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UBC Theses and Dissertations

Positrons : practical plasma probe? Ziemelis, Ugis Oskars 1976

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POSITRONS: PRACTICAL PLASMA PROBE? by UG.IS OSKARS ZIEMELIS B . S c , M c M a s t e r U n i v e r s i t y , 1 972* A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n The F a c u l t y o f G r a d u a t e S t u d i e s D e p a r t m e n t o f 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 to t h e r e q u i r e d s t a n d a r d The U n i v e r s i t y of B r i t i s h Co lumbia S e p t e m b e r , 1976 © i s O s k a r s Z i e m e l i s , 1976 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 r e q u i r e -ments 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 C o l u m b i a , I ag ree 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 s t u d y . I f u r t h e r ag ree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g of t h i s t h e s i s f o r s c h o l a r l y pu rposes may be g r a n t e d by the Head o f my Department or by h i s r e p r e -s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n of 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 hot be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f P h y s i c s The U n i v e r s i t y o f B r i t i s h Co lumb ia 2075 Wesbrook P l a c e Vancouve r , Canada V6T 1W5 Date : September 27, 1976 A B S T R A C T The f e a s i b i l i t y o f u s i n g p o s i t r o n s t o probe p lasmas i s i n v e s t i g a t e d . A n a l y s i s of the y r a y s r e s u l t i n g f rom the a n n i h i l a t i o n of p o s i t r o n s w i t h p lasma e l e c t r o n s may y i e l d i n f o r -ma t i on about the momentum d i s t r i b u t i o n , t e m p e r a t u r e and number d e n s i t y o f these e l e c t r o n s . A number of ' p o s i t r o n p r o b e 1 s u c ce s s c r i t e r i a a re i n t r o d u c e d and a wide range of p o s i t r o n - p l a s m a systems a re e v a l u a t e d i n l i g h t o f t he se c r i t e r i a . Of s p e c i a l i m p o r t a n c e a r e p o s i t r o n a n n i h i l a t i o n t ime ( x , ) and t h e r m a l i z a t i o n t ime a (T ^ ) c a l c u l a t i o n s , , wh ich i n d i c a t e t h a t the most i m p o r t a n t s u c c e s s c r i t e r i o n , x. < x,., i s s a t i s f i e d by the c l a s s of p lasmas t a c h a r a c t e r i z e d by k T g > 10 eV. Two p o t e n t i a l f u s i o n p l a smas , namely Tokamak and l a s e r c ompre s s i on p l a smas , be long to t h i s c l a s s (where some p o s i t r o n d i a g n o s t i c t e c h n i q u e s may be p r a c t i c a l ) . D e t a i l e d f r a c t i o n a l t h e r m a l i z a t i o n c a l c u l a t i o n s f o r p o s i t r o n s 22 f rom a Na sou rce i n h y p o t h e t i c a l , f u l l y i o n i z e d H 2 p lasmas o f t he se t ype s i n d i c a t e t h a t most p o s i t r o n s w i l l t h e r m a l i z e and a n n i h i l a t e w h i l e the plasma env i r onmen t i s in e x i s t e n c e . i i P o s i t r o n s o u r c e s , i n c l u d i n g p a i r c r e a t i o n , wh ich i s p r e d i c t e d to o c c u r i n both Tokamak and l a s e r c ompre s s i on p l a sma s , and y r ay d e t e c t o r s a re d i s c u s s e d . A n n i h i l a t i o n y ray c o u n t i n g r a t e s a re e s t i m a t e d under a v a r i e t y of c i r c u m -s t a n c e s f o r both N a I ( T £ ) and G e ( L i ) d e t e c t o r s to p r e p a r e the way f o r the a s ses sment o f s p e c i f i c p o s i t r o n d i a g n o s t i c t e c h -n i q u e s . Four such t e c h n i q u e s are e v a l u a t e d w i t h r e f e r e n c e to bo th Tokamak and l a s e r c ompre s s i on p l a smas . I t i s c o n c l u d e d t h a t d i a g n o s t i c t e c h n i q u e s i n v o l v i n g measurements o f s h i f t and b r oaden i n g of the a n n i h i l a t i o n l i n e appear most p r o m i s i n g i n both c a s e s . The measurement o f p o s i t r o n l i f e t i m e s may be p o s s i b l e i n some Tokamak p l a smas , but i s not f e a s i b l e i n l a s e r c o m p r e s s i o n p l a smas . ' S l o w ' p o s i t r o n p r o b i n g of p lasmas would be i d e a l , but s l ow p o s i t r o n s o u r c e s o f s u f f i c i e n t i n t e n s i t y do not y e t e x i s t . A n g u l a r c o r r e l a t i o n measurements do not appear f e a s i b l e a t t h i s t i m e . i i i \ TABLE OF CONTENTS Page ABSTRACT '. i i LIST OF TABLES v i i LIST OF FIGURES v i i i ACKNOWLEDGMENTS x i Chapte r I INTRODUCTION 1 II POSITRON REVIEW 5 A. P o s i t r o n P rev i ew . . 5 B. The P o s i t r o n : B a s i c P h y s i c a l P r o p e r t i e s 5 C. E s t a b l i s h e d D i a g n o s t i c T e c h n i q u e s . . . . . 10 I I I POSITRONS IN PLASMAS 19 A. P o s i t r o n - P l a s m a Su rvey . 19 B. P lasma Measurement O b j e c t i v e s 20 C. The P o s i t r o n as a Plasma P r o b e : F e a s i b i 1 i t y C r i t e r i a 21 D. The L i f e of a P o s i t r o n i n a P lasma . . . . 26 E. P o s i t r o n Energy Loss Rate 32 i v \ Chap te r Page IV POSITRON ANNIHILATION/THERMALIZATION TIMES AND RANGES 36 A. C a l c u l a t i o n Summary 36 B. Range and Time E x p r e s s i o n s ' . . 37 C. Coulomb C o r r e c t i o n 42 D. P o s i t r o n D i s t r i b u t i o n s - F r a c -t i o n a l Thermal i z a t i on 4 7 V POSITRONS IN TOKAMAK AND LASER COMPRESSION PLASMAS 58 A. P lasma P a r a m e t e r s . . 58 B. Thermal i z a t i on Times and Ranges 59 C. P o s i t r o n Sou rce s . . 62 VI GAMMA RAY DETECTORS 73 A. S c i n t i l l a t i o n C o u n t e r s 73 B. S e m i c o n d u c t o r Coun te r s 77 C. M u l t i - W i r e P r o p o r t i o n a l Coun te r s 78 VI I ANALYSIS TECHNIQUES 82 A. Techn ique Survey 82 B. Gene ra l C o u n t i n g Rate C a l c u l a t i o n s . . . . 82 C. 2y A n g u l a r C o r r e l a t i o n 86 D. Dopp l e r B r o a d e n i n g and S h i f t o f the A n n i h i l a t i o n L i n e . 89 E. P o s i t r o n L i f e t i m e Measurements 99 F. P o s i t r o n Beam B roaden i n g ( ' S l o w ' P o s i t r o n s ) • 102 v Chap te r Page V I I I CONCLUSION 104 A. Summary and R e s u l t s 104 B. S u g g e s t i o n s f o r F u t u r e Work . . . 105 BIBLIOGRAPHY 108 APPENDICES A A n n i h i l a t i o n Modes I l l B Energy and Momentum B a l a n c e i n A n n i h i l a t i o n 114 C P o s i t r o n i u m F o r m a t i o n . 118 D P o s i t r o n Energy Loss Rate i n a Plasma 121 v i \ L I S T OF T A B L E S T a b l e P a g e I P l a s m a D i a g n o s t i c T e c h n i q u e F e a s i b i l i t y C r i t e r i a 22 I I R a d i o n u c l i d e P o s i t r o n S o u r c e s 63 I I I y R a v S c i n t i 1 1 a n t s 76 IV Summary o f C o n c l u s i o n s , 1 0 6 v i i LIST OF FIGURES F i g u r e Page l a S chemat i c r e p r e s e n t a t i o n o f l y , 2y and 3y ann i h i 1 a t i on 2 l b Anni h i 1 a t i on f rom r e s t 9 l c A n n i h i 1 a t i o n i n mot i on 9 2 Energy s pec t rum o f photons from 3y a n n i h i l a t i o n 11 3 2 y a n g u l a r c o r r e l a t i o n e x p e r i m e n t a l a r rangement 13 4 A n g u l a r c o r r e l a t i o n r e s u l t s of S t e w a r t ( 1 957 ) f o r some P e r i o d IV m e t a l s 15 5 D e u t s c h ' s p o s i t r o n l i f e t i m e appa r a t u s 17 6a P o s i t r o n s i n p a r t i a l l y i o n i z e d p lasmas 28 6b P o s i t r o n s i n f u l l y i o n i z e d plasmas 31 7 P o s i t r o n energy l o s s r a t e c u r v e s f o r f u l l y i o n i z e d H 2 34 8 Equal t h e r m a l i z a t i o n t ime c u r v e s f o r 500 KeV p o s i t r o n s i n f u l l y i o n i z e d H-2 4 0 9 Equal range c u r v e s f o r 500 KeV p o s i t r o n s i n f u l l y i o n i z e d H 2 41 v i i i F i g u r e Page 10 , Equal a n n i h i l a t i o n t ime, c u r v e s ( W o l f e r -T o p t y g i n ) f o r p o s i t r o n s i n f u l l y i o n i z e d H 2 . . . . 45 11 R e l a t i v e a n n i h i l a t i o n and t h e r m a l i z a t i o n t i m e s f o r 500 KeV p o s i t r o n s i n f u l l y i o n i z e d H 2 46 12 V a r i a t i o n of t h e r m a l i z a t i o n t ime w i t h i n i t i a l p o s i t r o n energy i n f u l l y i o n i z e d H 2 48 13 V a r i a t i o n of range w i t h i n i t i a l p o s i t r o n energy i n f u l l y i o n i z e d H 2 49 14 V a r i a t i o n of dE/dt w i t h p o s i t r o n energy i n f u l l y i o n i z e d H 2 50 15 V a r i a t i o n o f p o s i t r o n energy w i t h t i m e i n f u l l y i on i zed H 2 51 2 2 16 D i s t r i b u t i o n f u n c t i o n s N ( t ) and M( r ) f o r Na p o s i t r o n s i n f u l l y i o n i z e d H 2 (kT = 10 KeV, N = 1 0 1 " c m - 3 ) . . . 7 54 e 22 17 D i s t r i b u t i o n f u n c t i o n s N ( t ) and M( r ) f o r Na p o s i t r o n s i n f u l l y i o n i z e d H 2 (kT = 1 KeV, N = 1 0 2 " c m - 3 ) . . . . 7 55 22 18 T h e r m a l i z a t i o n . of Na p o s i t r o n s i n f u l l y i o n i z e d H 2 ( k T g = 10 KeV, N = l O 1 * c m " 3 ) 56 22 19 T h e r m a l i z a t i o n of Na p o s i t r o n s i n f u l l y i o n i z e d H 2 ( k T g = 1 KeV, N g = 1 0 2 " c m " 3 ) 57 2 2 20 Decay scheme o f Na 65 2 2 21 Energy d i s t r i b u t i o n of Na p o s i t r o n s 67 22a> S c i n t i l l a t i o n t ype d e t e c t o r . . 74 22b S e m i c o n d u c t o r t ype d e t e c t o r 74 22c M u l t i w i r e d r i f t t ype d e t e c t o r 74 i x \ F i q u r e Page 23 V a r i a t i o n o f d e t e c t o r r e s o l u t i o n w i t h Y r a Y e n e r g y 79 24 V a r i a t i o n o f d e t e c t o r e f f i c i e n c y w i t h Y r a y e n e r g y 80 25 A n n i h i l a t i o n y r a y a n a l y s i s t e c h n i q u e s 83 26 V a r i a t i o n o f D o p p l e r b r o a d e n i n g and s h i f t ( o f t h e a n n i h i l a t i o n l i n e ) w i t h p l a s m a k T g . . . . 92 27 S c h e m a t i c a n n i h i l a t i o n gamma r a y s p e c t r u m 94 28 A n n i h i l a t i o n d y n a m i c s 115 x ACKNOWLEDGMENTS W r i t i n g a t h e s i s i s q u i t e an e x p e r i e n c e - an e x p e r i e n c e t h a t i s e n r i c h e d immensely by o t h e r peop l e - at e ve r y s t a g e . Fo r t h i s e n r i c h m e n t , thank you to a l l ! But s p e c i a l thanks t o Dr. Boye A h l b o r n who sugge s ted t he f o u r p ' s ( p l a sma , p o s i -t r o n s , p r a c t i c a l , p robe) and who has been a f i n e s u p e r v i s o r . D i s c u s s i o n s were numerous and a lways h e l p f u l - h e a r t y thanks to Dr. A . J . B a r n a r d , Dr . G. Jones and Dr. A .T . S t e w a r t . The f i n a n c i a l a s s i s t a n c e o f the N a t i o n a l Re sea rch C o u n c i l o f Canada was d e f i n i t e l y i n v a l u a b l e and i s i ndeed a p p r e c i a t e d . Some of t h i s f i n a n c i a l a s s i s t a n c e ended up w i t h S h a r i H a l l e r , who i n r e t u r n d i d a superb j o b of t y p i n g t h i s t h e s i s . There i s one pe r son i n a f a r away p l a c e , who p l a y e d a r a t h e r m y s t i c a l r o l e i n a l l t h i s . . . thank y o u , Ma rT te . \ Kas visu grib zinat . . . tas a t r i vali-ek vecs [He who wants to know i t a l l . . . q u i c k l y grows o l d . ] L a t v i a n p r o v e r b x i i \ Chap te r I INTRODUCTION "On 2 August 1932, Dr. C a r l D. Ander son d i s c o v e r e d t he p o s i t i v e e l e c t r o n . " T h i s i s i n d e e d , as Hanson (1963) pu t s i t , a s t a tement whose " i n n o c e n t , m a t t e r - o f - f a c t tone c o n -c e a l s one of the most i n t r i c a t e and i n t e r e s t i n g c h a p t e r s i n the h i s t o r y o f s c i e n t i f i c d i s c o v e r y " - a c h a p t e r wh i ch i s s t i l l be ing a c t i v e l y w r i t t e n . A n d e r s o n ' s ' p o s i t i v e l y charged e l e c t r o n ' was i d e n t i f i e d soon a f t e r i t s d i s c o v e r y as the ' n e g a t i v e energy e l e c t r o n ' of D i r a c ' s h o l e t h e o r y ( D i r a c , 1930) . The new p a r t i c l e , or p o s i t r o n , as i t was named, became the f o c u s o f much b a s i c r e s e a r c h wh ich c o n t i n u e s even today to be f r u i t f u l . I n i t i a l e xpe r imen t s were d e s i g n e d t o d e t e r m i n e the p r o p e r t i e s of the p o s i t r o n and i n a d d i t i o n , t o s tudy i t s un i que i n t e r a c t i o n w i t h m a t t e r . C h a r a c t e r i s t i c o f t h i s i n t e r -a c t i o n i s the a n n i h i l a t i o n r e a c t i o n (between a c o l l i d i n g e l e c t r o n and p o s i t r o n ) wh ich g e n e r a t e s one, two o r t h r e e y photons as i l l u s t r a t e d s c h e m a t i c a l l y i n F i g u r e l a . 1 The momenta and e n e r g i e s of the a n n i h i l a t i o n y photons are r e l a t e d , t h r ough the a c t i o n of c o n s e r v a t i o n l a w s , to the energy and c e n t r e o f mass momentum o f the p r e - a n n i h i 1 a t i o n p a i r . I t i s p o s s i b l e t o d e t e r m i n e the d i s t r i b u t i o n of e l e c t r o n - p o s i t r o n p a i r c e n t r e of mass v e l o c i t i e s by o b s e r v i n g the a n n i h i l a t i o n y p h o t o n s . w h i c h escape the sample under s c r u t i n y . Such o b s e r -v a t i o n s form the b a s i s o f a number o f i n t r i g u i n g ' p o s i t r o n p r o b e ' d i a g n o s t i c t e c h n i q u e s wh ich have been used t o s tudy * s o l i d s , l i q u i d s and ga se s . I t s h o u l d be n o t e d t h a t d u r i n g t h e c o u r s e o f t h e e a r l y p o s i t r o n e x p e r i m e n t s , a p o s i t r o n - e 1 e c t r o n b o u n d s t a t e w a s i d e n t i f i e d ( D e u t s c h , 1951) a n d a new l i n e o f p o s i t r o n r e s e a r c h i n i t i a t e d . T h i s b o u n d s t a t e , w h o s e e x i s t e n c e w a s o r i g i n a l l y s u g g e s t e d by M o h o r o v i c i c ' (1934) c a m e t o be k n o w n a s p o s i t r o n i u m ( R u a r k , 1 9 4 5 ) . In s ome c a s e s p o s i t r o n i u m ( P s ) p l a y s a p a r t i n t h e d i a g n o s t i c t e c h n i q u e s w h i c h u t i l i z e ' f r e e ' o r u n b o u n d p o s i t r o n s a s p r o b e s . \ 3 The use o f p o s i t r o n s t o probe p lasmas was p roposed i n i t i a l l y by V . I . G o l d a n s k i i (1961) and i n d e p e n d e n t l y by B.P. K o n s t a n t i n o v ( i n T o p t y g i n , 1962 ) . However, w i t h the e x c e p t i o n of the work of L o h n e r t and S c h n e i d e r (1971) no e x p e r i m e n t a l i n v e s t i g a t i o n s i n v o l v i n g p o s i t r o n s and plasmas * have been c a r r i e d out (based on a 1975 l i t e r a t u r e s e a r c h ) . I t i s the purpose of t h i s work t o a s s e s s the f e a s i b i l i t y o f u s i n g p o s i t r o n s to probe p l a smas . The r e c e n t i n t e r e s t i n p o s i t r o n s has v a r i o u s c a u s e s : f i r s t , the s ucce s s of p o s i t r o n t e c h n i q u e s i n o t h e r s t a t e s of m a t t e r ; s e c o n d l y , the p o s s i b i l i t y t h a t p o s i t r o n t e c h n i q u e s may be u s e f u l i n the h i g h l y compressed l a s e r p lasma reg ime where s t a n d a r d o p t i c a l d i a g n o s t i c t e c h n i q u e s f a i l because the p lasma d e n s i t i e s a re g r e a t e r than the c u t - o f f d e n s i t i e s f o r v i s i b l e r a d i a t i o n ( a n n i h i l a t i o n gamma photons p e n e t r a t e t he se dense p lasmas e a s i l y ) ; t h i r d l y , t he p o s s i b i l i t y t h a t p o s i t r o n s may be produced th rough n a t u r a l p a i r p r o d u c t i o n i n some plasma d e v i c e s ( e . g . by run-away e l e c t r o n s i n Tokamak t ype p l a s m a s ) , thus r a i s i n g the q u e s t i o n of whethe r t he se p o s i t r o n s (as opposed to p o s i t r o n s i n t r o d u c e d f rom an e x t e r n a l s o u r c e ) or t h e i r a n n i h i l a t i o n p r o d u c t s c o u l d r e v e a l any i n f o r m a t i o n about The work o f L o h n e r t and S c h n e i d e r does not c o n -s t i t u t e a t r u e p o s i t r o n a p p r o a c h b e c a u s e t h e i r t e c h n i q u e does no t demand the use o f p o s i t r o n s ; in f a c t , i t i s c o m p l i c a t e d by t he use o f p o s i t r o n s as o p p o s e d to e l e c t r o n s ( to be d i s c u s s e d i n C h a p t e r V I I ) . 4 the plasma i t s e l f ; and f o u r t h l y , t he r e c e n t deve lopment of h i gh r e s o l u t i o n gamma ray a n a l y s i s t e c h n i q u e s . In s c o p e , t h i s t h e s i s i s meant to be a s u r v e y , y e t i t i s a l s o meant to p r e s e n t s p e c i f i c c o n c l u s i o n s c o n c e r n i n g the f e a s i b i l i t y of u s i n g p o s i t r o n s t o probe p l a smas . I t r e v i e w s the n e c e s s a r y t h e o r e t i c a l m a t e r i a l and d i s c u s s e s the b a s i c a s p e c t s o f p lasma d i a g n o s t i c s u s i n g p o s i t r o n s , i n c l u d i n g o b j e c t i v e s , s ucce s s c r i t e r i a , p o s i t r o n s ou r ce s and gamma ray d e t e c t o r s . The f e a s i b i l i t y of f o u r p o s i t r o n d i a g n o s t i c t e c h -n i que s (measurements i n v o l v i n g : 2y a n g u l a r c o r r e l a t i o n , a n n i h i l a t i o n l i n e b r o a d e n i n g , p o s i t r o n l i f e t i m e s , and s low p o s i t r o n beam sp read ) i s s t u d i e d i n d e t a i l and c o u n t i n g r a t e c a l c u l a t i o n s a re c a r r i e d ou t f o r p o s i t r o n s a n n i h i l a t i n g i n Tokamak and l a s e r c ompre s s i on p l a s m a s . R e s u l t s i n d i c a t e t h a t s p e c i f i c t e c h n i q u e s a re f e a s i b l e under c e r t a i n plasma c o n d i t i o n s . See Chap te r V I I I f o r a summary o f the r e s u l t s . \ 5 C h a p t e r II POSITRON REVIEW A. P o s i t r o n P r e v i e w T h i s c h a p t e r d e t a i l s t he p h y s i c a l c h a r a c t e r i s t i c s of the p o s i t r o n w i t h a s p e c i a l emphas i s on those a s p e c t s wh ich make i t such a un ique d i a g n o s t i c t o o l . In t h i s l i g h t , some e s t a b l i s h e d d i a g n o s t i c t e c h n i q u e s a re d e s c r i b e d and a number o f i l l u s t r a t o r y e x p e r i m e n t s c i t e d . Those f a m i l i a r w i t h p o s i t r o n t e c h n i q u e s may bypass t h i s c h a p t e r w i t h o u t l o s s o f c o n t i n u i t y . B. The P o s i t r o n : B a s i c P h y s i c a l P r o p e r t i e s The p o s i t r o n i s the a n t i p a r t i c l e of the e l e c t r o n , the a n t i - e l e c t r o n , thus a n n i h i l a t i o n can t ake p l a c e between the two. T h i s a n n i h i l a t i o n r e s u l t s i n the e m i s s i o n of one, two, or t h r e e y p h o t o n s , depend ing on the r e l a t i v e s p i n o r i e n t a t i o n of the i n t e r a c t i n g p a r t i c 1 e s ( F i g u r e l a ) . V a r i o u s c o n s e r v a t i o n r e q u i r e m e n t s g u a r a n t e e t h a t a n n i h i l a t i o n s where the r e s p e c t i v e s p i n s are a n t i p a r a l l e l ( t o t a l a n g u l a r momentum: 6 J = 0) r e s u l t i n the e m i s s i o n o f an even number of gamma photons (2y a n n i h i l a t i o n ) . On t he o t h e r hand, the p a r a l l e l s p i n s i t u a t i o n (J = 1) must be accompanied by the e m i s s i o n of an odd number o f quanta ( l y and 3y a n n i h i l a t i o n ) . One y a n n i h i l a t i o n can o c c u r o n l y i n the p re sence o f a t h i r d body c a p a b l e of a b s o r b i n g the r e c o i l momentum a s s o c i a t e d w i t h the e m i s s i o n o f a s i n g l e y pho ton . The a n n i h i l a t i o n p r o c e s s can be e i t h e r bound a n n i h i l a t i o n (bound s t a t e formed between the p o s i t r o n and the e l e c t r o n p r i o r to a n n i h i l a t i o n ) or f r e e a n n i h i l a t i o n (no bound s t a t e f o r m e d ) . The f i r s t c a l c u l a t i o n o f an a n n i h i l a t i o n c r o s s - s e c t i o n was c a r r i e d out by D i r a c (1930) f o r the 2 y - f r e e a n n i h i l a t i o n c a s e , w i t h the r e s u l t : a 2 y r + 1 r2 + 4r + l , r2 - l 1 n r + /rz - l r+3 ( i ) where r= 1/ A - v z / c 2 v i s the r e l a t i v e p o s i t r o n - e l e c t r o n v e l o c i t y ( D i r a c assumes the e l e c t r o n at r e s t ) ; and r 0 = e 2 / m c 2 i s the c l a s s i c a l e l e c t r o n r a d i u s . In the n o n - r e l a t i v i s t i c case where v << c e x p r e s s i o n (1) reduces to ° 2 Y " ^ • < 2 ' S i n c e a n n i h i l a t i o n i s a c o l l i s i o n i nduced p r o c e s s and e l e c t r o n must be i n the immediate v i c i n i t y o f i t i s to o c c u r ) , t he c r o s s . s e c t i o n can be w r i t t e n ( t he each as p o s i t r o n o t h e r i f 7 I t f o l l o w s t h a t . t h e e x p r e s s i o n f o r the 2y a n n i h i l a t i o n r a t e , V2^, i s energy i n d e p e n d e n t : 1 2 Y ^ 2 y T T r 0 cn (3) where i s the t ime r e q u i r e d f o r a 2y a n n i h i l a t i o n to o c c u r ( p o s i t r o n l i f e t i m e ) and n i s the e l e c t r o n number d e n s i t y o f the medium i n wh ich the a n n i h i l a t i o n t a ke s p l a c e . The 2y a n n i h i l a t i o n mode i s t he most p r o b a b l e , t h e r e f o r e d e t a i l e d d i s c u s s i o n w i l l be r e s t r i c t e d to t h i s mode ( S p e c i f i c a l l y o^/o^ i s of o r d e r a1* and 02y^°2y 1 S ° ^ o r d e r a where a = e V f i c - 1/137). A more comp le te d i s c u s s i o n o f * a n n i h i l a t i o n modes can be found i n Append i x A. In the case of 2y a n n i h i l a t i o n f rom r e s t , the spect rum i s a d e l t a f u n c t i o n a t a y energy of E(y) = m c 2 , As p r e v i o u s l y i n d i c a t e d , in some c a s e s an e l e c t r o n -p o s i t r o n bound s t a t e ( p o s i t r o n i u r n ) f o rms b e f o r e a n n i h i l a t i o n o c c u r s . T h i s 1 h y d r o g e n - 1 i k e a t o m 1 has a b i n d i n g e n e r g y o f 6.76 eV and e x i s t s in two s t a t e s : s i n g l e t o r p a r a - p o s i t r o n i u r n ( p a r t i c l e s p i n s o p p o s e d ; J = 0) and t r i p l e t o r o r t h o - p o s i t r o n i u r n ( p a r t i c l e s p i n s p a r a l l e l ; J = 1 ) . A l t h o u g h many f a c t o r s can i n f l u e n c e t h e l i f e t i m e ( a n n i h i l a t i o n t i m e ) o f p o s i t r o n i u m , i t does n o t , l i k e the f r e e a n n i h i l a t i o n t ime ( e q u a t i o n 3 ) , depend on t h e e l e c t r o n number d e n s i t y o f t he medium in a s t r a i g h t f o r w a r d way. A d d i t i o n a l i n f o r m a t i o n can be f o u n d in A p p e n d i x A. \ 8 where m i s the p o s i t r o n or e l e c t r o n r e s t mass. Each y photon c a r r i e s away one e l e c t r o n r e s t mass energy and to c o n s e r v e l i n e a r momentum they e x i t f rom the a n n i h i l a t i o n s i t e i n o p p o s i t e d i r e c t i o n s ( s e p a r a t i o n ang l e <$> = 180° - see F i g u r e l b ) . T h i s , of c o u r s e , i s a s p e c i f i c c a s e ; i n g e n e r a l , t he p o s i t r o n -e l e c t r o n p a i r i s moving when i t a n n i h i l a t e s . The c . of m. mot i on i s r e f l e c t e d i n the mot ions of t he a n n i h i l a t i o n photons i n t h a t i t i n t r o d u c e s (among o t h e r e f f e c t s ) a d e p a r t u r e f rom c o l l i n e a r e m i s s i o n of t he se pho ton s ; t hey e x i t f rom the a n n i h i l a t i o n s i t e s e p a r a t e d by some ang l e <}> < 1 8 0 ° . C o n s i d e r F i g u r e l c . As v i ewed f rom the l a b o r a t o r y r e f e r e n c e f r ame , j u s t p r i o r t o a n n i h i l a t i o n the p o s i t r o n and e l e c t r o n have a c e n t r e of mass momentum: p ( e + + e ) = p ( e + ) + p (e~) and a t o t a l energy £ t ( e + + e~) = 2mc 2 + E ( e + ) + E(e ), where p r e p r e s e n t s l i n e a r momentum and E, k i n e t i c ene rgy . A f t e r a n n i h i l a t i o n , the y photons e x i t w i t h momenta p(y - j ^ and e n e r g i e s E ( y ] 2 ) . In most ca se s the magn i tude o f the c e n t r e o f mass momentum of the p a i r , p ( e + + e ) << mc and hence 0, the ang l e of d e v i a t i o n from 180° photon s e p a r a t i o n i s s m a l l . In such c a s e s , i t i s a good a p p r o x i m a t i o n to w r i t e p ( y i ) = p ( y 2 ) = mc. Under t he se c i r c u m s t a n c e s , P j j e + + e~ ) - mc s i n 0 (4) or p i ( e + + e~) = mc 8 ( f o r sma l l 0 ) , (5) \ 2 Fig. 1b. Annihilation from rest. et(e++e") = 2mc E (^ 2 )=mc 2 % , = mC © = 180 rr~. : '7z Fig. 1c. Annihilation in motion (not to scale). 10 where pj^ i s t he magn i tude o f the component o f p c m p e r p e n d i c u l a r to the d i r e c t i o n o f e m i s s i o n of the y photon under c o n s i d e r a -t i o n (see F i g u r e l c ) . Both photon e n e r g i e s remain a lmo s t equa l t o m c 2 ; d e v i a t i o n s r e s u l t f rom the i n i t i a l p o s i t r o n - e l e c t r o n k i n e t i c energy c o n t r i b u t i o n s : E ( e + ) and E(e ). A d d i t i o n a l d e t a i l s and e x a c t e x p r e s s i o n s f o r energy and momentum s h a r i n g between the two a n n i h i l a t i o n gamma photons can be found i n Append i x B. The energy spec t rum a s s o c i a t e d w i t h 3y a n n i h i l a t i o n i s shown i n F i g u r e 2 (Ore and P o w e l l , 1949) . (On t h i s p l o t , the 2y spec t rum would be a d e l t a f u n c t i o n at E ( y ) = m c 2 ) , In a 3y p r o c e s s , the energy d i v i d e s t h r e e ways; any one g i v e n photon can c a r r y away an energy r a n g i n g f rom ze r o to m c 2 . C. E s t a b l i s h e d D i a g n o s t i c Techn ique s Two d i a g n o s t i c t e c h n i q u e s become i m m e d i a t e l y obv i ou s from the p r e v i o u s d i s c u s s i o n . P o s i t r o n l i f e t i m e a n a l y s i s and 2y a n g u l a r c o r r e l a t i o n t e c h n i q u e s are be i ng a p p l i e d s u c e s s -f u l l y to s t udy s o l i d s , l i q u i d s and ga se s . ( i ) 2y A n g u l a r C o r r e l a t i o n Measurements of the d e v i a t i o n a n g l e , 6 y i e l d i n f o r -mat i on about the p o s i t r o n - e l e c t r o n c e n t r e o f mass momentum. The 2y a n g u l a r c o r r e l a t i o n t e c h n i q u e i n v o l v e s ' c o i n c i d e n c e c o u n t i n g ' photons t h a t have o r i g i n a t e d f rom a s i n g l e a n n i h i l a t i o n \ Fig. 2. Energy spectrum of photons from 3/ annihilation. \ 1 2 as a f u n c t i o n o f the d e v i a t i o n a n g l e of one d e t e c t o r f rom c o l i n e a r i t y w i t h the o t h e r d e t e c t o r . (A t y p i c a l a r rangement i s t h a t of Lang and D e B e n e d i t t i ( 1 957 ) , see F i g u r e 3 ) . Note t h a t the a p p a r a t u s does not measure the d e v i a t i o n a n g l e 6, but 8^, the p r o j e c t i o n o f 6 on the yz p l a n e . T h i s and the v e r t i c a l s l i t geometry make the i n s t r u m e n t s e n s i t i v e to o n l y one c e n t r e o f mass momentum component of the a n n i h i l a t i n g p a i r ( i n t h i s case the z component, p z ) . I f the r e l a t i o n b e t w e e n t h e c e n t r e of mass momentum d i s t r i b u t i o n i n the z d i r e c t i o n (wh ich s h o u l d be d i r e c t l y r e f l e c t e d by the c o i n c i d e n c e c o u n t i n g r a t e , c ) and 6 z i s known, then N ( p ) , t he d i s t r i b u t i o n i n magni tude o f c e n t r e o f mass momentum can be f o u n d . Here , p = | p ( e + + e )|, the magn i tude o f the p o s i t r o n - e l e c t r o n c e n t r e of mass momentum. c i d e n c e c o u n t i n g r a t e can be w r i t t e n as the f o l l o w i n g one-d i m e n s i o n a l i n t e g r a l : where p(p) i s the p r o b a b i l i t y of f i n d i n g an a n n i h i l a t i n g p a i r w i t h momentum p i n dp. The d i s t r i b u t i o n i n magn i tude o f momentum i s u s u a l l y e x p r e s s e d as : With the geometry s k e t c h e d i n F i g u r e 3, t he c o i n -p (p )p dp, (6) N(p) = 4TT p 2 p (p) (7) Fig.3. 2J angular correlation experimental arrangement 14 S u b s t i t u t i n g f o r p(p) i n ( 6 ) and d i f f e r e n t i a t i n g w i t h r e s p e c t t o p y i e l d s : dc(8 7) N ( P ) « p o r d c ( e _ ) NCP ) a e z d e z , (8) u s i n g e x p r e s s i o n (5) to r e l a t e p and 6 . Thus , N(p) can be c a l c u l a t e d i n a s t r a i g h t f o r w a r d manner once c ( 0 z ) has been measured. In m e t a l s , where e l e c t r o n e n e r g i e s a r e of the o r d e r of e l e c t r o n v o l t s , the d e v i a t i o n a n g l e 0 f a l l s i n the m i l l i -r a d i a n range. Even so, the 2 y a n g u l a r c o r r e l a t i o n t e c h n i q u e i s p o w e r f u l . An e x c e l l e n t example o f the a p p l i c a t i o n of t h i s t e c h n i q u e i s the e a r l y work of S t e w a r t (1957) w i t h m e t a l s . F i g u r e 4 shows some r e s u l t s o f t h i s work: p ( k ) and N(k ) c u r v e s f o r a n n i h i l a t i o n s i n P e r i o d IVb m e t a l s (k a p h e r e ) . C o n t r i b u t i o n s f rom both low momentum c o n d u c t i o n e l e c t r o n s (nar row component) and h i gh momentum c o r e e l e c t r o n s (broad component) a re c l e a r l y v i s i b l e . The Fermi ene r g y , Ep, marks the boundary between ' t he se two components as e x p e c t e d . With The momentum s c a l e in F i g u r e k s h o u l d be i n t e r p r e t e d as i n d i c a t i n g e l e c t r o n momentum", a l t h o u g h , s t r i c t l y s p e a k i n g i t i n d i c a t e s p o s i t r o n - e 1 e c t r o n c e n t r e o f mass momentum. T h i s i s a v e r y good a p p r o x i m a t i o n b e c a u s e t h e p o s i t r o n s t h e r m a l i z e t o mean e n e r g i e s o f t h e o r d e r o f 0.025 eV ( a t room t e m p e r a t u r e ) and a n n i h i l a t e w i t h e l e c t r o n s o f t h e s o l i d w h i c h a r e h u n d r e d s o f t i m e s more e n e r g e t i c , w i t h t h e r e s u l t t h a t a l m o s t a l l o f t he momentum p a s s e d on to t he a n n i h i l a t i o n gamma p h o t o n s comes f r om t h e e l e c t r o n s . T h i s i s not t he c a s e in p l a s m a s , e s s e n t i a l l y b e c a u s e p l a sma e l e c t r o n s a r e not b o u n d . \ 1 5 MOMENTUM IN UNITS OF 2 mc X I0"3 I , , i i . | . i . . | I i • i | 1 1 I | ' | • I • I • • 1 i 1 • • • • l 1 I ' I - i " 0 1 2 5 10 20 30 40 50 ev 0 12 5 10 20 30 40 50 ev ENERGY - ONE PARTICLE MOVING Fig. A. Angular correlation results of Stewart (1957) for some Period IVb metals. 1 6 e x p e r i m e n t a l r e s u l t s l i k e t h e s e , S t e w a r t and o t h e r s have e s t a b l i s h e d 2y a n g u l a r c o r r e l a t i o n as a v a l u a b l e d i a g n o s t i c t e c h n i q u e o f the s o l i d s t a t e . ( i i ) P o s i t r o n L i f e t i m e Measurements D i r a c 1 s e x p r e s s i o n (.3) i n d i c a t e s t h a t t he p o s i t r o n l i f e t i m e ( a n n i h i l a t i o n t i m e ) i s i n v e r s e l y p r o p o r t i o n a l t o e l e c t r o n d e n s i t y , thus p o s i t r o n l i f e t i m e measurements s hou ld y i e l d i n f o r m a t i o n about e l e c t r o n d e n s i t i e s . Note t h a t t h i s i s so because a n n i h i l a t i o n i s a c o l l i s i o n i nduced p r o c e s s wh ich can be d e s c r i b e d by a c r o s s s e c t i o n of the t y p e d e f i n e d i n e x p r e s s i o n ( 2 a ) . The f i r s t l i f e t i m e measurements ( S hea re r and D e u t s c h , 1949) and subsequent work ( D e u t s c h , 1951) e s t a b l i s h e d the mu11i -component n a t u r e o f p o s i t r o n l i f e t i m e s p e c t r a f o r a number of g a se s . One component was c o n s i s t e n t w i t h the D i r a c e x p r e s s i o n (3) and was a t t r i b u t e d t o f r e e a n n i h i l a t i o n s ; the o t h e r s were a t t r i b u t e d to a n n i h i l a t i o n of * v a r i o u s bound s t a t e s . F i g u r e 5 shows D e u t s c h ' s a p p a r a t u s ( D e u t s c h , 1953 ) . A gamma photon e m i t t e d a lmos t s i m u l t a n e o u s l y w i t h each p o s i t r o n 22 f rom the Na sou rce was used to t r i g g e r the c o u n t i n g c i r -c u i t r y and a s top p u l s e was s u p p l i e d by one of the a n n i h i l a t i o n gamma pho ton s . Deutsch employed a d e l a y e d c o i n c i d e n c e c o u n t i n g It i s g e n e r a l l y c o n s i d e r e d t h a t t h i s work v e r i f i e d t h e e x i s t e n c e o f t h e bound s t a t e p o s i t r o n i u m . 1 7 test gas source 1-3 MeV phosphor absorber (prevents direct e+ counting ) •51 MeV phosphor Fig. 5. Deutsch's positron lifetime apparatus. 18 t e c h n i q u e to b u i l d up the l i f e t i m e s p e c t r a . W i th t he advent o f improved c o u n t i n g t e c h n i q u e s , p o s i t r o n l i f e t i m e s have been measured i n a v e r y l a r g e number o f s u b s t a n c e s i n g a seou s , s o l i d and l i q u i d s t a t e s . A l t h o u g h q u a n t i t a t i v e l y , some c o r r e c t i o n s have been a p p l i e d to (3) t o g a i n agreement w i t h e x p e r i -menta l r e s u l t s ( C h a p t e r IV, S e c t i o n C ) , q u a l i t a t i v e l y , l i f e -t imes do v a r y i n v e r s e l y w i t h the e l e c t r o n d e n s i t y . In s o l i d s and l i q u i d s , f r e e a n n i h i l a t i o n l i f e t i m e s a re of the o r d e r o f 10" 1 0 sec as compared t o 1 0 " 7 sec i n ga se s . A l i f e t i m e l i s t i n g can be found i n Hogg et al. ( 1968 ) . * A s u r v e y o f t e c h n i q u e s has not been a t t e m p t e d h e r e . The t e c h n i q u e s ment i oned a re by way of example : t o i l l u s t r a t e s u c c e s s f u l app roache s and to e s t a b l i s h p h y s i c a l c o r r e l a t e s to the c o n c e p t s i n t r o d u c e d i n S e c t i o n B. These t e c h n i q u e s and o t h e r s w i l l be examined i n terms of t h e i r u s e f u l n e s s w i t h r e s p e c t to plasma d i a g n o s t i c s i n Chap te r V I I . An u p - t o - d a t e d i s c u s s i o n o f p o s i t r o n t e c h n i q u e s b e i a p p l i e d t o p r o b e the s o l i d s t a t e can be f o u n d in a r e v i e w a r t i c l e by West ( 1 9 7 3 ) . \ Chap te r 1 1 1 POSITRONS IN PLASMAS A. P o s i t r o n - P l a s m a Survey What a re the b a s i c p lasma measurement o b j e c t i v e s ? What c o n d i t i o n s must be s a t i s f i e d i f p o s i t r o n s a r e to be used t o he lp r e a l i z e t he se o b j e c t i v e s ? What i s the f a t e o f a p o s i t r o n i n a plasma? Answers t o t he se t h r e e q u e s t i o n s form the b a s i s o f t h i s c h a p t e r . I f p o s i t r o n s c o u l d somehow be i n t r o d u c e d i n t o a p lasma (see Chap te r V) t h e i r e f f e c t i v e t e m p e r a t u r e ( ene rgy ) w o u l d , i n most c a s e s , be much g r e a t e r than the plasma e l e c t r o n t e m p e r a t u r e ( e n e r g y ) . Thus , the p o s i t r o n s would be e x p e c t e d to l o s e ene rgy t o the plasma and i n so d o i n g , t o s low to average p lasma e l e c t r o n v e l o c i t i e s i n a c h a r a c t e r i s t i c t h e r m a l i z a t i o n t ime T ^ . The a n n i h i l a t i o n p r o c e s s , c h a r a c -t e r i z e d by an a n n i h i l a t i o n t i m e , T , would compete as an a energy l o s s mechanism w i t h the t h e r m a l i z a t i o n p r o c e s s . The l a t t e r p a r t o f t h i s c h a p t e r d e a l s q u a l i t a t i v e l y and q u a n t i t a -t i v e l y w i t h t h i s mu l t imechan i sm e n e r g y - l o s s p r o c e s s . A 1 9 \ 20 number of s l o w i n g down mechanisms are d i s c u s s e d and a p p r o -p r i a t e energy l o s s e q u a t i o n s a re i n t r o d u c e d . B. Plasma Measurement O b j e c t i v e s Mean t e m p e r a t u r e and mean number d e n s i t y a re the two pr ime pa ramete r s wh i ch c h a r a c t e r i z e a g i v e n p l a sma . Due t o the mu1t icomponent n a t u r e of a p l a sma , t h e r e are i n f a c t t h r e e mean t e m p e r a t u r e s and d e n s i t i e s t h a t must be c o n s i d e r e d -t ho se d e s c r i b i n g the e l e c t r o n s , i o n s and n e u t r a l s . In some ca se s ( ' w e l l - b e h a v e d p l a s m a s ' ) t he mean component t e m p e r a t u r e s a re i d e n t i c a l , or a lmo s t so , and i n o t h e r s ( ' bad p l a s m a s ' ) t h i s i s not the c a s e . More comp le te c h a r a c t e r i z a t i o n s i n c l u d e i n f o r m a t i o n of v a r y i n g s p a t i a l and t e m p o r a l r e s o l u t i o n on component v e l o c i t y d i s t r i b u t i o n s (o r the d e v i a t i o n t h e r e o f f rom the i d e a l Maxwel1 - B o l t z m a n n d i s t r i b u t i o n s ) , p a r t i c l e c o l l i s i o n f r e q u e n c i e s ( c o l l i s i o n c r o s s - s e c t i o n s ) and some m a c r o s c o p i c p r o p e r t i e s ( e . g . p r e s s u r e ) . A l t h o u g h i t i s not i n c o n c e i v a b l e t h a t p o s i t r o n s c o u l d y i e l d i n f o r m a t i o n about the i on s and n e u t r a l s o f a p lasma ( t h r ough a n n i h i l a t i o n w i t h t h e i r e l e c t r o n s ) , t hey a re u n i q u e l y s u i t e d to measure a l l of the above ment ioned e l e c t r o n p a r a m e t e r s : t e m p e r a t u r e , d e n s i t y , d i s t r i b u t i o n of v e l o c i t i e s and c o l l i s i o n f r e q u e n c i e s . 21 C. The P o s i t r o n as a P lasma P r o b e : F e a s i b i l i t y C r i t e r i a I f a ' p o s i t r o n p r o b e ' method i s to be an i d e a l plasma d i a g n o s t i c t e c h n i q u e , then i t must s a t i s f y a number of b a s i c c r i t e r i a . These f a l l i n t o two c a t e g o r i e s : g e n e r a l ( a p p l i c a b l e to a l l d i a g n o s t i c app roache s ) and s p e c i f i c ( a p p l i c a b l e t o p o s i t r o n approaches o n l y ) . A summary appear s i n T a b l e I. ( i ) Gene ra l C r i t e r i a AT Macro > T. > AT X M i c r o (10) The i n t e r r o g a t i o n or s amp l i n g t ime (T^) s hou ld be s h o r t e r than A T M a c r 0 » the t ime t h a t c h a r a c t e r i z e s m a c r o s c o p i c p lasma changes such as bu l k f l o w , but l o n g e r than the c h a r a c t e r i s t i c m i c r o s c o p i c change t ime ( r e l a x a t i o n t i m e ) , "AT^- Q . * t i s r e a s o n a b l e to demand the degree of t ime r e s o l u t i o n sugge s ted by the LHS (A) of the i n e q u a l i t y to escape a v e r a g i n g ove r l a r g e s c a l e f l u c t u a t i o n s . The LHS can be w r i t t e n a s T^ < d D l a s m a where AT^ has been app rox ima ted by the r a t i o of s the average plasma d i me'n s i on ( s i z e ) , d p - j a s m a , t o the a p p r o -* p r i a t e sound speed c c . The RHS (B) of the i n e q u a l i t y T h i s y i e l d s , f o r e x a m p l e , AT.. ~ 1 0 8 s e c w i t h 1 ' r Macro d = 1 mm and T ( e l e c t r o n t e m p e r a t u r e ) = 1KeV; A T U ~ 1 0 e Macro sec w i t h d = 1 m and T = 1 0 0 e V . e T a b l e I Plasma D i a g n o s t i c Techn ique F e a s i b i l i t y C r i t e r i a -Genera l C r i t e r i a ( A l l D i a g n o s t i c Techn i que s ) S p e c i f i c C r i t e r i a ( P o s i t r o n D i a g n o s t i c T e c h n i q u e s ) Time . A x M a c r o > T i > A x M i c r o Pi a t Di s t a n c e d f > d P r o b e > XD t plasma Reasonable y Ray Count ing Rates A T A T Mac ro M i c r o probe ' D PI p1asma C h a r a c t e r i s t i c M a c r o s c o p i c Plasma V a r i a t i o n Time M i c r o s c o p i c Plasma I n t e r r o g a t i o n ( Samp l ing ) Time C h a r a c t e r i s t i c Large S c a l e Plasma F l u c t u a t i o n D imens ion Probe D imens ion Debye Length Plasma L i f e t i m e P o s i t r o n L i f e t i m e ( A n n i h i l a t i o n Time) P o s i t r o n T h e r m a l i z a t i o n Time P o s i t r o n Range Du r i ng T h e r m a l i z a t r o n Mean PIasma S i ze IN3 ro 23 g u a r a n t e e s t h a t t he system be ing ' i n t e r r o g a t e d ' has a c h i e v e d l o c a l u n i f o r m i t y and hence can be m e a n i n g f u l l y d e s c r i b e d by a mean t e m p e r a t u r e pa ramete r ( T , f o r e x a m p l e ) . The t ime r e q u i r e d f o r a few e l e c t r o n - e l e c t r o n c o l l i s i o n s ( say 1 0 ) i s u s u a l l y i n d i c a t i v e of A x ^ j c r 0 -VJith r e f e r e n c e to p o s i t r o n t e c h n i q u e s : each i n d i v i d u a l a n n i h i l a t i o n c o n s t i t u t e s an i n s t a n t a n e o u s s amp l i n g e v e n t , however t i m e a v e r a g i n g i s t o be e x p e c t e d due to t he sp read ( i n t i m e ) of a n n i h i l a t i o n even t s when more than one p o s i t r o n i s be ing c o n s i d e r e d . The magn i tude o f the sp read depends on the c h a r a c t e r i s t i c s o f the a n n i h i l a t i o n medium and the t ype o f p o s i t r o n s o u r c e be ing used ( e . g . p u l s e d or c o n t i n u o u s ) . T h i s sp read g u a r a n t e e s t h a t the e f f e c t i v e x i i s g r e a t e r i n magn i tude than A x ^ . c r Q , thus s a t i s f y i n g r e q u i r e m e n t ( B ) . In some ca se s the sp read may be so l a r g e t h a t the e f f e c t i v e x^ i s i n f a c t g r e a t e r i n magn i tude than A x ^ a c r 0 , w i t h the r e s u l t : poor t ime r e s o l u t i o n ( r e q u i r e m e n t A not s a t i s f i e d ) . T h i s i s a p o t e n t i a l p r ob l em i f good t ime r e s o l u t i o n i s r e q u i r e d . b ' , d f > , d P r o b e , > XD A I The probe d i m e n s i o n , d p ^ ^ , s hou ld be l e s s than the c h a r a c -t e r i s t i c d i m e n s i o n of l a r g e s c a l e plasma f l u c t u a t i o n s , d ^ , but g r e a t e r t han the Debye l e n g t h , The LHS (A) o f t h i s i n e q u a l i t y g u a r a n t e e s a s p a t i a l r e s o l u t i o n f i n e r than the l a r g e s c a l e f l u c t u a t i o n s , thus e l i m i n a t i n g a v e r a g i n g over such \ 24 f l u c t u a t i o n s ( c h a r a c t e r i s t i c a l l y o f s i z e ~ c i - , , . , / 1 0 ) . On p I d S III d the o t h e r hand, the RHS (B) i n s u r e s t h a t a v e r a g i n g w i l l t a k e p l a c e ove r d i s t a n c e s s h o r t e r than the Debye l e n g t h : k T c t 4TT V n " q 2 ' ( 1 2 ) •NJ a a a where a i s t he component s u b s c r i p t ( e l e c t r o n s or i o n s ) k i s the Bo l tzman c o n s t a n t T i s the t e m p e r a t u r e n i s the number d e n s i t y and q i s t he c h a r g e , thus a v e r a g i n g out the n o n - u n i f o r m i t i e s t h a t beg in to appear on t h i s s i z e s c a l e . A g a i n , w i t h r e f e r e n c e t o p o s i t r o n app roache s : a l t h o u g h i n d i v i d u a l p o s i t r o n s , i n a n n i h i 1 a t i n g w i t h i n d i v i d u a l e l e c t r o n s , sample ' v e r y sma l l r e g i o n s ' , a l a r g e number of p o s i t r o n s d i s t r i b u t e d t h r oughou t a p lasma samples e s s e n t i a l l y the e n t i r e p l a sma. Thus, a v e r a g i n g o ve r d i s t a n c e s of the o r d e r c e r t a i n l y o c c u r s ; however, a v e r a g i n g a l s o o c c u r s o ve r d i s t a n c e s of the o r d e r d f , making s p a t i a l r e s o l u t i o n p o o r . The s p a t i a l r e s o l u t i o n s i t u a t i o n c o u l d be improved by l o c a l i z i n g a l l the p o s i t r o n s to a p a r t i c u l a r r e g i o n of the p lasma or by l o o k i n g a t o n l y t ho se a n n i h i l a t i o n gamma ray s o r i g i n a t i n g from a s p e c i f i c volume of p l a sma. Of c o u r s e , i f \ \ 25 s i t u a t i o n s a r i s e where a v e r a g i n g o ve r the plasma i s d e s i r e d , p o s i t r o n t e c h n i q u e s a r e , i n t h i s r e s p e c t , i d e a l . ( i i ) S p e c i f i c C r i t e r i a A V > "V Tt (13) O b v i o u s l y , t he p o s i t r o n a n n i h i l a t i o n t ime T must be l e s s a than the plasma l i f e t i m e , T , (A) i f p o s i t r o n s a re t o a n n i -h i l a t e i n p lasma c o n d i t i o n s . In a d d i t i o n , the p o s i t r o n s s hou l d be t h e r m a l i z e d w i t h the plasma e l e c t r o n s a t the p o i n t of or p r i o r to a n n i h i l a t i o n w i t h the se e l e c t r o n s ( t h e r m a l i z a -t i o n t ime l e s s than a n n i h i l a t i o n t ime - B ) . T h i s r e q u i r e m e n t t h a t mean p o s i t r o n e n e r g i e s be of the o r d e r of (o r l e s s than ) mean e l e c t r o n e n e r g i e s a t a n n i h i l a t i o n i n s u r e s t h a t the p e r -t u r b a t i o n of the a n n i h i l a t i o n gammas a s s o c i a t e d w i t h e l e c t r o n mot ion i s comparab le i n magn i tude (and hence d e t e c t a b l e ) t o the p e r t u r b a t i o n a s s o c i a t e d w i t h the p o s i t r o n m o t i o n . I f a 1 KeV p o s i t r o n a n n i h i l a t e s w i t h a 1 eV e l e c t r o n , , the p e r t u r -b a t i o n o f the a n n i h i l a t i o n y photons due to the p r e - a n n i h i l a t i o n e l e c t r o n mot ion i s c e r t a i n l y not d e t e c t a b l e . R + < d , (14) t plasma 26 In the most d e s i r a b l e s i t u a t i o n , the m a j o r i t y o f p o s i t r o n s i n t r o d u c e d i n t o a plasma f o r d i a g n o s t i c purposes would a n n i h i l a t e i n t h e p lasma and not i n s u r r o u n d i n g m a t e r i a l s . I f t h i s i s to be the c a s e , the p o s i t r o n range w i t h i n the plasma ( R t ) must be l e s s than" the average plasma d i m e n s i o n d , plasma c . A l l d i a g n o s t i c t e c h n i q u e s i n v o l v i n g the use of p o s i t r o n s a re based on the c o u n t i n g of the a n n i h i l a t i o n y pho ton s , u s i n g one o f a number of methods ( e . g . 2 y - c o i n c i d e n c e ) , t h e r e f o r e , c o u n t i n g s t a t i s t i c s must be c o n s i d e r e d . Such a c o n s i d e r a t i o n i n v o l v e s many f a c t o r s (see Chap te r V I I ) , however the key r e q u i r e m e n t i s t h a t c o u n t i n g t imes be r e a s o n a b l y s h o r t ( r e a s o n a b l y l a r g e c o u n t i n g r a t e s ) . P r e c i s e l i m i t i n g magn i tudes w i l l o f t e n be d i c t a t e d by i n d i v i d u a l e x p e r i m e n t a l a r r angemen t s . E v a l u a t i o n of v a r i o u s p o s i t r o n d i a g n o s t i c t e c h -n i que s w i t h r e f e r e n c e to t he se ' s p e c i f i c f e a s i b i l i t y c r i t e r i a ' forms the bu l k of t he l a t t e r p a r t of t h i s s t u d y . D. The L i f e o f a P o s i t r o n i n a Plasma The l i f e o f an e n e r g e t i c p o s i t r o n i n a p lasma c o n s i s t s e s s e n t i a l l y of energy l o s s p r o c e s s e s wh ich i t unde r -goes i n an a t t empt to r each e q u i l i b r i u m w i t h i t s s u r r o u n d i n g s . These p r o c e s s e s w i l l d i f f e r f o r p o s i t r o n s i n a p a r t i a l l y \ 27 i o n i z e d env i r onment and p o s i t r o n s i n a f u l l y i o n i z e d e n v i r o n -ment. A l t h o u g h l a t e r c h a p t e r s d e a l o n l y w i t h f u l l y i o n i z e d p l a smas , a b r i e f s k e t c h o f t h e p a r t i a l l y i o n i z e d s i t u a t i o n w i l l be p r e s e n t e d he r e . ( i ) P a r t i a l l y I o n i z e d P lasmas A p o s i t r o n s l o w i n g down i n a p a r t i a l l y i o n i z e d p lasma w i l l pass t h rough t h r e e c h a r a c t e r i s t i c i n t e r a c t i o n s t a ge s ( F i g u r e 6 a ) . The f i r s t ( I ) i n v o l v e s a r a p i d energy l o s s t h rough i n e l a s t i c c o l l i s i o n s , r e s u l t i n g i n the d e g r a d a -t i o n o f the p o s i t r o n energy from the s ou r ce l e v e l (0 .5 to a few MeV) to ~50 or a few hundred eV. Most of the c o l l i s i o n s t h a t o c cu r d u r i n g t h i s h i gh energy s tage r e s u l t i n i o n i z a t i o n o f n e u t r a l p lasma p a r t i c l e s or a d d i t i o n a l i o n i z a t i o n o f e x i s t i n g plasma i o n s . P o s i t r o n s w i t h e n e r g i e s l e s s than a few hundred eV beg in t o undergo e l a s t i c c o l l i s i o n s w i t h the v a r i o u s s p e c i e s p r e s e n t i n the p l a sma. In s t age I I ( p o s i t r o n e n e r g i e s f rom a few hundred eV to the minimum e x c i t a t i o n energy - a l l s p e c i e s c o n s i d e r e d ) , energy l o s s e s a s s o c i a t e d w i t h e l a s t i c c o l l i s i o n s a re comparab le to tho se a s s o c i a t e d w i t h i n e l a s t i c c o l l i s i o n s . P o s i t r o n s w i t h e n e r g i e s l e s s than V.j (minimum i o n i z a t i o n energy - a l l s p e c i e s c o n s i d e r e d ) can no l o n g e r i n i t i a t e i o n i z a t i o n r e a c t i o n s (where the i o n i -z a t i o n e l e c t r o n e s c a p e s ) . Energy l o s s i n the energy gap: V. to V g (minimum e x c i t a t i o n energy - a l l s p e c i e s c o n s i d e r e d ) o c c u r s by e x c i t a t i o n c o l l i s i o n s and e l a s t i c c o l l i s i o n s . \ 28 M o s t l y i n e l a s t i c c o l l i s i o n s ( I o n i -z a t i o n ) : few e l a s t i c c o l l i s i o n s I o n i z a t i o n / E x c i -t a t i on c o l l i -s i o n s comparab le w i t h e l a s t i c c o l l i s i o n s i r t E l a s t i c c o l l i s i o n s o n l y : some gene-r a t i on of p iasma waves mi sou rce forms f r e e Ps w i t h ENERGY -550 KeV 'A forms J .bound e M ' 2y 3 Y (bound) (bound) —*— 2y/3y f r e e 1 a t i o n anni h i -p r o b a b i - + 1 i t y i n c r e a s e s as e energy dec rea se s 50 eV V i Mean e " ene rgy (p la sma kT ) V . : l o w e s t i o n i z a t i o n p o t e n t i a l V : l o w e s t e x c i t a t i o n p o t e n t i a l e V : Ps f o r m a t i o n t h r e s h o l d w i t h bound e P Fig. 6a. e in partially ionized plasmas. 29 Stage I I I ( p o s i t r o n e n e r g i e s f rom V to the back -ground e l e c t r o n t e m p e r a t u r e ) i s c h a r a c t e r i z e d by energy l o s s t h r o u g h e l a s t i c c o l l i s i o n s , i n c l u d i n g a component a t t r i b u t a b l e t o e x c i t a t i o n of plasma o s c i l l a t i o n s ( f o r d e t a i l s see d i s -c u s s i o n . o f f u l l y i o n i z e d p l a s m a s ) . Super imposed on t h i s energy d e g r a d a t i o n scheme a re the p o s s i b l e a n n i h i l a t i o n modes. At p o s i t r o n e n e r g i e s g r e a t e r than r o u g h l y 50 eV, e l a s t i c and i n e l a s t i c c o l l i s i o n c r o s s s e c t i o n s are much g r e a t e r than a n n i h i l a t i o n c r o s s s e c t i o n s , thus the f o rmer p r o c e s s e s p redomina te a lmos t e x c l u s i v e l y . Some f r e e a n n i h i l a t i o n and p o s i t r o n i u m f o r m a t i o n ( t h i s p o s s i -b i l i t y i s d i s c u s s e d i n Append ix C) i s e x p e c t e d to o c c u r i n the low energy s t age s II and I I I . The p r o b a b i l i t y t h a t t h e s e r e a c t i o n s w i l l o c cu r p r i o r to t h e r m a l i z a t i o n ( u n d e s i r a b l e ) as opposed to a f t e r t h e r m a l i z a t i o n ( d e s i r a b l e ) , depends on the s p e c i f i c n a t u r e o f the plasma i n q u e s t i o n ( s p e c i e s p r e s e n t , T , n ). T h i s t o p i c w i l l be c o n s i d e r e d i n d e t a i l i n the next e e two c h a p t e r s . ( i i ) F u l l y I o n i z e d Plasmas A f u l l y i o n i z e d plasma c o n s i s t s of o n l y two s p e c i e s : f r e e e l e c t r o n s and bare n u c l e i . A p o ' s i t r o n l o s i n g energy i n F o r e x a m p l e : i o n i z a t i o n c r o s s s e c t i o n s in t he KeV r ange a r e o f t he o r d e r o f 1 0 " 1 6 t o 1 0 - 1 7 c m 2 as compared to a n n i h i l a t i o n c r o s s s e c t i o n s ( D i r a c ) o f t h e o r d e r o f 1 0 - 2 5 to 1 0 _ 2 l t c m " 2 . 30 such a medium ( F i g u r e 6b) can do so o n l y t h rough e l a s t i c c o l l i s i o n s , a l l i n e l a s t i c energy l o s s o p t i o n s hav i ng been e l i m i n a t e d . As i n the case of p a r t i a l l y i o n i z e d - p l a s m a s , some p r e - t h e r m a l i z a t i o n a n n i h i l a t i o n and p o s i t r o n i u m f o r m a t i o n may t a k e p l a c e (see a l s o Append ix C ) . A l t h o u g h the p o s i t r o n energy l o s s p r o c e s s c o n s i s t s e s s e n t i a l l y o f e l a s t i c c o l l i s i o n s , i t i s u s e f u l f o r the sake of a n a l y s i s , t o d i v i d e these c o l l i s i o n s i n t o two t y p e s . (I and I I ) , a c c o r d i n g to the energy (momentum) t r a n s f e r per c o l l i s i o n . Type I a re termed ' h a r d ' or b i n a r y c o l l i s i o n s , i n the sense t h a t t hey are c h a r a c t e r i z e d by a c l o s e ( s m a l l impact pa r amete r ) i n t e r a c t i o n between two p a r t i c l e s w i t h l a r g e momentum t r a n s f e r . Type I I , on the o t h e r hand, a re ' s o f t ' c o l l i s i o n s c h a r a c t e r i z e d by d i s t a n t ( l a r g e impact pa ramete r ) i n t e r a c t i o n s between more than two p a r t i c l e s , w i t h a sma l l t o t a l momentum t r a n s f e r . A c o l l i s i o n of the l a t t e r t ype c o r r e s p o n d s to the c r e a t i o n of a plasma wave o r waves ( p l a smon s ) , t h a t i s , the c r e a t i o n of a quantum (or a number o f quanta ) o f d i r e c t e d c o l l e c t i v e momentum (by some d i r e c t e d i n d i v i d u a l momentum). A rough d i v i d i n g l i n e between t h e s e two t ype s of c o l l i s i o n can be drawn a t a momentum t r a n s f e r of App = fikp = fiMn, (where i s the Debye r a d i u s ) , c o r r e spond i -ng to the c r e a t i o n of a plasma wave d e s c r i b e d by a w a v e v e c t o r of magn i tude k n = l A n . The two r e g i o n s thus d e f i n e d a r e : 31 ENERGY 550 KeV I -50 eV Mean e energy (p iasma kT ) s ou r ce forms f r e e s i n g l e t t r i p i e t 2Y bound 3y a n n i h i 1 a t i o n o r b reak up due i on i z a t i on to ( r a r e ) r I E l a s t i c c o l l i s i o n s w i t h e " and n u c l e i ( m o s t l y b i n a r y , s h o r t range c o l l i -s i o n s ) H I J 2 Y / 3 Y f r e e a n n i h i -l a t i o n ( 2 y / 3 Y = 3 7 2 ) : a n n i h i l a t i o n p r o b a -b i l i t y i n c r e a s e s as e energy dec rea se s Long range e l a s t i c c o l l i s i o n s (genera-t i o n o f p lasma waves) Fig. 6b. e in fully ionized plasmas. 32 A p D < Ap < A p m a x Type I (15) He re , k (= Ap/fi) > kp, c o r r e s p o n d s to hard c o l l i s i o n s . A c t u a l l y , t he wave i d e a b reak s down f o r Ap > A p ^ , wh i ch c o r r e s p o n d s to the c r e a t i o n o f plasma waves s h o r t e r than the Debye r a d i u s -a p h y s i c a l i m p o s s i b i l i t y s i n c e m e a n i n g f u l p a r t i c l e c o r r e l a t i o n s ( such as waves) a re p o s s i b l e o n l y on s c a l e s l a r g e r than Ap. A p m a x r e p r e s e n t s the l a r g e s t p o s s i b l e s i n g l e - c o l 1 i s i on momentum t r a n s f e r . A p m i n < A p < A p D T y p e 1 1 ( 1 6 ) Here , k < k ^ , c o r r e s p o n d s t o s o f t c o l l i s i o n s or the g e n e r a -t i o n of p lasma waves w i t h w a v e l e n g t h s : A > A^ (a m e a n i n g f u l s i t u a t i o n ) . - n r e p r e s e n t s the s m a l l e s t p o s s i b l e momentum t r a n s f e r . E. P o s i t r o n Energy Loss Rate C a l c u l a t i o n of the p o s i t r o n energy l o s s r a t e by summing the c o n t r i b u t i o n s of Type I and Type II c o l l i s i o n s y i e l d s ( d e t a i l s i n Append ix D): 2 2 e a) d E - e - In where to e dt v f4TT n e 2 l * 2mv-e (17) m i s the p lasma f r e q u e n c y , 33 n i s the plasma e l e c t r o n number d e n s i t y , e i s the magn i tude o f the p o s i t r o n o r e l e c t r o n e l e m e n t a r y c h a r g e , m i s the mass o f t he e l e c t r o n o r p o s i t r o n , v i s t he v e l o c i t y o f the p o s i t r o n , v . i s the v e l o c i t y o f t he e l e c t r o n , e J fi = h/2-rr (h i s P l a n c k ' s c o n s t a n t ) , and v > v e T h i s r e s u l t was o r i g i n a l l y o b t a i n e d by Bethe (1930) and i s s u b s t a n t i a t e d by the r e s u l t s o b t a i n e d by L a r k i n (1959) u s i n g quantum f i e l d t e c h n i q u e s . Note t h a t the energy l o s s as e xp re s s ed by (17) i s i n d e p e n d e n t of the plasma t e m p e r a t u r e , (mean plasma e l e c t r o n v e l o c i t y ) . Hu s s e i n y and Fo r sen ( 1970 ) , u s i n g an a l t e r n a t e t e c h n i q u e ( d e t a i l s i n Append ix D) , d e r i v e a plasma t e m p e r a t u r e dependent e x p r e s s i o n f o r the energy l o s s r a t e : d_E dt 1 4 T T E I 2 2 2 e to — — In 2TTV , m v v 2fi a) e (18) ( r a t i o n a l i z e d MKS u n i t s - e 0 i s the pe rm i t -t i v i t y of f r e e space ) The appearance o f v g v as opposed to v 2 i n the argument of the l o g a r i t h m does not g r e a t l y a l t e r the energy l o s s r a t e ( F i g u r e 7 ) , nor does i t g r e a t l y a l t e r t h e r m a 1 i z a t i o n t ime v a l u e s based on e x p r e s s i o n s wh ich f o l l o w f rom (17) and (18) by i n t e g r a t i o n (Chap te r I V ) . 34 Fig. 7. Positron energy loss rate curves for fully ionized h^. 35 A number of t y p i c a l energy l o s s r a t e c u r v e s a re p l o t t e d i n F i g u r e 7 f o r purposes of i l l u s t r a t i o n and com-p a r i s o n . These c a l c u l a t i o n s have been c a r r i e d out f o r p o s i t r o n s ( i n i t i a l ene rgy - 500 KeV) i n a f u l l y i o n i z e d hydrogen p l a sma , a t v a r i o u s e l e c t r o n t e m p e r a t u r e s . The energy l o s s a s s o c i a t e d w i t h p o s i t r o n - i o n c o l l i s i o n s i s i n s i g n i f i c a n t and has been i g n o r e d ( s e e A p p e n d i x D ) . \ Chap te r IV POSITRON ANN IHILATION/THERMALIZATI ON TIMES AND RANGES A. C a l c u l a t i o n Summary Of the f e a s i b i l i t y c r i t e r i a ment ioned i n t he p r e v i o u s c h a p t e r , t ho se i n v o l v i n g p o s i t r o n t h e r m a l i z a t i o n t ime s and ranges a re the most i m p o r t a n t . T h i s c h a p t e r i s devo ted t o d e t a i l e d c a l c u l a t i o n s of t h e r m a l i z a t i o n t ime s (T ^ ) , a n n i h i l a -t i o n t ime (T ) and t h e r m a l i z a t i o n ranges ( R^ ) . C a l c u l a t i o n s a t are c a r r i e d out ove r a wide range of plasma p a r a m e t e r s , assuming a f u l l y i o n i z e d hydrogen p la sma: 1 0 8 < n ( c m " 3 ) < 1 0 2 6 and 1 0 " 1 < Te(eV) < 10 1 4 C a l c u l a t e d v a l u e s o f T . and T a re used to d e l i m i t t h o s e t a r e g i o n s o f the n-T e p l ane where p o s i t r o n d i a g n o s t i c s might be s u i t a b l e ( i . e . t ho se r e g i o n s where T . < T ). Two r e g i o n s L • a S t r i c t l y s p e a k i n g , a f u l l y i o n i z e d h y d r o g e n p l a s m a w i l l no t e x i s t f o r t h e l o w e s t T v a l u e s b e i n g c o n s i d e r e d ( T e < ~ 5 e V ) . In t h i s r a n g e , trie a p p r o x i m a t i o n o f a l l c o l l i s i o n s as e l a s t i c w i l l y i e l d u p p e r l i m i t v a l u e s f o r t h e r m a l i z a t i o n t i m e s and r a n g e s . E x a c t c a l c u l a t i o n s f o r t he p a r t i a l l y i o n i z e d c a s e s have not been c a r r i e d o u t . 3 6 37 o f i n t e r e s t a re a n a l y z e d i n d e t a i l , w i t h r e f e r e n c e t o . t h e ' 22 t h e r m a l i z a t i o n b e h a v i o u r of p o s i t r o n s f rom a Na s ou r ce and cu rve s o f per cen t o f p o s i t r o n s t h e r m a l i z e d as a f u n c t i o n o f t ime and range a re c a l c u l a t e d . B. Range and Time E x p r e s s i o n s The mean range and mean t h e r m a l i z a t i o n t ime of a p o s i t r o n w i t h an i n i t i a l energy E^ can be e x p r e s s e d i n i n t e g r a l form as : t h E . dE dx dE (19) th dE dt dE (20) where E ^ i s the mean t he rma l energy of the plasma e l e c t r o n s . S u b s t i t u t i n g f o r dE/dt ( e x p r e s s i o n 17) i n (19) and (20) y i e l d s the f o l l o w i n g e x p r e s s i o n s f o r p o s i t r o n range and t h e r m a l i z at i on t i me: f. 2 2 m 1 i 2 E. i 2 f l 0) - l i 2 E • n  t t h f i (21) In t h e s e e x p r e s s i o n s l i i s t h e l o g a r i t h m i c i n t e g r a l , d e f i n e d as f o 1 1 ows : 1 i (x ) = x dx 1 n x \ 38 and 1 2 e ^ fl moj. 1 i 2 E ^ 3 / 2 fi OJ 1 i >2 E t h , 3 / 2 fl OJ (22) The ana logous e x p r e s s i o n s based on the Hu s se i n y e q u a t i o n (18) f o r dE/dt a r e : R t = nu 8fi 2 1 mE th 1 i ( 4^E ) 2  v o 7 i th 2f. z OJ " - l i (4TT£ 0 ) E t h 2 2 f i z OJ * e (23) T . = ( 4 T T £ 0 ) 8 f i 3 e 2 (m E t h 3 ) ^ 1 i (4TTE O) 3/2 E- E, , i t h 2 f i 2 o) 2 e 3/2 1 i (4TT£ J 3 / 2 t h 2 f i 2 OJ 2 e 3/2 (24) E x p r e s s i o n s (21) t o (24) a re v a l i d f o r p o s i t r o n s t h e r m a l i z i n g i n a f u l l y i o n i z e d hydrogen p la sma. The b a s i c D i r a c a n n i h i l a t i o n t ime (3) can be w r i t t e n i n terms of OJ as e 4c D r 0 OJ 2 (25) Comprehens ive c a l c u l a t i o n s were c a r r i e d out by o v e r l a y i n g the T g - n p l ane o f i n t e r e s t w i t h a g r i d and comput ing v a l u e s o f H H R^, R^, T^, and a t each g r i d p o i n t . Equal t i m e and \ \ 39 equa l range c u r v e s were c o n s t r u c t e d by j o i n i n g g r i d p o i n t s w i t h i d e n t i c a l v a l u e s of the pa r amete r o f i n t e r e s t . F i g u r e 8 shows cu r ve s of equa l t h e r m a l i z a t i o n t ime based on e x p r e s s i o n s (22) and (24) as w e l l as s t a n d a r d F o k k e r - P l a n c k equa l r e l a x a -t i o n t ime c u r v e s f o r c o m p a r i s o n . The D i r a c equa l a n n i h i l a -t i o n t ime c u r v e s a re s t r a i g h t h o r i z o n t a l l i n e s on t h i s p l o t . In o r d e r to a v o i d c o n f u s i o n o n l y t he l i n e l e v e l s a re i n d i c a t e d a l ong the r i g h t hand marg in o f the p l o t . Equal range c u r v e s based on e x p r e s s i o n s (21) and (23) appear i n F i g u r e 9. A l l c u r v e s have been c a l c u l a t e d f o r p o s i t r o n s w i t h i n i t i a l e n e r g i e s : E i = 500 KeV. I t i s e v i d e n t f rom F i g u r e s 8 and 9 , t h a t on the s c a l e s of i n t e r e s t , r e s u l t s based on the Hus se iny e x p r e s s i o n (18) f o r dE/dt do n o t , on the w h o l e , d i f f e r s i g n i f i c a n t l y f rom r e s u l t s based on e x p r e s s i o n (17) f o r dE/dt (nor do F o k k e r - P l a n c k r e s u l t s d i f f e r g r e a t l y ) . F o r t h com ing t h e r m a l i -z a t i o n t ime and range c a l c u l a t i o n s w i l l t h e r e f o r e be based on e x p r e s s i o n ( 1 7 ) . The F o k k e r - P l a n c k e x p r e s s i o n f o r t h e t h e r m a l i z a t i o n t i m e : ( t h i s can a l s o be t e r m e d a r e l a x a t i o n t i m e , where t h e h i g h e n e r g y p o s i t r o n d i s t r i b u t i o n i s c o n s i d e r e d t o r e l a x to t he p l a sma e l e c t r o n d i s t r i b u t i o n ) p 3/2 t v m ; e z UJ 2 1 nA e where InA i s the Cou lomb l o g a r i t h m and 3/2 3 E t h A = i 2 2 m e OJ e f o l l o w s f r om F o k k e r - P l a n c k t r a n s p o r t t h e o r y in f u l l y i o n i z e d p l a s m a s ( see f o r e x a m p l e : K r a l l and T r i v e l p i e c e , 1973, C h a p t e r 6) 40 27 25 23 21 i i . _ T 19 £ <L> cn o 17 15 13 11 F3-10 = n - 3 , 0 - 1 2 k i d 3*10 r10 h310 r-3-10 — 3 > i Q -6 h310 3-10 .-4 3-10 -2 3-10 3-irAec j L r12 .-10 ,-8 r6 .-4 r2 h3-10 h3 h3-10' r-3-10^  Dirac times (sec.) -1 0 1 2 3 A log 1 Q(kT e) [kT e in ev] :ig. 8. Equal thermalization time curves for e+in fully ionized (22) (24) Fokker-Planck Initial e+ energy: 500 KeV. 41 -1 0 1 2 3 U log1 0(kTe) [kTe in eV] :ig.9. Equal range curves for 500 KeV e+ in fully ionized H 9 (21) (23) \ 42 I t i s a l s o c l e a r f rom F i g u r e 8 t h a t x t < x D (26) f o r a l l v a l u e s of T e and n. T h i s i s as e x p e c t e d f rom a com-p a r i s o n of the two c r o s s s e c t i o n s i n v o l v e d : = r 0 2 c / v , the 2 2 D i r a c a n n i h i l a t i o n c r o s s s e c t i o n and a (90°) = ir(-^-o-) , -c ' mv ^  the 90° Coulomb s c a t t e r i n g c r o s s s e c t i o n wh ich d e s c r i b e s ( a p p r o x i m a t e l y ) the t y p i c a l t h e r m a 1 i z a t i o n i n t e r a c t i o n . C l e a r l y , a c << 1 (27) f o r the energy ranges o f i n t e r e s t ( r e c a l l t h a t v i s the r e l a t i v e p o s i t r o n - e l e c t r o n v e l o c i t y ) . C. Coulomb C o r r e c t i o n ( i ) C a l c u l a t i o n s The key r e l a t i o n of the p r e c e e d i n g s e c t i o n ( i n -e q u a l i t y (26) ) i s based on D i r a c ' s f i r s t o r d e r e x p r e s s i o n f o r the a n n i h i l a t i o n c r o s s s e c t i o n (and t he a n n i h i l a t i o n t i m e ) . The d e r i v a t i o n of t h i s e x p r e s s i o n i s i n t u r n based on a p l ane wave model f o r both p o s i t r o n s and e l e c t r o n s wh ich i g n o r e s the d i s t o r t i n g e f f e c t s o f l ong range Coulomb i n t e r a c t i o n s . S i n c e l ong range Coulomb i n t e r a c t i o n s a re ve r y impor -t a n t i n plasmas - moreso than i n o t h e r s t a t e s of m a t t e r - t he \ 43 u s e o f O p t o c a l c u l a t e a n n i h i l a t i o n t i m e s i s q u e s t i o n a b l e . A n u m b e r o f a t t e m p t s h a v e b e e n made t o c a r r y o u t a n n i h i l a t i o n r a t e c a l c u l a t i o n s f o r f u l l y i o n i z e d h y d r o g e n p l a s m a s , t a k i n g i n t o a c c o u n t t h i s l o n g r a n g e C o u l o m b a t t r a c t i o n b e t w e e n p o s i t r o n s a n d e l e c t r o n s ( T o p t y g i n , 1962; W o l f e r , 1969) . T h e r e s u l t s o f t h e s e a t t e m p t s a r e s i m i l a r a n d i n d i c a t e a v e r y l a r g e e n h a n c e m e n t o f t h e a n n i h i l a t i o n r a t e a t l o w p l a s m a e l e c t r o n t e m p e r a t u r e s . W o l f e r d e r i v e s t h e n o n r e l a t i v i s t i c H a m i l t o n i a n f o r a m a n y - b o d y s y s t e m o f p o s i t r o n s a n d e l e c t r o n s i n t e r a c t i n g v i a a C o u l o m b f o r c e a n d c a p a b l e o f a n n i h i l a t i n g w i t h e a c h o t h e r ( i . e . c a p a b l e o f g i v i n g r i s e t o a n n i h i l a t i o n p h o t o n s ) . T h i s H a m i l t o n i a n i s t h e n u s e d t o c a l c u l a t e , w i t h t h e a i d o f Q.E.D. c o v a r i a n t p e r t u r b a t i o n f o r m a l i s m , t h e 2y a n n i h i l a t i o n p r o b a b i l i t y a n d t h e 2y a n n i h i l a t i o n r a t e i n a f u l l y i o n i z e d h y d r o g e n p l a s m a . W o l f e r ' s r e s u l t f o r t h e a n n i h i l a t i o n r a t e c a n be w r i t t e n a s : v w = 4 r 0 2 c n [ ^ - j f A \ ) e L e k T g e _ me w h e r e I = i s t h e i o n i z a t i o n p o t e n t i a l o f p o s i t r o n i u m , n, T g a r e p l a s m a e l e c t r o n d e n s i t y a n d t e m p e r a t u r e r e s p e c t i v e l y , f i ( T ) , fz(J ) a r e c o m p l i c a t e d f u n c t i o n s o f T g ( s e e W o l f e r , 1969 ) . a n d E 0 = m c 2 kT (28) 44 B o t h W o l f e r ' s a n d T o p t y g i n ' s r e s u l t s p r e d i c t a d r a s t i c v a r i a t i o n i n the' a n n i h i 1 a t i o n t i m e w i t h p l a s m a e l e c t r o n t e m p e r a t u r e . V a l u e s o f x = 1/v a r e w i t h i n an o r d e r o f W W m a g n i t u d e o f t h e D i r a c v a l u e s , T d , f o r 50eV < k T g < 10 KeV, h o w e v e r T d e c r e a s e s a l m o s t s i x o r d e r s o f m a g n i t u d e w i t h w 3 r e s p e c t t o x^ i n t h e r a n g e l e V < k T g < 50eV. T h i s d r a m a t i c e f f e c t i s e v i d e n t i n F i g u r e 10, w h e r e c u r v e s o f e q u a l a n n i h i l a t i o n t i m e ( c o n s t r u c t e d u s i n g W o l f e r ' s e x p r e s s i o n (28)) a r e p l o t t e d on t h e T g - n p l a n e . I t i s o b v i o u s a f t e r c o m p a r i n g F i g u r e s 8 and 10 t h a t t h e r m a l i z a t i o n t i m e s a r e no l o n g e r l e s s t h a n a n n i h i l a t i o n t i m e s f o r a l l T g and n v a l u e s . T h e T g - n p l a n e c a n be d i v i d e d i n t o r e g i o n s w h e r e < x w a n d x^ . > x w by s u p e r i m p o s i n g t h e p l o t s o f F i g u r e 8 and 10. The r e s u l t s o f s u c h a d i v i s i o n , f o r p o s i t r o n s w i t h i n i t i a l energy:, 500 KeV, a n n i h i l a t i n g i n a f u l l y i o n i z e d h y d r o g e n p l a s m a a r e shown i n F i g u r e 11. U n d e r t h e s e c o n d i -t i o n s , t h e r e g i o n o f i n t e r e s t (x^ < T ) i s d e f i n e d by t h e l i n e x w = x t a t k T g ~ l O e V . T h u s ' h o t ' p o s i t r o n s i n t r o d u c e d i n t o p l a s m a s w h e r e kT > ~ l O e V s h o u l d t h e r m a l i z e b e f o r e v e a n n i h i l a t i n g . T h i s may n o t be t h e c a s e i n ' c o o l e r ' p l a s m a s , w h e r e l o n g - r a n g e a t t r a c t i v e f o r c e s b etween, p o s i t r o n s and e l e c t r o n s become s i g n i f i c a n t e n o u g h t o e n h a n c e m u t u a l l o c a l i z a -t i o n and h e n c e a n n i h i l a t i o n . ( i i ) P r o m i s i n g P l a s m a s Of m a j o r i n t e r e s t f r o m t h e v i e w p o i n t o f d i a g n o s t i c s i s t h e f a c t t h a t two p o t e n t i a l f u s i o n p l a s m a s a r e l o c a t e d i n t h e 45 27 log 1 Q(kT e) [kT e in eV] Fig. 10. Equal annihilation time curves (Wolfer-Toptygin) for e+ in fully ionized h^-46 -I I I I ! L - 1 0 1 2 3 4 log (kT e) (N e in crrf3 ; kT e in ev) Fig. 11. Relative annihilation and thermalization times: 500 KeV e+ in fully ionized H ? . 47 f a v o u r a b l e x. < x a r e a : Tokamak o r t o r o i d a l c o n f i n e m e n t t w p l a sma s : 1 0 1 2 c m " 3 < n < 1 0 1 6 c m " 3 ; 1 0 3 eV < k T g < 1 0 5 eV and l a s e r c ompre s s i on p l a smas : 1 0 2 3 c m - 3 < n < 1 0 2 6 c m " 3 ; 1 0 3 eV < k T g < 10 4 eV. By way o f i n t r o d u c i n g Chap te r V, wh ich d e a l s e x c l u s i v e l y w i t h t h e s e p l a sma s , a number of p r e -l i m i n a r y c a l c u l a t i o n s are p r e s e n t e d h e r e . These dea l w i t h p o s i t r o n s t h e r m a l i z i n g i n the f o l l o w i n g h y p o t h e t i c a l p lasmas: Tokamak Type: n = 10 1"* c m " 3  J y e k T e = 10 KeV \ (29) L a s e r Compres s ion Type: n g = 10 2"* c m " 3 kT = 1 KeV e F i g u r e 12 i s a p l o t of t h e r m a l i z a t i o n t ime as a f u n c t i o n of i n i t i a l p o s i t r o n energy ( i . e . energy o f a p o s i t r o n when i t f i r s t appear s i n the p lasma) under the above c o n d i t i o n s . A s i m i l a r p l o t of t h e r m a l i z a t i o n range as a f u n c t i o n of i n i t i a l p o s i t r o n energy appear s i n F i g u r e 13. F i g u r e s 14 and 15 a r e l o g a r i t h m i c p l o t s of p o s i t r o n energy l o s s r a t e as a f u n c t i o n of i n s t a n t a n e o u s p o s i t r o n energy and p o s i t r o n energy as a f u n c t i o n of t ime r e s p e c i v e l y , f o r both s e t s of pa ramete r s d e f i n e d above. D. P o s i t r o n D i s t r i b u t i o n s - F r a c t i o n a 1 T h e r m a l i z a t i o n I t must be r e a l i z e d t h a t the above p l o t s a re mean-i n g f u l o n l y i n terms of s i n g l e p o s i t r o n s (o r p e r f e c t l y mono-e n e r g e t i c p o s i t r o n d i s t r i b u t i o n s ) and thus g i v e o n l y a g e n e r a l 48 "kTe=1 KeV N = 102/» c m fkT p = 10 KeV N e=10 W - c m - 3 Fig. 12. Variation of thermalization time with initial e+ energy in fully ionized h^. 49 kTe= 1 KeV L N e = 10 2 A c m - 3 kTe=10 KeV LN e= 10 Mcm~ 3 Fig. 13. Variation of range with initial e+energy in fully ionized H r \ 50 kTe=1 KeV N e = 102Z>cm~3 fkTe = 10 KeV Ne= 101A cm - 3 Fig. H. Variation of dE/dt with e +energy in fully ionized H 2 . 51 sec. x 10 -2 sec. x 10 a 1 kTe = 10 KeV N e=10 1 Z ,cm" 3 ,"kTp= 1 KeV M N e = 1024cm-3 Fig. 15. Variation of e+energy with time in fully ionized 52 d e s c r i p t i o n o f the b e h a v i o u r o f p o s i t r o n s f rom a r e a l s o u r c e -I f the sou rce i s a r a d i o a c t i v e n u c l e u s u n d e r g o i n g 3 + d e c a y , then the e m i t t e d p o s i t r o n s a r e d i s t r i b u t e d i n energy a c c o r d -i ng t o : (see Chap te r V, S e c t i o n C f o r d e t a i l s ) P ( E + ) d E + = K F ( Z , E + ) ( E + 2 - m 2c"y ( E + a x - E + ) 2 E + d E + (30) where E + i s the t o t a l p o s i t r o n ene r g y , i n c l u d i n g the r e s t mass energy and P ( E + ) d E + i s the f r a c t i o n o f a l l p o s i t r o n s e m i t t e d w i t h an energy E + i n the range d E + . S i n c e t h i s i s the c a s e , i t i s u s e f u l to d e f i n e two d i s t r i b u t i o n f u n c t i o n s , namely t he t h e r m a l i z a t i o n t ime d i s t r i b u t i o n : N ( t ) d t = P ( E ^ d E + (31 ) where N ( t ) d t i s t he f r a c t i o n of a l l p o s i t r o n s t h e r m a l i z i n g i n a t ime t i n the range dt and the t h e r m a l i z a t i o n range d i s t r i b u t i o n : H ( r ) d r - = ^ > " V . E l f M - L (32) 53 where M ( r ) d r i s the f r a c t i o n o f a l l p o s i t r o n s t h e r -m a l i z i n g i n a d i s t a n c e r i n t he range d r , v + i s the- i n s t a n t a n e o u s p o s i t r o n v e l o c i t y , and i t has been assumed t h a t a l l e m i t t e d p o s i t r o n s e v e n t u a l l y t he rma1 i z e . The d i s t r i b u t i o n f u n c t i o n s N ( t ) and M( r ) have been c a l c u l a t e d f o r the c o n d i t i o n s s p e c i f i e d by (29) w i t h p o s i t r o n s f rom a 22 + Na Source ( E m a x ( e ) - 542 KeV) . The c u r v e s , c o n s t r u c t e d p o i n t by p o i n t u s i n g e x p r e s s i o n s (17) and ( 3 0 ) , appear i n F i g u r e 16 (Tokamak) and F i g u r e 17 ( l a s e r c o m p r e s s i o n ) . S i n c e t he area under t he se c u r v e s r e p r e s e n t s the t o t a l number of p o s i t r o n s t h e r m a l i z e d , n u m e r i c a l i n t e g r a t i o n o f N ( t ) and M( r ) a l l o w s the d e t e r m i n a t i o n of the f r a c t i o n o f p o s i t r o n s t h a t t h e r m a l i z e i n a t ime t and a d i s t a n c e r. F i g u r e s 18 and 19 show the r e s u l t s of such c a l c u l a t i o n s : p e r c e n t a g e of p o s i t r o n s t h e r m a l i z e d as a f u n c t i o n o f (a) t ime i n the p lasma and (b) d i s t a n c e t r a v e l l e d t h r o u g h the p lasma ( f o r both plasmas be i n g c o n s i d e r e d ) . 54 Fig. 16. Distribution functions N(t) and M(r) for Na e in fully ionized kT e=10KeV N e = 1 0 U c m - 3 \ 55 sec. —»• 10'A 10'3 10"2 10" cm. —<~ Fig.17. Distribution functions N(t) and M(r) for 2 2 N a e+in fully ionized H^1 kTe= 1 KeV N e=10 2 Z ;cm" 3 22 Fig. 18. Thermalization of Na e+ in fully ionized h^: kTe = 10 KeV N e =10 U cm" 3 T Q~1sec. 57 cm. — ICL! 1 0 ^ i o " 2 ip" 1 T 1 — 1 ; T^— Fig. 19. Thermalization of " N a e+ in fully ionized H 2 -kT e = 1 KeV N e =l0 2 / | cm~ 3 T a - 10"10 sec Chap te r V POSITRONS IN TOKAMAK AND LASER COMPRESSION PLASMAS A. Plasma Pa r amete r s A number o f i m p o r t a n t f a c t o r s r e l a t e d t o the use of p o s i t r o n s to probe plasmas a re d i s c u s s e d i n t h i s c h a p t e r w i t h s p e c i f i c r e f e r e n c e to the two h y p o t h e t i c a l p lasmas i n t r o d u c e d i n the p r e v i o u s c h a p t e r . For the purposes of t h i s c h a p t e r a number of a d d i t i o n a l p a r a m e t e r s , t o b e t t e r c h a r a c t e r i z e t he se p l a smas , w i l l be i n t r o d u c e d : Tokamak Type Laser Compression Type n = l O 1 " cm" 3 n = 10 2 " cm" 3 e e kT =10 KeV kT = 1 KeV e e 10" 2 - 10" 1 sec x , = l O " 1 0 - 1 0 - 9 3 plasma volume: V ~ 7 x 10 6 cm3 plasma volume: V ~ 9 x 10~ 7 cm (minor radius ~ 0.5m; d , c m a ~ 120u . . ~ , ,- \ p i a b l l l d major radius ~ 1.5m) r The r e s u l t s p r e s e n t e d i n the p r e v i o u s c h a p t e r a re d i s c u s s e d and t h e i r i m p l i c a t i o n s o u t l i n e d . P o s i t r o n s ou r ce s a re c o n s i d e r e d , w i t h s p e c i a l emphasis ( q u a l i t a t i v e and 58 59 q u a n t i t a t i v e ) on the un ique p o s s i b i l i t y t h a t some p o s i t r o n s a r e a c t u a l l y c r e a t e d w i t h i n the p lasmas o f i n t e r e s t . B. T h e r m a l i z a t i o n Times and Ranges ( i ) Tokamak: F i g u r e 18 i n d i c a t e s t h a t most o f the 22 p o s i t r o n s e m i t t e d f rom a Na s ou r ce would t h e r m a l i z e i n a t i m e o f -55 msec and a d i s t a n c e o f ~ 2.5 x 1 0 8 cm. T h i s i s c o n s i s t e n t w i t h F i g u r e s 12 and 13. wh ich c o n f i r m t h a t ' t h e s e v a l u e s c o r r e s p o n d to comp le te t h e r m a l i z a t i o n of the most 22 * e n e r g e t i c Na p o s i t r o n s (542 KeV) . A l t h o u g h an i n d i v i d u a l p o s i t r o n l o s e s most o f i t s energy d u r i n g the l a t t e r s t age s o f t h e r m a 1 i z a t i o n ( d u r i n g the l a s t 15 msec f o r a p o s i t r o n w i t h an i n i t i a l ene rgy o f 500 KeV) , as i n d i c a t e d by F i g u r e s 14 and 15, the m a j o r i t y of p o s i t r o n s t h e r m a l i z e i n the 1.6 - 18 msec r ange , or i n terms o f d i s t a n c e t r a v e l l e d , i n t h e 4 x 1 0 5 cm to 7 x 10 6 cm range . T h i s i s i l l u s t r a t e d 2 2 by F i g u r e 16 and i s a r e s u l t o f the c h a r a c t e r i s t i c Na p o s i t r o n spect rum ( F i g u r e 2 1 ) . + T h e e f f e c t o f t h e m a g n e t i c f i e l d on dE / d t a n d h e n c e T a n d R h a s n o t b e e n t a k e n i n t o a c c o u n t h e r e . H o n d a + et al. (1963) h a v e c a l c u l a t e d dE / d t , t a k i n g i n t o a c c o u n t t h i s e f f e c t w i t h t h e r e s u l t : d E d t 2 2 1 n 2mv 2 1 f lU) roj 2 + 00 2 e B w h e r e u)g i s t h e p o s i t r o n g y r o f r e q u e n c y i n t h e e x i s t i n g m a g n e t i c f i e l d . T h e e n e r g y l o s s r a t e d e c r e a s e s , t h u s T a n d R^ i n c r e a s e ; b u t t h e s e a r e n o t m a j o r p e r t u r b a t i o n s . \ 60 The c a l c u l a t e d t h e r m a l i z a t i o n t imes a re l e s s than o r o f the o r d e r of t he p lasma l i f e t i m e , t h e r e f o r e f rom a t ime p o i n t of v i e w , most of t he p o s i t r o n s w i l l t h e r m a l i z e w h i l e the plasma i s s t i l l i n e x i s t e n c e . The s i t u a t i o n f rom a ' d i s t a n c e t r a v e l l e d ' p o i n t of v i ew may seem h o p e l e s s because o f the l a r g e p o s i t r o n range a t t h i s e l e c t r o n d e n s i t y ; how-e v e r , t h i s i s not the c a s e . Any p o s i t r o n s p r e s e n t w i l l be h e l d w i t h i n the plasma by the same f i e l d s wh ich c o n f i n e the plasma i t s e l f . The p o s i t r o n s , l i k e the e l e c t r o n s , w i l l f o l l o w h e l i c a l t r a j e c t o r i e s ( o n l y i n o p p o s i t e d i r e c t i o n s ), r o t a t i n g around f i e l d , l i n e s at the p o s i t r o n g y r o f r e q u e n c y : - e B t w i t h a r o t a t i o n r a d i u s of v I r, = — , (Larmour r a d i u s ) (34) where B i s the t o r o i d a l f i e l d s t r e n g t h and i s the p o s i t r o n v e l o c to the t o r o i d a l f i e l d Vj^ v i t y p e r p e n d i c u l a r The Larmour r a d i u s of a 500 keV p o s i t r o n , t r a v e l l i n g p e r -p e n d i c u l a r to a 10 KG t o r o i d a l f i e l d i s r^ = 2.4 mm - c o n -s i d e r a b l y s m a l l e r than mean plasma d i m e n s i o n s . A number of d r i f t mot ions are a l s o e x p e c t e d t o o c c u r , but t h e r e i s good rea son to b e l i e v e (see d i s c u s s i o n on runaway e l e c t r o n d r i f t 61 i n ORMAK-Knoepfel et al.-t 1975) t h a t t h e s e w i l l not r e s u l t i n t he l o s s of a l a r g e number o f p o s i t r o n s . With t h e s e f a c t s i n mind i t i s not u n r e a s o n a b l e to expec t p o s i t r o n s t o be c o n f i n e d w i t h i n the plasma f o r t h e e n t i r e d u r a t i o n o f t h e i r e x i s t e n c e ( c o r r e s p o n d i n g to a 2 2 maximum d i s t a n c e t r a v e l l e d o f ~ 2.5 x 1 0 8 cm f o r Na p o s i t r o n s ). A n n i h i l a t i o n t imes a re p r e d i c t e d to be i n the range 1-2 sec by both t he D i r a c (25) and the W o l f e r (28) e x p r e s s i o n s , i n t he plasma be ing c o n s i d e r e d . T h i s i s at l e a s t one o r d e r o f magn i tude l a r g e r than t h e ' p l a s m a l i f e t i m e and t h e r e f o r e r e p r e s e n t s some l o s s of u s e f u l gamma r a y s i g n a l . A l t h o u g h most of the p o s i t r o n s w i l l t h e r m a l i z e w i t h i n the plasma l i f e -t i m e , some w i l l a n n i h i l a t e o n l y a f t e r the plasma has ceased t o e x i s t . T h i s s i t u a t i o n c o u l d be improved by i n c r e a s i n g the plasma d e n s i t y o r the plasma l i f e t i m e or b o t h . ( i i ) L a s e r Compre s s i on : Complete t h e r m a l i z a t i o n 2 2 of Na p o s i t r o n s i n t h i s h y p o t h e t i c a l plasma shou ld o c cu r i n a t ime of ~ 1.3 x 1 0 " 1 1 sec and a d i s t a n c e o f ~ 1 0 " 1 cm (based on F i g u r e 19 ) . Energy l o s s c h a r a c t e r i s t i c s are s i m i l a r t o t ho se o u t l i n e d above and are i l l u s t r a t e d i n F i g u r e s 12-15. The m a j o r i t y o f p o s i t r o n s t h e r m a l i z e i n the 0.5-5 psec r ange , or a f t e r hav ing t r a v e l l e d between 1.4 x 10 _ 1 * cm and 1 0 " 2 cm. C a l c u l a t e d t imes f o r comp le te t h e r m a l i z a t i o n a re much l e s s t han the 0.1 t o 1 nsec plasma l i f e t i m e f i g u r e . In a d d i t i o n , c a l c u l a t e d a n n i h i l a t i o n t imes (~ 1.3 x 1 0 ~ 1 0 s ec ) a re of the \ 6 2 r i g h t o r d e r o f magn i tude - g r e a t e r than the t h e r m a l i z a t i o n t i m e s , but l e s s than the plasma l i f e t i m e . Maximum range v a l u e s a re somewhat l a r g e r than the mean plasma d i m e n s i o n , thus some l o s s e s a r e t o be e x p e c t e d , however t he m a j o r i t y of p o s i t r o n s s h o u l d t h e r m a l i z e w i t h i n the p l a sma . Range l o s s e s c o u l d be reduced or e l i m i n a t e d by i n c r e a s i n g the p h y s i c a l e x t e n t and/or d e n s i t y o f the p l a sma. 2 2 As w i l l be d i s c u s s e d i n the next s e c t i o n , a Na s ou r ce i s not t he most e f f i c i e n t p o s i t r o n s ou r ce f o r use w i t h l a s e r c ompre s s i on t ype p l a smas , thus the c a l c u l a t i o n s c i t e d i n t h i s s e c t i o n s h o u l d be t aken o n l y as a g e n e r a l i n d i c a t i o n of the t h e r m a l i z a t i o n c h a r a c t e r i s t i c s of KeV to MeV p o s i t r o n s i n t h i s p la sma. ' C. P o s i t r o n Sources P o s i t r o n s ou r ce s f o r p lasma d i a g n o s t i c s work can be c l a s s e d as e x t r i n s i c and i n t r i n s i c , an e x t r i n s i c s ou rce be i ng one t h a t i s not i n h e r e n t to the plasma and an i n t r i n s i c s ou r ce one t h a t i s . ( i ) E x t r i n s i c S ou r ce s : The most common e x t r i n s i c s o u r c e s a re r a d i o n u c l i d e s which undergo B + decay . A p a r t i a l l i s t o f some p o s i t r o n e m i t t i n g r a d i o n u c l i d e s , s p e c i f y i n g h a l f - l i v e s and maximum ( c u t o f f ) k i n e t i c e n e r g i e s a t t a i n e d by e m i t t e d p o s i t r o n s , i s p r e s e n t e d i n T a b l e - I I . The sou rce 63 T a b l e I I R a d i o n u c l i d e e + Sou rces Radi o n u c l i de H a l f L i f e Max e + Energy (KeV) 2.6 y 542 7.4 x 1 0 5 y 1 1 60 48 23 V 16 d 698 2 7 C o 71 d 510 6 4 r  2 g C u 12.8 h 650 6 5 Z n 3 0 ^ n 244 d 325 > 33 d 780 8 8 v 39 T 107 d 760 \ 64 22 most commonly used i n s o l i d s t a t e s t u d i e s i s Na, f o l l o w e d . 6 4 r by Cu. Copper -64 i s s h o r t l i v e d ( h a l f - l i f e ~ 1 2 . 8 h ) , but i t has the advantage t h a t i n t e n s e s o u r c e s of any c o n f i g u r a t i o n can be o b t a i n e d q u i c k l y by t h e r m a l n eu t r on i r r a d i a t i o n of Cu (wh ich i s a b u n d a n t ) . Sod ium-22 , due to i t s c h e m i c a l 22 p r o p e r t i e s , i s u s u a l l y a v a i l a b l e i n the form of NaCl s o l u t i o n s . 2 2 The u.se of Na has the advantage t h a t the b i r t h of each p o s i t r o n i s marked i n t i m e , to w i t h i n l e s s than l C r 1 1 s e c , by the e m i s s i o n o f a 1.28 MeV gamma ray a s s o c i a t e d w i t h the t r a n s i t i o n of neon-22 ( the p r o d u c t of the 8 + dacay) from the f i r s t e x c i t e d s t a t e to the ground s t a t e (see F i g u r e 20 -2 2 t he decay scheme of Na). P o s i t r o n s e m i t t e d by r a d i o n u c l i d e s have a c h a r a c -t e r i s t i c energy d i s t r i b u t i o n , r e p r e s e n t e d here by P(w)dw, E+ the f r a c t i o n of a l l p o s i t r o n s e m i t t e d w i t h energy W = i n the range dw: P(w)dw = K F(Z,w) ( w - 1 ) * ( w 0 - w ) 2 w dw (35) where E + = mc 2 + E ( e + ) i s the t o t a l p o s i t r o n ene rgy : r e s t mass energy p l u s k i n e t i c ene rgy , K i s a c o n s t a n t , F(.Z,w) i s the Fermi f u n c t i o n , a sma l l c o r r e c t i o n f a c t o r to accoun t f o r the p e r t u r b a t i o n of the B + spec t rum by the n u c l e u s , Z i s t he a tom ic number o f the p r oduc t or daugh te r n u c l e u s , \ \ 66 E + a and w0 = m 2 x i s the maximum or c u t o f f p o s i t r o n / m c energy f o r a g i v e n r a d i o n u c l i d e . + 2 2 + A p l o t o f the t h e o r e t i c a l 6 s pec t rum of Na ( E m a x ( e ) = 542 KeV) appea r s i n F i g u r e 21. 2 2 64 T y p i c a l NaCl or Cu s ou r ce s t r e n g t h s range f rom a few m i c r o c u r i e s t o hundreds of m i l l i c u r i e s (1 C u r i e ( C i ) = 3.7 x 1 0 1 0 d i s i n t e g r a t i o n s per second ( d p s ) ) . A l t h o u g h s t r o n g e r s o u r c e s can be p r o d u c e d , h a n d l i n g and p r o t e c t i v e s h i e l d i n g become cumbersome. I t i s o b v i ou s t h a t s ou r ce s of t h i s n a t u r e ( e x t r i n s i c ) w i l l not be v e r y e f f i c i e n t i n terms of p r o d u c i n g enough p o s i t r o n s to p robe l a s e r c ompre s s i on plasmas w i t h l i f e t i m e s o f the o r d e r o f 1 0 ~ 9 sec o r l e s s . I f a 1 C i r a d i o n u c l i d e s ou r ce c o u l d somehow be i n c o r p o r a t e d i n t o a l a s e r c o m p r e s s i o n p l a sma , o n l y 37 or so p o s i t r o n s would be p roduced d u r i n g the l i f e t i m e o f the p la sma. No s t a t i s t i c a l l y s i g n i f i c a n t a n n i h i l a t i o n y s i g n a l c o u l d be measured under t h e s e c i r c u m -s t a n c e s . T h i s i s not the case i n Tokamak t ype p l a smas , wh ich have l i f e t i m e s about seven o r d e r s of magn i tude l o n g e r , a l l o w i n g f o r the p r o d u c t i o n of s t a t i s t i c a l l y s i g n i f i c a n t numbers of p o s i t r o n s d u r i n g the e x i s t e n c e of the p lasma ( e . g . ~ 3.7 x 1 0 6 p o s i t r o n s would be produced by a 10 mCi source d u r i n g t h e 1 0 " 2 sec l i f e t i m e of a t o r o i d a l p l a s m a ) . I n t r i n s i c S o u r c e s : P o s i t r o n - e l e c t r o n p a i r not o c c u r under normal c i r c u m s t a n c e s i n or g a s e s , but i t i s e xpec ted to o c c u r , under ( i i ) p r o d u c t i o n does s o l i d s , l i q u i d s 68 \ normal c i r c u m s t a n c e s , i n some p l a sma s . Such p a i r p r o d u c t i o n would a c t as a n a t u r a l , i n t r i n s i c s ou r ce of p o s i t r o n s -p o s i t r o n s wh ich c o u l d t hen be used f o r d i a g n o s t i c p u r p o s e s . (a) Tokamak Type P l a smas : P a i r p r o d u c t i o n i n t o r o i d a l p lasmas s hou l d be m e d i a t e d by c o l l i s i o n s between h i gh energy ' r u n a w a y ' e l e c t r o n s and ' c o o l e r ' e l e c t r o n s , i o n s or n e u t r a l s . In a f u l l y i o n i z e d hydrogen plasma o f the t ype be ing c o n s i d e r e d , the t o t a l c r o s s s e c t i o n f o r p a i r p r o d u c t i o n by e l e c t r o n - e l e c t r o n c o l l i s i o n s i s g i v e n a p p r o x i -mate l y by: ( H e i t l e r , 1954. C h a p t e r V) PP 2 2 In BE' mc • ( 3 6 ) where and B i s a c o n s t a n t of o r d e r one; t a ke 3=1 h e r e , P 2 r o mc-i s the c l a s s i c a l e l e c t r o n r a d i u s , a = ^— i s the f i n e s t r u c t u r e c o n s t a n t , Tic E" = mc 2 + E ( e " ) The t o t a l number of p a i r s p roduced per u n i t t i m e , per u n i t plasma volume i s : r 2 n 2 „ - c n n — 1 n p r TT mc : ( 3 . 7 ) where n i s t he t he rma l e l e c t r o n number d e n s i t y and n r i s the runawy e l e c t r o n d e n s i t y . 69 Choos ing c o n s e r v a t i v e v a l u e s f o r n (= 1 0 7 c m - 3 ) and f o r the runaway e l e c t r o n k i n e t i c energy (~ 10 MeV), ba sed .on e x p e r i -ments by K n o e p f e l et al. (1975) w i t h the Oak R idge Tokamak (ORMAK), Np can be c a l c u l a t e d . These v a l u e s a re deemed c o n -s e r v a t i v e because ORMAK i s c o n s i d e r a b l y s m a l l e r (majo r r a d i u s = 80 cm, m inor r a d i u s = 23 cm) than t he h y p o t h e t i c a l t o r o i d a l p lasma under c o n s i d e r a t i o n here and runaway e l e c t r o n d e n s i t i e s and e n e r g i e s a re e x p e c t e d t o i n c r e a s e as plasma d i m e n s i o n s a re s c a l e d up. The c a l c u l a t e d v a l u e , u s i n g n = 1 0 1 4 c m - 3 i s : Np - 1.1 x 1 0 3 s e c - 1 c m - 3 . Thus, the t o t a l number of p a i r s produced d u r i n g the e x i s t e n c e of the plasma (runaway e l e c t r o n s e x i s t d u r i n g most of the plasma l i f e t i m e of X p ^ = 1 0 ~ 2 s ec ) i n a volume o f 1 0 3 cm 3 i s ~ 1.1 x 10 1*, or summed over the e n t i r e volume o f the plasma (~ 7.4 x 10 6 c m 3 ) : ~ 8.4 x 1 0 7 p a i r s . I f a l l t h e p o s i t r o n s a n n i h i l a t e i n the p l a sma , t h i s c o r r e s p o n d s t o an e q u i v a l e n t a c t i v i t y of ~ 30uCi per l i t r e . (b) L a s e r Compres s ion P l a smas : P a i r s c o u l d t h e o r e t i c a l l y be produced by t h r e e mechansims i n a l a s e r plasma s i t u a t i o n : vacuum p o l a r i z a t i o n by i n t e n s e l a s e r f i e l d s , c o l l i s i o n l e s s o s c i l l a t i o n of a f r e e e l e c t r o n d r i v e n by i n t e n s e l a s e r f i e l d s and c o l l i s i o n of e l e c t r o n s a c c e l e r a t e d by i n t e n s e l a s e r f i e l d s w i t h o t h e r plasma components . The t h r e s h o l d i n t e n s i t i e s f o r the f i r s t two p r o c e s s e s u s i n g neodymium g l a s s - l a s e r s (A. = 1.06y) are ~ 1 0 2 6 W c m " 2 and ~ 3 x 1 0 2 " W c m - 2 r e s p e c t i v e l y ( H o r a , 1973) - out o f t he range o f e x i s t i n g 70 t e c h n o l o g y . The t h i r d p r o c e s s , however , does not have such a h i g h t h r e s h o l d . E x p r e s s i o n (37) can be used to c a l c u l a t e the t o t a l number of p a i r s produced by t h i s c o l l i s i o n p r o c e s s . A c a l -c u l a t i o n of t h i s n a t u r e was o r i g i n a l l y c a r r i e d out by Bunk i n and Kazakov ( 1 970 ) , who wro te E~ as the t o t a l ( n o n - r e 1 a t i v i s t i c ) ene rgy o f an e l e c t r o n i n the f i e l d o f a p l a n e monochromat ic wave of f r e q u e n c y w and vacuum a m p l i t u d e E^: E " - + 7 ( 3 8 ) The c a l c u l a t i o n was r e p e a t e d r e l a t i v i s t i c a l l y by Hora ( 1973 ) , who c o n s i d e r e d o n l y the energy a s s o c i a t e d w i t h the e l e c t r o n o s c i l l a t i o n i n t he l a s e r f i e l d , w r i t i n g : ec L E" = 4 (39) a) n In t h i s way, he was a b l e to c o n s i s t e n t l y t a k e i n t o account the enhancement of e l e c t r o n e n e r g i e s due to the v a r i a t i o n o f the complex r e f r a c t i v e i ndex o f t he p l a s m a , n, w i t h e l e c t r o n c o l l i s i o n f r e q u e n c y . The r e s u l t s o f Np c a l c u l a t i o n s u s i n g (38) and (39) do not d i f f e r by more than an o r d e r of magn i tude because E appear s i n the l o g a r i t h m i c term o f e x p r e s s i o n ( 37 ) . U s i ng the e x p r e s s i o n f o r t o t a l e l e c t r o n energy ( 3 8 ) , t h e t o t a l p a i r p r o d u c t i o n c r o s s s e c t i o n can be w r i t t e n i n te rms of l a s e r i n t e n s i t y , I, a s : 71 2 2 r 0 z o r P P In 1 + 2TT r 0 ) I mc uv (40) where co i s t he l a s e r f r e q u e n c y . The t o t a l number o f p a i r s p roduced per u n i t t i m e , per u n i t p lasma volume i s t h e n : c n 2 2 2 r o o r TT In 2TT r. mc I U)' (41) where n i s the e l e c t r o n number d e n s i t y of the p la sma. T h i s c a l c u l a t i o n has been c a r r i e d out f o r a Nd g l a s s l a s e r p r o -duced plasma w i t h n = 1 0 2 1 c m - 3 . The v a l u e of n chosen f o r t h i s c a l c u l a t i o n i s a p p r o x i m a t e l y the c r i t i c a l d e n s i t y , above wh i ch the r e f r a c t i v e i ndex of the plasma f o r 1 . 0 6 u l i g h t becomes i m a g i n a r y and the l i g h t no l o n g e r p e n e t r a t e s the * p1 a sma. R e s u l t i n g N - v a l u e s a r e : ~ 1.4 x 1 0 2 2 s e c - 1 c m " 3 w i t h I = 5 x 1 0 1 8 W c m - 2 and - 5.6 x 1 0 2 3 s e c - 1 c m " 3 w i t h I = 5 x 1 0 1 9 W c m - 2 . In terms of p a i r s p roduced w i t h i n the f i n i t e p lasma be ing c o n s i d e r e d , t h i s t r a n s l a t e s i n t o : H i g h e r d e n s i t y p l a s m a s a r e p r e d i c t e d , however-e n e r g y t r a n s f e r by some method o t h e r t h a n by e l e c t r o m a g n e t i c f i e l d s i s e x p e c t e d to t a k e p l a c e f r o m the o u t e r ' l a s e r i n t e r -a c t i o n s h e l l 1 t o the i n n e r , h i g h d e n s i t y c o r e . The h i g h e n e r g y p a r t i c l e s in the f i e l d - f r e e c o r e s o f s u c h p l a s m a s s h o u l d make some c o n t r i b u t i o n to p a i r p r o d u c t i o n by c o l l i s i o n s , h o w e v e r , t h i s c o n t r i b u t i o n i s not b e i n g c o n s i d e r e d q u a n t i -t a t i v e l y h e r e . Such c o n s i d e r a t i o n s a r e c o m p l i c a t e d by t he f a c t t h a t t h e ' s h e l l ' to ' c o r e ' e n e r g y t r a n s f e r p r o c e s s i s not w e l l u n d e r s t o o d . 72 - 1 .4 x 1 0 5 w i t h I = 5 x 1 O 1 8 W c m " 2 and ~ 5.6 x 10 6 w i t h I = 5 x 1 0 J 9 W c m " 2 f o r a plasma w i t h a volume V = 1 0 " 6 c m 3 and a l a s e r p u l s e l e n g t h x ^ = 1 0 " 1 1 s e c . Note t h a t the r e l e v a n t t ime pa ramete r he re i s x^ (which d e f i n e s how long the l a s e r f i e l d s a re p r e s e n t ) and not x , the plasma l i f e t i m e , Which i s somewhat l o n g e r . C h a p t e r VI GAMMA RAY DETECTORS An a s s o r t m e n t o f d e t e c t o r s can be used to measure the a n n i h i l a t i o n r a d i a t i o n wh ich escapes from the p l a sma . Each type has i t s advan tage s and d i s a d v a n t a g e s and i s u s u a l l y s u i t e d f o r use w i t h a s p e c i f i c a n a l y s i s t e c h n i q u e (see Chap te r V I I ) . A. S c i n t i l l a t i o n C o u n t e r s Counte r s of t h i s t ype c o n s i s t of a s c i n t i l l a n t (phosphor ) wh ich i s o p t i c a l l y c o u p l e d to a p h o t o m u l t i p l i e r ( F i g u r e 22a, 0 u s e p h , 1 9 7 5 ) . An a n n i h i l a t i o n gamma wh i ch e n t e r s the s c i n t i l l a n t can l o s e energy to i t v i a the p h o t o -e l e c t r i c e f f e c t and the Compton e f f e c t , l e a v i n g beh i nd a t r a i l of i on p a i r s . R e c o m b i n a t i o n o c c u r s ( a l t h o u g h energy t r a n s f e r mechanisms a s s o c i a t e d w i t h i t d i f f e r f rom one phosphor to a n o t h e r ) w i t h the e m i s s i o n of v i s i b l e l i g h t pho ton s . S i n c e the phosphors a re chosen to be t r a n s p a r e n t to t h e i r own r a d i a t i o n , some f r a c t i o n o f the e m i t t e d photons i s d e t e c t e d by the p h o t o m u l t i p l i e r , wh ich a m p l i f i e s the s i g n a l 73 \ 74 Mognef ic Sh ie ld Reflector Crysta l Opticol window Photomultiplier Fig. 22 a. Scintillation type detector. Fig. 22 b. Semiconductor type detector. @-0 -J M I L l ' l ) « 7 - Lithium-diffused loyer * V P-type s i l i c o n — D r i l l e d region Evoporoled gold enlronce. window Sensitive diameter MPC 75 and t r i g g e r s a c o u n t e r . The main s c i . n t i 11 an t s s u i t a b l e f o r y r ay d e t e c t i o n a r e l i s t e d i n T a b l e I I I . Sodium i o d i d e a c t i v a t e d w i t h t h a l l i u m ( t he a c t i v a t o r d e f i n e s the e m i s s i o n w a v e l e n g t h ) i s t he most common s c i n t i l l a -t i o n type gamma ray d e t e c t o r . ' The e f f i c i e n c y of t he se d e t e c t o r s v a r i e s w i t h the energy o f the i n com ing gamma r a d i a -t i o n and w i t h the p h y s i c a l s i z e o f the c r y s t a l . A c y l i n d r i c a l c r y s t a l of Nal (T&) , 4 cm in d i a m e t e r and 2.5 cm i n l e n g t h has an e f f i c i e n c y of e - 50% f o r 500 KeV gamma pho ton s . The energy r e s o l u t i o n i s l i m i t e d by s t a t i s t i c a l f l u c t u a t i o n s a s s o c i a t e d w i t h the d e t e c t i o n and a m p l i f i c a t i o n p r o c e s s e s . A d e t e c t o r u t i l i z i n g a Nal (11) c r y s t a l w i t h the d imen s i on s quoted i s c a p a b l e o f p - 8% energy r e s o l u t i o n at gamma ray e n e r g i e s o f E(.y) - 600 KeV, wh ich c o r r e s p o n d s t o a gamma r ay l i n e w i d t h of A E p w H M - 48 KeV (Burcham, 1973, Chap te r 6 ) . Average re sponse t imes of the common s c i n t i l l a n t s a re i n d i c a t e d i n Tab l e I I I . E f f i c i e n c y , e, i s d e f i n e d as t he p e r c e n t a g e o f Y r a y s e n t e r i n g t h e d e t e c t o r w h i c h a r e d e t e c t e d in ( c o n t r i b u t e t o ) t he ' f u l l e n e r g y ' o r a n n i h i l a t i o n p h o t o p e a k ( F i g u r e 27). P e r c e n t a g e r e s o l u t i o n , p, i s d e f i n e d a s : M p W H M x 100% ETY) where AE,.,,.... i s the f u l l w i d t h a t h a l f maximum o f F WHM t he gamma ray ' l i n e ' c o r r e s p o n d i n g to a gamma ray o f e n e r g y E ( Y ) ( s ee C h a p t e r V I I , S e c t i o n D f o r more d e t a i l s ) . T a b l e I I I y Ray S c i n t i l l a n t s M a t e r i a l Form Main E m i s s i o n X (A) Response Time ( n sec ) Na l ( TA ) H y g r o s c o p i c C r y s t a l 41 00 250 A n t h r a c e n e / S t i 1 bene O r g a n i c C r y s t a l 4000 -4450 8-30 O r g a n i c P l a s t i c s Geometry as R e q u i r e d 4250 2 ( a f t e r Burcham, 1973, C h a p t e r \ 77 B. Sem iconduc to r Coun te r s A number of methods e x i s t to p roduce s e m i c o n d u c t o r s w i t h r e g i o n s o f low (or z e r o ) c o n d u c t i v i t y (no e l e c t r o n s i n t he c o n d u c t i o n band ) . A h i gh e l e c t r i c f i e l d can be p l a c e d a c r o s s such r e g i o n s of low c o n d u c t i v i t y w i t h o u t s i g n i f i c a n t c u r r e n t f l o w . Gamma r a y s p a s s i n g t h r ough t he se r e g i o n s e l e v a t e e l e c t r o n s i n t o the c o n d u c t i o n band c r e a t i n g f r e e e l e c t r o n - h o i e p a i r s . The energy r e q u i r e d to a c c o m p l i s h t h i s i s l ow: 3.6 eV and 0.67 eV per e l e c t r o n - h o i e p a i r i n s i l i c o n and germanium r e s p e c t i v e l y . Under the i n f l u e n c e of the e l e c t r i c f i e l d , the e l e c t r o n s and ho l e s m i g r a t e i n o p p o s i t e d i r e c t i o n s , the net r e s u l t be i ng a s m a l l c u r r e n t wh i ch can be d e t e c t e d and a m p l i f i e d e x t e r n a l l y and used to t r i g g e r a c o u n t e r . L i t h i u m d r i f t e d germanium d e t e c t o r s ( thn l i t h i u m compensates the a c c e p t o r c e n t r e s i n the germanium to c r e a t e r e g i o n s of low c o n d u c t i v i t y ) a re the most s u i t a b l e f o r h i gh r e s o l u t i o n gamma ray work ( F i g u r e 22b, Coche and S i f f e r t , 1968 ) . A l t h o u g h t h i s t ype o f d e t e c t o r must be c o o l e d to l i q u i d n i t r o g e n t e m p e r a t u r e s t o m i n i m i z e t he rma l e x c i t a t i o n o f e l e c t r o n s t o t h e c o n d u c t i o n band, the energy r e s o l u t i o n i s e x c e l l e n t because the unwanted ' n o i s e 1 due to s t a t i s t i c a l f l u c t u a t i o n s i s s i g n i f i c a n t l y l o w e r . An energy r e s o l u t i o n * of p = 0.6% at gamma e n e r g i e s o f 500 KeV i s t y p i c a l . T h i s H i g h e r r e s o l u t i o n ( n a r r o w e r r e s p o n s e f u n c t i o n ) i s p o s s i b l e i f p r e a m p l i f i e r n o i s e ( t h e r e s o l u t i o n l i m i t i n g f a c t o r a t gamma ray e n e r g i e s o f t h i s o r d e r ) can be r e d u c e d . F o r e x a m p l e , a r e s o l u t i o n o f p - 0.25% at E ( y ) = 500 KeV has been r e p o r t e d (Hea th et a l . 3 1966) . 78 c o r r e s p o n d s to a l i n e w i d t h o f A E r . . u y . - 3 KeV and i s more than FWHM an o r d e r of magn i tude improvement o ve r s c i n t i l l a t i o n c o u n t e r s . F i g u r e 23 ( H o l l a n d e r , 1966) i s a c ompa r i s on of a number o f t y p i c a l d e t e c t i o n systems i n te rms o f a t t a i n a b l e r e s o l u t i o n ( w i d t h of the re sponse f u n c t i o n ) . The c r y s t a l d i f f r a c t i o n c u r v e i s i n c l u d e d f o r c o m p l e t e n e s s o n l y - t h i s i s not a u s e f u l t e c h n i q u e where weak s o u r c e s a re i n v o l v e d , d u e . t o i t s i n h e r e n t i n e f f i c i e n c y ( c o u n t i n g r a t e s o f the o r d e r of 5 x 1 0 " 9 o f the s ou r ce s t r e n g t h f o r 500 KeV y r a y s ) . The e f f i c i e n c y of s e m i c o n d u c t o r d e t e c t o r s i s l ower than t h a t of s c i n t i l l a t i o n t ype d e t e c t o r s because of the d i f f e r e n c e i n s t o p p i n g power ( w h i c h . v a r i e s w i t h a tomic number) and because the p h y s i c a l s i z e o f u s e a b l e s e m i c o n d u c t o r s i s l i m i t e d by f a b r i c a t i o n p r o c e d u r e s . Depend ing p r i m a r i l y on t he volume of the d e t e c t o r , the e f f i c i e n c y v a r i e s f rom e -1% to c - 10% f o r 500 KeV gamma r a y s , as shown i n F i g u r e 24 ( C a p p e l l a n i and R e s t e l l i , 1968) . One Nal (Ta) c u r ve i s p l o t t e d f o r c o m p a r i s o n . The r e spon se t ime of s e m i c o n d u c t o r d e t e c t o r s i s i n the nanosecond r ange . C. M u l t i - W i r e P r o p o r t i o n a l C o u n t e r s R e c e n t l y , an i n gen i ou s ' gamma ray d e t e c t o r w i t h s p a t i a l r e s o l u t i o n i n two d i m e n s i o n s has been deve l oped ( Jeavons et al.3 1975) . Some p r e l i m i n a r y work w i t h a n n i h i l a -t i o n r a d i a t i o n has been c a r r i e d out and S t e w a r t (Queen ' s \ 79 Fjg. 23. Variation of detector resolution with / energy. 80 0-11 ' 1 i I i L2. 0 500 1000 1500 7 Energy (KeV) Fig. 24. Variation of detector efficiency with / energy. \ 81 U n i v e r s i t y ) i s i n c o r p o r a t i n g t h i s t ype of., d e t e c t o r i n an a n g u l a r c o r r e l a t i o n system wh ich i s to have an a n g u l a r r e s o l u -t i o n of 0.1 nirad x 0.1 mrad. T h i s i s e q u i v a l e n t t o an energy r e s o l u t i o n of p - 0 .01% at E ( y ) = 511 KeV ( c o r r e s p o n d i n g to a l i n e w i d t h o f ~ 50 eV ) . The d e v i c e c o n v e r t s i n coming gamma photons t o e l e c t r o n s v i a the p h o t o e l e c t r i c i n t e r a c t i o n . These e l e c t r o n s e scape i n t o one of many pa r a l 1 e l h o i e s d r i l l e d t h r o u g h the c o n v e r t e r and d r i f t out o f the c o n v e r t e r ( gu i ded by the h o l e s ) under the i n f l u e n c e of an e x t e r n a l e l e c t r i c f i e l d . Hav ing l e f t the c o n v e r t e r the e l e c t r o n s are counted by a m u l t i - w i r e p r o p o r t i o n a l c o u n t e r . The l a t t e r i s e s s e n t i a l l y a g r i d of i n d i v i d u a l , c l o s e l y spaced p r o p o r t i o n a l c o u n t e r s . U s i n g a g a s - s o l i d h y b r i d c o n v e r t e r ( F i g u r e 22c , Jeavons et al.> 1 975 ) , 60% of the gamma photons ( a t 1.8 MeV) t h a t i n t e r a c t a re c o u n t e d . T h i s e f f i c i e n c y i s p r o j e c t e d f o r the e n t i r e 0.1 MeV to 2.0 MeV range of i n coming gamma ray e n e r g i e s . A l t h o u g h the s i n g l e - e v e n t energy r e s o l u t i o n i s poo r , the s p a t i a l r e s o l u t i o n (wh ich i s the i m p o r t a n t f a c t o r i n a n g u l a r c o r r e l a t i o n e x p e r i m e n t s ) i s good. The l a t t e r i s ~ 1.3 mm, a p p r o x i m a t e l y equal t o the ho l e s i z e (1 .0 mm) f o r i n c i d e n t gamma ray e n e r g i e s l e s s than about 1.0 MeV. Response t ime s a r e of the o r d e r o f 500 n sec . C h a p t e r V I I ANALYSIS TECHNIQUES A. Techn i que Su rvey As a f i n a l gauge o f t h e common a n n i h i l a t i o n r a d i a t i o n a n a l y s i s t e c h n i q u e s , and i n l i g h t o f the p r e c e d i n g d i s c u s s i o n s c o n c e r n i n g s o u r c e s ( s t r e n g t h s ) and d e t e c t o r s ( r e s o l u t i o n s ; e f f i c i e n c y ) , gamma ray c o u n t i n g r a t e c a l c u l a t i o n s a r e c a r r i e d o u t . The i n i t i a l c a l c u l a t i o n s f o r both Tokamak and l a s e r com-p r e s s i o n p lasmas a r e g e n e r a l i n n a t u r e , however subsequent c a l -c u l a t i o n s r e l a t e t o s p e c i f i c t e c h n i q u e s . The f o u r t e c h n i q u e s c o n s i d e r e d (measurements o f : 2y a n g u l a r c o r r e l a t i o n , a n n i h i l a -t i o n l i n e w i d t h , p o s i t r o n l i f e t i m e and s l ow p o s i t r o n beam sp r ead ) a r e i l l u s t r a t e d s c h e m a t i c a l l y i n F i g u r e 25. In each c a s e , a s c h e m a t i c raw da ta cu r ve i s shown and d e r i v e a b l e f u n c t i o n s or pa r amete r s i n d i c a t e d . He re , c i s the coun t r a t e , N g _ i s t he e l e c t r o n number d e n s i t y and f (v) i s t he e l e c t r o n v e l o c i t y d i s t r i b u t i o n . B. Gene r a l C o u n t i n g Rate C a l c u l a t i o n s C o n s i d e r i n g o n l y p o s i t r o n s p roduced i n a p lasma *and assuming t h a t n e a r l y a l l t h e se p o s i t r o n s a n n i h i l a t e i n 82 83 2J ANGULAR CORRELATION Nal detectors fp-(v) = g(0) ANNIHILATION LINE WIDTH plasma Ge(Li) detector fe-(v) = g(AE) POSITRON LIFETIME POSITRON BEAM SPREAD birth signal death signal slow positron beam counter N e r = g W Ne-=g(r) Fig. 25. Analysis techniques. 8 4 t h i s p l a sma , the number o f coun t s (gamma photons ) r e g i s t e r e d d u r i n g one plasma l i f e t i m e can be w r i t t e n as : 2A e GM a (42) where A i s the number o f p o s i t r o n s produced d u r i n g the plasma l i f e t i m e i n the volume be ing c o n s i d e r e d , e i s the d e t e c t o r e f f i c i e n c y , G i s the s o l i d ang l e f a c t o r t h a t i n d i c a t e s what p e r c e n t a g e of gamma r a y s p roduced i n the plasma volume be ing c o n s i d e r e d e n t e r s the d e t e c t o r , and M i s a m i s c e l l a n e o u s a t t e n u a t i o n f a c t o r t h a t i s meant t o a c coun t f o r : * (a) p o s i t r o n s e s c a p i n g f rom the p l a sma , (b) p o s i t r o n s a n n i h i l a t i n g a f t e r the plasma has ceased to e x i s t , ( c ) a b s o r p t i o n of gamma r a y s by i n t e r -ven i n g m a t e r i a l , (d) v a r i o u s gamma ray s c a t t e r i n g c o r r e c t i o n s . The f a c t o r o f 2 a c c o u n t s f o r the f a c t t h a t each a n n i h i l a t i o n even t c r e a t e s two gamma pho ton s . F o r t h e p u r p o s e o f t h e s e o r d e r o f m a g n i t u d e c a l -c u l a t i o n s M_ v a l u e s o f 0 . 1 (Tokamak) and 0.5 ( l a s e r c o m p r e s s i o n ) have been c n o s e n , a s s u m i n g t h a t m i s c e l l a n e o u s l o s s e s a s s o c i a t e d '* w i t h the w e l l l o c a l i z e d l a s e r c o m p r e s s i o n p l a sma a r e l e s s t h a n t h o s e a s s o c i a t e d w i t h t he more d i f f u s e Tokamak p l a s m a . E x a c t M a v a l u e s can be d e t e r m i n e d o n l y when t h e c o m p l e t e e x p e r i m e n t a l l a y o u t i s known. 85 ( i ) Tokamak Plasma C i s c a l c u l a t e d f o r t he f o l l o w i n g h y p o t h e t i c a l a r r angement : D e t e c t o r E f f i c i e n c y e = 10% Plasma Volume V i s i b l e t o D e t e c t o r T o t a l number o f A n n i h i l a t i o n s i n t h i s Volume d u r i n g Plasma 1 i f e t i me (10~ 2 sec ) D e t e c t o r D i s t a n c e f rom C e n t r e o f P lasma S o l i d Ang l e F a c t o r M i s c e l l a n e o u s A t t e n u a t i o n F a c t o r 50 cm x 50 cm x 50 cm = 1 . 25 x 1 0 5 cm 3 (~ 1.7% o f t o t a l vo lume) A = 1.4 x 1 0 6 ( f rom Chap te r V, S e c t i o n C) 100 cm G = 2% M. 0.1 D e t e c t o r D imens ions << P lasma D imens i on s C a l c u l a t e d C o u n t i n g Rate C - 560 c o u n t s per d i s c h a r g e T h i s count r a t e i s s u f f i c i e n t l y h i g h , t h a t o f t he a n a l y s i s t e c h n i q u e s o u t l i n e d i n t h i s c h a p t e r , a t l e a s t the a n n i h i l a t i o n l i n e b r oaden i n g t e c h n i q u e s hou l d be f e a s i b l e . ( i i ) L a s e r Compres s i on Plasma The pa ramete r s used to c a l c u l a t e C a r e : D e t e c t o r E f f i c i e n c y D e t e c t o r S u r f a c e A rea e - 10% 7 cm 2 86 P l a s m a - D e t e c t o r D i s t a n c e 20 cm S o l i d Ang l e F a c t o r G = 0.14% T o t a l Number of A n n i h i l a t i o n s A = 1.4 x 1 0 5 ( f o r I -i n E n t i r e Plasma Volume d u r i n g 5 x 1 0 1 8 W c m - 2 , C h a p t e r V, P lasma L i f e t i m e ( 1 0 ~ 9 s ec ) S e c t i o n C) M i s c e l l a n e o u s A t t e n u a t i o n F a c t o r M = 0.5 Plasma D imens ions << D e t e c t o r D imens ions C ~ 20 coun t s per plasma d e t o n a t i o n The c o u n t i n g r a t e i n c r e a s e s l i n e a r l y w i t h the s u r f a c e a rea of the d e t e c t o r , thus the use o f a number of d e t e c t o r s i n p a r a l l e l would s i g n i f i c a n t l y enhance the c o u n t i n g r a t e . Fo r a 2TT d e t e c t o r (G = 50% ) , w i t h the plasma at t he c e n t r e , C -7,000 coun t s per p lasma d e t o n a t i o n . Any i n c r e a s e i n l a s e r power and hence A w i l l a l s o i n c r e a s e C. A n n i h i l a t i o n gamma r a d i a t i o n from l a s e r c o m p r e s s i o n p lasmas s hou l d be d e t e c t a b l e and i n t e n s e enough to f a c i l i t a t e at l e a s t one t y p e o f a n a l y s i s . C. 2y A n g u l a r C o r r e l a t i o n The t h e o r y of t h i s t e c h n i q u e has been o u t l i n e d i n Chap te r I I , S e c t i o n ( C ) ( i ) , thus i t rema in s to e v a l u a t e i t s a p p l i c a b i l i t y t o the Tokamak or l a s e r c o m p r e s s i o n s y s tems . a re ( i ) TOKAMAK Plasma I f two d e t e c t o r s a re u s ed , and o n l y y c o u n t e d , then the c o i n c i d e n c e c o u n t i n g r a t e c o i n c i d e n c e s i s a p p r o x i m a t e l y : 8 7 C c = 4A £ l e 2 G XG 2 , ( 4 3 ) where the s u b s c r i p t s i d e n t i f y the d e t e c t o r s . No s p a t i a l ( o r a n g u l a r ) r e s o l u t i o n i s p o s s i b l e w i t h the d e t e c t o r a r r a n g e -ment quoted i n S e c t i o n ' (B) ( i ) o f t h i s c h a p t e r . In o r d e r t o measure ang l e s i n the mi 1 1 i r a d i a n range , t he g e o m e t r i c a l r e s o l u t i o n o f the system must be a t l e a s t o f t h i s o r d e r . T h i s i m p l i e s : (a) a d e t e c t o r w i t h s l i t w i d t h : - 1 mm s a m p l i n g a plasma c h a r a c t e r i z e d by l i n e a r d i m e n s i o n s o f the o r d e r of mm, from a d i s t a n c e o f 100 cm o r (b) a s l i t w i d t h o f - 1 cm at a d i s t a n c e of 10 m s amp l i n g a p lasma c h a r a c t e r i z e d by c e n t i m e t r e d i m e n s i o n s . ( R e c a l l t h a t s o u r c e and d e t e c t o r need be w e l l d e f i n e d i n o n l y one d i m e n s i o n s i n c e o n l y one component of the e l e c t r o n - p o s i t r o n c e n t r e of mass i s be i n g measu red . ) F o r a d e t e c t o r 10 cm l o n g , the s o l i d a n g l e f a c t o r s a r e G :, 2 - 1 0 - 3 % and G i , 2 - 10 _ 1*% r e s p e c t i v e l y f o r g e o m e t r i e s (a) and (b) ment i oned above. In case ( a ) , A ^ 1 0 1 ; t h u s , even c h o o s i n g £i = e 2 = 100%, the t o t a l number of c o i n c i d e n c e coun t s per Tokamak d i s c h a r g e i s C - 1 0 - 8 ( m i s c e l l a n e o u s a t t e n u a t i o n i g n o r e d ) . C l e a r l y , a n g u l a r c o r r e l a t i o n measu re -ments of the t ype d e s c r i b e d here a re not f e a s i b l e a t t he p o s i t r o n d e n s i t y c a l c u l a t e d t o e x i s t d u r i n g a t y p i c a l Tokamak d i s c h a r g e . T h i s d e n s i t y c o u l d be i n c r e a s e d by somehow i n j e c t -i n g p o s i t r o n s i n t o the plasma f rom o u t s i d e , or dop i ng the d i s c h a r g e gas w i t h a 3 + decay r a d i o n u c l i d e , however i t i s d o u b t f u l whether i t c o u l d be i n c r e a s e d by t he r e q u i s i t e e i g h t 8 8 o r so o r d e r s of magn i tude to y i e l d a r e a s o n a b l e number of c o i n c i d e n c e coun t s per d i s c h a r g e . ( i i ) L a s e r Compres s i on Plasma R e p l a c i n g the ex tended Tokamak s ou r ce by the l a s e r c o m p r e s s i o n p o i n t s ou rce (A = 5.6 x 1 0 5 c o r r e s p o n d i n g to a Nd l a s e r i n t e n s i t y : I - 5 x 1 0 1 9 W c m - 2 ) , but r e t a i n i n g the e and G f a c t o r s of t he p r e v i o u s c a l c u l a t i o n (same a n g u l a r r e s o l u t i o n r e q u i r e d ) , e x p r e s s i o n (43) i n d i c a t e s C - 2 x 1 0 " 3 c o i n c i d e n c e coun t s per d e t o n a t i o n . T h i s i s the optimum count r a t e wh ich would i n p r a c t i c e be reduced by the l e s s than p e r f e c t d e t e c t o r e f f i c i e n c i e s and by m i s c e l l a n e o u s l o s s e s . As . p o i n t e d out i n Chap te r V, S e c t i o n ( C ) ( i ) , i t i s not p o s s i b l e t o s i g n i f i c a n t l y i n c r e a s e the p o s i t r o n d e n s i t y i n t he plasma by any c o n v e n t i o n a l means ( e . g . r a d i o n u c l i d e d o p i n g ) , due to the s h o r t t ime s c a l e s i n v o l v e d . S i n c e i t i s not p r a c t i c a l ( a t t h i s t i m e ) t o sum o ve r t he o r d e r of t h o u s a n d s - o f d e t o n a t i o n s , one d i m e n s i o n a l a n g u l a r c o r r e l a t i o n a n a l y s i s of l a s e r c o m p r e s s i o n p lasmas must be r u i e d o u t . The e x a c t s i t u a t i o n w i t h r e s p e c t to t w o - d i m e n s i o n a l a n g u l a r c o r r e l a t i o n s u s i n g m u l t i - w i r e p r o p o r t i o n a l c o u n t e r s i s not c l e a r , s i n c e such a system does not y e t e x i s t . One o f t he advan tage s of u s i n g t w o - d i m e n s i o n a l m u l t i - w i r e p r o p o r t i o n a l c o u n t e r s i s t h a t the c o u n t i n g r a t e i n c r e a s e s s i g n i f i c a n t l y due to t h e i r i n h e r e n t s e n s i t i v i t y and the i n c r e a s e i n t h e 89 s o l i d a n g l e f a c t o r . I f an e f f i c i e n t 2D a n g u l a r c o r r e l a t i o n sys tem can be deve l oped to r e p l a c e t he c u r r e n t ID, ' l o n g s l i t ' s y s t e m s , a n g u l a r c o r r e l a t i o n s t u d i e s may be p o s s i b l e w i t h some f u s i o n p l a smas . An a d d i t i o n a l advantage o f the 2D t e c h n i q u e i s t h e f a c t t h a t i t would make a v a i l a b l e i n f o r m a t i o n about two components of the p o s i t r o n - e l e c t r o n c e n t r e o f mass momentum s i m u l t a n e o u s l y . In p lasmas ( u n l i k e s o l i d s ) the i n d i v i d u a l p o s i t r o n and e l e c t r o n c o n t r i b u t i o n s t o the c e n t r e of mass momentum would be r o u g h l y e q u a l , n e c e s s i t a t i n g the a p p l i c a t i o n of some t y p e o f u n f o l d i n g p r o c e d u r e t o s e p a r a t e the c o n t r i b u t i o n s (more d e t a i l s i n the nex t s e c t i o n ) . D. Dopp l e r B r oaden i n g and S h i f t o f the A n n i h i l a t i o n L i n e Perhaps the most p r o m i s i n g p o s i t r o n t e c h n i q u e t h a t c o u l d be used to s t udy p lasmas i n v o l v e s the a n a l y s i s o f t he D o p p l e r b r o a d e n i n g o f the 511 KeV a n n i h i l a t i o n l i n e . A number o f " l i n e - b r o a d e n i n g " s t u d i e s have been c a r r i e d out w i t h p o s i t r o n s a n n i h i l a t i n g i n m e t a l s (Du Mond et al. , 1949; Mu r r a y , 1967) . A l t h o u g h Dopp l e r b r oaden i n g was measured i n t he se e x p e r i m e n t s , e x c e s s i v e l y h i gh s ou r ce a c t i v i t y (Du Mond et al.) and low d e t e c t o r r e s o l u t i o n (Mur ray ) l i m i t t he u s e f u l n e s s o f t he se t e c h -n i que s f o r s o l i d - s t a t e d i a g n o s t i c p u r p o s e s . T h i s i s not t h e ca se i n many p l a s m a s , where the r e s o l u t i o n p r ob l em does not e x i s t . T h i s i s an a d v a n t a g e o n l y in c a s e s where t h e momentum d i s t r i b u t i o n i s e x p e c t e d t o be a n i s o t r o p i c . In i s o t r o p i c s i t u a -t i o n s a p a r a l l e l w i r e p r o p o r t i o n a l c o u n t e r ( m u l t i - s l i t s y s t e m ) wo u l d s u f f i c e . N o n e t h e l e s s , some work o f t h i s n a t u r e has been a t t e m p t e d ; s e e , f o r e x a m p l e , Ho tz et al. (1968) and H a c K e n z i e (1969) -90 ( i ) Theory The t o t a l energy of one a n n i h i l a t i o n gamma p h o t o n , as measured by an o b s e r v e r at r e s t w i t h r e s p e c t to t he p l a sma , i s g i v e n by e x p r e s s i o n (.61) (Append ix B ) . I t i s p o s s i b l e t o a n a l y z e t h i s energy i n te rms of t h r e e e f f e c t s c o n t r i b u t i n g to i t s magn i t ude . I t s a p p r o x i m a t e magn i tude i s d e t e r m i n e d by E 0 = m c 2 , one h a l f of the p a i r r e s t mass e n e r g y . The t he rma l mot i on of the p r e - a n n i h i 1 a t i o n p a i r m a n i f e s t s i t s e l f i n two d i f f e r e n t ways: (a) i t causes an a b s o l u t e s h i f t i n t h e energy o f each gamma photon (away f rom E 0 = m c 2 ) o f t he o r d e r o f : <5E - = ^ mv , (44) 2 cm ' v y cm (see e x p r e s s i o n ( 6 3 ) , Append i x B) where v c m i s the mean c e n t r e o f mass v e l o c i t y of t he p r e - a n n i h i 1 a t i o n p a i r ; and (b) i t i n t r o d u c e s a Dopp l e r s h i f t i n the energy of a g i v e n gamma photon wh ich depends on t he magn i tude of t he v e l o c i t y component o f the p r e - a n n i h i 1 a t i o n p a i r i n the d i r e c t i o n o f e m i s s i o n o f t h a t gamma photon ( v c m ) - The magn i tude o f t h i s s h i f t i s : ss. AE D = Eo ^ , ' • (45) where v c m i s t a ken r e l a t i v e t o a f i x e d d e t e c t o r ( see e x p r e s s i o n ( 6 4 ) , Append ix B ) . 91 The E 0 te rm l o c a t e s the ' a n n i h i l a t i o n l i n e ' a t an energy of about 511 KeV, w h i l e SE ( e x p r e s s i o n ( 44 ) ) m a n i f e s t s i t s e l f as a s h i f t o f the e n t i r e l i n e t o a h i g h e r ene rgy . The mean a b s o l u t e s h i f t , as a f u n c t i o n of the mean plasma e l e c t r o n energy ( k T g ) i s p l o t t e d i n F i g u r e 26 as the ' s h i f t ' c u r ve ( d e t a i l s i n p a r t ( i i ) o f t h i s , s e c t i o n ) . C l e a r l y , i t i s i m p o s s i b l e t o measure the s h i f t a t low plasma t e m p e r a t u r e s and at e l e c t r o n t e m p e r a t u r e s c h a r a c t e r i s t i c o f s o l i d s ( e . g . the f r a c t i o n a l s h i f t <5E/E0 i s ~ 4 x 1 0 _ 3 % a t a mean plasma t e m p e r a t u r e o f ~ 1 0 5 0 K o r ~ 10 eV ) . A l t h o u g h i t may be p o s s i b l e to measure a b s o l u t e s h i f t s i n more e n e r g e t i c p lasmas and thus d e t e r m i n e mean plasma t e m p e r a t u r e s , t h i s i s not t he most p r o m i s i n g a p p r o a c h . The AE D te rm ( e x p r e s s i o n ( 45 ) ) d e s c r i b e s the b r o a d e n i n g o f the v c m = 0 d e l t a f u n c t i o n p r o f i l e . W r i t i n g t h e f r a c t i o n a l l i n e b r o a d e n i n g a s : A E n v D _ cm / Eo c ' 1 i t i s e v i d e n t t h a t i n s o l i d s , where mean e l e c t r o n e n e r g i e s a r e o f t he o r d e r of e l e c t r o n v o l t s , t he mean a n n i h i l a t i o n l i n e b r oaden i n g w i l l be l e s s than 1% ( ^ E Q o f t he o r d e r o f a few KeV) . The r e s o l u t i o n o f the p r e s e n t day s e m i c o n d u c t o r F i g u r e 2 6 , w h i c h i s a p l o t o f 2 / l n 2 AE^ a g a i n s t kT ( b r o a d e n i n g c u r v e ) , g i v e s an i n d i c a t i o n o f how AEn v a r i e s w i t h e l e c t r o n t e m p e r a t u r e , T ( see p a r t ( i i ) o f t h i s s e c t i o n f o r mo re d e t a i 1 s ) . Fig. 26. Variation of Doppler broadening and shift with plasma kT e . 93 d e t e c t o r s wh ich c o u l d be used to measure a n n i h i l a t i o n l i n e p r o f i l e s i s of t he same o r d e r of m a g n i t u d e , mak ing e x a c t d e t e r m i n a t i o n of b r oaden i ng and l i n e p r o f i 1 e s d i f f i c u 1 t . The s i t u a t i o n i n p lasmas i s much more f a v o u r a b l e , because mean e l e c t r o n t e m p e r a t u r e s a re c o n s i d e r a b l y h i g h e r t han i n s o l i d s ; i n f a c t , i n some f u s i o n p lasmas e l e c t r o n t e m p e r a t u r e s may r each 1 0 8 ° K (mean e l e c t r o n e n e r g i e s o f t he o r d e r o f 10 KeV) At t h e s e t e m p e r a t u r e s , the mean f r a c t i o n a l l i n e b r o a d e n i n g , AEp/Eo amounts to ~ 20% - e a s i l y d e t e c t a b l e . Even a t e l e c t r o n t e m p e r a t u r e s as low as 1 0 6 ° K ( k T g ~ 100 eV) t h e l i n e b r o a d e n -i n g i s e a s i l y d e t e c t a b l e . In a d d i t i o n t o the a n n i h i l a t i o n l i n e w i d t h , t he shape i s a l s o of i n t e r e s t . When the l y c o n t r i b u t i o n s can be i g n o r e d ( low p r o b a b i l i t y - see Append i x A ) , t he t h e o r e t i c a l 2y a n n i h i l a t i o n spec t rum f o r p o s i t r o n s and e l e c t r o n s a n n i h i l a t i n g a t r e s t i s a d e l t a f u n c t i o n a t E (y ) = m c 2 . I f s u c h . s t a t i c s i t u a t i o n s e x i s t ( i t seems t h i s may be the case i n i c e , d e Z a f r a and J o y n e r , 1958 ) , then the measured w i d t h o f the a n n i h i l a t i o n l i n e i n t h e s e ca se s i s e n t i r e l y a r e s u l t o f i n s t r u m e n t b r oaden i n g ( i . e . i t i s j u s t t he i n s t r u m e n t r e s p o n s e f u n c t i o n ) . In t he u sua l s i t u a t i o n where t he p o s i t r o n - e l e c t r o n c e n t r e of mass v e l o c i t y i s not z e r o , t he measured p r o f i l e i s a c o n v o l u t i o n of t he i n s t r u m e n t p r o f i l e and t he D o p p l e r broadened d e l t a f u n c t i o n p r o f i l e . S i n c e the D o p p l e r p r o f i l e P r a c t i c a l l y s p e a k i n g , t he a n n i h i l a t i o n s p e c t r u m ( F i g u r e 27) c o n s i s t s o f a Compton c o n t i n u u m ( w h i c h r e s u l t s b e c a u s e some o f t he a n n i h i l a t i o n gammas s u f f e r Compton s c a t t e r -i ng and t h e a s s o c i a t e d e n e r g y l o s s b e f o r e b e i n g d e t e c t e d ) photopeak possible enhancement of Compton continuum by 3/ annihilation. . -^ = » possible enhancement of —*/ AE r -/ F W H M \ 1 high energy wing by annihilation in flight. / A > 511 Kev gamma ray energy —*> •*>—5E Fig. 27. Annihilation gamma ray spectrum. 95 r e s u l t s f rom the p o s i t r o n - e l e c t r o n c e n t r e of mass m o t i o n , i t i s e s s e n t i a l l y a r e c o r d o f the p o s i t r o n - e 1 e c t r o n c e n t r e o f mass v e l o c i t y d i s t r i b u t i o n and t h e r e f o r e i m p l i c i t l y o f the e l e c t r o n v e l o c i t y d i s t r i b u t i o n . ( i i ) A n a l y s i s A s i d e from the s c a l e f a c t o r E 0 / c , the a n n i h i l a t i o n l i n e D o p p l e r p r o f i l e , I D ( F - h i s j u s t w c m ( v ) , the p o s i t r o n -e l e c t r o n c e n t r e o f mass v e l o c i t y d i s t r i b u t i o n i n one component ( d e f i n e d by the d e t e c t o r l o c a t i o n ) . I f w i s i s o t r o p i c , then f c m ( v ) > the d i s t r i b u t i o n i n magn i tude of c e n t r e o f mass v e l o c i t y , f o l l o w s by i n t e g r a t i o n ove r v e l o c i t y s pace . I t i s i m p o r t a n t to note t h a t both w (v) and f (v ) r cm cm ' a r e a c t u a l l y c o n v o l u t i o n s o f the two c o r r e s p o n d i n g p o s i t r o n and e l e c t r o n f u n c t i o n s : w + ( v ) , w _ ( v ) or f + ( v ) , f _ ( v ) . G G G G I f f + and f _ a re i d e n t i c a l , as one would expec t them to be e e r a f t e r c omp le te p o s i t r o n t h e r m a l i z a t i o n , d e c o n v o l u t i o n or u n f o l d i n g t o d e t e r m i n e f ( v ) , t he e l e c t r o n v e l o c i t y d i s t r i b u -t i o n o f i n t e r e s t i s s t r a i g h t f o r w a r d . O t h e r w i s e , t he f u n c t i o n a l fo rm o f f + must be known i n o r d e r t o f i n d f e T e -and the p h o t o p e a k ( t h e a n n i h i l a t i o n l i n e i t s e l f ) , s e p a r a t e d by a w e l l d e f i n e d v a l l e y . Any gamma p h o t o n s r e s u l t i n g f r o m 3y a n n i h i l a t i o n ( e n e r g i e s d i s t r i b u t e d f r o m 0 to 511 KeV, F i g u r e 2) w i l l i n c r e a s e t h e i n t e n s i t y o f t h e Compton d i s t r i b u t i o n and t h e v a l l e y r e l a t i v e to t h a t o f t h e p h o t o p e a k . A n n i h i l a t i o n o f p o s i t r o n s in f l i g h t ( p r i o r t o t h e r m a l i z a t i o n ) w o u l d be \ s i g n a l l e d by an i n t e n s i t y i n c r e a s e o f t h e h i g h e n e r g y w ing o f t h e a n n i h i l a t i o n l i n e . The l a t t e r e f f e c t i s e x p e c t e d t o be n e g l i g i b l e in p l a s m a s . 96 The f o l l o w i n g i s an o u t l i n e o f the c o m p l e t e p r o -c e d u r e to d e t e r m i n e f ( v ) : e (a) Measure t he a n n i h i l a t i o n p r o f i l e : I m ( E ) . T h i s can be w r i t t e n as : where I.. i s t he i n s t r u m e n t r e spon se f u n c t i o n and I ^ i s t he t r u e Dopp l e r p r o f i l e . (b) U n f o l d I m ( E ) t o d e t e r m i n e I p ( e ) . T h i s i s s t r a i g h t -f o r w a r d u s i n g F o u r i e r t e c h n i q u e s . ( c ) Make the a p p r o p r i a t e s c a l e change to c o n v e r t I n ( E ) (d) C a l c u l a t e f c m ( v ) f rom w c m ( v ) by i n t e g r a t i n g o ve r v e l o c i t y space . The f u n c t i o n f" ( v ) can now be w r i t t e n a s : l.(E - E ' ) I D ( E ' ) dE ' (47) — oo t o w (V) cm v ' cm f e + ( v " v ' ) f e - ( v ' ) d v ' » (48) — CO where f + e (magn i tud e) d i s t r i but and f e~ a r e ons r e s p e c t i v e l y . i n d i v i d u a l p o s i t r o n and e l e c t r o n v e l o c i t y 97 U n f o l d f (v) t o d e t e r m i n e f _ ( v ) . U n l e s s cm / e , f + m u s t be known, e A somewhat l e s s d e t a i l e d a n a l y s i s can be c a r r i e d out by l o o k i n g a t l i n e w i d t h s o n l y . The HWHM of I^  (E) or w (v) c a n b e t a k e n a s an i n d i c a t i o n of t h e mean k i n e t i c cm energy a s s o c i a t e d w i t h the c e n t r e o f mass mo t i on o f t h e p o s i t r o n e l e c t r o n p a i r s . T h i s i s i n t u r n r e l a t e d to the mean k i n e t i c energy a s s o c i a t e d w i t h e l e c t r o n mo t i on s and hence to the mean e l e c t r o n t e m p e r a t u r e . Fo r examp le , i f w ( v ) i s a G a u s s i a n w i t h a FWHM r cm o f Av and both f + and f _ a re G a u s s i a n s w i t h FWHM of Av cm • e + e e and Av then A v c m = A v e + 2 + A v g _ 2 (49) I f f + = f _ then e + e Av 2 A v e - 2 = - f 1 - < 5°) The FWHM of I D ( E ) i s u s u a l l y w r i t t e n i n terms o f the D o p p l e r w i d t h a s : A EFWHM - 2 / W ^ A E D (51) 9 8 Us i n g e x p r e s s i o n (46) f o r A E ^ , e x p r e s s i o n (50) t o l i n k p a i r c e n t r e o f mass and i n d i v i d u a l e l e c t r o n v e l o c i t i e s , and w r i t i n g : A v e -v = — % 2k T ej m ( 5 2 ) f o r t h e HWHM v e l o c i t y a s s o c i a t e d w i t h f ( v ) , e x p r e s s i o n ( 5 1 ) can be s o l v e d f o r T e 2 T - , A E FHHM • ( t . . ' e - (4 In 2). 4k mc* { 0 *> where k i s B o l t z m a n ' s c o n s t a n t . The ' b r o a d e n i n g 1 c u r v e o f F i g u r e 26 i s a p l o t of M F W H M a g a i n s t kT g _ - i t s p e c i f i e s how the a n n i h i l a t i o n l i n e w i d t h changes w i t h plasma e l e c t r o n t e m p e r a t u r e . The v a r i a -t i o n o f the a b s o l u t e s h i f t of the a n n i h i l a t i o n l i n e w i t h kT g _ has a l s o been p l o t t e d i n F i g u r e 26, w i t h t h e a i d o f e x p r e s s i o n s ( 4 4 ) , (50) and ( 5 2 ) , wh i ch i n d i c a t e ( i f v c m i s t a k e n to be Av /2) t h a t f o r a g i v e n v a l u e of T _ , the cm ' 3 e a p p r o x i m a t e s h i f t i s j u s t 2 k T g _ . ( i i i ) A s sessment v. The s u c ce s s of the 1 i n e - b r o a d e n i n g t e c h n i q u e depends on b e i n g a b l e t o a c c u r a t e l y measure the a n n i h i l a t i o n l i n e shape , o r at l e a s t i t s w i d t h . D e t e c t o r s w i t h t he r e q u i r e d 9 9 r e s o l u t i o n o f 1 to 3 KeV at FWHM e x i s t . The c a l c u l a t i o n s o f Chap te r V I I , S e c t i o n B, i n d i c a t e t h a t c u r r e n t l y a v a i l a b l e G e ( L i ) d e t e c t o r s s hou l d be s e n s t i v e enough t o d e t e c t s t a -t i s t i c a l l y s i g n i f i c a n t amounts o f a n n i h i l a t i o n r a d i a t i o n f rom p o s i t r o n s c r e a t e d i n Tokamak and l a s e r c o m p r e s s i o n p l a smas . I t s h o u l d be p o s s i b l e to measure a c c u r a t e l i n e p r o f i l e s by i n t e g r a t i n g over a few c y c l e s o f e i t h e r o f t he se p lasmas o r by c o u n t i n g w i t h more than one d e t e c t o r s i m u l -t a n e o u s l y (based on c a l c u l a t e d c o u n t i n g r a t e s ) . The l i n e -b r o a d e n i n g t e c h n i q u e appear s f e a s i b l e . E. P o s i t r o n L i f e t i m e Measurements The e s s e n t i a l s of measu r i ng p o s i t r o n l i f e t i m e s have been o u t l i n e d i n Chap te r I I , S e c t i o n ( C ) ( i i ) . Measu re -ments o f p o s i t r o n l i f e t i m e s i n p lasmas of known e l e c t r o n d e n s i t y c o u l d r e s o l v e an i m p o r t a n t t h e o r e t i c a l q u e s t i o n : how l a r g e i s the Coulomb c o r r e c t i o n to the a n n i h i 1 a t i o n c r o s s s e c t i o n (and hence to the a n n i h i l a t i o n t i m e ) o f p o s i t r o n s i n t e r a c t i n g w i t h a ' s e a ' o f unbound e l e c t r o n s , . t hemse l ve s e x h i b i t i n g b i n a r y and c o l l e c t i v e i n t e r a c t i o n s ? Is W o l f e r ' s t h e o r e t i c a l p r e d i c t i o n o f t he Coulomb c o r r e c t i o n a c c u r a t e ? ( C h a p t e r IV, S e c t i o n C ) . P o s i t r o n l i f e t i m e measurements c o u l d be used to a s c e r t a i n plasma e l e c t r o n d e n s i t i e s once t h e s e q u e s t i o n s have been an swe red , t h a t i s , once the c o r r e c t f u n c t i o n a l r e l a t i o n s h i p between a n n i h i l a t i o n t i m e s and p lasma e l e c t r o n d e n s i t i e s has been d e t e r m i n e d . 1 00 A l t h o u g h the measurement of 1 i f e t i m e s i n vo1ve s d e l a y e d c o i n c i d e n c e c o u n t i n g ( c o u n t i n g p o s i t r o n c r e a t i o n and a n n i h i l a t i o n p u l s e s i n c o i n c i d e n c e and keep i ng t r a c k of the e l e c t r o n i c a l l y i n t r o d u c e d d e l a y t imes t o o b t a i n c o i n c i d e n c e ) , i t does not r e q u i r e the ext reme s p a t i a l r e s o l u t i o n demanded by a n g u l a r c o r r e l a t i o n methods , t h u s , c o u n t i n g r a t e s a re h i g h e r . An a c c u r a t e t ime mark s i g n a l l i n g the c r e a t i o n o f each p o s i t r o n i s n e c e s s a r y f o r the s u c ce s s of t h i s method. S i n c e t h i s i s d i f f i c u l t t o a r r a n g e i n a l a s e r p lasma system ( s i z e and t ime s c a l e s be ing what they a r e ) , p o s i t r o n l i f e -t i m e measurements i n such a sys tem do not seem f e a s i b l e . In a Tokamak p l a s m a , p o s i t r o n s f rom e x t r i n s i c s o u r c e s c o u l d be u sed , c r e a t i o n ' p u l s e s be ing s u p p l i e d by a t h i n s c i n t i l l a t o r p l a c e d between the s ou r ce and t he plasma o r by the n a t u r a l 1.28 MeV gamma ray wh ich s i g n a l s the 3 + 2 2 decay of Na ( Chap te r V, S e c t i o n ( C ) ( i ) ) . The e x p e c t e d count r a t e can be c a l c u l a t e d a p p r o x i m a t e l y u s i n g e x p r e s s i o n (43) w i t h : e\ = e2 = 50% ( h i g h e f f i c i e n c y N a I ( T £ ) d e t e c t o r s : t he r e l a t i v e l y s l ow t ime re sponse i s not a d e t e r r e n t here because a n n i h i l a t i o n t i m e s s h o u l d be o r d e r s of magn i tude l o n g e r ) G i - 2% ( the s o l i d a n g l e f a c t o r f o r the d e t e c t o r g e n e r a t i n g p o s i t r o n c r e a t i o n p u l s e s can v a r y g r e a t l y depend ing on the e x a c t d e t e c t i o n method and geometry c h o s e n ; f o r example Gi - 50% f o r a s ou r ce nex t t o the d e t e c t o r -a common a r r angement ) 101 G2 - 2% ( s o l i d a n g l e f a c t o r f o r the d e t e c t o r genera t -ing p o s i t r o n a n n i h i l a t i o n p u l s e s - same as i n f i r s t c a l c u l a t i o n ) A ~ 3 x 1 0 6 (number o f p o s i t r o n s a n n i h i l a t i n g i n a 50 cm 50 cm x 50 cm volume d u r i n g the plasma l i f e -t i m e , assuming t h a t a l l t he p o s i t r o n s f rom a 500 mCi e x t r i n s i c s ou r ce a re r e t a i n e d by the p lasma and homogeneously d i s t r i b u t e d t h e r e i n ) ~ 0.1 (assume a one o r d e r of magn i tude s i g n a l l o s s due to m i s c e l l a n e o u s causes as i n f i r s t c a l c u l a t i o n ) . These v a l u e s g i v e a c o i n c i d e n c e count r a t e o f C - 120 c oun t s 3 c pe r d i s c h a r g e . I f t h i s s i t u a t i o n i s r e a l i z e a b l e , p o s i t r o n l i f e t i m e measurements i n t h i s t y p e o f p lasma a re f e a s i b l e ; however, i m p o r t a n t t e c h n i c a l prob lems r e m a i n . B e f o r e measure-ments o f t h i s n a t u r e can be c o n d u c t e d , t h e f o l l o w i n g p rob lems must be s o l v e d : ( a ) i n t r o d u c i n g t h e p o s i t r o n s in.to t h e p l a s m a w i t h o u t m a j o r l o s s e s ( i t m i g h t b e i p o s s i b ' l e t o dope t h e p l a s m a ' f u e l ' w i t h t h e a p p r o p r i a t e amount o f a B + r a d i o n u c l i d e b e f o r e t h e d i s c h a r g e -b u t t h i s comp l i c a t e s (b ) b e l o w ) (b ) d i f f e r e n t i a t i n g w i t h c e r t a i n t y b e t w e e n p o s i t r o n c r e a t i o n and a n n i h i l a t i o n p u l s e s ( t h e d e g r e e o f d i f f i c u l t y h e r e d e p e n d s on t h e s o l u t i o n t o ( a ) a b o v e ) ( c ) a r r a n g i n g t h e e x p e r i m e n t so t h a t t h e random c o i n c i d e n c e c o u n t i n g r a t e ( n o i s e ) i s a t an a c c e p t a b l e low l e v e l " T h e p o s i t r o n s c r e a t e d in t he p l a s m a by c o l l i s i o n s a r e o f no use i n t h i s c a s e b e c a u s e t h e r e i s no d e t e c t a b l e s i g n a l to mark t h e i r b i r t h s in t i m e - t h e y o n l y add to t h e random c o i n c i d e n c e c o u n t r a t e . 102 F. P o s i t r o n Beam B roaden i ng ( ' S l o w ' P o s i t r o n s ) I t i s p o s s i b l e t o pass a c o l l i mated beam of ' s l o w ' p o s i t r o n s t h r ough a p lasma and deduce t h e e l e c t r o n d e n s i t y f rom the a n g u l a r b r oaden i ng of the beam (wh ich i s due to the c u m u l a t i e f f e c t o f s m a l l a n g l e p o s i t r o n s c a t t e r i n g ) . T h i s t ype o f e x p e r i m e n t has been c a r r i e d ou t by L o h n e r t and S c h n e i d e r (1971) who d e s i g n e d a s low p o s i t r o n ' g u n ' c a p a b l e o f p r o d u c i n g - 1850 ' s l o w ' ( k T g + = 3 KeV) p o s i t r o n per second at a beam r a d i u s 22 o f 1.9 mm. A 2 mCi Na s ou r ce s u p p l i e d the p o s i t r o n s , wh ich were d e c e l e r a t e d , f o c u s s e d and c o l 1 imated to p roduce t he beam. A f t e r p a s s i n g t h r ough a 6" s l a b o f p lasma and an i r i s and a n n i h i l a t i n g i n a t a r g e t , i n d i v i d u a l p o s i t r o n s were d e t e c t e d by ' c o i n c i d e n c e c o u n t i n g 1 t he a n n i h i l a t i o n r a d i a t i o n . The r a d i a l p o s i t r o n i n t e n s i t y and hence t he beam sp read ang l e c o u l d be c a l c u l a t e d , knowing the c o i n c i d e n c e count r a t e a t v a r i o u s i r i s d i a m e t e r s . E l e c t r o n d e n s i t y measurements were c a r r i e d out i n a g low d i s c h a r g e plasma (n v a r i a b l e f rom ~ 1 0 1 1 c m " 3 t o - 1 0 1 3 • c m " 3 ) . Beam sp read r e s u l t s ag reed r e a s o n a b l y w e l l w i t h s p e c t r o -s c o p i c and probe measurements , a l t h o u g h a s y s t e m a t i c e r r o r o f 12-15% was e v i d e n t . The advantage o f u s i n g a p o s i t r o n beam i n s t e a d o f an e l e c t r o n beam f o r t h i s t ype of measurement i s not e v i d e n t . ( I n t e n s e e l e c t r o n beams a re r e a d i l y a v a i l a b l e and e l e c t r o n d e t e c t i o n i s much l e s s c o m p l i c a t e d than the method used by L o h n e r t and S c h n e i d e r t o d e t e c t p o s i t r o n s ) . A l t h o u g h the un ique p r o p e r t i e s of p o s i t r o n s were not d i r e c t l y e x p l o i t e d i n t h i s work, i t makes a v a l u a b l e 103 c o n t r i b u t i o n , i n t h a t i t s u g g e s t s t h e i d e a o f u s i n g ' s l o w ' p o s i t r o r i s • t o p r o b e p l a s m a s . I f a n e f f i c i e n t s l o w p o s i t r o n s o u r c e c o u l d be d e v e l o p e d , p o s i t r o n s c o u l d be i n t r o d u c e d i n t o p l a s m a s a l r e a d y ' t h e r m a 1 i z e d ' t o t h e mean e n e r g y o f t h e p l a s m a e l e c t r o n s . T h e y w o u l d a n n i h i l a t e a l m o s t i m m e d i a t e l y ( d u r i n g t h e f i r s t o r s e c o n d c o l l i s i o n' w i t h an e l e c t r o n ) , t h u s d e f i n i n g a n ' a c t i v e ' p l a s m a v o l u m e f r o m w h i c h h i g h c o u n t r a t e s c o u l d be e x p e c t e d . L o s s e s a s s o c i a t e d w i t h l o n g t h e r m a l i z a t i o n t i m e s a n d r a n g e s i n some p l a s m a s w o u l d be e l i m i n a t e d . E x c e p t f o r f i n i t e p l a s m a l i f e t i m e s , t h e s i t u a t i o n w o u l d b e s i m i l a r t o t h a t i n s o l i d s a n d l i q u i d s , v i z . , a l l p o s i t r o n s a n n i h i l a t i n g v e r y * s o o n a f t e r e n t e r i n g t h e m a t e r i a l a n d n e a r t h e p o i n t o f e n t r y . U n f o r t u n a t e l y , h i g h i n t e n s i t y s l o w p o s i t r o n s o u r c e s do n o t , a s y e t , e x i s t . T h e L o h n e r t - S c h n e i d e r d e v i c e y i e l d s o n e s l o w (3 K e V ) p o s i t r o n f o r e v e r y 4 x 10 1* f a s t p o s i t r o n s 22 e m i t t e d b y t h e Na s o u r c e . O t h e r s l o w p o s i t r o n s o u r c e s h a v e b e e n c o n s t r u c t e d ( s e e f o r e x a m p l e C a n t e r , et a l . , 1 9 7 2 ) a n d a l t h o u g h t h e i r c o n v e r s i o n e f f i c i e n c i e s a r e no b e t t e r , t h e y d o y i e l d l o w e r e n e r g y ' s l o w ' p o s i t r o n s ( a b o u t two 1 eV p o s i t r o n s 2 2 f o r e v e r y 1 0 5 f a s t p o s i t r o n s e m i t t e d by a Na s o u r c e ) . U n t i l h i g h i n t e n s i t y s 1 o w p o s i t r o n s o u r c e s a r e d e v e l o p e d , e x p e r i m e n t s t a k i n g f u l l a d v a n t a g e o f t h e u n i q u e p r o p e r t i e s o f s l o w p o s i t r o n s a r e d i f f i c u l t . No s i g n i f i c a n t a n n i h i l a t i o n o c c u r s in t h e L o h n e r t -' S c h n e i d e r e x p e r i m e n t b e c a u s e t he ' s l o w ' , 3 KeV p o s i t r o n s a r e s t i l l c o n s i d e r a b l y ' f a s t e r ' t h a n t h e a v e r a g e p l a s m a e l e c t r o n s (by a b o u t 3 o r d e r s o f m a g n i t u d e in e n e r g y ) . C h a p t e r V I I I C O N C L U S I O N A. S u m m a r y a n d R e s u l t s T h i s t h e s i s h a s d i s c u s s e d t h e p o s i t r o n a n d i t s b e h a v i o u r i n a p l a s m a w i t h t h e v i e w o f e v a l u a t i n g t h e f e a s i b i l i t y o f u s i n g p o s i t r o n s t o p r o b e p l a s m a s a n d m e a s u r e c h a r a c t e r i s t i c p l a s m a p a r a m e t e r s . O u t l i n e s o f ( a ) t h e b a s i c p h y s i c a l p r o -p e r t i e s o f t h e p o s i t r o n a n d ( b ) t h e o b j e c t i v e s o f p l a s m a d i a g n o s t i c s w e r e p r e s e n t e d a s n e c e s s a r y b a c k g r o u n d m a t e r i a l . A f t e r a d i s c u s s i o n o f t h e c r i t e r i a t h a t m u s t be s a t i s f i e d i f p o s i t r o n p r o b e s a r e t o be f e a s i b l e , t h e l i f e o f a p o s i t r o n i n p a r t i a l l y i o n i z e d a n d f u l l y i o n i z e d p l a s m a s was c o n s i d e r e d q u a l i t a t i v e l y . A q u a n t i t a t i v e a n a l y s i s - o f p o s i t r o n b e h a v i o u r i n a f u l l y i o n i z e d h y d r o g e n p l a s m a f o l l o w e d . B a s e d o n t h i s , i t was a s c e r t a i n e d t h a t p o s i t r o n t e c h n i q u e s may be a p p l i c a b l e t o t w o p o t e n t i a l f u s i o n p l a s m a s : t h e T o k a m a k t y p e a n d t h e l a s e r c o m p r e s s i o n t y p e . P o s i t r o n t h e r m a l i z a t i o n c a l c u l a t i o n s f o r t h e s e t w o h y p o t h e t i c a l p l a s m a s w e r e c a r r i e d o u t . A d i s c u s s i o n o f gamma r a y d e t e c t o r s a n d p o s i t r o n s o u r c e s ( i n c l u d i n g c a l c u l a t i o n s o f p a i r p r o d u c t i o n i n t h e two p l a s m a s b e i n g 104 105 c o n s i d e r e d ) was n e c e s s a r y b e f o r e c o u n t i n g r a t e c a l c u l a t i o n s c o u l d be c a r r i e d o u t . Four gamma ray a n a l y s i s t e c h n i q u e s were d e s c r i b e d , and based p a r t l y on c o u n t i n g r a t e c a l c u l a t i o n s , e v a l u a t e d as to t h e i r s u i t a b i l i t y f o r use i n p o t e n t i a l ' p o s i t r o n plasma p r o b e ' s y s t ems . The r e s u l t s of t h i s e v a l u a -t i o n , a l ong w i t h c h a r a c t e r i s t i c t ime and d i s t a n c e s c a l e s a re summarized i n T a b l e IV. These r e s u l t s can be t a ken as a g e n e r a l i n d i c a t i o n of the u s e f u l n e s s of p o s i t r o n t e c h n i q u e s (at t h i s p o i n t i n t i m e ) to probe p lasmas where kT g _ > - 10 eV. Some of t he se t e c h n i q u e s c o u l d be adapted f o r use w i t h l owe r t e m p e r a t u r e p lasmas when h i gh i n t e n s i t y , s l ow p o s i t r o n s o u r c e s become a v a i l a b l e . B. S u g g e s t i o n s f o r F u t u r e Work The f o l l o w i n g r e s e a r c h ( deve l opment ) p r o j e c t s a re s ugge s ted on the b a s i s o f m a t e r i a l p r e s e n t e d i n t h i s t h e s i s : (a) A s e a r c h f o r a n n i h i l a t i o n r a d i a t i o n i n c u r r e n t f u e l p e l l e t d e t o n a t i o n e x p e r i m e n t s t o d e t e r m i n e i f p a i r p r o d u c t i o n o c c u r s i n l a s e r produced p l a smas , f o l l o w e d by a d e t a i l e d a n a l y s i s o f the a n n i h i l a t i o n l i n e w i d t h and shape to a s c e r t a i n i f the p o s i t r o n s have t h e r m a l i z e d p r i o r t o a n n i h i l a t i n g . (b) r a d i a t i o n f l u x A s e a r c h f o r a n n i h i l a t i o n gamma r ay s i n t he e x i t i n g from e x i s t i n g Tokamak d e v i c e s t o Table IV Summary of C o n c l u s i o n s Tokamak Plasma Laser Compression Plasma Time Scales T, < T > T t a ~ p£ (0.002-0.018) (1) (0.01-1.0) (sec) T. < T < T t a p£ (0.0005-0.005) (0.-13) (0.1-1.0) (nsec) Distance Scales R. < d „ t p£ ((0.4-7) x 10 6) (co, o rb i t ing ) (m) R, : d „ t p£ . (1-100) (120) (ym) 2Y Angular . Correlat ion Not Feasible at This Time (Counting Rate Too Low) Not Feasible at This Time (Counting Rate Too Low) Energy Sh i f t & Line Broadening Measurements Feasible Feasible Positron Lifet ime Measurements Marginal at This Time Not Feasible (Plasma Dimensions Too Small) Slow Positron Probing Feasible (Depending on Intense Slow Positron Source Development) Marginal (Depending on Intense Slow Positron Source Development) Tj. : Positron Thermalization Time R t : Positron Thermalization Range 22 dp^: Character i s t i c Plasma Dimension NOTE: x t and R^  f igures denote ranges in which most Na positrons T , : Positron Ann ih i l a t ion Time would thermalize. a T „: Plasma Lifet ime 107 d e t e r m i n e i f p a i r p r o d u c t i o n o c c u r s i n h i gh t e m p e r a t u r e , h i g h d e n s i t y t o r o i d a l p l a smas , f o l l o w e d by a l i n e a n a l y s i s as i n (a) above; a l s o , a check f o r c o r r e l a t i o n between p o s i t r o n s and runaway e l e c t r o n s to t e s t the p a i r c r e a t i o n by c o l l i s i o n h y p o t h e s i s . (c ) Development of i n t e n s e , s l ow p o s i t r o n s o u r c e s ( i . e . deve lopment o f e f f i c i e n t f a s t to s low p o s i t r o n con v e r t e r s ) t o improve the e f f e c t i v e n e s s o f most p o s i t r o n d i a g -n o s t i c t e c h n i q u e s and ex tend the n and T g ranges o ve r wh i ch t hey a r e u s e f u l . (d) Development of a comp le te t h e o r y o f p o s i t r o n b e h a v i o u r i n p lasmas to complement the e x p e r i m e n t a l work. I t seems t h a t y e t a n o t h e r a r ea of a p p l i c a t i o n has opened to A n d e r s o n ' s ' p o s i t i v e l y cha rged e l e c t r o n ' . To t he l i s t o f s t a t e s o f m a t t e r wh i ch the p o s i t r o n i s h e l p i n g to e l u c i d a t e : s o l i d s , l i q u i d s and g a s e s , one may soon be a b l e t o add the f o u r t h s t a t e o f m a t t e r : the p l a sma. BIBLIOGRAPHY B e t h e , H. A. (1 930). Ann. P h y s i k 5, 325. B u n k i n , F.V. and Kazakov , A . E . ( 1970 ) . D o k l . Akad. Nauk SSSR 1 93 , 1 274. [ Sov . P h y s . - D o k l . 1_5, 758 (1 971 ) ] Burcham, W.E. ( 1973 ) . Nucl eo.r Physics an Introduction, Longman, London. C a n t e r , K .F . , Co leman, P .G . , G r i f f i t h , T.C. and H e y l a n d , G.R, ( 1 972 ). J . Phys . B 5, L l 67. C a p p e l l a n i , F. and R e s t e l l i , G. ( 1968 ) . In Semiconductor Detectors. (G. B e r t o l i n i and A. Coche, e d s . ) , p. 365. N o r t h - H o l l a n d P u b l . C o . , Amsterdam. Coche, A . , and S i f f e r t , P. ( 1968 ) . In Semiconductor Detectors. (G. B e r t o l i n i .and A. Coche, e d s . ) , p. 149. N o r t h - H o l l a n d P u b l . C o . , Amsterdam. D e B e n e d e t t i , S. (1 956 ) . Nuovo C imento [10 ] 4, Suppl . 3, 1 209. D e u t s c h , M. ( 1951 ) . Phys . Rev. 82^, 455. D e u t s c h , M. ( 1953 ) . P r o g r . N u c l . Phy s . 3, 131. d e Z a f r a , R.L. and J o y n e r , W.T. (1 958 ) . Phys . Rev. 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S t e w a r t and L.O. Ro e 11 i g , eds . ). Academic P r e s s , New York ( 1967 ) . T o p t y g i n , I.N. ( 1 962 ) . Zhu r . E k s p e r i m . i T e o r e t . F i z . 43 , 1031. [ Sov . Phys . - JETP 1_6, 728 (1 963 ).] T o p t y g i n , I.N. ( 1964 ) . Zhur . Tekh . F i z . 34, 645. [Sov. Phys . - T e c h . Phys . 9, 499~["1 9 6 4 ) . ] West, R.N. ( 1973 ) . Advances i n P h y s i c s 22, 263. W o l f e r , W.G. ( 1 969 ) . Ph.D. T h e s i s , U n i v e r s i t y o f F l o r i d a , Department of N u c l e a r E n g i n e e r i n g S c i e n c e s . APPENDIX A ANNIHILATION MODES l y F ree Anni hi1 a t i on In t he l y c a s e , t he a n n i h i l a t i o n c r o s s s e c t i o n i s r e l a t e d t o the a tom i c number (Z) o f t he r e c o i 1 - a b s o r b i n g n u c l e u s . I f t he r e l a t i v e p o s i t r o n - e l e c t r o n v e l o c i t y i s s m a l l (v << c ) , t he a n n i h i l a t i o n c r o s s s e c t i o n can be w r i t t e n as ( H e i t l e r , 1954; Chap te r V ) : (54) T h i s i s a p p r o x i m a t e l y e i g h t o r d e r s o f magn i tude l e s s than t h e c o r r e s p o n d i n g 2y c r o s s s e c t i o n , and i n d i c a t e s t h a t l y a n n i h i l a t i o n s do not c o n t r i b u t e s i g n i f i c a n t l y t o t h e a n n i h i l a t i o n r a d i a t i o n f l u x . T h e s t r o n g Z d e p e n d e n c e d o e s enhance t he l y c o n t r i b u t i o n somewhat, however even a t Z = 82, t h r e e o r d e r s o f magn i tude s t i l l s e p a r a t e t he l y , 2 y a n n i h i l a t i o n c r o s s s e c t i o n s . I l l 112 2 y , 3 y F ree A n n i h i l a t i o n The a n n i h i l a t i o n c r o s s s e c t i o n f o r p o s i t r o n s (v << c ) i n a medium w i t h e l e c t r o n d e n s i t y n can be w r i t t e n i n terms o f a compound a n n i h i l a t i o n r a t e v ' , w h i c h i n c l u d e s both 2 y and 3y c o n t r i b u t i o n s . 2 Y + 3 Y v n v w i t h C 2 Y ^2y + C 3 Y * 3 Y ' (55) (56) where ^ 1 S t n e d e n s i t y of e l e c t r o n s w i t h a n t i p a r a l l e l s p i n s ( w . r . t . the p o s i t r o n ) at the average p o s i t i o n o f the 2 p o s i t r o n and 1 s the d e n s i t y o f e l e c t r o n s w i t h p a r a l l e l s p i n s at the average p o s i t i o n of the p o s i t r o n . C ^ , and C, d e s c r i b e fundamenta l i n t e r a c t i o n r a t e s : 3y C0 = 4TTC 2y ,2 1 mc (57) C 0 = 4TTC 3y 2 ^ m c 4 a 9TT TT - 9 1115 (58) In the i d e a l f r e e a n n i h i l a t i o n c a s e , ij;^ = ^ n and ip^ = 3 4- n. The f a c t o r of t h r e e r e f l e c t s the f a c t t h a t a 0 = 1 c o n f i g u r a t i o n can o c c u r i n t h r e e ways: m s = - 1 , 0, + 1 , where m s i s the s p i n a n g u l a r momentum of the e l e c t r o n - p o s i t r o n s y s t e m , as opposed to the one p o s s i b l e J = 0 c o n f i g u r a t i o n (m s = 0 ) . E x p r e s s i o n (b6) can now be w r i t t e n as 1 1 3 v ir r i cn 1 + 4 a 3rr TI - 9 (59) the second te rm i n d i c a t i n g the 3y c o n t r i b u t i o n . The f i r s t te rm i s j u s t the D i r a c r e s u l t quo ted i n C h a p t e r I I ( e q u a t i o n 3) 2y -3y Bound A n n i h i l a t i o n Bound a n n i h i l a t i o n c o r r e s p o n d s to a n n i h i l a t i o n o f p o s i t r o n i u m . The e x p r e s s i o n f o r the a n n i h i l a t i o n r a t e (and the l i f e t i m e ) i s i d e n t i c a l to ( 5 6 ) , however a n d ^2y n i u s t now e x p r e s s the a p p r o p r i a t e bound s t a t e wave f u n c t i o n s . In the most s t a b l e c o n f i g u r a t i o n , and a r e hydrogen-1 i k e , S s t a t e wave f u n c t i o n s . C a l c u l a t i o n o f a n n i h i l a t i o n t i m e s y i e l d s the r e s u l t s : ( D e B e n e d e t t i , 1956) 2y 1.25 x 10" 1 0 sec T 3 y = 1.41 x TO " 7 sec (60) U n l i k e the f r e e p o s i t r o n t h a t e x p e r i e n c e s the i n f l u e n c e of a l a r g e number o f e l e c t r o n s , t he bound p o s i t r o n i n t e r a c t s e s s e n t i a l l y w i t h one p a r t i c u l a r e l e c t r o n , hence no e l e c t r o n d e n s i t y term appear s i n the e x p r e s s i o n f o r the l i f e t i m e ( a n n i h i l a t i o n t i m e ) T „ . APPENDIX B ENERGY AND MOMENTUM BALANCE IN ANNIHILATION The p o s i t r o n - e l e c t r o n c e n t r e o f mass momentum ( P c m ) m a n i f e s t s i t s e l f i n two d i f f e r e n t ways a f t e r the p a i r has a n n i h i l a t e d . The component of momentum p e r p e n d i c u l a r t o the d i r e c t i o n of y r a y e m i s s i o n (pj) causes a s l i g h t d e v i a -t i o n from c o l l i n e a r t r a j e c t o r i e s (and a s m a l l s h i f t i n the y ray e n e r g i e s ) and the component of momentum p a r a l l e l t o the d i r e c t i o n of y ray e m i s s i o n (P||) causes unequa l d i s t r i -b u t i o n of energy to the a n n i h i l a t i o n gamma photons ( the geometry of a t y p i c a l a n n i h i l a t i o n i s i l l u s t r a t e d i n F i g u r e 2 8a ) . These e f f e c t s a re summarized by the energy p a r t i t i o n i n g e x p r e s s i o n ( s e e , f o r examp le , G o l d a n s k i i , 1968 ) : r v _< E ( Y i , 2 ) = - ^ | ' C 1 1 (61) 1 cm cos CXi , 2 where cm 2mc 2 + E ( e " ) + E ( e + ) , t he t o t a l energy o f the p a i r , i s the magn i tude of the v e l o c i t y of the c e n t r e o f mass of the a n n i h i l a t i n g p a i r , 114 Fig.28. Annihilation dynamics. r a 2 b2 (not to scale) 116 and c t i , a 2 a re the ang l e s between the d i r e c t i o n s o f Y ray e m i s s i o n and t he d i r e c t i o n d e f i n e d by P, -cm The magn i tudes o f the a n n i h i l a t i o n y r ay momenta a r e j u s t : P . ( Y I , 2 ) = iWJ- (62) A l ook at the l i m i t i n g cases (components ) i s i n f o r -m a t i v e . When y ray e m i s s i o n o c c u r s a lmos t p e r p e n d i c u l a r t o Bern' « i = ct 2 - 9 0 ° , e x p r e s s i o n (61) reduces t o : E ( Y i ) = E ( Y 2 ) mc i 1 cm 1 " 2 C 2 (63) T h i s c o r r e s p o n d s t o the equa l p a r t i t i o n i n g s i t u a t i o n p i c t u r e d i n F i g u r e 2.8b. The second term r e p r e s e n t s the k i n e t i c energy of the p r e - a n n i h i 1 a t i o n p a i r . I f the Y r a y s a re e m i t t e d a lmos t p a r a l l e l to P c m > then cti ~ 180° f a 2 ~ 0° and e x p r e s s i o n (61) i n d i c a t e s : E ( Y i ) E ( Y 2 ) 1 + cm cm (64) T h i s e x p r e s s i o n i s meant o n l y to i n d i c a t e t h e a p p r o x i mate m a g n i t u d e o f t he s m a l l s h i f t in E ( Y I , 2 ) away f r o m m c 2 when a l > 2' - 9 0 ° . The minus s i g n i s a r e s u l t o f t he a p p r o x i m a t i o n p r o c e d u r e u sed (o f c o u r s e , v__ = 0 when ex i , 2 i s e x a c t l y e q u a l cm to 9 0 ° ) . In f a c t t he s i g n o f the s h i f t in e n e r g y i s p o s i t i v e . 1 1 7 c o r r e s p o n d i n g to the unequal p a r t i t i o n i n g s i t u a t i o n i l l u -t r a t e d i n F i g u r e 28c. The second terms here r e p r e s e n t Dopp l e r s h i f t s . Thus, a l t h o u g h the energy of any a n n i h i l a t i o n y ray i s g i v e n by e x p r e s s i o n ( 6 1 ) , i t i s u s e f u l t o keep i n mind the component makeup of p (and hence o f p ( y i , 2 ) ) and what ~ C ill ~ t h i s means ( i . e . t h a t p e r t u r b a t i o n s i n y ray e n e r g i e s - away f rom mc 2 - can be t hough t of as a r i s i n g from two d i f f e r e n t m o t i o n s ) . The magn i tudes o f y ray energy s h i f t s a s s o c i a t e d w i t h pj^ and p|| ( v a l u e s i n te rms of v c m appear i n e x p r e s s i o n s ( 6 3 ) a n d ( 6 4 ) ) a r e a p p r o x i m a t e l y : AE AE 1 2 P 1 a 2 P l (65) where P|| and pj^ are the magn i tudes o f P|| and p t r e s p e c t i v e l y . 1 Oil y pj^ p l ays a r o l e i n p e r t u r b i n g c o l l i n e a r y r ay e m i s s i o n , thus i t i s pj^ t h a t appear s i n e x p r e s s i o n s (4) and (5) wh ich d e s c r i b e the magn i tude o f t h i s p e r t u r b a t i o n . APPENDIX C POSITRONIUM FORMATION P o s i t r o n i u m f o r m a t i o n i s e x p e c t e d t o t a k e p l a c e between f r e e plasma e l e c t r o n s and p o s i t r o n s r a n g i n g i n energy f rom about 50 eV down t o and i n c l u d i n g the mean p lasma e l e c t r o n ene r g y . P o s i t r o n i u m f o r m a t i o n i n v o l v i n g p r e v i o u s l y bound e l e c t r o n s would be l i m i t e d by a t h r e s h o l d p o s i t r o n ene r g y , V : ( l owe r l i m i t ) P V p = V . - I p , (66) where I = 6.8 eV i s the p o s i t r o n i u m b i n d i n g (o r i o n i z a t i o n ) e n e r g y . A p o s i t r o n w i t h V p would have j u s t enough energy to i o n i z e (and c a p t u r e ) the most weak l y bound e l e c t r o n and s t i l l m a i n t a i n the energy b a l a n c e ( s i n c e the p o s i t r o n - e l e c t r o n sys tem r e l e a s e s 6.8 eV upon f o r m i n g p o s i t r o n i u m ) . By ana l o g y w i t h the s i t u a t i o n i n g a s e s , most o f t he p o s i t r o n i u m f o r m a t i o n i n v o l v i n g a p r e v i o u s l y bound e l e c t r o n might be e x p e c t e d t o 118 1 I 9 occu r i n the energy gap between V and V (see F i g u r e 6 a ) . e p T h i s energy range i s known as the 'O re g a p ' . P o s i t r o n i u m f o r m a t i o n i n a f u l l y i o n i z e d medium c o u l d o c c u r o n l y w i t h f r e e e l e c t r o n s , as i n d i c a t e d i n F i g u r e 6b. The i m p o r t a n t q u e s t i o n i s : w i l l t he P , once formed remain bound long enough to a n n i h i l a t e ? I t i s n a t u r a l t o t a k e k T g = I as a d i v i d i n g energy as d i d T o p t y g i n (1964) i n d i s c u s s i n g a s i m i l a r q u e s t i o n . In p lasmas where kT • > I ( t he m a j o r i t y of f u l l y i o n i z e d p l a s m a s ) , P would be s c a r c e , because even i f i t d i d f o r m , i t would be b roken up ( i o n i z e d ) i m m e d i a t e l y by c o l l i s i o n s . T o p t y g i n i n d i c a t e s t h a t even P g formed i n p lasmas where k T g < I ( t he m a j o r i t y would be p a r t i a l l y i o n i z e d p la smas ) would not e x i s t much l o n g e r than about 1 0 ~ 1 0 s e c , t he l i f e t i m e of the s i n g l e t P s s t a t e , due to (a) s t r a i g h t f o r w a r d s i n g l e t a n n i h i l a t i o n , (b) c o l l i s i o n s r e s u l t i n g i n t r i p l e t to s i n g l e t c o n v e r s i o n f o l l o w e d by s i n g l e t a n n i h i l a t i o n and (c ) c o l l i s i o n s r e s u l t i n g i n P s breakup ( i o n i z a t i o n ) . I t s h o u l d be noted t h a t c o l l i s i o n a l b reakup may p l a y an even more s i g n i f i c a n t r o l e than s ugge s ted by T o p t y g i n -a t a l l v a l u e s of k T g , but e s p e c i a l l y f o r k T g < I p . The e l e c t r o n s i n the h i gh energy t a i l o f t he t h e r m a l Maxwe l l d i s t r i b u t i o n (whatever the mean t e m p e r a t u r e ) s h o u l d enhance n o t i c e a b l y P $ i o n i z a t i o n , as they enhance c o n v e n t i o n a l i o n i z a t i o n . In H 2 gas , f o r examp le , comp le te d i s s o c i a t i o n and i o n i z a t i o n has o c c u r r e d when kT - 1.5 eV, however t h e 1 2 0 d i s s o c a t i o n - i o n i z a t i o n p o t e n t i a l c o r r e s p o n d s to k T g - 15 eV. The f a c t o r o f ten r e f l e c t s the i o n i z i n g power of the f a s t e l e c t r o n s a s s o c i a t e d w i t h a t he rma l d i s t r i b u t i o n w i t h a r e l a t i v e l y low mean energy (1.5 e V ) . A l t h o u g h i t s e x a c t magn i tude i s d i f f i c u l t t o p r e d i c t , i t does seem t h a t t h e r e w i l l be a bound s t a t e c o n t r i b u t i o n ( m o s t l y f rom 2 y decay of s i n g l e t P ) t o the o v e r a l l a n n i h i l a -t i o n r a d i a t i o n spec t rum of c oo l ( k T g < ~ I ) p l a s m a s . T h i s c o n t r i b u t i o n has not been c o n s i d e r e d i n d e t a i l i n t h i s s t udy because the m a j o r i t y of p lasmas of i n t e r e s t f a l l i n t o the r e g i o n where P g i s e x p e c t e d to be s c a r c e , i f a t a l l p r e s e n t -t h a t i s , the r e g i o n where the mean t h e r m a l p lasma t e m p e r a t u r e s a re such t h a t : kT > ~ I . APPENDIX D POSITRON ENERGY LOSS RATE IN A PLASMA The s t o p p i n g power (and energy l o s s r a t e ) o f a s i n g l e component ( e l e c t r o n ) p lasma can be w r i t t e n as an i n t e g r a l over the p r o d u c t o f t h e energy t r a n s f e r , AE ( p o s i t r o n to e l e c t r o n ) and t he d i f f e r e n t i a l Coulomb s c a t t e r i n g c r o s s s e c t i o n , da a s : d_E = 1 dE dx v dt n AE da (67) where E i s the p o s i t r o n k i n e t i c energy and v i t s v e l o c i t y and n i s the plasma e l e c t r o n number d e n s i t y . W r i t i n g AE i n terms o f Ap, the momentum t r a n s f e r r e d i n a g i v e n c o l l i s i o n , and da i n terms of Ap as : T h i s i s j u s t R u t h e r f o r d ' s Cou lomb s c a t t e r i n g f o r m u l a w r i t t e n in te rms o f Ap ( i n t he Born a p p r o x i m a t i o n l i m i t ) . It i s u s u a l l y w r i t t e n in te rms o f t he c e n t r e o f mass s c a t t e r i n g a n g 1 e 8 as • c da = 2 TF m v • c s c " ( i 6 ) s i n e d6 c c c 121 1 22 d a 8TT fe2 d(Ap) (Ap )3 " ( 6 8 ) i t f o i l o w s t h a t : dJE _ 4TT n e ** dt mv d(Ap) Ap ( 6 9 ) where the i n t e g r a t i o n i s c a r r i e d out o ve r a l l p o s s i b l e v a l u e s o f Ap i n the range Ap . t o Ap max The l a r g e s t p o s s i b l e momentum t r a n s f e r , Ap , o c c u r s d u r i n g head-on c o l l i s i o n s III Q A and has a v a l u e : Ap max 2 mv (70) The s m a l l e s t p o s s i b l e momentum t r a n s f e r , A p m i n , c o r r e s p o n d s to the g e n e r a t i o n of a s i n g l e plasma wave of energy f i w e : No e x c h a n g e te rms a p p e a r b e c a u s e t h e p o s i t r o n i s d i s t i n c t f r om the e l e c t r o n . E x p r e s s i o n (68) f o l l o w s f r o m the above i f t h e f r a c t i o n a l e n e r g y t r a n s f e r ( f o r a g i v e n c o l l i s i o n ) i s wr i t t e n as AE . , , n •> — = s i n ( i 9 ) . t c If e n e r g y i s w r i t t e n as E = ^ , t h e n momentum t r a n s f e r can be w r i t t e n a s : Ap = m AE w h i c h y i e l d s , w i t h = fioo , a Ap . va 1 ue o f e min f lu) Ap im n 1 23 (71) R e c a l l t h a t the momentum t r a n s f e r v a l u e t h a t s e p a r a t e s ' h a r d ' and ' s o f t ' c o l l i s i o n s i s : A p D = ^— , where X^ i s the Debye l e n g t h : k T e ^ 4TT n e 2 (72) where i s B o l t z m a n ' s c o n s t a n t , and e,n,T are the e l e c t r o n c h a r g e , number d e n s i t y and t e m p e r a t u r e r e s p e c t i v e l y . The c o n t r i b u t i o n f rom ' s o f t ' c o l l i s i o n s (Ap . < mi n Ap < A p D ) f o l l o w s from (69) upon i n t e g r a t i o n f rom A p m i n t o Ap dt 4TT n e 4 SOFT mv In r A P , I Ap . 4 i n e mv In f / 2 v l (73) I n t e g r a t i o n of (69) f rom A p n t o A p m 3 v y i e l d s the ' h a r d ' u max c o l l i s i o n c o n t r i b u t i o n : d_i d t 4TT n e ** HARD mv In •Ap max A p n J 4TT n e 4 r/2 m v v >i mv In fi w (74) 1 24 Note t h a t ' s o f t ' and ' h a r d ' c o l l i s i o n c o n t r i b u t i o n s are s e p a r a t e d o n l y t o l o g a r i t h m i c a c c u r a c y . The t o t a l energy l o s s r a t e i s the sum of t he se c o n t r i b u t i o n s ( e s s e n t i a l l y the r e s u l t of i n t e g r a t i n g (69) f rom Ap t o Ap ): K m i n ^max' dE dt TOTAL m v 4TT n e1* , - 1 n Ap •max [Ap min 2 2 e 0 3 . -S- i n f2 m v2 l e (75) The c o n t r i b u t i o n t o dE/dt f rom p o s i t r o n - i o n c o l l i s i o n s has been i g n o r e d here s i n c e i t i s s m a l l e r than the c o n t r i b u t i o n f rom p o s i t r o n - e l e c t r o n c o l l i s i o n s by a f a c t o r o f m/iru , where m. i s the i o n mass. R e l a t i v i s t i c e f f e c t s have not been c o n s i d e r e d i n t h i s d e r i v a t i o n s i n c e t h e i r magn i tudes a re s m a l l i n the energy range of i n t e r e s t ( E ( e + ) < - 500 KeV) . The approach of Hu s s e i n y and S a b r i (1974) i s s i m i l a r , but d i f f e r s i n one e s s e n t i a l r e s p e c t . They d e f i n e t he ' s o f t ' c o l l i s i o n c r o s s s e c t i o n u s i n g a Debye l e n g t h s h i e l d e d p o t e n t i a l and use t he pure i n v e r s e Coulomb p o t e n t i a l t o . d e f i n e o n l y the ' h a r d ' c o l l i s i o n c r o s s s e c t i o n . The g e n e r a l r e s u l t i s c o m p l i c a t e d , but i n t he s l o w i n g down range (v > v ) i t r educes t o : 1 25 d_E dt 1 e2 a) 2 e TOTAL (4TT e 0 )2 2 TT v I n (4TT E O ) , m v v i e 2 1) w (76) an e x p r e s s i o n d e r i v e d by Hu s se i n y and Fo r sen i n 1970. Note t h a t , a s i d e from u n i t s , t h i s e x p r e s s i o n i s s i m i l a r t o the ' h a r d ' c o l l i s i o n c o n t r i b u t i o n ( 7 4 ) , i n the sense t h a t dE/dt i s f u n c t i o n a l l y r e l a t e d to v and v i n a s i m i l a r manner. J e I t i s , as y e t , u n c l e a r wh i ch e x p r e s s i o n (75 o r 76) y i e l d s the most a c c u r a t e v a l u e f o r dE/dt and hence f o r the t h e r m a l i z a t i o n t i m e . 

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