a t t r i b u t e d to a decrease i n the  - Ill detrapping  frequency,  arising  -  i n t u r n from the h y d r o l y s i s p r o c e s s .  r e s u l t s c l e a r l y i n d i c a t e t h a t the s u r f a c e h y d r o x y l s in  p l a y an important  the d e p o l a r i z a t i o n of the \i s p i n f o r muonium on the s i l i c a  interaction.  A few  prepared  at 600  Si0 (3)  prepared  2  The  however, may  not be  simply  role  surface.  +  p r e c i s e r o l e played,  These  to p r o v i d e a d i p o l e - d i p o l e  data p o i n t s were a l s o taken w i t h sample S i 0 ( 4 ) , 2  °C, which reproduce the same behavior at the  as observed f o r sample  same temperature.  temperature dependence of the average muonium h y p e r f i n e - s t r u c t u r e  interval v g 0  about 110  was  a l s o s t u d i e d , u s i n g the same s i l i c a  powder prepared  °C, over the temperature range 17 K < T < 300  measurements were made i n h i g h t r a n s v e r s e F i g u r e IV.2.  Above ~100  the vacuum v a l u e m a j o r i t y of t h e i r  K,  the v a l u e s  (~4463.3 MHz),  about -0.6%.  field  obtained  indicating  time i n the e x t r a g r a n u l a r  decreases r a p i d l y to a v a l u e T h i s e f f e c t has  (~500 for v  K [5].  G) and Q 0  at  These  are shown i n  are c o n s i s t e n t  region. at 17.0  Below ~100 ± 0.1  K,  K,  v  0 0  a change of on  be compared to the room temperature r e s u l t s  of  -0.12% and  -0.13% observed f o r hydrogen and  deuterium, r e s p e c t i v e l y (see  s e c t i o n I.D.2).  These measurements are s e n s i t i v e only to the i s o t r o p i c  of the h y p e r f i n e  i n t e r a c t i o n and  thus p r o v i d e  However, i f one  observed d i s t o r t i o n i s due  Operating the h y d r o x y l  no i n f o r m a t i o n r e g a r d i n g  i s c o r r e c t i n assuming that  to muonium adsorbed onto the s i l i c a  e a s i l y argued t h a t the r e s u l t i n g h y p e r f i n e  a n i s o t r o p i c , and  the  been a t t r i b u t e d to a d s o r p t i o n of muonium  s u r f a c e and may  a n i s o t r o p i c components.  with  that the muonium atoms spend  of 4437 ± 4 MHz  the s i l i c a  is  The  thus induce  i n t e r a c t i o n would  a r e l a x a t i o n of the \i s p i n +  part any  the  surface, i t be  polarization.  under the assumption that a d i p o l e - d i p o l e i n t e r a c t i o n between  protons  and  the muonium atoms i s a major c o n t r i b u t o r to  the  - 112  4470 4465 4460 N  4455 4450 -  oo 4445 -  4440 4435 4430 0  100  200  300  T (K)  F i g u r e IV.2 H y p e r f i n e - s t r u c t u r e i n t e r v a l v e r s u s temperature f o r muonium i n t e r a c t i n g w i t h the s i l i c a s u r f a c e (~110 C p r e p a r a t i o n ) . The curves through the p o i n t s a r e f i t s t o the data u s i n g E q u a t i o n IV.6. a  - 113 s p i n d e p o l a r i z a t i o n of the muonium ensemble, a t h r e e - s t a t e model was developed  [6,7].  T h i s model u t i l i z e s a p r e v i o u s l y developed m u l t i - s t a t e  d i f f u s i o n and t r a p p i n g theory  [8-10], which i s d e s c r i b e d  To d e s c r i b e the t r a n s v e r s e  field  data  shown i n F i g u r e IV.1, one chooses  a model which c h a r a c t e r i z e s the muonium o c c u p a t i o n terms of e q u i v a l e n t remaining finite  i n Chapter I I I .  s i t e s on the s u r f a c e i n  t r a p p i n g s i t e s of r e l a t i v e c o n c e n t r a t i o n C  f r a c t i o n l ~ C of e q u i v a l e n t host  sites.  t  Since  the s i t u a t i o n can  by a t h r e e - s t a t e model i n which muonium atoms have the  p o s s i b i l i t y of occupying state.  and a  there i s a l s o a  temperature dependent p r o b a b i l i t y f o r d e s o r p t i o n ,  be r e p r e s e n t e d  t  By denoting  e i t h e r of the two adsorbed s t a t e s or the desorbed  the o c c u p a t i o n  p r o b a b i l i t i e s f o r the host  t r a p s i t e s and the desorbed s t a t e as N  Q  , N  F C  s i t e s , the  and N ^ , r e s p e c t i v e l y (which obey  the n o r m a l i z a t i o n c o n d i t i o n N + N + N , . = 1) one can d e f i n e the f o l l o w i n g o t f set of coupled r a t e e q u a t i o n s : •  N  -K t C  o  K  +  v C o  of  V  Here V  q  and v  t  fo^- t) ft t  V  \^~ t'  of)  V  C  -[v (l-C ) t  t  +  tf  V  tfJ  V  -[v  V  f o  t  +  a r e the s u r f a c e hop r a t e and d e t r a p p i n g  In E q u a t i o n  assumed to be equal  Q  V  between host  f  detrapping  IV.2.  Assuming A r r h e n i u s  rates are r e s p e c t i v e l y  behavior,  t  (IV.2)  C ]_ t  rate,  and v ^  t  are  a r e the d e s o r p t i o n  s i t e to a t r a p s i t e i s sites.  n o r m a l i z a t i o n c o n d i t i o n , one can then o b t a i n the L a p l a c e to E q u a t i o n  o t  N  and v  IV.2, the hop r a t e from a host to the hop r a t e v  N  C  (l-C )  the host and t r a p s i t e a d s o r p t i o n r a t e s and rates.  C  A p p l y i n g the  transform  solutions  the s u r f a c e hopping and  - 114 V  q  = v exp(-E /kT) 1  where E  and E  Q  and  Q  of  V  =  3  e x p  = v exp(-E /kT) 2  (IV.3)  t  are the r e s p e c t i v e a c t i v a t i o n e n e r g i e s .  t  d e s o r p t i o n r a t e s are d e f i n e d V  v  -  H o E  +  Similarly,  the  as  E )/kT}  and  f  v  = v exp{-(E  t f  4  t  + E )/kT}  (IV.4)  f  th where the q u a n t i t y E^+E^ Finally, v, fo  the a s s o c i a t e d a d s o r p t i o n r a t e s are d e f i n e d  = F(T) P (T) o  Here P ( T )  and P ( T )  Q  t  trap s i t e s , atoms w i t h model was °C  i s the d e s o r p t i o n energy f o r the i  and  v  £  ft  = F(T) P (T) t  F(T)  the g r a i n s u r f a c e s , g i v e n i n E q u a t i o n used to f i t the t r a n s v e r s e  field  data  and  q  =  of the muonium three-state ° C and  = 0 and  the f i t t e d  c o n d i t i o n s assumed.  the f i t t e d  Because of the expected low  This  and  f o r both the 110  c o n d i t i o n s were t r i e d as w e l l , but  i s shown i n F i g u r e IV.1  frequency  IV.1.  c o n d i t i o n of N  found to be independent of the i n i t i a l  IV.1(a).  (IV.5)  i s the c o l l i s i o n  p r e p a r a t i o n s , assuming an i n i t i a l  curve  as  are the t r a p p i n g p r o b a b i l i t i e s f o r the host  r e s p e c t i v e l y , and  Other I n i t i a l  state.  =  600 1.  parameters were The  resulting  parameters are g i v e n i n T a b l e  c o n c e n t r a t i o n of t r a p s i t e s ,  the  Mu effect  on \  (i.e.,  (T) of d i r e c t d e s o r p t i o n from the  = 0).  the 600  the f i t t e d  assumed  In a d d i t i o n , the t r a p p i n g p r o b a b i l i t i e s P ( T )  were both s e t equal for  traps was  Q  to u n i t y .  ° C preparation.  T h i s model has The  resulting  parameters are g i v e n i n Table  negligible  and  P (T) t  a l s o been used to f i t the  curve  IV.1(b).  i s shown i n F i g u r e IV.1 In t h i s f i t , some of  parameters were not w e l l determined, owing to the l a c k of data above 85 Because of t h i s , only a few e r r o r estimates were o b t a i n e d . As can be seen from F i g u r e IV.1, t h i s simple model d e s c r i b e s the  data and the K.  data  - 115 Table IV.1(a) Parameter  F i t R e s u l t s f o r Sample S i 0 ( l ) Prepared a t 110 °C 2  Value  Error  ( / d e g . f r . = 10.5/11) 2  X  v  l  87  +86 / -37  us  v  2  11.2  +6.6 / -2.8  us'  v  3  441  +889 / -230  US  E  o  63  +10 / -8  K  E  t  118  +25 / -17  K  E  f  212  +108 / -43  K  1.02  +0.06 / -0.06  us'  18.9  +3.6 / -5.8  us'  c  t  Table IV.1(b) Parameter  %  0.66  F i t R e s u l t s f o r Sample S i 0 ( 3 ) Prepared a t 600 °C 2  Value  Error  ( x / d e g . f r . = 11.6/8) 2  0.548 v  2  us  - 1  4.21  +0.42 /  us  - 1  - 1  v  3  1557  +2.3 /  us  E  o  8.37  +0.021 /  K  E  t  93  +3.5 /  K  E  f  97 0.61  c  t  K +0.12 / -0.12  US  - 2  ^  4.39  us --2  0.52  %  £  - 116 quite well. simpler  Of course,  the r e l a x a t i o n i n t e r a c t i o n assumed here i s much  than the a c t u a l i n t e r a c t i o n , and  the p h y s i c a l parameters (hop t h i s model.  some of i t s f e a t u r e s may  rates, a c t i v a t i o n energies,  obscure  e t c . ) deduced from  However, the q u a l i t a t i v e e x p l a n a t i o n a f f o r d e d by t h i s model i s  satisfactory. two  -  For example, by comparing the f i t r e s u l t s obtained  preparations  (see Table  I V . l ) , one  are s i g n i f i c a n t l y reduced f o r the 600 °C p r e p a r a t i o n . f o r the 600  f o r the  observes that the a c t i v a t i o n e n e r g i e s °C p r e p a r a t i o n as compared to the  This i s p a r t i c u l a r l y noticeable for E . Q  110  In a d d i t i o n ,  °C p r e p a r a t i o n i s s m a l l e r than t h a t f o r the 110  °C  preparation.  To understand what i m p l i c a t i o n s t h i s has with r e s p e c t to the motion of muonium atoms on the s i l i c a hop  rates v  (defined i n Equation  Q  both sample p r e p a r a t i o n s . IV.3.  The  Comparison of the two  presence of the h y d r o x y l Mu  s u r f a c e , i t i s i n s t r u c t i v e to p l o t the  atoms at low  significantly  IV.3)  temperatures.  to i n h i b i t  Another important  s m a l l e r f o r the 600  are are shown i n F i g u r e  i n F i g u r e IV.3  groups serves  suggests that  to h i g h e r The  the  s u r f a c e d i f f u s i o n of d i f f e r e n c e i s that v  is  °C p r e p a r a t i o n as opposed to the 110  °C  rate.  The  reduced  r a t e i s r e s p o n s i b l e f o r the observed s h i f t of the t r a p p i n g peak temperatures, as suggested  data i n F i g u r e IV.2,  earlier.  showing the temperature dependence of  the  h y p e r f i n e - s t r u c t u r e i n t e r v a l , have been f i t assuming the model of a Mu thermalized  i n a system w i t h  s i t u a t i o n i s represented ~\o  "  the  2  p r e p a r a t i o n , i n d i c a t i n g a r e d u c t i o n i n the d e t r a p p i n g detrapping  surface  as a f u n c t i o n of temperature, f o r  r e s u l t i n g curves  curves  the  +  t o t a l area A and  by the equation  j{U\f)e*p[-*/K)]  + vJ  t o t a l f r e e volume V f .  atom  This  [2] l  + £  (^ )exp(E/kT)]. U  (IV.6)  - 117  1.5  H  1  0.  L  10.  5. T  15.  (K)  F i g u r e IV.3 S u r f a c e hop r a t e v v e r s u s temperature, c a l c u l a t e d from E q u a t i o n IV.3, u s i n g the f i t t e d parameters of T a b l e IV.1. The s o l i d l i n e corresponds to the 110 °C p r e p a r a t i o n , and the dashed curve corresponds to the 600 °C p r e p a r a t i o n . 0  - 118 where V  g Q  IS  Q  surface, V  q  -  the h y p e r f i n e - s t r u c t u r e i n t e r v a l f o r muonium on the q  Mu  i s the vacuum v a l u e ,  silica  i s the thermal de B r o g l i e wavelength g  and E i s the a c t i v a t i o n energy at  4437 MHz,  the v a l u e measured a t the lowest  i s o b t a i n e d by f i x i n g  In the f i t s , V  for desorption.  the r a t i o A/Vf  temperature.  a t a v a l u e of 6.24  ( c a l c u l a t e d u s i n g the model of Appendix I I I ) and energy  E.  T h i s f i t gave a Chi-square  an a c t i v a t i o n energy setting  of 76  of 5.66  (+35.4/-12.8) K.  the  f o r 5 degrees The  solid  was  o  The  x 10  fitting  q  dashed cm  5  fixed curve  -1  activation of freedom and  l i n e i s obtained  the a c t i v a t i o n energy E equal to the sum E^ + E  Q  (= 275  by  K),  c a l c u l a t e d u s i n g the v a l u e s from the t h r e e - s t a t e model f i t g i v e n i n T a b l e I V . 1 ( a ) , and 5 degrees  fitting  the r a t i o A/V . f  T h i s f i t gave a Chi-square  of freedom and a v a l u e of A/V  of 4 f o r  = 3587 (+2865/-1960) cm , -1  f  o r d e r s of magnitude l e s s than t h a t g i v e n by the model c a l c u l a t i o n . result  i s not s u r p r i s i n g  owing to the tendency  IV.A.2  to  zero and  longitudinal field  sample S i 0 ( 4 ) , prepared 2  overestimate,  aggregate.  a t 110  asymmetry s p e c t r a taken at 7.0 °C, are shown i n F i g u r e IV.4.  the i n t e r p r e t a t i o n of the a s s o c i a t e d t r a n s v e r s e f i e l d  IV.1,  these data correspond  spectrum  to the s t a t i c l i m i t .  N o t i c e a l s o t h a t the r e l a x a t i o n i s almost  l o n g i t u d i n a l f i e l d s of o n l y a few Gauss.  Chapter  I I I one  recalls  ± 0.2  data of F i g u r e field  tends to zero a t completely  decoupled  From the d i s c u s s i o n s i n  t h a t , f o r r e l a x a t i o n s due  K  According  N o t i c e t h a t the zero  e x h i b i t s an i n i t i a l e x p o n e n t i a l - l i k e decay and  long times. for  f o r the s i l i c a powder g r a i n s to  This  Zero and L o n g i t u d i n a l F i e l d R e s u l t s  The for  s i n c e the model c a l c u l a t i o n i s an  two  to random l o c a l magnetic  - 119 -  F i g u r e IV.4 Zero and l o n g i t u d i n a l f i e l d asymmetry s p e c t r a f o r muonium on the s i l i c a s u r f a c e (110 "C p r e p a r a t i o n ) a t 7.0 ± 0.2 K. The zero f i e l d data are r e p r e s e n t e d by the square symbols and are compared to data taken a t t h r e e d i f f e r e n t l o n g i t u d i n a l f i e l d s ; the c i r c l e s correspond to 1.0 G, the t r i a n g l e s to 3.0 G and the diamonds to 10.0 G. The curve through the zero f i e l d data i s a f i t to the data u s i n g the s t a t i c zero f i e l d r e l a x a t i o n f u n c t i o n of E q u a t i o n I I I . 2 1 .  - 120 fields  (RLMF), the  u n l e s s one  i s i n the  o r d e r e d " moments. limit,  any  relaxation function  I t has  r e c o v e r y of the  time r e c o v e r y , one  i n i t i a l Gaussian shape l i m i t of  "randomly  a l s o been shown i n Chapter I I I t h a t i n the  Since the data f o r muonium on long  e x h i b i t s an  f a s t hopping l i m i t or i n the  form of r e l a x a t i o n due  e x h i b i t a 1/3  -  has  to random l o c a l magnetic f i e l d s would  initial u the  only  static  +  silica two  s p i n p o l a r i z a t i o n at long surface  below about 7 K  times.  shows no  possibilities:  (1)  The and  d i p o l e moments ( h y d r o x y l protons) are randomly ordered, the muonium atoms are not s t a t i c on the s i l i c a s u r f a c e .  (2)  A d i p o l e - d i p o l e c o u p l i n g i s not the p r i n c i p a l i n t e r a c t i o n governing the time e v o l u t i o n of the \i s p i n p o l a r i z a t i o n . +  The  first  that  of these p o s s i b i l i t i e s i s d i f f i c u l t  there are about 4 h y d r o x y l groups per  translates  i n t o one  concentration, justify.  2  on  the  h y d r o x y l f o r every other S i atom.  surface,  With such a  d i s t r i b u t i o n of random l o c a l magnetic f i e l d s ,  in longitudinal f i e l d .  w i t h the  whereas f o r a dynamic system, the  to e x h i b i t a decay at long data shown i n F i g u r e  IV.4  decoupled f o r v e r y s m a l l  the  R e c a l l from Chapter I I I that  c o m p l e t e l y decoupled i n a l o n g i t u d i n a l f i e l d field;  which  large  on  the  postulation  for a  static  spin relaxation  the order of the  r e l a x a t i o n would  r e l a x a t i o n to be  almost  f i e l d s , which i s i n c o n s i s t e n t  expect f o r a dynamic probe.  From t h i s argument one  random d i p o l a r i n t e r a c t i o n i s not  the  may  of  relaxation  can  local  continue  times, even In h i g h l o n g i t u d i n a l f i e l d s . show the  to  concomitant  observed shape of the  muonium atom i n t e r a c t i n g w i t h a random l o c a l f i e l d ,  dipolar  fact  the l i m i t of randomly ordered moments would be d i f f i c u l t  d i f f u s i n g muonium i s i n c o n s i s t e n t  be  silica  Moreover, even i f t h i s were accepted, along w i t h the  Lorentzian  function  nm  to r e c o n c i l e w i t h the  The  completely  w i t h what one conclude t h a t  would a  p r i n c i p a l r e l a x a t i o n mechanism f o r  - 121 muonium on the s i l i c a  surface.  The  -  fact  that the data i n F i g u r e IV.4  the r e l a x a t i o n to be e a s i l y quenched i n low suspect  field,  however, leads one  a random a n i s o t r o p i c h y p e r f i n e d i s t o r t i o n (RAHD) as a  candidate.  One  would a l s o e x h i b i t a decay at long times, assumes a r e l a x a t i o n due  longitudinal field static  limit,  L e t us now alone  to RAHD, the low  fields.  temperature zero  and  i n d i c a t e s muonium to be i n the  the i n t e r p r e t a t i o n of the t r a n s v e r s e  c o n s i d e r whether a random a n i s o t r o p i c h y p e r f i n e  can adequately  e x p l a i n the d a t a .  field  interaction  In t h i s case, an e x p o n e n t i a l - l i k e  as l o n g as the f r e q u e n c i e s are d i s t r i b u t e d a c c o r d i n g  Lorentzian distribution function. a 1/3  interaction,  IV.1.  decay i s expected  III,  likely  even i n h i g h l o n g i t u d i n a l  data shown i n F i g u r e IV.4  i n agreement w i t h  data of F i g u r e  to  can e a s i l y argue that the r e l a x a t i o n f u n c t i o n f o r a dynamic  muonium atom, i n t e r a c t i n g v i a a random a n i s o t r o p i c h y p e r f i n e  Thus i f one  shows  As d i s c u s s e d i n Appendix I and  r e s i d u a l p o l a r i z a t i o n i s expected  i n the s t a t i c  to a  i n Chapter  limit  for a  cylindrically distorted  random h y p e r f i n e i n t e r a c t i o n .  of random l o c a l f i e l d s ,  a c y l i n d r i c a l d i s t o r t i o n of the muonium h y p e r f i n e  i n t e r a c t i o n i s not  sufficient  to e x p l a i n the d a t a .  I f , however,  i n c l u d e s a p l a n a r d i s t o r t i o n component as w e l l , one has  the r e q u i r e d e x p o n e n t i a l - l i k e i n i t i a l decay, and  l o n g times  (Equation III.21).  The  curve  Thus, as i n the  of 45.1  f o r the c y l i n d r i c a l and were found  to be 12.1  respectively.  one  o b t a i n s a f u n c t i o n which a l s o tends to zero at  i n F i g u r e IV.4  i s a f i t of  11.10, assuming the r e l a x a t i o n f u n c t i o n of I I I . 2 1 , to the d a t a . gave a Chi-square  case  f o r 28 degrees of freedom, and  Equation  The f i t  the f i t t e d  results  p l a n a r d i s t o r t i o n f r e q u e n c i e s ( d i s t r i b u t i o n widths)  (+1.59/-1.33) u s  - 1  and 0.86  (+0.085/-0.090)  The muonium asymmetry ( f o r t r i p l e t muonium) was  us  - 1  allowed  , to  - 122 vary i n the f i t and was is  found  -  to equal 0.103  (+0.0047/-0.0042).  c o n s i s t e n t w i t h t h a t o b t a i n e d f o r the c o r r e s p o n d i n g  This value  transverse f i e l d  data  ( t a k e n w i t h the same sample and p r e p a r a t i o n ) . As has  a l r e a d y been d i s c u s s e d , the e f f e c t s of a random a n i s o t r o p i c  h y p e r f i n e i n t e r a c t i o n can be e f f e c t i v e l y decoupled r e s u l t s of the zero f i e l d  f i t , this  for  »  translates into a f i e l d  a few Gauss, which i s c o n s i s t e n t w i t h the data i n F i g u r e  o^.  From the  on the order of  IV.4.  M  the v a l u e s of O^Q  By s u b s t i t u t i n g fit,  into  the t r a n s v e r s e f i e l d  a transverse f i e l d  A N <  * 22 a  0  approximation  relaxation rate \  t  a  i  n  e  <  i  i  n  t  n  e  zero  field  of E q u a t i o n I I I . 2 5 , one  of 3.1  M u  D  ± 0.38  us  - 1  .  obtains  This result i s  c o n s i s t e n t w i t h the r e l a x a t i o n r a t e determined f o r the a s s o c i a t e d t r a n s v e r s e field  data, shown i n F i g u r e The  zero and  IV.1.  longitudinal field  u s i n g sample S i 0 ( 3 ) prepared 2  asymmetry s p e c t r a taken a t 3.6  at 600  the case of the data f o r the 110  U  C  U  C,  p r e p a r a t i o n , the zero f i e l d  e x h i b i t s an e x p o n e n t i a l - l i k e decay and The  curve through the zero f i e l d  data, assuming the s t a t i c of  Equation III.21.  freedom, and  The  the f i t t e d  (+0.2/-0.15) [ i s ,  vary i n the f i t and  - 1  also  f i t gave a Chi-square  of 87.3  r e s u l t s f o r the c y l i n d r i c a l and to be equal to 4.4  respectively.  found  As i n  spectrum  tends to zero at long  data i s a f i t of E q u a t i o n 11.10  K,  times. to the  random a n i s o t r o p i c h y p e r f i n e d i s t o r t i o n f u n c t i o n  d i s t o r t i o n parameters were found 1.8  are shown i n F i g u r e IV.5.  ± 0.2  f o r 53 degrees of the p l a n a r  (+0.8/-0.9) | i s , - 1  The muonium asymmetry was  to equal 0.069 (+0.0023/-0.0022).  c o n s i s t e n t w i t h t h a t o b t a i n e d f o r the c o r r e s p o n d i n g but i t i s s i g n i f i c a n t l y l e s s than t h a t found  and  allowed  to  This value i s  transverse f i e l d  data,  f o r sample S i 0 ( 4 ) prepared 2  at  - 123  -  T  (N T  ^  00 •  O  1  CO •  O  1  ^ •  O  1  CN O  CD  O  •  /Q:J.8UJUJA'SV p a p a j J O Q  F i g u r e IV.5 Zero and l o n g i t u d i n a l f i e l d asymmetry s p e c t r a f o r muonium on the s i l i c a s u r f a c e (600 "C p r e p a r a t i o n ) , a t 3.6 ± 0.2 K. The zero f i e l d data are r e p r e s e n t e d by the square symbols and are compared to data taken a t two d i f f e r e n t l o n g i t u d i n a l f i e l d s ; the c i r c l e s correspond to 0.2 G and the t r i a n g l e s to 0.5 G. The curve through the zero f i e l d data i s a f i t to the data u s i n g the zero f i e l d s t a t i c r e l a x a t i o n f u n c t i o n of E q u a t i o n I I I . 2 1 .  - 124 110  °C.  T h i s d i f f e r e n c e a r i s e s simply  -  because the window on sample  i s 25 \im t h i c k , whereas the window on sample S i 0 ( 3 ) 2  i s 50 um  Si0 (4) 2  thick.  More  muons are t h e r e f o r e stopped i n the window f o r sample S i 0 ( 3 ) , thereby  adding  2  to  the diamagnetic f r a c t i o n observed.  parameters o b t a i n e d  here w i t h  those  A comparison of the  obtained  f o r the 110  distortion  "C  preparation  i n d i c a t e s a correspondence between the muonium h y p e r f i n e d i s t o r t i o n and c o n c e n t r a t i o n of s u r f a c e h y d r o x y l  the  groups; the c y l i n d r i c a l component O^Q M  increases, while hydroxyl  the p l a n a r component  concentration.  decreases,  with i n c r e a s i n g  T h i s r e s u l t i s r a t h e r i n t e r e s t i n g because i t  suggests t h a t the presence of the h y d r o x y l  groups a f f e c t s the  environment of the muonium atom i n a manner which induces d i s t o r t i o n symmetry i n the muonium h y p e r f i n e I n t e r a c t i o n . fact  that the observed d i s t o r t i o n f o r the 110  have an enhanced c y l i n d r i c a l component and r e l a t i v e to the 600  U  C  preparation,  a diminished  the h y d r o x y l  Although t h i s i n t e r p r e t a t i o n does adequately one  cannot exclude  the p o s s i b i l i t y  an a s s o c i a t e d Moreover,  the  °C p r e p a r a t i o n i s shown to  suggests that the  i n t e r a c t i o n between the muonium atom and  local  planar  component,  electrostatic groups i s r e p u l s i v e .  e x p l a i n the observed  behavior,  that t h i s r e s u l t could merely be  a  m a n i f e s t a t i o n of a combined r e l a x a t i o n i n v o l v i n g both random h y p e r f i n e a n i s o t r o p i c s and By  s u b s t i t u t i n g the v a l u e s of O^Q  into Equation to  random d i p o l a r f i e l d s .  be 1.5  transverse  III.25,  ± 0.25  u-s .  field  data.  -1  A R U  M * 22  the t r a n s v e r s e f i e l d This r e s u l t  Data were a l s o obtained  with  a  ^  o r  t  n  preparation Mu  e  relaxation rate \  i s again consistent with  sample S i 0 ( 3 ) 2  (prepared  i s calculated the  at 600  associated  °C), i n  - 125 the dynamic r e g i o n from 6 K to 20 K. taken at 16.0  ± 0.1  -  The  zero and  K (where the muonium atoms are b e l i e v e d to be  between the host s i t e s ) are shown i n F i g u r e IV.6. this  temperature  longitudinal f i e l d  e x h i b i t a s l i g h t decrease  The  zero f i e l d  i n the i n i t i a l  data  hopping data f o r  decay ( m o t i o n a l  narrowing) as compared to the s t a t i c case of F i g u r e IV.5,  but  there i s no  M i n d i c a t i o n of the 0 ^ 2 minimum. dynamical model d e r i v e d from  The  absence of a minimum i s p r e d i c t e d by  the s t a t i c  formula of E q u a t i o n III.21 and  s t r o n g c o l l i s i o n model of E q u a t i o n III.43 however, the observed function.  m o t i o n a l narrowing  T h i s i n c o n s i s t e n c y merely  f r e q u e n c i e s are o n l y approximately d i s t r i b u t i o n , and The sites  (host and  the host s i t e s  (see F i g u r e s III.10 - I I I . 1 2 ) ; effect  i s i n c o n s i s t e n t with  r e f l e c t s the f a c t  breaks  t h a t the  r e s u l t s i n d i c a t e two  different  t h a t the d e p o l a r i z a t i o n of the u  ( a t low temperatures)  distortion  surface.  Thus f a r i t  s p i n f o r muonium i n  +  i s l a r g e l y due  types of a d s o r p t i o n  to random a n i s o t r o p i c  hyperfine d i s t o r t i o n s , with p o s s i b l y a small c o n t r i b u t i o n a r i s i n g random l o c a l magnetic f i e l d s  this  down when motion i s i n t r o d u c e d .  t r a p s i t e s ) f o r muonium on the s i l i c a  has been concluded  the  d i s t r i b u t e d a c c o r d i n g to a L o r e n t z i a n  t h i s approximation  transverse f i e l d  the  produced  by the h y d r o x y l  from  the  protons.  To d e c i p h e r which r e l a x a t i o n mechanism(s) are o p e r a t i n g at the t r a p sites,  zero and  longitudinal field  data were taken a t the h i g h  peak, where muonium i s presumed trapped. sample S i 0 ( 4 ) prepared 2  of  at 110  data taken a t 25 ± 0.5  °C, are shown i n F i g u r e IV.7.  t h i s data w i t h the low temperature  atoms are thought  The  data of F i g u r e IV.4,  to be p r i m a r i l y i n the host s i t e s ,  d i f f e r e n t d e c o u p l i n g b e h a v i o r s f o r the two r e l a x a t i o n a t the host s i t e s i s almost  temperature  sites.  A  comparison  where the muonium  shows two  distinctly  Specifically,  c o m p l e t e l y decoupled  K, f o r  the  (quenched) f o r a  - 126 -  CN ~!  R  ~:  00 CD  r  CD  CD  ^-  CD  CM  CD  o  F i g u r e IV.6 Zero and l o n g i t u d i n a l f i e l d asymmetry s p e c t r a f o r muonium on the s i l i c a s u r f a c e (600 C p r e p a r a t i o n ) , a t 16.0 ± 0.1 K. The zero f i e l d data are r e p r e s e n t e d by the square symbols and are compared to data taken a t three d i f f e r e n t l o n g i t u d i n a l f i e l d s ; the c i r c l e s correspond to 0.2 G, the t r i a n g l e s to 0.5 G and the diamonds to 1.0 G. U  - 127 -  1  1  1  1  1  1  co  F i g u r e IV.7 Zero and l o n g i t u d i n a l f i e l d asymmetry s p e c t r a f o r muonium on the s i l i c a s u r f a c e (110 "C p r e p a r a t i o n ) , a t 25.0 ± 0.5 K. The zero f i e l d data are r e p r e s e n t e d by the square symbols and are compared to data taken a t f o u r d i f f e r e n t l o n g i t u d i n a l f i e l d s ; the c i r c l e s correspond to 4.0 G, the t r i a n g l e s to 10.0 G, the diamonds to 25.0 G and the c r o s s e s to 45.0 G. The curve through the zero f i e l d data i s a f i t to the data u s i n g the zero f i e l d s t a t i c r e l a x a t i o n f u n c t i o n of E q u a t i o n I I I . 2 1 .  - 128 longitudinal field  of o n l y 2.0  -  G, whereas a t the t r a p s i t e s  s m a l l unquenched component, even up to 45 G. field  data i s a f i t of E q u a t i o n 11.10  field  r e l a x a t i o n f u n c t i o n of E q u a t i o n I I I . 2 1 .  83.8  f o r 28 degrees  were found  to be equal to 13  respectively. 0.11  of freedom, and  The  sample S i 0 ( 3 ) prepared 2  and  IV.8  quenched f o r the 600 curve through static 39.1  The  also f i t t e d  °C, i s shown i n F i g u r e IV.8.  The  found  and  t r i p l e t muonium asymmetry was  to be  A comparison of  be more e a s i l y The the  f i t gave a Chi-square  1.04  ,  planar  distortion  (+0.075/-0.074)  also f i t t e d  associated transverse f i e l d  of  and  us  found  relaxation  - 1  ,  to be  rates,  from E q u a t i o n I I I . 2 5 , are a l s o c o n s i s t e n t w i t h the r e s p e c t i v e  transverse f i e l d  data f o r both the 110  These r e s u l t s suggest  °C and 600  U  C  preparations.  t h a t the nature of the r e l a x a t i o n i n the t r a p  be a f u n c t i o n of the s u r f a c e p r e p a r a t i o n .  A paramagnetic  ion, for  i n s t a n c e , which i s somehow n e u t r a l i z e d by baking at h i g h temperatures, explain t h i s data. the C a b - 0 - S i l EH-5 ppm  - 1  K, f o r  "C p r e p a r a t i o n .  the c y l i n d r i c a l and - 1  of  us  data i s a f i t of E q u a t i o n 11.10, assuming  to be 7 (+1/-0.9) u s  The  and  c o r r e s p o n d i n g data taken at 30 ± 0.5  of freedom, and  zero  p l a n a r components (+0.098/0.096)  i n d i c a t e s t h a t the r e l a x a t i o n may  0.076 (+0.0063/-0.0059).  s i t e s may  1.47  "C p r e p a r a t i o n than f o r the 110  parameters were found  the zero  f i t gave a Chi-square  and  - 1  f u n c t i o n of E q u a t i o n I I I . 2 1 .  f o r 38 degrees  calculated  The  the c y l i n d r i c a l and  (+1.4/-1.2) u s  at 600  the zero f i e l d  zero f i e l d  respectively.  The  curve through  to the d a t a , assuming the s t a t i c  t r i p l e t muonium asymmetry was  (+0.0043/-0.0041).  F i g u r e s IV.7  The  there remains a  The most l i k e l y c a n d i d a t e f o r t h i s i s an F e m a t e r i a l , i r o n i m p u r i t i e s are quoted  3 +  IV.C.  ion.  In  as being l e s s than 2  [11]; however, r e c e n t measurements have set t h i s l e v e l at ~6  This p o s s i b i l i t y i s discussed further i n section  might  ppm  [12].  - 129  CM ^  -;  00 CD •  CD  CD •  -  ^-  CD •  CM  CD  o  •  F i g u r e IV.8 Zero and l o n g i t u d i n a l f i e l d asymmetry s p e c t r a f o r muonium on the s i l i c a s u r f a c e (600 "C p r e p a r a t i o n ) , at 30.0 ± 0.5 K. The zero f i e l d data are r e p r e s e n t e d by the square symbols and are compared to data taken a t two d i f f e r e n t l o n g i t u d i n a l f i e l d s ; the c i r c l e s correspond to 0.5 G and the t r i a n g l e s to 2.0 G. The curve through the zero f i e l d data i a a f i t to the d a t a u s i n g the s t a t i c zero f i e l d r e l a x a t i o n f u n c t i o n of E q u a t i o n I I I . 2 1 .  - 130 IV.B  -  Muonium on the S u r f a c e of Helium Coated Gas  a d s o r p t i o n isotherms were measured u s i n g ^He  concomitant r a t e and  w i t h measurements of the t r a n s v e r s e f i e l d  the muonium f o r m a t i o n p r o b a b i l i t y .  performed  IV.B.l  Silica  a c c o r d i n g to the procedure  a t 6.0  ± 0.1  K,  muonium r e l a x a t i o n  The gas d e p o s i t i o n was  g i v e n i n Chapter I I .  R e l a x a t i o n Rate Versus **He Coverage a t 6 K Mu  The 110  transverse f i e l d  °C and  f o r sample S i 0 ( 3 ) prepared g  f o r sample S i 0 ( 4 ) prepared  i n F i g u r e IV.9.  U  C  i s p l o t t e d as a f u n c t i o n of  By d e f i n i t i o n ,  the volume of gas, measured a t STP,  the s p e c i f i c volume  d i v i d e d by the s u r f a c e area of Mu  target.  From t h i s d a t a , i t i s obvious  coverage  i s a s t r o n g f u n c t i o n of the sample p r e p a r a t i o n .  110  °C data are observed  w h i l e the 600  at  2  a t 600  2  the s p e c i f i c volume V is  relaxation rate \  t h a t the dependence of \  on s u r f a c e  In p a r t i c u l a r ,  to decrease m o n o t o n i c a l l y w i t h i n c r e a s i n g  °C data show a peak i n the coverage  the  dependence.  the  coverage,  Furthermore,  t h i s peak has a maximum which i s equal ( w i t h i n the u n c e r t a i n t i e s ) to the t r a p s i t e r e l a x a t i o n r a t e f o r the 600 I n t e r p r e t a t i o n of the 110  U  C  d a t a , shown i n F i g u r e  "C data i s s t r a i g h t f o r w a r d .  At  IV.1. zero  coverage,  the muonium atoms are s t a t i o n a r y i n the host s i t e s on the  surface.  As  the coverage  i s i n c r e a s e d from  atom i n t e r a c t i n g w i t h the s i l i c a  silica  z e r o , the p r o b a b i l i t y of a  s u r f a c e decreases  Mu  because there i s l e s s  exposed s u r f a c e a r e a . I n t e r p r e t a t i o n of the 600 developed in act  °C data i s not so t r i v i a l .  around the assumption  the s u r f a c e s of the s i l i c a  t h a t the baking  grains.  procedure  A model can produces  be  fissures  These f i s s u r e s are f u r t h e r assumed to  as deep p o t e n t i a l w e l l s which have the same r e l a x a t i o n mechanism as  the  - 131  -  3.5  12  3.0  2.5  CO  3  -  10  -  8  -  6  < O  2.0  w  1.5 CD  1.0  -  O -  0.5  2  0  0.0 .0  .1  .2  V  .3  (10"  4  s  .4  .5  cm)  F i g u r e IV.9 T r a n s v e r s e f i e l d muonium r e l a x a t i o n r a t e a t 6.0 ± 0.1 K v e r s u s He coverage (measured i n terms of s p e c i f i c volume V ) f o r s i l i c a prepared at 110 °C ( c i r c l e s ) and at 600 C ( s q u a r e s ) . The f i l l e d symbols correspond to the r e l a x a t i o n r a t e and the open symbols correspond to the vapor p r e s s u r e ( r i g h t hand s c a l e ) . N o t i c e that the vapor p r e s s u r e i n c r e a s e s r a p i d l y a t monolayer c o m p l e t i o n . H  U  - 132 host s i t e s .  With these assumptions, one can adopt  the f o l l o w i n g model:  zero coverage the muonium atoms are presumed s t a t i o n a r y , but i n t h i s the  muonium atoms may occupy e i t h e r  wells.  As the coverage i s i n c r e a s e d  At  case  the host s i t e s o r the deep p o t e n t i a l from zero, the helium i s adsorbed  p r e f e r e n t i a l l y i n t o the deep p o t e n t i a l w e l l s . which l o o k s to be about 20% of a monolayer,  At some c r i t i c a l  coverage,  the helium atoms f i l l  up the  f i s s u r e s s u f f i c i e n t l y to form " b r i d g e s " over which a muonium atom may diffuse rapidly until  i t reaches a "normal" t r a p s i t e .  As the coverage i s Mu  i n c r e a s e d beyond  t h i s p o i n t , the b e h a v i o r mimics  the 110 °C data; \  decreases m o n o t o n i c a l l y w i t h i n c r e a s i n g coverage because e n c o u n t e r i n g the s i l i c a IV. B. 2  s u r f a c e decreases w i t h i n c r e a s i n g  the chance of coverage.  Muonium Asymmetry V e r s u s **He Coverage  Measurements of the muonium asymmetry were a l s o made as a f u n c t i o n o f s u r f a c e coverage a t 6.0 ± 0.1 K.  The r e l a t i v e asymmetry ( f o r one of the  p o s i t r o n telescopes) i s plotted against adsorbed onto the s i l i c a  the s p e c i f i c volume V  s u r f a c e i n F i g u r e IV.10.  g  of ^He  The data show that the  muonium asymmetry decreases w i t h i n c r e a s i n g s u r f a c e coverage, s u g g e s t i n g that the charge exchange c r o s s s e c t i o n i s s i g n i f i c a n t a t the h e l i u m - s i l i c a interface.  U n f o r t u n a t e l y , i t i s not p o s s i b l e to draw any c o n c l u s i o n s  these data r e g a r d i n g the o r i g i n s and mechanics silica  powders ( i . e . ,  from  o f muonium f o r m a t i o n i n the  s u r f a c e or b u l k f o r m a t i o n ) , s i n c e the p r e c i s e  role  played by the adsorbed helium atoms i n the charge exchange i n t e r a c t i o n a t the  He-Si0  2  i n t e r f a c e i s not as y e t known.  phenomenon are put f o r t h i n Chapter V.  Two p o s s i b i l i t i e s  for this  - 133 -  10  10  h +->  .09  8  o  -H  CD  co  < >  .08  CD CO CO  H  CD  -rH -+->  "a;  07  H  h  OH  O 2  0  .06 .0  .1  .2  V  (10"  .3  cm)  4  s  .4  F i g u r e IV.10 T r a n s v e r s e f i e l d muonium asymmetry versus ^He coverage ( c l o s e d squares) f o r sample S i 0 ( 4 ) prepared a t 110 C . The c o r r e s p o n d i n g vapor p r e s s u r e data are r e p r e s e n t e d by the open c i r c l e s . U  2  - 134  IV.C  -  Muonium on the Surface of Supported Platinum Catalysts The  behavior of muonium on the s u r f a c e of p l a t i n u m loaded s i l i c a  was  Mu s t u d i e d by measuring  the t r a n s v e r s e f i e l d muonium r e l a x a t i o n r a t e \  f u n c t i o n of temperature,  over the temperature  as a  range 5 K < T < 100 K, f o r  samples prepared with v a r y i n g l e v e l s of Pt l o a d i n g .  S i n c e a g r e a t d e a l of  i n f o r m a t i o n has a l r e a d y been a c q u i r e d c o n c e r n i n g the b e h a v i o r of muonium on the s u r f a c e of the 35 A C a b - O - S i l (EH-5) powder, t h i s m a t e r i a l was as the support m a t e r i a l f o r the Pt loaded c a t a l y s t were prepared  following  the procedures  i n c l u d e r e d u c t i o n i n a mixture of H hour (see s e c t i o n I I . C . 3 ) . initial  tests;  d e s c r i b e d elsewhere  and He a t 500  2  0.001%, 0.01%, 0.1%  a l s o measured.  and 1.0%,  As a l r e a d y d i s c u s s e d i n Chapter  the experiment. surface.  of about  T h i s was  by weight.  A control  following  one  f o r these sample  the same  100  I I , these samples were evacuated "C f o r a p e r i o d of ten hours  done to remove p h y s i s o r b e d water from  and  the  done, the supported  and  p r i o r to silica platinum  10 A or l e s s i n r a d i u s , were presumed to be  with a p p r o x i m a t e l y a monolayer of chemisorbed  IV.C.l  "C f o r a p e r i o d of  11.4(b).  S i n c e no o t h e r s u r f a c e treatment was  p a r t i c l e s , which average  [13], which  Some of the s p e c i f i c s of these samples  the t a r g e t v e s s e l s are g i v e n i n T a b l e  warmed to a temperature  These samples  Four l e v e l s of Pt l o a d i n g were chosen  c o n t a i n i n g no Pt ( u n l o a d e d ) , but otherwise prepared procedures, was  samples.  selected  covered  oxygen.  Unloaded S i l i c a Support Mu  The  temperature  dependence of the t r a n s v e r s e f i e l d  relaxation rate \  was  measured f o r the H-reduced c o n t r o l sample, c o n t a i n i n g no Pt l o a d i n g .  The  r e s u l t s of these measurements are shown i n F i g u r e IV. 11, along w i t h  the  - 135 -  4.0 3.5  J  L  -  3.0  3  S  2.5  -  2.0  -  1.5 1.0 0.5  -  0.0  •  H—reduced  •  unreduced  T  0.  20.  T  40.  60.  80  T e m p e r a t u r e (K) F i g u r e IV. 11 T r a n s v e r s e f i e l d muonium r e l a x a t i o n r a t e v e r s u s temperature f o r unreduced and hydrogen-reduced s i l i c a . The f i l l e d c i r c l e s r e p r e s e n t the data taken w i t h the reduced m a t e r i a l (sample P t ( l ) prepared a t 100 C ) , and the open squares a r e the data taken w i t h unreduced s i l i c a (sample S i 0 ( l ) prepared a t 110 ° C ) . U  2  - 136 corresponding r e s u l t s 110  °C).  Comparison  f o r the unreduced  for  Without  Si0 (l)  prepared a t  2  of these data suggests roughly the same d i f f u s i o n and  t r a p p i n g b e h a v i o r f o r the H-reduced sample.  m a t e r i a l (sample  sample as observed f o r the unreduced  evidence to the c o n t r a r y , the two s u r f a c e s i t e s  observed  the reduced sample can be assumed to be of the same n a t u r e as the  corresponding  s i t e s of the unreduced  width o f the h i g h temperature hydrogen  reduction affects  low temperature interpreted unreduced  sites  following  m a t e r i a l , but the g r e a t l y  peak i n the l a t t e r i s i n d i c a t i v e  the h i g h temperature  (host s i t e s ) .  increased t h a t the  s i t e s ( t r a p s ) more than the  The data i n F i g u r e IV.11 a r e  the same l i n e of r e a s o n i n g as presented f o r the  silica.  The assumption  that the muonium atoms are s t a t i o n a r y a t low (< 8 K)  temperatures, f o r both the reduced and the unreduced observations.  silica,  i s based on two  F i r s t , the p h y s i s o r p t i o n of helium gas a t 6 K s h a r p l y  Mu decreases X^ as one nears monolayer  completion, i n d i c a t i n g  that the muonium  atoms a r e o u t s i d e the powder g r a i n s and spending a l a r g e p o r t i o n of t h e i r Mu l i v e s on the s u r f a c e .  Second, X^ (T) f o r the H-reduced  be t o t a l l y independent  of the Pt l o a d i n g a t low temperatures.  of a Pt l o a d i n g dependence a t low temperatures lifetime,  the muonium atoms cannot d i f f u s e  mean s e p a r a t i o n between Pt p a r t i c l e s .  With  implies  diffusing  These  that d u r i n g t h e i r to the  the l o a d i n g s that have been corresponds to spacings of  d i s t a n c e s can e a s i l y be spanned  muonium atom moving a t thermal v e l o c i t i e s ,  as 5 K; t h e i r  The absence  over d i s t a n c e s comparable  s t u d i e d , the mean s e p a r a t i o n between Pt p a r t i c l e s 50 or more SiO-H groups.  s i l i c a was found t o  by a  f o r temperatures  as low  f a i l u r e to do so p r o v i d e s f u r t h e r evidence that the muonium  atoms a r e indeed s t a t i o n a r y on s i l i c a  s u r f a c e s a t low temperatures.  - 137 IV.C.2  0.001% and  P l a t i n u m Loaded S i l i c a :  The  e f f e c t s of extremely l i g h t  transverse  field  reasons d i s c u s s e d  0.01%  (0.001% and  0.01%) Pt l o a d i n g on  muonium r e l a x a t i o n r a t e i s shown i n F i g u r e previously,  f r o z e n i n a s u r f a c e host increased  -  the muonium atoms are again  s i t e at low  above about 10 K,  temperatures.  As  the  IV.12.  For  assumed to  the  be  temperature i s  Mu X^ (T) decreases, presumably due  narrowing, i n the same manner as f o r the unloaded sample.  the  to m o t i o n a l  At  higher  Mu temperatures, one  observes that f o r both samples X^  decrease m o n o t o n i c a l l y  with only  s l i g h t h i n t of a t r a p p i n g eventually  peak.  the 0.001% l o a d i n g The  c a t a l y s t can H  2  first  recalling  I t i s c l e a r a l s o t h a t the high  trapping  .  that only w i t h a  reduced, unloaded samples, can be expected  that t h i s t r a p s i t e might be  Bearing  t h i s i n mind,  step of the  I f t h i s model i s c o r r e c t , i t would e x p l a i n the l a c k of a  peak f o r the Pt loaded  samples, i n which l a r g e q u a n t i t i e s of atomic  unloaded H-reduced sample, where there i s very generated, the t r a p p i n g  little  techniques,  of p a l l a d i u m  peak i s q u i t e pronounced. hydrogen  have been observed i n magnetic s u s c e p t i b i l i t y  loaded  the  atomic hydrogen  S i m i l a r l y pronounced e f f e c t s of c a t a l y s t l o a d i n g , u s i n g  [14]  one  loading  hydrogen can be generated upon H - r e d u c t i o n , whereas f o r the case of  reduction  to  f i l l e d , or otherwise n e u t r a l i z e d ,  the atomic hydrogen generated i n the r e d u c t i o n  procedure.  - i  temperature peak observed at about  be a trap f o r atomic hydrogen as w e l l as muonium. can p o s t u l a t e  us  samples  molecules be d i s s o c i a t e d to form atomic  25 K i n both the unreduced and  by  indicating a  become i n d i s t i n g u i s h a b l e , l e v e l i n g o f f at about 0.5  silica  hydrogen.  still  to  r e l a x a t i o n r a t e s f o r these two  These r e s u l t s can be understood by loaded  (T) continues  silica  c a t a l y s t s , as w e l l as i n ESR  studies  studies [15]  of  - 138 -  J  3.0  L  I  2.5 H  •  0.001%  •  0.01%  2.0  Pt  Pt  3. 1.5 3 S  * 5  1.0 H  KM  0.5 0.0 0  20  T  r  40  60  80  100  T e m p e r a t u r e (K)  F i g u r e IV. 12 T r a n s v e r s e f i e l d muonium r e l a x a t i o n r a t e v e r s u s temperature f o r 0.001% ( f i l l e d c i r c l e s ) and 0.01% (open squares) p l a t i n u m loaded Si0->  - 139 p l a t i n u m loaded s i l i c a e f f e c t was silica  attributed  catalysts. to F e  3 +  In both of these two  i m p u r i t i e s (10 - 100  support, which were reduced  treatment.  As has  C a b - O - S i l EH-5  -  ppm)  cases, the  present i n the  to m e t a l l i c i r o n by the hydrogen  a l r e a d y been mentioned, the i r o n c o n t a m i n a t i o n  m a t e r i a l has been measured to be ~6  might a l s o expect a s i m i l a r e f f e c t unreduced s i l i c a  i s baked at 600  h i g h temperature  trap s i t e  observed  ppm  [12] .  f o r the  Since  one  ( a l t h o u g h not as pronounced) when the  °C, i t may  be p o s s i b l e to a t t r i b u t e  to i r o n i m p u r i t i e s .  The  beauty  h y p o t h e s i s i s twofold; not o n l y does i t e x p l a i n why,  of  the  this  upon hydrogen  r e d u c t i o n , the t r a p p i n g peak d i s a p p e a r s (H atoms occupy the t r a p s ) , but i t a l s o p r o v i d e s a paramagnetic through  i o n which can i n t e r a c t w i t h the Mu  s p i n exchange p r o c e s s e s .  Because r e l a x a t i o n s a r i s i n g  exchange i n t e r a c t i o n s are i n g e n e r a l not decoupled fields, in  data of F i g u r e IV.7.  of  P l a t i n u m Loaded S i l i c a :  0 . 1 %  and  1 . 0 %  f o r platinum l o a d i n g s of 0.1%  are the data o b t a i n e d f o r the 0.1%  Mu X^ (T) i s e s s e n t i a l l y  p o s s i b i l i t y of an  exists.  F i g u r e IV.13, the muonium r e l a x a t i o n r a t e \  temperature  interest  spin  i n small l o n g i t u d i n a l  The  i n t e r a c t i o n w i t h an e l e c t r o n i c d i p o l e moment a l s o  In  from  t h i s h y p o t h e s i s p r o v i d e s an e x p l a n a t i o n f o r the unquenched component  the l o n g i t u d i n a l f i e l d  IV.C.3  atoms  Mu  i s shown as a f u n c t i o n  and 1.0%. sample.  Of  At low  particular temperatures,  the same as f o r the other f o u r samples,  that the muonium atoms are s t a t i o n a r y .  As  the temperature  Mu beyond about 10 K, X^ (T) e x p e r i e n c e s a sharp decrease  and  indicating  i s increased reaches a minimum  Mu at  about 20 K.  Between 20 K and 30 K, X  c o n t i n u e s to r i s e s l o w l y w i t h temperature  (T) r i s e s  s h a r p l y , and  thereafter  toward the v a l u e o b t a i n e d f o r the  - 140 -  4.0  J  L  3.5 3.0 T  2.5  IP s  —1  <  2.0 1.5 1.0 0.5 0.0  i  0  •  0.1%  Pt  •  1.0%  Pt  r  10 20 30 40 50 60 70  T e m p e r a t u r e (K)  F i g u r e IV.13 T r a n s v e r s e f i e l d muonium r e l a x a t i o n r a t e versus temperature f o r 0.1% ( f i l l e d c i r c l e s ) and 1.0% (open squares) p l a t i n u m loaded S i 0 • 2  - 141 1.0%  sample i n the same temperature  -  region.  In a s i m i l a r  f a s h i o n , the data Mu  o b t a i n e d f o r the 1.0%  sample i n d i c a t e a s l i g h t minimum i n  K, which i s f o l l o w e d by a sharp r i s e Above 30 K, \ The  M u  (T)  (T) a t about  to a v a l u e of about 3.6  f l u c t u a t e s w i t h an average  us *  a t 30  -  v a l u e of about 3.5  i n t e r p r e t a t i o n of these r e s u l t s i s r e l a t i v e l y  us  - 1  20  K.  .  straightforward i f  one  assumes the p o s s i b i l i t y of a chemical r e a c t i o n between the muonium atoms  and  the oxygen-covered p l a t i n u m s u r f a c e .  assume t h a t at 0.1%  about 30 K,  to encounter a p l a t i n u m p a r t i c a l u n t i l Mu  the  enough temperature  above which \^ (T) c o n t i n u e s to i n c r e a s e i n a manner  which can be d e s c r i b e d by the T ^ 1  One  can  l o a d i n g , the muonium atoms do not d i f f u s e f a s t  during t h e i r l i f e t i m e  reaches  B e a r i n g t h i s i n mind one  2  b e h a v i o r expected  can c o n t i n u e t h i s l i n e of r e a s o n i n g and  f o r thermal  diffusion.  assume then t h a t the 1.0%  sample  c o n t a i n s a s u f f i c i e n t l y h i g h c o n c e n t r a t i o n of p l a t i n u m p a r t i c l e s t h a t the muonium atoms have a v e r y high p r o b a b i l i t y of e n c o u n t e r i n g a platinum atom or  aggregate  even a t extremely  low hop  rates.  Believing  a c c u r a t e d e s c r i p t i o n of the r e l e v a n t p h y s i c s , one e f f e c t i v e l y c o n s t a n t r e l a x a t i o n r a t e above 30 K,  t h i s model to be  then concludes observed  an  that the  f o r the  1.0%  sample, a r i s e s from a chemical r e a c t i o n of muonium at the p l a t i n u m s u r f a c e . The  r a t e f o r t h i s r e a c t i o n i s found  the i s o t o p i c  to be 3.5  r e l a t i o n s h i p between muonium and  upper bound f o r the r e a c t i o n r a t e of atomic  ± 0.15  us  -1  which, owing to  hydrogen, should c o n s t i t u t e  hydrogen w i t h oxygen-coated  platinum. T h i s i n t e r p r e t a t i o n should be s u s c e p t i b l e to e x p e r i m e n t a l zero and  tests  l o n g i t u d i n a l f i e l d methods, s i n c e (as mentioned i n Chapter  c h e m i c a l r e a c t i o n s l e a d i n g to diamagnetic  molecular  III)  s p e c i e s do not cause  r e l a x a t i o n of the a"" s p i n p o l a r i z a t i o n i n zero or l o n g i t u d i n a l 1  with  field.  an  - 142 CHAPTER V —  V.A.  CONCLUSIONS AND FUTURE DIRECTIONS  Summary o f R e s u l t s The r e s u l t s and d i s c u s s i o n s presented  provided  information regarding  on the s u r f a c e of f i n e s i l i c a r e l a x a t i o n mechanisms  i n t h i s d i s s e r t a t i o n have  the d i f f u s i o n and t r a p p i n g of muonium atoms powders, as w e l l as the nature  involved.  of the s p i n  Experiments have a l s o p r o v i d e d  i n d i c a t i n g charge exchange processes  o c u r r i n g at the s i l i c a  evidence  surfaces f o r  sub-monolayer ^He coverages.  V.A.I  D i f f u s i o n and T r a p p i n g Measurements of the temperature dependence of the t r a n s v e r s e  field  muonium r e l a x a t i o n r a t e , have i n d i c a t e d the e x i s t e n c e of two d i f f e r e n t of s i t e s  (host and t r a p s i t e s ) f o r muonium on the s i l i c a  s i t e s were d e f i n e d  to be the most common.  surface.  types  The host  The i n t e r p r e t a t i o n of the data  f o l l o w s a c c o r d i n g l y ; a t low temperatures, two-dimensional d i f f u s i o n and t r a p p i n g of muonium i s observed, w i t h d e s o r p t i o n o c c u r i n g a t h i g h (>100 K) temperatures.  T h i s d i f f u s i o n and t r a p p i n g behavior  a s t r o n g f u n c t i o n of the s u r f a c e h y d r o x y l model was subsequently every  site  of the data  developed  to be due e n t i r e l y  concentration.  A three-state  [1,2], which assumes the r e l a x a t i o n ' a t  to random l o c a l magnetic f i e l d s .  f o r h i g h and low s u r f a c e h y d r o x y l  u s i n g t h i s model.  was f u r t h e r shown to be  One of the more important  was t h a t as the s u r f a c e h y d r o x y l  concentrations observations  A comparison  was then made  arising  from  this  c o n c e n t r a t i o n i s reduced the s u r f a c e hop  r a t e f o r muonium i s enhanced a t low temperatures ( s e e F i g u r e I V . 4 ) . i n t e r p r e t a t i o n of t h i s suggests t h a t the s u r f a c e h y d r o x y l s  serve  The  to i n h i b i t  - 143 s u r f a c e d i f f u s i o n of the muonium atoms.  This hypothesis  f u r t h e r by a r e c e n t hydrogen chromotography study behavior The  f o r hydrogen atoms on s i l i c a host  d e s o r p t i o n of a Mu summing the host to  be  the a c t i v a t i o n energy f o r  s i t e on the s i l i c a  s i t e a c t i v a t i o n energy E  q  IV.1,  °C p r e p a r a t i o n s  the host are 275  similar  reported.  and  s u r f a c e ) , obtained  the energy E ^ , was  s t r o n g l y dependent upon the sample p r e p a r a t i o n .  given i n Table 600  atom from a host  [3] i n which  s u r f a c e s was  s i t e d e s o r p t i o n energy ( i . e . ,  i s substantiated  From the  s i t e desorption energies  (+118/-51) K and ~105  be  by  the t h r e e - s t a t e model proved q u i t e v a l u a b l e .  a l s o found results  f o r the 110  muonium at both s u r f a c e s i t e s .  The  and  later  somewhat i n a p p r o p r i a t e , the s e m i - q u a n t i t a t i v e understanding  conducted to a s c e r t a i n the t r u e nature  °C  K, r e s p e c t i v e l y .  A l t h o u g h the assumption of random l o c a l magnetic f i e l d s was to  by  shown  afforded  S t u d i e s were l a t e r  of the r e l a x a t i o n mechanisms f o r  r e s u l t i n g c o n c l u s i o n s are summarized i n  the f o l l o w i n g s e c t i o n . •  V.A.2  Relaxation Mechanisms Using  zero and  longitudinal field  uSR  techniques,  i t was  shown t h a t a  d i p o l e - d i p o l e i n t e r a c t i o n (presumably between the muonium atom and hydroxyl  protons) i s i n f a c t not  muonium on the s i l i c a  surface.  the predominant r e l a x a t i o n mechanism f o r T h i s was  r e l a x a t i o n f o r muonium i n the host longitudinal field possibilities;  on the order  deduced by f i r s t n o t i n g  of a few  Gauss, thereby  leaving only  the  a two  (RLMF) or random a n i s o t r o p i c  F u r t h e r d i s c r i m i n a t i o n was  c o n s i d e r i n g the observed zero and  that  s i t e s can be e a s i l y decoupled by  random l o c a l magnetic f i e l d s  h y p e r f i n e d i s t o r t i o n s (RAHD).  the  longitudinal field  then done by  long time b e h a v i o r s  in  - 144  -  the context of a s t a t i c versus dynamic muonium atom; i t was one  assumes RLMF, the zero f i e l d  s t a t i c muonium atom, and  found  that i f  long time behavior i s i n c o n s i s t e n t w i t h a  the l o n g i t u d i n a l f i e l d  I n c o n s i s t e n t w i t h a dynamic system.  d e c o u p l i n g behavior i s  From these arguments then, a random  a n i s o t r o p i c d i s t o r t i o n of the muonium h y p e r f i n e i n t e r a c t i o n was  deduced to  be the p r i n c i p a l c o n t r i b u t o r to the r e l a x a t i o n , e s p e c i a l l y f o r muonium In the host  sites.  A theory was  developed  which d e s c r i b e s the time e v o l u t i o n of the u  +  s p i n p o l a r i z a t i o n f o r a completely a n i s o t r o p i c muonium h y p e r f i n e interaction.  The  approach  taken here i n v o l v e s expanding  tensor i n terms of s p h e r i c a l harmonics and to  parameterize  the d i s t o r t i o n .  L o r e n t z i a n and  d i s t r i b u t i o n s of the d i s t o r t i o n parameters.  to f u l l y  temperature  e x p l a i n the d a t a .  By comparing t h i s theory w i t h  The  °C and 600°C p r e p a r a t i o n s was  vary and was The  found  The  zero f i e l d  to the  relaxation function,  p l a n a r d i s t o r t i o n s , was  ( s t a t i c l i m i t ) data.  < 2 ) , but not e x c e l l e n t .  data.  were  Lorentzian-like  shown t h a t both a c y l i n d r i c a l d i s t o r t i o n (normal  assuming both c y l i n d r i c a l and  to  spin  s u r f a c e ) and a p l a n a r d i s t o r t i o n ( i n the plane of the s u r f a c e ) are  required  the 110  +  coefficients  " h i g h " e x t e r n a l magnetic f i e l d ,  then c a l c u l a t e d assuming "zero average"  silica  u s i n g the expansion  E x p r e s s i o n s f o r the s t a t i c u  r e l a x a t i o n f u n c t i o n s , both i n zero and  the d a t a , i t was  the h y p e r f i n e  used  to f i t the  low  q u a l i t y of the f i t s o b t a i n e d f o r both reasonable  In these f i t s ,  (typically, 1 <  x /deg.fr. 2  the muonium asymmetry was  allowed  to be c o n s i s t e n t w i t h the a s s o c i a t e d t r a n s v e r s e f i e l d  r e l a t i v e l y high %  L o r e n t z i a n approximation  2  v a l u e s may  adopted  allows i n f i n i t e d i s t o r t i o n s .  be d i r e c t l y r e l a t e d  to the  f o r the frequency  d i s t r i b u t i o n s , which  I t i s a l s o important  to remember t h a t the  - 145  -  L o r e n t z i a n d i s t r i b u t i o n s assumed i n the c a l c u l a t i o n s had the a c t u a l d i s t r i b u t i o n s have non-zero averages, o s c i l l a t i o n superimposed on the r e l a x a t i o n . tell  one would expect  The  f o r the h i g h y.  2  f i t s of the zero f i e l d  If  an  I t i s , however, i m p o s s i b l e to  from the data whether there i s a s m a l l amplitude  p r e s e n t i t c o u l d a l s o account  zero averages.  oscillation,  but i f  values.  random a n i s o t r o p i c h y p e r f i n e  relaxation  f u n c t i o n s to the data i n d i c a t e t h a t the c y l i n d r i c a l component O Q I n c r e a s e s 2  M while the p l a n a r component o" concentration. electrostatic  22  decreases  Assuming t h i s phenomenon to be due  r e g a r d i n g the e f f e c t of the induced  observed  e n t i r e l y to the  i n t e r a c t i o n between the muonium e l e c t r o n and  the neighboring, h y d r o x y l groups, one  s i t e symmetry.  with i n c r e a s i n g surface hydroxyl  can i n p r i n c i p l e e x t r a c t i n f o r m a t i o n e l e c t r o s t a t i c i n t e r a c t i o n on the muonium  For i n s t a n c e , the f a c t  t h a t the h y p e r f i n e d i s t o r t i o n  f o r h i g h h y d r o x y l c o n c e n t r a t i o n s (110  have an enhanced c y l i n d r i c a l component and  °C p r e p a r a t i o n ) i s shown to  a diminished  p l a n a r component, as  compared to the case of low h y d r o x y l c o n c e n t r a t i o n s (600 suggests  °C p r e p a r a t i o n ) ,  t h a t the e l e c t r o s t a t i c i n t e r a c t i o n between the muonium atom and  h y d r o x y l groups i s r e p u l s i v e .  Although  t h i s i n t e r p r e t a t i o n does  p r o v i d e a s a t i s f a c t o r y e x p l a n a t i o n f o r the observed disregard of  the e l e c t r o n s of  the p o s s i b i l i t y  b e h a v i o r , one  the  indeed cannot  t h a t t h i s r e s u l t c o u l d merely be a m a n i f e s t a t i o n  a combined r e l a x a t i o n i n t e r a c t i o n i n v o l v i n g both random a n i s o t r o p i c  h y p e r f i n e d i s t o r t i o n s and protons  random l o c a l magnetic f i e l d s due  to the  hydroxyl  a t the s u r f a c e .  Dynamical r e l a x a t i o n f u n c t i o n s were a l s o c a l c u l a t e d by s u b s t i t u t i n g static  f u n c t i o n s i n t o the s t r o n g c o l l i s i o n model [ 4 ] .  The  resulting  the  dynamic  f u n c t i o n s , f o r L o r e n t z i a n and L o r e n t z i a n - l i k e d i s t r i b u t i o n s , were found  to  - 146 be c o m p l e t e l y  independent  of the hop  frequency  i n c o n s i s t e n t with the m o t i o n a l narrowing bulk fused S i 0  2  and  the d i s t r i b u t i o n cannot yield  The  behavior observed  t h e r e f o r e t e s t e d and  a p p r o p r i a t e m o t i o n a l narrowing  cannot  a t e a r l y times, which i s  on the s u r f a c e of f i n e s i l i c a  L o r e n t z i a n d i s t r i b u t i o n was  of  -  behavior.  powders. found  f o r muonium i n A modified  to p r o v i d e  However, s i n c e the " c o r r e c t "  be deduced from e x i s t i n g knowledge, the  to be p a r t i a l l y  However, l o n g i t u d i n a l  d e c o u p l i n g measurements have i n d i c a t e d a s m a l l component of the which i s l a r g l y u n a f f e c t e d by the f i e l d s a p p l i e d .  U  C  preparation.  to  field  relaxation  T h i s component i s f u r t h e r  to be more prominent i n the data taken w i t h the 110  t h a t o b t a i n e d f o r the 600  also  c o n s i s t e n t w i t h r e l a x a t i o n due  random a n s i s o t r o p i c h y p e r f i n e d i s t o r t i o n s .  in  rates.  nature of the r e l a x a t i o n mechanism(s) at the t r a p s i t e s was found  form  data  " a b s o l u t e l y c a l i b r a t e d " q u a n t i t a t i v e v a l u e s f o r the hop  i n v e s t i g a t e d and  seen  the  "C p r e p a r a t i o n than  The most l i k e l y c a n d i d a t e f o r  t h i s unquenched component i s a s p i n exchange or perhaps a d i p o l e - d i p o l e i n t e r a c t i o n between the Mu reduced  to m e t a l l i c  atom and  an F e  3 +  ion.  i r o n upon hydrogen r e d u c t i o n .  employed i n the present s i g n i f i c a n t l y reduce  The  baking  3 +  content i n the s i l i c a  s t u d i e s , which are summarized i n s e c t i o n V.A.4.  V.A.3  Muonium Formation P r o b a b i l i t y  coverage  ( a t 6.0  field ± 0.1  hydrogen to  i n the d e c o u p l i n g behavior i n l o n g i t u d i n a l  silica  be  powder which  T h i s p o s s i b i l i t y has been f u r t h e r s u b s t a n t i a t e d by the p l a t i n u m  Transverse  would  procedures  study might w e l l produce enough atomic  the paramagnetic  would then be r e f l e c t e d  Moreover, F e  loaded  measurements of the muonium asymmetry versus K) have shown t h a t the muonium asymmetry  field.  helium  decreases  - 147 w i t h i n c r e a s i n g s u r f a c e coverage.  Although  these d a t a suggest t h a t the  charge exchange c r o s s s e c t i o n i s s i g n i f i c a n t a t the s i l i c a s u r f a c e s , i t i s i m p o s s i b l e t o say a t t h i s time what r o l e the h e l i u m atoms p l a y i n the charge exchange p r o c e s s .  One p o s s i b i l i t y i s t h a t the h e l i u m atoms a r e r e l a t i v e l y  p a s s i v e and o n l y s e r v e t o cover up the s u r f a c e , thereby muonium f o r m a t i o n .  impairing surface  I f t h i s i n t e r p r e t a t i o n i s c o r r e c t , these d a t a  clearly  show t h a t muonium f o r m a t i o n i s p a r t i a l l y s u r f a c e r e l a t e d , as p o s t u l a t e d i n Chapter I . There i s , however, another p o s s i b i l i t y which c a s t s the h e l i u m atoms i n a more a c t i v e r o l e , where they might a c t t o d i s s o c i a t e the muonium atoms a t the s u r f a c e .  Consider,  f o r i n s t a n c e , the s c e n a r i o i n which the  h e l i u m i o n s , produced i n the i o n i z a t i o n t r a i l of the s t o p p i n g u , a r e a b l e +  to  capture  indeed  the e l e c t r o n s of newly formed muonium atoms.  l e a v e the muons i n a diamagnetic  the p r e c e s s i n g muonium ensemble.  s t a t e , thereby  T h i s process  would  removing them from  Because of these two p o s s i b i l i t i e s , one  can say n o t h i n g about the o r i g i n s of muonium f o r m a t i o n , s i n c e one cannot d i s t i n g u i s h between muonium formed a t the s i l i c a s u r f a c e and  subsequently  d i s s o c i a t e d , o r muonium which i s formed i n the g r a i n s and d i f f u s e s t o the s u r f a c e where i t i s then d i s s o c i a t e d .  V.A.4  C a t a l y t i c Chemistry  These i n v e s t i g a t i o n s were a l s o extended t o the study of the i n t e r a c t i o n s of muonium w i t h the s u r f a c e of a s i l i c a - s u p p o r t e d p l a t i n u m c a t a l y s t . the r e s u l t s o b t a i n e d  f o r the temperature dependence of the t r a n s v e r s e  muonium r e l a x a t i o n r a t e , an upper l i m i t of 3.5 ± 0.15 u s  - i  From field  was deduced f o r  the r e a c t i o n r a t e of muonium w i t h an oxygen-covered p l a t i n u m s u r f a c e . experiments have a l s o p r o v i d e d  information concerning  the n a t u r e of the  These  - 148 t r a p p i n g s i t e observed f o r muonium.  S p e c i f i c a l l y , i t was observed t h a t the  r e l a x a t i o n a t the t r a p s i t e s i s c o m p l e t e l y platinum  n e u t r a l i z e d f o r s m a l l (~0.01% P t )  l o a d i n g s , whereas f o r an unloaded sample ( 0 % P t ) , the t r a p p i n g peak  i s q u i t e pronounced.  T h i s e f f e c t was found t o be c o r r e l a t e d w i t h the l a r g e  amount of atomic hydrogen which i s generated by the hydrogen r e d u c t i o n methods employed i n the p r e p a r a t i o n of the p l a t i n u m loaded  silica  catalyst.  S i m i l a r l y pronounced e f f e c t s of c a t a l y s t s l o a d i n g , u s i n g hydrogen r e d u c t i o n techniques,  have been observed both i n magnetic s u s c e p t i b i l i t y and ESR  s t u d i e s of m e t a l loaded  s i l i c a catalysts.  e f f e c t was a t t r i b u t e d t o F e  3 +  i m p u r i t i e s , which were reduced t o m e t a l l i c  i r o n by the hydrogen t r e a t m e n t . the t r a p s i t e i s an F e  3 +  I n these c a s e s , the observed  The u t i l i t y of a d o p t i n g  the h y p o t h e s i s  that  i o n not o n l y a f f o r d s one w i t h an e x p l a n a t i o n why,  upon hydrogen r e d u c t i o n , the t r a p p i n g peak d i s a p p e a r s , but i t a l s o  provides  a paramagnetic i o n which can i n t e r a c t w i t h the muonium atoms through s p i n exchange o r d i p o l e - d i p o l e ( e l e c t r o n i c d i p o l e ) p r o c e s s e s .  This  hypothesis  can thus account f o r the unquenched r e l a x a t i o n component observed a t t h e t r a p s i t e f o r the sample prepared  a t 110 "C, and perhaps a l s o e x p l a i n why  the e f f e c t may be l e s s prominent f o r the sample prepared  a t 600 °C, where  enough atomic hydrogen may be generated by the baking procedure t o p a r t i a l l y n e u t r a l i z e the F e  V.B.  3 +  centers.  Future Directions  The work p r e s e n t e d for  i n t h i s d i s s e r t a t i o n has p r o v i d e d  many new i n v e s t i g a t i o n s , both e x p e r i m e n t a l  surface c a t a l y s i s to surface p h y s i c s . a c c e s s i b l e avenues a r e d i s c u s s e d  here.  the ground work  and t h e o r e t i c a l , r a n g i n g  A few of t h e more  immediately  from  - 149 V.B.I.  Theoretical  The  s p i n r e l a x a t i o n theory  i n t e r a c t i o n (RAHD), d e r i v e d obtained  f o r muonium i n bulk  fine s i l i c a  powders.  "high"  f o r a random a n i s o t r o p i c  i n Appendix I, was fused  quartz  However, o n l y a few  symmetries were c o n s i d e r e d , and  -  external f i e l d  and  and  hyperfine  developed to e x p l a i n the  data  f o r muonium on the s u r f a c e  of  appropriately selected  distortion  the f u n c t i o n s were d e r i v e d o n l y f o r the  limits.  Further  c a l c u l a t i o n s assuming a non-zero  u>2^ component should  t h e r e f o r e be done, and  i n v e s t i g a t e d , before  a complete understanding of the s t a t i c r e l a x a t i o n  functions  f o r a random a n i s o t r o p i c h y p e r f i n e  Moreover, these f u n c t i o n s should  zero  the low  d i s t o r t i o n can be  be d e r i v e d a l l o w i n g  f o r the d i s t o r t i o n parameter d i s t r i b u t i o n s ,  external f i e l d  limit  obtained.  f o r non-zero averages  s i n c e some of the data do  to e x h i b i t s m a l l amplitude o s c i l l a t i o n s which are superimposed on  seem  the  relaxation. The  dynamic zero  be u n s u i t a b l e  for f i t t i n g  dependence on hop motional  f i e l d RAHD model d i s c u s s e d  i n c r e a s i n g hop  of assuming L o r e n t z i a n  f o r the d i s t o r t i o n parameters.  narrowing), w h i l e  rate.  T h i s behavior  The  modified  to  the data e x h i b i t was  found to  More thought must t h e r e f o r e be g i v e n narrowing  Lorentzian d i s t r i b u t i o n , discussed  Chapter I I I does have the r e q u i r e d f e a t u r e s ( f i n i t e  to  the  theory in  second moment and  decay),  t h i s i s done, e x t e n s i o n s  can be made to a m u l t i - s t a t e model f o r d i f f u s i o n i n  traps.  c o u l d be used f o r t h i s purpose.  an  exponential-like i n i t i a l  the presence of  and  be  and L o r e n t z i a n - l i k e d i s t r i b u t i o n s ,  c h o i c e of d i s t r i b u t i o n f u n c t i o n so that a proper m o t i o n a l can be developed.  found  because i t e x h i b i t s no e a r l y time  r a t e ( i . e . , no m o t i o n a l  narrowing w i t h  a manifestation  the data  i n Chapter I I I was  Once  - 150  V.B.2  Experimental T h i s r e s e a r c h has l a i d  a l o n g two of  -  the f o u n d a t i o n s f o r f u t u r e e x p e r i m e n t a l s t u d i e s  complimentary paths; one  i n v o l v i n g the study of chemical r e a c t i o n s  muonium w i t h v a r i o u s r e a c t a n t s on the s i l i c a  ( o r other) powder s u r f a c e s  and another  c o n c e r n i n g the i n t e r a c t i o n s of p o s i t i v e muons and muonium w i t h  macroscopic  surfaces.  Consider f i r s t the s i l i c a  the study of muonium r e a c t i o n s .  powder p l a y s the r o l e of an i n e r t  p r o v i d e s a way  s u b s t r a t e which s i m u l t a n e o u s l y  There are two  t h i s study which should be c o n s i d e r e d .  The  first  r e a c t i o n of muonium w i t h p h y s i s o r b e d m o l e c u l e s , w i t h the prime g o a l being to measure and three-dimensional r e a c t i o n r a t e s . muonium w i t h metal  The  field.  such as e t h y l e n e and  the  oxygen, and  reaction  Because of the enormous i n t e r e s t i n  probe such as muonium i n t e r a c t i n g w i t h a metal  a r a p i d l y expanding  a s p e c t s of  of these a s p e c t s i s the  aspect concerns  loaded c a t a l y s t s ,  p r o v i d e an e x c e l l e n t o p p o r t u n i t y f o r uSR  distinct  compare the two-dimensional  second  loaded c a t a l y s t s .  hydrogen c a t a l y s i s w i t h metal  to  study,  of producing muonium i n vacuum as w e l l as an I n e r t s u r f a c e  on which r e a c t a n t s can be s t a b i l i z e d .  of  In t h i s l i n e of  the study of a h y d r o g e n - l i k e loaded c a t a l y s t  to make a s i g n i f i c a n t  An immediate b e n e f i t  should contribution  t h a t can be f o r s e e n i s  that muonium can p r o v i d e i n f o r m a t i o n about the i n t e r m e d i a t e r e a c t i o n s t h a t o c c u r a t s h o r t times, which are c u r r e n t l y not o b s e r v a b l e by any  other  technique. P r e l i m i n a r y s t u d i e s of both of these a s p e c t s have a l r e a d y been made, w i t h the r e s u l t s o b t a i n e d f o r muonium on the s u r f a c e of a p l a t i n u m c a t a l y s t being r e p o r t e d i n t h i s d i s s e r t a t i o n . however, were made i n a low (< 10 G)  loaded  A l l of the s t u d i e s to date,  t r a n s v e r s e magnetic f i e l d .  I f one's  - 151  -  g o a l i s to measure the r e a c t i o n r a t e of muonium w i t h a r e a c t a n t s t a b i l i z e d on the s i l i c a r e l a x a t i o n due  s u r f a c e , i t w i l l be n e c e s s a r y to c h e m i c a l r e a c t i o n s and  of muonium w i t h the s i l i c a  observed  on the f r a c t i o n a l determining two  r e l a x a t i o n due  substrate i t s e l f .  r e p e a t i n g these measurements i n zero and The  to d i s t i n g u i s h between to the  T h i s c o u l d be accomplished  longitudinal  s u r f a c e coverage  f o r the r o l e played by the  helium atoms are both v e r y i n t e r e s t i n g , i f o n l y from an atomic  charge  probability  begs f u r t h e r i n v e s t i g a t i o n aimed at  the o r i g i n s of muonium f o r m a t i o n i n these powders.  To determine  by  field.  f u n c t i o n a l dependence of the muonium f o r m a t i o n  p o s s i b i l i t i e s mentioned e a r l i e r  of view.  interactions  Indeed, the physisorbed physics point  the t r u e r o l e played by the p h y s i s o r b e d atoms i n the  exchange p r o c e s s , i t w i l l be n e c e s s a r y  using d i f f e r e n t adsorbates.  to repeat these  I t would a l s o be i n t e r e s t i n g  s u b s t r a t e i n a s y s t e m a t i c way,  such as changing  experiments  to a l t e r  the  the s u r f a c e h y d r o x y l  concentration. Now  c o n s i d e r the p o s s i b i l i t y  and muonium atoms w i t h macroscopic interesting  crystalline  the i n t e r a c t i o n s of  surfaces.  A  are implanted  [5].  into ionic  equal to the band gap a l s o observed +  T h i s experiment single crystals,  shows that when e  of keV  +  they are r e e m i t t e d  energy  to be e m i t t e d .  ( t y p i c a l l y ~10-20 eV).  energies  isotropically approximately  P o s i t r o n i u m (Ps)  was  R e s u l t s a l s o i n d i c a t e t h a t the e m i s s i o n of  and Ps i s a s s o c i a t e d w i t h p o s i t r o n i u m d i f f u s i n g  these c r y s t a l s where there e x i s t s some branching Ps e m i s s i o n .  +  particularly  the s o l i d s w i t h a continuum of e n e r g i e s having a maximum  both e  u  t o p i c to l e a d of these i n v e s t i g a t i o n s a r i s e s from a r e c e n t  p o s i t r o n experiment  from  of i n v e s t i g a t i n g  ratio  to the s u r f a c e of for e  +  as opposed to  An e x p l a n a t i o n f o r t h i s phenomenon has been proposed  which  - 152 assumes a f i n i t e c o n c e n t r a t i o n such t h a t the e site.  +  of acceptor  s i t e s a t the c r y s t a l l i n e  i s Auger-emitted when the Ps e l e c t r o n combines w i t h  Assuming that one can draw c e r t a i n a n a l o g i e s  surface, such a  between the behavior of  p o s i t r o n s and p o s i t i v e muons, and t h a t the mechanism(s) r e s p o n s i b l e f o r the re-emission  of p o s i t r o n s would a l s o be i n v o l v e d i n the analogous phenomenon  f o r p o s i t i v e muons, t h i s l i n e of r e s e a r c h would have the added b e n e f i t t h a t one  can draw guidance from e a r l i e r  research (0-10  could conceivably  keV) u  +  beam.  p o s i t r o n experiments.  l e a d the way to producing  In addition, this  an u l t r a - l o w  A d e t a i l e d d e s c r i p t i o n of the p h y s i c s  energy  involved,  w i t h a p o s s i b l e t e s t case experiment, i s g i v e n i n Appendix I I .  along  - 153 APPENDIX I —  THE TIME EVOLUTION OF THE u+ SPIN POLARIZATION IN MUONIUM FOR A GENERALLY ANISOTROPIC HYPERFINE INTERACTION  I n t h i s Appendix, the time e v o l u t i o n of the u p o s i t i v e muon i n the n e u t r a l atomic  +  spin polarization for a  s t a t e (muonium, u e ~ ) i s d i s c u s s e d f o r +  the case o f an a n i s o t r o p i c a l l y d i s t o r t e d muonium h y p e r f i n e i n t e r a c t i o n .  The  a s s o c i a t e d s t a t i c s p i n r e l a x a t i o n f u n c t i o n i s a l s o c a l c u l a t e d f o r a few s e l e c t e d symmetries.  Because t h e r e s e a r c h p r e s e n t e d  i n this dissertation i s  p r i m a r i l y concerned w i t h muonium on the s u r f a c e o f powders, where t h e r e i s no w e l l d e f i n e d c r y s t a l o r i e n t a t i o n , t h e d i s c u s s i o n w i l l be d i r e c t e d accordingly.  AI.A  O b s e r v a b l e s - C r y s t a l and D e t e c t o r Frames  There a r e two frames o f r e f e r e n c e a s s o c i a t e d w i t h s o l i d s t a t e \iSR experiments, and  t h e c r y s t a l frame ( d e s i g n a t e d by t h e c o o r d i n a t e s x', y', z')  t h e d e t e c t o r frame ( x , y, z ) .  Observations  a r e o f course made i n t h e  d e t e c t o r frame; however, t h e time e v o l u t i o n o f t h e o b s e r v a b l e s  i s generally  more r e a d i l y d e s c r i b e d i n terms o f t h e c r y s t a l frame, where the symmetries of t h e problem can be e x p l i c i t l y i n c o r p o r a t e d .  These two r e f e r e n c e frames  are r e l a t e d by t h e E u l e r angles (a,B,y) through  the second rank r o t a t i o n  tensor R ( Q ) = exp(-iJ a) • exp(-iJ B) • e x p ^ i J j ) z  and  y  the i n v e r s e r o t a t i o n t e n s o r R ( Q ) = exp(+iJ , ) z  Y  (AI.l)  • exp(+iJ S) y I  •  exp(+iJ ,a) z  where the J . a r e t h e r e s p e c t i v e i n f i n i t e s i m a l r o t a t i o n g e n e r a t o r s  [1].  With  - 154 these d e f i n t i o n s , one can d e f i n e the u n i t v e c t o r t r a n s f o r m a t i o n s and i n v e r s e t r a n s f o r m a t i o n s as x^ = R(Q)»x^ and x^ = R(Q)»x^, r e s p e c t i v e l y .  AI.A.l  Spin Relaxation Functions  As f o r any o b s e r v a b l e ,  the time  e v o l u t i o n o f the u  +  spin polarization  f o r a g e n e r a l l y a n i s o t r o p i c muonium h y p e r f i n e i n t e r a c t i o n i s , i n the Heisenberg P  p i c t u r e , g i v e n by the e q u a t i o n exp[i(2it/h)Htl  (t) =  ~°P  expl"-if 2rc/h)Htl  P  (AI.2)  ~°P ' = 2 exp[i(2n:/h)Ht] s£ exp[-i(2n;/h)Ht] p  where H i s the s p i n H a m i l t o n i a n operator.  of the system and S^  p  i s the muon s p i n  The time e v o l u t i o n of the s p i n p o l a r i z a t i o n f o r an i n d i v i d u a l  muon, i s then r e p r e s e n t e d  by the second rank time a u t o c o r r e l a t i o n t e n s o r  d e f i n e d as g(t) = i §  4 =  Defining  TrlP  T  (t) P  l~op  }  ~op<  Tr{exp[i(2Tr/h)Ht]  (  P  Q p  exp[-i(2Tc/h)Ht]  the e i g e n s t a t e s of the H a m i l t o n i a n  as  (h/2-it)w^, and r e c a l l i n g  operator  E q u a t i o n AI.3 can be w r i t t e n as  product,  g(t) = \ X e ^ i i o ^ t )  P  o p  KjX+jl  where we d e f i n e the t r a n s i t i o n f r e q u e n c i e s  I  3  )  Q p  w i t h the c o r r e s p o n d i n g  the d e f i n i t i o n of a t r a c e of an  P  o p  |* > ±  = (oo^ - w..).  (AI.4) By s e p a r a t i n g  the e x p r e s s i o n g i v e n i n E q u a t i o n AI.4 i n t o the d i a g o n a l and o f f d i a g o n a l p a r t s one o b t a i n s  #  P }  \tyj>  eigenenergies  A  - 155  »  v  '  4 £ +  v  i ' ~op  |  Y  ~op  |  V R e l e x p f l w . . t l <cp.| P i j ^ l ' ~op L  K V  ;  By d e f i n i t i o n , <<|>. I P ' ^ i ~op 3  of  i  l  N  |  -  M  |(|».X<|>. I P j ~op  |  |  Y  \<\> •> } j J  |<|> .> = x P . + y P . + z P ., j i j i j i j ' X  1  y  S  j  5  such t h a t , i n terms  Z  J  i t s c a r t e s i a n components, §(t) can be w r i t t e n as  g ( t ) - l H S +  (xy  2  yxjP^P^ + (S  +  I f  2 |  +  (P__) + yy ( P ^ )  c  0  r  8  1  + -  K j  t  T  ^8in(i  + (yz -  U  l  j  +  ± j  X  t){(xy  (P  ^KAi  +  \ i-*-* , X ,2 H - l* l  + y^)Re(P .P^)  + «  2  + (S  Z I ±  )  2  &  +  ^ ) i i i i }  +  p  •+-»• , V ,2 -»-*• , z y y | l j l + " 1 ^ 1  p  ,2 (AI.6)  p  + lx)Re(p .P *) + & X  - yxllmCP^py.)  Z  +  + (xz - zx) I m ^  ^y)Re(P*.P *)} Z  .P .) ±  zy)lm(py.P .)} 1  S i n c e §(t) i s a second rank t e n s o r i n t h r e e - d i m e n s i o n a l space, i t can be expanded i n terms of (1) the second rank u n i t  t e n s o r U , (2) a s e t of  t r a c e l e s s a n t i s y m m e t r i c second rank t e n s o r , c o n s t r u c t e d of  the L e v i - C e v i t a  t e n s o r £ and the d e t e c t o r frame —  (3) the t r a c e l e s s symmetric The r e s u l t i n g g(t) = s ' 00 v  The  B  U »  from the dot product  spherical vectors E \ ~m  second rank d e t e c t o r frame  s p h e r i c a l tensors  and 2 E^.  expansion g i v e s -  1  1  T g, E nr ,1m ~m /2 m=-l L  t r a n s f o r m a t i o n to s p h e r i c a l  c o n v e n i e n t s e t of v e c t o r s and  1  • £ =  +  2  L  y g„ E _ 2m »m m=-2 B  2 v  (AI.7)  t e n s o r s i s made i n order to p r o v i d e a more  t e n s o r s f o r the r o t a t i o n a l t r a n s f o r m a t i o n s  - 156 between  the c r y s t a l  frame and the d e t e c t o r  frame.  In terms of t h e i r  c a r t e s i a n components  (x, y and z ) , the s p h e r i c a l c o v a r i a n t v e c t o r s E^" and ~m lm r \\* contravariant vectors E = E are g i v e n as ~ -~m  the corresponding S}  >  -  l  ;  = - i ( x  U  /2 EJ  ;  E  it  =  1 0  /2  • §  Ej  . i  = - i [ ( x y - y£)]  E \  • £  = - z [ ( y z - zy) + /2  = Uy /2  =  - zy) - i ( z x - x z j j  z  (AI.9) - xz)]  The symmetric s p h e r i c a l t e n s o r s c a r t e s i a n components  with  E  Q  = /2/3  [- zz +  g  ± 1  - ± J  [ (  2  L  ± 2  At  z x  j (xx + y y ) ] )  + i(y  \  ,  ~  XX  y y  the r e s u l t  ^  +  A  _ ^  z +  z  y)l  (ALIO)  c •*"*• x y  An  +  From  AI.9, one can w r i t e i (*J)  = "I (E /J  1  1  J -  z+  can a l s o be w r i t t e n i n terms i f t h e i r  t h i s p o i n t i t i s convenient to d e f i n e a few i d e n t i t i e s .  Equation  X  x  1 r( " 2 <-  =  -  §, the antisymmetric s p h e r i c a l  can be w r i t t e n as  Si  z  (AI.8)  /2  By d e f i n i t i o n of the L e v i - C e v i t a operator  1  i y )  /2  - - iz  tensors  +  I ( E j /2  = - E ~  1  .)  -  _ 1  +  E \ )  =  - i (£ /2  1 0  ( E  I ( E /2  1  ) - E " ) ~  U  1  1  E - ) 1  +  1  1  (AI.ll)  - 157 Combining  the i d e n t i t i e s g i v e n i n E q u a t i o n A I . l l w i t h the d e f i n i t i o n s of  E q u a t i o n AI.10, the second rank symmetric s p h e r i c a l t e n s o r s can be c o n s t r u c t e d from the s p h e r i c a l v e c t o r s such that one has  il - m KsS B5) 1 s i i 4 all)] s \ - ~ f(sj six) ( 4 sj)] +  +  <«- >  +  ±  i2  /2  §±2 =  5_.)]  ^  The c o v a r i a n t and c o n t r a v a r i a n t s p h e r i c a l v e c t o r s , which now  d e f i n e the  p h y s i c a l space, obey the f o l l o w i n g c r o s s product r e l a t i o n s (E (E  1  0  *E  1  0  )  x E  =  )  ( E  U  x E  1  1  )  =  ( E ^ x E  "  1  1  )  = i z x ( - ) ( x + xy) = _{xy + x) /2 /2  =  0  =  E (AI.13)  (E  XE  ) = i z x ( - ) ( x - i y ) = ±(iy - x) /2 /2  = -E  (E  XE  J =  = -E  ( x + xyj x [x - xyj = - i z  By combining E q u a t i o n s AI.6 and AI.7, and u t i l i z i n g AI.8 - AI.12,  the r e s u l t s of E q u a t i o n s  the nine d e t e c t o r frame components S-^Ct) °f the  relaxation  tensor are d e f i n e d as W  = T 2  = »<t> =  I [(p* )  + (p^) +  2  2  + 2j_cos( ..t)[|P .| X  2  W  8 <t) 10 1 0  = " Z (£ /2  1 0  ' §) : gCO "  = " Z I sin( /2 i < j  +  W 1 3  | y.| P  (p') ] 2  |P .| ]}  2  Z  +  t)[lm(p 1 J  (AI.14)  X  2  P^*)] 1 3  (AI.15)  - 158  g  2 0 Z U  (t)  - E  {I  : g ( t ) = j Sift *  2 0  ±  [^[(P^) + z i i  U  t)[I(|P  + 2 I cos( i<j  W  «  -  i  2  ±  1  §  :  ( t )  -  " ' i f ? !  (Pj ) ] " i i  2  |  X  (P ) ] i i  2  Z  ±  |P  2 +  y  2  1 ±  | ) -  (AI.17)  |P*| ]}  2  2  J  t  p  i i  p  i i  *  l  p  i i  u ]  p  (AI.18)  X  ± 2 X cos(o,..)[Re(p .P *) X  g  2 ± 2  (t,  - E  2 ± 2  :  S  (t) - - i  ( I I x  [(P*/-  ±  Z  (P^)  iRe(p .P *)]} y  2 5  Z  IP^P^J (AI.19)  . j ^ . t i t l P - . ,  2  -  IPj.l ] 2  T  IRe(P^)]!  Because a l l o b s e r v a t i o n s are made i n the d e t e c t o r frame, whereas the symmetries are d e f i n e d i n the c r y s t a l frame, i t i s necessary to r e l a t e d e t e c t o r frame o b s e r v a b l e s S - ^ C t )  t 0  the c r y s t a l frame o b s e r v a b l e s  the  8^ (t)m  T h i s i s done v i a the r o t a t i o n o p e r a t i o n , such t h a t  8  LM  ( t )  "  I Lm m 8  ( t )  * £ ™  (  where R^*^(Q) are the m a t r i x elements mM  A  I  '  2  0  )  of the r o t a t i o n tensor R(Q), d e f i n e d »  i n Equation A I . l . With these d e f i n i t i o n s ,  the dynamics of the u  +  s p i n i s found  to i n v o l v e  nine o b s e r v a b l e d e t e c t o r frame r e l a x a t i o n f u n c t i o n s , which are c o n s t r u c t e d from the nine d e t e c t o r frame components 8  l m  ( t ) of the r e l a x a t i o n tensor  - 159 [2,3]. in  Specifically,  there are three l o n g i t u d i n a l (1) r e l a x a t i o n f u n c t i o n s ,  the d i r e c t i o n of the l o c a l f i e l d ,  *1  (  t  )=  g  00  "  ( t )  /  2  /  3  8  20  d e f i n e d as  ( t )  g* (t) = l m { g ( t ) } + Re{g (t)}  (AI.21)  c  n  2 1  g ^ ( t ) = - R e { g ( t ) } + Im{g (t)} S  n  21  three c o p l a n a r - t r a n s v e r s e d e f i n e d by the f i e l d  ( c t ) r e l a x a t i o n f u n c t i o n s , which a r e i n the plane  d i r e c t i o n and the incoming muon s p i n p o l a r i z a t i o n  g^ (t) = - Im{g (t)} + Re{g (t)} t  8  ct  u  (  t  )  =  8  00  ( t )  2 1  \  +  g°*(t) = - /T7T g  /  2  1 Q  7  T  8  20  "  ( t )  Re  {s 2  (t)  2  >"  (AI.22)  ( t ) - im{g (t)} 2 2  and three p e r p e n d i c u l a r - t r a n s v e r s e ( p t ) r e l a x a t i o n f u n c t i o n s , which a r e directed perpendicular  to both the f i e l d  and the incoming muon s p i n  polarization. g£ (t) = R e { g ( t ) } + lm{g (t)} t  n  2 1  g ^ ( t ) = /1/2 g  8  pt  ( t )  - 00 8  ( t )  +  1 Q  ( t ) - Im{g (t)}  (AI.23)  22  \  /  2  /  3  8  20  ( t )  +  R e  {g  2 2  (t)}  - 160 It  i s convenient  to work i n the v e c t o r space spanned by the Im^m^  d e t e c t o r frame e i g e n f u n c t i o n s muonium i s o t r o p i c to be along |1>  =  of the unperturbed  hyperfine Hamiltonian.  the magnetic f i e l d ,  (vacuum)  D e f i n i n g the a x i s of q u a n t i z a t i o n  one has  |+,+>  ;  |2> =  |-,-> (AI.24)  |3> = s|+,-> + c|-,+>  ;  |4> = c|+,-> - s|-,+>  where c = c o s ( x / 2 ) , s = s i n ( \ / 2 ) and X = a r c s i n [ l / ( l + x ) 2  dimensionless  quantity x = |B|/BQ  (= 1585 G) i s the h y p e r f i n e i n vacuum. Equation [4-6]  The l a b e l s  field  i s the s p e c i f i c f i e l d  ].  parameter where B  of the second and t h i r d e i g e n f u n c t i o n s with respect  that the h y p e r f i n e H a m i l t o n i a n  i n b l o c k d i a g o n a l form. hyperfine Hamiltonian  In general  Q  given i n  to the standard  notation  can be p a r t i a l l y w r i t t e n down  the e i g e n f u n c t i o n s  can be expressed  i s o t r o p i c muonium b a s i s v e c t o r s  The  f o r i s o t r o p i c muonium i n the ground s t a t e  AI.24 have been i n t e r c h a n g e d  i n order  1 / 2  |(|>j> of a s p e c i f i c  as l i n e a r combinations of the  |k> g i v e n i n E q u a t i o n  AI.24, such that one  may w r i t e = l  H|(|>.> = S j d , ^ Utilizing  I  ±  this result,  c  i  |k>  k  (AI.25)  one can then w r i t e  <(b. IP U.> = [ x P . + y P . + z P . l = V c * c . <k|P k l > ^l'^op'^j i j i j i j £ i k jk '~op X  y  Z  J  +  I [ !i c  c  -^ i  <  k  l '~op l*> p  1  c  +  * iiM,l j kl C  <  k  S o l v i n g f o r the three d e t e c t o r frame p o l a r i z a t i o n P j  -  X  cos(\/2)[A  1 3  P . = i{cos(\/2)[B y  P  ij  =  C 12^ " C  c  o  s  1 3  !!>*] '~op'  p  2 3  -  - B  2 3  + B ]}  J - sin(\/2)[B  M [ 3 4 ] + sin(x)[A J C  3  J  components, one then has  + A _ J + sin(\/2)[A  2  (AI.26)  A ^  u  (AI.27)  - 161 -  I^JL C. = kl  L  r * * i c . . c . - c. c „ . i k jk i l j l n  AI.B  J  c..c. + c ^ c . , B, . = ik j l 11 j k - " k l 1  L  c . , c . , - c ^ c . and ik j l i i jk J  Note t h a t i n zero f i e l d , \ = n/2.  The S p i n H a m i l t o n i a n f o r I s o l a t e d Muonium The  spin Hamiltonian  f o r an i s o l a t e d muonium atom i n a magnetic  field  B, assuming a g e n e r a l l y a n i s o t r o p i c muonium h y p e r f i n e i s g i v e n by H = H  2 6 6  +  H  h f  (AI.28) = (h/2Tc)(y S - y ^ ) ' B ' ^ e ~op u ~op ~ 6  1  where Y  E  and Y  e the muon, S ~op (in  y  a  r  e  u  6  y  the r e s p e c t i v e magnetogyric r a t i o s of the e l e c t r o n and  u and S a r e the s p i n o p e r a t o r s and W i s a second rank tensor ~op »  three-dimensional  Although  + (h/2Ti) W : ( s 1 « ^~op ~ o p  space) r e p r e s e n t i n g the c o n t a c t h y p e r f i n e  t h i s Hamiltonian  interaction.  i s d i s c u s s e d i n some d e t a i l elsewhere [ 7 ] , a  somewhat d i f f e r e n t but e q u i v a l e n t e v a l u a t i o n i s g i v e n here which the c a l c u l a t i o n o f the time e v o l u t i o n of the u  +  facilitates  spin polarization, f o r a  g e n e r a l l y a n s i s o t r o p i c muonium h y p e r f i n e i n t e r a c t i o n , and the c o r r e s p o n d i n g zero, l o n g i t u d i n a l and t r a n s v e r s e f i e l d  AI.B.l  spin relaxation functions.  E v a l u a t i o n o f the H y p e r f i n e Term  Because the i s o t r o p i c muonium b a s i s g i v e n i n E q u a t i o n AI.24 i s d e f i n e d w i t h r e s p e c t to the d e t e c t o r frame, i t i s e a s i e s t Hamiltonian  i n t h i s frame.  dimensional  space,  unit  to e v a l u a t e the h y p e r f i n e  Since W i s a second rank tensor i n t h r e e  i t can a l s o be expanded i n terms of (1) the second rank  tensor U, (2) a s e t o f t r a c e l e s s antisymmetric  second rank t e n s o r s ,  - 162 constructed  from the dot product  of the L e v i - C e v i t a tensor £ and the  d e t e c t o r frame s p h e r i c a l v e c t o r s E^, and (3) the t r a c e l e s s symmetric second 2 rank d e t e c t o r frame s p h e r i c a l tensors E^.  However, because the h y p e r f i n e  tensor W i n v o l v e s only d i p o l e - d i p o l e and c o n t a c t have r e f l e c t i o n symmetry, the antisymmetric identically  zero ( i . e . , W^Q  = 0).  = w^ ^ +  p a r t of the h y p e r f i n e  2  +  (AI.29)  0  L  tensor W i s d e f i n e d by the  c r y s t a l frame c o e f f i c i e n t s co , which are r e l a t e d to the d e t e c t o r Lm T  c o e f f i c i e n t s v i a the r o t a t i o n a l  W  tensor i s  The r e s u l t i n g expansion g i v e s  2 I w E „ 2m «m m=-2 where the symmetry of the a n i s o t r o p i c h y p e r f i n e W = w„„ U » 00 »  i n t e r a c t i o n s , both of which  frame  transformation  L M " I "Lm * £ ™ m  < - > A I  The s p i n o p e r a t o r s  3 0  f o r the e l e c t r o n and the muon can be w r i t t e n i n  terms of the c o n t r a v a r i a n t s p h e r i c a l v e c t o r s as  a  S =S ~°P =  +  +  x+S  X  I S /2  where a = {p,,e}.  «  =  ~op where we  1 1  (AI.31)  - E  1 _ 1  ) +  I S /I  a  ( E  y  By combining l i k e  L (E ) « 1 1  s  J2  -  I  (E  1 _ 1  U  + E ~  1 _ 1  ) - iS E ~ a  ) S?  -  i (E  1 0  z  terms i n  • /2  and E^" \ 1 0  ) S  one o b t a i n s  (AI.32)  a Z  define  S = (S - iS ) x y' a  z  Z  (E ~  a  x  S  •*•  y + S  y  a  a  and  S? - ( s + iS ) + ^ x y a  a  (AI.33)  - 163 The  three o p e r a t o r s  and  t u r n out  product ( ~op e  v  S  Using  and  ;  [s«,  They obey the commutation  S«] = -S«  ;  [s«,  x ^ ) = {i ( E ~op 2 ~ S  x E  U  1 _ 1  ~  v  k  +  I  (E  1 _ 1  +  -  (E  1 0  (~ o p x ~op *) = {i2 e  S  +  i 2  +  antisymmetric  S«] = 2S«  x  )S  Sj +  e  ' —  E  U  x E  1 1  ±  /T  +  )S?  -  )S  -  e  ( E  I (E /2 -  (-E ^ ~  )S  10  (+E  +  y  U  Sjj +  6  )S?  1 0  ~  v  x E  U  cross  x  1 _ 1  (E  E  1 0  z  ) f  (AI.35)  S  x E  1 0  6 S  1 _ 1  )s  S^}  e  of the s p h e r i c a l v e c t o r s  )S  —  -  -  6  Z  +  "  1  (-E  I  (-HS )S?  /o" /2  y^"  )S  11  z  (AI.36)  1_1  ' +  I /2  (-E ~  1 _ 1  )s  Z S^}  e  Z  +  p a r t of the h y p e r f i n e tensor W i s i d e n t i c a l l y  t h i s p o i n t we  have a s e t of convenient  v e c t o r s , the second rank u n i t +  (E. E ~ l ~-1 1  v  1  1  the  zero.  s p i n o p e r a t o r s , expressed so we  i n terms of these o p e r a t o r s .  i s o t r o p i c part of the h y p e r f i n e H a m i l t o n i a n .  - EJ EI ~o ~0  given  6  r e l a t i o n i s not used i n the present work because  hyperfine Hamiltonian  =  (AI.34)  o b t a i n s the  )  1 0  ~  terms of the c r y s t a l frame s p h e r i c a l v e c t o r s , and  U »  relations  can r e w r i t e E q u a t i o n AI.35 as  - (+E /I ~  T h i s c r o s s product  At  operator  of the e l e c t r o n and muon s p i n o p e r a t o r s , namely  the i d e n t i t i e s f o r the c r o s s products  S  d e f i n e the v e c t o r  the o p e r a t i o n s g i v e n i n E q u a t i o n AI.13, one  i n E q u a t i o n AI.13, one  v  completely  to be more c o n v e n i e n t .  [ S ^ , S j ] - S° Utilizing  s",  -  can r e w r i t e  First  consider  in  the the  In terms of the s p h e r i c a l  tensor i s w r i t t e n +  E , 1  ~—1  E. )  (AI.37)  ~1 1  Thus the t r a c e p a r t of the h y p e r f i n e H a m i l t o n i a n  becomes  - 164 H ^ = fh/2iO w 00 00 v  [- EJ E i + (E. E . + E . fih] : ( S ~0 ~0 ~l ~ - l ~—1 ~ 1 ~op ~op 1  n nL  1  1  1  )  6  K  ;J  v  ;  (AI.38)  and s u b s t i t u t i n g the e x p r e s s i o n s f o r the e l e c t r o n and muon s p i n o p e r a t o r s g i v e n i n E q u a t i o n AI.32 i n t o E q u a t i o n AI.38, y i e l d s the r e s u l t H J J = (h/2„)  W  q  I  [ S * S£ +  o  (S*  +  Sf s £ ) ]  By combining the d e f i n i t i o n s of E q u a t i o n AI.12  (AI.39) and the e x p r e s s i o n s f o r  the e l e c t r o n and muon s p i n o p e r a t o r s , g i v e n i n E q u a t i o n AI.32, the f i v e terms o f the symmetric t r a c e l e s s p a r t of the h y p e r f i n e H a m i l t o n i a n become  H  22  H  21  " ~ 7 (  =  H_J  I (  h / 2 l t  h / 2 L T  ) 22 ( - -) W  S  ) 21 ( z W  S  = /273(h/2 ) w 7l  H  2-l  =  H  2-2  =  " I  ( ^ h/2  W  2Q  (AI.40)  S  S  +  - z)  S  [- S* S^ +  2-l^ z S  S  i +  (AI.41)  S  S  (S*  + Sf J ) ]  (AI.42)  S  + z^ S  (  A  I  " J (n/27t) W _ (S^ sj) 2  *  4  3  )  (AI.44)  2  The r e s u l t of o p e r a t i n g on the space spanned by the i s o t r o p i c muonium eigenvectors by f i r s t  |k> w i t h the o p e r a t o r s s",  S^ and s" can be e a s i l y  c o n s i d e r i n g t h e i r e f f e c t on the s i n g l e p a r t i c l e e i g e n k e t s ,  ( e q u a l to ly, -j >^ o r |y, ~ >^ f o r s p i n 1/2). S  a  z  S" i-  1  understood  |j,m> = m -" a a  1  a  Thus one can w r i t e [1]  |j,m> a  |j,m> = / j ( j + l ) - m (m +1) a a a  |j,m>  (AI.45)  = |j,m+l>  -1 y  ; for m a a ; for m  =  y  (AI.46)  165 a 0 Returning the  now  ; for m = a  +1 2  (AI.47)  i 2  to the two-spin e i g e n s t a t e s of E q u a t i o n AI.24, one can d e f i n e  operations z z  z z  H> •  z  3> = - T  |3>  1> =  |2>  z  z z  2>  14  4>  1 4  ti*>  2> -  0  |2>  (AI.48)  (AI.49)  s  e  s^  1  3> =  0  4> =  1> =  0  2> =  0  |1> (AI.50)  s s  e  z  3> =  0  1> =  £  |> - | 3  |4>  -  e  z  -  -  z  -  z  3> = - T  1> 3>  |2>  T  |3> + f  T  |2>  l  4  >  z  -  z  -  -  z  -  z  4> =  0  2> =  0 (AI.51)  4> = - £  2> =  |2>  0 (AI.52)  4> =  £  |2>  - 166 -  s s  S  J  1>  S  z  +  1  s  ;  3>  S z  +  1  1>  Sf S^ I2> + z '  7  S? S^ |4> + z  *|1> 2  e  z e  z  +  +  z  s f S^ +  e  +  sf  z  e  f  3>  |1>  |4>  |2> = - -I |3> 2 1  |4>  1  (AI.53)  f  1°  2  |3> + f  l> 4  (AI.54)  S® S^ |2> =  1> -  (AI.55) 3> =  sc|3> - s |4>  ;  |4> =  sf  s!  sj  1> =  0  sf  sj  3> =  sc|3> + c |4>  S  J  c |3> - sc|4>  |2> = (AI.56)  With these o p e r a t i o n s d e f i n e d , anisotropic can  |4> = - s |3> - sc|4>  2  hyperfine  the m a t r i x elements of the g e n e r a l l y  H a m i l t o n i a n i n the i s o t r o p i c muonium  representation  We begin w i t h the i s o t r o p i c part HQQ and c a l c u l a t e the  be d e r i v e d .  m a t r i x elements to o b t a i n  w,00  1  0  0  0  0  1  0  0  0  0  -(l-4sc)  2(c -s )  0  0  2(c -s )  -(l+4sc)_  2  2  2  In a s i m i l a r manner, the m a t r i x elements of the f i v e symmetric can  t r a c e l e s s part  be c a l c u l a t e d  giving  of the g e n e r a l l y  anisotropic  (AI.57) 2  terms of the  hyperfine  Hamiltonian  - 167 -  *22  hf 21  (h/2n)  4  W  (h/2Tt)  21 4 _  h /2/3  W  2%  w  (h/2n)  20 4  2-l 4  0  0  0  0  -1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  (c+s)  0  0  0  (c-s)  0  0  0  -1  0  0  0  0  -1  0  0  0  0  (l+2sc)  ( c -s )  0  0  r  (l-2sc)_  0  0  -(c+s)  0  0  0  0  0  (c+s)  0  0  0  (c-s)  0  0  0  0  0 w  2-2  AI.B.2  (h/2u)  2-2 4  -1  -(c+s)  i  2 2  (c -s j  (AI.58)  -(c-s)  -(c-s)  0  0  0  0  0  0  0  0  0  0  0  0  (AI.59)  (AI.60)  (AI.61)  (AI.62)  E v a l u a t i o n o f the Zeeman Term  To e v a l u a t e the Zeeman H a m i l t o n i a n , i t i s again e a s i e r to work i n the  - 168 d e t e c t o r frame w i t h the magnetic f i e l d Recalling  the e x p r e s s i o n s  r  e  Operating  along  the z - a x i s .  f o r the e l e c t r o n and muon s p i n o p e r a t o r s g i v e n i n  E q u a t i o n AI.32, the Zeeman H a m i l t o n i a n zee H - ~ = (h/2*) [ y S*  B directed  -  Y ( 1  becomes (AI.63)  Sjj] |B|  on the e i g e n s t a t e s of the i s o t r o p i c muonium b a s i s , g i v e n i n  E q u a t i o n AI.24, one has f o r the e l e c t r o n s p i n o p e r a t o r  S_  |1> " +  J  S  |1>  | > = ( i - s ) | 3 > - sc|4> 2  e s  3  6  z  1  |2> = - i |2> 2 1  S* 14> = ( I - c ) 14> - sc13>  ;  2  (AI.64)  and f o r the muon s p i n o p e r a t o r  s  z  |1> - + J |3> = ( | -  With these is  |2> = - j  |1> 2  |4> = ( I - s ) | 4 > + sc|3> 2  o p e r a t i o n s d e f i n e d , the Zeeman H a m i l t o n i a n  i n the d e t e c t o r frame  g i v e n by %tu H  Z e e  where  AI.C  0  = (h/2*)  1 = - (y_  0 -0),  Mu  0  0  0  0  0  0  0  0  a)., - a Mu  (AI.65)  - s c f y +V ) *• e ' p/ 1  e 'ii  2 2 - Y , J |£|» a = ( y s + y, c )|B| and b = ( o  Y e  2 2 c + y^s )|B|.  I s o l a t e d Muonium i n Zero F i e l d In zero f i e l d ,  to  S»  c ) | 3 > + sc|4>  |2>  the c r y s t a l  one has s = c = 1//2.  I n t h i s case, one can t r a n s f o r m  frame and w r i t e the t o t a l H a m i l t o n i a n  i n terms of the c r y s t a l  - 169 frame c o e f f i c i e n t s w^  namely  m>  Ho~ 2o)  -w  H = (h/2it) \  w  -u)  (wnrT^n) 00 ~20'  22  oo,21  2-1  0  " 2-l  2-2  U  Ko  0  21  + 2 u )  0  2o)  0 where we have d e f i n e d oo^ = /2 Deriving  s p e c i f i c hyperfine  -3w  w i t h the e x c e p t i o n  the c r y s t a l and d e t e c t o r  (AI.66)  -I  00  that UJ^Q = /2/3 O^Q •  frame r e l a x a t i o n f u n c t i o n s  Hamiltonian i s i n general  s t r a i g h t forward.  the r e l a x a t i o n due to some random d i s t o r t i o n of the muonium  fora  To c a l c u l a t e  hyperfine  i n t e r a c t i o n , one averages over the a s s o c i a t e d OJ ' s . However, s i n c e u . i s Lm UU t  i n general ignore the  nr  q u i t e l a r g e and unobservable due to timing  the o s c i l l a t o r y  terms c o n t a i n i n g WQQ,  l i m i t a t i o n s , one can  which simply  s i n g l e t component of the muonium ensemble, and only  the a p p r o p r i a t e  s  '  e  a  c  n  implies  take an average over  °^ which have some d i s t r i b u t i o n  the data presented i n t h i s d i s s e r t a t i o n , i t i s i n g e n e r a l d i s c r i m i n a t e between an T h i s being  d i s t o r t i o n as opposed t o  +  ^2TXS 2TS? * w  With  not p o s s i b l e to contributions.  the case, the p o s s i b l e c o n t r i b u t i o n of an  r e l a x a t i o n of the u  ignoring  component to the  s p i n p o l a r i z a t i o n i s omitted i n the remaining  discussions. I n the case of powders i n zero the h y p e r f i n e  applied f i e l d ,  the a x i s of symmetry of  d i s t o r t i o n f o r each muonium atom I s o r i e n t e d  randomly.  When  t a k i n g an ensemble average one averages over the E u l e r angles which, by d e f i n i t i o n of s p h e r i c a l harmonics, f o r c e s a l l of the d e t e c t o r to zero detector  except g^^Ct).  Because of t h i s a v e r a g i n g ,  only  frame o b s e r v a b l e r e l a x a t i o n f u n c t i o n s , g i v e n  frame 8  L m  (t)  three of the n i n e  i n E q u a t i o n s AI.21 -  - 170 c sc ss AI.23, s u r v i v e ; g-^(t), g . ( t ) and g p ( t ) . c{  t  Furthermore, these  three  f u n c t i o n s a r e i d e n t i c a l and equal to g ^ ^ C t ) , which simply r e f l e c t s  the f a c t  t h a t the s p i n r e l a x a t i o n f u n c t i o n i s i s o t r o p i c i n the d e t e c t o r frame. t h i s understanding,  the g e n e r a l form of the zero f i e l d  With  spin relaxation  f u n c t i o n , due to random a n i s o t r o p i c h y p e r f i n e d i s t o r t i o n s ,  f o r the case of  powders, i s then g i v e n by the e q u a t i o n CO  rh g (t) = / d(0 n n  OO  2 m  f  2 m  (o) J 2  ^ m,' / d uS ,m ' f (^ w m" ) 0 g 0(^t , a2) m ' ,..,u) 6  7 t r i I  9 m  Since a l l of the zero and t r a n s v e r s e f i e l d t h i s d i s s e r t a t i o n appear q u a l i t a t i v e l y e a r l y times,  , U J  nn  9ni  7Tn  , ) (AI.67)  s p i n r e l a x a t i o n data presented i n  to e x h i b i t an e x p o n e n t i a l decay a t  the k> 's a r e assumed to be d i s t r i b u t e d a c c o r d i n g to L o r e n t z i a n 2m  and L o r e n t z i a n - l i k e d i s t r i b u t i o n s . p u r e l y phenomenological,  AI.C.l  9 t n f  T h i s c h o i c e of d i s t r i b u t i o n f u n c t i o n i s  s i n c e the a c t u a l f u n c t i o n i s not known.  R e l a x a t i o n Due t o a C y l i n d r i c a l  Distortion  Take as an example a c y l i n d r i c a l d i s t o r t i o n o f the muonium hyperine as i n the case of anomolous muonium [ 8 ] . is  cylindrically  I n t h i s case,  the h y p e r f i n e  symmetric about some g i v e n a x i s .  Assuming t h a t a l l of the  muonium atoms i n the ensemble have h y p e r f i n e d i s t o r t i o n s about the z - a x i s , the H a m i l t o n i a n  ' K"00 r T"20 ^J H = (h/2*) ^  As  Hamiltonian  0  0  0  0  0  0  stated e a r l i e r ,  0  °  °  (" +2u, )  the e i g e n f u n c t i o n s  can be expressed  t h a t a r e symmetric  of E q u a t i o n AI.66 becomes  Ko^o)  0  distortion  00  20  0  0 -3u>,00  |c|^> of a s p e c i f i c  as l i n e a r combinations  (AI.68)  -I hyperfine  of the i s o t r o p i c muonium  - 171 basis vectors  |k> g i v e n i n E q u a t i o n AI.24.  E q u a t i o n AI.68  i s d i a g o n a l i n the i s o t r o p i c muonium r e p r e s e n t a t i o n , one has  the c o e f f i c i e n t s elements, l  S i n c e the H a m i l t o n i a n i n  c ^ = 6^  w i t h the e i g e n v a l u e s £^ equal to the d i a g o n a l  namely  x  = (h/8Ti)(a)  / 2 / 3 a) )  ;  l  3  = (h/BiOfaoo + 2 / 2 7 3 t o )  ;  ^  -  00  20  2  = {h/8n){  - /2?3" a> )  %Q  20  (AI.69) l  2Q  w i t h the c o r r e s p o n d i n g  transition  U  13  to14  k  =  =  (-  T (  3 / I  2 0  4a)  00 "  /  I  Using E q u a t i o n AI.27 components, P . X  p P  0  '  y 12  ^13  p  c  ,  P *14 =  -s  . '  P  X  23 =  s  P *24  c  P *34  0  X  X  "34  =  (-3/2/3 u ) ) 20  T ( Q0 " 4 w  = T 4 (^S"i n0 0  one can then c a l c u l a t e  13 = X  9 n2CV )  which s i m p l i f y  0  ' * i i  12 X  T u  = i  +  /  2  /  3  ™20 )  u  (AI.70)  J  ' *"*"'"' *>9n) "20' 2  /  2  /  3  the c r y s t a l frame  polarization  namely  0  x  7  23  co'24  7 3 o) )  00  frequencies  03  "12 " °  = (h/8u)(-3a> )  P  P  ' . ' . ' further  S i n c e a l l of  P  P  =  ;  P  ; *  P = - cosX, P,,= cosX 33 44  U  - I  y = -is 14 y = -is 23 y = -ic 24 y = 0 34  Z  22  = - 1  Z  0 ic  P  0 > '  P  13p  u  "  0 0  (AI.71)  0 0 sinX  i n zero f i e l d where X = n/2, thus c = s = l//2~.  the c r y s t a l frame c o e f f i c i e n t s i n the  cylindrically  - 172 distorted  H a m i l t o n i a n are s e t equal to zero except  =  ( t ) 0 0  T  N  E  O  I  V  By  o f E q u a t i o n AI.70, the former  U  l  +  C  ° ^ 00 S  U  +  I  u  2 0 ^ ]  +  2  c  becomes  ° t^ S  W  2 0 ^  , (AI.72)  1  +  Similarly  N  2  the v a l u e s o f E q u a t i o n AI.71 i n t o E q u a t i o n AI.14, and u s i n g the  definitions  S  W Q>  f u n c t i o n s are g ^ C t ) and g ^ C t ) .  non-zero c r y s t a l frame r e l a x a t i o n Substituting  and  OJQQ  substituting  2 c o s  t( 00 " 4 20^^ u  w  the v a l u e s o f E q u a t i o n AI.71 i n t o E q u a t i o n AI.17,  along w i t h the t r a n s i t i o n f r e q u e n c i e s d e f i n e d i n E q u a t i o n AI.70, the e x p r e s s i o n f o r the l a t t e r can be d e r i v e d , namely g' (t) = I  /2T3 {-1 - c o s [ ( o )  0  + i ' ) t ] + cos[(|  00  U  0  U  ' )t] 0  (AI.73)  l  + cos[(o)  00  - 4 o) )t]}  To d e r i v e an e x p r e s s i o n f o r the observable r e l a x a t i o n hyperfine distortions, averaged  we t r e a t  i s one-dimensional  due t o random  the terms o s c i l l a t i n g a t o r near  t o zero and only average  distribution  2 0  over W Q . 2  OJQQ  as  I n t h i s case, the frequency  and, assuming a L o r e n t z i a n d i s t r i b u t i o n , i s  g i v e n by  f  20^2 ) - k C 2 20 0  u  +  2 ] °20  < ' > AI  where O"Q i s the width parameter o f the d i s t r i b u t i o n , 2  i s a normalization constant.  and the f a c t o r  74  o f 1/n  I n the case of a powder, a v e r a g i n g over the  E u l e r angles f o r c e s a l l o f the d e t e c t o r frame S 2 ( ) components t o zero t  m  except g Q g ( t ) .  By combining  Equations AI.72 and AI.74 w i t h the d e f i n i t i o n  of E q u a t i o n AI.67, one can then w r i t e f o r powders  - 173 -  O  fc)  I"-f-l \ o  =  o  Performing 8  00  Notice  t  +  t  1+ 2 C  4o^  °^ " S  /I73  " 2 0 ^  (  A  I  *  7  5  )  the i n t e g r a t i o n i n E q u a t i o n AI.81, then y i e l d s  i  = (  Ho  )  J exp(-  +  I/273  cr  20  t)  (AI.76)  t h a t as t •* , t h i s f u n c t i o n tends t o 1/6 ( o r 1/3 of the i n i t i a l 00  p o l a r i z a t i o n of the t r i p l e t muonium ensemble).  The time independent 1/3  component of the ensemble s p i n p o l a r i z a t i o n ( r e s i d u a l p o l a r i z a t i o n ) a r i s e s because there e x i s t s a n o n - t r i v i a l zero frequency.  T h i s can be understood  i n t u i t i v e l y by n o t i n g that f o r a random h y p e r f i n e i n t e r a c t i o n , 1/3 o f the time the c y l i n d r i c a l d i s t o r t i o n a x i s i s d i r e c t e d along d e t e c t o r frame, ( i . e . ,  AI.C.2  along  the z - a x i s of the  the i n i t i a l muon s p i n p o l a r i z a t i o n ) .  R e l a x a t i o n Due t o a P l a n a r D i s t o r t i o n  Now c o n s i d e r  the time e v o l u t i o n of the u  s p i n f o r the case of a p l a n a r  +  d i s t o r t i o n o f the muonium h y p e r f i n e i n t e r a c t i o n .  I n t h i s case,  one has the  Hamiltonian -oo  00 H = {h/2n)  -to.  to,  0  0  0  0  00  '22  \  2-2  Since t h i s H a m i l t o n i a n  0  0  0  0  (AI.77)  0  00 0  -3co "00  -I  f  i s b l o c k d i a g o n a l , the f i r s t  two energy  eigenvalues  can be c a l c u l a t e d by d i a g o n a l i z i n g the 2x2 b l o c k  Ko  -  -to  ^ 22  where we d e f i n e  "2-2 (w oo =  0 0  = 0  =  to  -  2(O> OJ) +  oo „  =  /(to„ )(oo„_ )  00  to  -  00 f t n  M >2 ) "22^  (w  9 9  (AI.78)  -  £(8it/h), and  M  "22  22' 9  "2-2'  9  =  R >2 "22-  [(oo,,)'' +  I -,2 l/2 "22-  (to,-)^]  n  - 174 Solving Equation  h,2  (  =  h / 2 l t  AI.78 f o r to then g i v e s (AI.79)  ) 7 K o * 22^ a  To compute the e i g e n f u n c t i o n s eigenvalues, U  12  (" 2-2^ 2  +  u  u  (AI.80)  1  00  are the components of the e i g e n v e c t o r s .  2  d) 22  i(j> '22  on  "  — to.  to and  u^ = - e  u  u  - +  2  i<D  2 2  — W  l  u  =  +  e  22  2 2  f o r both  |C|J^> and  Thus one can w r i t e  2  i V  =  1  Z [|1> + * /2  H  2  |2>]  2  ;  3  the c o r r e s p o n d i n g  l  = (h/2u) \ ( t o  I  = (h/2n) \  x  '3  0Q  nn  w  +  M W  2  ;  The t r a n s i t i o n f r e q u e n c i e s I f ,  -  1  =  "  2  I V  = [i /2  x  -  >  e i < f > 2 2  2  >  ]  (AI.82)  = |4>  )  ;  '  C  9  = (h/2*) \ (to, 00  M> to,22'  (AI.83)  v."'-*; 4  ^4  are t h e r e f o r e g i v e n as  -  r i  1  r  ^  M  M  X (  l  to23 - 7 1-^22^  to13 - r 1^22^ to 14  ,. V  eigenvalues  (to )  4 ^ 00  to12  ;  Z  l * > - |3> with  (AI.81)  From E q u a t i o n AI.81 and t h e  2 2  u s u a l n o r m a l i z a t i o n c o n d i t i o n s , one can o b t a i n e x p r e s s i o n s |CJJ>.  22  l  u  22  where <t> = a r c c o s ( t o / t o ) = a r c s i n ( t o / t o ) . 22  Equation  L  22 22  Solving  respectively  2  =  equations  °  =  AI.80 f o r to = to^ and to = to , one o b t a i n s  2  linear  + (w -u)u2 = 0  where u^ and u  U  to these  2  one s o l v e s the s e t of coupled  ^00" K ( -^ )M  |(|>i> and |4>> c o r r e s p o n d i n g  4a)  00  W  +  u  22'  24  =  7 (  to'34 = r  4a)  00 " 22^  (  W  4 a  W  (AI.84)  - 175 Again  utilizing  the e x p r e s s i o n s  -  g i v e n i n E q u a t i o n AI.27, one  the c r y s t a l frame p o l a r i z a t i o n  components.  To do  the e i g e n s t a t e expansion c o e f f i c i e n t s  ll  c  c  c  n  J  " =  3k  ;  ±  =  6  C  12  ;  3k  c  °4k  ;  Substituting  " 2  e  1  *  2  2  5  6  coefficients  0  ;  P „ 12  =  0  ; P ' 1 2  y  ;  y  >  1 z [[cc + + s s ee" " 2^2] /I -1 22 ] = - i [s [ s —- c e - ^ 2  P  X  P  X  i(1) 1  -  X 3  =  3  =  P4 ^  =  P  =  X  X 4  P  P  y  i i  n ii n - i  +  +  ce "  c e  _il|>  14  - 0  =  ;  0  c  = 0  2 4  (AI.85)  22 ]  0  ;  P , = i i  0  =  0  :  pf„ = ' 1 2  1  z  ± 1[ cc  -- ssee " ^2 2 ] _ i < t , 1  Z  .  8  .  *  l 1 > 2 2  gives  =  ce~^22 -= "I " i [ [ s ++ ce-^22] /2  ;  _ i < t ,  s  C  •2  — e 22 se-^22] [[cc - s  = t //2I  ;  ^  ==  3  _i<t,  = 4  °  =  i n t o E q u a t i o n AI.24 then  =  X  0  =  P  24  ;  Z  P  4  =  =  4  s  s " c  ;  P^  =  0  ;  P^  =  0  ss ee " 2^2] _ i < | 1)  c  = t /2 •2  =  y  [c +  1 /2  e  ce  _i<1)  $ 2 2  ]  22  '  0  P  24 ;  c = s = 1//2.  = Z  P  * Z 4  g i v e n i n E q u a t i o n AI.86 i n t o E q u a t i o n AI.14, and  definitions  of the  S 0 0  f o r zero  ( t )  =  \  t  C  transition  (AI.86)  ° =  sinX  By  expressions  obtains  w r i t e down  namely  13  ;  first  calculate  4k  P . ii X  C  - - ^ e ^ 2 2  2  =  these  J  +  t h i s we  can  recalling  f r e q u e n c i e s g i v e n i n E q u a t i o n AI.84,  the  one  field  O  S  ^ I  U  22^]  +  2  c  o  s  t ^  w  22^] Z  + COS[(OJ - i 0()  o) )t] + 2 2  cos[(w + i u 0 0  (AI.87)  Z  2 2  )t] +  cos[(oj )t]} 0 0  - 176  -  S i m i l a r l y s u b s t i t u t i n g the v a l u e s of E q u a t i o n AI.86 i n t o E q u a t i o n AI.19, the zero f i e l d  8  22  ( t )  e x p r e s s i o n f o r g^(t)  =  ^  c o s  t( 00 W  +  i s found  I 22^]  "  w  c o s  to be  C( 00 " 7 22^ ^ A )  u  t  e X p  ^"  i ( | >  22^  <  A I  '  8 8  To o b t a i n the r e l a x a t i o n f u n c t i o n f o r random p l a n a r a n i s o t r o p i c s , once a g a i n i g n o r e s the O)QQ  terms and M  t  the frequency  o n l y averages  over  *  this  n  a)  two-dimensional  S  and,  W  YiX  2  J  case,  assuming a  L o r e n t z i a n - l i k e d i s t r i b u t i o n , i s of the form g i v e n by M M M . > 4 22 22 >-l 2 2 ^ 2 2 ' * 2 ) = £U „ 7M.2.2 r c N \L r M NZ-IL ( u22^) + K^22' , j J M O where -*width of the frequency d i s t r i b u t i o n , and 2/it i s a a  one  i  d i s t r i b u t i o n f22^ 22*^22' *  f  )  'W  '  (AI  ( n  89)  +  9 9  s  t n e  z  normalization constant. a zero average,  The  even though  seem i n c o n s i s t e n t , except  d i s t r i b u t i o n d e f i n e d i n E q u a t i o n AI.89 assumes M  i s positive-definate.  At f i r s t  t h i s might  t h a t E q u a t i o n AI.89 i s the d i s t r i b u t i o n f u n c t i o n M  for  a complex number; a l t h o u g h the magnitude  i s positive-def inate,  the  o r i e n t a t i o n s of the a s s o c i a t e d v e c t o r are d i s t r i b u t e d  over a l l d i r e c t i o n s i n  a plane.  i s made by averaging  In the case of powders, an ensemble average  over the E u l e r a n g l e s , which as e a r l i e r  s t a t e d , f o r c e s a l l of the d e t e c t o r  frame r e l a x a t i o n t e n s o r components to zero except  for g ^ C t ) .  By  E q u a t i o n s AI.87 and AI.89 w i t h the d e f i n i t i o n of E q u a t i o n AI.67, i g n o r i n g the U)QQ  terms, one  M rh,^ 8  00  ( T )  r,  4 °22 =  a  H —  >  d w  r  M r  and  then o b t a i n s f o r powders l  22 ,  combining  M ,2 [l" ) 22  "22  ,  i  M  2  2 + Ml 2,2 2) J {  f  rl  r  J  0  n  c  O  S  %  M  -i  "22^ L  (AI.90)  M  + 2cos[x w t ] } 4 "22' 9 9  where the i n t e g r a t i o n over the phase angle c j ^ has been done, and  the e x t r a  M f a c t o r of  1 S  t  n  e  Jacobian a r i s i n g  from the c o o r d i n a t e t r a n s f o r m a t i o n  - 177 from  R  the c o o r d i n a t e s  I > °°22  i n t e g r a t i o n , one f i n a l l y  -U  (t)  t  0  t0  M 22*  ^  y  performing the  M  M  ~°-¥' W - ~ir ) j t ~ ~r  l  fc  N o t i c e t h a t as t fact  * ^22  obtains  M  «£  a n c  +  1  «>, t h i s f u n c t i o n tends  L  to z e r o .  M aj  r  This result  t h a t , u n l i k e i n the case of a c y l i n d r i c a l d i s t o r t i o n ,  frequency arises  terms i n g ^ C t ) .  from  the assumption  a t r u e two-dimensional  AI.C.3  The simple a n a l y t i c  -  (AI  result  91)  r e f l e c t s the  there a r e no zero  of E q u a t i o n AI.91  of a " L o r e n t z i a n - l i k e " d i s t r i b u t i o n .  L o r e n t z i a n i s assumed, the r e s u l t  I f instead  i s not a n a l y t i c .  Cylindrical and Planar Distortions Combined  Now c o n s i d e r the time e v o l u t i o n of the u c y l i n d r i c a l and a p l a n a r d i s t o r t i o n . (o)  0 0  -  U 2 0  )  U  Ko- 2 ) w  0  0  0  °  °  "00 "200  0  0  s p i n assuming both a  I n t h i s case, one has the H a m i l t o n i a n  - „ „ 2-2  -to.22 0  H = (h/2*) i  +  (AI.92)  -3oj, "00  -I  Because t h i s H a m i l t o n i a n i s of the same form as t h a t of the simple  g i v e n i n E q u a t i o n AI.77, i t s e i g e n f u n c t i o n s \fy^> a r e the same  distortion, i V  -  I V  = [|1> + e * •2 1  2 2  |2>]  - |3>  ; ;  I V " I V  z [|1> - e * /2 1  can of course be c a l c u l a t e d by s o l v i n g ( OO" 2o)W  _ u  22  w  —OJ,2-2 Ko~ 2o)~ W  = 0 = OJ U  2 2  |2>] (AI.93)  = |4>  Since t h i s H a m i l t o n i a n i s b l o c k - d i a g o n a l , the f i r s t  w  planar  two energy  eigenvalues  the s e c u l a r e q u a t i o n  - 2a)(oj„ -oj' ) 00 2 0 ^  (AI.94)  n w  .f  , >,2 "00 "20 ^  , M ^2 ^22'  S o l v i n g E q u a t i o n AI.94, the f o u r e i g e n v a l u e s o f t h i s H a m i l t o n i a n a r e  - 178 h,2  =  5  [  h  /  2  7  l  )  £  [  (  U  = (W2n) j ( u  3  00  12  W  13  U  14  7  3  + 2 /273  0 0  w i t h the c o r r e s p o n d i n g  U  ~ ^  ^  co )  7 (" 73 20 "22^ =  ^  4 (  w  °00~  +  20  / I 7 J w  (AI.95) = (h/2*) i (-3 a )  4  Q 0  )  t r a n s i t i o n frequencies  4 ^"223/I  4j C  ;  20  -1«) =  ±(  W  23  =  7 (-  w  24  =  7 (  4(0  W  34  =  7 ^  4 u )  22^  +  W  3 / I 7 J  u  00" 00  +  20" 22^ W  /  I  7  2  /  1  J  7  "20" 2 2 ^ w  (  A I  -  9 6  )  "2C-)  1  Because the e i g e n s t a t e s of the combined H a m i l t o n i a n a r e the same as those of the p l a n a r d i s t o r t i o n H a m i l t o n i a n , the e i g e n s t a t e expansion  coefficients c ^  of the combined H a m i l t o n i a n a r e e q u a l t o those quoted i n E q u a t i o n AI.85. Consequently,  the c r y s t a l  frame p o l a r i z a t i o n components, which a r e d e r i v e d  through  the use of E q u a t i o n AI.27, a r e the same as those g i v e n i n E q u a t i o n  AI.86.  By s u b s t i t u t i n g  transition  the the r e s u l t s o f E q u a t i o n AI.86, as w e l l as the  f r e q u e n c i e s of E q u a t i o n AI.96, i n t o E q u a t i o n AI.14, one o b t a i n s  an e x p r e s s i o n f o r g ^ C t ) i n zero f i e l d ,  8  00  ( t )  I i i(j cos  =  u  22^^  +  c o s  + c o s [ - | ( u - co + u 0 0  +  the c r y s t a l  2 0  (t)  rm  +  "22^^  ) t ] + cos[-i{-3co - (i>2 )t] 20  0  ^  n  a  2  2  s  i  m  i ^  00  a  r  (AI.97)  {-cos[(I u  + cos[(w 0 0  M  (t),  such t h a t one has  ) t ] + c o s [ i { - 3 u ) + o> )t]  \ w + i u 20  0  2 2  M  2  cos[(co - \ o)' - £ w 00  2Q  manner one can o b t a i n e x p r e s s i o n s f o r  frame components g ^ C t ) and g  = \  +  20  - \ <o' - \ u ) ) t ] + c o s [ ( o , + \ c o ) t ] }  2  2  2 2  3 w  M  W ( ) 0  = /2/3 w o*  where W'Q  g  cos[(  20  ti(~  namely  2  20  2 2  2 2  ) t ] + cos[i(-3u^ 0  u> )t]  ) t ] + cos[(u) + j  o> )t]}  00  22  20  (AI.98)  - 179 and g  2 2  j{- c o s [ ( - I OJ + i u* )t] + c o s [ a ) - i  (t) =  2  20  + cos[(- I u x  2 0  -  w  M 2  £ a, )t] M  00  ) t ] - cos[(o) 0()  u  2 0  2  - -J- w ) t ] }  (AI.99)  2 2  {exp(-i(|> )} 22  respectively. The  r e l a x a t i o n f u n c t i o n f o r the combination o f random c y l i n d r i c a l and  planar a n i s o t r o p i c s  can be c a l c u l a t e d by i g n o r i n g  the unobservable  OJ terms 00  M and averaging over OJ2Q and w In t h i s case, the frequency d i s t r i b u t i o n t M •> M f (OJ2Q ,OJ J i s simply g i v e n by the product o f the o j ^ a n d OJ22 d i s t r i b u t i o n s as 22  22  defined  i n Equations AI.74 and AI.89, r e s p e c t i v e l y , namely  f ( u ) , o ) ) = f(o> ) 20  22  20  . f(o> ,<D ) 22  1  For  22  °20  powders, one averages  4  a  M 22  w  M 22  -1  over the E u l e r a n g l e s , and by combining  (AI.100)  Equations  AI.97 and A I . 1 0 0 w i t h the d e f i n i t i o n o f E q u a t i o n AI.67, and i g n o r i n g  the WQQ  terms, one then o b t a i n s the e x p r e s s i o n  g  00  ( t )  =  TT 3it  (°20 °22^ / -»  d a >  20 ^ 2 0 ^  +  u  ^ O ^ "  1  OO  x / doo o  22  (u  2 2  )  [ 0  + cos[i(-3/273 u )  2  2  ) + (cr ) ] ~ {cosfj w 22  t]  22  + 0 3 ) t ] + c o s [ i ( - 3 / 2 7 3 a> M  2 0  2  2()  where the i n t e g r a t i o n over the angle <t>  22  (AI.101)  - co^Jt]}  has a l r e a d y been performed, and  - 180 M where the e x t r a t o  f a c t o r i s again the J a c o b i a n of the c o o r d i n a t e  22  transformation. simple  Performing  analytic  rh,^ % ) (  t  )  1 6 I  1 M \ " 2 22 >  1  0  U  +  22  y i e l d s the  result  f l  =  the i n t e g r a t i o n s over U^Q  M and OJ  z  ~ T*22  1  ^  e X p  e X  _  s  r 1 M 2 22 ^  (AI.102)  a  Pt" ^ 2 2  ^  +  20^  a  M A check of the l i m i t i n g  cases  shows that as  0, E q u a t i o n AI.102  approaches the e x p r e s s i o n f o r a c y l i n d r i c a l d i s t o r t i o n g i v e n i n E q u a t i o n AI.76, and t h a t as OJ^Q -*• 0> E q u a t i o n AI.102 s i m p l i f i e s  to E q u a t i o n AI.91 f o r  a planar d i s t o r t i o n .  Furthermore, because of the p l a n a r c o n t r i b u t i o n to the  hyperfine d i s t o r t i o n ,  the s p i n p o l a r i z a t i o n of the muonium ensemble tends to  zero a t long  AI.D  times.  Isolated Muonium i n an External Magnetic Field In the presence of an e x t e r n a l magnetic f i e l d ,  as the amplitudes  the f r e q u e n c i e s as w e l l  a r e dependent upon the E u l e r a n g l e s .  Assuming both a  c y l i n d r i c a l and a p l a n a r d i s t o r t i o n of the muonium h y p e r f i n e  interaction,  the t o t a l H a m i l t o n i a n i s  "Ko  + 4 a  ~ 22 w  H =  8-rc  where r =  -w  1  W 2o) W  _ w  2-2  - 2-l w  ^00- °Mu 20) 4 t  _ W  n  -w _  0  0  2  1  _ W  0  21  °  (u> +4r+(l+2sc)w )  2 1/2 <J0QQ[(1 + x ) - l],  0()  20  Cw  2Q  Cw  (AI.103) 2Q  (-3a) -4r+(l-2sc)w }_ 00  2 2 £ = 2 ( c - s ) and the w  20  2m ?m  a r e the d e t e c t o r  - 181  -  frame h y p e r f i n e c o e f f i c i e n t s which are r e l a t e d coefficients io AI.30.  2 m  through  Calculating  to the c r y s t a l frame  the r o t a t i o n a l t r a n s f o r m a t i o n d e f i n e d i n E q u a t i o n  the a p p l i e d f i e l d  s p i n r e l a x a t i o n f u n c t i o n from  the  H a m i l t o n i a n of E q u a t i o n AI.103, f o r a l l a p p l i e d f i e l d s i s a somewhat difficult  problem.  certain limiting Consider  However, t h i s problem becomes c o n s i d e r a b l y e a s i e r f o r  cases  the problem i n the l i m i t of "high f i e l d s " ,  In t h i s l i m i t , one  ( i . e . , u>  »  2m^*  a  can approximate the t o t a l H a m i l t o n i a n by i t s d i a g o n a l  components and w r i t e  Ho  + 4 a  W- 2 ) 0  ^00- Mu- 20)  0  8¥  H  where WQ 2  °  W  4w  0  0  0  0  W  °  °  °  °  (co +4r+(l+2sc)w ) 0()  20  (AI.104)  0  (-3o) -4r+(l-2sc)w l_  0  00  20  i s the d e t e c t o r frame c o e f f i c i e n t which i s r e l a t e d  frame c o e f f i c i e n t s through  to the  crystal  the r o t a t i o n a l t r a n s f o r m a t i o n s d e f i n e d i n  Equation AI.30, namely w  2Q  = /273  I co m  ,(2) R  2m  (AI.105)  2 m  S i n c e t h i s H a m i l t o n i a n i s d i a g o n a l , the energy by the d i a g o n a l elements,  5  , and  - (h/2*) 7 K o  X  + 4 a  namely  W" 2 ] W  ;  0  h  =  = (h/2*) i [ o , + 4 r + ( l + 2 s c ) w ] ; l ^ „ .,x . w - Q j » s, nn  3  4 L  0 ( )  the corresponding  e i g e n v a l u e s are simply g i v e n  9n  2  L  4  ( ^ h/2  7Ko- Mu- 2o] 4u  W  ,  a t  i n  = (h/2*) ^ [ - 3 c o - 4 r + ( l - 2 s c ) w ] v. L —  t r a n s i t i o n f r e q u e n c i e s are  nn  4  then  0 0  „  (AI.106) 20  - 182 -  W  12  =  U  13  =  W  14  =  2w  V u  23  Mu " " l(  1  1 + S C  OO Mu ~ + U  ^ 20 W  T ^2*i-- s c^j " w ,2 0  + r  The e l g e n f u n c t l o n s  of t h i s Hamiltonian  eigenstates give i n Equation coefficients  W  24  =  W  34  =  • P '^ i i  ii  Y  a )  -a,  00  Mu  ) 20 w  +r- j ( l - s c ) w  + 2 r + S C W  • '  0  Z  the i s o t r o p i c  P  1,  (AI.107)  20  expansion  the e x p r e s s i o n s f o r  given i n Equation  P *11  2 0  Z  22  AI.27,  gives  =  z  P  12 =  P  13 =  X  c  ; ' ; '  P = 12 P 13 =  P *14  s  ; *  P 14 = - i s  P  s  ; '  P 23 = - i s  c  • '  0  • *  0  X  X  X  23 =  P 24 X  =  r  P *34 X  Substituting  33 = - c o s \ ,  P* =  0  =  0  =  0  P = 23  0  P = -ic 24  =  0  P = 34  =  sinX  Y  Y  ;  0  2  ic  Y  '  Y  H  P  ' '  Z  Y  Y  0  the r e s u l t s of E q u a t i o n  AI.17, one o b t a i n s  f o r the d e t e c t o r  'I-  (AI.108)  AI.114 i n t o Equations  AI.14, AI.15 and  frame  ( t ) = \ ( l + c o s \ ) + \ f c o s ( \ / 2 ) [ c o s ( , t ) + cos ( w t ) ] "24' "13" 2  Q 0  00  Combining t h i s w i t h  P  g  U  1 + s c  AI.24 and so the e i g e n f u n c t i o n  are just c ^ =  0  x  11  a r e o f course  the d e t e c t o r frame p o l a r i z a t i o n components p  -^Mu- - j (  =  2  W l  +  7 A  (AI.109)  2 X 2 s i n ( \ / 2 ) [ c o s ( o j ^ t ) + c o s ( c o ^ t ) ] + y s i n \ [ c o s ( i o ^ ^ t ) ]} "34" 23" 9  and g  ( t ) = y / 2 7 T {-(l + cos \) - s i n \ [ c o s ( o ) , t ) ] + 34' 2  2 Q  2  A  cos (\/2) 2  (AI.110)  x [cosloo^^t) + c o s ( o o t ) ] + s i n ( \ / 2 ) [ c o s ( u ) ^ t ) + c o s f i o ^ t ) ]} 24  Evaluating Equation  AI.112, one w r i t e s  - 183 w  = /273 {to  2Q  20  R<  2 )  (Q) + o  + (. _  R< (a) 2 )  2  2  2  From the d e f i n i t i o n o f the r o t a t i o n matrix  R^(Q)}  2  (AI.lll)  elements R ^ ^ Q ) , E q u a t i o n  AI.lll  becomes . w  *  < )  = /27T {a>  2Q  ^"t99  M  OJ A u / 5  P (cos8) +  22  2  [e  (-f3,-o)  Y  (AI.112) -1  where P ( c o s B ) i s the Legendre p o l y n o m i a l  *22  *  and the Y  2  M  ,  ,  (-8,-a) are the  s p h e r i c a l harmonics which d e f i n e the r o t a t i o n a l t r a n s f o r m a t i o n . the a s s o c i a t e d complex  w  20  =  T  "20 (  conjugate  3 c o s  Recalling  r e l a t i o n s h i p s , E q u a t i o n AI.112 becomes  (P) ~ ) 1  u  +  22  P  s i n 2 (  )  cos(<p + 22  2a)}  (AI.113)  Because we now have c y l i n d r i c a l symmetry, the r e l a x a t i o n f u n c t i o n s f o r a random a n i s o t r o p i c h y p e r f i n e i n t e r a c t i o n f o r powders a r e c a l c u l a t e d by OO  g j j ( t ) = / da)  00  J do) (a) ) f ( a ) , a) ) / dQ g I  20  M  2 2  M  2  2 0  2  r  (AI.114)  i s the combined d i s t r i b u t i o n f u n c t i o n  22  2  d e f i n e d i n E q u a t i o n AI.100.  I f i t i s f u r t h e r assumed that the a p p l i e d f i e l d  i s low w i t h r e s p e c t to the h y p e r f i n e f i e l d r = 0 and c o s \ = 0.  (i.e.,  (t)  M ->  where L = 0 and 2 and f[oo Q, w J  1//2,  L Q  O  —oo  i g n o r e the  (i.e., x «  1),  one has c = s =  With t h i s one can ignore the s i n g l e t s t a t e ,  terms i n the g Q ( t ) ' s ) , and d e r i v e an e x p r e s s i o n f o r L  the t r i p l e t muonium r e l a x a t i o n f u n c t i o n .  F o r the case o f L=0,  Equation  AI.114 becomes 8  00  ( t )  =  6 3 J 20 +  J 22 C« J  dw  du  22  -oo  o  x / dQ c o s ( o ) t ) Mu  cos(j w  f  l 20» 22J w  w  (AI.115) 2 Q  t)  - 184 and  f o r the case L=2 i n Equations g^(t) = - I  /T73  +  /T73  T  AI.114 g i v e s  J d ^ -oo 3  x / dQ c o s ( t o t ) c o s ^ w M u  By  i n t e g r a t i n g over 8  00  ( t )  =  k  +  the ^m'  8  2 Q  J do> o  (o)^)  f(co ,co ) I  2Q  2  2  (AI.llo)  f-"- » E q u a t i o n AI.115 becomes rst  ^QISCOS^B  exp{- ^ t/2/3  2  t)  (^r^r^os^t)  J  M  J*dB sinB  111  —  / da  -g- s i n ' p a  —  /  2 2  (AI.117)  d ^  | cos( <t> + 2a) |t} 22  which can be w r i t t e n as ,  -,  ,  Tl  COsfoi. t )  Tt/2  o {exp[- | s i n B  o  o v ^ t cos9] exp[- |  2  In a s i m i l a r manner, the case L=2  u  1  (AI.118)  1  c o s  JTJl o ^ t |3cos p - 1|]} 2  gives  (  ( j J M  0  11  o 3 2 ^1 {exp[- -g- s i n B a t 2 2  3  cose] e x p [ -  o  /2/3 a  2 Q  t  (AI.119) 2 |3cos p - 1|]}  Now one can c o n s i d e r the motion of the | i s p i n p o l a r i z a t i o n i n the c o n t e x t +  of  the c o n v e n t i o n a l f i e l d  AI.D.l The  geometries;  l o n g i t u d i n a l and t r a n s v e r s e  field.  Longitudinal F i e l d Relaxation Function l o n g i t u d i n a l r e l a x a t i o n f u n c t i o n can be e a s i l y c a l c u l a t e d from the  d e f i n i t i o n s of the l o n g i t u d i n a l r e l a x a t i o n f u n c t i o n s g i v e n i n E q u a t i o n AI.21.  In the l i m i t i n g  case under d i s c u s s i o n , the o n l y non-zero  l o n g i t u d i n a l r e l a x a t i o n f u n c t i o n i s g..(t).  Since t h i s f u n c t i o n i s a  - 185 of both &QQ{^)  l i n e a r combination gj(t)  = g j j ( t ) - /ITT g ^ ( t )  and g2Q( )>  one has  t  =  j  (AI.120)  which simply means t h a t the h y p e r f i n e i n t e r a c t i o n of the t r i p l e t muonium ensemble i s completely  AI.D.2  decoupled  f o r co^ »  o" ^.  u  2  Transverse Field Relaxation Function  The  d e t e c t o r frame o b s e r v a b l e  defined i n Equations  AI.22 and AI.23.  under c o n s i d e r a t i o n , one observes coplanar-transverse  c t  F o r the "high f i e l d "  limiting  case  t h a t there i s o n l y one non-zero  and one non-zero p e r p e n d i c u l a r - t r a n s v e r s e r e l a x a t i o n  sc function, g  transverse r e l a x a t i o n functions are  ss ( t ) and g p ( t ) , r e s p e c t i v e l y , which a r e e q u i v a l e n t . t  Thus one  can w r i t e 8  ct  (  t  )  =  8  D t  (  t  )  =  (  2 l l  )~  l c o s  ^Mu^  S  o  O  {exp[- -g- s i n 6 0  2 2  g ^ ( t ) = g£<t>  =  n  P  o  Tuf  t  cose] e x p ( -  V  )  O  /iTS a t  |3cos B - 1|)}  2Q  F o r e a r l y times  to o b t a i n the approximate  \cos(  (AI.121)  _____  O  which can be c a l c u l a t e d n u m e r i c a l l y . expand the i n t e g r a n d  l  o  v  ex [-( P  / 2  a  2  Q  ( t •*• 0 ) , one can  expression  + - L . af )t]  (AI.122)  2  7t/6 As a matter of c o n v e n t i o n , omitted By  the cos(co^ t) p a r t of E q u a t i o n AI.122 i s u s u a l l y u  from the d e f i n i t i o n of the t r a n s v e r s e f i e l d comparing the i n i t i a l  slope (  m z  f u n c t i o n g i v e n i n E q u a t i o n AI.102, w i t h transverse f i e l d  relaxation function.  f ) °f the zero f i e l d r e l a x a t i o n the i n i t i a l  slope (  m t  f ) o f the  f u n c t i o n of E q u a t i o n AI.122, one can d e f i n e the r a t i o  - 186 -  —  = /3  =—J  [i  +  2  Thus, one f i n d s t h a t i s greater  i n zero  -  /TTi(^ /  O 2 0  > m £> t  /3  (AI.123)  )] i n d i c a t i n g that the r a t e of d e p o l a r i z a t i o n  than i n t r a n s v e r s e  field.  - 187 -  APPENDIX II  ~ ULTRA-LOW ENERGY MUON PRODUCTION (uSOL)  In recent it  years,  the development of high  p o s s i b l e to study atomic s c a t t e r i n g c r o s s  positrons with surfaces vacuum [1-4] .  f l u x p o s i t r o n beams has made s e c t i o n s , the i n t e r a c t i o n o f  as w e l l as the s p e c t r o s c o p y of p o s i t r o n i u m  atoms i n  The analogous experiments w i t h p o s i t i v e muons a r e a t present  i m p r a c t i c a l s i n c e comparable  beams do not as yet e x i s t .  a r t method f o r producing slow \x i n v o l v e s  tuning  +  The s t a t e o f the  the secondary channel to  lower momenta P, thereby c o l l e c t i n g and t r a n s p o r t i n g p, which o r i g i n a t e from +  7t  +  decaying i n s i d e the p i o n p r o d u c t i o n  a better  term, c a l l e d  target.  s u b s u r f a c e muons.  These muons a r e ,  I t can be e a s i l y argued t h a t the  s u b s u r f a c e \x r a t e R i s p r o p o r t i o n a l t o p /2 m u l t i p l i e d by an +  decay f a c t o r .  f o r l a c k of  7  appropriate  For 10 keV (1.5 MeV/c), the s u b s u r f a c e u"*" r a t e a t 100 \ik f o r  the M13 secondary channel a t TRIUMF would be a p p r o x i m a t e l y _. R =  1.4 x lO^sec ^ -7/2 YTT (29.8  MeV/c)  /  t  not  T  u  Q  *  „  8  / » IN (AII.l)  -1 s  e  T  c  through the channel (3.66 x 1 0  i s the mean muon l i f e t i m e  acceptable The  /  T  Z  where t f i s the time of f l i g h t M13) and  f  e  (~ 2.2 \xs).  from a p r a c t i c a l e x p e r i m e n t a l  sec f o r  C l e a r l y , this rate i s  perspective.  need f o r an u l t r a - l o w energy (0 to ~10 k e V ) , h i g h  somewhat s e l f e v i d e n t .  - 6  f l u x \x beam i s +  Such a beam i f developed can be immediately  utilized  i n the study o f : (1) Electron-Muon Capture S p e c t r o s c o p y - By s c a t t e r i n g slow u"" a t g r a z i n g i n c i d e n c e to a s u r f a c e , one can study e l e c t r o n pick-up processes i n v o l v e d i n u e ~ and u e ~ e ~ f o r m a t i o n as a f u n c t i o n o f the s u r f a c e p r o p e r t i e s and magnetic o r d e r i n g . T h i s type of experiment has a l r e a d y been done f o r D ions [ 5 ] . 1  +  +  +  - 188 (2) Adatom A d s o r p t i o n on S u r f a c e s - The a b i l i t y to adsorb \i or Mu on w e l l characterized surfaces w i l l provide information regarding s u r f a c e p r o p e r t i e s as w e l l as the e f f e c t of a reduced d i m e n s i o n a l i t y on the e v o l u t i o n of the muon s p i n p o l a r i z a t i o n . +  (3) Charge Exchange Cross S e c t i o n s - Measurements o f the muonium f o r m a t i o n p r o b a b i l i t y as a f u n c t i o n of i n c i d e n t | i energy (down to thermal e n e r g i e s ) would p r o v i d e v a l u a b l e i n f o r m a t i o n to h e l p d i s c r i m i n a t e between spur and hot atom mechanisms. +  (4) M o l e c u l a r I o n Formation - Muon m o l e c u l a r i o n f o r m a t i o n has been observed i n He and Ne [ 6 ] , but i t i s not as y e t known a t what stage of t h e r m a l i z a t i o n the m o l e c u l a r Ion i s formed. The c o r r e l a t i o n between the i n c i d e n t energy (down to thermal e n e r g i e s ) and m o l e c u l a r i o n f o r m a t i o n would help d e c i p h e r the mechanisms involved. A slow Mu beam c o u l d be produced by p a s s i n g the low energy \x beam +  through  a thin f o i l  ( o r gas j e t ) and t a k i n g advantage of the l a r g e e l e c t r o n  capture  cross s e c t i o n .  The r e s u l t i n g  slow Mu beam c o u l d then be u t i l i z e d i n  experiments such as: (1) Muonium Lamb S h i f t - R e c e n t l y measured to an accuracy  of 1%  at TRIUMF [ 7 ] , t h i s experiment becomes a s i g n i f i c a n t t e s t of QED i f an a c c u r a c y of 100 ppm can be o b t a i n e d . Slow | i f l u x e s i n excess of 1 0 / s e c a t 1 MeV/c would make a p r e c i s i o n experiment possible. +  3  (2) Muonium t o Anti-Muonium C o n v e r s i o n - A low energy Mu beam would of course b e n e f i t these s t u d i e s , but to be c o m p e t i t i v e w i t h e x i s t i n g e x p e r i m e n t a l s c e n a r i o s the Mu f l u x would have to be i n excess of l O ^ / s e c . In what f o l l o w s , the i n v e s t i g a t i o n of p o s s i b l e \i e m i s s i o n +  s u r f a c e s i s proposed which u t i l i z e s  the knowledge gained  from  i n low energy  p o s i t r o n p r o d u c t i o n r e s e a r c h w i t h the a p p r o p r i a t e a n a l o g i e s drawn between p o s i t r o n s and p o s i t i v e muons.  The u l t i m a t e g o a l of t h i s r e s e a r c h would of  course be the development of an u l t r a - l o w energy (0 to ~10 keV) \i beam. +  AII.l  Current Status of Slow Positron Production The  beam moderation techniques  employed i n the p r o d u c t i o n of slow e"  - 189 beams i n v o l v e s i n g l e c r y s t a l metal moderators and u t i l i z e n e g a t i v e work f u n c t i o n f o r p o s i t r o n s backscattering  moderator s u r f a c e . and  2 2  N a f o r instance)  The beta-decay p o s i t r o n s  become t h e r m a l i z e d  exponential  a t the moderator s u r f a c e .  geometry i s used where the h i g h  from a r a d i o a c t i v e source (  w i t h a stopping  attenuation  law [ 8 ] .  d i f f u s e back to the s u r f a c e  beams, d e f i n e d positrons By  energy p o s i t r o n s  of a  Normally a originate  which i s mounted f a c i n g the a r e implanted i n t o the c r y s t a l  d i s t r i b u t i o n c o r r e s p o n d i n g to an  Some o f the implanted e  +  a r e a b l e to  b e f o r e a n n i h i l a t i n g , and a f r a c t i o n of those a r e  then emitted from the moderator s u r f a c e f u n c t i o n mechanism.  the e x i s t e n c e  The c o n v e r s i o n  as a r e s u l t of a n e g a t i v e work  e f f i c i e n c y £ f o r present day moderated  as the r a t i o of the slow e  +  yield  emitted from the source, i s g e n e r a l l y  to the t o t a l number of f a s t on the order of 1 0  - 3  .  analogy w i t h the e l e c t r o n case, the work f u n c t i o n c j of the +  p o s i t r o n i n a metal i s g i v e n by *. = - D - u. + *p  (All.2)  where D i s a p o t e n t i a l due to the s u r f a c e  d i p o l e l a y e r and \x-p i s the  p o s i t r o n c h e m i c a l p o t e n t i a l i n s i d e the m e t a l . potential incorporates from the p o s i t r o n - i o n along w i t h the s u r f a c e metal.  two terms.  The f i r s t  The p o s i t r o n  chemical  c o n t r i b u t i o n to p,p a r i s e s  i n t e r a c t i o n ( B l o c h wave e n e r g y ) . d i p o l e l a y e r a c t to expel  This i n t e r a c t i o n  the p o s i t r o n from the  The second c o n t r i b u t i o n to |a.p i s the e l e c t r o n - p o s i t r o n c o r r e l a t i o n  energy, which i s of course an a t t r a c t i v e p o t e n t i a l a c t i n g to bind the p o s i t r o n to the metal Considering one  surface.  the mass of the muon i n comparison to that of the p o s i t r o n ,  can conclude that  f o r a metal moderator a \x n e g a t i v e work f u n c t i o n i s +  - 190 not such a l i k e l y candidate to be used  -  i n the p r o d u c t i o n of an u l t r a - s l o w \i  +  beam, p r i m a r i l y because the a t t r a c t i v e e ~ - u about 1 Roo r a t h e r than the 1/2  c o r r e l a t i o n energy would  +  as i s the case f o r e - e -  A l s o , the added k i n e t i c energy  arising  becomes n e g l i g i b l e f o r the  +  From t h i s one  can conclude  t h a t , at l e a s t f o r metal  the \x a f f i n i t i e s are probably not n e g a t i v e . +  a f f i n i t i e s may  very w e l l be n e g a t i v e because i n t h i s case the  c o r r e l a t i o n s are i n g e n e r a l s m a l l . observed  for e  +  implanted  Another  e~-|i  +  emission p r o c e s s , which has  i n i o n i c s i n g l e c r y s t a l s , produces e  k i n e t i c e n e r g i e s on the order of the band gap energy mechanism and  moderators,  For i n s u l a t o r s , however, the  +  \i  correlations.  from B l o c h wave k i n e m a t i c s , which i s  on the order of a few e l e c t r o n v o l t s f o r e , heavier  +  be  +  been  having  of the s o l i d .  This  i t s p o s s i b l e a p p l i c a t i o n to \i emission i s d i s c u s s e d i n d e t a i l +  i n the f o l l o w i n g pages.  All.2  Band Gap  E m i s s i o n of e  +  from I o n i c C r y s t a l S u r f a c e s  Recent p o s i t r o n experiments implanted  [9] show that when e  of keV  +  e n e r g i e s are  i n t o i o n i c s i n g l e c r y s t a l s they are r e e m i t t e d i s o t r o p i c a l l y  from  the s o l i d s w i t h a continuum of e n e r g i e s having a maximum approximately to the band gap  energy  under the assumption  ( t y p i c a l l y on the order of 10 to 20 eV).  experiments,  +  Operating  that the mechanism(s) r e s p o n s i b l e f o r the r e e m i s s i o n of  p o s i t r o n s would a l s o be i n v o l v e d i n the analogous s y n o p s i s of the e  equal  phenomena f o r a"", a b r i e f 1  along w i t h the c u r r e n t understanding  of the  mechanism(s) i n v o l v e d , are g i v e n here. F i v e a l k a l i - h a l i d e s ( L i F , NaF, solids  ( S i 0 , A 1 0 , MgO, 2  2  3  NaCl, KC1,  KBr)  C a F ) were s t u d i e d i n a l l . 2  samples were o r i e n t e d w i t h the (100)  and  four other  The  ionic  alkali halide  a x i s normal to the e m i t t i n g s u r f a c e , as  - 191 was  the MgO  crystal.  The  S10  -  c r y s t a l was  2  z-cut,  o r i e n t e d w i t h the c - a x i s normal to the  emitting  o r i e n t a t i o n was  The  believed  vacuum of 5 x 1 0 surface  to be  (110).  T o r r , w i t h the  - 1 0  c o n t a m i n a t i o n was  A beam of 500  estimated to be  the a x i a l component of the r e e m i t t e d  The  energy s p e c t r a obtained  statistically  was  the  2  CaF  experiments were performed i n a °C, and  the  somewhat l e s s than a monolayer. the  s u r f a c e of the  p o s i t r o n spectrum was  samples  measured.  of the  w i t h the maximum energy a p p r o x i m a t e l y  individual solids.  These experiments were  i n c i d e n t p o s i t r o n s , w i t h the r e s u l t s showing  s i g n i f i c a n t d e v i a t i o n from the 500  d i s t r i b u t i o n of the e m i s s i o n s p e c t r a was approximately  sample  f o r each of the nine i o n i c c r y s t a l s show a  c h a r a c t e r i s t i c continuum of e n e r g i e s  eV  3  s u r f a c e and  i n c i d e n t on  and  repeated f o r 1500  2  samples heated to about 330  eV p o s i t r o n s was  equal to the band gap  the A 1 0  eV  data.  a l s o s t u d i e d and  The  no  angular  found to  be  isotropic.  I n a d d i t i o n to e m i t t i n g a l s o emit p o s i t r o n i u m emission processes, p r o b a b i l i t y on  the  p o s i t r o n s , i t was  (Ps).  To  f o r both e incident e  +  +  found that a l l n i n e samples  d i s c r i m i n a t e between d i f f e r e n t p o s s i b l e and  Ps,  the dependence of the Ps  energy was  studied.  Results  formation  from these  s t u d i e s as w e l l as from p o s i t r o n d i f f r a c t i o n experiments, have l e a d to conclusion  that at l e a s t f o r L i F and  be a s s o c i a t e d w i t h Ps Approximately 60% 60%  of the Ps  the e m i s s i o n of both e  incident e  +  form Ps  atoms t h a t d i f f u s e back to the  i t was  of l e a v i n g a s u r f a c e  +  and  Ps  can  d i f f u s i n g to the s u r f a c e of the c r y s t a l moderator.  of the  thereby r e - e m i t t i n g In 1972  NaF,  the  the  i n these samples, w i t h about s u r f a c e being  dissociated,  positron.  postulated [10].  that Ps  could  be f i e l d - i o n i z e d  T h i s , however, does not  i n the  e x p l a i n the  process  anomalously  - 192 l a r g e e m i s s i o n e n e r g i e s or the c o r r e l a t i o n w i t h the band gap energy solid.  of the  An a l t e r n a t e e x p l a n a t i o n [9] i s t h a t the p o s i t r o n i s Auger-emitted  when the Ps e l e c t r o n f a l l s model, the maximum energy e l e c t r o n recombining  i n t o an a c c e p t o r s t a t e a t the s u r f a c e . E  o f the emitted e" , c o r r e s p o n d i n g  &  x  this  to the Ps  r  m  With  w i t h a h o l e a t the bottom of the v a l e n c e band i s g i v e n  by the e x p r e s s i o n E  = (E + AE ) - ( E g v b  6  max  J  P  K  S  + «f) +  where Eg i s the band gap energy, Ps E^ i s the b i n d i n g energy p o s i t r o n work f u n c t i o n . gives E of  m  a  (All.3)  J  AE  V  i s the width of the v a l e n c e band,  e of Ps on the s u r f a c e o f the s o l i d and <3? i s the +  P l u g g i n g the v a l u e s f o r NaF i n t o E q u a t i o n  = 11.8(6) eV, which agrees w e l l w i t h the e x p e r i m e n t a l  x  12.3 +/- 0.7 eV.  S i n c e one does not normally expect  the v a l e n c e band, however, E q u a t i o n A l l . 3 The  All.3  r e s u l t [9]  l o n g - l i v e d holes i n  r e p r e s e n t s an o v e r e s t i m a t e .  o r i g i n o f the s u r f a c e a c c e p t o r s t a t e s i s not yet known.  one would not expect h o l e s below the the Fermi  energy  Normally,  E^ (about 4.5 eV  below the bottom of the c o n d u c t i o n band f o r a i r c l e a v e d NaF a t 300 °C [ 9 ] ) . However, e l e c t r o n - h o l e p a i r s a r e produced incident e migrate  +  i n the i o n i z a t i o n t r a i l  of the  beam, and p o s s i b l y some of the h o l e s s u r v i v e long enough t o  to the s u r f a c e .  I n any case, the branching  ratio for e  +  e m i s s i o n as  opposed to Ps e m i s s i o n i s equal to the s u r f a c e d e n s i t y of a c c e p t o r s t a t e s multiplied branching  by the e l e c t r o n c a p t u r e c r o s s s e c t i o n . ratio for e  +  F o r these i n i t i a l by c l e a v i n g i n a i r .  As mentioned e a r l i e r the  vs Ps e m i s s i o n has been found experiments  the a l k a l i h a l i d e samples were prepared  Subsequent experiments  and L i F were a l s o performed  [9] to be about 60%.  on vacuum-cleaved samples of NaF  [9] w i t h the r e s u l t s i n d i c a t i n g  the same g e n e r a l  - 193 p o s i t r o n e m i s s i o n s p e c t r a as found f o r the a i r - c l e a v e d except f o r a few s l i g h t d i f f e r e n c e s . NaF  In p a r t i c u l a r , $  samples, +  f o r the a i r - c l e a v e d  and L i F c r y s t a l s i s equal to +0.5 eV and -0.7 eV, r e s p e c t i v e l y [ 9 ] .  However, f o r the vacuum-cleaved samples of both NaF and L i F , $ be p o s i t i v e .  Thus even though  i s expected  All.3  was found t o  the p o s i t r o n work f u n c t i o n i s not n e g a t i v e ,  p o s i t r o n e m i s s i o n of band gap e n e r g i e s i s s t i l l to note i n l i g h t of the f a c t  +  t h a t the u  +  observed.  T h i s i s important  work f u n c t i o n f o r these m a t e r i a l s  to be p o s i t i v e .  Comparison of e and u WRT Band Gap Emission +  Although  +  some tend to view  the muon as a heavy e l e c t r o n , the b e h a v i o r  of muonium (Mu) i n s o l i d s i s more r e m i n i s c e n t of hydrogen  r a t h e r than P s .  The d i f f u s i o n c o n s t a n t D f o r Mu i n these m a t e r i a l s can be e s t i m a t e d by c o n s i d e r i n g experiments  i n v o l v i n g Mu e m i s s i o n from f i n e S i 0  U s i n g a d i f f u s i o n model [12] o r i g i n a l l y that D ~ 1 0  powders [ 1 1 ] .  a p p l i e d to p o s i t r o n i u m , i t was found  cm /s a t room temperature.  - 7  2  2  The d i f f u s i v i t y i n s i n g l e  would of course be g r e a t e r than t h i s w i t h a good room temperature being D ~ I O  - 5  cm /s. 2  In comparison,  i n the same m a t e r i a l s i s about 10 , - 5  10  the d i f f u s i o n l e n g t h ( T , , D )  h i g h e r temperatures  - 3  1 / 2  crystals  estimate  the d i f f u s i o n c o n s t a n t f o r p o s i t r o n s  cm /s. 2  With a d i f f u s i o n c o n s t a n t of  i s then about  5 x 10  - 6  would of course enhance the d i f f u s i o n .  cm.  H e a t i n g to  The s t o p p i n g  d i s t r i b u t i o n of s u r f a c e \i i n these m a t e r i a l s has a range of about 0.05 cm. +  Clearly, u  +  tuning to s u b s u r f a c e momenta would i n c r e a s e the p r o b a b i l i t y f o r the  to r e a c h the s u r f a c e w i t h i n t h e i r  incident  l i f e t i m e , but c o n s i d e r i n g  the l o s s i n  f l u x as the beam momentum i s reduced, no net i n c r e a s e i n the  - 194 r e e m i t t e d beam f l u x i s f o r s e e n . In a d d i t i o n to d i f f u s i v i t y , to Ps must be c o n s i d e r e d .  As mentioned e a r l i e r ,  p o s i t r o n s form p o s i t r o n i u m . probabilities deduced from  I n comparison the muonium f o r m a t i o n temperature,  the observed m i s s i n g f r a c t i o n s i n the p. s p e c t r a , a r e 98 + 5% +  i n the slow u  +  respectively. moderation  The muonium f r a c t i o n w i l l be  efficiency.  To estimate the maximum e m i s s i o n energy the muonium b i n d i n g energy E ^ \x work f u n c t i o n  u  for u  +  one r e q u i r e s v a l u e s f o r  on the s u r f a c e of the s o l i d  which a r e as y e t not w e l l known.  +  analogy  roughly 60% o f the i n c i d e n t  f o r s i n g l e c r y s t a l s of NaF and L i F a t room  [13] and 44 ± 6% [13,14], reflected  the f o r m a t i o n p r o b a b i l i t y f o r Mu as opposed  w i t h Ps s t u d i e s , the maximum Mu k i n e t i c energy  and the  However, i n i s the n e g a t i v e o f  i t s work f u n c t i o n <3?M, which i s g i v e n by U  d  M u  = (E^  where band.  U  - R j + ($! + *J)  (AII.4)  i s the e l e c t r o n a f f i n i t y a t the bottom of the c o n d u c t i o n Using E q u a t i o n AII.4,  E^ = (E + AE ) - $ max g v v  ;  M u  one can r e w r i t e E q u a t i o n A l l . 3 as  - R + $ ao -  (All.5)  e  A n e g a t i v e muonium work f u n c t i o n has been p o s t u l a t e d to e x p l a i n the e m i s s i o n of Mu from f i n e l y d i v i d e d S i 0  2  powders.  A c o n s e r v a t i v e e s t i m a t e of $  M u  Mu would be $  « 0±1 eV.  respectively  [15].  For NaF and L i F , E  +  i s 11.5 eV and 13.7 eV,  The width of the v a l e n c e band i s AE^ = 4.0(5) eV [9]  « 0±1 eV f o r both c r y s t a l s . emitted u  g  and  Thus, the maximum k i n e t i c e n e r g i e s of the  f o r NaF and L i F a r e estimated  to be  - 195  max  (NaF)  =  1.9  E^ (LiF) = max The  ± 2 eV  4.1+2  (All.6)  eV  branching r a t i o y  difficult  -  to e s t i m a t e .  f o r \i e m i s s i o n as opposed to Mu +  Q  emission i s  However, the problem i s somewhat s i m p l i f i e d  since  the s u r f a c e d e n s i t y of a c c e p t o r s t a t e s i s l i k e l y to be a p r o p e r t y of the sample p r e p a r a t i o n .  Thus one  is left  only w i t h e v a l u a t i n g the e l e c t r o n  capture c r o s s s e c t i o n at the s u r f a c e f o r Mu as opposed to Ps. b i n d i n g energy  of Mu  i s approximately  twice t h a t of Ps, one would expect  e l e c t r o n capture c r o s s s e c t i o n at the s u r f a c e to decrease has a l r e a d y been mentioned, the h i g h e r b i n d i n g energy compared to Ps a l s o s h i f t s  the maximum energy  e n e r g i e s w i t h r e s p e c t to the e branching r a t i o y the branching  f o r | i vs Mu  ratio for e  y  Q  the u  +  branching  +  e m&X  LL  - E^  ID. 3.X  ) to E  e 3.X  ffl  r a t i o would then be 5-10%.  lower  times  by the f r a c t i o n of  .  Thus, a good  e  +  estimate  In any case, measurement of  One  l a s t p o i n t to be made i s  +  e l e c t r o n with neighboring  nuclei.  C a l c u l a t i o n s f o r P o s i t i v e Muon E m i s s i o n Y i e l d To make a t h e o r e t i c a l estimate of the slow \x c o n v e r s i o n +  E  as  and L i F the \x s p i n i n the muonium s t a t e d e p o l a r i z e s due  s u p e r h y p e r f i n e i n t e r a c t i o n s of the Mu  All.4  +  As  From these c o n s i d e r a t i o n s the  i s one of the g o a l s of these experiments.  that i n both NaF  and mass of Mu  f o r L I emission to  vs Ps e m i s s i o n weighted  range (E  the  accordingly.  should be r o u g h l y equal to one h a l f  +  Q  emitted i n the energy for  spectra.  +  Because the  for u (1) (2) (3) (4)  +  efficiency  emission, one needs to know: The The The The  stopping d i s t r i b u t i o n f ( x ) of the i n c i d e n t beam bulk d i f f u s i o n constant D f o r t h e r m a l i z e d muonium f r a c t i o n F(T) of muonium formed ( f o r m a t i o n p r o b a b i l i t y ) branching r a t i o y f o r e n e r g e t i c \x e m i s s i o n +  Q  to  - 196 Consider of  the random walk problem  t h e r m a l i z e i n the muonium s t a t e .  atoms d i f f u s e a p p r o x i m a t e l y one +  of  where the muons are i n c i d e n t  on the s u r f a c e  a homogeneous moderator of t h i c k n e s s d as shown i n F i g u r e A I I . l ,  subsequently  |i  -  randomly through  Once t h e r m a l i z e d , the the l a t t i c e  of the moderator s u r f a c e s where there e x i s t s emission.  With t h i s geometry and  until  some f i n i t e  t a k i n g i n t o account  they  Mu reach  probability for  the f i n i t e  lifetine  the muon, the c o n v e r s i o n e f f i c i e n c y £^ can be w r i t t e n [16]  « l  u ^  =  I  (  F ( T ) y  where R ( t ; x )  surface.  o) / o  d  -tlx t  * f  6  d d  x  f  (  x  )  R  (  The  f a c t o r of 1/2  I t can be shown that  R(t;x) = (2t)  (4uDt)  1  /  t  ;  x  (All.7)  )  o  i s the r a t e at which a Mu  the s u r f a c e .  arises  starting  at X  s i n c e we  are n e g l e c t i n g the o t h e r  a t time  q  approximately mean ranges  -x /4Dt  2  appears  at  -(d-x) /4Dt  2  [x e  3  +  (d-x) e  ]  (All.8) » 10% w i l l  stop  u n i f o r m l y over a d i s t a n c e r d e f i n e d by the minimum and maximum  of the beam p a r t i c l e s ,  namely  r « Range(30 MeV/c) - Range(27 MeV/c) = 2.35 With t h i s  t=0  [16,17]  A beam of 30 MeV/c muons w i t h a momentum spread of AP/P  approximation,  x 10~  2  cm  (All.9)  the s t o p p i n g d i s t r i b u t i o n f ( x ) i n E q u a t i o n A l l . 7  can be assumed to be uniform and g i v e n by f ( x ) = 1/r.  S i n c e the mean  s t o p p i n g d i s t a n c e r i s l a r g e compared to the d i f f u s i o n l e n g t h the e x p r e s s i o n f o r R ( t ; x ) g i v e n i n E q u a t i o n A l l . 8 two  and  (T D)l  c l e a r l y breaks  n  / 2  ,  down i n t o  components of equal magnitude c o r r e s p o n d i n g to the a r r i v a l r a t e of Mu  at  - 197 -  REFLECTED  BEAM  [  BEAM  Figure AII.l Target geometry showing both r e f l e c t i o n and t r a n s m i s s i o n modes. I n t r a n s m i s s i o n mode degrading i s p r o v i d e d by the t a r g e t i t s e l f .  - 198 each of the two s u r f a c e s .  Thus R(t;x)  f o r each s u r f a c e can be s i m p l i f i e d to  give R(t;x) = (2t)  -1/2 [x e (4nDt)  ]  (All.10)  Substituting t h i s expression into Equation A l l . 7 i g n o r i n g the second s u r f a c e ) , the slow | i  +  and l e t t i n g d •*• »  conversion e f f i c i e n c y £  (i.e., is  written  o  o (All.11)  = | [ F ( T ) y ] ± (x o  D)  = y  1 / 2  = y This e f f i c i e n c y  (~10  -b  (9.78xl0- );  f o r NaF  (4.4xl0~ )  for L i F  5  Q  5  ;  f o r a c o n s e r v a t i v e v a l u e of y ) i s of course not Q  v e r y good, i t does, however, t r a n s l a t e i n t o 10 such muons per second f o r muon i n t e n s i t i e s such as a v a i l a b l e from M13 or M20 and ~ 1 0  3  per second i f  the moderator c o u l d be p l a c e d c l o s e to the p i o n p r o d u c t i o n t a r g e t . proposed e m i s s i o n process  exists for u  +  i t should be observable  TRIUMF's s u r f a c e muon beams, and once observed the above e f f i c i e n c y .  I f the  u s i n g one o f  steps can be taken to improve  These steps c o u l d i n c l u d e the development of h i g h  s u r f a c e area moderators, i n v e s t i g a t i n g ways of i n c r e a s i n g the d e n s i t y of s u r f a c e a c c e p t o r s t a t e s , i n c r e a s i n g the muonium d i f f u s i o n l e n g t h as w e l l as investigating utilizes  other moderators.  Producing  surface emission processes, w i l l  (~10 G) magnetic f i e l d interactions.  a polarized u  +  beam, which  r e q u i r e the a p p l i c a t i o n of a l a r g e  to quench the e f f e c t s of s u p e r h y p e r f i n e  - 199  All.5  -  Prototype Apparatus The  apparatus designed f o r the i n i t i a l  from s o l i d  surfaces  chamber and  search  i s shown i n F i g u r e A l l . 2 .  f o r the e m i s s i o n of | i  I t c o n s i s t s of a s c a t t e r i n g  t a r g e t assembly, combined w i t h a DQQ  spectrometer which i s  designed to momentum s e l e c t the e x t r a c t e d muons and channeltron  detector.  designed to be  either a transmission  respect  as  - 9  to a l l o w  Torr. the  The  spectrometer s e c t i o n to be mounted i n by  rotating  to a grounded g r i d  t a r g e t and 2  the  180°.  t a r g e t assembly w i l l be h e l d at a p o t e n t i a l  potential V  thereby p r o v i d i n g  from the  surface.  of about +10  an e l e c t r i c  field  to  the grounded g r i d w i l l have an i n d e p e n d e n t l y v a r i a b l e order measurement of  P r o v i s i o n s have a l s o been made f o r h e a t i n g  targets  diffusion rate.  using  to enhance the Mu simple and  dry n i t r o g e n .  ladder  w i l l be  with  A second g r i d , which l i e s between  a p p l i e d to i t which w i l l a l l o w a f i r s t  relatively  kV  accelerate  e m i s s i o n energy s p e c t r a .  be  limit  s c a t t e r i n g chamber i s  or r e f l e c t i o n geometry, simply  muons which are emitted the  initial  employ borrowed, non-bakable components which w i l l  designed i n such a way  The  them onto a  A l t h o u g h a l l of the vacuum components have been  the a t t a i n a b l e vacuum to about 1 0  apparatus by  focus  c o n s i s t e n t w i t h u l t r a - h i g h vacuum requirements, the  experiments w i l l  +  The  t a r g e t changes w i l l  initial  the  t a r g e t assembly  require venting  the  the  will  system  In the f u t u r e a bakable, remotely c o n t r o l l a b l e t a r g e t  introduced  with more s o p h i s t i c a t e d temperature c o n t r o l  capability.  All.6  Measurements In t r a n s m i s s i o n  mode the \i beam i s i n c i d e n t on a moderator of s u i t a b l e +  -  200  -  5 0 cm I i iii iii iiI  Figure A l l . 2 transmission  LISOL s c a t t e r i n g geometry).  chamber and DQQ  spectrometer (shown here i n  -  201 -  t h i c k n e s s such t h a t the muons a r e stopped a t or near the downstream In  r e f l e c t i o n mode the i n c i d e n t \x beam must be degraded +  e x t r a c t i o n g r i d s to such an energy t h a t upstream  s u r f a c e of the moderator.  the | i  will  +  upstream  surface.  of the  stop a t or near the  M u l t i p l e s c a t t e r i n g i n the degrader  will  c l e a r l y reduce the e f f e c t i v e i n c i d e n t beam r a t e i n r e f l e c t i o n geometry, however background reduced.  r a t e s , e s p e c i a l l y due to beam p o s i t r o n s , may a l s o be  I n some c a s e s , s i n g l e c r y s t a l  samples  having the a p p r o p r i a t e  t h i c k n e s s f o r t r a n s m i s s i o n geometry may not be r e a d i l y a v a i l a b l e . of  In l i g h t  these c o n s i d e r a t i o n s , both t r a n s m i s s i o n and r e f l e c t i o n geometries  will  have to be t e s t e d . The b a s i c measurement to be made i s the time of f l i g h t i n c i d e n t beam L I which  fires  +  of  the beam counter and the subsequent  a slow \x i n the c h a n n e l t r o n . +  characteristic  spectrum which  From E q u a t i o n A l l . 1 1 ,  R(t)  = — (D/n)  t  /  T  +  begins about 3 7 0 ns a f t e r  the u  +  spectrum i s expected to be  • exp(-Xt)  ;  (All.12)  C = [ F ( T ) y„ i ] °  /t  r  w i t h the number of events observed w i t h i n a f i n i t e gate width %  N(T  ) = C / 8  dt R ( t ) = £ D  o  1 / 2  [\ + -  ] "  1  /  2  (\+1/T  erf [ A  a  Here the \ parameter  start pulse.  Ll  -  Z  detection  S u r f a c e e m i t t e d L I should g i v e a  the shape of t h i s "  1 / 2  (TOF) between an  8  )]  O  g i v e n by  (All.13)  *  r e p r e s e n t s the r a t e o f l o s s of Mu from the d i f f u s i n g  muonium ensemble, and i s i n c l u d e d to account f o r p o s s i b l e l o s s e s of Mu due to c h e m i c a l r e a c t i o n s , e t c . , i n the c r y s t a l . E p i t h e r m a l L I or u e ~ e ~ which are produced +  +  by m u l t i p l e s c a t t e r i n g and  charge exchange p r o c e s s e s i n the t a r g e t and accepted by the spectrometer system, w i l l g e n n e r a l l y be d i s t r i b u t e d at h i g h e r v e l o c i t i e s as compared t o LI  +  a r i s i n g from s u r f a c e e m i s s i o n , and w i l l  therefore give r i s e  to prompt  - 202 events.  Measurement of the e p i t h e r m a l \i of Mu +  -  (|i e~e ) y i e l d w i l l +  likely  _  r e q u i r e the development of an e l e c t r o s t a t i c l e n s i n j e c t i o n system t o i n c r e a s e the acceptance  o f the spectrometer  section.  F i n a l i z a t i o n o f an  i n j e c t i o n system d e s i g n w i l l depend on the s p e c t r a observed p r e s e n t l y proposed histogram,  apparatus.  a |i -decay histogram +  Simultaneous  +  expected  a l s o be  accumulated. +  c  to be ~0.75, but t h i s needs to be b e t t e r determined.  energy,  of the TOF  d e t e c t i o n e f f i c i e n c y , e ^ o f the c h a n n e l t r o n f o r 10 keV u i s  measurement of t h i s w i l l  of  w i t h the c o l l e c t i n g  between the c h a n n e l t r o n and the p o s i t r o n  t e l e s c o p e s , gated by an i n c i d e n t u , w i l l The  w i t h the  A  t h e r e f o r e be made, p o s s i b l y as a f u n c t i o n of  u s i n g the three N a i d e t e c t o r s p l a c e d s t r a t e g i c a l l y around the cone  the c h a n n e l t r o n to p r o v i d e the maximum p o s s i b l e d e t e c t i o n s o l i d  The N a i d e t e c t o r s w i l l d e t e c t the decay p o s i t r o n s from u c h a n n e l t r o n and thus determine  +  stopped  angle. i n the  the a b s o l u t e f l u x of stopped muons.  These  measurements w i l l of course r e q u i r e a p p r o p r i a t e veto and c o i n c i d e n c e scintillators  t o c o r r e c t l y d e f i n e the s o l i d  The N a i d e t e c t o r s can a l s o be used to  angles and s e n s i t i v e volume.  i n c o i n c i d e n c e w i t h the c h a n n e l t r o n  reduce backgrounds i n the TOF spectrum, but w i t h a l o s s i n event W i t h these c o n s i d e r a t i o n s , the e x p e r i m e n t a l r a t e R R  = R  exp  where R  o  I  e r f ( A Ix ) e, e . e. g u d ch N a i v  TT  ;  i s the i n c i d e n t \x f l u x , +  Q  not decay i n f l i g h t ,  0.2 and  (All.14) t h a t the | i  do  +  e ^ i s the d e t e c t i o n e f f i c i e n c y of the c h a n n e l t r o n and  experiment  ~ 0.85.  p i s then  ; for \ = 0  i s the p r o b a b i l i t y  e„ _ i s the d e t e c t i o n e f f i c i e n c y and s o l i d Nai For the proposed  e x  rate.  angle o f the N a i c r y s t a l a r r a y .  and apparatus, R  ~ 10 /s, 6  q  ~ 0.75, ^ j - j - ~ G  a  F o r a gate width of 2 |j.s, erf(/T^/T^) i s a p p r o x i m a t e l y  - 203 equal to 0.84. LiF,  From t h i s , c o n s e r v a t i v e estimates of R  exp  , f o r both NaF and  are then  R  « y e  x  (10/second) ; f o r NaF °  p  (calculated  «* y  where the v a l u e of y  All.7  (All.15)  0  i s of course d i f f e r e n t  f o r the two c r y s t a l s  Backgrounds There  are t h r e e major sources of backgrounds to be c o n s i d e r e d ; beam  p o s i t r o n s , p o s i t r o n s from muons stopped muons which decay i n f l i g h t w i l l pass through bremsstrahlung will  f o r \=0)  ( 5/second) ; f o r L i F  reflect  through  i n the moderator and p o s i t r o n s from  the spectrometer  system.  Beam p o s i t r o n s  the moderator and s c a t t e r downstream producing  and a n n i h i l a t i o n r a d i a t i o n .  These beam r e l a t e d  the RF s t r u c t u r e of the c y c l o t r o n and w i l l  i n t r a n s m i s s i o n than i n r e f l e c t i o n geometry. beam i s h i g h l y d e s i r a b l e .  backgrounds  probably be g r e a t e r  Because of t h i s , a s e p a r a t e d  The p o s i t r o n s which a r i s e from LI decaying i n the +  moderator have some p r o b a b i l i t y of being emitted i n t o the acceptance of t h e spectrometer.  These p o s i t r o n s a r e too e n e r g e t i c to be t r a n s p o r t e d through  the spectrometer, produce  but c o l l i s i o n s w i t h the w a l l s of the vacuum chamber  background r a d i a t i o n which i s f l a t  decaying i n f l i g h t  will  background r a d i a t i o n . flat,  but w i l l  i n time.  P o s i t r o n s from \x  +  a l s o c o l l i d e w i t h the vacuum chamber w a l l s producing I n t h i s case, however, the background w i l l not be  decay w i t h the muon mean l i f e t i m e m u l t i p l i e d  dependent f u n c t i o n .  will  These backgrounds a r e d i f f i c u l t  c l e a r l y have to be minimized  by some p o s i t i o n  to e s t i m a t e but w i l l  by s h i e l d i n g the d e t e c t o r s from a l l sources  other than the t a r g e t , r e d u c i n g the beam c o n t a m i n a t i o n and i f f e a s i b l e r e d u c i n g the momentum b i t e of the beam.  - 204 T h i s experiment at  first  u s i n g NaF  -  should c l o s e l y p a r a l l e l the e a r l i e r p o s i t r o n  and L i F i n the <100> o r i e n t a t i o n .  experiment  Depending on  r e s u l t s o b t a i n e d w i t h the a l k a l i h a l i d e s , these i n v e s t i g a t i o n s may extended  the be  to o t h e r c r y s t a l o r i e n t a t i o n s as w e l l as other m a t e r i a l s such as  q u a r t z (Eg « 9 eV) and T h i s appendix  solid  r a r e gases  ( w i t h m o d i f i c a t i o n s ) was  p r o p o s a l (E-325) to the December 1984 E v a l u a t i o n Committee and was [18] of the f i r s t low energy  (<10  such as argon  experiments  eV)  (Eg » 19  submitted as an  meeting  eV).  experimental  of the TRIUMF E x p e r i m e n t a l  accepted a t high p r i o r i t y .  Preliminary results  have indeed shown p o s i t i v e i n d i c a t i o n s of a  component f o r L i F .  - 205 APPENDIX I I I —  AIII.A  -  COLLISION FREQUENCY OF THERMAL MUONIUM  Derivation  C o n s i d e r a p o i n t p a r t i c l e of mass m and mean thermal v e l o c i t y v, moving f r e e l y i n a u n i f o r m d i s t r i b u t i o n of N s p h e r i c a l p a r t i c l e s of r a d i u s R. one  d e f i n e s the number d e n s i t y to be N/V,  If  where V i s the t o t a l volume of  the  sample, the mean f r e e path L i s then w r i t t e n L = V (TT R By d i v i d i n g  N)  2  the mean f r e e path L by  the average time  and  the mean thermal v e l o c i t y v, one  t between c o l l i s i o n s ,  t = 3 = V [% R v Taking  (AIII.l)  - 1  v N)  2  (AIII.2)  - 1  v  =  N r „2 8kTil/2 - (n R ) [ — ]  Low  T i s the  r  A  T  T  T  temperature.  ( n e g l e c t i n g the volume of the s o l i d ) ,  number d e n s i t y i s simply g i v e n by the  (  has  Density Limit  For low packing d e n s i t i e s  «  frequency,  , (AIII.3)  u  where k i s Boltzmann's c o n s t a n t and  AIII.A.1  then g i v e s the c o l l i s i o n  the d e f i n i t i o n of the mean thermal v e l o c i t y , one  ^/mN 1 N r Jl\ F(T) = - = - ( * R )  v  namely  the r e c i p r o c a l of E q u a t i o n AIII.2  substituting  obtains  ?  '  equation  p  <  4n R o where M i s the mass of one  the  A  I  I  I  -  4  K  (i.e.,  g r a i n ( p a r t i c l e ) , p i s the mass packing d e n s i t y  a f t e r compression) of the t a r g e t p a r t i c l e s and p  d e n s i t y of the b u l k m a t e r i a l ( f o r S i 0 ; 2  p  Q  = 2.2  Q  g/cm ). 3  i s the mass Thus i n the  low  )  - 206 density l i m i t ,  - ®  the c o l l i s i o n  §- ^  m  frequency i s  1 1 / 2  < A m  - > 5  "o T h i s e q u a t i o n i s , however, not c o r r e c t i f the volume of the s o l i d  ( i . e . , the  volume of the N p a r t i c l e s ) i s s i g n i f i c a n t w i t h r e s p e c t to the t o t a l of  volume  the sample.  AULA.2 In  High D e n s i t y L i m i t the h i g h packing d e n s i t y l i m i t ,  negligible,  f  =  (  ~ solid)  V  V  By combining f V  - 4it R  Using  =  V  " (T* N  " f r e e volume" V^, namely  r 3  Equations AIII.4  3  f  i s no l o n g e r  so t h a t one must r e d e f i n e the number d e n s i t y to be the number of  p a r t i c l e s ( g r a i n s ) per u n i t  V  the volume of the s o l i d  T J  [ ^ "  l ] "  )  =  V  ^  " f f T * ) ]  (AIII.6)  3  and A I I I . 6 , one o b t a i n s (AIII.7)  1  P  t h i s " c o r r e c t e d " number d e n s i t y , the c o l l i s i o n  frequency  f o r the h i g h  density l i m i t i s F  < > = l ^ ) T  1  /  2  lir- T l  l  < ->  t 1 / 2  N o t i c e t h a t f o r low d e n s i t i e s , E q u a t i o n AIII.8  aiii  reduces  8  to the e x p r e s s i o n o f  Equation AIII.5.  Q.E.D.  - 207 APPENDIX IV —  TABULATED TRANSVERSE FIELD DATA  S i 0 ( l ) Prepared a t 110 "C; X  M u  2  Vs Temperature  T (K)  AT (K)  \  4.1  0.10  2.59  0.180  5.8  0.20  2.49  0.137  9.0  1.00  2.11  0.186  9.5  0.20  2.02  0.118  10.2  0.20  2.00  0.154  11.5  1.50  1.82  0.142  12.5  0.20  1.72  0.112  14.0  0.30  1.50  0.101  16.8  0.20  1.63  0.118  19.3  0.20  2.48  0.135  22.0  0.20  2.58  0.145  25.0  0.20  2.99  0.336  32.5  3.50  2.55  0.135  40.3  2.30  2.13  0.270  47.5  12.50  1.85  0.153  59.0  11.00  1.38  0.153  60.0  2.00  1.02  0.072  86.0  1.00  0.62  0.046  128.0  1.00  0.51  0.037  300.0  3.00  0.40  0.028  M  u  (us  - 1  )  A^  i U  (ns  - i  )  - 208 S i 0 ( 3 ) Prepared a t 600 °C; \  M  2  u  Vs Temperature  T (K)  AT (K)  \  4.6  0.05  1.18  0.033  6.0  0.10  1.08  0.052  8.0  0.10  1.01  0.045  10.0  0.10  0.90  0.060  12.0  0.10  0.73  0.035  16.0  0.10  0.51  0.026  18.0  0.10  0.57  0.035  20.0  0.30  0.84  0.036  22.0  0.10  1.13  0.053  24.0  0.30  1.42  0.086  25.0  0.20  1.38  0.052  26.0  0.10  1.68  0.098  28.0  0.10  1.85  0.107  30.0  0.20  2.11  0.108  40.0  2.00  1.97  0.082  50.0  3.00  1.85  0.065  85.0  5.00  1.37  0.074  M  u  (us  - 1  )  AX  M u  (us  - 1  )  - 209 S i 0 ( 2 ) Prepared a t 110 °C; X  M u  2  Vs Temperature  T (K)  AT (K)  \  5.8  0.20  2.49  0.137  10.1  0.20  2.13  0.089  13.5  0.20  1.76  0.086  25.0  0.20  3.31  0.206  45.0  0.20  1.59  0.089  58.5  5.50  0.99  0.078  64.0  0.20  1.03  0.057  M  u  (us  - 1  )  A\  M u  (us  - 1  )  - 210  -  REFERENCES  CHAPTER I . 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[2]  D.R. Harshman, R. K e i t e l , M. Senba, R.F. K i e f l , J.H. Brewer, Phys. L e t t . 104A, 472 (1984).  Brewer,  E . J . Ansaldo  and  - 217  -  [3  L. Anthony,  B e l l Laboratories, Private  [4  R. Kubo, J . Phys. Soc. Japan 9_, 935  [5  A.P. M i l l s , J r . and W.S.  Communication.  (1954).  Crane, Phys. Rev. L e t t . 53, 2165  (1984),  APPENDIX I [1  A. Messiah, Quantum Mechanics, V o l . 2, (Wiley and Sons, New 1958).  [2  R.E. T u r n e r , accepted f o r p u b l i c a t i o n i n H y p e r f i n e  [3  R.E. T u r n e r , Phys. Rev. B 31.. 112  [4  J.H. Brewer, K.M. Crowe, F.N. Gygax and A. Schenck, i n Muon P h y s i c s , V o l . I l l , Chapter 7, e d i t e d by V.W. Hughes and C.S. (Academic P r e s s , New York, 1975).  York,  Interactions.  (1985).  [5  J.H. Brewer and K.M.  Crowe, Ann. Rev. P a r t . S c i . 28, 239  [6  A. Schenck, i n N u c l e a r and P a r t i c l e P h y s i c s a t I n t e r m e d i a t e E n e r g i e s , e d i t e d by J.B. Warren, (Plenum, New York, 1976).  [7  D.P. Spencer, Ph.D. (1985).  [8  A. Hinterman, P.F. Meier and B.D. (1980).  T h e s i s , U n i v e r s i t y of B r i t i s h  P a t t e r s o n , Am.  Wu,  (1978).  Columbia  J . Phys. 48,  956  APPENDIX I I [1]  A.P. M i l l s , J r . , S. Berko and K.F. Canter, i n Atomic P h y s i c s , V o l . 5, e d i t e d by R. Marrus, M. P r i o r and H. Shugart, (Plenum, New York, 1977).  [2]  T.C. G r i f f i t h and G.R.  [3]  S. Berko and H.N. (1980).  [4]  A.P. M i l l s , J r . , S c i e n c e 218, 335  [5]  C a r l Rau, J . Magnetism  [6]  D.G.  [7]  C.J. Oram, J.M. B a i l e y , P.W. Schmor, C A . F r y , R.F. K i e f l , J.B. Warren, G.M. M a r s h a l l and A. O l i n , Phys. Rev. L e t t . 52_, 910 (1984).  Heyland, P h y s i c s Reports 39, 169  (1978).  P e n d l e t o n , Ann. Rev. N u c l . P a r t . S c i . 30,  F l e m i n g , et a l . ,  543  (1982).  and Magnetic M a t e r i a l s 3£, 141  (1982).  Chem. Phys. 82, 75 (1983).  - 218 W.  -  Brandt and R. P a u l i n , Phys. Rev. B 1_5, 2511  A.P. M i l l s , J r . and W.S.  (1977).  Crane, Phys. Rev. L e t t . 53_, 2165  K.F. Canter, P.G. Coleman, T.C. G r i f f i t h and G.R. Phys. B _5, L167 (1972).  (1984).  Heyland, J .  G.M. M a r s h a l l , 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 , Phys. L e t t . 65A, 351 (1978). R. P a u l i n and G. Ambrosino, G.G. V.G.  J . Physique 29, 263  (1968).  Myasishcheva, Yu.V. Obukhov, V.S. Roganov, L.Ya. Suvorov F i r s o v , S o v i e t High Energy Chemistry 3^, 460 (1969).  and  E.V. M i n a i c h e v , G.G. Myasishcheva, Yu.V. Obukhov, V.S. Roganov, G.I. Savel'ev and V.G. F i r s o v , S o v i e t P h y s i c s JETP 30_, 230 (1970). A.B. Kunz, i n Elementary E x c i t a t i o n s i n S o l i d s , M o l e c u l e s and Atoms, P a r t B, e d i t e d by J.T. Devreese, A.B. Kunz and T.C. C o l l i n s , 343 (Plenum, London, 1974). S. Chandrasekhar, Rev. Mod. G.M. M a r s h a l l , (1981).  Ph.D.  Phys. L5, 1 (1943).  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  D.R. Harshman, J.B. Warren, J . L . B e v e r i d g e , K.R. K e n d a l l , R.F. K i e f l , C.J. Oram, A.P. M i l l s . J r . , W.S. Crane, A.S. Rupaal and J.H. T u r n e r , (submitted to Phys. Rev. L e t t . ) .  PUBLICATION LIST f o r D.R. Harshman as of  JAN. 1986 (Chronological Order)  Papers Published i n Refereed Journals Surface Interactions of Muonium i n Oxide Powders at Low Temperatures, R.F. K i e f l , J.B. Warren, C.J. Oram, G.M. M a r s h a l l , J.H. Brewer, Harshman, and C.W. Clawson, P h y s i c a l Review B 26_, 2432 (1982).  D.R.  Giant Muon Knight S h i f t s i n Antimony and Antimony A l l o y s , J.H. Brewer, D. R. Harshman, E. K o s t e r , H. S c h i l l i n g , D.LI. W i l l i a m s and M.G. S o l i d S t a t e Communications 46_, 863 (1983).  Positronium i n S i 0  2  Priestly,  Powder at Low Temperature, R.F. K i e f l and D.R.  Harshman, P h y s i c s L e t t e r s 98A,  447 (1983).  Zero-Field Muon-Spin Relaxation i n CuMn Spin-Glasses Compared With Neutron and S u s c e p t i b i l i t y Measurements, Y . J . Uemura, D.R. Harshman, M. Senba, E. J . Ansaldo and A.P. Murani, P h y s i c a l Review B (Rap. Comm.) 30, 1606 (1984) .  D i f f u s i o n and Trapping of Muonium on S i l i c a Surfaces, D.R. Harshman, R. K e i t e l , M. Senba, R.F. K i e f l , E . J . Ansaldo and J.H. Brewer, Phys. L e t t . 104A, 472 (1984).  Muon Spin Relaxation i n AuFe and CuMn Spin Glasses, Y . J . Uemura, T. Yamazaki, D.R. (1985) .  Harshman, M. Senba and E . J . A n s a l d o , Phys. Rev. B 31,  Study of the Hybrid State of Y C o 9  7  546  (2 < T < 6 K) by Means of Zero F i e l d  Muon Spin Relaxation, E . J . A n s a l d o , D.R. Noakes, J.H. Brewer, R. D.R. Harshman, M. Senba, C.Y. Huang and B.V.B. S a r k i s s i a n , S o l i d Communications 55, 193 (1985).  Keitel, State  Level-Crossing Resonance Muon Spin Relaxation i n Copper, S.R. Kreitzman, J.H. Brewer, D.R. Harshman, R. K e i t e l , D . L l . W i l l i a m s and E . J . A n s a l d o , ( a c c e p t e d f o r p u b l i c a t i o n i n Phys. Rev. L e t t . ) .  MSR Measurement of the Reaction Rate of Muonium with a Supported Platinum Catalyst, R.F. Marzke, W.S. G l a u n s i n g e r , D.R. Harshman, J.H. Brewer, R. K e i t e l , M. Senba, E . J . A n s a l d o , D.P. Spencer and D.R. Noakes, (accepted f o r p u b l i c a t i o n i n Chem. Phys. L e t t . ) .  Observation of Muon-Fluorine "Hydrogen Bonding" i n Ionic Crystals, J.H. Brewer, Keitel,  S.R. K r e i t z m a n , D.R. Noakes, E . J . Ansaldo, D.R. Harshman and (submitted to Phys. Rev. B, r a p i d communications).  Anisotropic Muonium With Random Hyperfine Anisotropics: A New Relaxation Theory, R.E. Turner and D.R.  R.  Static  Harshman, (sub. to Phys. Rev.  B.)  Observation of Low Energy u+ Emission from Solid Surfaces, D.R. Harshman, J.B. Warren, J . L . B e v e r i d g e , K.R. K e n d a l l , R.F. K i e f l , C.J. Oram, A.P. M i l l s , J r . , W.S. Crane, A.S. Rupaal and J.H. T u r n e r , (submitted to Phys. Rev. L e t t . ) .  Conference  Proceedings  (Refereed)  Muonium In A1 2 0 3 Powder at Low Temperatures, R.F. K i e f l , J.B. Warren, C. J . Oram, J.H. Brewer and D.R. Harshman, i n P o s i t r o n A n n i h i l a t i o n , e d i t e d by P.G. Coleman, S.C. Sharma and L.M. Diana, 693 ( N o r t h - H o l l a n d P u b l i s h i n g Company, 1983). Magnetic S u s c e p t i b i l i t y , Proton NMR and Muon Spin Rotation (uSR) Studies of an Unsupported Platinum Catalyst with Adsorbed H and 0, R.F. Marzke, W.S. G l a u n s i n g e r , K.B. Rawlings, P. Van Rheenen, M. McKelvy, J.H. Brewer, D. R. Harshman and R.F. K i e f l , i n E l e c t r o n i c S t r u c t u r e and P r o p e r t i e s o f Hydrogen i n M e t a l s , e d i t e d by P. Jena and C.B. S a t t e r t h w a i t e , (Plenum P u b l i s h i n g C o r p o r a t i o n , 1983). Dynamical Behavior of Muonium on S i l i c a Surfaces, D.R. Harshman, R. K e i t e l , M. Senba, E . J . Ansaldo and J.H. Brewer, H y p e r f i n e I n t e r a c t i o n s 17-19, 557 (1984). Hyperfine S p l i t t i n g of Muonium i n Si0 2 Powder, R.F. K i e f l , B.D. P a t t e r s o n , E. H o l z s c h u h , W. Odermatt and D.R. Harshman, H y p e r f i n e I n t e r a c t i o n s 17-19, 563 (1984). Zero-Field pSR i n a Spin Glass CuMn (1.1 a t . % ) : Precise Measurement of S t a t i c and Dynamic E f f e c t s Below Tg, Y.J. Uemura, T. Yamazaki, D.R. Harshman, M. Senba, J.H. Brewer, E . J . Ansaldo and R. K e i t e l , H y p e r f i n e I n t e r a c t i o n s 17-19, 453 (1984). u SR Study of Some Magnetic Superconductors, C.Y. Huang, E . J . Ansaldo, J.H. Brewer, D.R. Harshman, K.M. Crowe, S.S. Rosenblum, C.W. Clawson, Z. F i s k , S. Lambert, M.S. T o r i k a c h v i l i and M.B. Maple, H y p e r f i n e I n t e r a c t i o n s 17-19, 509 (1984). +  P o s i t i v e Muon Knight S h i f t i n Graphite and G r a f o i l , F.N. Gygax, A. Hintermann, A. Schenck, W. Studer, A . J . van der Wal, J.H. Brewer and D.R. Harshman, H y p e r f i n e I n t e r a c t i o n s 17-19, 383 (1984). P o s i t i v e Muons i n Antimony Bismuth A l l o y s , F.N. Gygax, A. Hintermann, A. Schenck, W. Studer, A . J . van der Wal, J.H. Brewer, D.R. Harshman, E . K o s t e r , H. S c h i l l i n g , D.LI. W i l l i a m s and M.G. P r i e s t l y , H y p e r f i n e I n t e r a c t i o n s 17-19, 387 (1984). Muon Spin Relaxation i n ErRh^B^, D.R. Noakes, E . J . Ansaldo, J.H. Brewer, D.R. Harshman, C.Y. Huang, M.S. T o r i k a c h v i l i , S.E. Lambert and M.B. Maple, J . A p p l . Phys. 57_, 3197 (1985). Muonium on Surfaces, D.R. Harshman, ( I n v i t e d P a p e r ) , Proceedings of the European Workshop on the Spectroscopy of Sub-Atomic Species i n N o n - M e t a l l i c S o l i d s , 3-7 Sept. 1985, V i t r y - s u r - S e i n e , France, ( i n p r e p a r a t i o n ) .  Papers Published i n Unrefereed Journals Study of the Hybrid State of Y C o (2 < T < 6K) by means of Zero F i e l d Muon Spin Relaxation, E . J . Ansaldo, D.R. Noakes, J.H. Brewer, R. K e i t e l , D.R. Harshman, M. Senba, C.Y. Huang and B.V.B. S a r k i s s i a n , LiSR-Newsletter 30, 1661 (1984). 9  7  LF-uSR Quadrupolar Leval Crossing Resonance i n Copper at 20K, S.R. Kreitzman, J.H. Brewer, D.R. Harshman, R. K e i t e l , D.LI. W i l l i a m s , K.M. Crowe and E . J . Ansaldo, LtSR-Newsletter 30, 1675 (1984). Observation of Muon-Fluorine "Hydrogen" Bonding i n Ionic C r y s t a l s , J.H. Brewer, S.R. Kreitzman, D.R. Noakes, E . J . Ansaldo, D.R. Harshman and R. K e i t e l , iiSR-Newsletter 31, 1747 (1985).  Abstracts P o s i t i v e Muon Spin Rotation i n Magnetic-Superconducting SmRh^B^, C.Y. Huang, Z. F i s k , C.W. Clawson, K.M. Crowe, S.S. Rosenblum, J.H. Brewer, D.R. Harshman, S.E. Lambert, M.S. T o r i k a c h v i l i and M.B. Maple, APS Meeting, D a l l a s , Texas, USA, March 1982. Magnetic S u s c e p t i b i l i t y , Proton NMR and Muon Spin Rotation (uSR) Studies of Unsupported Platinum Catalysts with Adsorbed H and 0, R.F. Marzke, W.S. G l a u n s i n g e r , K.B. Rawlings, P. Van Rheenan, M. McKelvy, J.H. Brewer, D.R. Harshman and R.F. K i e f l , APS Meeting, D a l l a s , Texas, USA, March 1982. Muonium on Bare S i l i c a Surfaces, D.R. Harshman, R.F. K i e f l and J.H. Brewer, Western R e g i o n a l N u c l e a r P h y s i c s Conference, B a n f f , A l b e r t a , Canada, February 1982. Muonium on Bare S i l i c a Surfaces, D.R. Harshman, R. K e i t e l , R.F. K i e f l , M. Senba and J.H. Brewer, 2'nd T r i e s t e I n t e r n a t i o n a l Symposium on S t a t i s t i c a l Mechanics of A d s o r p t i o n , T r i e s t e , I t a l y , J u l y 1982. Observation of H NMR and of Muon Spin Rotation i n Unsupported Platinum, R.F. Marzke, W.S. G l a u n s i n g e r , K.B. Rawlings, P. Van Reenen, M. McKelvy, J.H. Brewer, D.R. Harshman and R.F. K i e f l , C a l i f o r n i a C a t a l y s i s S o c i e t y , Annual F a l l Meeting, I r v i n e , C a l i f o r n i a , USA, October 1982. Behavior of Muonium on S i l i c a Surfaces, D.R. Harshman, J.H. Brewer, M. Senba, R. K e i t e l and E . J . Ansaldo, CAP Meeting, U n i v e r s i t y o f V i c t o r i a , V i c t o r i a , B.C., Canada, June 1983. Spin Relaxation i n YRh,^ and SmRh^B^, E . J . Ansaldo, D.R. Harshman, J.H. Brewer, C.Y. Huang, K.M. Crowe and S.S. Rosenblum, CAP Meeting, U n i v e r s i t y of V i c t o r i a , B.C., Canada, June 1983. D i f f u s i o n and Trapping of Muonium on S i l i c a Surfaces, D.R. Harshman, J.H. Brewer, M. Senba, R. K e i t e l and E . J . Ansaldo, C a l i f o r n i a C a t a l y s i s S o c i e t y Annual F a l l Meeting, Brea, C a l i f o r n i a , USA, October 1983.  Muonium on Amorphous S i 0 S u r f a c e s , D.R. Harshman, J.H. Brewer, R. K e i t e l , M. Senba, J.M. B a i l e y and E . J . Ansaldo, CAP Meeting, U n i v e r s i t y de Sherbrooke, Sherbrooke, Quebec, Canada, June 1984. 2  G i a n t Muon K n i g h t S h i f t s i n Antimony A l l o y s , J.H. Brewer, D.R. Harshman, E . K o s t e r , S.R. Kreitzman and D.LI. W i l l i a m s , CAP Meeting, U n i v e r s i t y de Sherbrooke, Sherbrooke, Quebec, Canada, June 1984. Muon S p i n R e l a x a t i o n and C r y s t a l l i n e E l e c t r i c F i e l d E f f e c t s i n RERh^B^, R.R. Noakes, E . J . Ansaldo, J.H. Brewer, C.Y. Huang and D.R. Harshman, CAP Meeting, U n i v e r s i t y de Sherbrooke, Sherbrooke, Quebec, Canada, June 1984. Muonium D i f f u s i o n on Amorphous S i 0 S u r f a c e s , D.R. Harshman, J.H. Brewer, R. K e i t e l , M. Senba, J.M. B a i l e y and E . J . Ansaldo, I n t e r n a t i o n a l Chemical Congress of P a c i f i c B a s i n S o c i e t i e s , Honolulu, Hawaii, USA, December 1984. 2  Measurement of the R e a c t i o n Rate of Muonium w i t h the S u r f a c e of a Supported P l a t i n u m C a t a l y s t by Muonium S p i n R o t a t i o n (MSR), R.F. Marzke, W.S., G l a u n s i n g e r , D.R. Harshman, E . J . Ansaldo, R. K e i t e l , D.R. Noakes, M. Senba and J.H. Brewer, APS Meeting, B a l t i m o r e , Maryland, USA, March 1985. Measurement of the R e a c t i o n Rate o f Muonium w i t h the S u r f a c e o f a Supported P l a t i n u m C a t a l y s t by Muonium S p i n R o t a t i o n (MSR), R.F. Marzke, W.S. G l a u n s i n g e r , D.R. Harshman, E . J . Ansaldo, R. K e i t e l , D.R. Noakes, M. Senba and J.H. Brewer, C a l i f o r n i a C a t a l y s i s S o c i e t y Annual S p r i n g Meeting, Menlo Park, C a l i f o r n i a , USA, A p r i l 1985.  Special Oral Presentations (Invited Talks) Muonium on S i l i c a S u r f a c e s , D.R. Harshman, I n v i t e d T a l k , uSR J o u r n a l Seminar, Dept. of P h y s i c s , Univ. of C a l i f o r n i a at B e r k e l e y , B e r k e l e y , USA, 16 February 1983.  Ca.,  Behavior of Muonium on S i 0 S u r f a c e s , D.R. Harshman, I n v i t e d T a l k , S o l i d S t a t e Seminar, Department of P h y s i c s , The U n i v . of B r i t i s h Columbia, Vancouver, B.C., Canada, 28 February 1985. 2  Muonium on S i l i c a S u r f a c e s , D.R. Harshman, I n v i t e d T a l k , presented at A T & T - B e l l L a b o r a t o r i e s , Murray H i l l , New J e r s e y , USA, 4 A p r i l 1985. The I n t e r a c t i o n of Muonium With S i l i c a S u r f a c e s , D.R. Harshman, I n v i t e d T a l k , Physics-Astronomy Seminar, Department of P h y s i c s , Western Washington U n i v e r s i t y , Bellingham, Washington, USA, 15 May 1985. Muonium on S u r f a c e s , D.R. Harshman, I n v i t e d T a l k , European Workshop on the Spectroscopy of Sub-Atomic Species i n N o n - M e t a l l i c S o l i d s , 3-7 Sept. 1985, V i t r y - s u r - S e i n e , France. The I n t e r a c t i o n s of Muonium w i t h S u r f a c e s , D.R. Harshman, I n v i t e d T a l k , presented at Lawrence Livermore N a t i o n a l L a b o r a t o r y , Livermore, C a l i f o r n i a , USA, 21 October 1985.  

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