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An improved upper limit for muonium conversion to antimuonium Marshall, Glen Murray 1981

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AN IMPROVED UPPER LIMIT FOR MUONIUM CONVERSION TO ANTIMUONIUM by GLEN MURRAY MARSHALL B . S c , M c G i l l U n i v e r s i t y , 1974 .Sc., U n i v e r s i t y of B r i t i s h Columbia, 1977 THESIS SUBMITTED IN THE REQUIREMENTS DOCTOR OF PARTIAL FULFILLMENT OF FOR THE DEGREE OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES ( P h y s i c s ) We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA F e b r u a r y , 1981 © Glen Murray M a r s h a l l , 1981 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be gran t e d by the head o f my department o r by h i s or her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of C\y^-J S e *->S  The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date DE-6 (2/79) i i ABSTRACT An experiment r e s u l t i n g i n the r e d u c t i o n of the upper l i m i t f o r muonium (yU +e*) c o n v e r s i o n t o antimuonium (yuc'e*) i s d e s c r i b e d . The l i m i t o b t a i n e d f o r the e f f e c t i v e f o u r f e r m i o n c o u p l i n g c o n s t a n t i s G < 42G p (95% c o n f i d e n c e l e v e l ) . The muon i n a system i n i t i a l l y formed as muonium and e v o l v i n g under the most f a v o r a b l e c o n d i t i o n s w i l l thus be i d e n t i f i a b l e as a n e g a t i v e p a r t i c l e i n l e s s than 4% of the observed decays. The r e s u l t s improve by over one o r d e r of magnitude the be s t p r e v i o u s l i m i t o b t a i n e d from a s e a r c h f o r e~e~ ->yu.^ u.- i n t e r a c t i o n s . N e i t h e r p r o c e s s i s e x p e c t e d t o e x i s t i f an a d d i t i v e c o n s e r v a t i o n law i s obeyed by muon number. The p r e s e n t s t a t u s of the t h e o r y of e l e c t r o w e a k i n t e r a c t i o n s , as i t p e r t a i n s t o muonium c o n v e r s i o n , i s r e v i e w e d . I t i s shown t h a t muon number n o n c o h s e r v a t i o n can be accommodated i n a v a r i e t y of ways, some of which might a l l o w a v a l u e f o r G of 0.1G F. The s t e p s t h a t were t a k e n t o make the p r e s e n t experiment as s e n s i t i v e as p o s s i b l e a r e d e t a i l e d . The major improvement over p r e v i o u s c o n v e r s i o n e x p e r i m e n t s i s the use of f i n e s i l i c a powder i n c a r e f u l l y a r r a n g e d l a y e r s t o a l l o w muonium t o e x i s t f o r a l a r g e f r a c t i o n of i t s l i f e t i m e i n vacuum, where c o n v e r s i o n i s not h i g h l y s u p p r e s s e d . Another i m p o r t a n t f a c e t of the t e c h n i q u e , which i s d e s c r i b e d i n d e t a i l , i s the use of an i n t e n s e beam of s u r f a c e muons w i t h a s t o p p i n g d e n s i t y p r e v i o u s l y u n a t t a i n a b l e . A c h a p t e r on t h e a n a l y s i s of the d a t a c o n t a i n s a q u a n t i t a t i v e d i s c u s s i o n of the p r o c e s s e s which must o c c u r f o r i i i c o n v e r s i o n t o be d e t e c t e d . The numbers d e r i v e d t h e r e a r e e s s e n t i a l t o the e s t a b l i s h m e n t of a r e a l i s t i c l i m i t on the c o u p l i n g of muonium t o antimuonium. i v TABLE OF CONTENTS 1. INTRODUCTION 1 1.1. Muon Number C o n s e r v a t i o n 3 1.2. I n c o r p o r a t i o n of the T h i r d Lepton G e n e r a t i o n 6 2. THEORY OF THE MUONIUM-ANTIMUONIUM INTERACTION 9 2.1. The Theory B e f o r e 1967 10 2.1.1. The Four Fermion C u r r e n t - C u r r e n t I n t e r a c t i o n 10 2.1.2. Vacuum E i g e n s t a t e s of the Muonium-Antimuonium System 15 2.1.3. E v o l u t i o n of Antimuonium from Muonium 16 2.1.4. The E f f e c t of E l e c t r o m a g n e t i c F i e l d s on C o n v e r s i o n 18 2.1.5. Muonium i n the Presence of M a t t e r 20 2.2. U n i f i e d Gauge T h e o r i e s and Muonium-Antimuonium C o n v e r s i o n 23 2.2.1. Gauge T h e o r i e s of Weak and E l e c t r o m a g n e t i c I n t e r a c t i o n s 25 2.2.2. Muon Number V i o l a t i o n and Muonium C o n v e r s i o n i n Extended T h e o r i e s ' 31 3. DETAILS OF THE CONVERSION EXPERIMENT 37 3.1. A Review of R e l a t e d E x p e r i m e n t s : 39 3.1.1. Other Muonium-Antimuonium Exp e r i m e n t s 39 3.1.2. E x p e r i m e n t s w i t h Other Systems 42 3.2. The Apparatus and Techniques Used 44 3.2.1. M13 and S u r f a c e Muons 44 3.2.2. The T a r g e t : P r o d u c t i o n of Muonium i n Vacuum . 52 3.2.3. Magnetic F i e l d Measurement and C o n t r o l 57 3.2.4. D e t e c t o r s and Hardware i n the T a r g e t Region . 62 3.3. Data A c q u i s i t i o n 66 3.3.1. P r i m a r y Sources of Background and T h e i r M i n i m i z a t i o n 66 3.3.2. E l e c t r o n i c s C o n f i g u r a t i o n Used 72 3.3.3. A c c u m u l a t i o n and St o r a g e of Data 77 4. ANALYSIS AND INTERPRETATION OF THE DATA 79 4.1. The R e s u l t s : D e t e r m i n a t i o n of a L i m i t on the Number of Events Observed 80 4.2. R e l a t i o n s h i p of E v e n t s Observed t o the Upper L i m i t f o r C o n v e r s i o n 88 4.2.1. Muonium F o r m a t i o n 91 4.2.2. P r o b a b i l i t y of E j e c t i o n i n t o Vacuum; S i n g l e F o i l s 93 4.2.3. C o n v e r s i o n P r o b a b i l i t y . . . j 103 4.2.4. N e g a t i v e Muonic X-ray P r o b a b i l i t y 105 4.2.5. D e t e c t i o n E f f i c i e n c y 109 4.2.6. D e t e c t a b l e E v e n t s i n Terms of the C o u p l i n g Constant G 117 5. CONCLUSION 120 5.1. The L i m i t on the Muonium-Antimuonium C o u p l i n g Constant 120 5.2. F e a s i b i l i t y of an Improved Experiment 123 APPENDICES 126 A l . MUONIUM-ANTIMUONIUM CONVERSION VIA THE FOUR FERMION CURRENT-CURRENT INTERACTION 126 A2. MOTION OF MUONIUM ATOMS IN SPHERICAL SILICA PARTICLES 132 A3. MOTION OF MUONIUM ATOMS IN FINE POWDER LAYERS 139 BIBLIOGRAPHY 143 VI LIST OF FIGURES F i g u r e 2.1.4.1. S p l i t t i n g of m,=±l s t a t e s of muonium and antimuonium i n a magnetic f i e l d 19 F i g u r e 2.2.1. Diagrammatic e x p a n s i o n of the n e u t r i n o -l e p t o n s c a t t e r i n g o p e r a t o r 24 F i g u r e 2.2.2. Order e 4 (or Gp) l e p t o n - l e p t o n s c a t t e r i n g i n (a) QED, and (b) weak i n t e r a c t i o n s 25 F i g u r e 2.2.2.1. C o n v e r s i o n p r o c e s s v i a non-degenerate n e u t r i n o s 32 F i g u r e 2.2.2.2. C o n v e r s i o n v i a non-minimal Higgs c o u p l i n g . 34 F i g u r e 3.2.1.1. The M13 pion/muon c h a n n e l a t TRIUMF . ... 46 F i g u r e 3.2.1.2. M13 p o s i t i v e p a r t i c l e f l u x e s from a 1.45 mm g r a p h i t e t a r g e t 48 F i g u r e 3.2.1.3. E f f e c t of h o r i z o n t a l s l i t s on M13 p a r t i c l e f l u x and beam spot d i m e n s i o n 49 F i g u r e 3.2.1.4. E f f e c t of h o r i z o n t a l jaws on M13 p a r t i c l e f l u x and beam spot d i m e n s i o n 50 F i g u r e 3.2.1.5. S u r f a c e muon r a t e v e r s u s p r o t o n beam p o s i t i o n on the p r o d u c t i o n t a r g e t 51 F i g u r e 3.2.1.6. S u r f a c e muon i n t e g r a l range c u r v e 52 F i g u r e 3.2.2.1. I l l u s t r a t i o n of the t a r g e t used f o r muonium p r o d u c t i o n i n vacuum 53 F i g u r e 3.2.2.2. A s i n g l e l a y e r of the t a r g e t , i l l u s t r a t i n g t he mechanism of muonium p r o d u c t i o n i n vacuum 54 F i g u r e 3.2.3.1. S a t u r a b l e i n d u c t o r magnetometer waveforms (a) i n z e r o ambient f i e l d . (b) w i t h a nonzero f i e l d component B. 61 F i g u r e 3.2.4.1. Schematic of muonium c o n v e r s i o n a p p a r a t u s . 63 F i g u r e 3.2.4.2. Photopeak e f f i c i e n c i e s v e r s u s energy f o r the two d e t e c t o r s used 64 F i g u r e 3.3.2.1. E l e c t r o n i c s diagram f o r c i r c u i t used (see t e x t ) 73 F i g u r e 4.1.1. Gamma s p e c t r a from s i m p l e summation of a l l d a t a 81 F i g u r e 4.1.2. Summation of s h i f t e d s p e c t r a i n the Ca X-ray r e g i o n . Note s u p p r e s s i o n of the z e r o of the y a x i s . .. 83 F i g u r e 4.1.3. X-ray d a t a f o r p o s i t i v e muons i n argon gas t a r g e t a t room temperature and one atmosphere 85 F i g u r e 4.2.1.1. Muonium s p i n r o t a t i o n s i g n a l i n the c o n v e r s i o n t a r g e t 92 F i g u r e 4.2.2.1. MSR p r e c e s s i o n s i g n a l r e l a x e d by oxygen gas 95 F i g u r e 4.2.2.2. E x p e c t e d time d i s t r i b u t i o n of decays from muonium d r i f t i n g t h e r m a l l y i n vacuum from a s i n g l e f o i l . 99 F i g u r e 4.2.2.3. Muon decay c u r v e o b t a i n e d w i t h narrow t e l e s c o p e c e n t r e d 1 cm downstream of s i l i c a l a y e r 100 F i g u r e 4.2.2.4. Enhancement i n the muon decay spectrum. .102 F i g u r e 4.2.5.1. X-ray s p e c t r a from n e g a t i v e muons i n the c o n v e r s i o n t a r g e t . 114 F i g u r e 4.2.5.2. X-ray spectrum from n e g a t i v e muons i n s i l i c o n d i o x i d e 116 F i g u r e 4.2.5.3. X-ray spectrum from n e g a t i v e muons i n c a l c i u m o x i d e 117 v i i i F i g u r e A2.1. Geometry f o r muonium e m i s s i o n from a sphere. 133 F i g u r e A3.1. Geometry f o r muonium e m i s s i o n from a l a y e r . 139 i x LIST OF TABLES T a b l e 1.1.1. O r i g i n a l le'pton number a s s i g n m e n t s , w i t h the p o s i t i v e muon c o n s i d e r e d as an a n t i p a r t i c l e 3 T a b l e 1.1.2. Muon number assignments 4 Tab l e 1.2.1. S e q u e n t i a l l e p t o n number assignments 6 T a b l e 4.1.1. V a l u e s o b t a i n e d by MINUIT f o r d i f f e r e n t ranges of d a t a a n a l y z e d 88 X ACKNOWLEDGMENT I t i s a p l e a s u r e t o e x p r e s s my g r a t i t u d e t o the many people whose encouragement and a s s i s t a n c e were e s s e n t i a l t o the r e s e a r c h d e s c r i b e d h e r e i n . My t h e s i s s u p e r v i s o r , Dr. John B. Warren, d i d not t i r e of r e a s s u r i n g me of the worth of the expe r i m e n t ; he i s a l s o r e s p o n s i b l e f o r the c o n c e p t i o n of u s i n g f i n e powder t a r g e t s f o r p r o d u c i n g muonium i n vacuum. Dr. J e s s H. Brewer has been a p a t i e n t and e x c e l l e n t source of a d v i c e , i d e a s , and e x p e r i m e n t a l e x p e r t i s e , p a r t i c u l a r l y i n the ar e a of muon s p i n r o t a t i o n . For the smoothe, o r d e r l y p r o g r e s s of the experiment I am e s p e c i a l l y i n d e b t e d t o Dr. C h r i s J . Oram, Mr. Robert F. K i e f l , and Mr. George S. C l a r k , w i t h o u t whose a s s i s t a n c e i t would not have been s u c c e s s f u l l y c o m p l e t e d . Dr. Da v i d M. Garner k i n d l y s u p p l i e d and s u p p o r t e d s e v e r a l computer r o u t i n e s used i n the a n a l y s i s , and p l a y e d a dynamic r o l e i n the o r i g i n a l e s t a b l i s h m e n t of muonium i n vacuum. The a p p a r a t u s was c o n s t r u c t e d w i t h the e x t r e m e l y c a p a b l e t e c h n i c a l a s s i s t a n c e of Mr. A l a n Morgan, Mr. C h r i s S t e v e n s , and Mr. B r i a n S m ith, who, a l o n g w i t h the o p e r a t o r s and s t a f f of TRIUMF, deserve my s i n c e r e t h a n k s . I w i s h t o acknowledge s e v e r a l p e r s o n s not i n v o l v e d i n the experiment f o r t h e i r f r i e n d s h i p and s u p p o r t . Dr. Pamela A. Maher s u r v i v e d the o r d e a l of l i v i n g w i t h the odd hours of a TRIUMF grad u a t e s t u d e n t most a m i a b l y and a d m i r a b l y . Mr. H. Dean R o l f s o n was the f i r s t t o demonstrate t o me the o r d e r and l o g i c t h a t i s the magic of p h y s i c s . My b r o t h e r , G r e g o r y , was a l s o r e s p o n s i b l e f o r my e a r l i e s t academic a s p i r a t i o n s , w h i l e my x i f a t h e r , G e r a l d , p r o v i d e d the encouragement and o p p o r t u n i t y t o a t t a i n them. But the l e s s o n s t h a t I v a l u e most a r e not those of p h y s i c s , but of p e o p l e , of r i g h t and wrong, and of the need t o be i n e v e r y sense an i n d i v i d u a l ; f o r these and many o t h e r t h i n g s I thank my mother, the l a t e Grace E l i z a b e t h M a r s h a l l , t o whom t h i s t h e s i s i s d e d i c a t e d . G.M.M. 1 1. INTRODUCTION P h y s i c s i n the t w e n t i e t h c e n t u r y can be d i s t i n g u i s h e d from the o t h e r s c i e n c e s p a r t l y by i t s concern not o n l y w i t h i n t e r a c t i o n s whose e f f e c t s are e v i d e n t i n everyday l i f e , namely those of e l e c t r o m a g n e t i c and g r a v i t a t i o n a l o r i g i n , but the so-c a l l e d s t r o n g and weak phenomena as w e l l . The l a t t e r p l a y no e s t a b l i s h e d r o l e i n o t h e r n a t u r a l or l i f e s c i e n c e s except p o s s i b l y as t o o l s which have been deve l o p e d t h r o u g h an u n d e r s t a n d i n g of the p h y s i c s . C o n s i d e r a t i o n of the apparent c l a s s i f i c a t i o n of n a t u r e ' s p r o c e s s e s i n t o f o u r d i v i s i o n s has been a common o c c u p a t i o n of modern p h y s i c i s t s , who would seek some reason or c o n n e c t i o n w i t h i n what seems (or seemed, u n t i l r e c e n t l y ) a r a t h e r a r b i t r a r y scheme. We are not s a t i s f i e d t o t h i n k of n a t u r e as a r b i t r a r y . An advance toward the i d e a l of a u n i f i e d approach t o two or more of the fundamental p r o c e s s e s was made p o s s i b l e by the development of gauge f i e l d t h e o r i e s . I t now appears t h a t t h e r e i s an i n h e r e n t c o n n e c t i o n between the weak and e l e c t r o m a g n e t i c f o r c e s , t o the e x t e n t t h a t they a r e now c o n s i d e r e d s e p a r a t e m a n i f e s t a t i o n s of something known as the e l e c t r o w e a k i n t e r a c t i o n . The analogy between the elementary p a r t i c l e s which do not t a k e p a r t i n the s t r o n g i n t e r a c t i o n , the l e p t o n s , w i t h those p h y s i c a l l y e l u s i v e but w i d e l y a c c e p t e d hadron c o n s t i t u e n t s , the q u a r k s , i n s p i r e the hope t h a t the s t r o n g i n t e r a c t i o n can be s o l i d l y i n t e g r a t e d w i t h the e l e c t r o w e a k by 2 f u r t h e r development of the gauge f i e l d approach. Many, many problems have y e t t o be s o l v e d b e f o r e i t can be s a i d t h a t even the e l e c t r o w e a k s e c t o r i s u n d e r s t o o d . P r o g r e s s i s made p a r t l y by s e a r c h i n g f o r p r o c e s s e s a l l o w e d (but not n e c e s s a r i l y demanded) by a u n i f i e d t h e o r y . In many ca s e s these p r o c e s s e s are r a r e , and one measure of p r o g r e s s has been the a b i l i t y of the e x p e r i m e n t e r t o e s t a b l i s h j u s t how r a r e they must b e 1 , so t h a t a t l e a s t a few v e r s i o n s of the t h e o r y can be d e f i n i t e l y o v e r r u l e d . One of the b y p r o d u c t s of t h i s s i t u a t i o n i s the e n d l e s s c h a l l e n g e i t o f f e r s t o the w i l e s of the e x p e r i m e n t e r . The s u b j e c t of t h i s t h e s i s i s the improvement of the l i m i t on muonium ( juCe~) c o n v e r s i o n t o antimuonium ( y U ~ e + ) . The h i s t o r y of i n t e r e s t i n t h i s r e a c t i o n p r e d a t e s the a c c e p t a n c e of gauge f i e l d t h e o r i e s because of an a m b i g u i t y i n the form of muon number c o n s e r v a t i o n . Recent e v i d e n c e has done much t o r e s o l v e the a m b i g u i t y , but i t w i l l be demonstrated i n t h i s t h e s i s t h a t the l a t i t u d e of the' t h e o r y s t i l l j u s t i f i e s the c o s t and e f f o r t of a t t e m p t i n g t o reduce the upper l i m i t f o r c o n v e r s i o n . The o u t s t a n d i n g problem i n a c o n v e r s i o n experiment has been, f o r reasons which w i l l be d i s c u s s e d , the d i f f i c u l t y i n p r o d u c i n g ample muonium i n an environment c o n d u c i v e t o c o n v e r s i o n : the s u c c e s s of the p r e s e n t experiment i s l a r g e l y due t o the d i s c o v e r y of a method by which t h i s may be a c c o m p l i s h e d . Another i m p o r t a n t f a c t o r has been the development i n the p a s t f i v e y e a r s , l a r g e l y a t TRIUMF, of muon beams w i t h i n t e n s i t i e s *The concept of "never" i s r a r e i n i t s e l f , s i n c e i t i s an i m p o s s i b l e l i m i t t o a t t a i n e x p e r i m e n t a l l y . 3 and s t o p p i n g d e n s i t i e s which make a c o n v e r s i o n s e a r c h p r a c t i c a b l e . The o t h e r f a c e t s of the experiment might be d e s c r i b e d as c o n v e n t i o n a l . Some s u g g e s t i o n s w i l l be made i n the c o n c l u d i n g c h a p t e r f o r improvements which may i n f u t u r e a l l o w the l i m i t on muonium c o n v e r s i o n t o be reduced s t i l l f u r t h e r . 1.1. Muon Number C o n s e r v a t i o n The f i r s t s u g g e s t i o n t h a t muonium might s p o n t a n e o u s l y c o n v e r t t o antimuonium was made by P o n t e c o r v o ( P o n t e c o r v o , 1957), who c o n s i d e r e d the p r o c e s s as an analogue t o the weak n e u t r a l k a o n - a n t i k a o n m i x i n g . There had p r e v i o u s l y been d i s c u s s e d ( K o n o p i n s k i and Mahmoud, 1953) the p o s s i b i l i t y of an a d d i t i v e law g o v e r n i n g the c o n s e r v a t i o n of a l e p t o n number L, which i n one form d i d not d i s a l l o w muonium c o n v e r s i o n (see T a b l e 1.1.1). In another form, f a v o r e d by the a u t h o r s a t t h a t t i m e , + — + — others Particle L +1 -1 +1 -1 0 T a b l e 1.1.1. O r i g i n a l l e p t o n number a s s i g n m e n t s , w i t h the p o s i t i v e muon c o n s i d e r e d as an a n t i p a r t i c l e . R e a c t i o n s must co n s e r v e £L. 4 i t d i d d i s a l l o w c o n v e r s i o n because the n e g a t i v e muon was c o n s i d e r e d as an a n t i p a r t i c l e r a t h e r than a p a r t i c l e , and the l e p t o n numbers f o r the muonic p a r t i c l e s were of the o p p o s i t e s i g n . P a r e n t h e t i c a l l y i t i s i n t e r e s t i n g t h a t t h i s p a r t i c u l a r form i s the o n l y one, even t o d a y , w i t h i n which the nomenclature of muonium and antimuonium makes sense. I f the p o s i t i v e muon i s an a n t i p a r t i c l e , the /M*e' system s h o u l d be known as antimuonium, c o n t r a r y t o p o p u l a r usage. At any r a t e , almost c o n c u r r e n t w i t h P o n t e c o r v o ' s s u g g e s t i o n were the f i r s t ( N i s h i j i m a , 1957; Schwinger, 1957) p r o p o s a l s t h a t t h e r e might be a f u r t h e r quantum number, a muon number Ly* (see Table 1.1.2), t h a t a l s o must be c o n s e r v e d . I f i t were, p r o c e s s e s such as Particle e",v e e+,v"e u'.v^ y +,v others L y 0 0 + 1 - 1 0 Table 1.1.2. Muon number a s s i g n m e n t s . R e a c t i o n s must conserve ZTLyu. ( N i s h i j i m a , 1957; Schwinger, 1 9 5 7 ) i n an a d d i t i v e scheme, or (-l)cu/*- i n a m u l t i p l i c a t i v e scheme ( F e i n b e r g and Weinberg, 1961a). yu e¥, JJ~ eee, and y"-"p e"p, which i n v o l v e a muon c o n v e r t i n g t o an e l e c t r o n w i t h o u t n e u t r i n o e m i s s i o n or a b s o r p t i o n , would not take p l a c e ; they had not been observed (and s t i l l haven't) i n e x p e r i m e n t s s e n s i t i v e t o r a t e s below those o t h e r w i s e e x p e c t e d . C o n s i s t e n c y of the muon number concept i n t h i s form a l s o demanded the acc e p t a n c e of a second type of n e u t r i n o , the muon n e u t r i n o , which i s d i s t i n g u i s h a b l e 5 from i t s e l e c t r o n c o u n t e r p a r t by i t s muon number. S e v e r a l y e a r s l a t e r a n o ther scheme was proposed ( F e i n b e r g and Weinberg, 1961a), based on i d e a s which w i l l be e l a b o r a t e d upon i n s e c t i o n 2.1.1, i n which muon number was co n s e r v e d not as the sum £ Iy* but as the pro d u c t (-1) f, a l o n g w i t h the u s u a l a d d i t i v e l e p t o n c o n s e r v a t i o n . O b v i o u s l y any p r o c e s s c o n s e r v i n g £ I y i , thus s a t i s f y i n g the a d d i t i v e scheme, would a l s o s a t i s f y the m u l t i p l i c a t i v e scheme, but the c o n v e r s e i s not t r u e . The m u l t i p l i c a t i v e scheme a l s o demanded a d i s t i n g u i s h a b l e muon n e u t r i n o ; i t s e x i s t e n c e was proven i n an experiment u s i n g n e u t r i n o s from p i o n decay ( m o s t l y produced i n a s s o c i a t i o n w i t h p o s i t i v e muons) i n t e r a c t i n g w i t h n e u t r o n s . The absence of e l e c t r o n p r o d u c t i o n s i g n i f i e d the "muonness" of the n e u t r i n o s , and a l s o s u p p o r t e d some form of the muon c o n s e r v a t i o n law. No r e a l r e s o l u t i o n t o the a m b i g u i t y i n the form of muon c o n s e r v a t i o n o c c u r r e d f o r more than one and a h a l f decades. Only r e c e n t l y have e x p e r i m e n t a l d a t a been o b t a i n e d which tend t o d i s c o u n t the m u l t i p l i c a t i v e approach. A t e s t must be made by s e a r c h i n g f o r i n t e r a c t i o n s f o r which EL^. changes by a t l e a s t two u n i t s between i n i t i a l and f i n a l s t a t e s , w h i l e s a t i s f y i n g o t h e r c o n s e r v a t i o n laws f o r charge and l e p t o n number. C a n d i d a t e s f o r the t e s t a r e t h e r e a c t i o n s yu*e +7J?»v^ , e" e" -*JJCJ*T , and yu*e~-»y"~e*. The f i r s t , a charged c u r r e n t i n t e r a c t i o n , has r e c e n t l y been se a r c h e d f o r ( W i l l i s e_t a l . , 1980), and i t s n o n o b s e r v a t i o n s t r o n g l y f a v o r s the a d d i t i v e a ssignment. The l a t t e r two, both mediated by a n e u t r a l c u r r e n t i n t e r a c t i o n , have a l s o been se a r c h e d f o r (Barber e t a l . , 1 9 6 9 ; Amato e t a l . , 1968, and t h i s work) but the s e n s i t i v i t y i s too low and the r e s u l t s a r e i n c o n c l u s i v e f o r a d e t e r m i n a t i o n of the 6 c h a r a c t e r of muon number c o n s e r v a t i o n . 1.2. I n c o r p o r a t i o n of the T h i r d Lepton G e n e r a t i o n Any modern d i s c u s s i o n of l e p t o n i n t e r a c t i o n s must i n c l u d e some mention of the newest g e n e r a t i o n , the t a u p a r t i c l e or tauon, and i t s n e u t r i n o ( P e r l e_t a l . , 1975). The same i s t r u e h e r e , f o r i t s e x i s t e n c e and decay modes must be ac c o u n t e d f o r by the c o n s e r v a t i o n l a w s . A l l e v i d e n c e i s c o m p a t i b l e w i t h the s e q u e n t i a l l e p t o n model, i n which each type of l e p t o n - n e u t r i n o p a i r ( e l e c t r o n , muon, tauon, and p o s s i b l y h e a v i e r u n d i s c o v e r e d g e n e r a t i o n s ) i s a s s o c i a t e d w i t h a d i s t i n c t l e p t o n number, as i n T a b l e 1.1.3 (see P e r l , 1978, f o r a r e v i e w ) . Note t h a t t h i s i s Particle e \ v e + — e , v e y ' , v y H + ' V y + — others L e +1 -1 0 0 0 0 0 L y 0 0 +1 -1 0 0 0 L T 0 0 0 0 +1 -1 0 T a b l e 1.2.1. S e q u e n t i a l l e p t o n number a s s i g n m e n t s . R e a c t i o n s must c o n s e r v e S L e , E L ^ , and E L T s e p a r a t e l y , i n the a d d i t i v e scheme. e q u i v a l e n t f o r muons and e l e c t r o n s t o the numbers a l r e a d y 7 a s s i g n e d , s i n c e i n an a d d i t i v e scheme any l i n e a r c o m b i n a t i o n of quantum numbers (such t h a t the t r a n s f o r m a t i o n m a t r i x has nonzero d e t e r m i n a n t ) w i l l g i v e e q u i v a l e n t s e l e c t i o n r u l e s (L = L^+L^); the m u l t i p l i c a t i v e law needs o n l y t o be accompanied by the c o n d i t i o n t h a t I C f L ^ L ^ ) ( i . e . , p a r t i c l e s minus a n t i p a r t i c l e s i n the l e p t o n s e c t o r ) must be c o n s e r v e d , t o o b t a i n the p r e v i o u s s e l e c t i o n r u l e s . A r e f o r m u l a t i o n of the m u l t i p l i c a t i v e scheme t o i n c l u d e the tauon might, i n an ad hoc manner, w i t h o u t r e c o u r s e t o t h e o r e t i c a l j u s t i f i c a t i o n s , demand s e p a r a t e c o n s e r v a t i o n of 2 (l± + L / u + L x ) , (-1) £' L/ /, and e i t h e r ( - l ) E L * o r the more s t r i n g e n t EL . On the o t h e r hand, the m u l t i p l i c a t i v e law i s the r e s u l t of an assumed i n v a r i a n c e under p e r m u t a t i o n of p r i m i t i v e l e p t o n s (see s e c t i o n 2.1.1). M a i n t a i n i n g and e x t e n d i n g t h a t approach has l e d t o a t h e o r y (Derman, 1978) which r e q u i r e s , g i v e n the n o n e x i s t e n c e o f e ^ , the c o n s e r v a t i o n E (I-e+L^+L^) and (-1 ) E ( L y + L ^ o n l y ; the decay -c-» may take p l a c e , a l b e i t a t a reduced r a t e , w e l l below p r e s e n t l i m i t s . Muonium may a l s o c o n v e r t t o antimuonium. A more d e t a i l e d e x p l a n a t i o n of muon number and muonium c o n v e r s i o n w i l l be g i v e n i n the second c h a p t e r , i n which the aim i s t o c o n v i n c e the reader of a s o l i d j u s t i f i c a t i o n f o r an improvement of e x i s t i n g l i m i t s . The e xperiment performed a t TRIUMF t o a c h i e v e t h i s g o a l w i l l be d e s c r i b e d i n the t h i r d c h a p t e r , w i t h f u r t h e r r e f e r e n c e t o the e x p e r i m e n t s which have gone b e f o r e . The a n a l y s i s of the d a t a , a l o n g w i t h a q u a n t i t a t i v e d e s c r i p t i o n of the p r o c e s s e s which must be u n d e r s t o o d f o r the r e d u c t i o n of the numbers o b t a i n e d i n t o the form of a l i m i t f o r muonium c o n v e r s i o n , c o m p r i s e the f o u r t h c h a p t e r . The f i f t h c h a p t e r c o n t a i n s the f i n a l r e s u l t , and a 8 b r i e f d i s c u s s i o n of the i n f l u e n c e of v a r i o u s e x p e r i m e n t a l f a c t o r s on the s e n s i t i v i t y a c h i e v e d . 9 2. THEORY OF THE MUONIUM-ANTIMUONIUM INTERACTION A p a r t i a l l y s u c c e s s f u l t h e o r y of weak i n t e r a c t i o n s was f i r s t d e veloped by Fermi ( F e r m i , 1934) t o d e s c r i b e the beta decay of n u c l e i . The proposed i n t e r a c t i o n was v e c t o r a l o n e , a l l o w i n g t r a n s i t i o n s between n u c l e a r s t a t e s of e q u a l a n g u l a r momentum. S h o r t l y t h e r e a f t e r a more g e n e r a l i n t e r a c t i o n was proposed (Gamow and T e l l e r , 1936) which was a l i n e a r c o m b i n a t i o n of v e c t o r as w e l l as s c a l a r , p s e u d o s c a l a r , a x i a l v e c t o r , and t e n s o r terms, and c o u l d c o u p l e n u c l e a r s t a t e s d i f f e r i n g by one u n i t of a n g u l a r momentum. N u c l e a r beta decay was s t u d i e d from t h i s f o u n d a t i o n u n t i l the m i d d l e f i f t i e s , when the t a c i t a s s umption of p a r i t y c o n s e r v a t i o n i n the i n t e r a c t i o n was q u e s t i o n e d and found t o be i n c o r r e c t (Wu et a l . , 1957). F u r t h e r e x p e r i m e n t s were c o n s i s t e n t w i t h an i n t e r a c t i o n H a m i l t o n i a n w i t h both v e c t o r and a x i a l v e c t o r p a r t s , which now forms the b a s i s of the s o - c a l l e d v e c t o r minus a x i a l v e c t o r (V-A) t h e o r y of weak i n t e r a c t i o n s . T h i s t h e o r y was s t i l l not p e r f e c t because of the u n p h y s i c a l h i g h energy b e h a v i o r ; i t i s now g e n e r a l l y r e g a r d e d as a p h e n o m e n o l o g i c a l l y c o r r e c t d e s c r i p t i o n of the low energy l i m i t of a c l a s s of more w i d e l y a p p l i c a b l e f o r m u l a t i o n s , known g e n e r i c a l l y as gauge t h e o r i e s . They d e s c r i b e the ob s e r v e d weak f o r c e i n terms of massive exchange p a r t i c l e s , t he W and Z bosons. B e f o r e g e t t i n g i n t o more d e t a i l s of t h e s e l a t e s t a p proaches, however, i t i s u s e f u l t o p r o v i d e a framework f o r muonium-antimuonium c o n v e r s i o n based on a s i m p l e V-A i n t e r a c t i o n 10 H a m i l t o n i a n . 2.1. The Theory B e f o r e 1967 The c a l c u l a t i o n s i n t h i s s e c t i o n f o l l o w those d e s c r i b e d i n an e a r l y paper on the s u b j e c t ( F e i n b e r g and Weinberg, 1961b), when a muonium-antimuonium c o n v e r s i o n experiment was l e s s than a gleam i n the e x p e r i m e n t e r ' s eye. That r e f e r e n c e s t r e s s e d the severe i n f l u e n c e e x e r t e d by the atom's environment on any o b s e r v a b l e muonium c o n v e r s i o n r a t e . I n c l u d e d here i s some t h e o r e t i c a l background which w i l l prove u s e f u l i n the a n a l y s i s of the experiment performed a t TRIUMF. 2.1.1. The Four Fermion C u r r e n t - C u r r e n t I n t e r a c t i o n Low energy weak i n t e r a c t i o n s have been w e l l d e s c r i b e d by an i n t e r a c t i o n H a m i l t o n i a n of the form H w ( x ) = 2 - ( 1"> GF J * ( x ) J * ( x ) , 2.1.1.1 where summation over r e p e a t e d Greek i n d i c e s i s assumed. J ^ ( x ) i s the weak f o u r - c u r r e n t , a f u n c t i o n of the spacetime v a r i a b l e x = (x°=t,x). G p i s the weak c o u p l i n g c o n s t a n t , w i t h an e x p e r i m e n t a l l y d e t e r m i n e d v a l u e of 1.03 x 10" 5 nip 2 (mp = p r o t o n 11 mass) or 1.4 x 10""' erg*cm 3 i n cgs u n i t s . The most g e n e r a l f o u r - c u r r e n t i s a sum of l e p t o n i c and h a d r o n i c p a r t s ; however, f o r t he f o l l o w i n g d i s c u s s i o n , t h e . h a d r o n i c p a r t w i l l be i g n o r e d . In. p u r e l y l e p t o n i c p r o c e s s e s such as muon decay or n e u t r i n o s c a t t e r i n g on e l e c t r o n s , c o m p l i c a t i o n s due t o s t r o n g i n t e r a c t i o n s do not a r i s e . A c c o r d i n g t o the V-A t h e o r y , the l e p t o n i c c u r r e n t has the form j j * (x) = T ^ i l ~ y*)Vt , 2.1.1.2 where the f ' s a re f i e l d o p e r a t o r s f o r the p a r t i c l e s concerned (1= e or y U ). For the moment, the t h i r d t a u l e p t o n g e n e r a t i o n w i l l not be c o n s i d e r e d , s i n c e i t p l a y e d no p a r t i n t h e o r e t i c a l developments a t t h i s s t a g e . E q u a t i o n 2.1.1.2 denotes the charged c u r r e n t of a l e p t o n and i t s n e u t r i n o . There a l s o e x i s t n e u t r a l c u r r e n t weak i n t e r a c t i o n s , such as muon n e u t r i n o i n e l a s t i c s c a t t e r i n g on p r o t o n s ( B a r i s h e t a l • , 1974; t h i s of co u r s e i n v o l v e s hadrons) and muon a n t i n e u t r i n o e l a s t i c s c a t t e r i n g on e l e c t r o n s ( H a s e r t e t a l . , 1973). A l l p r o c e s s e s so f a r o b s e r v e d , whether charged or n e u t r a l , obey a muon number c o n s e r v a t i o n law. The assumed form of the l e p t o n i c c u r r e n t , c o n t a i n i n g o n l y f i e l d s of one type or g e n e r a t i o n of l e p t o n s (muon or e l e c t r o n ) , l e a d s t o the p r e d i c t i o n of an a d d i t i v e c o n s e r v a t i o n of muon number. T h i s can be f o r m a l i z e d as f o l l o w s (see B a i l i n , 1977). For the l e p t o n number assignments of the f i r s t c h a p t e r and the f i e l d s as d e f i n e d i n appendix A l , the l e p t o n numbers of a s t a t e a r e the e i g e n v a l u e s of an o p e r a t o r L., which then s a t i s f i e s t he 12 commutation r e l a t i o n s 2.1.1.3a 2.1.1.3b and commutes w i t h o t h e r f i e l d o p e r a t o r s ( i . e . , those of o t h e r l e p t o n g e n e r a t i o n s ) . The r e s u l t i s t h a t [t~ , J ^ ( x ) ] = 0, 2.1.1 . 4 so t h a t a l s o commutes w i t h the weak H a m i l t o n i a n ( e q u a t i o n 2.1.1.1). S i m i l a r l y , commutes w i t h the f r e e f i e l d and e l e c t r o m a g n e t i c f i e l d H a m i l t o n i a n s , and thus w i t h the f u l l H a m i l t o n i a n . I t i s then apparent t h a t the e i g e n v a l u e i s c o n s e r v e d i n t r a n s i t i o n s between p h y s i c a l e i g e n s t a t e s of . I f we do not r e s t r i c t t he form of the c u r r e n t , such an a d d i t i v e scheme does not n e c e s s a r i l y p r e v a i l , and a more g e n e r a l m u l t i p l i c a t i v e one w i l l a c count f o r p h y s i c a l o b s e r v a t i o n s . A m u l t i p l i c a t i v e muon number f o l l o w s , f o r i n s t a n c e , from assuming i n v a r i a n c e of n a t u r e under t h e p e r m u t a t i o n of two p r i m i t i v e l e p t o n s , say, e 1 and JJC ( F e i n b e r g and Weinberg, 1961a; see a l s o Cabibbo and G a t t o , 1960) which can t r a n s f o r m i n t o each o t h e r , and w i l l m a n i f e s t themselves p h y s i c a l l y as l i n e a r c o m b i n a t i o n s e = (/(! + e ' ) / 2 ( 1 / 2 ) and yw-= (/*! - e ' ) / 2 ( 1 / J ) of d i f f e r e n t mass. The e'-yn' p e r m u t a t i o n i n v a r i a n c e under weak i n t e r a c t i o n s r e q u i r e s two k i n d s of n e u t r i n o s , v and vM, which of c o u r s e a r e now known t o e x i s t (Danby et. a l . , 1962), and i m p l i e s o n l y a m u l t i p l i c a t i v e s e l e c t i o n r u l e (but does not r u l e out a s t r o n g e r a d d i t i v e l a w ) . 13 I t was not u n t i l r e l a t i v e l y r e c e n t l y t h a t e x p e r i m e n t a l i n f o r m a t i o n became a v a i l a b l e which p e r m i t t e d some c h o i c e between the two schemes. An experiment t o dete r m i n e the type of e l e c t r o n n e u t r i n o i n p o s i t i v e muon decay showed t h a t the r a t i o of v t o v> was l e s s than 0.25 w i t h 90% c o n f i d e n c e ( E i c h t e n e t c e — a l • , 1973). An a d d i t i v e law p r e d i c t s e x a c t l y z e r o , whereas a m u l t i p l i c a t i v e law p r e d i c t s 0.5 . A more s e n s i t i v e experiment ( W i l l i s e t a l . , 1980) measured the same r a t i o t o be l e s s than 0.098 w i t h 90% c o n f i d e n c e . T h e r e f o r e , any u n i v e r s a l muon number c o n s e r v a t i o n law i s u n l i k e l y t o be m u l t i p l i c a t i v e . The q u e s t i o n s t h a t must now be asked about the fundamental i n t e r a c t i o n s a r e q u i t e d i f f e r e n t from th o s e of 1961, because of the s u c c e s s e s of u n i f i c a t i o n t h e o r i e s (see s e c t i o n 2.2). The p r e s e n t emphasis i s on whether t h e r e i s a u n i v e r s a l muon (or l e p t o n , or baryon) c o n s e r v a t i o n law a t a l l . Assuming t h a t t h e r e i s n o t , i t i s then i m p o r t a n t t o determine the l e v e l a t which i t i s v i o l a t e d i n or d e r t o draw f u r t h e r c o n c l u s i o n s about the pro p e r f o r m u l a t i o n of the t h e o r y . For the case of muonium-antimuonium c o n v e r s i o n (and e"e" t o ^f*-' s c a t t e r i n g ) the p o i n t of c o n t a c t between t h e o r y and experiment i s an e f f e c t i v e c o u p l i n g c o n s t a n t G, u s u a l l y e x p r e s s e d i n terms of Gp , f o r the i n t e r a c t i o n H a m i l t o n i a n d e n s i t y , H w ( x ) = '2-<\"> G ( ^ * * ( l - y M £ M ^ ^ W ) ^ ). 2.1.1.5 T h i s has the c u r r e n t - c u r r e n t form of e q u a t i o n 2.1.1.1, and the 14 c u r r e n t s a r e V-A, but o t h e r w i s e t he n a t u r e of the e x p r e s s i o n i s q u i t e d i f f e r e n t from e q u a t i o n 2.1.1.2. No n e u t r i n o f i e l d s a r e p r e s e n t , and the c u r r e n t s do not change the charge but r a t h e r the g e n e r a t i o n (muon or e l e c t r o n ) of the p a r t i c l e . Each c u r r e n t changes muon number by +1 and e l e c t r o n number by - 1 , so t h a t t he o v e r a l l changes f o r t h e i n t e r a c t i o n a r e +2 and -2 r e s p e c t i v e l y . T h i s means t h a t £ ( L e + L / J and ( - l F ^ A are c o n s e r v e d , and a m u l t i p l i c a t i v e muon number c o n s e r v a t i o n i s s a t i s f i e d , whereas an a d d i t i v e one i s n o t . I t i s shown i n appendix A l t h a t the m a t r i x element f o r c o n v e r s i o n from IS muonium (|Mu>) .to IS antimuonium (|Mu>) s t a t e s i s <Mu| H w ( x ) |Mu> = S/2 . 2.1.1.6 D i s r e g a r d i n g s p i n s e l e c t i o n r u l e s and an unimportant s i g n , S = 16G/(2 ( 1 ,/2'TTaJ ) = 2.1 x 10" 1 2 (G/G p) eV, 2.1.1.7 where a 0 i s the Bohr r a d i u s of IS muonium . A u n i v e r s a l ( i . e . , t o a l l o r d e r s of p e r t u r b a t i o n t h e o r y ) a d d i t i v e law c o r r e s p o n d s t o G = 0, whereas the e x i s t e n c e of a f i r s t o r d e r i n t e r a c t i o n , d e s c r i b e d by e q u a t i o n 2.1.1.6 and unsuppressed by any a d d i t i v e muon c o n s e r v a t i o n , might imply G •~ ,G F. 15 2.1.2. Vacuum E i g e n s t a t e s of the Muonium-Antimuonium System I f G * 0, muonium and antimuonium a r e not energy e i g e n s t a t e s of the f u l l ( t h a t i s , i n c l u d i n g H w) H a m i l t o n i a n , H. Le t H i n c l u d e a term of e l e c t r o m a g n e t i c o r i g i n which causes the p a r t i a l H a m i l t o n i a n , H - H w, t o have nondegenerate e i g e n v a l u e s f o r the s t a t e s |Mu> and |Mu> . I f the d i f f e r e n c e of e i g e n v a l u e s i s A , then the f u l l H a m i l t o n i a n H can be r e p r e s e n t e d by a 2x2 m a t r i x of the form ( t a k i n g S t o be r e a l ) H = H e + Hj , w i t h H = ( E 0 / 2 ) 1 0 0 IJ , and 2.1.2.1 '2.1.2.2 H = 1/2 S -A ~  Een+ H w 2.1.2.3 a c t i n g on the s t a t e s |Mu> = and Mu> = H wi t h e i g e n v a l u e E G , i n c l u d e s a l l terms i n H d e s c r i b i n g the energy of the atom which a r e the same f o r muonium and antimuonium. The e i g e n v a l u e s E, and E a of H d i f f e r by W = ( S 2 + A 2 ) ( 1 / 2 ' , and the n o r m a l i z e d e i g e n v e c t o r s become: | M, > = (2W(W - 4 )')-< > (+£|Mu> + (W -A)|Mu>), |Ma> = (2W(W + A ) ) " ( 1 / 2 ) ("S|Mu> + (W +A)|MU>). 2.1.2.4 16 These reduce, f o r A « S , t o : |M,> = 2"' 1 / 2 1(+|Mu> + |Mu>), |MX> = 2"(»/»>(-|Mu> + |Mu>). 2.1.2.5 2.1.3. E v o l u t i o n of Antimuonium from Muonium A nonzero v a l u e of S l e a d s t o energy e i g e n s t a t e s which are o r t h o g o n a l l i n e a r c o m b i n a t i o n s of muonium and antimuonium s t a t e s . The consequence i s t h a t a s t a t e \{f(t)>, i n the S c h r o e d i n g e r p i c t u r e , e v o l v i n g from a s t a t e |V(0)> = |Mu> a t time z e r o , has a n o n v a n i s h i n g p r o b a b i l i t y of b e i n g i d e n t i f i e d as |Mu> a t some l a t e r t i m e . The p r o b a b i l i t y t h a t the s t a t e w i l l be i d e n t i f i e d as antimuonium r a t h e r than muonium ( f o r t h e s e a r e s t a t e s which w i l l be o b s e r v e d v i a e l e c t r o m a g n e t i c s c a t t e r i n g or weak muon decay p r o c e s s e s ) a t time t i s Expanding the s t a t e | r ' ( t ) > = e x p ( - i H t ) |Mu> i n terms of the energy e i g e n s t a t e s |M:> of e i g e n v a l u e E: g i v e s : I f o ( t ) = | < M u | f ( t ) > | 2 . 2.1.3.1 2.1.3.2 17 A p a r t of the f u l l H a m i l t o n i a n H i n v o l v e s the weak muon decay f o r both |Mu> and |Mu> . T h i s can be acco u n t e d f o r by c o n s i d e r i n g H t o have complex e i g e n v a l u e s . P u t t i n g Im(E 0) =\, the muon decay r a t e , the i m a g i n a r y p a r t s of E, and E x a r e 'X/2, and e q u a t i o n 2.1.3.1 becomes Prj- ( t ) = e x p ( - ^ t ) (S/W) 2 s i n 2 ( W t / 2 ) . 2.1.3.3 For W << A , the domain f o r which Ej^ ( t ) i s g r e a t e s t s a t i s f i e s Wt << 1, so Pp^ ( t ) ~ exp(->t) ( S t / 2 ) 2 . 2.1.3.4 I f an experiment were performed which d e t e c t e d c o n v e r s i o n by means of the o b s e r v a t i o n of a f a s t e l e c t r o n from n e g a t i v e muon decay, an e x p r e s s i o n f o r the p r o b a b i l i t y of the muon d e c a y i n g as /t" would be r e q u i r e d . M u l t i p l y i n g e q u a t i o n 2.1.3.3 by the decay r a t e A, and i n t e g r a t i n g over a l l time y i e l d s P/l- = 6 2 / 2 ( 8 2 + A 2 + ^ 2 ) , 2.1.3.5 which i s a p p r o x i m a t e l y S 2/2^ 2 = 2.5 x 10" s » ( G / G p ) 2 f o r A « A . I f , however, a muonic X-ray were used f o r the s i g n a t u r e , the r e l e v a n t e x p r e s s i o n would be 2.1.3.3 e v a l u a t e d a t the time of c o l l i s i o n of antimuonium w i t h the atom p r o d u c i n g the X-ray. The r i g h t hand s i d e of e q u a t i o n 2.1.3.3 has a maximum a t t = 2/%, or 4.4 x 10"' s, of 6.6 x 1 0 " ( G / G F ) 2 i n the a p p r o x i m a t i o n of 2.1.3.4. T h i s w i l l be a p p l i e d i n the f o u r t h c h a p t e r i n o r d e r t o e s t i m a t e the e x p e c t e d r a t e of the TRIUMF 18 muonium-antimuonium experiment i n terms of ( G / G F ) . 2.1.4. The E f f e c t of E l e c t r o m a g n e t i c F i e l d s on C o n v e r s i o n L i t t l e has been s a i d up t o t h i s p o i n t about the o r i g i n or v a l u e of A; one might wonder whether i t i s i m p o r t a n t , or under what c i r c u m s t a n c e s the a p p r o x i m a t i o n A << 7i might be v a l i d . Such q u e s t i o n s were a d d r e s s e d i n the f i r s t d e t a i l e d p u b l i c a t i o n on muonium-antimuonium c o n v e r s i o n , from which most of t h i s c h a p t e r has been d e r i v e d ( F e i n b e r g and Weinberg, 1961b). For the case of muonium i n vacuum, 4 a r i s e s from the macr o s c o p i c e l e c t r o m a g n e t i c f i e l d s p r e s e n t . Two p o i n t s s h o u l d be kept i n mind: 1. Not a l l f i e l d s or f i e l d d e r i v a t i v e s c o n t r i b u t e t o A. 2. Not a l l of the IS muonium h y p e r f i n e s t a t e s a r e a f f e c t e d i n the same way. The f i r s t p o i n t i s e x e m p l i f i e d by c o n s i d e r i n g p o s s i b l e c o n t r i b u t i o n s t o A from a u n i f o r m e l e c t r i c f i e l d . For a s t a t e of p a r t i c u l a r a n g u l a r momentum (an a x i a l v e c t o r ) the o n l y s c a l a r s upon which can depend (E»E and (J»E) J) a r e even i n the e l e c t r i c f i e l d and a r e the same f o r muonium and antimuonium. S i n c e the energy s h i f t i s , by d e f i n i t i o n ( e q u a t i o n 2.1.2.3), p o s i t i v e f o r muonium and n e g a t i v e f o r antimuonium , i t cannot depend on a u n i f o r m e l e c t r i c f i e l d . I f t h e r e i s a l s o a u n i f o r m magnetic f i e l d ( another a x i a l v e c t o r ) p r e s e n t , f u r t h e r s c a l a r s can be formed, but o n l y those c o n t a i n i n g H«J a r e not even i n 19 bo t h E and H. T h i s s e r v e s t o i l l u s t r a t e t he second p o i n t ; H»l? i s z e r o f o r the two hyper f i n e s t a t e s of J = S^ .+S<. f o r which mJ, the p r o j e c t i o n of If a l o n g H, i s z e r o . These two s t a t e s a re c o n v e n t i o n a l l y l a b e l l e d (J,mj.) = (1,0) and ( 0 , 0 ) . A n a i v e but u s e f u l p i c t u r e of the source of A can be g l e a n e d from a B r e i t -Rabi diagram f o r the e n e r g i e s of the h y p e r f i n e s t a t e s of muonium i n a magnetic f i e l d , and the r e f l e c t i o n of the diagram t h r o u g h the H=0 a x i s (see F i g u r e 2.1.4.1). I t must be remembered t h a t , MUONIUM ENERGY LEVELS "-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 FIELD (UNITS OF H0=1585 GAUSS) F i g u r e 2.1.4.1. S p l i t t i n g of mj=±l s t a t e s of muonium and antimuonium i n a magnetic f i e l d . f o r S*0, these a r e not the e i g e n s t a t e s of the f u l l H a m i l t o n i a n 20 (see Morgan, 1967, f o r a more c o r r e c t and complete t r e a t m e n t ) . S w i t c h i n g the s i g n of H has the same e f f e c t on a g i v e n l e v e l as s w i t c h i n g the s i g n s of the magnetic moments ( i . e . , c h a r g e s ) of the muon and the e l e c t r o n , as happens i n muonium t o antimuonium c o n v e r s i o n . The nondegeneracy of the (1,±1) s t a t e s of muonium and antimuonium r e s u l t s . The v a l u e of A one o b t a i n s i s eH/m, where m i s the e l e c t r o n mass. Thus A ~ 1.2 x 10" 8 x H eV, i f H i s i n gauss. Comparing t h i s t o "X, fX ~ 40 H , 2.1.4.1 and.the c o n c l u s i o n i s t h a t H must be kept w e l l below 0.025 gauss i n o r d e r f o r c o n v e r s i o n of these s t a t e s t o be unsuppressed. E s t i m a t e s of c o n t r i b u t i o n s t o A from e l e c t r i c f i e l d g r a d i e n t s have a l s o been made ( F e i n b e r g and Weinberg, 1961b). These can be i g n o r e d f o r the case of muonium i n vacuum. 2.1.5. Muonium i n the Presence of M a t t e r I t i s not easy t o o b t a i n abundant muonium i n vacuum, so the p o s s i b i l i t y of o b s e r v i n g the c o n v e r s i o n i n gases and s o l i d s has been c a r e f u l l y i n v e s t i g a t e d ( F e i n b e r g and Weinberg, 1961b). The c o n c l u s i o n reached was t h a t such e x p e r i m e n t s would not be v e r y s e n s i t i v e because of the l a r g e v a l u e s of A e n c o u n t e r e d i n the i n t e r a c t i o n s of muonium w i t h t h e t a r g e t s u b s t a n c e . In a gas, c o l l i s i o n s of the muonium-antimuonium system w i t h the s u r r o u n d i n g atoms i n v o l v e Coulomb p r o c e s s e s and the 21 a s s o c i a t e d l a r g e {A, » /\ ) energy d i f f e r e n c e s . D u r i n g a c o l l i s i o n the energy e i g e n s t a t e s approach: |M,> = ( S 2 ( l + S 2 / 4 A 2 ) ) - ' (S|Mu> + S 2 / 2 A | M U > ) Mu> , and |Ma> = (S1 + 4 a 2 ) - ( 1 / 2 ) (-S|MU> + 2A|MU > ) ~ |Mu> . 2.1.5.1 The |M,> component w i l l s c a t t e r from, say, an argon atom l i k e |Mu>, t h a t i s , e l a s t i c a l l y , but the |Ma> component has a h i g h p r o b a b i l i t y f o r s c a t t e r i n g i n e l a s t i c a l l y l i k e |Mu>, f o r m i n g a muonic argon atom and a f r e e p o s i t r o n or p o s i t r o n i u m atom (Morgan, 1967). D u r i n g the c o l l i s i o n any |MX> component i s i n e f f e c t removed, and when A i s s w i t c h e d o f f as the s c a t t e r i n g p r o c e s s ends, the system i s i n the |Mu> s t a t e as i t was a t t=0, the time of muonium f o r m a t i o n . In e f f e c t , the c o l l i s i o n i n h i b i t s the coherent growth of the |Mu> component (somewhat i n an a l o g y w i t h the r e g e n e r a t i o n of the s h o r t - l i v e d component of a n e u t r a l kaon beam as i t passes t h r o u g h m a t t e r ) . The net r e s u l t i s t h a t the p r o b a b i l i t y of the muon d e c a y i n g as /*" ( e q u a t i o n 2.1.3.5) i s reduced by a f a c t o r of 1/N, where N i s the mean number of c o l l i s i o n s d u r i n g the muon l i f e t i m e . For argon gas a t normal temperature and p r e s s u r e , N i s of the o r d e r of 1 0 5 . In s o l i d s the s i t u a t i o n i s more c o m p l i c a t e d s t i l l . C o n t r i b u t i o n s t o A a r i s e from the o v e r l a p of the muonium wave f u n c t i o n w i t h those of the atoms of a m o l e c u l a r c r y s t a l , o r from 22 h i g h e l e c t r i c f i e l d s and g r a d i e n t s w i t h i n an i o n i c c r y s t a l . Depending on the n a t u r e of the m a t e r i a l , the p r o b a b i l i t y of the muon i n muonium d e c a y i n g as i s 1 0 ' 1 4 t o 1 0 " 2 0 times i t s v a l u e i n vacuum ( e q u a t i o n 2.1.3.5; see F e i n b e r g and Weinberg, 1961b). I t has been suggested by B. Be r g e r s e n t h a t t h i s may be o v e r l y p e s s i m i s t i c . For example, i n a me t a l the e l e c t r o n d e n s i t y a t the p o s i t i v e muon i s n e a r l y as l a r g e as f o r a muonium atom, even though no s i n g l e e l e c t r o n i s c o r r e l a t e d f o r l o n g w i t h the muon. The energy band of the c o n d u c t i o n e l e c t r o n s might p r o v i d e a way around the problem of non-degeneracy of i n i t i a l and f i n a l s t a t e s : f o r any i n i t i a l - s t a t e energy, t h e r e w i l l be a band of n e a r l y degenerate f i n a l s t a t e e n e r g i e s . The c o n v e r s i o n r a t e then depends upon the l o c a l d e n s i t y of s t a t e s . These r a m i f i c a t i o n s a r e not s i m p l e t o c a l c u l a t e , so the use of a metal t a r g e t t o e s t a b l i s h an a c c e p t a b l e upper l i m i t f o r G i s not a p p e a l i n g . Moreover, the d e n s i t y of s t a t e s i s l i k e l y not l a r g e enough t h a t an e x p e r i m e n t a l advantage would a c c r u e . I t i s t h e r e f o r e d e s i r a b l e , i f d i f f i c u l t , t o a l l o w the muonium-antimuonium system t o e v o l v e i n vacuum or a r a r e f i e d gas a f t e r muonium f o r m a t i o n . A t e c h n i q u e f o r a c c o m p l i s h i n g t h i s i s d e s c r i b e d i n s e c t i o n s 3.2.2 and 4.2.2. 23 2.2. U n i f i e d Gauge T h e o r i e s and Muonium-Ant imuonium C o n v e r s i o n The f o r c e s of n a t u r e a r e u s u a l l y s u b d i v i d e d i n t o the s t r o n g , e l e c t r o m a g n e t i c , weak, and g r a v i t a t i o n a l i n t e r a c t i o n s , each w i t h unique a t t r i b u t e s . Attempts t o u n i f y two or more of t h e s e , t h a t i s , t o c h a r a c t e r i z e them by common s t r e n g t h and symmetry p r o p e r t i e s , has been a major g o a l of p h y s i c i s t s i n t h i s c e n t u r y . The Fermi t h e o r y of beta decay ( F e r m i , 1934) was an e a r l y example i n which the v e c t o r n a t u r e of e l e c t r o m a g n e t i s m was adapted t o the weak l e p t o n - h a d r o n i n t e r a c t i o n t h r o u g h a change i n c o u p l i n g c o n s t a n t . Some f u r t h e r development i n the Fermi t h e o r y , l e a d i n g t o the s u c c e s s f u l V-A i n t e r p r e t a t i o n of some low energy weak p r o c e s s e s , was mentioned i n s e c t i o n 2.1.1. Of cou r s e the d i s c o v e r y t h a t weak i n t e r a c t i o n s were not pure v e c t o r b l i g h t e d the apparent a n a l o g y w i t h e l e c t r o m a g n e t i s m . There were o t h e r problems w i t h the f o u r f e r m i o n i n t e r a c t i o n . Even b e f o r e the V-A form was assumed, d i v e r g e n c e s e x i s t e d i n the c a l c u l a t i o n of some weak p r o c e s s e s such as n e u t r i n o - l e p t o n s c a t t e r i n g . When expanding the s c a t t e r i n g o p e r a t o r S i n the u s u a l p e r t u r b a t i o n s e r i e s i n G , shown d i a g r a m m a t i c a l l y i n F i g u r e 2.2.1, one hopes t h a t the f i r s t o r d e r term w i l l dominate because of the weakness of the i n t e r a c t i o n . However, the p r o p a g a t o r f o r the i n t e r m e d i a t e l e p t o n s from the second o r d e r term, c o n t a i n i n g i n s u f f i c i e n t i n v e r s e powers of the l e p t o n momenta, d i v e r g e s i n the momentum i n t e g r a l , so t r u n c a t i o n of the s e r i e s i s o n l y p o s s i b l e by i n s e r t i n g an ad hoc c u t o f f A . 24 F i g u r e 2.2.1. Diagrammatic e x p a n s i o n of the n e u t r i n o - l e p t o n s c a t t e r i n g o p e r a t o r The second o r d e r term then becomes p r o p o r t i o n a l t o G F ( G F A 2 ) and the d i m e n s i o n a l i t y of the e x t r a f a c t o r G F i s b a l a n c e d by A . R e n o r m a l i z a t i o n of the masses and the c o u p l i n g c o n s t a n t are not p o s s i b l e as i n quantum e l e c t r o d y n a m i c s . The i n c l u s i o n of an i n t e r m e d i a t e exchange boson W improves the s i t u a t i o n w i t h r e s p e c t t o the u n i t a r i t y v i o l a t i o n . R e n o r m a l i z a t i o n i s s t i l l a problem. I f the e l e m e n t a r y c o u p l i n g of the l e p t o n c u r r e n t i s t o a massive s p i n one boson, the a n a l o g y w i t h QED i s c l o s e r . Assume, f o r the moment, a weak c o u p l i n g of about the same s t r e n g t h as f o r e l e c t r o m a g n e t i c i n t e r a c t i o n s (the boson mass must then s a t i s f y G p / v / e 2 / n i w 2 , so nu^ ~ 30 GeV). Then, f o r the e 4 ( f o u r v e r t i c e s ) c o n t r i b u t i o n s t o l e p t o n - l e p t o n s c a t t e r i n g ( f i g u r e 2.2.2), the weak boson p r o p a g a t o r has terms p r o p o r t i o n a l t o k^k^/m^, from l o n g i t u d i n a l degrees of freedom, which do not c o n t r i b u t e i n the photon p r o p a g a t o r s of QED due t o gauge i n v a r i a n c e . These terms dominate a t h i g h e r e n e r g i e s , and the " e x t r a " convergence g u a r a n t e e d by the gauge i n v a r i a n c e i n QED i s absent from the weak p r o c e s s . P e r t u r b a t i o n t h e o r y thus breaks down. 25 F i g u r e 2.2.2. Order e* (or GF2) l e p t o n - l e p t o n s c a t t e r i n g i n (a) QED, and (b) weak i n t e r a c t i o n s . A.remedy f o r these o u t s t a n d i n g d e f i c i e n c i e s of the o l d approach was proposed i n 1967, i n the form of a gauge t h e o r y u n i f y i n g weak and e l e c t r o m a g n e t i c i n t e r a c t i o n s . 2.2.1. Gauge T h e o r i e s of Weak and E l e c t r o m a g n e t i c I n t e r a c t i o n s S i n c e 1967 t h e r e have appeared numerous p u b l i c a t i o n s which have attempted t o e x p l a i n the n a t u r e of the u n i f i e d gauge model f o r weak and e l e c t r o m a g n e t i c i n t e r a c t i o n s , now known as the Weinberg-Salam (WS) t h e o r y (Weinberg, 1967; Salam, 1968). In a d d i t i o n t o the o r i g i n a l r e f e r e n c e s , s o u r c e m a t e r i a l f o r t h i s s e c t i o n i n c l u d e s t e x t b o o k s ( B a i l i n , 1977; T a y l o r , 1976) and s e v e r a l f i n e review a r t i c l e s (Weinberg, 1974 and 1977; Abers and Lee, 1973; B e r n s t e i n , 1974). Only a b r i e f i n t r o d u c t o r y e x p l a n a t i o n of the t h e o r y w i l l be g i v e n h e r e . The s u g g e s t i o n of the p r e v i o u s s e c t i o n i s t h a t a convergent t h e o r y might be based on a gauge i n v a r i a n t L a g r a n g i a n , as i s 26 QED, but i n v o l v i n g a charged v e c t o r f i e l d . T h i s i s a c c o m p l i s h e d by the WS model i n the f o l l o w i n g way, c o n s i d e r i n g f o r the moment o n l y f i r s t g e n e r a t i o n ( i . e . , e l e c t r o n - t y p e ) l e p t o n s . The f e r m i o n f i e l d s appear i n the L a g r a n g i a n as l e f t - h a n d e d d o u b l e t s and r i g h t - h a n d e d s i n g l e t s , L = 1/2 ( i - y s) R = 1/2 ( l + y») [e] , 2.2.1.1 where ue and e are n e u t r i n o and e l e c t r o n f i e l d s (analogous t o % 6 and f£ of appendix A l ) . I f ( i = l , 2 , 3 ) a re the P a u l i m a t r i c e s f o r SU(2) l e p t o n i c i s o s p i n T = nr/2, and T+= 1/2 (T,±T a), then T + L = 1/2 ( i - y») 2.2.1.2 and the charged c u r r e n t s can be w r i t t e n 1/2 ue * * t l - y*)e = L ^ t t L , and 1/2 e - /5)i/e= L/'V.L. 2.2.1.3 To complete the group of which these c u r r e n t s a re members ( u s u a l l y known as SU(2) l e p t o n i c i s o s p i n ) one needs 1/2 [Uj^i - r>)»e - eP\l - T)e] 2.2.1.4 A major s u c c e s s of the t h e o r y was the e x p e r i m e n t a l c o n f i r m a t i o n 27 of i t s p r e d i c t i o n of n e u t r a l c u r r e n t s as d e s c r i b e d by t h i s l a s t e x p r e s s i o n . The e l e c t r o m a g n e t i c c u r r e n t can be i n c l u d e d i n the u s u a l form: e Y'e = -1/2 l y ' r , L + 1/2 L y*L + R . 2.2.1.5 The l a s t two terms on the r i g h t hand s i d e c o n s e r v e l e p t o n i c h y percharge Y = NL/2 + N R , the N's r e f e r r i n g t o the e i g e n v a l u e s of a number o p e r a t o r f o r l e p t o n s of the i n d i c a t e d h e l i c i t y . In t h i s way the group U ( l ) of l e p t o n i c h ypercharge e n t e r s the t h e o r y , and a L a g r a n g i a n can be w r i t t e n down which i s i n v a r i a n t under SU(2)^x U ( l ) i n f i n i t e s i m a l l o c a l gauge t r a n s f o r m a t i o n s . The t r a n s f o r m a t i o n s a r e L ->L' = ( l + ig4 (x)«T+i ( g 1 / 2 ) A ( x ) ) L , and R ->R' = ( l + i g ' A ( x ) ) R , 2.2.1.6 where g and g' a r e c o n s t a n t s and A ( x ) and A (x) are gauge f u n c t i o n s . The i n c l u s i o n of g' i n t h i s p a r t i c u l a r way i n s u r e s t h a t the h y p e r c h a r g e - c o n s e r v i n g terms of e q u a t i o n 2.2.1.5 c o u p l e t o the gauge f i e l d B a s s o c i a t e d w i t h U ( l ) . The gauge f i e l d s f o r S U ( 2 ) L and U ( l ) t r a n s f o r m as - » A J , = + ^ A - gAxA^ B A-> By = B^ A 2.2.1.7 and the r e n o r m a l i z a b l e L a g r a n g i a n i s ( i n the n o t a t i o n and phase c o n v e n t i o n of Weinberg, 1967) 28 Jio = -1/4 ( ? r A>^A^+gA^xA^ ) 2 - 1/4 ( 2 - E K t ^ - i g ' B^jR 2.2.1.8 -L^(^ l-igf»A / l-(i/2)g'B 4 < t)L . There are more terms i n Weinberg's L a g r a n g i a n , but these s e r v e t o i l l u s t r a t e some p o i n t s of the model, i f the p h y s i c a l l e p t o n s a re i n s e r t e d a c c o r d i n g t o e q u a t i o n s 2.2.1.1: 1. The n o r m a l i z e d n e u t r a l v e c t o r f i e l d c o u p l i n g t o the ^ c u r r e n t i s lr = (g 2+g f 2) - 1 1 / 2 ' (gA^+g'B^) . 2.2.1.9 2. The f i e l d o r t h o g o n a l t o Z^ . i s = ( g 2 + g ' 2 ) " ' 1 / 2 > (-g'A^j+gB^) , 2.2.1.10 which c o u p l e s o n l y t o t h e e e c u r r e n t ( the A^ used here i s not t o be c o n f u s e d w i t h A^ as used i n e q u a t i o n s 2.2.1.7 and 2.2.1.8) . 3. The c o e f f i c i e n t of ie ^ A _ which i s t o be i d e n t i f i e d w i t h e l e c t r o d y n a m i c c o u p l i n g , i s g g ' / ( g 2 + g ' 2 ) 1 1 / 2 ' , and i s e q u a l t o the e l e c t r o n i c charge e. 4. The charged v e c t o r f i e l d a s s o c i a t e d withez^ and*£e terms appears as w = 2 - ( 1 / 2 » (A^, + i A / t a ) . 2.2.1.11 Note t h a t i t has become common t o d e f i n e a m i x i n g a n g l e 0W (the 29 Weinberg a n g l e ) as t a n Bw = g'/g (Weinberg, 1972), and t o w r i t e the f i e l d s and c o u p l i n g s i n terms of t h i s a n g l e . The concept of i s o t o p i c l o c a l gauge i n v a r i a n c e r e p r e s e n t e d by the SU(2) c h a r a c t e r of e q u a t i o n 2.2.1.8 was not a new i n v e n t i o n i n WS; i t had been f o r m u l a t e d much e a r l i e r (Yang and M i l l s , 1954) but convergence of the t h e o r y demanded u n p h y s i c a l m a s s l e s s quanta of the SU(2) gauge f i e l d . The new f e a t u r e was t h a t e x p l i c i t mass terms (and n o n - r e n o r m a l i z a b i l i t y ) c o u l d be a v o i d e d by i n t r o d u c i n g a s c a l a r meson d o u b l e t whose vacuum e x p e c t a t i o n v a l u e broke the symmetry of the L a g r a n g i a n under T and Y (a p r o c e s s r e f e r r e d t o as the H i g g s - K i b b l e mechanism, a f t e r H i g g s , 1964 and K i b b l e , 1967). There i s no e x p e r i m e n t a l guide t o the e x a c t s t r u c t u r e of the Higgs s c a l a r mesons t h a t a r e i n t r o d u c e d , but the s o - c a l l e d m i n i m a l s t r u c t u r e of the o r i g i n a l 1967 model i s a d o u b l e t of the form z e r o v a l u e ) ; t h i s ^ i s not t o be c o n f u s e d w i t h the muon decay r a t e of s e c t i o n 2.1. The L a g r a n g i a n of e q u a t i o n 2.2.1.8 becomes 2.2.1.12 where the vacuum e x p e c t a t i o n v a l u e <4>"> of f i s r e t a i n s a XL = 2.2.1.13 R e p l a c i n g masses by i t s vacuum e x p e c t a t i o n v a l u e w i l l l e a d t o the 30 me = ^ Ge' m w = ^9/2, and 2.2.1.14 m z = ( g 2 + g' 2 ) ( 1 / 2 '•(2-< 3 / 2» } ) , w i t h the photon r e m a i n i n g m a s s l e s s . The low energy l i m i t of the e l e c t r o n - n e u t r i n o c u r r e n t c o u p l i n g v i a W exchange must be t h a t of the V-A t h e o r y , so g 2/8m w J = 1 / 4 V = GF/2< 1 / 2 ' . 2.2.1.15 The model can be extended t o second and h i g h e r ( i . e . , yw. and T ) g e n e r a t i o n s of l e p t o n s by the a d d i t i o n of a p p r o p r i a t e terms c o n t a i n i n g L ^ , t and R A ^ T f i e l d s t o e x p r e s s i o n s 2.2.1.8 and 2.2.1.13. The parameters and G T must be chosen such t h a t G / < , T / G e = ^ . . T ^ e -The WS t h e o r y i n t h i s form has s u r v i v e d a l l the t e s t s t o which i t has been s u b j e c t e d w i t h p r e s e n t l y a v a i l a b l e t e c h n i q u e s . There a r e f e a t u r e s , however, which l a c k a e s t h e t i c a p p e a l ( e . g . , the need t o i n c l u d e the Higgs bosons and a d j u s t t h e i r c o u p l i n g s by hand t o g i v e c o r r e c t l e p t o n masses). I t i s s a f e r t o r e g a r d SU(2) t >x U ( l ) as a subgroup of the t r u e e l e c t r o w e a k gauge group, whose f u r t h e r p r o p e r t i e s might not be c l e a r l y r e v e a l e d by p r e s e n t knowledge. Much e f f o r t i s b e i n g c h a n n e l l e d i n t o so-c a l l e d grand u n i f i c a t i o n schemes, gauge t h e o r i e s u n i f y i n g s t r o n g , e l e c t r o m a g n e t i c , and weak i n t e r a c t i o n s w i t h c o r r e s p o n d i n g l y l a r g e r gauge groups such as SU(5) and SO(10). 31 2.2.2. Muon Number V i o l a t i o n and Muonium C o n v e r s i o n i n Extended T h e o r i e s The WS t h e o r y i n i t s s i m p l e s t form (sometimes r e f e r r e d t o as the s t a n d a r d model), as d e f i n e d by the l e p t o n i c e l e c t r o w e a k L a g r a n g i a n of e q u a t i o n 2.2.1.13 w i t h e x t e n s i o n t o JJL and t f i e l d s , does not a l l o w a v i o l a t i o n of muon number. T h i s i s apparent from the o b s e r v a t i o n t h a t the fe r m i o n c u r r e n t s c o u p l i n g t o gauge v e c t o r or Higgs s c a l a r bosons do not change l e p t o n f l a v o r . The c o n s e r v a t i o n i s not r e a l l y b u i l t i n t o the t h e o r y , but i s a consequence of the assumptions made from the o u t s e t . In the f i r s t p l a c e , the n e u t r a l l e f t - h a n d e d l e p t o n s ( n e u t r i n o s ) were assumed m a s s l e s s . I f t h i s were not so, and i f the muon n e u t r i n o and e l e c t r o n n e u t r i n o masses were not de g e n e r a t e , t h e i r weak e i g e n s t a t e s would not be mass e i g e n s t a t e s , and a s i t u a t i o n much l i k e the Cabibbo m i x i n g of the quark s e c t o r would o c c u r . A GIM mechanism (Glashow, I l i o p o u l o s , and M a i a n i , 1970) c o u l d l e a d t o l e p t o n f l a v o r - c h a n g i n g weak n e u t r a l c o u p l i n g s s u ppressed by a f a c t o r of the or d e r of Gp.sin0cosP (mf - m^) (becoming second o r d e r i n G F ) , where m, and ma a r e the n e u t r i n o masses and 9 a m i x i n g a n g l e . For the l a r g e s t mass d i f f e r e n c e a l l o w e d by c u r r e n t d i r e c t measurements of the muon n e u t r i n o mass of about 0.5 MeV (Daum e_t a l . , 1980), the s u p p r e s s i o n i s of the o r d e r of 1 0 " 1 2 . U s i n g a more s t r i n g e n t e s t i m a t e of (m,2 - mfj = 25 eV 2 (Mann and P r i m a k o f f , 1977), based on c o s m o l o g i c a l arguments, the s u p p r e s s i o n would be more l i k e 1 0 - 2 2 . In some sense, t h e n , a p p r o x i m a t e muon number c o n s e r v a t i o n can be reg a r d e d as due t o the near degeneracy of 32 n e u t r i n o masses, compared w i t h o t h e r t y p i c a l masses. I f t h e r e F i g u r e 2.2.2.1. C o n v e r s i o n p r o c e s s v i a non-degenerate n e u t r i n o s . i s no o t h e r , more dominant mechanism f o r c o n v e r s i o n , the muonium-antimuonium e f f e c t i v e c o u p l i n g as d e s c r i b e d by F i g u r e 2.2.2.1 s h o u l d s a t i s f y G/GF < 1 0 - 1 4 2.2.2.1 f o r a mass d i f f e r e n c e of 1 MeV. Very r e c e n t l y , a r e a c t o r n e u t r i n o experiment has shown some e v i d e n c e f o r n e u t r i n o i n s t a b i l i t y or m i x i n g ( R e i n e s e t a l . , 1980), and the lower l i m i t on the squared mass d i f f e r e n c e i s about 0.3 e V 2 . There i s a t t h i s w r i t i n g . m u c h c o n t r o v e r s y r e g a r d i n g s e v e r a l n e u t r i n o mass or mass d i f f e r e n c e e x p e r i m e n t s , and no r e s u l t has as y e t been w i d e l y a c c e p t e d . S e c o n d l y , the Higgs boson c o n t e n t of the t h e o r y i s m i n i m a l . 33 The s i n g l e Higgs d o u b l e t was i n t r o d u c e d i n the s i m p l e s t way p o s s i b l e t o account f o r the observed mass s t r u c t u r e , and a l s o p r e d i c t s the r e l a t i v e s t r e n g t h of charged and n e u t r a l c u r r e n t s . A s l i g h t l y more g e n e r a l s e t of Higgs d o u b l e t s was c o n s i d e r e d ( B j o r k e n and Weinberg, 1977) on the s t r e n g t h of a rumour of the o b s e r v a t i o n of the decay (which, though l a t e r shown t o be unfounded, caused much a c t i v i t y among t h e o r i s t s p o s t u l a t i n g mechanisms f o r muon number v i o l a t i o n ) . The e l e c t r o n and muon mass terms i n a g e n e r a l i z a t i o n of X H of e q u a t i o n 2 .2 .1 .13, -G c ( L e ^ R e ) - G ^ ( I ^ R ^ J , become -9, I ^ f c R A " 9x^ e ft 9* - 9 3 l J " fV^e " 9 A A Rc' 2:2.2.2 Assuming and g a = g 3 i s the same o r d e r as g, (m^GjL 1 ' 2 ' f o r < ^ > - v G p n / 1 1 ) , i t i s p o s s i b l e t h a t G/Gpfv ny2/mH2 ryj 10* * 2.2.2.3 f o r m^'vlO GeV (see F i g u r e 2 .2 .2 .2 ) . There i s , however, l i t t l e b a s i s f o r such a s s u m p t i o n s . Note t h a t e x p r e s s i o n 2.2.2.2 can be extended t o i n c l u d e the t a u g e n e r a t i o n , i n which the muon mass i n e q u a t i o n 2 .2.2.3 would be r e p l a c e d by the t a u mass, and the r a t i o of c o u p l i n g c o n s t a n t s might approach 10" A t h i r d e x t e n s i o n of the s t a n d a r d model p o s t u l a t e s a m o d i f i c a t i o n of t h e fe r m i o n f i e l d s t o i n c l u d e r i g h t - h a n d e d d o u b l e t s and a s s o c i a t e d V+A c u r r e n t s i n v o l v i n g massive n e u t r a l r i g h t - h a n d e d f e r m i o n s N l > a L (Cheng and L i , 1977). One p r e d i c t i o n of t h i s e x t e n s i o n i s t h a t the e l e c t r o n i c n e u t r a l c u r r e n t i s p u r e l y v e c t o r , w i t h no a x i a l p a r t as i n the s t a n d a r d model. 34 F i g u r e 2.2.2.2. C o n v e r s i o n v i a non-minimal Higgs c o u p l i n g . Some e a r l y e x p e r i m e n t s on atomic p a r i t y v i o l a t i o n by n e u t r a l c u r r e n t s i n heavy atoms d i d not agree w i t h V-A p r e d i c t i o n s ( f o r a r e v i e w , see B o u c h i a t , 1977), but more r e c e n t meaurements w i t h systems more e a s i l y u n d e r s t o o d show the e x p e c t e d p a r i t y v i o l a t i o n s , and the model w i t h V+A c u r r e n t s i s not s u p p o r t e d . The diagram f o r p o s s i b l e muonium-antimuonium c o n v e r s i o n i n t h i s e x t e n s i o n i s t h a t of F i g u r e 2.2.2.1, w i t h *VI>JL r e p l a c e d by the heavy f e r m i o n s N,^. The GIM mechanism i s s t i l l a p p l i c a b l e , but the mass d i f f e r e n c e i s not n e c e s s a r i l y s m a l l . Cheng and L i o b t a i n G/GF = (GF/1677-2 ) s i n 2 0 c o s 2 0 (m,2 - m^) 2.2.2.4 when m, > >m a i. For the maximal m i x i n g a n g l e and a mass squared d i f f e r e n c e of 1 GeV 2, one o b t a i n s G / G F ^ 1 0 _ * . Note t h a t s i n c e r i g h t - h a n d e d c u r r e n t s a r e p a r t i c i p a t i n g , the a x i a l p a r t s of the e f f e c t i v e H a m i l t o n i a n ( e q u a t i o n 2.1.1.5) must change s i g n , (1 -35 / 5 ) t o (1 + ^ 5 ) . T h i s has no e f f e c t on the f i n a l r e s u l t f o r of appendix A l . The l a s t e x t e n s i o n of the s t a n d a r d model which w i l l be mentioned d i d not r e s u l t from s p e c u l a t i o n on the rumoured yu.-> et^ decay, but r a t h e r from the d i s c o v e r y of the t h i r d t a u g e n e r a t i o n of l e p t o n s ( P e r l e t a l . , 1975). The c o n v e r s i o n scheme i s based on the p r o p o s i t i o n (Derman, 1978 and 1979) of i n v a r i a n c e of the L a g r a n g i a n under p e r m u t a t i o n of the t h r e e l e p t o n l a b e l s , e, JJL, and TT, a symmetry s p o n t a n e o u s l y broken by the H i g g s - K i b b l e mechanism by which the l e p t o n s a c q u i r e t h e i r g r o s s l y d i s s i m i l a r masses. P e r m u t a t i o n i n v a r i a n c e r e q u i r e s a minimum of t h r e e Higgs d o u b l e t s ; the demand of mass non-degeneracy of the charged l e p t o n s l e a d s t o the c o n c l u s i o n t h a t some of the Higgs p a r t i c l e s c a r r y l e p t o n f l a v o r , l e a d i n g t o a n o n - c o n s e r v a t i o n of a d d i t i v e muon (or tauon) number. I n s t e a d , a m u l t i p l i c a t i v e p a r i t y - l i k e c o n s e r v a t i o n law i s i m p l i e d , the T and one of JUL or e p o s s e s s i n g a p o s i t i v e p a r i t y , the o t h e r (e or /*.) a n e g a t i v e one. T h i s p a r t i c u l a r assignment i n s u r e s the apparent absence of y< -> etf. The t h e o r y makes some s p e c i f i c p r e d i c t i o n s on decay phenomena which might be o b s e r v e d , but as u s u a l the Higgs masses o f t e n appear i n m a t r i x e l e m e n t s , making e s t i m a t e s of r a t e s r a t h e r i m p r e c i s e . I t i s i m p o r t a n t t o note t h a t , w h i l e the H i g g s p a r t i c l e s of the p o s i t i v e l e p t o n p a r i t y a r e r e q u i r e d t o be v e r y massive t o reproduce known phenomena, those of n e g a t i v e p a r i t y c o u l d be of the o r d e r of 10 GeV. Diagrams s i m i l a r t o f i g u r e 2 . 2 . 2 . 2 c o u l d then r e s u l t i n G/GF as l a r g e as 0.1, e s s e n t i a l l y the same as f o r the B j o r k e n and Weinberg model extended t o t h i r d g e n e r a t i o n l e p t o n s . I t i s a l s o p o s s i b l e , w i t h i n t h i s model, t h a t charged c u r r e n t i n t e r a c t i o n s 36 may e x h i b i t a m u l t i p l i c a t i v e c o n s e r v a t i o n law a t a much lower l e v e l than n e u t r a l c u r r e n t ones, making muonium c o n v e r s i o n an i n t e r e s t i n g t e s t . The purpose of t h i s s e c t i o n has been t o show t h a t modern t h e o r i e s of weak and e l e c t r o m a g n e t i c i n t e r a c t i o n s can a t p r e s e n t encompass a broad range of p o s s i b i l i t i e s ; v a r i a t i o n s on the WS theme a r e d i v e r s e and m a n i f o l d , because e x p e r i m e n t a l r e s u l t s can not y e t d i f f e r e n t i a t e among s e v e r a l of the p o s s i b i l i t i e s . The a t t i t u d e of the p a r t i c l e p h y s i c s community must then be t h a t e f f o r t s toward an improvement i n our knowledge of the e l e c t r o w e a k phenomenology, toward an i n c r e a s e i n the l e v e l of c o n f i d e n c e i n (or d e m o n s t r a t i o n of the breakdown o f ) laws such as t h a t of muon number c o n s e r v a t i o n , are w o r t h w h i l e . 37 3. DETAILS OF THE CONVERSION EXPERIMENT The p r i n c i p l e of the muonium-antimuonium c o n v e r s i o n experiment i s q u i t e s i m p l e ; one needs o n l y t o observe muonium and determine whether i t does or does not c o n v e r t t o antimuonium. In p r a c t i c e t h e r e are s e v e r a l problems of v a r y i n g s e v e r i t y making t h i s d e t e r m i n a t i o n d i f f i c u l t . The muon i s not s t a b l e , so t h a t o b s e r v a t i o n s must be made d u r i n g i t s l i f e t i m e , and the p r o b a b i l i t y of c o n v e r s i o n i s reduced because of the competing p r o c e s s of muon decay. The environment of muonium can d r a s t i c a l l y i n f l u e n c e the i n t e r a c t i o n r a t e , as e x p l a i n e d i n s e c t i o n 2.1. D e t e c t i o n of p o s s i b l e c o n v e r s i o n e v e n t s must be a c h i e v e d w i t h r e a s o n a b l y h i g h e f f i c i e n c y w h i l e m a i n t a i n i n g a low background r a t e . The f o r m a t i o n of muonium atoms from a beam of muons w i l l t a k e p l a c e d u r i n g the passage through and energy l o s s i n some moderator ( s o l i d , l i q u i d , or g a s ) . The e l e c t r o n c a p t u r e and l o s s p r o c e s s e s i n v o l v e d a r e analogous t o those of p r o t o n s s l o w i n g i n m a t t e r , and f o r m a t i o n of a n e u t r a l atom i s thought t o be most l i k e l y when the muon (or p r o t o n ) has a v e l o c i t y comparable t o t h a t of the v a l e n c e atomic e l e c t r o n s ( s e e , f o r example, Tawara and Russek, 1973). A f t e r f u r t h e r energy l o s s , muonium s t a t e s may p e r s i s t or d i s a p p e a r , depending upon the moderator. In some m a t e r i a l s , the p r e c e s s i o n of the s p i n s of p o l a r i z e d muonium atoms may be observed a f t e r t h e r m a l i z a t i o n i s c o m p l e t e , and l i m i t s on the amount of muonium formed a r e e a s i l y 38 o b t a i n e d (Brewer et a l . , 1975) by the methods of muon and muonium s p i n r o t a t i o n . The moderator, n e c e s s a r y f o r muonium f o r m a t i o n i n a c o n v e r s i o n e x p e r i m e n t , s u b s e q u e n t l y i n h i b i t s c o n v e r s i o n by i n t e r a c t i n g e l e c t r o m a g n e t i c a l l y w i t h the charges and magnetic moments of muonium. H e r e i n l i e s the reason t h a t v e r y s e n s i t i v e s e a r c h e s f o r muonium-antimuoniurn c o n v e r s i o n have not y e t o c c u r r e d ; muonium f o r m a t i o n and c o n v e r s i o n cannot t a k e p l a c e a t t h e i r r e s p e c t i v e optimum r a t e s i n any one environment, and compromises must be made. D e t e c t i o n of the c o n v e r s i o n i s a r e l a t i v e l y l e s s d i f f i c u l t problem. There a r e two ways t o approach i t . One can s e a r c h e i t h e r f o r a f a s t n e g a t i v e e l e c t r o n r e s u l t i n g from the normal decay of the muon i n antimuonium, or f o r a c h a r a c t e r i s t i c muonic X-ray from the c a p t u r e of the n e g a t i v e muon i n an atomic o r b i t a l about a n u c l e u s from the t a r g e t environment. P r e s e n t e d i n t h i s c h a p t e r w i l l be the d e t a i l s of the methods used t o o p t i m i z e the p r o b a b i l i t y of d e t e c t i n g muonium c o n v e r s i o n i n the experiment a t TRIUMF . In o r d e r t o put t h i s most r e c e n t e f f o r t i n p e r s p e c t i v e , i t i s w o r t h w h i l e t o comment on p r e v i o u s muonium-antimuonium c o n v e r s i o n e x p e r i m e n t s , b o t h s u c c e s s f u l and u n s u c c e s s f u l , as w e l l as on some r e l a t e d i n v e s t i g a t i o n s r e g a r d i n g the m u l t i p l i c a t i v e muon number c o n s e r v a t i o n scheme. 39 3.1. A Review of R e l a t e d E x p e r i m e n t s 3.1.1. Other Muonium-Antimuonium Ex p e r i m e n t s The o n l y r e l i a b l e l i m i t which has been s e t by a c o n v e r s i o n experiment on the c o u p l i n g c o n s t a n t G of e q u a t i o n 2.1.1.5 i s due to a Y a l e group w o r k i n g a t the N e v i s s y n c h r o c y c l o t r o n (Amato et  a l . , 1968). A c o n v e n t i o n a l beam of muons from p i o n decay i n f l i g h t was degraded so t h a t some muons would s t o p i n an argon gas t a r g e t a t one atmosphere p r e s s u r e . V a r i o u s p l a s t i c s c i n t i l l a t o r s c o m p l e t e l y surrounded the gas r e g i o n f o r the i d e n t i f i c a t i o n of s t o p p i n g muons. The s c i n t i l l a t o r s a l s o d e t e c t e d subsequent decay e l e c t r o n s . Two sodium i o d i d e s c i n t i l l a t i o n c o u n t e r s , s e n s i t i v e t o argon muonic 2P-1S X-rays a t 644 keV, were used t o i d e n t i f y the c o n v e r s i o n . To reduce background, a c c e p t e d e v e n t s i n c l u d e d o n l y t h o s e photon c o u n t s f o l l o w i n g a muon s t o p f o r which no e l e c t r o n of e i t h e r charge was d e t e c t e d . T h i s meant t h a t some r e a l c o n v e r s i o n e v e n t s c o u l d be r e j e c t e d due t o n e g a t i v e muon decay from the IS atomic s t a t e i n argon (as opposed t o n u c l e a r muon c a p t u r e ) , but the l o s s was more than o f f s e t by the r e s u l t a n t background r e d u c t i o n . However, the e f f e c t of the argon environment on the muonium produced was such t h a t the be s t upper l i m i t o b t a i n a b l e was G < 5800G F. The assignment of t h i s l i m i t r e l i e d upon s e v e r a l numbers, 40 one of which was the muonium f o r m a t i o n p r o b a b i l i t y i n argon gas at one atmosphere. T h i s was taken t o be u n i t y , c o n s i s t e n t w i t h c a l c u l a t i o n s and o b s e r v a t i o n s a t the time of the experiment. More r e c e n t measurements have shown t h i s p r o b a b i l i t y t o be somewhat lower a t 0.63 ± 0.07 ( M i k u l a e t a l . , 1979; see a l s o B a r n e t t e t a l . , 1975). Much e f f o r t was d i r e c t e d toward the c a l c u l a t i o n of the p r o b a b i l i t y t h a t a muonium atom i n the t a r g e t would l e a d t o an argon muonic atom (Morgan, 1967), a r e s u l t c r u c i a l i n the i n t e r p r e t a t i o n of the d a t a . T h i s p r o b a b i l i t y , s t a t e d here f o r f u t u r e r e f e r e n c e , i s (1.0 ± 0.2) x 1 0 " 1 0 ( G / G p ) 2 , i n c l u d i n g the e f f e c t of muonium-argon c o l l i s i o n s which d r a s t i c a l l y reduce the c o n v e r s i o n r a t e . For comparison, the c o n v e r s i o n p r o b a b i l i t y i n vacuum (see s e c t i o n 2.1.3) i s 2.5 x 10- 5 (G/Gp.) 2 . An attempt was made by a U n i v e r s i t y of A r i z o n a group, wo r k i n g a t the B e r k e l e y 184" c y c l o t r o n , t o use a t a r g e t of t h i n , hot m e t a l f o i l s i n vacuum. An inhomogeneous t a r g e t was used i n o r d e r t o reduce the e f f e c t of the moderator on the c o n v e r s i o n r a t e . One component of the t a r g e t was t o s e r v e as an e l e c t r o n donor, the o t h e r as a r e g i o n where the c o n v e r s i o n p r o b a b i l i t y was h i g h ( t h e vacuum). I t was e x p e c t e d t h a t muons might d i f f u s e i n the f o i l s q u i c k l y enough t o r e a c h the s u r f a c e , where they c o u l d p i c k up an e l e c t r o n t o form muonium b e f o r e e n t e r i n g the vacuum between f o i l s . Some i n d i r e c t e v i d e n c e f o r muonium f o r m a t i o n was o b t a i n e d (Bowen e t a l . , u n p u b l i s h e d ; see a l s o K e n d a l l , 1972) from muon s p i n r e l a x a t i o n measurements, but no d i r e c t muonium s p i n . r o t a t i o n s i g n a l was o b s e r v e d . D e t e c t i o n of a p o s s i b l e muonium-antimuonium c o n v e r s i o n was t o be by h e l i c a l s c i n t i l l a t o r t e l e s c o p e s i n a magnetic f i e l d such t h a t o n l y f a s t 41 n e g a t i v e e l e c t r o n s from antimuonium decay would be o b s e r v e d . One v e r y i m p o r t a n t a s p e c t of t h i s experiment was the a s s o c i a t e d development of a h i g h s t o p p i n g d e n s i t y beam of low momentum (~29 MeV/c) " A r i z o n a " muons, now u s u a l l y r e f e r r e d t o as s u r f a c e muons because of t h e i r o r i g i n from p i o n s stopped i n the s u r f a c e of a p r o d u c t i o n t a r g e t ( P i f e r e t a l . , 1976). The use of such beams i s now common f o r stopped p o s i t i v e muon ex p e r i m e n t s a t meson f a c i l i t i e s , and i s e s s e n t i a l t o the c o n v e r s i o n experiment a t TRIUMF (see s e c t i o n 3.2.1). Another unique type of muon beam has been used i n an i n v e s t i g a t i o n of muonium f o r m a t i o n i n low p r e s s u r e (~10~ 2 mm Hg) argon gas. P o s i t i v e p i o n s of 39.5 MeV/c momentum w i l l , upon decay i n f l i g h t , produce some muons of energy l e s s than 10 keV from decay i n the d i r e c t i o n o p p o s i t e the p i o n momentum. In a subsequent c o l l i s i o n w i t h the gas, muonium c o u l d be formed. A group from the U n i v e r s i t y of Berne, working a t CERN, at t e m p t e d t o d e t e c t the f o r m a t i o n by a p p l y i n g a magnetic f i e l d a l o n g the i n c i d e n t beam, t o t r a p the charg e d p a r t i c l e s , 'while s e a r c h i n g f o r muons d e c a y i n g a t some d i s t a n c e from the beam a x i s (Hofer e t  a l . , 1972). Muons e m i t t e d i n p i o n decay w i t h a v e l o c i t y t r a n s v e r s e t o the a p p l i e d f i e l d would s p i r a l u n t i l f o r m i n g muonium, when they would be a b l e t o escape the magnetic f i e l d t o be d e t e c t e d i n a r i n g c o n c e n t r i c w i t h the i n c i d e n t beam. No s i g n a l a t t r i b u t a b l e t o muonium f o r m a t i o n was d e t e c t e d . The most r e c e n t i d e a f o r fo r m i n g muonium u s e f u l f o r a c o n v e r s i o n experiment i s t h a t of a V i r g i n i a - M a r y l a n d group working a t SREL ( B a r n e t t e t a l . , 1977), u s i n g a t a r g e t s i m i l a r t o t h a t of the A r i z o n a experiment and the o r i g i n a l A r i z o n a beamline t r a n s p o r t e d from B e r k e l e y . In t h i s c a s e , a s t a c k of 42 200 (unheated) g o l d . f o i l s , each of 10" 5 cm t h i c k n e s s , was used. Muonium s p i n r o t a t i o n i n d i c a t e d t h a t the stopped muons formed muonium i n vacuum w i t h a c l a i m e d p r o b a b i l i t y of 0.28 ± 0.05. A l t h o u g h t h i s r e s u l t i n d i c a t e d t h a t a v a s t improvement i n muonium c o n v e r s i o n e x p e r i m e n t s was i m m e d i a t e l y p o s s i b l e , i t c o u l d not be reproduced a t LAMPF (Beer e t a l . , 1979) or SIN ( A r n o l d e t a l . , 1979a), where s i m i l a r e x p e r i m e n t s of h i g h e r s e n s i t i v i t y were u n d e r t a k e n ; no e f f e c t s a t t r i b u t a b l e t o muonium f o r m a t i o n i n vacuum were observed i n e i t h e r i n s t a n c e . A t a r g e t c o n s i s t i n g of c o l l o d i o n f i l m s w i t h g o l d e v a p o r a t e d onto the s u r f a c e s was s t u d i e d a t TRIUMF, but i t a l s o showed no measurable muonium f o r m a t i o n . Thus i t seems p r o b a b l e t h a t t h i s method i s not w o r k a b l e . T h i s completes the l i s t of p r i o r a t t e m p t s , m o s t l y u n s u c c e s s f u l , t o l a y a f o u n d a t i o n upon which a s e n s i t i v e muonium-antimuonium c o n v e r s i o n experiment c o u l d be based. As an a s i d e , the p r o d u c t i o n of muonium i n vacuum i s a l s o a p r e r e q u i s i t e f o r some s e n s i t i v e t e s t s of quantum e l e c t r o d y n a m i c e f f e c t s i n muonium ( e . g . , i n measurements of the f i n e s t r u c t u r e , Lamb s h i f t , and h y p e r f i n e s t r u c t u r e of the n=2 e x c i t e d s t a t e ) (Hughes, 1979). 3.1.2. E x p e r i m e n t s w i t h Other Systems A non-zero v a l u e f o r the c o u p l i n g c o n s t a n t G of e q u a t i o n 2.1.1.5 w i l l r e s u l t i n p r o c e s s e s o t h e r than muonium c o n v e r s i o n . In p a r t i c u l a r , a c o l l i s i o n of two e n e r g e t i c n e g a t i v e e l e c t r o n s 43 c o u l d r e s u l t i n two n e g a t i v e muons by the same i n t e r a c t i o n . U s i n g t h i s f a c t , an experiment at the P r i n c e t o n - S t a n f o r d e l e c t r o n s t o r a g e r i n g s has s e t a l i m i t of G £ 610G F (Barber et a l . , 1969), n e a r l y an o r d e r of magnitude b e t t e r than the Y a l e -N e v i s muonium-antimuonium upper l i m i t . In a d d i t i o n t o t h i s , a r e c e n t experiment on n e u t r i n o s from p o s i t i v e muon decay has search e d f o r e l e c t r o n a n t i n e u t r i n o s , which would be a l l o w e d o n l y i f muon c o n s e r v a t i o n obeyed a t most a m u l t i p l i c a t i v e law ( W i l l i s e t a l . , 1980). The r e s u l t s t r o n g l y d i s f a v o r s the m u l t i p l i c a t i v e scheme, i n t h a t the r a t i o of "wrong" muon decays t o a l l decays i s measured t o be R < 0.098, whereas R = 0.5 might be e x p e c t e d from the m u l t i p l i c a t i v e r u l e . The i n t e r a c t i o n which c o u l d l e a d t o a nonzero v a l u e of R, however, i s not the n e u t r a l c u r r e n t one of e q u a t i o n 2.1.1.5, but r a t h e r i n v o l v e s charged c u r r e n t s and the a s s o c i a t e d exchange bosons. For t h i s r e a s o n , the l i m i t on the r a t i o R cannot be used t o c a l c u l a t e a l i m i t on the r a t i o G/GP w i t h o u t r e c o u r s e t o a s p e c i f i c model t o determine the elementary c o u p l i n g s , exchange p a r t i c l e s , and t h e i r i n t e r r e l a t i o n s h i p s . F o r i n s t a n c e , i n the model of Derman d i s c u s s e d i n s e c t i o n 2.2.2, the masses of the cha r g e d and n e u t r a l Higgs p a r t i c l e s would determine the r e s p e c t i v e r a t i o s , and t h e r e i s no a p r i o r i reason t o assume t h a t the masses a r e e q u a l . The upper l i m i t on "wrong" muon decay, t h e r e f o r e , does not a b s o l u t e l y r u l e out muonium-antimuonium c o n v e r s i o n w i t h a s t r o n g e r c o u p l i n g . On the o t h e r hand, i t i s c o n v i n c i n g e v i d e n c e t h a t a muon c o n s e r v a t i o n law, i f i t does e x i s t a t a l l , i s a d d i t i v e i n n a t u r e . W i t h the knowledge of the h i s t o r y of muonium c o n v e r s i o n e x p e r i m e n t s and the r e l a t e d m u l t i p l i c a t i v e law r e s e a r c h , the 44 o u t s t a n d i n g problems become more c l e a r l y d e f i n e d . The attempt at TRIUMF t o f i n d s o l u t i o n s t o t h e s e w i l l be the s u b j e c t of the f o l l o w i n g s e c t i o n . 3.2. The Apparatus and Techniques Used An attempt was made a t TRIUMF t o d e s i g n an a p p a r a t u s which would maximize the s e n s i t i v i t y t o muonium c o n v e r s i o n . T h i s e f f o r t , i n i t i a t e d i n 1975, l a r g e l y c oncerned the muonium p r o d u c t i o n t a r g e t i t s e l f , but p r e c i s e magnetic f i e l d c o n t r o l and a s i m p l e , e f f i c i e n t scheme f o r d e t e c t i o n were a l s o sought. W i t h i n the l i m i t a t i o n s thus imposed, i t was i m p e r a t i v e t h a t a beamline w i t h a w e l l d e f i n e d , h i g h l u m i n o s i t y , h i g h s t o p p i n g d e n s i t y muon beam be a v a i l a b l e . 3.2.1. M13 and S u r f a c e Muons Such a beamline was b u i l t and commissioned a t TRIUMF by the s p r i n g of 1979. I t was a low energy c h a n n e l , d e s i g n e d t o be as s h o r t as p o s s i b l e (9.4 m) so t h a t p i o n s down t o about 10 MeV ( w i t h a mean decay l e n g t h of about t h r e e metres) might s u r v i v e passage t h r o u g h i t . The o p t i c a l and beam c o n t r o l ( v i a s l i t s and jaws) p r o p e r t i e s a l s o make i t e x c e l l e n t f o r s u r f a c e muons, e s p e c i a l l y s i n c e the muon s o u r c e , the p r i m a r y p r o d u c t i o n t a r g e t , 45 can be made s m a l l . The o r i g i n s of s u r f a c e muons, both h i s t o r i c a l l y and p h y s i c a l l y ( P i f e r e t a l . , 1976), were mentioned b r i e f l y i n the p r e c e d i n g s e c t i o n . When 500 MeV p r o t o n s i n t e r a c t w i t h n u c l e o n s of a t a r g e t such as ca r b o n , p o s i t i v e p i o n s w i l l be produced. The energy d i s t r i b u t i o n of these p i o n s , o n l y r e c e n t l y d e t e r m i n e d ( C r a w f o r d e t a l . , 1980) down t o the low regime i m p o r t a n t f o r s u r f a c e muon p r o d u c t i o n , makes i t p o s s i b l e f o r some t o come t o r e s t c l o s e t o the s u r f a c e of the p r o d u c t i o n t a r g e t (a 13 MeV p i o n has a range c l o s e t o 1.0 g * c n r 2 ) . These p i o n s may a r i s e from p r o t o n i n t e r a c t i o n s anywhere i n the t a r g e t , as l o n g as the p i o n energy c o r r e s p o n d s t o a range which i s the amount of t a r g e t m a t e r i a l which must be t r a v e r s e d i n o r d e r t h a t i t j u s t reaches the s u r f a c e . One does not want the t a r g e t t o be much l a r g e r than the p r o t o n beam p r o f i l e , t h e n , because the s u r f a c e muon r a t e depends on the s o l i d a n g l e subtended by the s u r f a c e averaged over the p r o t o n i n t e r a c t i o n r e g i o n ( n e g l e c t i n g any a n i s o t r o p y i n the p i o n p r o d u c t i o n a n g l e s ) . There are two o t h e r reasons t h a t s m a l l p r o d u c t i o n t a r g e t s a r e d e s i r a b l e . The f i r s t i s o b v i o u s ; a s m a l l e r muon source w i l l l e a d t o a s m a l l e r muon beam spot i n a c h a n n e l w i t h r e a s o n a b l e o p t i c s . The second i n v o l v e s c o n t a m i n a t i o n of the beam by p o s i t r o n s . Below a momentum of 52.8 MeV/c, the maximum p o s i t r o n energy from muon decay a t r e s t , the p o s i t r o n s i n the beam come both from c o n v e r s i o n of gammas from n e u t r a l p i o n decay and from muons d e c a y i n g i n t h e p r o d u c t i o n t a r g e t . The former a r e much reduced by the use of s m a l l , low Z t a r g e t s s i n c e c o n v e r s i o n of gammas i s l e s s l i k e l y . The l a t t e r may be d e a l t w i t h by beam s e p a r a t i o n t e c h n i q u e s . 46 One such s e p a r a t i o n t e c h n i q u e which has been a p p l i e d f o r s u r f a c e muons ( R e i s t e t a l . , 1978; A r n o l d et a l . , 1979b) i s the use of a few mg»cnr 2 of degrader m a t e r i a l a t an a p p r o p r i a t e f o c u s b e f o r e the f i n a l bend i n the beamlin e . S i n c e the muons l o s e more energy i n t h i s degrader than do p o s i t r o n s , the two components of the beam w i l l be s p a t i a l l y s e p a r a t e d a f t e r the next bending element. T h i s t e c h n i q u e has been observed a t TRIUMF t o have some adverse e f f e c t on f i n a l muon beam spot s i z e s . A schematic diagram of the M13 cha n n e l i s shown i n F i g u r e 3.2.1.1. V i e w i n g the p r i m a r y p r o d u c t i o n t a r g e t a t an a n g l e of horizontal & vertical jaws .vacuum valve beam blocker forget ^ I A T I horizontal slit-•cal* X final focus lm*tr* 0 I 2 3 f t i t F i g u r e 3.2.1.1. The M13 pion/muon c h a n n e l a t TRIUMF . 135° t o the 500 MeV p r o t o n beam, i t c o n s i s t s of n i n e magnetic elements; two q u a d r u p o l e s , a 60° r i g h t b e n d i n g d i p o l e , t h r e e more q u a d r u p o l e s , the second 60° l e f t bending d i p o l e , and the f i n a l two q u a d r u p o l e s . A vacuum box w i t h r e m o t e l y c o n t r o l l a b l e 47 v e r t i c a l and h o r i z o n t a l jaws, t o d e f i n e the beamline acceptance and determine f l u x , i s p o s i t i o n e d between Q2 and B I . Between BI and Q3 i s the f i r s t d i s p e r s e d f o c u s , F l . H o r i z o n t a l s l i t s d e t e rmine the segment of the d i s p e r s i o n p l a n e t o be used, t h e r e b y a l l o w i n g c o n t r o l of the momentum a c c e p t a n c e , w h i l e v e r t i c a l jaws can be used f o r f l u x l i m i t a t i o n and the r e d u c t i o n of s c a t t e r e d beam p a r t i c l e s . Two f o u r - p o s i t i o n wheels, s u p p o r t i n g v a r i o u s t h i c k n e s s e s of p o l y e t h y l e n e s h e e t , a r e a l s o near F l , and can be used t o reduce c o n t a m i n a t i o n by unwanted p a r t i c l e s ( p o s i t r o n s i n a s u r f a c e muon beam, or p r o t o n s i n a p i o n beam). An e q u i v a l e n t (by the symmetry of the c h a n n e l ) d i s p e r s e d f o c u s F2, between Q5 and B2, i s equipped w i t h a n other s e t of v e r t i c a l jaws and h o r i z o n t a l s l i t s . Though t o some e x t e n t redundant, t h i s s e t can reduce the t a i l s of the f i n a l beam p r o f i l e a r i s i n g from p a r t i c l e s s c a t t e r e d i n the beam tube f u r t h e r upstream. A l l magnet power s u p p l i e s (except B2, a t t h i s w r i t i n g ) as w e l l as s l i t s , jaws, and a b s o r b e r a r e c o n t r o l l e d v i a microcomputer through CAMAC i n t e r f a c e s or th r o u g h the remote c o n t r o l s system m a i n t a i n e d by TRIUMF . F i g u r e 3.2.1.2 shows the p a r t i c l e f l u x e s measured from a 1.45 mm g r a p h i t e t a r g e t a v a i l a b l e when the c h a n n e l was f i r s t commissioned. Note the d r a m a t i c s u r f a c e muon peak a t 29 MeV/c. The low e l e c t r o n c o n t a m i n a t i o n i s o b t a i n e d by the use of t h i n , b a r e , p r o d u c t i o n t a r g e t s , w i t h no e x t r a n e o u s s u r r o u n d i n g m a t e r i a l s . P r e s e n t l y a v a i l a b l e t a r g e t s (used i n the c o n v e r s i o n e x p e r i m e n t ) i n c l u d e 2.0 mm and 10.0 mm g r a p h i t e p i e c e s , and s c a l i n g of muon f l u x e s w i t h t a r g e t t h i c k n e s s i s i n t h i s case r e l i a b l e . F i g u r e s 3.2.1.3 and 3.2.1.4 show the e f f e c t of h o r i z o n t a l s l i t s and jaws on p a r t i c l e f l u x and beam spot 48 50,000 40,000 30,000 20,000 10 20 30' 40 50 60 70 80 90 100 MeV/c F i g u r e 3.2.1.2. M13 p o s i t i v e p a r t i c l e f l u x e s from a 1.45 mm g r a p h i t e t a r g e t . 4 9 i 1 1 i i i 0 25 5 7-5 10 FULL OPEN HORZ. SLITS cm F i g u r e 3.2.1.3. E f f e c t of h o r i z o n t a l s l i t s on M13 p a r t i c l e f l u x and beam spot d i m e n s i o n . d i m e n s i o n , i n terms of f u l l w i d t h a t h a l f , q u a r t e r , and t e n t h maximum. F i n a l l y , the e f f e c t on s u r f a c e muon r a t e s of the p o s i t i o n i n g of the p r i m a r y p r o t o n beam on the p r o d u c t i o n t a r g e t i s shown i n F i g u r e 3.2.1.5, a l o n g w i t h a c u r v e e s t i m a t e d by C. J . Oram based on s i m p l e a s sumptions on the mechanism of s u r f a c e muon p r o d u c t i o n ; s p e c i f i c a l l y , t he p r o b a b i l i t y t h a t a p i o n o r i g i n a t i n g a t any p o i n t i n the t a r g e t s t o p s i n a s u r f a c e l a y e r i s ta k e n t o be p r o p o r t i o n a l t o the p r o d u c t of the s o l i d a n g l e subtended by, and the l e n g t h of the p i o n t r a c k i n , the 50 HORZ. JAWS FULL APERTURE cm F i g u r e 3.2.1.4. E f f e c t of h o r i z o n t a l jaws on M13 p a r t i c l e f l u x and beam spot d i m e n s i o n . l a y e r . The w i d t h of the p r o t o n beam was r e a l i s t i c a l l y assumed t o be 1.0 mm (FWHM). The g r o s s f e a t u r e s of the da t a a r e reprod u c e d s u r p r i s i n g l y w e l l . F u r t h e r d e t a i l s on p r o c e d u r e s used i n t u n i n g the c h a n n e l and measuring the beam p r o p e r t i e s have been p u b l i s h e d (Oram e t a l . , 1980). The beam tube i s h e l d under vacuum by pumps on the p r i m a r y p r o t o n beam l i n e , which i n t u r n i s c o n t i g u o u s w i t h the main c y c l o t r o n vacuum. No windows e x i s t between the p r i m a r y p r o d u c t i o n t a r g e t and the end of the beam tube. The end c o n s i s t e d , f o r t h i s e x p e r i m e n t , of a 0.127 mm (0.005 n) M y l a r window of 7.6 cm (3.0") d i a m e t e r , t h r o u g h which s u r f a c e muons 51 o PRIMARY BEAM POSITION F i g u r e 3.2.1.5. S u r f a c e muon r a t e v e r s u s p r o t o n beam p o s i t i o n on the p r o d u c t i o n t a r g e t . w i l l e a s i l y p a s s . A l t h o u g h a f l u x of 10' s u r f a c e muons per second c o u l d be o b t a i n e d f o r 100 ^ uA p r o t o n c u r r e n t i n c i d e n t on a one c e n t i m e t e r p r o d u c t i o n t a r g e t , the experiment was run a t about 1 0 s s~1. The r e d u c t i o n was due t o a lower p r o t o n c u r r e n t and/or t h e use of a 2.0 mm p r o d u c t i o n t a r g e t , as w e l l as a s l i g h t c l o s u r e of the h o r i z o n t a l jaws t o reduce the p o s i t r o n s i n t h e t a i l s of the beam p r o f i l e . The range of the muons ( f i g u r e 3.2.1.6) was 135 mg«cnr 2 M y l a r , w i t h an apparent range spr e a d of about 25 mg«cnr 2 52 SURFACE MUON RANGE IN MYLAR 70 90 110 130 150 170 TOTAL MASS IN BEAM (MG/CM2) 190 F i g u r e 3.2.1.6. S u r f a c e muon i n t e g r a l range curve, (FWHM). 3.2.2. The T a r g e t : P r o d u c t i o n of Muonium i n Vacuum I t has been p o i n t e d out t h a t a s e n s i t i v e c o n v e r s i o n experiment cannot be performed w i t h o u t the p r o d u c t i o n of muonium i n vacuum. I t i s the purpose of t h i s s e c t i o n t o d e s c r i b e the t a r g e t and i n d i c a t e , i n a q u a l i t a t i v e way, how muonium may be produced i n vacuum. More d e t a i l e d r a t e e s t i m a t e s can be found i n s e c t i o n 4.2.2, w i t h r e f e r e n c e t o a p p e n d i c e s A2 and A3. 53 The t a r g e t c o n s i s t e d of a s l o p i n g s t a c k of seventeen i d e n t i c a l e l l i p t i c a l expanded p o l y s t y r e n e frames s u p p o r t i n g t h i n c o l l o d i o n f i l m s . Each f i l m i n t u r n s u p p o r t e d , on the upper downstream s u r f a c e , a t h i n f l u f f y l a y e r of s i l i c a powder f o r muonium p r o d u c t i o n . The l o w e r , upstream s u r f a c e was c o a t e d w i t h c a l c i u m o x i d e i n which antimuonium c o u l d g i v e r i s e t o an e a s i l y r e c o g n i z e a b l e muonic X-ray (see F i g u r e s 3.2.2.1 and 3.2.2.2). Each frame was a r i n g , 3 mm t h i c k by 8 t o 10 mm wide, of F i g u r e 3.2.2.1. I l l u s t r a t i o n of the t a r g e t used f o r muonium p r o d u c t i o n i n vacuum. r o u g h l y e l l i p t i c a l shape (20 cm major by 10 cm minor a x i s l e n g t h s ) . The frames were heat t r e a t e d f o r s t r e n g t h and vacuum c o m p a t i b i l i t y . The c o l l o d i o n f i l m was c r e a t e d by f i r s t d i s s o l v i n g a c h i p of c o l l o d i o n ( c e l l u l o s e n i t r a t e , t r a d e name P a r l o d i o n , from M a l l i n c k r o d t C hemical) i n amyl a c e t a t e t o produce a t h r e e t o f i v e per c e n t s o l u t i o n , then f l o a t i n g a m i l l i l i t e r of s o l u t i o n on a c l e a n water s u r f a c e , a l l o w i n g the amyl a c e t a t e t o e v a p o r a t e . The r e s i d u a l c o l l o d i o n f i l m c o u l d 54 F i g u r e 3.2.2.2. A s i n g l e l a y e r of the t a r g e t , i l l u s t r a t i n g the mechanism of muonium p r o d u c t i o n i n vacuum. then be p i c k e d up by s l i d i n g the frame i n the water beneath the f i l m and g e n t l y removing i t from the water s u r f a c e w i t h f i l m a t t a c h e d . The mean weight of c o l l o d i o n a t t a c h e d t o a frame was 17 ± 6 mg, c o r r e s p o n d i n g t o an average f i l m t h i c k n e s s of 0.11 mg»cnr 2 . The o n l y f u n c t i o n of the c o l l o d i o n was t o support the a c t i v e c o n s t i t u e n t s of the t a r g e t , the s i l i c a and the c a l c i u m o x i d e . The s i l i c a , i n the form of a v e r y f i n e ( p a r t i c l e d i a m e t e r 7 x 10" 7 cm, d e n s i t y 0.032 g«cnr 3 , or 0.015 tim e s t h a t of b u l k s i l i c a ) powder, i s known t o produce muonium c o p i o u s l y i n the i n t e r g r a n u l a r v o i d s ( M a r s h a l l e t a l . , 1978; K i e f l e t a l . , 55 1979). However, the e s t i m a t e d t h e r m a l muonium c o l l i s i o n r a t e w i t h powder p a r t i c l e s , about 2 x l O 1 0 s " 1 , makes the powder i t s e l f u n s u i t a b l e f o r antimuonium c o n v e r s i o n . By u s i n g a t h i n l a y e r of powder, t h e r e i s some chance t h a t muonium can escape the v o i d s i n t o a t r u e empty vacuum (the a c t u a l p r o b a b i l i t y of escape w i l l be c a l c u l a t e d i n the f o u r t h c h a p t e r ) . The t a r g e t employing the c o l l o d i o n s u b s t r a t e was c o n s t r u c t e d t o take advantage of t h i s . The reason f o r the s l o p e of the t a r g e t i s twofold.. In the f i r s t p l a c e , i t means t h a t the s i l i c a powder can r e s t on the s u r f a c e w i t h o u t f a l l i n g o f f ; the a l t e r n a t i v e s would be t o use a v e r t i c a l muon beam, p l a c i n g the l a y e r s h o r i z o n t a l l y , or t o s t i c k the s i l i c a t o the c o l l o d i o n by o t h e r means, p o s s i b l y d e s t r o y i n g i t s muonium p r o d u c t i o n p r o p e r t i e s . N e i t h e r seemed p r a c t i c a l . The second reason i s t h a t i t d o u b l e s the l e n g t h of m a t e r i a l per l a y e r i n the beam d i r e c t i o n f o r a g i v e n l a y e r t h i c k n e s s , t h e r e b y s t o p p i n g more muons near the l a y e r s u r f a c e . P r i o r t o the a p p l i c a t i o n of the powder l a y e r and f i n a l assembly of the t a r g e t s t a c k , two t h i n g s were done. F i r s t , the i n d i v i d u a l f o i l s were l i g h t l y m o istened on one s u r f a c e w i t h a spray of an acetone-water m i x t u r e and d u s t e d w i t h s i l i c a powder. Only a t i n y amount (0.04 ± 0.02 mg»cnr 2) of the s i l i c a adhered t o the c o l l o d i o n a f t e r the l i q u i d e v a p o r a t e d , but i t s e r v e d t o roughen the s u r f a c e c o n s i d e r a b l y . T h i s was i m p o r t a n t f o r reasons which w i l l soon become a p p a r e n t . S e c o n d l y , a t h i n c o a t i n g of c a l c i u m m e t a l was e v a p o r a t e d onto the o t h e r s u r f a c e and a l l o w e d t o o x i d i z e i n a d r y atmosphere. The r e s u l t i n g c a l c i u m o x i d e f i l m was e s t i m a t e d t o average f i v e t o s i x micrograms per square c e n t i m e t e r , but p r e c i s e w e i g h i n g was 56 d i f f i c u l t s i n c e the heat from e v a p o r a t i o n of the c a l c i u m metal tended t o d r i v e m o i s t u r e from the c o l l o d i o n f i l m , w h i l e the o x i d e would c o n v e r t t o h y d r o x i d e and/or c a r b o n a t e i n a p e r i o d of minutes d u r i n g w e i g h i n g . Comparisons of w e i g h t s b e f o r e and a f t e r e v a p o r a t i o n were thus o n l y a p p r o x i m a t e . The c o n v e r s i o n r e a c t i o n of the o x i d e was e l i m i n a t e d by keeping the t a r g e t f o i l s i n a d r y , i n e r t environment u n t i l i n s e r t i o n i n t o the evacuated t a r g e t r e g i o n a t the end of the M13 beam l i n e . The f i n a l s t e p was the b u i l d i n g of the s t a c k w i t h powder l i g h t l y d u s t e d on one s u r f a c e of each f o i l . The f i r s t frame was mounted, on an expanded p o l y s t y r e n e support s t r u c t u r e , w i t h the major a x i s making a 30° a n g l e w i t h the h o r i z o n t a l , c a l c i u m o x i d e s i d e down. On the upper, roughened s u r f a c e was s p r i n k l e d ( w i t h the a i d of f i n e mesh scr e e n ) s i l i c a powder. A second frame was s i m i l a r l y p l a c e d a d j a c e n t t o the f i r s t and l i g h t l y g l u e d i n p l a c e , then s p r i n k l e d w i t h more powder. T h i s procedure was c o n t i n u e d u n t i l a l l seventeen f o i l s were i n p l a c e , each s u p p o r t i n g a p p r o x i m a t e l y 0.85 mg»cnr 2 of s i l i c a . The f i n a l f o i l was not s p r i n k l e d . The roughness of the c o l l o d i o n f i l m , a f t e r the t r e a t m e n t mentioned p r e v i o u s l y , made the powder s t a b l e t o s m a l l v i b r a t i o n s ; i t would not s l i d e on the 30° i n c l i n e when c a r e f u l l y h a n d l e d . The average p e r p e n d i c u l a r d i s t a n c e between f i l m s was measured t o be 4.4 mm, w h i l e the p h y s i c a l t h i c k n e s s of the powder l a y e r was 0.3 mm. W i t h t h i s i n f o r m a t i o n , and r e a s o n a b l e a s s u m p t i o n s about the motion of muonium w i t h i n the powder l a y e r , an e s t i m a t e can be made of the p r o b a b i l i t y of muonium c o n v e r s i o n t o antimuonium. T h i s w i l l be c a l c u l a t e d i n s e c t i o n 4.2.2, u s i n g the formulae d e r i v e d i n appendix A3. I t i s s u f f i c i e n t a t t h i s 57 p o i n t t o have a more g e n e r a l i d e a of the mechanism. Muons t h e r m a l i z i n g i n a t a r g e t c o n s i s t i n g of l a y e r s of evacuated s i l i c a powder (as used on the c o l l o d i o n f i l m s ) form muonium atoms i n the i n t e r g r a n u l a r v o i d w i t h h i g h p r o b a b i l i t y . The muonium atoms are f r e e t o move a t t h e r m a l v e l o c i t y among the g r a i n s , undergoing c o l l i s i o n s and m i g r a t i n g m a c r o s c o p i c d i s t a n c e s i n the 2.2 x 10"' s muon l i f e t i m e . A f r a c t i o n of the atoms, e s p e c i a l l y those formed near the s u r f a c e of a l a y e r , a c t u a l l y l e a v e the v i c i n i t y of the powder and t r a v e l w i t h o u t c o l l i s i o n a c r o s s the gap between f i l m s . The t r a n s i t times approach the mean muon l i f e t i m e f o r s p a c i n g s of the o r d e r of one c e n t i m e t e r . For t h i s s m a l l f r a c t i o n of the i n c i d e n t muons, the o p p o r t u n i t y f o r c o n v e r s i o n i s n e a r l y maximized. Moreover, any atom t h a t does c o n v e r t t o antimuonium can be t o r n a p a r t on c o l l i s i o n w i t h the atoms or m o l e c u l e s on the f a c i n g s u r f a c e ( c a l c i u m o x i d e , i n t h i s c a s e ) , which can i n t u r n l e a d t o a muonic 2P-1S X-ray, the s i g n a t u r e of a c o n v e r s i o n e v e n t . 3.2.3. Magnetic F i e l d Measurement and C o n t r o l A f a c t o r of two i n the s e n s i t i v i t y of the experiment i s g a i n e d by f o r m i n g muonium i n a r e g i o n where magnetic f i e l d s a r e of the o r d e r of t e n m i l l i g a u s s or l e s s (see e q u a t i o n 2.1.4.1 and F i g u r e 2.1.4.1). T h i s was a c c o m p l i s h e d a t TRIUMF by the use of t h r e e m u t u a l l y p e r p e n d i c u l a r H e l m h p l t z p a i r s and a sensor which c o n t r o l l e d t h e i r power s u p p l i e s t o compensate a u t o m a t i c a l l y f o r changes i n the ambient f i e l d . 58 The problem of c r e a t i n g a u n i f o r m magnetic f i e l d (or g r a d i e n t ) can be approached from s e v e r a l p o i n t s of v i e w . The most e l e g a n t and e f f i c i e n t p r o c e d u r e i s the c o n s t r u c t i o n of a g e o m e t r i c a l l y c o r r e c t a r r a y of c u r r e n t l o o p s , c r e a t i n g a c e n t r a l f i e l d f r e e of u n d e s i r e d g r a d i e n t s t o the r e q u i r e d o r d e r ( G a r r e t t , 1967). C o n s i d e r p a i r s of c i r c u l a r c o a x i a l c u r r e n t l o o p s above and below a p l a n e of symmetry, such t h a t the c u r r e n t i n each member of a p a i r t r a v e l s i n the same sense ( f o r the c r e a t i o n of a g r a d i e n t , the c u r r e n t would be i n the o p p o s i t e s e n s e ) . The c u r r e n t s w i l l c r e a t e an a x i a l f i e l d a t the i n t e r s e c t i o n of the p l a n e of symmetry and the a x i s , which, by the symmetry, has no f i r s t or h i g h e r odd d e r i v a t i v e a l o n g the a x i s . W ith o n l y one p a i r , the s e p a r a t i o n of the c o i l s can be s p e c i f i e d t o n u l l the second d e r i v a t i v e , and the H e l m h o l t z c o n d i t i o n r e s u l t s . F u r t h e r p a i r s w i t h d i f f e r e n t s e p a r a t i o n s and/or c u r r e n t s can be added t o n u l l f o u r t h , s i x t h , e t c . , d e r i v a t i v e s , p r o v i d e d t h a t problems i n computation of the c o r r e c t parameters can be overcome. I f one can t o l e r a t e a s a c r i f i c e i n e f f i c i e n c y , i n terms of f i e l d produced per k i l o w a t t of power consumed, amount of m a t e r i a l s needed, and o v e r a l l s i z e , a l a r g e volume of u n i f o r m f i e l d can a l s o be produced by making the i n i t i a l H e l m h o l t z p a i r l a r g e , w i t h no n u l l i n g of h i g h e r d e r i v a t i v e s . An advantage of t h i s approach, f o r a p r a c t i c a l system w i t h i n which an experiment can be e r e c t e d , i s t h a t the r e g i o n of u n i f o r m i t y i s w e l l away from the c o i l w i n d i n g , so t h a t l a r g e d e t e c t o r s can be accommodated e n t i r e l y w i t h i n the s t r u c t u r e q u i t e e a s i l y . S i n c e the l i q u i d n i t r o g e n r e s e r v o i r dewars on t h e germanium c r y s t a l s used were b u l k y , and a l a r g e f i e l d was not r e q u i r e d , i t was 59 d e c i d e d t o opt f o r a l a r g e H e l m h o l t z assembly. The t h r e e m u t u a l l y p e r p e n d i c u l a r p a i r s which were b u i l t were square r a t h e r than c i r c u l a r , t o make f a b r i c a t i o n more simple'. The x,. y, and z c o i l s measured 189, 196, and 204 cm on a s i d e , r e s p e c t i v e l y . B l o c k s e n a b l e d attachment of one p a i r t o each of the o t h e r two a t the t w e n t y - f o u r p o i n t s ( f o u r on each f a c e of the cube) of p r o x i m i t y of the frames. For square c o i l geometry, the H e l m h o l t z c o n d i t i o n i s t h a t the s e p a r a t i o n s h o u l d be 0.855 times the dimension of the square. By f a r the l a r g e s t f i e l d component t h a t had t o be e l i m i n a t e d was due t o the c y c l o t r o n f r i n g e f i e l d , a p p r o x i m a t e l y t h r e e gauss i n the v e r t i c a l (z) d i r e c t i o n . Moreover, i t was a t some t i m e s d e s i r e a b l e t o a p p l y a v e r t i c a l f i e l d f o r muonium s p i n r o t a t i o n t e s t s on v a r i o u s samples, i n c l u d i n g the c o n v e r s i o n t a r g e t . T h e r e f o r e , a l l o w a n c e was made f o r a ±10 gauss range v e r t i c a l l y and a ±1 gauss range both i n the beam d i r e c t i o n (+x) and p e r p e n d i c u l a r t o i t ( y ) . The z c o i l s c o n s i s t e d of 2 x 260 t u r n s of AWG 12 copper c o n d u c t o r , c a p a b l e of s u s t a i n e d o p e r a t i o n a t 140 V and 5 A. The x and y c o i l s were each 2 x 52 t u r n s of AWG 14, o p e r a t i n g a t up t o 36 V and 5 A. The c o i l s were wound on frames of k i l n - d r i e d oak. The reason t h a t m e t a l l i c forms were not used i s t h a t the c u r r e n t s i n d u c e d i n them by v a r i a t i o n s i n the c o i l c u r r e n t a t up t o 200 Hz would r e s t r i c t the fr e q u e n c y bandwidth f o r the p r o d u c t i o n of a.c. magnetic f i e l d s . The aim was f o r the system t o be c a p a b l e of e l i m i n a t i n g a t l e a s t 60 Hz v a r i a t i o n s i n the ambient f i e l d , when used i n c o n j u n c t i o n w i t h the magnetometer c o n t r o l l e r and feedback system. The c o i l s were powered by Kepco b i p o l a r o p e r a t i o n a l power 60 s u p p l i e s . A 36 V, 5 A s u p p l y was used f o r each of the x and y p a i r s , and two 72 V, 5 A u n i t s e n e r g i z e d the z c o i l s . The o u t p u t s were governed by a S c h o n s t e d t HCM-3 t r i a x i a l magnetic f i e l d c o n t r o l u n i t , which a l s o c o n s t a n t l y m o n i t o r e d the t h r e e magnetic f i e l d components produced a t a t r i a x i a l magnetometer probe p l a c e d on the z a x i s near the c e n t e r of the H e l m h o l t z a r r a y . In t h i s manner, the c o n t r o l l e r , power s u p p l i e s , c o i l s , and magnetometer probe formed a l o o p such t h a t v a r i a t i o n s i n the ambient magnetic f i e l d components were a u t o m a t i c a l l y measured and compensated f o r . A l t h o u g h most a c c e l e r a t o r - o r i e n t e d p h y s i c i s t s have a w o r k i n g knowledge of the p r i n c i p l e s of measurement of h i g h e r magnetic f i e l d s ( e . g . , NMR, H a l l e f f e c t ) , the o p e r a t i o n of the s a t u r a b l e i n d u c t o r magnetometer, a d e v i c e u s e f u l a t z e r o and v e r y low f i e l d s , may be l e s s f a m i l i a r . I f a magnetic f i e l d i s induced i n an e a s i l y s a t u r a b l e m a t e r i a l w i t h a s i n u s o i d a l d r i v i n g c u r r e n t of f r e q u e n c y f t h r o u g h a s o l e n o i d a l p r i m a r y c o i l , t h a t f i e l d can be sensed by a secondary c o i l . I n z e r o ambient magnetic f i e l d , the output of the secondary w i l l be a s i n e f u n c t i o n w i t h f l a t t ops and bottoms c o r r e s p o n d i n g t o the s a t u r a t i o n of the m a t e r i a l . A F o u r i e r d e c o m p o s i t i o n of the f u n c t i o n w i l l i n c l u d e f r e q u e n c i e s 3 f , 5 f , and so on. I f a nonzero f i e l d component e x i s t s a l o n g t h e s o l e n o i d a x i s , the waveform on the secondary c o i l w i l l f l a t t e n a t p o i n t s on the s i n e c u r v e which are d i s p l a c e d from the z e r o f i e l d v a l u e s by an amount c o r r e s p o n d i n g t o the a p p l i e d f i e l d component (see F i g u r e 3.2.3.1). The F o u r i e r spectrum w i l l then a l s o c o n t a i n harmonic terms of the even f r e q u e n c y m u l t i p l e s 2 f , 4 f , and so on. C i r c u i t r y t o sense one of these f r e q u e n c i e s can be made t o cause 61 primary B secondary a) b) F i g u r e 3.2.3.1. S a t u r a b l e i n d u c t o r magnetometer waveforms (a) i n z e r o ambient f i e l d , (b) w i t h a nonzero f i e l d component B. a d.c. c u r r e n t t h r o u g h the p r i m a r y c o i l ( i n a d d i t i o n t o the s i n u s o i d a l d r i v i n g c u r r e n t ) such t h a t the even components a r e e l i m i n a t e d ; the v a l u e of the r e q u i r e d d.c. c u r r e n t i s then p r o p o r t i o n a l t o the ambient f i e l d component a l o n g the s o l e n o i d a x i s . W i t h fundamental f r e q u e n c i e s i n the kHz range, s l o w l y v a r y i n g ( i . e . , up t o 200 Hz) a.c. magnetic f i e l d s can a l s o be measured. S e t t i n g the t h r e e f i e l d components t o z e r o was a c c o m p l i s h e d v i a an independent, u n i a x i a l s a t u r a b l e i n d u c t o r magnetometer ( H e w l e t t P a ckard model 3529A probe and 428BR c o n t r o l l e r ) . The z e r o i n g of t h i s probe was i n t u r n a c c o m p l i s h e d by i n s u r i n g t h a t a c c u r a t e r o t a t i o n s of the probe by 180° caused e q u a l but o p p o s i t e f i e l d measurements. A f t e r some i t e r a t i o n s , each of t h e t h r e e f i e l d components a t the c o i l c e n t r e c o u l d be reduced t o one m i l l i g a u s s or l e s s , w e l l w i t h i n the t o l e r a n c e of an antimuonium c o n v e r s i o n e x p e r i m e n t . 62 F i e l d g r a d i e n t s , however, p r e s e n t e d a n other problem. S i n c e the t a r g e t was a p p r o x i m a t e l y 10 cm l o n g , a g r a d i e n t of o n l y two m i l l i g a u s s cm" 1 was s u f f i c i e n t t o exceed the c r i t e r i o n of l e s s than t e n m i l l i g a u s s i n the s t o p p i n g r e g i o n . A l t h o u g h the t h e o r e t i c a l a p p l i e d f i e l d u n i f o r m i t y s a t i s f i e d t h i s demand, the ambient f i e l d u n i f o r m i t y d i d n o t . I t was thus d i f f i c u l t (and p o i n t l e s s ) t o determine whether the c o i l s were o p e r a t i n g a t the d e s i g n e d f i e l d homogeneity. The lowe s t maximum g r a d i e n t t h a t c o u l d be o b t a i n e d under beam c o n d i t i o n s , u s i n g i r o n b l o c k s e x t e r n a l t o the c o i l s f o r tr i m m i n g p u r p o s e s , was f o u r m i l l i g a u s s c m - 1 . The s o l u t i o n t o the g r a d i e n t problem was a sheet of mumetal, 0.25 mm t h i c k and 35 cm l o n g , wrapped around the vacuum p i p e c o n t a i n i n g the t a r g e t (see F i g u r e 3.2.4.1). With t h i s i n p l a c e , a l o n g w i t h the compensation p r o v i d e d by the c o i l s , the maximum g r a d i e n t was reduced t o l e s s than one m i l l i g a u s s cm" 1, and f i e l d s u r v e y s a t the b e g i n n i n g and end of the da t a t a k i n g showed no component i n e x c e s s of 3 ± 1 m i l l i g a u s s over the e n t i r e t a r g e t r e g i o n . 3.2.4. D e t e c t o r s and Hardware i n the Tar g e t Region The gamma r a y s from a m y r i a d of p r o c e s s e s i n the t a r g e t r e g i o n (and els e w h e r e ) were d e t e c t e d v i a two independent germanium d e t e c t o r s on e i t h e r s i d e of the t a r g e t e n c l o s u r e , a p p r o x i m a t e l y 8 cm from i t s c e n t r e . A diagram of the l a y o u t c o m p r i s e s F i g u r e 3.2.4.1. Both d e t e c t o r s were l a r g e , h i g h e f f i c i e n c y , c o a x i a l , 6 3 M13Q7 ~ y Helmholtz co i l (x,z not shown) paraffin lead J : lead-lead mumetal A M13Q7 CH 2 collimator •t: Mylar window variable degrader 1 • JZ T 1 \ V n c U * upstream veto(VU) fixed degrader MU counter Ge detector front veto(VF) S i 0 2 / f o i l target to vacuum pump IO" ion gauge F i g u r e 3.2.4.1. Schematic of muonium c o n v e r s i o n a p p a r a t u s , 64 cooled-FET germanium d e v i c e s , but were o t h e r w i s e d i s s i m i l a r . One was an O r t e c l i t h i u m - d r i f t e d c r y s t a l of 51.55 mm diameter by 55.5 mm l e n g t h ( a c t i v e volume 103.2 cm 3) w h i l e the o t h e r , an i n t r i n s i c germanium d e t e c t o r of 48.53 mm d i a m e t e r by 40 mm depth ( a c t i v e volume 71.5 c m 3 ) , was manufactured by Ap t e c . Curves r e p r e s e n t i n g the photopeak e f f i c i e n c i e s of each d e t e c t o r , as a f u n c t i o n of photon energy, a re d i s p l a y e d i n F i g u r e 3.2.4.2. The MEASURED DETECTOR EFFICIENCIES 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 GAMMA ENERGY (MEV) F i g u r e 3.2.4.2. Photopeak e f f i c i e n c i e s v e r s u s energy f o r the two d e t e c t o r s used. c u r v e s w i l l be r e f e r r e d t o i n the a n a l y s i s p r e s e n t e d i n the next c h a p t e r . Both d e t e c t o r s were w a r r a n t e d f o r a r e s o l u t i o n of b e t t e r than 1.9 keV (FWHM) a t 1.33 MeV, and i n p r a c t i c e 65 s u r p a s s e d t h a t f i g u r e . Other f e a t u r e s of the t a r g e t r e g i o n may be noted from f i g u r e 3.2.4.1. For- i n s t a n c e , a 0.25 mm s c i n t i l l a t i o n c o u n t e r (MU) of 5 cm d i a m e t e r i d e n t i f i e d incoming muons, d i s t i n g u i s h e d by p u l s e h e i g h t from the p o s i t r o n c o n t a m i n a t i o n i n the beam. A s u r f a c e muon w i l l d e p o s i t more than s i x times the energy i n a t h i n s c i n t i l l a t o r t h a t a beam e l e c t r o n w i l l (Trower, 1966). With good l i g h t c o l l e c t i o n , the p u l s e h e i g h t d i f f e r e n c e makes f o r easy d i s c r i m i n a t i o n between the two p a r t i c l e s . Immediately upstream of t h i s c o u n t e r , a 0.20 mm f i x e d M y l a r degrader ranged the muons t o s t o p i n the s i l i c a / c o l l o d i o n t a r g e t . F i n e range a d j u s t m e n t s were made by s e l e c t i n g a p p r o p r i a t e a d d i t i o n a l t h i c k n e s s e s of M y l a r from s e v e r a l mounted on two frames which c o u l d be a d j u s t e d by m e c h a n i c a l vacuum f e e d t h r o u g h s . In a d d i t i o n t o a 0.12 mm (7.5 cm d i a m e t e r ) M y l a r window s e p a r a t i n g t a r g e t from beamline vacuum systems, a f u r t h e r 0.15 mm was r e q u i r e d t o maximize the stopped muon r a t e . Two v e t o s c i n t i l l a t i o n c o u n t e r s (VF) d e t e c t e d charged p a r t i c l e s a p p r o a c h i n g the germanium d e t e c t o r s . Muons d e c a y i n g t o p o s i t r o n s of up t o 52.8 MeV were one i n e v i t a b l e source of background r a d i a t i o n problems which c o u l d be a l l e v i a t e d by the v e t o c o u n t e r s . F u r t h e r s u p p r e s s i o n of u n d e s i r a b l e r a d i a t i o n i n the germanium c r y s t a l s was a c c o m p l i s h e d by s h i e l d i n g w i t h p o l y e t h y l e n e , p a r a f f i n , and l e a d . An e x p l a n a t i o n of the background problems e n c o u n t e r e d i s c o n t a i n e d i n the f o l l o w i n g s e c t i o n . 66 3.3. Data A c q u i s i t i o n Data were c o l l e c t e d i n a v e r y s i m p l e f a s h i o n . Most of the pro c e d u r e s employed o r i g i n a t e d i n p i o n i c and muonic X-ray e x p e r i m e n t s . T h i s i s not t o say t h a t no p a r t i c u l a r problems emerged; on the c o n t r a r y , the o p e r a t i o n of l a r g e germanium d e t e c t o r s i n c l o s e p r o x i m i t y t o a s t o p p i n g beam of over 1 0 s muons per second r e q u i r e d some s p e c i a l t a c t i c s . In t h i s s e c t i o n , a d e s c r i p t i o n of the s o u r c e s of r a d i a t i o n e n t e r i n g the d e t e c t o r s w i l l be g i v e n . The s t e p s taken t o m i n i m i z e the impact of background s o u r c e s on the d a t a a r e a l s o d i s c u s s e d . The ha r d w i r e d p r o c e s s i n g of i n f o r m a t i o n from c o u n t e r s and d e t e c t o r s i s d e s c r i b e d i n d e t a i l , w i t h an e x p l a n a t i o n of the l o g i c of the e l e c t r o n i c s used. F i n a l l y , a s u b s e c t i o n i s devoted t o a summary of the c o n d i t i o n s under which the da t a were accumulated and s t o r e d , i n c l u d i n g e x p e r i m e n t a l parameters such as r a t e s . 3.3.1. P r i m a r y Sources of Background and T h e i r M i n i m i z a t i o n The s e n s i t i v i t y of the TRIUMF experiment t o a p o s s i b l e muonium-antimuonium c o u p l i n g depends i n p a r t on the l i k e l i h o o d of p r o c e s s e s r e s u l t i n g i n background gamma r a d i a t i o n of s i m i l a r energy t o the muonic X-ray p r o c e s s e s of i n t e r e s t . The r u l e of thumb s t a t e s t h a t the square r o o t of the number of background gammas, accumulated w i t h i n the f u l l w i d t h a t h a l f maximum (FWHM) of the exp e c t e d X - r a y , d e t e r m i n e s the minimum number of 67 f o r e g r o u n d events n e c e s s a r y f o r o b s e r v i n g an e f f e c t of one s t a n d a r d d e v i a t i o n (68% c o n f i d e n c e l e v e l ) . T h e r e f o r e an u n d e r s t a n d i n g of the backgrounds p r e s e n t and the pr o c e d u r e s f o r t h e i r r e d u c t i o n i s of some i m p o r t a n c e . The s o u r c e s of the r a d i a t i o n s of i n t e r e s t may be c a t e g o r i z e d as f o l l o w s : 1. Muon decay p o s i t r o n s : I t i s e s s e n t i a l l y c e r t a i n t h a t a p o s i t i v e muon w i l l decay t o a p o s i t r o n w i t h an energy f o l l o w i n g the M i c h e l spectrum, up t o a maximum of 52.8 MeV. B r e m s s t r a h l u n g energy l o s s e s of the p o s i t r o n s r e s u l t i n a continuum of gamma r a y s , up t o the f u l l energy. Some gammas are w i t h i n the muonic X-ray energy r e g i o n ( e i t h e r i n i t i a l l y or a f t e r subsequent Compton s c a t t e r i n g ) . The p o s i t r o n s e v e n t u a l l y a n n i h i l a t e w i t h e l e c t r o n s , g i v i n g m o s t l y two 0.511 MeV photons per a n n i h i l a t i o n . The p o s i t r o n s themselves can e n t e r the d e t e c t o r , d e p o s i t i n g energy d i r e c t l y by r a d i a t i o n and i o n i z a t i o n : most such p a r t i c l e s l e a v e more than a few MeV i n the germanium c r y s t a l ( u n l i k e the gammas of i n t e r e s t ) s i n c e the r a d i a t i o n l e n g t h i s about 2.2 cm (or 12 g * c i r r 2 ; see P a r t i c l e Data Group, 1976). 2. Beam p o s i t r o n s : A p p r o x i m a t e l y one h a l f t o two t h i r d s of the beam e n t e r i n g the t a r g e t i s p o s i t r o n s , the s o u r c e s of which were e x p l a i n e d i n s e c t i o n 3.2.1. L i k e t h e muon decay p o s i t r o n s , t hey can r a d i a t e and produce the troublesome b r e m s s t r a h l u n g background. On the o t h e r hand, they t e n d t o remain a beam, p a s s i n g through the v e r y l i g h t t a r g e t and out of the a p p a r a t u s 68 where they do l i t t l e harm. A v e r y s m a l l number undergo l a r g e a n g l e s c a t t e r i n g , c r e a t i n g problems s i m i l a r t o those caused by decay p o s i t r o n s . 3. Other s o u r c e s : I n c l u d e d i n t h i s c a t e g o r y a r e the induced n u c l e a r gammas from m a t e r i a l i n the v i c i n i t y of the d e t e c t o r s , d i r e c t n e u t r o n a c t i v a t i o n of the d e t e c t o r s , and cosmic r a y s . Some energy s t r u c t u r e i s of c o u r s e p r e s e n t i n t h i s background; most n o t a b l y , a 0.835 MeV 5 4Mn l i n e appeared c l o s e t o the 0.784 MeV c a l c i u m muonic 2P-1S l i n e which was searc h e d f o r as a c o n v e r s i o n s i g n a l . Rather than p r e s e n t i n g a problem, t h i s l i n e p r o v e d t o be a v a l u a b l e c a l i b r a t i o n d u r i n g the a n a l y s i s . The decay p o s i t r o n background was the most severe and the most d i f f i c u l t t o d e a l w i t h . T h i s f a c t had been a n t i c i p a t e d ; s e v e r a l f e a t u r e s were i n c o r p o r a t e d i n t o the a p p a r a t u s t o m i t i g a t e the problem. F i r s t of a l l , i t was apparent t h a t muons s t o p p i n g near or i n the t a r g e t , but not i n the a c t i v e r e g i o n s , s h o u l d be e l i m i n a t e d . The low t h i c k n e s s and s t o p p i n g power of the c o l l o d i o n f i l m s was i m p o r t a n t f o r t h i s r e a s o n . C o n t r o l of the muon beam spot s i z e v i a M13 jaws was an a s s e t , e s p e c i a l l y as i t reduced t h e f l u x i n the t a i l s of the beam d i s t r i b u t i o n . The degrader ( i n c l u d i n g the d e f i n i n g s c i n t i l l a t o r ) n e c e s s a r y f o r r a n g i n g muons i n t o the t a r g e t was as c l o s e t o i t as p o s s i b l e . T h i s was found t o be c r u c i a l because of the l a r g e m u l t i p l e s c a t t e r i n g s u f f e r e d by the slow muons. With o u t the c o l l o d i o n t a r g e t i n p l a c e , the muons s t o p m o s t l y i n the w a l l s of the vacuum tube i n the t a r g e t r e g i o n or s l i g h t l y downstream of i t , 69 so the background due t o decay p o s i t r o n s was not much reduced by i t s r e moval. The second f e a t u r e b u i l t i n t o the a p p a r a t u s i n i t i a l l y was a l e n g t h of aluminum tube spun t o a w a l l t h i c k n e s s of 1 mm, which c o n s t i t u t e d the vacuum p i p e i n which the t a r g e t would be p l a c e d . I t was ex p e c t e d t h a t t h i s would m i n i m i z e the r a d i a t i v e energy l o s s by the p o s i t r o n s i n the vacuum tube w a l l , and hence the b r e m s s t r a h l u n g e n t e r i n g the d e t e c t o r ( s i m i l a r r e a s o n i n g l e d t o the use of expanded p o l y s t y r e n e as a support m a t e r i a l f o r the c o l l o d i o n ) . However, the spun aluminum tube was abandoned f o r a p i p e of 3 mm w a l l t h i c k n e s s , a f t e r a measurement of gamma count r a t e s i n the energy range of i n t e r e s t , as a f u n c t i o n of the amount of aluminum between t a r g e t and d e t e c t o r , r e v e a l e d a d i f f e r e n c e of l e s s than f i v e per c e n t . I t s h o u l d a l s o be noted t h a t t h i s was performed b e f o r e e x t e n s i v e s h i e l d i n g of the d e t e c t o r s was u n d e r t a k e n , and o t h e r s o u r c e s of ( p o s i t r o n independent) background were dominant. The e x p e r i m e n t e r s were not unhappy a t the change, s i n c e many p r e c a u t i o n s had t o be obs e r v e d i n u s i n g the t h i n t ube: an i d e n t i c a l one had b u c k l e d w h i l e b e i n g used under vacuum by an o t h e r group a t TRIUMF! In a d d i t i o n , a f l a n g e c o u l d be welded on the t h i c k e r p i p e which a l l o w e d the f i n a l degrader and d e f i n i n g s c i n t i l l a t o r t o be i n s t a l l e d much c l o s e r t o the t a r g e t , f u r t h e r r e d u c i n g the beam l o s s due t o m u l t i p l e s c a t t e r i n g . T h i r d l y , s c i n t i l l a t i o n v e t o c o u n t e r s were i n s t a l l e d . These c o u l d be used i n f a s t c o i n c i d e n c e (see s e c t i o n 3.3.2 f o l l o w i n g ) w i t h p u l s e s from t h e c r y s t a l s t o v e t o e v e n t s a s s o c i a t e d w i t h c h a r g e d p a r t i c l e s i n and near i t . I n i t i a l l y a s e t of t h r e e s c i n t i l l a t o r s c o v e r e d a l l but the r e a r f a c e of each d e t e c t o r . 70 T h i s arrangement was m o d i f i e d when a h i g h background, u n c o r r e l a t e d w i t h beam i n M13 ( i . e . , " o t h e r s o u r c e s " ) , n e c e s s i t a t e d more e x t e n s i v e l e a d s h i e l d i n g s u r r o u n d i n g the d e t e c t o r s and thus c o n f l i c t i n g . w i t h the s c i n t i l l a t o r s p h y s i c a l l y . E v e n t u a l l y a l a r g e , 20 x 20 cm 2 c o u n t e r , 0.65 cm t h i c k , was used i n f r o n t of each d e t e c t o r . The l a r g e c o u n t e r s i z e a i d e d the r e d u c t i o n of background e v e n t s due t o the b r e m s s t r a h l u n g of p o s i t r o n s i n the l e a d s h i e l d i n g . As an a s i d e , the c h o i c e of a c a l c i u m o x i d e c o a t i n g f o r the p r o d u c t i o n of p o s s i b l e muonic X-rays was a l s o m o t i v a t e d p a r t i a l l y by background c o n s i d e r a t i o n s . The 2P-1S energy of 0.784 MeV (Engfer e t a l . , 1974) i s above the p o s i t r o n a n n i h i l a t i o n background y e t s t i l l low enough t o be d e t e c t e d w i t h r e a s o n a b l e e f f i c i e n c y . Another reason f o r the c h o i c e i s the low e v a p o r a t i o n temperature of Ca, which the f r a g i l e c o l l o d i o n f i l m s can a t l e a s t sometimes w i t h s t a n d (many were r u p t u r e d d u r i n g p r e p a r a t i o n ) . The p r o c e d u r e s adopted because of muon decay p o s i t r o n background were o f t e n a l s o e f f e c t i v e f o r the m i n i m i z a t i o n of beam p o s i t r o n e f f e c t s . Some f u r t h e r e f f o r t i n t h i s d i r e c t i o n i n c l u d e d the trimming of s l i t s and jaws t o reduce the beam h a l o . P o s i t r o n s , l i g h t p a r t i c l e s t h a t they a r e , can undergo l a r g e d e v i a t i o n s d u r i n g Coulomb s c a t t e r i n g from a n u c l e u s . T h i s t a k e s p l a c e a l o n g the l e n g t h of the beam l i n e , c a u s i n g those not o r d i n a r i l y w i t h i n i t s momentum and c o o r d i n a t e a c c e ptance t o emerge i n a broad smear. By c l o s i n g o t h e r w i s e unnecessary s l i t s and jaws t o the p o i n t j u s t s h o r t of r e d u c i n g the muon f l u x , some of the smear was removed. In a d d i t i o n , p a r a f f i n and p o l y e t h y l e n e s h i e l d i n g i m m e d i a t e l y upstream of the muon t a r g e t 71 removed some of the h a l o w i t h m i n i m a l r a d i a t i o n r e l e a s e , which was i n t u r n d e a l t w i t h by s u b s t a n t i a l l e a d s h i e l d i n g (see s e c t i o n 3.2.4). By r e m o t e l y i n s e r t i n g a s m a l l amount (about 30 mg»cnr 2 CH^) of degrader a t F l , one c o u l d d i s p l a c e the muon beam a t t h e end of M13 such t h a t i t d i d not pass the upstream p o l y e t h y l e n e c o l l i m a t i o n . The beam p o s i t r o n s s t i l l remained more or l e s s u n d e v i a t e d , so i t was q u i t e easy t o see t h e i r c o n t r i b u t i o n t o the gamma spectrum. I t t u r n e d out t o be much s m a l l e r than t h a t due t o decay p o s i t r o n s , as was e x p e c t e d . The background due t o s o u r c e s o t h e r than decay or beam p o s i t r o n s was i n i t i a l l y l a r g e r than e x p e c t e d . I t was not c l e a r a t the o u t s e t whether optimum s e n s i t i v i t y would be a c h i e v e d w i t h or w i t h o u t e x t e n s i v e l e a d s h i e l d i n g of the d e t e c t o r s ; i n some c a s e s , e s p e c i a l l y w i t h r e l a t i v i s t i c e l e c t r o n s or p o s i t r o n s p r e s e n t , i t can a c t as a gamma source because of shower p r o d u c t i o n . By t e s t i n g the e f f e c t of s h i e l d i n g , i t was det e r m i n e d t h a t , a t the moderate r a t e s used, about h a l f of the gamma background was not beam r e l a t e d and c o u l d be d r a s t i c a l l y reduced by c a r e f u l placement of l e a d b l o c k s . Neutrons were p a r t of the problem, p a r t i a l l y a l l e v i a t e d by the c o n s t r u c t i o n of a c o n c r e t e and p a r a f f i n w a l l a t t h e maze e n t r a n c e t o the p r o t o n beam t u n n e l a d j a c e n t t o M13, where some n e u t r o n s seemed t o o r i g i n a t e . C h a r a c t e r i s t i c n e u t r o n l i n e s ( B u n t i n g and K r a u s h a a r , 1974) were o b s e r v e d i n t h e gamma s p e c t r a . One l i n e came v e r y c l o s e i n energy t o the 0.784 MeV c a l c i u m l i n e ( m o r e d i s c u s s i o n on t h i s w i l l appear i n s e c t i o n 4.1). A s u b s t a n t i a l p o s i t r o n a n n i h i l a t i o n peak was observed even w i t h M13 shut o f f , p r o b a b l y from p r i m a r y p r o t o n i n t e r a c t i o n s such as n e u t r a l p i o n gamma 72 c o n v e r s i o n i n the p r o d u c t i o n t a r g e t . Some f u r t h e r l i n e s were i d e n t i f i a b l e , but o n l y the a f o r e m e n t i o n e d s 4Mn l i n e was c l o s e i n energy t o the r e g i o n of i n t e r e s t . I t s u f f i c e s t o say t h a t the background problem was d e a l t w i t h m o s t l y by t r i a l and e r r o r . By cha n g i n g v a r i o u s p a r a m e t e r s , one c o u l d d e t e r m i n e t h e b e s t method of l e s s e n i n g a p a r t i c u l a r background r a t e . S e v e r a l d i f f e r e n t a s p e c t s of the a p p a r a t u s were a t t h i s s tage d e s i g n e d , t e s t e d , or a l t e r e d . T h i s e v o l u t i o n was p a r a l l e l e d by s i m i l a r development of the e l e c t r o n i c hardware arrangement, the s u b j e c t of the f o r t h c o m i n g s e c t i o n . 3.3.2. E l e c t r o n i c s C o n f i g u r a t i o n Used The f u n c t i o n of the data a c q u i s i t i o n system was t o o b t a i n energy i n f o r m a t i o n about the spectrum of gamma r a y s below a p p r o x i m a t e l y 1.0 MeV, o r i g i n a t i n g i n the t a r g e t . The t a s k of the e l e c t r o n i c s was t o ensure t h a t the events o b t a i n e d s a t i s f i e d some s i m p l e c r i t e r i a . A p a r t from the s u p p r e s s i o n of background e v e n t s , the most i m p o r t a n t a s p e c t of t h i s s e l e c t i o n r e l a t e d t o the maintenance of the h i g h r e s o l u t i o n c a p a b i l i t y of the germanium s p e c t r o m e t e r . The s e n s i t i v i t y of the experiment depended on, among o t h e r t h i n g s , the r e s o l u t i o n o b t a i n a b l e . A d e t a i l e d diagram of the c i r c u i t r y t h a t was used i n t a k i n g d a t a i s shown i n F i g u r e 3.3.2.1. Some u n i t s , denoted by an x, a r e d u p l i c a t e d i n the a c t u a l system t o handle two d e t e c t o r s . An attempt w i l l be made t o e x p l a i n , i n the remainder of t h i s s e c t i o n , the l o g i c a l arrangement of the i m p o r t a n t components. F i g u r e 3.3.2.1. E l e c t r o n i c s diagram f o r c i r c u i t used (see t e x t ) . 74 The e l e c t r o n - h o l e p a i r s , c r e a t e d i n the germanium c r y s t a l by the passage of i o n i z i n g r a d i a t i o n , cause a sha r p (~50 ns) change i n the v o l t a g e output of the charge s e n s i t i v e p r e a m p l i f i e r , on the d e t e c t o r . T h i s s i g n a l , denoted as Ge(E) i n the f i g u r e , i s shaped and a m p l i f i e d f u r t h e r , so t h a t the peak v o l t a g e (something l e s s than 10 V) i s p r o p o r t i o n a l t o the energy d e p o s i t e d i n the germanium, f o r the energy range of i n t e r e s t . The BIASED A M P l i f i e r f u r t h e r a m p l i f i e s a s e l e c t e d range of the a n a l o g s i g n a l , i f a l o g i c a l g a t i n g p u l s e has been p r o p e r l y a p p l i e d . From t h e r e , the s i g n a l i s f e d through a MIXER ROUTER i n t o a p u l s e h e i g h t a n a l y z e r (PHA). I t i s d i g i t i z e d by t h a t u n i t ' s a n a l o g t o d i g i t a l c o n v e r t e r (ADC) and the a p p r o p r i a t e channel of one of the two 2048 c h a n n e l h i s t o g r a m s i s increment e d , depending upon which d e t e c t o r saw the e v e n t . The a m p l i f i e r output a l s o feeds a l i n e a r f a n - i n f a n - o u t (LIN FAN) which i n v e r t s the p o s i t i v e a n a l o g p u l s e f o r i n p u t i n t o a NIM d i s c r i m i n a t o r (DISC) o p e r a t i n g i n the t i m e - o v e r - t h r e s h o l d mode. T h i s u n i t performs two f u n c t i o n s , the f i r s t b e i n g t o p u l s e the GATE i n p u t of the BIASED AMP i f s e v e r a l g a t i n g c o n d i t i o n s have been met. The second i s t o d e f i n e the end of the a n a l o g p u l s e . The t r a i l i n g edge s e r v e s t o f i r e a f u r t h e r D i s c r i m i n a t o r which i n t u r n s t o p s a GATE GENerator t h a t e v e n t u a l l y i n h i b i t s g a t i n g of the BIASED AMP . x T h i s second f u n c t i o n i s an i m p o r t a n t one. I t was found t h a t the l a r g e energy d e p o s i t e d i n a germanium c r y s t a l by the f a s t e l e c t r o n s c o u l d s a t u r a t e the p r e a m p l i f i e r , c a u s i n g the *The au t h o r i s g r a t e f u l t o J . A. Macdonald f o r s u g g e s t i o n s r e g a r d i n g t h i s p a r t i c u l a r p o i n t . 75 a m p l i f i e r t o "hang up" a t a n o n - q u i e s c e n t v a l u e of about e i g h t v o l t s f o r v a r y i n g p e r i o d s of up t o 40 /us. T h i s was a r e a l s o u r c e of dead t i m e , e s p e c i a l l y f o r the Aptec i n t r i n s i c germanium d e t e c t o r whose p r e a m p l i f i e r was not as c a p a b l e of h a n d l i n g the h i g h l y i o n i z i n g e n e r g e t i c p o s i t r o n s . F u r t h e r m o r e , a c e r t a i n l o s s of energy r e s o l u t i o n and photopeak symmetry was a consequence of the poor b a s e l i n e d e f i n i t i o n i n the l o n g p u l s e t a i l s . The c i r c u i t d e a l s w i t h t h i s i n the f o l l o w i n g way. Whenever a l a r g e p u l s e o c c u r s , i t i s a m p l i f i e d (by the T.F. AMP) t o f i r e a d i s c r i m i n a t o r , s t a r t i n g a GATE GENerator t h a t i n h i b i t s g a t i n g of the BIASED AMP. The i n h i b i t i s removed o n l y when a q u i e s c e n t ( l e s s than about 50 mV) l e v e l i s sensed by the t i m e - o v e r - t h r e s h o l d D i s c r i m i n a t o r . Three o t h e r s i g n a l s s e r v e t o i n h i b i t g a t i n g of the BIASED AMP, each f a n n i n g i n t o the SUM INHIBIT u n i t . One i s d e r i v e d from the BUSY s i g n a l of the PHA through the MIXER ROUTER, w h i l e a n o t h e r (INH) comes from s p e c i a l p i l e - u p c i r c u i t r y of the main a m p l i f i e r (572 AMP). The l a t t e r m a i n t a i n s r e s o l u t i o n a t the expense of a c c e p t e d r a t e f o r h i g h count r a t e s . Both must pass th r o u g h a TTL-NIM l e v e l c o n v e r t e r (CON). The t h i r d i n h i b i t o c c u r s when e i t h e r v e t o s c i n t i l l a t o r , f r o n t (VF) or upstream (VU), f i r e s i n f a s t ( i . e . , w i t h i n about 40 ns) c o i n c i d e n c e w i t h the a m p l i f i e d t i m i n g output (Ge(T)) of the d e t e c t o r . As mentioned i n the p r e v i o u s s e c t i o n , t h i s s e r v e s t o reduce c o u n t s due t o showers from charged p a r t i c l e s i n the l e a d s h i e l d i n g around each d e t e c t o r . Constant f r a c t i o n d i s c r i m i n a t i o n (CFD) i s used on d e t e c t o r t i m i n g s i g n a l s . Other t i m i n g i n the c i r c u i t i s s l o w , and gate l e n g t h s a r e i n the m i c r o s e c o n d range. The SUM INHIBIT fan b l a n k s any o u t p u t from an AND gate 7 6 e n a b l e d by an u p d a t i n g 12JJLS gate (U.D. GATE) i n t u r n i n i t i a t e d by an i n c i d e n t muon f i r i n g the d e f i n i n g c o u n t e r (MU). At the r a t e s used i n the e x p e r i m e n t , the 1 2 ^ 5 gate was on v i r t u a l l y whenever the beam was on, and t h e r e was a r e a s o n a b l y h i g h p r o b a b i l i t y of two muons b e i n g i n the t a r g e t a t one t i m e . The major f u n c t i o n of t h i s g a t i n g i s t o p r o t e c t the spectrum from background o c c u r r i n g d u r i n g o c c a s i o n a l c y c l o t r o n t r i p s of anywhere from a f r a c t i o n of a second t o s e v e r a l hours d u r a t i o n . Most of the remainder of the c i r c u i t r y measures numbers v i t a l t o the a n a l y s i s of the ex p e r i m e n t , the t o t a l muons i n c i d e n t and the f r a c t i o n a l l i v e t i m e . I n c i d e n t muons a r e i d e n t i f i e d by p u l s e s from the MU d e f i n i n g c o u n t e r , and a r e counted i n a s c a l e r e n a b l e d whenever the PHA i s c o l l e c t i n g e v e n t s . The l i v e time measurement i s s l i g h t l y more complex. One s c a l e r measures a q u a n t i t y l a b e l l e d gated m o n i t o r (GAT MON), which i s accumulated when the PHA i s c o l l e c t i n g e v e n t s and the u p d a t i n g muon gate (U.D. GATE) i s on ( n e c e s s a r y f o r an event t o be ga t e d through the BIASED AMP and measured by the PHA). A second and t h i r d s c a l e r ( o n l y one of which i s shown i n the diagram) measure l e f t (LGM) and r i g h t (RGM) c o u n t s . These r e q u i r e the same g a t i n g as GAT MON, but a r e i n h i b i t e d by SUM INHIBIT, which a l s o i n h i b i t s g a t i n g of the BIASED AMP, t h a t i s , the measurement of an e v e n t . The f r a c t i o n a l l i v e time f o r a p a r t i c u l a r d e t e c t o r i s j u s t the r a t i o of the c o r r e s p o n d i n g l e f t or r i g h t gated m o n i t o r t o GAT MON. The m o n i t o r used i n the experiment c o n s i s t e d of p e r i o d i c p u l s e s d e r i v e d from a p u l s e g e n e r a t o r . T h i s completes the d e s c r i p t i o n of the o p e r a t i o n of the 77 e l e c t r o n i c hardware; the r e m a i n i n g s e c t i o n w i l l d e a l w i t h the a c t u a l procedure of data a c c u m u l a t i o n u s i n g the d e s c r i b e d system. 3.3.3. A c c u m u l a t i o n and Stora g e of Data W i t h a l l the equipment s e t up, the magnetic f i e l d a d j u s t e d t o z e r o , and the t a r g e t q u i c k l y and c a r e f u l l y t r a n s f e r r e d from i t s d r y s t o r a g e t o the vacuum (5 x 10" 7 mm Hg) of the t a r g e t tube, the s u r f a c e muon beam was a l l o w e d i n t o the t a r g e t r e g i o n . The f i r s t t a s k was t o a d j u s t the degrader f o r a maximum s t o p p i n g r a t e i n the f o i l t a r g e t . T h i s was a c c o m p l i s h e d by s c a l i n g the e n e r g e t i c p o s i t r o n e v e nts i n the d e t e c t o r s i n c o i n c i d e n c e w i t h the f r o n t v e t o (VFL and VFR) s c i n t i l l a t o r s , and m a x i m i z i n g the r a t e . A f t e r a s h o r t c a l i b r a t i o n run u s i n g s o u r c e s w i t h the beam on, the f i r s t of t e n runs of a c t u a l d a t a t a k i n g began. The i n c i d e n t muon r a t e was t y p i c a l l y about 1.7 x 1 0 5 s ' 1 , u s i n g 25 j*& of pr o t o n c u r r e n t c a r e f u l l y s t e e r e d onto a g r a p h i t e p r o d u c t i o n t a r g e t of 10 mm t h i c k n e s s . S e v e r a l hours of a c c u m u l a t i o n t i m e , c o r r e s p o n d i n g t o a few b i l l i o n i n c i d e n t muons, made up each r u n , a f t e r which the d a t a were r e c o r d e d , a l o n g w i t h the r e l e v a n t s c a l e r t o t a l s . I n t e r s p e r s e d w i t h these were c a l i b r a t i o n runs ( u s i n g m o s t l y 1 3 7 C s and 5*Mn s o u r c e s ) t o check f o r p o s s i b l e g a i n s h i f t s , p l u s some n e g a t i v e muon n o r m a l i z a t i o n c h e c k s , the r e s u l t s of which w i l l be used i n the f o r t h c o m i n g c h a p t e r on a n a l y s i s . The gamma s p e c t r a were t r a n s f e r r e d v i a B.C. Telephone 78 l i n e s t o temporary d i s k s t o r a g e on the U.B.C. Amdahl 470 V/6 computer f o r subsequent a n a l y s i s and permanent s t o r a g e on magnetic t a p e . A program s p e c i f i c a l l y f o r the t r a n s f e r , PHARUN ( C l a r k , u n p u b l i s h e d ) , c r e a t e d a d a t a f i l e on the d i s k , c o p i e d the i n f o r m a t i o n i n e a s i l y u s a b l e c h a r a c t e r f o r m a t , and checked t h a t the d a t a were s u c c e s s f u l l y s t o r e d . D u r i n g the time r e q u i r e d t o w r i t e the c o n t e n t s of 4096 c h a n n e l s t o the f i l e , any n e c e s s a r y changes i n the e x p e r i m e n t a l parameters were made, and the vacuum was checked. When t h i s had been completed, the spectrum was e r a s e d and the c u m u l a t i v e s c a l e r s z e r o e d . T h i s c o n c l u d e s the d e s c r i p t i o n of the muonium t o antimuonium c o n v e r s i o n experiment performed a t TRIUMF . The i n f o r m a t i o n thus o b t a i n e d must be used i n c o n j u n c t i o n w i t h t h a t from some p e r i p h e r a l e x p e r i m e n t s , i n o r d e r t h a t a m e a n i n g f u l upper l i m i t f o r the p r o c e s s can be d e t e r m i n e d . The a c t u a l r e s u l t s of a l l of t h e s e , the methods by which they a r e deduced, and the way i n which they can be combined t o c a l c u l a t e a v a l u e f o r G, a r e the s u b j e c t s of the next c h a p t e r . 79 4. ANALYSIS AND INTERPRETATION OF THE DATA There are two b a s i c s t e p s which must be t a k e n , a f t e r the d a t a have been c o l l e c t e d , t o d e r i v e a r e s u l t f o r muonium t o antimuonium c o n v e r s i o n i n the form of a v a l u e (or l i m i t ) on the c o u p l i n g c o n s t a n t G d e f i n e d by e q u a t i o n 2.1.1.5. The f i r s t c o n s i s t s of a p p l y i n g a w e l l - u n d e r s t o o d s t a t i s t i c a l p rocedure t o the energy spectrum of gamma r a y s o b t a i n e d . The f r u i t of t h i s e f f o r t , and the g o a l of the f i r s t s e c t i o n of t h i s c h a p t e r , w i l l be the d e t e r m i n a t i o n of a number and i t s a s s o c i a t e d u n c e r t a i n t y r e p r e s e n t i n g the c o u n t s i n a ( p o s s i b l y n o n e x i s t e n t ) photopeak i n the spectrum from a c a l c i u m 2P-1S muonic X-ray t r a n s i t i o n . The second s t e p i s more complex. An e s t i m a t e must be made f o r the p r o b a b i l i t y t h a t a muon, upon e n t e r i n g the t a r g e t r e g i o n , w i l l e v e n t u a l l y l e a d t o d e t e c t i o n of the f u l l energy of the muonic X-ray a f t e r c o n v e r s i o n has taken p l a c e . Each s t e p i n the sequence of e v e n t s r e q u i r e d f o r the s u c c e s s f u l c o u n t i n g of the X-ray must be u n d e r s t o o d and a s s i g n e d a p r o b a b i l i t y , i n o r d e r t h a t the o n l y unmeasurable or i n c a l c u l a b l e q u a n t i t y i n the o v e r a l l p r o b a b l i l i t y i s the c o u p l i n g c o n s t a n t G. There i s a l s o an u n c e r t a i n t y i n the p r o b a b i l i t y which s h o u l d be e s t i m a t e d . S i n c e t h e r e are s e v e r a l independent s t e p s between muon i n c i d e n c e and event d e t e c t i o n , the o v e r a l l p r o b a b i l i t y w i l l be the product of p r o b a b i l i t i e s or e f f i c i e n c i e s f o r the i n d i v i d u a l p r o c e s s e s . Some f a c t o r s a r e e a s i l y measured, some must be computed, w h i l e 80 o t h e r s may be more a c c u r a t e l y e v a l u a t e d as a subproduct w i t h o t h e r terms. The meaning of t h i s w i l l become more c l e a r i n the second s e c t i o n of t h i s c h a p t e r , which d e a l s w i t h the c a l c u l a t i o n of the e x p e r i m e n t a l s e n s i t i v i t y i n terms of G. Meanwhile, the time i s r i p e f o r p r e s e n t a t i o n of the da t a upon which t h i s t h e s i s depends. 4.1. The R e s u l t s : De te rmina t ion of a L i m i t on the Number of Events Observed Ten p u l s e h e i g h t s p e c t r a f o r each of the two germanium d e t e c t o r s comprise the r e s u l t s of the s e a r c h f o r the n e g a t i v e muonic X-ray i n the c a l c i u m c o n s t i t u e n t of the t a r g e t . The a c c u m u l a t i o n t i m e s f o r each spectrum were t y p i c a l l y t h r e e t o f i v e h o u r s , c o r r e s p o n d i n g t o two or t h r e e b i l l i o n i n c i d e n t muons. F i g u r e 4.1.1 shows the sums of each s e t of ten s p e c t r a , from 0.55 t o 1.0 MeV. The d a t a c o r r e s p o n d t o a t o t a l of 2.32 x I O 1 0 i n c i d e n t muons. The f r a c t i o n a l l i v e t i m e s f o r the Aptec and Or t e c s p e c t r a were 0.82 and 0.92 r e s p e c t i v e l y . Because of s l i g h t l o n g term d r i f t s i n t h e g a i n and o f f s e t ( c a u s i n g s h i f t s of ±0.6 keV maximum, a t 0.835 MeV) among the s p e c t r a , the r e s o l u t i o n o b t a i n e d by a s i m p l e summation i s not o p t i m a l . One c a n , however, a m e l i o r a t e the s i t u a t i o n f o r a p a r t i c u l a r r e g i o n of i n t e r e s t by c a l c u l a t i n g the e f f e c t i v e d r i f t UJ 2 5 0 0 0 2 0 0 0 0 h 5 15000 UJ o_ o o 10000 5000 40000 £ 3 0 0 0 0 h cr x o £5 2 0 0 0 0 Q_ CO o 10000 u' IN CONVERSION TARGET (APTEC) T uir IN CONVERSION TARGET (ORTEC) T 0 500 550 600 650 700 750 800 850 900 950 100 ENERGY ( 0 . 4 KEV/CHANNEL) , F i g u r e 4.1.1. Gamma s p e c t r a from s i m p l e summation of a l l dat 82 i n o f f s e t u s i n g known l i n e s and a d j u s t i n g t he c h a n n e l numbers a c c o r d i n g l y b e f o r e a d d i t i o n of the s p e c t r a . I f the g a i n (as opposed t o o f f s e t ) d r i f t s a r e s m a l l , t h i s p r o c e d u r e w i l l r e s t o r e much of the s h o r t term r e s o l u t i o n over l i m i t e d r e g i o n s of i n t e r e s t . The adjustment of c h a n n e l numbers was a c c o m p l i s h e d i n the f o l l o w i n g way. The l a r g e (Doppler broadened) a n n i h i l a t i o n peak a t 0.511 MeV (not shown i n the f i g u r e s ) and the s m a l l background 5 4Mn peak a t 0.835 MeV i n each of the twenty h i s t o g r a m s were f i t t e d t o a g a u s s i a n form t o determine the c h a n n e l numbers of the c e n t r o i d s (gamma photopeak e n e r g i e s were o b t a i n e d from M a r i o n , 1968). From t h i s c a l i b r a t i o n , the c h a n n e l numbers f o r the 0.7837 MeV c a l c i u m X-ray were deduced, and the s p e c t r a were summed over a r e g i o n encompassing the peak such t h a t the c a l c u l a t e d p o s i t i o n s , t o the n e a r e s t c h a n n e l , s h o u l d c o i n c i d e . The r e s u l t s of the summation a r e shown i n F i g u r e 4.1.2. The f e a t u r e s t o note from the h i s t o g r a m s a r e : 1. the prominent 5 4Mn peak a t 0.8348 MeV. I t i s p o s s i b l e t h a t a 0.8345 MeV l i n e from n e u t r o n a c t i v a t i o n of 7 2Ge i s a l s o p r e s e n t ( B u n t i n g and Kra u s h a a r , 1974), but the l i k e l i h o o d of the manganese assignment was shown t o be c o r r e c t d u r i n g background runs w i t h the c y c l o t r o n o f f . Moreover, the peak shape i s not c o n s i s t e n t w i t h t h a t of a n e u t r o n induced l i n e . The i n t e n s i t y of t h e l i n e r e l a t i v e t o the smooth background d i f f e r s f o r the two s p e c t r a presumably because of the geometry. 2. a r e s o l v e d peak a t 0.804 MeV, of unknown o r i g i n . A l t h o u g h the shape of the peak i n t h e Aptec d a t a i s 83 RPTEC CR 2P-1S (784 KEV) REGION 15000 14000 _ i LU I 13000 cr x <_> cr 12000 LU CL £ 11000 2 IOOOO A . .-I •1- >•-•>-.-W-L 9000 21000 20000 h _ L J _ ORTEC CR 2P-1S (784 KEV) REGION cr x (_> or LU Q_ 19000 18000 h £ 17000 S 16000 15000 740 750 760 770 780 790 800 810 820 830 840 850 ENERGY (KEV) F i g u r e 4.1.2. Summation of s h i f t e d s p e c t r a i n the Ca X-ray r e g i o n . Note s u p p r e s s i o n of the z e r o of the y a x i s . 84 not w e l l d e t e r m i n e d , i t l o o k s t o be skewed f o r the Or t e c spectrum toward h i g h e r e n e r g i e s , a w e l l e s t a b l i s h e d p r o p e r t y of n e u t r o n induced l i n e s r e s u l t i n g from a v a r i a b l e f r a c t i o n of the n u c l e a r r e c o i l energy b e i n g c o n v e r t e d t o e l e c t r o n - h o l e p a i r s i n germanium ( i b i d ) . 3. d e f i n i t e n e u t r o n induced peaks a t 0.569, 0.585, 0.596, and 0.609 MeV ( f i g u r e 4.1.1 o n l y ) , which were a l s o observed by B u n t i n g and K r a u s h a a r . They were a b l e t o a t t r i b u t e the l a t t e r two t o n e u t r o n i n e l a s t i c s c a t t e r i n g on , 4 G e , but c o u l d not s p e c i f y the o r i g i n of the lower energy p a i r . 4. a r a t h e r f e a t u r e l e s s r e g i o n near the 0.784 MeV c a l c i u m muonic 2P-1S energy. T h i s i s c o n v i n c i n g g r a p h i c e v i d e n c e t h a t no muonium c o n v e r s i o n t o antimuonium has been observed i n the p r e s e n t e x p e r i m e n t . As a p r e l i m i n a r y check of the a p p a r a t u s and d e t e c t i o n systems, some data were a l s o a c q u i r e d from an argon gas t a r g e t a t room temperature and one atmosphere p r e s s u r e , s i m i l a r t o t h a t used i n the p r e v i o u s s e a r c h f o r muonium c o n v e r s i o n (Amato e t  a l . , 1968). They a r e shown i n F i g u r e 4.1.3. The c o r r e s p o n d i n g number of i n c i d e n t muons was 3.4 x 10'. An a n a l y s i s of t h i s i n f o r m a t i o n a l l o w s some comparison of t a r g e t t e c h n i q u e s , which appears i n the f i n a l c h a p t e r . Q u a n t i t a t i v e l i m i t s on the number of e v e n t s t h a t c o u l d be p r e s e n t i n a c a l c i u m photopeak were d e r i v e d as f o l l o w s . From a g a u s s i a n f i t t o the summed 5*Mn peaks of F i g u r e 4.1.2, an e x p e r i m e n t a l r e s o l u t i o n f o r each d e t e c t o r i n terms of f u l l w i d t h 85 9000 8000 h <x X CJ cn LU Q_ CO 7000 6000 5000 4000 3000 14000 APTEC RR 2P-1S (644 KEV) REGION ~1 1 r~ T T , - *J| _L _L 12000 h cr x CJ £ 10000 o_ 3 o CJ 8000 6000 ORTEC FIR 2P-1S (644 KEV) REGION ~~I T 600 610 620 630 640 650 660 670 680 690 700 ENERGY (0.33 KEV/CHflNNEL) F i g u r e 4.1.3. X-ray d a t a f o r p o s i t i v e muons i n argon gas t a r g e t a t room temperature and one atmosphere. 86 a t h a l f maximum (1.67 ± 0.05 keV f o r A p t e c , 1.73 ± 0.07 keV f o r O r t e c ) was o b t a i n e d a t 0.835 MeV. The ex p e c t e d l i n e w i d t h was c a l c u l a t e d f o r the lower c a l c i u m energy by assuming t h a t i t s c a l e s w i t h the square r o o t of the energy, which h o l d s when the s t a t i s t i c a l f l u c t a t i o n s i n the number of e l e c t r o n - h o l e p a i r s dominates the c o n t r i b u t i o n s t o the w i d t h ; i n any c a s e , the c o r r e c t i o n i s o n l y about 3%, s i m i l a r i n magnitude t o the u n c e r t a i n t y i n the FWHM. The w i d t h s a t 0.784 MeV were taken as 1.62 keV (Aptec) and 1.68 keV ( O r t e c ) . A g a u s s i a n f u n c t i o n of the form N = 0.939(N o/FWHM)exp[-(l/2)»{(E-E 0)/(0.425»FWHM)} 2] + A + B(E) 4.1.1 was used t o f i t a range of da t a over a r e g i o n c e n t r e d on the 0.7837 MeV energy. The same form was used t o f i t 5 4Mn peaks, w i t h good s u c c e s s . When f i t t i n g a n n i h i l a t i o n l i n e s , which were w e l l d e f i n e d s t a t i s t i c a l l y , i t c o u l d not reproduce the wings of the peaks. T h i s was not a problem s i n c e an a c c u r a t e d e t e r m i n a t i o n of the are a (N„) was not r e q u i r e d . In t h i s e x p r e s s i o n , N e i s the number of co u n t s i n , E 0 the c e n t r e , and FWHM the w i d t h o f , the peak. The c o n s t a n t s a r i s e from a c o n v e r s i o n of the s t a n d a r d d e v i a t i o n of the peak i n t o FWHM, a l o n g w i t h f a c t o r s u s u a l l y seen i n a n o r m a l i z e d g a u s s i a n e x p r e s s i o n . The numbers A and B r e p r e s e n t the c o n s t a n t and l i n e a r ( i n energy) terms f o r the background, r e s p e c t i v e l y . The m u l t i p a r a m e t e r m i n i m i z a t i o n r o u t i n e MINUIT (James and Roos, 1971) was used t o m i n i m i z e % 2 and a s s e s s from the shape of the h y p e r s u r f a c e the v a l u e s and u n c e r t a i n t i e s of the par a m e t e r s . 87 For the n o r m a l i z a t i o n , d e t e r m i n a t i o n s of u n c e r t a i n t i e s (which determine i n t u r n the f i n a l l i m i t on the number of e v e n t s observed) were a c c o m p l i s h e d w i t h the MINUIT s u b r o u t i n e MINOS, which s e a r c h e s the "X2 h y p e r s u r f a c e f o r the change i n the v a l u e of the parameter (both p o s i t i v e and n e g a t i v e ) f o r which t h a t s t a t i s t i c i s i n c r e a s e d by one, thus d e f i n i n g a one s t a n d a r d d e v i a t i o n u n c e r t a i n t y . F i t s t o a p o s s i b l e c a l c i u m l i n e were o b t a i n e d by f i x i n g E and FWHM t o the c a l c u l a t e d v a l u e s and a l l o w i n g N c, A and B t o v a r y . S i n c e A and B a r e h i g h l y c o r r e l a t e d and t h e r e f o r e not i n d e p e n d e n t l y w e l l d e f i n e d over s m a l l energy r e g i o n s , p r e l i m i n a r y e s t i m a t e s based on l i n e a r f i t s of the d a t a from 0.740 t o 0.800 MeV were used as i n i t i a l i n p u t , i n o r d e r t h a t r e a l i s t i c minima c o u l d be found. N„ was i n i t i a l l y s e t t o z e r o . The v a l u e s o b t a i n e d were q u i t e i n s e n s i t i v e t o the v a l u e of E© i n a range ±1.2 keV from the e x p e c t e d peak p o s i t i o n . However, the answers o b t a i n e d f o r N 0 and i t s u n c e r t a i n t y d i d depend on the range of c h a n n e l s around the 0.784 MeV p o s i t i o n t h a t were used f o r the f i t , as shown i n T a b l e 4.1.1. The source of t h i s dependence i s the s t a t i s t i c a l improvement i n the d e t e r m i n a t i o n of the c o n t i n u o u s background as a broader energy range i s i n c l u d e d . The v a l u e of y2 (per degree of freedom) i s r a t h e r s m a l l u n l e s s more d a t a i s i n c l u d e d on e i t h e r s i d e of a peak. I t i s e s p e c i a l l y t r u e f o r the Aptec r e s u l t s , where the number of e v e n t s per c h a n n e l i s o n l y about 60% of t h a t of the O r t e c spectrum. The one s t a n d a r d d e v i a t i o n l i m i t s on N 0 can be taken t o be 277 (Aptec) and 371 ( O r t e c ) . The l a s t e n t r i e s from T a b l e 4.1.1 a r e used s i n c e they s h o u l d i n t h e o r y be more r e p r e s e n t a t i v e of 88 Energy Range Degrees of Aptec Ortec Analyzed (keV) Freedom N + o (counts) N + a (counts) x2 ±3 12 -249+333 7.04 +186+141 7.34 ±4 17 -182+310 8.50 +17+406 21 .4 ±5 22 -206+296 11 .2 -134+389 24.4 ±6 27 -146+289 23.7 -51+378 34.7 ±7 32 -159+284 32.9 -44+371 35.5 ±8 37 -153+277 42.4 Tab l e 4.1.1. V a l u e s o b t a i n e d by MINUIT f o r d i f f e r e n t ranges of da t a a n a l y z e d . an a c c u r a t e e s t i m a t e ; by i n c l u d i n g a l a r g e r segment of the spectrum i n the f i t , the background underneath a p o s s i b l e photopeak i s b e t t e r d e f i n e d , and t2 i s more r e a s o n a b l e . The s t a n d a r d d e v i a t i o n i n the sum of ^ from the two d e t e c t o r s i s o b t a i n e d by ad d i n g the v a r i a n c e s , g i v i n g an upper l i m i t of 463. 4.2. R e l a t i o n s h i p of Events Observed t o the Upper L i m i t f o r C o n v e r s i o n With a l i m i t on the number of events o b s e r v e d i n hand, i t remains t o t r a n s l a t e t h a t number i n t o a l i m i t on t h e c o u p l i n g c o n s t a n t G of the f o u r f e r m i o n H a m i l t o n i a n of e q u a t i o n 2.1.1.5, which c o u l d d e s c r i b e muonium-antimuoniurn c o n v e r s i o n . To t h i s end, a number P can be d e f i n e d f o r each d e t e c t o r such t h a t the r e l a t i o n between the number of ob s e r v e d e v e n t s , 8 9 N 0 , i s the p r o d u c t of P and the t o t a l number of i n c i d e n t muons, N, w i t h i n the l i v e time of t h a t d e t e c t o r , i . e . , N 0 = PN . 4.2.1 P i s j u s t the p r o b a b i l i t y t h a t , g i v e n a p o s i t i v e muon e n t e r i n g the t a r g e t , the c a l c i u m 2P-1S muonic X-ray f o l l o w i n g i t s c o n v e r s i o n t o antimuonium w i l l c o n t r i b u t e t o the photopeak a t 0.784 MeV i n the d e t e c t o r under c o n s i d e r a t i o n . P, of c o u r s e , depends on G. V a l u e s f o r P can be d e r i v e d by f i r s t w r i t i n g i t as a p r o d u c t of the p r o b a b i l i t i e s , f r a c t i o n s , or e f f i c i e n c i e s of each member of the s e r i e s of p r o c e s s e s which must occur f o r the d e t e c t i o n of the X-ray. These numbers and t h e i r r e s p e c t i v e d e f i n i t i o n s a r e l i s t e d below, i n the o r d e r i n which they take p l a c e : 1. F ( f o i l s ) : the f r a c t i o n of muons which, h a v i n g passed through the t h i n MU c o u n t e r , e v e n t u a l l y s t o p i n the a c t i v e s i l i c a component of the s l o p i n g t a r g e t s t a c k . I t w i l l be assumed i n s e c t i o n 4.2.5 t h a t t h i s number does not depend on whether the muons ar e p o s i t i v e or n e g a t i v e , i n o r d e r t o f a c i l i t a t e c a l c u l a t i o n s . 2. F(Mu): the f r a c t i o n of p o s i t i v e muons which, h a v i n g stopped i n the s i l i c a powder component, form muonium. 3. F ( v a c ) : the f r a c t i o n of muonium atoms formed i n the powder g r a i n s which r e a c h the i n t e r g r a n u l a r v o i d s b e f o r e d e c a y i n g . I t depends on, among o t h e r t h i n g s , the g r a i n s i z e , and has been measured by muonium s p i n 90 r o t a t i o n t e c h n i q u e s . 4. F ( g a p ) : t h e f r a c t i o n of those muonium atoms i n the i n t e r g r a n u l a r v o i d s i n a s i l i c a powder l a y e r which m i g r a t e by a random walk among t h e g r a i n s and e v e n t u a l l y i n t o the gap on the upper, downstream s i d e of the l a y e r . T h i s f r a c t i o n i s a f u n c t i o n of the l a y e r t h i c k n e s s . 5. P(Mu): the p r o b a b i l i t y t h a t a t h e r m a l muonium atom l e a v i n g one s i d e of the gap between t a r g e t l a y e r s w i l l have c o n v e r t e d t o antimuonium by the time i t reaches o p p o s i t e s i d e . T h i s number c o n t a i n s the v a l u e of G (or more p r e c i s e l y , G 2 ) . 6. P ( c a p t ) : the p r o b a b i l i t y t h a t an antimuonium atom i m p i n g i n g on a c a l c i u m o x i d e s u r f a c e w i l l r e s u l t i n the i n e l a s t i c s c a t t e r i n g p r o c e s s whereby the n e g a t i v e muon i s c a p t u r e d by the atoms of the m o l e c u l e ( e i t h e r c a l c i u m or oxygen). 7. E: the o v e r a l l d e t e c t i o n e f f i c i e n c y , d e f i n e d as the p r o b a b i l i t y t h a t a n e g a t i v e muon from antimuonium i n c a l c i u m o x i d e w i l l r e s u l t i n a c a l c i u m 2P-1S X-ray which d e p o s i t s i t s f u l l 0.784 MeV energy i n the germanium d e t e c t o r under c o n s i d e r a t i o n . Two v a l u e s w i l l be d e r i v e d , E ( A p t ) and E ( O r t ) , f o r use i n the c a l c u l a t i o n s . The d e t e r m i n a t i o n of each of t h e s e numbers (or i n one c a s e , a subproduct F(foils)«E) w i l l a l l o w the c a l c u l a t i o n of P: P = F(foils)»F(Mu)»F(vac)»F(gap)»P(Mu)«P(capt)»E . 4.2.2 91 The f o l l o w i n g s e c t i o n s w i l l d e a l w i t h the terms i n d i v i d u a l l y , e x p l a i n i n g the b a s i s f o r the assignment of a v a l u e t o each. 4.2.1. Muonium Fo r m a t i o n I t has been p r e v i o u s l y mentioned t h a t the c h o i c e of s i l i c a as a t a r g e t m a t e r i a l was based p a r t i a l l y on i t s penchant f o r muonium f o r m a t i o n . The most c a r e f u l and s y s t e m a t i c measurement of the muonium f r a c t i o n i n the s i l i c a powder used i n t h i s experiment i s t h a t of K i e f l e t a l . (1979), who accounted f o r muons not s t o p p i n g i n the powder. From a d i r e c t o b s e r v a t i o n of muonium s p i n r o t a t i o n , the f r a c t i o n was measured t o be F(Mu) = 0.61 ± 0.03. 4.2.1.1 Muonium s p i n r o t a t i o n ( f o r a d e s c r i p t i o n of the t e c h n i q u e , see, f o r example, G a r n e r , 1979) was a l s o o b s e r v e d d u r i n g the c o n v e r s i o n experiment i n t h e s i l i c a / c o l l o d i o n t a r g e t , u s i n g a s m a l l s c i n t i l l a t o r t e l e s c o p e f o r p o s i t r o n d e t e c t i o n ( F i g u r e 4.2.1.1). By n o r m a l i z i n g the muonium asymmetry o b t a i n e d t o the muon s p i n r o t a t i o n s i g n a l i n copper ( c l o s e t o 100% muon s p i n s i g n a l ) , a v a l u e of 0.29 ± 0.02 was o b t a i n e d f o r the muonium f r a c t i o n i n the e n t i r e t a r g e t r e g i o n . S i n c e the s i l i c a powder was the o n l y substance c a p a b l e of c a u s i n g the muonium p r e c e s s i o n s i g n a l , t h i s lower muonium f r a c t i o n p r o v i d e s a measure of the pr o d u c t F(foils)»F(Mu), and an e s t i m a t e of 92 MUONIUM IN CONVERSION TARGET cm i — UJ t o CL -0.1 0 .0 0 .2 0 . 4 0 .6 0 . 8 1.0 1.2 1.4 1.6 1.8 TIME (uSECJ F i g u r e 4.2.1.1. Muonium s p i n r o t a t i o n s i g n a l i n the c o n v e r s i o n t a r g e t . F ( f o i l s ) = 0.48 ± 0.04 4.2.1.2 . r e s u l t s . The a p p a r a t u s , c o n s t r u c t e d f o r muonium c o n v e r s i o n r a t h e r than s p i n r o t a t i o n , was not apt t o e l i m i n a t e s y s t e m a t i c e r r o r s i n muonium f o r m a t i o n f r a c t i o n s , and the quoted s t a t i s t i c a l u n c e r t a i n t y may be m i s l e a d i n g . R e g a r d l e s s , the d e t e r m i n a t i o n of F ( f o i l s ) i s b e t t e r a c c o m p l i s h e d by o t h e r t e c h n i q u e s d e s c r i b e d i n s e c t i o n 4.2.5. 93 4.2.2. P r o b a b i l i t y of E j e c t i o n i n t o Vacuum; S i n g l e F o i l s The d e r i v a t i o n s of v a l u e s f o r F ( v a c ) and F(gap) w i l l be the conc e r n of the f o l l o w i n g d i s c u s s i o n . The c a l c u l a t i o n s r e q u i r e d a r e i n some cases l e n g t h y , and t h e r e f o r e have been r e l e g a t e d t o app e n d i c e s A2 and A3. S i m i l a r t e c h n i q u e s can be a p p l i e d f o r the e s t i m a t i o n of both f r a c t i o n s , w i t h a p p r o p r i a t e changes i n geometry, muonium v e l o c i t y , and the d i s t a n c e s c a l e . The major d i f f e r e n c e i n approach t o the e x t r a c t i o n of the f i n a l v a l u e s i s t h a t F(gap) r e l i e s s o l e l y on r e a s o n a b l e (and p o s s i b l y p e s s i m i s t i c ) c o n j e c t u r e s on the motion of muonium among the f i n e s i l i c a p a r t i c l e s of the l a y e r s i n the c o n v e r s i o n t a r g e t . Only s k e t c h y , q u a l i t a t i v e e x p e r i m e n t a l e v i d e n c e c o u l d be o b t a i n e d f o r the e f f e c t s t o be d e s c r i b e d ; the s e n s i t i v i t y r e q u i r e d f o r a d i r e c t , unambiguous measurement of F(gap) c o u l d not be reached w i t h the methods employed. The s i t u a t i o n f o r F ( v a c ) i s more c l e a r . The r e s u l t s a re on a f i r m e x p e r i m e n t a l b a s i s , e x p l i c a b l e i n terms of a model y e t l a r g e l y independent of i t s v a l i d i t y . I t i s e x p e c t e d t h a t an e n e r g e t i c p o s i t i v e muon s l o w i n g down i n a low d e n s i t y s i l i c a powder t a r g e t w i l l behave much the same as i n b u l k s i l i c a , a t l e a s t i n s o f a r as s u r f a c e e f f e c t s p l a y no r o l e . That i s , e l e c t r o n c a p t u r e and l o s s w i l l h e l p t o slow the muon, the former d o m i n a t i n g a t lower e n e r g i e s so t h a t the ob s e r v e d t h e r m a l i z e d muonium f r a c t i o n i s l a r g e . One e x p e c t s the atoms t o come i n t o t h e r m a l e q u i l i b r i u m w i t h the s i l i c a g r a i n s ( a l t h o u g h not n e c e s s a r i l y a t the ambient t e m p e r a t u r e , s i n c e the muon's energy may have caused h e a t i n g of the g r a i n ) i n a u n i f o r m d i s t r i b u t i o n 94 throughout the powder g r a i n s . Any f u r t h e r t h e r m a l motion would cause some of the atoms t o rea c h the g r a i n s u r f a c e s and e n t e r the v o i d between p a r t i c l e s . G i v e n the s i z e and s p h e r i c i t y of the g r a i n s , and the assumption t h a t t h e r e e x i s t s some b a r r i e r i n h i b i t i n g muonium from r e - e n t e r i n g a s i l i c a p a r t i c l e , the r a t e a t which muonium reaches the v o i d s can be c a l c u l a t e d i n terms of a parameter D ( h e n c e f o r t h r e f e r r e d t o , r i g h t l y or wrongly, as a d i f f u s i o n p a r a m e t e r ) . An experiment t o t e s t the a p p l i c a b i l i t y of t h i s approach has been c a r r i e d out ( M a r s h a l l e t a l . , 1978) u t i l i z i n g the s p i n exchange r e l a x a t i o n of the muonium p o l a r i z a t i o n w i t h oxygen gas t o determine the r a t e of e x p u l s i o n from the p a r t i c l e s and e x t r a c t v a l u e s f o r D. I t s u p p o r t e d the mechanism j u s t d e s c r i b e d ; moreover, the e v i d e n c e t h a t muonium does i n f a c t r e a c h the v o i d s and move e s s e n t i a l l y f r e e l y a t t h e r m a l e n e r g i e s between c o l l i s i o n s w i t h g r a i n s u r f a c e s seems unshakable (appendix A2 c o n t a i n s more d i s c u s s i o n of the t e c h n i q u e s and r e s u l t s ) , whether or not the mechanism i s v a l i d . E v i d e n c e t h a t muonium i n the c o n v e r s i o n t a r g e t i s e s c a p i n g the s i l i c a p a r t i c l e s t o vacuum i s p r o v i d e d by F i g u r e 4.2.2.1, where the muonium s p i n r o t a t i o n s i g n a l of F i g u r e 4.2.1.1 has been removed by the a d d i t i o n of oxygen gas t o the t a r g e t r e g i o n . The f r a c t i o n of muonium r e a c h i n g the v o i d s b e f o r e d e c a y i n g , based on the model, i s F ( v a c ) = 0.93 ± 0.01, 4.2.2.1 where the e x p r e s s i o n s A2.12 and A2.14 have been a p p l i e d . Assuming no p a r t i c u l a r model, but u s i n g the measured e x p o n e n t i a l muonium s p i n r e l a x a t i o n r a t e i n a powder sample w i t h oxygen 95 MUONIUM IN TARGET WITH OXYGEN t o 0 . 4 0 . 3 0 . 2 h 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1.0 1.2 1 .4 1.6 TIME (uSEC) 1 . 8 F i g u r e 4.2.2.1. MSR p r e c e s s i o n s i g n a l r e l a x e d by oxygen gas, p r e s e n t , one g e t s an e s t i m a t e f o r F ( v a c ) of 0.97 ± 0.01 ( i b i d ) . Note t h a t t h i s number i s d e r i v e d from a d i f f e r e n t r e l a x a t i o n spectrum, f o r a d i f f e r e n t powder s i z e , than t h a t from which the v a l u e of D i n A2.14 r e s u l t s . G r a i n h e a t i n g ( K i e f l , 1981) may be the cause of the d i s c r e p a n c y , s i n c e D, a f u n c t i o n of te m p e r a t u r e , may depend on g r a i n r a d i u s . In subsequent c a l c u l a t i o n s , the more p e s s i m i s t i c f r a c t i o n of 4.2.2.1 w i l l be used. The e s t i m a t i o n of F(gap) can be a c c o m p l i s h e d v i a s i m i l a r m a t h e m a t i c a l methods, w i t h the major d i f f e r e n c e b e i n g t h a t a v a l u e f o r D (now r e f e r r i n g t o the d i f f u s i o n parameter c h a r a c t e r i z i n g muonium motion i n the v o i d s i n a l a y e r c o n s i s t i n g 9 6 of hard s i l i c a spheres surrounded by vacuum, r a t h e r than i n s i d e a s i l i c a sphere) has not been d e t e r m i n e d by e x p e r i m e n t , but must be i n f e r r e d from s i m p l e , r e a s o n a b l e e s t i m a t e s on the p r o p e r t i e s of the s i l i c a l a y e r . T h i s has been c a r r i e d out i n appendix A3 ( e q u a t i o n A3.8), assuming a homogeneous d i s t r i b u t i o n of s i l i c a p a r t i c l e s i n a powder such as the one used i n the c o n v e r s i o n t a r g e t . The d i f f u s i o n parameter can then be i n s e r t e d i n t o e q u a t i o n A3.3 w i t h the muon decay r a t e "\ = (2.2 x 10'' s ) " 1 and the average powder l a y e r t h i c k n e s s of 0.3 mm ( s e c t i o n 3.2.2) t o g i v e an e s t i m a t e of F(gap) = 0.078 . 4.2.2.2 Note t h a t the second ( e x p o n e n t i a l ) term i n A3.3 i s n e g l i g i b l e f o r the l a r g e v a l u e of the powder l a y e r t h i c k n e s s d, and t h a t F(gap) i s then i n v e r s e l y p r o p o r t i o n a l t o d. T h i s makes sense; o n l y a s m a l l p a r t of the l a y e r , t h a t n e a r e s t the gap, i s " a c t i v e " , or w i l l emit the muonium t h a t can e n t e r the gap b e f o r e decay. As d i s i n c r e a s e d , the p r o b a b i l i t y t h a t a muonium atom w i l l b e g i n i t s random m i g r a t i o n from the " a c t i v e " p o r t i o n d e c r e a s e s as d " 1 , e x p l a i n i n g t he dependence of F ( g a p ) . The p o i n t i s t h a t a n o n u n i f o r m i t y i n the v a l u e of d over the 150 cm 2 a r e a of the c o l l o d i o n f i l m does not r a d i c a l l y change F(gap) c a l c u l a t e d from an average v a l u e of d, as l o n g as coverage of the s u r f a c e i s complete. V i s u a l e x a m i n a t i o n of the t a r g e t s t a c k a f t e r a l l runs were completed i n d i c a t e d t h a t coverage was g r e a t e r than 2/3. A c c o r d i n g l y , the e s t i m a t e which w i l l be a p p l i e d f o r the f i n a l a n a l y s i s i s 97 F(gap) > 0.052 . 4.2.2.3 F(gap) c o u l d not be p r e c i s e l y measured because of the d i f f i c u l t y i n d e t e r m i n i n g the whereabouts of muonium atoms near a s i n g l e c o l l o d i o n f o i l (or f i l m ) s u p p o r t i n g a s i l i c a l a y e r , s i m i l a r t o the f o i l s making up the t a r g e t s t a c k . However, a q u a l i t a t i v e i n d i c a t i o n of the presence of muon decays a t one c e n t i m e t e r from a s i n g l e f o i l was obs e r v e d . A p l a s t i c d e t e c t o r t e l e s c o p e c o n s i s t i n g of two p i e c e s of s c i n t i l l a t o r m a t e r i a l 5 cm l o n g by 0.3 cm wide was a r r a n g e d so as t o d e t e c t muon decays w i t h i n a narrow sheet of about one c e n t i m e t e r t h i c k n e s s . The t e l e s c o p e was aimed a t a r e g i o n one c e n t i m e t e r from and p a r a l l e l t o the upper, downstream f a c e of a s i l i c a - c o v e r e d f o i l i n the evac u a t e d t a r g e t a r e a of f i g u r e 3.2.4.1. The o b j e c t of the e x e r c i s e was t o observe some enhancement i n the muon decay r a t e , as measured by a time h i s t o g r a m , a t t i m e s when muonium d r i f t i n g away from the f o i l a t t h e r m a l v e l o c i t y would pass t h r o u g h the r e g i o n viewed by the s c i n t i l l a t o r t e l e s c o p e . C o n s i d e r a non-decaying atom of v e l o c i t y v e m i t t e d a t t=0 i s o t r o p i c a l l y from a s u r f a c e x=0 i n t o 2 i r s r . The component of v p e r p e n d i c u l a r t o the s u r f a c e i s , by a v e r a g i n g over a l l d i r e c t i o n s , v/2, and the number dN i n a sheet of t h i c k n e s s dx a t a d i s t a n c e x from the s u r f a c e i s dN = Six - ( v / 2 ) t ) d x , 4.2.2.4 where t h e D i r a c d e l t a f u n c t i o n has t h e u s u a l d e f i n i t i o n . I f t h e e x p o n e n t i a l decay r a t e of the atom i s ~\t the obser v e d decay r a t e 98 of the dN atoms i n the sheet can be w r i t t e n as -d(dN)/dt = }exp(-*t)»£(x - ( v / 2 ) t ) d x . 4.2.2.5 I f a t h e r m a l M a x w e l l i a n d i s t r i b u t i o n P ( v ) = 4 T T (m/2^kT) < »/* » v 2 exp(-mv 2/2kT) 4.2.2.6 i s assumed, the decay r a t e must be i n t e g r a t e d over v t o g i v e A p l o t of -d(dN)/dtdx v e r s u s t a t room temperature f o r s e v e r a l v a l u e s of x, n o t a b l y 1.0 cm, com p r i s e s F i g u r e 4.2.2.2. The data c o l l e c t e d from a t o t a l of about 3 x 10' i n c i d e n t muons ( i n an e l a p s e d time of about f o u r t e e n hours) a r e shown i n F i g u r e 4.2.2.3. An unknown number of the muons ( p o s s i b l y 50% or g r e a t e r , as e s t i m a t e d from F ( f o i l s ) i n e q u a t i o n 4.2.1.2) were l o s t from the severe m u l t i p l e s c a t t e r i n g of the beam i n the i n c i d e n t d e t e c t o r (MU) and d e g r a d e r . Of tho s e r e m a i n i n g , o n l y about f o u r per cent would s t o p i n the s i l i c a l a y e r , e s t i m a t e d from the l a y e r t h i c k n e s s (1.8 mg»cnr 2) d i v i d e d by the FWHM range s p r e a d f o r the s u r f a c e muon beam (25 mg*cm - 2). F u r t h e r m o r e , muonium forms w i t h about 61% p r o b a b i l i t y ( e q u a t i o n 4.2.1.1). An e s t i m a t e of the number of muonium atoms formed i n the s i l i c a d u r i n g the c o u r s e of the run i s then -d(dN)/dt = > e x p ( - X t ) d x » 4 T ( m / 2 T k T ) < 3 / 2 ' ( 8 x 2 / t 3 ) • e x p ( - 2 m x 2 / k T t 2 ) 4.2.2.7 N(Mu) = 4.6 x 1 0 7 4.2.2.8 99 MUON DECAYS AWAY FROM A SINGLE FOIL 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 TIME (uSEC) F i g u r e 4.2.2.2. Expected time d i s t r i b u t i o n of decays from muonium d r i f t i n g t h e r m a l l y i n vacuum from a s i n g l e f o i l . The t e l e s c o p e had a g e o m e t r i c a l s o l i d a n g l e e f f i c i e n c y of 3.7 x 10"*. At the i n c i d e n t r a t e s used (6 x 1 0 4 s - 1 ) , a f r a c t i o n of 0.38 of the muon decay e v e n t s d e t e c t e d were r e c o r d e d , h a v i n g passed a c r i t e r i o n of b e i n g " a l o n e " i n the t a r g e t (meaning the muon e n t r a n c e time was both preceded and f o l l o w e d by an e i g h t m i c r o s e c o n d i n t e r v a l i n which no o t h e r muon e n t e r e d ) . T h i s c u t i s n e c e s s a r y t o p r e v e n t d i s t o r t i o n from muon p i l e - u p ( G a r n e r , 1979). The d e t e c t i o n e f f i c i e n c y f o r the s i n g l e f o i l e x p e r i m e n t , E ( s f ) , may be taken r o u g h l y as the pr o d u c t of the two, namely 100 U J t— <x 5000 4000 3000 h S INGLE S I L I C A LAYER IN VACUUM T g 2000 U J Q 1000 h 0 . 0 1.0 2 . 0 3 . 0 4 . 0 5 . 0 6 . 0 TIME (uSEC) 7 . 0 8 . 0 F i g u r e 4.2.2.3. Muon decay c u r v e o b t a i n e d w i t h narrow t e l e s c o p e c e n t r e d 1 cm downstream of s i l i c a l a y e r . E ( s f ) = 1.4 x 1 0 " 4 . 4.2.2.9 The enhancement i n the number of e v e n t s above an e x p o n e n t i a l background i n the muon decay c u r v e s h o u l d be, i f the assumptions made a r e r e a s o n a b l e , N(enh) = N(Mu)«E(sf )»exp(-'>t)»F(vac)»F(gap) . 4.2.2.10 The e x p o n e n t i a l f a c t o r makes a l l o w a n c e f o r t h e decay of the atoms b e f o r e r e a c h i n g the r e g i o n of d e t e c t i o n . 101 A f i v e parameter f i t t o the d a t a of F i g u r e 4.2.2.3 was a t t e m p t e d u s i n g the f u n c t i o n N ( t ) = N 0 e x p ( - ^ t ) + B + 0.939(N,/FWHM) 4.2.2.11 • e x p [ - ( l / 2 ) { ( t - t0 )/(0.425»FWHM)}2] where "X was f i x e d a t the muon e x p o n e n t i a l decay r a t e . The g a u s s i a n form was assumed f o r ease of a n a l y s i s ; a more a c c u r a t e and somewhat i n t r a c t a b l e f u n c t i o n a l dependence, d e r i v a b l e from e q u a t i o n 4.2.2.7, was judged unnecessary f o r the s i z e and s t a t i s t i c a l s i g n i f i g a n c e of the enhancement o b s e r v e d . The v a l u e of N, o b t a i n e d was +160 w i t h p o s i t i v e and n e g a t i v e s t a n d a r d d e v i a t i o n s of 85 and 77 r e s p e c t i v e l y . The c e n t e r (t„) and FWHM of the f i t t e d enhancement were 2.8 ± 0.1 and 0.7 ± 0.3 m i c r o s e c o n d s , so the shape i s s i m i l a r t o the 1.0 cm c u r v e of F i g u r e 4.2.2.1. The reduced % 2 v a l u e was 0.845 f o r 33 degrees of freedom. With N, s e t t o z e r o , the f i t o b t a i n e d had a reduced 7fJ v a l u e of 0.913 f o r 36 degrees of freedom. The d a t a , t h e n , do show some enhancement a t a p p r o x i m a t e l y the r i g h t p o s i t i o n i n the decay c u r v e f o r e m i s s i o n of t h e r m a l muonium, a l t h o u g h the s e n s i t i v i t y of the method i s not adequate f o r a p r e c i s e measurement of the r a t e . A p l o t of the d a t a and the f i t a r e d i s p l a y e d i n F i g u r e 4.2.2.4, where the f i r s t two terms of the r i g h t hand s i d e of e q u a t i o n 4.2.2.11 have been s u b t r a c t e d . Muonium e j e c t e d e p i t h e r m a l l y from a g r a i n would be e x p e c t e d t o t h e r m a l i z e i n a few subsequent c o l l i s i o n s w i t h n e i g h b o r i n g powder p a r t i c l e s , b e f o r e r e a c h i n g the gap between l a y e r s ( a l s o i n d i c a t e d by the M S R r e l a x a t i o n r a t e s i n oxygen gas, as e x p l a i n e d i n appendix A 2 ) . T a k i n g the v a l u e f o r N, as 102 ENHANCEMENT (MUON DECRY SUBTRACTED) CD or LU 0_ , CO o CJ -150 0.0 1.0 2.0 3.0 4.0 5.0 6.0 TIME IN uSEC (200 NSEC/BIN) 7.0 8.0 F i g u r e 4.2.2.4. Enhancement i n the muon decay spectrum. an e s t i m a t e of N(enh) of e q u a t i o n 4.2.2.10, s e t t i n g t = 2.8 x 10"' s and F ( v a c ) = 0.93 ( e q u a t i o n 4.2.2.1), and s o l v i n g f o r F ( g a p ) , i t i s found t h a t F(gap) = 0.095 4.2.2.12 w i t h a s t a n d a r d d e v i a t i o n of about 50%. Because of the l a r g e u n c e r t a i n t y and the d i f f i c u l t y i n a s s e s s i n g s y s t e m a t i c e r r o r s , t h i s r e s u l t i s by no means an u n e q u i v o c a l measurement of F ( g a p ) . I t i s a t l e a s t not i n c o n s i s t e n t w i t h the c a l c u l a t e d v a l u e of 0.078 of e q u a t i o n 4.2.2.3. T h i s c o n c l u d e s the d i s c u s s i o n of the v a l u e s which w i l l be 103 taken f o r F ( v a c ) and F(gap) i n the d e r i v a t i o n of a l i m i t on muonium c o n v e r s i o n t o antimuonium. The next s t e p i s t o examine what may happen t o a muonium atom once i t has reached the gap between f o i l s and f i n d s the vacuum environment where c o n v e r s i o n i s not s t r o n g l y s u p p r e s s e d . 4.2.3. C o n v e r s i o n P r o b a b i l i t y As a muonium atom l e a v e s a f l a t s u r f a c e and moves i n vacuum toward a n o t h e r f l a t s u r f a c e , the s p l i t A i n the muonium and antimuonium energy e i g e n v a l u e s ( i n t r o d u c e d i n s e c t i o n 2.1.2) i s g r e a t l y reduced and i s no l o n g e r d e t r i m e n t a l t o p o s s i b l e c o n v e r s i o n mechanisms. The e x p r e s s i o n f o r the p r o b a b i l i t y of c o n v e r s i o n as a f u n c t i o n of t i m e , e q u a t i o n 2.1.3.4, w i l l h o l d t o a h i g h l e v e l of a c c u r a c y , so i t can be s a i d t h a t P(Mu;t) = e x p ( - ^ t ) ( S t / 2 ) 2 . 4.2.3.1 Assuming a speed v f o r the atom as i t l e a v e s one s u r f a c e ( t h e s i l i c a l a y e r ) d e s t i n e d f o r a n other s u r f a c e a d i s t a n c e s away, a t an a n g l e 9 w i t h the p e r p e n d i c u l a r , the p r o b a b i l i t y can be e x p r e s s e d i n terms of v and 9 as P ( M u ; v , 0 ) = { e x p ( - ^ s / v cos0) S2s>}/4v2cos2& . 4.2.3.2 An i n t e g r a t i o n over a n g l e s can be c a r r i e d o u t , assuming 104 i s o t r o p y of the v e l o c i t y v e c t o r i n the h a l f space 0 < 9 < -jr/2, t o get P(Mu;v) = ( S 2/4> 2)»(>s/v)«exp(-^s/v) . 4.2.3.3 I n s e r t i n g the v a l u e s of S and ^ , s e t t i n g v = 7.4 x 1 0 5 cm»s"', (mean t h e r m a l muonium v e l o c i t y a t room t e m p e r a t u r e , e q u a t i o n A3.8) and s = 0.41 cm (see s e c t i o n 3.2.2), the r e s u l t i s A more c a r e f u l c a l c u l a t i o n u s i n g n u m e r i c a l i n t e g r a t i o n over the v e l o c i t y d i s t r i b u t i o n ( e q u a t i o n 4.2.2.6) w i l l g i v e which i s the number t h a t w i l l be a p p l i e d i n the e v a l u a t i o n of the r i g h t hand s i d e of e q u a t i o n 4.2.2. B e f o r e l e a v i n g the t o p i c of P(Mu), two p o i n t s s h o u l d be emphasized: 1. The v a l u e d e r i v e d here i s e x a c t l y one o r d e r of magnitude lower than the t i m e - i n t e g r a t e d p r o b a b i l i t y f o r decay as antimuonium under i d e a l c i r c u m s t a n c e s ( e q u a t i o n 2.1.3.5), and i s s l i g h t l y l e s s than h a l f the maximum of the u n i n t e g r a t e d e x p r e s s i o n 2.1.3.4. 2. P(Mu) depends on the square of the c o u p l i n g . E x p e r i m e n t a l l y t h i s means t h a t i n c r e a s i n g the s e n s i t i v i t y t o G by some f a c t o r n e c e s s i t a t e s r e d u c i n g P(Mu) = 2.4 x 10-' (G/Gp) 2 . 4.2.3.4 P(Mu) = 2.5 x 10" 6 ( G / G p ) 2 , 4.2.3.5 105 the l i m i t on the number of e v e n t s o b s e r v e d , or a l t e r n a t i v e l y , i n c r e a s i n g the p r o d u c t of f a c t o r s i n the combined d e t e c t i o n p r o b a b i l i t y of e q u a t i o n 4.2.2, by the square of t h a t f a c t o r . One way of a c h i e v i n g the former i s t o t a k e more d a t a , but the r e d u c t i o n i n the l i m i t of e v e n t s observed i t s e l f depends on the square r o o t of the i n c r e a s e i n d a t a . T h i s means t h a t a r e d u c t i o n of the l i m i t on G by a f a c t o r of two r e q u i r e s 2* or s i x t e e n times as much d a t a , o t h e r t h i n g s b e i n g e q u a l . More w i l l be s a i d on t h i s p o i n t i n the c o n c l u d i n g c h a p t e r . The next f a c t o r t o be e v a l u a t e d i n the e x p r e s s i o n f o r P d e s c r i b e s the b e h a v i o r of an antimuonium atom i m p i n g i n g on a s u r f a c e of c a l c i u m o x i d e and the l i k e l i h o o d of c a p t u r e of the n e g a t i v e muon i n t o an atomic o r b i t a l , a p r o c e s s which must occur f o r the o b s e r v a t i o n of muonium c o n v e r s i o n by the method employed i n t h i s e x p e r i m e n t . 4.2.4. N e g a t i v e Muonic X-ray P r o b a b i l i t y T h i s s e c t i o n w i l l attempt t o p r e s e n t a c l u e t o the s o l u t i o n of the q u e s t i o n , "What happens when antimuonium i n vacuum i n t e r a c t s w i t h a c a l c i u m o x i d e s u r f a c e ? " . Very l i t t l e t h e o r e t i c a l work e x i s t s on the i n t e r a c t i o n of atoms such as antimuonium (a n e g a t i v e n u c l e u s surrounded by a p o s i t i v e e l e c t r o n d i s t r i b u t i o n ) w i t h normal m a t t e r . E x p e r i m e n t a l l y , i t i s i m p o s s i b l e a t t h i s s t a g e t o measure the l i k e l i h o o d of 106 f o r m a t i o n of a muonic atom from an i n i t i a l s t a t e c o n t a i n i n g antimuonium. A l t h o u g h much i s known about p o s i t r o n i u m , which might be thought of as an antimuonium a n a l o g u e , the e q u a l i t y of mass of the e l e c t r o n and p o s i t r o n makes i t s atomic p r o p e r t i e s v a s t l y d i f f e r e n t ; a l s o , because of the e x c l u s i o n p r i n c i p l e , the e l e c t r o n w i l l not be a f f e c t e d by an atomic c o l l i s i o n as a n e g a t i v e muon would. The problem has been approached f o r the case of antimuonium i n atomic hydrogen and i n e r t gases, e s p e c i a l l y a rgon, f o r the a n a l y s i s of the p r e v i o u s s u c c e s s f u l muonium c o n v e r s i o n experiment d e s c r i b e d i n s e c t i o n 3.1.1 (Amato e t a l . , 1968). The methods employed and the c o n c l u s i o n s reached (Morgan, 1967) w i l l be summarized h e r e , i n s o f a r as they r e l a t e t o the problem a t hand. For a t omic hydrogen, an i n t e r a c t i o n p o t e n t i a l f o r the s c a t t e r i n g of antimuonium was d e r i v e d u s i n g a p e r t u r b a t i o n e x p a n s i o n f o r the ground s t a t e e i g e n v a l u e of the S c h r o e d i n g e r e q u a t i o n f o r the system. The pr o c e d u r e i s s i m i l a r t o t h a t f o r dedu c i n g the Van der Waals d i s p e r s i o n energy f o r two hydrogen atoms (Margenau, 1939), but i s c l a i m e d t o be r e l e v a n t f o r s m a l l e r atomic s e p a r a t i o n s . The r e s u l t i s a p p l i c a b l e i n the a d i a b a t i c a p p r o x i m a t i o n where the n u c l e i (muon and p r o t o n ) can be c o n s i d e r e d f i x e d f o r the c a l c u l a t i o n of e l e c t r o n wave f u n c t i o n s , and h o l d s f o r t h e r m a l e n c o u n t e r s a t room t e m p e r a t u r e . A c r i t i c a l r a d i u s , R c, i s d e f i n e d as t h a t s e p a r a t i o n of the muon and p r o t o n below which the f o r m a t i o n of p o s i t r o n i u m a t r e s t , l e a v i n g the muon and p r o t o n c a p a b l e of f o r m i n g muonic hydrogen, i s e n e r g e t i c a l l y f a v o r e d . The assumption i s made t h a t t h i s r e a c t i o n w i l l t a k e p l a c e i f the muon-proton s e p a r a t i o n i n a 107 c o l l i s i o n of antimuonium w i t h hydrogen i s l e s s than R t . I t i s then a q u e s t i o n of d e t e r m i n i n g the t u r n i n g p o i n t , or d i s t a n c e of c l o s e s t approach, of an o r b i t f o r a g i v e n c o l l i s i o n and impact parameter, u s i n g the i n t e r a t o m i c p o t e n t i a l . I t was found t h a t f o r a p a r t i c u l a r c o l l i s i o n energy t h e r e e x i s t e d a d e f i n i t e v a l u e of impact parameter, R,' , below which the t u r n i n g p o i n t i s s i g n i f i c a n t l y l e s s than R c ( p o s s i b l e rearrangement of the p a r t i c l e s was i g n o r e d f o r the c a l c u l a t i o n ) . At t h e r m a l c o l l i s i o n e n e r g i e s , R t / > j 0 . 5 a o , R J = 6.5a 0, and the p a t h l e n g t h of antimuonium w i t h i n R c i s about a,, f o r impact parameters l e s s than R J ( a 0 i s the Bohr r a d i u s ) . The e l e c t r o n v e l o c i t y i s about f i v e t i m e s the r e l a t i v e muon-proton v e l o c i t y , so the f o r m a t i o n of the lower energy p o s i t r o n i u m s t a t e i s a c e r t a i n t y . Moreover, i t s f o r m a t i o n w i t h i n the volume r < R c i n d i c a t e s , by the u n c e r t a i n t y p r i n c i p l e , t h a t i t w i l l escape r a p i d l y , and a r e v e r s a l of the i n t e r a c t i o n i f the muon-proton s e p a r a t i o n were a g a i n t o become g r e a t e r than R c i s not l i k e l y . Subsequent photon e m i s s i o n by muonic hydrogen t a k e s p l a c e on a time s c a l e much s h o r t e r than the muon l i f e t i m e , and the muon w i l l u s u a l l y decay from the IS atomic s t a t e . The case f o r antimuonium s c a t t e r i n g by argon i s s i m i l a r e xcept t h a t the a c c u r a c y of the d e t e r m i n a t i o n of the r a d i i R and R,' i s reduced because of a h i g h e r u n c e r t a i n t y i n the i n t e r a t o m i c p o t e n t i a l f o r some s e p a r a t i o n s . The f o r c e s t e n d i n g t o break up antimuonium a r e g r e a t e r due t o the h i g h e r charge of the argon c o r e f o r s e p a r a t i o n s l e s s than ^ . A g a i n , muonic argon f o r m a t i o n i s a near c e r t a i n t y f o r impact parameters l e s s than a p a r t i c u l a r v a l u e , c a l c u l a t e d t o be 7.4a a f o r the mean t h e r m a l c o l l i s i o n energy a t room t e m p e r a t u r e . The c o r r e s p o n d i n g 108 i n e l a s t i c s c a t t e r i n g c r o s s s e c t i o n i s a~z = 55.1-iraJ . 4.2.4.1 An e s t i m a t i o n of t h i s q u a n t i t y by use of an o p t i c a l model app r o a c h , which i s not s u s c e p t i b l e t o e r r o r from u n c e r t a i n t y i n the i n t e r a t o m i c p o t e n t i a l , g i v e s i n c l o s e agreement. A l a r g e d i s c r e p a n c y between t h i s c r o s s s e c t i o n and one f o r anot h e r s i m p l e atom i s not l i k e l y (the c r o s s s e c t i o n f o r atomic hydrogen i s c a l c u l a t e d t o be about 37-fra^ ). T h i s can p r o v i d e a c l u e t o the b e h a v i o r of antimuonium i n a c o l l i s i o n w i t h a c a l c i u m o x i d e s u r f a c e . I t i s v e r y d i f f i c u l t t o r i g o r o u s l y c a l c u l a t e a t h e o r e t i c a l c r o s s s e c t i o n , and the o n l y r e c o u r s e i s t o p l a u s i b i l i t y arguments. The reduced de B r o g l i e wavelength of t h e r m a l antimuonium i s ^ a„/6, so the s u r f a c e can be c o n s i d e r e d as an a r r a y of atoms r a t h e r than a continuum. Moreover, a v a l u e of the i n e l a s t i c c r o s s s e c t i o n s of anywhere near the or d e r of magnitude of e q u a t i o n 4.2.4.1, c o u p l e d w i t h the t h i c k n e s s of the o x i d e l a y e r used, i n s u r e s the c e r t a i n t y t h a t antimuonium e n c o u n t e r i n g the c o a t i n g w i l l r e s u l t i n the n e g a t i v e muon b e i n g c a p t u r e d by e i t h e r a c a l c i u m or an oxygen atom. Thus i t i s r e a s o n a b l e t o s e t (Tj = (56.5 ± 2 .3) ij- a£ 4.2.4.2 F ( c a p t ) = 1.0 4.2.4.3 109 f o r use i n the o v e r a l l d e t e c t i o n p r o b a b i l i t y . Some assumptions w i l l have t o be made r e g a r d i n g the r e l a t i v e c a p t u r e p r o b a b i l i t i e s f o r c a l c i u m and oxygen. W h i l e the s e can be measured f o r f a s t n e g a t i v e muons s t o p p i n g i n o x i d e s , the a p p l i c a t i o n of the r a t i o of p r o b a b i l i t i e s t o atomic muon c a p t u r e from t h e r m a l antimuonium r e q u i r e s some j u s t i f i c a t i o n . As d e f i n e d i n s e c t i o n 4.2, the r a t i o i s c o n t a i n e d w i t h i n the f a c t o r E, the d e t e c t i o n e f f i c i e n c y , which i s the s u b j e c t of the f o l l o w i n g s e c t i o n . 4.2.5. D e t e c t i o n E f f i c i e n c y I t has been s t a t e d t h a t the d e t e c t i o n e f f i c i e n c y i s the p r o b a b i l i t y t h a t a n e g a t i v e muon from antimuonium, when c a p t u r e d i n c a l c i u m o x i d e , w i l l r e s u l t i n a c a l c i u m X-ray which d e p o s i t s i t s f u l l 0.784 MeV i n the germanium d e t e c t o r . Two v a l u e s , one f o r each d e t e c t o r , w i l l be deduced i n t h i s s e c t i o n t o account f o r the d i f f e r e n c e i n response t o the r a d i a t i o n . I t i s more c o n v e n i e n t and p r e c i s e t o e v a l u a t e the e f f i c i e n c i e s i n the form of a p r o d u c t w i t h F ( f o i l s ) , the f r a c t i o n of muons p a s s i n g t h r o u g h the t h i n c o u n t e r which s t o p i n the s i l i c a p o r t i o n of the c o n v e r s i o n t a r g e t . The method of e x t r a c t i n g the numbers r e l i e s on the use of a beam of n e g a t i v e muons a t 29 MeV/c, the momentum of p o s i t i v e s u r f a c e muons. Because of the s t r o n g a b s o r p t i o n of n e g a t i v e p i o n s produced and stopped i n the p r o t o n t a r g e t which i s viewed by M13, t h e r e a r e none (or v e r y few) which decay t o muons i n the s u r f a c e of 110 the t a r g e t . Hence, n e g a t i v e s u r f a c e muons do not e x i s t . Some n e g a t i v e p i o n s , however, escape the t a r g e t w i t h a v e l o c i t y t h a t a l l o w s t h e i r decay i n t o n e g a t i v e muons which s a t i s f y the phase space a c c e p t a n c e of the secondary beamline when i t i s s e t up f o r 29 MeV/c n e g a t i v e p a r t i c l e s . Rates a r e t y p i c a l l y (Oram e t a l . , 1980) about 1.5% of the p o s i t i v e s u r f a c e muon f l u x , which i s low but n o n e t h e l e s s u s e f u l f o r the purposes of t h i s e x p e r i m e n t . Because of i t s h i g h s t o p p i n g d e n s i t y the slow n e g a t i v e beam s h o u l d f i n d a p p l i c a t i o n s , f o r i n s t a n c e i n muonic X-ray s t u d i e s of r a r e t a r g e t s , or low p r e s s u r e gases, but i t has not y e t been e x p l o i t e d . One p o i n t t o note i s t h a t t h e h i g h p o l a r i z a t i o n c h a r a c t e r i z i n g the p o s i t i v e beam i s reduced t o 0.45 ± 0.20 f o r the n e g a t i v e one (as measured by J.H. Brewer a t TRIUMF ). No d i f f e r e n c e i n q u a l i t y (except f l u x and thus r e l a t i v e c o n t a m i n a t i o n ) c o u l d be d i s t i n g u i s h e d between n e g a t i v e and p o s i t i v e muon beams a t 29 MeV/c, nor i s one e x p e c t e d when the sharp edge a t 29.8 MeV/c of the p o s i t i v e p a r t i c l e momentum d i s t r i b u t i o n i s not w i t h i n the c h a n n e l a c c e p t a n c e , and the f i n a l beam spot s i z e i s not d e t e r m i n e d by the muon source s i z e ( m u l t i p l e s c a t t e r i n g i s r e s p o n s i b l e f o r the e n l a r g e d beam a t the c o n v e r s i o n t a r g e t , as e v i d e n c e d by the low v a l u e of F ( f o i l s ) e s t i m a t e d i n e q u a t i o n 4.2.1.2). The s t o p p i n g power f o r n e g a t i v e and p o s i t i v e beams i s the same f o r e n e r g i e s d e t e r m i n i n g the range and range spr e a d of a 4 MeV muon beam; o n l y a t e n e r g i e s comparable t o atomic b i n d i n g p o t e n t i a l s does the charge of the beam l e a d t o a d i f f e r e n c e i n the s l o w i n g mechanism (atomic c a p t u r e v e r s u s muonium f o r m a t i o n ) . I t i s t h e r e f o r e a c c u r a t e t o assume t h a t , i n t h e p r e s e n t c o n t e x t , the muon s t o p p i n g d i s t r i b u t i o n i s the same f o r p o s i t i v e and n e g a t i v e beams. In I l l p a r t i c u l a r , the f r a c t i o n of muons s t o p p i n g i n the a c t i v e s i l i c a component of the t a r g e t i s independent of the charge of the beam. With t h i s i n mind, i t i s e v i d e n t t h a t the p r o b a b i l i t y of o b s e r v i n g a n e g a t i v e muonic 2P-1S X-ray from s i l i c o n a t 0.400 MeV ( a l l muonic X-ray e n e r g i e s quoted a re from the c o m p i l a t i o n of E n g f e r et. a l . , 1974), g i v e n t h a t a n e g a t i v e muon has e n t e r e d the t a r g e t , i s P ( S i ) = F(foils)»R(Si)»eff(Si) . 4.2.5.1 F ( f o i l s ) has been d e f i n e d p r e v i o u s l y , and: 1. R ( S i ) i s the r a t i o of s i l i c o n 2P-1S X-ray i n t e n s i t y t o the t o t a l muonic K X-ray i n t e n s i t y from s i l i c o n and oxygen r e s u l t i n g from muons stopped i n the s i l i c a powder of the c o n v e r s i o n t a r g e t . I t i s the r e l a t i v e c a p t u r e r a t i o of s i l i c o n t o oxygen m u l t i p l i e d by the r a t i o of t o a l l K X - r a y s , and can be measured u s i n g a s i l i c a powder t a r g e t . 2. e f f ( S i ) i s the e f f i c i e n c y f o r o b s e r v i n g the f u l l energy of a s i l i c o n 2P-1S X-ray i n a p a r t i c u l a r d e t e c t o r . I n c l u d e d a r e the photopeak e f f i c i e n c y a t 0.400 MeV, the s o l i d a n g l e , and an a t t e n u a t i o n f a c t o r f o r the 0.32 cm aluminum vacuum p i p e , mumetal s h i e l d , and 0.62 cm f r o n t v e t o s c i n t i l l a t o r ( n e g l e c t i n g the t a r g e t i t s e l f , the s c i n t i l l a t o r w r a p p ing, and a i r , which a r e n e g l i g i b l e ) . The s o l i d a n g l e e f f e c t i s the o n l y one which cannot be a c c u r a t e l y and i n d e p e n d e n t l y e s t i m a t e d . The e f f i c i e n c y and a t t e n u a t i o n depend on 112 the energy of the X-ray. Both the e f f i c i e n c y and the s o l i d a n g l e w i l l be d i f f e r e n t f o r each d e t e c t o r . The g o a l of t h i s s e c t i o n i s t o e s t i m a t e the q u a n t i t i e s E f o r each d e t e c t o r . From the way i n which i t - h a s been d e f i n e d , i t i s e v i d e n t t h a t E = R(Ca)»eff(Ca) , 4.2.5.2 where the q u a n t i t i e s on the r i g h t a r e analogous t o those of e q u a t i o n 4.2.5.1, f o r c a l c i u m o x i d e r a t h e r than s i l i c o n d i o x i d e . The r e a d e r s h o u l d by now u n d e r s t a n d the g i s t of t h i s approach. M u l t i p l y i n g e q u a t i o n 4.2.5.2 by F ( f o i l s ) and u s i n g 4.2.5.1: F(foils)»E = P(Si)»{R(Ca)/R(Si)}»{eff(Ca)/eff(Si)} . 4.2.5.3 P ( S i ) i s measured u s i n g the 29 MeV/c n e g a t i v e muon beam and the c o n v e r s i o n t a r g e t , R(Ca) and R ( S i ) a r e measured i n d e p e n d e n t l y w i t h n e g a t i v e muons i n s i l i c a and c a l c i u m o x i d e t a r g e t s , and the r a t i o e f f ( C a ) / e f f ( S i ) i s j u s t the r a t i o of the d e t e c t o r photopeak e f f i c i e n c i e s t i m e s the a t t e n u a t i o n f a c t o r s a t 0.784 and 0.400 MeV. The f a c t o r s from F ( f o i l s ) and the s o l i d a n g l e e f f i c i e n c i e s have been absorbed i n P ( S i ) and a r e thus d e t e r m i n e d as a pr o d u c t e x p e r i m e n t a l l y . T h i s approach s h o u l d be r e l a t i v e l y f r e e of s y s t e m a t i c e r r o r s , i f c a r r i e d out w i t h c a r e . The use of R(Ca) i s made under the assumption t h a t i t i s independent of whether t h e muon i s c a p t u r e d from a t h e r m a l n e u t r a l antimuonium atom or an e n e r g e t i c n e g a t i v e muon beam. The q u a l i t a t i v e u n d e r s t a n d i n g of muon c a p t u r e i n atoms and 113 m o l e c u l e s has i n c r e a s e d w i t h the growth of a c t i v i t y i n mesic c h e m i s t r y (Schneuwly, 1979; D a n i e l , 1979), but no t h e o r y e x i s t s which can a c c u r a t e l y p r e d i c t the e f f e c t on R(Ca) of c a p t u r e from a t h e r m a l antimuonium s t a t e . A measurement of R(Ca) u s i n g t h e r m a l antimuonium i s not e x p e r i m e n t a l l y f e a s i b l e , but a rough a n a l o g y e x i s t s w i t h t h e r m a l muonic hydrogen. T h i s has been s t u d i e d i n the o x i d i z e d s u r f a c e l a y e r s of t h i n aluminum f o i l s i n a hydrogen gas t a r g e t ( B e r t i n e_t a l . , 1978). A r e d u c t i o n of about 40% was observed i n the r e l a t i v e 2P-1S aluminum muonic X-ray i n t e n s i t y , presumably because the s m a l l s i z e (some 200 times s m a l l e r than muonium) and h i g h b i n d i n g energy a l l o w i t t o p e n e t r a t e w e l l w i t h i n the o u t e r e l e c t r o n o r b i t a l s b e f o r e b e i n g t o r n a p a r t . The n e g a t i v e muon w i l l then p o p u l a t e lower a n g u l a r momentum s t a t e s of lower p r i n c i p a l quantum number, enhancing nP-1S w i t h r e s p e c t t o 2P-1S t r a n s i t i o n s . No mention i s made of any e f f e c t on the r e l a t i v e aluminum and oxygen c a p t u r e r a t i o s , so i t i s presumably n e g l i g i b l e . The l a r g e r , more l o o s e l y bound antimuonium atom s h o u l d behave more l i k e a f r e e ' n e g a t i v e muon than muonic hydrogen, s i n c e i t cannot p e n e t r a t e the atom as e a s i l y w i t h o u t breakup. T h i s s u p p o r t s the assumption of the s u i t a b i l i t y of R(Ca) as d e r i v e d from n e g a t i v e beam da t a f o r c a p t u r e from the antimuonium s t a t e . The P ( S i ) a r e o b t a i n a b l e from d a t a summarized by F i g u r e 4.2.5.1, the s p e c t r a fom the two d e t e c t o r s w i t h a n e g a t i v e muon beam i n the c o n v e r s i o n t a r g e t . The number of muons e n t e r i n g t h r o u g h the d e f i n i n g (MU) c o u n t e r was (2.53 ± 0.25) x 1 0 1 . The l a r g e u n c e r t a i n t y r e s u l t s from the h i g h e l e c t r o n c o n t a m i n a t i o n (>100 e" per muon) of t h e beam, w h i c h , a l t h o u g h counted w i t h low e f f i c i e n c y i n the c o u n t e r , was r e s p o n s i b l e f o r 46 ± 5 per cen t u" IN CONVERSION TARGET (APTEC) u" IN CONVERSION TARGET (ORTEC) 500 600 700 800 900 ENERGY (0.4 KEV/CHANNEL) 1000 F i g u r e 4.2.5.1. X-ray s p e c t r a from n e g a t i v e muons i n the c o n v e r s i o n t a r g e t . 115 of t h e r a t e . T h i s number was de t e r m i n e d by i n s e r t i n g a t h i n CH a b s o r b e r a t F l (see s e c t i o n 3.2.1) which a f f e c t s the e l e c t r o n s v e r y l i t t l e but slows the muons such t h a t they have too low a momentum t o s u r v i v e the bend between F2 and F3. The number of c o u n t s i n the s i l i c o n 2P-1S photopeaks (from g a u s s i a n f i t s ) were 2961 ± 63 (Aptec) and 3439 ± 68 ( O r t e c ) , l e a d i n g t o the assignments P ( S i ; A p t ) = (1.17 ± 0.13) x 1 0 " 4 , 4.2.5.4a P ( S i ; O r t ) = (1.36 ± 0.15) x 1 0 ' 4 . 4.2.5.4b F i g u r e s 4.2.5.2 and 4.2.5.3 show muonic s p e c t r a from s i l i c a powder and c a l c i u m o x i d e t a r g e t s r e s p e c t i v e l y . The c a l c i u m o x i d e was i n the form of ,a c o a r s e powder and was heated o v e r n i g h t p r i o r t o b e i n g p l a c e d hot i n the t a r g e t chamber vacuum, t o i n s u r e the absence of h y d r o x i d e which w i l l a l t e r the c a p t u r e r a t i o . A f t e r c o r r e c t i n g f o r d e t e c t o r e f f i c i e n c y and a b s o r p t i o n , the l i n e i n t e n s i t i e s were deduced and summed a p p r o p r i a t e l y . Note the p r o x i m i t y of the oxygen 3P-1S l i n e (158 keV) and the c a l c i u m 3D-2P d o u b l e t (157 keV, 158 keV). Rather than t r y i n g t o measure the oxygen l i n e i n t e n s i t y , i t was assumed t h a t the r e l a t i v e i n t e n s i t i e s of the 3P-1S t o the 4P- and 5P-1S sums were e q u a l i n s i l i c a and c a l c i u m o x i d e . The r e s u l t s were R ( S i ) = 0.17 ± 0.01 , and 4.2.5.5a R(Ca) = 0.48 ± 0.03 . 4.2.5.5b The r a t i o of a t t e n u a t i o n f a c t o r s f o r 0.784 and 0.400 MeV 116 n" IN SILICON DIOXIDE POWDER 7000 LU Z C L X C J cn LU o_ CO ZD O C J 100 200 300 400 500 600 ENERGY (0.32 KEV/CHANNEL) • F i g u r e 4.2.5.2. X-ray spectrum from n e g a t i v e muons i n s i l i c o n d i o x i d e . r a d i a t i o n i n aluminum, mumetal, and s c i n t i l l a t o r between t a r g e t and d e t e c t o r s i s 1.05. The r a t i o s of photopeak e f f i c i e n c i e s a r e e s t i m a t e d from F i g u r e 3.2.4.2 t o be 0.52 ± 0.03 (Aptec) and 0.57 ± 0.04 ( O r t e c ) . Combining t h i s i n f o r m a t i o n w i t h t h a t of e q u a t i o n 4.2.5.4 and 4.2.5.5 i n 4.2.5.3, the d e s i r e d r e s u l t s a r e : F(foils)»E(Apt) = (1.80 ± 0.26) x 1 0 " 4 , 4.2.5.6a F(foils)»E(Ort) = (2.30 ± 0.35) x 10' 4 . 4.2.5.6b A l l numbers r e q u i r e d f o r a d e t e r m i n a t i o n of the s e n s i t i v i t y of 117 u" IN CRLCIUM OXIDE POWDER X (_> CC LU LL. CO o CJ 5000 4000 h 3000 h 2000 h 1000 h 0 100 200 300 400 500 600 700 800 900 1000 1100 1200 ENERGY (0.32 KEV/CHflNNEL) F i g u r e 4.2.5.3. X-ray spectrum from n e g a t i v e muons i n c a l c i u m o x i d e . the experiment a r e now i n hand; the e v a l u a t i o n w i l l be c a r r i e d out i n the next s e c t i o n . 4.2.6. D e t e c t a b l e Events i n Terms of the C o u p l i n g Constant G Because of the d i f f e r e n c e s of the two d e t e c t o r s , two v a l u e s f o r e q u a t i o n 4.2.2 w i l l be d e t e r m i n e d , P ( A p t ) and P ( O r t ) . A p o l o g i e s a r e extended f o r the p r o l i f e r a t i o n of F's and P's i n t h i s c h a p t e r . The au t h o r has attemp t e d t o make the l a b e l l i n g unambiguous, but t h a t does not mean i t i s not c o n f u s i n g ; the 118 p l e t h o r a of p r o c e s s e s which must be pondered c a l l s f o r the i n c l u s i o n of a c o l l o s s a l c o l l e c t i o n of f r a c t i o n s and p r o b a b i l i t i e s . Numbers f o r the f a c t o r s on the r i g h t hand s i d e of e q u a t i o n 4.2.2 can be found i n the e x p r e s s i o n s 4.2.1.1, 4.2.2.1, 4.2.2.3 (a lower l i m i t ) , 4.2.3.5, 4.2.4.3, and 4.2.5.6. They imply t h a t P ( A p t ) > (1.33 ± 0.20) x l O ^ M G / G p ) 2 and 4.2.6.1a P ( O r t ) > (1.70 ± 0.27) x l O ^ M G / G p ) 2 . 4.2.6.1b Data from the two s e p a r a t e d e t e c t o r s may be combined by m o d i f y i n g e q u a t i o n 4.2.1 t o N e ( A p t ) + N e ( O r t ) = P(Apt)»N(Apt) + P(Ort)»N(Ort) . 4.2.6.2 The v a l u e of N d i f f e r s f o r the two s i n c e dead times were not e q u a l . For 2.32 x l O 1 0 i n c i d e n t muons and l i v e t imes of 0.82 and 0.92 r e s p e c t i v e l y , N(Apt) = 1.90 x 1 0 1 0 and 4.2.6.3a N ( O r t ) = 2.13 x 1 0 1 0 . 4.2.6.3b The r i g h t hand s i d e of 4.2.6.2 then g i v e s N e ( A p t ) + N,(Ort) > (0.61 ± 0.09)«(G/G F) 2, 4.2.6.4 which e x p r e s s e s the minimum number of d e t e c t a b l e e v e n t s , i n 119 terms of the c o u p l i n g c o n s t a n t G f o r muonium c o n v e r s i o n t o antimuonium. I t i s s t r a i g h t f o r w a r d t o use e q u a t i o n 4.2.6.4 i n c o n j u n c t i o n w i t h l i m i t s on H© as d e r i v e d i n s e c t i o n 4.1 to dete r m i n e an upper l i m i t f o r G. T h i s w i l l be undertaken i n the next c h a p t e r . 5. CONCLUSION 120 "Nature shows us o n l y s u r f a c e s , but she i s a thousand fathoms deep." -Ralph Waldo Emerson The f o u r t h c h a p t e r d e s c r i b e d q u a n t i t a t i v e l y the s t e p s r e q u i r e d f o r the o b s e r v a t i o n of the c a l c i u m muonic X-ray s i g n a t u r e of muonium c o n v e r s i o n t o antimuonium i n the TRIUMF ex p e r i m e n t . In t h i s c o n c l u d i n g c h a p t e r a l i m i t f o r c o n v e r s i o n w i l l be e x t r a c t e d , r e p r e s e n t i n g a c o n s i d e r a b l e improvement over p r e v i o u s l i m i t s . In a d d i t i o n , the r e s u l t s of the f o u r t h c h a p t e r w i l l be s c r u t i n i z e d i n an attempt t o e v a l u a t e p r e s e n t t e c h n i q u e s and suggest improvements t h e r e o f . 5.1. The L i m i t on the Muonium-Antimuoniurn C o u p l i n g C o n stant In s e c t i o n 4.1 i t was shown t h a t the number of X-ray e v e n t s o b s e r v e d which c o u l d have r e s u l t e d from muonium c o n v e r s i o n i s c o n s i s t e n t w i t h z e r o , and t h a t the one s t a n d a r d d e v i a t i o n l i m i t of the sum of e v e n t s from both d e t e c t o r s i s 463. I t can t h u s be s a i d t h a t 121 N e = N.(Apt) + N o ( 0 r t ) < 926 (95% C . L . ) , 5.1.1 where a 95% c o n f i d e n c e l i m i t f o r the u s u a l assumption of g a u s s i a n s t a t i s t i c s c o r r e s p o n d s v e r y c l o s e l y t o two s t a n d a r d d e v i a t i o n s . From e q u a t i o n 4.2.6.4, t h e n , The u n c e r t a i n t y of 15% i n 4.2.6.4 i s h a l v e d by t a k i n g the square r o o t ; i n c l u d i n g t h i s , the f i n a l r e s u l t of the experiment i s which, u s i n g the v a l u e of GF, i s about 6 x 1 0 " 4 " erg»cm 3. I t i s p o s s i b l e t o d e f i n e the l i m i t i n more m e a n i n g f u l e x p e r i m e n t a l terms. A muon i n a system i n i t i a l l y formed as muonium, e v o l v i n g i n a f i e l d - f r e e vacuum where c o n v e r s i o n i s most l i k e l y ( A = 0 ) , w i l l decay as antimuonium i n t o a n e g a t i v e e l e c t r o n w i t h b r a n c h i n g r a t i o R, e q u a l t o the r i g h t hand s i d e of e q u a t i o n 2.1.3.5 w i t h the v a l u e of $ d e t e r m i n e d from G by e q u a t i o n 2.1.1.7. The s t a t e d l i m i t f o r G i m p l i e s a v a l u e f o r R of l e s s than 0.04. The e x p e r i m e n t a l r e s u l t i s an improvement of more than one o r d e r of magnitude on the l i m i t G < 610G F (R < 0.47) d e t e r m i n e d by a s e a r c h f o r a f i n a l s t a t e c o n s i s t i n g of two n e g a t i v e muons from t h e c o l l i s i o n of e n e r g e t i c e l e c t r o n s ( B a r b e r e t a l . , 1969). I t a l s o r e p r e s e n t s an improvement of over two o r d e r s of magnitude on the l i m i t G < 5800G F (R < 0.50) s e t u s i n g an argon (G/GF) 2 < 1.51 x 1 0 3 , or G/GF < 38.8 . 5.1.2 5.1.3 G < 42 G F (95% C.L.) 5.1.4 122 gas t a r g e t a t one atmosphere (Amato e_t a l . , 1968); t h a t e xperiment i s the o n l y o t h e r one which measures G by s e a r c h i n g f o r muonium c o n v e r s i o n . I t was mentioned i n s e c t i o n 4.1 t h a t i n p r e p a r a t i o n f o r the TRIUMF experiment an argon gas t a r g e t was i n s t a l l e d f o r e a r l y c hecks of the a p p a r a t u s and d e t e c t i o n systems. The p r o c e d u r e and a n a l y s i s were s i m i l a r t o the o l d muonium-antimuonium e f f o r t , u t i l i z i n g gas a t one atmosphere and room t e m p e r a t u r e , and s e a r c h i n g f o r an argon muonic 2P-1S X-ray a t 644 keV. In the o l d e x p e r i m e n t , a muonium f o r m a t i o n p r o b a b i l i t y of u n i t y was assumed, whereas a v a l u e of 0.63±0.07 ( M i k u l a e t a l . , 1979) i s used h e r e . An upper l i m i t of 1066 c o u n t s (95% c o n f i d e n c e l i m i t ) i s o b t a i n e d f o r a p o s s i b l e argon muonic photpeak i n the d a t a of F i g u r e 4.1.3, which i m p l i e s a l i m i t of G < 190 G F (95% C . L . ) , 5.1.5 or R < 0.32, independent of the s i l i c a r e s u l t . In the sense t h a t the p r e s e n t powder t a r g e t experiment a l l o w s a s u b s t a n t i a l improvement i n the l i m i t of the c o u p l i n g c o n s t a n t , i t was undoubtedly s u c c e s s f u l . To have impact on p r e s e n t q u e s t i o n s about the r e a l n a t u r e of l e p t o n i c e l e c t r o w e a k i n t e r a c t i o n s w i l l r e q u i r e an improvement of a f u r t h e r two ( a t l e a s t ) o r d e r s of magnitude, which may or may not be w i t h i n the realm of p r e s e n t t e c h n o l o g y . In the o p i n i o n of the a u t h o r , the l i m i t must be reduced t o l e s s than 0.01 G F b e f o r e i t can be s a i d t h a t muonium c o n v e r s i o n can p r o v i d e an e f f e c t i v e t e s t of some u n i f i e d t h e o r i e s . 123 5.2. F e a s i b i l i t y of an Improved Experiment C e r t a i n l y some f a c t o r s d e t e r m i n i n g the e x p e r i m e n t a l s e n s i t i v i t y can be improved upon. I t i s the i n t e n t of t h i s f i n a l s e c t i o n t o p o i n t t o the most p r o m i s i n g methods f o r f u r t h e r r e d u c t i o n of the l i m i t on G, w i t h i n the framework of known p r o c e d u r e s and a v a i l a b l e equipment. When a s s e s s i n g the v a l u e of v a r i o u s improvements, i t i s h e l p f u l t o u n d e r s t a n d the dependence of the l i m i t on e x p e r i m e n t a l p a r a m e t e r s . In s e c t i o n 4.2.3 i t was p o i n t e d out t h a t i n c r e a s i n g the amount of d a t a , t h a t i s , the number of i n c i d e n t muons, l o w e r s the l i m i t o n l y as the f o u r t h r o o t of the i n c r e a s e ; the same i s t r u e f o r any e x p e r i m e n t a l change which i n c r e a s e s both the s i g n a l p r o b a b i l i t y P and the background r a t e by s i m i l a r f a c t o r s . The use of more and/or h i g h e r e f f i c i e n c y germanium d e t e c t o r s ( t o i n c r e a s e E) w i t h o u t more e x t e n s i v e background s u p p r e s s i o n f a l l s i n t o the same c a t e g o r y . An e x p e r i m e n t a l change which i n c r e a s e s P by some f a c t o r w i t h o u t c h a n g i n g the background r a t e w i l l reduce the l i m i t on G by the square r o o t of the f a c t o r . T a k i n g o p t i m i s t i c improvements of f a c t o r s of two i n F ( f o i l s ) , one and a h a l f i n F(Mu), t e n i n F ( g a p ) , and t h r e e i n P(Mu), a r e d u c t i o n i n the l i m i t of G by a f a c t o r of t e n might be p o s s i b l e . T h i s i s not s u f f i c i e n t i n i t s e l f t o a l l o w a t e s t of t h e o r y . I f the experiment i s not r e s t r i c t e d t o d e t e c t i o n of muonic X - r a y s , but uses i n s t e a d f a s t e l e c t r o n s from the h i g h energy 124 p a r t of the M i c h e l spectrum f o r n e g a t i v e muon decays t o s i g n a l a c o n v e r s i o n e v e n t , s e v e r a l t h i n g s change: 1. P(Mu) can i n c r e a s e one o r d e r of magnitude to 2.5 x 10" 5 ( G / G F ) 2 as shown by e q u a t i o n 2/1.3.5. 2. P ( c a p t ) does not e n t e r i n t o e q u a t i o n 4.2.2 f o r P. 3. The d e t e c t i o n e f f i c i e n c y E c o u l d be i n c r e a s e d s u b s t a n t i a l l y by l a r g e s o l i d a n g l e d e t e c t i o n of the c u r v a t u r e of e l e c t r o n s i n a magnetic f i e l d . I t i s d i f f i c u l t a t t h i s j u n c t u r e t o e s t i m a t e background l e v e l s which would be e n c o u n t e r e d , but an i n c r e a s e i n E by two or more o r d e r s of magnitude i s not beyond re a s o n . I t i s p r o b a b l e t h a t , w i t h the magnetic f i e l d r e q u i r e d , P(Mu) would d e c r e a s e by one h a l f i n l i g h t of the e x p l a n a t i o n s of s e c t i o n 2.1.4, but t h a t s h o u l d be more than o f f s e t by the g a i n i n d e t e c t i o n e f f i c i e n c y . T a k i n g o n l y t h e s e g a i n s i n t o a c c o u n t , P might be improved by a f a c t o r of 1 0 3 , and G might be l i m i t e d t o a v a l u e l e s s than G . However, i t i s o n l y s p e c u l a t i o n , and a g r e a t e r i n c r e a s e i n s e n s i t i v i t y c o u l d be the p r o d u c t of c l e v e r e x p e r i m e n t a t i o n . The t e c h n i q u e c r u c i a l t o the s u c c e s s of the p r e s e n t e x p e r i m e n t , t h a t of u s i n g t h i n l a y e r s of f i n e powder t o produce a u s e f u l f r a c t i o n of muonium i n vacuum, i s a t l e a s t e n c o u r a g i n g . There i s ample room f o r development and r e f i n e m e n t of the i d e a . Most i m p o r t a n t l y , f o r c o n v e r s i o n e x p e r i m e n t s and a l s o f o r some i n v e s t i g a t i o n s of the atomic s t r u c t u r e of muonium i n vacuum, a r e l i a b l e measurement of the p r o b a b i l i t y of muonium e s c a p i n g the s i l i c a l a y e r p l u s an a c c u r a t e e s t i m a t e of the r a t e a t which i t i s e x p e l l e d a r e of h i g h p r i o r i t y . I f s u f f i c i e n t s e n s i t i v i t y can be r e a l i z e d by any means 125 whatsoever, be they s i m i l a r t o t h o s e of t h i s experiment or r a d i c a l l y d i f f e r e n t , the c o n v e r s i o n of muonium t o antimuonium c o u l d p r o v i d e a c l u e t o the n a t u r e of the fundamental i n t e r a c t i o n s . 126 A l . MUONIUM-ANTIMUONI UM CONVERSION VIA THE FOUR FERMION CURRENT-CURRENT INTERACTION In t h i s appendix a v a l u e f o r the muonium-antimuonium c o n v e r s i o n m a t r i x element w i l l be c a l c u l a t e d assuming a v e c t o r minus a x i a l v e c t o r (V-A) c u r r e n t c o n t a i n i n g the muon (Y^u.) and e l e c t r o n ( f e ) f i e l d s . T h i s l e p t o n f l a v o r c hanging n e u t r a l c u r r e n t , when c o u p l e d t o i t s e l f , v i o l a t e s an a d d i t i v e muon number but p r e s e r v e s a m u l t i p l i c a t i v e one (as d e f i n e d i n the f i r s t c h a p t e r ) . The H a m i l t o n i a n d e n s i t y t a k e s the form ( F e i n b e r g and Weinberg, 1961b) H(x) = 2-< 1 / J> G « [ ^ (x)C? Te ( x ^ U J O ^ t e U ) + H.C. ] , A l . l where G i s the e f f e c t i v e c o u p l i n g c o n s t a n t , 0* = y ^ ( l - Xs) i s the V-A form f o r the c o u p l i n g , and H.C. stan d s f o r the H e r m i t i a n c o n j u g a t e . G i s t o be compared t o the Fermi c o u p l i n g c o n s t a n t G P = 1.4 x 1 0 - 4 ' erg*cm 3 (1.03 x 1 0' s nip 2, mp b e i n g the p r o t o n mass). 0^ and the f i e l d o p e r a t o r s fix) a r e d e f i n e d t o conform w i t h t h e c o n v e n t i o n s and n o r m a l i z a t i o n s of B j o r k e n and D r e l l (1964) (except t h a t "R = c = 1 ) , and the r e p r e s e n t a t i o n of the D i r a c m a t r i c e s which w i l l be used e x p l i c i t l y i s : 127 I 0 0 - I { X1} - ?- 0 <r\ I- O 1 0 0 1 1 0 0 - i 1 0 <r3 = y 5 = i j e ^ y y = y 5 = A1.2 Note a l s o that a»b = a^ly = a ° b 0 - a»S . The f i e l d ope ra to r s s a t i s f y i n g the D i r a c equa t ion are ( fo r JL = / A o r e) (2-ir)"« 3 / 2 > £ j V p (rn^/E)' 1 / J » [ b|_ ( p , r ) u^ (p, r ) exp(-ip»x) + cy ( p , r ) v (p,r)exp(+ip»x)] A l . 3a ^ ( x ) = Y+(x)y° = ( 2 T T ) - (  3/2)frf d 3 p ( m ^ E ) ' 1 ' 2 ' [b£(p, r ) u ^ ( p , r )exp( + ip»x) + dJL(p,r)vjL(p,r)exp(-ip»x) ] . A l . 3b Here, b^(b£) and d^(d£) a r e the u s u a l anticonunuting f e r m i o n and a n t i f e r m i o n a n n i h i l a t i o n ( c r e a t i o n ) o p e r a t o r s , r e s p e c t i v e l y . The D i r a c s p i n o r s a r e g i v e n by ( s e e , f o r example, Commins, 1973; note t h a t h i s n o r m a l i z a t i o n d i f f e r s ) 128 u £ ( p , s ) = {(E + m A)/2m A}< 1 " > A1.4a cr- p (p,s) = { (E + m^  )/2m1} < 1 7 2 ) where ^ i s a two component P a u l i s p i n o r s a t i s f y i n g A1.4b cr»§^ = *i and -^y = 1/2 (I +(r»§) . A1.5 Here £ i s the s p i n p o l a r i z a t i o n u n i t v e c t o r i n the r e s t frame of the p a r t i c l e . For £ = ( 0 , 0 , 1 ) , ^) ~ f f o r u(p,s) and ^ f o r v ( p , s ) , where ^ and a r e the b a s i s v e c t o r s A1.6 For % - ( 0 , 0 , - 1 ) , the r o l e s of fi and K a r e r e v e r s e d . Making the n o n - r e l a t i v i s t i c a p p r o x i m a t i o n , the s m a l l components of the D i r a c s p i n o r s can be i g n o r e d and E approaches m^ i n the n o r m a l i z a t i o n c o e f f i c i e n t s . The g o a l i s t o e v a l u a t e the energy S , d e f i n e d by the e x p r e s s i o n 6/2 = <Mu(p'+q',r',s') | H(x) |Mu(p+q,r,s)> , A1.7 129 where |Mu(p+q,r,s)> and |Mu(p'+q*, r', s') > a r e p a r t i c u l a r s p i n s t a t e s of the muon and e l e c t r o n i n a IS atomic s t a t e , w i t h p (p*) and q Cq) the ju* (jut) and e" ( e + ) momenta r e s p e c t i v e l y . The muonium and antimuonium s t a t e s a r e c o n s t r u c t e d i n o c c u p a t i o n number space by w e i g h t i n g the o c c u p a t i o n s t a t e s w i t h the IS momentum wave f u n c t i o n . For example, |Mu(K,r,s)> = j d 3 k f ( k ) d£( (m^/MjK + k,r) •b^((m e/M)K - k*,s) |0> , A1.8 where K = p + q , k = (m^p - nyqJ/M , and M = rn^M. + m e . A l . 9 C o n s t r u c t e d i n t h i s way, the s t a t e s a r e n o r m a l i z e d t o d e l t a f u n c t i o n s i n K, r , and s; the f (k) a r e momentum space wave f u n c t i o n s and s h o u l d not be c o n f u s e d w i t h the f i e l d o p e r a t o r s i n H ( x ) . E v a l u a t i n g e q u a t i o n A1.7, u s i n g A l . l w i t h e x p a n s i o n s such as A l . 3 and A1.8, and a p p l y i n g the a n t i c o m m u t a t i o n r e l a t i o n s f o r the c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s , i t can be shown t h a t S/2 = ( 2 i 7 - ) - ' y y r d 3 k 6>k'Y*{1Z')f (k) e x p ( - i ( K - K ' )»x) • 2 ( 1 / z ) G • B ( r , s, r s O , A L I O where B<r rs,r'fiO = u^(p',r ')6*v e(cj',s')^ t (p, r XD^  u £ (q, s) 130 - "u^(p',r')0 Qu e ( q , s ) v ^ ( p , r ) 0 ^ v e (q',s') . B can be e v a l u a t e d u s i n g the e x p l i c i t r e p r e s e n t a t i o n , by i n s e r t i n g v a l u e s f o r r , s, r ' , and s'. The r e s u l t s can be s u c c i n t l y w r i t t e n as B(r,s,r',s') = 4 ( - l ) , , r ^ ' / ! L r . S S f ) . A l . l l Then, a t t = 0 i n the r e s t frame of muonium (K = 0 ) , S/2 = 2"' 1 / 2 ' G » 8 ( - l ) " r + S ) / 2 g £ ') • •*> s (2T)-ff d 3k d ^ T ^ k O f (k) A1.12 a f t e r i n t e g r a t i o n over the t o t a l f i n a l momentum K. Expanding the f i n a l momentum space wave f u n c t i o n s i n terms of c o o r d i n a t e space wave f u n c t i o n s V (x) ( a g a i n , not t o be c o n f u s e d w i t h the f i e l d o p e r a t o r s ) Ytil) = <t \y> = y " d 3 x <k | x><x i y > = (2T^- ( 3. / 2 ) J~ d 3 x exp(-ik»x) f (t) , A1.13 and the e x p r e s s i o n f o r & becomes g/2 = 2 - < 1 / ^ G » 8 ( - 1 ) ^ + S ) " 6^- S"5)S-_yjd3x d 3x ' Y*(x^ j f ^ x j S f x ' ) ^ * ) , or S/2 = 2 - ( 1 " > G»8(-1) ( f " + $ ) /2S"r,r«Ss>t» •(ira03)-1 A1.14 f o r t he IS wave f u n c t i o n , where a0 i s the Bohr r a d i u s . The d e l t a f u n c t i o n s denote the s p i n s e l e c t i o n r u l e s f o r t r a n s i t i o n s 131 between p a r t i c u l a r i n i t i a l and f i n a l s p i n s t a t e s . A l l q u a n t i t i e s of i n t e r e s t c o n t a i n £ 2 , p o s s i b l y averaged over i n i t i a l v a l u e s of r and s, so the s i g n f a c t o r can be d i s r e g a r d e d . I n s e r t i n g n u m e r i c a l v a l u e s , £ = 2.1 x 10" 1 2 (G/Gp) eV. A l .15 132 A2. MOTION OF MUONIUM ATOMS IN SPHERICAL SILICA PARTICLES In t h i s appendix e x p r e s s i o n s w i l l be d e r i v e d t o p r e d i c t the r a t e a t which an ensemble of muonium atoms, formed i n f i n e s i l i c a powder, can escape the powder p a r t i c l e s and move i n an i n t e r s t i t i a l vacuum. The s t a r t i n g p o i n t f o r the f o l l o w i n g c a l c u l a t i o n s i s an e x p r e s s i o n f o r the r a t e a t which p a r t i c l e s appear a t an a b s o r b i n g i n t e r f a c e per u n i t a r e a , per u n i t time (Chandrasekhar, 1943), R ( t ) = -D(f) » V W ) , A2.1 where ft i s a u n i t v e c t o r , normal t o the s u r f a c e and W(?)dr = U ^ D O ' ' 3 / 2 , e x p ( - | r | 2 / 4 D t ) d f A2.2 i s the p r o b a b i l i t y per u n i t a r e a of f i n d i n g a p a r t i c l e between r and f + d ? a f t e r movement by random walk from r = 0 a t t = 0. The boundary c o n d i t i o n W = 0 a p p l i e s t o e q u a t i o n A2.1 a t an a b s o r b i n g i n t e r f a c e . The d i f f u s i o n c o e f f i c i e n t i s d e f i n e d here as D = n<y 2>/6 A2.3a f o r a p a r t i c l e undergoing n independent d i s p l a c e m e n t s of mean square d i s t a n c e <y 2> per u n i t t i m e . An e q u i v a l e n t d e f i n i t i o n i n 133 terms of the mean speed c and the mean f r e e p a t h s i s D = cs/3 . A2.3b The words " a b s o r b i n g i n t e r f a c e " i n d i c a t e t h a t a p a r t i c l e r e a c h i n g t h a t s u r f a c e w i l l pass t h r o u g h i t w i t h z e r o p r o b a b i l i t y of r e - e n t r y , p r e s u p p o s i n g a mechanism l i k e a work f u n c t i o n i n h i b i t i n g the p o s s i b i l i t y . C o n s i d e r a sphere of r a d i u s a, whose s u r f a c e i s the a b s o r b i n g i n t e r f a c e , and f i n d R ( t ; x ) f o r a p a r t i c l e s t a r t i n g i t s random motion a t x a t t = 0 (see F i g u r e A2.1). Then W = 0 when F i g u r e A2.1. Geometry f o r muonium e m i s s i o n from a sphere. r = a-x, and 134 R ( t ; x ) = (7TDt)-( 3 / 2 >/16t d s ( a ) n(a)»(a - x) •expt-|3 - x| 2/4Dt) . A2.4 The s u r f a c e i n t e g r a t i o n can be performed by n o t i n g t h a t d s ( a ) = a 2 sin© d© d ^ , |a - x | 2 = a 2 + x 2 - 2ax cos0 , and A2.5 n (a) • (3 - x) = | a - x | cos ot = a - x cos© . C a r r y i n g out the s u r f a c e i n t e g r a t i o n y i e l d s R ( t ; x ) = (17-/2 ) a 2D(r rDt) - ' 3 / 2 » exp( - (a 2+x 2 )/4Dt) • [ x - 2 ( l + 2 D t / a 2 ) s i n h ( a x / 2 D t ) - a " x c o s h ( a x / 2 D t ) ] . A2.6 Now assume t h a t the p a r t i c l e ensemble i s i n i t i a l l y d i s t r i b u t e d u n i f o r m l y throughout the sphere w i t h a c o n s t a n t d e n s i t y of 1 / V = 3 / 4 T B 3 , i n o r d e r t o p r e s e r v e the p r o b a b i l i s t i c n o r m a l i z a t i o n of e q u a t i o n A2.2. Then, i n t e g r a t i n g over the volume of the sphere, R ( t ) = 3 a ' 3 / dx x 2 R ( t ; x ) , A2.7 which i s the r a t e a t which p a r t i c l e s appear a t the s u r f a c e of the s phere, g i v e n a u n i f o r m d i s t r i b u t i o n a t t = 0. T h i s may be e v a l u a t e d u s i n g the e x p o n e n t i a l form f o r the h y p e r b o l i c f u n c t i o n s , a l t h o u g h the pr o c e d u r e becomes q u i t e t e d i o u s . The s u b s t i t u t i o n z = -(x±a)/2(Dt) ( 1 / 2> i n i n t e g r a t i o n s i n v o l v i n g exp(±ax/2Dt), a l o n g w i t h the knowledge t h a t 135 dx x n e x p ( - x 2 ) = (-ifj dx x " e x p ( - x 2 ) , l e a d s t o an i n t e g r a n d i n A2.7 p r o p o r t i o n a l t o e x p ( - z 2 ) times terms c o n s t a n t , l i n e a r , and q u a d r a t i c i n z. F o r t u n a t e l y , the c o e f f i c i e n t of the c o n s t a n t term v a n i s h e s f o r a l l t . The q u a d r a t i c term may be i n t e g r a t e d by p a r t s , g i v i n g a f u r t h e r l i n e a r term i n the i n t e g r a n d p l u s a c o n t r i b u t i o n p r o p o r t i o n a l t o t ( 1 ' 2 >ex p ( - a 2/ D t ) . The l i n e a r term can then be e v a l u a t e d , and the r e s u l t o b t a i n e d i s R ( t ) = ( T ) - ( l / 2 ) ( 3 / 2 t ) ( D t / a 2 ) < 1 / 2 > [1 - 2 ( D t / a 2 ) + ( l + 2 ( D t / a 2 ) ) e x p ( - a 2 / D t ) ] . A2.8 As time approaches z e r o , R ( t ) grows as f 1 / 2 , which i s a r e s u l t of the non-zero p a r t i c l e d e n s i t y a t the s u r f a c e of the sphere a t t = 0 and the assumption of a p e r f e c t l y a b s o r b i n g i n t e r f a c e . As t grows w i t h o u t l i m i t , R ( t ) behaves as t ~ 5 / 2 . I t i s now p o s s i b l e t o c a l c u l a t e the p r o b a b i l i t y t h a t a p a r t i c l e has passed through the s u r f a c e by time t (assuming t h a t i t w i l l not d e c a y ) . E v a l u a t i n g P ( t ) = / dt* R ( t ' ) A2.9 g i v e s the d e s i r e d r e s u l t : P ( t ) = 1 - e r f [ ( D t / a 2 ) - ' 1 / 2 ' ] - (17-)"( 1 / 2 ' ( D t / a 2 ) ' 1 / 2 » • •[3 - 2 D t / a 2 - ( l - 2 D t / a 2 ) e x p ( - ( D t / a 2 ) " 1 ) ] . A2.10 136 The e r r o r f u n c t i o n e r f ( x ) has the u s u a l d e f i n i t i o n : e r f (x) = 2 ( T ^ ) - « 1 / 2 > o dt e x p ( - t 2 ) A 2 . l l Another u s e f u l r e s u l t i n c l u d e s the decay of a p a r t i c l e of mean l i f e t i m e "X'1, so t h a t the p r o b a b i l i t y of i t p a s s i n g t h r ough the s u r f a c e b e f o r e d e c a y i n g i s (see a l s o Brandt and P a u l i n , 1968; note t h a t t h e i r f o r m u l a c o n t a i n s a t y p o g r a p h i c a l e r r o r ) Formula A2.10 has been used i n the a n a l y s i s of a muonium s p i n r o t a t i o n experiment i n a s i l i c a powder sample ( M a r s h a l l e t  a l . , 1978). The s i l i c a p a r t i c l e s , whose r a d i i c o u l d be i n d e p e n d e n t l y i n f e r r e d , were surrounded w i t h oxygen a t v a r i o u s c o n c e n t r a t i o n s . The muonium p o l a r i z a t i o n was r e l a x e d by s p i n exchange a t a r a t e which depended on the c o n c e n t r a t i o n ( F l e m i n g et aJU, 1980). When t h i s r a t e was h i g h compared t o the r a t e a t which muonium appeared a t the s u r f a c e of the s i l i c a p a r t i c l e s , the time dependence of the muonium p o l a r i z a t i o n was assumed t o have the form P = A2.12 where p = (D/>a J) A ( t ) = A(0) (1 - P ( t ) ) A2.13 In o t h e r words, the r e l a x a t i o n p r o c e s s was c o n t r o l l e d by the r a t e a t which the muonium atoms c o u l d get t o the v o i d s between 137 p a r t i c l e s , where the p o l a r i z a t i o n was then d e s t r o y e d i n a c o m p a r a t i v e l y s h o r t t i m e . That the d a t a s u p p o r t e d the model proved the l i k e l i h o o d t h a t muonium was r e a c h i n g the v o i d s between s i l i c a p a r t i c l e s b e f o r e decay. A f u r t h e r r e s u l t of the experiment was the c l o s e s i m i l a r i t y of the dependence of the r e l a x a t i o n r a t e on the oxygen c o n c e n t r a t i o n , g i v e n by the r a t e c o n s t a n t k, t o t h a t i n a t a r g e t c o n s i s t i n g of oxygen i n an argon gas moderator ( w i t h no s i l i c a p r e s e n t ) . In the low c o n c e n t r a t i o n regime, where the r e l a x a t i o n r a t e was determined by the oxygen c o n c e n t r a t i o n r a t h e r than the motion of muonium i n s i d e the s i l i c a s p h e r e s , a r a t e c o n s t a n t k = (2.55 ± 0.13) x 1 0 ~ 1 0 cm 3»mol~ l»s" 1 was measured, i n agreement w i t h k = (2.52 ± 0.18) x 1 0 - 1 0 cm 3»mol-^s" 1 from an argon-oxygen t a r g e t a t one atmosphere. For s p i n exchange, the r a t e c o n s t a n t can be w r i t t e n as k = cr^v>, where o~ i s a g e o m e t r i c a l c r o s s s e c t i o n and <v> i s the mean r e l a t i v e speed. S i n c e the mean t h e r m a l v e l o c i t y of muonium i s 16.8 ti m e s t h a t of oxygen, t h i s e q u a l i t y of r a t e c o n s t a n t s s t r o n g l y s u p p o r t s the h y p o t h e s i s t h a t muonium i s moving t h e r m a l l y w i t h i n the v o i d s , r a t h e r than a t t a c h i n g i t s e l f a t or near the powder s u r f a c e and d e p o l a r i z i n g t h e r e . The d i f f u s i o n parameter o b t a i n e d f o r muonium i n the s i l i c a powder p a r t i c l e s , assuming the v a l i d i t y of the f o r e g o i n g model, was D = (2.2 ± 0.4) x 1 0 - ' c m ^ S " 1 . A2.14 I t must be s t r e s s e d t h a t t h i s may not be a p p l i c a b l e t o s i l i c a i n g e n e r a l , nor even t o powders w i t h d i f f e r i n g p a r t i c l e s i z e s . 138 There a r e s e v e r a l p o i n t s which must be c o n s i d e r e d : 1. There may e x i s t f i s s u r e s i n the p a r t i c l e s through which muonium c o u l d escape more q u i c k l y than by d i f f u s i o n . 2. The r a d i u s assumed i n the c a l c u l a t i o n s i s d e r i v e d from measurements of the s p e c i f i c s u r f a c e a r e a , and may not be u n i f o r m . 3. The temperature of the g r a i n i n which muonium i s moving may have been s h a r p l y e l e v a t e d by the energy of the i n c i d e n t muon d e p o s i t e d t h e r e d u r i n g the s l o w i n g p r o c e s s ( R i e f l e t a l . , 1981). Because of the model dependent n a t u r e of the a n a l y s i s , the v a l u e e x t r a c t e d f o r the d i f f u s i o n parameter D, which i s c e r t a i n l y a f u n c t i o n of t e m p e r a t u r e , would l i k e l y depend on the s i z e of the g r a i n . S canning e l e c t r o n m i c r o g r a p h s do show the approximate s p h e r i c i t y of the p a r t i c l e s , and a r e a s o n a b l y u n i f o r m s i z e d i s t r i b u t i o n c o n s i s t e n t w i t h the r a d i u s assumed (Cabot C o r p o r a t i o n , u n p u b l i s h e d t e c h n i c a l r e p o r t ) . The m i c r o g r a p h s a l s o show the tendency of the p a r t i c l e s t o form c h a i n - l i k e a g g r e g a t e s , which m e c h a n i c a l l y e n t a n g l e t o form a g g l o m e r a t e s . W h i l e t h i s has l i t t l e consequence f o r the motion of muonium w i t h i n the p a r t i c l e s , i t c o u l d p l a y a p a r t i n the t h e r m a l motion a f t e r muonium reaches the v o i d s , t h e s u b j e c t of appendix A3. 139 A3. MOTION OF MUONIUM ATOMS IN FINE POWDER LAYERS The purpose of t h i s appendix i s t o e s t i m a t e the r a t e at which an ensemble of t h e r m a l muonium atoms may escape a t h i n powder l a y e r . The proced u r e i s e s s e n t i a l l y i d e n t i c a l t o t h a t of the p r e v i o u s appendix, w i t h o n l y the geometry and the parameter D r e q u i r i n g m o d i f i c a t i o n . C o n s i d e r a homogenous l a y e r of t h i c k n e s s d, bounded on both s i d e s by an a b s o r b i n g s u r f a c e , as i n F i g u r e A3.1. E q u a t i o n s F i g u r e A3.1. Geometry f o r muonium e m i s s i o n from a l a y e r . A2.1 and A2.2 a p p l y , and the e x p r e s s i o n f o r R ( t ; x ) becomes, 140 a f t e r i n t e g r a t i o n over the s u r f a c e s (assumed i n f i n i t e i n a r e a ) , R ( t ; x ) = ( 2 t ) " 1 (4T7-Dt)"( 1 / 2 ' [x«exp(-x 2/4Dt) + (d-x)»exp(-(d-x) 2/4Dt)] . A3.1 For an i n c i d e n t p a r t i c l e d i s t r i b u t i o n of 1/d per u n i t t h i c k n e s s a t t = 0, the r a t e of e m i s s i o n can be i n t e g r a t e d over x t o g i v e the e f f e c t i v e r a t e from the e n t i r e volume: R ( t ) = J dx d " 1 R ( t ; x ) = (4-irDt)"' 1 / 2 > d' xD [1 - e x p ( - d 2 / 4 D t ) ] . A3.2 The chosen n o r m a l i z a t i o n p e r m i t s the c a l c u l a t i o n of the p r o b a b i l i t y t h a t a p a r t i c l e , d e c a y i n g a t a r a t e of *X s _ 1 , w i l l r e a c h one p a r t i c u l a r s u r f a c e of the l a y e r : P = ( 1 / 2 ) y ' d t exp ( - ' X t)R ( t ) = (l/2 )ad 2/D)-< 1 / 2 ) [1 - exp ( - a d 2/D)\/ 2)] , A3.3 where the f a c t o r of 1/2 comes from the n e g l e c t of the second s u r f a c e . In o r d e r t o e s t i m a t e t h i s v a l u e n u m e r i c a l l y , some r e a s o n a b l e assumptions must be made f o r the v a l u e of D. U s i n g the e x p r e s s i o n A2.3b, and the f o r m u l a ( K i t t e l , 1969) f o r the mean f r e e p a t h s of a p o i n t p a r t i c l e (a muonium atom) moving f r e e l y i n a u n i f o r m random c o l l e c t i o n of n c m - 2 s t a t i o n a r y s p heres ( s i l i c a p a r t i c l e s ) of r a d i u s r , s = (TTr'n)- 1 , A3.4 141 i t i s found t h a t D = c ( 3 f ! n ) - J . A3. 5 The average number d e n s i t y n can be w r i t t e n i n terms of the d e n s i t y of s i l i c o n d i o x i d e , ^ , and the d e n s i t y of the s i l i c a powder used i n the l a y e r , j>' , as n = 3f'/brr* y , A3. 6 so t h a t D = ( 4 c r / 9 ) (f/j>' ) . A3.7 The mean t h e r m a l v e l o c i t y c i s 7.37 x 1 0 s c m « s _ 1 f o r muonium a t room t e m p e r a t u r e . The d e n s i t y of s i l i c a powder used i n the experiment was p' =0.032 g»cm" 3, whereas f o r b u l k s i l i c a i t i s p = 2.2 g»cm" 3. A p a r t i c l e r a d i u s of 3.5 x 1 0 " 7 cm then l e a d s t o an e s t i m a t e of D = 7.9 cm 2»s" 1 , A3.8 which i s more than seven o r d e r s of magnitude g r e a t e r than the c o n s t a n t of A2.14 f o r muonium i n s i d e a s i l i c a p a r t i c l e . The a g g l o m e r a t i o n of c h a i n - l i k e a g g r e g a t e s of the powder spheres make the assumption of u n i f o r m i t y of the l a y e r a l i t t l e t e nuous. Without t h a t a s s u m p t i o n , a c a l c u l a t i o n u s i n g the approach which has been taken becomes much more f o r m i d a b l e , and at some p o i n t must r e l y on a d e t a i l e d knowledge of the powder 142 s t r u c t u r e . I t i s i m p o r t a n t t o r e a l i z e , though, t h a t u n l e s s the a g g l o m e r a t i o n were t o r e s u l t i n v e r y dense r e g i o n s of the l a y e r e x t e n d i n g f o r d i s t a n c e s comparable i n magnitude t o i t s t h i c k n e s s , the e f f e c t i s t o i n c r e a s e the r a t e a t which muonium i s e m i t t e d . The reason f o r t h i s statement i s t h a t the p r o b a b i l i t y f o r escape from a s m a l l agglomerate (of t y p i c a l d i m e n s i o n << ( D / ^ ) ( 1 . / 2 ) w i t h D the a p p r o p r i a t e d i f f u s i o n parameter) i s c l o s e t o one; a f t e r l e a v i n g t h a t a g glomerate, the mean square d i s t a n c e between c o l l i s i o n s i s much l o n g e r than i n a u n i f o r m l a y e r , and the p r o b a b i l i t y of r e a c h i n g the s u r f a c e i s c o r r e s p o n d i n g l y h i g h e r . In t h a t c a s e , the numbers d e r i v e d from the methods used here w i l l be v a l i d as a lower l i m i t , and as such a r e good enough f o r the purposes of the f o u r t h c h a p t e r . The e x p r e s s i o n s a r e used i n s e c t i o n 4.2.2 t o f i n d the p r o b a b i l i t y of escape of a muonium atom from one s u r f a c e of a s i l i c a powder l a y e r . 143 BIBLIOGRAPHY Amato, J . J . , P. Crane, V.W. Hughes, J.E. R o t h b e r g , and P.A. Thompson, 1968, Phys. Rev. L e t t . 21, 1709. A r n o l d , K.-P., P.O. Egan, M. G l a d i s c h , W. J a c o b s , H. O r t h , J . V e t t e r , and P. Zimmerman, 1979b, i n SIN N e w s l e t t e r No. 12, ed. G.H. E a t o n , 4. A r n o l d , K.P., P.O. Egan, M. G l a d i s c h , D. H e r l a c h , V.W. Hughes, W. J a c o b s , H. Metz, H. O r t h , G. zu P u l i t z , J . V e t t e r , W. Wahl, and M. Wigand, 1979a, i n SIN N e w s l e t t e r No. 11, ed. G.H. Eaton, 48. B a i l i n , D. , 1977, Weak I n t e r a c t i o n s , Sussex U n i v e r s i t y P r e s s . B a r b e r , W.C, B. G i t t e l m a n , D.C. Cheng, and G.K. O ' N e i l l , 1969, Phys. Rev. L e t t . 22, 902. B a r i s h , S .J., Y. Cho, M. D e r r i c k , L.G. Hyman, J . R e s t , P. S c h r e i n e r , R. S i n g e r , R.P. Smith, H. Y u t a , D. Koetke, V.E. Barues, D.D. Carmony, and A.F. G a r f i n k e l , 1974, Phys. Rev. L e t t . 33, 448. B a r n e t t , B.A., C.Y. Chang, G.B. Yodh, J.B. C a r r o l l , M. Eckhause, C.S. H s i e h , J.R. Kane, and C.B. Spence, 1975, Phys. Rev. A 11, 39. B a r n e t t , B.A., C.Y. Chang, P. S t e i n b e r g , G.B. Yodh, H.D. O r r , J.B. C a r r o l l , M. Eckhause, J.R. Kane, C.B. Spence, and C.S. H s i e h , 1977, Phys. Rev. A 15, 2246. Beer, W., P.R. B o l t o n , P.O. Egan,'V.W. Hughes, D.C. Lu, F.G. Mariam, P.A. Souder, J . V e t t e r , M. G l a d i s c h , G. zu P u l i t z , U. Moser, L . J . T e i g , R.H. Holmes, P.H. S t e i n b e r g , J.R. Kane, and R. Hartmann, 1979, a b s t r a c t s u b m i t t e d t o 8 t h I n t . Conf. on H i g h Energy P h y s i c s and N u c l e a r S t r u c t u r e , Vancouver, Canada. B e r n s t e i n , J . , 1974, Rev. Mod. Phys. 46, 7. B e r t i n , A., F. F e r r a r i , I . Massa, M. P i c c i n i n i , G. V a n n i n i , and A. V i t a l e , 1978, Phys. L e t t . 68A, 201. B j o r k e n , J.D., and S. Weinberg, 1977, Phys. Rev. L e t t . 38, 622. B j o r k e n , J.D., and S.D. D r e l l , 1964, R e l a t i v i s t i c Quantum  Me c h a n i c s , M c g r a w - H i l l . B o u c h i a t , C , 1977, P r o c . 7 t h I n t . Conf. on High Energy P h y s i c s and N u c l e a r S t r u c t u r e , E x p e r . S u p p l . 31, 353. Bowen, T., K.R. K e n d a l l , K .J. N i e l d , and A.E. P i f e r , u n p u b l i s h e d , U n i v e r s i t y of A r i z o n a I n t e r n a l R e p o r t . 144 B r a n d t , W., and R. P a u l i n , 1968, Phys. Rev. L e t t . 21, 193. Brewer, J.H., K.M. Crowe, F.N. Gygax, and A. Schenck, 1975, Muon P h y s i c s , V o l . I l l , eds. C S . Wu and V.W. Hughes, Academic P r e s s , New York, Chapter 7. B u n t i n g , R.L., and J . J K r a u s h a a r , 1974, N u c l . I n s t r . and Meth. 118, 565. Cabibbo, N., and R. G a t t o , 1960, Phys. Rev. L e t t . 5, 114. Cabot C o r p o r a t i o n , t e c h n i c a l r e p o r t , Cab-Q-Si1 P r o p e r t i e s and  F u n c t i o n s , a v a i l a b l e from Cabot C o r p o r a t i o n , 125 High S t r e e t , B o s t o n , MA 02110, USA. Chandrasekhar, S., 1943, Rev. Mod. Phys. 1_5, 1. Cheng, T.-P., and L.-F. L i , 1977, Phys. Rev. D 1_6, 1425. C l a r k , G.S., u n p u b l i s h e d , PHARUN: A Program f o r T r a n s f e r r i n g PHA  Dat a . Commins, E.D., 1973, Weak I n t e r a c t i o n s , M c G r a w - H i l l . C r a w f o r d , J.F., M. Daum, G.H. Ea t o n , R. F r o s c h , H. Hirschmann, R. H o r i s b e r g e r , J.W. M c C u l l o c h , E. S t e i n e r , R. Hausammann, R. Hess, and D. Werren, 1980, Phys. Rev. C 22, 1184. Danby, G., J.M. G a i l l a i r d , K. G o u l i a n o s , L.M. Lederman, N. M i s t r y , M. Schwartz, and J . S t e i n b e r g e r , 1962, Phys. Rev. L e t t . 9, 36. D a n i e l , H., 1979, Z. Phys. A 291, 29. Daum, M., G.H. Ea t o n , R. F r o s c h , H. Hirschmann, J.M. M c C u l l o c h , R.C. M i n e h a r t , and E. S t e i n e r , 1979, Phys. Rev. D 20, 2692. Derman, E., 1978, Phys. L e t t . 78B, 497. Derman, E., 1979, Phys. Rev. D 19, 317; see a l s o Derman, E., and H.-S. Tsao, 1979, Phys. Rev. D 20, 1207. E i c h t e n , T., H. Deden, F . J . H a s e r t , W. K r e n z , J . Von Krogh, D. Lanske, J . M o r f i n , H. Weerts, G.H. Bertrand-Coremans, J . S a c t o n , W. Van Doninck, P. V i l a i n , D.C. Cundy, D. H a i d t , M. J a f f r e , G. K a l b f l e i s c h , S. N a t a l i , P. Musset, J.B.M. P a t t i s o n , D.H. P e r k i n s , A. P u l l i a , A. R o u s s e t , W. Venus, H.W. Wachsmuth, V. B r i s s o n , B. Degrange, M. Haguenauer, L. K l u b e r g , U. Nguyen-Khac, P. P e t i a u , E. B e l l o t t i , S. B o n e t t i , D. C a v a l l i , C. Conta, E. F i o r i n i , C. F r a n z i n e t t i , M. R o l l i e r , B. A u b e r t , L.M. Chounet, P. Heusse, A.M. L u t z , J.P. V i a l l e , F.W. B u l l o c k , M.J. E s t e n , T.W. J o n e s , J . McKenzie, G. M y a t t , and J.L. P i n f o l d , 1973, Phys. L e t t . 46B, 281. E n g f e r , R., H. Schneuwly, J . L . V u i l l e u m i e r , H.K. W a l t e r , and 145 A. Zehnder, 1974, Atomic Data and N u c l e a r Data T a b l e s 14, 509. F e i n b e r g , G., and S. Weinberg, 1961a, Phys. Rev. L e t t . 6, 381. F e i n b e r g , G., and S. Weinberg, 1961b, Phys. Rev. 123, 1439. F e r m i , E., 1934, Z. Phys. 88, 161. F l e m i n g , D.G., R.J. M i k u l a , and D.M. G a r n e r , 1980, J . Chem. Phys. 73, 2751. Gamow, G., and E. T e l l e r , 1936, Phys. Rev. 49, 895. G a r n e r , D.M., 1979, Ph.D T h e s i s , U n i v e r s i t y of B r i t i s h C o lumbia, u n p u b l i s h e d . G a r r e t t , M.W., 1967, J . A p p l . Phys. 38, 2563. Glashow, S.L., J . I l i o p o u l o s , and L. M a i a n i , 1970, Phys. Rev. D 2, 1285. H a s e r t , F . J . , H. F a i s s n e r , W. K r e n z , J . Von Krogh, D. Lanske, J . M o r f i n , K. S c h u l t z e , H. Weerts, G.H. B e r t r a n d -Coremans, J . Lemonne, J . S a c t o n , W. Van Doninck, P. V i l a i n , C. B a l t a y , D.C. Cundy, D. H a i d t , M. J a f f r e , P. Musset, A. P u l l i a , S. N a t a l i , J.B.M. P a t t i s o n , D.H. P e r k i n s , A. R o u s s e t , W. Venus, H.W. Wachsmuth, V. B r i s s o n , B. Degrange, M. Haguenauer, L. K l u b e r g , U. Nguyen-Khac, P. P e t i a u , E. B e l l o t t i , S. B o n e t t i , D. C a v a l l i , C. C onta, E. F i o r i n i , M. R o l l i e r , B. A u b e r t , L.M. Chounet, P. Heusse, A. L a g a r r i g u e , A.M. L u t z , J.P. V i a l l e , F.W. B u l l o c k , M.J. E s t e n , T. J o n e s , J . McKenzie, A.G. M i c h e t t e , G. M y a t t , J . P i n f o l d , and W.D. S c o t t , 1973, Phys. L e t t . 46B, 121. H i g g s , P.W., 1964, Phys. L e t t . 12» 1 3 2 a n d Phys. Rev. L e t t . 13, 508. H o f e r , H., K. B o r e r , P. J e n n i , P. Le C o u l t r e , P.G. S e i l e r , and P. W o l f f , 1972, CERN P r o p o s a l P H I I I -72/20, u n p u b l i s h e d . Hughes, V.W., P r o c . E r i c e S c h o o l on E x o t i c Atoms, 1979, eds. G. F i o r e n t i n i and G. T o r e l l i , Plenum P r e s s . James, F., and M. Roos, 1971, MINUIT, Cern Computer Program L i b r a r y Documentation, u n p u b l i s h e d . K e n d a l l , K.R., 1972, Ph.D. T h e s i s , U n i v e r s i t y of A r i z o n a . K i b b l e , T.W.B., 1967, Phys. Rev. 155, 1554. K i e f l , R.F., J.B. Warren, G.M. M a r s h a l l , C.J. Oram, J.H. Brewer, D.J. Judd, and L.D. S p i r e s , 1979, H y p e r f i n e I n t e r a c t i o n s 6, 185. K i e f l , R.F., 1981, P r o c . 2nd I n t . T o p i c a l C onference on Muon 146 S p i n R o t a t i o n , t o be p u b l i s h e d i n H y p e r f i n e I n t e r a c t i o n s . K i t t e l , C , 1969, Thermal P h y s i c s , J . W i l e y and Sons. K o n o p i n s k i , E . J . , and H.M. Mahmoud, 1953, Phys. Rev. 92, 1045. Mann, A., and H. P r i m a k o f f , 1977, Phys. Rev. D 15, 655. Margenau, H., 1939, Rev. Mod. Phys. 11, 1. M a r i o n , J.B., 1968, N u c l e a r Data A4, 301. M a r s h a l l , G.M., J.B. Warren, D.M. Ga r n e r , G.S. C l a r k , J.H. Brewer, and D.G. F l e m i n g , 1978, Phys. L e t t . 65A, 351; see a l s o M a r s h a l l , G.M., 1977, M.Sc. T h e s i s , U n i v e r s i t y of B r i t i s h Columbia, u n p u b l i s h e d . M i k u l a , R.J., D.M. G a r n e r , D.G. F l e m i n g , G.M. M a r s h a l l , and J.H. Brewer, 1979, H y p e r f i n e I n t e r a c t i o n s 6, 379. Morgan, D.L., 1967, Ph.D. T h e s i s , Y a l e U n i v e r s i t y . N i s h i j i m a , K., 1957, Phys. Rev. 108, 907. Oram, C.J., J.B. Warren, G.M. M a r s h a l l , J . Doornbos, and D. O t t e w e l l , 1980, TRIUMF Report TRI-80-1. See a l s o Oram, C.J., J.B. Warren, G.M. M a r s h a l l , And J . Doornbos, 1981, N u c l . I n s t r . and Meth. 179, 95. P a r t i c l e Data Group, 1976, Review of P a r t i c l e P r o p e r t i e s , Rev. Mod. Phys. 48, S I . P e r l , M.L., G.S. Abrams, A.M. B o y a r s k i , M. B r e i d e n b a c h , D.D. B r i g g s , F. B u l o s , W. Chinowsky, J.T. D a k i n , G.J. Feldman, C E . F r i e d b e r g , D. F r y b e r g e r , G. Goldhaber, G. Hanson, F.B. H e i l e , B. J e a n - M a r i e , J.A. Kadyk, R.R. L a r s e n , A.M. L i t k e , D. L i i k e , B.A. L u l u , V. L u e t h , D. Lyon, C C . Morehouse, J.M. P a t e r s o n , F.M. P i e r r e , T.P. Pun, P.A. R a p i d i s , B. R i c h t e r , B. S a d o u l e t , R.F. S c h w i t t e r s , W. Tanenbaum, G.H. T r i l l i n g , F. V a n u c c i , J.S. W h i t a k e r , F.C. Winkelman, and J.E. Wi s s , 1975, Phys. Rev. L e t t . 35, 1489. P e r l , M.L., 1978, Nature 275, 273. P i f e r , A.E., T. Bowen, and K.R. K e n d a l l , 1976, N u c l . I n s t r . and Meth. 135, 39. P o n t e c o r v o , B., 1958, Zhur. Eksp. i T e o r e t . F i z . 3_3, 549 (1957); see Sov. Phys. JETP 6 ( 3 3 ) , 429 (1958) f o r t r a n s l a t i o n . R e i n e s , F., H.W. S o b e l , and E. P a s i e r b , 1980, Phys. Rev. L e t t . 45, 1307. R e i s t , H.-W., D.E. C a s p e r s o n , A.B. D e n i s o n , P.O. Egan, V.W. Hughes, F.G. Mariam, G. Zu P u l i t z , P.A. Souder, P.A. Thompson, and J . V e t t e r , 1978, N u c l . I n s t r . and Meth. 153, 147 61. Salam, A., 1968, "Elementary P a r t i c l e P h y s i c s " , ed. by N. S v a r t h o l m , A l m q v i s t and W i k s e l l s , Stockholm, p. 367. Schneuwly, H., 1979, P r o c . E r i c e S c h o o l on E x o t i c Atoms, eds. G. F i o r e n t i n i and G. T o r e l l i , Plenum P r e s s . Schwinger, J . , 1957, Ann. Phys. 2, 407. Tawara, H., and A. Russek, 1973, Rev. Mod. Phys. £5, 178. T a y l o r , J.C., 1976, Gauge T h e o r i e s of Weak I n t e r a c t i o n s , Cambridge U n i v e r s i t y P r e s s . Trower, W.P., 1966, Range-Energy and dE/dx P l o t s of Charged  P a r t i c l e s i n M a t t e r , UCRL-2426, V o l . I I . Weinberg, S., 1967, Phys. Rev. L e t t . 19, 1264. Weinberg, S., 1972, Phys. Rev. D 5, 1962. Weinberg, S., 1974, Rev. Mod. Phys. 46, 255. Weinberg, S., 1977, P r o c . 7 t h I n t . Conf. on High Energy P h y s i c s and N u c l e a r S t r u c t u r e , Exper. S u p p l . 33^, 339. W i l l i s , S.E., V.W. Hughes, P. Nemethy, R.L. Burman, D.R.F. Cochran, J.S. Frank, R.P. Redwine, J . D u c l o s , H. Kaspar, C K . Hargrove, and U. Moser, 1980, Phys. Rev. L e t t . 44, 522. E r r a t a , 1980, Phys. Rev. L e t t . 45, 1370. Wu, C.S., E. Ambler, R.W. Hayward, D.D. Hoppes, and R.P. Hudson, 1957, Phys. Rev. 105, 1413. Yang, C N . , and R.L. M i l l s , 1954, Phys. Rev. 96, 191. 

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