THE INTERACTIONS OF MUONIUM WITH SILICA SURFACES by DALE RICHARD HARSHMAN B.Sc, P a c i f i c Lutheran University, 1978 M.Sc, Western Washington University, 1980 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES Department of Physics We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA January 1986 © Dale Richard Harshman, 1986 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 DE-6(3/81) ABSTRACT The behavior of muonium on the surface of f i n e (35 A mean radius) S i 0 2 powders has been studied using the techniques of muon spin r o t a t i o n (uSR). Results indicate d i f f u s i o n and trapping behavior of the muonium atoms on the s i l i c a surface, which i s strongly influenced by the concentration of surface hydroxyl groups. S p e c i f i c a l l y , the presence of the surface hydroxyl groups has been shown to i n h i b i t the motion of muonium on the s i l i c a surface. These studies have also provided information regarding the o r i g i n of the r e l a x a t i o n of the muon spin p o l a r i z a t i o n for muonium on the s i l i c a surface. S p e c i f i c a l l y , a random anisotropic d i s t o r t i o n of the muonium hyperfine i n t e r a c t i o n , induced by the l o c a l surface environment of the muonium atom, has been shown to be a p r i n c i p a l contributor to the relaxation of the muon ensemble spin 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 due to the neighboring hydroxyl protons were found to play only a minor r o l e . From this r e s u l t , the observed strong dependence of the relaxation on the surface hydroxyl concentration has been at t r i b u t e d to an associated hyperfine d i s t o r t i o n , induced by the neighboring hydroxyls. A new spin r e l a x a t i o n theory, for the case of random anisotropic hyperfine d i s t o r t i o n s , has also been developed to explain the data. Gas adsorption isotherm studies were also performed, with ^He at 6 K, which show the muonium asymmetry to be strongly influenced by the f r a c t i o n a l surface coverage. These r e s u l t s c l e a r l y indicate that the muonium formation p r o b a b i l i t y decreases with increasing surface coverage, suggesting that the charge exchange cross section at the s i l i c a surface i s s i g n i f i c a n t . The implication of these r e s u l t s with regard to the o r i g i n s of muonium formation - i i i -( i . e . , surface or bulk formation) i s as yet unclear, however, since the precise role played by the adsorbed helium atoms i s not known. These investigations have also been extended to platinum loaded s i l i c a , where the f i r s t surface reaction of muonium has been observed; the reaction rate of muonium with the surface of oxygen-covered platinum microcrystals was found to be 3.5 ± 0.15 [is~^ • - i v -TABLE OF CONTENTS ABSTRACT i i LIST OF TABLES i x LIST OF FIGURES x ACKNOWLEDGEMENTS x i i i CHAPTER I. INTRODUCTION 1 A. Muons and Muonium 3 1. Muon C h a r a c t e r i s t i c s 4 2. Thermalization of P o s i t i v e Muons i n Matter 6 3. Muonium Formation and Ch a r a c t e r i s t i c s 8 B. Time Evolution of the Muonium Spin State 12 1. Muonium i n Vacuum 12 2. Interactions with the Environment 15 C. The Interactions of Muonium with 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 Surfaces 19 3. Muonium Formation i n Fine S i l i c a Powders 20 4. Extragranular Muonium Production 25 D. The Interactions of Hydrogen and Deuterium with S i l i c a 27 1. Hydrogen D i f f u s i o n i n Bulk S i l i c a 28 - v -2. Hydrogen and Deuterium on S i l i c a Surfaces 28 CHAPTER I I . EXPERIMENTAL TECHNIQUE 31 A. Accelerators and Beamlines 31 1. The TRIUMF Cyclotron F a c i l i t y 31 2. Muon Production and Transport 33 B. The uSR / MSR Technique 41 1. Zero and Longitudinal F i e l d (ZF and LF) 42 2. Transverse F i e l d (TF) 42 C. Experimental 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 Logic 48 3. Targets 52 4. Cryogenics 58 5. Gas Handling 60 D. Data Analysis 63 1. Transverse F i e l d Spectra 63 2. Zero and Longitudinal F i e l d Spectra 64 CHAPTER I I I . THEORY OF MUONIUM RELAXATION 65 A. Spin Relaxation Functions 65 - v i -1. Random Local Magnetic F i e l d s (RLMF) 66 2. Random Anisotropic Hyperfine D i s t o r t i o n s (RAHD) 71 3. Chemical Reactions (CH) 83 4. Spin Exchange (SE) 84 5. Superhyperfine Interactions (SHF) 85 B. Dynamical Relaxation Functions 88 1. Gaussian-Markovian Process 89 2. Strong Collision-Markovian Process 91 3. D i f f u s i o n i n the Presence of Traps 102 CHAPTER IV. EXPERIMENTAL RESULTS AND INTERPRETATIONS 106 A. Muonium on S i l i c a Surfaces 107 1. Transverse F i e l d Results 108 2. Zero and Longitudinal F i e l d Results 118 B. Muonium on the Surface of Helium Coated S i l i c a 130 1. Relaxation Rate Versus ^He Coverage 130 2. Muonium Asymmetry Versus ^He Coverage 132 C. Muonium On the Surface of Supported Platinum Catalysts 134 1. Unloaded S i l i c a Support 134 2. Platinum Loaded S i l i c a : 0.001% and 0.01% 137 3. Platinum Loaded S i l i c a : 0.1% and 1.0% 139 - v i i -CHAPTER V. CONCLUSIONS AND FUTURE DIRECTIONS 142 A. Summary of Results 142 1. D i f f u s i o n and Trapping 142 2. Relaxation Mechanisms 143 3. Muonium Formation P r o b a b i l i t y 146 4. C a t a l y t i c Chemistry 147 B. Future Directions 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 153 A. Observables - C r y s t a l and Detector Frames 153 1. Spin Relaxation Functions 154 B. The Spin Hamiltonian for Isolated Muonium 161 1. Evaluation of the Hyperfine Term 161 2. Evaluation of the Zeeman Term 167 C. Isolated Muonium i n Zero F i e l d 168 1. Relaxation 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. Relaxation Due to a Planar D i s t o r t i o n 173 - v i i i -3. C y l i n d r i c a l and Planar D i s t o r t i o n s Combined 177 D. Isolated Muonium i n an External Magnetic F i e l d 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. Current Status of Slow Positron Production 188 2. Band Gap Emission of e + from Ionic C r y s t a l Surfaces 190 3. Comparison of e + and \i+ WRT Band Gap Emission 193 4. Calculations for P o s i t i v e Muon Emission Y i e l d 195 5. Prototype Apparatus 199 6. Measurements 199 7. Backgrounds 203 APPENDIX I I I . COLLISION FREQUENCY OF THERMAL MUONIUM 205 A. Derivation 205 1. Low Density Limit 205 2. High Density Limit 206 APPENDIX IV. TABULATED TRANSVERSE FIELD DATA 207 REFERENCES 210 - i x -LIST OF TABLES CHAPTER I. INTRODUCTION 1. Properties of Muons (u+.u -) 5 2. Properties of Muonium (Mu) 10 3. Muonium Fractions F M„ and Transverse F i e l d Relaxation M u " - U Rates \ j _ l u for Bulk and Powdered S i l i c a 21 CHAPTER I I . EXPERIMENTAL TECHNIQUE 1. Beam Parameters for the M20 Secondary Channel 37 (a) Backward Decay Muons at 75° (M20-A) and 37.5° (M20-B) v (b) Simultaneous Decay Muons on M20 2. Surface Muons at 75° (M20-A) and 37.5° (M20-B) 39 3. Physical Properties of the S i 0 2 Powder 54 4. Experimental Target C h a r a c t e r i s t i c s 59 (a) S i 0 2 Targets (b) Platinum Loaded S i 0 2 Targets CHAPTER IV. EXPERIMENTAL RESULTS AND INTERPRETATIONS 1. F i t Results of Three-State Model for Mu on S i 0 2 Surfaces .... 115 (a) Sample S i 0 2 ( l ) prepared at 110 °C (b) Sample Si0 2(3) Prepared at 600 °C APPENDIX IV. TABULATED TRANSVERSE FIELD DATA 1. S i 0 2 ( l ) Prepared at 110 °C; v s Temperature 207 2. Si0 2(3) Prepared at 600 °C; \ M u vs Temperature 20b 3. Si0 2(2) Prepared at 110 °C; vs Temperature 209 - x -LIST OF FIGURES CHAPTER I. INTRODUCTION 1. Muon Decay Parameters 7 CHAPTER I I . EXPERIMENTAL TECHNIQUE 1. TRIUMF Cyclotron F a c i l i t y 32 2. M20 Secondary Channel 35 3. Schematic Representations of u.SR Techniques 43 (a) Zero and Longitudinal F i e l d (b) Transverse F i e l d 4. The "Eagle" \iSR Spectrometer 45 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. Logic Level Diagram for a "Good" Event 53 7. Thermogravimetric Plot for Cab-0-Sil 56 8. Gas Handling System 61 CHAPTER I I I . THEORY OF MUONIUM RELAXATION 1. S t a t i c Longitudinal F i e l d Gaussian Kubo-Toyabe 70 2. S t a t i c Longitudinal F i e l d Lorentzian Kubo-Toyabe 72 3. S t a t i c Zero F i e l d Random Anisotropic Hyperfine In t e r a c t i o n Relaxation Function (Lorentzian) 77 4. S t a t i c Transverse F i e l d Random Anisotropic Hyperfine In t e r a c t i o n Relaxation Function (Lorentzian) 79 5. S t a t i c Zero F i e l d Random Anisotropic Hyperfine Interaction Relaxation Function (Modified Lorentzian) ... 80 6. Zero and Longitudinal F i e l d Data for Bulk Fused S i 0 2 .... 82 7. Superhyperfine Interaction Diagram 87 - x i -8. Dynamic Transverse F i e l d Gaussian Kubo-Tomita (Gaussian-Markovian) 90 9. Dynamic Zero F i e l d Gaussian Kubo-Toyabe (Strong-Collision) 94 10. Dynamic Zero F i e l d Lorentzian Kubo-Toyabe (Strong C o l l i s i o n ) 96 11. Dynamic Zero F i e l d Lorentzian Random Anisotropic Hyperfine Interaction Relaxation function (Strong C o l l i s i o n ) 98 12. Dynamic Zero F i e l d Lorentzian Random Anisotropic 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 13. Dynamic Zero F i e l d Lorentzian Random Anisotropic Hyperfine Interaction Relaxation Function (Strong C o l l i s i o n ) - Planar Component 101 14. Dynamic Zero F i e l d Modified Lorentzian Random Anisotropic 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 103 CHAPTER IV. EXPERIMENTAL RESULTS AND INTERPRETATIONS 1. Transverse F i e l d Mu Relaxation Rate vs Temperature fo r Muonium on S i l i c a Surfaces Prepared at 110 °C and at 600 °C 109 2. Hyperfine-Structure I n t e r v a l v 0 0 vs Temperature f or Muonium on S i l i c a Surfaces Prepared at 110 °C 112 3. Surface Hop Rate v 0 vs Temperature for Muonium on S i l i c a Surfaces Prepared at 110 UC and 600 "C 117 4. Zero and Longitudinal F i e l d Spectra for Muonium on the S i l i c a Surface (110 UC preparation) at 7.0 ± 0.2 K 119 5. Zero and Longitudinal F i e l d Spectra for Muonium on the S i l i c a Surface (600 °C preparation) at 3.6 ± 0.2 K 123 6. Zero and Longitudinal F i e l d Spectra for Muonium on the S i l i c a Surface (600 "C preparation) at 16.0 ± 0.1 K 126 7. Zero and Longitudinal F i e l d Spectra for Muonium on the S i l i c a Surface (110 °C preparation) at 25 ± 0.5 K 127 - x i i -8. Zero and Longitudinal F i e l d Spectra for Muonium on the S i l i c a Surface (600 °C preparation) at 30 ± 0.5 K 129 9. Transverse F i e l d Mu Relaxation Rate vs ^He Coverage for S i l i c a Prepared at 110 "C and at 600 "C 131 10. Transverse F i e l d Mu Asymmetry vs ^He Coverage for S i l i c a Prepared at 110 °C 133 11. Transverse F i e l d Mu Relaxation Rate vs Temperature fo r unreduced and H-reduced, 0.0% Pt Loaded S i l i c a 135 12. Transverse F i e l d Mu Relaxation Rate vs Temperature for 0.001% and 0.01% Pt Loaded S i l i c a 138 13. Transverse F i e l d Mu Relaxation Rate vs Temperature for 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 Orientation WRT Incident \x+ beam 197 2. Scattering Chamber and DQQ Spectrometer 200 - x i l i -ACKNOWLEDGEMENTS It i s a pleasure to acknowledge the help and support of my research supervisor, J.H. Brewer, who has provided me with many new and esthetic insights into the physical world. I wish also to express my gratitude to J.B. Warren, D.LI. Williams and the UBC Physics department, the members of my Ph.D. committee and the TRIUMF s t a f f . I am p a r t i c u l a r l y g r a t e f u l to D.J. Arseneau, K.M. Crowe, D.G. Fleming, D.M. Garner, R. K e i t e l , R.F. K i e f l , R.F. Marzke, M. Senba, D.P. Spencer, J . Stewart, Y.J. Uemura and J . Worden for th e i r assistance with some of the experiments and for enlightening discussions, and e s p e c i a l l y to R.E. Turner for his help and guidance i n the development of the RAHD relaxation theory. I would also l i k e to thank my wonderful wife, Sandra, who's • understanding nature was frequently put to the test by my many lat e nights spent writing this work. Above a l l , I wish to express my sincerest gratitude to my parents, Richard and LaVonne Harshman, for t h e i r support and encouragement, and i t i s to them that t h i s work i s l o v i n g l y dedicated. - 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 int 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) with the surfaces of f i n e l y divided s i l i c a powders (35 A mean radius). This research represents the f i r s t d e t a i l e d 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 trapping behavior, and the relaxation mechanisms, for muonium on surfaces. 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 well 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]. The motivation for the present work arises because of these features, and the desire to extend the studies of muons and muonium to int e r a c t i o n s with a surface environment. This work develops a basic q u a l i t a t i v e understanding of the behavior of muonium on surfaces and could conceivably lead to the study of surface magnetism and the extensive use of muonium as an i s o t o p i c probe of hydrogen c a t a l y s i s . Experimental methods such as NMR, ESR, LEED, etc., which are widely used i n the study of adatom adsorption, generally require a f a i r l y high density of atoms, which has obvious ramifications with regard to the s t a t i s t i c a l mechanics of adsorption. In contrast, the uSR (muon spin rotation) techniques [1-3] employed i n the present work require observation of one muon (or Mu atom) at a time. This feature of the experimental method allows no p o s s i b i l i t y for any study of u +-u +, u+-Mu or Mu-Mu in t e r a c t i o n s ; indeed, at presently achievable stopped muon de n s i t i e s , the mutual encounter of two muons or muonium atoms must be an extremely rare occurrence. S p e c i f i c a l l y , the present work has provided information concerning the d i f f u s i o n and trapping behavior of muonium on the s i l i c a surface, as well as - 2 -the e f f e c t of the l o c a l surface environment on the hyperfine 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 describing the time evolution of the \i+ spin p o l a r i z a t i o n i n muonium for a generally anisotropic hyperfine i n t e r a c t i o n , which can adequately explain the muonium relaxation data. The behavior of muonium on the s i l i c a surface was also found to exhibit a strong dependence on the concentration of surface hydroxyl groups. A great deal of i n t e r e s t has been generated concerning the i n t e r a c t i o n of hydrogen atoms with c a t a l y t i c surfaces. The s i l i c a powder used i n the present study i s t y p i c a l of those used as support materials for c a t a l y s t s . Because muonium can be thought of as a l i g h t chemical isotope of hydrogen, i t i s i d e a l l y suited for this type of study. In f a c t , the present work has provided the f i r s t study of muonium on platinum loaded s i l i c a surfaces, i n which the reaction rate of muonium with the oxygen-coated surfaces of the platinum microcrystals was measured. This d i s s e r t a t i o n i s organized into f i v e chapters and two appendices. The present chapter, Chapter I, i s primarily introductory. It provides background information regarding the properties of muons and muonium, the e f f e c t of the l o c a l environment on the time evolution of the n + spin p o l a r i z a t i o n f or both charge states, a b r i e f synopsis of previous experimental and t h e o r e t i c a l \iSR studies which are pertinent to the present work and a discussion of relevant hydrogen atom experiments. In Chapter I I , the discussion focuses on the s p e c i f i c experimental techniques employed i n the present i n v e s t i g a t i o n . Included are descriptions of muon production and transport, the |iSR apparatus, e l e c t r o n i c s and data a c q u i s i t i o n , target preparation and the methods of data analysis employed. - 3 -Chapter III provides a general t h e o r e t i c a l discussion concerning the time evolution of the p.+ spin p o l a r i z a t i o n for the f i v e known muonium spin "relaxation" mechanisms, and the associated spin r e l a x a t i o n functions. Of p a r t i c u l a r importance are the relaxation functions, for both zero and applied magnetic f i e l d , which ar i s e from a random anisotropic d i s t o r t i o n of the muonium hyperfine i n t e r a c t i o n (see Appendix I ) . These functions are used i n the analysis and i n t e r p r e t a t i o n of some of the data. The experimental r e s u l t s are presented and discussed i n Chapter IV. These r e s u l t s indicate d i f f u s i o n and trapping behavior of muonium on the s i l i c a surface and suggest a random anisotropic d i s t o r t i o n of the muonium hyperfine i n t e r a c t i o n as a p r i n c i p a l contributor to the depolarization of the u-+ spin on the s i l i c a surface. F i n a l l y , Chapter V provides a b r i e f summary of the subject to date, along with a discussion of possible future d i r e c t i o n s . Appendix I contains the detai l e d derivations of the time evolution of the u + spin p o l a r i z a t i o n i n muonium subject to a generally anisotropic hyperfine i n t e r a c t i o n , along with the associated spin r e l a x a t i o n functions described i n Chapter I I I . L a s t l y , Appendix II outlines 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 with "macroscopic" surfaces, and draws heavily on knowledge already gained i n the study of positrons ( e +) and positronium ( e + e ~ , Ps). I.A Muons and Muonium Some of the c h a r a c t e r i s t i c s of muons and muonium are discussed i n the following few pages. - 4 -I.A.I Muon C h a r a c t e r i s t i c s The muon was f i r s t observed [4,5] as a component of cosmic rays i n 1937. Muons are unstable leptons, having a rest mass of about 105.7 MeV/c2, and apart from t h e i r f i n i t e l i f e t i m e can i n nearly every respect be considered heavy electrons (or positrons). Some of the properties of muons are given i n Table 1.1. The most common source of muons i s from the decay of charged pions ( i t + , i t ~ ) . Pions (spin = 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 nonconserving processes [6] i t + •> u . + + v and 7 t - > L i + v (I«l) with a free mean l i f e t i m e of 26.030(23) nanoseconds (ns). In the rest frame of the pion, the decay i s s p a t i a l l y i s o t r o p i c with the muon and neutrino being emitted i n opposite d i r e c t i o n s . In the case of 7t + decay the muon and neutrino both have negative h e l i c i t y ( i . e . , spin a n t i p a r a l l e l to momentum), whereas for n - decay they are both emitted with p o s i t i v e h e l i c i t y ( i . e . , spin p a r a l l e l to momentum). Since neutrinos possess zero (or near zero) rest mass, the momentum of the emitted muon i n the rest frame i s 29.8 MeV/c, which translates into a k i n e t i c energy of 4.1 MeV. Like the pion, the muon also decays v i a the weak i n t e r a c t i o n , according to the p a r i t y v i o l a t i n g reactions [6] u . + -> e + + v + v and u - > e + v + v (I«2) e n ^ e ^ i with a free mean l i f e t i m e of T = 2.19695(6) \is [7]. In contrast to pion decay, muon decay i s s p a t i a l l y anisotropic i n the center of mass frame; the muon provides a preferred d i r e c t i o n ( i t s spin orientation) as a reference. Table 1.1 Properties of Muons (n +,fi-) Property (symbol) Value Charge e H+,u- = ±1.60225 x 10" 1 9 Coulombs Spin s 1/2 Rest Mass \ 105.6596 MeV/c2 = 206.76859(29) mfi (a) 0.7570 m^ 0.1126123(6) m (a) Mean Free Lifetime 2.19695(6) \is (b) g-Factor -2[l.001165895(27)] (c) Magnetogyric Ratio K e 2 V 8.5165 x 10 4 s 1 G 1 2% x 13.5544 kHz/G Magnetic Moment 28.0272(2) x 10" 1 8 MeV/G 0.00484 n e 3.1833417(39) u. (a) Compton Wavelength h v 1.86758 fm de Broglie Wavelength Mi h(2nm [ ikT)" 1 / 2 = 2.99 A (300 K) 25.29 A (4.2 K) (a) D.E. Casperson, et a l . , Phys. Rev. L e t t . 38.. 9 5 6 (1977). (b) K.L. Giovanetti, et a l . , Phys. Rev. D 29_, 343 (1984). (c) J.M. Bailey, et a l . , Phys. L e t t . 55B, 420 (1975). - 6 -The maximum momentum of the decay electron i s given by the r e l a t i o n m c 2 = [ p 2 c 2 + m 2 c 4 ] 1 / 2 + p c ; p m a X = 52.827 MeV/c (1.3) u. e e e e This maximum occurs when both the neutrino and antineutrino are emitted i n the same d i r e c t i o n , opposite to that of the decay electron. For th i s case the spins of the neutrino (negative h e l i c i t y ) and the antineutrino ( p o s i t i v e h e l i c i t y ) cancel, leaving the positron(electron) to balance the spin of the u + ( u ~ ) . In weak i n t e r a c t i o n s , the momentum of the e +(e~) tends strongly to be ( a n t i ) p a r a l l e l to i t s spin, so that the high energy e +(e~) tends to ex i t along(opposite) the |i +(u~) spin. Since the topic of th i s d i s s e r t a t i o n concerns only p o s i t i v e muons, the discussions henceforth w i l l be constrained accordingly. By neglecting the mass of the positron i n comparison with the mass m of the muon, the p r o b a b i l i t y per unit time dW(e,9) for the emission of a positron of energy E i n the elemental s o l i d angle dco at an angle 9 with respect to the muon spin d i r e c t i o n can be expressed as [6] dW(e,9) = ^ [ e 2 ( 3 - 2 e ) ] [ l + P cos(9)] dedu (1.4) where e = ^ / ^ m a x = ^^myL°2 a n ( * p represents the degree of p o l a r i z a t i o n of the u + ensemble. Equation 1.4 i s written i n terms of an i s o t r o p i c average energy spectrum C(e)=e 2(3-2e) and an asymmetry factor D(e)=P(2e-l)/(3-2e), both of which are shown i n Figure 1.1 for 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 of a u"1" i n matter involves several stages of energy - 7 1.0 0.8 -0.6 0.4 -0.2 0.0 -.2 --.4 I I / D ( £ ) I I I I 0.0 0.2 0.4 0.6 0.8 e = E / E 1.0 max Figure 1.1 Positron energy spectrum from muon decay (upper curve) and energy dependence of the asymmetry factor for 100% polarized (P=l) muons (lower curve). The positron energy i s given i n units of the maximum possible emission energy E_„„ = 52.827 MeV. - 8 -loss mechanisms [1]. A high energy \i+ i n t e r a c t i n g with matter w i l l f i r s t lose energy by scattering with electrons. When the | i + v e l o c i t y approaches that of the valence electrons of the target atoms (corresponding to a | i + k i n e t i c energy of 2-3 keV), the energy loss per unit time occurs primarily through i o n i z a t i o n , i n accordance with the Bethe equation [8]. Below ~2 keV, energy loss s t i l l occurs through c o l l i s i o n s with electrons, except i n t h i s case the Bethe equation does not hold since the electrons now behave as a degenerate gas. In t h i s energy region, a muon can also capture and lose electrons i n i t s i n t e r a c t i o n s with the target medium, forming s h o r t - l i v e d neutral hydrogen-like muonium (fi +e~) atoms; i n many cases, the neutral muonium atom i s the favored charge state as the \i+ v e l o c i t y drops below the threshold for t h i s capture/loss cycle. The f i n a l muonium atom then slows down through subsequent non-ionizing c o l l i s i o n s with atoms and/or molecules. The e f f e c t of the slowing down process on the u.+ spin p o l a r i z a t i o n has been given extensive consideration by many authors [9,10], and found to be n e g l i g i b l e i n s o l i d s , where the charge exchange cycles are much shorter than the hyperfine period. This i s , of course, good news i f one wishes to study the i n t e r a c t i o n of the \i+ spin with i t s environment. In gases, however, depolarization can indeed occur since the charge exchange cycles may be comparable to the hyperfine period [11,12]. I.A.3 Muonium Formation and Characteristics The d e t a i l s of muonium (Mu) formation were f i r s t discussed i n 1952 [13], but i t was not u n t i l 1960 that d i r e c t experimental evidence of i t s formation was obtained [14]. Some of the properties of muonium are given i n - 9 -Table 1.2. The reduced mass of the electron i n muonium i s about 0.996 that for hydrogen, making the Bohr r a d i i and i o n i z a t i o n potentials of muonium and hydrogen e s s e n t i a l l y the same. Consequently, muonium behaves chemically l i k e a l i g h t isotope of hydrogen [1,15,16], having a rest mass equal to 0.1131 the rest mass m^ of hydrogen. Unlike hydrogen, however, muonium i s a purely leptonic system whose properties are calculable to extreme p r e c i s i o n e n t i r e l y from f i r s t p r i n c i p l e s . As a r e s u l t , the muonium atom i s an i d e a l system to be used for tests of quantum electrodynamics, and has been widely employed as such. A completely general Hamiltonian for the hyperfine i n t e r a c t i o n between the \i+ and e~ spins, i n the presence of a magnetic f i e l d B and allowing for e f f e c t s due to an anisotropic environment, can be written K. = (h/2Tt)( Y S e - y ^ ) • B + (h/2n) W : (s e ) (1.5) Mu ^'e ~op u ~op ; ~ » v~op ~op ; where y £ = 2n:(Ye) a n a Y^ = 2 T C ( Y ) are the respective magnetogyric r a t i o s of e u the electron and the muon, S and S r are the corresponding dimensionless ~op ~op spin operators and W i s a second rank tensor representing the contact hyperfine i n t e r a c t i o n , which has been e x p l i c i t l y generalized here to include the p o s s i b i l i t y of an anisotropic Mu atom, as might be imposed by a s o l i d medium. In vacuum, of course, W reduces to a constant multiplying the unit second rank 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 hyperfine i n t e r a c t i o n ) , the eigenvalues of the spin Hamiltonian (Equation 1.5) are given i n terms of t h e i r respective weak-field quantum numbers (F,mp) by the Breit-Rabi formula [17], namely - 10 -Table 1.2 Properties of Muonium (Mu) Property (symbol) Value Rest Mass mMu 0.1131 mR = 207.8 ni e Reduced Mass mMu 0.9952 me = 0.9956 m£ Bohr Radius ^ao^Mu 0.5315 A = 1.0044 ( a Q ) H Ground State Energy ( R O T)M U -13.54 eV = 0.9956 ( R J H Magnetogyric Ratio YMu t r i p l e t ; = 8.8 x 106 s - 1 G"1 « 2it x 1.4 MHz/G Hyperfine Frequency voo theo. = 4463.3185(6.5) MHz expt. = 4463.30235(52) MHz (a) (a) de Broglie Wavelength ,Mu 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). E . . = - 7- E ~ (g v?) |B|m_ ± i E ( l + 2m X + X 2 ) 1 / 2 _ rl . l - i 4 o KOu ^oJ '~' F 2 o*- F ; F - ( j ± >*F (1.6) ] BI X - ( g e H* + g |# and = 1, 0, -1 r o where E Q i s the zero f i e l d hyperfine energy s p l i t t i n g between the t r i p l e t (F=l) and s i n g l e t (F=0) states, g and g are the respective g-factors and e (i, e u ii and u r are the electron and muon Bohr magnetons. Denoting v = E /h to 0 0 0 00 o be the hyperfine-structure i n t e r v a l (~4463.3 MHz [18]), evaluation of Equation 1.6 for the four e x i s t i n g spin coupling states then gives 1 „ 1 , 1 r 1 ^ r l 2 ^ 2-.1/2 v i = TT E i = T + v ; v 0 = T- E = - T v + I 7- v + v . l 1 h 1 4 00 - 2 h 2 4 00 L4 00 + J 1 „ 1 1 „ 1 rl 2 ^ 2-11/2 v 0 = T- E„ = -r v - v ; v. = 7- E. = - 7- v - 7- v + v. 3 h 3 4 00 - 4 4 4 4 00 L 4 o o + J with the d e f i n i t i o n (1.7) V I ^ ^ ^ D T = i C l v . l ± \\\) (1.8) Choosing the axis of quantization to be along the magnetic f i e l d , the energy eigenstates |j> of i s o t r o p i c muonium can be represented i n terms of the i n d i v i d u a l spin eigenfunctions |m ,m > as 1 LI e (1.9) |1> = |+,+> ; |2> = s|+,-> + c|-,+> |3> = |-,-> ; |4> = c|+,-> - s|-,+> where the amplitudes s (sine) and c (cosine) are defined as 1 Ti x -|l/2 , 1 r. . x -,1/2 s = - [1 TTPy\ a n d c = - L 1 + 9 i 10 J /2 (1 + x 2 ) 1 / 2 /2 (1 + x 2 ) 1 / 2 |B| |B| x = ( g e - g = = S p e c i f i c F i e l d Parameter ^ 0 0 where one has the normalization condition s 2 + c 2 = 1, and B (» 1585 G) i s ' o the hyperfine f i e l d . Note that i n zero f i e l d , s = c = 1//2 and v 2 i l = v 0 0 . (I.10) - 12 -I.B Time Evolution of the Muonium Spin State The four hyperfine states are, i n general, unequally populated since the muons a r r i v e with a preferred p o l a r i z a t i o n (directed opposite to the d i r e c t i o n of emission from pion decay), while the captured electrons are normally unpolarized. Thus, choosing the spin quantization axis along the i n i t i a l u + spin p o l a r i z a t i o n d i r e c t i o n , half of the Mu ensemble forms i n the state |a0> = |+,+> and the other half i n the state |bQ> = |+,->. In a l o n g i t u d i n a l f i e l d , where the external magnetic f i e l d B i s directed along the i n i t i a l muon p o l a r i z a t i o n , the o r i e n t a t i o n energy i s quantized along this same d i r e c t i o n such that the state |a0> = |+,+> = |1> i s an eigenstate of the Hamiltonian, but the state |b0> = |+,-> = s|2> + c|4> i s not. With these designations, half of the muonium ensemble i s formed i n the "polarized t r i p l e t " state |a0> while the remaining half i s formed i n the "mixed" state |bQ>, and the i n i t i a l r e l a t i v e populations of the four hyperfine states are given by the p r o b a b i l i t i e s 1 s 2 c 2 p l = 2 ; p2 = f 5 p3 = ° ; P4 = f ( I- U ) Although the Coulomb i n t e r a c t i o n which governs the electron capture process has a n e g l i g i b l e e f f e c t on the spin p o l a r i z a t i o n i n s o l i d s , the hyperfine i n t e r a c t i o n between the \x+ spin and the spin of the electron i n muonium does give r i s e to phase o s c i l l a t i o n s i n the superposition |bQ> of the hyperfine states at frequencies on the order of the hyperfine-structure i n t e r v a l V Q Q . I.B.I Muonium i n Vacuum Consider the time evolution of the muon spin p o l a r i z a t i o n i n free muonium. In l o n g i t u d i n a l f i e l d , the polarized t r i p l e t eigenstate |aQ> i s a - 13 -stationary state, while the mixed state |bn> i s a superposition of two eigenstates. Since the state |aQ> i s stationary, one must only determine the time dependence of state |bQ>. Recalling Equation 1.7, and defining oa . = 2nv . and u>. . = 2n(v. .) = 2it(v. - v.)» one then finds [1] J 3 i j i j i 3 |b(t)> = e ~ i u 2 t { [ s 2 + c 2 e x p ( i u ) 2 4 t ) ] | + , - > | | + sc[ l-exp( i o ^ t ) ] |-,+>(|} (1.12) In zero f i e l d , where s = c = 1//2 and - oo 0 0 = 2 T C V 0 0 , the state |b(t)> o s c i l l a t e s with a frequency to0Q between the i n i t i a l hyperfine state |+>-n and the state |-,+>||, i n which the muon spin d i r e c t i o n i s reversed. The spin p o l a r i z a t i o n of the muons i n the state |b(t)> i s given by the r e l a t i o n p£l|(x,t)z = <b(t) | |b(t)>z, where 0-3 i s the muon P a u l i spin matrix for projections along the quantization axis (z-axis) and z i s the corresponding unit vector. By combining t h i s with the 100% p o l a r i z a t i o n of the muons i n the stationary state |aQ>, the time dependence of the t o t a l muon ensemble i n l o n g i t u d i n a l f i e l d i s given by [1] 1 i 1 x ^ + c o s ( w o / . t ) P)f(x,t) = \ [1 + P£„(x,t)]z = J + J [ 2 ^ — ] (1.13) 1 + x In transverse f i e l d (TF), where the external magnetic f i e l d B i s applied perpendicular to the i n i t i a l muon p o l a r i z a t i o n , the states |aQ> and |bQ> are not eigenstates and neither one i s stationary. In t h i s case, the i n i t i a l state vectors |aQ> and |bQ> can be written i n terms of the long i t u d i n a l f i e l d basis |m^ ,me>|| . By expanding these states i n terms of the i s o t r o p i c muonium energy eigenstates |j>, given i n Equation 1.9, the time dependence for each of these two states i s found to be [1] |a 0(t)> = i [ e " 1 " ! 1 1 ! ^ + (s + c)e" i u2 t|2> + e" i u3 t|3> - (s - c^e'^l^] (1.14) |b 0(t)> = i [ - e " 1 " ! ^ ^ + (s - c)e _ i a )2 t|2> + e" i a 33 t| 3> + (s + c)e" l u'» t :|4>] Because the magnetic f i e l d (B z) i s oriented perpendicular to the i n i t i a l muon p o l a r i z a t i o n , and since the muonium electron i s only i n t e r a c t i n g with the muon spin, a l l of the motion of the \i+ spin i n the muonium state i s confined to the x-y plane. This being the case, the time evolution of the muon p o l a r i z a t i o n for the enti r e muonium ensemble i s given by the complex quantity P^(x,t) - l[< a(t)|(o-£ + io-£)|a(t)> + <b(t)|(o-f + ia£)|b(t)>] (1.15) where and are the x and y P a u l i spin operators. Here the r e a l part i s the \i+ p o l a r i z a t i o n along the i n i t i a l x d i r e c t i o n and the imaginary part represents the \i+ p o l a r i z a t i o n along the y d i r e c t i o n , perpendicular to both x and z ( i . e . , x x y = z ) . Substituting the expressions for the state vectors given i n Equation 1.14 into Equation 1.15 then gives the r e s u l t = exp(iw_t) c o s ( - ^ t) [ c o s ( - | ^ + o)t - i6sin(-22. + Q ) t ] (1.16) 6 = ( c 2 - s 2 ) = X 2 / 2 and Q = | < O J 2 3 - w ) = - ^ ( l + x 2 ) 1 / 2 - l ] (1 + x J Most of the experiments reported i n th i s d i s s e r t a t i o n were performed i n the low f i e l d l i m i t (x « 1). In th i s l i m i t i n g case, the r e a l part of - 15 -Equation 1.16 s i m p l i f i e s to give [1] Re{p^(t)} « j cos(u_t) [cos(fit) + C O S ( O D q o + Q)t] (1.17) where u_ = 2TCV_ as defined i n Equation 1.8. Since the frequency (COQ 0+ Q ) i s i n general too high to be observed experimentally, except i n high transverse f i e l d s , Equation 1.17 describes a s i g n a l with half of the i n i t i a l | 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 at the Larmor frequency modulated at a beat frequency equal to Q. A more elaborate formalism i s developed elsewhere [19,20] for cases where the muonium electron i n t e r a c t s with i t s environment. I.B.2 Interactions with the Environment The i n t e r a c t i o n of the u-+ spin with i t s environment may i n some cases r e s u l t i n a d e p o l a r i z a t i o n or a r e l a x a t i o n of the u + spin ensemble. I t i s Instructive at t h i s point to define what i s meant by depolarization versus rel a x a t i o n . The term "depolarization" encompasses a l l v a r i e t i e s of spin dynamics, including i n t e r a c t i o n s i n which the phase coherence of the spin ensemble could i n p r i n c i p l e be recovered at some l a t e r time (e.g., by spin echo techniques); whereas "relaxation" applies to those interactions which re 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 loss of ensemble p o l a r i z a t i o n , such as i n the case of a d i f f u s i n g magnetic probe. Conventionally, however, the term "relaxation" i s applied i n a somewhat generic fashion and w i l l generally be applied herein i n the same manner. In the case of a bare u + , spin relaxation occurs v i a the i n t e r a c t i o n of the \x+ spin with the l o c a l magnetic f i e l d d i s t r i b u t i o n . In the case of muonium, however, the \i+ i s strongly coupled to the electron so that i n weak - 16 -magnetic f i e l d s the f r a c t i o n (50%) of muonium that forms i n the polarized t r i p l e t (F=l, m^ = +1) state behaves magnetically l i k e a polarized spin-one object with a magnetic moment on the order of the electron's and i s thus about 103 times more s e n s i t i v e to l o c a l magnetic f i e l d s than a bare u + . Because of the rather strong hyperfine coupling, the u + spin p o l a r i z a t i o n i s also s e n s i t i v e to e l e c t r i c f i e l d gradients or other mechanisms that may d i s t o r t the muonium electron wavefunction and thereby induce anisotropics into the muonium hyperfine i n t e r a c t i o n . It was for this reason that W was generalized i n Equation 1.5. Thus, i n addition to interactions with the l o c a l magnetic f i e l d s , which can cause depolarization for both bare u.+ and Mu, there are four other mechanisms that can induce depolarization or relaxation of the u-+ spin i n a Mu atom. The f i v e known mechanisms are: (1) Random Local Magnetic F i e l d s (depolarization) (2) Random Anisotropic Hyperfine D i s t o r t i o n s (depolarization) (3) Chemical Reactions ( r e l a x a t i o n , for TF) (4) Spin Exchange (relaxation) (5) Superhyperfine Interactions (depolarization) Here the designation of "depolarization" applies only i n the case of a s t a t i c (non-diffusing) probe (Mu atom). These mechanisms, along with the corresponding spin relaxation functions, are discussed i n more d e t a i l i n Chapter I I I . Of p a r t i c u l a r importance to the present study are the s t a t i c r e l axation functions associated with random anisotropic hyperfine i n t e r a c t i o n s . These functions are derived i n d e t a i l i n Appendix I. Owing to i t s r e l a t i v e l y l i g h t mass, the muon (or Mu atom) may be very mobile i n the stopping medium. This motion or hopping may r e s u l t i n an e f f e c t i v e r elaxation rate which d i f f e r s i n magnitude i n comparison to the s t a t i c value. This difference comes about because the e f f e c t s of the i n t e r a c t i o n ( s ) governing the time evolution of the \i+ spin p o l a r i z a t i o n are averaged by the motion, hence the term "motional averaging". Depending upon the s p e c i f i c i n t e r a c t i o n ( s ) and the time scales involved, the motion can produce an e f f e c t i v e relaxation rate that has either a reduced magnitude ("motional narrowing") or an increased magnitude ("motional broadening") i n comparison to the s t a t i c value. Relaxations due to chemical reactions or spin exchange are not affected by motional averaging, but relaxations a r i s i n g from random dipolar f i e l d s , random anisotropic hyperfine d i s t o r t i o n s and superhyperfine in t e r a c t i o n s are indeed affected. The 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 presence of s t a t i c nuclear dipoles. Assuming a Gaussian d i s t r i b u t i o n of random l o c a l f i e l d s , and defining % c to be the c o r r e l a t i o n time of the f i e l d f l u c t u a t i o n s as sensed by the L I + , the spin relaxation rate i n the l i m i t of fast f l u c t u a t i o n s becomes [21] X « <AOJ 2> x (1.18) Ll C 2 where <Aco > i s the second moment of the frequency d i s t r i b u t i o n for the random l o c a l f i e l d [22]. The e f f e c t of hopping on the shape of the r e l a x a t i o n functions a r i s i n g from random anisotropic hyperfine d i s t o r t i o n s or superhyperfine i n t e r a c t i o n s i s not as straightforward to determine. However, a det a i l e d discussion of t h i s i n the case of the former i s given i n Chapter I I I . I.C The Interactions of Muonium with S i l i c a Much of the work presented i n t h i s d i s s e r t a t i o n stems from e a r l i e r studies involving both muonium i n bulk s i l i c a and on s i l i c a surfaces. A b r i e f summary of these studies i s therefore presented here, along with - 18 -discussions on those points of p a r t i c u l a r relevance to the present work. I.C.I Muonium i n Bulk S i l i c a Extensive studies 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 anisotropic d i s t o r t i o n s of the muonium hyperfine i n t e r a c t i o n . Zero f i e l d measurements of muonium i n sin g l e c r y s t a l quartz have revealed three frequencies at low temperatures (< 77 K). These frequencies, which obey the sum rule v 1 3 = v 1 2 + v23» r e m a * n constant but have amplitudes that vary, as the c r y s t a l i s rotated about the i n i t i a l muon spin p o l a r i z a t i o n . This r e s u l t i s consistent with an e f f e c t i v e spin Hamiltonian i n which the hyperfine tensor has three p r i n c i p a l axes of symmetry. With this picture, the three observed frequencies then correspond to t r a n s i t i o n s between three l e v e l s , and as such are l a b e l l e d accordingly. At higher temperatures (near room temperature), the muonium hyperfine i n t e r a c t i o n has an anisotropy which i s symmetric about the c-axis of the c r y s t a l due to motional averaging. In t h i s (high temperature) case, the hyperfine tensor W can be broken down into an i s o t r o p i c part <W> = O)Q 0 and a term 6W£ = 6u c associated with a d i s t o r t i o n along the c-axis. By denoting S £ and S£ as the projections of the e and u"1" spins along the c-axis, respectively, the a x i a l l y symmetric contact hyperfine Hamiltonian becomes H h f = (h/2ii) W:(S6 S^ ) = (h/2it){<W>(se • sM + 6W ( S e S^)} (1.19) ' » v~op ~op' 1 ^~op ~op' c^ c CJ ' In zero f i e l d , with the c-axis oriented perpendicular to the i n i t i a l muon po 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 i s observed; however, with the c-axis oriented p a r a l l e l to the i n i t i a l muon p o l a r i z a t i o n , the o s c i l l a t i o n - 19 -disappears, as predicted by Equation 1.19. In fused quartz, the zero f i e l d hyperfine o s c i l l a t i o n s are suppressed and the depolarization of the spin i s enhanced v i a ensemble dephasing, owing to the the random magnitude, symmetry and o r i e n t a t i o n of the muonium hyperfine d i s t o r t i o n with respect to the i n i t i a l muon spin. A more general discussion of random anisotropic hyperfine d i s t o r t i o n s and t h e i r e f f e c t on the time evolution of the \i+ spin p o l a r i z a t i o n for s t a t i c muonium i s given i s Appendix I. I.C.2 Muonium on S i l i c a Surfaces It has long been known that f i n e i n s u l a t i n g powders, such as MgO and S i 0 2 , can be used i n the production of positronium i n vacuum [28], even at low temperatures [29] . It i s thought that positronium i s formed i n the powder grains, d i f f u s e s r a p i d l y to the surface and f i n a l l y escapes into the void between the grains. The analogous phenomena for muonium was f i r s t reported i n 1978 for f i n e (35 A mean radius) evacuated S i 0 2 powder [30], where the emergence of muonium into the extragranular region was v e r i f i e d by the introduction of 0 2 gas. The Mu spin depolarization rate was observed to increase l i n e a r l y with 0 2 concentration, due to spin exchange int e r a c t i o n s with the paramagnetic 0 2 molecules, i n a manner consistent with r e s u l t s obtained with 0 2 i n an argon gas moderator at one atmosphere [31] , thus demonstrating that the S i 0 2 powder acts l i k e a very coarse moderator gas. Later investigations [32-34] concerning Mu i n f i n e oxide powders, namely S i 0 2 , A1 20 3 and MgO, show that a c e r t a i n f r a c t i o n of the muonium formed finds i t s way to the extragranular region for a l l three oxides. Of a l l the oxides tested, S i 0 2 was found to - 20 -have the highest formation p r o b a b i l i t y for Mu, (possibly because i t was av a i l a b l e i n the smallest grain s i z e ) ; and the 35 A SiOj powder was found to produce the highest y i e l d of extragranular muonium (>97% of Mu formed [30]), regardless of the ambient temperature of the powder. This l a s t point made the 35 A s i l i c a powder the obvious candidate for further studies of the i n t e r a c t i o n of \i+ and muonium with surfaces, the subject of t h i s t h e s i s . I.C.3 Muonium Formation i n Fine S i l i c a Powders The muonium fr a c t i o n s for bulk fused quartz as well as for 35 A and 70 A mean radius s i l i c a powders are given i n Table 1.3. As i n the case of positronium formation, muonium formation i n fin e oxide powders may involve thermal, epithermal, spur and/or surface processes. Because the atomic binding energy of positronium i s about half that of muonium, i t i s d i f f i c u l t to draw a simple analogy between the formation p r o b a b i l i t i e s for the two atoms. F i r s t l e t us ask whether muonium formation i n f i n e oxide powders i s a bulk or a surface phenomenon; surface formation of positronium has, for instance, been observed for low energy positrons incident on metal and metal-oxide surfaces [35]. If muonium formation i s indeed surface related, one would expect the Mu f r a c t i o n to increase with increasing s p e c i f i c surface area. From the values given i n Table 1.3, t h i s e f f e c t does not appear to be p a r t i c u l a r l y dramatic, i f i t exists at a l l , suggesting that the formation of muonium i n s i l i c a powders takes place primarily i n the bulk. The p o s s i b i l i t y of some charge exchange ocurring at the s i l i c a surface i s not however ruled out. The next question i s whether Mu formation occurs v i a thermal, epithermal or spur processes. In the spur model [36], muonium formation - 21 -Table 1.3 Muonium Fractions ( F M u ) and Transverse F i e l d Relaxation Rates (\ M u) for Bulk and Powdered S i l i c a . Sample T (K) F M u (%) X ™ (us" 1) Bulk fused S i 0 2 6 79 + 3 3.3 + 0.5 (a) 295 79 + 3 0.20 + 0.05 (a) S i 0 2 powder (70 A) 6 bulk 35 + 5 4.1 + 0.7 (b) surface 35 + 5 0.16 + 0.05 (b) 295 45 + 20 0.18 + 0.03 (c) S i 0 2 powder (35 A) 6 49 + 3 0.46 + 0.03 (b) 295 61 + 3 0.18 + 0.03 (b) (a) J.H. Brewer, Hyperfine Interactions 8^, 375 (1981). (b) R.F. K i e f l , Ph.D. Thesis, University of B r i t i s h Columbia (1982). (c) G.M. Marshall, et a l . , Phys. L e t t . 65A, 351 (1978). The measurements on the 70 A mean diameter powder were performed i n a helium atmosphere. - 22 -comes about when a thermalized LI + combines with an electron 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, for positronium formed v i a a spur mechanism, that the a p p l i c a t i o n of an e l e c t r i c f i e l d i n h i b i t s the combination of e + with the spur e - [37]. Similar experiments concerning muonium formation have shown the Mu formation p r o b a b i l i t y i n bulk S i 0 2 to be independent of applied e l e c t r i c f i e l d s of up to 60 kV/cm [38], suggesting that Mu formation i n bulk S i 0 2 i s probably not governed by a spur mechanism. However, the analogous experiments using fin e s i l i c a powders have not as yet been performed. For the case of epithermal (or hot atom) formation, the LI + undergoes a serie s of charge-exchange processes as i t slows down, as discussed e a r l i e r . Recent r e s u l t s [39] on the formation of muonium and "muonated" r a d i c a l s i n l i q u i d s , where the spur model i s most popular [40], suggest that epithermal processes play a s i g n i f i c a n t r o l e , even i n the presence of spurs. F i n a l l y , recent experiments [41] on muonium formation i n A1 20 3 show clear evidence that \i+ •*• Mu on a thermal basis over times as long as microseconds at low temperatures (T < 10 K), shortening to picoseconds near room temperature. This process, however, does not seem l i k e l y for s i l i c a powders since, from Table 1.3, the muonium f r a c t i o n i s observed to be temperature independent i n bulk fused quartz. One l a s t point can be made by drawing attention to the fact that there exists a s t a t i s t i c a l l y s i g n i f i c a n t discrepancy i n the muonium formation p r o b a b i l i t y between f i n e s i l i c a powders and bulk fused quartz. From Table 1.3, the muonium fr a c t i o n s F w measured at 295 K for the 35 A radius S i 0 o Mu ^ powder and bulk fused S i 0 2 are 61 ± 3% and 79 ± 3%, r e s p e c t i v e l y . In addition, F M f o r bulk fused S i 0 2 i s found to be independent of temperature - 23 -whereas for the 35 A powder F., decreases to 49 ± 3% at 6 K. These r e s u l t s Mu may be explained by the fact that i n powders the Mu atoms have the p o s s i b i l i t y of in t e r a c t i n g with the grain surfaces. There are two possible mechanisms associated with the surface that might account for the reduction of the muonium f r a c t i o n ; covalent bonding, which i n zero and l o n g i t u d i n a l f i e l d causes no depolarization of the \i+ spin but which i n transverse f i e l d removes muons from the muonium ensemble, or i o n i z a t i o n of the muonium atom at the surface, which has the same e f f e c t . F i r s t consider the p o s s i b i l i t y of covalent bonding. Generally, the s i l i c a surfaces are covered with hydroxyl groups [42,43] and are l i k e l y chemically i n e r t for muonium of thermal energies. I t may be e n e r g e t i c a l l y possible for a stopping \i+ to exchange with a hydroxyl proton; because this type of process requires non-thermal energies, however, one would not expect i t to be temperature dependent, making i t inconsistent with observations. Now consider the p o s s i b i l i t y of i o n i z a t i o n at the grain surfaces. Recent positron experiments [44] show that when e + of keV energies are implanted into i o n i c c r y s t a l s they are reemitted i s o t r o p i c a l l y from the so l i d s with a continuum of energies having a maximum approximately equal to the band gap energy of the s o l i d . This phenomenon has further been shown to be associated with positronium d i f f u s i n g to the surface and subsequently d i s s o c i a t i n g . In 1972 i t was postulated that Ps could be f i e l d - i o n i z e d i n the process of leaving a surface [45] . This, however, does not adequately account for the anomalously large emission energies or the c o r r e l a t i o n with the band gap energy of the s o l i d . An alternate explanation [44] i s that the positron i s Auger-emitted when the Ps electron f a l l s into an acceptor state at the - 24 -surface of the c r y s t a l . I t i s quite possible that the same mechanism(s) governing e + emission may be involved i n the interactions of muonium with i o n i c surfaces such as f i n e s i l i c a powders. A detai l e d discussion of th i s p a r t i c u l a r phenomenon i s given i n Appendix I I , and thus no elaborate explanations w i l l be given here. S u f f i c e i t to say that with the model just described the maximum energy of the emitted corresponding to the Mu electron recombining with a hole at the bottom of the valence band, can be written as = (E + AE ) - $ M u - R + $ e (1.20) max v g v y oo -where E i s the band gap energy, AE i s the width of the valence band, R i s g V °o Q the binding energy of muonium i n vacuum, §?_ i s the electron a f f i n i t y at the Mu bottom of the conduction band and $ i s the muonium work function at the surface. In analogy with positronium studies, the maximum k i n e t i c energy for Mu emission i s the negative of i t s work function, which i s given by SMU = ( E ^ U - R j + (*f + *J) (1.21) where E ^ U i s the binding energy of muonium at the surface and i s the L I + work function. A negative work function has been postulated for muonium on S i 0 2 surfaces [30,33,34]. For S i 0 2 , E = 10.7 eV [46] and one may assume a Mu conservative estimate for $ of 0 ± 1 eV. Substituting these values, along e with R and rather conservative estimates for AE and $ , into Equation GO 1/ — 1.20, one can conclude that Mu i o n i z a t i o n at the surface of fine s i l i c a powders i s e n e r g e t i c a l l y f e a s i b l e . However, since one does not expect long-lived holes i n the valence band, Equation 1.20 i s an overestimate. Assuming t h i s model to be correct, i t can be e a s i l y argued that this process would indeed be temperature dependent simply because at lower - 25 -temperatures a Mu atom w i l l spend a larger f r a c t i o n of i t s l i f e on the surface, thereby enhancing the p r o b a b i l i t y of encountering a hole. More extensive measurements of the muonium formation p r o b a b i l i t y have been made as part of t h i s d i s s e r t a t i o n , for S i 0 2 powders as well as for helium coated S i 0 2 powders. This i s discussed i n Chapter IV. I.C.4 Extragranular Muonium Production Two models concerning the production of Mu i n the extragranular region i n f i n e oxide powders have been put forth; one termed the thermal d i f f u s i o n (TD) model [30] and another which w i l l be referred to as the d i r e c t thermalization (DT) model [33,34]. Both models assume Mu formation to be a bulk phenomenon, but present d i f f e r i n g explanations of how the Mu atoms end up i n the extragranular region. The TD model i s an adaptation of a model o r i g i n a l l y applied to positronium d i f f u s i o n [47], which assumes that the Mu atoms thermalize i n the powder grains and then d i f f u s e to the surface where they may be ejected from the surface v i a a negative work function mechanism. As mentioned e a r l i e r , Mu i s s t a t i c i n bulk fused S i 0 2 below about 50 K [26] . If one assumes that the s i l i c a grains are of the same structure as bulk fused S i 0 2 and that the grain i n which the u + stops remains at the ambient temperature, then the reduced d i f f u s i o n expected at low temperatures appears to cast doubt on the TD model since i t would predict a temperature dependence i n the p r o b a b i l i t y for the production of extragranular Mu, i n contradiction with e x i s t i n g data. However, l o c a l heating of the grains due to the energy deposited by the stopping may play an important role i n the bulk d i f f u s i o n and subsequent e j e c t i o n of the Mu atoms from the oxide grains. Calculations [48] of t h i s e f f e c t estimate an energy deposition of 0.3 keV for muons stopping i n a 35 A radius S i 0 2 powder grain; assuming a uniform temperature d i s t r i b u t i o n within the grain, t h i s translates into an average temperature increase of ~300 K. Bear i n mind that these c a l c u l a t i o n s are crude and thus only indicate an order of magnitude. At this temperature, Mu i s known to d i f f u s e very r a p i d l y (at least i n c r y s t a l l i n e quartz [26]), Thus the muonium has a high p r o b a b i l i t y of a r r i v i n g at the surface i n a shorter period of time. Furthermore, t h i s temperature corresponds to an average energy of E^= (3/2)kT « 0.04 eV for the muonium atom, which may a s s i s t i n the e j e c t i o n of Mu from grain surfaces. If the muonium work Mu function $ i s indeed negative at the s i l i c a surface, the muonium atoms w i l l escape the powder grains with k i n e t i c energy E, + |$^ u|. Once outside the powder grains (extragranular region), a muonium atom w i l l l i k e l y remain outside since i t would require only a few e l a s t i c c o l l i s i o n s for the condition E^ « |$ M u| to be met. Thus the TD model can indeed explain the e x i s t i n g data on extragranular muonium production, subject only to the v a l i d i t y of the grain heating hypothesis. The DT model was o r i g i n a l l y proposed to circumvent the question of temperature dependent d i f f u s i o n . This model postulates the existence of a Mu r e l a t i v e l y large (<S> ~ -2 eV) negative Mu work function at the surface of the powder grains, which provides the p o s s i b i l i t y of d i r e c t thermalization of the Mu atoms i n the extragranular region. This model predicts the extragranular Mu f r a c t i o n to be temperature independent, i n better agreement with experiments, but i t i s d i f f i c u l t to explain the o r i g i n of such a large negative work function. - 27 -Mu Recall the expression for the muonium work function $ given i n Mu Equation 1.21. The negativity of i s of course influenced by many fac t o r s , however there are two phenomena which are of p a r t i c u l a r i n t e r e s t . One involves the d i s t o r t i o n of the muonium hyperfine i n t e r a c t i o n and the other involves what i s termed the "photoelectric s i z e e f f e c t " . Let us f i r s t consider the e f f e c t of the former. If the muonium hyperfine i n t e r a c t i o n i s di s t o r t e d by v i r t u e of being on the s i l i c a grain surfaces, so that the i s o t r o p i c part of the hyperfine i n t e r a c t i o n i s reduced ( i . e . , v Q 0 less than Mu the vacuum value), the atomic binding energy of muonium on the surface E^ would decrease accordingly with respect to the vacuum value R , thereby 00 Mu enhancing the neg a t i v i t y of $ . Now consider the l a t t e r case of photoelectric s i z e e f f e c t . Both the electron work function and photoelectric y i e l d for small (< 50 A radius) Ag p a r t i c l e s were studied with r e s u l t s i n d i c a t i n g an decrease of a few percent i n the electron work function along with a corresponding increase i n the photoelectric y i e l d by a factor of 10 2 over the macroscopic surface value for the smallest p a r t i c l e sizes [49]. 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 the electron work function may act to enhance the negativity of the muonium Mu work function $ for the same material. Thus this e f f e c t may also a s s i s t i n increasing the p r o b a b i l i t y of extragranular muonium production; however, no conclusion can be drawn at t h i s time. I.D The Interactions of Hydrogen and Deuterium with Silica Although muons are considered to be heavy electrons (or positrons), the behavior of p o s i t i v e muons and muonium i n matter i s more reminiscent of - 28 -protons and hydrogen than of positrons and positronium. The interactions of hydrogen and deuterium with s i l i c a surfaces has been extensively studied; a b r i e f synopsis of what i s presently known about the behavior of both hydrogen and deuterium i n bulk s i l i c a (fused and single c r y s t a l ) and on s i l i c a surfaces w i l l be presented here. I.D.I Hydrogen Diffusion i n Bulk S i l i c a Results [50] obtained for hydrogen i n single c r y s t a l quartz at low (< 120 K) temperatures, indicate hyperfine anisotropies along three p r i n c i p l e axes. Like the observations made for muonium i n quartz [23-27], the observed frequencies for hydrogen are assumed to a r i s e from t r a n s i t i o n s between three l e v e l s . This correspondence between muonium and hydrogen suggests that they occupy the same s i t e at low temperatures. Experiments (ESR) involving the d i f f u s i o n of protons i n single c r y s t a l quartz have shown that the recovery of a gamma pulse-induced frequency s h i f t at 306 K follows a t ~ ^ / 2 dependence over an extended period of time (up to 50 seconds), i n d i c a t i v e of one-dimensional d i f f u s i o n [51]. Analysis of t h i s data [52] indicates 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 o p t i c a l axis (c-axis) with an a c t i v a t i o n energy of about 0.25 eV and a d i f f u s i o n constant of about 5 x 1 0 - 6 cm 2/s. As mentioned e a r l i e r , one-dimensional d i f f u s i o n i s suspected for muonium i n single c r y s t a l quartz at this temperature [23-27]. I.D.2 Hydrogen and Deuterium on S i l i c a Surfaces The 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 surface properties of s i l i c a - g e l have been extensively investigated using ESR [53,54]. S i l i c a - g e l has a rather d i f f e r e n t structure than powdered s i l i c a ; i t i s comprised of large porous p a r t i c l e s , whereas the powdered material i s composed of non-porous S i 0 2 grains, which are i n general much smaller i n s i z e . In these studies, r a d i a t i o n induced d i s s o c i a t i o n of the surface hydroxyl (OH) groups was observed along with the subsequent formation of hydrogen 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 surface at low temperatures. As the temperature was raised from 123 K to 153 K the hydrogen ESR signal i n t e n s i t y decreased, corresponding to a reduction i n the stable H atom population. The adsorbed H atoms were also found to be highly r e a c t i v e . In p a r t i c u l a r , chemical reactions with oxygen and ethylene were observed i n the temperature range from 123 K to 153 K, which suggests that the H-ethylene reaction involves the formation of an ethyl r a d i c a l . Measurements of the. s p i n - l a t t i c e r e laxation time and l i n e width i n the presence of oxygen indicate that the average separation between adsorbed H and an oxygen molecule i s about 10 A. The authors suggest that the hydrogen atoms are located i n deep " m i c r o s l i t s " where the oxygen molecules cannot penetrate. Weak ESR sidebands, possibly due to hyperfine in t e r a c t i o n s between unpaired electrons and the hydroxyl protons, were also observed and found to be dependent upon pretreatment of the s i l i c a - g e l . In p a r t i c u l a r , samples degassed at 200-300 °C for 8 hours p r i o r to i r r a d i a t i o n were found to exhibit a s i g n a l with 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 hydrazyl r a d i c a l , and a structure apparently due to the hyperfine i n t e r a c t i o n between the odd electron of the r a d i c a l and the hydroxyl protons. This was v e r i f i e d by replacing the hydrogen atoms of the surface OH groups by deuterium. The signal obtained for samples degassed at 500 °C prior to i r r a d i a t i o n was i n d i c a t i v e of an enhancement i n the bulk formation - 30 -of F-centers due to the capture of electrons by oxygen l a t t i c e vacancies. In 1975, measurements [55] were made of the hyperfine i n t e r v a l v Q 0 for both hydrogen and deuterium adsorbed on the surface of fused quartz at room temperature. Results ind i c a t e reductions i n v 0 0 of 0.12% and 0.13% for hydrogen and deuterium, r e s p e c t i v e l y . In addition to the reduction i n the i s o t r o p i c hyperfine i n t e r a c t i o n , an anisotropic hyperfine i n t e r a c t i o n ( d i s t o r t i o n ) was also introduced, producing hyperfine s p l i t t i n g s d i f f e r i n g by < 0.4% from the vacuum values; t h i s anisotropy has been a t t r i b u t e d to an e l e c t r i c f i e l d at the surface. The perturbation of the hydrogen atom hyperfine i n t e r a c t i o n due to an e l e c t r i c f i e l d has been discussed i n some d e t a i l elsewhere [56,57]. The in t e r a c t i o n s of gas-phase deuterium atoms with s i l i c a surfaces have been studied [58] with r e s u l t s showing evidence for a chemical reaction of D atoms with with these surfaces, signaled by the formation of SiO-D bonds. Both Cab-O-Sil and porous Vycor glass (amorphous) surfaces were studied. For Cab-O-Sil, the formation of SiO-D groups was accompanied by a corresponding decrease i n SiO-H groups, suggesting an exchange reaction favoring l i b e r a t i o n of the l i g h t e r isotope. In the case of Vycor glass, however, no s i g n i f i c a n t decrease i n the SiO-H group population was observed. The Cab-O-Sil surface used i n the ESR studies of deuterium on s i l i c a was the same surface used i n the present work on muonium. The text up to this point has been a general introduction to muons and muonium atoms and th e i r c h a r a c t e r i s t i c s , and has provided a review of the r e s u l t s of previous studies involving muonium and hydrogen i n bulk s i l i c a and on s i l i c a surfaces. The present work w i l l now be considered i n more d e t a i l , beginning with a discussion of the experimental technique. - 31 -CHAPTER II — EXPERIMENTAL TECHNIQUE II.A Accelerators and Beamllnes At the present time, there are three "meson f a c t o r i e s " i n existence: (1) Los Alamos Meson Physics F a c i l i t y (LAMPF), (2) Schweizerisches I n s t i t u t fur Nuklearforschung (SIN) and (3) T r i - U n i v e r s i t y Meson F a c i l i t y (TRIUMF). A l l three f a c i l i t i e s currently support rather large scale \xSR research programs. In addition, CERN, Brookhaven, JINR, Gatchina (Leningrad) and the BOOM f a c i l i t y at KEK also support ongoing \iSR research. The experiments described i n t h i s d i s s e r t a t i o n were conducted on the M9 and M20 secondary channels of the TRIUMF cyclotron f a c i l i t y . II.A.1 The TRIUMF Cyclotron F a c i l i t y The present layout of the TRIUMF f a c i l i t y i s shown i n Figure II.1. The TRIUMF accelerator [1-6] i s a sector focussed H~ cyclotron capable of accelerating protons to energies ranging from 180 to 520 MeV at maximum currents of 170 uA at 520 MeV [7]. The proton beam has a 100% "macroscopic" duty cycle and a microscopic time structure consisting (normally) of a 5 ns burst every 43 ns. The proton beam i s extracted with e s s e n t i a l l y 100% e f f i c i e n c y by passing the H - ions through a t h i n carbon " s t r i p p e r " f o i l thus s t r i p p i n g o f f the two electrons and e f f e c t i v e l y reversing the charge of the ions. The extracted proton energy i s selected by the r a d i a l position.of the stripp e r f o i l s . The r e s u l t i n g protons then swerve out of the machine through three available extraction ports. A more detai l e d discussion of the recombination magnet and beam optics associated with the extraction system i s given REMOTE HANDLING FACILITY PROTON HALL EXTENSION CHEMISTRY ANNEX BL IB (P ) MESON HALL ,Ml3(n/u) (Ml1(n) M9(n/u) M20(g) SERVICE ^ ANNEX EXTENSION ION SOURCE 3 42 MeV' ISOTOPE PRODUCTION CYCLOTRON INTERIM -RADIOISOTOPE LABORATORY NEUTRON /-ACTIVATION I ANALYSIS XM15(ij) BATHO BIOMEDICAL LABORATORY - 33 -elsewhere [8]. During high i n t e n s i t y (unpolarized) operation, a proton beam i s extracted down Beam Line 4 and then transmitted at currents as low as 1 nA down Beam Line 4B (maximum 1 LIA) or 4A (maximum 10 LIA) . Both of these channels are u t i l i z e d for nucleon experiments at energies between 180 and 520 MeV. In addition, a 20 to 30 LIA proton beam can be extracted down Beam Line 2C for isotope research and production. F i n a l l y , a 130 to 140 LLA, 500 MeV proton beam i s extracted down Beam Line IA and passed through two pion production targets (1A-T1 and 1A-T2) and ultimately dumped into a molten lead target at the T.N.F. (Thermal Neutron F a c i l i t y ) . Low i n t e n s i t y (polarized) beam operation i s usually shared between 4B, 4A and IB. T y p i c a l l y , the pion production target at 1A-T1 i s a 10 mm thick (long) water cooled p y r o l i t i c graphite s t r i p and the one at 1A-T2 i s a 100 mm thick (long) water cooled beryllium s t r i p . Pions are produced at these targets v i a nuclear reactions. Six secondary channels are currently operational along Beam Line IA. Three channels simultaneously extract it-mesons or muons at 1A-T1: M13 [9], Mil and a newly commissioned channel, M15, that extracts p o s i t i v e muons i n the momentum range 21 - 29.8 MeV/c. The length of the M15 channel p r o h i b i t s pion transport. Three secondary channels simultaneously extract it-mesons or muons at 1A-T2: M8, i s dedicated to TC cancer therapy, M9 [10] pr i m a r i l y ( i n recent years) to the TPC (Time Projection Chamber) and M20 [14] i s used p r i n c i p a l l y f or LISR experiments. II.A.2 Muon Production and Transport To begin t h i s discussion of muon production and transport, we focus our attention on the M20 secondary channel [14] . The M20 channel (shown i n - 34 -Figure II.2) i s mainly a decay muon channel which views the 1A-T2 target at 55° with respect to the primary proton beam d i r e c t i o n . It consists of a i n j e c t i o n system incorporating two quadrupole doublets (Q1&2, Q3&4) separated by a 42.5° bending magnet ( B l ) , which has an acceptance of 12 msr. In addition to providing the p a r t i c l e c o l l e c t i o n , t h i s system selects the momentum of the p a r t i c l e s emitted from the pion production target at 1A-T2 and focuses them at the s l i t s (SL1). The i n j e c t i o n system Is followed by a ten quadrupole decay section which i s designed to c o l l e c t and transport muons produced by pions that decay i n f l i g h t along i t s length. The p a r t i c l e s that emerge from the decay section are co l l e c t e d by a quadrupole doublet (Q7 & Q8) and focused through a second bending magnet (B2) which has two exit ports, one (M20-A) at 75° and the other (M20-B) at 37.5° to the secondary beam d i r e c t i o n before the B2 bender. A Wien f i l t e r or cro s s e d - f i e l d v e l o c i t y separator i s incorporated into M20-B and used to reduce the positron contamination of the beam and may also be used as a "spin rotator". Muons can be transported through M20 i n any one of three operational modes; "conventional", "cloud", or "surface / subsurface". Conventional muons (LI + or LI -) are produced by pions decaying i n f l i g h t along the length of the decay section between Bl and B2. In i t s rest frame, the pion decay i s 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. The decay muons born i n the d i r e c t i o n of the pion momentum i n the lab frame are termed "forward muons" and those born i n a d i r e c t i o n opposite to the pion momentum are c a l l e d "backward muons". From r e l a t i v i s t i c kinematics, the lab frame momenta are t y p i c a l l y ~140 MeV/c and ~86 MeV/c, resp e c t i v e l y . The decay section of M20 i s designed to transmit only those Figure II.2 The M20 secondary channel showing both legs (A and B). muons having a small angular divergence from the i n i t i a l pion momentum ( i . e . , backward and forward muons). This feature not only narrows the two ava i l a b l e conventional momenta, but also gives r i s e to a high (85%) p o l a r i z a t i o n . The M20 channel can be tuned to transport backward muons through either M20-A or M20-B with low positron contamination. In addition, a "simultaneous" tune i s available which simultaneously d e l i v e r s "low -contamination" backward and forward decay muons to M20-A and M20-B, res p e c t i v e l y . Because conventional muons are produced from an extended source (pions decaying i n f l i g h t along the length of the decay section) the beam spot size at the f i n a l focus i s generally rather large. The measured beam parameters [11] for M20 operating i n backward mode and simultaneous mode are given i n Tables II.1(a) and II.1(b), r e s p e c t i v e l y . Cloud muons (p.+ or u~) are produced by pions decaying i n f l i g h t between the production target at 1A-T2 and B l . In th i s mode, both backward and forward muons are present. However, because the i n j e c t i o n system does not discriminate on the angular divergence of decay muons as much as does the decay section between Bl 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 present, there are no calculated or measured beam parameter values f o r cloud muons on the recently r e b u i l t version of M20; but on M9 the beam p o l a r i z a t i o n for cloud muons at 77 MeV/c i s ~30%. Surface muons [12] (only u + ) are produced from TC + that decay at rest on the surface of the pion production target. Muons produced i n th i s manner have several advantages over cloud or conventional muons. Unlike cloud muons, for example, surface muons include only the forwardly-decaying component. This feature, coupled with the acceptance of the i n j e c t i o n system and the kinematics of Tt + decay, gives r i s e to two important - 37 -Table I I . 1 ( a ) Backward Decay Muons at 75° (M20-A) and 37.5° (M20-B) Beam Parameters M20-A M20--B T o t a l F l u x 2.5 x 10 6/sec 5.9 x 10 5/sec C e n t r a l Luminosity H + H~ 2.9 x 10Vsec/cm 2 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 n 0.3% 1.3% P o l a r i z a t i o n 85% Beam Spot (fwhm) X y 7.2 cm 9.5 cm Divergence X y 63 mr 70 mr Table II.1(b) Simultaneous Decay Muons on M20 Beam Parameters M20-A M20--B T o t a l F l u x 1.6 x 10 6/sec 3.6 x 10 5/sec 1.6 4.6 X X 10 6/sec 10 5/sec C e n t r a l Luminosity n + 1.6 x lO'Vsec/cm 2 3.0 X IO1*/ sec/cm 2 C e n t r a l Momentum (P) 85.5 MeV/c Momentum Spread (AP/P) 6.7% A l l r ates are f o r a 100 Liamp proton beam i n c i d e n t on a 10 cm Be ta r g e t - 38 -c h a r a c t e r i s t i c s of surface u + : (1) surface u + are at least 99.9% polarized i n a d i r e c t i o n opposite to the beam momentum and (2) the surface n + beam momentum d i s t r i b u t i o n has a sharp "edge" at a momentum of 29.8 MeV/c, which translates into a k i n e t i c energy of 4.1 MeV. Because of t h e i r low and well defined energy, surface \i+ have a high (~140 mg/cm2) stopping density and a rather small range spread. Another advantage of surface \x+ arises because the u + originate d i r e c t l y from the pion production target. This feature provides a rather small source for surface u + i n comparison to the extended conventional muon source. In the case of surface the production target i s imaged at the f i n a l focus thereby producing a small (~2 cm diameter) beam spot. Owing to the low energy and monochromatic nature of the surface muon beam, surface muons have a high stopping density as w e l l . The small beam spot and high stopping density of surface | i + make i t possible to stop muons i n small and/or low density targets. In p a r t i c u l a r , the work described i n this d i s s e r t a t i o n involves the use of low density S i 0 2 powder targets and could not have been c a r r i e d out without a high i n t e n s i t y surface u + beam. The measured beam parameters for M20 operating i n surface mode are given i n Table II.2. By tuning the channel to lower momenta, i t i s possible to c o l l e c t and transport which are produced by TC + decaying inside the pion production target. These muons are, for lack of a better term, c a l l e d subsurface muons. Consider the cross section i n the pion stopping d i s t r i b u t i o n having a width Ay and located inside the production target a distance y from the emitting surface. The decay u~*" emitted from TC + stopped i n t h i s region, must traverse the distance y through the production target before reaching the Table II.2 Surface Muons at 75° (M20-A) and 37.5° (M20-B) Beam Parameters M20-A M20-B Total Flux 2.7 x 10 6/sec 1.5 x 10 6/sec Central Luminosity 1.6 x 10^/sec/cm 2 Central Momentum (P) 29.4 MeV/c Momentum Spread (AP/P) 6.4% 7.1% Electron Contamination ( e + / u + ) 40/1 40/1 P o l a r i z a t i o n 100% 100% Beam Spot (fwhm) x y 4.5 cm 4.3 cm Divergence x y ——— A l l rates are for a 100 uamp proton beam incident on a 10 cm Be target - 40 -emitting surface. Thus the LI + are degraded by the production target before being c o l l e c t e d and transported down the beamline. The range R Q of the decay u + i s approximated by [13] 7/2 R = k P ' (II.1) o o o where P Q i s the i n i t i a l \i+ momentum ( i n this case, 29.8 MeV/c) and k Q i s a constant that depends upon the stopping medium ( i . e . , the production ta r g e t ) . I t can then be e a s i l y argued that the subsurface LI + rate i s proportional to the range R Q m u l t i p l i e d by the appropriate decay f a c t o r . The u t i l i t y of subsurface muons i s e a s i l y understood by now considering the stopping d i s t r i b u t i o n of the transported subsurface u + beam of momentum P < 29.8 MeV/c. Similar to LI + i n the production target, the range R of the transported beam i s given approximately 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/cm2 (29.8 M e V / c ) - 7 / 2 , for surface u + ) . For a given spread i n momentum and taking range straggling into consideration ( t y p i c a l l y ~10% of the range [14]), an estimate of the t o t a l stopping spread AR i s then given by AR « [ ( 0 . 1 ) 2 + (3.5 A P / P ) 2 ] 1 / 2 R = k [ ( 0 . 1 ) 2 + (3.5 A P / P ) 2 ] 1 / 2 P 7 / 2 From this i t i s obvious that decreasing the momentum bi t e r e s u l t s i n only a li m i t e d reduction i n the stopping spread AR, while decreasing the momentum yi e l d s a dramatic decrease i n AR. Thus the use of subsurface muons makes i t - 41 -possible to tune for a desired stopping 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 positron production research and drawing the appropriate analogies. Recent measurements [15] show that when e + of keV energies are implanted into L i F and NaF, they are reemitted i s o t r o p i c a l l y from the sol i d s with a continuum of energies having a maximum approximately equal to the a l k a l i halide band gap. De t a i l s regarding the physics involved and a d e s c r i p t i o n of the prototype apparatus to test f o r the analogous a"1" phenomenon are given i n Appendix I I . II.B The iiSR / MSR Technique The experiments reported i n th i s d i s s e r t a t i o n were performed using the conventional t i m e - d i f f e r e n t i a l |iSR technique [16-18] . In th i s type of measurement one records the a r r i v a l time t of the u + and i t s subsequent decay at time t e , and then constructs a time histogram for the i n t e r v a l s defined by At = ( t g - t ^ ) . This technique requires the a b i l i t y to unambiguously associate a given e + with the muon from which i t was emitted. Normally, t h i s requirement i s s a t i s f i e d by allowing only one muon at a time to be present i n the sample. In general, the t i m e - d i f f e r e n t i a l uSR spectrum observed i n a d i r e c t i o n defined by the unit vector n with respect to the beam momentum can be expressed as -tlx N(t) = N e ^ f l + A P ( t ) ' n + A,, p\, ( t ) . n l + B (II.4) o L u u Mu Mu J where N Q i s a normalization constant, x u i s the mean muon l i f e t i m e , A^ and Aj^ u are the i n i t i a l instrumental asymmetries for the \x+ and muonium - 42 -sign a l s , r e spectively, P^(t) represents the muon spin p o l a r i z a t i o n for the u + s i g n a l and p M u ( t ) * s t n e corresponding quantity f o r the muonium state and the constant B i s a time independent background. I I . B . l Zero and Longitudinal F i e l d (ZF and LF) In zero f i e l d , one acquires information regarding the time dependence of the | i + spin p o l a r i z a t i o n by observing and comparing the LISR spectra at angles 9=0° and 9=180° with respect to the i n i t i a l muon spin d i r e c t i o n . A schematic representation of th i s i s given i n Figure 11.3(a). In these d i r e c t i o n s , the observed uSR spectra are 9 - 0°; N°(t) = N° e _ t / ^ [ l + A° (t) + A° G M u ( t ) ] + B° O L Ll zz Mu ZZ J (II.5) 9 = 180°; N 1 8 ° ( t ) = N 1 8 0 e _ t / ^ [ l - A 1 8 V (t) - A ^ G ^ t ) ] + B 1 8 0 O Ll zz Mu ZZ J where the time evolution of the muon spin for the \x+ and Mu signals are Li Mu represented by the two zero f i e l d r e l axation functions, G j ^ t ) a n <* G z z ^ t ^ » res p e c t i v e l y . II.B.2 Transverse F i e l d (TF) In weak transverse f i e l d s (B^ « B Q ), the evolution of the muon spin p o l a r i z a t i o n i n muonium can be treated using perturbation theory [16]. In t h i s approximation, t r i p l e t muonium |a(t)> = |1> precesses i n a sense opposite to that of a free L I + i n the same f i e l d , with a Larmor frequency u>Mu= - 103OJ^, while for Mu i n the mixed state |b(t)> = s|2> + c|4> the u + spin p o l a r i z a t i o n o s c i l l a t e s at a high frequency which i s on the order of - 43 -Figure 11.3(a) Figure II.3 Schematic representations of the |iSR techniques; Figure 11.3(a) shows the zero and l o n g i t u d i n a l f i e l d configuration and Figure 11.3(b) shows the corresponding diagram for transverse magnetic f i e l d . Figures taken from Y.J. Uemura, Ph.D. Thesis, University of Tokyo, 1981. - 44 -the hyperfine frequency (DQ'Q ; However, since the experimental timing reso l u t i o n i s t y p i c a l l y about 2 ns, the hyperfine o s c i l l a t i o n i s generally not observable, making this half of the muonium ensemble appear to be completely depolarized. The transverse f i e l d geometry i s schematically represented i n Figure 11.3(b). In very weak f i e l d s (B^ < 10 G), the t r a n s i t i o n frequencies v 1 2 a n c^ v23 a r e a P P r o x i m a t e l y equal and thus the |iSR positron spectra take the simple form N (t) = N e * [ l + A G^ (t) cosfw t - $ ) u xx u ( I I > 6 ) + A M u G x x ^ > C 0 S K u t + 4 j ] + B where and c&jj^ are the muon and muonium phase angles, defined by the o r i e n t a t i o n of the p a r t i c u l a r telescope with respect to the i n i t i a l muon spin d i r e c t i o n , G^ (t) i s the transverse f i e l d r e l axation function for the xx \x+ signal and G^(t:) i s the corresponding function for muonium. II.C Experimental Apparatus and Data A c q u i s i t i o n During the course of these experiments, both the experimental apparatus and data a c q u i s i t i o n system have evolved. Since i t i s not possible to describe the d i f f e r e n t stages of development here, only the present state of a f f a i r s i s discussed i n any d e t a i l . I I . C . l The |iSR Spectrometer The uSR ("Eagle") spectrometer, shown i n Figure II.4, was designed to take f u l l advantage of the properties of surface muons. One of the most important design constraints arises because of the r e l a t i v e l y short range - 45 -Figure I I.4 The "Eagle" L I S R spectrometer. - 46 -(~140 mg/cm2) and large multiple scattering of surface \i+, which dict a t e s the requirement of minimizing the amount of mass i n the beam path. To t h i s end, the spectrometer i s evacuated to a pressure of ~5 microns with both the counter array and sample situated i n s i d e . The spectrometer vacuum i s i s o l a t e d from the beam l i n e and cyclotron vacuum by a 76 Lim (0.003") mylar window through which the \i+ enters the spectrometer. Aft e r entering the spectrometer, the p o s i t i v e muons pass through a variable collimator and are detected by a 0.305 mm (0.012") thick s c i n t i l l a t o r ("D" counter) before f i n a l l y stopping i n the target assembly. Four positron telescopes, each comprised of two 6.35 mm (0.25") p l a s t i c s c i n t i l l a t o r s (B1-B2, F1-F2, R1-R2, L1-L2), are arranged around the target assembly, perpendicular (B and F telescopes) and p a r a l l e l (R and L telescopes) to the beam d i r e c t i o n . The B telescope (up stream) i s provided with a 5 cm hole i n i t s center to pass the \i+ beam. The F telescope (down stream) i s also provided with a 5 cm hole, primarily intended to pass beam positrons thereby reducing possible backgrounds due to beam contamination. 2 The positron telescope array subtends a t o t a l s o l i d angle of about -j(4n) steradians. The two counters comprising each of the four positron telescopes are separated by a 2.5 cm thick graphite moderator. This has the e f f e c t of increasing the positron asymmetry by cutting off the low energy end of the of the Michel spectrum [16] and helps prevent scattered beam (29.8 MeV/c) positrons from f i r i n g one of the positron telescopes. The l i g h t produced i n the counters ( s c i n t i l l a t o r s ) i s transmitted through the bottom of the vacuum chamber v i a UVT Lucite l i g h t guides and then detected and amplified by RCA 8575 photomultipliers. The variable collimator (2.5 cm thick brass), immediately upstream of - 47 -the D-counter, provides four e a s i l y selectable collimator diameters (5.2, 8.0, 10.8 and 18.0 mm) and serves to define the incoming \i+ beam. The collimator i s positioned between the Bl and B2 counters such that decay positrons from L I + stopped i n the collimator have only a small p r o b a b i l i t y of f i r i n g both counters of the B-telescope, which would r e s u l t i n a bad event. The present spectrometer has four pairs of c o i l s which provide magnetic f i e l d s i n three orthogonal d i r e c t i o n s . A pair of water-cooled Helmholtz c o i l s , having a mean diameter of 56 cm and a B/I factor of 4.63 G/A, provides a magnetic f i e l d 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 , transverse to the incident muon momentum and v e r t i c a l i n the lab frame). In p r i n c i p l e , these c o i l s can produce 6.5 kG, however because of the small turning radius of surface muons i n a magnetic f i e l d and the geometry of the spectrometer i t s e l f , the apparatus i s l i m i t e d to f i e l d s below about 500 G. A pair of air-cooled c o i l s , not i n Helmholtz configuration, provide a f i e l d 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 . , along the incident muon momentum) from 0-12 G. When connected i n se r i e s , the c o i l s have a B/I factor of about 1 G/A. With t h i s arrangement, one can study the long i t u d i n a l p o l a r i z a t i o n of the L I + spin as a function of time by observing the decay spectrum i n the F and B telescopes. The remaining two pairs of c o i l s are air-cooled and provide small (~1 G) bucking f i e l d s i n the "transverse-horizontal" ( i . e . , transverse to the incident muon momentum and hori z o n t a l i n the lab frame) and lo n g i t u d i n a l d i r e c t i o n s . These bucking c o i l s are used to achieve zero f i e l d plus or minus ~100 mG; however, they do not automatically compensate for time dependent d r i f t s which may introduce small f i e l d f l u ctuations (~0.2 G) over several hours. The l i m i t a t i o n s of the Eagle spectrometer, namely the i n a b i l i t y to do research i n a high (> 500 G) f i e l d or a stable zero f i e l d environment, w i l l be overcome with the commissioning of a new apparatus dubbed "Omni" which i s currently under construction. This apparatus has three pairs of water-cooled Helmholtz c o i l s that produce magnetic f i e l d s i n three orthogonal d i r e c t i o n s ; l o n g i t u d i n a l , t r a n s v e r s e - v e r t i c a l and transverse-horizontal. The l o n g i t u d i n a l c o i l s are capable of producing high f i e l d s (< 6 kG) while the other two can produce maximum f i e l d s of only ~ 100 G. The l o n g i t u d i n a l o r i e n t a t i o n of the high f i e l d c o i l s makes possible research i n high f i e l d s , since the | i + enter the spectrometer along the f i e l d l i n e s and thus t h e i r t r a j e c t o r i e s , except for focusing e f f e c t s , remain unaffected. The two low f i e l d Helmholtz pairs can of course be operated independently to produce f i e l d s of up to ~ 100 G i n the two transverse d i r e c t i o n s ; however, they w i l l be u t i l i z e d p r i m a r i l y as bucking c o i l s . A feedback system coupling a l l three Helmholtz pairs along with two s t r a t e g i c a l l y placed 3-dimensional H a l l probes w i l l be used to produce a stable, time independent zero f i e l d environment with a s t a b i l i t y of ~10 - 1* G, lim i t e d by the s e n s i t i v i t y of the H a l l probes. II.C.2 Electronics 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 evolved during the course of the present study, primarily due to the Introduction of the LeCroy 4204 TDC. From a uSR point of view, the 4204 possesses several a t t r a c t i v e features. Two of the more important a t t r i b u t e s are i t s buffer memory and a nominal time r e s o l u t i o n of 156.25 ps. With the incorporation of a buffer memory, the 4204 TDC e f f e c t i v e l y combines a l l the functions - 49 -performed by the TRIUMF B080 1 GHz TDC and EG&G C212 pattern unit, as employed i n the previous system [19]. This feature greatly reduces the event processing dead time. The signals from the photomultipliers are transmitted along ~30 m of coaxial cable before being discriminated and routed through the NIM l o g i c shown i n Figure II.5. A | i + entering the spectrometer f i r s t passes through the D-counter thereby generating a pulse at a time t which both s t a r t s r* the TDC and also triggers a pileup gate that defines the time window T for the subsequent decay e + event. At a l a t e r time t e the \i+ decays, emitting a positron p r e f e r e n t i a l l y along i t s spin d i r e c t i o n . If the decay e + i s detected by one of the four positron telescopes, within the preselected time window T ( t y p i c a l l y 10 [is), a pulse i s generated which stops the TDC. If a decay e + i s not detected within the time T, the TDC i s automatically reset. Constant f r a c t i o n discriminators (CFD) were used on the c r i t i c a l timing s i g n a l s . The discriminated pulses from the counters comprising the four positron telescopes are routed into four separate coincidence u n i t s . The two-fold coincidences (e+E^ = E.^l'E.^2, etc.) ensure that accepted events correspond to decay positrons that pass through both telescope counters and the carbon degrader separating them. The t h i r d coincidence, shown i n Figure II.5, was not employed i n the present study. The outputs of the four coincidence units are l o g i c a l l y "OR-ed", with the r e s u l t i n g pulse serving as the TDC stop. Simultaneous with stopping the TDC, pulses from each of the four telescopes are also routed to set i d e n t i f i c a t i o n b i t s i n NIM-ECL converter u n i t s , thereby i d e n t i f y i n g which e + telescope was f i r e d . The data are written Into the PDP-11 memory v i a a CAMAC computer-logic i n t e r f a c e which i s serviced by a Bi-Ra Microprogrammed Branch Driver (MBD-11). The - 50 -T A R G E T !x"i • [ D ] IT"! ! B ! — ! ; \ £ F D V \ C F D / I I • J \ / \ / ' • IDISO D©6 V bisd ^ S T O P M U S T C A R R I V E B E F O R E T ITS B U S Y F O R A D D I T I O N A L P O S I T R O N C O U N T E R C I R C U I T S I S TART ) Figure II.5 Data a c q u i s i t i o n e l e c t r o n i c s . Only the c i r c u i t for one positron telescope i s shown. - 51 -MBD reads the memory buffer of the 4204 TDC, i d e n t i f i e s the telescope that generated the event, and then performs the necessary functions required to increment the histogram bin corresponding to the measured time i n t e r v a l (t„ - t..). To guard against possible loss of data due to computer f a i l u r e , the data are p e r i o d i c a l l y updated on an RL02 disk. The requirement of having only one \x+ present i n the sample at a time i s f u l f i l l e d with the use of the pileup gate (model: GP 100/N EG&G Ortec). The pileup gate (data gate) i s triggered on an incident u + pulse i n the D-counter and latched for a preset time T (normally set to 4-8 muon l i f e t i m e s ) . I f a positron i s detected during the data gate period T, a "good" event i s logged and the appropriate histogram updated. There are however "bad" events that, i f l e f t unsuppressed, would introduce d i s t o r t i o n s into the spectra. The two most important processes that produce "bad" events are early second [i+ events ( u - u - e ) , where the second u + a r r i v e s during the period T but before t e , and l a t e second u + events (|j,-e-^), where the second p.+ a r r i v e s a f t e r t e . In e a r l i e r versions of the data a q u i s i t i o n e l e c t r o n i c s [19] the u - u - e events were rejected i n l o g i c by vetoing multiple clock stops, thereby causing the clock to time out. The u - e - u events on the other hand, which must be rejected a f t e r the clock has stopped, were rejected i n software by s e t t i n g a fake pattern i n the now obsolete C212 unit when the second \x+ was detected. The C212 and TDC were then read and cleared by the MBD. With the new system incorporating the 4204 TDC, the u-u-e events are s t i l l rejected i n the same manner as i n the e a r l i e r version. The u-e-p. events, however, are rejected i n the TDC i t s e l f which rej e c t s multiple h i t events. Thus the incorporation of the 4204 TDC into 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 possible to process a l l - 52 -"bad" event r e j e c t i o n i n hardware thereby greatly reducing the event processing dead time. The r e j e c t i o n of the Li-Li-e events places a r e s t r i c t i o n on the maximum rate at which one can take data using the t i m e - d i f f e r e n t i a l technique. Since the time structure of the TRIUMF cyclotron has a period of 43 ns, which i s much smaller than the muon l i f e t i m e , one can assume that the incident muons a r r i v e with a time d i s t r i b u t i o n c l o s e l y described by Poisson s t a t i s t i c s . From t h i s assumption, and denoting the incident rate | i + by RQ the "good" event rate R i s given by [19] O R = R exp(-2R T) (II.7) g o ^ o ; v ' For R Q = ( 2 T ) - 1 , the "good" event rate R g i s maximized. With a t y p i c a l gate width of 8 LIS, the maximum "good" event rate occurs for a LI + stop rate R Q of about 62 kHz. This translates into a positron event rate of 2k-3k e +/s per positron telescope. The l o g i c l e v e l diagram for a "good" event i s shown i n Figure II.6, and a more de t a i l e d discussion of bad events and th e i r e f f e c t on LISR spectra can be found elsewhere [19]. II.C .3 Targets The SiO^ powder used i n these experiments was chosen because of i t s high s p e c i f i c surface area (390 ± 40 m2/g [20]) and high y i e l d (61 ± 3%) of extragranular muonium previously observed at 300 K [21] and at 6 K [22,23]. Some of the physical c h a r a c t e r i s t i c s of t h i s powder are given i n Table II.3. The surfaces of these powders normally have ~4.5 chemisorbed hydroxyl (OH) groups per nm2, corresponding to about half of the surface S i atoms being associated with a surface hydroxyl [20] . When evacuated at room temperature - 53 -Good Event T-B — | , P e Gate 1 Routing Bits [J Reject Dead Time Computer Busy /i.Stop y Gated e ^j-Figure II.6 Logic l e v e l diagram for a "good" event. Table II.3 Physical C h a r a c t e r i s t i c s of the SiO~ Powder Property Value Supplier Cabot Corporation, 125 High Street, Boston, MA., 02110 (U.S.A.) S e r i a l Number EH-5 Density (unpacked) 0.033 gram/cm3 S p e c i f i c Surface Area 390 +/- 40 m2/gram Mean Grain Size 35 Angstrom (mean radius) Major Impurities Na 20-40 ppm P < 300 ppm Other < 30 ppm Hydroxyl Concentration average 3.5-4.5 groups per nm2 maximum (calc.) 7.8 groups per nm2 The above values are taken from reference [20]. - 55 -( i n a vacuum equal to 1 0 - 2 Torr) or heated above 110°C, the powder surfaces undergo "r e v e r s i b l e dehydration". In this process, the surface hydroxyls combine to form H 20 which, when released, leaves behind a d d i t i o n a l siloxane groups (S i - O - S i ) . Above about 800 °C th i s hydrolysis i s completed and the powder begins to s i n t e r . The term r e v e r s i b l e dehydration means that the powder surfaces can be restored to t h e i r o r i g i n a l state by either exposure to a i r or immersion i n water; with the target geometry used i n the present study, t h i s r e s t o r a t i o n process takes about 24 hours i n a i r , subject to the ambient atmospheric moisture. I t i s therefore possible to vary the surface density of hydroxyls, and indeed study the reactions of various molecules with the surface hydroxyl groups [24-26] . The thermogravimetric analysis curve (measured at one atmosphere), for the S i 0 2 powder used i n the present study, i s shown i n Figure II.7. Four targets were prepared with the S i 0 2 powder e s s e n t i a l l y unaltered from the manufacturer's s p e c i f i c a t i o n s . Five other targets were prepared with the same S i 0 2 powder, but i n th i s case hydrogen-reduced, with four of these having a non-zero platinum loading. The platinum loaded samples were prepared at Arizona State U n i v e r s i t y , according to procedures described elsewhere [27] . B r i e f l y , the loading procedure involves f i r s t physisorbing H 2 P t C l 6 onto the S i 0 2 powder surfaces. This molecule i s then reduced i n a hydrogen atmosphere at 500 °C, v i a the reaction H 2 P t C l 6 + 2H 2 -> Pt° + 6HC1, to produce surface Pt atoms. Because of the high temperatures, the Pt atoms move about on the s i l i c a surface and eventually begin aggregating. Five l e v e l s of platinum loading were chosen for the c a t a l y s t samples; 0.0%, 0.001%, 0.01%, 0.1% and 1.0% by weight. A l l samples were characterized by well-known gas adsorption techniques at Stanford U n i v e r s i t y . The s p e c i f i c - 56 -T E M P E R A T U R E (°C) Figure II.7 Thermogravimetric analysis of Cab-O-Sil. Above 110 °C, the weight loss i s caused by a-gradual loss of water as the hydroxyls undergo condensation. Figure taken from reference [20]. - 57 -surface area of the S i 0 2 support, which had been hydrogen-reduced during the sample preparation, was measured using the B.E.T. adsorption isotherm technique [28] ( i n t h i s case, N 2 at 77 K) and found to be 320 ± 20 m2/g. This i s somewhat smaller than the manufacturer's s p e c i f i c a t i o n of 390 ± 40 m2/g for the unreduced S i 0 2 powder. Platinum dispersions (# of Pt atoms at surface / t o t a l # of Pt atoms In sample) were measured i n both the 0.1% and 1.0% Pt loaded samples by hydrogen chemisorption [29], and were found to be 1.0 ± 0.02 and 0.39 ± 0.02, re s p e c t i v e l y . The percentage of the t o t a l surface area of the loaded ca 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 i s then 0.08% for the 0.1% sample and 0.34% for the 1.0% sample. The targets were prepared by compressing the S i 0 2 powder into s t a i n l e s s s t e e l vacuum vessels, onto which a 25 Lim or 50 Lim s t a i n l e s s window was then TIG (Tungsten Inert Gas) welded or electron beam welded. Welded s t a i n l e s s s t e e l targets were used both for cleanliness and because of the need to prepare some of the samples by baking i n vacuum at high temperatures. A l l heat treatments were performed i n vacuum for a period of 10-12 hours p r i o r to the experiments. The targets were evacuated through a 110 cm length of 0.635 cm (0.25") outer diameter (0.4 cm I.D.) st a i n l e s s s t e e l tubing, using a d i f f u s i o n pump. The pumping system was i s o l a t e d from the target assembly by a l i q u i d nitrogen cold trap, to reduce the p o s s i b i l i t y of contamination a r i s i n g from backstreaming. The pressure at the input of the d i f f u s i o n pump was measured to be 1 0 - 6 Torr, whereas the ultimate pressure at the target ( a f t e r baking at T > 110 °C) was measured to be 10*"5 - 10~ 6 Torr. Although "low-magnetic" steels (types 316-L and 321) were used, there s t i l l existed some remnant magnetization, which was found to introduce a small relaxation of the muonium spin due to induced f i e l d inhomogeneity. S p e c i f i c d e t a i l s - 58 -regarding the targets used i n this work are given i n Table II.4. A l l but one of the targets used i n the present study had 25 |im TIG welded windows. The exception was target S i 0 2 ( 3 ) , which had a 50 |im electron beam welded window. I I . C .4 C r y o g e n i c s The evacuated samples were inserted through the top of a Janis ^He gas-flow cryostat (model: 10DT Super-VariTemp) which provides a uniform low-temperature environment, variable from 1.8 to 300 K. The Janis cryostat i s mounted through the top of the Eagle spectrometer with the cryostat t a i l extending down between the four positron telescopes. To minimize the mass i n the beam path the t a i l outer vacuum s h i e l d i s removed, making the cryostat i n s u l a t i n g vacuum contiguous with the Eagle vacuum chamber. The muons enter the cryostat by f i r s t passing through a 76.2 urn (0.0003") aluminized mylar heat shield and then a 0.127 mm (0.005") mylar window separating the helium gas thermal bath of the "sample space" from the i n s u l a t i n g vacuum. The temperature i s regulated by adjusting the vaporizer heater current and the ^He flow rate through a needle valve. The heater i s incorporated into a PID [30] temperature feedback system along with the thermometer (mounted on the outside of the target vessel) which monitors the sample temperature. For temperatures i n the range 300 K - 75 K, a c a l i b r a t e d platinum r e s i s t o r was used, and for the range 100 K - 1.8 K, a c a l i b r a t e d germanium r e s i s t o r was employed. Some question may a r i s e as to whether the thermometers measure the "true" temperature of the sample ( i . e . , i s the sample at thermal equilibrium with the helium thermal bath). This was - 59 -Table 11.4(a) S i 0 2 Targets Targets C h a r a c t e r i s t i c Value S i 0 2 ( l ) ( V o l . = Si0 2(2) (Vo l . = Si0 2(3) ( V o l . = Si0 2(4) (Vo l . = * Mass of Powder 1.50 + 0.05 g 4.48 + 0.05 cm3 ) Packing Density 0.335 + 0.015 g/cm3 Surface Area 585 + 79.5 m2 cm3) Mass of Powder 0.50 + 0.05 g 4.48 + 0.05 Packing Density 0.112 + 0.012 g/ cm3 Surface Area 195 + 39.5 m2 Mass of Powder 0.72 + 0.05 g 6.63 + 0.05 cm3) Packing Density 0.109 + 0.008 g/cm3 Surface Area 281 + 48.3 m2 Mass of Powder 1.76 + 0.05 g 6.08 + 0.05 cm3) Packing Density 0.289 + 0.011 g/cm3 Surface Area 686 + 89.8 m2 Table 11.4(b) Platinum Loaded SiO, Targets Targets C h a r a c t e r i s t i c Value P t ( l ) (0.0% loading) (Vo l . = 4.38 ± 0.05 cm3) Pt(2) (0.001% loading) (Vo l . = 4.38 ± 0.05 cm3) Pt(3) (0.01% loading) (Vo l . = 4.38 ± 0.05 cm3) Pt(4) (0.1% loading) (Vo l . = 4.38 ± 0.05 cm3) Pt(5) (1.0% loading) (Vo l . = 4.38 ± 0.05 cm3) Mass of Powder 1.50 + 0.08 Packing Density 0.61 + 0.03 Surface Area 480 + 25.6 Mass of Powder 1.50 + 0.08 Packing Density 0.61 0.03 Surface Area 480 + 25.6 Mass of Powder 1.50 + 0.08 Packing Density 0.61 + 0.03 Surface Area 480 + 25.6 Mass of Powder 1.50 + 0.08 Packing Density 0.61 + 0.03 Surface Area 480 + 25.6 Mass of Powder 1.50 + 0.08 Packing Density 0.61 + 0.03 Surface Area 480 + 25.6 g g/cm3 m2 g g/cm3 m2 g g/cm3 m2 g g/ cm3 m2 g g/cm3 m2 - 60 -tested experimentally by f i r s t stepping through the temperature range In increasing steps, and then repeating these measurements i n reverse order. The data were found to be quite i n s e n s i t i v e to the order i n which the temperature points were taken, i n d i c a t i n g that the sample was indeed i n thermal equilibrium with the helium bath of the cryostat. III.0.5 Gas Handling Figure II.8 shows the gas handling system used for physisorbing or chemisorbing controlled amounts of d i f f e r e n t gases onto the sample surface. By a l t e r i n g the surface c h a r a c t e r i s t i c s i n this fashion, and observing the associated change i n the muonium behavior, one gains further ins i g h t into the relaxation and d i f f u s i o n behavior of muonium on surfaces. In the present study, ^He i s deposited on the s i l i c a surfaces. The gas handling apparatus i s quite t y p i c a l for the purpose at hand and consists of a doser volume (three d i f f e r e n t doser volumes were used for these experiments, 41.9 ± 0.16 cm3, 36.5 ± 0.6 cm3 and 28.2 ± 0.5 cm 3), a standard volume V s (1331 ± 28 cm3) and a metering valve on the output. The pressure i n the system i s monitored by a Baratron gauge (MKS model: 220BHS-2A1-B-1000) capable of measuring pressures In the range 0-100 Torr. This gauge i s accurate to within 0.15% of the reading and i s also temperature compensated with an associated error of 0.01% F.S./A°C. The l a t t e r source of error was not taken into account i n the data analysis, and no "zero d r i f t " i n the gauge was observed. The dead volume i n the output section a f t e r valves and Vg was measured to be 35 ± 2 cm3, not including the targets. During gas deposition, the sample temperature was kept low enough to ensure that any gas atoms reaching the surface would be adsorbed; for ^He on - 61 -PRESSURE RELIEF VALVE w BYPASS £ VALVE GAS I N P U T r = ^ V \ Figure I I . 8 Gas handling system. S i 0 2 > the temperature was t y p i c a l l y kept below 10 K. To estimate the amount of gas required for f r a c t i o n a l or complete monolayer coverage, one calcu l a t e s the change i n pressure A P i n the doser volume, according to the simple equation AP = f ^rz— RT (II.8) x d where f i s the f r a c t i o n of the surface area to be covered, A i s the surface area of the sample, R i s the gas constant (1.036 x 10 -* 9 t o r r cm3 K - 1 ) , aK i s the area covered by an adsorption atom or molecule X (~10-^-5 cm2, for helium on s i l i c a ) , i s the doser volume of the system and T i s the doser volume temperature. For physisorption of atoms and molecules, the following gas handling procedure was used: (1) With V^, V 3, V^, V 5 and Vg closed, open V 2 and evacuate the system using the turbo pump. (2) Once the system i s evacuated, close V 2. (3) Open to pressurize doser volume to desired l e v e l . (4) Once.the desired pressure i s . attained, close and note pressure on the Baratron gauge. (5) Open or V 5 (depending on requirements placed on flow r a t e ) . (6) Open V 6 and wait for the system to come to equilibrium, (may take 30 minutes). (7) Note pressure on Baratron gauge. (8) Take data. (9) Note pressure once again to ensure that the system was near equilibrium. (10) Repeat this procedure as many times as necessary to achieve the desired surface coverage. Because one i s normally dealing with small quantities of gas and small flow rates, the temperature of the sample i s only b r i e f l y perturbed by t h i s procedure; such temperature fl u c t u a t i o n s were normally unobservable. II.D Data Analysis The MINUIT [31] minimization package was used to provide least-squares f i t s to the data and to generate the s t a t i s t i c a l errors on the function parameters. II.D.l Transverse Field Spectra For the transverse f i e l d data, the raw spectra were f i t separately, with the function given i n Equation II.6. In these f i t s , the transverse f i e l d muonium relaxation function was assumed to be of the form G ^ ( t ) = e x P [ - ( \ M u + \ Q ) t ] (II.9) Mu where \ q ( « ) Is the relaxation rate due to f i e l d inhomogeneity and was determined by a measurement at low temperature (~6 K) with several monolayers of 4He on the grain surfaces; e a r l i e r experiments [22,23] on f i n e alumina powders have shown that Mu i s protected from the depolarization centers on the oxide surface by just such a helium f i l m . The two parameters of i n t e r e s t to the present study are the i n i t i a l muonium asymmetry A^ and Mu the relaxation rate \^ . The f i t t e d asymmetries for each positron telescope were treated independently, whereas the relaxation rates were combined i n a weighted average. - 6 4 -II.D.2 Zero and Longitudinal Field Spectra For the zero and l o n g i t u d i n a l f i e l d data, the F and B spectra were f i r s t treated separately by removing the respective backgrounds using the " t < 0 " time bins. After t h i s , they were combined to form an asymmetry spectrum defined by A S Y , r»v> - B°I - rAt> - Bp [N B(t> - B ] + [N (t) - B ] i n which the muon l i f e t i m e i s automatically divided out. The r e s u l t i n g spectrum was then f i t using Equation 1 1 . 1 0 with the appropriate relaxation function assumed. In p r a c t i c e , the two telescopes do not i n general have the same s o l i d angle, with respect to the target, or e f f i c i e n c i e s . These differences 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 aQ = N Q / N q . The associated correction i s given by the equation ( 1 + a )ASY - ( 1 - a ) o o A S Y ( c o r r ) = ( 1 + a Q) - ( 1 - a Q)ASY ( H - H ) The r e s u l t i n g asymmetry spectrum Is then referred to as a "corrected asymmetry". - 65 -CHAPTER III — THEORY OF MUONIUM RELAXATION To obtain a clear i n t e r p r e t a t i o n of the experimental r e s u l t s , i t i s important to understand the d i f f e r e n t relaxation mechanisms through which the a"1" spin may be depolarized. For muonium, the ensemble spin p o l a r i z a t i o n can be l o s t through i n t e r a c t i o n s with the environment by means of the f i v e relaxation mechanisms: (1) Random Local Magnetic F i e l d s (2) Random Anisotropic Hyperfine D i s t o r t i o n s (3) Chemical Reactions (4) Spin Exchange (5) Superhyperfine Interactions In t h i s chapter, these mechanisms, along with the associated spin r e l a x a t i o n functions, are discussed. The e f f e c t of d i f f u s i o n on these r e l a x a t i o n functions i s also considered. Since the mixed state |bQ> i s normally unobservable, t h i s discussion w i l l be r e s t r i c t e d to the half of the muonium ensemble i n i t i a l l y i n the polarized t r i p l e t state |aQ>. Ill.A Spin Relaxation Functions For a muonium atom, the expectation value of the LI + spin p o l a r i z a t i o n i s defined by the equation <S^ (t)> = Tr{S^ (t) [ i + (S^ (0) • ) ] U 6 p (0)} ~op l~op L4 v~op ~ i n ' J "s ' •>op ^ p L4 ( I I I . l ) = i-Trfs^J (t) p (0)} + Tr{(s^(t) (0) p (0)) . s|M H ~op s 1 "-~op ~op s ~ i n J u e I where IT' and U are the unit operators for the yr and e~ spins, r e s p e c t i v e l y , + (S^CO) • £^n)]ue i s the i n i t i a l spin density operator associated with the spin dynamics of the muon, P g(0) Is the i n i t i a l spin density matrix for the environment and pj£n = 2»S^ n i s the incoming muon spin - 66 -p o l a r i z a t i o n vector. In general, the f i r s t term i n Equation I I I . l reduces to zero because the trace over any spin vector operator i s zero. This makes i t possible to express the expectation value, <S^ p(t)>, i n terms of a second rank spin-spin autocorrelation tensor §(t), defined as g(t) = Tr{s^ (t) (0) p (0)} ' ° P ~ ° P - (III.2) = Tr{exp[iHt(2*/h)] s£p(0) exp[-iHt(2*/h)] s£p(0) P g ( 0 ) } where H i s the spin Hamiltonian of the system. With t h i s , Equation I I I . l can be written as <S^ (t)> - g(t) • s|J (III.3) ~op e ~ i n ' where the time evolution of the u + spin p o l a r i z a t i o n i s completely determined by the motion tensor §(t). In zero and lo n g i t u d i n a l f i e l d , | ( t ) i s defined as the re l a x a t i o n tensor. In transverse f i e l d , however, t h i s tensor includes o s c i l l a t o r y terms corresponding to Larmor precession which are omitted i n the d e f i n i t i o n of the transverse f i e l d r elaxation tensor. I I I . A . l Random L o c a l Magnetic F i e l d s (RLMF) In the context of the spin-spin i n t e r a c t i o n between a magnetic probe and some weak dipolar f i e l d d i s t r i b u t i o n , the spin Hamiltonians for t r i p l e t muonium and for p o s i t i v e muons are mathematically equivalent, except that the former has a magnetic moment which i s ~103 times greater than the l a t t e r . The general spin Hamiltonian for a muonium atom, i n the polarized t r i p l e t state, i n t e r a c t i n g with N nuclear spins i s given by the equation H = f (H^ + jfl) + H Z (III.4) j = l - 67 -where represents the dipolar i n t e r a c t i o n s between the muonium atom and the N neighboring n u c l e i , represents the quadrupolar in t e r a c t i o n s of the nuclear spins (due for instance to e l e c t r i c f i e l d gradients induced by the presence of the muonium atom) and H are the Zeeman terms. Note that any dipolar interactions among the nuclear spins have been neglected. Because of the desire to treat t r i p l e t muonium the same as a p o s i t i v e muon, one defines to be the spin operator of the magnetic probe (LI + or Mu, etc.) and y (= 2 I C Y ) as the corresponding magnetogyric r a t i o . By further s s i tti *— defining J as the spin operator of the j nucleus, and y T ( = 2icy T) as the corresponding magnetogyric r a t i o , the Zeeman term i s written as N H Z = (h/2u) y (S~ • B) - T (h/2*) Y t ( j J ' B) (III-5) ' S ~ O P ~J ^ J ^ O D ~ J=l V In addition, the dipolar terms can be written H d = (h/2it) 2 ( Y Y . ) ( r . ) ~ 3 [(S • J j ) - 3(n.. S )(n . . J j )] (III.6) j ^ s ' j ' * 1 j ' L1-~op ~op^ v j ~op ; v j ~op ' J and the quadrupole terms are H<j = (h/2*) ^ 1 [3(n. • j j p ) 2 - J ( J + 1)] (III.7) where n^ i s the unit vector i n the d i r e c t i o n from the muonium spin to the j 1 " * 1 nucleus located at a distance r_., J ( J + 1) i s the eigenvalue of the operator jjj and OJ^ represents the strength of the quadrupolar i n t e r a c t i o n . The zero and low f i e l d (external f i e l d « l o c a l f i e l d ) spin relaxation functions for a magnetic probe i n t e r a c t i n g with a random l o c a l magnetic f i e l d were f i r s t discussed i n 1967 [1,2]. In t h i s formulation, the quadrupole int 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 dipolar i n t e r a c t i o n s , which are i n general described by a quantal l o c a l magnetic f i e l d operator, are approximated by a s t a t i c (continuous) e f f e c t i v e l o c a l - 68 -dipole f i e l d H. With these assumptions, the dipolar i n t e r a c t i o n term takes the simple form N 7 H. = (h/2*) Y (S • H) (III.8) J 's^~op ; The e f f e c t i v e f i e l d d i s t r i b u t i o n i s further assumed to be i s o t r o p i c , with the magnitude of each component being d i s t r i b u t e d according to a continuous Gaussian d i s t r i b u t i o n function of width A/y , given by s Y Y ^ f G ( H ) = exp[- - i - ^ i ] ; i = x, y, z (III.9) /2TI A 2A and f G(|H|) = [ ^ - ] 3 exp[- Y s ] [4* |H|2] (III.10) /2TC A 2A where A i s the second moment of the f i e l d d i s t r i b u t i o n [3]. Assuming this d i s t r i b u t i o n , the zero f i e l d spin relaxation function of a magnetic probe i n a system of s t a t i c l o c a l dipolar f i e l d s i s found to be [1,2] 8 z z ( t ) = J + f t 1 - A 2* 2) e x P ( " k (III.11) which i s the f a m i l i a r s t a t i c Gaussian Kubo-Toyabe function. In Equation III.11, 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-axis), while the damped o s c i l l a t i o n of the 2/3 component arises from the x and y components ( i . e . , normal to the incident muons pol a r i z a t i o n ) of the random l o c a l f i e l d . The a p p l i c a t i o n of a lo n g i t u d i n a l f i e l d B (directed along the z-axis) can be used to e f f e c t i v e l y "decouple" the magnetic probe spin from the s t a t i c l o c a l f i e l d s , thereby quenching the depolarizing e f f e c t s of the dipolar i n t e r a c t i o n . For small l o n g i t u d i n a l f i e l d s , one obtains the expression [1,2] 2 8 z z ( t , U L ) = 1 2~ L* " 6 X P ^ " 2 A fc ) c o s ( t 0 L t ^ W L (III.12) 2A^ fc 1 2 2 + — j - J dt exp(- A T ) sin(o) Lx) uT o Li where, OJT = y B . This function i s shown for various values of OJ i n Figure Li S LI LI I I I . l . 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 studying relaxation functions i n any (or zero) external magnetic f i e l d , and thus provided the f i r s t experimental observation [4] of the Kubo-Toyabe function given i n Equation III.11. In the l i m i t of "randomly ordered" moments ( i . e . , d i s t r i b u t e d randomly i n the l a t t i c e ) , the l o c a l magnetic f i e l d d i s t r i b u t i o n at the u + s i t e approximates a Lorentzian d i s t r i b u t i o n [5-7]. The Lorentzian f i e l d d i s t r i b u t i o n (HWHM = a/y^), can be written as f L ( H i ) = IT L 2 3 2 2^ 5 i = x, y, z (III.13) a + Y H, ' s i and 3 f L(|H|) = - ^ § [ - 2 ~2 j-*"] t ^ | H | 2 ] (III.14) * l a + Y g |H| J For the case of s t a t i c l o c a l f i e l d s , the zero f i e l d spin relaxation function takes the form [8] 8 z z ( t ) = J + f t 1 " a t ) e x P ( - a t ) (III.15) As i n the case of the Gaussian Kubo-Toyabe function, given i n Equation I I I . 11, the s t a t i c Lorentzian function g ^ C t ) e x n i b i t s the c h a r a c t e r i s t i c time dependent 1/3 component. Furthermore, an applied l o n g i t u d i n a l f i e l d Figure I I I . l S t a t i c l o n g i t u d i n a l f i e l d spin relaxation function for a Gaussian random l o c a l f i e l d , plotted for various values of av/A. > w i l l e f f e c t i v e l y "decouple" the spin-spin i n t e r a c t i o n between the magnetic probe and the random f i e l d s according to the equation [8] where j n and j 1 denote Spherical Bessel functions, and u = y B . This U J. JJ s Li function i s shown i n Figure III.2 for selected values of oo . Lt The exact quantum mechanical so l u t i o n for the time dependence of the spin p o l a r i z a t i o n , assuming the Hamiltonian of Equation III.4, has also been investigated [9-11]. The zero f i e l d r e l axation functions 8 z z ( t ) obtained are found to deviate from the Kubo-Toyabe function, given i n Equation III.11, at long times ( t » 2/A). This deviation, which manifests i t s e l f as extra o s c i l l a t i o n s i n the long time t a i l , can be understood i n t u i t i v e l y by noting that the exact quantum mechanical solutions allow s p i n - f l i p t r a n s i t i o n s involving the p.+ and neighboring nuclei which, i n the d i l u t e l i m i t , appear as coherent o s c i l l a t i o n s . This type of r e l a x a t i o n function has recently been observed experimentally for \i+ i n a l k a l i f l u o r i d e s [12], with r e s u l t s suggesting the u + to be l o c a l i z e d along the <110> axis, between two 1 9 F n u c l e i . The corresponding transverse f i e l d function, f or a magnetic probe i n t e r a c t i n g with l o c a l magnetic f i e l d s , i s discussed i n both the s t a t i c and dynamic l i m i t s i n section III.B. - a[l + ) 2 ] / d x [j 0(o) Lx) exp(-aT)] (III.16) L o III.A.2 Random Anis o t r o p i c Hyperfine D i s t o r t i o n s (RAHD) In condensed media, the muonium hyperfine coupling may be perturbed due 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 electron and the - 72 -Figure III.2 S t a t i c l o n g i t u d i n a l f i e l d spin relaxation function for a Lorentzian random l o c a l f i e l d , plotted for various values of u^/a. - 73 -l a t t i c e . As discussed i n section I.C.I, a c l a s s i c example of t h i s i n t e r a c t i o n i s the case of muonium i n bulk quartz. The r e s u l t i n g change i n the ground state wavefunction i s transmitted to the u + v i a the magnetic dipole-dipole coupling of the u + and e - spins. The contribution to the t o t a l Hamiltonian due 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 included i n the W tensor of the hyperfine term of the Hamiltonian H = (h/2Tc)(y S e - y 1 • B + (h/2Ti) W : (S& S*1 ) (III.17) e ~op LI ~op ; ~ » ^~op ~opJ If the d i s t o r t i o n of the hyperfine coupling i s i s o t r o p i c , a s h i f t i n the hyperfine-structure i n t e r v a l v 0 0 w i l l occur, along with a corresponding s h i f t i n the energy eigenvalues. The zero f i e l d eigenfunctions for the system w i l l however remain f u n c t i o n a l l y unaltered from the vacuum hyperfine states so that no a d d i t i o n a l time dependence of the LI + spin p o l a r i z a t i o n i s induced. In general, however, the d i s t o r t i o n may have some anisotropic components, and i n t h i s case one observes dramatic e f f e c t s even i n zero f i e l d . The time evolution of the \i+ spin p o l a r i z a t i o n for a generally anisotropic muonium hyperfine i n t e r a c t i o n i s discussed i n some d e t a i l i n Appendix I. The approach that i s taken involves expanding the hyperfine tensor W i n terms of s p h e r i c a l harmonics and using the expansion c o e f f i c i e n t s u) L m to parameterize the d i s t o r t i o n . Because the hyperfine tensor W involves only dipole-dipole and contact i n t e r a c t i o n s , both of which have r e f l e c t i o n symmetry, the antisymmetric part of the expansion i s i d e n t i c a l l y zero. Although in 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 electron wavefunction of each muonium atom, they do hot lead to a true ( 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 spin vector for the i n d i v i d u a l muonium atoms. For an ensemble of muonium atoms, however, - 74 -depolarization can occur v i a ensemble dephasing, provided that there 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 of i n d i v i d u a l muonium atoms i n the enemble. To describe the ensemble rela x a t i o n , an approximation of Tr{p g(0)} i s adopted, where each of the w^'s * s assumed to be d i s t r i b u t e d according to some d i s t r i b u t i o n function ^ n j ^ m ^ " w * t n t h i s approximation, one has Tr{p s(0)} - H / dco 2 m f 2 m ( c o 2 J (111.18) —OO The motion tensor f o r the ensemble i s then approximated by CO g(t) - II / dw, f_ (OJ0 ) Tr{s^ (t) } (III.19) s m _o> 2 m 2 m 2m l~op ~op J where the trace over the muon spin operators i s included i n the i n t e g r a l . The spin relaxation functions associated with a s p e c i f i c d i s t o r t i o n symmetry are then calculated by averaging over the appropriate oo 2 m d i s t r i b u t i o n s . Of p a r t i c u l a r i n t e r e s t to the present study are the zero and transverse f i e l d spin r e l a x a t i o n functions for a randomly oriented system such as i n the case of muonium i n bulk fused quartz or on the surface of fine s i l i c a powders. To ca l c u l a t e these functions, one must also average over a l l possible o r i e n t a t i o n s . Consider the combination of a c y l i n d r i c a l d i s t o r t i o n coupled with a planar d i s t o r t i o n , which are parameterized by the frequencies M O ) 2 Q and u 2 2 » r e s p e c t i v e l y . Assuming each of these frequencies to be di s t r i b u t e d according to a Lorentzian or Loren 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 discussed i n Appendix I, one has M M f ^ o , ^ ) = - f — ^ H - M 2 M 2-2 (2n) } (111.20) ( U20J + (°20J L(u 2 2 ) + ^22^ •! M where O^Q and a 2 2 represent the respective widths (HWHM) of the frequency d i s t r i b u t i o n s . Ignoring the mixed state component of the muonium ensemble, - 75 -the polarized t r i p l e t muonium s t a t i c r e laxation function i n zero f i e l d i s found to be rh,,. 1 r i 1 M r 1 M \ 8 z z ( t ) " 6 t 1 ~ 2 ^ E X P ( " 2 ^ (111.21) + 1 ^ _ J ° 2 2 ^ e X p t " ^ ° 2 2 + 3 / 2 7 1 02O>] This r e s u l t can be better understood by considering the two cases of a purely c y l i n d r i c a l d i s t o r t i o n and a purely planar d i s t o r t i o n , separately. If one neglects the planar component of the hyperfine d i s t o r t i o n , Equation III.21 becomes * £ ( t ' ° 2 0 > = J + T e x p t " T / 2 7 T °2o^ <III'22> Notice that as t •» «, th i s function tends to 1/6, (or 1/3 of the i n i t i a l 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 1/6 component of the t o t a l ensemble spin 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 exists a n o n - t r i v i a l zero frequency. This can be understood i n t u i t i v e l y by drawing an analogy with random dipolar f i e l d s and noting that for a random hyperfine i n t e r a c t i o n , the c y l i n d r i c a l d i s t o r t i o n axis w i l l be directed along the z-axis ( i . e . , along the i n i t i a l muon spin pol a r i z a t i o n ) 1/3 of the time on average. If on the other hand one neglects the c y l i n d r i c a l component of the hyperfine d i s t o r t i o n , Equation 11.21 becomes rh._ M > 1 r i 1 M •> r 1 M \ g z z ( t ; C T 2 2 ) = 6 C 1 " 2 °22^ e X p ^ 2 °22^ \ J.J.1 • ^ J ) , 1 11 1 M -v t 1 M -v + 3 t 1 " 4 a22^ e x p ^ " 4 0 2 2 ^ Notice here that as t -> », th i s function approaches zero. This r e s u l t simply r e f l e c t s the fact that, unlike a c y l i n d r i c a l d i s t o r t i o n , a planar d i s t o r t i o n generates no n o n - t r i v i a l zero frequencies. - 76 -From considering these two l i m i t i n g cases, i t i s obvious that the planar component of the d i s t o r t i o n i s responsible for d r i v i n g the function rh g z z ( t ) i n Equation III.21 to zero at long times. A t y p i c a l example of t h i s function, along with the two l i m i t i n g cases Is plotted i n Figure III.3. In an external magnetic f i e l d B, 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 functions for a random anisotropic hyperfine i n t e r a c t i o n becomes somewhat more d i f f i c u l t , e s p e c i a l l y for a randomly oriented system. There are, however, a few simple l i m i t i n g cases that can be treated. Consider, for example, t r i p l e t muonium i n the l i m i t of "high f i e l d s " ( i . e . , co^ » o"2m and x « 1). In t h i s l i m i t , the i s o t r o p i c frequencies dominate so that the Hamiltonian can be approximated by i t s diagonal elements alone (secular approximation). These c a l c u l a t i o n s are given i n Appendix I for the case of a randomly oriented system. Results for the l o n g i t u d i n a l f i e l d case show that, i n this l i m i t , a l o n g i t u d i n a l l y applied f i e l d w i l l completely decouple the random anisotropic hyperfine i n t e r a c t i o n . In the transverse f i e l d case, one obtains (omitting the Larmor precession part) the relaxation function g ! \ t ) = [2n)~l j'dB sinB fdQ o o (III.24) 3 2 M 3 _____ 2 x {exp[- -g- s i n B o 2 2 c o s ( 9 ) t ] exp[- j /2/3 a 2 Q |3cos B - 111]} For early times (t -> 0), one can expand the integrand to obtain the short time behavior and approximate the r e l a x a t i o n function with the expression g_Ct) - \ ex P[-( % o2Q + - 2 - 4 2 ) t ] (111.25) TC/6 The exact s o l u t i o n of Equation III.24, calculated numerically, as well as the expansion approximation given i n Equation III.25 are plotted for 1.0 —.2 H ' r < r r 0. 1. 2. 3 . t (JMS) Figure III.3 S t a t i c zero f i e l d RAHD relaxation functions assuming Lorentzian frequency d i s t r i b u t i o n s with 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 represented by the long-dashed curve (^20 = 10 u s - 1 ) , whereas the planar d i s t o r t i o n component (022 = 10 u s - 1 ) i s represented by the short-dashed curve. The s o l i d l i n e i s the combined relaxation function for equal c y l i n d r i c a l and planar components (o"2o = 022 = 10 u s - 1 ) . A l l three curves have been normalized to equal 1 at t=0. - 78 -comparison as a function of time i n Figure III.4 . Notice that at early times, these two functions are v i r t u a l l y i n d i s t i n g u i s h a b l e i n shape. The assumption of a Lorentzian (or Lorentzian-like) d i s t r i b u t i o n may not i n general be appropriate, simply because a Lorentzian d i s t r i b u t i o n has an i n f i n i t e second moment. A more appropriate approximation may be made by assuming a modified Lorentzian, with a Gaussian damping. The a f f e c t of assuming t h i s frequency d i s t r i b u t i o n w i l l of course be r e f l e c t e d i n the shape of the calculated relaxation function, and can most e a s i l y be understood by considering the simple example of a purely c y l i n d r i c a l hyperfine d i s t o r t i o n . In th i s case one defines the d i s t r i b u t i o n f ( c o 2 Q ) = [n e ^ e r f c a ) ) " 1 [ 2 °20 2 ] e x p ^ u ^ / a ^ j x 2 ] ( 1 11 . 26 ) u 2 0 + CT20 where X i s a damping parameter ( t y p i c a l l y less than one), and erf c ( X ) i s a complimentary error function. Using t h i s d e f i n i t i o n , the s t a t i c zero f i e l d r elaxation function i s found to be g ^ ( t , \ ; a 9 n ) = i + (6 e r f c ( X ) ) " 1 {e ° 2 0 erfc ( x - a' t / ( 2 X ) ) Z Z Z U 6 Z U (III. 2 7 ) + a 2 0 t + e erfc ( X + a 2 Q t / ( 2 X ) ) } where O^Q = /2/3 O " 2 Q. This function has been calculated and i s plotted i n Figure III. 5 f o r X = 0.0 and 0 . 1 . Note that for small values of X, t h i s function i s v i r t u a l l y i n d i s t i n g u i s h a b l e from the function derived using a standard Lorentzian d i s t r i b u t i o n (Figure I I I . 3 ) . However, as X i s increased, the i n i t i a l decay begins to mimic a Gaussian shape. Because of t h i s , one should be able to put an upper l i m i t on X for 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. t ( / x s ) Figure III.4 S t a t i c transverse f i e l d RAHD relaxation function. The s o l i d curve i s the exact so l u t i o n of Equation 11.24, calculated numerically, and the dashed curve i s the expansion approximation of Equation 11.25. Both functions have been evaluated f or O^Q = °"22 = ^ u s - 1 and have been nomalized to equal 1 at t=0. 0.0 0.2 0.4 0.6 0.8 1.0 t (/us) Figure III.5 S t a t i c zero f i e l d RAHD spin relaxation function for 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 Equation III.27. The function i s plotted for two values of the damping parameter \, 0.0 and 0.1, for the case of c?20 = 10 j i s - 1 . The curves have been normalized to equal 1 at t=0. - 81 -An i n t e r e s t i n g point can now be made by comparing the i n i t i a l slopes of the zero f i e l d r e l axation function given i n Equation III.21 and the high transverse f i e l d function of Equation III.24. By defining m^ and m^ to be the i n i t i a l slopes of the zero and transverse f i e l d relaxation functions, one can define the r a t i o • z f _ [1 + ( a M /a )] — - /3 = — ~ — — — • /3 (III.28) Thus, one finds that m^ > i n d i c a t i n g that the depolarization rate i s fas t e r i n zero f i e l d than i n transverse f i e l d . This important r e s u l t can be understood by considering the problem i n terms of dimensionality; i n zero f i e l d , a l l three components (x,y,z) of the hyperfine d i s t o r t i o n contribute to the relaxation of the u + spin, whereas i n high transverse f i e l d , one i s able to make a secular approximation to the Hamiltonian and e f f e c t i v e l y ignore 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 (z-axis) components. For Mu i n bulk s i l i c a , the \& spin p o l a r i z a t i o n relaxes v i a random anisotropic hyperfine d i s t o r t i o n s (RAHD) [13]; interactions with 2 9 S i n u c l e i (4.6%, i s o t o p i c ) are r e l a t i v e l y i n s i g n i f i c a n t . In an amorphous environment such as bulk fused s i l i c a , the hyperfine d i s t o r t i o n s are 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. I t i s also known that muonium i s s t a t i c i n bulk quartz below about 50 K [13]. Because of these two features, Mu i n bulk fused s i l i c a provides an excellent test case for the zero and transverse f i e l d RAHD spin r e l a x a t i o n functions developed here. The zero and lo n g i t u d i n a l f i e l d spectra for muonium i n bulk fused quartz at 7.0 ± 0.1 K are plotted i n Figure III.6. The curve through the zero f i e l d data i s a f i t of Equation 11.10 to the data, assuming the s t a t i c RAHD function of Equation III.21. The f i t gave a Chi-square of 86.2 for 53 - 82 -!_ __ d d 6 — 00 CD OJ O o o o o • • • • o o o o Figure III.6 Zero and lo n g i t u d i n a l f i e l d data for muonium i n bulk fused quartz at 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 function to the data. The l o n g i t u d i n a l f i e l d data shown are for 0.5 G ( t r i a n g l e s ) , 1.0 G (diamonds) and 2.0 G (squares). - 83 -degrees of freedom and the f i t t e d r e s u l t s for the c y l i n d r i c a l and the planar d i s t o r t i o n parameters were 5.7 (+1.5/-1.2) u s - 1 and 6.2 (+0.54/-0.52) u s - 1 , r e s p e c t i v e l y . The t r i p l e t muonium asymmetry parameter was allowed to vary i n the f i t and found to be 0.118 (+0.0030/-0.0030). Recall that 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 r e s u l t obtained here i s consistent with the corresponding value obtained i n low transverse f i e l d . As has been discussed, a random hyperfine i n t e r a c t i o n can be almost completely decoupled i n l o n g i t u d i n a l f i e l d f o r » 0*2 m« F r o m the f i t of the zero f i e l d 7 K data, t h i s translates into a l o n g i t u d i n a l f i e l d on the order of a few Gauss. As can be seen i n Figure III.6, the decoupling behavior i s consistent with t h i s . M F i n a l l y , by su b s t i t u t i n g the values for O^Q and obtained i n the zero f i e l d f i t into the approximation for the transverse f i e l d function given i n Equation III.25, one obtains a transverse f i e l d r e l axation rate \ M u of 2.9 + 0.5 u s - 1 , which i s consistent with e x i s t i n g transverse f i e l d data [13]. Although a Lorentzian d i s t r i b u t i o n of frequencies does reproduce the basic features of the quartz data, i t i s probably not the most r e a l i s t i c assumption. Perhaps using the type of d i s t r i b u t i o n defined i n Equation III.26 would be better, but there i s at present no obvious physical argument or model that can be used to help decide what the correct d i s t r i b u t i o n should be. Ill.A.3 Chemical Reactions (CH) The coherence of the spin p o l a r i z a t i o n of the u + i n the muonium state can ( i n transverse f i e l d ) be l o s t through c o l l i s i o n s of the muonium atoms with molecules which r e s u l t i n chemical reactions forming diamagnetic products [14]. With t h i s mechanism, the muon spin p o l a r i z a t i o n must decay exponentially. In the gas phase, the transverse f i e l d r e l axation function i s given by the expression g ^ ( t ) = exp[- xf t] = exp[-(n 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 unit 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 i s the cross section for the reactio n . In transverse f i e l d , a u + precesses about 103 time slower than t r i p l e t muonium i n the same f i e l d , such that i f a chemical reaction produces a u + i n a diamagnetic environment, the u + i s e f f e c t i v e l y removed from the precessing muonium ensemble. In zero and l o n g i t u d i n a l f i e l d , however, such reactions produce no observable relaxation since the u + spin remains polarized along the z-axis regardless of whether the u + i s i n the muonium state or i n a diamagnetic state. Ill.A.4 Spin Exchange (SE) In c o l l i s i o n s with paramagnetic molecules, hyperfine t r a n s i t i o n s such as |m^ ,me> = |+,+> •*• | + can take place. As i n the case of chemical reactions i n transverse f i e l d , the decay of the muon spin p o l a r i z a t i o n due to spin exchange i s found to vary exponentially with time so that, i n transverse f i e l d , the rel a x a t i o n function i s written g ^ ( t ) = exp[- XSLe t] = exp[-(f T, v a j t ] (III.30) - 85 -where TJ i s again the number density of i n t e r a c t i n g molecules, 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, a i s the se s p i n exchange cross s e c t i o n and f i s a f a c t o r which depends on both the s p i n of the paramagnetic molecules and the o r i e n t a t i o n of the e x t e r n a l magnetic f i e l d w i t h respect to 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 . In l o n g i t u d i n a l f i e l d , where the q u a n t i z a t i o n a x i s i s along the i n i t i a l muon spin p o l a r i z a t i o n , s p i n exchange causes hyperfine t r a n s i t i o n s w i t h a p r o b a b i l i t y of s 2 c 2 = (1 + x 2 ) - 1 . Thus i n terms of the s p e c i f i c f i e l d parameter x, the 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 l o n g i t u d i n a l f i e l d i s w r i t t e n as [14,15] g__(t ,x ) = exp[- \ J e t ] = e x p [ - ( | - v a g e ) ( l + x 2 ) - 1 t ] ( I I I . 3 1 ) Transverse f i e l d s t u d i e s [16] of the temperature dependence of the s p i n exchange r e a c t i o n of muonium w i t h 0 2 (S=l) and NO (S=l/2) have shown that the f a c t o r f to equal 8/9 f o r 0 2 and 3/4 f o r NO, whereas i n l o n g i t u d i n a l f i e l d s t udies [15] the f a c t o r f was found to equal 64/27 and 2 f o r 0 2 and NO, r e s p e c t i v e l y . Ill.A.5 Superhyperfine Interactions (SHF) The time 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 i n the muonium s t a t e can a l s o be i n f l u e n c e d by the superhyperfine i n t e r a c t i o n s between the unpaired e l e c t r o n of the muonium atom and neigboring n u c l e i w i t h non-zero magnetic moments. To describe t h i s i n t e r a c t i o n , one f i r s t defines a coordinate system with the muon located at the o r i g i n , the e l e c t r o n p o s i t i o n e d at radius s, the i n t e r a c t i n g nucleus l o c a t e d at radius ^ ( d i r e c t e d along the z ' - a x i s ) , and the distance between the e l e c t r o n and the - 86 -nucleus defined by the vector r which originates from the electron. A schematic diagram of t h i s i s shown i n Figure I I I . 7 . With these designations, the Hamiltonian for the superhyperfine (SHF) i n t e r a c t i o n between the electron and a nucleus of spin J can be written as [171 ~op 1 ' H S h f = (a - b) ( S e • J ) + 3b ( S e , J ,) (III.32) v~op ~op' ; z z ' g where S z, and J , are the respective z'-components of the electron and the nuclear spins and the superhyperfine parameters (a & b), corresponding to a "contact"-like term and dipole-dipole term, re s p e c t i v e l y , are defined as ,2 and (III.33) 2 , If e * J> f ,3+ ,2 r3 cos ( t ) - In b = 2 ^ 8e ^o 8 J ^ J d r I ? ( S ) I [ 3 ^ r Here g and are the g-factor and Bohr magneton of the neighboring nucleus, respectively, and i Is the angle defined by the vectors r and R. Take as an example an i s o t r o p i c superhyperfine i n t e r a c t i o n of muonium with a single nucleus of spin J (>1), i n zero magnetic f i e l d . In this case one has b=0 and z=z' such that the t o t a l Hamiltonian can be written i n the form H = ( H h f + H s h f ) - W : ( S e ) + a(se . J ) (III.34) ' a v~op ~ O p ' ^ ~ O p ~ 0 p The four eigenvalues of this Hamiltonian are X = \ ( h / 2 * ) u + § J ; X = \ ( h / 2 « ) U o ( J - f ( j '+ l ) 1 4 uu L u u L (III.35) X2,4 " " \ [ ( h / 2 i r > 0 0 + a] ± \ { [ ( h / 2 x ) U ( J 0 - f ] 2 + j ( j + l ) a 2 } 1 / 2 where o) 0 0 i s the hyperfine-structure i n t e r v a l of the perturbed muonium atom, Figure III.7 Diagram representing the superhyperfine i n t e r a c t i o n . Taken from reference [17]. - 88 -and the corresponding t r a n s i t i o n frequencies between the eigenstates are defined as u>^s= ( 2 i t / h ) ( \ ^ - Since to a f i r s t approximation, \^ « \ 2 " X 3 « (l/4)(h/2Tc)a)Qg, the t r a n s i t i o n frequencies oo^, u 2 i t and u>23 are of order (JJQQ. Frequencies of order u 0 0 are generally not experimentally observable i n zero f i e l d due to timing l i m i t a t i o n s . Bearing t h i s i n mind, an approximation can be made by ignoring the o s c i l l a t o r y terms involving these frequencies, which simply implies ignoring the s i n g l e t s tate. Assuming a r e l a t i v e l y large value of J , one then obtains an approximate r e l a t i o n for the zero f i e l d r e l axation function of the observable muonium ensemble, namely g z z f ( t ) - \ {1 + cos(o) 1 2t) + c o s ( u 2 3 t ) } (III.36) Comparison to muonium i n vacuum shows that the time-independent part of shf g (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 to 1/6. zz The contact term of the superhyperfine i n t e r a c t i o n requires the i n t e r a c t i n g nucleus to be within about one Bohr radius of the muon. Because of t h i s , one would expect a superhyperfine i n t e r a c t i o n to be much stronger than a simple dipole-dipole i n t e r a c t i o n , and therefore 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 the discussions on spin depolarization for a magnetic probe i n a s o l i d have assumed the magnetic probe to be s t a t i c with respect to i t s environment. Owing to i t s r e l a t i v e l y l i g h t mass, however, the muon (or muonium atom) may be very mobile i n the stopping medium. This motion or hopping may a l t e r the shape of the r e l a x a t i o n function i n comparison to the s t a t i c case. This phenomenon comes about because the e f f e c t s of the - 89 -i n t e r a c t i o n ( s ) which govern the time evolution of the u + spin p o l a r i z a t i o n i n the s o l i d are averaged by the motion of the probe ( u + , Mu, e t c . ) , hence the term "motional averaging". I I I . B . l Gaussian-Markovian P r o c e s s In the case of a magnetic probe hopping s t o c h a s t i c a l l y i n the presence of s t a t i c nuclear dipoles, the motion induces a modulation or f l u c t u a t i o n of the l o c a l f i e l d as sensed by the magnetic probe. In the o r i g i n a l research of Kubo and Toyabe [1,2] the modulation of the l o c a l f i e l d i s assumed to follow a "Gaussian-Markovian" process, where the c o r r e l a t i o n of the f l u c t u a t i n g f i e l d i s characterized by the equation .2 < H ^ t ) H ±(0) > = -A_ exp(-t/T C) (III.37) where T c = 1/v i s the c o r r e l a t i o n time of the f i e l d f l u c t u a t i o n . Assuming a Gaussian random process automatically implies that the f l u c t u a t i o n of the l o c a l f i e l d i s determined by the cumulative e f f e c t of a large number of random processes, each of which induces a gradual change i n the l o c a l f i e l d . The Gaussian-Markovian assumption has also been applied to the case of a magnetic probe i n t e r a c t i n g with a Gaussian l o c a l f i e l d i n a strong external transverse external magnetic f i e l d , y i e l d i n g the a n a l y t i c r e s u l t G x x ( t , v ) = exp{-A 2/v 2 [exp(-tv) - 1 + tv]} (III.38) which i s the f a m i l i a r formula of Kubo and Tomita [18]. I t i s obvious that t h i s expression behaves properly i n the slow modulation l i m i t (v/A « 1), where the relaxation function exhibits a Gaussian shape, as well as i n the fa s t modulation l i m i t (v/A » 1), where the shape resembles an exponential decay. This function i s plotted for various values of A/v i n Figure III.8. Figure I I I .8 Dynamic Kubo-Tomita high transverse f i e l d spin r e l a x a t i o n function (Gaussian-Markovian process) plotted for various values 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 for the case of d i f f u s i o n i n the presence of traps. This i s because a magnetic probe jumping from s i t e to s i t e with a hopping frequency v would l i k e l y sense a sudden change i n the l o c a l f i e l d d i s t r i b u t i o n and not an adiabatic or gradual one. This type of behavior i s i d e a l i z e d by the "Strong C o l l i s i o n Model" [19]. III.B .2 Strong C o l l i s i o n Model In the strong 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 abruptly upon c o l l i s i o n , with the l o c a l f i e l d d i s t r i b u t i o n before and a f t e r t h i s c o l l i s i o n being completely uncorrelated. In t h i s approximation, the time evolution of the dynamical relaxation function G..(t,v) i s constructed from an i n f i n i t e series of discr e t e s t a t i c x i r e l a x a t i o n functions according to the equation G i ± ( t , v ) = I g ^ C t . v ) (111.39) n=o where g ^ (t,v) i s the re l a x a t i o n function for the magnetic probes that jump n times i n time t. The f i r s t term (n=0) i n the series of Equation 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 exp(-vt) i s the p r o b a b i l i t y that the magnetic probe does not hop i n time t, and Sj^ C O 1 S the s t a t i c r e laxation function. The following term i n the series describes the process i n which the magnetic probe hops at time t^ (0 < t^ < t) and i s expressed as OO g ^ ( t , v ) = v / d t L e ~ v ( t " t l ) g i i ( t - t 1 ) e ~ v t l g . ^ ^ ) (111.41) By introducing the Laplace transforms CO CO f ± 1 ( s ) = / dt e ~ S t 8 i l ( t ) and F . ^ s ) = / dt e " S t G±±(t) (III.42) the exact so l u t i o n i n the frequency domain becomes i f (s+v) F^CB.-V)- I v n ffif v) • , _** f (111.43) n=o i i To obtain the time domain dynamical relaxation function G ^ ( t , v ) , for a s p e c i f i c s t a t i c r e laxation function 8^^(t)» o n e must ca l c u l a t e the inverse Laplace transform of Equation III.43. The time domain dynamical transverse f i e l d r e l axation function Q G ( t , v ) , f o r the case of a magnetic probe i n t e r a c t i n g with a Gaussian XX random l o c a l f i e l d , has been numerically calculated [20] using a strong c o l l i s i o n formula s i m i l a r to that of Equation III.43. This c a l c u l a t i o n was performed using the "Korrektur-Verfahren" ( I t e r a t i o n Procedure) method [21,22]. Comparison of the relaxation functions obtained i n this manner with the Gaussian-Markovian approximation of the Kubo-Tomita formalism [18], given i n Equation III.38, reveals that the two cases are nearly i d e n t i c a l except that the strong c o l l i s i o n function exhibits a s l i g h t l y slower decay rate. This discrepancy i s p a r t i c u l a r l y noticable i n the l i m i t of slow hopping (v/A or v/a « 1, for a Gaussian or Lorentzian d i s t r i b u t i o n , r e s p e c t i v e l y ) . However, the difference between the relaxation function obtained using the strong c o l l i s i o n approximation and that obtained assuming a Gaussian-Markovian process i s so small that the simple a n a l y t i c expression of Equation III.38 i s generally preferred for data a n a l y s i s . Q The dynamical zero f i e l d spin relaxation function G z z ( t , v ) for the case of a magnetic probe i n t e r a c t i n g with a Gaussian random l o c a l f i e l d has also been calculated [4] using the strong c o l l i s i o n model given i n Equation III.43. This function i s plotted i n Figure III.9 f or various values of A/v. Comparison of Figure III.9 with the Gaussian-Markovian curves of Kubo-Toyabe [1,2] indicates that, as i n the case of the transverse f i e l d function Q G^ C t.v), the zero f i e l d curves generated with the strong c o l l i s i o n model decay at a s l i g h t l y slower rate 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 « 1). The modulation of the l o c a l f i e l d has a marked e f f e c t on the shape of the long time t a i l of the relaxation function as w e l l . As has already been discussed, the zero f i e l d s t a t i c r e laxation functions for both Gaussian and Lorentzian random l o c a l f i e l d d i s t r i b u t i o n s , exhibit a 1/3 recovery of the asymmetry at long times. For slow modulations of the l o c a l f i e l d , t h i s recovery i s suppressed, and for Gaussian random l o c a l f i e l d , follows the asymptotic form G z z ( t , v ) <* j exp(- j vt) ; for t » 3/A (III.44) where the factor of 2/3 i n the exponent can be understood i n t u i t i v e l y by noting that, on average, 1/3 of the l o n g i t u d i n a l (z-axis) p o l a r i z a t i o n i s preserved for each hop. In the l i m i t of fast 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 exponential, due to motional narrowing, such that G z z ( t , v ) « exp(-2A 2t/v) (III.45) which tends to zero at long times. A p p l i c a t i o n of the strong c o l l i s i o n model of Equation III.43 to the problem of a magnetic probe i n t e r a c t i n g with a Lorentzian l o c a l f i e l d - 94 -Figure III.9 Dynamic zero f i e l d Gaussian Kubo-Toyabe spin r e l a x a t i o n function (Strong C o l l i s i o n Process) plotted for various values of A/v. y i e l d s the corresponding dynamical zero f i e l d r e l axation function G z z ( t , v ) . This function i s shown i n Figure III.10 for various values of a/v. A major feature to make note of i s that, unlike i t s Gaussian counterpart, the function G z z ( t , v ) does not exhibit a motional narrowing e f f e c t . In f a c t , the i n i t i a l decay rate i s quite independent of the hop frequency. In the l i m i t of slow hopping, and for t » 3/A, one obtains an equation s i m i l a r to Equation III.44, whereas i n the l i m i t of fast f l u c t u a t i o n s , (v/a » 1), the relaxation function takes the form G z z ( t , v ) * exp(-4at/3) (III.46) This, l i k e the Gaussian case, tends to zero at long times; however, i t has the peculiar feature of being independent of v, as explained above. Thus, the zero f i e l d technique has been shown to be a powerful t o o l f o r studying the d i f f u s i o n and trapping behavior of a magnetic probe i n t e r a c t i n g with a random l o c a l f i e l d (Gaussian or Lorentzian) d i s t r i b u t i o n . Comparison of Figures III.9 and III.8 also reveals the advantage provided by the zero f i e l d technique, as opposed to the transverse f i e l d (precession) method, for such studies. 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 also be obtained using l o n g i t u d i n a l f i e l d , where the relaxation function would s t i l l e xhibit an exponential decay at long times, even i n r e l a t i v e l y high magnetic f i e l d s . This behavior can be understood by considering the case of muonium i n the intermediate and high f i e l d l i m i t s , i n the context of the strong c o l l i s i o n assumption of Equation III.43. The s t a t i c l o n g i t u d i n a l f i e l d r e l a x a t i o n function can be obtained by combining Equations AI.109 and AI.110 with the d e f i n i t i o n s of Equation AI.21. For l o n g i t u d i n a l f i e l d s of Figure I I I . 10 Dynamic zero f i e l d Lorentzian Kubo-Toyabe spin r e l a x a t i o n function (Strong C o l l i s i o n Process) plotted for various values of a/v. intermediate strength (0 < X < Tt/2), the observable s t a t i c l o n g i t u d i n a l "relaxation function" (omitting the modulating time part) i s written g z z ( t , X ) « j ( l + cos 2X) ; X = a r c s i n [ l / ( l + x 2 ) 1 / 2 ] (III.47) where x (= |B|/B Q) i s the s p e c i f i c f i e l d parameter, defined i n Equation I.10. From the strong c o l l i s i o n model, one then writes i n frequency space F (s+v) = 1 + C ° s 2 x , (111.48) Z Z 2(s+v) - v(l-cos Z x ) and by taking the inverse Laplace transform of Equation III.48, one obtains the r e l a x a t i o n function G (t,v,X) = y(l+cos X) exp[- J ( l - c o s X)t] (III.49) ZZ _j «_ Notice that only for the extreme high f i e l d l i m i t (X ->• 0, x -v <»), i s the re l a x a t i o n completely decoupled for muonium, unless v = 0. This argument can be extended to any relaxation mechanism, as long as the "high f i e l d " (secular approximation) l i m i t applies. In the present work, x « 0.01. Since the strong c o l l i s i o n model can be applied to any s t a t i c r e l a x a t i o n function, the dynamical spin r e l a x a t i o n functions for the case of a random hyperfine i n t e r a c t i o n can also be obtained using Equation III.43. Taking the Laplace transform of the s t a t i c zero f i e l d r e l a xation function of Equation III.21 gives .rh, N l r 1 M - i - l l r l M i f . 1 M 1-2 f z z ( s ) = 6 ^ + 2 a2 2 J " 6<2 °2 2 )L S + J a22^ (11.50) + i [ s + + 3/273 o2Q)Tl - 1 ( 1 a ^ ) [ s + 3/271 c ^ ) ] " 2 Substituting t h i s expression into Equation III.43, one can numerically c a l c u l a t e the inverse Laplace transform to obtain the time domain re l a x a t i o n function G ( t , v ) . This function i s shown i n Figure III.11 for selected ZZ - 9 8 -1.0 -4 1 L 0.8 -0.6 -I T +i 0.4 -0. 1. 2. 3. t (yUs) Figure I I I . 1 1 Dynamic zero f i e l d Lorentzian RAHD spin relaxation function (Strong C o l l i s i o n Process) plotted for selected values of the hop rate v, where O^Q = = i U u s - 1 . The function i s normalized to equal 1 at t = 0 . - 99 -M i values of v, where both O^Q A N A a22 a r e e c l u a l t o ^ u u s - x . As i n the case of a magnetic probe i n t e r a c t i n g with a Lorentzian random l o c a l f i e l d d i s t r i b u t i o n (Figure III.10), the i n i t i a l decay rate of the function shown i n Figure III.11 i s completely unaffected by the hop rate v. This feature, which i s somewhat c o u n t e r - i n t u i t i v e , arises d i r e c t l y from the assumption of Lorentzian or L o r e n t z i a n - l i k e frequency d i s t r i b u t i o n s . This behavior i s also independent of the dimensionality of the d i s t r i b u t i o n since i n the case of a Lorentzian random l o c a l f i e l d the d i s t r i b u t i o n i s three-dimensional, whereas for random hyperfine d i s t o r t i o n s one has 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 plus a two-dimensional planar component d i s t r i b u t i o n . Thus the behavior shown i n Figure III.11 implies that for Lorentzian 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 (of a l l dimensions), the shape at early times i s independent of the motion of the magnetic probe. It i s p a r t i c u l a r l y i n s t r u c t i v e to consider the two l i m i t i n g cases of either a t o t a l l y c y l i n d r i c a l or t o t a l l y planar d i s t o r t i o n . In Figure III.12 the relaxation function generated by assuming only 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 hyperfine i n t e r a c t i o n ( i . e . , = 0) i s p l o t t e d . The e f f e c t of motion on this component of the relaxation function i s to suppress the long time t a i l , even for small hop rates; for s u f f i c i e n t l y high hop rates, this long time t a i l tends to zero. The case of a purely planar d i s t o r t i o n ( i . e . , °20 = ^ * S s t l o w n * n f i g u r e III.13. Since i n the s t a t i c l i m i t this function already tends to zero, the e f f e c t of hopping i s not very noticeable at long times. Instead, the e f f e c t of hopping on the relaxation function i s more evident at early times, where i t serves to reduce the depth of the minimum. This same procedure can be applied to calculate the dynamical rh transverse f i e l d r e l axation function G ( t , v ) . Taking the Laplace transform - 100 -1.0 H • L 0. 1. 2. 3. t (yU-S) Figure I I I . 12 Dynamic zero f i e l d Lorentzian RAHD spin r e l a x a t i o n function for a pure c y l i n d r i c a l d i s t o r t i o n (Strong C o l l i s i o n Process) plotted for selected values of the hop rate v, where a20 = 10 u s - 1 . The curves have been normalized to equal one at t=0. - 101 -1.0 H 1 L 0.8 H 0. 1. 2. 3. t ( / x s ) Figure I I I . 13 Dynamic zero f i e l d Lorentzian RAHD spin relaxation function for a pure planar d i s t o r t i o n (Strong C o l l i s i o n Process) plotted for selected values of the hop rate v, where 0^2 = 10 u s - 1 . The curves have been normalized to equal one at t=0. - 102 -of the approximation i n Equation III.25, one obtains £<•> • T i - + [ 1 ° 2 o + <&]rl ( " L S I ) TC/O Substituting this into Equation III.43 and numerically c a l c u l a t i n g the inverse Laplace transform y i e l d s a time domain function which i s completely independent of the hop frequency v. The dynamical relaxation functions corresponding to random anisotropic hyperfine interactions have thus far been calculated assuming the d i s t o r t i o n parameters to be d i s t r i b u t e d according to Lorentzian (or Lorentzian-like) d i s t r i b u t i o n s of dimension less than three. These functions have been found to exhibit no motional dependence at early times, owing to the assumption of Lorentzian 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 . If instead one assumes the same d i s t o r t i o n symmetries, but chooses a d i f f e r e n t frequency d i s t r i b u t i o n which has a f i n i t e second moment, such as the d i s t r i b u t i o n defined i n Equation III.26, one would expect the r e s u l t i n g functions to eventually motionally narrow. The strong c o l l i s i o n dynamical function derived by assuming the "modified Lorentzian" d i s t r i b u t i o n of Equation III.26 has been calculated numerically and i s shown i n Figure III.14 for various values of the hop rate v. Notice that for small hop rates, t h i s function exhibits no v dependence at early times, but as v i s increased, motional narrowing becomes more apparent. As mentioned, however, there i s no obvious physical argument that can 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 . III.B .3 D i f f u s i o n i n the Presence of Traps Thermally activated d i f f u s i o n and trapping of p o s i t i v e muons at defects and impurities has been observed by many authors [23,24]. Two theories have - 103 -1.0 0. 1. 2. 3. t ( y l c - S ) Figure III.14 Dynamic zero f i e l d modified Lorentzian RAHD spin r e l a x a t i o n function for a pure c y l i n d r i c a l d i s t o r t i o n (Strong C o l l i s i o n Process) plotted for selected values of the hop rate v, where a2Q - 10 u s - 1 and the parameter X = 0.1. The functions are normalized to equal one at t=0. - 104 -been proposed to explain the e f f e c t of such phenomena on the u + spin p o l a r i z a t i o n , one suggested by Kehr et a l . [20] and another put f o r t h by McMullen and Zaremba [25] and Petzinger [26,27] . Both of these theories can be extended to a muon i n the muonium state as w e l l . In the case of the former theory, a strong c o l l i s i o n process i s assumed, making this formalism applicable to any relaxation mechanism, whereas the l a t t e r case only applies to muons or t r i p l e t muonium i n t e r a c t i n g with a l o c a l dipolar f i e l d . The l a t t e r theory has been used i n the present work i n the analysis of some of the transverse f i e l d data. It assumes a Gaussian approximation for the frequency d i s t r i b u t i o n due to dipolar f i e l d s and expresses the spin r e l a x a t i o n function i n terms of time dependent s i t e occupation 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 time dependent s i t e occupation p r o b a b i l i t i e s allows for the p o s s i b i l i t y that the muons are not i n thermal equilibrium with respect to t h e i r s i t e occupancy. Using second-order time dependent perturbation theory, the transverse f i e l d spin relaxation function for a multi-state system can be written as [26,27] M TI 9 t t 1 G ™ ( t ) = exp[-r(t)] = exp[- £ a\ / dt'/ dt" N ±( f ' H ^ C f - t " ) ] (III.52) i= l o o 2 Here the sum extends over n states; i s the second moment of the frequency th d i s t r i b u t i o n for the i state, i s the time dependent p r o b a b i l i t y for the occupation of the i state and $ ^ ( t ) ^ s t n e corresponding s i t e autocorrelation function. The evaluation of Equation III.52 can be f a c i l i t a t e d by introducing a dimensionless linewidth parameter a [28] , defined as CO a = / dt exp(-t/T )[dT(t)/dt] (III.53) o ^ - 105 -where x i s the mean muon l i f e t i m e . For s i m p l i c i t y , stochastic hopping i s assumed so that the functions 't'^^Ct) a r e given by an exponential of the form exp(-t/x^), where x^ i s the mean dwell time i n the i * " * 1 state. By combining Equations III.52 and III.53, one obtains the simple r e s u l t a = I a\ £{N ±(t)} [\ + \ ) ~ l (111.54) 1=1 u i where £{N^(t)} i s the Laplace transform of the state occupation p r o b a b i l i t y , with the i m p l i c i t transform variable s = 1/x... For the case of an exponential relaxation, T(t) = at/x^, and for a Gaussian re l a x a t i o n , 1 2 2 F( t ) = j at lx^m Thus, the problem of c a l c u l a t i n g the relaxation function for a multi-state system has been reduced to determining the £{N^(t)}, which are i n general the solutions to a s p e c i f i e d set of rate equations. - 106 -CHAPTER IV — EXPERIMENTAL RESULTS AND INTERPRETATIONS Previous to the present work i t was shown [1-3] that muonium escapes the grains of f i n e oxide powders, including the s i l i c a powders used i n the present study, and resides i n the extragranular region and on the grain surfaces. This phenomenon was further shown to be t o t a l l y independent of the ambient temperature of the powder grains (see section I.C). Because of the large s p e c i f i c surface area (390 ± 40 m2/g [4]) and the high y i e l d of extragranular muonium, provided by the 35 A s i l i c a powder, this material was chosen for the present study of the i n t e r a c t i o n s ( i . e . , surface d i f f u s i o n , desorption and spin relaxation mechanisms) of muonium with surfaces. Zero, l o n g i t u d i n a l and transverse f i e l d uSR techniques have been used i n the present work to study the behavior of muonium on s i l i c a surfaces (section IV.A). In the i n i t i a l stages of t h i s work (section IV.A.l), transverse f i e l d (< 10 G) data were taken to investigate the temperature dependence of the transverse f i e l d muonium relaxation rate for several surface hydroxyl concentrations. These studies were prompted by the hypothesis that a dipole-dipole i n t e r a c t i o n between muonium and the hydroxyl protons might be a p r i n c i p a l contributor to the relaxation of the u + spin p o l a r i z a t i o n for muonium on the s i l i c a surface. Assuming this hypothesis to be correct, a three-state model was also developed, which describes the d i f f u s i o n and trapping of muonium on the s i l i c a surface and includes the p o s s i b i l i t y of desorption. A second set of experiments were then performed (section IV.A.2), using zero and l o n g i t u d i n a l f i e l d techniques, to obtain information on the shape of the relaxation as well as the decoupling behavior. This information was used to discriminate 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 relaxation theory, involving random hyperfine anisotropies, which i s used to i n t e r p r e t some of the data. A t h i r d set of experiments (section IV.B), again i n transverse f i e l d , were done to study the e f f e c t of f r a c t i o n a l surface coverages of helium on the surface d i f f u s i o n and trapping behavior of muonium. F i n a l l y (section IV.C), transverse f i e l d data were taken to investigate the interactions of muonium with the surfaces of platinum loaded c a t a l y s t s . These experiments provided the f i r s t observation of the chemical reaction of muonium with surfaces ( i n t h i s case platinum), and also suggested a possible o r i g i n for one of the surface s i t e s for muonium on the s i l i c a surface. The r e s u l t s of these experiments are i n d i v i d u a l l y subtle; however, they do allow one to construct an unbroken chain of l o g i c leading to some clear deductions concerning which spin r e l a x a t i o n mechanism(s) are operable for muonium on the s i l i c a surface. IV.A Muonium on S i l i c a Surfaces As i n bulk quartz, the spin p o l a r i z a t i o n for muonium on the s i l i c a surface may experience relaxation due to random anisotropic d i s t o r t i o n s of the muonium hyperfine i n t e r a c t i o n . However, s i g n i f i c a n t contributions to the t o t a l spin relaxation may also a r i s e from other relaxation mechanisms such as random l o c a l magnetic f i e l d s (due mainly to the surface hydroxyl protons) or perhaps spin exchange int e r a c t i o n s (with any paramagnetic i m p u r i t i e s ) . In t h i s section, data are presented and arguments are put f o r t h to extract information concerning the motion of the muonium atoms on the s i l i c a surface as well as the o r i g i n of the relaxation i n t e r a c t i o n . - 108 -IV.A.l Transverse Field Results Mu The transverse f i e l d muonium relaxation rate \ i s shown as a function of inverse temperature for two sample preparations i n Figure IV.1; the c i r c l e s are the data obtained for sample S i 0 2 ( l ) prepared at 110 °C and the squares represent the date taken with sample Si0 2(3) prepared at 600 °C. Let us f i r s t consider the 110 °C data. Q u a l i t a t i v e l y , these data are interpreted as follows: The plateau below about 8 K i s due to muonium " l o c a l i z e d " i n a host adsorption s i t e (by which i s meant a very common shallow p o t e n t i a l w e l l ) , and the peak which occurs at about 25 K i s taken to be due to trapping at less common depolarization centers (trap s i t e s ) . From Mu the low temperature plateau to the minimum at about 16 K, \ (T) decreases because of motional narrowing due to hopping of the muonium atom between host s i t e s . Between the minimum and the 25 K peak, the hopping becomes s u f f i c i e n t l y rapid f o r the muonium atom to reach the trap s i t e s before i t Mu decays. As the temperature i s increased beyond the peak temperature, \ (T) i s seen to decrease monotonically. This decrease i s at t r i b u t e d to detrapping and eventual desorption of the muonium atom from the grain surfaces. If one pictures the s i l i c a powder target as a uniform d i s t r i b u t i o n of sph e r i c a l p a r t i c l e s of radius R and mass density p Q , packed to an o v e r a l l mass packing density p, the maximum c o l l i s i o n frequency of the Mu atoms with the grain surfaces i s e a s i l y shown to be (see Appendix III) where N i s the number of spher i c a l p a r t i c l e s i n the sample, V^uis the free - 109 -.00 .05 .10 .15 .20 .25 1 /T ( K " ' ) Figure IV.1 Transverse f i e l d muonium relaxation rate as a function of inverse temperature for muonium on the surfaces of fine s i l i c a powders (mean grain radius 35 A) with samples prepared at 110 °C ( f i l l e d c i r c l e s ) and at 600 °C (open squares). The l i n e s shown are f i t s of the three-state model described i n the text. - 110 -volume of the sample (V - V a n c* v 1 S the mean thermal v e l o c i t y of the muonium atom. In the case of S i 0 2 , p Q « 2.2 g/cm3. From Equation IV. 1, one Mu might expect \^ (T) to exhibit some dependence on the packing density p, which would presumably be more evident at higher temperatures. In order to Mu test for a possible packing density dependence of \ (T), transverse f i e l d measurements were made using sample S i 0 2 ( 2 ) , which has a mass packing density of about 1/3 that of sample S i 0 2 ( l ) , prepared at 110 °C. Comparison of the these r e s u l t s with those obtained for the higher packing density reveals no s i g n i f i c a n t differences below ~65 K, leading one to conclude that the r e l a x a t i o n rate i s l a r g e l y independent of the target packing density i n the temperature and packing ranges studied. This r e s u l t i s consistent with the idea that at low temperatures, the muonium atoms are constrained primarily to motion on the surfaces of the s i l i c a grains. These data are tabulated i n Appendix IV. Mu Now consider the temperature dependence of \ for sample Si0 2(3) prepared at 600 °C, also shown i n Figure IV.1, for which the surface hydroxyl concentration i s reduced. These data indicate the same general d i f f u s i o n and trapping behavior as o r i g i n a l l y observed for sample S i 0 2 ( l ) prepared at 110 °C; however, there are some important d i f f e r e n c e s . In p a r t i c u l a r , one observes that the reduction i n the concentration of surface Mu hydroxyl groups i s accompanied by a decrease i n \ . Moreover, this e f f e c t seems to be more evident at lower (< 30 K) temperatures. In addition to the Mu observed general reduction i n \ , the p o s i t i o n of the "trapping peak" i s seen to s h i f t to higher temperatures with reduced hydroxyl concentration. This s h i f t i n p o s i t i o n can for instance be a t t r i b u t e d to a decrease i n the - I l l -detrapping frequency, a r i s i n g i n turn from the hydrolysis process. These re s u l t s c l e a r l y indicate that the surface hydroxyls play an important r o l e i n the depolarization of the \i+ spin for muonium on the s i l i c a surface. The precise role played, however, may not be simply to provide a dipole-dipole i n t e r a c t i o n . A few data points were also taken with sample S i 0 2 ( 4 ) , prepared at 600 °C, which reproduce the same behavior as observed for sample Si0 2(3) prepared at the same temperature. The temperature dependence of the average muonium hyperfine-structure i n t e r v a l v 0g was also studied, using the same s i l i c a powder prepared at about 110 °C, over the temperature range 17 K < T < 300 K [5]. These measurements were made i n high transverse f i e l d (~500 G) and are shown i n Figure IV.2. Above ~100 K, the values obtained for v Q 0 are consistent with the vacuum value (~4463.3 MHz), i n d i c a t i n g that the muonium atoms spend the majority of t h e i r time i n the extragranular region. Below ~100 K, v 0 0 decreases r a p i d l y to a value of 4437 ± 4 MHz at 17.0 ± 0.1 K, a change of about -0.6%. This e f f e c t has been at t r i b u t e d to adsorption of muonium on the s i l i c a surface and may be compared to the room temperature r e s u l t s of -0.12% and -0.13% observed for hydrogen and deuterium, respectively (see section 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 part of the hyperfine i n t e r a c t i o n and thus provide no information regarding any a n i s o t r o p i c components. However, i f one i s correct i n assuming that the observed d i s t o r t i o n i s due to muonium adsorbed onto the s i l i c a surface, i t i s e a s i l y argued that the r e s u l t i n g hyperfine i n t e r a c t i o n would be anisotropic, and thus induce a r e l a x a t i o n of the \i+ spin p o l a r i z a t i o n . Operating under the assumption that a dipole-dipole i n t e r a c t i o n between the hydroxyl protons and the muonium atoms i s a major contributor to the - 112 N o o 4470 4465 -4460 -4455 -4450 -4445 -4440 -4435 -4430 0 100 200 300 T ( K ) Figure IV.2 Hyperfine-structure i n t e r v a l versus temperature for muonium in t e r a c t i n g with the s i l i c a surface (~110 aC preparation). The curves through the points are f i t s to the data using Equation IV.6. - 113 -spin depolarization of the muonium ensemble, a three-state model was developed [6,7]. This model u t i l i z e s a previously developed multi-state d i f f u s i o n and trapping theory [8-10], which i s described i n Chapter I I I . To describe the transverse f i e l d data shown i n Figure IV.1, one chooses a model which characterizes the muonium occupation s i t e s on the surface i n terms of equivalent trapping s i t e s of r e l a t i v e concentration C t and a remaining f r a c t i o n l ~ C t of equivalent host s i t e s . Since there i s also a f i n i t e temperature dependent p r o b a b i l i t y for desorption, the s i t u a t i o n can be represented by a three-state model i n which muonium atoms have the p o s s i b i l i t y of occupying either of the two adsorbed states or the desorbed state. By denoting the occupation p r o b a b i l i t i e s for the host s i t e s , the trap s i t e s and the desorbed state as N Q , N F C and N ^ , respectively (which obey the normalization condition N + N + N , . = 1) one can define the following o t f set of coupled rate equations: • N o -KCt + Vof) \^~Ct' Vfo^ -Ct) N o K v o C t - [ v t ( l - C t ) + VtfJ VftCt Nt Vof Vtf - [ v f o ( l - C t ) + V f t C t ] _ (IV.2) Here V q and v are the surface hop rate and detrapping rate, and v ^ t are the host and trap s i t e adsorption rates and and v are the desorption rates. In Equation IV.2, the hop rate from a host s i t e to a trap s i t e i s assumed to be equal to the hop rate v Q between host s i t e s . Applying the normalization condition, one can then obtain the Laplace transform solutions to Equation IV.2. Assuming Arrhenius behavior, the surface hopping and detrapping rates are res p e c t i v e l y - 114 -V q = v 1exp(-E Q/kT) and v = v 2exp(-E t/kT) (IV.3) where E Q and E t are the respective a c t i v a t i o n energies. S i m i l a r l y , the desorption rates are defined as V o f = V 3 e x p H E o + E f)/kT} and v t f = v 4exp{-(E t + E f)/kT} (IV.4) th where the quantity E^+E^ i s the desorption energy for the i state. F i n a l l y , the associated adsorption rates are defined as v, = F(T) P (T) and v £ = F(T) P (T) (IV.5) fo o f t t Here P Q(T) and P t(T) are the trapping p r o b a b i l i t i e s for the host and trap s i t e s , r e s pectively, and F(T) i s the c o l l i s i o n frequency of the muonium atoms with the grain surfaces, given i n Equation IV.1. This three-state model was used to f i t the transverse f i e l d data for both the 110 ° C and 600 ° C preparations, assuming an i n i t i a l condition of N q = = 0 and = 1. Other I n i t i a l conditions were t r i e d as w e l l , but the f i t t e d parameters were found to be independent of the i n i t i a l conditions assumed. The r e s u l t i n g curve i s shown i n Figure IV.1 and the f i t t e d parameters are given i n Table IV.1(a). Because of the expected low concentration of trap s i t e s , the Mu e f f e c t on \ (T) of d i r e c t desorption from the traps was assumed n e g l i g i b l e ( i . e . , = 0). In addition, the trapping p r o b a b i l i t i e s P Q(T) and P t(T) were both set equal to unity. This model has also been used to f i t the data for the 600 ° C preparation. The r e s u l t i n g curve i s shown i n Figure IV.1 and the f i t t e d parameters are given i n Table IV.1(b). In t h i s f i t , some of the parameters were not well determined, owing to the lack of data above 85 K. Because of t h i s , only a few error estimates were obtained. As can be seen from Figure IV.1, t h i s simple model describes the data - 115 -Table IV.1(a) F i t Results for Sample S i 0 2 ( l ) Prepared at 110 °C Parameter Value Error ( X 2 / d e g . f r . = 10.5/11) v l 87 +86 / -37 us v2 11.2 +6.6 / -2.8 us' v3 441 +889 / -230 U S 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 0.66 % Table IV.1(b) F i t Results for Sample Si0 2(3) Prepared at 600 °C Parameter Value Er r o r ( x 2/deg.fr. = 11.6/8) 0.548 u s - 1 v 2 4.21 +0.42 / u s - 1 v3 1557 +2.3 / u s - 1 E o 8.37 +0.021 / K E t 93 +3.5 / K E f 97 K 0.61 +0.12 / -0.12 - 2 U S ^ 4.39 - 2 us £-c t 0.52 % - 116 -quite w e l l . Of course, the relaxation i n t e r a c t i o n assumed here i s much simpler than the actual i n t e r a c t i o n , and some of i t s features may obscure the physical parameters (hop rates, a c t i v a t i o n energies, etc.) deduced from t h i s model. However, the q u a l i t a t i v e explanation afforded by t h i s model i s s a t i s f a c t o r y . For example, by comparing the f i t r e s u l t s obtained for the two preparations (see Table I V . l ) , one observes that the a c t i v a t i o n energies are s i g n i f i c a n t l y reduced for the 600 °C preparation as compared to the 110 °C preparation. This i s p a r t i c u l a r l y noticeable for E Q . In addition, for the 600 °C preparation i s smaller than that for the 110 °C preparation. To understand what implications this has with respect to the motion of the muonium atoms on the s i l i c a surface, i t i s i n s t r u c t i v e to plot the surface hop rates v Q (defined i n Equation IV.3) as a function of temperature, for both sample preparations. The r e s u l t i n g curves are are shown i n Figure IV.3. Comparison of the two curves i n Figure IV.3 suggests that the presence of the hydroxyl groups serves to i n h i b i t surface d i f f u s i o n of the Mu atoms at low temperatures. Another important difference i s that v 2 i s s i g n i f i c a n t l y smaller for the 600 °C preparation as opposed to the 110 °C preparation, i n d i c a t i n g a reduction i n the detrapping rate. The reduced detrapping rate i s responsible for the observed s h i f t of the trapping peak to higher temperatures, as suggested e a r l i e r . The data i n Figure IV.2, showing the temperature dependence of the hyperfine-structure i n t e r v a l , have been f i t assuming the model of a Mu atom thermalized i n a system with t o t a l area A and t o t a l free volume Vf. This s i t u a t i o n i s represented by the equation [2] ~\o " + j{U\f)e*p[-*/K)] + v J l + £ (^ U)exp(E/kT)]. (IV.6) - 117 -1.5 H 1 L 0. 5. 10. 15. T ( K ) Figure IV.3 Surface hop rate v 0 versus temperature, calculated from Equation IV.3, using the f i t t e d parameters of Table IV.1. The s o l i d l i n e corresponds to the 110 °C preparation, and the dashed curve corresponds to the 600 °C preparation. - 118 -g where V Q Q I S the hyperfine-structure i n t e r v a l for muonium on the s i l i c a Mu surface, V q q i s the vacuum value, i s the thermal de Broglie wavelength g and E i s the a c t i v a t i o n energy for desorption. In the f i t s , V q o was f i x e d at 4437 MHz, the value measured at the lowest temperature. The dashed curve i s obtained by f i x i n g the r a t i o A/Vf at a value of 6.24 x 10 5 cm - 1 (calculated using the model of Appendix III) and f i t t i n g the a c t i v a t i o n energy E. This f i t gave a Chi-square of 5.66 for 5 degrees of freedom and an a c t i v a t i o n energy of 76 (+35.4/-12.8) K. The s o l i d l i n e i s obtained by setting the a c t i v a t i o n energy E equal to the sum E^ + E Q (= 275 K), calculated using the values from the three-state model f i t given i n Table IV.1(a), and f i t t i n g the r a t i o A/V f. This f i t gave a Chi-square of 4 for 5 degrees of freedom and a value of A/V f = 3587 (+2865/-1960) cm - 1, two orders of magnitude less than that given by the model c a l c u l a t i o n . This r e s u l t i s not surprising since the model c a l c u l a t i o n i s an overestimate, owing to the tendency for the s i l i c a powder grains to aggregate. IV.A.2 Zero and Longitudinal F i e l d Results The zero and l o n g i t u d i n a l f i e l d asymmetry spectra taken at 7.0 ± 0.2 K for sample S i 0 2 ( 4 ) , prepared at 110 °C, are shown i n Figure IV.4. According to the i n t e r p r e t a t i o n of the associated transverse f i e l d data of Figure IV.1, these data correspond to the s t a t i c l i m i t . Notice that the zero f i e l d spectrum exhibits an i n i t i a l exponential-like decay and tends to zero at long times. Notice also that the r e l a x a t i o n i s almost completely decoupled for l o n g i t u d i n a l f i e l d s of only a few Gauss. From the discussions i n Chapter III one r e c a l l s that, for relaxations due to random l o c a l magnetic - 119 -Figure IV.4 Zero and l o n g i t u d i n a l f i e l d asymmetry spectra for muonium on the s i l i c a surface (110 "C preparation) at 7.0 ± 0.2 K. The zero f i e l d data are represented by the square symbols and are compared to data taken at 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 1.0 G, the triang 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 using the s t a t i c zero f i e l d r e l axation function of Equation III.21. - 120 -f i e l d s (RLMF), the relaxation function exhibits an i n i t i a l Gaussian shape unless one i s i n the fast hopping l i m i t or i n the l i m i t of "randomly ordered" moments. It has also been shown i n Chapter III that i n the s t a t i c l i m i t , any form of relaxation due to random l o c a l magnetic f i e l d s would exhibit a 1/3 recovery of the i n i t i a l u + spin p o l a r i z a t i o n at long times. Since the data for muonium on the s i l i c a surface below about 7 K shows no long time recovery, one has only two p o s s i b i l i t i e s : (1) The dipole moments (hydroxyl protons) are randomly ordered, and the muonium atoms are not s t a t i c on the s i l i c a surface. (2) A dipole-dipole coupling 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 evolution of the \i+ spin p o l a r i z a t i o n . The f i r s t 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 to reconcile with the f a c t that there are about 4 hydroxyl groups per nm2 on the s i l i c a surface, which translates into one hydroxyl for every other Si atom. With such a large concentration, the l i m i t of randomly ordered moments would be d i f f i c u l t to j u s t i f y . Moreover, even i f this were accepted, along with the concomitant Lorentzian 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 , the postulation of d i f f u s i n g muonium i s inconsistent with the observed shape of the relaxation function i n l o n g i t u d i n a l f i e l d . R e c a l l from Chapter III that for a s t a t i c muonium atom i n t e r a c t i n g with a random l o c a l f i e l d , the spin relaxation can be completely decoupled i n a l o n g i t u d i n a l f i e l d on the order of the l o c a l dipolar f i e l d ; whereas for a dynamic system, the relaxation would continue to exhibit a decay at long times, even In high l o n g i t u d i n a l f i e l d s . The data shown i n Figure IV.4 show the relaxation to be almost completely decoupled for very small f i e l d s , which i s inconsistent with what one would expect for a dynamic probe. From t h i s argument one may conclude that a random dipolar i n t e r a c t i o n i s not the p r i n c i p a l relaxation mechanism for - 121 -muonium on the s i l i c a surface. The fact that the data i n Figure IV.4 shows the relaxation to be e a s i l y quenched i n low f i e l d , however, leads one to suspect a random anisotropic hyperfine d i s t o r t i o n (RAHD) as a l i k e l y candidate. One can e a s i l y argue that the relaxation function for a dynamic muonium atom, i n t e r a c t i n g v i a a random anisotropic hyperfine i n t e r a c t i o n , would also exhibit a decay at long times, even i n high l o n g i t u d i n a l f i e l d s . Thus i f one assumes a relaxation due to RAHD, the low temperature zero and l o n g i t u d i n a l f i e l d data shown i n Figure IV.4 indicates muonium to be i n the s t a t i c l i m i t , i n agreement with the i n t e r p r e t a t i o n of the transverse f i e l d data of Figure IV.1. Let us now consider whether a random anisotropic hyperfine i n t e r a c t i o n alone can adequately explain the data. In t h i s case, an exponential-like decay i s expected as long as the frequencies are d i s t r i b u t e d according to a Lorentzian d i s t r i b u t i o n function. As discussed i n Appendix I and i n Chapter I I I , a 1/3 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 l i m i t for a c y l i n d r i c a l l y d i s t o r t e d random hyperfine i n t e r a c t i o n . Thus, as i n the case 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 hyperfine i n t e r a c t i o n i s not s u f f i c i e n t to explain the data. I f , however, one includes a planar d i s t o r t i o n component as well, one obtains a function which has the required exponential-like i n i t i a l decay, and also tends to zero at long times (Equation III.21). The curve i n Figure IV.4 i s a f i t of Equation 11.10, assuming the relaxation function of III.21, to the data. The f i t gave a Chi-square of 45.1 for 28 degrees of freedom, and the f i t t e d r e s u l t s for the c y l i n d r i c a l and planar d i s t o r t i o n frequencies ( d i s t r i b u t i o n widths) were found to be 12.1 (+1.59/-1.33) u s - 1 and 0.86 (+0.085/-0.090) u s - 1 , r e s p e c t i v e l y . The muonium asymmetry (for t r i p l e t muonium) was allowed to - 122 -vary i n the f i t and was found to equal 0.103 (+0.0047/-0.0042). This value i s consistent with that obtained for the corresponding transverse f i e l d data (taken with the same sample and preparation). As has already been discussed, the e f f e c t s of a random anisotropic hyperfine 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 for » o ^ . From the r e s u l t s of the zero f i e l d f i t , t h i s translates into a f i e l d on the order of a few Gauss, which i s consistent with the data i n Figure IV.4. M By su b s t i t u t i n g the values of O^Q A N < * a22 0 D t a i n e < i i n t n e zero f i e l d f i t , into the transverse f i e l d approximation of Equation III.25, one obtains a transverse f i e l d r e l axation rate \ M u of 3.1 ± 0.38 u s - 1 . This r e s u l t i s consistent with the relaxation rate determined for the associated transverse f i e l d data, shown i n Figure IV.1. The zero and l o n g i t u d i n a l f i e l d asymmetry spectra taken at 3.6 ± 0.2 K, using sample Si0 2(3) prepared at 600 UC, are shown i n Figure IV.5. As i n the case of the data for the 110 UC preparation, the zero f i e l d spectrum exhibits an exponential-like decay and also tends to zero at long times. The curve through the zero f i e l d data i s a f i t of Equation 11.10 to the data, assuming the s t a t i c random anisotropic hyperfine d i s t o r t i o n function of Equation III.21. The f i t gave a Chi-square of 87.3 for 53 degrees of freedom, and the f i t t e d r e s u l t s for the c y l i n d r i c a l and the planar d i s t o r t i o n parameters were found to be equal to 4.4 (+0.8/-0.9) | i s - 1 , and 1.8 (+0.2/-0.15) [ i s - 1 , r e s p e c t i v e l y . The muonium asymmetry was allowed to vary i n the f i t and found to equal 0.069 (+0.0023/-0.0022). This value i s consistent with that obtained for the corresponding transverse f i e l d data, but i t i s s i g n i f i c a n t l y less than that found for sample Si0 2(4) prepared at - 123 -T 1 1 1 CD (N ^ 00 CO ^ CN O T O O O O • • • • /Q:J.8UJUJA'SV papajJOQ Figure IV.5 Zero and l o n g i t u d i n a l f i e l d asymmetry spectra for muonium on the s i l i c a surface (600 "C preparation), at 3.6 ± 0.2 K. The zero f i e l d data are represented by the square symbols and are compared to data taken at 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 trian 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 using the zero f i e l d s t a t i c r e laxation function of Equation III.21. - 124 -110 °C. This difference arises simply because the window on sample Si0 2(4) i s 25 \im thick, whereas the window on sample Si0 2(3) i s 50 um thick. More muons are therefore stopped i n the window for sample S i 0 2 ( 3 ) , thereby adding to the diamagnetic f r a c t i o n observed. A comparison of the d i s t o r t i o n parameters obtained here with those obtained for the 110 "C preparation indicates a correspondence between the muonium hyperfine d i s t o r t i o n and the concentration of surface hydroxyl groups; the c y l i n d r i c a l component O^Q M increases, while the planar component decreases, with increasing hydroxyl concentration. This r e s u l t i s rather i n t e r e s t i n g because i t suggests that the presence of the hydroxyl groups a f f e c t s the l o c a l environment of the muonium atom i n a manner which induces an associated d i s t o r t i o n symmetry i n the muonium hyperfine Interaction. Moreover, the fact that the observed d i s t o r t i o n for the 110 °C preparation i s shown to have an enhanced c y l i n d r i c a l component and a diminished planar component, r e l a t i v e to the 600 UC preparation, suggests that 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 the hydroxyl groups i s repulsive. Although t h i s i n t e r p r e t a t i o n does adequately explain the observed behavior, one cannot exclude the p o s s i b i l i t y that this r e s u l t could merely be a manifestation of a combined relaxation involving both random hyperfine anisotropics and random dipolar f i e l d s . M By s u b s t i t u t i n g the values of O^Q A R U * a22 ^ o r t n e preparation Mu into Equation III.25, the transverse f i e l d r e l a x a t i o n rate \ i s calculated to be 1.5 ± 0.25 u-s - 1. This r e s u l t i s again consistent with the associated transverse f i e l d data. Data were also obtained with sample Si0 2(3) (prepared at 600 °C), i n - 125 -the dynamic region from 6 K to 20 K. The zero and l o n g i t u d i n a l f i e l d data taken at 16.0 ± 0.1 K (where the muonium atoms are believed to be hopping between the host s i t e s ) are shown i n Figure IV.6. The zero f i e l d data for t h i s temperature exhibit a s l i g h t decrease i n the i n i t i a l decay (motional narrowing) as compared to the s t a t i c case of Figure IV.5, but there i s no M i n d i c a t i o n of the 0 ^ 2 minimum. The absence of a minimum i s predicted by the dynamical model derived from the s t a t i c formula of Equation III.21 and the strong c o l l i s i o n model of Equation III.43 (see Figures III.10 - III.12); however, the observed motional narrowing e f f e c t i s inconsistent with t h i s function. This inconsistency merely r e f l e c t s the fact that the d i s t o r t i o n frequencies are only approximately d i s t r i b u t e d according to a Lorentzian d i s t r i b u t i o n , and t h i s approximation breaks down when motion i s introduced. The transverse f i e l d r e s u l t s indicate two d i f f e r e n t types of adsorption s i t e s (host and trap s i t e s ) for muonium on the s i l i c a surface. Thus far i t has been concluded that the depolarization of the u + spin for muonium i n the host s i t e s (at low temperatures) i s l a r g e l y due to random anisotropic hyperfine d i s t o r t i o n s , with possibly a small contribution a r i s i n g from the random l o c a l magnetic f i e l d s produced by the hydroxyl protons. To decipher which relaxation mechanism(s) are operating at the trap s i t e s , zero and l o n g i t u d i n a l f i e l d data were taken at the high temperature peak, where muonium i s presumed trapped. The data taken at 25 ± 0.5 K, for sample Si0 2(4) prepared at 110 °C, are shown i n Figure IV.7. A comparison of t h i s data with the low temperature data of Figure IV.4, where the muonium atoms are thought to be primarily i n the host s i t e s , shows two d i s t i n c t l y d i f f e r e n t decoupling behaviors for the two s i t e s . S p e c i f i c a l l y , the relaxation at the host s i t e s i s almost completely decoupled (quenched) for a - 126 -CN R~: 00 CD ^- CM o r~! CD CD CD CD Figure IV.6 Zero and l o n g i t u d i n a l f i e l d asymmetry spectra for muonium on the s i l i c a surface (600 UC preparation), at 16.0 ± 0.1 K. The zero f i e l d data are represented by the square symbols and are compared to data taken at 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 triang l e s to 0.5 G and the diamonds to 1.0 G. - 127 -1 1 1 1 1 1 co Figure IV.7 Zero and l o n g i t u d i n a l f i e l d asymmetry spectra for muonium on the s i l i c a surface (110 "C preparation), at 25.0 ± 0.5 K. The zero f i e l d data are represented by the square symbols and are compared to data taken at four 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 triang l e s to 10.0 G, the diamonds to 25.0 G and the crosses to 45.0 G. The curve through the zero f i e l d data i s a f i t to the data using the zero f i e l d s t a t i c r e l axation function of Equation III.21. - 128 -l o n g i t u d i n a l f i e l d of only 2.0 G, whereas at the trap s i t e s there remains a small unquenched component, even up to 45 G. The curve through the zero f i e l d data i s a f i t of Equation 11.10 to the data, assuming the s t a t i c zero f i e l d r e l a x a t i o n function of Equation III.21. The f i t gave a Chi-square of 83.8 for 28 degrees of freedom, and the c y l i n d r i c a l and planar components were found to be equal to 13 (+1.4/-1.2) u s - 1 and 1.47 (+0.098/0.096) u s - 1 , r e s p e c t i v e l y . The t r i p l e t muonium asymmetry was also f i t t e d and found to be 0.11 (+0.0043/-0.0041). The corresponding data taken at 30 ± 0.5 K, for sample Si0 2(3) prepared at 600 °C, i s shown i n Figure IV.8. A comparison of Figures IV.7 and IV.8 indicates that the relaxation may be more e a s i l y quenched for the 600 "C preparation than for the 110 "C preparation. The curve through the zero f i e l d data i s a f i t of Equation 11.10, assuming the s t a t i c zero f i e l d function of Equation III.21. The f i t gave a Chi-square of 39.1 for 38 degrees of freedom, and the c y l i n d r i c a l and planar d i s t o r t i o n parameters were found to be 7 (+1/-0.9) u s - 1 and 1.04 (+0.075/-0.074) u s - 1 , r e s p e c t i v e l y . The t r i p l e t muonium asymmetry was also f i t t e d and found to be 0.076 (+0.0063/-0.0059). The associated transverse f i e l d relaxation rates, calculated from Equation III.25, are also consistent with the respective transverse f i e l d data for both the 110 °C and 600 UC preparations. These r e s u l t s suggest that the nature of the relaxation i n the trap s i t e s may be a function of the surface preparation. A paramagnetic ion, for instance, which i s somehow neutralized by baking at high temperatures, might explain this data. The most l i k e l y candidate for t h i s i s an F e 3 + ion. In the Cab-0-Sil EH-5 material, i r o n impurities are quoted as being less than 2 ppm [11]; however, recent measurements have set t h i s l e v e l at ~6 ppm [12]. This p o s s i b i l i t y i s discussed further i n section IV.C. - 129 -CM - ; 00 CD ^- CM o ^ CD CD CD CD • • • • Figure IV.8 Zero and l o n g i t u d i n a l f i e l d asymmetry spectra for muonium on the s i l i c a surface (600 "C preparation), at 30.0 ± 0.5 K. The zero f i e l d data are represented by the square symbols and are compared to data taken at 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 trian 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 data using the s t a t i c zero f i e l d r e l axation function of Equation III.21. - 130 -IV.B Muonium on the Surface of Helium Coated S i l i c a Gas adsorption isotherms were measured using ^He at 6.0 ± 0.1 K, concomitant with measurements of the transverse f i e l d muonium rel a x a t i o n rate and the muonium formation p r o b a b i l i t y . The gas deposition was performed according to the procedure given i n Chapter I I . IV.B.l Relaxation Rate Versus **He Coverage at 6 K Mu The transverse f i e l d r e l a x a t i o n rate \ for sample Si0 2(4) prepared at 110 °C and for sample Si0 2(3) prepared at 600 UC i s plotted as a function of the s p e c i f i c volume V g i n Figure IV.9. By d e f i n i t i o n , the s p e c i f i c volume i s the volume of gas, measured at STP, divided by the surface area of the Mu target. From this data, i t i s obvious that the dependence of \ on surface coverage i s a strong function of the sample preparation. In p a r t i c u l a r , the 110 °C data are observed to decrease monotonically with increasing coverage, while the 600 °C data show a peak i n the coverage dependence. Furthermore, t h i s peak has a maximum which i s equal (within the uncertainties) to the trap s i t e r e l a x a t i o n rate for the 600 UC data, shown i n Figure IV.1. Interpretation of the 110 "C data i s straightforward. At zero coverage, the muonium atoms are stationary i n the host s i t e s on the s i l i c a surface. As the coverage i s increased from zero, the p r o b a b i l i t y of a Mu atom i n t e r a c t i n g with the s i l i c a surface decreases because there i s less exposed surface area. Inter p r e t a t i o n of the 600 °C data i s not so t r i v i a l . A model can be developed around the assumption that the baking procedure produces f i s s u r e s i n the surfaces of the s i l i c a grains. These f i s s u r e s are further assumed to act as deep p o t e n t i a l wells which have the same relaxation mechanism as the - 131 -CO 3 3 . 5 3 . 0 2 . 5 -2 . 0 1.5 1.0 -0 . 5 0 . 0 12 - 10 - 8 - 6 - 2 < O w CD O 0 . 0 .1 . 2 . 3 . 4 . 5 V s (10"4 cm) Figure IV.9 Transverse f i e l d muonium relaxation rate at 6.0 ± 0.1 K versus HHe coverage (measured i n terms of s p e c i f i c volume V ) for s i l i c a prepared at 110 °C ( c i r c l e s ) and at 600 UC (squares). The f i l l e d symbols correspond to the r e l a x a t i o n rate and the open symbols correspond to the vapor pressure (r i g h t hand s c a l e ) . Notice that the vapor pressure increases r a p i d l y at monolayer completion. - 132 -host s i t e s . With these assumptions, one can adopt the following model: At zero coverage the muonium atoms are presumed stationary, but i n th i s case the muonium atoms may occupy either the host s i t e s or the deep p o t e n t i a l wells. As the coverage i s increased from zero, the helium i s adsorbed p r e f e r e n t i a l l y into the deep p o t e n t i a l wells. At some c r i t i c a l coverage, which looks to be about 20% of a monolayer, 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 "bridges" over which a muonium atom may di f f u s e r a p i d l y u n t i l i t reaches a "normal" trap s i t e . As the coverage i s Mu increased beyond t h i s point, the behavior mimics the 110 °C data; \ decreases monotonically with increasing coverage because the chance of encountering the s i l i c a surface decreases with increasing coverage. IV. B. 2 Muonium Asymmetry Versus **He Coverage Measurements of the muonium asymmetry were also made as a function of surface coverage at 6.0 ± 0.1 K. The r e l a t i v e asymmetry ( f o r one of the positron telescopes) i s plotted against the s p e c i f i c volume V g of ^He adsorbed onto the s i l i c a surface i n Figure IV.10. The data show that the muonium asymmetry decreases with increasing surface coverage, suggesting that the charge exchange cross section i s s i g n i f i c a n t at the h e l i u m - s i l i c a i n t e r f a c e . Unfortunately, i t i s not possible to draw any conclusions from these data regarding the ori g i n s and mechanics of muonium formation i n the s i l i c a powders ( i . e . , surface or bulk formation), since the precise role played by the adsorbed helium atoms i n the charge exchange i n t e r a c t i o n at the He-Si0 2 i n t e r f a c e i s not as yet known. Two p o s s i b i l i t i e s for this phenomenon are put f o r t h i n Chapter V. - 133 -10 +-> CD c o < > -rH -+-> "a; OH . 0 9 -H . 0 8 H 0 7 H . 0 6 10 h 8 h 2 o CD CO CO CD O 0 . 0 .1 . 2 . 3 V s (10"4 cm) .4 Figure IV.10 Transverse f i e l d muonium asymmetry versus ^He coverage (closed squares) for sample Si0 2(4) prepared at 110 UC. The corresponding vapor pressure data are represented by the open c i r c l e s . - 134 -IV.C Muonium on the Surface of Supported Platinum Catalysts The behavior of muonium on the surface of platinum loaded s i l i c a was Mu studied by measuring the transverse f i e l d muonium relaxation rate \ as a function of temperature, over the temperature range 5 K < T < 100 K, for samples prepared with varying l e v e l s of Pt loading. Since a great deal of information has already been acquired concerning the behavior of muonium on the surface of the 35 A Cab-O-Sil (EH-5) powder, th i s material was selected as the support material for the Pt loaded ca t a l y s t samples. These samples were prepared following the procedures described elsewhere [13], which include reduction i n a mixture of H 2 and He at 500 "C for a period of one hour (see section II.C.3). Four l e v e l s of Pt loading were chosen for these i n i t i a l tests; 0.001%, 0.01%, 0.1% and 1.0%, by weight. A control sample containing no Pt (unloaded), but otherwise prepared following the same procedures, was also measured. Some of the s p e c i f i c s of these samples and the target vessels are given i n Table 11.4(b). As already discussed i n Chapter I I , these samples were evacuated and warmed to a temperature of about 100 "C for a period of ten hours p r i o r to the experiment. This was done to remove physisorbed water from the s i l i c a surface. Since no other surface treatment was done, the supported platinum p a r t i c l e s , which average 10 A or less i n radius, were presumed to be covered with approximately a monolayer of chemisorbed oxygen. IV.C.l Unloaded Silica Support Mu The temperature dependence of the transverse f i e l d r e l a x a t i o n rate \ was measured for the H-reduced control sample, containing no Pt loading. The r e s u l t s of these measurements are shown i n Figure IV. 11, along with the 3 S 4.0 0.0 3.5 -3.0 2.5 -2.0 -1.5 1.0 -0.5 -- 135 -J L • H — r e d u c e d • u n r e d u c e d T T 0. 20. 40. 60. 80 T e m p e r a t u r e (K) Figure IV. 11 Transverse f i e l d muonium re l a x a t i o n rate versus temperature for unreduced and hydrogen-reduced s i l i c a . The f i l l e d c i r c l e s represent the data taken with the reduced material (sample P t ( l ) prepared at 100 U C ) , and the open squares are the data taken with unreduced s i l i c a (sample S i 0 2 ( l ) prepared at 110 °C). - 136 -corresponding re s u l t s for the unreduced material (sample S i 0 2 ( l ) prepared at 110 °C). Comparison of these data suggests roughly the same d i f f u s i o n and trapping behavior for the H-reduced sample as observed for the unreduced sample. Without evidence to the contrary, the two surface s i t e s observed for the reduced sample can be assumed to be of the same nature as the corresponding s i t e s of the unreduced material, but the greatly increased width of the high temperature peak i n the l a t t e r i s i n d i c a t i v e that the hydrogen reduction a f f e c t s the high temperature s i t e s (traps) more than the low temperature s i t e s (host s i t e s ) . The data i n Figure IV.11 are interpreted following the same l i n e of reasoning as presented for the unreduced s i l i c a . The assumption that the muonium atoms are stationary at low (< 8 K) temperatures, for both the reduced and the unreduced s i l i c a , i s based on two observations. F i r s t , the physisorption of helium gas at 6 K sharply Mu decreases X^ as one nears monolayer completion, i n d i c a t i n g that the muonium atoms are outside the powder grains and spending a large portion of the i r Mu l i v e s on the surface. Second, X^ (T) for the H-reduced s i l i c a was found to be t o t a l l y independent of the Pt loading at low temperatures. The absence of a Pt loading dependence at low temperatures implies that during t h e i r l i f e t i m e , the muonium atoms cannot d i f f u s e over distances comparable to the mean separation between Pt p a r t i c l e s . With the loadings that have been studied, the mean separation between Pt p a r t i c l e s corresponds to spacings of 50 or more SiO-H groups. These distances can e a s i l y be spanned by a d i f f u s i n g muonium atom moving at thermal v e l o c i t i e s , for temperatures as low as 5 K; the i r f a i l u r e to do so provides further evidence that the muonium atoms are indeed stationary on s i l i c a surfaces at low temperatures. - 137 -IV.C.2 Platinum Loaded S i l i c a : 0.001% and 0.01% The e f f e c t s of extremely l i g h t (0.001% and 0.01%) Pt loading on the transverse f i e l d muonium relaxation rate i s shown i n Figure IV.12. For the reasons discussed previously, the muonium atoms are again assumed to be frozen i n a surface host s i t e at low temperatures. As the temperature i s Mu increased above about 10 K, X^ (T) decreases, presumably due to motional narrowing, i n the same manner as for the unloaded sample. At higher Mu temperatures, one observes that for both samples X^ (T) continues to decrease monotonically with only the 0.001% loading s t i l l i n d i c a t i n g a s l i g h t hint of a trapping peak. The relaxation rates for these two samples eventually become in d i s t i n g u i s h a b l e , l e v e l i n g off at about 0.5 u s - i . These re s u l t s can be understood by f i r s t r e c a l l i n g that only with a loaded s i l i c a c a t a l y s t can H 2 molecules be dissociated to form atomic hydrogen. I t i s clear also that the high temperature peak observed at about 25 K i n both the unreduced and reduced, unloaded samples, can be expected to be a trap for atomic hydrogen as well as muonium. Bearing t h i s i n mind, one can postulate that t h i s trap s i t e might be f i l l e d , or otherwise neutralized, by the atomic hydrogen generated i n the reduction step of the loading procedure. If t h i s model i s correct, i t would explain the lack of a trapping peak for the Pt loaded samples, i n which large quantities of atomic hydrogen can be generated upon H-reduction, whereas for the case of the unloaded H-reduced sample, where there i s very l i t t l e atomic hydrogen generated, the trapping peak i s quite pronounced. 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 loading, using hydrogen reduction techniques, have been observed i n magnetic s u s c e p t i b i l i t y studies [14] of palladium loaded s i l i c a c a t a l y s t s , as well as i n ESR studies [15] of 3. 3 S 3.0 2.5 H 2.0 1.5 1.0 H 0.5 0.0 - 138 -J I L • 0 . 0 0 1 % Pt • 0 . 0 1 % Pt * 5 KM T r 0 20 40 60 80 100 T e m p e r a t u r e (K) Figure IV. 12 Transverse f i e l d muonium relaxation rate versus temperature for 0.001% ( f i l l e d c i r c l e s ) and 0.01% (open squares) platinum loaded Si0-> - 139 -platinum loaded s i l i c a c a t a l y s t s . In both of these two cases, the observed e f f e c t was a t t r i b u t e d to F e 3 + impurities (10 - 100 ppm) present i n the s i l i c a support, which were reduced to m e t a l l i c i r o n by the hydrogen treatment. As has already been mentioned, the i r o n contamination for the Cab-O-Sil EH-5 material has been measured to be ~6 ppm [12] . Since one might also expect a s i m i l a r e f f e c t (although not as pronounced) when the unreduced s i l i c a i s baked at 600 °C, i t may be possible to a t t r i b u t e the high temperature trap s i t e to i r o n impurities. The beauty of t h i s hypothesis i s twofold; not only does i t explain why, upon hydrogen reduction, the trapping peak disappears (H atoms occupy the traps), but i t also provides a paramagnetic ion which can i n t e r a c t with the Mu atoms through spin exchange processes. Because relaxations a r i s i n g from spin exchange int e r a c t i o n s are i n general not decoupled i n small l o n g i t u d i n a l f i e l d s , this hypothesis provides an explanation for the unquenched component i n the l o n g i t u d i n a l f i e l d data of Figure IV.7. The p o s s i b i l i t y of an i n t e r a c t i o n with an e l e c t r o n i c dipole moment also e x i s t s . IV.C.3 Platinum Loaded S i l i c a : 0 . 1 % and 1 . 0 % Mu In Figure IV.13, the muonium relaxation rate \ i s shown as a function of temperature for platinum loadings of 0.1% and 1.0%. Of p a r t i c u l a r i n t e r e s t are the data obtained for the 0.1% sample. At low temperatures, Mu X^ (T) i s e s s e n t i a l l y the same as for the other four samples, i n d i c a t i n g that the muonium atoms are stationary. As the temperature i s increased Mu beyond about 10 K, X^ (T) experiences a sharp decrease and reaches a minimum Mu at about 20 K. Between 20 K and 30 K, X (T) r i s e s sharply, and thereafter continues to r i s e slowly with temperature toward the value obtained for the - 140 -4.0 3.5 3.0 T 2.5 2.0 IP s —1 1.5 < 1.0 0.5 0.0 i r J L • 0 . 1 % Pt • 1.0% Pt 0 10 20 30 40 50 60 70 T e m p e r a t u r e (K) Figure IV.13 Transverse f i e l d muonium relaxation rate versus temperature for 0.1% ( f i l l e d c i r c l e s ) and 1.0% (open squares) platinum loaded S i 0 2 • - 141 -1.0% sample i n the same temperature region. In a s i m i l a r fashion, the data Mu obtained for the 1.0% sample indicate a s l i g h t minimum i n (T) at about 20 K, which i s followed by a sharp r i s e to a value of about 3.6 us -* at 30 K. Above 30 K, \ M u ( T ) fluctuates with an average value of about 3.5 u s - 1 . The 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 straightforward i f one assumes the p o s s i b i l i t y of a chemical reaction between the muonium atoms and the oxygen-covered platinum surface. Bearing t h i s i n mind one can assume that at 0.1% loading, the muonium atoms do not d i f f u s e fast enough during t h e i r l i f e t i m e to encounter a platinum p a r t i c a l u n t i l the temperature Mu reaches about 30 K, above which \^ (T) continues to increase i n a manner which can be described by the T 1 ^ 2 behavior expected for thermal d i f f u s i o n . One can continue t h i s l i n e of reasoning and assume then that the 1.0% sample contains a s u f f i c i e n t l y high concentration of platinum p a r t i c l e s that the muonium atoms have a very high p r o b a b i l i t y of encountering a platinum atom or aggregate even at extremely low hop rates. Believing this model to be an accurate d e s c r i p t i o n of the relevant physics, one then concludes that the e f f e c t i v e l y constant relaxation rate above 30 K, observed for the 1.0% sample, arises from a chemical reaction of muonium at the platinum surface. The rate for t h i s reaction i s found to be 3.5 ± 0.15 us - 1 which, owing to the i s o t o p i c r e l a t i o n s h i p between muonium and hydrogen, should constitute an upper bound for the reaction rate of atomic hydrogen with oxygen-coated platinum. This i n t e r p r e t a t i o n should be susceptible to experimental tests with zero and l o n g i t u d i n a l f i e l d methods, since (as mentioned i n Chapter III) chemical reactions leading to diamagnetic molecular species do not cause relaxation of the a"1" spin 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 f i e l d . - 142 -CHAPTER V — CONCLUSIONS AND FUTURE DIRECTIONS V.A. Summary of Results The re s u l t s and discussions presented i n th i s d i s s e r t a t i o n have provided information regarding the d i f f u s i o n and trapping of muonium atoms on the surface of fine s i l i c a powders, as well as the nature of the spin r e l a x a t i o n mechanisms involved. Experiments have also provided evidence i n d i c a t i n g charge exchange processes ocurring at the s i l i c a surfaces for sub-monolayer ^He coverages. V.A.I D i f f u s i o n and Trapping Measurements of the temperature dependence of the transverse f i e l d muonium relaxation rate, have indicated the existence of two d i f f e r e n t types of s i t e s (host and trap s i t e s ) for muonium on the s i l i c a surface. The host s i t e s were defined to be the most common. The i n t e r p r e t a t i o n of the data follows accordingly; at low temperatures, two-dimensional d i f f u s i o n and trapping of muonium i s observed, with desorption occuring at high (>100 K) temperatures. This d i f f u s i o n and trapping behavior was further shown to be a strong function of the surface hydroxyl concentration. A three-state model was subsequently developed [1,2], which assumes the relaxation'at every s i t e to be due e n t i r e l y to random l o c a l magnetic f i e l d s . A comparison of the data for high and low surface hydroxyl concentrations was then made using t h i s model. One of the more important observations a r i s i n g from this was that as the surface hydroxyl concentration i s reduced the surface hop rate for muonium i s enhanced at low temperatures (see Figure IV.4). The in t e r p r e t a t i o n of th i s suggests that the surface hydroxyls serve to i n h i b i t - 143 -surface d i f f u s i o n of the muonium atoms. This hypothesis i s substantiated further by a recent hydrogen chromotography study [3] i n which s i m i l a r behavior for hydrogen atoms on s i l i c a surfaces was reported. The host s i t e desorption energy ( i . e . , the a c t i v a t i o n energy for desorption of a Mu atom from a host s i t e on the s i l i c a surface), obtained by summing the host s i t e a c t i v a t i o n energy E q and the energy E ^ , was also found to be strongly dependent upon the sample preparation. From the r e s u l t s given i n Table IV.1, the host s i t e desorption energies for the 110 °C and 600 °C preparations are 275 (+118/-51) K and ~105 K, r e s p e c t i v e l y . Although the assumption of random l o c a l magnetic f i e l d s was l a t e r shown to be somewhat inappropriate, the semi-quantitative understanding afforded by the three-state model proved quite valuable. Studies were l a t e r conducted to ascertain the true nature of the relaxation mechanisms for muonium at both surface s i t e s . The r e s u l t i n g conclusions are summarized i n the following section. • V.A.2 Relaxation Mechanisms Using zero and l o n g i t u d i n a l f i e l d uSR techniques, i t was shown that a dipole-dipole i n t e r a c t i o n (presumably between the muonium atom and the hydroxyl protons) i s i n fact not the predominant relaxation mechanism for muonium on the s i l i c a surface. This was deduced by f i r s t noting that the relaxation for muonium i n the host s i t e s can be e a s i l y decoupled by a l o n g i t u d i n a l f i e l d on the order of a few Gauss, thereby leaving only two p o s s i b i l i t i e s ; random l o c a l magnetic f i e l d s (RLMF) or random anisotropic hyperfine d i s t o r t i o n s (RAHD). Further d i s c r i m i n a t i o n was then done by considering the observed zero and l o n g i t u d i n a l f i e l d long time behaviors i n - 144 -the context of a s t a t i c versus dynamic muonium atom; i t was found that i f one assumes RLMF, the zero f i e l d long time behavior i s inconsistent with a s t a t i c muonium atom, and the l o n g i t u d i n a l f i e l d decoupling behavior i s Inconsistent with a dynamic system. From these arguments then, a random anisotropic d i s t o r t i o n of the muonium hyperfine 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 contributor to the relaxation, e s p e c i a l l y for muonium In the host s i t e s . A theory was developed which describes the time evolution of the u + spin p o l a r i z a t i o n for a completely anisotropic muonium hyperfine i n t e r a c t i o n . The approach taken here involves expanding the hyperfine tensor i n terms of spherical harmonics and using the expansion c o e f f i c i e n t s to parameterize the d i s t o r t i o n . Expressions for the s t a t i c u + spin relaxation functions, both i n zero and "high" external magnetic f i e l d , were then calculated assuming "zero average" Lorentzian and Lorentzian-like 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. By comparing t h i s theory with the data, i t was shown that both a c y l i n d r i c a l d i s t o r t i o n (normal to the s i l i c a surface) and a planar d i s t o r t i o n ( i n the plane of the surface) are required to f u l l y explain the data. The zero f i e l d r e l a x a t i o n function, assuming both c y l i n d r i c a l and planar d i s t o r t i o n s , was used to f i t the low temperature ( s t a t i c l i m i t ) data. The q u a l i t y of the f i t s obtained for both the 110 °C and 600°C preparations was reasonable ( t y p i c a l l y , 1 < x 2/deg.fr. < 2), but not e x c e l l e n t . In these f i t s , the muonium asymmetry was allowed to vary and was found to be consistent with the associated transverse f i e l d data. The r e l a t i v e l y high % 2 values may be d i r e c t l y related to the Lorentzian approximation adopted for the frequency d i s t r i b u t i o n s , which allows i n f i n i t e d i s t o r t i o n s . It i s also important to remember that the - 145 -Lorentzian 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 zero averages. If the actual d i s t r i b u t i o n s have non-zero averages, one would expect an o s c i l l a t i o n superimposed on the r e l a x a t i o n . I t i s , however, impossible to t e l l from the data whether there i s a small amplitude o s c i l l a t i o n , but i f present i t could also account for the high y.2 values. The f i t s of the zero f i e l d random anisotropic hyperfine r e l a x a t i o n functions to the data indicate that the c y l i n d r i c a l component O 2 Q Increases M while the planar component o"22 decreases with increasing surface hydroxyl concentration. Assuming t h i s phenomenon to be due e n t i r e l y 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 electron and the electrons of the neighboring, hydroxyl groups, one can i n p r i n c i p l e extract information regarding the e f f e c t of the induced 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 s i t e symmetry. For instance, the fact that the hyperfine d i s t o r t i o n observed for high hydroxyl concentrations (110 °C preparation) i s shown to have an enhanced c y l i n d r i c a l component and a diminished planar component, as compared to the case of low hydroxyl concentrations (600 °C preparation), suggests that 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 the hydroxyl groups i s repulsive. Although t h i s i n t e r p r e t a t i o n does indeed provide a s a t i s f a c t o r y explanation for the observed behavior, one cannot disregard the p o s s i b i l i t y that this r e s u l t could merely be a manifestation of a combined relaxation i n t e r a c t i o n involving both random anisotropic hyperfine d i s t o r t i o n s and random l o c a l magnetic f i e l d s due to the hydroxyl protons at the surface. Dynamical re l a x a t i o n functions were also calculated by s u b s t i t u t i n g the s t a t i c functions into the strong c o l l i s i o n model [4]. The r e s u l t i n g dynamic functions, for Lorentzian 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 completely independent of the hop frequency at early times, which i s inconsistent with the motional narrowing behavior observed for muonium i n bulk fused S i 0 2 and on the surface of f i n e s i l i c a powders. A modified Lorentzian d i s t r i b u t i o n was therefore tested and found to provide the appropriate motional narrowing behavior. However, since the "correct" form of the d i s t r i b u t i o n cannot be deduced from e x i s t i n g knowledge, the data cannot y i e l d "absolutely c a l i b r a t e d " quantitative values for the hop rates. The nature of the relaxation mechanism(s) at the trap s i t e s was also investigated and found to be p a r t i a l l y consistent with r e l a x a t i o n due to random ansisotropic hyperfine d i s t o r t i o n s . However, l o n g i t u d i n a l f i e l d decoupling measurements have indicated a small component of the relaxation which i s l a r g l y unaffected by the f i e l d s applied. This component i s further seen to be more prominent i n the data taken with the 110 "C preparation than i n that obtained for the 600 UC preparation. The most l i k e l y candidate for t h i s unquenched component i s a spin exchange or perhaps a dipole-dipole i n t e r a c t i o n between the Mu atom and an F e 3 + ion. Moreover, F e 3 + would be reduced to m e t a l l i c i r o n upon hydrogen reduction. The baking procedures employed i n the present study might well produce enough atomic hydrogen to s i g n i f i c a n t l y reduce the paramagnetic content i n the s i l i c a powder which would then be r e f l e c t e d i n the decoupling behavior i n l o n g i t u d i n a l f i e l d . This p o s s i b i l i t y has been further substantiated by the platinum loaded s i l i c a studies, which are summarized i n section V.A.4. V.A.3 Muonium Formation Probability Transverse f i e l d measurements of the muonium asymmetry versus helium coverage (at 6.0 ± 0.1 K) have shown that the muonium asymmetry decreases - 147 -w i t h i n c r e a s i n g surface coverage. Although these data suggest that the charge exchange cross s e c t i o n i s s i g n i f i c a n t at the s i l i c a s u r f a c e s , i t i s impossible to say at t h i s time what r o l e the helium atoms play i n the charge exchange process. One p o s s i b i l i t y i s that the helium atoms are r e l a t i v e l y passive and only serve to cover up the surface, thereby i m p a i r i n g surface muonium formation. 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 data c l e a r l y show that muonium formation i s p a r t i a l l y surface r e l a t e d , as postulated i n Chapter I . There i s , however, another p o s s i b i l i t y which casts the helium atoms i n a more a c t i v e r o l e , where they might act to d i s s o c i a t e the muonium atoms at the surface. Consider, f o r ins t a n c e , the scenario i n which the helium 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 stopping u + , are able to capture the e l e c t r o n s of newly formed muonium atoms. This process would indeed leave the muons i n a diamagnetic s t a t e , thereby removing them from the precessing muonium ensemble. Because of these two p o s s i b i l i t i e s , one can say nothing about the o r i g i n s of muonium formation, since one cannot d i s t i n g u i s h between muonium formed at the s i l i c a surface and subsequently d i s s o c i a t e d , or muonium which i s formed i n the grains and d i f f u s e s to the surface where i t i s then d i s s o c i a t e d . V.A.4 Catalytic Chemistry These i n v e s t i g a t i o n s were al s o extended to the study of the i n t e r a c t i o n s of muonium w i t h the surface of a s i l i c a - s u p p o r t e d platinum c a t a l y s t . From the r e s u l t s obtained f o r the temperature dependence of the transverse f i e l d 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 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 platinum surface. These experiments have al s o provided i n f o r m a t i o n concerning the nature of the - 148 -trapping 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 that the r e l a x a t i o n at the trap s i t e s i s completely n e u t r a l i z e d f o r small (~0.01% Pt) platinum loadings, whereas f o r an unloaded sample (0% P t ) , the trapping peak i s q u i t e pronounced. This e f f e c t was found to 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 re d u c t i o n methods employed i n the preparation of the platinum loaded s i l i c a c a t a l y s t . 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 , using hydrogen re 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 stud i e s of metal loaded s i l i c a c a t a l y s t s . In these cases, the observed e f f e c t was a t t r i b u t e d to F e 3 + i m p u r i t i e s , which were reduced to m e t a l l i c i r o n by the hydrogen treatment. The u t i l i t y of adopting the hypothesis that the trap s i t e i s an F e 3 + i o n not only a f f o r d s one w i t h an explanation why, upon hydrogen r e d u c t i o n , the trapping peak disappears, but i t a l s o provides a paramagnetic i on which can i n t e r a c t w i t h the muonium atoms through s p i n exchange or 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 ) processes. This hypothesis can thus account f o r the unquenched r e l a x a t i o n component observed at the trap s i t e f o r the sample prepared at 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 at 600 °C, where enough atomic hydrogen may be generated by the baking procedure to p a r t i a l l y n e u t r a l i z e the F e 3 + c e n t e r s . V.B. Future Di r e c t i o n s The work presented i n t h i s d i s s e r t a t i o n has provided the ground work f o r many new i n v e s t i g a t i o n s , both experimental and t h e o r e t i c a l , ranging from surface c a t a l y s i s to surface p h y s i c s . A few of the more immediately a c c e s s i b l e avenues are discussed here. - 149 -V.B.I. T h e o r e t i c a l The spin r e l a x a t i o n theory for a random anisotropic hyperfine i n t e r a c t i o n (RAHD), derived i n Appendix I, was developed to explain the data obtained for muonium i n bulk fused quartz and for muonium on the surface of fin e s i l i c a powders. However, only a few appropriately selected d i s t o r t i o n symmetries were considered, and the functions were derived only for the zero and "high" external f i e l d l i m i t s . Further c a l c u l a t i o n s assuming a non-zero u>2^ component should therefore be done, and the low external f i e l d l i m i t investigated, before a complete understanding of the s t a t i c r e l axation functions for a random anisotropic hyperfine d i s t o r t i o n can be obtained. Moreover, these functions should be derived allowing for non-zero averages for 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 , since some of the data do seem to exhibit small amplitude o s c i l l a t i o n s which are superimposed on the r e l a x a t i o n . The dynamic zero f i e l d RAHD model discussed i n Chapter III was found to be unsuitable for f i t t i n g the data because i t exhibits no early time dependence on hop rate ( i . e . , no motional narrowing), while the data exhibit motional narrowing with increasing hop rate. This behavior was found to be a manifestation of assuming Lorentzian 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 , for the d i s t o r t i o n parameters. More thought must therefore be given to the choice of d i s t r i b u t i o n function so that a proper motional narrowing theory can be developed. The modified Lorentzian d i s t r i b u t i o n , discussed i n Chapter III does have the required features ( f i n i t e second moment and an exponential-like i n i t i a l decay), and could be used for this purpose. Once thi s i s done, extensions can be made to a multi-state model for d i f f u s i o n i n the presence of traps. - 150 -V.B.2 Experimental This research has l a i d the foundations for future experimental studies along two complimentary paths; one involving the study of chemical reactions of muonium with various reactants on the s i l i c a (or other) powder surfaces and another concerning the in t e r a c t i o n s of p o s i t i v e muons and muonium with macroscopic surfaces. Consider f i r s t the study of muonium reactions. In t h i s l i n e of study, the s i l i c a powder plays the role of an i n e r t substrate which simultaneously provides a way of producing muonium i n vacuum as well as an Inert surface on which reactants can be s t a b i l i z e d . There are two d i s t i n c t aspects of t h i s study which should be considered. The f i r s t of these aspects i s the reaction of muonium with physisorbed molecules, such as ethylene and oxygen, with the prime goal being to measure and compare the two-dimensional and three-dimensional reaction rates. The second aspect concerns the reaction of muonium with metal loaded c a t a l y s t s . Because of the enormous i n t e r e s t i n hydrogen c a t a l y s i s with metal loaded c a t a l y s t s , the study of a hydrogen-like probe such as muonium i n t e r a c t i n g with a metal loaded c a t a l y s t should provide an excellent opportunity for uSR to make a s i g n i f i c a n t contribution to a r a p i d l y expanding f i e l d . An immediate benefit that can be forseen i s that muonium can provide information about the intermediate reactions that occur at short times, which are currently not observable by any other technique. Preliminary studies of both of these aspects have already been made, with the r e s u l t s obtained for muonium on the surface of a platinum loaded c a t a l y s t being reported i n t h i s d i s s e r t a t i o n . A l l of the studies to date, however, were made i n a low (< 10 G) transverse magnetic f i e l d . If one's - 151 -goal i s to measure the reaction rate of muonium with a reactant s t a b i l i z e d on the s i l i c a surface, i t w i l l be necessary to d i s t i n g u i s h between re l a x a t i o n due to chemical reactions and relaxation due to the interactions of muonium with the s i l i c a substrate i t s e l f . This could be accomplished by repeating these measurements i n zero and l o n g i t u d i n a l f i e l d . The observed functional dependence of the muonium formation p r o b a b i l i t y on the f r a c t i o n a l surface coverage begs further i n v e s t i g a t i o n aimed at determining the or i g i n s of muonium formation i n these powders. Indeed, the two p o s s i b i l i t i e s mentioned e a r l i e r for the role played by the physisorbed helium atoms are both very i n t e r e s t i n g , i f only from an atomic physics point of view. To determine the true role played by the physisorbed atoms i n the charge exchange process, i t w i l l be necessary to repeat these experiments using d i f f e r e n t adsorbates. It would also be i n t e r e s t i n g to a l t e r the substrate i n a systematic way, such as changing the surface hydroxyl concentration. Now consider the p o s s i b i l i t y of i n v e s t i g a t i n g the i n t e r a c t i o n s of u + and muonium atoms with macroscopic c r y s t a l l i n e surfaces. A p a r t i c u l a r l y i n t e r e s t i n g topic to lead of these investigations arises from a recent positron experiment [5]. This experiment shows that when e + of keV energies are implanted into i o n i c single c r y s t a l s , they are reemitted i s o t r o p i c a l l y from the s o l i d s with a continuum of energies having a maximum approximately equal to the band gap energy ( t y p i c a l l y ~10-20 eV). Positronium (Ps) was also observed to be emitted. Results also indicate that the emission of both e + and Ps i s associated with positronium d i f f u s i n g to the surface of these c r y s t a l s where there exists some branching r a t i o for e + as opposed to Ps emission. An explanation for this phenomenon has been proposed which - 152 -assumes a f i n i t e concentration of acceptor s i t e s at the c r y s t a l l i n e surface, such that the e + i s Auger-emitted when the Ps electron combines with such a s i t e . Assuming that one can draw c e r t a i n analogies between the behavior of positrons and p o s i t i v e muons, and that the mechanism(s) responsible for the re-emission of positrons would also be involved i n the analogous phenomenon for p o s i t i v e muons, th i s l i n e of research would have the added benefit that one can draw guidance from e a r l i e r positron experiments. In addition, t h i s research could conceivably lead the way to producing an ultra-low energy (0-10 keV) u + beam. A detailed d e s c r i p t i o n of the physics involved, along with a possible test case experiment, i s given i n Appendix I I . - 153 -APPENDIX I — THE TIME EVOLUTION OF THE u+ SPIN POLARIZATION IN MUONIUM FOR A GENERALLY ANISOTROPIC HYPERFINE INTERACTION In t h i s Appendix, 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 p o s i t i v e muon i n the n e u t r a l atomic s t a t e (muonium, u +e~) i s discussed f o r the case of 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 hyperfine i n t e r a c t i o n . The asso 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 sel e c t e d symmetries. Because the research presented i n t h i s d i s s e r t a t i o n i s p r i m a r i l y concerned w i t h muonium on the surface of powders, where there i s no w e l l defined c r y s t a l o r i e n t a t i o n , the d i s c u s s i o n w i l l be d i r e c t e d a c c o r d i n g l y . AI.A Observables - C r y s t a l and Detector Frames There are two frames of reference associated w i t h s o l i d s t a t e \iSR experiments, the c r y s t a l frame (designated by the coordinates x', y', z') and the detector frame (x, y, z ) . Observations are of course made i n the detector frame; however, the time e v o l u t i o n of the observables i s g e n e r a l l y more r e a d i l y described i n terms of the c r y s t a l frame, where the symmetries of the problem can be e x p l i c i t l y i n c orporated. These two reference frames are r e l a t e d by the Eu l e r angles (a,B,y) through the second rank r o t a t i o n tensor R ( Q ) = e x p ( - i J z a ) • e x p ( - i J y B ) • e x p ^ i J j ) and the inverse r o t a t i o n tensor ( A I . l ) R ( Q ) = e x p ( + i J z , Y ) • e x p ( + i J y I S ) • exp(+iJ z,a) where the J . are the 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 generators [1 ]. With - 154 -these d e f i n t i o n s , one can define the unit vector transformations and inverse transformations as x^ = R(Q)»x^ and x^ = R(Q)»x^, re s p e c t i v e l y . AI.A.l Spin Relaxation Functions As for any observable, the time evolution of the u + spin p o l a r i z a t i o n for a generally anisotropic muonium hyperfine i n t e r a c t i o n i s , i n the Heisenberg picture, given by the equation P (t) = exp[i (2 i t/h)Htl P expl"-if 2rc/h)Htl ~°P ~°P ' (AI .2) = 2 exp[i(2n:/h)Ht] s£ p exp[-i(2n;/h)Ht] where H i s the spin Hamiltonian of the system and S^ p i s the muon spin operator. The time evolution of the spin p o l a r i z a t i o n for an i n d i v i d u a l muon, i s then represented by the second rank time autocorrelation tensor defined as g(t) = i TrlP (t) P } § 4 l~op ~op< ( A I # 3 ) = T Tr{exp[i(2Tr/h)Ht] P Q p exp[-i(2Tc/h)Ht] P Q p} Defining the eigenstates of the Hamiltonian as \tyj> with the corresponding eigenenergies (h/2-it)w^, and r e c a l l i n g the d e f i n i t i o n of a trace of an operator product, Equation AI.3 can be written as g(t) = \ X e ^ i i o ^ t ) P o p K j X + j l P o p |*±> (AI.4) where we define the t r a n s i t i o n frequencies = (oo^ - w..). By separating the expression given i n Equation AI.4 into the diagonal and off diagonal parts one obtains - 155 -» v ' 4 £ v i ' ~op | Y i ~op | N l 5 j + V Relexpflw. . t l <cp.| P |(|».X<|>. I P \<\> •> } L K V i j ; ^ l ' ~op | M j ~op | Y j J By d e f i n i t i o n , <<|>. I P |<|> .> = x P X. + y P y. + z P Z ., such that, i n terms 3 ' ^ i 1 ~op | S j i j J i j i j ' of i t s cartesian components, §(t) can be written as g ( t ) - l H S (P__) 2 + yy ( P ^ ) 2 + « ( P Z I ± ) 2 + (xy + y x j P ^ P ^ + ( S + ^KAi + & + ^ ) p i i p i i } I f r \ i-*-* , X ,2 •+-»• , V ,2 -»-*• , z ,2 + 2 1 | c 0 8 K j t H - l * ± j l + y y | p l j l + " 1 ^ 1 (AI.6) + + y^)Re(P X.P^) + ( S + l x ) R e ( p X . P Z * ) + & + ^y)Re(P*.P Z*)} - T ^ 8 i n ( i U l j t ) { ( x y - yxllmCP^py.) + (xz - zx) I m ^ .P± .) + (yz - zy)lm(py.P 1.)} Since §(t) i s a second rank tensor i n three-dimensional space, i t can be expanded i n terms of (1) the second rank unit tensor U , (2) a set of traceless antisymmetric second rank tensor, constructed from the dot product of the Levi-Cevita tensor £ and the detector frame sph e r i c a l vectors E \ and — ~m 2 (3) the traceless symmetric second rank detector frame spherical tensors E^. The r e s u l t i n g expansion gives 1 1 1 2 2 g(t) = U - - T g, E • £ + y g„ E (AI.7) s v ' B00 » nr L ,1m ~m = L _B2m »m v /2 m=-l m=-2 The transformation to spherical tensors i s made i n order to provide a more convenient set of vectors and tensors for the r o t a t i o n a l transformations - 156 -between the c r y s t a l frame and the detector frame. In terms of t h e i r cartesian components (x, y and z ) , the spherical covariant vectors E^ " and ~m l m r \\* the corresponding contravariant vectors E = E are given as ~ >-~m S} - ; lU = - i ( x + i y ) /2 /2 E J - - i z ; E 1 0 = it (AI.8) /2 /2 By d e f i n i t i o n of the Levi-Cevita operator §, the antisymmetric s p h e r i c a l tensors can be written as S i • § = = U y z - zy) - i ( z x - x z j j /2 Ej . i = - i [(xy - y£)] (AI.9) E \ • £ = - z [(yz - zy) + - xz)] /2 The symmetric spherical tensors can also be written i n terms i f th e i r cartesian components with the r e s u l t E Q = /2/3 [- zz + j (xx + yy)] g ± 1 - ± J [ ( x z + z x ) + i ( y z + z y ) l (ALIO) 2 1 r ( , \ _ c •*"*• A n 1 ± 2 = " 2 L<- X X ~ y y ^ + A ^ x y + At this point i t i s convenient to define a few i d e n t i t i e s . From Equation AI.9, one can write z - i (*J) = - i ( £ 1 0 ) X = " I ( E 1 - E 1 . ) - - ( E U - E 1 " 1 ) ( A I . l l ) / J 1 ~ _ 1 /2 ~ J - I ( E j + E \ ) = I ( E 1 1 + E 1 - 1 ) /2 /2 - 157 -Combining the i d e n t i t i e s given i n Equation A I . l l with the d e f i n i t i o n s of Equation AI.10, the second rank symmetric spherical tensors can be constructed from the spherical vectors 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 covariant and contravariant spherical vectors, which now define the physical space, obey the following cross product r e l a t i o n s ( E 1 0 * E 1 0 ) = ( E U x E 1 1 ) = ( E ^ x E 1 " 1 ) = 0 (AI.13) (E x E ) = i z x ( - ) ( x + xy) = _{xy + x) = E /2 /2 (E X E ) = i z x ( - ) ( x - iy) = ±(iy - x) = -E /2 /2 (E X E J = (x + xyj x [x - xyj = - i z = -E By combining Equations AI.6 and AI.7, and u t i l i z i n g the r e s u l t s of Equations AI.8 - AI.12, the nine detector frame components S-^Ct) °f the relaxation tensor are defined as W = T 2 = »<t> = I [ ( p * ) 2 + ( p ^ ) 2 + ( p ' ) 2 ] + 2 j _ c o s ( W . . t ) [ | P X . | 2 + | P y . | 2 + |P Z.| 2]} (AI.14) 8 1 0<t) = " Z ( £ 1 0 ' §) : gCO = " Z I s i n ( W t ) [ l m ( p X P^*)] (AI.15) 10 /2 " /2 i<j 1 3 1 J 1 3 - 158 -g 2 0 ( t ) - E 2 0 : g(t) = j Sift { I [ ^ [ ( P ^ ) 2 + ( P j ± ) 2 ] " ( P Z 1 ± ) 2 ] Z U * ± z i i i i i i (AI.17) + 2 I c o s ( U t ) [ I ( | P X | 2 + |Py | 2) - |P*| 2]} i<j J W « - i 2 ± 1 : § ( t ) " ' i f ? ! t p i i p i i * l p i i p u ] X ± 2 X cos(o,..)[Re(p X.P Z*) ± iRe(p y.P Z*)]} (AI.18) g 2 ± 2 ( t , - E 2 ± 2 : S ( t ) - - i ( I I [ ( P * / - ( P ^ ) 2 5 I P ^ P ^ J x (AI.19) . j ^ . t i t l P - . , 2 - IPj.l 2] T I R e ( P ^ ) ] ! Because a l l observations are made i n the detector frame, whereas the symmetries are defined i n the c r y s t a l frame, i t i s necessary to r e l a t e the detector frame observables S - ^ C t ) t 0 the c r y s t a l frame observables 8 ^ m ( t ) -This i s done v i a the r o t a t i o n operation, such that 8 L M ( t ) " I 8 L m ( t ) * £ ™ ( A I ' 2 0 ) m where R^*^(Q) are the matrix elements of the r o t a t i o n tensor R(Q), defined mM » i n Equation A I . l . With these d e f i n i t i o n s , the dynamics of the u + spin i s found to involve nine observable detector frame relaxation functions, which are constructed from the nine detector frame components 8 l m ( t ) of the r e l a x a t i o n tensor - 159 -[2,3]. S p e c i f i c a l l y , there are three l o n g i t u d i n a l (1) relaxation functions, i n the d i r e c t i o n of the l o c a l f i e l d , defined as *1 ( t ) = g 0 0 ( t ) " / 2 / 3 8 2 0 ( t ) g * c ( t ) = l m { g n ( t ) } + Re{g 2 1(t)} (AI.21) g ^ S ( t ) = - R e { g n ( t ) } + Im{g 2 1(t)} three coplanar-transverse (ct) relaxation functions, which are i n the plane defined by the f i e l d d i r e c t i o n and the incoming muon spin p o l a r i z a t i o n g ^ t ( t ) = - Im{g u(t)} + Re{g 2 1(t)} 8 c t ( t ) = 8 0 0 ( t ) + \ / 2 7 T 8 2 0 ( t ) " R e{s22(t)>" (AI.22) g°*(t) = - /T7T g 1 Q ( t ) - im{g 2 2(t)} and three perpendicular-transverse (pt) relaxation functions, which are directed perpendicular to both the f i e l d and the incoming muon spin p o l a r i z a t i o n . g£ t(t) = R e { g n ( t ) } + lm{g 2 1(t)} g ^ ( t ) = /1/2 g 1 Q ( t ) - Im{g 2 2(t)} (AI.23) 8 p t ( t ) - 8 0 0 ( t ) + \ / 2 / 3 8 2 0 ( t ) + R e { g 2 2 ( t ) } - 160 -It i s convenient to work i n the vector space spanned by the detector frame eigenfunctions Im^m^ of the unperturbed (vacuum) muonium i s o t r o p i c hyperfine Hamiltonian. Defining the axis of quantization to be along the magnetic f i e l d , one has |1> = |+,+> ; |2> = |-,-> (AI.24) |3> = s|+,-> + c|-,+> ; |4> = c|+,-> - s|-,+> where c = cos(x/2), s = sin(\/2) and X = a r c s i n [ l / ( l + x 2 ) 1 / 2 ] . The dimensionless quantity x = |B | /BQ i s the s p e c i f i c f i e l d parameter where B Q (= 1585 G) i s the hyperfine f i e l d for i s o t r o p i c muonium i n the ground state i n vacuum. The labels of the second and t h i r d eigenfunctions given i n Equation AI.24 have been interchanged with respect to the standard notation [4-6] i n order that the hyperfine Hamiltonian can be p a r t i a l l y written down i n block diagonal form. In general the eigenf unctions |(|>j> of a s p e c i f i c hyperfine Hamiltonian can be expressed as l i n e a r combinations of the i s o t r o p i c muonium basis vectors |k> given i n Equation AI.24, such that one may write H|(|>.> = S j d , ^ = l± I c i k |k> (AI.25) U t i l i z i n g this r e s u l t , one can then write <(b. IP U.> = [ x P X. + y P y. + z P Z . l = V c* c . <k|P kl> ^ l ' ^ o p ' ^ j i j i j i j J £ i k jk '~op (AI.26) + I [ c ! i c - i < k l p l*> + c * i CM, < k l p !!>*] ^ '~op1 i l jk '~op' J Solving for the three detector frame p o l a r i z a t i o n components, one then has P X j - co s ( \ / 2 ) [ A 1 3 + A _ J + s i n ( \ / 2 ) [ A 2 3 - A ^ P y. = i { c o s ( \ / 2 ) [ B 1 3 - B 2 J - s i n ( \ / 2 ) [ B 2 3 + B u ] } (AI.27) P i j = CC12^ " c o s M [ C 3 4 ] + s i n ( x ) [ A 3 J - 161 -c..c. 1+ c ^ c . , B, . = c.,c.,- c ^ c . and I^JL i k j l 11 jk-" k l L i k j l i i j k J r * * i C. = c..c. - c. n c „ . Note that i n zero f i e l d , \ = n/2. k l L i k jk i l j l J AI.B The Spin Hamiltonian f o r Isolated Muonium The spin Hamiltonian for an i s o l a t e d muonium atom i n a magnetic f i e l d B, assuming a generally anisotropic muonium hyperfine i s given by H = H 2 6 6 + H h f (AI.28) = (h/2Tc)(y S 6 - y ^ ) ' B + (h/2Ti) W : ( s 6 1 ' ^ 1 e ~op u ~op y ~ « ^~op ~op y where Y E and Y u a r e the respective magnetogyric r a t i o s of the electron and e u the muon, S and S are the spin operators and W i s a second rank tensor ~op ~op » ( i n three-dimensional space) representing the contact hyperfine i n t e r a c t i o n . Although t h i s Hamiltonian i s discussed i n some d e t a i l elsewhere [7], a somewhat d i f f e r e n t but equivalent evaluation i s given here which f a c i l i t a t e s the c a l c u l a t i o n of the time evolution of the u + spin p o l a r i z a t i o n , for a generally ansisotropic muonium hyperfine i n t e r a c t i o n , and the corresponding zero, l o n g i t u d i n a l and transverse f i e l d spin relaxation functions. AI.B.l Evaluation of the Hyperfine Term Because the i s o t r o p i c muonium basis given i n Equation AI.24 i s defined with respect to the detector frame, i t i s easiest to evaluate the hyperfine Hamiltonian i n th i s frame. Since W i s a second rank tensor i n three dimensional space, i t can also be expanded i n terms of (1) the second rank unit tensor U, (2) a set of traceless antisymmetric second rank tensors, - 162 -constructed from the dot product of the Levi-Cevita tensor £ and the detector frame sph e r i c a l vectors E^, and (3) the traceless symmetric second 2 rank detector frame spherical tensors E^. However, because the hyperfine tensor W involves only dipole-dipole and contact i n t e r a c t i o n s , both of which have r e f l e c t i o n symmetry, the antisymmetric part of the hyperfine tensor i s i d e n t i c a l l y zero ( i . e . , W^Q = w +^^ = 0 ) . The r e s u l t i n g expansion gives 2 2 W = w„„ U + I w0 E (AI.29) » 00 » L „ 2m «m m=-2 where the symmetry of the anisotropic hyperfine tensor W i s defined by the c r y s t a l frame c o e f f i c i e n t s coT , which are related to the detector frame Lm 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 transformation WLM " I "Lm * £ ™ < A I- 3 0> m The spin operators for the electron and the muon can be written i n terms of the contravariant spherical vectors as a + + •*• S = S x + S y + S z ~°P X y Z (AI.31) = I S a ( E 1 1 - E 1 _ 1 ) + I S a ( E U + E 1 _ 1 ) - i S a E 1 0 /2 x ~ / I y ~ z ~ where a = {p,,e}. By combining l i k e terms i n and E^ " \ one obtains S« = L (E 1 1) s « - I (E 1 _ 1) S? - i ( E 1 0 ) S a (AI.32) ~op J2 • /2 Z where we define S a = ( S a - i S a ) and S? - ( s a + i S a ) (AI.33) - x y' + ^ x y - 163 -The three operators s", and completely define the vector operator and turn out to be more convenient. They obey the commutation r e l a t i o n s [S^, Sj] - S° ; [s«, S«] = -S« ; [s«, S«] = 2S« (AI.34) U t i l i z i n g the operations given i n Equation AI.13, one obtains the cross product of the electron and muon spin operators, namely ( S e x S ^ ) = { i ( E U x E 1 _ 1 ) S e Sj + ± ( E U x E 1 0 ) S 6 v~op ~op k2 v ~ ~ ' — + / T ~ z + I ( E 1 _ 1 x E U )S? - I ( E 1 _ 1 x E 1 0 ) S f (AI.35) / 2 + - ( E 1 0 x E 1 1 ) S e - - ( E 1 0 x E 1 _ 1 ) s e S^ } Using the i d e n t i t i e s for the cross products of the s p h e r i c a l vectors given i n Equation AI.13, one can rewrite Equation AI.35 as ( S e x S*) = {i (-E 1 0 ) S 6 Sjj + 1 (-E 1 1 ) S 6 v~op ~op 2 ^ ~ + /o" z / 2 + i (+E 1 0 )S? - I (-HS 1 _ 1)S? (AI.36) 2 v ~ y + — y^ " ' + Z + - ( + E U ) S 6 - I ( - E 1 _ 1 ) s e S^ } / I ~ Z " /2 ~ Z + This cross product r e l a t i o n i s not used i n the present work because the antisymmetric part of the hyperfine tensor W i s i d e n t i c a l l y zero. At t h i s point we have a set of convenient spin operators, expressed i n terms of the c r y s t a l frame spherical vectors, and so we can rewrite the hyperfine Hamiltonian i n terms of these operators. F i r s t consider the i s o t r o p i c part of the hyperfine Hamiltonian. In terms of the s p h e r i c a l vectors, the second rank unit tensor i s written U = - EJ EI + (E.1 E 1 1 + E 1, E.1) (AI.37) » ~o ~0 v ~ l ~-1 ~—1 ~1 Thus the trace part of the hyperfine Hamiltonian becomes - 164 -H ^ = fh /2 iO w n n [- EJ E i + (E.1 E 1 . + E 1 . fih] : ( S 6 ) (AI.38) 00 v 1 00 L ~0 ~0 K~l ~ - l ~—1 ~1 ; J v~op ~op ; and s u b s t i t u t i n g the expressions for the electron and muon spin operators given i n Equation AI.32 into Equation AI.38, y i e l d s the r e s u l t H J J = (h/2„) W q o [S* S£ + I (S* + Sf s £ ) ] (AI.39) By combining the d e f i n i t i o n s of Equation AI.12 and the expressions f o r the electron and muon spin operators, given i n Equation AI.32, the f i v e terms of the symmetric traceless part of the hyperfine Hamiltonian become H22 " ~ 7 ( h / 2 l t) W22 ( S- S-) (AI.40) H21 = I ( h / 2 L T) W21 ( Sz S - + S - S z ) (AI.41) H_J = /273(h/2 7 l) w 2 Q [- S* S^ + (S* + Sf S J ) ] (AI.42) H 2 - l = " I (h/2^ W 2 - l ^ S z S i + S+ S z ^ ( A I * 4 3 ) H2-2 = " J (n/27t) W2_2(S^ sj) (AI.44) The r e s u l t of operating on the space spanned by the i s o t r o p i c muonium eigenvectors |k> with the operators s", S^ and s" can be e a s i l y understood by f i r s t considering th e i r e f f e c t on the single p a r t i c l e eigenkets, |j,m>a (equal to ly, -j >^ or |y, ~ >^ for spin 1/2). Thus one can write [1] S a |j,m> = m |j,m> (AI.45) z 1 -" a a 1 a -1 S" |j,m> = / j ( j + l ) - m (m +1) = |j,m+l> ; for m - y i - a a a a a (AI.46) ; for m = y 165 -0 +1 a 2 (AI.47) ; for m = i a 2 Returning now to the two-spin eigenstates of Equation AI.24, one can define the operations H> • z z z z s e s^ 1 s e z -s e z -- z - z 1> 3> 3> = - T |3> 1> = |2> 3> = 0 1> = 0 3> = 0 z z 1> = £ |3> - | |4> 3> = - T |2> T |3> + f l 4 > T |2> z z z -z -- z - z 2> 4> 1 |2> 4 1 4 ti*> 2> - 0 4> = 0 2> = |1> 4> = 0 2> = 0 4> = - £ |2> 2> = 0 4> = £ |2> (AI.48) (AI.49) (AI.50) (AI.51) (AI.52) - 166 -s e S J z + s e s ; z + s f + z s f S^ + z s ! s j s f s j 1> 3> 1> 3> f |1> sc|3> - s |4> ; |4> = c |3> - sc|4> 1> -3> = 1> = 0 3> = sc|3> + c 2|4> s f S J S e |2> = - -I |3> z + 1 2 1 |4> S e |4> z + 1 Sf S^ I2> + z ' S? S^ |4> + z 1 S® S^ |2> = f 1 ° 7 |3> + f l4> 2 2 *|1> |2> = |4> = - s |3> - sc|4> (AI.53) (AI.54) (AI.55) (AI.56) With these operations defined, the matrix elements of the generally anisotropic hyperfine Hamiltonian i n the i s o t r o p i c muonium representation can be derived. We begin with the i s o t r o p i c part HQQ and ca l c u l a t e the matrix elements to obtain w, 00 1 0 0 0 0 1 0 0 0 0 0 0 -( l - 4 s c ) 2 ( c 2 - s 2 ) 2 ( c 2 - s 2 ) - ( l + 4 s c ) _ (AI.57) In a s i m i l a r manner, the matrix elements of the f i v e terms of the symmetric traceless part of the generally anisotropic hyperfine Hamiltonian can be calculated giving - 167 -*22 hf 21 2-2 0 0 0 0 (h/2n) 4 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (h/2Tt) W21 4 0 0 -(c+s) -(c-s) (c+s) 0 0 0 _ (c-s) 0 0 0 -1 0 0 0 h /2/3 W20 0 -1 0 0 2% 4 0 0 (l+2sc) (c -s ) 0 0 r 2 2 i (c -s j ( l - 2 s c ) _ 0 0 -(c+s) -(c-s) -(h/2n) w 2 - l 4 0 0 0 0 0 (c+s) 0 0 0 (c-s) 0 0 0 -1 0 0 (h/2u) w 2 - 2 4 0 0 0 0 0 0 0 0 0 0 0 0 (AI.58) (AI.59) (AI.60) (AI.61) (AI.62) AI.B.2 Evaluation of the Zeeman Term To evaluate the Zeeman Hamiltonian, i t i s again easier to work i n the - 168 -detector frame with the magnetic f i e l d B directed along the z-axis. R e c a l l i n g the expressions for the electron and muon spin operators given i n Equation AI.32, the Zeeman Hamiltonian becomes (AI.63) rzee H-~ = (h/2*) [ y e S* - Y ( 1 Sjj] |B| Operating on the eigenstates of the i s o t r o p i c muonium basis, given i n Equation AI.24, one has for the electron spin operator S_ |1> "+ J |1> S 6 |2> = - i |2> z 1 2 1 se |3> = ( i - s 2 ) | 3 > - sc|4> ; S* 14> = ( I - c 2) 14> - sc13> and for the muon spin operator s z |1> - + J |1> |3> = (|- c 2)|3> + sc|4> (AI.64) |2> = - j |2> S» |4> = ( I - s 2)|4> + sc|3> With these operations defined, the Zeeman Hamiltonian i n the detector frame i s given by H Z e e = (h/2*) % t u 0 0 0 -0), 0 Mu 0 0 0 0 0 0 a)., -a -scfy +V ) Mu *•1 e ' p/ e 'ii (AI.65) 1 2 2 2 2 where = - (y_ - Y,J |£|» a = ( y o s + y, c )|B| and b = ( Y e c + y s^ )|B|. AI.C Isolated Muonium i n Zero F i e l d In zero f i e l d , one has s = c = 1//2. In this case, one can transform to the c r y s t a l frame and write the t o t a l Hamiltonian 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^m> namely H = (h/2it) \ H o ~ U 2 o ) 2-2 -w22 (wnrT^n) 00 ~20' " w 2 - l - u ) 2 1 oo, 21 2-1 K o + 2 u ) 2 o ) 0 0 0 0 -3w 00 -I (AI.66) where we have defined oo^ = /2 with the exception that UJ^Q = /2/3 O^ Q • Deriving the c r y s t a l and detector frame relaxation functions for a s p e c i f i c hyperfine Hamiltonian i s i n general straight forward. To calculate the relaxation due to some random d i s t o r t i o n of the muonium hyperfine i n t e r a c t i o n , one averages over the associated OJt 's. However, since unr. i s Lm UU i n general quite large and unobservable due to timing l i m i t a t i o n s , one can ignore the o s c i l l a t o r y terms containing WQQ, which simply implies ignoring the s i n g l e t component of the muonium ensemble, and only take an average over the appropriate s ' e a c n °^ which have some d i s t r i b u t i o n ^2TXSw2TS? * With the data presented i n this d i s s e r t a t i o n , i t i s i n general not possible to discriminate between an d i s t o r t i o n as opposed to contributions. This being the case, the possible contribution of an component to the relaxation of the u + spin p o l a r i z a t i o n i s omitted i n the remaining discussions. In the case of powders i n zero applied f i e l d , the axis of symmetry of the hyperfine d i s t o r t i o n for each muonium atom Is oriented randomly. When taking an ensemble average one averages over the Euler angles which, by d e f i n i t i o n of sphe r i c a l harmonics, forces a l l of the detector frame 8 L m ( t ) to zero except g^^Ct). Because of th i s averaging, only three of the nine detector frame observable r e l a x a t i o n functions, given i n Equations AI.21 -- 170 -c sc s s AI.23, survive; g-^(t), g c {.(t) and g p t ( t ) . Furthermore, these three functions are i d e n t i c a l and equal to g^^Ct), which simply r e f l e c t s the fact that the spin relaxation function i s i s o t r o p i c i n the detector frame. With t h i s understanding, the general form of the zero f i e l d spin relaxation function, due to random anisotropic hyperfine d i s t o r t i o n s , for the case of powders, i s then given by the equation CO OO rh g n n ( t ) = / d ( 0 2 m f 2 m ( o ) 2 J / du, 7 t r i If 9 m,(w 9 t n f) g n n(t,a) 9 n i,..,u) 7 T n,) (AI.67) Sm'^m'^m" 600^ , U J2m' Since a l l of the zero and transverse f i e l d spin relaxation data presented i n thi 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 to exhibit an exponential decay at early times, the k>2m's are assumed to be d i s t r i b u t e d according to Lorentzian 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 . This choice of d i s t r i b u t i o n function i s purely phenomenological, since the actual function i s not known. AI.C.l Relaxation Due to a C y l i n d r i c a l D i s t o r t i o n 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 of the muonium hyperine as i n the case of anomolous muonium [8]. In th i s case, the hyperfine d i s t o r t i o n i s c y l i n d r i c a l l y symmetric about some given axis. Assuming that a l l of the muonium atoms i n the ensemble have hyperfine d i s t o r t i o n s that are symmetric about the z-axis, the Hamiltonian of Equation AI.66 becomes ' K r T ^ J 0 0 0 H = (h/2*) ^ "00 "20 0 K o ^ o ) ° ° 0 0 ("00+2u,20) 0 0 0 0 -3u>, 00 -I (AI.68) As stated e a r l i e r , the eigenf unctions |c|^ > of a s p e c i f i c hyperfine Hamiltonian can be expressed as l i n e a r combinations of the i s o t r o p i c muonium - 171 -basis vectors |k> given i n Equation AI.24. Since the Hamiltonian i n Equation AI.68 i s diagonal i n the i s o t r o p i c muonium representation, one has the c o e f f i c i e n t s c ^ = 6^ with the eigenvalues £^ equal to the diagonal elements, namely lx = (h/8Ti)(a) 0 0 - / 2 / 3 a) 2 0) ; l2 = {h/8n){%Q - /2?3" a>20) l3 = (h/BiOfaoo + 2 / 2 7 3 to 2 Q) ; ^ = (h/8u)(-3a> 0 0) with the corresponding t r a n s i t i o n frequencies (AI.69) "12 " ° U13 = k ( - 3 / I 7 3 o) 2 0) to 14 = T ( 4 a )00 " / I 7 T u9n) 2CV 03 23 = i (-3/2/3 u) 2 0) co '24 = T ( 4 wQ0 " / 2 / 3 u ™ ) 20 J (AI.70) = T ( S i n + 2 / 2 / 3 *>9n) "34 4 ^"00 ' *"*"'"' "20'Using Equation AI.27 one can then c a l c u l a t e the c r y s t a l frame p o l a r i z a t i o n components, namely P X. 0 ' * i i 0 ; P U - I ; P Z = -* 33 P Z = - 1 22 cosX, P,,= cosX 44 p x 12 0 p y = ' 12 0 0 P X 13 = c , ^13 i c > P 1 3 - 0 P X *14 = -s . P y = ' 14 - i s ' p u " 0 (AI.71) P X 23 = s P y = ' 23 - i s 0 P X *24 c . P y = ' 24 - i c 0 P X *34 0 . P y = ' 34 0 sinX which simplify further i n zero f i e l d where X = n/2, thus c = s = l//2~. Since a l l of the c r y s t a l frame c o e f f i c i e n t s i n the c y l i n d r i c a l l y - 172 -dis t o r t e d Hamiltonian are set equal to zero except OJQQ and W 2 Q > T N E O N I V non-zero c r y s t a l frame relaxation functions are g^Ct ) and g^ C t ) . By Substituting the values of Equation AI.71 into Equation AI.14, and using the d e f i n i t i o n s of Equation AI.70, the former becomes S 0 0 ( t ) = U l + C ° S ^ U 0 0 + I u 2 0 ^ ] + 2 c ° S t ^ W 2 0 ^ , 1 (AI.72) + 2 c o s t ( w 0 0 " 4 u 2 0 ^ ^ S i m i l a r l y s u b s t i t u t i n g the values of Equation AI.71 into Equation AI.17, along with the t r a n s i t i o n frequencies defined i n Equation AI.70, the expression for the l a t t e r can be derived, namely g ' 0 ( t ) = I /2T3 {-1 - cos[(o) 0 0 + i U ' 0 ) t ] + c o s [ ( | U ' 0 ) t ] l (AI.73) + cos[(o) 0 0 - 4 o) 2 0)t]} To derive an expression for the observable relaxation due to random hyperfine d i s t o r t i o n s , we treat the terms o s c i l l a t i n g at or near OJQQ as averaged to zero and only average over W 2 Q . In this case, the frequency d i s t r i b u t i o n i s one-dimensional and, assuming a Lorentzian d i s t r i b u t i o n , i s given by f 2 0 ^ 2 0 ) - k C 2 2 ] <AI'74> u20 + °20 where O"2Q i s the width parameter of the d i s t r i b u t i o n , and the factor of 1/n i s a normalization constant. In the case of a powder, averaging over the Euler angles forces a l l of the detector frame S 2 m ( t ) components to zero except gQg(t). By combining Equations AI.72 and AI.74 with the d e f i n i t i o n of Equation AI.67, one can then write for powders - 173 -O f c ) = I "-f- l \ o Ho + 4o^ t1 + 2 C° S^ / I73" " 2 0 ^ ( A I * 7 5 ) o Performing the inte g r a t i o n i n Equation AI.81, then y i e l d s 8 0 0 ( t ) = i + J exp(- I / 273 cr 2 0 t) (AI.76) Notice that as t •* 0 0, this function tends to 1/6 (or 1/3 of the i n i t i a l 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 spin 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. This can be understood i n t u i t i v e l y by noting that for a random hyperfine i n t e r a c t i o n , 1/3 of the time the c y l i n d r i c a l d i s t o r t i o n axis i s directed along the z-axis of the detector frame, ( i . e . , along the i n i t i a l muon spin p o l a r i z a t i o n ) . AI.C.2 Relaxation Due to a Planar D i s t o r t i o n Now consider the time evolution of the u + spin for the case of a planar d i s t o r t i o n of the muonium hyperfine i n t e r a c t i o n . In this case, one has the Hamiltonian H = {h/2n) \ -to. 0 0 00 '22 -oo to, 2-2 00 0 0 0 0 0 00 0 0 0 -3co f (AI.77) "00 -I Since t h i s Hamiltonian i s block diagonal, the f i r s t two energy eigenvalues can be calculated by diagonalizing the 2x2 block K o - ^ -to 22 "2-2 ( w 0 0 -M >2 = 0 = to - 2(O>00OJ) + t o f t n - ( w 9 9 ) 00 where we define oo = £(8it/h), and oo M„ = / ( t o „ 9 ) ( o o „ _ 9 ) = [(oo,,)'' + ( t o , - ) ^ ] "22^ R >2 (AI.78) I - , 2 n l / 2 "22 2 2 ' " 2 - 2 ' "22- "22-(AI.79) - 174 -Solving Equation AI.78 for to then gives h,2 = ( h / 2 l t) 7 K o * a22^ To compute the eigenf unctions |(|>i> and |4>2> corresponding to these eigenvalues, one solves the set of coupled l i n e a r equations ^ 0 0 " U K + (" u2-2^ u2 = ° ( -^12)M1 + (w 0 0-u)u2 = 0 where u^ and u 2 are the components of the eigenvectors. Solving Equation (AI.80) AI.80 for to = to^ and to = to 2, one obtains respectively d) o n i(j> U2 = " to. '22 and u 2 - + to 2 2 i<D22 — u l = + e u l (AI.81) 22 — u^ = - e u L 22 W22 where <t>22 = arccos(to 2 2/to 2 2) = a r c s i n ( t o 2 2 / t o 2 2 ) . From Equation AI.81 and the usual normalization conditions, one can obtain expressions for both |C|J^> and |CJJ2>. Thus one can write i V = Z [|1> + * H 2 2 |2>] ; ,. 2 1 /2 Z /2 l* 3> - |3> ; I V = |4> with the corresponding eigenvalues lx = (h/2u) \ (to 0 Q + W M 2 ) ; C 9 = (h/2*) \ (to, V " = [ i x > - e i < f > 2 2 l 2 > ] (AI.82) 00 to, M > 22' I = (h/2n) \ (to n n) (AI.83) '3 4 ^ w00 ; ' ^4 v."'-*; 4 The t r a n s i t i o n frequencies are therefore given as to 12 I f , to 13 - 1 r M i - r 1^22^ to 14 = X ( 4 a )00 + u22' to 23 - 1 r M ^ - 7 1-^22^ W24 = 7 ( 4 a )00 " W22^ (AI.84) to '34 = r ( 4 a W - 175 -Again u t i l i z i n g the expressions given i n Equation AI.27, one can calculate the c r y s t a l frame p o l a r i z a t i o n components. To do t h i s we f i r s t write down the eigenstate expansion c o e f f i c i e n t s namely c l l " J ; C12 " + J e 1 * 2 2 5 C13 = ° ; C14 = 0 c n = ± ; c 2 2 - - ^ e ^ 2 2 ; ^ - 0 ; c 2 4 = 0 (AI.85) c3k = 63k ; °4k = 64k Substituting these c o e f f i c i e n t s into Equation AI.24 then gives P X. = 0 ; P y = 0 ; P Z, = 0 i i i i i i P X„ = 0 ; P y = 0 : pf„ = 1 12 ' 1 2 ' 1 2 >X3 - z [c + s e " 1 ^ ] ; P y 3 = z [c - s e " 1 ^ ] . P X 4 = - i [s - ce-^22] ; - "I [ 8 + ce-^22] ; P ^ = 0 (AI.86) 1 c se" i ( 1 )22 / I = -1 [s — _ i < t , n i [c — se _ i < t ,22 n = - i + ce _ i l | >22 / I P X 3 = i [c - -^22] . = 1 [ c + s e " 1 ^ ] ; P ^ = 0 = ± 1 c - se _ i < t ,22 •2 = " i [ s ce~^ 2 /2 c _ i < | )22 /2 = s - ce _ i < 1 )22 •2 = 0 c = s = 1//2. By P X4 = = t s + c e " l 1 > 2 2 ] * P24 = = t s " c e $ 2 2 ] ' P24 = ° ^ 2 Z 4 /2 Z * P X 4 = 0 ; P y 4 = ; P Z 4 = sinX expressions given i n Equation AI.86 into Equation AI.14, and r e c a l l i n g the d e f i n i t i o n s of the t r a n s i t i o n frequencies given i n Equation AI.84, one obtains for zero f i e l d S 0 0 ( t ) = \ t C O S ^ I U 2 2 ^ ] + 2 c o s t ^ w 2 2 ^ ] Z Z (AI.87) + COS[(OJ0()- i o ) 2 2 ) t ] + cos[(w 0 0+ i u 2 2 ) t ] + cos[(oj 0 0)t]} - 176 -S i m i l a r l y s u b s t i t u t i n g the values of Equation AI.86 into Equation AI.19, the zero f i e l d expression for g^(t) i s found to be 8 2 2 ( t ) = ^ c o s t ( W 0 0 + I w 2 2 ^ ] " c o sC( A )00 " 7 u 2 2 ^ t ^ e X p ^ " i ( | > 2 2 ^ < A I' 8 8) To obtain the relaxation function for random planar anisotropics, one once again ignores the O)QQ terms and only averages over * n t h i s case, t M i the frequency d i s t r i b u t i o n f22^ a )22*^22' * S two-dimensional and, assuming a Lorentzian-like d i s t r i b u t i o n , i s of the form given by M M f22^22'*2 2) = £ „ YiX 7M.2.2 ' W (AI'89) J22^ + ^22' M . > 4 a22 W22 ( n >-l U r c N \L r M NZ-I-L ( u 9 9 ) + K , j J M O where -*-s t n e width of the frequency d i s t r i b u t i o n , and 2/it z i s a normalization constant. The d i s t r i b u t i o n defined i n Equation AI.89 assumes M a zero average, even though i s p o s i t i v e - d e f i n a t e . At f i r s t t h i s might seem inconsistent, except that Equation AI.89 i s the d i s t r i b u t i o n function M for a complex number; although the magnitude i s positive-def inate, the orientations of the associated vector are d i s t r i b u t e d over a l l di r e c t i o n s i n a plane. In the case of powders, an ensemble average i s made by averaging over the Euler angles, which as e a r l i e r stated, forces a l l of the detector frame re l a x a t i o n tensor components to zero except for g ^ C t ) . By combining Equations AI.87 and AI.89 with the d e f i n i t i o n of Equation AI.67, and ignoring the U)QQ terms, one then obtains for powders M r M i 2 r h , ^ 4 °22 ra, M l"22 J r r l M -i 8 0 0 ( T ) = H — > d w22 r, M ,2 , f M ,2 n2 { c O S % "22^ [l"22) + l 022) J L M (AI.90) 4 "22' + 2cos[x w 9 9t]} where the i n t e g r a t i o n over the phase angle c j ^ has been done, and the extra M factor of 1 S t n e Jacobian a r i s i n g from the coordinate transformation - 177 -R I M from the coordinates > °°22 a n c * ^22 t 0 t022* ^ y performing the integration, one f i n a l l y obtains M M M M «£(t) - Ul ~ °-¥ ' W - ~ir fc) + j t 1 ~ ~rL ajr (AI-91) Notice that as t «>, this function tends to zero. This r e s u l t r e f l e c t s the fact that, unlike 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 , there are no zero frequency terms i n g ^ C t ) . The simple a n a l y t i c r e s u l t of Equation AI.91 ari s e s from the assumption of a "Lorentzian-like" d i s t r i b u t i o n . If instead a true two-dimensional Lorentzian i s assumed, the r e s u l t i s not a n a l y t i c . AI.C.3 Cylindrical and Planar Distortions Combined Now consider the time evolution of the u + spin assuming both a c y l i n d r i c a l and a planar d i s t o r t i o n . In th i s case, one has the Hamiltonian ( o ) 0 0 - U 2 0 ) - U „ „ 0 0 H = (h/2*) i 2-2 -to. 0 0 22 K o - w 2 0 ) ° ° -3oj , 0 "00 "20-0 (AI.92) "00 -I Because t h i s Hamiltonian i s of the same form as that of the simple planar d i s t o r t i o n , given i n Equation AI.77, i t s eigenfunctions \fy^> are the same i V - = [|1> + e 1 * 2 2 |2>] ; I V " z [|1> - e 1 * 2 2 |2>] •2 /2 (AI.93) I V - |3> ; I V = |4> Since t h i s Hamiltonian i s block-diagonal, the f i r s t two energy eigenvalues can of course be calculated by solving the secular equation ( wOO" W2o)- w _ u22 —OJ, 2-2 K o ~ W 2 o ) ~ U = 0 = OJ - 2a)(oj„n-oj' ) 00 w20^ . f , >,2 , M ^2 (AI.94) "00 "20 ^ ^22' Solving Equation AI.94, the four eigenvalues of th i s Hamiltonian are - 178 -h,2= [ h / 2 7 l ) £ [ ( U 0 0 ~ ^ 7 3 ^ ± (4j 5 3 = (W2n) j ( u 0 0 + 2 /273 co 2 0) ; C 4 = (h/2*) i (-3 a) Q 0) with the corresponding t r a n s i t i o n frequencies (AI.95) - 1 « ) U12 4 ^"22-W13 = 7 (" 3 / I73 w20 + "22^ U14 = ^ 4 ( ° 0 0 ~ / I 7 J w 2 0 + W22^ W23 = 7 ( - 3 / I 7 J u20" W22^ w24 = 7 ( 4 ( 000" / I 7 J "20" w22^ ( A I - 9 6 ) W34 = 7 ^ 4 u )00 + 2 / 1 7 1 "2C-) Because the eigenstates of the combined Hamiltonian are the same as those of the planar d i s t o r t i o n Hamiltonian, the eigenstate expansion c o e f f i c i e n t s c ^ of the combined Hamiltonian are equal to those quoted i n Equation 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 are derived through the use of Equation AI.27, are the same as those given i n Equation AI.86. By subst i t u t i n g the the r e s u l t s of Equation AI.86, as well as the t r a n s i t i o n frequencies of Equation AI.96, into Equation AI.14, one obtains an expression for g ^ C t ) i n zero f i e l d , namely 8 0 0 ( t ) = I icosi(j u22^^ + c o s t i ( ~ 3 w 2 0 + "22^^ + cos[-|(u 0 0- co20+ u 2 2 ) t ] + cos[-i{-3co 2 0- (i>22)t] (AI.97) + c o s [ ( W ( ) 0 - \ <o'0- \ u) M 2 ) t ] + cos[(o, 0 0+ \ co 2 Q)t]} where W'2Q = /2/3 w 2o* ^ n a s i m i ^ a r manner one can obtain expressions for the c r y s t a l frame components g ^ C t ) and g 2 2 ( t ) , such that one has g 2 0 ( t ) = \ rm {-cos [ ( I u M 2 ) t ] + cos[i{-3u) 2 0+ o>M 2)t] + c o s [ ( w 0 0 - \ w 2 0+ i u 2 2 ) t ] + c o s[i(-3u^ 0- u> 2 2)t] (AI.98) + cos[(co 0 0 - \ o) ' 0 - £ w 2 2 ) t ] + cos[(u) 0 0+ j o> 2 0)t]} - 179 -and g 2 2 ( t ) = j{- cos[(- I OJ20+ i u*2)t] + cos[a) 0 0- i £ a, M 2)t] + cos[(- I u 2 0 - w M 2 ) t ] - cos[(o) 0 ( )- u 2 0 - -J- w 2 2 )t]} (AI.99) x {exp(-i(|>22)} re s p e c t i v e l y . The r e l a x a t i o n function for the combination of random c y l i n d r i c a l and planar anisotropics can be calculated by ignoring the unobservable OJ00 terms M and averaging over OJ2Q and w22- In th i s case, the frequency d i s t r i b u t i o n t M •> M f (OJ2Q ,OJ22J i s simply given by the product of the oj^and OJ22 d i s t r i b u t i o n s as defined i n Equations AI.74 and AI.89, respectively, namely f(u) 2 0,o) 2 2) = f(o> 2 0) . f(o>22,<D22) M M 1 °20 4 a 2 2 w 2 2 -1 (AI . 100) For powders, one averages over the Euler angles, and by combining Equations AI.97 and AI . 100 with the d e f i n i t i o n of Equation AI.67, and ignoring the WQQ terms, one then obtains the expression g 0 0 ( t ) = TT (°20 °22^ / d a > 2 0 ^ u 2 0 ^ + ^ O ^ " 1 3it - » OO x / doo22 ( u 2 2 ) [ 0 2 2 ) + (cr 2 2) ]~ { c o s f j w 2 2 t] (AI . 101) o + cos [ i ( -3/273 u ) 2 0 + 0 3 M 2)t] + cos [ i ( -3/273 a>2() - co^Jt]} where the int e g r a t i o n over the angle <t>22 has already been performed, and - 180 -M where the extra to 2 2 factor i s again the Jacobian of the coordinate M transformation. Performing the integrations over U^ Q and OJ22 y i e l d s the simple a n a l y t i c r e s u l t r h , ^ 1 f l 1 M \ r 1 M s % ) ( t ) = 6 I1 " 2 022 z> e X p ^ _ 2 a22 ^ (AI.102) + U1 ~ T*22 e X P t " ^ 2 2 + ^ a20^ M A check of the l i m i t i n g cases shows that as 0, Equation AI.102 approaches the expression for a c y l i n d r i c a l d i s t o r t i o n given i n Equation AI.76, and that as OJ^Q -*• 0> Equation AI.102 s i m p l i f i e s to Equation AI.91 for a planar d i s t o r t i o n . Furthermore, because of the planar contribution to the hyperfine d i s t o r t i o n , the spin p o l a r i z a t i o n of the muonium ensemble tends to zero at long times. AI.D Isolated Muonium in an External Magnetic Field In the presence of an external magnetic f i e l d , the frequencies as well as the amplitudes are dependent upon the Euler angles. Assuming both a c y l i n d r i c a l and a planar d i s t o r t i o n of the muonium hyperfine i n t e r a c t i o n , the t o t a l Hamiltonian i s H = 8-rc (AI.103) " K o + 4 a W W 2 o ) _ w2-2 - w 2 - l 0 ~ w22 ^00- 4 t°Mu _ W20) _ W 2 1 ° -w n -w2_1 (u> 0 ( )+4r+(l+2sc)w 2 0) Cw 2 Q 0 0 Cw 2 Q (-3a ) 0 0-4r+(l-2sc)w 2 0}_ 1 2 1/2 2 2 where r = <J0QQ[(1 + x ) - l ] , £ = 2(c - s ) and the w ? m are the detector 2m - 181 -frame hyperfine c o e f f i c i e n t s which are related to the c r y s t a l frame c o e f f i c i e n t s i o 2 m through the r o t a t i o n a l transformation defined i n Equation AI.30. Calculating the applied f i e l d spin relaxation function from the Hamiltonian of Equation AI.103, for a l l applied f i e l d s i s a somewhat d i f f i c u l t problem. However, this problem becomes considerably easier for c e r t a i n l i m i t i n g cases Consider the problem i n the l i m i t of "high f i e l d s " , ( i . e . , u> » a2m^* In t h i s l i m i t , one can approximate the t o t a l Hamiltonian by i t s diagonal components and write H o + 4 a W - W 2 0 ) ° ° ° 0 ^00- 4 wMu- W20) ° ° 0 0 (co 0 ( ) +4r+(l+2sc)w 2 0) 0 0 0 0 (-3o) 0 0-4r+(l-2sc)w 2 0l_ where W2Q i s the detector frame c o e f f i c i e n t which i s related to the c r y s t a l frame c o e f f i c i e n t s through the r o t a t i o n a l transformations defined i n Equation AI.30, namely ,(2) H 8¥ (AI.104) w 2 Q = / 2 7 3 I co 2 m R 2 m (AI.105) m Since this Hamiltonian i s diagonal, the energy eigenvalues are simply given by the diagonal elements, namely 5X - (h/2*) 7 K o + 4 a W " W 2 0 ] ; h = (h/2^ 7Ko- 4 uMu- W 2o] , a t i n „ (AI.106) = (h/2*) i[o, n n+4r+(l+2sc)w 9 n] ; lL = (h/2*) ^[-3co n n -4r+(l-2sc)w 2 0] , 3 ^ 4 L „ 0 ( ) . , x . w - 2 Q j » s,4 v. 4 L — 0 0 and the corresponding t r a n s i t i o n frequencies are then - 182 -W12 = 2 wMu U13 = V 1 " " l ( 1 + S C ^ W 2 0 W14 = uOO + UMu + r~ T^i-scjw, 23 = -^Mu-11- j ( 1 + s c ) w 2 0 +r- j ( l - s c ) w 2 0 (AI.107) W24 = U00 - a ,Mu W34 = a ) 0 0 + 2 r + S C W 2 0 2*- ^ " 2 0 The elgenfunctlons of this Hamiltonian are of course the i s o t r o p i c eigenstates give i n Equation AI.24 and so the eigenfunction expansion c o e f f i c i e n t s are just c ^ = Combining this with the expressions for the detector frame p o l a r i z a t i o n components given i n Equation AI.27, gives p x i i 0 • P Y ' ^ i i 0 • P Z ' *11 Pz 33 = 1, -cos\, P Z = 22 'I-P X 12 = 0 ; P Y ' 12 = 0 ; P* 2 = 0 P X 13 = c ; PY ' 13 = i c = 0 P X *14 s ; P Y * 14 = - i s ' P H = 0 P X 23 = s ; PY ' 23 = - i s ' P Z ' 23 = 0 P X r24 = c • P Y ' 24 = - i c = 0 P X *34 0 • P Y * 34 = 0 = sinX (AI.108) Substituting the re s u l t s of Equation AI.114 into Equations AI.14, AI.15 and AI.17, one obtains for the detector frame g Q 0 ( t ) = \ ( l + cos 2\) + \ f c o s 2 ( \ / 2 ) [ c o s ( W l , t ) + cos ( w 7 A t ) ] "13" "24' 2 X 2 + s i n (\/2)[cos(oj^t) + cos(co 9^t)] + y s i n \[cos(io^^t) ]} (AI.109) 23" "34" and g 2 Q ( t ) = y / 2 7 T {-(l + cos 2\) - s i n 2 \ [ c o s ( o ) , A t ) ] + c o s 2 ( \ / 2 ) 34' (AI.110) x [cosloo^^t) + cos(oo 2 4t)] + s i n (\/2)[cos(u)^t) + co s f i o ^ t ) ]} Evaluating Equation AI.112, one writes - 183 -w 2 Q = /273 {to 2 0 R < 2 ) ( Q ) + o 2 2 R < 2 ) ( a ) + ( . 2_ 2 R ^ ( Q ) } ( A I . l l l ) From the d e f i n i t i o n of the ro t a t i o n matrix elements R ^ ^ Q ) , Equation A I . l l l becomes . M "^<t)99 * w 2 Q = /27T {a> P 2(cos8) + OJ22 Au/5 [e Y (-f3,-o) (AI.112) - 1*22 * , M , where P 2(cosB) i s the Legendre polynomial and the Y (-8,-a) are the spherical harmonics which define the r o t a t i o n a l transformation. R e c a l l i n g the associated complex conjugate r e l a t i o n s h i p s , Equation AI.112 becomes w20 = T "20 ( 3 c o s ( P ) ~ 1 ) + u22 s i n 2 ( P ) cos(<p22+ 2a)} (AI.113) Because we now have c y l i n d r i c a l symmetry, the relaxation functions f o r a random anisotropic hyperfine i n t e r a c t i o n for powders are calculated by OO 00 g j j ( t ) = / da) 2 0 J do) 2 I 2(a)M 2) f ( a ) 2 0 , a)M 2) / dQ g L Q ( t ) (AI.114) —oo O r M -> where L = 0 and 2 and f[oo2Q, w 2 2J i s the combined d i s t r i b u t i o n function defined i n Equation AI.100. If i t i s further assumed that the applied f i e l d i s low with respect to the hyperfine f i e l d ( i . e . , x « 1), one has c = s = 1//2, r = 0 and cos\ = 0. With this one can ignore the s i n g l e t state, ( i . e . , ignore the terms i n the g L Q ( t ) ' s ) , and derive an expression for the t r i p l e t muonium relaxation function. For the case of L=0, Equation AI.114 becomes 8 0 0 ( t ) = 6 + 3 J d w20 J d u22 C« 2 2J f l w 20» w 22J -oo o (AI.115) x / dQ cos ( o) M ut) cos(j w 2 Qt) - 184 -and for the case L=2 i n Equations AI.114 gives g ^ ( t ) = - I / T 7 3 + T / T 7 3 J d ^ J do>M2 (o)^) f(co 2 Q,co 2 I 2) -oo o ( A I . l l o ) 3 x / dQ cos(to M ut) c o s ^ w 2 Qt) By integrating over the ^ m'8 f-"-rst» Equation AI.115 becomes 8 0 0 ( t ) = k+ J ( ^ r ^ r ^ o s ^ t ) J*dB sinB / da / d ^ (AI.117) exp{- ^ t/2/3 ^QISCOS^B — 111 — -g- sin'p a 2 2 | cos( <t>22+ 2a) |t} which can be written as , -, , COsfoi. t) Tl Tt/2 o o (AI.118) {exp[- | s i n 2 B ov^t cos9] exp[- | JTJl o ^ t |3cos 2p - 1|]} In a sim i l a r manner, the case L=2 gives u 1 1 c o s ( ( j J M 0 1 1 o o (AI.119) 3 2 1^ 3 2 {exp[- -g- s i n B a 2 2 t cose] exp[- /2/3 a 2 Q t |3cos p - 1|]} Now one can consider the motion of the | i + spin p o l a r i z a t i o n i n the context of the conventional f i e l d geometries; l o n g i t u d i n a l and transverse f i e l d . AI.D.l Longitudinal F i e l d Relaxation Function The l o n g i t u d i n a l relaxation function can be e a s i l y calculated 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 laxation functions given i n Equation AI.21. In the l i m i t i n g case under discussion, the only non-zero lo n g i t u d i n a l r e l a x a t i o n function i s g..(t). Since this function i s a - 185 -l i n e a r combination of both &QQ{^) and g2Q( t)> one has gj ( t ) = g j j ( t ) - /ITT g ^ ( t ) = j (AI.120) which simply means that the hyperfine 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 decoupled for co^u » o"2^. AI.D.2 Transverse Field Relaxation Function The detector frame observable transverse relaxation functions are defined i n Equations AI.22 and AI.23. For the "high f i e l d " l i m i t i n g case under consideration, one observes that there i s only one non-zero coplanar-transverse and one non-zero perpendicular-transverse r e l a x a t i o n sc ss function, g c t ( t ) and g p t ( t ) , respectively, which are equivalent. Thus one can write 8 c t ( t ) = 8 D t ( t ) = ( 2 l l ) ~ l c o s ^ M u ^ S l n P v o o (AI.121) o O Tuf O _____ O {exp[- -g- s i n 6 0 2 2 t cose] exp(- /iTS a2Qt |3cos B - 1|)} which can be calculated numerically. For early times ( t •*• 0), one can expand the integrand to obtain the approximate expression g ^ ( t ) = g£<t> = \ c o s ( V ) e x P [ - ( / 2 a 2 Q + - L . a f 2 ) t ] (AI.122) 7t/6 As a matter of convention, the cos(co^ ut) part of Equation AI.122 i s usually omitted from the d e f i n i t i o n of the transverse f i e l d r e l a x a t i o n function. By comparing the i n i t i a l slope ( m z f ) °f the zero f i e l d r e l axation function given i n Equation AI.102, with the i n i t i a l slope ( m t f ) of the transverse f i e l d function of Equation AI.122, one can define the r a t i o - 186 -— = /3 =—J - /3 (AI.123) [ i + / T T i ( ^ 2 / O 2 0 ) ] Thus, one finds that > m t£> i n d i c a t i n g that the rate of depo l a r i z a t i o n i s greater i n zero than i n transverse f i e l d . - 187 -APPENDIX II ~ ULTRA-LOW ENERGY MUON PRODUCTION (uSOL) In recent years, the development of high f l u x positron beams has made i t possible to study atomic scattering cross sections, the i n t e r a c t i o n of positrons with surfaces as well as the spectroscopy of positronium atoms i n vacuum [1-4] . The analogous experiments with p o s i t i v e muons are at present impractical since comparable beams do not as yet e x i s t . The state of the art method for producing slow \x+ involves tuning the secondary channel to lower momenta P, thereby c o l l e c t i n g and transporting p,+ which or i g i n a t e from 7t+ decaying inside the pion production target. These muons are, for lack of a better term, c a l l e d subsurface muons. I t can be e a s i l y argued that the subsurface \x+ rate R i s proportional to p 7/2 m u l t i p l i e d by an appropriate decay f a c t o r . For 10 keV (1.5 MeV/c), the subsurface u"*" rate at 100 \ik for the M13 secondary channel at TRIUMF would be approximately _. 1.4 x lO^sec ^ -7/2 t f / T u Q „ -1 / » T T I N R = YTT e * 8 s e c ( A I I . l ) (29.8 MeV/c) / Z where t f i s the time of f l i g h t through the channel (3.66 x 1 0 - 6 sec for M13) and i s the mean muon l i f e t i m e (~ 2.2 \xs). C l e a r l y , t h i s rate i s not acceptable from a p r a c t i c a l experimental perspective. The need for an ultra-low energy (0 to ~10 keV), high f l u x \x+ beam i s somewhat s e l f evident. Such a beam i f developed can be immediately u t i l i z e d i n the study of: (1) Electron-Muon Capture Spectroscopy - By scattering slow u"1" at grazing incidence to a surface, one can study electron pick-up processes involved i n u +e~ and u +e~e~ formation as a function of the surface properties and magnetic ordering. This type of experiment has already been done for D + ions [5]. - 188 -(2) Adatom Adsorption on Surfaces - The a b i l i t y to adsorb \i+ or Mu on well characterized surfaces w i l l provide information regarding surface properties as well as the e f f e c t of a reduced dimensionality on the evolution of the muon spin p o l a r i z a t i o n . (3) Charge Exchange Cross Sections - Measurements of the muonium formation p r o b a b i l i t y as a function of incident | i + energy (down to thermal energies) would provide valuable information to help discriminate between spur and hot atom mechanisms. (4) Molecular Ion Formation - Muon molecular ion formation has been observed i n He and Ne [6], but i t i s not as yet known at what stage of thermalization the molecular Ion i s formed. The c o r r e l a t i o n between the incident energy (down to thermal energies) and molecular ion formation would help decipher the mechanisms involved. A slow Mu beam could be produced by passing the low energy \x+ beam through a thi n f o i l (or gas jet) and taking advantage of the large electron capture cross section. The r e s u l t i n g slow Mu beam could then be u t i l i z e d i n experiments such as: (1) Muonium Lamb S h i f t - Recently 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 test of QED i f an accuracy of 100 ppm can be obtained. Slow | i + fluxes i n excess of 10 3/sec at 1 MeV/c would make a pr e c i s i o n experiment possi b l e . (2) Muonium to Anti-Muonium Conversion - A low energy Mu beam would of course benefit these studies, but to be competitive with e x i s t i n g experimental scenarios the Mu flux would have to be i n excess of lO^/sec. In what follows, the i n v e s t i g a t i o n of possible \i+ emission from surfaces i s proposed which u t i l i z e s the knowledge gained i n low energy positron production research with the appropriate analogies drawn between positrons and p o s i t i v e muons. The ultimate goal of th i s research would of course be the development of an ultra-low energy (0 to ~10 keV) \i+ beam. A I I . l Current Status of Slow Positron Production The beam moderation techniques employed i n the production of slow e" - 189 -beams involve single c r y s t a l metal moderators and u t i l i z e the existence of a negative work function for positrons at the moderator surface. Normally a backscattering geometry i s used where the high energy positrons originate from a radioactive source ( 2 2Na for instance) which i s mounted facing the moderator surface. The beta-decay positrons are implanted into the c r y s t a l and become thermalized with a stopping d i s t r i b u t i o n corresponding to an exponential attenuation law [8]. Some of the implanted e + are able to di f f u s e back to the surface before a n n i h i l a t i n g , and a f r a c t i o n of those are then emitted from the moderator surface as a r e s u l t of a negative work function mechanism. The conversion e f f i c i e n c y £ for present day moderated beams, defined as the r a t i o of the slow e + y i e l d to the t o t a l number of fast positrons emitted from the source, i s generally on the order of 1 0 - 3 . By analogy with the electron case, the work function c j + of the positron i n a metal i s given by *. = - D - u. ( A l l . 2 ) + *p where D i s a p o t e n t i a l due to the surface dipole layer and \x-p i s the positron chemical po t e n t i a l inside the metal. The positron chemical p o t e n t i a l incorporates two terms. The f i r s t contribution to p,p ar i s e s from the positron-ion i n t e r a c t i o n (Bloch wave energy). This i n t e r a c t i o n along with the surface dipole layer act to expel the positron from the metal. The second contribution to |a.p i s the electron-positron 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 acting to bind the positron to the metal surface. Considering the mass of the muon i n comparison to that of the positron, one can conclude that for a metal moderator a \x+ negative work function i s - 190 -not such a l i k e l y candidate to be used i n the production of an ultra-slow \i+ beam, primarily because the a t t r a c t i v e e~-u + c o r r e l a t i o n energy would be about 1 Roo rather than the 1/2 as i s the case for e - - e + c o r r e l a t i o n s . Also, the added k i n e t i c energy a r i s i n g from Bloch wave kinematics, which i s on the order of a few electron v o l t s for e +, becomes n e g l i g i b l e for the heavier From t h i s one can conclude that, at least for metal moderators, the \x+ a f f i n i t i e s are probably not negative. For i n s u l a t o r s , however, the \i+ a f f i n i t i e s may very well be negative because i n t h i s case the e ~ - | i + c o r r e l a t i o n s are i n general small. Another emission process, which has been observed for e + implanted i n i o n i c single c r y s t a l s , produces e + having k i n e t i c energies on the order of the band gap energy of the s o l i d . This mechanism and i t s possible a p p l i c a t i o n to \i+ emission i s discussed i n d e t a i l i n the following pages. A l l . 2 Band Gap Emission of e + from Ionic C r y s t a l Surfaces Recent positron experiments [9] show that when e + of keV energies are implanted into i o n i c single c r y s t a l s they are reemitted i s o t r o p i c a l l y from the s o l i d s with a continuum of energies having a maximum approximately equal to the band gap energy ( t y p i c a l l y on the order of 10 to 20 eV). Operating under the assumption that the mechanism(s) responsible for the reemission of positrons would also be involved i n the analogous phenomena for a"1", a b r i e f synopsis of the e + experiments, along with the current understanding of the mechanism(s) involved, are given here. Five a l k a l i - h a l i d e s ( L i F , NaF, NaCl, KC1, KBr) and four other i o n i c s o l i d s ( S i 0 2 , A1 20 3, MgO, CaF 2) were studied i n a l l . The a l k a l i halide samples were oriented with the (100) axis normal to the emitting surface, as - 191 -was the MgO c r y s t a l . The S10 2 c r y s t a l was z-cut, the A1 20 3 sample was oriented with the c-axis normal to the emitting surface and the CaF 2 o r i e n t a t i o n was believed to be (110). The experiments were performed i n a vacuum of 5 x 1 0 - 1 0 Torr, with the samples heated to about 330 °C, and the surface contamination was estimated to be somewhat less than a monolayer. A beam of 500 eV positrons was incident on the surface of the samples and the a x i a l component of the reemitted positron spectrum was measured. The energy spectra obtained for 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 energies with the maximum energy approximately equal to the band gap of the i n d i v i d u a l s o l i d s . These experiments were repeated for 1500 eV incident positrons, with the r e s u l t s showing no s t a t i s t i c a l l y s i g n i f i c a n t deviation from the 500 eV data. The angular d i s t r i b u t i o n of the emission spectra was also studied and found to be approximately i s o t r o p i c . In addition to emitting positrons, i t was found that a l l nine samples also emit positronium (Ps). To discriminate between d i f f e r e n t possible emission processes, for both e + and Ps, the dependence of the Ps formation p r o b a b i l i t y on the incident e + energy was studied. Results from these studies as well as from positron d i f f r a c t i o n experiments, have lead to the conclusion that at least for L i F and NaF, the emission of both e + and Ps can be associated with Ps d i f f u s i n g to the surface of the c r y s t a l moderator. Approximately 60% of the incident e + form Ps i n these samples, with about 60% of the Ps atoms that d i f f u s e back to the surface being dissociated, thereby re-emitting the positron. In 1972 i t was postulated that Ps could be f i e l d - i o n i z e d i n the process of leaving a surface [10]. This, however, does not explain the anomalously - 192 -large emission energies or the c o r r e l a t i o n with the band gap energy of the s o l i d . An alternate explanation [9] i s that the positron i s Auger-emitted when the Ps electron f a l l s into an acceptor state at the surface. With t h i s model, the maximum energy E m & x of the emitted e"r, corresponding to the Ps electron recombining with a hole at the bottom of the valence band i s given by the expression E 6 = (E + AE ) - ( E P S + «f) (Al l . 3 ) max g vJ K b + J where Eg i s the band gap energy, AE V i s the width of the valence band, Ps e E^ i s the binding energy of Ps on the surface of the s o l i d and <3?+ i s the positron work function. Plugging the values for NaF into Equation A l l . 3 gives E m a x = 11.8(6) eV, which agrees well with the experimental r e s u l t [9] of 12.3 +/- 0.7 eV. Since one does not normally expect long-lived holes i n the valence band, however, Equation A l l . 3 represents an overestimate. The o r i g i n of the surface acceptor states i s not yet known. Normally, one would not expect holes below the the Fermi energy E^ (about 4.5 eV below the bottom of the conduction band for a i r cleaved NaF at 300 °C [9]). However, electron-hole pairs are produced i n the i o n i z a t i o n t r a i l of the incident e + beam, and possibly some of the holes survive long enough to migrate to the surface. In any case, the branching r a t i o for e + emission as opposed to Ps emission i s equal to the surface density of acceptor states m u l t i p l i e d by the electron capture cross section. As mentioned e a r l i e r the branching r a t i o for e + vs Ps emission has been found [9] to be about 60%. For these i n i t i a l experiments the a l k a l i halide samples were prepared by cleaving i n a i r . Subsequent experiments on vacuum-cleaved samples of NaF and L i F were also performed [9] with the re s u l t s i n d i c a t i n g the same general - 193 -positron emission spectra as found for the air-cleaved samples, except for a few s l i g h t d i f f e r e n c e s . In p a r t i c u l a r , $ + for the air-cleaved NaF and L i F c r y s t a l s i s equal to +0.5 eV and -0.7 eV, respectively [9]. However, for the vacuum-cleaved samples of both NaF and L i F , $ + was found to be p o s i t i v e . Thus even though the positron work function i s not negative, positron emission of band gap energies i s s t i l l observed. This i s important to note i n l i g h t of the fact that the u + work function for these materials i s expected to be p o s i t i v e . All . 3 Comparison of e + and u + WRT Band Gap Emission Although some tend to view the muon as a heavy electron, the behavior of muonium (Mu) i n s o l i d s i s more reminiscent of hydrogen rather than Ps. The d i f f u s i o n constant D for Mu i n these materials can be estimated by considering experiments involving Mu emission from f i n e S i 0 2 powders [11]. Using a d i f f u s i o n model [12] o r i g i n a l l y applied to positronium, i t was found that D ~ 1 0 - 7 cm2/s at room temperature. The d i f f u s i v i t y i n single c r y s t a l s would of course be greater than this with a good room temperature estimate being D ~ I O - 5 cm 2/s. In comparison, the d i f f u s i o n constant for positrons i n the same materials i s about 1 0 - 3 cm 2/s. With a d i f f u s i o n constant of 1 0 - 5 , the d i f f u s i o n length ( T , , D ) 1 / 2 i s then about 5 x 1 0 - 6 cm. Heating to higher temperatures would of course enhance the d i f f u s i o n . The stopping d i s t r i b u t i o n of surface \i+ i n these materials has a range of about 0.05 cm. Cl e a r l y , tuning to subsurface momenta would increase the p r o b a b i l i t y for the u + to reach the surface within t h e i r l i f e t i m e , but considering the loss i n incident flux as the beam momentum i s reduced, no net increase i n the - 194 -reemitted beam fl u x i s forseen. In addition to d i f f u s i v i t y , the formation p r o b a b i l i t y for Mu as opposed to Ps must be considered. As mentioned e a r l i e r , roughly 60% of the incident positrons form positronium. In comparison the muonium formation p r o b a b i l i t i e s for single c r y s t a l s of NaF and L i F at room temperature, deduced from the observed missing f r a c t i o n s i n the p.+ spectra, are 98 + 5% [13] and 44 ± 6% [13,14], r e s p e c t i v e l y . The muonium f r a c t i o n w i l l be re f l e c t e d i n the slow u + moderation e f f i c i e n c y . To estimate the maximum emission energy for u + one requires values for the muonium binding energy E ^ u on the surface of the s o l i d and the \x+ work function which are as yet not well known. However, i n analogy with Ps studies, the maximum Mu k i n e t i c energy i s the negative of i t s work function <3?MU, which i s given by d M u = ( E ^ U - R j + ($! + *J) (AII.4) where i s the electron a f f i n i t y at the bottom of the conduction band. Using Equation AII.4, one can rewrite Equation A l l . 3 as E^ = (E + AE ) - $ M u - R + $ e ( A l l . 5 ) max v g v ; ao -A negative muonium work function has been postulated to explain the emission of Mu from f i n e l y divided S i 0 2 powders. A conservative estimate of $ M u Mu would be $ « 0±1 eV. For NaF and L i F , E i s 11.5 eV and 13.7 eV, g respectively [15]. The width of the valence band i s AE^ = 4.0(5) eV [9] and « 0±1 eV for both c r y s t a l s . Thus, the maximum k i n e t i c energies of the emitted u + for NaF and L i F are estimated to be - 195 -(NaF) = 1.9 ± 2 eV max ( A l l . 6 ) E^ (LiF) = 4.1+2 eV max The branching r a t i o y Q for \i+ emission as opposed to Mu emission i s d i f f i c u l t to estimate. However, the problem i s somewhat s i m p l i f i e d since the surface density of acceptor states i s l i k e l y to be a property of the sample preparation. Thus one i s l e f t only with evaluating the electron capture cross section at the surface for Mu as opposed to Ps. Because the binding energy of Mu i s approximately twice that of Ps, one would expect the electron capture cross section at the surface to decrease accordingly. As has already been mentioned, the higher binding energy and mass of Mu as compared to Ps also s h i f t s the maximum energy for L I + emission to lower energies with respect to the e + spectra. From these considerations the branching r a t i o y Q for | i + vs Mu should be roughly equal to one half times the branching r a t i o for e + vs Ps emission weighted by the f r a c t i o n of e + e LL e emitted i n the energy range (E - E^ ) to E . Thus, a good estimate m&X ID. 3.X ffl 3.X for the u + branching r a t i o would then be 5-10%. In any case, measurement of y Q i s one of the goals of these experiments. One l a s t point to be made i s that i n both NaF and L i F the \x+ spin i n the muonium state depolarizes due to superhyperfine in t e r a c t i o n s of the Mu electron with neighboring n u c l e i . A l l . 4 Calculations f o r P o s i t i v e Muon Emission Y i e l d To make a t h e o r e t i c a l estimate of the slow \x+ conversion e f f i c i e n c y E for u + emission, one needs to know: (1) The stopping d i s t r i b u t i o n f(x) of the incident beam (2) The bulk d i f f u s i o n constant D for thermalized muonium (3) The f r a c t i o n F(T) of muonium formed (formation p r o b a b i l i t y ) (4) The branching r a t i o y Q for energetic \x+ emission - 196 -Consider the random walk problem where the muons are incident on the surface of a homogeneous moderator of thickness d as shown i n Figure A I I . l , and subsequently thermalize i n the muonium state. Once thermalized, the Mu atoms d i f f u s e approximately randomly through the l a t t i c e u n t i l they reach one of the moderator surfaces where there exists some f i n i t e p r o b a b i l i t y for | i + emission. With this geometry and taking into account the f i n i t e l i f e t i n e of the muon, the conversion e f f i c i e n c y £^ can be written [16] « -tlx d lu = I ( F ( T ) y o ) / d t 6 * f d x f ( x ) R ( t ; x ) ( A l l . 7 ) ^ o o where R(t;x) i s the rate at which a Mu s t a r t i n g at X q at time t=0 appears at the surface. The factor of 1/2 arises since we are neglecting the other surface. It can be shown that [16,17] - 1 / 2 -x 2/4Dt -(d-x) 3/4Dt R(t;x) = (2t) (4uDt) [x e + (d-x) e ] ( A l l . 8 ) A beam of 30 MeV/c muons with a momentum spread of AP/P » 10% w i l l stop approximately uniformly over a distance r defined by the minimum and maximum mean ranges of the beam p a r t i c l e s , namely r « Range(30 MeV/c) - Range(27 MeV/c) = 2.35 x 10~ 2 cm ( A l l . 9 ) With t h i s approximation, the stopping d i s t r i b u t i o n f(x) i n Equation A l l . 7 can be assumed to be uniform and given by f(x) = 1/r. Since the mean stopping distance r i s large compared to the d i f f u s i o n length ( T n D ) l / 2 , the expression for R(t;x) given i n Equation A l l . 8 c l e a r l y breaks down into two components of equal magnitude corresponding to the a r r i v a l rate of Mu at - 197 -REFLECTED BEAM [ B E A M Figure A I I . l Target geometry showing both r e f l e c t i o n and transmission modes. In transmission mode degrading i s provided by the target i t s e l f . - 198 -each of the two surfaces. Thus R(t;x) for each surface can be s i m p l i f i e d to give R(t;x) = (2t) (4nDt) -1/2 [x e ] (All.10) Substituting t h i s expression into Equation A l l . 7 and l e t t i n g d •*• » ( i . e . , ignoring the second surface), the slow | i + conversion e f f i c i e n c y £ i s This e f f i c i e n c y (~10 - b for a conservative value of y Q) i s of course not very good, i t does, however, translate into 10 such muons per second for muon i n t e n s i t i e s such as ava i l a b l e from M13 or M20 and ~10 3 per second i f the moderator could be placed close to the pion production target. If the proposed emission process exists for u + i t should be observable using one of TRIUMF's surface muon beams, and once observed steps can be taken to improve the above e f f i c i e n c y . These steps could include the development of high surface area moderators, in v e s t i g a t i n g ways of increasing the density of surface acceptor states, increasing the muonium d i f f u s i o n length as well as in v e s t i g a t i n g other moderators. Producing a polarized u + beam, which u t i l i z e s surface emission processes, w i l l require the a p p l i c a t i o n of a large (~10 G) magnetic f i e l d to quench the e f f e c t s of superhyperfine i n t e r a c t i o n s . written o o = | [ F ( T ) y o ] ± (x D ) 1 / 2 = y Q (9.78xl0- 5); for NaF = y (4.4xl0~ 5) ; for L i F (All.11) - 199 -All.5 Prototype Apparatus The apparatus designed for the i n i t i a l search for the emission of | i + from s o l i d surfaces i s shown i n Figure A l l . 2 . It consists of a scattering chamber and target assembly, combined with a DQQ spectrometer which i s designed to momentum select the extracted muons and focus them onto a channeltron detector. Although a l l of the vacuum components have been designed to be consistent with u l t r a - h i g h vacuum requirements, the i n i t i a l experiments w i l l employ borrowed, non-bakable components which w i l l l i m i t the attainable vacuum to about 1 0 - 9 Torr. The scattering chamber i s designed i n such a way as to allow the spectrometer section to be mounted i n either a transmission or r e f l e c t i o n geometry, simply by rotating the apparatus by 180°. The target assembly w i l l be held at a p o t e n t i a l of about +10 kV with respect to a grounded g r i d thereby providing an e l e c t r i c f i e l d to accelerate muons which are emitted from the surface. A second g r i d , which l i e s between the target and the grounded g r i d w i l l have an independently v a r i a b l e p o t e n t i a l V 2 applied to i t which w i l l allow a f i r s t order measurement of the emission energy spectra. Provisions have also been made for heating the targets to enhance the Mu d i f f u s i o n rate. The i n i t i a l target assembly w i l l be r e l a t i v e l y simple and target changes w i l l require venting the system using dry nitrogen. In the future a bakable, remotely c o n t r o l l a b l e target ladder w i l l be introduced with more sophisticated temperature control c a p a b i l i t y . All.6 Measurements In transmission mode the \i+ beam i s incident on a moderator of s u i t a b l e - 200 -5 0 c m I i i i i i i i i i I Figure A l l . 2 LISOL scattering chamber and DQQ spectrometer (shown here i n transmission geometry). - 2 0 1 -thickness such that the muons are stopped at or near the downstream surface. In r e f l e c t i o n mode the incident \x+ beam must be degraded upstream of the extraction grids to such an energy that the | i + w i l l stop at or near the upstream surface of the moderator. Multiple scattering i n the degrader w i l l c l e a r l y reduce the e f f e c t i v e incident beam rate i n r e f l e c t i o n geometry, however background rates, e s p e c i a l l y due to beam positrons, may also be reduced. In some cases, single c r y s t a l samples having the appropriate thickness for transmission geometry may not be re a d i l y a v a i l a b l e . In l i g h t of these considerations, both transmission and r e f l e c t i o n geometries w i l l have to be tested. The basic measurement to be made i s the time of f l i g h t (TOF) between an incident beam L I + which f i r e s the beam counter and the subsequent detection of a slow \x+ i n the channeltron. Surface emitted L I + should give a c h a r a c t e r i s t i c spectrum which begins about 3 7 0 ns aft e r the u + s t a r t pulse. From Equation A l l . 1 1 , the shape of th i s spectrum i s expected to be " t / T L l R(t) = — ( D / n ) 1 / 2 - • exp(-Xt) ; C = [F(T) y„ i ] ( A l l . 1 2 ) Z / t ° r with the number of events observed within a f i n i t e gate width % given by O N ( T ) = C / dt R(t) = £ D 1 / 2 [\ + - ] " 1 / 2 erf [ A ( \ + 1 / T )] ( A l l . 1 3 ) 8 o a 8 * Here the \ parameter represents the rate of loss of Mu from the d i f f u s i n g muonium ensemble, and i s included to account for possible losses of Mu due to chemical reactions, etc., i n the c r y s t a l . Epithermal L I + or u +e~e~ which are produced by multiple scattering and charge exchange processes i n the target and accepted by the spectrometer system, w i l l gennerally be d i s t r i b u t e d at higher v e l o c i t i e s as compared to L I + a r i s i n g from surface emission, and w i l l therefore give r i s e to prompt - 202 -events. Measurement of the epithermal \i+ of Mu- (|i +e~e _) y i e l d w i l l l i k e l y require the development of an e l e c t r o s t a t i c lens i n j e c t i o n system to increase the acceptance of the spectrometer section. F i n a l i z a t i o n of an i n j e c t i o n system design w i l l depend on the spectra observed with the presently proposed apparatus. Simultaneous with the c o l l e c t i n g of the TOF histogram, a |i +-decay histogram between the channeltron and the positron telescopes, gated by an incident u +, w i l l also be accumulated. The detection e f f i c i e n c y , e c ^ of the channeltron for 10 keV u + i s expected to be ~0.75, but this needs to be better determined. A measurement of th i s w i l l therefore be made, possibly as a function of energy, using the three Nai detectors placed s t r a t e g i c a l l y around the cone of the channeltron to provide the maximum possible detection s o l i d angle. The Nai detectors w i l l detect the decay positrons from u + stopped i n the channeltron and thus determine the absolute f l u x of stopped muons. These measurements w i l l of course require appropriate veto and coincidence s c i n t i l l a t o r s to c o r r e c t l y define the s o l i d angles and s e n s i t i v e volume. The Nai detectors can also be used i n coincidence with the channeltron to reduce backgrounds i n the TOF spectrum, but with a loss i n event rate. With these considerations, the experimental rate R e xp i s then R = R I erf ( A Ix ) e, e . e.T T ; for \ = 0 (All.14) exp o v g u ; d ch Nai where R Q i s the incident \x+ f l u x , i s the p r o b a b i l i t y that the | i + do not decay i n f l i g h t , e ^ i s the detection e f f i c i e n c y of the channeltron and e„ _ i s the detection e f f i c i e n c y and s o l i d angle of the Nai c r y s t a l array. Nai For the proposed experiment and apparatus, R q ~ 10 6/s, ~ 0.75, G ^ j a - j - ~ 0.2 and ~ 0.85. For a gate width of 2 |j.s, erf(/T^/T^) i s approximately - 203 -equal to 0.84. From t h i s , conservative estimates of R , for both NaF and exp L i F , are then R « y (10/second) ; for NaF e x p ° (calculated for \=0) (All.15) «* y ( 5/second) ; for L i F where the value of y 0 i s of course d i f f e r e n t for the two c r y s t a l s All.7 Backgrounds There are three major sources of backgrounds to be considered; beam positrons, positrons from muons stopped i n the moderator and positrons from muons which decay i n f l i g h t through the spectrometer system. Beam positrons w i l l pass through the moderator and scatter downstream producing bremsstrahlung and a n n i h i l a t i o n r a d i a t i o n . These beam related backgrounds w i l l r e f l e c t the RF structure of the cyclotron and w i l l probably be greater i n transmission than i n r e f l e c t i o n geometry. Because of t h i s , a separated beam i s highly d e s i r a b l e . The positrons 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 into the acceptance of the spectrometer. These positrons are too energetic to be transported through the spectrometer, but c o l l i s i o n s with the walls of the vacuum chamber w i l l produce background r a d i a t i o n which i s f l a t i n time. Positrons from \x+ decaying i n f l i g h t w i l l also c o l l i d e with the vacuum chamber walls producing background r a d i a t i o n . In th i s case, however, the background w i l l not be f l a t , but w i l l decay with the muon mean l i f e t i m e m u l t i p l i e d by some p o s i t i o n dependent function. These backgrounds are d i f f i c u l t to estimate but w i l l c l e a r l y have to be minimized by shiel d i n g the detectors from a l l sources other than the target, reducing the beam contamination and i f f e a s i b l e reducing the momentum b i t e of the beam. - 204 -This experiment should c l o s e l y p a r a l l e l the e a r l i e r positron experiment at f i r s t using NaF and L i F i n the <100> or i e n t a t i o n . Depending on the r e s u l t s obtained with the a l k a l i halides, these investigations may be extended to other c r y s t a l orientations as well as other materials such as quartz (Eg « 9 eV) and s o l i d rare gases such as argon (Eg » 19 eV). This appendix (with modifications) was submitted as an experimental proposal (E-325) to the December 1984 meeting of the TRIUMF Experimental Evaluation Committee and was accepted at high p r i o r i t y . Preliminary r e s u l t s [18] of the f i r s t experiments have indeed shown p o s i t i v e i n d i c a t i o n s of a low energy (<10 eV) component for L i F . - 205 -APPENDIX I I I — COLLISION FREQUENCY OF THERMAL MUONIUM AIII.A Derivation Consider a point 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 uniform d i s t r i b u t i o n of N sph e r i c a l p a r t i c l e s of radius R. If one defines the number density to be N/V, where V i s the t o t a l volume of the sample, the mean free path L i s then written L = V (TT R 2 N ) - 1 ( A I I I . l ) By d i v i d i n g the mean free path L by the mean thermal v e l o c i t y v, one obtains the average time t between c o l l i s i o n s , namely t = 3 = V [% R 2 v N ) - 1 (AIII.2) v Taking the r e c i p r o c a l of Equation AIII.2 then gives the c o l l i s i o n frequency, and su b s t i t u t i n g the d e f i n i t i o n of the mean thermal v e l o c i t y , one has ^ / m N 1 N r J l \ - N r „2 u8kTil/2 , A T T T F(T) = - = - ( * R ) v = - (n R ) [ — ] (AIII.3) where k i s Boltzmann's constant and T i s the temperature. AIII.A.1 Low Density Limit For low packing densities (neglecting the volume of the s o l i d ) , the number density i s simply given by the equation v « ( r ? ' p < A I I I - 4 ) 4n R K o where M i s the mass of one grain ( p a r t i c l e ) , p i s the mass packing density ( i . e . , after compression) of the target p a r t i c l e s and p Q i s the mass density of the bulk material (for S i 0 2 ; p Q = 2.2 g/cm 3). Thus i n the low - 206 -density l i m i t , the c o l l i s i o n frequency i s - ® §- ^ m 1 1 / 2 < A m - 5 > "o This equation i s , however, not correct 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 with respect to the t o t a l volume of the sample. AULA.2 High Density Limit In the high packing density l i m i t , the volume of the s o l i d i s no longer n e g l i g i b l e , so that one must redefine the number density to be the number of p a r t i c l e s (grains) per unit "free volume" V^, namely V f = ( V ~ V s o l i d ) = V " N ( T * r 3 ) = V ^ " f f T * 3 ) ] (AIII.6) By combining Equations AIII.4 and AIII.6, one obtains f - - 3 - T [ ^ " l ] " 1 (AIII.7) V f 4it R J P Using this "corrected" number density, the c o l l i s i o n frequency for the high density l i m i t i s F< T> = l ^ ) 1 / 2 lir- lTl t 1 / 2 <aiii-8> Notice that for low de n s i t i e s , Equation AIII.8 reduces to the expression of Equation AIII.5. Q.E.D. - 207 -APPENDIX IV — TABULATED TRANSVERSE FIELD DATA S i 0 2 ( l ) Prepared at 110 "C; X M u Vs Temperature T (K) AT (K) \ M u ( u s - 1 ) A ^ i U ( n s - i ) 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 - 208 -Si0 2(3) Prepared at 600 °C; \ M u Vs Temperature T (K) AT (K) \ M u ( u s - 1 ) AX M u ( u s - 1 ) 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 - 209 -Si0 2 ( 2 ) Prepared at 110 °C; X M u Vs Temperature T (K) AT (K) \ M u ( u s - 1 ) A \ M u ( u s - 1 ) 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 - 210 -REFERENCES CHAPTER I. 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Oram, J.M. Bailey, P.W. Schmor, C A . Fry, R.F. K i e f l , J.B. Warren, G.M. Marshall and A. O l i n , Phys. Rev. L e t t . 52_, 910 (1984). - 218 -W. Brandt and R. Paulin, Phys. Rev. B 1_5, 2511 (1977). A.P. M i l l s , J r . and W.S. Crane, Phys. Rev. L e t t . 53_, 2165 (1984). K.F. Canter, P.G. Coleman, T.C. G r i f f i t h and G.R. Heyland, J . Phys. B _5, L167 (1972). G.M. Marshall, J.B. Warren, D.M. Garner, G.S. Clark, J.H. Brewer and D.G Fleming, Phys. L e t t . 65A, 351 (1978). R. Paulin and G. Ambrosino, J . Physique 29, 263 (1968). G.G. Myasishcheva, Yu.V. Obukhov, V.S. Roganov, L.Ya. Suvorov and V.G. Firsov, Soviet High Energy Chemistry 3^ , 460 (1969). E.V. Minaichev, G.G. Myasishcheva, Yu.V. Obukhov, V.S. Roganov, G.I. Savel'ev and V.G. Firsov, Soviet Physics 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 , Molecules and Atoms, Part B, edited 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. Phys. L5, 1 (1943). G.M. Marshall, Ph.D. Thesis, U n i v e r s i t y of B r i t i s h Columbia (1981). D.R. Harshman, J.B. Warren, J.L. Beveridge, K.R. Kendall, 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. Turner, (submitted to Phys. Rev. L e t t . ) . PUBLICATION LIST for 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. Marshall, J.H. Brewer, D.R. Harshman, and C.W. Clawson, Physical Review B 26_, 2432 (1982). Giant Muon Knight Shifts i n Antimony and Antimony Alloys, J.H. Brewer, D. R. Harshman, E. Koster, H. S c h i l l i n g , D.LI. Williams and M.G. P r i e s t l y , S o l i d State Communications 46_, 863 (1983). Positronium i n Si0 2 Powder at Low Temperature, R.F. K i e f l and D.R. Harshman, Physics Letters 98A, 447 (1983). Zero-Field Muon-Spin Relaxation i n CuMn Spin-Glasses Compared With Neutron and Susceptibility Measurements, Y.J. Uemura, D.R. Harshman, M. Senba, E. J . Ansaldo and A.P. Murani, Physical Review B (Rap. Comm.) 30, 1606 (1984) . Diffusion 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. Harshman, M. Senba and E.J. Ansaldo, Phys. Rev. B 31, 546 (1985) . Study of the Hybrid State of Y 9Co 7 (2 < T < 6 K) by Means of Zero Field 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. Sarkissian, S o l i d State Communications 55, 193 (1985). 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.Ll. Williams and E.J. Ansaldo, (accepted for 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. Glaunsinger, D.R. Harshman, J.H. Brewer, R. K e i t e l , M. Senba, E.J. Ansaldo, D.P. Spencer and D.R. Noakes, (accepted for 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, S.R. Kreitzman, D.R. Noakes, E.J. Ansaldo, D.R. Harshman and R. K e i t e l , (submitted to Phys. Rev. B, rapid communications). Anisotropic Muonium With Random Hyperfine Anisotropics: A New Static Relaxation Theory, R.E. Turner and D.R. Harshman, (sub. to Phys. Rev. B.) Observation of Low Energy u+ Emission from Solid Surfaces, D.R. Harshman, J.B. Warren, J.L. Beveridge, K.R. Kendall, 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. Turner, (submitted to Phys. Rev. L e t t . ) . Conference Proceedings (Refereed) Muonium In A1203 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 Positron A n n i h i l a t i o n , edited by P.G. Coleman, S.C. Sharma and L.M. Diana, 693 (North-Holland Publishing Company, 1983). Magnetic Susceptibility, Proton NMR and Muon Spin Rotation (uSR) Studies of an Unsupported Platinum Catalyst with Adsorbed H and 0, R.F. Marzke, W.S. Glaunsinger, 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 Structure and Properties of Hydrogen i n Metals, edited by P. Jena and C.B. Satterthwaite, (Plenum Publishing Corporation, 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, Hyperfine Interactions 17-19, 557 (1984). Hyperfine Splitting of Muonium i n Si0 2 Powder, R.F. K i e f l , B.D. Patterson, E. Holzschuh, W. Odermatt and D.R. Harshman, Hyperfine Interactions 17-19, 563 (1984). Zero-Field pSR i n a Spin Glass CuMn (1.1 at. % ) : Precise Measurement of Static and Dynamic Effects 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 , Hyperfine Interactions 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. Fisk, S. Lambert, M.S. T o r i k a c h v i l i and M.B. Maple, Hyperfine Interactions 17-19, 509 (1984). Positive Muon Knight Shift i n Graphite and Grafoil, F.N. Gygax, A. Hintermann, A. Schenck, W. Studer, A.J. van der Wal, J.H. Brewer and D.R. Harshman, Hyperfine Interactions 17-19, 383 (1984). Positive Muons i n Antimony Bismuth Alloys, F.N. Gygax, A. Hintermann, A. Schenck, W. Studer, A.J. van der Wal, J.H. Brewer, D.R. Harshman, E. Koster, H. S c h i l l i n g , D.LI. Williams and M.G. P r i e s t l y , Hyperfine Interactions 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 . Appl. Phys. 57_, 3197 (1985). Muonium on Surfaces, D.R. Harshman, (Invited Paper), Proceedings of the European Workshop on the Spectroscopy of Sub-Atomic Species i n Non-Metallic Sol i d s , 3-7 Sept. 1985, Vitry-sur-Seine, France, ( i n preparation). Papers Published i n Unrefereed Journals Study of the Hybrid State of Y 9Co 7 (2 < T < 6K) by means of Zero Field 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. Sarkissian, LiSR-Newsletter 30, 1661 (1984). 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. Williams, K.M. Crowe and E.J. Ansaldo, LtSR-Newsletter 30, 1675 (1984). Observation of Muon-Fluorine "Hydrogen" Bonding i n Ionic Crystals, 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 Positive Muon Spin Rotation i n Magnetic-Superconducting SmRh^ B^ , C.Y. Huang, Z. Fisk, 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, Dallas , Texas, USA, March 1982. Magnetic Susceptibility, Proton NMR and Muon Spin Rotation (uSR) Studies of Unsupported Platinum Catalysts with Adsorbed H and 0, R.F. Marzke, W.S. Glaunsinger, K.B. Rawlings, P. Van Rheenan, M. McKelvy, J.H. Brewer, D.R. Harshman and R.F. K i e f l , APS Meeting, Dallas, 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 Regional Nuclear Physics Conference, Banff, Alberta, 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 International Symposium on S t a t i s t i c a l Mechanics of Adsorption, T r i e s t e , I t a l y , July 1982. Observation of H NMR and of Muon Spin Rotation i n Unsupported Platinum, R.F. Marzke, W.S. Glaunsinger, 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 Society, Annual F a l l Meeting, Irvine, 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, University of 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, Un i v e r s i t y of V i c t o r i a , B.C., Canada, June 1983. Diffusion 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 Society 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 2 Surfaces, D.R. Harshman, J.H. Brewer, R. K e i t e l , M. Senba, J.M. Bailey and E.J. Ansaldo, CAP Meeting, University de Sherbrooke, Sherbrooke, Quebec, Canada, June 1984. Giant Muon Knight S h i f t s i n Antimony A l l o y s , J.H. Brewer, D.R. Harshman, E. Koster, S.R. Kreitzman and D.LI. Williams, CAP Meeting, University de Sherbrooke, Sherbrooke, Quebec, Canada, June 1984. Muon Spin Relaxation 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, University de Sherbrooke, Sherbrooke, Quebec, Canada, June 1984. Muonium D i f f u s i o n on Amorphous S i 0 2 Surfaces, D.R. Harshman, J.H. Brewer, R. K e i t e l , M. Senba, J.M. Bailey and E.J. Ansaldo, International Chemical Congress of P a c i f i c Basin S o c i e t i e s , Honolulu, Hawaii, USA, December 1984. Measurement of the Reaction Rate of Muonium with the Surface of a Supported Platinum Catalyst by Muonium Spin Rotation (MSR), R.F. Marzke, W.S., Glaunsinger, D.R. Harshman, E.J. Ansaldo, R. K e i t e l , D.R. Noakes, M. Senba and J.H. Brewer, APS Meeting, Baltimore, Maryland, USA, March 1985. Measurement of the Reaction Rate of Muonium with the Surface of a Supported Platinum Catalyst by Muonium Spin Rotation (MSR), R.F. Marzke, W.S. Glaunsinger, 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 Society Annual Spring 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 Surfaces, D.R. Harshman, Invited Talk, uSR Journal Seminar, Dept. of Physics, Univ. of C a l i f o r n i a at Berkeley, Berkeley, Ca., USA, 16 February 1983. Behavior of Muonium on S i 0 2 Surfaces, D.R. Harshman, Invited Talk, S o l i d State Seminar, Department of Physics, The Univ. of B r i t i s h Columbia, Vancouver, B.C., Canada, 28 February 1985. Muonium on S i l i c a Surfaces, D.R. Harshman, Invited Talk, presented at AT&T-Bell Laboratories, Murray H i l l , New Jersey, 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 Surfaces, D.R. Harshman, Invited Talk, Physics-Astronomy Seminar, Department of Physics, Western Washington Un i v e r s i t y , Bellingham, Washington, USA, 15 May 1985. Muonium on Surfaces, D.R. Harshman, Invited Talk, European Workshop on the Spectroscopy of Sub-Atomic Species i n Non-Metallic Solids, 3-7 Sept. 1985, Vitry-sur-Seine, France. The Interactions of Muonium with Surfaces, D.R. Harshman, Invited Talk, presented at Lawrence Livermore National Laboratory, Livermore, C a l i f o r n i a , USA, 21 October 1985.
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The interactions of muonium with silica surfaces Harshman, Dale Richard 1986
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Title | The interactions of muonium with silica surfaces |
Creator |
Harshman, Dale Richard |
Publisher | University of British Columbia |
Date Issued | 1986 |
Description | The behavior of muonium on the surface of fine (35 Å mean radius) SiO₂ powders has been studied using the techniques of muon spin rotation (μSR). Results indicate diffusion and trapping behavior of the muonium atoms on the silica surface, which is strongly influenced by the concentration of surface hydroxyl groups. Specifically, the presence of the surface hydroxyl groups has been shown to inhibit the motion of muonium on the silica surface. These studies have also provided information regarding the origin of the relaxation of the muon spin polarization for muonium on the silica surface. Specifically, a random anisotropic distortion of the muonium hyperfine interaction, induced by the local surface environment of the muonium atom, has been shown to be a principal contributor to the relaxation of the muon ensemble spin polarization, whereas the random local magnetic fields due to the neighboring hydroxyl protons were found to play only a minor role. From this result, the observed strong dependence of the relaxation on the surface hydroxyl concentration has been attributed to an associated hyperfine distortion, induced by the neighboring hydroxyls. A new spin relaxation theory, for the case of random anisotropic hyperfine distortions, has also been developed to explain the data. Gas adsorption isotherm studies were also performed, with ⁴He at 6 K, which show the muonium asymmetry to be strongly influenced by the fractional surface coverage. These results clearly indicate that the muonium formation probability decreases with increasing surface coverage, suggesting that the charge exchange cross section at the silica surface is significant. The implication of these results with regard to the origins of muonium formation (i.e., surface or bulk formation) is as yet unclear, however, since the precise role played by the adsorbed helium atoms is not known. These investigations have also been extended to platinum loaded silica, where the first surface reaction of muonium has been observed; the reaction rate of muonium with the surface of oxygen-covered platinum microcrystals was found to be 3.5 ± 0.15 μs⁻¹. |
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Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-08-06 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0085069 |
URI | http://hdl.handle.net/2429/27108 |
Degree |
Doctor of Philosophy - PhD |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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