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Formation of positronium in helium gas Albrecht, Robert Stephen 1973

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FORMATION OF POSITRONIUM IN HELIUM GAS by ROBERT STEPHEN ALBRECHT B.S . j U n i v e r s i t y of B r i t i s h Columbia, 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the department of PHYSICS We accept t h i s t h e s i s as conforming t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1973 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver 8, Canada Date ABSTRACT The f o r m a t i o n of p o s i t r o n i u m i n helium gas under an a p p l i e d e l e c t r i c f i e l d has been i n v e s t i g a t e d i n order t o d e r i v e a value f o r the momentum-transfer c r o s s s e c t i o n , (j m> of p o s i t r o n s on helium atoms. The major i n c r e a s e i n p o s i t r o n i u m f o r m a t i o n was n o t i c e d at f/D's of 60 V cm -1 c i -1 -1 amagats . C-/D v a l u e s up t o 100 V cm amagats were o b t a i n e d . A minor i n c r e a s e i n p o s i t r o n i u m f o r m a t i o n was found at 30 V cm""'"amagats ~~ and was a t t r i b u t e d t o i m p u r i t i e s i n the helium gas, s p e c i f i c a l l y , , and/or H 2. U s i n g the t h e o r y of Teutsch and Hughes a value of 2 (j m of ( 0 . 2 5 + 0 . 0 3 ) 7Ta o was found by f i t t i n g t o the upper i n c r e a s e . T h i s corresponds t o p o s i t r o n e n e r g i e s near 17*7 eV. Although the t h e o r y i s s e n s i t i v e t o the p o s i t r o n i u m f o r m a t i o n c r o s s s e c t i o n , the combined e f f e c t of i m p u r i t i e s and c o u n t i n g s t a t i s t i c s l e f t t h i s parameter u n d e f i n e d . A f i t was made t o the lower p o s i t r o n i u m , .A: f o r m a t i o n i n c r e a s e under the assumption t h a t t h i s i n c r e a s e i s due e n t i r e l y t o the i m p u r i t y atoms, whereas the p o s i 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 s due t o the helium atoms. T h i s gave a v a l u e f o r t h e p o s i t r o n i u m t h r e s h o l d f o r the i m p u r i t i e s of 8 + 1 eV. i i i TABLE OF CONTENTS page ABSTRACT i i LIST OF TABLES v i LIST OF FIGURES . . v i i ACKNOWLEDGEMENTS v i i i CHAPTER ONE POSITRON-ATOM INTERACTIONS 1 I n t r o d u c t i o n 1 CHAPTER TWO POSITRONIUM FORMATION 4 2.1 I n t r o d u c t i o n 4 2.2 P o s i t r o n s i n a Gas 5 2.21 Slowing Down of P o s i t r o n s i n a Gas . 5 2.22 E q u i l i b r i u m P i c t u r e f o r P o s i t r o n s i n a Gas Under an A p p l i e d E l e c t r i c F i e l d 9 2.3 P r o p e r t i e s of P o s i t r o n i u m . . . . 14 2.31 S t r u c t u r e of P o s i t r o n i u m i n Vacuum .. 14 2.32 P o s i t r o n i u m i n a Gas 15 2.4 Formation of P o s i t r o n i u m i n Helium . . . . 19 2.41 P o s i t r o n i u m Formation Cross S e c t i o n . 19 2.411 T h e o r e t i c a l 19 2.412 E x p e r i m e n t a l 21 2.42 T h e o r i e s of P o s i t r o n i u m Formation . . 22 2.43 Momentum-Transfer Cross S e c t i o n . . . 27 2.431 T h e o r e t i c a l 27 2.432 E x p e r i m e n t a l 30 2.44 A n n i h i l a t i o n Cross S e c t i o n f o r Free P o s i t r o n s i n Helium 30 i v page 2.45 The D i f f u s i o n E q u a t i o n and P o s i t r o n i u m Formation . . 33 2.46 E f f e c t of I m p u r i t i e s on P o s i t r o n i u m Formation . 35 2.461 T h e o r e t i c a l 35 2.462 Experimental 37 CHAPTER THREE EXPERIMENTAL PROCEDURE 38 3.1 Technique f o r Measuring P o s i t r o n i u m Formation 3 8 3.2 E x p e r i m e n t a l Apparatus 45 3.21 Chamber, Source, and E l e c t r i c F i e l d . 45 3.22 D e t e c t o r s and Data A c q u i s i t i o n System. 47 3.3 Gas H a n d l i n g . 48 3.31• F i l l i n g . . . . . . . . . . 48 3.32 T i t a n i u m P u r i f i e r 50 3.33 E s t i m a t i o n of P u r i t y 51 3.4 E l e c t r o n i c s 55 3.41 NOVA Computer f o r Spectrum Accumulation 55 3 .42 D i s p l a y 56 3.43 L i n e a r i t y 58 3.44 Energy S t a b i l i t y . . . . . . 58 3.5 Data A n a l y s i s 58 3.51 A n a l y s i s of Energy S p e c t r a 58 3.52 F i t t o the V a l l e y - t o - P e a k R a t i o s . . . 65 CHAPTER FOUR PRESENTATION OF RESULTS . . 69 V page 4.1 Scope of R e s u l t s 69 4.2 A c c e p t a b i l i t y of R e s u l t s 69 4.21 F i t s t o the Energy S p e c t r a 69 4-22 F i t s t o the P o s i t r o n i u m Formation Curve 70 4.23 Gas P u r i t y 70 4.231 I n i t i a l Runs A f t e r F i l l i n g . . 70 4.232 A f t e r S p a r k i n g 72 4.233 E s t i m a t i o n of P u r i t y 72 4-3 E x p e r i m e n t a l V a l l e y - t o - P e a k R e s u l t s . . . . 73 4.4 D i s c u s s i o n of R e s u l t s 74 4.41 Comparison wi t h Other Workers . . . . 74 4.42 Comparison t o Theory 75 4.5 D i s c u s s i o n of E r r o r s 76 4.51 E l e c t r o n i c S t a b i l i t y 76 4.52 A p p l i e d E l e c t r i c F i e l d 76 4.53 D e n s i t y and £/D 76 4.6 Recommendations f o r Future Work 77 CHAPTER FIVE CONCLUSIONS 79 BIBLIOGRAPHY 8 l APPENDIX A 8-PARAMETER FIT TO THE PEAK AND VALLEY . 83 APPENDIX B ADC INTERFACE OPERATION, SCHEMATICS . . 85 APPENDIX C NOVA PROGRAM - LOGARITHMIC DISPLAY . . . 94 APPENDIX D FIT TO (j LARGE 96 v i LIST OF TABLES page Table I P o s i t r o n i u m Formation T h r e s h o l d s of 7 Common Gases Table I I P r o p e r t i e s of n=l P o s i t r o n i u m 17 Table I I Ex p e r i m e n t a l S c a t t e r i n g Cross S e c t i o n s 31 Near E t h r Table IV I n i t i a l and F i n a l Gas P u r i t y 52 v i i LIST OF FIGURES page F i g u r e 1 Helium Energy L e v e l Scheme 7 F i g u r e 2 T h e o r e t i c a l P o s i t r o n D i s t r i b u t i o n 12 F i g u r e 3 Energy Spectrum f o r Three-Gamma A n n i h i l a t i o n s 16 F i g u r e 4 P o s i t r o n Formation Cross S e c t i o n s 20 F i g u r e 5 P o s i t r o n i u m Formation R e s u l t s of Teutsch and Hughes 25 F i g u r e 6 S c a t t e r i n g Cross S e c t i o n s 29 F i g u r e 7 N a 2 2 Spectrum 40 F i g u r e 8 Peak and V a l l e y Regions 44 F i g u r e 9 Breakdown of 8-Parameter F i t 46 F i g u r e 10 E x p e r i m e n t a l Setup 49 F i g u r e 11 Simple DAC 57 F i g u r e 12 D i f f e r e n t i a l L i n e a r i t y 59 F i g u r e 13 I n t e g r a l L i n e a r i t y 60 F i g u r e 14 V a l l e y - t o - P e a k R a t i o s Versus £/D 63 F i g u r e 15 Approximation Made t o the Impurity Formation 67 F i g u r e 16 E f f e c t of P u r i f i e r on V a l l e y - t o - P e a k R a t i o s 71 F i g u r e B l B l o c k Diagram of I n t e r f a c e 86 F i g u r e B2 Device S e l e c t i o n and I n t e r r u p t S e r v i c e 89 F i g u r e B3 Data Channel S e r v i c e and Data Word Strobe 90 F i g u r e B4 Data Word S e r v i c e and ADC C l e a r 91 F i g u r e DI I t e r a t i v e Search F i t t o (jm Large 99 v i i i ACKNOWLEDGEMENTS I wish t o express my s i n c e r e thanks t o Dr. G. Jones f o r h i s gre a t h e l p and p a t i e n c e d u r i n g the course of t h i s r e s e a r c h . H i s a b i l i t y and w i l l i n g n e s s t o communicate h i s e x t e n s i v e knowledge, p h y s i c a l i n s i g h t and e x a c t i n g s t andards made the completion of t h i s p r o j e c t immeasurably e a s i e r . I a l s o extend my most s i n c e r e a p p r e c i a t i o n t o Dr. P.H.R. Orth f o r the many d i s c u s s i o n s on p h y s i c s and u n r e l a t e d s u b j e c t s . My thanks t o the people of room 100 f o r t h e i r f r i e n d s h i p over the years and t o others i n the department who made t h i s work more e n j o y a b l e . Thanks are a l s o due t o M.J. f o r i n t r o d u c i n g me t o the second g r e a t e s t s p o r t . In a d d i t i o n , I wish t o thank my parents f o r t h e i r encouragement over the y e a r s , even though, t o them, g o l f i n g i s much more i n t e r e s t i n g than p h y s i c s . F i n a l l y , t he N a t i o n a l Research C o u n c i l of Canada S c h o l a r s h i p over the past few ye a r s i s g r a t e f u l l y acknowledged. 1 CHAPTER ONE POSIT RON-ATOM INTERACTIONS I n t r o d u c t i o n The e x i s t e n c e of the p o s i t r o n was p r e d i c t e d by D i r a c i n 1931 and observed by Anderson i n 1932. Sin c e then, i t s i n t e r a c t i o n w i t h a l l t y p e s of matter has been e x t e n s i v e l y s t u d i e d . Of p a r t i c u l a r i n t e r e s t are the i n e r t gases w i t h t h e i r s p h e r i c a l l y f i l l e d e l e c t r o n s h e l l s , s i n c e these are the gases t h a t permit study of the i n t e r a c t i o n of the p o s i t r o n w i t h s i n g l e atoms. Even f o r t h i s case, however, the a n a l y s i s i s s t i l l a many-body-problem. The s i m p l e s t i n t e r a c t i o n , t h a t of a p o s i t r o n w i t h a hydrogen atom i s , u n f o r t u n a t e l y , i m p o s s i b l e t o study e x p e r i m e n t a l l y , at the gas d e n s i t i e s r e q u i e d f o r s t o p p i n g p o s i t r o n s . Most of the t h e o r e t i c a l work p e r t a i n s t o the e l a s t i c s c a t t e r i n g r e g i o n . Some c a l c u l a t i o n s , however, r e f e r t o the next h i g h e r energy r e g i o n where p o s i t r o n i u m f o r m a t i o n i s e n e r g e t i c a l l y p o s s i b l e . I t i s only below the t h r e s h o l d f o r p o s i t r o n i u m f o r m a t i o n t h a t the c o l l i s i o n s are f r e e of any c o m p l i c a t i o n s due t o rearrangements of the p a r t i c l e s and 1 i s t h e r e f o r e the energy r e g i o n most amenable t o d e t a i l e d t h e o r e t i c a l a n a l y s i s . Up t i l l the presen t time, the experiments c o n s i s t e d of i n t r o d u c i n g h i g h energy p o s i t r o n s i n t o the sample gas and a n a l y s i n g the a n n i h i l a t i o n r a d i a t i o n e i t h e r d u r i n g the s l o w i n g of the p o s i t r o n s or a f t e r they had a t t a i n e d a 2 time-independent v e l o c i t y d i s t r i b u t i o n . These swarm type experiments have been used e x t e n s i v e l y t o measure the 2-5 p o s i t r o n - g a s atom i n t e r a c t i o n . The need t o know the p o s i t r o n v e l o c i t y d i s t r i b u t i o n as a f u n c t i o n of time c o m p l i c a t e s the experiment and the r e s u l t i n g comparison w i t h t h e o r y . Thus, i n t e r e s t c e n t e r e d on ways of m o d i f y i n g the s t e a d y - s t a t e v e l o c i t y d i s t r i b u t i o n of the p o s i t r o n s . P o s i t r o n energy v a r i a t i o n i s o b t a i n e d by v a r y i n g the temperature of the gas and/or by a p p l y i n g an e l e c t r i c f i e l d . The f i r s t i n f l u e n c e s both the p o s i t r o n s and the gas, the second o n l y the p o s i t r o n s . T h i s type of experiment w i l l be d i s c u s s e d more t h o r o u g h l y i n c h a p t e r two. On t h e o t h e r hand the use of monenergetic low energy p o s i t r o n beams, f i r s t suggested by D.F. H e r r i n g i n 1964 t o e x p l o r e t h e v e l o c i t y dependence of the i n t e r a c t i o n between a p o s i t r o n and an atom, i s s t i l l i n i t s i n f a n c y . The beams are produced by moderating ( t o about 1 eV) and c o l l i m a t i n g e i t h e r t h e p o s i t r o n s a r i s i n g from bremsstralung r a d i a t i o n 22 6 — 8 or t h o s e from a Na source of about 100/tfCi. V a r i a b l e e n e r g i e s are o b t a i n e d by a c c e l e r a t i n g the moderated p o s i t r o n s w i t h a s u i t a b l e e l e c t r i c f i e l d . The e n e r g i e s are then measured e i t h e r w i t h an e l e c t o s t a t i c a n a l y s e r of by use of t i m e - o f - f l i g h t . For t h e experiment i t s e l f , a low d e n s i t y of s c a t t e r i n g gas i s s u s t a i n e d by continuous pumping on the sample area w h i l e l e a k i n g i n the gas. The e f f e c t of the i n t e r a c t i o n i s t h e n i n f e r r e d from a measurement of the number of p o s i t r o n s 3 t r a v e r s i n g the length of f l i g h t path as a function of gas density. An a x i a l magnetic f i e l d i s usually applied to confine the unscattered positrons to the axis. Detection of i n d i v i d u a l positrons i s straightforward because of t h e i r convenient "signature", the production of a pair of 0.511 MeV a n n i h i l a t i o n gamma rays, when they are stopped i n a metal f o i l . At present, the beams, although of very low in t e n s i t y , are producing preliminary t o t a l s c a t t e r i n g cross sections f o r helium and argon gases as a function of energy up to 6 8 about 27 eV. Of course the comparison with theory i s more d i r e c t f o r t h i s type of experiment. In f a c t , i t was due to the disagreement between old r e s u l t s of a swarm type experiment^, where the sc a t t e r i n g cross se c t i o n (for momentum-transfer) was measured, with both t h e o r e t i c a l estimates'''^''''''" and the preliminary r e s u l t s of recent beam measurements^ -^ ( t o t a l cross sections), which prompted repeating t h i s swarm type measurement f o r helium. In addition the role of impurities i n swarm experiments was investigated. Interestingly, the presence of impurities can, on the one hand, hinder the task at hand, namely, e x t r a c t i n g s c a t t e r i n g cross sections, or, on the other hand, add another degree of freedom to the experiment and thus broaden the scope of the experimental technique. 4 CHAPTER TWO POSITRONIUM FORMATION 2.1 I n t r o d u c t i o n The use of swarm experiments t o study the be h a v i o r 2_ c of p o s i t r o n s i n gases has been continued i n t h i s woek. In p a r t i c u l a r , the i n v e s t i g a t i o n c e n t e r e d on the enhancement of p o s i t r o n i u m p r o d u c t i o n i n helium at 300 K by a D.C. e l e c t r i c f i e l d . T h i s e f f e c t , f i r s t observed by Deutsch and 12 Brown who a p p l i e d a r a d i o frequency f i e l d i n t h e i r study of the Zeeman e f f e c t of p o s i t r o i u m , was subsequently s t u d i e d i n d e t a i l i n v a r i o u s gased by Marder et a l . ^ From the enhancement of p o s i t r o n i u m f o r m a t i o n as a f u n c t i o n of a p p l i e d e l e c t r i c f i e l d the magnitude of t h e v a r i o u s c r o s s s e c t i o n s p e r t i n e n t t o the f o r m a t i o n p r o c e s s can be obtained, ( s e c t . 2 .42) . In t h i s way Marder et a l , by s t u d y i n g the energy spectum of the gamma r a d i a t i o n which r e s u l t s from the a n n i h i l a t i o n of the p o s i t r o n - e l e c t r o n system, d e r i v e d a va l u e f o r the momentum-transfer c r o s s s e c t i o n , (j m > ^ o r p o s i t r o n s on helium of 0. o23TTaQ + 25%-\ T h i s value f o r the c r o s s s e c t i o n i s an average over a range of e n e r g i e s i n the v i c i n i t y of 17.7 eV, the t h r e s h o l d f o r p o s i t r o n i u m f o r m a t i o n i n helium. T h i s c r o s s s e c t i o n f o r helium i s c u r r e n t l y c o n s i d e r e d t o o s m a l l by an order of magnitude. More re c e n t experiments by Lee and Paul i n d i c a t e d a lower l i m i t f o r the c r o s s s e c t i o n of about 0.1357Ta . Recent t o t a l c r o s s s e c t i o n o 5 measurements using monenergetic beams^-^ indicate values of 0.24 to 0.37Tfao2 at 17-7 eV. Theoretical e s t i m a t e s 1 0 ' 1 1 are also more consistent with the larger values. Lee^ at t r i b u t e d the small values obtained i n the early work to the r e s u l t of impurities i n the helium gas. As was mentioned i n chapter one, the purpose of the present work was to repeat the measurements of Marder et a l f o r helium gas i n an attempt to determine whether the d i f f u s i o n technique would y i e l d a value consistent with the more recent values me nt i one d ab ove . 2.2 Positrons i n a Gas 2.21 The Slowing down of Positrons i n a Gas 2 2 The positrons are produced by the beta decay of Na , having a h a l f l i f e of 2.58 years. The emission of a 1.2 74 MeV gamma ray less than I O - 1 1 sec l a t e r , although useful f o r positron l i f e t i m e measurements, complicates the energy spectrum of the a n n i h i l a t i o n gamma rays. The positrons themselves have a maximum energy of 542 keV and a mean energy of 190 keV. In a gas, these energetic positrons slow down by i n e l a s t i c c o l l i s i o n s t o about 20 eV and below i n a very short time^, and with l i t t l e a n n i h i l a t i o n i n f l i g h t . ^ 4 - 1 ° 2-4 * This time i s about 3 nsec f o r helium at 20 amagats . * 1 amagat = 2.687 x l O 1 ^ molecules cm~^ and i s the molecular density of 1 mole of i d e a l gas at STP. 6 Now i t i s known t h a t the bound system of a p o s i t r o n and an e l e c t r o n , denoted p o s i t r o n i u m or mnemonically as 17 Ps, can e x i s t xn many m a t e r i a l s and e s p e c i a l l y i n gases. The energy t h r e s h o l d , ^ n r > f ° r the f o r m a t i o n of t h i s system i s r e l a t e d t o the i o n i z a t i o n energy, E ^ o n , of the sample gas and the b i n d i n g energy of the Ps atom (6.78 eV), by the e x p r e s s i o n E t h r = E i o n ~ 6 - 7 8 ' For h elium, the r e l e v e n t energy l e v e l s are d i s p l a y e d i n F i g . 1. E-{-nr and E ^ Q n , ..are shown i n Table I f o r other common gases. I n o r d e r t o e s t i m a t e the f r a c t i o n of Ps formed on s l o w i n g down, an assumption must be made about the v e l o c i t y d i s t r i b u t i o n of the p o s i t r o n s a f t e r t h e i r l a s t i o n i z i n g c o l l i s i o n . L i m i t s can then be put on t h i s f r a c t i o n by f i r s t i g n o r i n g Ps f o r m a t i o n i n r e g i o n I I I of F i g . 1 and secondly by i g n o r i n g i n e l a s t i c c o l l i s i o n s i n r e g i o n I I . The i n e l a s t i c c o l l i s i o n s reduce the number of p o s i t r o n s a v a i l a b l e i n r e g i o n I I . Region I I i s the s o - c a l l e d "Ore gap". T r a d i t i o n a l l y t h e d i s t r i b u t i o n has been c o n s i d e r e d 17 u n i f o r m i n energy between 0 and E ^ Q n . The f r a c t i o n i s t h e n between ( l - ( E t h r / E e x c ^ > w h e r e E e x c i s t h e e n e r S Y of the f i r s t e x c i t e d l e v e l f o r the gas atom, and ( l - (E,, /E. )) or, f o r helium, 0.11 and 0.28 r e s p e c t i v e l y , t h r ' i o n f J I f , however, the d i s t r i b u t i o n i s u n i f o r m i n phase space then the l i m i t s are ( l - E,, 3 / 2 / E 3 ^ 2 ) and v t h r ' exc ' 7 Gas E i o n ( e V ) ^ E t h r ( e He 24.48 17.7 Ar 15.76 9.0 Kr 14.00 7.2 N 2 15.58 8.8 °2 12 .06 5-3 c o 2 13.77 7.0 H 2 0 12.6 5.8 H 2 15.43 8.65 C 2 H 2 11.4 4.6 C 7Hg 8.5 1.7 Table I Po s i t r o n i u m Formation Thr e s h o l d s of Common Gases. * from Handbook of  Chemistry and P h y s i c s , 51st. ed., Chemical Rubber P u b l i s h i n g Co. IV 24.48 eV I o n i z a t i o n ^ 17,7 e V Th r e s h o l d f o r Ps Formation E l a s t i c C o l l i s i o n Region Helium L e v e l Scheme Showing Region Predominating i n P o s i t r o n i u m Formation I (shaded). 0.0 F i g u r e 1 eV 8 ( l - E 3 / V E . 3 / / 2 ) or, 0.16 and 0.38 f o r helium. On t h r xon simple s t a t i s t i c a l arguments, one would expect the l a t t e r l i m i t s . In any case the ex p e r i m e n t a l value f o r helium of 18 0.23 + .02 found by Pond i s w i t h i n both s e t s of l i m i t s . However, the p u r i t y of the gas was not g i v e n and no mention -1 o was made of gas pressure or the e f f e c t of quenching ( s e c t . 2.32). A l s o the quenching e f f e c t of NO was used i n the experiment without c o n s i d e r a t i o n of the e f f e c t of t h i s i m p u r i t y on the Ps f o r m a t i o n . In t h i s t h e s i s , the e f f e c t of 32 ppm NO on Ps f o r m a t i o n was determined, as g i v e n i n s e c t . 2.32. I f Ps i s not formed, the p o s i t r o n s f a l l below the Ore gap and tend t o reach e q u i l i b r i u m v i a e l a s t i c c o l l i s i o n s w i t h the gas atoms. Of course, i m p u r i t y molecules having l o w - l y i n g v i b r a t i o n a l and r o t a t i o n a l l e v e l s w i l l c o n s i d e r a b l y s h o r t e n the time t o e q u i l i b r i u m . In any case these p o s i t r o n s o n l y have the chance t o a n n i h i l a t e f r e e l y (or sometimes spoken of as d i r e c t l y ) . The r a d i a t i o n from t h i s channel i s p r i m a r i l y composed of two photons each w i t h an energy of 0.511 MeV, and em i t t e d c l o s e t o 180° w i t h r e s p e c t t o each other i n order t o conserve momentum. The preponderance of the two photon a n n i h i l a t i o n i s due t o the r e l a t i v e l i f e t i m e s of the s i n g l e t and t r i p l e t c o n f i g u r a t i o n ( s e c t . 2.31) 19 of 1 t o 372. A n n i h i l a t i o n of the p o s i t r o n s d u r i n g the time t h a t they approach e q u i l i b r i u m has been e x p e r i m e n t a l l y demonstrated by the e x i s t e n c e of a sh o u l d e r p r e c e d i n g the 9 e x p o n e n t i a l p a r t s of the time spectrum.^ T h i s shoulder i m p l i e s a v e l o c i t y - d e p e n d e n t a n n i h i l a t i o n r a t e . 2.22 The E q u i l i b r i u m P i c t u r e f o r P o s i t r o n s i n a Gas Under an A p p l i e d E l e c t r i c F i e l d . Assume now t h a t the p o s i t r o n s are i n thermal e q u i l i b r i u m w i t h the gas. The p o s i t r o n s are then d e s c r i b e d by a v e l o c i t y d i s t r i b u t i o n v a r y i n g i n amplitude due t o d i r e c t a n n i h i l a t i o n s but not i n shape. By a p p l y i n g a D.C. e l e c t r i c f i e l d , the average energy of the p o s i t r o n s i s i n c r e a s e d and t h u s , as w i l l become clear,, the s c a t t e r i n g c r o s s s e c t i o n i n d i f f e r e n t energy r e g i o n s can, i n t h e o r y , be sampled. I f one i g n o r e s a n n i h i l a t i o n s , t h e n the i s o t r o p i c p a r t of the p r o b a b i l i t y 2 20 21 d e n s i t y i n v e l o c i t y space, f Q ( v , S / D ) , can be w r i t t e n ' ' , t o a good approximation, as f D ( v , g / D ) A exp mv'dv T 1 M e (£/D) kT + 3 (mv Cjv*))2 (1) where k = Boltzmann's constant, m = mass of the p o s i t r o n , M = mass of the gas atom, e = e l e c t r i c charge, v = v e l o c i t y of the p o s i t r o n , £ = e l e c t r i c f i e l d s t r e n g t h , 10 D = d e n s i t y of gas, and A = a n o r m a l i z a t i o n constant d e f i n e d below. The a n i s o t r o p i c p a r t of the v e l o c i t y d i s t r i b u t i o n i s i 21 n e g l i g i b l e i f (m/M) 2 « 1. Indeed, the next term of the expansion f = f Q + e j f . "v f x + . . . , — (2) m where f i s the complete v e l o c i t y d i s t r i b u t i o n , i s r e l a t e d t o f by Chapman and Cowling, i n the approximation (m/M) << 1, by the e x p r e s s i o n f 1 = 3 m 3v f Q , — ( 3 ) M e 2 6 2 ^ ( v ) where ^ ( v ) i s the mean f r e e path of the p o s i t r o n s . For an i s o t r o p i c d i s t r i b u t i o n , f 0 ( v ) i s r e l a t e d t o the v e l o c i t y d i s t r i b u t i o n per u n i t v e l o c i t y i n t e r v a l , Y ( v ) , by Y(v) = 4 7 T v 2 f 0 ( v ) . (4) Thus the n o r m a l i z a t i o n constant A i s def.ined by p u t t i n g r-OO 4 77/ v 2 f Q ( v ) dv = 1. I t s h o u l d be noted t h a t the shape of Y(v) i s , i n g e n e r a l , dependent on the v e l o c i t y dependence of Om* F a l k c a l c u l a t e d f 0 ( v , c R / D ) < f o r two cases, (j m a c o n s t a n t , and p r o p o r t i o n a l t o l / v . More important e x p e r i m e n t a l l y i s the dependence of the d i s t r i b u t i o n on the r a t i o (£/D)/ (jm-For example i t i s reasonable t o expect t h a t the mean of 11 "the d i s t r i b u t i o n occurs approximately where the average energy gained between c o l l i s i o n s due t o the e l e c t r i c f i e l d , m 2 2 ' m (6) m where ^ m ~ the average of the momentum-transfer r a t e , note 7 m ( v ) = D 0 m ( v ) ' v J , i s balanced by the average energy l o s s , mi " I E, I M J d u r i n g an e l a s t i c c o l l i s i o n , where E of the d i s t r i b u t i o n . ^ Thus 1 < (7) E ss ,3/2 l , M \ 2 m = the average energy (8) m so t h a t i f the p o s i t i o n of the maximum of the d i s t r i b u t i o n m o d i f i e d by the f i e l d can be measured , t h e n a value of (j m which i s averaged over some energy r e g i o n near E can be i n f e r r e d . Lee^ u s i n g the t h e o r e t i c a l c r o s s s e c t i o n s of Drachman 1 1 , c a l c u l a t e d the d i s t r i b u t i o n f o r £/D = 27.7 V cm'^amagats""^. T h i s d i s t r i b u t i o n f o r helium i s shown i n F i g . 2 i n r e l a t i o n t o the Ps f o r m a t i o n t h r e s h o l d . Thus from F i g . 2. i t i s c l e a r t h a t i f an i n c r e a s e i n Ps f o r m a t i o n can be "detected as a f u n c t i o n of a p p l i e d f i e l d t hen at l e a s t the o v e r l a p of the p o s i t r o n d i s t r i b u t i o n above the Ps f o r m a t i o n t h r e s h o l d can be d e t e c t e d . An estimate of the momentum-transfer c r o s s s e c t i o n i s then o b t a i n a b l e from E / D = -1 -1 27.7 volts-cra-amagat E t h r in Helium 12~ —t— 14 16 8 10 ENERGY (ev) T h e o r e t i c a l P o s i t r o n Energy D i s t r i b u t i o n U s i n g R e s u l t s of Drachman. Ps f o r m a t i o n t h r e s h o l d i s w e l l above upper l i m i t of d i s t r i b u t i o n . 13 the average energy and the a p p l i e d f i e l d by u s i n g eq. (8). Of c o u r s e , i n the f i n a l a n a l y s i s , a more e x t e n s i v e a n a l y s i s based on d i f f u s i o n t h e o r y i s used than t h a t l e a d i n g up t o eq. (8). The present treatment i s u s e f u l , however, as a q u a l i t a t i v e argument. To summarize, at 300 K, most p o s i t r o n s at thermal e q u i l i b r i u m have e n e r g i e s below 0.1 eV (re p r e s e n t e d by a ver y narrow l i n e near 0 eV i n F i g . 2 ) . At t h i s j u n c t u r e , the d i s t r i b u t i o n i s independent of (jm('r) > whereas once an e l e c t r i c f i e l d i s a p p l i e d the e q u i l i b r i u m d i s t r i b u t i o n does depend on (jm(v) a s d i s c u s s e d above (e.g. eq. l ) . I t then f o l l o w s t h a t above a c e r t a i n a p p l i e d e l e c t r i c f i e l d t h e r e w i l l be an i n c r e a s e i n Ps f o r m a t i o n due t o the t a i l of the e q u i l i b r i u m d i s t r i b u t i o n moving up i n t o the Ore gap. I t i s t h i s i n c r e a s e i n Ps f o r m a t i o n t h a t i s measured i n t h i s work, and from i t an estimate of the momentum-t r a n s f e r c r o s s s e c t i o n averaged over the d i s t r i b u t i o n can be made. In the d i s c u s s i o n below of the v a r i o u s c r o s s s e c t i o n s , the magnitude of the Ps f o r m a t i o n c r o s s s e c t i o n , ( j f , a l s o appears as a parameter. However, because of the combined e f f e c t s of gas i m p u r i t i e s and c o u n t i n g s t a t i s t i c s , the e x p e r i m e n t a l r e s u l t s were i n s e n s i t i v e t o assumed v a r i a t i o n s of ( j f of up t o t h r e e o r d e r s of magnitude around 7Tao^ •• In the next s e c t i o n , the p r o p e r t i e s of Ps and Ps f o r m a t i o n are d i s c u s s e d i n order t o determine those p r o p e r t i e s which make the f o r m a t i o n of p o s i t r o n i u m d e t e c t a b l e . 14 2.3 P r o p e r t i e s of P o s i t r o n i u m 2.31 S t r u c t u r e of P o s i t r o n i u m i n Vacuum P o s i t r o n i u m i s s i m i l a r t o the hydrogen atom, b e i n g composed of an e l e c t r o n and a p o s i t i v e p a r t i c l e , but d i f f e r i n g i n the mass of the p o s i t i v e p a r t i c l e , which, i n the case of p o s i t r o n i u m , has the same mass as the e l e c t r o n . T h i s r e s u l t s i n a reduced mass very n e a r l y one-half t h a t of hydrogen y i e l d i n g a b i n d i n g energy of 0.5 Ry ( i . e . 6.78 e V ) . However, the mean r a d i u s of e i t h e r system i n -8 i t s c e n t e r of mass i s 0.794 x 10 cm. The ground s t a t e of Ps i s uns t a b l e a g a i n s t a n n i h i l a t i o n i n t o two 6r more photons, ' the number depending on whether the e l e c t r o n or p o s i t r o n combine i n t o a s i n g l e t ( p a r a p o s i t r o n i u m , d e s i g n a t e d p-Ps) J=0 s t a t e or t r i p l e t ( o r t h o p o s i t r o n i u m , o-Ps) J=l s t a t e . A n n i h i l a t i o n from the s i n g l e t s t a t e , p r i m a r i l y a two photon channel, has been mentioned i n c o n n e c t i o n w i t h the process of d i r e c t 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 from the t r i p l e t s t a t e c o n s i s t s b a s i c l y of t h r e e photons. The a n n i h i l a t i o n energy spectrum 19 f o r t h r e e photons has been c a l c u l a t e d by Ore and Powell * The mean r a d i u s i s g i v e n by <r> = P V'*(r)r 3 ^ ( r ) dr, where I iff * ( r ) r 2 iff ( r ) dr - 1. In t h e c e n t e r of mass, f o r the ground s t a t e s , ^ h y d r o g e n ( r ) = ^ P s ( r ) = 2 a 0 " 3 / 2 , e x p ( - r / a 0 ) , where a o = Bohr r a d i u s . 15 and c o n s i s t s of a continuous d i s t r i b u t i o n up t o 0.511 MeV, where i t i s s i g n i f i c a n t l y peaked, ( F i g . 3 ) . T h i s suggests a h i g h p r o b a b i l i t y t h a t two of the t h r e e photons leave the p o i n t of a n n i h i l a t i o n t o g e t h e r . Table I I shows the a p p r o p r i a t e mean l i f e t i m e , a n n i h i l a t i o n r a t e , and allowed number of a n n i h i l a t i o n photons f o r the two s p i n s t a t e s . 1 9 ' 1 4 2.32 Positonium i n a Gas I f the Ps i s formed i n a gas the r a t i o of t r i p l e t t o s i n g l e t i s expected t o be 3 t o 1, i n the r a t i o of the r e l a t i v e s t a t i s t i c a l weights. T h i s r e p r e s e n t s a g r e a t l y i n c r e a s e d r a t e of t h r e e photon a n n i h i l a t i o n s compared t o 19 t h a t of t h r e e photon a n n i h i l a t i o n s of f r e e p o s i t r o n s , which i s reduced by a f a c t o r of 372 from the two photon f r e e p o s i t r o n a n n i h i l a t i o n s . Thus i n p r i n c i p l e an i n c r e a s e i n the f r a c t i o n of p o s i t r o n s f o r m i n g Ps can be d e t e c t e d by the r e s u l t i n g change i n the energy spectrum of the gamma r a y s . T h i s f o l l o w s from the f a c t t h a t the t h r e e photon energy spectrum i s a continuous d i s t r i b u t i o n peaked at 0.511 MeV, whereas the two photon spectrum c o n s i s t s of a l i n e c e n t e r e d at 0.511 MeV. The p i c t u r e d e s c r i b e d above n e g l e c t s the c o m p l i c a t i n g p rocess known as quenching. In t h i s p r o c e s s , the a n n i h i l a t i o n of the t r i p l e t s t a t e by the normally f o r b i d d e n two photon channel can occur when the p o s i t r o n i n o-Ps encounters a s u i t a b l y o r i e n t e d second e l e c t r o n bound t o the gas atoms 16 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 energy (mc 2) Energy Spectrum f o r Three Gamma Annih i l a t i o n s , (ref. 11). P r o p e r t i e s of n=l P o s i t r o n i u m D e s i g n a t i o n s p i n / s t a t e l i f e t i m e (sec ) 11 a n n i h i l a t i o n r a t e ( s e c " 1 ) number of a n n i h i l a t i o n photons o-Ps 1 t r i p l e t 1.4 x 10 -7 71 x 10' odd minimum of 3 p-Ps s i n g l e t 1.25 x 10 -10 80 x 10 10 even minimum of 2 Table I I ••Hmsciv±he Ps atom encounters d u r i n g c o l l i s i o n s . I f i t s s p i n i s a n t i p a r a l l e l t o the p o s i t r o n ^ , a much more r a p i d a n n i h i l a t i o n may ensue. The v a r i o u s types of quenching 17 are w e l l d i s c u s s e d by F r a s e r . The most e f f i c i e n t quenchers are H^, NO, and 0^. In the type of experiment r e p o r t e d here, where the p o s i t r o n i u m " s i g n a t u r e " i s t h a t of a t h r e e .photon a n n i h i l a t i o n , i t i s c l e a r t h a t the sample gas sh o u l d be as f r e e as p o s s i b l e of such quenching agents. For example, f o r t h e r m a l i z e d p o s i t r o n s , NO has a quenching c r o s s s e c t i o n of 0 . 167Ta Q 2. In comparison, the s m a l l s e l f - q u e n c h i n g r a t e f o r helium as c i t e d by Lee^ i s (0 .086 + . 0 0 8 ) x 10^ sec ''"amagats \ T h i s corresponds t o a s e l f - q u e n c h i n g c r o s s s e c t i o n of 0 . 0 0 2 7 T 7 " a o 2 at thermal e n e r g i e s . Such a va l u e means t h a t the o-Ps l i f e t i m e i s _7 o n l y reduced t o 1 . 2 x 10 sec at 15 amagats of helium whereas 32 ppm NO would y i e l d the same e f f e c t . Thus, al t h o u g h quenching i n helium does e x i s t , i t s e f f e c t i s r e l a t i v e l y u n i m p i r t a n t i n terms of d e c r e a s i n g the photon r a t e from the t h r e e t o one r a t i o . In any event because o n l y the i n c r e a s e of Ps f o r m a t i o n as a f u n c t i o n of e l e c t r i c f i e l d need be measured t o o b t a i n a momentum-transfer c r o s s s e c t i o n , seIf-quenchingjis unimportant i f the d e n s i t y remains c o n s t a n t . The e f f e c t of other quenchers w i l l depend on t h e i r i m p u r i t y l e v e l s . 19 2.4 Formation of P o s i t r o n i u m i n Helium 2.41 P o s i t r o n i u m Formation Cross S e c t i o n 2.411 T h e o r e t i c a l The Ps f o r m a t i o n c r o s s s e c t i o n , (j f, f o r helium i s zero below E^. n r. Above E^. n r, d e t a i l e d t h e o r e t i c a l u n d e r s t a n d i n g i s very incomplete. However, re c e n t c a l c u l a t i o n s by F e l s and M i t t l e m a n 1 0 f o r (j f f o r helium g i v e a value which r i s e s v ery s h a r p l y at E ^ n r and remains at about 1.6 x 10' " 3 TTa J up t o E i o n where the c a l c u l a t i o n was terminated., P r e v i o u s t o the F e l s and Mittleman work, c a l c u a t i o n s 9 T were based on the f i r s t Born approximation, J whereas the l a t e r work employed a v a r i a t i o n a l t e c h n i q u e . The Born c a l c u l a t i o n y i e l d e d a peak f o r i n helium of 0.4 77*ao at 28 eV. ^comparison i s g i v e n i n F i g . 4. Because of the rearrangement nature of the c o l l i s i o n s the Born approximation i s of d o u b t f u l v a l i d i t y . S i m i l a r r e s u l t s have a l s o been found f o r hydrogen 4 ' 26 where the impulse approximation, d i s t o r t e d wave 2 A 27 approximation H or the two-state approximation have a l l been employed i n an attempt t o improve on the Born approximation i n l i e u of a " s a t i s f a c t o r y " t h e o r y f o r 2 8 29 rearrangement c o l l i s i o n s ' " ( c o l l i s i o n s i n which p a r t i c l e s , i n t h i s case e l e c t r o n s , are t r a n s f e r r e d between the c o l l i d i n g systems). The r e s u l t of F e l s and M ittleman g i v e s a value of (j ^ f o r hydrogen about 60 times t h a t f o r helium, 20 o H < O H § Oi H H CO O fe S3 O H H O w CO CO CO o o 0 .1 .2 .4 .6 1.4 1.5 1.6 1.7 B 1.307448 Rydbergs 2 _ p o s i t r o n energy, Ps energy. A : Massey and Moussa (1961). (Born a p p r o x i m a t i o n ) . B : F e l s and Mittleman (1969). ( V a r i a t i o n a l ) f i g u r e 4 P o s i t r o n i u m Formation Cross S e c t i o n s 21 t h i s difference being attributed to the t i g h t binding energy of the helium atom.^ The corresponding difference 9 1 27 i n the Born c a l c u l a t i o n i s only a factor of 10. J ' ' A l l c a l c u l a t i o n s used various polarzation potentials fo r the atom and showed some s e n s i t i v i t y to the form of the p o l a r i z a t i o n . 2.412 Experimental Experimentally, very l i t t l e i s known about O f • However, some idea of the r e l a t i v e size ofO'f to (j m and (ja, the momentum-transfer and d i r e c t a n n i h i l a t i o n cross sections respectively, i s required f o r a thorough analysis o f experimental r e s u l t s . T h e r e f o r e , t h e o r e t i c a l estimates must be used, at present, f o r O f b e f o r e any analysis of Ps f o r m a t i o n data can be made. The reason f o r t h i s i s discussed l a t e r (sect. 2 .42). 20 Teutsch and Hughes i n t h e i r analysis of the data of Marder et a l ^ , based t h e i r estimate of (j f on the e x i s t i n g theories which consisted of Born approximations as mentioned i n sect. 2.411. Thus they assumed that O'-p was as large as 7Ta 2 and thus as large or larger than O'm* "*"n t h i s case a l l positrons passing above can be assumed to form Ps. Since, experimentally, one knows that (j & I s much less than (jmAJ (sect. 2.44)* only the energy region below E ^ needs to be analysed by the d i f f u s i o n theory i n order to in t e r p r e t the point at which Ps formation begins. In the l i g h t of the more recent c a l c u l a t i o n s , however, 22 i n d i c a t i n g a much s m a l l e r value f o r (j ^ ( s e c t . 2.41l)> i t i s p o s s i b l e i n p r i n c i p l e , ( s e c t . 2.41l)> t o determine the order of magnirude of (j f from the e x p e r i m e n t a l e v i d e n c e . The c o n s i d e r a t i o n of the r e g i o n above E-thr* of course, depends on the r e l a t i v e s i z e of (j^ t o (j m . 2.42 T h e o r i e s of P o s i t r o n i u m Formation For some reason, which i s not e n t i r e l y c l e a r t o the author^ i t was thought t h a t a (j f of the order of 10 3 7Ta Q would not c o n t r i b u t e s u b s t a n t i a l l y t o Ps f o r m a t i o n . F e l s and Mittleman, however, p u r e l y on t h e o r e t i c a l grounds, have concluded t h a t even w i t h t h e i r s m a l l (j f of 1.6 x 10 3 7Ta 02, the a n n i h i l a t i o n r a t e above t h r e s h o l d d e r i v e s almost c o m p l e t e l y from the Ps formed t h e r e . That i s , such a s m a l l value of (j £ i s q u i t e c o n s i s t e n t w i t h a r e l a t i v e l y l a r g e i n c r e a s e of Ps at the a p p r o p r i a t e v a l u e s of e l e c t r i c f i e l d . In the a n a l y s i s p r e s e n t e d here, both (j f > " l a r g e " and " s m a l l " , have been assumed. Because of d i f f i c u l t i e s e x p e r i e n c e d i n s o l v i n g the d i f f u s i o n e q u a t i o n , ( s e c t . 2.45)> at the t i m e , the l i m i t s of (j £ cz (j m , and (j f « (jm had t o be assumed. A more g e n e r a l s o l u t i o n s h o u l d c e r t a i n l y be p o s s i b l e n u m e r i c a l l y . However } the s e n s i t i v i t y of the r e s u l t s t o the value* of (j ^ was not s u f f i c i e n t l y g r e a t t o warrent more d e t a i l e d treatment i n t h i s p r e l i m i n a r y work. 2 0 1 Since Teutsch and Hughes had c a l c u l a t e d r e s u l t s f o r both cases of (jf, t h e i r method of a n a l y s i s was used. T h i s 23 procedure has been f o l l o w e d here even though a more d e t a i l e d 31 model has r e c e n t l y been developed by Brandt and F e i b u s . J The r e a s o n f o r t h i s i s t h a t the l a t t e r c a l c u l a t i o n i s l i m i t e d i n a p p l i c a b i l i t y s i n c e i t assumed onl y ..i t h a t t h e f o r m a t i o n of Ps i s a s m a l l p e r t u r b a t i o n on the e l e c t r i c f i e l d - d e p e n d e n t t h e r m a l i z e d d i s t r i b u t i o n . Thus, t h i s assumption does not permit v a l u e s of (j ^ as l a r g e as CTjn' ^ n such a case the v e l o c i t y d i s t r i b u t i o n can be s e v e r e l y d i s t o r t e d . I t was f e l t t h a t f i t s t o experiment w i t h i n t h e framework of a s i n g l e t h e o r y would be more m e a n i n g f u l . I n f a c t , the f u n c t i o n a l dependence of I, the f r a c t i o n of t h e r m a l i z e d p o s i t r o n s f o r m i n g Ps, on £ Y D appeared s i m i l a r f o r the case of (jf^(jm i n both c a l c u l a t i o n s . In terms of the th e o r y of Teutsch and Hughes the f o l l o w i n g r e s u l t s were o b t a i n e d . For (j f l a r g e , both the p o i n t at which I i n c r e a s e s and the value of the slope of I depend on CTm. There i s a l s o a dependence on (j a -However, s i n c e t h i s c r o s s s e c t i o n i s determined a c c u r a t e l y from l i f e t i m e experiments^ i t was a known parameter, ( s e c t . 2.442). In g e n e r a l , f o r the average value of (ja i n c r e a s i n g or t h a t of Cfm d e c r e a s i n g , as parameters, the sl o p e of I (at t h e p o i n t of maximum s l o p e ) d e c r e a s e s . Thus f o r (j m s m a l l t h e r i s e i s g r a d u a l . For (j ^ s m a l l , the p o s i t i o n at which the maximum s l o p e o f I occ u r s as a f u n c t i o n of < £ / D i s e q u i v a l e n t t o t h a t d e f i n e d f o r (j^ l a r g e . However, now the s l o p e , at 24 maximum, i n c r e a s e s with (jf/ Cf a' Both s e t s of curves are shown i n F i g . 5- Note t h a t I i s n o r m a l i z e d t o the value 1 at " s a t u r a t i o n " and 0 at zero f i e l d . A more complete d e s c r i p t i o n of the parameters a p p e a r i n g i n F i g . 5 f o l l o w s . These are the g e n e r a l r e s u l t s p r e s e n t e d by Teut s c h and Hughes and now a p p l i e d t o the case of helium h e r e . For (jf l a r g e , the curves are paramete r i z e d by Ca a Q 2 o M A, 4 (j v., m w m t h r (9) where v^. n r = v e l o c i t y of the p o s i t r o n at t h r e s h o l d , a Q = Bohr r a d i u s , c = v e l o c i t y of l i g h t , m, M are d e f i n e d w i t h eq. ( l ) , p.9> and rf,,* and Qm are d e f i n e d i n s e c t . 2.412. >"' cl 111 For helium^ / C T a or A; A, i f CJ. 6 x 1.85 x 10""3 s m i / (2.01 + .25) x 1 0 - 3 , ,-6 (10) = (1.1 + .12) x 10" u 7 f a Q 2 i s used, ( s e c t . 2.442), (j has u n i t s of 77* ac? • Eq. (10), t h u s , d i s p l a y s the dependence of the slo p e of I o n <Jm' For (jm s m a l l , f o r helium, the parameter k, i n F i g . S, 25 I I n c r e a s e i n the Ps f o r m a t i o n f r a c t i o n , I as a f u n c t i o n of the d i m e n s i o n l e s s parameter €. , where e. « ( £ /D)/ (f m (jf l a r g e L i i i i _ 0 0.5 1.0 c 1.5 2.0 Cjf s m a l l R e s u l t s of the a n a l y s i s of Teutsch and Hughes f i g u r e 5 26 i s r e l a t e d t o the c r o s s s e c t i o n s as f o l l o w s or 10 10* 1 cfa x 3 .88 x 10 -7 - (jf x (0.35 + 0.03 ) , (11) as f o r eq. ( 1 0 ) . Now, eq. ( l l ) d i s p l a y s the dependence of I on Cjf For helium, the d i m e n s i o n l e s s parameter € i s r e l a t e d t o (j m and by / 1 \ D x 1.34 x 10 -3 (12) ml Thus, j u s t from eq. 12, a rough estimate of (jm can be made, i f the value of £/D at which an i n c r e a s e i n Ps f o r m a t i o n i s d e t e c t a b l e , i s known. The s i m i l a r i t y i n the sl o p e and p o s i t i o n of the curves i n F i g . 5 f o r Aa* s m a l l or k l a r g e i s q u i t e e v i d e n t In such a case t h e r e i s no p r e f e r r e d value f o r (j f and t h i s parameter remains q u i t e u n d e f i n e d at l e a s t u n t i l the ex p e r i m e n t a l r e s u l t s are s u f f i c i e n t l y p r e c i s e t h a t the s m a l l d i f f e r e n c e i n the curves f o r the two cases are s i g n i f i c a n t . T h i s would r e q u i r e an accuracy i n I of about 1 % and i m p u r i t y l e v e l s of l e s s than 4 ppm. Brandt and Feibus i n t h e i r c a l c u l a t i o n f o r ( j f s m a l l , have d e r i v e d the e f f e c t due t o the f i e l d on the p o s i t r o n s t h a t are found i n the Ore gap a f t e r t h e i r l a s t i n e l a s t i c c o l l i s i o n . They f i n d t h a t i n g e n e r a l t h e r e i s an i n i t i a l decrease i n (^ ) , the t o t a l f r a c t i o n of Ps formed ( i n c l u d i n g 27 that formed on slowing down) at moderate f i e l d s p r i o r t o the r i s e at larger f i e l d s , the e f f e c t of I changing with £! /D discussed above. By t h i s means, they explain the anomaly which has been observed i n SF^. 9 This decrease i s discussed more f u l l y i n connection with the d i f f u s i o n equation describing positrons i n a gas, (sect. 2 .45) . The point i s raised i n t h i s section because i f such an e f f e c t i s indeed observable, one has i n p r i n c i p l e a way of measuring (j £ to within an order of magnitude or better. In the experiment reported here, any possible decrease would have been overshadowed by the e f f e c t of impurities as discussed i n sect. 2.46. 2.43 Momentum-Transfer Cross Section 2.431 Theoretical I f xs the d i f f e r e n t i a l e l a s t i c s c a t t e r i n g cross s e c t i o n then the momentum-transfer cross section, 0 m> i s defined as _ m whereas the more "common" t o t a l e l a s t i c s c a t t e r i n g cross section, (j m> xs Cfe = J ± ( Q ) d Q . (14) J ° Q In t h i s work, (j' , i s the more relevent of the two cross sections as glancing deflections do not decrease the amount of energy gained by the e l e c t r i c f i e l d , ( i . e . "eq". (8) ), 28 as much as l a r g e angle d e f l e c t i o n s do. E q u i v a l e n t l y , (jm i s r e l a t e d t o the average l o s s of momentum or energy per 3 2 e l a s t x c c o l l i s i o n . However, i f only S-wave s c a t t e r i n g i s predominant then the two c r o s s s e c t i o n s are e q u a l . T h i s i s 2 ? most e a s i l y seen from the p a r t i a l wave r e p r e s e n t a t i o n s , Cfe = \ E ( 2 / + 1) s i n 2 (5 (15) and (Jm .= V E U + D s i n 2 ( ( 5 - < 5 / + 1 ) , - ( 16) where k i s the p o s i t r o n energy i n Rydbergs and the (j f s are i n u n i t s of 77* a c • A r e c e n t c a l c u l a t i o n by F e l s and Mittleman"'"^ p r o v i d e d the (j and p a r t i a l waves shown i n F i g . 6 f o r e n e r g i e s from E^. n r t o 28.6 eV. (Some e x p e r i m e n t a l r e s u l t s , next s e c t i o n , are a l s o shown.) These r e s u l t s ^ would suggest t h a t about 90 % of the t o t a l s c a t t e r i n g c r o s s s e c t i o n d e r i v e s from the //=0 p a r t i a l wave. In t h i s case the two c r o s s s e c t i o n s s h o u l d be e q u a l t o w i t h i n 10 %. The v a l u e s found f o r (j e v a r y from (0.22 t o 0 .40) 77/ a Q depending on the degree of atomic p o l a r i z a t i o n assumed. For helium, they used the p o l a r i z a t i o n p o t e n t i a l of hydrogen t o g e t h e r w i t h a simple s c a l i n g f a c t o r . An a r b i t r a r y c u t o f f parameter s e t the s t r e n g t h of the p o l a r i z a t i o n . Drachman"'""'" g i v e s v a l u e s f o r both ( j m and (j & f o r 1.0-—J I I I 1.4 1.5 1.6 1.7 E t h r P i Rydbergs defined, figure 4-Fels and Mittleman (1969), theor. A- with weaker p o l a r i z a t i o n of the atom Drachman (1965), theor., extrapolated above E t h r , spread due to strength of p o l a r i z a t i o n . Canter et a l (1972), exper. beam work. Marder et a l (1956), exper. swarm-type. E l a s t i c Scattering and Momentum-Transfer Cross Sections. 30 helium f o r p o s i t r o n e n e r g i e s from zero t o E^ . ^ . For low v e l o c i t i e s , where S-waves dominate, (jm and (je are e q u a l . Near E^- n r, (j m i s g r e a t e r than (j e by 16 %. Thus the comparison of (j e and Cfm i s s e n s i t i v e t o the p a r t i a l waves c a l c u l a t e d . The spread of v a l u e s i s again due t o the s t r e n g t h of the p o l a r i z a t i o n . These r e s u l t s are a l s o shown i n F i g . 6. 2.432 Experimental Monenergetic beam experiments measure t o t a l s c a t t e r i n g whereas the swarm experiments measure momentum t r a n s f e r i n a c o l l i s i o n . These, as mentioned, are r e p r e s e n t e d by the .Ma-c o r r e s p o n d i n g c r o s s s e c t i o n s (je and (jm> r e s p e c t i v e l y . Values f o r both c r o s s s e c t i o n s are g i v e n i n Table I I I . L e e 4 concluded t h a t the low value o b t a i n e d by Marder et a l r e s u l t e d from i m p u r i t i e s as a r e s u l t of h i s study of the e f f e c t of the i m p u r i t i e s on the p o s i t r o n l i f e t i m e s p e c t r a of helium performed i n co n d u c t i o n w i t h a s e r i e s of v a l l e y t o peak r a t i o measurements. 2.44 A n n i h i l a t i o n Cross S e c t i o n f o r Free P o s i t r o n s i n Helium I n s e c t . 2.412, i t was mentioned t h a t a value f o r (j a , : the s i n g l e t a n n i h i l a t i o n c r o s s s e c t i o n , the predominant mode f o r f r e e a n n i h i l a t i o n s , was r e q u i r e d f o r a complete t h e o r y of Ps f o r m a t i o n . (ja i s normally r e l a t e d t o z e f f ( E ) , the e f f e c t i v e number of e l e c t r o n s per atom. Thus <J a(E) = (1.43 x 1 0 - 6 ) Z e f f ( E ) / E 5 7Ta Q 2 , — ( 1 7 ) Experimental S c a t t e r i n g Cross S e c t i o n s Near E^. n r, f o r Helium (TTa02) Cm (7Ta02) Approximate P o s i t r o n Energy (eV) C o s t e l l o et a l Leung and Paul Marder et a l 9 L e e 4 J a d u s z l i v e r e t a l * 8 Canter e t a l 0.35 + .04 0.37 + .04 0.24 ± .02 > 0.115 0.023 + .006 ^0.135 16 .5 near E t h r E t h r E t h r Note: E t h r = 17.7 eV, Table I I I 32 where E = energy of the p o s i t r o n i n eV. 2 0 Since the a n a l y s i s of Teutsch and Hughes i s r a t h e r i n s e n s i t i v e t o t h i s parameter as f a r as d e r i v i n g a value f o r (j m i s concerned, Marder et a l ^ used the simple estimate Z e £ f ( E ) = Z = 2, (the plane wave approximation), as no b e t t e r estimate was a v a i l a b l e at the time. E x p e r i m e n t a l l y , Lee^ measured Aa> ^ n e v e l o c i t y averaged f r e e a n n i h i l a t i o n r a t e , as a f u n c t i o n of £/D. From eq. (17), A a i s r e l a t e d t o Z e f f , the v e l o c i t y averaged Z e f f ( v ) , where v i s the v e l o c i t y of the p o s i t r o n , as f o l l o w s A a = (2.01 x 10 5) D Z e f f s e c " 1 , — . . — (18) where D = the d e n s i t y i n amagats. At f i e l d s above 10 V cm-•'•amagats-^, Aa w a s found t o be f a i r l y c o n s t a n t . T h i s y i e l d s a value f o r ^eff of 3.2 + .4 a t f i e l d s from 10 t o 35 V cm"''"amagats""'. I t i s thus reasonable t o use t h i s value f o r Z i n e r r the a n a l y s i s of Ps f o r m a t i o n data r a t h e r than Z, the atomic at t h r e s h o l d , (compared t o 0.71 x 10"° TTaQ f o r %eff = Z ) . number. With t h i s v alue of %eff} (ja i s 1.1 x 10 Tf a = _ £ z e f f ( v ) Y ( y ) d v where Y(v) i s d e f i n e d i n eq.(4). * i . e . Z e f f 33 2.45 D i f f u s i o n Equation and Positronium Formation The d i f f u s i o n equation which governs the time-dependent positron v e l o c i t y d i s t r i b u t i o n under an applied e l e c t r i c f i e l d 2 3 3 i s discussed i n d e t a i l by Falk. ' The equation f o r the is o t o p i c part of t h i s d i s t r i b u t i o n , f Q ( v , t ) , i s written by 31 Brandt and Feibus as follows i f . a t -2 (e£k) : + 7v^ a k [ 3H27m a k - (Ja + Jf) f G > where k = wave vector of the positron, = (positron momentum)/ R, "]f *s = rates, = D <J v, R-= Planck fs constant/ 277"> and the other symbols are as defined f o r eq. ( l ) , p. 9. Thus m i s the momentum-transfer rate, If ^  ~ 2m 1£m i s the f r a c t i o n a l rate of energy loss (here c h a r a c t e r i s t i c of the energies near the Ore gap), and fa anc* a r e ^ e free positron a n n i h i l a t i o n rate and the Ps formation rate re s p e c t i v e l y . Note, (m/M) i s 1.37 x 10~^ for helium. 2 0 Teutsch and Hughes have written eq. (19) i n terms of the positron v e l o c i t y , v, and the mean free path between c o l l i s i o n s , X = The equation was solved numerically f o r the Ps formation f r a c t i o n I, defined below, i n the l i m i t s of "large" and "small" (j f r e l a t i v e to (j m (sect. 2.42), along with the assumptions that (j and (j a are both 34 p r o p o r t i o n a l t o l / v . They f i n d t h a t i f the r e l a x a t i o n time of the p o s i t r o n s t o e q u i l i b r i u m a f t e r t h e i r l a s t i n e l a s t i c c o l l i s i o n i s much s h o r t e r than the f r e e a n n i h i l a t i o n l i f e t i m e t h e n I depends only on the e q u i l i b r i u m s i t u a t i o n and i s , t h e r e f o r e , independent of the i n i t i a l d i s t r i b u t i o n . Brandt and Feibus s o l v e eq. (19) wit h the assumption t h a t Ps f o r m a t i o n a c t s merely as a p e r t u r b a t i o n on the f i e l d t h e r m a l i z e d v e l o c i t y d i s t r i b u t i o n , but again keeping the r a t e s c o n s t a n t . T h i s i s , of course, e q u i v a l e n t t o T r u t s c h and Hughes* s o l u t i o n f o r (j £ s m a l l and g i v e s a s i m i l a r r e s u l t . 21 However, Brandt and Feibus a l s o d e r i v e the e f f e c t of the f i e l d on the p o s i t r o n s which drop i n t o the Ore gap from h i g h e r e n e r g i e s f o r the case when the f i e l d i s not l a r g e enough t o heat the t h e r m a l i z e d p o s i t r o n s up t o E ^ n r , ( F i g . 2 ) . The r e s u l t i s a s p r e a d i n g out of the p o s i t r o n s i n t he Ore gap, due t o the g r e a t e r s t e p s i z e f o r the p o s i t r o n s i n v e l o c i t y space under an a p p l i e d e l e c t r i c f i e l d . Thus the p o s i t r o n s " s p i l l " out of the Ore gap i n the d i r e c t i o n of i n c r e a s i n g and d e c r e a s i n g energy r e s u l t i n g i n a decrease i n the p o s i t r o n s f o r m i n g Ps. Those t h a t s p i l l out at h i g h e r e n e r g i e s are assumed l o s t t o the t h e r m a l i z e d d i s t r i b u t i o n . The amount of decrease i s a d i r e c t measure of the r a t i o f *Yv a n d thus allows one t o i n f e r t he magnitude of If ^ or(jf i f If v (°r Cjm) i s known. As one i s a t t e m p t i n g t o deci d e between v a l u e s of (j f 35 d i f f e r i n g by f a c t o r s of 250 ( s e c t . 2.411), t h i s s hould be a v e r y s e n s i t i v e t e s t (as f a r as cho o s i n g a p a r t i c u l a r t h e o r y ) . Since (j f/(jm determines the a c t u a l magnitude of the minimum, t h e n i f (J ^ <C< & m> the minimum w i l l most l i k e l y not be seen. The r e l a t i o n s h i p between l ( £ / D ) , the f r a c t i o n of p o s i t r o n s f o r m i n g Ps, and f Q ( v , t , £ / D ) , the time-dependent v e l o c i t y d i s t r i b u t i o n of the p o s i t r o n s , i g n o r i n g d i r e c t a n n i h i l a t i o n s , can be w r i t t e n as f o l l o w s . ( P r o b a l i l i t y of a p o s i t r o n w i t h v e l o c i t y v, dv at time t , dt) x ( P r o b a b i l i t y t h a t a p o s i t r o n w i t h a v e l o c i t y v w i l l form Ps) dt dv . (20) So t h a t 00 Co I ( £ / D ) = / / f 0 ( v , t , £ / D ) 7f dv d t . (21) •/» -'a The e f f e c t of fa and (for(jf s m a l l ) on f Q i s assumed n e g l i g i b l e . For the case of Ej-h r a c t i n g as an a b s o r b i n g b a r r i e r f o r p o s i t r o n s above E^^j,, only v e l o c i t i e s up t o v-^hr need be c o n s i d e r e d . 2.46 E f f e c t of I m p u r i t i e s on P o s i t r o n i u m Formation 2.461 T h e o r e t i c a l Brandt and F e i b u s 3 4 have c o n s i d e r e d the e f f e c t of the presence of an i m p u r i t y gas p o s s e s s i n g an Ore gap s i t u a t e d lower i n energy than t h a t of the host gas. They show, t h a t by u s i n g the c o n c e n t r a t i o n of the i m p u r i t y as 36 a v a r i a b l e , a value f o r the r a t i o of CTf of the i m p u r i t y t o (j m of the host gas ( a c t u a l l y t o (j r which depends on ( j m , s e c t . 2.45) near the i m p u r i t y Ore gap can be found. For example, i f the above r a t i g ^ i s about 10 , which i s t y p i c a l f o r l a r g e i m p u r i t y a t o m s ^ , then a c o n c e n t r a t i o n of the i m p u r i t y of 0.2'% woujd double the Ps f o r m a t i o n at zero f i e l d . I f a f i e l d i s a p p l i e d the t h e o r e t i c a l r e s u l t ^ p r e d i c t s t h a t the i n c r e a s e i n Ps f o r m a t i o n , as a f u n c t i o n of the a p p l i e d f i e l d , w i l l occur at a s i g n i f i c a n t l y lower f i e l d . For e x a m p l e , ^ f o r 200 ppm of argon i n helium the i n c r e a s e o c c u r s at an £*/D value which i s about \ t h a t f o r pure helium. Note t h a t these f i e l d e f f e c t s occur f o r much lower c o n c e n t r a t i o n s than t h a t r e q u i r e d t o i n c r e a s e the zero f i e l d v a l u e of Ps f o r m a t i o n . T h i s , of course, r e s u l t s from the f a c t t h a t the zer o f i e l d t h e r m a l i z e d d i s t i b u t i o n i s much lower i n energy t h a n any p o s s i b l e Ore gaps ( s e c t . 2.22, Table I ) . A f i n a l p r i d i c t i o n of the work of Brandt and F e i b u s 3 ^ i s the i n f e r e n c e of the r a t i o of the (jm*s) f ° r the i m p u r i t y and the host gas } from the manner by which the value of the f i e l d f o r a 50 % Ps f o r m a t i o n i n c r e a s e v a r i e s w i t h c o n c e n t r a t i o n . Massey e t al^ 5 have c a l c u l a t e d the e f f e c t of the presence of m olecules on the a n n i h i l a t i o n p r o c e s s . Since most molecules have at l e a s t one e x c i t a i o n l e v e l below t h e i r E-thr 5 ^ n e 37 e f f e c t of the molecule on Ps f o r m a t i o n as a f u n c t i o n of a p p l i e d f i e l d i s p r i m a r i l y t h a t of i n f l u e n c i n g the energy d i s t r i b u t i o n , v i a the i n c r e a s e i n energy l o s s t o v i b r a t i o n a l and r o t a i o n a l modes of the molecule. Thus h i g h e r f i e l d s would be r e q u i r e d t o measurably i n c r e a s e Ps f o r m a t i o n above the zero f i e l d v a l u e . T h i s e f f e c t o f f s e t s the e f f e c t due t o other molecules c o n t r i b u t i o n lower Ore gaps mentioned above. 2.462 Experimental Lee's r e s u l t s 4 which i n d i c a t e t h a t the Ps f o r m a t i o n i n c r e a s e i n helium i s s m a l l up t o 35 V cm-^amagats - 1, c o n s i d e r i n g the care taken by him t o e l i m i n a t e and monitor i m p u r i t i e s i n h i s work, p o i n t s t o i m p u r i t i e s as the reason f o r the low value of (jm f o r helium obtained by Marder et a l ( s e c t . 2.432). T h i s emphasizes the need f o r s t r i c t c o n t r o l of gas p u r i t y . I m p u r i t i e s r e s u l t mainly from o u t g a s s i n g of chamber w a l l s and a l s o t h a t r e l e a s e d by s p a r k i n g of the e l e c t r i c f i e l d . 38 CHAPTER THREE EXPERIMENTAL PROCEDURE 3.1 Technique f o r Measuring P o s i t r o n i u m Formation I n s e c t . 2.45j the f r a c t i o n a l i n c r e a s e i n Ps f o r m a t i o n , l ( £ ) , was d e f i n e d i n terms of f 0 ( v , t ) , the time dependent p o s i t r o n v e l o c i t y d i s t r i b u t i o n , and the Ps f o r m a t i o n r a t e . For the i n c r e a s e i n Ps fo r m a t i o n , only f Q ( v ) , t h e t h e r m a l i z e d d i s t r i b u t i o n i s important. The f r a c t i o n l(£) i s r e l a t e d t o the amount of o-Ps d e t e c t e d as f o l l o w s . Let N e+ = t o t a l number of p o s i t r o n s r e l e a s e d i n t o the gas, and Np g = t o t a l number of p o s i t r o n s t h a t form Ps. Now N p g = N Q + N p, where N Q = number of o-Ps atoms, and N = number of p-Ps atoms, = N D / 3 , i f the f o r m a t i o n of Ps i s s t a t i s t i c a l w i t h r e s p e c t t o s p i n s , and quenching ( s e c t . 2.32) i s n e g l i g i b l e . Let N w = number of w a l l a n n i h i l a t i o n s . Then K8) = N P s ( £ > - NPs(°> N + " N w - N P s ( 0 ) 39 Or 4 ( N Q(€) - N (0) ) 1 ( g ) = - - 2 . (22) 3 ( N , - N - N (0) ) J v e+ w Ps where o n l y the p o s i t r o n s r e a c h i n g t hermal q u i l i b r i u m are c o n s i d e r e d f o r Ps f o r m a t i o n . Thus I ranges from 0 t o 1, based on the assumption t h a t a l l p o s i t r o n s i n e q u i l i b r i u m can form Ps under a h i g h enough f i e l d . T h i s i s based on the assumption, ( j a < ^ < (jfs ( s e c t . 2.42, 2.44) • Now e x p e r i m e n t a l l y a measure of (NQ(£.) - N Q (0)) i n eq. (22) i s o b t a i n e d by measuring the i n c r e a s e i n the number of 2~f a n n i h i l a t i o n s . The i d e a l energy spectrum f o r if and zfwas d i s c u s s e d i n s e c t . 2.3, t o g e t h e r with the i n c r e a s e i n 2~]f events due t o i n c r e a s e d Ps f o r m a t i o n . An a c t u a l energy spectrum f o r a N a l ( T l ) d e t e c t o r i s shown i n F i g . 7 C o m p l i c a t i n g t h i s spectrum are the Compton events from the 0.511 MeV f u l l energy peak s i t u a t e d below the Compton edge atO.34 MeV, the Compton events from the 1.27 MeV f u l l energy peak, ( s e c t . 2.21), u n d e r l y i n g both the zfand if i d e a l spectrums, and the degraded 2 ^ e v e n t s d i s t r i b u t e d i n some manner below the peak at 0.511 MeV. The r e g i o n between 0.34 MeV and 0.511 MeV, denoted the " v a l l e y " r e g i o n and c o n t a i n i n g the g r e a t e r r a t i o of 2f t o if e v e n t s , i s t h e r e f o r e a n a l y s e d t o o b t a i n a measure of the number of 2fannihilations and thus the amount of Ps formed. The a c t u a l spectrum i n F i g . 7 i s t h a t of p o s i t r o n s a .511 tyev Photopeak descr imanator sett i ng for hel ium + Na22 SPECTRUM (annihilations in A l ) U*X3" Nal detector) f igure 7 1.27 Mev •Photopeak. 128.0 192.0 256.0 CHANNEL NUMBER 320.0 3B4.0 448.0 41 a n n i h i l a t i n g i n A l and i s expected t o c o n t a i n n e g l i g i b l e "yf a n n i h i l a t i o n s i n the v a l l e y . Shown f o r comparison are the a p p r o p r i a t e v a l l e y counts f o r p o s i t r o n s i n helium f o r £*/D rs of 0 and 100 V cm - 1amagats 1 . Only the change i n Ps f o r m a t i o n as a f u n c t i o n of f i e l d i s r e q u i r e d t o d e r i v e a value f o r the momentum-transfer c r o s s s e c t i o n . T h e r e f o r e the 3fannihilations at 0 f i e l d , t h e degraded 0.511 MeV r a d i a t i o n , and the 1.2 7 MeV Compton event s i n the v a l l e y r e g i o n are r e l a t i v e l y unimportant. Of t h e s e j o n l y the degraded i f a n n i h i l a t i o n s change wi t h an a p p l i e d f i e l d . The number of these events decreases with i n c r e a s i n g f i e l d as a r e s u l t of the r e d u c t i o n i n the number of t o t a l i f events due t o Ps f o r m a t i o n i n c r e a s i n g w i t h t h e a p p l i e d f i e l d . Thus, i f V(£) i s the number of counts i n the. v a l l e y , t h e n i t i s c l e a r t h a t i s approximately g i v e n by V(£) V(0) g N(£) N(0) V(£)/N(£) V(0)/N(0) - 1 (23) g 'max Ob v i o u s l y , one can w r i t e (V/N), _V(0)/N(0) as I i s n o r m a l i z e d so t h a t I - 1 -1 as (V/N)( ) 1 (V/N) max N(£) i s a n o r m a l i z a t i o n c o n s t a n t e q u i v a l e n t t o 42 ( N e + - N w - N P s ) i n eq. ( 2 2 ) . V/N (or V/p) i s denoted the " v a l l e y - t o - p e a k " r a t i o . I f N(£) i s approximately independent of £ then eq. (23) can be w r i t t e n v ( £ ) - v(o) Vmax " V ( ° ) x = A v 3 y ( ^ + A v 2 y ( S ) ^ ( 2 4 ) AV3er,max + A V 2 2T,max where the A^ Ts a r e the i n c r e a s e s i n the 2f and llf a n n i h i l a t i o n s i n the v a l l e y w i t h a p p l i e d f i e l d . Thus the approximation i n eq. (23) i s exact i f ti\^2t c a n be i g n o r e d . The dependence of N(£) on £ i s a r e s u l t of n o r m a l i z i n g the V(£) t o the number of counts i n the 0.511 MeV peak r a t h e r than t o the t o t a l time t o c o l l e c t a spectrum. Since w a l l a n n i h i l a t i o n s were 85 % of the t o t a l p o s i t r o n p o p u l a t i o n , N(£.) changes w i t h f i e l d at most by 15 %, thus making the simple, approximate e x p r e s s i o n , eq. (24), a reason a b l e approxomation. The e x t e n t t o which Av2 y(£) and N(£) a f f e c t t h e shape of l(£) can be e s t a b l i s h e d by c o n s i d e r i n g the f u l l e x p r e s s i o n f o r I : "A V 3 ^ + Av2*r + [N(°) - N(8)] v(o) / N(0) N _ max N(£)- _AV3*-,max + A V 2 r,max + { N (° ) - N M A X J V(0)/N(0). (25) where V(0) i n c l u d e s a l l counts i n the V a l l e y except the 1.27 MeV Compton ( s e c t . 2 . 5 1 ) . 43 T a k i n g (N (0 ) - N m a x ) / N ( 0 ) = .15 and u s i n g the amplitudes i n the v a l l e y of F i g . 7, one o b t a i n s a value from eq. ( 25 ) t h a t i s 6 % lower than t h a t c a l c u l a t e d from the exact e x p r e s s i o n As the d e v i a t i o n u n c e r t a i n t i e s i n t r o d u c e d due t o i m p u r i t i e s was much l a r g e r t han t h i s 6 %, the simple approximation, eq. ( 2 3 ) , w i t h N(£.) the peak amplitude, was used i n most of the subsequent a n a l y s i s . I f i m p u r i t y l e v e l s can be lowered and the s t a t i s t i c s improved t o the p o i n t at which a Ps f o r m a t i o n c r o s s s e c t i o n i s o b t a i n a b l e , as d i s c u s s e d i n s e c t . 2 . 4 2 and 2 . 4 6 1 , t h e n the decrease i n the 2 g events would have t o be c o n s i d e r e d . As was mentioned, the spectrum i n F i g . 7 i s t h a t of a N a l ( T l ) d e t e c t o r . I f a G e ( L i ) d e t e c t o r i s used then the r e s o l u t i o n i s about 5 keV, r a t h e r than 60 keV, ( f o r a l a r g e N a l ) . (A comparison i s shown i n F i g . 8 . ) D e t e r m i n a t i o n of v a l l e y - t o - p e a k r a t i o s i s then q u i t e simple, amounting merely t o summing over the v a l l e y and peak r e g i o n s and u s i n g eq. ( 2 3 ) • However, much poorer r e s o l u t i o n and g a i n s t a b i l i t y of the N a l ( T l ) make the above p r e s c r i p t i o n d i f f i c u l t t o f o l l o w . In t h i s case both r e g i o n s are p o o r l y d e f i n e d , making a f u n c t i o n a l f i t t o the data b e n e f i c i a l . I t was f u r t h e r f e l t t h a t a v a l l e y amplitude (see Appendix A) was the best measure of the counts i n the v a l l e y . However the I (26 ) o a . in CO a . x cn ZD rvi a g / D V Counts in chs' 155-225 cm amagats A 43.5 1.37x10 B 95.9 1.30x10 C o 6 0 Compton 1.27 MeV Compton f t I T > - H i I ± a I I | | I r I 155.0 165.0 175.0 185.0 195.0 203.0 215.0 225.0 CHANNEL NUMBER f i g u r e 8 : 0.511 MeV f u l l energy peak and v a l l e y r e g i o n showing e f f e c t of e l e c t r i c f i e l d . Curves are 8-parameter f i t s . 45 v a l l e y amplitude from an a n a l y t i c f i t w i l l be affected by the 1.27 MeV Compton increase, below 0.511 MeV? (Fig. 7 and 8) as the v a l l e y amplitude increases with f i e l d r e l a t i v e to t h i s Compton. Therefore the 1.27 MeV Compton was approximated by a Co^^ spectrum having two almost equal i n t e n s i t y peaks at 1.17 and 1.33 MeV and subtracted (after appropriate normalization) from the Nal specta, p r i o r to f i t t i n g to the f u n c t i o n given i n Appendix A. A schematic of the f u n c t i o n i s given i n F i g . 9- In Fig. 8 are shown t y p i c a l f i t s t o the v a l l e y and peak regions. The marked e f f e c t of high f i e l d s on the depletion of the 0.511 MeV peak and the corresponding increase i n the v a l l e y region i s quite evident there. 3.2 Experimental Apparatus 3.21 Chamber, Source and E l e c t r i c F i e l d The chamber used, was the high pressure vessel, constructed by Falk , (and subsequently used by Lee 4), which w i l l s a f e l y take densities up to 50 amagats at room temperature. The one modification involved putting t e f l o n between the higher voltage f i e l d rings and the chamber walls. (A schematic of the chamber i s given i n r e f . 4 ) . Unfortunately, these modifications did not permit any higher D.C. f i e l d s larger than those used by Lee^, before breakdown occured. This l i m i t was probably due to i o n i z i n g the helium gas at the higher f i e l d points i n s i d e the chamber. This phenomena was not explored further f i g u r e 9 : S i m p l i f i e d Diagram of Energy Spectrum For P o s i t r o n s A n n i h i l a t i n g i n a Gas. Shown i s a Breakdown of an 8-Parameter Function Used t o F i t the V a l l e y and Peak Regions. 47 at the time, however, as adequate 5"/D va l u e s c o u l d be o b t a i n e d by l o w e r i n g the gas d e n s i t y . T h i s of course c a r r i e d w i t h i t the disadvantage of h i g h w a l l a n n i h i l a t i o n s (about 85 %)• The e f f e c t of w a l l a n n i h i l a t i o n s on the measure-ment . of Ps f o r m a t i o n was d i s c u s s e d i n s e c t . 3 - 1 . The p o s i t r o n source was 2 2 N a C l d r i e d on .75 micron n i c k e l f o i l . The a c t i v i t y , 2 . 9 / ^ C i at the s t a r t , was determined by comparing the c o u n t i n g r a t e t o t h a t of a 22 known Na s o u r c e . A 32 kV power supply ( U n i v e r s a l V o l t r o n i c s , model # BAP - 3 2 - 1 . 5 ) was used t o generate the e l e c t r i c f i e l d . The maximum v o l t a g e used was 22 kV, at which p o i n t breakdown occured i n the gas. T h i s breakdown v o l t a g e c o r r e s p o n d s t o a f i e l d i n s i d e the chamber of about 1600 V cm - 1, or an & /D r a t i o of about 120 V cm-"'"amagats-"'" at a d e n s i t y of 13 amagats. 3 . 2 2 D e t e c t o r s and Data A c q u i s i t i o n System Two t y p e s of d e t e c t o r s were t r i e d i n the experiment. I n i t i a l l y a 30 cm ( t a p e z o i d a l ) G e ( L i ) d e t e c t o r was used because of i t s b e t t e r energy r e s o l u t i o n , (5 keV), ( F i g . 8 ) . T h i s s e p a r a t e d the v a l l e y r e g i o n from the Compton edge q u i t e w e l l . However, i t was not c o n t i n u o u s l y a v a i l a b l e f o r t h i s experiment. F i n a l l y two N a l ( T l ) d e t e c t o r s were employed; a 4" ( d i a ) x 3" and a 5" ( d i a ) x 4" c r y s t a l . Most of the data was t a k e n w i t h t h i s second l a r g e r c r y s t a l . The maximum 48 number of counts,6 5 , 5 3 6 ( f o r a NOVA computer) i n the 0 .51 MeV peak,was ob t a i n e d i n about 27 min. The specrum a c q u i s i t i o n procedure was as f o l l o w s . The s c i n t i l l a t i o n p u l s e from the p h o t o m u l t i p l i e r c o l l e c t o r of the Nal d e t e c t o r was a m p l i f i e d u s i n g an Ortec 485 a m p l i f i e r and f e d i n t o a Northern NS 622, 1024 channel ADC, of which 512 channels were used. The lower d i s c r i m i n a t o r l e v e l of the ADC was s e t t o e l i m i n a t e the l a r g e number of counts at s m a l l e n e r g i e s , ( F i g . 7), t h e r e b y keeping the dead time between 10 % t o 20 %. The d i g i t a l address s i g n a l was t h e n f e d through an i n t e r f a c e , ( s e c t . 3.41> Appendix B), t o a NOVA computer f o r storage and i n i t i a l a n a l y s i s , ( s e c t . 3 « 5 l ) . A l s o a v a i l a b l e was a H.V. s h u t o f f which stopped a n a l y s i s when the H.V. sparked. T h i s allowed runs t o be taken near the p o i n t at which the gas broke down. F i g . 10 shows the e x p e r i m e n t a l setup. 3.3 Gas H a n d l i n g 3.31 F i l l i n g The chamber was c l e a n e d w i t h acetone and the 0-rings were greased w i t h Apiezon type L vacuum greased. I t was t h e n outgased by pumping down t o 8 x 10 ^ t o r r and heated t o about 180 C. In order t o outgas hydrogen from the t i t a n i u m c h i p s d u r i n g pumping, the p u r i f i e r was kept at 450 C, ( s e c t . 3 . 3 2 ) . The chamber was then f l u s h e d and heated w i t h gas i n , a t p r e s s u r e s of 270 and 300 p s i , 49 AO V. wa-fer Chamber P u r i f i e r 32 kV H.V. 3 kV Fluke H.V. on off sense. +5 V *-999.3 M relay + 5 v coxnc, H.V. shutoff Amp. maximum 4 V input A D C d i g i t a l out Interface Terminator board D A C 0 - -IV display pulse spectrum t r i g g e r N O V A 'Computer n IScope) \ / Spectrum display figure 10 : Experimental setup (Nal(Tl) detector) 50 over about 3 days, a l e n g t h of time set by past workers , and f i n a l l y f i l l e d t o 700 p s i . T h i s procedure was c a r r i e d out once u s i n g the G e ( L i ) and the s m a l l Nal d e t e c t o r s and a second time f o r the l a r g e r Nal c r y s t a l . For the f i r s t f i l l i n g the t i t a n i u m p u r i f i e r was heated f o r about 6 days b e f o r e any runs were attempted. The second time runs were t a k e n b e f o r e and d u r i n g the i n i t i a l p u r i f i c a t i o n . 3.32 T i t a n i u m P u r i f i e r The p u r i f i e r used was s i m i l a r t o t h a t used p r e v i o u s l y by L e e 4 who d e s c r i b e d the e f f i c i e n c y of t i t a n i u m f o r p u r i f y i n g i n e r t gases. The p u r i f i e r was m o d i f i e d f o r s a f e t y reasons ( t o e l i m i n a t e p r e v i o u s l y observed p r e s s u r e -heat deformation) by i n c r e a s i n g the O.D. of the h e a t i n g pipe t o 1^" by adding a t i g h t f i t t i n g pipe t o the o u t s i d e of the e x i s t i n g 3/4" one. A l s o added was a thermocouple which was s e t into the middle of the t i t a n i u m c h i p s . The p u r i f i e r was then run c o n t i n u o u s l y w i t h power from an i s o l a t i o n t r a n s f o r m e r i n order t o e l i m i n a t e problems i n v o l v i n g s h o r t i n g of the h e a t i n g c o i l t o the chamber. The temperature was kept at 650 C, ( L e e 4 ) , the most e f f i c i e n t "getting"* temperature . Since t i t a n i u m absorbs hydrogen at 3 50 C and reemits i t at 400 C, i t i s important t o outgas any hydrogen from the c h i p s b e f o r e f i l l i n g . The need f o r t h i s i s e v i d e n t f r o n the f a c t t h a t the Ps f o r m a t i o n c r o s s s e c t i o n f o r hydrogen appears t o be 10 t o 60 times l a r g e r than t h a t f o r helium, 51 (sect. 2.411). In addition hydrogen having an E-j-^r of 8.9 eV, w i l l present a second Ore gap, producing an e f f e c t somewhat s i m i l a r to that discussed f o r argon, (sect. 2 . 4 6 l ) . 3.33 Estimation of Purity Two estimates were made. The f i r s t imvolved a mass spectrometry scan to a s e n s i t i v i t y of 4 ppm performed by Gollob A n a l y t i c a l Service Corp. (New Jersey). The r e s u l t s are shown i n Table IV. The purity of the i n i t i a l gas i s also shown. This t e s t was made half-way through the l a s t set of runs using the large Nal c r y s t a l . The 53 ppm " a i r " i s very d i f f i c u l t to understand as the l i n e from the gas supply to the chamber and that to the sample t e s t chamber were pumped down along with the chamber i t s e l f , (sect. 3-31). Table II shows that t h i s impurity has an Ore gap s i m i l a r to hydrogen, although hydrogen i s the expected impurity, (sect. 3-32). Note also that O2 i s an e f f i c i e n t quencher of o-Ps, (sect. 2.32). The need f o r p u r i t y with respect to hydrocarbons was mentioned i n sect. 2 . 4 6 L Lee 4, using an approximate impurity cross s e c t i o n of 30 TTaJ has shown that at an impurity l e v e l of 10 ppm, about 10 c o l l i s i o n s of the positron w i l l occur with the impurity during i t s l i f e t i m e (as compared to 10 4 t o t a l c o l l i s i o n s ) . The e f f e c t of impurity l e v e l s of 10 ppm i s thus p r i m a r i l y one of a d d i t i o n a l Ps formation ( i f the Ps formation cross section i s much larger than the host gas) and not so much a d i s t o r t i o n of the positron v e l o c i t y . 52 I n i t i a l Gas Supplied by Matheson of Canada Ltd, Constituent Concentration (ppm) N 2 3 °2 2 Ar .1 CO .1 2 H 2 .1 He balance Methane .1 H n 0 .8 2 Mass Spectrometer Analysis Constituent Concentration (ppm) N 2 42 0 2 11 Ar < 4 C0 2 <4 H 2 <4 He balance T o t a l hydrocarbons as methane 4 H„0 not determined Table IV 53 d i s t r i b u t i o n ( u n l e s s the s c a t t e r i n g c r o s s s e c t i o n of the i m p u r i t y atom i s much l a r g e r than 3^TfaQ^), ( s e c t . 2 . 4 6 1 ) . Lee's gas^ was a p p a r e n t l y f r e e of methane; the 4 ppm found i n the gas here i s c o n s i d e r e d i n s i g n i f i c a n t i n the l i g h t of the above d i s c u s s i o n and the r e s u l t s of s e c t . 2 . 4 6 1 . U n l i k e our method of e v a c u a t i n g the chamber and then f i l l i n g i t w i t h helium, Lee used a f l u s h i n g technique whereby the chamber was f i l l e d a few times w i t h helium gas at h i g h p r e s s u r e s and then leaked out u n t i l l the o r i g i n a l a i r i n the chamber was at a l e v e l l e s s than the i m p u r i t y l e v e l of the gas s u p p l i e d . In view of the r e s u l t s , i t i s recommended t h a t the f l u s h i n g technique be adapted i n any f u t u r e work of t h i s k i n d . The second p u r i t y e s t i m a t e c o n s i s t e d of measurements of the l i f e t i m e s of p o s i t r o n s i n the helium gas, f o r the comparison w i t h the r e s u l t s of L e e 4 f o r helium. In the present case a Z e f f , w i t h no a p p l i e d f i e l d , of 3 -75 + 0 .15 was found which compares f a v o u r a b l y w i t h 3 -63 + .04 ^. Both v a l u e s compare f a v o u r a b l y w i t h other worker's r e s u l t s , ( r e f . 4 ) . A f u r t h e r s e n s i t i v e t e s t of the i m p u r i t y l e v e l i s the width of the s h o u l d e r i n the time spectrum, which i s r e l a t e d t o the s l o w i n g down time of p o s i t r o n s t o thermal v e l o c i t i e s , ( s e c t . 2 . 2 l ) . Thus the absence o f j o r a b b r e v i a t i o n of t h i s s h o u l d e r width i m p l i e s a more r a p i d t h e r m a l i z a t i o n due t o " l a r g e " i m p u r i t i e s . Thus the time-width should be a maximum f o r the pure gas, and i n p a r t i c u l a r the s h o u l d e r -54 width d e n s i t y product s h o u l d be a constant as a f u n c t i o n of d e n s i t y . A value of 1380 +_300 ns amagats was found here and compares w e l l w i t h 1490 + 280 ns amagats d e r i v e d from one of Lee's runs ( a s s o c i a t e d w i t h r e f . 4 ) . Thus, based on the above two e s t i m a t e s , one can s a f e l y assume t h a t the i m p u r i t y l e v e l f o r a l l gases, except N 2 and Gv,, i s l e s s t han 10 ppm. I f the 53 ppm " a i r " i s p r e s e n t , i t s e f f e c t on Ps f o r m a t i o n i s expected t o be s m a l l e r than t h a t f o r monatomic atoms due t o c o m p e t i t i o n from l o w - l y i n g v i b r a t i o n a l and r o t a t i o n a l modes of d i a t o m i c m o l e c u l e s . In any case i t s presence cannot be accounted f o r . However, i f the h i g h f i e l d runs f o r the v a l l e y - t o - p e a k measurements performed by Lee^ are compared w i t h the present r e s u l t s ( F i g . 14), then i t i s e v i d e n t t h a t the p u r i t y l e v e l i n t h i s work i s somewhat b e t t e r than p r e v i o u s workers. T h i s f o l l o w s from the f a c t t h a t on t h e o r e t i c a l grounds ( s e c t . 2.42) the v a l l e y - t o - p e a k curve should be constant and e q u a l t o the z e r o f i e l d v a l u e u n t i l v ery near the £ / D value at which Ps f o r m a t i o n from the t h e r m a l i z e d p o s i t r o n s takes p l a c e . The s l i g h t improvement i s l a r g e l y a t t r i b u t a b l e t o the co n t i n u o u s r u n n i n g of the p u r i f i e r . 55 3•4 E l e c t r o n i c s 3.41 NOVA Computer f o r Spectrum Accumulation A NOVA computer was used i n t h i s work t o s t o r e energy s p e c t r a . The t r a n s f o r m a t i o n from a computer t o a m u l t i c h a n n e l a n a l y s e r r e q u i r e s an a n a l o g u e - t o - d i g i t a l c o n v e r t e r (ADC), an i n t e r f a c e which c l e a r s s i g n a l s between the de v i c e and the computer, and software t o manipulate memory, and s t a r t and stop the a n a l y s i s . The o p e r a t i o n of the i n t e r f a c e i s presented i n Appendix B. Such a computer sustem has the p o t e n t i a l f o r continuous data a c q u i s i t i o n and simultaneous data a n a l y s i s . I n t h i s work the b a s i c programs: c l e a r i n g of memory, s w i t c h i n g h a l v e s of a spectrum and adding or s u b t r a c t i n g two s p e c t r a , were w r i t t e n i n machine language f o r the sake of e f f i c i e n c y . The o p e r a t i o n s were s i g n a l e d t o the CPU by t e l e t y p e key l e t t e r s . There was a c h o i c e of e i t h e r 3 s e c t i o n s of memory of 1024 channels each or 6 of 512 channels. The data a n a l y s i s c o n s i s t e d of a simple v a l l e y - t o - p e a k c a l c u l a t i o n performed by adding up channel counts between v a r i a b l e l i m i t s and computing the f l o a t i n g p o i n t r a t i o . I n f u t u r e , the present 4k of memory w i l l be expanded t o 8k, a l l o w i n g f o r more program c o n t r o l . M o n i t o r i n g the p o s i t i o n of the 0.51 MeV peak, d u r i n g a run i s an o b v i o u s l y important concern. 56 3.42 D i s p l a y In order t o v i s i b l y d i s p l a y the s p e c t r a , a very simple d i g i t a l t o analogue c o n v e r t e r (DAC) with an o s c i l l o s c o p e d i s p l a y (shown i n F i g . l l ) was s e t up. I t 16 c o n s i s t s e s s e n t i a l l y of a weighted r e s i s t o r network"' and diode switches which operate d u r i n g the p e r i o d t h a t d i g i t a l s i g n a l s are p r e s e n t on the i n p u t . The channels were outputed s e r i a l l y under program c o n t r o l onto the i n t e r f a c e l i n e s (data channel) and the scope was swept on the 2-5 ms/cm range. The scope was t r i g g e r e d by a pulse sent out on one of the s p e c i a l p u l s e output l i n e s by the program at the b e g i n n i n g of each spectrum. Each channel was r e p r e s e n t e d by a 0 . 8 s p u l s e i n the range 0 t o -1 V, w i t h a channel s e p a r a t i o n of 4/^s. The l i n e a r d i s p l a y was more than s a t i s f a c t o r y t o d i s t i n g u i s h separate channels on a 512 channel d i s p l a y . A simple l o g a r i t h m i c d i s p l a y was achieved through the development of a program which operated by l o o k i n g f o r the p o s i t i o n of the f i r s t non-zero b i t of a channel of memory, approximating the remaining b i t s w i t h a one-to-one correspondence t a b l e of l e n g t h 15, s t o r i n g the r e s u l t u n t i l a l l channels had been con v e r t e d and then d i s p l a y i n g i t , (Appendix C ) . Use of the computer o n - l i n e thus permits d i s p l a y of the data as i t was b e i n g r e c o r d e d . Of course the l o g d i s p l a y i s very u s e f u l f o r d i s p l a y i n g time s p e c t r a c o n s i s t i n g of e x p o n e n t i a l l i f e t i m e components. 57 +24 V o b i t s : 2 H - 215 Lines of Data Channel - 2 diode array TID 22 T 154 figure 11 : Display, Simple DAC. 58 3.43 I n t e g r a l and D i f f e r e n t i a l L i n e a r i t y The l i n e a r i t y of the ADC was checked u s i n g the setups d e s c r i b e d by Lee,^ ( a c o n u c l e a r C-126 TAC was used). The r e s u l t s are g i v e n i n F i g s . 12 and 13. Such t e s t s are more important i n time spectrum work where the r e l a t i v e channel widths p l a y an important r o l e i n the a n a l y s i s . In our case, however, o n l y the i n t e g r a l l i n e a r i t y was important as o n l y m a c r o s c o p i c changes w i t h r e s p e c t t o the average channel width are i m p o r t a n t . T h i s i s a r e s u l t of b e i n g i n t e r e s t e d o n l y i n t o t a l counts i n a r e g i o n of the spectrum. In any case i t i s c l e a r t h a t d e v i a t i o n s from l i n e a r i t y are s m a l l compared t o s t a t i s t i c a l e r r o r s , e s p e c i a l l y t o those i n the v a l l e y r e g i o n . 3 . 4 4 Energy S t a b l i t y The G e ( L i ) d e t e c t o r was s t a b l e t o b e t t e r than a f r a c t i o n of a channel i n 512 over the d u r a t i o n of the runs. The N a l ( T l ) system, however, s l o w l y s h i f t e d 2 channels i n 2 days. Since runs were about 30 min l o n g t h i s was o n l y a problem f o r d e f i n i n g the v a l l e y and peak r e g i o n s and was t a k e n account of i n the data a n a l y s i s , ( s e c t . 3 .5) 3.5 Data A n a l y s i s 3.51 A n a l y s i s of Energy S p e c t r a f o r V a l l e y - t o - P e a k R a t i o s . F i g . 8, (p. 4 4) , shows two s e t s of energy s p e c t r a , one s e t f o r t h e G e ( L i ) d e t e c t o r and the other f o r the l a r g e N a l ( T l ) . The N a l s p e c t r a have the 1.27 MeV Compton 12B.0 192.0 2 5 S . 0 CHANNEL NUMBER 3 2 0 . 0 3B4 .0 0 . 0 64 .0 figure 12 : D i f f e r e n t i a l Linearity. Channel Number Represents, Time 4 4 0 J ] I I 1 1 I 1 I I 1 1 1— 0 100 2 0 0 3 0 0 4 0 0 5 0 0 CANN EL NUMBER f i g u r e 13 : Integral L i n e a r i t y . 61 s u b t r a c t e d i n order t o show the r e l e v e n t v a l l e y counts as d i s c u s s e d i n s e c t . 3-1 As was mentioned i n s e c t . 3.1> the G e ( L i ) energy s p e c r a were analysed by simply summing over a p p r o p r i a t e c h a n n e l s i n the v a l l e y and 0.51 MeV peak regions. The constancy of the coun t s / c h a n n e l i n the v a l l e y r e g i o n i s q u i t e e v i d e n t from F i g . 8. The Ps i n c r e a s e f o r the s m a l l N a l ( T l ) c r y s t a l , w i t h a r e s o l u t i o n of about 40 keV, was a l s o computed i n t h i s manner. A more d e t a i l e d a n a l y s i s was not f e l t worthwhile at the time as the u n c e r t a i n t i e s i n the r e s u l t s were dominated by i m p u r i t i e s r a t h e r than by the approximations employed i n the a n a l y s i s (as d i s c u s s e d i n s e c t . 3»l)« With the a v a i l a b i l i t y of the l a r g e r N a l ( T l ) c r y s t a l and i t s even poorer r e s o l u t i o n of 60 keV, i t was f e l t t h a t an a n a l y t i c f i t t o the peak and v a l l e y r e g i o n s would f a c i l i t a t e e r r o r d e t e r m i n a t i o n , run a c c e p t a b i l i t y and i n g e n e r a l , a b e t t e r d e f i n i t i o n of the v a l l e y and peak r e g i o n s . The p o s i t i o n of the peak from such a f i t would then e s t a b l i s h the peak and v a l l e y r e g i o n s . The peak width s e t the c r i t e r i u m f o r a c c e p t a b i l i t y of the run. The peak value from the f i t , ( F i g . 9, p. 46), was t a k e n as a measure of the area of the 0.51 MeV peak. For, i f the width of the peak remains c o n s t a n t , t h e n the peak amplitude i s d i r e c t l y p r o p o r t i o n a l t o the a r e a . As mentioned i n s e c t . 3.1j the v a l l e y - t o - p e a k , ( n o r m a l i z e d ) , was r a t h e r i n s e n s i t i v e t o the form of the 62 f u n c t i o n chosen f o r f i t t i n g the v a l l e y r e g i o n . To f i r s t order then, a s t e p f u n c t i o n , w i t h the r e s o l u t i o n of the 0.51 MeV peak f o l d e d i n , was used. The v a l l e y value was then the amplitude of t h i s s t e p . The lower c u t o f f f o r the f i t was tak e n near the minimum of the v a l l e y r e g i o n a f t e r the 1.27 MeV Compton had been s u b t r a c t e d T h i s s u b t r a c t i o n was performed s i n c e the i n c r e a s i n g amplitude of the 1.27 MeV Compton and the d e c r e a s i n g amplitude of the 3 gamma spectrum (with d e c r e a s i n g energy) wouldjadd s y s t e m a t i c e r r o r s t o both the p o s i t i o n of the minimum and the ste p f u n c t i o n amplitude. D u r i n g the experiment, zero f i e l d runs were taken at r e g u l a r i n t e r v a l s between f i e l d runs. T h i s allowed f o r a check on the c o n s i s t e n y of the spectrum f i t s as the n o n - s e n s i t i v i t y of the zero f i e l d runs t o i m p u r i t i e s was d i s c u s s e d i n s e c t . 3.461, and w i l l be f u r t h e r d i s c u s s e d i n s e c t . 4*321. Thus any change i n the z e r o - f i e l d V/P va l u e s would be the r e s u l t of e l e c t r o n i c s h i f t s and the i n a b i l i t y of the f i t t i n g procedure t o d e t e c t them ( an example i s i l l u s t r a t e d i n F i g . 16 and d i s c u s s e d i n s e c t . 4.321). The non-zero f i e l d runs were then n o r m a l i z e d t o the zero f i e l d r u n s. These r e s u l t s are p l o t t e d i n F i g . 14, f o r a l l t h r e e d e c t e c t o r s . A l s o shown i s the curve obtained 20 by T e u t s c h and Hughes and the h i g h e r f i e l d r e s u l t s of Lee^ . 1.6 1.5 1.4 >] 1.2 1.1 1.0 hi.3 (see next page f o r d e f i n i t i o n s of curves and d a t a ) . Marder et a l s helium r e s u l t T I f 1 4<lTj % I "1 *' i T A -4L A - " b e s t " f i t 5 10 Jl 20 30 40 50 60 £/D (V c m - 1 a m a g a t s - 1 ) f i g u r e 14 ' V a l l e y - t o - P e a k R a t i o s , V/P, Normalized t o Zero F i e l d s as a Fu n c t i o n of c/D. Curves are F i t s t o Theory of Teutsch and Hughes. 64 D e f i n i t i o n of the Re s u l t s of F i g . 14. Data : De t e c t o r D e n s i t y (amagats) G e ( L i ) 14.8 4 " x 3 " N a l ( T l ) 14.0 B V • - e a r l y work 5"x4" N a l ( T l ) ' 14.8 A,e> - l a t e r work 5"x4" N a l ' T l ) 13 .1 Curves : Curve (jf (j Comments 2 (/Ta02) (JTa02) A " l a r g e " .25 + .03 f i t , up from 75 V c m _ x a m a g a t s - x B l a r g e .22 + .03 f i t , up from 65 V cm - amagats" C 1.6x10""3 .25 computed w i t h (jm and Anax from A. D 1.6x10"3 .22 same as C E l a r g e .25 f i t 20 - 63 V cm" 1 amagats" E t h r v a r i e d -« F Highest f i e l d r e s u l t s of L e e 4 G see s e c t . 4 • 3 • 65 3.52 F i t t o the V a l l e y - t o - P e a k R a t i o s Since the expected i m p u r i t i e s , e i t h e r H^, Gv,, or N 2, have Ore gaps s i t u a t e d near 8 eV whereas t h a t f o r helium i s 17.6 eV, the d e v i a t i o n of the e x p e r i m e n t a l Ps enhancement curve from the pure helium r e s u l t i s expected t o be n o t i c e a b l e at about \ the f i e l d v a lue where the Ps f o r m a t i o n due t o helium alone t a k e s p l a c e . F i g . 14 does i n f a c t show such an i n c r e a s e at lower f i e l d s . The f i t of the V/P r a t i o s f o r (jm f o r helium should t h e r e f o r e be made w i t h emphasis put on the h i g h e r f i e l d p o i n t s . A f i t t o the lower i n c r e a s e would g i v e an average Ore gap f o r the i m p u r i t i e s , assuming the i m p u r i t y l e v e l s are low enough t h a t the shape of the v e l o c i t y d i s t r i b u t i o n of the p o s i t r o n s i s not i n f l u e n c e d by t h e i r presence. As the e x i s t i n g t h e o r y permits f i t t i n g the V/P data w i t h e i t h e r very l a r g e or very s m a l l v a l u e s of (jf, ( s e c t . 2.42), both c h o i c e s were i n v e s t i g a t e d . I t was p o i n t e d out i n s e c t 2.42, however, t h a t the number found f o r (jm i s independent of t h i s c h o i c e of (j ^ . In any case the i m p u r i t y e f f e c t a r i s i n g at low f i e l d s w i l l d i s t o r t the V/P data at h i g h f i e l d s making only the d e t e r m i n a t i o n of the f i e l d v a l u e f o r the p o s i t i o n of the l a r g e r i n c r e a s e p o s s i b l e , not the comparison of shapes of the V/P c u r v e s . The method of f i t t i n g when (j ^  i s l a r g e i s g i v e n i n Appendix D. The problem here i s the dependence on (j m of both the slope of the V/P versus curve and the 66 p o s i t i o n of the r i s e . The f i t with (j small was made with the Ps formation, parameter, fimax> (shown i n Fig. 14, and defined i n Appendix D), taken from the "best" (j f large re s u l t and the curve made to cross the "large" f i t at about the $ m a x / 2 p o s i t i o n . Thus, the (j^ small f i t i s just a consistency check of the i n s e n s i t i v i t y of Q to the value of predicted from m f 1 the r e s u l t s of the theory i n sect. 2.42. I f the Ps formed s o l e l y from the impurity atoms has reached i t s saturation value at 60 V cm-•'"amagats-"'", i t i s conceivable that t h i s Ps formation can be subtracted from the t o t a l Ps formation or V/P, Fig. 14. This was done i n Fig. 15. (Note that the amount of Ps formed due to the impurity depends on & £ ^ ^ v ± * where i = impurity. Thus fo r (j£ constant, the amount of Ps production should increase as Thus the constant l i n e i n Fig. 15 f o r high f i e l d s i s an approximation to V"""^ , since, i n any case the positron energy d i s t r i b u t i o n i s not known.) Thus the lower increase was assumed to saturate at a = 1 . 1 . max The curve drawn fo r the upper f i e l d points was the the "best" f i t scaled to extend from $ m a x = l - l to fimax= 1.55, ( i n order to pass through most of the points). Note also, however^ that the upper points f i t quite n i c e l y to a straight l i n e , i f not better! In any case, since i t i s hard to know the actual shape of the impurity Ps formation curve, no more can be said at t h i s time. Also i t i s possible that the 68 saturation f i e l d f o r the Ps production i n the helium has not been reached or that the high f i e l d points are being affected by undetected sparking of the e l e c t r i c f i e l d , (sect. 2 . 4 6 2 ) . 69 CHAPTER FOUR PRESENTATION OF RESULTS 4 . 1 Scope of Results A l l of the runs were made with applied voltages under 22 kV, at which point sparking occurred. 20 kV was then set as the spark-free l i m i t . This l i m i t e d £"/D to 3 5 V cm"•'-amagats 1 f o r densities of about 3 7 amagats. In order, then, to obtain the highest £/D values, the density-was dropped to 1 3 . 5 amagats (l8 kV at a density of 13.5 amagats corresponds to an £/Ti of 1 0 0 V cm-"'"amagats 1 ) . This was the f i n a l density at which the large Nal(Tl) c r y s t a l was used. Densities of 3 5 . 1 , 2 4 - 9 and 14.8 amagats were also used. 4 . 2 A c c e p t a b i l i t y of Results 4 . 2 1 F i t s t o the Energy Spectra In sect. 3 « 5 1 i t was mentioned that the width of the peak f o r the Nal(Tl) spectra was the crit e r i u m for a c c e p t a b i l i t y of these runs. Since the widths of these runs d i f f e r e d by at most, 1 . 2 5 % from the average, a l l runs were accepted. Note that the s t a t i s t i c a l error i n the V/P was about 4 %. Also, i t was f e l t that r e s u l t s were i n s e n s i t i v e to the channel p o s i t i o n of the 0 . 5 1 MeV peak, since a s h i f t i n gain should e f f e c t both the v a l l e y and peak amplitudes by the same f a c t o r . 70 4.22 F i t s t o the Po s i t r o n i u m Formation Curve I t was mentioned i n s e c t . 3-52 t h a t i m p u r i t i e s would produce an i n c r e a s e i n Ps at lower £/D va l u e s than t h a t f o r helium. T h e r e f o r e , most of the f i t s were made t o the h i g h e r f i e l d runs. F i t s t o these were made wit h the lower l i m i t 65 t o 75 V cm""'"amagats "*", i n order t o f i n d an optimum f i t t o the h i g h e r f i e l d s . The f i t from the upper l i m i t gave the best f i t . However, i f the Ps f o r m a t i o n due t o the i m p u r i t y w i t h a lower Ore gap reaches s a t u r a t i o n and simply adds t o t h a t formed o f f the helium atoms, then i t would be of i n t e r e s t t o f i t t o t h i s s i t u a t i o n . T h e r e f o r e a con s t a n t s a t u r a t i o n value of 1.1 was assumed f o r the lower i n c r e a s e and a f i t was then made t o the remaining curve. 4.23 Gas P u r i t y 4.231 I n i t i a l Runs A f t e r F i l l i n g The e f f e c t of the p u r i f i e r on the V/P r a t i o s , both f o r medium and zero f i e l d runs i s presented i n F i g . 16. E v i d e n t i s the i n i t i a l h e a t i n g of the p u r i f i e r t o i t s f i n a l working temperature and the decay of the r a t i o s t o con s t a n t v a l u e s a f t e r 70 hrs f o r zero f i e l d s and 150 hrs f o r £/D of 55 V craf'-amagats"''". The l o n g e r time f o r the hi g h f i e l d i s an example of the s e n s i t i v i t y of Ps enhancement t o the e x i s t e n c e of i m p u r i t i e s , ( s e c t . 2 . 4 6 1 ) . The i n c r e a s e d v a l u e s at zero f i e l d s i s probably due t o 0 50 Time(hrs) 100 150 f i g u r e 16 : E f f e c t of P u r i f i e r on V/P as a F u n c t i o n of Time A f t e r Turn-On 72 Ps b e i n g formed by p o s i t r o n s s l o w i n g down t o ther m a l e n e r g i e s and p a s s i n g through the Ore gaps of the i m p u r i t i e s . T h i s number w i l l of course be s m a l l e r and the e l i m i n a t i o n of i t by p u r i f i c a t i o n not as n o t i c e a b l e . The f i n a l drop shown i n F i g . 16 i s due t o a s h i f t i n t he g a i n of the e l e c t r o n i c s , a p p a r e n t l y over about 2 days. T h i s would not show up i n the f u n c t i o n a l f i t s , u n l e s s t h e spectrum had been taken d u r i n g a s h i f t . In t h i s case t h e spread i n the peak w i l l be g r e a t e r and the run d i s c a r d e d , ( s e c t . 4 . 2 l ) . The r e s u l t s i n F i g . 16 were computed w i t h a NOVA program which assumed no s h i f t of the peak, the c a l c u l a t i o n of V/P b e i n g made simply by i n t e g a t i n g a p p r o p r i a t e constant r e g i o n s and computing th e r a t i o , ( s e c t . 3 .51) . 4.232 A f t e r S p a r k i n g An e s t i m a t e of the amount of i m p u r i t y added by a s p a r k i n g e l e c t r i c f i e l d was attempted but f a i l e d t o show any p o s i t i v e r e s u l t . As i t i s prob a b l y as s m a l l as or s m a l l e r t h a n the s t a t i s t i c a l e r r o r shown i n F i g . 14, the e f f e c t i s not deemed s i g n i f i c a n t at the present l e v e l of ac c u r a c y . 4.233 E s t i m a t i o n of P u r i t y The f i n a l e s t i m a t e of p u r i t y depended on a l l i n d i c a t o r s : t h e G o l l o b mass spectrometer a n a l y s i s , the time spectrum r e s u l t s and a comparison of the decreased V/P v a l u e s found 73 here t o those of L e e 4 , ( F i g . 14). Note t h a t t h i s l a t e r i n d i c a t i o n i s . not t h a t s i g n i f i c a n t . The impuity l e v e l i s t h e r f o r e esimated t o be about 10 ppm (not i n c l u d i n g the 53 ppm of " a i r " found i n the G o l l o b a n a l y s i s ) , about the same as L e e f s ^ . T h i s l i m i t was obtained by c o n t i n u o u s l y running of the t i t a n i u m p u r i f i e r at 650 C i n order t o overcome o u t g a s s i n g from the chamber w a l l s . 4•3 E x p e r i m e n t a l V a l l e y - t o - P e a k R e s u l t s The f i t s t o the h i g h f i e l d v a l l e y - t o - p e a k r a t i o s f o r the " t o t a l " V/P r e s u l t s are p l o t t e d i n F i g . 14. In F i g . 15 are p l o t t e d the f i t s t o the V/P r e s u l t s , minus the s m a l l e r i n c r e a s e at lower f i e l d s . Note t h a t t h i s i s j u s t a s c a l e d curve of the " b e s t " f i t , t h a t t o the t o t a l curve from 75 V cm ^amagats - 1 up. I t i s thus apparent from F i g s . 14 and 15 t h a t the r e s u l t s f o r (j m are i n s e n s i t i v e t o whichever method i s used. In a d d i t i o n , i t i s not c l e a r whether the a d d i t i o n model i s v a l i d or t h a t even the s a t u r a t i o n p o i n t f o r Ps f o r m a t i o n i n helium has been reached. In any case, i t was mentioned t h a t , i n the f i n a l f i t , 20 the r e s u l t s of the a n a l y s i s of Teutsch and Hughes were used t o o b t a i n a value f o r (jm- Thus, independent of (j^, as i t appears t o be from f i r s t order c o n s i d e r a t i o n s , ( s e c t . 2 2.22), a value of (0.25 + 0.03) 77" a Q was found f o r the momemtum-transfer c r o s s s e c t i o n f o r p o s i t r o n s on helium atoms at e n e r g i e s near 17-7 eV, The e r r o r t a k e s i n t o account the e f f e c t of i m p u r i t i e s , whether they simply 74 add t o t h e h e l i u m Ps f o r m a t i o n o r a c t u a l l y d i s t o r t t h e Ps f o r m a t i o n c u r v e . (A c u r v e f o r Qc s m a l l and 0 r m = .22 7"7"a02 i s t a k e n as t h e l o w e r l i m i t i n F i g . 14). A c u r v e f o r (j f s m a l l , (jm = . 25 7Ta 0 2 and $ m a x f r o m t h e " b e s t " f i t , ( F i g . 14), y i e l d s as good a f i t as t h e " b e s t " . The d i f f e r e n c e between t h e two c u r v e s , (A and C i n F i g . 14) i s t o o minor t o e n a b l e a d e t e r m i n a t i o n o f t h e magnitude o f (j^ and i s u n i m p o r t a n t as f a r as d e f i n i n g a number f o r (j m a t E^_ n r, i s c o n c e r n e d . The d e v i a t i o n of t h e a c t u a l Ps f o r m a t i o n f r a c t i o n , I , f r o m t h e V/P r a t i o s , (as d i s c u s s e d i n s e c t . 3 .1) i s a l s o shown i n F i g . 14, ( c u r v e G). I t i s c l e a r t h a t t h e e r r o r i n (J due t o t h i s 6 % d e v i a t i o n i s n e g l i g i b l e compared t o t h e d e v i a t i o n due t o i m p u r i t i e s and s t a t i s t i c a l e r r o r s . A f i t t o t h e l o w e r i n c r e a s e was made assuming t h a t t h e e n ergy l o s s i s due e n t i r e l y t o t h e h e l i u m atoms. T h i s t h e n y i e l d s a v a l u e f o r E.^ f o r t h e i m p u r i t i e s of 8.0 + 1.0 eV. 4.4 D i s c u s s i o n of R e s u l t s 4-41 Comparison W i t h Other Workers The main purpose o f t h i s e x p e r i m e n t was t o dete r m i n e t h e f i e l d dependent e f f e c t i n h e l i u m gas when some e f f o r t was made t o m a i n t a i n t h e p u r i t y o f t h e gas. The r e s u l t s o f 9 Marder e t a l y i e l d e d much l o w e r momentum-transfer c r o s s s e c t i o n s t y p i c a l l y 0.023 7TaQ2 . T h e i r s m a l l v a l u e i s 75 suspected of being the result of excessive impurity l e v e l s . It i s f e l t here, that the impurity e f f e c t has been resolved from the Ps formation e f f e c t i n helium, although the e f f e c t i s far from n e g l i g i b l e . It was mentioned i n sect. 2.431 that (j , the t o t a l e l a s t i c cross section d i f f e r s from (jm> the momentum-tra n s f e r cross section by 10 % to 16 % depending on the 10,11 , . 11 rl . , , , theory , and i n the one case U m i s shown to be larger than (j e . If t h i s l a t t e r i s the case then the (j r e s u l t of Canter et a l 8 of 0.22 7Ta o 2 scales to 0.256 T[aQ' and thus compares favourably with the r e s u l t , (0.25 + 0.03) 77" a Q 2 j found here. In turn, the (j & r e s u l t s of Costello et a l and Jaduszliver et a l of 0.35 and 0.37TfaQ2 respectively, can not be said to be i n disagreement with the res u l t found here. Of course, the present res u l t i s above the lower l i m i t of about 0.12 7Ta G 2 given by Lee 4and P a u l 1 3 . 4.42 Comparison to Theory Drachman's re s u l t s are the only ones givi n g values f o r both Cj m a n d (j e U P "to E^- n r. At E ^ n r these are 0.195 2 and 0.167 7 7 ^ respectively. It i s not known whether (j m y (j e l n S e n e r a l - l n a n Y case the (jm r e s u l t at E^_ n r i s 30 % lower than the average r e s u l t of t h i s work. If the (j r e s u l t of 0.22 77a 2 of Fels and Mittleman 1 0 e o i s scaled by the r a t i o C T M / (j & from Drachman's r e s u l t s , i t w i l l agree with the present res u l t here. This r e s u l t 1 ^ 76 uses a "strong" p o l a r i z a t i o n of the gas atom. 4.5 Dscussion of Errors 4.51 E l e c t r o n i c S t a b i l i t y The o v e r a l l s t a b i l i t y was quite acceptable, the peak p o s i t i o n being stable to about 1 %/ week. S h i f t s when they did occur, occurred slowly over about 40 hrs. As runs were 30 mins long t h i s was not c r i t i c a l , since v a r i a t i o n s i n peak p o s i t i o n was allowed f o r i n the analysis. Faster s h i f t s , which would worsen resolution and cause r e j e c t i o n of the run, appeared not to occur. Since the width of the 0.51 MeV peak changed at most by 1.25 %, the error i n the peak amplitude due to s h i f t s i s expected to be of t h i s s i z e . Note that f o r an anal y t i c f i t to the peak, the error i n the area i s also of t h i s size . 4.52 Applied F i e l d The method of measuring the f i e l d was the same as that used by Lee 4, see Fig . 10. The accuracy of the r e s i s t o r chain was 1 % or less and the meter could be read to about 3 % at f i e l d s of 20 kV. 4.53 Density and £/D The pressure was measured to about 3 % with a "master t e s t " type 200 (Marsh Co., 111.) pressure gauge, previously 2 c a l i b r a t e d by Falk . The density was then related to the pressure by the i d e a l gas law. Since the l i n e a r i t y of 77 .0 / , J d e n s i t y and pressure f o r argon i s b e t t e r than 1% , t h a t f o r helium i s expected, on the b a s i s of a comparison of the r e l a t i v e s i z e of Van der Waal's c o n s t a n t s f o r helium and argon (Rubber Handbook, see r e f e r e n c e i n Table I ) , t o be even b e t t e r . Background d e t a i l s concerned w i t h the measurement of £ and D are gi v e n i n r e f . 4. The f i n a l e s t i mate of the e r r o r i n £/D i s thus 4.5 %, the same as t h a t deduced by L e e 4 . 4.6 Recommendations f o r Future Work I t should-be p o s s i b l e t o do a more accurate experiment t o b e t t e r d e f i n e the V/P curve i n order t o e s t a b l i s h t o what extent i m p u r i t i e s a f f e c t the d e t e r m i n a t i o n of h e l i u m - p o s i t r o n c r o s s s e c t i o n s . An improved aim would be a b e t t e r p u r i f i c a t i o n t e chnique e i t h e r u s i n g a completely c l o s e d system with o i l f r e e pumping, the f l u s h i n g technique of L e e 4 , or a system l i k e t h a t used i n spark chambers, wi t h a continuous slow b l e e d i n g of u l t r a - p u r e gas. The hope would then be the e l i m i n a t i o n of the m i d - f i e l d i n c r e a s e and maybe even the d e t e c t i o n of the p r e d i c t e d d e c r e a s e 3 4 at these f i e l d s , although the second aim i s l i k e l y t o f a i l , ( s e c t . 2 .45) . A d e t e r m i n a t i o n of (j f > the Ps f o r m a t i o n c r o s s s e c t i o n may a l s o be p o s s i b l e . An a l t e r n a t e program c o u l d be the c o n s i d e r a t i o n of hi g h e r i m p u r i t y l e v e l s , ( s e c t . 2.461). Thus some of Brandt and Fei b u s ' s " c o n c e n t r a t i o n c u r v e s " c o u l d be sought. T h i s , of course, would again r e q u i r e accurate d e t e r m i n a t i o n 78 of gas c o n c e n t r a t i o n s , but now at h i g h e r " i m p u r i t y " l e v e l s . T h i s c o u l d be accomplished by gas a n a l y s i s , as was performed on the h e l i u m gas, or by u s i n g the vapor p r e s s u r e of c e r t a i n gases at l i q u i d n i t r o g e n temperatures. Both methods would use d i l u t i o n of the i n i t i a l mixture by the pure gas t o o b t i a n lower c o n c e n t r a t i o n s than the i n i t i a l one. I t might a l s o be of i n t e r e s t t o study the 3-gamma energy spectrum u s i n g the 8-parameter f i t s t o the v a l l e y and peak r e g i o n s i n an attempt t o separate the 0.51 MeV peak from t h e v a l l e y , (Appendix A). T h i s would r e q u i r e a G e ( L i ) d e t e c t o r w i t h i t s b e t t e r r e s o l u t i o n of 5 keV i n c o i n c i d e n c e w i t h a N a l ( T l ) d e t e c t o r s et on the 1.27 MeV peak i n order t o e l i m i n a t e the r e s u l t i n g Compton events from the 3_gamma spectrum. Although other sources without competing gammas c o u l d be used, t h e i r c o s t and/or sho r t l i f e t i m e makes them p r o h i b a t i v e . I t might a l s o be p o s s i b l e t o gate the energy spectrum w i t h the l o n g l i v e d component of a l i f e t i m e spectrum r a t h e r t han the 1.27 MeV s i g n a l . 79 CHAPTER FIVE CONCLUSIONS T h i s work was undertaken i n order t o f i n d a more r e l i a b l e v a l u e of the momentum-transfer c r o s s s e c t i o n f o r p o s i t r o n s on helium near the p o s i t r o n i u m f o r m a t i o n t h r e s h o l d u s i n g a swarm type experiment i n order t o t e s t Q the r e l i a b i l i t y of e a r l i e r r e s u l t s (Marder et a l ) which y i e l d e d an anomalously low v a l u e . I t i s f e l t t h a t t h i s r e s u l t has been a c h i e v e d and furthermore, the new r e s u l t i s s i m i l a r t o those of e x i s t i n g t h e o r i e s and v a l u e s of the t o t a l e l a s t i c e l a s t i c c r o s s s e c t i o n found from beam experiments. S p e c i f i c a l l y a momentum-transfer c r o s s s e c t i o n 2 of (0.25 ± 0.03) Tfan has been found by d e t e c t i n g the i n c r e a s e o f 3-gamma events over 2-gamma events r e s u l t i n g from the f o r m a t i o n and subsequent a n n i h i l a t i o n of p o s i t r o n i u m i n h e l i u m gas when an e l e c t r i c f i e l d was a p p l i e d . T h i s v a l u e , t h e r e f o r e corresponds t o p o s i t r o n e n e r g i e s near 17.6 eV. The r e s u l t t u r n s out t o be independent of (j f i n t h i s experiment, but the p r e d i c t i o n s of the th e o r y of Teutsch and Hughes are not co m p l e t e l y i n s e n s i t i v e t o (j £. In f u t u r e a more d e t a i l e d computer s o l u t i o n w i t h fewer i n i t i a l s i m p l i f y i n g assumptions should be a p p l i e d . I t has a l s o been found t h a t i m p u r i t i e s c o n t r i b u t e c o n s i d e r a b l e Ps f o r m a t i o n under an a p p l i e d e l e c t r i c f i e l d down t o l e v e l s of 10 ppm. 0„, N , and/or H 2 are the main 80 c o n t r i b u t o r s . Since the t i t a n i u m p u r i f i e r outgasses H 2 at i t s normal working temperature, while other i m p u r i t i e s are absorbed, i t was expected t h a t would have been the primary contaminant. S u r p r i s i n g l y , the spectrometer t e s t i n d i c a t e d l i t t l e , but the presence of 0 2 and N 2 at a t o t a l l e v e l of 53 ppm. Note however t h a t the t h e o r e t i c a l Ps f o r m a t i o n c r o s s s e c t i o n f o r H 2 i s 10 t o 60 times t h a t of helium. I t was f e l t i n any case t h a t the i m p u r i t y c o n t r i b u t i o n had an of 8 + 1 e V. 81 REFERENCES 1 (1967). " P o s i t r o n A n n i h i l a t i o n " (ed. by A.T. Stewart and L.O. R o e l l i g ) . Academic P r e s s , New York. 2 Falk, W.R. (1965). Ph.D. T h e s i s , U n i v e r s i t y of B r i t i s h Columbia. 3 Orth, P.H. R. (1966). Ph.D. T h e s i s , U n i v e r s i t y of B r i t i s h Columbia. 4 Lee, G. (1969). M.Sc. T h e s i s , U n i v e r s i t y of B r i t i s h Columbia. 5 Lee, G. (1972). Ph.D. T h e s i s , U n i v e r s i t y of B r i t i s h Columbia. 6 J a d u s z l i v e r , B., Keever, W.C. and Paul, D.A.L. ( l 9 7 l ) . i n " P o s i t r o n A n n i h i l a t i o n , 2nd I n t e r n a t i o n a l Conference", Queen's U n i v e r s i t y , Kingston, Canada, unp u b l i s h e d . 7 C o s t e l l o , C.D., Groce, D.E., H e r r i n g , D.F. and McGowan, J . Win. (1972). Can. J . Phys. i O , 23. 8 Canter, K.F., Coleman, P.G., G r i f f i t h , T.C. and Heyland, G.R. (1972).. J . Phys. B jj.. L l67 . 9 Marder, S., Hughes, V.W. , Wu, C S . and Bennet, W. (1956). Phys. Rev. 103., 1258. 10 Drachman, R.J. (1966). Phys. Rev. 144, 25. 11 F e l s , M.F. and Mittleman, M.H. (1969). Phys. Rev. 182, 77 12 Deutsch,M. and Brown, S.C. (1952). Phys. Rev. 85, 1047. 13 Leung, C.Y. and Pa u l , D.A.L. (1969). Phys. L e t t e r s A 28, 674. 14 Green, J . and Lee, J . (1964). " P o s i t r o n i u m Chemistry", Academic Press, New York. 15 Gerhart, J.B., C a r l s o n , B.C. and Sherr, R. (1954). Phys. Rev. 2£, 917. 16 K e n d a l l , H.W. and Deutsch, M. (1954). Phys. Rev. £3_, 932. 17 F r a s e r , P.A. (1968). Advances i n Atomic and M o l e c u l a r P h y s i c s 4_, 63 . 18 Pond, T.A. (1949). Phys. Rev. 25, 489. 82 19 Ore, A. and Powell, J.L. (1949). Phys. Rev. 25., 1696. 20 T e u t s c h , W.B. and Hughes, V.W. (1956). Phys. Rev. 103, 1266. 21 Chapman, S. and Cowling, T.G. (1970). "The Mathematical Theory of Non-Uniform Gases" (3rd e d . ) . pp. 382-389. Cambridge U n i v e r s i t y Press, Cambridge. 22 Yang, C.N. (1949). Phys. Rev. 21, 242. 23 Massey, H.S.W. and Moussa, A.W. ( l 9 6 l ) . Proc. Phys. Soc. (London) 21,  8 1 1 • 24 F e l s , M.F. and Mittleman, M.H. (1967). Phys. Rev. 16_3_, 129-25 Massey, H.S.W. and Mohr, C.B.O. (1954). Proc. Phys. Soc. (London) A §J_, 695-26 C h e s i r e , I.M. (1964). Proc. Phys. Soc. (London) 8 l , 227. 27 Bransden, B.H. and Ju n d i , Z. (1967). Proc. Phys. Soc. (London) £2, 880. 28 M i t t l e m a n , M.H. (1964). Annals of Phys. 28, 430. 29 Chen, J.C.Y. and Mittleman, M.H. (1966). Annals of Phys. 11, 264. 30 F e l s , W.R. and Mittleman, M.H. (1971) . Phys. Rev. A _3_, 1827. 31 Brandt, W. and Feibus, H. (1968). Phys. Rev. ±2A, 454. 32 Morse, P.M. and Feshback, H. (1953). "Methods of T h e o r e t i c a l P h y s i c s " , pp. 188-200. McGraw-Hill, New York. 33 F a l k , W.R., Jones, G. and Orth, P.H.R. (1965). Phys. Rev. L e t t e r s 1£, 447. 34 Brandt, W. and Feibus, H. (1969). Phys. Rev. 184, 277. 35 Massey, H.S.W., Lawson, 0. and Hara, S. (1972). J . Phys. B 1, 599. 36 Malmstadt, N.V. and Enke, L.G. (1969). " D i g i t a l E l e c t r o n i c s f o r S c i e n t i s t s " , pp. 331-33 6. W.A. Benjamin,. Inc., New York. 37 N a t i o n a l Bureau of Standards, C i r c u l a r 5641 (1955). 83 APPENDIX A ANALYTIC FIT TO THE PEAK AND VALLEY AREAS OF A N a l ( T l ) ENERGY SPECTRUM The best f i t t o the v a l l e y and peak r e g i o n s minus the 1.27 MeV Compton approximation was obtained w i t h the f o l l o w i n g f u n c t i o n : N(x = y-P 2) = P x exp(-x 2/2P 4) + P (1 - erf(x/22P 4)/2 + P 3 + P 5 + P 6 x 2 + Pgx 3 . ( A l ) where N = number of counts/channel, P^ = standard d e v i a t i o n or " r e s o l u t i o n " , P = p o s i t i o n of the 0.51 MeV f u l l energy peak, mil y = channel number, P i = peak parameter, "P", (A2) oth e r P*s d e f i n e the v a l l e y parameter, "V", such t h a t 2 V = maximum of N - F± exp(-x /2P^), -(A3) and x = r e l a t i v e channel number from the peak p o s i t i o n . I t was found t h a t the 4-parameter f i t w i t h P^ = = P^ = Pg = 0 gave e q u i v a l e n t r e s u l t s f o r f>, (Appendix D). F i t s u s i n g the 8-parameter f u n c t i o n are t y p i f i e d i n F i g . 8. The v a l l e y p o s i t i o n i s the p o s i t i o n of the maximum d e f i n e d by ( A 3 ) . Of course the a b s o l u t e p o s i t i o n of t h i s p o i n t w i t h r e s p e c t t o the 0.51 MeV peak depends on the 84 symmetry of t h i s peak. (The peak may f a l l o f f slower f o r the lower energy s i d e due t o the manner of the l i g h t p r o d u c t i o n i n the N a l ( T l ) c r y s t a l and/or due t o the i n c r e a s e i n .. r e s o l u t i o n of the c r y s t a l w i t h lower energy.) 85 APPENDIX B ADC INTERFACE OPERATION AND SCHEMATICS A representative block diagram f o r the computer-shown in interface-ADC system i s A F i g . B l . The state of readiness of the analogue-to-digital converter, ADC, i s set by the BUSY-DONE block (consisting of two separate f l i p - f l o p s , f f , as shown i n Fig . B l ) . To set the ADC i n readiness f o r accepting analogue pulses, a START (pulse) i s sent from the NOVA computer under program contol, PC, (sequence ( l ) , Fig. Bl) and turns the BUSY f f on. This occurs i f the device s e l e c t i o n code f o r the ADC, (40) , i s also sent out simultaneously. (This gating i s represented by shaded blocks The START also resets the ADC, (shown at ( l T ) ) 5 so that analogue pulses can be accepted by i t . The next sequence of events i s the a r r i v a l of an analogue s i g n a l , ( 2 ) , of which the amplitude i s the paramete of i n t e r e s t . I f t h i s s a t i s f i e s the amplitude conditions of the ADC then a STORE l e v e l i s set by the ADC at (3) and a 1 0-bit binary representation of the amplitude (maximum 4 V and 5V i f the 25 % zero o f f s e t f o r the ADC i s used) i s held i n the ADC output buffer, ( 2 ? ) . The beginning of the l e v e l at STORE i s now used to send a pulse to the CONVERT DONE f f and t h i s i n turn requests service from the NOVA through the data channel, (DCH), by s e t t i n g the DCH REQUEST f f , (4) . The clock pulse, (5)> then synchronizes t h i s request with the NOVA program cycle and ac t u a l l y sends the 86 START OVERFLOW CLEAR < > o S3 DONE, BUSY (sense ) INTERRUPT REQUEST DCH REQUEST MEMORY FUCTION (increment DCH ACKNOWLEDC Get Address I/O Buffer ANALOGUE IN © o © © •O O off .on DON o ff on BUSY © f f ® f f © w © on C ONV f> DONE f f © DCH SEL © DCH REQUEST on off f f Priority-^ f f 0© 1WC STR (BE © on drf o o < STORE D i g i t a l Buffer (10-bit\ figure B l : Block Diagram of Interface For An ADC. 87 DCH REQUEST, ( 5 f ) . S i m u l t a n e o u s l y a pulse i s sent from (5) t o the 2.6/^s d e l a y at (8) and f i n a l l y t o the ADC as a r e s e t , i n order t o enable i t t o accept f u r t h e r p u l s e s . T h i s 2.6/^s d e l a y a l l o w s an overflow of memory t o be d e t e c t e d . I f such i s the case, then an e x t r a lO^u s d e l a y i s added t o the 2.6 /ws ,s and t h i s allows minor programing t o take care of the overflow a c c o r d i n g t o the d i s c r e t i o n of the programmer be f o r e the address of the overflow i s l o s t from the ADC output b u f f e r by the c l e a r at (8). (A memory l o c a t i o n has a c a p a c i t y of 2^-1.) In t u r n the overflow p u l s e t u r n s o f f BUSY and s e t s DONE. T h i s l a t t e r t r a n s i t i o n t u r n s CONVERT DONE o f f u n c o n d i t i o n a l l y and b l o c k s any STORE from r e q u e s t i n g a DCH INTERRUPT. The former sends an INTERRUPT REQUEST, which a l e r t s the program t o the presence of an overflow. The device code i s a l s o sent i n order t o d e f i n e the d e v i c e r e q u e s t i n g the i n t e r r u p t , ( u s e f u l i f t h e r e are more than one d e v i c e ) . A f t e r a DCH REQUEST has been r e c e i v e d , the NOVA r e t u r n s , a f t e r . 5ju s, a DCH ACKNOWLEDGE (DCHA), (6). T h i s gates through t o the NOVA p r o c e s s o r the number h e l d i n the ADC b u f f e r i f the DCH REQUEST i s s t i l l s e t . Otherwise, the number i s l o s t . The NOVA r e t r i e v e s the word from memory, at the addressed l o c a t i o n , increments the word and writes, i t back, then sending an o v e r f l o w i f n e c e s s a r y . The address of the overflow l o c a t i o n can be requested, (9")> by the program which i s s e t i n t o o p e r a t i o n , as mentioned, by the request at ( 9 T ) . 88 DCHA also resets CONVERT DONE and DCH REQUEST to prepare f o r the next pulse to be processed by the ADC. A RESET pulse, program generated, resets a l l the major f f T s . F i n a l l y , the ADC i s disabled by sending a CLEAR pulse out which turns off both DONE and BUSY, the l a t t e r thus blocking CONVERT DONE as mentioned. ADC INTERFACE LOGIC Shown i n Figs. B2, B3, B4 are the complete schematics f o r the ADC i n t e r f a c e . The l o g i c units shown are 1. F l i p - f l o p s : C —I D " R 3 i - 1 0 - a ground at S sets 1 output high, 0 low, a ground at R sets 0 high and, 1 low; both unconditional. - a p o s i t i v e t r a n s i t i o n at C, sets 1, (0), low, (high), i f D i s low, high, (low)', i f D i s high. 2. "Or" gates: — - c i r c l e at an input means that low input y i e l d s a high output, - a l l inputs must be high before input i s low. 3. Inverters: 89 DSO DS1 * IT LE S2_ IS. £3-±_3X PHX"| BLECI] / STRT \ 1 — iCLR. / PHA I /SEIECff ^'>..... -^ START]3-9 » / IORST j| OVFL(r j RQENB* -i_r I INTP •XT -+5V 330 390 INTA IORST 3 (r4 Z3_ BUSY 4 OVFLC PHA'I rx ,j— \ BUSY I T - n . 11 DONE 4 / "PHA7-.SELEGtf "PTTA" DONE 2 \—^SELJ^* -LT 4 H L INT REQ SELD, -vr 4 INT%> ,330 INT P> 3 90 DtATA y ,10 (40) f i g u r e B2 : Device S e l e c t i o n and I n t e r r u p t S e r v i c e 90 f i g u r e B3 : Data Channel S e r v i c e and Data Word Strobe. 92 4. "And" gates: - a l l inputs high produce a low output. - any input low, output i s high. The c i r c u i t shown i n Figs. B2, B3 , and B4 was taken from the NOVA i n s t r u c t i o n manual. However, the following modifications had to be made i n order to make i t compatible with the Northern NS-622 ADC. 1. Since t h i s ADC produces a high l e v e l for STORE when conversion of an analogue s i g n a l has been completed, t h i s l e v e l had to be d i f f e r e n t i a t e d so that only the t r a n s i t i o n to the high l e v e l reached the f i r s t gate i n Fi g . B3. 2. The CLEAR back to the ADC had to be delayed by at least 2.6/Us i n order to hold the converted number on the ADC output l i n e s long enough for the computer to send an overflow i f required, and the program to request the overflow address from the ADC i f necessary. I f such was the case then a further 1 0 ^ s delay was added by the overflow pulse, thus g i v i n g the programer time to receive the address before i t was l o s t . For convenience the START pulse from the computer, besides s e t t i n g BUSY, cleared the ADC. A l l these additions are shown i n the bottom of Fig. B4. 93 I.C's Used i n the I n t e r f a c e and Numbered i n F i g s . B2, B3. and B4. Number I.C Type. 3,4,5 ,6 MC-7479 f l i p f l o p 7 MC-7430 8 -input NAND 8 MC-836 i n v e r t e r 11 MC-7404 i n v e r t e r 10 MC-862 3-input NAND 18 MC-7410 3-input NAND 1,2 MC-74OI 2-input NAND 9,12 MC-846 2-input NAND 13,14,15 MC-7401 2-input NAND 16,17 MC-7400 2-input NAND 94 APPENDIX C NOVA. PROGRAM FOR A LOGARITHMIC DISPLAY LABEL AD RESS CONTENTS CODE COMMENTS LOGHV: 1100 020376 LDA 0,ZARMW K i c k s o r t e r l o c . 1 024441 LDA 1,HLF Set up 2 040026 STA 0,26 counters 3 044435 STA 1,LGCTR " 4 123000 ADD 1,0 " 5 040027 STA 0,27 » 6 020435 LDA 0,SXTN NLOG: 7 040432 STA 0,EXCTR 10 022026 LDA 0,@26 Load a number 1 024436 LDA 1,FAD I n i t i a l i z e 2nd. 2 044434 STA 1,FD order f a c t o r add. 3 024431 LDA 1 ,BITZR I n i t i a l i z e . 4 030431 LDA 2 ,BITFR 1st order 5 176400 SUB 3,3 f l a g s e t o f f 6 101122 MOVZL 0 , 0 ,SZC Search f o r 1st 7 000407 JMP .+7 (or 2nd) "1" 20 175014 MOV# 3,3,SZR F l a g ? 1 010425 ISZ FD ON - count d i f f 2 146420 SUBO 2,1 OFF - count "M" 3 014416 DSZ EXCTR 16 b i t s ? 4 000772 JMP .-6 YES 5 000405 JMP .+5 NO 6 175014 MOV# 3,3,SZR F l a g ? 7 000403 JMP .+3 ON 30 175400 INC 3,3 OFF - Set on. 1 000765 JMP .-13 Search f o r 2nd "1" 2 036414 LDA 3,@FD Load c o r r e c t i o n 3 167000 ADD 3,1 Add t o 1st order 4 046027 STA 1,@27 Store 2nd order 5 014403 DSZ LGCTR F i n i s h e d 512 ? 6 000750 JMP NLOG NO 7 002431 JMP ©display YES t o k i c k s o r t e r LGCTR: 40 0 LGCTR (512 s t o r e ) EXCTR: 1 0 EXCTR (16 s t o r e ) HLF: 2 001000 HLF 5 1 2 1 0 / SXTN: 3 000017 SXTN "16" (so t h a t BITZR: 4 160000* BITZR log ( 2 1 5 ) -BITFR: 5 010000* BITFR log ( 2 ) FD : 6 0 FD FAD s t o r e FAD: 7 001150 FAD Add. of f a c t o r s FACTORS: 50 006244 f a c t o r s log(l+ 2)=log ( 3 ) 1 011220 log ( 5 ) 2 014534 tr 9 3 020270 ti 17 4 024120 TT 33 5 030050 TT 65 95 1156 034024 7 040000 60 044000 1 050000 2 054000 3 060000 4 064000 5 070000 6 074000 log ( l 2 9 ) = log ( l 2 8 ) e t c . * so t h a t l o g ( l ) = log ( 0 ) = . 5 8 1 0 96 APPENDIX D TEUTSCH AND HUGHES'S FIT TO CS LARGE The normalized valley-to-peak r a t i o , fi, i s related to the v a l l e y to peak r a t i o , V/P, as follows. (V/P) (£/D) /KS/D) = _ . _ (DI) (V/P) ( 0 ) The Ps formation f r a c t i o n , I, i s then approximately defined i n terms of fi, ( i . e . eq. ( 2 3 ) ) , as follows. (fi(£/B) - 1) I(£/D) = . (D2) (fi -1) max 20 Thus the f i t to the theory of Teutsch and Hughes was made by t r a n s l a t i n g a l l the V/P r a t i o s to I. I i s written by Teutsch and Hughes i n terms of the f i r s t eigenvalue of the sol u t i o n of the Surm-Liouville equation, which r e s u l t s from the s i m p l i f i c a t i o n of the d i f f u s i o n equation under the assumptions given i n sect. 2.45. Thus A* A* + Aa* (D3) where A are eigenvalues calculated by Teutsch and Hughes and are fuctions of the dimensionless parameter,€ } (eq. (12)), and f\a i s the p a r t i c u l a r eigenvalue defined by eq. (9), or (10). Now the dependence of I on (j i s seen to arise from the dependence of A on € and the subsequent dependence of £ on (j m , eq. (12) . Eq- (D3) was s i m p l i f i e d by solving for - l ) i n terms of 1 / /\* Thus 1 _ 1 + Aa* /_!_ (D4) Now eq. (D4) i s l i n e a r i n the above two variables except that the slope depends on l / (jm, through Aa^ ' eq. (10), and the variable (l/A ) depends on l / (jm through i t s £ dependence, eq. (12). However an i t e r a t i v e search can be made to eq. (D4). This was done on an IBM 36O computer by tabulating - l ) versus £/D and l/A* versus £ . A value of (jm, lower than that expected was then picked as an i n i t i a l condition. £ was then calculated at the measured £/D values and the corresponding l/A > which had to be interpolated l i n e a r l y (using the log of l/A > since the values of A* ranged from 7-34 x 10~ 7 to 2.80) from i t s table, was matched with the value of - l ) possessing the same 6VD • A maximum l i k e l i h o o d f i t was then made to the r e s u l t i n g set of points. From the slope of the curve the value of (j m implied by eq. (lO) was calculated. According to the sign of disagreement from the i n i t i a l estimate of (j m , the i n i t i a l estimate was eithe r incremented or decremented by a set increment or the set increment decreased by a factor of 2 and the d i r e c t i o n of the search 9 8 f o r Q changed. The search was stopped when the present estimate from the previous i t e r a t i o n ^ a s equal to that calculated from the slope of the present i t e r a t i o n . A graph of the f i n a l f i t f o r eq. (D4), i s given i n Fig. DI. The dotted l i n e i s only a v i s i b l e f i t to the higher f i e l d points. Cf course the value of (j from the slope of t h i s l i n e i s d i f f e r e n t from that of the s o l i d l i n e , and thus d i f f e r e n t from the value of (j defing the "x"-axis. That is, the dotted l i n e i s not a f i t which has converged. 

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