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Annihilation of positrons in argon Orth, Paul Hans Robert 1966

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The University of British Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY PAUL HANS ROBERT ORTH B,Sc.,j University of Capetown^ i960 B o Sc., (Hons) 9 University of Capetown,, 1961 THURSDAY^  JANUARY 26, I967 AT 2 s 30 P* M. IN ROOM 301, HENNINGS BUILDING COMMITTEE IN CHARGE Chairmanz I» McT. Cowan External Examiner; D„ A. L . Paul Department of Physics University of Toronto Toronto of M 0 S c o S University of Capetown^  1963 G„ Jones L 0 de Sobrino J , M0 Kennedy F. W. Dalby J . H. Williamson D, C Frost Research Supervisor: G. Jones THE ANNIHILATION OP POSITRONS IK ARGON . ABSTRACT The annihilation of positrons in Argon has been investigated as a function of Argon density and j applied electric f ie ld using the technique of l l f e= time measurementso Lifetime spectra "were analyzed using the nMimum likelihood method of curve f i t t ing. Results obtained at zero electric f ie ld yielded a linear dependence on density for the direct amihi~ lation rate of (5.6 * O.l) x 10^ sec"1 amagat"""1^ with some evidence of. non° l inear i ty at densities greater than 10 amagats. .The density dependence of the long= lived component of the time spectra indicated a zero density intercept of (7.2 ± 0.1+) x 10 6 sec"1 in agree-ment with the theoretical value of the free orthopos-itronium annihilation raite (7.2 x 10^ sec"1), in addition, an orthopositronium quenching rate of (0.29 * o.O"+) x 10 6 sec"1 araagat™1 was obtained from the linear dependence of the orthopositronium anni-hilation rate on density. The electric f ie ld dependence of the direct annihilation rate and orthopositronium formation has been measured and is used to provide an internally consistent picture of the behaviour of positrons in a gas under the influence of an applied electric f ie ld. Furthermore3 these results have been compared with theo= retical results for the direct annihilation rate obtained from one parameter representations of the effective posi-tron=Argon atom interactions. It is shown that, while such potentials are successful in describing the low-energy elastic-scattering of electrons from noble gas atoms, they are inadequate for the case of positrons. However9 consideration of the way in which the direct annihilation rate changes as a function of electric f ie ld leads to an 2 upper limit of 157Ta0 for the momentum-transfer cross-section for positrons in Argon at thermal energies. Such an estimate is shown to be independent of any assumption concerning the effective positron=Argon atom interaction. GRADUATE STUDIES Field of Study % Positron Physics F, AQ Kaenrpffer Jo Bo Warren Htt Schmidt Bo L* White M0 McMillan J 6 B, Warren F , K . Bowers Elementary Quantum Mechanics Nuclear Physics Special Relativity Theory Physics of Nuclear Reactions Theoretical Nuclear Physics Cosmic Rays and High Energy Physics Electronic Instrumentation PUBLICATIONS AND PAPERS W„ Falk, G„ Jones and Rc Orth, A Random Time Gener<* ator for Timesorter Linearity Measurements* N u e l o Inst. & Methods 3J., 3^5 (1965 )-W0 Ro Falk,* PoH„Ro Orth and G,: Jones. Effect of Electric Field on Positron Lifetimes in Argon and H e l i u m o Phys* Rev* Letters 14, 507 (1965),. G„ Jones, W» R* Falk and P^H.A,-Orth,,, Positron Annihilation in the Noble Gases, p..-372, IVth Inter-national Conference on the Physics of Electronic and Atomic Collisions (abstracts of papers)3 Science Bookcrafters Inc., N * Y a (1965)0 G e Jones and P..H0R,> Orth* The Annihilation of Posi-trons in Argon. "Positron Annihilation" Proceedings of the Conference on Positron Annihilation held at Wayne State University* Detroit,, Michigan,, 1965» A o T o Stewart and L„0„ Roellig s ed« Academic Press (1966) N aYo Garth Jones and Robert Orth, Annihilation of Posi-trons in Argon Gas. Bull . Am* Phys* Soc. II, 11, 7"+9 (1966), THE ANNIHILATION OF POSITRONS IN ARGON by P a u l Hans R o b e r t O r t h B . S c , U n i v e r s i t y of Cape Town, 1960 B.Sc.( Hons.)., U n i v e r s i t y o f Cape Town, 1 961 M . S c , U n i v e r s i t y o f Cape Town, 1963 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF . DOCTOR OF PHILOSOPHY ' i n t h e Department of PHYSICS We a c c e p t 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 December, 1966 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h Columbia,, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study« I f u r t h e r agree t h a t p e r m i s s i o n - f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of pfr The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada ABSTRACT The a n n i h i l a t i o n o f p o s i t r o n s i n Argon has been i n v e s t i -g a t e d as a f u n c t i o n o f Argon d e n s i t y and a p p l i e d e l e c t r i c f i e l d u s i n g t h e t e c h n i q u e o f l i f e t i m e measurements. L i f e t i m e s p e c t r a were a n a l y z e d u s i n g thej maximum l i k e l i h o o d method o f cu r v e f i t t i n g . R e s u l t s o b t a i n e d a t z e r o e l e c t r i c f i e l d y i e l d e d a l i n e a r dependence on d e n s i t y f o r the d i r e c t a n n i h i l a t i o n r a t e o f (5.6 i 0.1) x 10^ s e c ~ 1 amagat , w i t h some e v i d e n c e o f n o n - l i n e a r i t y a t d e n s i t i e s g r e a t e r t h a n 10 amagats. The d e n s i t y dependence o f the l o n g - l i v e d com-ponent o f the time s p e c t r a i n d i c a t e d a z e r o d e n s i t y i n t e r c e p t o f (7 . 2 ioA) x 10 sec i n agreement w i t h the t h e o r e t i c a l v a l u e o f the 6 — 1 f r e e o r t h o p o s i t r o n i u m a n n i h i l a t i o n r a t e (7-2. x 10 sec ). I n a d d i t i o n an o r t h o p o s i t r o n i u m quenching r a t e o f (0 o 29 - 0.Oh) x 1 0 ^ -1 -1 sec amagat was o b t a i n e d from the l i n e a r dependence o f the o r t h o -p o s i t r o n i u m a n n i h i l a t i o n r a t e on d e n s i t y . The e l e c t r i c f i e l d dependence o f the d i r e c t a n n i h i l a t i o n r a t e and o r t h o p o s i t r o n i u m f o r m a t i o n have been measured and a r e used t o p r o v i d e an i n t e r n a l l y c o n s i s t e n t p i c t u r e o f the b e h a v i o u r o f p o s i t r o n s i n a gas under the i n f l u e n c e o f an a p p l i e d e l e c t r i c f i e l d . F u r t h e r m o r e , t h e s e r e s u l t s have been compared w i t h t h e o r e t i c a l r e s u l t s f o r the d i r e c t a n n i h i l a t i o n r a t e o b t a i n e d f r om one p a r a -meter r e p r e s e n t a t i o n s o f the e f f e c t i v e p o s i t r o n - A r g o n atom i n t e r -a c t i o n . I t i s shown t h a t , w h i l e such p o t e n t i a l s a r e s u c c e s s f u l i n d e s c r i b i n g the low-energy e l a s t i c - s c a t t e r i n g o f e l e c t r o n s from n o b l e gas atoms, t h e y a r e i n a d e q u a t e f o r the case o f p o s i t r o n s . However, c o n s i d e r a t i o n o f the way i n w h i c h the d i r e c t a n n i h i l a t i o n - i i i -r a t e changes as a f u n c t i o n of e l e c t r i c f i e l d l e a d s t o an upper 2 l i m i t o f l5 ira Q f o r 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 i n Argon a t t h e r m a l e n e r g i e s . Such an e s t i m a t e i s shown t o be independent o f any assumption c o n c e r n i n g the e f f e c t i v e p o s i t r o n - A r g o n atom i n t e r a c t i o n . - i v-TABLE OF CONTENTS page ABSTRACT i i LIST OF TABLES v l i i LIST OF FIGURES ' i x ACKNOWLEDGMENTS x i 1 . POSITRONS AND THEIR INTERACTION WITH GAS ATOMS . . . 1 1.1. I n t r o d u c t i o n 1 1.2. The f a t e o f p o s i t r o n s i n a gas 2 1.2.1. I n t r o d u c t o r y remarks 1.2.2. D e s c r i p t i o n o f the a n n i h i l a t i o n time spectrum i n terms o f a 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 1.2.3' The i n f l u e n c e ' o f i n e l a s t i c c o l l i s i o n s on the slowing-down time 1,2.h. The i n f l u e n c e o f e l a s t i c c o l l i s i o n s on the slowing-down time 1.2.5. Summary 1.3. D i r e c t a n n i h i l a t i o n r a t e and momentum-transfer c r o s s 8 - s e c t i o n s f o r p o s i t r o n s i n Argon 1.3*1• R e l a t i o n s h i p o f a n n i h i l a t i o n r a t e t o p o s i t r o n - e l e c t r o n o v e r l a p 1.3-2. D i r e c t a n n i h i l a t i o n r a t e i n Argon 1-3-3- V e l o c i t y dependence o f the d i r e c t a n n i h i l a t i o n r a t e i n Argon 1.3-^ - E l e c t r i c f i e l d dependence o f the d i r e c t a n n i h i l a t i o n r a t e i n Argon 1.3-5- D i r e c t a n n i h i l a t i o n r a t e o f p o s i t r o n s i n He l i u m 1.h. P o s i t r o n i u m f o r m a t i o n and a n n i h i l a t i o n i n Argon 13 1.^,1. S t r u c t u r e o f p o s i t r o n i u m 1.h.2. A n n i h i l a t i o n o f p o s i t r o n i u m 1A. 3- P o s i t r o n i u m f o r m a t i o n 1 A A . Quenching o f p o s i t r o n i u m l i f e t i m e s 1.M-.5. T h e o r e t i c a l s i t u a t i o n r e g a r d i n g f o r m a t i o n , quenching and e l a s t i c - s c a t t e r i n g c r o s s -s e c t i o n s 1.^.5.1. P o s i t r o n i u m f o r m a t i o n 1A.5.2. P o s i t r o n i u m quenching and e l a s t i c -s c a t t e r i n g c r o s s - s e c t i o n s 1.5- Summary o f work c o n t a i n e d i n the t h e s i s 18 1.5-1- T h e o r e t i c a l a s p e c t s 1.5-2. E x p e r i m e n t a l t e c h n i q u e s 1.5-3- P o s i t r o n i u m f o r m a t i o n 1.5A. O r t h o p o s i t r o n i u m a n n i h i l a t i o n r a t e s page EXPERIMENTAL INVESTIGATION OF POSITRON LIFETIMES IN .21 ARGON 2.1. Introduct ion 21 2.2. Experimental method 23 2.2.1. Li fet ime measurements 2.2.2. Va l l ey - to -peak r a t i o measurements 2.3- The exponential portions of the time spectra 27 2.3.1• The d i r e c t or free ann ih i la t ions 2.3*2. Orthopositronium annih i la t ions 2.3*3. Parapositronium annih i la t ions 2.3*'+. The observed spectrum i n the exponential region 2.*+. The va l l ey - to -peak r a t i o 29 2 .5* Analys i s of resu l t s 31 2.5.1. Analys is of time spectra 2.5*1*1. Maximum l i k e l i h o o d theory 2.5.1.2. I t era t ive so lu t ion of the maximum l i k e l i h o o d problem 2.5.1.3. I n i t i a l estimates of the four parameters 2.5.1."+. Est imation of channel widths, w, , and random background B 2 .5.1 .5. Est imation of variances 2.5*1*6. Goodness of f i t 2.5«2.Analysis of experimentally determined a n n i h i l a t i o n rates 2.6. Experimental resu l t s 38 2.6.1. C r i t e r i a for presentation of data 2.6.2. D irec t a n n i h i l a t i o n rate: zero e l e c t r i c f i e l d resu l t s 2.6.2.1. Results of f i t t i n g the a n n i h i l a t i o n rate to functions of the Argon density 2.6.2.2. Discuss ion of the f i t s to the data 2.6.2.3* Comparison of the l i n e a r term with previous resu l t s 2.6.3* D irec t a n n i h i l a t i o n rate and va l l ey - to -peak r a t i o : e l e c t r i c f i e l d resu l t s 2.6.k. The shoulder i n the time spectra 2.6.-+.1. Width of the shoulder 2.6A.2. The logarithmic slope of the shoulder 2.6."+.3. E f f e c t of the e l e c t r i c f i e l d on the shoulder 2.6.5. Orthopositronium a n n i h i l a t i o n rate 2.6.5.1. F i t t i n g of experimental data 2.6.5.2. Discussion of the l i n e a r densi ty . dependence; of 'the orthopositronium' .. , . a n n i h i l a t i o n rate . . . . . 2.6.5.3. Influence .of ..the- e l e c t r i c f i e l d i t . ; 2.6.5«l+. Summary of orthopositronium resu l t s - v i -page 2.6.6. Discuss ion of errors not re la ted to counting s t a t i s t i c s 2.6.6.1. E f f e c t of i n s t a b i l i t i e s i n the e l ec tron ic instrumentation 2.6.6.2. The 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 i e s of the t imesorter 2.6.6.3. Systematic error i n the a n n i h i l a t i o n rates 2.6.6.^ -. Appl ied e l e c t r i c f i e l d 2.6.6.5. Measurement of gas density 2.6.6.6. Uncertainty i n E/P 2.6.6.7. Gas composition 3. THEORETICAL CONSIDERATIONS OF THE POSITRON-ARGON ATOM 6k INTERACTION 3.1 • Introduct ion 6*+ 3-1.1. Discuss ion of the dif ferences between low-energy pos i tron and e lectron scat ter ing 3.1.2. Outl ine of procedure 3«2. The Schrodinger equation and i t s so lut ion 69 3.2.1. The Schrodinger equation 3.2.2. C a l c u l a t i o n of phase sh i f t s and wave funct ions 3.2.2.1. Asymptotic so lut ion for k^O 3.2.2.2. Asymptotic so lut ion for k=0 3.3. C a l c u l a t i o n of Z f f 75 3-*+. The pos i tron v e l o c i t y d i s t r i b u t i o n 76 3A.1 . The modified Wilkins equation 3.k.2. General computer so lu t ion of the d i f f e r e n t i a l equation 3A.3. Case of no i m p l i c i t time dependence 3A.1+. So lut ion of the'time-independent equation 3-5- Results 82 3.5.1. Discuss ion of the potent ia l s used 3.5.2. Comparison with experiment 3.5.3. Discussion of the break i n the dependence of a n n i h i l a t i o n rate on e l e c t r i c f i e l d Discuss ion of the experimental dependence of a n n i h i l a t i o n rate on e l e c t r i c f i e l d 3.6.. Conclusions 88 3.6.1. Summary of experimental resu l t s 3-6.2. Theore t i ca l conclusions REFERENCES 91' APPENDIX: Modif icat ions to the fast-s low coincidence 9^  c i r c u i t r y of W. Falk (1965). - v i i -page 1. P h o t o m u l t i p l l e r c i r c u i t r y 2. T i m i n g p u l s e g e n e r a t o r 3. A m p l i f i e r s and s i n g l e c h a n n e l a n a l y z e r s P i l e - u p r e j e c t o r s 5. ND 101 k i c k s o r t e r 6. D i f f e r e n t i a l and i n t e g r a l l i n e a r i t y o f the t i m e s o r t e r - v i i i -LIST OF TABLES page Table I . Gas p u r i t y analys is 2*+ Table I I . Results of chi-square test on the l i f e t i m e 37 spectra Table I I I . Dependence of A a on P ^0 Table IV. Published values of the d i r e c t a n n i h i l a t i o n h2 rate i n Argon Table V. Dependence of l 2 T 2 o n e ^ - e c ^ r ^ - c f i e l d M-6 Table V I . Dependence of ~*Q on P 53 Table V I I . Summary of published resu l t s for ortho- 57* positronium quenching i n Argon Table V I I I . Dependence of l i f e t imes on S . C . A . se t t ing 58* * Indicates page number preceding tab le . - i x -LIST OF FIGURES to fol low page 1 . A n n i h i l a t i o n mechanisms of positrons i n gases 3 2. Representative time spectrum of pos i tron a n n i h i l a t i o n 11 i n Argon 3. Block diagram of e l ec tronics used i n the l i f e t i m e 2*+ measurements h. Energy spectrum of 0.51 MeV gamma rays showing S . C . A . 25 sett ings 5- Representative time spectrum of pos i tron a n n i h i l a t i o n 26 i n Argon 6. Dependence of the l i k e l i h o o d funct ion on 1^ , I 2 , T 1 , T 2 36 7- D irec t a n n i h i l a t i o n rate, i n Argon at zero e l e c t r i c 39 f i e l d as a funct ion of density' 8. D irec t a n n i h i l a t i o n rate i n Argon as a funct ion of E/P hh 9- F r a c t i o n of positrons forming positronium as a funct ion of E /P 10. Time spectrum for positrons i n Argon at small E /P hj 11 . Comparison of d i r e c t - and ortho-enhanced time spectra *+8 obtained at P = h .9 amagats, E/P =0 V cm - 1amagat" 1 12. Comparison of d i r e c t - and ortho-enhanced time spectra *+9 obtained at P = 9.3 amagats, E/P =0 V cm"1 amagat"'1 13- Time spectrum for positrons at high E/P 51 \h. Orthopositronium a n n i h i l a t i o n rate i n Argon as a 52 funct ion of density 1 5- Theore t i ca l re su l t s for Zeff as a funct ion of pos i tron 83 wave number k 16. Theore t i ca l resu l t s for the momentum-transfer cross - 83 sect ion for positrons i n Argon as a funct ion of k 17. Comparison of t h e o r e t i c a l and experimental a n n i h i l a t i o n 8^ rates as a funct ion of E/P 18. Photomult lp l ier c i r c u i t 97 -x-to fo l low page 19- T i m i n g p u l s e g e n e r a t o r c i r c u i t 97 20. Prompt r e s o l u t i o n o f the e l e c t r o n i c system 97 21. C i r c u i t f o r d r i v i n g e l e c t r o m e c h a n i c a l r e g i s t e r 97 22. 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 o f the t i m e s o r t e r 97 - x i -ACKNOWLEDGMENTS. This thesis i s a report of research work performed under the aegis of Dr. G. Jones. I am much indebted to him for h i s enthusiasm, h i s exacting standards, and his deep phys ica l i n s i g h t . Thanks are also due to the many members of the Physics Department and Computing Centre, with whom I had f r u i t f u l d iscuss ions during the course of th i s projec t . Furthermore, I wish to acknowledge the assistance of members of the Physics Workshop and Van de Graaff t echnica l s ta f f . In a d d i t i o n , my sincere apprec iat ion goes to my wife and parents , who have been a source of encouragement and incent ive throughout the past years . F i n a l l y , I am indebted to the U n i v e r s i t y of B r i t i s h Columbia for two Graduate Fel lowships , and to the Nat ional Research Counc i l of Canada for a Studentship. -1 -1. POSITRONS AND THEIR INTERACTION WITH GAS ATOMS. 1.1. Introduct ion. Of a l l the predic t ions of modern r e l a t i v i s t i c quantum mechanics, surely one of the most sa t i s fy ing has been the de-duct ion of the existence of the pos i t ron . This achievement i s due to Dirac (1928), who proposed a r e l a t i v i s t i c wave equation for the e l ec t ron , which, i n add i t ion to pred ic t ing such dynamical c h a r a c t e r i s t i c s as the spin of the e l ec tron , also predicted the existence of a pos i t ive ly -charged a n t i - p a r t i c l e (D irac , 1931). This pos i t i ve p a r t i c l e , the pos i t ron , was discovered exper i -mentally by Anderson (1932), and was found to have a mass equal to that of the e l ec t ron . The study of the i n t e r a c t i o n of slow positrons with gas atoms was f i r s t undertaken by Shearer and Deutsch (19^9), who studied the slowing down of positrons i n various gases. As a d i r e c t r e s u l t of these experiments the existence of a p o s i t r o -nium atom postulated by Ruark 0 9^5) was v e r i f i e d . A bound system containing a pos i tron and an e l ec tron , positronium i s s i m i l a r to hydrogen i n many of i t s propert i e s . However, i t d i f f e r s r a d i c a l l y from hydrogen i n that the two p a r t i c l e s ann ih i la te each other with the emission of two or more quanta of gamma r a d i a t i o n . The a n n i h i l a t i o n of positrons with electrons forms the basis of most experiments deal ing with the atomic in teract ions of pos i trons . Use i s made of the a n n i h i l a t i o n r a d i a t i o n to study - 2 -e i ther the r e l a t i v e pos i t ron-e l ec tron v e l o c i t y at a n n i h i l a t i o n (Heinberg and Page, 1957; Ce l i tans and Green, 196M-), or the l i f e -time of the positrons i n a gas, l i q u i d or s o l i d (Fa lk , 1965). The l a t t e r technique forms the experimental basis of th is work. Under c e r t a i n condit ions to be made c lear i n the fo l low-ing sect ions , the l i f e t i m e of positrons i n a noble gas is. simply connected to the cross-sect ions for momentum t r a n s f e r , a n n i h i l a t i o n and positronium formation. I t i s the task of quantum mechanics to give precise predict ions of these cross - sec t ions , i n order that comparison, with experiment can be made. The t h e o r e t i c a l aspect of th i s thesis i s concerned both with the c a l c u l a t i o n of momentum transfer and a n n i h i l a t i o n cross-sect ions for some simple potent ia l s descr ib ing the positron-atom i n t e r a c t i o n , and with the reduction of these cross-sect ions i n order to make comparison with exper i -mental l i f e t i m e r e s u l t s . I t i s necessary to describe the pos i t ron-atom i n t e r a c t i o n i n an approximate way, siLnce, although the correct wave equation representing the i n t e r a c t i o n can be g iven, i t was not poss ib le to make a complete analys is because of the many p a r t i c l e aspect. Thus approximate models must be used and j u s t i f i e d by comparison with experiment. The current s i t u a t i o n regarding the theory of positron-atom interact ions i s enlarged upon i n Chapter 3-1.2. The fate of positrons i n a gas. 1.2.1. Introductory remarks. The study of positron-atom interact ions has been large ly confined to experiments using Na-22 or Cu-6 1* as pos i tron sources. - 3 -The energy-spectrum of positrons from such sources i s the continuous Fermi d i s t r i b u t i o n . In the case of Na-22, the maximum pos i tron energy i s 5^2 keV, and the d i s t r i b u t i o n i s peaked at 170 keV (Macklin,e,t a l . , 1950). The method by which these high energy posi trons lose energy and subsequently ann ih i la te i n a gas i s q u a l i t a t i v e l y understood. Figure 1 (Fa lk , 1965) i l l u s t r a t e s the various mechanisms involved i n th i s energy l o s s . In order to discuss these mechan-isms, d i scuss ion i s confined to the case of Argon at a density of the order of .10 amagats. (1 amagat=l+.I+589 x 10"^ moles/cc. Amer-lean I n s t i t u t e of Physics Handbook.) Argon i s the gas used t h r o u g h -out th i s work and the density i s such that the rate of the pro-cesses involved can be re la ted to the time r e s o l u t i o n , 7>k nsec, c h a r a c t e r i s t i c of the instrumentation used. 1.2.2. Descr ipt ion of the a n n i h i l a t i o n time spectrum i n terms of a pos i tron v e l o c i t y d i s t r i b u t i o n . A n n i h i l a t i o n l i f e time spectra of positrons are determined i n the fo l lowing way. The emission of a pos i tron from a Na-22 nucleus i s followed wi th in 10~ 1^ sees by a 1.28 MeV gamma ray necessary to de-excite the daughter nucleus Ne-22 to i t s ground s ta te . This gamma ray provides the pos i tron "b ir th" s i g n a l . The a n n i h i l a t i o n of the pos i tron with an e lec tron re su l t s i n a t o t a l of 1 .02 MeV of gamma r a d i a t i o n being emitted. The detect ion of th i s gamma r a d i a t i o n s i g n i f i e s the "death" of a pos i t ron . An a n n i h i l a t i o n time-spectrum displays the number of such events as Na-22 Positrons _ E m a x : 5^2keV Rapid Energy-Loss Ionization and I n e l a s t i c V C o l l i s i o n s N e g l i g i b l e A n n i h i l a t i o n Energy F i g u r e d . A n n i h i l a t i o n mechanisms of positrons i n gases. Ps-positronium. - l f -a funct ion of the l i f e t i m e of the pos i t ron . Such a time spectrum w i l l i n general contain components a r i s i n g from the a n n i h i l a t i o n of free positrons ( d i r e c t a n n i h i l a t i o n ) , and of para- and ortho-posi tronium. For a gas i n which there i s no positronium formation, the time spectrum consists of d i r e c t ann ih i la t ions only . In order to discuss the l i f e t i m e of a free pos i tron i n such a gas, i t i s convenient to consider the experiment as one i n which a l l the positrons detected were emitted simultaneously. Furthermore the pos i tron density i s thought of as so low that no one pos i t ron can in t erac t with another during i t s l i f e t i m e , which i s the case for the weak ( 1 0 / i C i ) sources used here. In such a case, the swarm of positrons has some energy d i s t r i b u t i o n , which i s modified as a funct ion of time by i n t e r a c t i o n of the positrons with the gas atoms. The logarithm of the slope of a time spectrum at any time t i s then Inversely proport iona l to the velocity-dependent a n n i h i l a t i o n rate averaged over the v e l o c i t y d i s t r i b u t i o n appropriate to the time t . Returning now to the ac tua l condit ions under which the experimental data i s gathered i t can be seen that there i s no d i f ference between these two descr ip t ions . The pos i tron v e l o c i t y d i s t r i b u t i o n y ( v , t ) consistent with e i ther of these descr ipt ions i s then the p r o b a b i l i t y that at a time t , the pos i tron has a v e l o c i t y between v and vtdv. For the case of a swarm of pos i trons , the pos i t ron v e l o c i t y d i s t r i b u t i o n i s equal ly defined as the f r a c t i o n of posi trons at time t which have a v e l o c i t y between v and vfdv. The above d i scuss ion appl ies equal ly to the case where -5-there i s positronium formation, except that positronium a n n i h i l a -t ions modify the shape of the time spectrum,somewhat. 1.2.3. The inf luence of i n e l a s t i c c o l l i s i o n s on the slowing-down time. At 10 amagats the positrons from the continuous energy d i s t r i b u t i o n lose energy r a p i d l y i n the region where i n e l a s t i c c o l l i s i o n s with the Argon atoms are poss ib le . Within about 0.7 nsec a l l the positrons have l o s t a l l but about 1 0 keV of t h e i r i n i t i a l energy (Fa lk , 1965). The c a l c u l a t i o n of th i s time r e l i e s on experimental and t h e o r e t i c a l values of dE/dx over the energy range from 5^0 keV to 10 keV (Nelms, 1956), and i s probably correct to w i th in 20$. The time involved for the remaining energy loss from 10 keV to the l a s t i n e l a s t i c l e v e l i n Argon at 11.6 eV i s less c e r t a i n . However, a r e a l i s t i c upper l i m i t to th i s time can be found by assuming that a minimum of 11.6 eV i s l o s t per i n -e l a s t i c c o l l i s i o n . For an average i n e l a s t i c c o l l i s i o n cross -2 sect ion of ^ir a Q , where a 0-=Bohr r a d i u s , th i s gives an upper l i m i t of ^ 5 x 1 0 ~ ^ seconds for a densi ty of 10 amagats. Thus i t i s c l ear that , given an experimental time r e s o l u t i o n of about 7 nsec, a l l a n n i h i l a t i o n s that occur during th i s i n i t i a l s lowing-down period occur wi th in the r e s o l u t i o n of the apparatus. In a d d i t i o n , . t h e pos i tron a n n i h i l a t i o n during th i s time i s n e g l i g i b l e 4 (Gerhart , et a l . , 195^5 Kendal l and Deutsch, 195*0--6-I, 2.h. The i n f l u e n c e o f t h e e l a s t i c c o l l i s i o n s on the s l o w i n g -down t i m e . The m a j o r i t y o f t h e p o s i t r o n s , t h e n , r e a c h e n e r g i e s below I I . 6 eV w i t h o u t a n n i h i l a t i n g . Below t h i s energy t h e p o s i t r o n s can o n l y be slowed down by e l a s t i c c o l l i s i o n s w i t h the Argon atoms. D u r i n g t h i s slowing-down p e r i o d , t h e p o s i t r o n d i s t r i b u t i o n w i l l be d e p o p u l a t e d by t h r e e main p r o c e s s e s . F i r s t l y , a p o s i t r o n can a n n i h i l a t e w i t h an at o m i c e l e c t r o n d u r i n g a d i r e c t c o l l i s i o n w i t h the atom, the s o - c a l l e d d i r e c t a n n i h i l a t i o n . S e c o n d l y , a bound Ar-e complex may be formed w h i c h s u b s e q u e n t l y a l s o a n n i h i l a t e s . These p r o c e s s e s w i l l be d i s c u s s e d i n S e c t i o n 1.3- T h i r d l y , a p o s i t r o n may c a p t u r e an at o m i c e l e c t r o n t o form a p o s i t r o n i u m atom w h i c h may s u b s e q u e n t l y a n n i h i l a t e w i t h one o f two c h a r a c t e r i s -t i c s pin-dependent l i f e t i m e s . T h i s p o s s i b i l i t y i s d i s c u s s e d i n S e c t i o n 1 ."+. S c a t t e r i n g e l a s t i c a l l y o n l y , t h e v e l o c i t y d i s t r i b u t i o n o f t h e p o s i t r o n s w i l l r e l a x t o an e q u i l i b r i u m d i s t r i b u t i o n c h a r a c -t e r i s e d o n l y by t h e te m p e r a t u r e T o f t h e h o s t gas. T h i s e q u i l i b -r i u m d i s t r i b u t i o n w i l l be M a x w e l l i a n except f o r a p o s s i b l e s m a l l d e v i a t i o n produced by t h e d i r e c t a n n i h i l a t i o n s . These d e v i a t i o n s o c c u r i f t h e a n n i h i l a t i o n r a t e ( p r o b a b i l i t y per u n i t t ime f o r a n n i h i l a t i o n o f a p o s i t r o n ) i s v e l o c i t y - d e p e n d e n t . F o r an a n n i h i l a t i o n r a t e w h i c h i s v e l o c i t y - i n d e p e n d e n t t h e r a t e o f removal o f p o s i t r o n s per u n i t v e l o c i t y i n t e r v a l from the v e l o c i t y d i s t r i b u t i o n i s a l s o v e l o c i t y - i n d e p e n d e n t . I t f o l l o w s t h a t f o r t h i s s p e c i a l case t h e shape o f t h e 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 u n a f f e c t e d by a n n i h i l a t i o n s and i s t h e r e f o r e M a x w e l l i a n -7-a t e q u i l i b r i u m . S i n c e the a n n i h i l a t i o n r a t e i s d i r e c t l y p r o p o r t i o n -a l t o t h e p r o d u c t o f v e l o c i t y o f t h e p o s i t r o n ( r e l a t i v e t o the atom) and the a n n i h i l a t i o n c r o s s - s e c t i o n a t t h a t v e l o c i t y , i t i s c l e a r t h a t a v e l o c i t y - i n 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 r e s u l t s from an a n n i h i l a t i o n c r o s s - s e c t i o n i n v e r s e l y p r o p o r t i o n a l t o t h e p o s i t r o n v e l o c i t y . F or t h e case o f an a n n i h i l a t i o n r a t e w h i c h i n c r e a s e s w i t h d e c r e a s i n g v e l o c i t y , p o s i t r o n s a r e removed p r e f e r e n t i a l l y from the low-energy end o f the d i s t r i b u t i o n . A t any i n s t a n t t h e average p o s i t r o n energy w i l l be h i g h e r t h a n i n the ca-s-© o f a c o n s t a n t o r z e r o a n n i h i l a t i o n r a t e . T h i s can o n l y o c c u r i f the c e n t r e o f g r a v i t y o f t h e v e l o c i t y d i s t r i b u t i o n i s s h i f t e d up i n v e l o c i t y compared w i t h t h e M a x w e l l i a n d i s t r i b u t i o n . The t ime t a k e n t o r e a c h t h e r m a l e q u i l i b r i u m depends on t h e shape o f t h e i n i t i a l v e l o c i t y d i s t r i b u t i o n below '11.6 eV and a l s o on t h e momentum-transfer c r o s s - s e c t i o n . T y p i c a l l y , f o r a momentum-transf er- c r o s s - s e c t i o n o f t h e o r d e r o f 7Fa02, the r e l a x -a t i o n t ime i s g r e a t e r t h a n about 10 nsec (Tao, e t a l . , 1963), w h i c h i s l a r g e r t h a n the e x p e r i m e n t a l time r e s o l u t i o n o f 7 n s e c . 1.2.5. Summary. For about 10 amagats o f Arg o n , and an e x p e r i m e n t a l time r e s o l u t i o n o f about 7 n s e c , p o s i t r o n a n n i h i l a t i o n s d u r i n g t h e i n i t i a l slowing-down p e r i o d t o the f i r s t e x c i t a t i o n l e v e l i n Argon a t 11.6 eV a r e n o t e x p e c t e d t o have any o b s e r v a b l e l i f e t i m e . The r e m a i n i n g p o s i t r o n s can l o s e energy o n l y by e l a s t i c c o l l i s i o n s , -8-t h e 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 b e i n g d e p o p u l a t e d by a n n i h i l a t i o n s and p o s i t r o n i u m f o r m a t i o n . The l a t t e r can o c c u r o n l y when the p o s i t r o n has s u f f i c i e n t energy t o make up t h e d i f f e r e n c e between the i o n i z a t i o n p o t e n t i a l s o f Argon and p o s i t r o n i u m . For Argon t h i s t h r e s h o l d energy i s 8.9 eV,below w h i c h p o s i t r o n i u m f o r m a t i o n i s n o t p o s s i b l e . The r e l a x a t i o n t i m e o f t h e d i s t r i b u t i o n t o e q u i -l i b r i u m i s e x p e c t e d t o be o f the o r d e r o f 10 nsec o r g r e a t e r . E f f e c t s due t o t h i s s l ow r e l a x a t i o n t ime have been p r e v i o u s l y o b s e r v e d and a r e d i s c u s s e d i n S e c t i o n 1.3* and i n C h a p t e r 2. 1 .3' D i r e c t a n n i h i l a t i o n r a t e And:-momentum-transfer c r o s s - S e c t i o n s f o r : . p o s i t r o n s i n cArgon. 1-3.1. R e l a t i o n s h i p o f a n n i h i l a t i o n r a t e t o p o s i t r o n - e l e c t r o n o v e r l a p . D u r i n g th e c o l l i s i o n o f a f r e e p o s i t r o n w i t h an atomic system, t h e r e i s t h e p o s s i b i l i t y t h a t the p o s i t r o n w i l l a n n i h i l a t e w i t h an a tomic e l e c t r o n w i t h o u t f o r m i n g p o s i t r o n i u m . F e r r e l l (1956) has shown t h a t the p o s i t r o n a n n i h i l a t i o n r a t e i s p r o p o r t i o n a l t o t h e gas d e n s i t y and t o t h e e l e c t r o n d e n s i t y a t t h e p o s i t r o n a v eraged over the p o s i t r o n p o s i t i o n . The d e t a i l e d e x p r e s s i o n i s g i v e n i n C h apter 3' I f the i n c oming p o s i t r o n i s d e s c r i b e d by a p l a n e wave, the i n t e g r a l i n v o l v i n g t h e p o s i t r o n - e l e c t r o n o v e r l a p : i s e q u a l t o t h e t o t a l number, Z, o f e l e c t r o n s c o m p r i s i n g the atom. Because o f t h e d e p a r t u r e o f the p o s i t r o n w a v e - f u n c t i o n from a p l a n e wave due t o t h e coulomb i n t e r a c t i o n , t h e o v e r l a p i n t e g r a l i s n o t i n g e n e r a l - 9 -equal to Z . The r e s u l t i n g e f fec t ive number of e lec trons , Z e f f , depends on the p a r t i c u l a r form of the positron-atom i n t e r a c t i o n assumed. An a l ternate mode of a n n i h i l a t i o n would r e s u l t i f a bound A r - e + s tate i s poss ib le . Annih i la t ions from such a system would be ind i s t ingu i shab le from the d i r e c t ann ih i la t ions by the experimental techniques employed here. The l i f e t i m e of th i s system w i l l be of the order, of the parapositronium l i f e t i m e (10~ 1^ sees). The observed a n n i h i l a t i o n rate c h a r a c t e r i s t i c of th i s process i s thus the capture rate of positrons into the bound Ar-e+ system and i s thus proport iona l to pressure. No t h e o r e t i c a l d i scuss ion of the a n n i h i l a t i o n rate so far has included th i s p o s s i b i l i t y . The a n n i h i l a t i o n rates ca lculated i n th i s thes is are those obtained by considering only the pos i t ron-e lec tron overlap during an e l a s t i c c o l l i s i o n . As ye t , there i s no way of t e l l i n g which of the two processes predominates. The ca l cu la t ions presented i n th i s thesis are based on the assumption that the contr ibut ion to the observed a n n i h i l a t i o n rate from th i s process i s n e g l i g i b l e compared with the d i r e c t rate considered. x. 1.3'2. D irec t a n n i h i l a t i o n rate i n Argon. Experimental re su l t s ind ica te (Fa lk , 1965; Tao, B e l l , and Green, ^^6U••, Osmon, 1965; Pau l , 196^) that the average 6 1 1 a n n i h i l a t i o n rate for Argon (about 5 x 10 sec amagat ) implies a Z e f f of about 30, s u b s t a n t i a l l y larger than the Dirac Z e f f of 18. -10-T h i s i n d i c a t e s t h a t t h e p o s i t r o n must he a t t r a c t e d t o the atom a t some s t a g e i n o r d e r f o r t h e w a v e - f u n c t i o n o f the p o s i t r o n a t the atom t o be s u b s t a n t i a l l y l a r g e r t h a n t h a t a p p r o p r i a t e t o a plane wave. The r e q u i r e d a t t r a c t i o n c o u l d be due, a t l e a s t p a r t i a l l y , t o the s t a n d a r d l o n g range p o l a r i z a t i o n term found n e c e s s a r y t o f i t low energy e l e c t r o n s c a t t e r i n g from n o b l e gas atoms (Mott and Massey, 1965). The p o l a r i z a t i o n p o t e n t i a l behaves a s y m p t o t i c a l l y as a/R1*, where R i s the p o s i t r o n - a t o m s e p a r a t i o n and <* i s t h e p o l a r i z a b i l i t y o f t h e Argon atom. A f u r t h e r c o n t r i b u t i o n t o t h e h i g h Z e f f c o u l d a r i s e f r o m d i s t o r t i o n o f the e l e c t r o n c l o u d around t h e Argon atom by t h e incoming p o s i t r o n , t h i s e f f e c t t e n d i n g t o i n c r e a s e the e l e c t r o n d e n s i t y a t t h e p o s i t r o n p o s i t i o n . Two c a l c u l a t i o n s d i r e c t e d s p e c i f i c a l l y t o t h e case o f Argon have been r e p o r t e d (Massey, e t a l . , 1966; Jones and O r t h , 1966). B o t h use an e m p i r i c a l p o l a r i z a t i o n p o t e n t i a l w i t h an a s y m p t o t i c a/R1+ b e h a v i o u r , w i t h a s i m p l e c u t o f f i n o r d e r t h a t the p o t e n t i a l r e m a i n f i n i t e a t t h e o r i g i n . The c u t o f f s a r e a l s o chosen such t h a t t h e H a r t r e e - F o c k p o t e n t i a l a p p r o p r i a t e t o t h e g r o u n d - s t a t e atom dominates i n t h e i n t e r i o r o f t h e atom. A s c a t t e r i n g p o t e n t i a l i s t a k e n t o be t h e sum o f t h e s t a t i c H a r t r e e - F o c k and p o l a r i z a t i o n p o t e n t i a l s . T h i s t y p e o f s c a t t e r i n g p o t e n t i a l i s t h e s i m p l e s t f o r w h i c h an a t t r a c t i o n i s p o s s i b l e , o c c u r r i n g when t h e p o l a r i z a t i o n p o t e n t i a l dominates t h e s t a t i c H a r t r e e - F o c k p o t e n t i a l , w h i c h i s r e p u l s i v e f o r a p o s i t r o n . I n b o t h c a l c u l a t i o n s r e p o r t e d no e f f e c t s due t o e l e c t r o n d i s t o r t i o n a r e c o n s i d e r e d . A l t h o u g h t h i s t y p e o f p o t e n t i a l i s adequate f o r d e s c r i b i n g t h e Ramsauer e f f e c t i n Argon 1 -11-( H o l t s m a r k , 1929; K i v e l , 1959; Labahn and C a l l a w a y , 1966) i t i s shown i n t h i s t h e s i s t o be i n a d e q u a t e f o r the p o s i t r o n - A r g o n s c a t t e r i n g c a s e . T h i s p o i n t i s d i s c u s s e d f u r t h e r i n Cha p t e r 3* 1.3•3• V e l o c i t y dependence o f the d i r e c t a n n i h i l a t i o n r a t e i n Argon. That t h e d i r e c t a n n i h i l a t i o n r a t e f o r p o s i t r o n s i n Argon i s v e l o c i t y - d e p e n d e n t i n the r e g i o n between 0 eV and about 10 eV f o l l o w s f r om t h e e x p e r i m e n t a l o b s e r v a t i o n t h a t t h e d i r e c t a n n i h i l a -t i o n r a t e i s n o t e x p o n e n t i a l , b u t i s c h a r a c t e r i s e d by a " s h o u l d e r " i n t h e p o s i t r o n a n n i h i l a t i o n time spectrum ( F a l k and J o n e s , 1964; F a l k , 1965; Tao, B e l l and Green, 1964; Osmon, 1965; P a u l , 1964). A t y p i c a l t i me spectrum showing t h i s f e a t u r e i s g i v e n i n F i g u r e 2. The s h o u l d e r i s f o l l o w e d by a s i n g l e e x p o n e n t i a l w h i c h i s t a k e n t o be t h e d i r e c t a n n i h i l a t i o n r a t e f o r t h e t h e r m a l i z e d p o s i t r o n s ( F a l k , 1965). The j u s t i f i c a t i o n f o r t h i s d e s c r i p t i o n o f the o b s e r v a t i o n s i s c o n s i d e r e d i n Chapter 2, S e c t i o n 2.1. 1.3.4. E l e c t r i c f i e l d dependence o f t h e d i r e c t a n n i h i l a t i o n r a t e i n Argon. The use o f an a p p l i e d e l e c t r i c f i e l d t o s t u d y t h e v e l o c i t y dependence o f t h e p o s i t r o n a n n i h i l a t i o n r a t e was i n t r o d u c e d by F a l k , O r t h and Jones (1965). T h i s method r e l i e s on t h e f a c t t h a t the v e l o c i t y d i s t r i b u t i o n o f t h e p o s i t r o n s a t e q u i l i b r i u m can be i n f l u e n c e d by t h e a p p l i c a t i o n o f a u n i f o r m dc e l e c t r i c f i e l d . S i n c e t h e p o s i t r o n s may g a i n energy from t h e f i e l d , t h e average v e l o c i t y o f t h e d i s t r i b u t i o n i s thus i n c r e a s e d . The observed a n n i h i l a t i o n r a t e , w h i c h i s t h e 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 o o o u~>' § o 55-CO \ — o § 3 O U. O O •O o o o I I III + 5 8 P = 8 . 9 E/P r .0 2.74 nsec/channel + Note: I n t h i s and a l l sub- + sequent time s p e c t r a the random c o i n c i d e n c e background has been s u b t r a c t e d . + + + ++ + + t j . 1 1 1 1 < r 60.000 85.000 110.000 135.000 160.000 185.000 210.000 235.000 CHANNEL NUMBER , F i g u r e 2. R e p r e s e n t a t i v e time spectrum p f p o s i t r o n a n n i h i l a t i o n i n Argon. P i s i n amagats, E/P i s i n V cm - 1 amagat" 1; I - p r o m p t peak; I I - s h o u l d e r r e g i o n ; I I I - d i r e c t / c o m p o n e n t ; IV - o r t h o p o s i t r o n i u m component.  -12-rate averaged over a l l pos i tron v e l o c i t i e s i n the d i s t r i b u t i o n , i s then e l e c t r i c f i e l d dependent. The de ta i l ed shape of the equi -l i b r i u m v e l o c i t y d i s t r i b u t i o n i s governed to f i r s t order by the magnitude of the appl ied e l e c t r i c f i e l d , and the v e l o c i t y dependence of the e l a s t i c scat ter ing momentum-transfer cros s - sec t ion . In order to test a s p e c i f i c model of the i n t e r a c t i o n with the exper i -mental r e s u l t s , i t i s therefore necessary to compute both the d i r e c t a n n i h i l a t i o n rate and the momentum-transfer cross - sec t ion for the model assumed. The f i r s t measurements of the e l e c t r i c f i e l d dependence of the d i r e c t a n n i h i l a t i o n i n Argon (Fa lk , 1965; F a l k , Orth and Jones, 1965) showed that the a n n i h i l a t i o n rate' decreased with increas ing e l e c t r i c f i e l d , reaching a value of about h a l f that at zero e l e c t r i c f i e l d when the f i e l d reached a value of about 1 -1 80 V cm" amagat 1.3«5- D irec t a n n i h i l a t i o n rate of positrons i n Helium. In order to test the general features of th i s p ic ture of the i n t e r a c t i o n of positrons with atoms of the noble gases, i t i s intended to extend such measurements to the other r e a d i l y obtained noble gases (eg. He, Ne, K r ) . Indeed pre l iminary measure-ments being performed at present ind ica te a s i g n i f i c a n t shoulder e f fec t i n the a n n i h i l a t i o n time spectrum for Helium i n contra-d i c t i o n to the e a r l i e r re su l t s of F a l k , Orth and Jones (1965), and Osmon (1965). In a d d i t i o n , the shape of the time spectrum i s found to be dependent on the presence of an e l e c t r i c f i e l d , much as i s observed for the case of Argon. In f a c t , a complete experimental inves t iga t ion of the a n n i h i l a t i o n rate i n Helium i s of p a r t i c u l a r s ign i f i cance since there appears to be considerable i n t e r e s t i n the t h e o r e t i c a l aspects of th i s problem at present (Drachman, 19665 Kestner, et a l . , 1965; Massey, et a l . , 1966). 1 .4. Positronium formation and a n n i h i l a t i o n i n Argon. 1.4.1. Structure of positronium. As mentioned i n Section 1.1., positronium i s the bound state of a pos i tron and e lec tron . In the gross s tructure of i t s energy l eve l s i t d i f f e r s from the hydrogen atom by a factor of two r e f l e c t i n g a smaller reduced mass. Thus the i o n i z a t i o n po tent ia l of positronium i s 6.78 eV instead of 13*6 eV. The t r a n s i t i o n p r o b a b i l i t i e s are halved as the average pos i t ron-e l ec tron distance i s twice that of the proton-e lectron dis tance . The f ine s tructure d i f f e r s considerably from that of hydrogen due to the large d i f f e r -ence i n mass between the proton and pos i t ron . An a d d i t i o n a l f ine s tructure term i s introduced by the p a r t i c l e - a n t i p a r t i c l e nature of the. e lec tron and pos i t ron . Only the gross s tructure of the energy l eve l s i s important i n th i s work. Deta i l s of the o v e r a l l s tructure of positronium have been summarized by Series (1957). 1.4.2. A n n i h i l a t i o n of positronium. Positronium i n the ground state has two spin s tates . Roughly speaking, orthopositronium contains the pos i tron and e lec tron with spins p a r a l l e l , whereas parapositronium has the spins -14-a n t i - p a r a l l e l . The p a r t i c l e - a n t i p a r t i c l e nature of the pos i t ron-e lec tron system causes i t to be unstable against a n n i h i l a t i o n into photons. Because of the s e l ec t ion rules governing the decay of a spin 1 p a r t i c l e into photons, orthopositronium can only decay into an odd number of photons. By the same token, parapositronium can only decay into an even number of photons. The general s i t u a t i o n regarding the number of photons for a n n i h i l a t i o n from the ground state and excited states of positronium has been summarized by Kaempffer (1965). The l i f e t i m e of positronium depends on the t o t a l angular momentum of the s tate . Orthopositronium i n i t s ground state has a ca lcu la ted mean l i f e t i m e of 1 .4 x 10-'7 seconds. Parapositronium i n the ground state decays far more r a p i d l y , i t s mean l i f e t i m e — 1 0 being ca lcu la ted to be 1.25 x 10 seconds (Ore and Powell , 1949). With reference to the experimental time reso lu t ion of 7 x 10~9 sees quoted i n Sect ion 1.2.5. i t i s c lear that the l i f e t i m e of ortho-positronium w i l l be reso lved , whereas that of parapositronium w i l l not be v i s i b l e . The l i f e t i m e of excited states of positronium has been ca lcu la ted for the S and P states by Alekseev (1959) •> and are a l l longer than for the ground states with the same spin alignments. In p a r t i c u l a r the 2S l e v e l should have a mean l i f e t i m e of about _Q 1 x 10 7 sec. Furthermore, i t appears that th is i s the only exci ted l e v e l i n positronium from which i t i s at a l l f eas ib l e to detect a n n i h i l a t i o n , since th i s state i s metastable against o p t i c a l t r a n s i t i o n s to the ground s tate . For a l l other states the o p t i c a l t r a n s i t i o n rates exceed the a n n i h i l a t i o n rates by a fac tor of at l eas t 1000. 1.h.3. Positronium formation. Formation of positronium i n i t s ground state i n a gas can occur i f the pos i tron energy i s greater than or equal to the d i f ference between the i o n i z a t i o n potent ia ls of the gas atom and the positronium atom. For Argon th i s threshold energy (Ethr) i s 8.9 eV. The threshold energy for formation into the f i r s t excited state i s Ih.'l eV. Because of the d i f f eren t spins of ground state ortho-and parapositronium, the r a t i o of the amounts formed i n each of these states i s expected to be the normal s t a t i s t i c a l r a t i o based on the r e l a t i v e m u l t i p l i c i t i e s of the s tates . Thus three ortho-positronium atoms should be formed for each parapositronium atom. The amount of n=2 positronium formed r e l a t i v e to ground state positronium w i l l be considerably less due to the expected smaller cross - sec t ion for formation (Massey and Mohr, 195*+) and the competition from the i n e l a s t i c c o l l i s i o n s with the Argon atom. Any positronium atom formed i n an excited state (except the 2S) should make the o p t i c a l t r a n s i t i o n to the ground state before a n n i h i l a t i o n . At 10 amagats of Argon, the 2S state i s more l i k e l y to suffer c o l l i s i o n a l de -exc i ta t ion to the 2P state than to ann i -h i l a t e . Furthermore, at these d e n s i t i e s , the c o l l i s i o n a l de-e x c i t a t i o n of any excited state into the ground state i s probably considerably more rap id than the o p t i c a l t r a n s i t i o n rate of about 8 1 10 sec" (Wallace, 1955)• This might explain to some extent the -16-unsuccessful attempts to detect the Lyman alpha spec tra l l i n e of positronium (Brock and S t r e i b , 1958; Bennett, et a l . , 1961; Duff and Heymann, 1963)• 1.h.h. Quenching of positronium l i f e t i m e s . The c o l l i s i o n s of a positronium atom with a gas atom can also r e s u l t i n the a n n i h i l a t i o n of the pos i tron with one of the atomic e lec trons . This "pick-off1"' quenching of the l i f e t i m e of the positronium atom i s the most important type of quenching encountered i n c o l l i s i o n s with noble gas atoms. This extra channel for a n n i h i l a t i o n of a positronium atom resu l t s i n a pressure-dependent a n n i h i l a t i o n ra te . The quenching rate has been measured for orthopositronium i n Argon and other noble gases by Heymann, 6 1 1 et al.,0 961) to be about 0.25 x 10 sec"'amagat" for Argon. At 10 amagats, the mean l i f e t i m e of orthopositronium Is thus expected to be reduced to about 1 .0 x 10-''7 sees. A quenching rate of th is magnitude w i l l have a n e g l i g i b l e e f fect on parapositronium l i f e t i m e . 1.^.5- Theore t i ca l s i t u a t i o n regarding formation, quenching and e l a s t i c scat ter ing cross - sec t ions . 1 A.5*1• Positronium formation. Very l i t t l e work has been done on the positronium forma-t i o n cross - sec t ion i n general . Cheshire 0 96'+) has considered positronium formation by fas t positrons i n atomic hydrogen, using the Born and impulse approximations. The cross-sect ions so obtained p are of the order of ifaQ and decrease with increas ing energy, i n -17-agreement to wi th in an order of magnitude with those of Massey and Mohr (195 l + ) ' S imi lar ca lcu la t ions using the Born approximation for the case of Helium hy Massey and Moussa (1961) ind ica te that the cros s - sec t ion i s of the order of 0.1^ao or less near threshold. The cross - sec t ion r i s e s to a maximum of 0.4 T a 0 2 at 27 eV. No ca lcu la t ions have been published for the case of many-electron atoms. 1.4.5.2. Positronium quenching and e l a s t i c - s c a t t e r i n g cross - sec t ions . The general problem of low energy positronium scat ter ing from atomic Hydrogen has been invest igated by Massey and Mohr (195^) using the Born approximation. In th i s case, quenching of ortho-positronium i s achieved by d i r e c t conversion of ortho- into para-positronium made poss ib le by the exchange of the s ing le atomic e l ec t ron . The cross - sec t ion for th i s type of quenching was found to be h i g h l y energy dependent. These ca lcu la t ions have been r e -peated i n more d e t a i l by Fraser (1961) who has considered the e l a s t i c scat ter ing of orthopositronium from Helium atoms (Fraser , 1962). Neglecting p o l a r i z a t i o n and e x c i t a t i o n , but inc luding the effects of e lec tron exchange, he has found an e l a s t i c scat ter ing cross -sect ion for positronium with Helium of 17-7^a 0 at zero k i n e t i c energy. The appropriate p i c k - o f f quenching cross - sec t ion has yet to be reported. Again no ca lcu la t ions exis t for other many-electron atoms. -18-1.5. Summary of work contained i n the thes i s . 1.5.1. Theore t i ca l aspects. That a v e l o c i t y dependent a n n i h i l a t i o n rate i s s u f f i c i e n t to give r i s e to the observed shoulder i n the time spectra has been shown by Falk (1965). One purpose of th i s thesis i s to present ca l cu la t ions of such v e l o c i t y dependent a n n i h i l a t i o n ra te s , based on a simple model of the positron-atom i n t e r a c t i o n . E las t i c— scat ter ing momentum-transfer cross-sect ions are also derived on the basis of these models. The appropriate v e l o c i t y averaged a n n i h i l a t i o n rates at equi l ibr ium as a funct ion of appl ied e l e c t r i c f i e l d have also been derived i n order to compare with experiment. The c a l c u l a t i o n of the v e l o c i t y dependent a n n i h i l a t i o n ra tes , momentum transfer cross - sec t ions , and appropriate v e l o c i t y averaged a n n i h i l a t i o n rates as a funct ion of e l e c t r i c f i e l d const i tute an o r i g i n a l contr ibut ion i n that there are no previous ly published reports . 1.5.2. Experimental techniques. Limitat ions i n the instrumentation used by Falk prevented the measurement of the l o n g - l i v e d orthopositronium component at the same time as the s h o r t e r - l i v e d d i r e c t component i n a n n i h i l a t i o n t ime-spectra . Consequently, there was some doubt as to the accuracy of the l i f e t i m e of the l o n g - l i v e d component. These inaccuracies were expected to have a not iceable ef fect on the estimation of the s h o r t - l i v e d (d irec t ) component. These problems were further com-pounded by errors i n the maximum l i k e l i h o o d method used i n f i t t i n g the experimental data. v The experimental resu l t s contained i n th i s thesis have been obtained with considerably modified instrumentation, i n that the time scale has been extended i n order that the orthopositronium component may be measured simultaneously with the d i r e c t component. Furthermore, the random background has been reduced by a factor of t e n , y i e l d i n g increased s t a t i s t i c a l accuracy. The errors inherent i n the o r i g i n a l method of curve f i t t i n g have been success fu l ly removed and, i n a d d i t i o n , the s ize and a p p l i c a b i l i t y of the standard deviations of the r e s u l t i n g parameters are discussed. The o r i g i n a l s t a t i s t i c a l analys is of Falk involved modif icat ion of the raw data to take into account instrumental e f fec t s . In the present case, these effects are taken into account by making appropriate modif ications to the funct ion to be f i t t e d to the raw data. 1 -5•3• Positronium formation. There i s some doubt as-to the i n t e r n a l consistency of the d iscuss ion i n F a l k ' s thesis which re lates to the e l e c t r i c f i e l d dependence of the positronium formation rate and to the ef fect of th i s positronium formation rate on the d i r e c t a n n i h i l a t i o n ra te . This ar i ses from the fact that the measurements of positronium formation as a funct ion of e l e c t r i c f i e l d due to Marder, et a l . , (1956) were used for that d i scuss ion . The present work contains such measurements for the same gas samples as were used i n the l i f e t i m e experiments presented here. The resu l t s d i f f e r consider--20-ably from those of Marder, et a l . There i s , therefore , reasonable doubt that the Marder, et a l . values were relevant to the Falk experiments. 1.5•4. Orthopositronium a n n i h i l a t i o n rates . The pressure dependent quenching rate of orthopositronium i n Argon i s discussed i n th i s thesis i n some d e t a i l . These measure-ments are considerably more accurate than any previous ly reported, a r e s u l t of increased s t a t i s t i c a l accuracy and improved curve-f i t t i n g techniques. -21 -2. EXPERIMENTAL INVESTIGATION OF  POSITRON LIFETIMES IN ARGON. 2.1. I n t r o d u c t i o n . Recent i n v e s t i g a t i o n s o f the l i f e t i m e o f p o s i t r o n s i n Argon have shown t h a t the f r e e p o s i t r o n a n n i h i l a t i o n r a t e cannot be d e s c r i b e d by a s i n g l e e x p o n e n t i a l (Tao, B e l l and Green, 196*+; P a u l , 196*+; Osmon, 1965; F a l k and J o n e s , 196*+). Time s p e c t r a o f t h e a n n i h i l a t i o n gamma ra y s show c l e a r e v i d e n c e o f a s h o u l d e r f o l l o w e d by a s i n g l e e x p o n e n t i a l whose l i f e t i m e i s presumed t o a r i s e f r om the d i r e c t a n n i h i l a t i o n a t t h e r m a l v e l o c i t i e s . I t has been shown t h a t the s h o u l d e r i s removed, and the l i f e t i m e o f the e x p o n e n t i a l changed, on t h e a p p l i c a t i o n of a moderate s t a t i c e l e c t r i c f i e l d ( F a l k , O r t h and J o n e s , 1965)" T y p i c a l l y , a f i e l d — 1 — 1 o f about 80 V cm amagat i s s u f f i c i e n t t o d e c r e a s e the d i r e c t l i f e t i m e by a f a c t o r o f two. I n v iew o f the i m p o r t a n c e of t h e s e r e s u l t s as the o n l y a v a i l a b l e e x p e r i m e n t a l t e s t o f the v a l i d i t y o f models d e s c r i b i n g the p o s i t r o n - A r g o n atom i n t e r a c t i o n , a f u r t h e r s e r i e s Of t h e s e measurements has been made w i t h improved i n s t r u -m e n t a t i o n , and t o a g r e a t e r degree o f s t a t i s t i c a l a c c u r a c y . These measurements a r e used t o t e s t the v a l i d i t y o f s e v e r a l e m p i r i c a l p o t e n t i a l s d e s c r i b i n g the e f f e c t i v e p o s i t r o n - A r g o n i n t e r a c t i o n ( C h a p t e r 3) -P o s i t r o n s e m i t t e d by a r a d i o a c t i v e s o u r c e i n t o a gas l o s e most o f t h e i r energy by i n e l s t i c c o l l i s i o n s w i t h the gas atoms. Once p o s i t r o n s i n Argon have t h e i r k i n e t i c energy d e c r e a s e d t o -22-11.6 eV, the lowest exc i ta t ion energy of Argon, they can only lose energy by e l a s t i c c o l l i s i o n s . In a d d i t i o n , positronium can be formed u n t i l the threshold energy for positronium formation, 8.9 eV, i s reached. Those positrons which terminate the i r rapid slowing-down at k i n e t i c energies below 11.6 eV, can be described by a p a r t i c u l a r i n i t i a l time-dependent v e l o c i t y - d i s t r i b u t i o n - -funct ion (Fa lk , Orth and Jones, 1965; F a l k , 1965). Once the positrons thermalize , the shape of the d i s t r i -bution funct ion becomes time independent ( e s s e n t i a l l y Maxwellian) and the depopulation by a n n i h i l a t i o n i s characterised by a s ingle exponential . A s ingle exponential w i l l also occur, regardless of the shape of the pos i tron d i s t r i b u t i o n funct ion , i f the a n n i h i l a t i o n rate i s v e l o c i t y independent (Chapter 3» Section 3 « ! ; f ° ) ° The observed shoulder i n the time spectrum from Argon, followed by a s ingle exponential , a l l superimposed on the orthopositronium decay, i s therefore a c l ear i n d i c a t i o n that here the d i r e c t a n n i h i l a t i o n rate of positrons i s v e l o c i t y dependent. That the d i r e c t a n n i h i l a t i o n rate changes on the a p p l i c a -t i o n of e l e c t r i c f i e l d further confirms th is point of view (Falk , Orth and Jones, 1965)° An appl ied e l e c t r i c f i e l d increases the average energy of the pos i trons , the equi l ibr ium d i s t r i b u t i o n being to f i r s t order a funct ion of the positron-Argon e l a s t i c -scat ter ing momentum-transfer cros s - sec t ion , and e l e c t r i c f i e l d . The observed a n n i h i l a t i o n rate i s thus changed since the v e l o c i t y -dependent a n n i h i l a t i o n rate i s averaged over a d i f f eren t equi l ibr ium v e l o c i t y d i s t r i b u t i o n . The experimental resul t s contained herein -23-are measurements of the a n n i h i l a t i o n rates for th is equi l ibr ium d i s t r i b u t i o n . Measurements of the r a t i o of three photon to two photon a n n i h i l a t i o n i n Argon have also been made by monitoring the "valley" and "peak" count rates of the 0.51 MeV gamma-ray spectra (Marder, et a l . , 1956). This r a t i o i s d i r e c t l y re la ted to the f r a c t i o n of pqsiftrons forming positronium i n Argon. 2.2. Experimental method. 2.2.1. Li fet ime measurements. A 10 ;uCi Na-22 source deposited on 30 / l i n c h aluminum f o i l served as the pos i tron source. The Instrumentation for the recording of the time spectra i s p r i m a r i l y that used previous ly by Falk (1965) (see also Falk and Jones, 196^ -5 F a l k , Orth and Jones, • 1965). To improve r e l i a b i l i t y , the l i m i t e r s and slow coincidence sections were replaced by t r a n s i s t o r i z e d equivalents . Deta i l s of these modif ications are given i n the Appendix. In order to improve the s t a t i s t i c a l accuracy of the measurements, the random coincidence background was reduced ( r e l a t i v e to the measurements of Falk) by a factor of ten by reducing the pos i tron source strength by the ' same f a c t o r . In order, however, to maintain the same o v e r a l l t r u e -coincidence count r a t e , detectors of s i g n i f i c a n t l y greater e f f i c i e n c y and s o l i d angle were required . To th i s end the gamma ray detectors used by F a l k , cons is t ing of 2 i n . x 2-g- i n . diameter Na l (T l ) c rys ta l s mounted on R . C . A . 6810 photomult ipl iers were replaced by h i n . x 3 i n . diameter Na l (T l ) c rys t a l s mounted on R . C . A . 70^6 -24-p h o t o m u l t i p l i e r s . I n a d d i t i o n , use o f a 256 c h a n n e l a n a l y z e r ( N D 1 0 1 ) made i t p o s s i b l e t o extend th e time range o f the measurements t o i n c l u d e t h e whole o r t h o p o s i t r o n i u m decay, thus f a c i l i t a t i n g t h e a n a l y s i s o f d a t a . The gas chamber used i s the same as t h a t used' p r e v i o u s l y i n t h i s l a b o r a t o r y ( F a l k , 1965; F a l k , O r t h and J o n e s , 1965). The gas was p u r i f i e d as b e f o r e by c o n t i n u o u s r e c i r c u l a t i o n over a h o t CaMg e u t e c t i c m i x t u r e ( C o l l i and F a c h i n i , 1952). A n a l y s i s o f the gas by The Matheson Co., I n c . N.J. i n d i c a t e s t h a t t h e main i m p u r i t y 4 p r e s e n t was N 2 a t about one p a r t i n 10 . The r e s u l t s o f the Matheson a n a l y s i s on b o t h the b o t t l e gas and chamber gas ( a f t e r p u r i f i c a t i o n ) a r e shown i n T a b l e I . TABLE I . Gas P u r i t y A n a l y s i s . I m p u r i t y gas B o t t l e gas Chamber gas N 2 190 ppm 119 ppm °2 5 ppm 3 ppm co2 < 4 ppm < 4 ppm H 2 23 ppm 32 ppm He 1 53 ppm <100 ppm The e f f e c t i v e n e s s o f the p u r i f i e r i n a t l e a s t m a i n t a i n i n g t h e p u r i t y l e v e l o f the Argon i s q u i t e a p p a r e n t . A b l o c k diagram of the e l e c t r o n i c i n s t r u m e n t a t i o n used i s shown i n F i g u r e 3. A t y p i c a l r u n extended over a p e r i o d o f two 1.28 MeV CHANNEL (PROMPT) 0 .5 \ MeV X CHANNEL (DELAYED) DYNODE PULSE P I L E - U P RETECTOR NO. I S I N G L E CHANNEL ANALYZER MO. I DyNODE PULSE TIME SORTER TS OUTPUT INVERTER-DRIVER NEGATIVE TIME ELIMINATOR COlNCIDENCt Ik TS S I G N A L > CATG S I M M . 256 CHANNEL KICKSORTER GATE PULSE A N T I -COINCIDENCE CIRCUIT PILE-UP REJECTOR NO. 2 GATE PULSE GATE PULSE GENERATOR SLOW COINCIDENCE CIRCUIT SINGLE C H A N N E L ANALYZER Ho. 2 Figure 3. Block diagram of electronics used i n l i f e t i m e measure-ments . -25-days i n w h i c h time some 2 x 10 1.28 MeV n u c l e a r gamma r a y s were counted i n the photo peak. The number o f th e s e gamma r a y s counted was used as n o r m a l i z a t i o n from r u n t o r u n i n o r d e r t o a v o i d d i f f i c u l t i e s caused by t h e sourc e decay and t h e s i n g l e c h a n n e l a n a l y z e r d r i f t s . Each s i n g l e c h a n n e l a n a l y z e r s e t t i n g was o b t a i n e d by u s i n g t h e two independent o u t p u t s o f the a p p r o p r i a t e l i n e a r a m p l i f i e r . The s i n g l e c h a n n e l a n a l y z e r a s s o c i a t e d w i t h the a n n i -h i l a t i o n gamma r a y s was u s u a l l y s e t a t the photopeak o f t h e 0.51 MeV gamma r a y . However, many runs were t a k e n w i t h the same c h a n n e l w i d t h but w i t h the b a s e l i n e s e t so as t o i n c l u d e o n l y the r e g i o n between the photopeak and the Compton edge, the s o - c a l l e d v a l l e y r e g i o n . A t t h i s s e t t i n g t h e r a t i o o f t h r e e photon t o two photon e v e n t s was enhanced r e l a t i v e t o t h e u s u a l s e t t i n g , s i n c e the t h r e e photon count r a t e i n the v a l l e y r e g i o n i s comparable t o t h a t i n the 0.51 MeV peak r e g i o n (Ore and P o w e l l , 1949)', w h i l e the two photon count r a t e i s s h a r p l y d i m i n i s h e d . T h i s s e t t i n g a l l o w e d much g r e a t e r a c c u r a c y f o r the o r t h o p o s i t r o n i u m l i f e t i m e . The energy spectrum o f the 0.51 MeV gamma r a d i a t i o n t o g e t h e r w i t h the two s e t t i n g s o f t h e s i n g l e c h a n n e l a n a l y z e r s a r e shown i n F i g u r e h> The o v e r a l l time r e s o l u t i o n f o r b o t h S.C.A. s e t t i n g s was 2.7 c h a n n e l s , or 7.k nsecs ( f u l l w i d t h a t h a l f maximum), measured u s i n g a Na-22 s o u r c e i n aluminum (see A p p e n d i x ) . A r e p r e s e n t a t i v e time spectrum i s shown i n F i g u r e 5« The peak i s due t o a n n i h i l a t i o n o f p o s i t r o n s i n t h e w a l l s o f t h e chamber and i n the so u r c e h o l d e r , o f p a r a p o s i t r o n i u m formed d u r i n g t h e i n i t i a l slowing-down p e r i o d , and o f p o s i t r o n s a n n i h i l a t i n g i n 8.0 7.0 6.0 t 5.0 o i 4 . 0 LU $ 3 . 0 ( J 01 LU C L 2.0 to I— 0.0 1 B to • ••• 9.0 keV/channel t s v t ^ t c - t r 40 1 JO 60 \ \ V J] *••«... 70 20 30  1CHANNEL NUMBER Figure h. Energy spectrum of 0.51 MeV gamma rays showing S.G.A. sett i n g s . V:-.* v a l l e y p o s i t i o n ; P - peak p o s i t i o n . A-; 2 ~ direct-enhanced se t t i n g ; 2 ~ ortho-enhanced se t t i n g ; A 2B - valley-to-peak r a t i o s e t t i n g . 80 -26-the gas wi th in about 7 nsecs of emission from the source. The end of the shoulder i s marked by a r e l a t i v e l y s h o r t - l i v e d exponential component, the d i r e c t component. The orthopositronium annih i la t ions give r i s e to the l o n g - l i v e d exponential . Omitted from the f igure i s the f l a t random background region which occupies that port ion of the k icksor ter up to the prompt peak. The spectrum i n th is region was obtained by recording events where a 0.51 MeV gamma-ray i s detected before a 1.28 MeV gamma ray . The counts per channel i n th i s part of the time spectrum, then, correspond to the random background counts per channel. The spectrum i n Figure 5 i s shown with th i s random coincidence background subtracted. 2.2.2. Va l l ey - to -peak r a t i o measurements. The energy spectrum of the a n n i h i l a t i o n gamma rays as a funct ion of the e l e c t r i c f i e l d was obtained using one k i n . x 3 i n . Nal c r y s t a l assembly and the 256 channel k i c k s o r t e r . A s ingle channel analyzer was used to gate the k i c k s o r t e r , and was set so that only pulses between the Compton edge for 0.51 MeV gamma rays and the high energy side of the P.51 MeV photopeak would be analyzed by the k i cksor ter (Figure >+). Each run lasted approximately h-5 minutes i n which some 2 x 1 0 and 8 x 1 0 counts per channel were obtained i n the v a l l e y and peak pos i t ions re spec t ive ly . 8 o i I I I I I "I 1 1 I 60,000 85.000 110.000 135.000 L60.000 165.000 210.000 23S.00O CHANNEL NUMBER F i g u r e 5» R e p r e s e n t a t i v e time spectrum o f p o s i t r o n a n n i h i l a t i o n i n Argon. P i s i n amagats,:E/P. i s - i n - V cm"1 a m a g a t - - 1 I o - ' - i p r b m p t peak; I I - "shoulder " r e g i o n ; I I I - d i r e c t : component"; I V ; - • o r t h o p o s i t r o n i u m component.  - 2 7 -2 . 3 - The exponential portions of the time spectra . The appearance of a s ingle exponential superimposed on the orthopositronium component i n the time spectra i s evidence that the v e l o c i t y d i s t r i b u t i o n of the positrons has become "static'" (see Chapter 1 , Sect ion 1 . 3 . and Chapter 3 , Sect ion 3 . 1 1 " . ) . That i s , the time dependence of the v e l o c i t y d i s t r i b u t i o n i s described by a s ing le exponential . The time spectrum under these conditions i s composed of the fol lowing components: 2 . 3 . 1 - The d i r e c t or free a n n i h i l a t i o n s . The population of free positrons at any time t i s given by = -C x f + x d ] N(t) ( l ) where Xf i s the ve loc i ty-averaged positronium formation r a t e , and X d i s the ve loc i ty-averaged d i r e c t a n n i h i l a t i o n r a t e . Assuming that the d i r e c t ann ih i la t ions are a l l by two photon decay whose e f f i c i e n c y for detect ion i s e2 • the observed d i r e c t a n n i h i l a t i o n rate i s : R d(t) = e 2X dN(0)e" ( Xf + X d } t sec"1 (2) 2 . 3 . 2 . Orthopositronium a n n i h i l a t i o n s . The orthopositronium population i s given by ^ M i i = -XoNo(t) + |* fN(t) (3) where xQ i s the sum of the free orthopositronium decay rate x 0 and the orthopositronium quenching rate Xq . It i s assumed that the f r a c t i o n of positronium atoms which are formed as orthopos i -tronium i s the s t a t i s t i c a l r a t i o 3: 1+« The population at any time -28-t i s then where N o ( 0 ) i s the number of orthopositronium atoms present at t=0. The observed orthopositronium decay r a t e , corresponding to X~0 N 0 ( t ) i s : RQ(t) = A 0e 3N 0(t) .(5)' where £3 i s the three photon detect ion e f f i c i e n c y . I t i s apparent that th i s contains two terms, one, R 0 ^ ( t ) , representing the decay of the orthopositronium atoms present at time t=0, the other, R Q F ( t ) , representing a n n i h i l a t i o n from the "delayed"1 positronium formation. Thus_ Rj(t) = A^NoCCDe"^ ( 6 )  Ro^> = *o*£• T ^ p ^ NCO) [ e"rot _ e - ( M ^ f )t j ( 7 ) I t should be noted that i f the positronium formation rate X f becomes appreciable under the inf luence of the e l e c t r i c f i e l d , and i f x d + x f »TQ , the observed orthopositronium spectrum w i l l exh ib i t a c h a r a c t e r i s t i c exponential growth, followed by a slow exponential decay (of l i f e t i m e l / T ) . 2 . 3 -3 • Parapositronium a n n i h i l a t i o n s . Since parapositronium has a mean l i f e (^  1 0~ 1 ^sec) considerably shorter than the experimental time re so lu t ion ava i lab l e ( 7 A nsec) i t s contr ibut ion to the spectrum i s governed e n t i r e l y by the parapositronium population present at any time t . Thus the decay rate i s given by Rp(t) =• e2 [ XqNG(t) + £f. N(t) ] (8) -29-The e f fect of orthopositronium quenching i s taken into account here noting that th i s i s equivalent to parapositronium formation at a rate given by the quenching rate A q . The separate formation of parapositronium from free positrons i s a lso taken into account. This component has the same detect ion e f f i c i e n c y as the d i r e c t ann ih i la t ions since the a n n i h i l a t i o n i s by two photons only (Yang, 1950). 2.3A. The observed spectrum i n the exponential reg ion . The t o t a l observed spectrum i s the sum of these three components, v i z . , R(t) = I xe " ( X d n f } t + (9) where and i 2 - [.,*„ • V q ] c Ne<o) • i x f T ^ s - ] m , I t i s c l ear that although there may be a growth i n the ortho-positronium component the observed spectrum i s always a simple sum of the two exponentials unless I-j happens to be a negative. Such a s i t u a t i o n has not yet been observed. 2.4. The va l l ey - to -peak r a t i o . The energy spectrum of three-photon a n n i h i l a t i o n i s continuous, with a maximum at 0.51 MeV (Ore and Powell , 1949). In the a n n i h i l a t i o n spectrum of positrons i n Argon, therefore , the r a t i o of counts i n the v a l l e y region (between the 0.51 MeV peak and Compton edge) to the counts i n the 0.51 MeV peak increases as -30-a funct ion of positronium formation. Consider a 0.51 MeV gamma-ray spectrum obtained with no three-photon events. The va l l ey - to -peak r a t i o , R , w i l l be less than that i n a,spectrum where three-photon events were also counted. I f , i n a spectrum of the l a t t e r type, the peak and v a l l e y count rates are Cp, C , r e s p e c t i v e l y , then C^, where C 3 = C v - ( C p - R3C3) R Q (12) w i l l be the count rate due to the three-photon events only , i n the v a l l e y reg ion . The c o e f f i c i e n t of R Q i s the count rate Cp i n the 0.51 MeV photopeak with the contr ibut ion due to three-photon events i n th i s reg ion , R3C3 , subtracted. In th i s case R^ represents the r e l a t i v e p r o b a b i l i t y of counting a gamma ray from a three-photon a n n i h i l a t i o n i n the 0.51 MeV peak to counting such a gamma ray at the v a l l e y p o s i t i o n . From a knowledge of the energy spectrum of three-photon ann ih i la t ions (Ore and Powell , 194-9), and the r e l a t i v e e f f i c i e n c i e s of the c r y s t a l for counting gamma rays whose energies correspond to the peak and v a l l e y regions re spec t ive ly , R^ can be determined. Equation (12) can be more u s e f u l l y rewri t ten i n the form: C3 = ( C v - C p R Q ) / (1 - R 3 R 0 ) . (13) In the case of positrons a n n i h i l a t i n g i n Argon under the inf luence of an appl ied e l e c t r i c f i e l d , where i t i s poss ib le to change only the r e l a t i v e number of three-photon orthopositronium and two-photon decays, C^ w i l l be d i r e c t l y proport ional to the f r a c t i o n of positrons forming orthopositronium. Thus f = k C 3 ( 1 4 ) where f i s the f r a c t i o n of positrons forming positronium and k i s -31-a p r o p o r t i o n a l i t y constant which i s independent of e l e c t r i c f i e l d , i f the f r a c t i o n of orthopositronium atoms which are quenched i s independent of e l e c t r i c f i e l d . In pract i ce the peak count rate has to be reduced by an a d d i t i o n a l fac tor of 1-W, where W represents the f r a c t i o n of positrons which ann ih i la t e i n the walls of the gas chamber. Thus f i n a l l y f = k [C v - Cp ( 1 - W) R Q ] / [ 1 - R 3 R Q ] ( 1 5 ) The p r o p o r t i o n a l i t y constant k i s determined i n Sect ion 2.6.3. from a knowledge of the f r a c t i o n of positrons forming positronium at zero f i e l d (Falk and Jones, 196*+), the f r a c t i o n (1-W) of positrons stopped i n the gas (Fa lk , 1965), and from the peak (C p ) and v a l l e y ( C y ) count rates at zero appl ied e l e c t r i c f i e l d . The expression (15) i s s i m i l a r to that deduced by Marder, et a l . , (1956), but i s simpler i n that no magnetic f i e l d quenching i s present i n th i s case. 2.5. Analys is of r e s u l t s . 2.5-1• Analys is of time spectra. 2.5-1-1- Maximum l i k e l i h o o d theory. In order to f i t the experimental data to a sum of two exponentials (as expected according to the d i scuss ion of Sect ion 2.3-), a computer programme was devised u t i l i z i n g maximum-likel i-hood theory (Orear, 1958). For f i n i t e channel widths, w^, and a constant random background B per un i t channel, the t h e o r e t i c a l spectrum shape i s -32-W 0 + W K yK = / C Ii'• expC-t/x ) + I 2 e x p(-t/T 2)] dt + wKB (16) wo where w i s the sum of the channel widths up to the s tar t of channel k. Since the counts i n i n d i v i d u a l channels are Poisson d i s t r i b u t e d , the p r o b a b i l i t y of observing counts i n channel k i s p k = tyy^expt-y*) I N k ! <17> while for the whole exponential region the j o i n t p r o b a b i l i t y i s the l i k e l i h o o d funct ion L =.j!Pk (18) The aim i s to determine the four paramenters, 1-j , l ^ i T1 • T2> ^ o r which th i s p r o b a b i l i t y L i s a maximum. It i s more convenient to deal with the logarithmic p r o b a b i l i t y W = I lnP k = I Nklny, - y, - ln(NR!) (18a) k k The programme which maximizes th is p r o b a b i l i t y follows the same l ine s as that of Fa lk (1965). The values of the paramenters for which W i s a maximum i s given by the set of four equations 9 W - n 8 W n 3 W n ™ n ' A O N 3TJ " °' 31, = °' 9^ = Q ; JT2 = °' ( 1 9 ) 2.5.1.2. I t e r a t i v e so lu t ion of the maximum l i k e l i h o o d problem. Since the equations (19).are extremely nonl inear , the method followed was an i t e r a t i v e procedure invo lv ing some i n i t i a l estimate of the parameters and a f i r s t order T a y l o r ' s ser ies expansion of each of the p a r t i a l der ivat ives (19) about these i n i t i a l estimates. I f the four parameters are denoted a-| , &21 a^, a^ _, r e p e c t i v e l y , then the four simultaneous equations above reduce to a set of four simultaneous l i n e a r equations represented -33-CA=V. (20) The h x h matrix C contains the elements (21) The vectors A and V are respec t ive ly (22) (23) Solut ion of the four simultaneous equations y i e lds the increments A a i which are to be added to the i n i t i a l estimates a^. The new a-± so obtained are then inserted i n the set cf equations (20) and the process repeated. Convergence for the case of two exponentials general ly requires less than s ix i t e r a t i o n s , once the i n i t i a l estimates are s u f f i c i e n t l y good that none of the parameters become negative. Should the l a t t e r occur i t was found convenient to f i x one of the parameters, usua l ly the long l i f e t i m e . In th i s way, the problem i s reduced to a problem invo lv ing so lu t ion of three equations i n three unknowns. Once convergence under th i s three parameter v a r i a t i o n was achieved, the four parameter i t e r a t i o n s , using the best values obtained from the three parameter so lu t ion as the i n i t i a l estimate, usua l ly converged. 2 . 5 - 1 - 3 - I n i t i a l estimates of the four parameters. The i n i t i a l estimates were made using a very fas t i t e r a t i v e procedure rased on the least squares method. The spectrum i s i. div ided in to three regions . The f i r s t region i s the range over w h i c h t h e s h o r t l i f e t i m e predominates and t h e second t h e range over w h i c h t h e l o n g l i f e t i m e i s most i m p o r t a n t . Between t h e s e i s t h e t h i r d r e g i o n , where c o n t r i b u t i o n s f r o m each o f the components a r e comparable. The i t e r a t i v e p r o c e d u r e i n v o l v e s making a s t r a i g h t l i n e f i t t o t h e l o g a r i t h m o f the c h a n n e l counts i n t h e second r e g i o n . T h i s f i t i s e x t r a p o l a t e d i n t o t h e f i r s t r e g i o n , and i t s c o n t r i b u t i o n t o the f i r s t r e g i o n s u b t r a c t e d o u t . The l o g a r i t h m o f t h e r e s u l t i n g c u r v e i n t h e f i r s t r e g i o n i s t h e n f i t t e d t o a s t r a i g h t l i n e . T h i s l i n e i s e x t r a p o l a t e d i n t o t h e second r e g i o n , and i t s c o n t r i b u t i o n s u b t r a c t e d from t h e second r e g i o n . The l o g a r i t h m o f the remainder i n t h e second r e g i o n i s once more f i t t e d t o a s t r a i g h t l i n e and the whole p r o c e d u r e r e p e a t e d . 2.5-1•^• E s t i m a t i o n o f c h a n n e l w i d t h s w^ and random background B. The r e l a t i v e c h a n n e l w i d t h s , w^, have been measured u s i n g t h e random time g e n e r a t o r (see S e c t i o n 2.6.6.2.).The r e s u l t s o f t h e measurement a r e shown i n t h e Appendix. E s t i m a t i o n o f t h e random background c o u n t s , B p e r u n i t c h a n n e l , f o l l o w s the p r o c e d u r e used by F a l k (1965) and i n v o l v e s the measurement o f t h e random c o i n c i d e n c e r a t e d i r e c t l y from the l i f e t i m e s p e c t r a ( S e c t i o n 2.2.1.) 2.5-1 • 5'• E s t i m a t i o n o f v a r i a n c e s . *K J | C j|e s(c Once the most p r o b a b l e s e t o f v a l u e s : a-j, a 2 , a^, a^, have been o b t a i n e d , an e s t i m a t e o f t h e s t a t i s t i c a l u n c e r t a i n t y i n each o f t h e s e v a l u e s i s d e s i r e d . I t has been shown ( O r e a r , 1958) t h a t a m a t r i x o f t h e type • 3 5 -when inver ted , y i e lds the so -ca l l ed error matrix (H _ 1 = ""(ai - a i ") (aj- aj") (25) where i s the most probable value of the parameter a^. I f i t i s assumed that the l i k e l i h o o d funct ion L i s Gaussian with respect to each of the parameters a^, and that the parameters are independent of each other, then L - r[ exp C- i ( a i*- a X) 2 ] (26) i 2 ° i where o| i s the variance of the Gaussian. I t follows that W = I - i ( a i a l ) + constant (27) i 2 0 i and 32W whence - 1/ai 2 Thus the diagonal elements of H are T T _ 3 ^ " i i " " Sip (29) In th i s case, since H^^ i s a diagonal matrix , (H- l )^ = (Hii)-l S O o±z = ( H i i ) - i y i e l d the variance of the parameters i f L i s Gaussian i n shape. I f H i s not d iagonal , invers ion of the ent i re matrix H -36-allows the c o r r e l a t i o n between d i f f eren t parameters to be taken in to account i n the estimation of the variances (Orear, 1958). Should any of the o f f -d iagonal elements be n e g l i g i b l e compared with the ma'in diagonal elements, then the relevant parameters have an appropr ia te ly small degree of c o r r e l a t i o n . In order to check the assumption that the l i k e l i h o o d funct ion descr ibing t y p i c a l experimental resu l t s i s near to Gaussian i n shape, the fol lowing analys is was performed. Each of the parameters i n turn was set at one and then two standard deviations on e i ther side of the best value obtained using the maximum l i k e l i h o o d programme. The remaining parameters were then var ied to maximize the l i k e l i h o o d once more. The r e s u l t i n g l i k e l i h o o d r e l a t i v e to the best maximum l i k e l i h o o d was then p lot ted as a funct ion of the parameter concerned. Figure 6 indicates that the shape of the l i k e l i h o o d funct ion i s indeed approximately Gaussian for each of the four parameters. 2.5.1.6. Goodness of f i t . In order to discuss the goodness - or otherwise - of the f i t , the usual chi-square test was made on each spectrum. However, s ince the chi-square test i s defined i n terms of the normalized p r o b a b i l i t y of gett ing a worse f i t for quant i t ies d i s t r i b u t e d according to Gaussian s t a t i s t i c s (Orear, 1958; Mathews and Walker, 1965) i t i s not s u f f i c i e n t to make a s tra ight channel"-by-channel computation i n the present case. This ar ises because of the low number of counts, usua l ly t h i r t y to f o r t y , i n the t a i l of the time — Fix^p VALUES OF PARAMETER a F i g u r e 6. Dependence o f the l i k e l i h o o d f u n c t i o n on l l , I 2» T 1 , xp« The v a l u e s o f t h e s e p a r a -meters w h i c h g i v e t h e maximum l i k e l i h o o d L* c o r r e s p o n d t o the p o i n t a*; o-^ * i s t h e v a r i a n c e o f a* as c a l c u l a t e d f r om E q u a t i o n 25. V a l u e s o f t h e G a u s s i a n c u r v e exp[-(£=£ )*3 a r e d e n o t e d - G a u s s i a n i n t h e l e g e n d . °*' -37-spectrum (Figure . . Although the variance of these Poisson-d i s t r i b u t e d counts N i s N, the best f i t i s not obtained by minimizing the mean square dev iat ion about the mean since the Poisson d i s t r i b u -t i o n i s skew about the mean for counts below about 1 0 0 . In f a c t , counts lower than the mean are more probable. However, i f the counts are summed over a cer ta in number of channels, such that the t o t a l number of counts i s above about 1 0 0 , then the chi-square test becomes a p p l i c a b l e . Table II shows the resu l t s of chi-square tests on a l l the spectra presented. In order to re in force the point concerning the n o n - a p p l i c a b i l i t y :bf the chi-square test for non-Gaussian d i s t r i b u t e d counts, Table II further contains resu l t s for the channel-by-channel chi-square t e s t . 2 . 5 - 2 . Analys is of experimentally-determined a n n i h i l a t i o n ra tes . In general the a n n i h i l a t i o n rates of positrons i n a gas are dependent on the gas dens i ty . In order to f i t the observed a n n i h i l a t i o n ra te s , d i r e c t or orthopositronium decays, to a p a r t i c u l a r polynomial dependence on dens i ty , the l eas t squares method was employed. The j u s t i f i c a t i o n for using the least squares method l i e s s o l e l y i n the fac t that the l i k e l i h o o d functions for the t y p i c a l time spectra reported here have been demonstrated to be near ly Gaussian i n shape (Sect ion 2 . 5 * 1 * 5 • ) • That they are so shows that the p r o b a b i l i t y d i s t r i b u t i o n governing the parameters I 1 , I , T 1 , T ^ , i s near ly Gaussian. Thus, i n order to analyze the l i f e t i m e s , i t i s consistent to use the maximum l i k e l i h o o d method assuming Gaussian s t a t i s t i c s , which gives r i s e to the least squares method of curve f i t t i n g (Orear, 1 9 5 8 ) . Furthermore, the standard TABLE I I . RESULTS OF CHI-SQUARE TEST ON LIFETIME SPECTRA. Q i s to be interpeted as the normalized (to 1) p r o b a b i l i t y of get t ing a worse f i t should the experiment be repeated. Q B i s the r e s u l t of the channel-by-channel computation, while Q A r e su l t s from the a l t e r n a t i v e i n t e g r a l method out l ined i n Sect ion 2 . 5 . 1 - 6 . -f iA-0.30 0.37 0.50 0.15 0.32 0.005 0.91 0.59 0 . 9 ^ 0.1*+ 0.70 0.00k 0.h6 0.12 0.91 0.0003 0.90 0.12 0.6k 0 . 3 ^ 0.k6 0.002 0.92 0 A 5 TABLE II (continued) RESULTS OF THE CHI-SQUARE TEST ON LIFETIME SPECTRA. - V - 2 B 0.75 0.39 0.22 0.003 0.77 0.06 0.58 0.06 0.24 0.11 0.79 0.000007 0.76 0.006 . 0.42 0.004 0.63 0.18 0.01 < 1 0 - 8 0.93 0.03 0.25 0.02 0.56 0.45 0.97 0.73 0.34 0.004 - 3 8 -chi-square test i s relevant i n th is case (Section 2.5.1.6.). The polynomial parameters were obtained using a computer s o l u t i o n for the set of ana ly t i c equations pert inent to the least squares problem for a funct ion l i n e a r i n these parameters (Orear, 1958; Rose, 1953). The funct ion which is l i n e a r i n the parameters a^ (to be determined) i s M y(x) = I a- fi(x) (30) i=l where the f-^(x) are any functions of x only. In a s i t u a t i o n where p experimental values N (XJ ) ± aj have been obtained as a funct ion of the p data points X j , the least squares so lut ions for the aj_ are M D ' a.* = H ^ j l ^ i j l (ITMW (31) k,j=l 0 J where = I a^k>4ji2kl (32) J k=l G k The error matrix i s given as usual by ( H - 1 ) ^ = (ai-a^) (aj-aj f t) (33) The chi-square test was also performed on each f i t . 2.6. Experimental r e s u l t s . 2.6.1. C r i t e r i a for presentat ion of data. The data presented here represent the resu l t s of runs made with four d i f f e r e n t Argon gas samples. A l l the spectra were f i t t e d to two exponentials by the maximum-likelihood technique (Sect ion 2.5-1•)• The resu l t s presented here s a t i s f i e d the fo l low-ing condit ions: -39-( a ) Convergence was o b t a i n e d v a r y i n g a i l f o u r parameters simultaneous-l y i n t h e m a x i m u m - l i k e l i h o o d programme. (b) No h i g h v o l t a g e breakdowns o c c u r r e d d u r i n g the a c t u a l r u n i n q u e s t i o n . I t was found t h a t t h e o c c u r r e n c e of such breakdowns c o u l d l e a d t o . s i g n i f i c a n t d e v i a t i o n s i n the r e s u l t s . These d e v i a t i o n s were a s c r i b e d t o the e f f e c t , o f c o n t a m i n a t i n g gases l i b e r a t e d by the breakdown p r i o r t o t h e a b s o r p t i o n o f s u c h c o n t a m i n a n t s by the p u r i f i e r . ( c ) The r e s u l t s o f a l l the c h i - s q u a r e t e s t s i n d i c a t e d t h a t the p r o b a b i l i t y o f g e t t i n g a worse f i t was g r e a t e r t h a n 0.1 (10$). Out o f a t o t a l o f 33 r u n s , 6 we r e r e j e c t e d on the b a s i s o f th e s e c r i t e r i a . None was r e j e c t e d on the b a s i s o f the c h i-square t e s t a l o n e , s i n c e a poor r e s u l t here s i g n i f i e d t h a t the a s s u m p t i o n o f two e x p o n e n t i a l s was poor f o r the c a s e c o n s i d e r e d . T h i s f a u l t c o u l d n o r m a l l y be remedied by a l t e r i n g the c h a n n e l number o f the t i m e s p e c t r a a t w h i c h a n a l y s i s was s t a r t e d . The c h i-square t e s t i n t h i s c a s e , thus s e r v e s as a u s e f u l t e s t f o r d e t e r m i n i n g a good s t a r t i n g p o i n t f o r t h e a n a l y s e s . 2 . 6 . 2 . D i r e c t a n n i h i l a t i o n r a t e : z e r o e l e c t r i c f i e l d r e s u l t s . 2 . 6 . 2 . 1 . R e s u l t s of - f i t t i n g t h e a n n i h i l a t i o n r a t e t o f u n c t i o n s o f the A r g o n d e n s i t y . F i g u r e 7 shows the dependence o f the d i r e c t a n n i h i l a t i o n r a t e a t z e r o e l e c t r i c f i e l d on Argon d e n s i t y . T a b l e I I I shows th e r e s u l t s o f f i t t i n g the r e s u l t s shown i n F i g u r e 7 t o v a r i o u s s i m p l e f u n c t i o n s o f the d e n s i t y . The r e s u l t s , Q, o f a c h i - s q u a r e 4 6 8 10 12 14 P - DENSITY (AMAGATS) Figure 7. Direct a n n i h i l a t i o n rate i n Argon at zero e l e c t r i c f i e l d as a function of density. The s t r a i g h t l i n e represents V » 5 . 6 P x 10°sec-1. -ko-c a l c u l a t i o n are also given. The value of Q i s to be interpreted as being the p r o b a b i l i t y of gett ing a worse f i t should the series of experiments be repeated. Unless otherwise s tated, the f i t s were made to a l l the points shown i n Figure 7-On the basis of the chi-square test alone i t would appear that form VI i s the best f i t to the experimental data. However, there i s no reason to suppose that the observed d i r e c t a n n i h i l a t i o n rate i s d i f f e r e n t from zero at zero Argon densi ty , unless a model invo lv ing an Argon-positron complex i s invoked. According to f i t VI such a system would have to have a h a l f - l i f e _Q of about *t2 x 1 0 sees, which i s considerably longer than that - 9 which would be expected ( 10 sec) , because of the higher e lectron densi ty at the pos i tron p o s i t i o n compared with the parapositronium atom (see Chapter 1 , Sect ion 1 A . 2 . ) . On the basis of th i s argu-ment f i t s V and VI can be re jec ted . Further examination of Table III shows that below 1 0 amagats a l i n e a r f i t IV to the data i s adequate. Inc lus ion of the data obtained at higher dens i t ies resu l t s i n a s i g n i f i c a n t l y worse l i n e a r f i t I . Some improvement i s obtained for f i t s containing a non- l inear dependence on density as we l l as the l i n e a r term ( f i t s II and I I I ) . I t i s apparent that a value for a 1 between 5-5 and 5-7 x 10 sec amagat describes we l l the l i n e a r dependence on Argon density of the d i r e c t a n n i h i l a t i o n ra te . Such a value for a 1 i s i n keeping with the observation that the l i n e a r term i n f i t s I to IV i s somewhat independent of the de ta i l ed nature of the f i t . Regarding the n o n - l i n e a r i t y at higher d e n s i t i e s , the tendency i s for the a n n i h i l a t i o n rate to be reduced r e l a t i v e to TABLE I I I . DEPENDENCE OF A a ON P. Summary of the r e s u l t s o f f i t t i n g the curve i n F i g u r e 7. Tvne of f i t R e s u l t s Q a 0 / 1 0 6 - af/106 a 2/10 6 a3/l0°" 1 0 6 s e c - 1 s e c - ^ -1 .-1 sec amagat -1 . -2 sec amagat s e c- ^ amagat I. A a = a 1 P 5.42 i 0.05 0.02 I I . X a = & 1 P + a 2 P 2 5.71 i 0.12 -0.027 3 1 0.011 0.08 I I I . X a = a 1 p * a 3 p 3 5.59 - 0.08 0.0013 - 0.0004 0.13 IV. X a = a 1 P P < 1 0 amagats 5.53 - 0.06 0.16 V. X a = a o + a 1 P 1 1 .44± 0.94 5.28 1 0.11 0.02 VI. X a = a o + a 2 p 2 23 .85 i 0.52 0.250 - 0.006 0.21 Note: Unless otherwise s t a t e d , the f i t s were made to a l l the p o i n t s i n F i g u r e 7-the l i n e a r f i t acceptable at lower d e n s i t i e s . The de ta i l ed nature of th i s n o n - l i n e a r i t y remains undetermined by these experiments, as examination of Q for f i t s II and III shows. 2 . 6 . 2 . 2 . Discussion of the f i t s to the data. The estimated p u r i t y of the Argon gas has been given i n Table I , Sect ion 2 . 2 . 1 . The small dev iat ion from l i n e a r i t y of the a n n i h i l a t i o n rate as a funct ion of density might be due i n some way to the Nitrogen present. However, as t h e ' d i r e c t a n n i h i l a t i o n rate i n Nitrogen i s expected to the f i r s t order to vary l i n e a r l y with dens i ty , i t i s d i f f i c u l t to see how the presence of Nitrogen or any other impurity gas could af fec t the l i n e a r i t y of the d i r e c t a n n i h i l a t i o n ra te . I t i s far more reasonable to suppose that the n o n - l i n e a r -i t y at high dens i t i es ar i ses from the i n t e r a c t i o n of the pos i tron with more than one Argon atom at a time. The p o s s i b i l i t y of such an e f fect has been ra i sed by Tao, B e l l and Green ( 1 9 6 l + ) ( s e e a lso K i v e l , 1 9 5 9 ) . At 1 5 amagats, the average interatomic distance - 7 i s about 1 0 cm, which i s of the same order as the de B r o g l i e wavelength of a thermalized pos i t ron . I t i s a lso poss ib le that interatomic distances of th i s magnitude could r e s u l t i n some screening of the f i e l d of the pos i tron at the scat ter ing atom, and thus reduce the magnitude of the very important a t t r a c t i v e p o l a r i z a t i o n p o t e n t i a l . C l e a r l y , th i s problem can only be resolved by further experiments at high pressures, and by ca lcu la t ions that take in to account the presence of more than one scat ter ing atom. - h 2 -From the resu l t s given here i t seems that the dev ia t ion from l i n e a r i t y of the dependence of the a n n i h i l a t i o n rate on density i s less than 10$ at about 17 amagats. 2 . 6 . 2 . 3 . Comparison of the l i n e a r term with previous r e s u l t s . The magnitude of the d i r e c t a n n i h i l a t i o n rate per un i t amagat i s compared i n Table IV with the values obtained by other workers. The value r e s u l t i n g from the present work was obtained from the f i t s to the data as indicated i n Sect ion . 2 . 6 . 2 . 1 . The error associated with the value r e f l e c t s the s t a t i s t i c a l errors and a lso the uncertainty i n the exact non- l inear behaviour of the a n n i h i l a t i o n ra te . The systematic errors (Sec; 2 . 6 . 6 . 3 } associated with th i s measurement are of the order of \%. This estimate i s not included i n the value given i n Table IV. TABLE IV. Published values of the D irec t A n n i h i l a t i o n Rate. D irec t a n n i h i l a t i o n , rate Author 1 0 sec" 1 amagat"1  5 - 9 0 * 0 . 2 3 Falk (1965) ^ . 9 6 Tao, B e l l and Green (196*0' 5.18 Duff and Heymann (1962) 3.0M- Osmon (1965) 5.78 Paul (196*0 5.6*0.1 Present work The values presented for other workers d i f f e r from the published 298 v a l u e s where n e c e s s a r y by the f a c t o r 273. T h i s i s t h e f a c t o r by w h i c h r e s u l t s o b t a i n e d a t room tem p e r a t u r e (25°C) have t o be -1 -1 m u l t i p l i e d i n o r d e r t o e x p r e s s them i n u n i t s o f sec amagat The l a c k o f agreement between a l l the r e s u l t s p u b l i s h e d t h u s f a r a r i s e s m a i n l y from the presence o f i m p u r i t i e s i n the gas, and i n a d e q u a t e a n a l y s i s o f r e s u l t s . F o r example, the r e s u l t g i v e n by Osmon (1965) i s s i g n i f i c a n t l y s m a l l e r t h a n t h e o t h e r v a l u e s p r o b a b l y because t h e l o n g - l i v e d o r t h o p o s i t r o n i u m component was c o m p l e t e l y i g n o r e d i n t h e l i f e t i m e a n a l y s i s . I f such a l o n g - l i v e d component i s n o t t a k e n i n t o a c c o u n t , i t can happen t h a t i t s c o n t r i b u t i o n w i l l r a i s e t he t a i l o f t h e s h o r t - l i v e d component r e l a t i v e t o t h e r e s t o f t h e spectrum. Subsequent a n a l y s i s as a s i n g l e e x p o n e n t i a l t h e n r e s u l t s i n an i n c r e a s e d l i f e t i m e f o r t h i s s h o r t - l i v e d component, and hence i n a reduced a n n i h i l a t i o n r a t e . The Argon used by Tao, B e l l and Green (196*+) has s u b s e q u e n t l y been shown (Tao and B e l l , 1966) t o c o n t a i n s i g n i f i c a n t i m p u r i t i e s . The agreement between d i f f e r e n t workers w i l l o n l y improve when r e p r o d u c i b l e r e s u l t s a r e o b t a i n e d w i t h gases c o n t a i n i n g l e s s t h a n a t o t a l o f 1 ppm o f i m p u r i t y . I n t h i s sense the r e s u l t s quoted i n T a b l e IV s h o u l d be assumed t o be t h o s e f o r Impure Argon. The a n n i h i l a t i o n r a t e o f (5.6*0.1) x 1 0 6sec" 1amagat" 1 p r e s e n t e d h e r e i s t h a t a p p r o p r i a t e t o Argon c o n t a i n i n g i m p u r i t i e s t o t h e e x t e n t g i v e n i n T a b l e I . S i n c e t h e main i m p u r i t y i s N i t r o g e n , and s i n c e t he d i r e c t a n n i h i l a t i o n r a t e i n N i t r o g e n i s about 5.5 x 10 sec"'amagat" ( F a l k , 1965), i t can h o p e f u l l y be assumed t h a t t h e a n n i h i l a t i o n r a t e g i v e n h e r e i s v e r y l i t t l e d i f f e r e n t from -44-t h e a n n i h i l a t i o n r a t e i n pure Argon. 2.6.3. D i r e c t a n n i h i l a t i o n r a t e and t h e v a l l e y - t o - p e a k r a t i o : e l e c t r i c f i e l d r e s u l t s . F i g u r e 8 i l l u s t r a t e s t h e e f f e c t o f a p p l i e d e l e c t r i c f i e l d p e r u n i t amagat on t h e d i r e c t a n n i h i l a t i o n r a t e . F o r s m a l l E/P t h e a n n i h i l a t i o n r a t e b e g i n s t o d e c r e a s e r a p i d l y r e a c h i n g a f a i r l y c o n s t a n t v a l u e o f about 2.8 x 1O^sec - 1amagat 1 a t 90 V cm~ 1amagat~ 1 A c o mparison o f t h e c u r r e n t r e s u l t s w i t h t h o s e o f F a l k (1965) i n d i c a t e s good agreement. The measurements o f t h e v a l l e y - t o - p e a k r a t i o ( i n d i c a t i n g as d i s c u s s e d ( S e c t i o n 2 .4.) t h e i n c r e a s e d f o r m a t i o n o f p o s i t r o n i u m ) as a f u n c t i o n o f E/P a r e p l o t t e d on t h e same graph f o r comparison. These d i f f e r f r o m th e measurements o f Marder, e t a l . , (1956) i n t h a t t h e r a t e o f r i s e o f the v a l l e y - t o - p e a k r a t i o w i t h i n c r e a s i n g a p p l i e d e l e c t r i c f i e l d i s c o n s i d e r a b l y l e s s i n t h e p r e s e n t c a s e . I n v e s t i g a t i o n o f t h e shape o f t h e c u r v e as a f u n c t i o n o f N i t r o g e n c o n t a m i n a t i o n o f up t o 1$ o f t h e t o t a l d e n s i t y i n d i c a t e d t h a t the r a t e o f r i s e can i n c r e a s e somewhat w i t h t h i s gas as an i m p u r i t y . However, the r e s u l t s o f Marder, e t a l . were n o t r e p r o d u c e d , p r o b a b l y because some o t h e r unknown i m p u r i t y was p r e s e n t i n t h a t A r g o n . That t h e r e was a tendency f o r t h e i r c u r v e t o r i s e more s t e e p l y as a f u n c t i o n o f t ime i s c o n s i s t e n t w i t h t h e e v o l u t i o n o f some i m p u r i t y . I t i s c l e a r f r o m F i g u r e 8 t h a t as E/P i s i n c r e a s e d , t h e r e i s an i n c r e a s e i n t h e t h r e e - p h o t o n component a t the expense o f t h e A a / P - D I R E C T . A N N I H I L A T I O N R A T E (1cfsec*amagatTl) V A L L E Y - T O - P E A K R A T I O ( N o r m a l i z e d t o 1 a t E / P = 0 ) two-photon component. Thus i t i s r e a s o n a b l e t o suppose t h a t some o f th e f l a t t e n i n g o f f o f the * a vs E/P c u r v e a r i s e s f rom th e i n -c r e a s e d p o s i t r o n i u m f o r m a t i o n . I n t h i s c a s e , p o s i t r o n i u m p r o d u c t i o n thus c o n t r i b u t e s an e x t r a c h a n n e l by w h i c h th e o v e r a l l e q u i l i b r i u m d i s t r i b u t i o n o f f r e e p o s i t r o n s can decay. As p o s i t r o n i u m f o r m a t i o n f r o m th e e q u i l i b r i u m d i s t r i b u t i o n i n c r e a s e s w i t h a p p l i e d e l e c t r i c f i e l d , so does t h a t c o n t r i b u t i o n t o the d i r e c t a n n i h i l a t i o n r a t e Ajp(see S e c t i o n 2.3.*+. E q u a t i o n 9) w h i c h r e s u l t s f rom the v e l o c i t y -a v e r a g e d p o s i t r o n i u m f o r m a t i o n r a t e . F i g u r e '9 shows t h e f r a c t i o n o f p o s i t r o n s f o r m i n g p o s i t r o n i u m as c a l c u l a t e d f rom E q u a t i o n 15, S e c t i o n 2 . T h e f r a c t i o n W o f p o s i t r o n s a n n i h i l a t i n g i n the w a l l s o f t h e chamber was c a l c u l a t e d f rom d a t a d e r i v e d f r o m the dE/dx d a t a f o r p o s i t r o n s i n Argon ( F a l k , 1965). The c o n s t a n t k was det e r m i n e d assuming t h a t 2,7% o f t h e p o s i t r o n s stopped i n the gas form p o s i -t r o n i u m a t z e r o f i e l d ( F a l k and J o n e s , 196*+). I t i s q u i t e a p p arent t h a t f o r t h e h i g h e s t e l e c t r i c f i e l d s used a m a j o r i t y o f the p o s i t r o n s f o r m p o s i t r o n i u m . T a b l e V shows the dependence o f the o v e r a l l number o f cou n t s i n t h e l o n g - l i y e d component as a f u n c t i o n o f e l e c t r i c f i e l d . Two s e t s o f d a t a a r e g i v e n c o r r e s p o n d i n g t o s p e c t r a o b t a i n e d w i t h t h e O.51 MeV s i n g l e c h a n n e l a n a l y z e r s e t a t the peak ( " d i r e c t -enhanced") and v a l l e y ("ortho-enhanced'*) p o s i t i o n s r e s p e c t i v e l y ( s e e S e c t i o n 2,2.1.) . The p r o d u c t I 2 T 2 i s found by i n t e g r a t i n g t h e l o n g - l i v e d e x p o n e n t i a l I 2 e x p ( r - t / t 2 ) (see S e c t i o n 2.3.*+.) from t =0 t o t=», and was c a l c u l a t e d u s i n g t h e a p p r o p r i a t e v a l u e s f o r I p and x found from m a x i m u m - l i k e l i h o o d f i t s t o the l i f e t i m e s p e c t r a . -1+6-I n t h e case o f the d a t a p r e s e n t e d i n T a b l e V t h e z e r o o f time has been a r b i t r a r i l y d e f i n e d as c h a n n e l 89 o f the k i c k s o r t e r , a t w h i c h p o i n t t h e m a x i m u m - l i k e l i h o o d f i t s were begun. I t i s n o t p o s s i b l e t o draw any c o n c l u s i o n s from T a b l e V as t o t h e e l e c t r i c f i e l d dependence o f the t o t a l number o f o r t h o -p o s i t r o n i u m a n n i h i l a t i o n s r e c o r d e d by the t i m e s o r t e r . The r e a s o n f o r t h i s i s t h a t t h e s t a t i s t i c a l u n c e r t a i n t y i n ^2X2 ^ s a ^ - * - e a s t as l a r g e as t h e s i z e o f the e f f e c t s e a r c h e d f o r ( s e e F i g u r e s 8 and .9)? evan a t the h i g h e s t f i e l d s used. To summarize, t h e r a p i d d e c r e a s e o f 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 o f a p p l i e d e l e c t r i c f i e l d shows t h a t t h e v e l o c i t y -dependent- a n n i h i l a t i o n r a t e d e c r e a s e s as the p o s i t r o n v e l o c i t y i n c r e a s e s . The e x p l i c i t v e l o c i t y dependence cannot be found from t h e e x p e r i m e n t a l d a t a u n l e s s t h e v e l o c i t y - d e p e n d e n t momentum-t r a n s f e r c r o s s - s e c t i o n i s known. Some i m p l i c a t i o n s o f t h e shape o f . a n n i h i l a t i o n r a t e dependence on- e l e c t r i c f i e l d a r e d i s c u s s e d i n C h a p t e r 3, S e c t i o n 3«5« 2.6.h. The s h o u l d e r i n t h e t i m e s p e c t r a . 2.6.l+.1. W i d t h o f t h e s h o u l d e r . I n t h e case o f z e r o a p p l i e d e l e c t r i c f i e l d , a l l the time s p e c t r a d i s p l a y e d , the u s u a l s h o u l d e r s t r u c t u r e (Tao, B e l l and Green, 196^; F a l k and J o n e s , 196!+; Osmon, 1965; P a u l , 196*+) ( s e e f o r example F i g u r e 5. ) • S i n c e the form o f the time spectrum o f the d i r e c t component f o l l o w i n g t h e s h o u l d e r i s w e l l f i t t e d , by a s i n g l e T A B L E V. DEPENDENCE OF I 2 T C ON E L E C T R I C F I E L D . Ortho-enhanced Direct-enhanced I 2 - T 2 E/P I 2 T 2 E/P 4 10 counts 77 _ 1 4--1 V cm amagat 10 counts V era amagat 6 . 0 * 0 . 3 0 4 . 7 - 0 . 3 0 5 .8 ± 0 . 3 1 4 . 4 4 . 4 i 0 . 4 14.0 5.6 ± 0 . 4 3 4 . 7 4 . 9 - 0 . 5 3^ . 7 6.3 t 0.5 50.7 5.1 t-0.7 4 8 . 9 6.1 ± 0 . 8 70.7 4 . 5 ± 0 . 8 70.7 6.5 - 1.1 125.1 Note: The resu l t s given above are for Argon at an average dens i ty of 9.0 amagats. The terms '"ortho-enhanced"3 and "direct-enhanced" are defined i n Sect ion 2 . 6 . 3 . The errors quoted were obtained by simply compounding the errors i n I 2 and t 2 that were found from the maximum l i k e l i h o o d f i t s . Each spectrum was normalized to the same t o t a l number of 1.28 MeV counts as indicated i n Sect ion 2.2.1. _ L 7 _ exponential , the end of the shoulder s i g n i f i e s e i ther a thermalized pos i tron v e l o c i t y d i s t r i b u t i o n , or that the a n n i h i l a t i o n rate becomes v e l o c i t y independent at low v e l o c i t i e s or some combination of these two. The a p p l i c a t i o n of a small e l e c t r i c f i e l d , however, reduces the a n n i h i l a t i o n rate s i g n i f i c a n t l y without perturbing the shoulder markedly. This i s shown i n Figure 10. Since the appl ied e l e c t r i c f i e l d increases the average pos i tron energy at equ i l ibr ium, the d i r e c t l i f e t i m e corresponds to a velocity-dependent a n n i h i l a t i o n rate averaged over v e l o c i t i e s higher than thermal. Therefore, i t seems consistent to in t erpre t the s ing le exponential a f ter the. shoulder i n terms of a thermalized pos i tron d i s t r i b u t i o n . The time width of the shoulder, then, i s the time taken for a pos i tron to thermalize from energies corresponding to E ^ . ^ (8.9 eV). I t has a lready been shown (Chapter 1, Sect ion 1.2.3*) that the time taken for theposi tron to reach such energies (about 10 eV) i s less than the time re so lu t ion of the apparatus used. The shoulder width i s inverse ly proport iona l to dens i ty . The width-density product was measured to be about 3*+0 nsec-amagat, i n agreement with previous resu l t s (Falk and Jones, 196*f; Pau l , 196k). A narrower shoulder width would indicate a more rapid slowing down of the pos i trons , which would occur through the agency of fore ign gas atoms. Thus the presence of fore ign molecules with low- ly ing exc i ta t ion levels, would be expected to narrow the shoulder s i g n i f i c a n t l y , simply by increas ing the energy loss per c o l l i s i o n and thereby the rate of thermal izat ion . A pos i tron scattered by the host atoms, with momentum-o o o I O o o o o o CO f - o Z ° 3 ~ O N O U_ O o -o 8 o + ++ 5 6 P = 9.0 E / P - 1 3 . 9 2.74 nsec/channel T r 1 1 1—• 60.000 85.000 110.000 135.000 160.000 185.000 210.000 235.000 CHANNEL NUMBER F i g u r e 1 0 . Time spectrum f o r p o s i t r o n s i n Argon a t s m a l l E/P. P i s i n amagats, E/P i s i n V cm" 'amagat - 1 . The c o n t i n u o u s c u r v e i s t h e maximum l i k e l i h o o d f i t t o t h e p o i n t s . T T -1+8-t rans fer cross-sect ions of the order of na 0 2 , w i l l suffer about 1 1 1 2 10 - 1 0 c o l l i s i o n s s e c - 1 at 10 amagats. Thus i n 3 0 nsec the. i h pos i tron suffers 1 0 J - 1 0 c o l l i s i o n s . An impurity concentration L. as low as a few parts i n 10 could be expected to have a s i g n i f i -cant e f fect on the shape of the shoulder. Throughout these experiments the shoulder width remained e s s e n t i a l l y constant showing that here there was no s i g n i f i c a n t change i n gas composition from run to run , e i ther due to evolut ion of impuri t ies from the w a l l s , or due to the d i f f eren t gas samples used. 2 . 6 . * + . 2 . The logarithmic slope of the shoulder. A second feature of the shoulder i s i t s reasonably constant logarithmic s lope. Furthermore i t appears that the shape of the time spectrum i n the region of the shoulder i s somewhat independent of the shape of the orthopositronium component i n that reg ion . This i s demonstrated by comparing the two time spectra i n Figure 11 which were obtained with the 0.51 MeV s ing le channel analyzer set at the peak (direct-enhanced) and v a l l e y (ortho-enhanced) pos i t ions i n turn (see Sect ion 2 . 2 . 1 . ) . In the region of the shoulder there i s very l i t t l e d i f ference i n the time spectra , despite the fac t that the r e l a t i v e i n t e n s i t i e s of the d i r e c t and orthopositronium components d i f f e r i n the two cases. Such a s i t u a t i o n could a r i s e i f the orthopositronium component under-l y i n g the shoulder has a logarithmic slope l i t t l e d i f f e r e n t from the logarithmic slope of the d i r e c t a n n i h i l a t i o n contr ibut ion to the shoulder. This indeed seems to be the case for Argon at 185.000 2 1 0 . 0 0 0 235.0001 1 i r 60-000 85.000 110.OOO 135.OOO 160.000 CHANNEL NUMBER F i g u r e 1 1 . Comparison o f d i r e c t - and ortho-enhanced t i m e s p e c t r a obtained at P=H-.9 amagats, E/P=0 V cm'Tamagat"1. The c o n t i n u o u s c u r v e s a r e t h e maximum l i k e l i h o o d f i t s to t h e p o i n t s . A - d i r e c t - e n h a n c e d ; C - ortho-enhanced.  -49-a densi ty of 4.9 amagats (Figure 1 V ) . For higher dens i t ies these logarithmic slopes should d i f f e r to a greater extent, because of the d i f f e r e n t pressure dependence, and hence the shoulder i n the direct-enhanced time spectrum should be somewhat d i f f e r e n t from that i n the ortho-enhanced time spectrum. Figure 12 shows such a pa ir of spectra obtained at 9«3 amagats. Because the shoulder i s much narrower i n th i s case, i t i s c l e a r l y d i f f i c u l t to draw any conclusions here concerning the previous argument. I f i t i s assumed, however, that a l l the positronium i s formed ( for zero e l e c t r i c f i e l d ) at the time corresponding to the prompt peak, i t i s poss ible to extrapolate the orthopositronium component i n the region of the two exponentials back into the region of the shoulder. Subtract ion of the orthopositronium component i n the shoulder region i n th i s fashion and f i t t i n g the r e s u l t i n g curve g r a p h i c a l l y to a s ingle exponential y i e lds an average a n n i h i l a t i o n rate i n the shoulder region of about 6 - 1 -1 1.5 x 10 sec amagat for the two spectra i n Figure 11. For 6 - 1 - 1 the two spectra i n Figure 12 the r e s u l t i s v K O x 10 sec amagat 6 1 — 1 This a g r e e s • f a i r l y we l l witn the f igure of 1.2 x 10 sec" amagat given by Osmon (1965) ( th i s value has been corrected as indicated i n Sect ion 2.6.2.3.). The s ize of th i s ""shoulder a n n i h i l a t i o n rate" , i n r e l a t i o n to the d i r e c t a n n i h i l a t i o n rates observed as a funct ion of e l e c t r i c f i e l d , i s ind icated i n Figure 8. A reasonably constant logarithmic slops of the d i r e c t a n n i h i l a t i o n contr ibut ion to the shoulder cannot be interpreted simply i n terms of a changing a n n i h i l a t i o n rate as a funct ion of o o o •. o o o 8 o •o o o o + + 5 1 P = 9 . 3 E/P = .0 2.74 nsec/channel i 1 1 — i 60.000 85.000 110.000 135.000 160.000 185.000 210.000 235.000 CHANNEL NUMBER F i g u r e 12. Comparison o f d i r e c t - and ortho-enhanced t i m e s p e c t r a o b t a i n e d a t P-9.3 amagats, E/P»0 V-cm" 1amagat"1; The c o n t i n u o u s c u r v e s a r e t h e maximum l i k e l i - l hood f i t s t o the p o i n t s . T T T -50-time. In order that a constant logarithmic slope appear i n the d i r e c t component of the time spectra , e i ther the pos i tron v e l o c i t y d i s t r i b u t i o n must be at equ i l ibr ium, or the d i r e c t a n n i h i l a t i o n rate must be e s s e n t i a l l y v e l o c i t y independent over the appropriate range of the shoulder. This i s discussed i n d e t a i l i n Chapter 3? Sect ion 3-4-. It has already been pointed out that the s ingle exponential fol lowing the shoulder corresponds to a n n i h i l a t i o n from the equi l ibr ium v e l o c i t y d i s t r i b u t i o n . Thus i t i s c lear that the constant logarithmic slope i n the shoulder region can only ar i s e from a reasonably v e l o c i t y independent a n n i h i l a t i o n rate at v e l o c i t i e s somewhat higher than thermal. Therefore, i t appears that i n a v e l o c i t y range appropriate to the time span defined by the shoulder, the d i r e c t a n n i h i l a t i o n rate i s approximately v e l o c i t y independent, subject to the assump-t i o n that a l l the positronium formation occurs at the time corres -ponding to the prompt peak. The small shoulder a n n i h i l a t i o n rate x should be taken as evidence that the v e l o c i t y dependent d i r e c t 6 -1 -1 a n n i h i l a t i o n rate can become as small as ^ 1 . 5 x 10 sec amagat Furthermore, since the shoulder has a constant logarithmic slope over the greater part of i t s extent, i t seems l i k e l y that the velocity-dependent d i r e c t a n n i h i l a t i o n rate i s constant over some of the energy range between 8.9 eV (the positronium formation threshold) and 1A0 eV (thermal energy). 2.6.4-.3- E f f e c t of the e l e c t r i c f i e l d on the shoulder. I t has been suggested i n the previous subsection that the v e l o c i t y averaged a n n i h i l a t i o n rate corresponding to the time i n t e r v a l occupied by the shoulder i s due to a somewhat v e l o c i t y independent a n n i h i l a t i o n rate at v e l o c i t i e s higher than thermal. Furthermore, th i s shoulder a n n i h i l a t i o n rate has been shown to be considerably less than the a n n i h i l a t i o n rate at thermal v e l o c i t i e s Reference to Figure 8 shows that the a p p l i c a t i o n of an e l e c t r i c f i e l d reduces the d i r e c t a n n i h i l a t i o n r a t e , but that th i s does not f a l l below the shoulder a n n i h i l a t i o n ra te . This i s so pre-sumably because of the onset of increased positronium formation as the e l e c t r i c f i e l d i s app l i ed . However, i t i s evident that the shoulder disappears as the e l e c t r i c f i e l d i s increased (Fig. 1 ' p r i n c i p a l l y because the d i r e c t a n n i h i l a t i o n rate averaged over the v e l o c i t y d i s t r i b u t i o n at high E /P tends towards the shoulder a n n i h i l a t i o n ra te . Lack of knowledge of the i n i t i a l v e l o c i t y d i s t r i b u t i o n charac ter i z ing the v e l o c i t y d i s t r i b u t i o n of positrons with energie below 1 1 . 6 eV (see Chapter 1 , Sect ion 1 . 2 A . ) prevents a more d e t a i l e d d iscuss ion of the way i n which the shoulder changes as a funct ion of e l e c t r i c f i e l d . I t can only be stated that the observations are e n t i r e l y consistent with the view of a v e l o c i t y -dependent a n n i h i l a t i o n rate which decreases as a funct ion of increas ing v e l o c i t y . o o o o o to f - o O o •o o o o o o + + ++ 5 5 P = 9.3 E / P z 7 0 . 6 2.74 nsec/channel + T T -+-t- •+-2 3 5 . 0 0 0 1-1 1 1 110.000 135.000 160.000 1 8 5 . 0 0 0 2 1 0 . 0 0 0 CHANNEL NUMBER Figure 13* Time spectrum for positrons at high E / P . The continuous l i n e i s the maximum l i k e l i h o o d f i t to the points . P i s i n amagats, E / P i s i n V cm"Tamagat" 1. 6 0 . 0 0 0 8 5 . 0 0 0 2.6.5. Orthopositronium a n n i h i l a t i o n ra te . 2.6.5.1. F i t t i n g of experimental data. Figure 14 shows the Argon-density dependence of the a n n i h i l a t i o n rate appropriate to the l o n g - l i v e d component of the time spectra . Only those resu l t s are presented which s a t i s f i e d the c r i t e r i a discussed i n Sect ion 2.6.1. and are thus for the same spectra as are the d i r e c t a n n i h i l a t i o n rates discussed i n Sect ion 2.6.2. The resu l t s of f i t t i n g the data i n Figure 14 to simple functions of the density are shown i n Table V I . The meaning of Q i s the same as i n Sect ion 2.6.2.1. Unless i n d i c a t e d , the f i t s were made to a l l the experimental values i n Figure 14. Of the 27 points presented, 11 were obtained with an appl ied e l e c t r i c f i e l d . The functions to which the data have been f i t t e d can be d iv ided into two categories . E i t h e r the parameter a Q i s determined by the data ( f i t s I to VII) or i t i s ( f i t s VIII to X) the t h e o r e t i -6 1 c a l free orthopositronium a n n i h i l a t i o n rate (7.2 x 10 sec" ; see Chapter 1, Sect ion 1.4.2.). The r e l a t i v e l y large scat ter of the resu l t s i n Figure 14 i s r e f l e c t ed i n the v poor f i t s obtained i n Table V I . However, some systematic trend i s apparent. F i t I , which contains no density dependent terms, i s c l e a r l y unacceptable. For dens i t ies less than 10 amagats, with E/P=0, a good l i n e a r f i t II i s obtained. Including the e l e c t r i c f i e l d r e s u l t s , the f i t VI i s obtained. The parameters a Q and a 1 are r e l a t i v e l y unchanged but the f i t i s judged worse by 6» ' l ' I » I i I i I i I i 1 i I » I 0 2 4 6 8 10 12 14 16 18 P — DENSITY (AMAGATS) Figure 14. Orthopositronium a n n i h i l a t i o n rate i n Argon as a function of density. The s t r a i g h t l i n e represents the function X » ( 7 . 2 * .0.29 P) x-10°sec"1. - 5 3 -the chi-square tes t . A s i m i l a r s i t u a t i o n holds for dens i t ies less than 20 amagats where the corresponding f i t s are III and V I I . As discussed prev ious ly , the l o n g - l i v e d component i s expected to be due to orthopositronium a n n i h i l a t i o n , while the change i n a n n i h i l a t i o n rate as a funct ion of gas density i s associated with the quenching of the free orthopositronium l i f e -time (Chapter 1, Sect ion 1 A A . ) . The parameter a 0 , then, i s the free orthopositronium a n n i h i l a t i o n rate as measured by these experiments. In view of the t h e o r e t i c a l p r e d i c t i o n of th i s l i f e -time (Chapter 1, Sect ion 1A.2.) and the associated a n n i h i l a t i o n 6 -1 rate (7.2 x 10 sec ) , f i t V has l i t t l e phys ica l j u s t i f i c a t i o n . The value of a 0 for th i s f i t , which contains only a quadratic dependence on dens i ty , i s considerably d i f f eren t from the t h e o r e t i c a l orthopositronium a n n i h i l a t i o n r a t e , i f compared with the a 0 from the l i n e a r f i t s to the data. A s i m i l a r argument holds for f i t IV where a l i n e a r term i s inc luded. However, i n view of the reason-able Q, and the fac t that the value for a Q i s s i g n i f i c a n t l y c loser to the t h e o r e t i c a l orthopositronium a n n i h i l a t i o n r a t e , the pos s i -b i l i t y of a quadratic densi ty dependence i n add i t ion to the l i n e a r term can not be ru led out by the set of data presented. In the f i t s VIII to X the parameter a 0 i s given the 6 1 value of 7.2 x 10 sec , the t h e o r e t i c a l orthopositronium l i f e -time. As expected the f i t s are considerably worse than those obtained where a Q was not f i x e d . This i s so since the value for a 0 from any of the l a t t e r y i e lds the best f i t to the data, and 6 — 1 these values for a 0 were not equal to the 7.2 x 10 sec . The TABLE V I . DEPENDENCE OF ON P. Summary o f th e r e s u l t s of f i t t i n g t h e c u r v e i n F i g u r e 14. R e s u l t s Q_ a Q/10 6 a i /106 a 2/10 6 » -1 sec s e c- ^ " 1 4.-1 sec amagat -1 -2 sec amagat ^o - 10.57* o . 05 <10 - 6 A o = a 0 -» a.,P 6.83+0. 50 0.339 t 0.059 0.88 E/P« o , P < 10 amagats r =. 0 a o + a P 7 . 5 5 1 0 . 18 0.240* 0.013 0.16 E/P = 0 r o - a o + a ^ + a 2 P 7-97* 0. 5 9 0.164* 0.102 0.0030* 0.0041 0.15 E/P = 0 r _ • - o a 0 + a 2 P 2 8.90±0. 11 0.0095* 0.0005 0.10 E/P = 0 TABLE VI (continued) DEPENDENCE OF ^ OJJ P. Type of f i t Results £_ a 0 / 1 0 6 a i / 1 0 6 a 2 / 1 0 6 1 0 6 sec" 1 s e c - 1 - 1 ^ - 1 sec amagat sec ^ amagat-^ V I . . X 0 = a o + a l P P < 1 0 amagats 7 . 0 7 ^ 0 . 1 + 8 o . 3 0 5 ± 0 . 0 5 6 0 . 0 2 V I I . x~ . 0 = a Q + a l P 7 . 5 ^ - 0 . 1 5 0.2h2± 0 . 0 1 1 0 . 0 0 * + V I I I . X 0 = 7 . 2 + & 1 P P < 1 0 amagats 0.29*+ ± 0 . 0 0 8 0 . 0 1 7 IX. X o = 7 . 2 + a P 0 . 2 6 7 ± 0 . 0 0 * + 0 . 0 0 0 * + X. x" o = 7 . 2 + a.,P t a 0 . 3 0 3 t 0 . 0 1 6 - 0 . 0 0 2 * + - 0 . 0 0 1 0 0 . 0 0 1 2 Note: Unless otherwise s ta ted , the f i t s were made to a l l the points i n Figure 1 * + . The e l e c t r i c f i e l d E/P is i n uni ts of V cm" 1amagat - 1 -54-v a l u e s f o r the parameter a-j i n f i t s V I I I t o X d i f f e r l i t t l e from t h e a-| o b t a i n e d i n the f i t s where a Q was a l l o w e d to v a r y , 2.6.5.2. D i s c u s s i o n o f the l i n e a r d e n s i t y dependence of the o r t h o p o s i t r o n i u m a n n i h i l a t i o n r a t e . As mentioned e a r l i e r , the a n n i h i l a t i o n r a t e a t z e r o gas d e n s i t y i s e x p e c ted t o be the f r e e o r t h o p o s i t r o n i u m a n n i h i l a t i o n r a t e . T h i s a n n i h i l a t i o n r a t e i n c r e a s e s by p i c k - o f f quenching as the d e n s i t y i n c r e a s e s (see C h apter 1, S e c t i o n 1.4,4.). The quenching c r o s s - s e c t i o n v / i l l i n g e n e r a l be v e l o c i t y dependent, and the v e l o c i t y - d e p e n d e n t quenching r a t e p r o p o r t i o n a l t o the gas d e n s i t y . The v e l o c i t y - a v e r a g e d quenching r a t e A ^ i s thus a l s o d e n s i t y dependent. On. the b a s i s o f the d i s c u s s i o n p r e s e n t e d here,, the d e n s i t y dependence of the quenching r a t e would be e x p e c ted t o be l i n e a r . The l a r g e s c a t t e r o f the e x p e r i m e n t a l p o i n t s i s i n t e r p r e t e d t o r e f l e c t s l i g h t d i f f e r e n c e s i n the t y p e and c o n c e n t r a t i o n o f i m p u r i t i e s i n the Argon gas from r u n t o r u n . On t h e average the quenched o r t h o p o s i t r o n i u m atom e x i s t s f o r about 100 nsec i n 10 amagats o f Argon. T h i s i s c o n s i d e r a b l y l o n g e r t h a n the w i d t h o f t h e s h o u l d e r o b s e r v e d i n the same time s p e c t r a ( S e c t i o n 2.6.4.). I n a d d i t i o n , t h e e l a s t i c - s c a t t e r i n g c r o s s - s e c t i o n f o r o r t h o p o s i -t r o n i u m i s e x p e c ted t o be l e s s than t h a t f o r f r e e p o s i t r o n s where the l o n g - r a n g e a t t r a c t i o n i s p r o p o r t i o n a l t o 1/R ( C h a p t e r 1, S e c t i o n 1.3*2.; Chapter 35 S e c t i o n 3*1 •) as compared w i t h the l o n g - r a n g e 1/R^ van der Waals a t t r a c t i o n e x p e c ted f o r t h e p o s i t r o n --55-ium case. For an e l a s t i c scat ter ing cross - sec t ion of the order p of T a 0 the orthopositronium atom would make of the order of 1 0 1 1 c o l l i s i o n s / s e c at 10 amagats. Thus the orthopositronium k atom makes of the order of 10 c o l l i s i o n s before a n n i h i l a t i n g . Should one of these c o l l i s i o n s involve an impurity atom with a large quenching cros s - sec t ion , the orthopositronium l i f e t i m e w i l l be considerably shortened. I t thus appears that the orthopos i tron-ium a n n i h i l a t i o n rate i s at l east as sens i t ive to impuri t ies as the shoulder width i n the Argon time spectra . The l e v e l of Nitrogen contamination found i n many of the Argon samples used i n these experiments (Table I) i s probably high enough to explain these d i screpancies . 2 . 6 . 5 . 3 * Influence of the e l e c t r i c f i e l d . Although there i s no phys ica l basis for expecting any e l e c t r i c f i e l d dependence i n the orthopositronium a n n i h i l a t i o n r a t e , the data has been checked for such an e f f ec t . An examination of the data i n Table VI shows no evidence for any e l e c t r i c f i e l d dependence. The increased scatter of the a n n i h i l a t i o n data, when the e l e c t r i c f i e l d resu l t s are included ( re f l ec ted i n the smaller value for Q) i s most probably due to the e f fect of fore ign gases l i b e r a t e d by undetected high-voltage breakdowns during a run. I t i s expected that fore ign gases added to the Argon i n th i s manner would not be immediately removed by the p u r i f i e r , and could thus a f f e c t the measured orthopositronium a n n i h i l a t i o n rate as discussed i n Sect ion 2 . 6 . 5 . 2 . -56-2.6.5. k . Summary of the orthopositronium r e s u l t s . From Table V I , Figure 1*+, and the above d i scuss ion i t i s c l ear that some density dependence of the orthopositronium .-. a n n i h i l a t i o n rate is~ necessary. A s t a t i s t i c a l analys is of the data indicates that a l i n e a r dependence on density i s s u f f i c i e n t i f the e l e c t r i c f i e l d resu l t s are excluded. Inc lus ion of the e l e c t r i c f i e l d resu l t s worsens the f i t but does not change the value of the parameters s i g n i f i c a n t l y . Comparison of the f i t s i n Table VI which contain a l i n e a r density dependence and a zero density intercept a 0 , 6 —1 indicates a value for a Q between 6.8 and 7.6 x 10 sec , i f the r e s u l t s of f i t IV are neglected (Sect ion 2.6 .5.1.). Such a range .for a Q takes into account the systematic error introduced by the e l e c t r i c f i e l d as discussed i n Section2.6.5.3* On th i s basis 6 -1 a free orthopositronium a n n i h i l a t i o n rate of ( 7.2 ± 0 . L ) x 10 sec i s given by these experiments, i n good agreement with the t h e o r e t i c a l 6 -1 p r e d i c t i o n of 7.2 x 10 sec The values for a 1 i n Table VI a l l l i e between 0.2 L and 0 . 3 k x 10° sec - 1 amagat" i f once again f i t IV i s neglected. Furthermore, the values for a-| i n f i t s VIII to X l i e wi th in these l i m i t s . F i n a l l y ^ t h i s range of values for a^  a lso allows for the systematic error introduced by the e l e c t r i c f i e l d . The l i n e a r quenching rate X q given as a r e s u l t of these experiments i s then X q = (0.29 * 0.05) x 10 6 sec" 1 amagat"1 . I t should be emphasized that the quenching rate has been obtained with about 100 ppm of Nitrogen present (Table I) and would be e x p e c t e d t o d e c r e a s e somewhat f o r Argon c o n t a i n i n g l e s s t h a n 1 ppm o f any i m p u r i t y . T a b l e V I I compares t h e p r e s e n t r e s u l t s w i t h p r e v i o u s l y p u b l i s h e d v a l u e s . B o t h o f the q u a d r a t i c f i t s ( C e l i t a n s , Tao and Green, 1964; C e l i t a n s and Green, 1964) were o b t a i n e d f rom d a t a where the s h o u l d e r w i d t h was about 90 nsec-amagats, w h i c h i s a t l e a s t a f a c t o r o f t h r e e s m a l l e r t h a n t h e c u r r e n t l y a c c e p t e d v a l u e . T h i s s m a l l e r s h o u l d e r w i d t h i s a r e s u l t o f a r e l a t i v e l y l a r g e i m p u r i t y c o n c e n t r a t i o n (Tao and B e l l , 1965), and f o l l o w i n g t h e arguments i n S e c t i o n 2 . 6 . 5 . 2 . , t h e o r t h o p o s i t r o n i u m l i f e t i m e s were p r o b a b l y a f f e c t e d t o a c o n s i d e r a b l e e x t e n t . The r e s u l t s o f Heymann e t al . ( 1 9 6 l ) were o b t a i n e d f rom a n a l y s i s o f 0..51 MeVi . gamma-ray s p e c t r a , and agree w e l l w i t h the v a l u e s p r e s e n t e d i n t h i s t h e s i s w h i c h were o b t a i n e d from l i f e t i m e measurements. 2 . 6 . 6 . D i s c u s s i o n o f e r r o r s n o t r e l a t e d t o c o u n t i n g s t a t i s t i c s . A l l t he s t a n d a r d d e v i a t i o n s quoted thus f a r , e xcept where i t i s e x p l i c i t l y s t a t e d o t h e r w i s e , a r e based on the e f f e c t o f c o u n t i n g s t a t i s t i c s o n l y . I t i s n e c e s s a r y a l s o t o d i s c u s s t h e r o l e o f e r r o r s I n t r o d u c e d by i n s t a b i l i t i e s i n t h e e l e c t r o n i c i n s t r u m e n t a t i o n , by i n a c c u r a c i e s a s s o c i a t e d w i t h t h e measurement o f the 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 i e s o f the t i m e s o r t e r , t h e a p p l i e d e l e c t r i c f i e l d and the gas d e n s i t y , and by changes i n t h e gas c o m p o s i t i o n f rom r u n t o r u n . TABLE V I I . SUMMARY OF PUBLISHED RESULTS FOR ORTHOPOSITRONIUM QUENCHING IN ARGON. Type of densi ty dependence 6 -1 10 sec A = 7.0 + a 0 P' o 2 . r =7.0 4 a0r O 2 A -=7.0 4 a.P _ ° A —a + a P o o 1 Results an/1O6 -1 sec 7.2-* 0.4 a-i/106 -1 -1 sec amagat 0.277 * 0.005 0.29 ± 0.04 a2/1 0£ -1 -2 sec amagat 0.017 - 0.002 0.015 - 0.002 Reference C e l i t a n s , Tao and Green (1964). Cel i tans and Green (1964). Heymann, et a l . (1961). Present work. Note: The resul ts due to other workers have been corrected by the fac tor 298/273 (see Section 2.6.2.3.) on the assumption that the atmosphere uni t s given by o them are at 25 C. The value of the free orthopositronium a n n i h i l a t i o n rate assumed by the other workers i s not i n accordance with the ca lcu la ted values (Ore and Powell , 1949; Alekseev, 1959). -58-2.6.6 .1. E f f e c t o f i n s t a b i l i t i e s i n t h e e l e c t r o n i c i n s t r u m e n t a t i o n . The d i s c u s s i o n o f t h e e f f e c t o f e l e c t r o n i c i n s t a b i l i t y c a n be d i v i d e d i n t o two s e c t i o n s . I n t h i s s u b s e c t i o n the e f f e c t s o f changes brought about i n the prompt r e s o l u t i o n and i n t h e prompt peak p o s i t i o n a r e c o n s i d e r e d . The f o l l o w i n g s u b s e c t i o n d e a l s w i t h t h e u n c e r t a i n t i e s i n t h e time c a l i b r a t i o n o f t h e t i m e s o r t e r . F i g u r e 20 ( A p p e n d i x ) , shows the e f f e c t on t h e t i m e r e s o l u t i o n o f making a l a r g e change i n the 0.51 MeV s i n g l e c h a n n e l a n a l y z e r s e t t i n g . One c u r v e was o b t a i n e d w i t h t h e s i n g l e c h a n n e l a n a l y z e r s e t a t t h e 0.51 MeV photopeak, t h e o t h e r w i t h t h e s i n g l e c h a n n e l a n a l y z e r s e t a t t h e v a l l e y p o s i t i o n (see S e c t i o n 2.2 . 1 . and F i g . 5). The r e s o l u t i o n i s somewhat d i f f e r e n t f o r the two s e t t i n g s . T a b l e V I I I shows t h e r e s u l t s o f l i f e t i m e measurements f o r some r e p r e s e n t a t i v e p a i r s o f time s p e c t r a where the main d i f f e r e n c e was the s i n g l e c h a n n e l a n a l y z e r s e t t i n g . Taking; n o t e of. t h e .standard d e v i a t i o n s i n ' t h e r e s u l t s , t h e r e i s l i t t l e e v i d e n c e f o r a marked t r e n d . To f i r s t o r d e r t h e n , changes i n t h e e l e c t r o n i c s w h i c h m o d i f y t h e time r e s o l u t i o n t o t h e e x t e n t i n d i c a t e d i n F i g u r e 2 0 , have no o b s e r v a b l e e f f e c t on t h e l i f e t i m e measurements. The e f f e c t i s ex p e c t e d t o be s i g n i f i c a n t o n l y when t h e l i f e t i m e s a r e o f the o r d e r o f , or l e s s t h a n , t h e prompt time r e s o l u t i o n . D u r i n g t h e c o u r s e o f the experiments t h e prompt peak ( s e e F i g u r e 6) o f t h e time s p e c t r a was obser v e d t o s h i f t a maximum of±1 c h a n n e l (2.7 n s e c ) . A s h i f t i n the prompt peak o f t h i s magnitude d u r i n g a r u n w i l l have no e f f e c t on the l i f e t i m e o f the e x p o n e n t i a l s but does m o d i f y the i n t e n s i t i e s . I n f a c t , any non-TABLE VIII. DEPENDENCE OF LIFETIME RESULTS ON S.C.A. SETTING. Direct-enhanced 0 r tho - enhanc ed T1 T 2 P T1 T 2 P nsec nsec amagats nsec nsec amagats 18.3 * 1 -7 97.8 * 2.6 9.3 19.4 0.7 101 .2 * 2.1 9.3 37.8 t 1.7 90.3 * 4.9 9.0 38.7 + 3.4 101 .0 t 5.0 9.0 27.9 ± 0.8 95.0 ± 4.0 9.0 29.3 1.5 105.5 * 3.3 9.0 22.0 t 0.5 107.0 t 4.0 9.0 24.2 1 .0 108.5 * 2.4 9.0 20.7 ± 0 . 5 104.3.± 3.0 8.9 18.6 + 1 .0 100.5 * 2.0 8.8 34.3 ± 1.4 90.3 * 4-.3 9.0 33-1 2.4 98.8 t 3 A 8.7 13-9 * 0.4 95.8 * 2.0 13.5 14.2 + 0.6 95.2 t 1 .4 13.5 35.2 * 1.5 112.9* 8.9 5.3 35.6 3-8 114.8 ± 5.9 5.3 -59-e x p o n e n t i a l " p o r t i o n of the time spectrum such as the shoulder r e g i o n i s m o d i f i e d by a s h i f t i n the prompt peak. Other i n s t a b i l i t i e s i n the e l e c t r o n i c s are expected to have an e f f e c t c o n s i d e r a b l y l e s s than the accuracy to which the 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 i e s of the t i m e s o r t e r have been measured. 2.6.6.2. The 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 i e s of the t i m e s o r t e r . The i n t e g r a l l i n e a r i t y of the t i m e s o r t e r was measured u s i n g the method d e s c r i b e d by Jones and F a l k (1965). The o v e r a l l e r r o r a s s o c i a t e d w i t h the measurement of the average time width per channel g i v e n by the i n t e g r a l l i n e a r i t y i s estimated to be l e s s than 1$, a r i s i n g mainly from the u n c e r t a i n t y i n r e a d i n g the time between pulses from the p u l s e r . The i n t e g r a l l i n e a r i t y measure-ment was repeated d u r i n g the course of the experiment i n order to check t h a t there were no s i g n i f i c a n t changes i n the average time width per channel. The second measurement, separated from the f i r s t by two months was w i t h i n \% of the f i r s t measurement (see Appendix). The d i f f e r e n t i a l l i n e a r i t y (or r e l a t i v e channel width) i s measured by the r e l a t i v e number of counts i n each channel of the t i m e s o r t e r , when the i n p u t to the t i m e s o r t e r i s a source of p u l s e p a i r s separated by time i n t e r v a l s of a random l e n g t h ( F a l k , Jones and Orth, 1965). The r e l a t i v e channel widths are d i r e c t l y p r o p o r t i o n a l to the r e l a t i v e number of counts i n each . channel. The accuracy of the d i f f e r e n t i a l l i n e a r i t y measurement -60-i s thus governed by the counting s t a t i s t i c s , and since about 1000 counts per channel were recorded, the r e l a t i v e channel widths are accurate to about 3%> The 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 i e s of the t imesorter are shown i n Figure 22 (Appendix). I t i s c l ear that the use of the d i f f e r e n t i a l l i n e a r i t y i n the f i t t i n g of the experimental data (see Sect ion 2.5.1.) already const i tutes a f i r s t order correc t ion 'to the estimation of the l i f e t i m e s . Any uncerta inty i n the r e l a t i v e channel widths must then be considered as a f f ec t ing th is correc t ion to the l i f e t i m e measurements (when compared with the uncertainty i n the i n t e g r a l l i n e a r i t y ) . From th i s point of view i t i s reasonable to suppose that the t o t a l uncertainty i n the l i f e t imes due to the c a l i b r a t i o n of the t imesorter i s of the order of 1$, th i s f igure representing the maximum systematic error due to the i n t e g r a l l i n e a r i t y measurement. 2.6.6.3. Systematic error i n the a n n i h i l a t i o n ra tes . Taking into account the above d i scuss ion , i t i s c lear that the systematic error i n the measurement of the a n n i h i l a t i o n rates i s of the order of 1$, and ar i ses from the uncerta inty i n the absolute time c a l i b r a t i o n of the t imesorter . 2.6.6.k. Appl ied e l e c t r i c f i e l d . Since much of the analys i s of the experiments r e l i e s on the assumption of a s p a t i a l l y uniform e l e c t r i c f i e l d , i t i s necessary to d i scuss , not only the uncerta inty i n the magnitude of th i s e l e c t r i c f i e l d , but also the degree to which th i s e l e c t r i c - 6 1 -f i e l d i s uniform wi th in the chamber. The voltage appl ied to the e l e c t r i c f i e l d r ings was found by measuring the current flowing through a selected 500 M ft r e s i s t o r connected i n p a r a l l e l with the e l e c t r i c f i e l d r i n g s . The value of the r e s i s t o r was measured to %^ at 20.0 kV and 0.5$ at 3 kV, and was found to have a n e g l i g i b l e voltage c o e f f i c i e n t . The Avometer used to measure the current through the r e s i s t o r was accurate to \% for a f u l l scale de f l ec t ion corresponding to the highest f i e l d s used. The distance between the ground plate and the high voltage r ing has been prev ious ly reported by Falk (1965) and i s known to less than \%. The o v e r a l l uncerta inty i n the magnitude of the e l e c t r i c f i e l d taking into account these factors i s of the order of 3% for a l l e l e c t r i c f i e l d s measured. The s p a t i a l uniformity of the e l e c t r i c f i e l d was i n -vest igated by Falk (1965) using a low-voltage two-dimensional analogue of the g r i d s tructure . The resu l t s indicated that the non-uniformit ies were confined to the region immediately surround-ing the e l e c t r i c f i e l d r i n g s , and occupied about 8% of the volume enclosed by these r i n g s . The ef fect of these non-uniformit ies w i l l be less marked at high Argon d e n s i t i e s , because of the reduced pos i tron range, compared with lower d e n s i t i e s . The points i n Figure 9 have been obtained for a v a r i e t y of d e n s i t i e s , and l i e on a continuous curve, depending on the E / P . In view of t h i s , i t i s considered that the small non-uniformit ies i n the e l e c t r i c f i e l d are unimportant. -62-2.6.6.5. Measurement of gas dens i ty . The density of the gas was found using the perfect gas law. Deviations from th i s law as expressed by the van der Waals equations are n e g l i g i b l e compared with the uncerta inty i n the pressure measurement. This uncerta inty \s associated with the c a l i b r a t i o n of the high pressure gauge and i s estimated to be 2% ( F a l k , 1965). The measurement of the absolute temperature of the gas was done using a mercury thermometer with the bulb placed against the chamber w a l l . There i s a p o s s i b i l i t y that the tempera-ture of the gas was higher than that of the chamber w a l l s , due to the heating ac t ion of the p u r i f i e r . However, there was no evidence that the temperature of the chamber walls increased when the p u r i f i e r was turned on, i n d i c a t i n g that most of the gas i s i n thermal equ i l ibr ium with the chamber wa l l s . . The average absolute temperature of the gas was known i n th i s way to wi th in \% during a run. The uncertainty i n the densi ty measurement i s thus of the order of 3%, taking into account uncerta int ies i n both the pressure and temperature measure-ments. 2.6.6.6. Uncertainty i n E / P . In the l i g h t of the discussions i n the previous two subsect ions, the uncerta inty i n the measurement of E/P i s of the order of 6% which i s obtained by compounding the uncerta int ies i n E and P. 2 . 6 . 6 . 7 . Gas c o m p o s i t i o n . I n e x periments o f the t y p e r e p o r t e d h e r e , i t i s d e s i r a h 1 t o be f a i r l y c e r t a i n t h a t t h e gas c o m p o s i t i o n remained c o n s t a n t o v e r th e l e n g t h o f time t h a t the e x periments were performed. The e f f e c t i v e n e s s of the Ca-Mg e u t e c t i c p u r i f i e r i n m a i n t a i n i n g t h e gas c o m p o s i t i o n over a p e r i o d of time has been d i s c u s s e d i n S e c t i o n 2 . 2 . 1 . The p o s s i b i l i t y t h a t s m a l l changes i n i m p u r i t y c o n c e n t r a t i o n have a f f e c t e d the l i f e t i m e r e s u l t s i s d i s c u s s e d i n S e c t i o n 2 . 6 . 2 . 3 . and S e c t i o n 2 . 6 . 5 . 2 . 3 - THEORETICAL CONSIDERATIONS  OF THE POSITRON-ARGON ATOM INTERACTION. 3.1. Introduct ion . The subject of low-energy positron-atom interac t ions has received considerable a t tent ion recent ly (Massey, et a l . , 1966; Drachman, 1966) due to the development of experimental techniques which make comparison between theory and experiment poss ib le (Fa lk , Orth and Jones, 1965; Pau l , 196 L ) . The o v e r a l l n o n - r e l a t i v i s t i c time-independent Hamiltonian H for a system of e lectrons bound to a nucleus, and an unbound pos i tron i s given by H = l E T ^ N ~ S T I - i " 2m~ lp~l | r . - r U + I l ^ ] r w v N e 1 e V 1 ' — l - N 1 1 3 — ; 7 2 2 i # j + , Z e 1 - I I e 1 (3Ha) where the subscr ipt N denotes the nucleus, p denotes the pos i tron and summation over i re fers to the Z electrons i n the atom (see Mott and Massey, 1965? p. 287). The wave equation descr ib ing the complete system i s given by Hr = EY. (34b) E i s the t o t a l energy of the system. The so lu t ion to a system such as (3*+a) and (3*+b) i s beyond the scope of the present day techniques, and rather d r a s t i c s impl i -f i c a t i o n s have to made i n order that the problem become at a l l t r a c t a b l e . The simplest approximation to th i s many-particle problem involves the use of an e f fec t ive two-body i n t e r a c t i o n . In the f i e l d of the low energy e lectron scat ter ing from noble gas atoms, such a s i m p l i f i c a t i o n has met with considerable success (Holtsmark, 1929; K i v e l , 1959; Labahn and Callaway, 1966). In th i s model the incoming e lectron scatters from the unperturbed atom, the I n t e r a c t i o n p o t e n t i a l being that due to the average e l ec tron ic charge d i s t r i b u t i o n i n the atom. The long range e l e c t r i c p o l a r i z a t i o n of the atom by the incoming e l ec tron , expected from f i r s t order perturbat ion theory (Crown and Russek, 1965), i s taken in to account by the a d d i t i o n , to the atomic p o t e n t i a l , of an a t t r a c t i v e term which behaves as ct/R^ for large electron-atom separations R. The parameter ct i s chosen to be the c l a s s i c a l e l e c t r i c p o l a r i z a b i l i t y of the atom. E a r l y work by Holtsmark (1929) showed that a two-body i n t e r a c t i o n cons i s t ing of the two potent ia l s indicated above was able to reproduce the Ramsauer ef fect i n Argon and other noble gases i n some d e t a i l . Much of the recent work i n the scat ter ing of electrons from noble gases has confirmed th is point of view ( K i v e l , 1959; Labahn and Callaway, 1966). 3.1.1. Discuss ion of the dif ferences between low-energy pos i tron and e lectron s c a t t e r i n g . In a recent review of the subject of approximate p o l a r i -za t ion potent ia l s for the electron-Helium i n t e r a c t i o n (Labahn and Callaway, 1966), the o v e r a l l lack of s e n s i t i v i t y of the v e l o c i t y --66-dependence of the t o t a l scat ter ing cross-sect ions to de t a i l s of the e f f ec t ive scat ter ing p o t e n t i a l i s c l e a r l y demonstrated. Good agreement with experiment i s obtained for a v a r i e t y of potent ia l s as long as they exhib i t a 1 / R dependence for large R, even though they d i f f e r s u b s t a n t i a l l y at the edge of the Helium atom. This chapter contains the resu l t s of ca lcu la t ions which, when compared with the experimental resu l t s of Chapter 2 i i n d i c a t e that the case of positron-Argon scat ter ing i s far less ambiguous. I t i s poss ible i n such a c a l c u l a t i o n to compute not only the scat ter ing cross - sec t ion as a funct ion of energy but a lso the p o s i t r o n - e l e c t r o n a n n i h i l a t i o n ra tes . The l a t t e r depend s o l e l y on the overlap of pos i tron and e lectron density i n the atom. An experiment which depends on both the scat ter ing cross-sect ions and a n n i h i l a t i o n rates as a funct ion of v e l o c i t y should thus impose more s tr ingent condit ions on the choice of p o t e n t i a l than experiments which depend on one alone. Such a s i t u a t i o n i s provided i n the a n n i h i l a t i o n of posi trons e l a s t i c a l l y scat ter ing i n Argon gas under the inf luence of an appl ied dc e l e c t r i c f i e l d . The pos i tron v e l o c i t y d i s t r i b u t i o n i s then determined to f i r s t order by the momentum-transfer cross-sec t ion and e l e c t r i c f i e l d . The observed a n n i h i l a t i o n rate i s given by the velocity-dependent a n n i h i l a t i o n rate averaged over th i s v e l o c i t y d i s t r i b u t i o n . Thus, i n general , the a n n i h i l a t i o n rate i s a funct ion of the appl ied e l e c t r i c f i e l d . Further , the scat ter ing of positrons from atoms d i f f e r s s i g n i f i c a n t l y from that for electrons i n that the pos i tron i s -67-di s t ingu i shab le and i s thus not prevented by the P a u l i p r i c i p l e from having s i g n i f i c a n t wave funct ion overlap with the electrons of the atom. For th is reason, short-range pos i t ron-e l ec tron c o r r e l a t i o n effects may be expected to play a more important ro l e than i n the corresponding case for e lec tron s ca t t er ing . The a n n i h i l a t i o n r a t e , i n p a r t i c u l a r , should r e f l e c t the de ta i l ed extent to which both the atomic and pos i tron wave functions are d i s tor ted by the i n t e r a c t i o n . F i n a l l y , i t may be expected that both long-range p o l a r i z a t i o n of the atom and short-range c o r r e l a t i o n of the pos i t ron-e l ec tron wave functions w i l l play an important part i n the e f fec t ive i n t e r a c t i o n . As mentioned i n Sect ion 3»1.» the problem of pos i t ron-atom scat ter ing i s complicated by the many p a r t i c l e aspect.- It has been pointed out that a very simple two-body approximation has been p a r t i c u l a r l y use fu l i n the e lectron scat ter ing case. It i s de s i rab le , then, to know the extent to which such two-body approximations apply to the case of pos i tron s ca t t er ing . Furthermore, i t i s of in teres t to e s tab l i sh whether the ambiguity which ar ises i n e lec tron sca t ter -ing occurs for pos i trons . Accordingly , the t h e o r e t i c a l resu l t s of th i s Chapter were obtained using the empir ica l p o l a r i z a t i o n potent ia l s of the type used i n e lec tron sca t t er ing . These take into account the long-range p o l a r i z a t i o n i n terms of a p o t e n t i a l whose asymptotic behaviour i s 1/R (see also Moussa, 1959). Some c u t - o f f parameter i s always used i n order that the p o l a r i z a t i o n p o t e n t i a l " Vp remain f i n i t e at the o r i g i n . The e f fec t ive i n t e r a c t i o n V i s taken to be the sum of V p and the p o t e n t i a l V H charac ter i z ing the Hartree-Fock s e l f consistent f i e l d for Argon i n the ground -68-state (Hartree and Hartree , 1936). The resu l t s ind ica te quite c l e a r l y that these simple two-body potent ia l s are inadequate to account for the experimental r e s u l t s . 3.1.2. Out l ine of procedure. In order to compute the e f fect of' the e l e c t r i c f i e l d on the a n n i h i l a t i o n rate of positrons i n a gas both the v e l o c i t y dependent a n n i h i l a t i o n rate \>a(v) and the momentum-transfer cross - sec t ion a d ( v ) have to be known. For a s p e c i f i c model of the positron-atom i n t e r a c t i o n , these can be found by obtaining the u s u a l \ p a r t i a l wave so lu t ion to the appropriate Schrodinger equation. I t should be emphasized that the analys is of th is Sect ion neglects any contr ibut ion to the a n n i h i l a t i o n rate from r a d i a t i v e ; capture of a pos i tron into a bound A r - e + system. Neither the existence nor p r o b a b i l i t y of formation of such bound states has been discussed i n any quant i ta t ive way i n the l i t e r a t u r e . Should i t e x i s t , i t i s expected that the l i f e t i m e would be of the order of the parapositronium l i f e t i m e (10~ 1 0 sec) . I f the capture rate i s comparable, then, to the d i r e c t a n n i h i l a t i o n rate due to the pos i t ron-e l ec tron overlap during e l a s t i c s c a t t e r i n g , competition from th i s channel could represent a s i g n i f i c a n t contr ibut ion to the o v e r a l l observed d i r e c t a n n i h i l a t i o n ra te . The v e l o c i t y d i s t r i b u t i o n of positrons under the inf luence of an e l e c t r i c f i e l d i n a gas at temperature T i s then determined using the ; 0 J (V ) and ^ ( v ) discussed above. The d i f f e r e n t i a l u a equation descr ib ing th i s s i t u a t i o n i s s i m i l a r to the Wilkins -69-equation used to describe thermal neutron d i f f u s i o n where neutron capture i s important (Sobrino and C l a r k , 1961), except that , i n t h i s case, there i s an a d d i t i o n a l term due to the appl ied e l e c t r i c f i e l d . In a d d i t i o n , the positrons i n the gas approximate a Lorentz gas to a very high degree by v i r t u e of t h e i r small mass and low dens i ty . The r e s u l t i n g d i f f e r e n t i a l equation has been reported previous ly i n pre l iminary work done on th i s problem ( F a l k , 1965; F a l k , Orth and Jones, 1965). Once the v e l o c i t y d i s t r i b u t i o n i s obtained for a p a r t i c u -l a r o J ( v ) , v (v) and e l e c t r i c f i e l d , the d i r e c t a n n i h i l a t i o n rate d a X i s found by averaging the velocity-dependent a n n i h i l a t i o n rate over the ent ire v e l o c i t y d i s t r i b u t i o n . 3.2. The two-body Schrodinger equation and i t s s o l u t i o n . 3.2.1. The Schrodinger equation. Because of the spher ica l symmetry of the e f f ec t ive posi tron-Argon atom i n t e r a c t i o n assumed, the problem i s reduced to so lv ing the r a d i a l part of the relevant Schrodinger equation. This i s , i n atomic uni t s (Wu and Ohmura, 1962) 2 C d W + k 2 - } - 2V(r)] X£ = 0 (35) where V(r) = h£± - V p « (35a) i s the e f f ec t ive i n t e r a c t i o n p o t e n t i a l at a distance r from the 2 o r i g i n , and k i s the k i n e t i c energy of the inc ident pos i tron ( i n Rydberg uni ts ) The term Z „ ( r ) takes into account the screening of the - 7 0 -nucleus by the atomic e lec trons . Thus, for Argon Z p ( 0 ) = 1 8 a 0 " 1 , and Z p C 0 0 ) = 0 a Q 1 , where a Q i s the Bohr rad ius . For a l l the potentials , used here, the Hartree-Fock part of the i n t e r a c t i o n ( Z p ( r ) / r ) , ascribed to the unperturbed Argon atom, i s due to Hartree and Hartree ( 1 9 3 6 ) . 3 . 2 . 2 . C a l c u l a t i o n of phase sh i f t s and wave funct ions . The so lut ion of ( 3 5 ) was undertaken using the Runge-Kutta method for so lv ing d i f f e r e n t i a l equations. The method of s o l u t i o n consisted of in tegrat ing the d i f f e r e n t i a l equation numerical ly from a point near the o r i g i n u n t i l the asymptotic form for x^ ? which i s known a n a l y t i c a l l y , i s obtained. In order to begin the i n t e g r a t i o n , the slope of x A , dx^/dr , was set to some a r b i t r a r y i n i t i a l value at a small distance r away from the o r i g i n . For non-zero A , -and kr<<A , the corresponding i n i t i a l value of obtained by so lv ing C d r ^ ] x * " 0 ( 3 6 ) Is given by ( 3 7 ) This ensures that xA has the correct r e l a t i o n s h i p to i t s slope near the o r i g i n . For 4 s 0 , and rZ (r) <,<. 1 ,' the i n i t i a l value of x n i s p u given by x 0 • r < 3 8 > -71-where x Q i s the so lu t ion to r d 2 2ZL(r)-| _ n [ dP" ~ - ^ - ] X Q = 0 « (39) i n the region where Z p ( r ) =Z. Expressions (37) and (38) can be combined to give v _ r dx0 for kr « £ i f * >0, and r Z p ( r ) « 1 i f l- 0. The absolute magnitude of was determined by matching the numerical so lu t ion for xA to the appropriate asymptotic expression for x £ . The value of the asymptotic x £ was found by _2 _ i normaliz ing the incoming pos i tron f lux to 1 pos i tron cm sec i n the usual manner (Wu and Ohmura, 1962). 3.2.2.1. Asymptotic so lu t ion for k i 0. For k unequal to zero, the asymptotic so lu t ion for x £ i n the region where the V ( r ) term i s n e g l i g i b l e compared with the others i n (35) i s X A ( r ) = kr CA £j J l(kr) - B £n £(kr)] (41) where j ^ k r ) , n^Ckr) are spher i ca l Bessel functions of the f i r s t and second k i n d . For s t i l l larger r where, i n a d d i t i o n , the iU+i ) ^ 2 — £ 2 — term i s n e g l i g i b l e compared with k , th i s becomes (Wu and Ohmura, 1962) X £ ( r ) = sin (kr - | ^ + 6 £) (42) where tan h = (43) -72-and Cj = Aj + Bj . (44) Thus i t i s unnecessary to compute the wave funct ion up to .values of r for which i t becomes s i n u s o i d a l , s ince the phase s h i f t and normal izat ion constant can be ca lcu la ted when the wave funct ion s a t i s f i e s (41). For any r sa t i s fy ing the asymptotic requirement • (V(r) i s n e g l i g i b l e ) , the constants Ajt, % can be ca lcu la ted from x& and . D i f f e r e n t i a t i n g (41 ) with respect to r gives jj*£ = (t+1) x £ / r - k r CA^j^Ckr) - B £ n £ + 1 (kr)] (45) where use has been made of the r e l a t i o n where f t i s any spher i ca l Besse l funct ion . Solving (41 ) and (45) for A£ , and B^ y i e lds A. = x . [(W) ^ t o ) - tol+1(kr)] - ^ ( k r ) . ^ B £ = xA CU+1) j 4 (kr) - kr j £ + 1 (kr ) ] - r S J ^ C k r ) In prac t i ce the r a t i o B jt/Ae, was continuously monitored as the d i f f e r e n t i a l equation (35) was solved. Once the r a t i o B £ / A £ converged, the c a l c u l a t i o n was .terminated, and the phase s h i f t and normal izat ion constant Cj_ found using equations (43) and (44) r e s p e c t i v e l y . The phase sh i f t s were then used to ca lcu la te the p t o t a l momentum-transfer cross-sect ions o d ( in units of i r a Q ) from (Bowe, 1960) o d = p I (4+1) sin2(6£ - 6 £ + 1 ) (47) 0 In p r a c t i c e , a l l the phase sh i f t s for p a r t i a l waves up to and -73-inc lud ing 4 =5 were c a l c u l a t e d , the higher order phase sh i f t s being n e g l i g i b l e . 3.2.2.2. Asymptotic so lut ion for k =0. For the case of very small inc ident v e l o c i t y , and large 2 2 r (that i s , k and V(r) <•< 4(4+1)/r ) , the Schrodinger equation (35) s i m p l i f i e s to r d 2 4U+D-, _ n ,.lQ. Cgpr- — r T " ] *£ ~ 0 (48) This free p a r t i c l e equation has as i t s general so lu t ion XA = C i r 1 + 1 + C 2/r l. (49) However, i t i s only necessary to compute the wave funct ion for 4=0 for th i s spec ia l case as a l l the other phase s h i f t s are small i n comparison with $Q. This occurs only because a l l the potent ia l s used here f a l l o f f more r a p i d l y than 1/r^ (Landau and L i f s h i t z , 1965; p.500). Therefore, x 0 i s appropriate ly normalized i f at large r XQ = r + C 2 (50) The constant C-j has been set to un i ty since asymptot ica l ly the wave funct ion R =x/r has uni ty amplitude. The momentum-transfer cross - sec t ion i s now i d e n t i c a l to the t o t a l e l a s t i c scat ter ing cros s - s ec t i on , the l a t t e r being expressed by (Landau and L i f s h i t z , 1965; p.500) a = 4 ( 0 ^ ) 2 (51) I f the wave funct ion is normalized (that i s , 0^1), th i s reduces to -74-o - 4C 2 2 (&2> LL F o r s m a l l e r v a l u e s o f r , f o r w h i c h t h e a / r t e r m i n V ( r ) i s 2 s i n i f l e a n t a n d t h e k t e r m s t i l l n e g l i g i b l e , t h e s o l u t i o n t o t h e S c h r o d i n g e r e q u a t i o n f o r A= 0 C | p XQ'» 0 (53) i s g i v e n b y ( L a n d a u a n d L i f s h i t z , 1965; p.504-) w h e r e Y S /2a F o r l a r g e r , t h i s e x p r e s s i o n f o r x Q ( r ) t e n d s t o t h e a s y m p t o t i c f o r m g i v e n i n e q u a t i o n (50), i f i n a d d i t i o n (51*) i s n o r m a l i z e d b y p u t t i n g C ^ = 1 . T h e c o n s t a n t s C- a n d C 2 c a n a l s o b e e x p r e s s e d i n t e r m s o f x Q , i i x o / d r ( s e e S e c t i o n 3.2.2.1.) a n d t h i s y i e l d s <=! ' ^ % < ¥ - £ * 0 ( r ) r . l ( l> T h e w a v e f u n c t i o n f o r v e r y s m a l l i n c i d e n t v e l o c i t y w a s t h u s f o u n d b y i n t e g r a t i n g (35) ( f o r k=0, a n d . •••fc«0) n u m e r i c a l l y u n t i l x Q w a s o f t h e f o r m g i v e n b y (5*+) • T h i s w a s c h e c k e d b y c o n t i n u o u s l y m o n i t o r i n g t h e r a t i o C 2 / C ' a s g i v e n b y (55) ' O n c e t h e r a t i o c o n v e r g e d , t h e c a l c u l a t i o n w a s t e r m i n a t e d a n d t h e m o m e n t u m - t r a n s f e r c r o s s - s e c t i o n w a s f o u n d u s i n g (51)« T h e w a v e f u n c t i o n w a s a l s o n o r m a l i z e d b y s e t t i n g C . * 1 , i n o r d e r t h a t t h e a p p r o p r i a t e Z e f f c o u l d b e c a l c u l a t e d . - 7 5 -3.3* C a l c u l a t i o n o f Z ? f f . As mentioned i n Chapter 1, S e c t i o n 1.3.1. t h e d i r e c t a n n i h i l a t i o n r a t e o f p o s i t r o n s i n a gas i s p r o p o r t i o n a l t o th e gas d e n s i t y , and t o t h e e l e c t r o n d e n s i t y I*-!2 a t the p o s i t r o n averaged o v e r t h e p o s i t r o n p o s i t i o n . T h i s a n n i h i l a t i o n r a t e , i s g i v e n by ( F e r r e l l , 1956) v = rrr 2cnJd3x | iT| 2|V"| 2 (56) a, O S ™" where r Q i s t h e c l a s s i c a l e l e c t r o n r a d i u s , and n s i s t h e gas d e n s i t y i n atoms cm J , and c i s t h e v e l o c i t y o f l i g h t . E q u a t i o n (56) may be r e w r i t t e n i n t h e form v a ( k ) = i r r o 2 c n s Z e f f ( k ) (57) where Ze^f i s the v a l u e o f t h e i n t e g r a l i n ( 5 6 ) . F o r a p l a n e wave r e p r e s e n t a t i o n o f th e p o s i t r o n wave <J<f t h e v a l u e o f Zggv i s j u s t t he atomic number Z o f t h e atom. When th e p o s i t r o n - a t o m i n t e r a c t i o n i s t a k e n i n t o a c c o u n t , 'l' +must be found by s o l v i n g t h e r e l e v a n t S c h r o d i n g e r e q u a t i o n ( see S e c t i o n 3.2.). I n terms o f the x a o f S e c t i o n 3.2., * + i s g i v e n by ( f o r k*0) 00 * + = 1 $ £ P » ( c o s 9) (58) where the a r e Legendre p o l y n o m i a l s . P e r f o r m i n g the i n t e g r a t i o n o v e r a n g l e s i n t h e e x p r e s s i o n f o r Zeff t h e n shows t h a t each p a r t i a l wave c o n t r i b u t e s t o t h e t o t a l Z e f - r , t he 4 t h p a r t i a l wave c o n t r i b u -t i o n b e i n g W i ^ C I * " ' 2 ^ 1 1 ^ * ( 5 9 ) The e l e c t r o n d e n s i t y I*""!2 I n a l l t h e c a l c u l a t i o n s p r e s e n t e d here -76-i s t h a t a p p r o p r i a t e t o the Argon atom i n the ground s t a t e as c a l c u l a t e d by H a r t r e e and H a r t r e e (1936). The n u m e r i c a l c a l c u l a t i o n o f ( Z e f f ) £ i n v o l v e s an o v e r l a p i n t e g r a l f o r w h i c h Simpson's r u l e f o r n u m e r i c a l i n t e g r a t i o n was found s u f f i c i e n t l y a c c u r a t e . The i n t e g r a t i o n was c a r r i e d out t o r=7a 0 a t w h i c h p o i n t t h e e l e c t r o n d e n s i t y k~| 2 i s l e s s t h a n -5 10 o f i t s maximum v a l u e w i t h i n t h e atom. Fo r k=0, t h e Z'ef+- was c a l c u l a t e d f r om Z e f f .= £ i n ' ^ d r (60) where x Q ( r ) was c a l c u l a t e d as i n d i c a t e d i n S e c t i o n 3'2.2.2. 3-l+' 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 . 3.4.1. The m o d i f i e d W i l k i n s e q u a t i o n . The d i f f e r e n t i a l e q u a t i o n d e s c r i b i n g the v e l o c i t y d i s t r i b u t i o n o f p o s i t r o n s i n a gas i n the energy range where o n l y e l a s t i c c o l l i s i o n s can t a k e p l a c e has been shown t o be ( F a l k , 1965; F a l k , O r t h and J o n e s , 1965) ^ = v - ! ^ ^ D (61) - Cv a(v)-+ v f ( v ) ] f ( v , t ) where: f ( v , t ) i s t h e p r o b a b i l i t y d e n s i t y i n v e l o c i t y space a t time a«eE/m i s the a c c e l e r a t i o n o f t h e p o s i t r o n due t o t h e e l e c t r i c f i e l d ; e i s the p o s i t r o n charge; E i s t h e a p p l i e d e l e c t r i c f i e l d j m i s t h e p o s i t r o n mass; -77-v d< v ) = V d ( v ) v 5 «jj i s the momentum-transfer cross - sec t ion for positron-gas atom c o l l i s i o n s ; n i s the densi ty of scat ter ing atoms; v^(v) = ur *cn Z ~.(v)j (see Equation 57) ct O S S i * r i s the c l a s s i c a l e lec tron rad ius ; o c i s the v e l o c i t y of l i g h t ; v f(v) = n ga f(v)v; f ! a . p ( v ) i s the cross - sec t ion for positronium formation; v »m/M; M i s the mass of scat ter ing atoms; T i s the temperature of the host g a s . i n ° K ; , K i s Boltzmann's constantj The v e l o c i t y v i s re la ted to wave" number k by v=k<*c, where ot i s the f ine s tructure constant. The funct ion y ( v , t ) - v 2 f ( v , t ) i s the v e l o c i t y d i s t r i b u t i o n of positrons per un i t v e l o c i t y i n t e r v a l . By analogy with the Maxwellian and Druyvesteyn d i s t r i b u t i o n s and from phys i ca l cons iderat ions , i t i s expected that the boundary condit ions for y ( v , t ) are y ( o , t ) * 0 (62) y ( - , t ) = 0 for a l l phys i ca l ly meaningful momentum-transfer, a n n i h i l a t i o n and formation cross - sec t ions . -78-3.k.2. General computer so lu t ion of the d i f f e r e n t i a l equation. The equation (61) can be transformed into at 9vL 13v, m 1 9v . t y v v d 3v,v - mv j y w » x < J d d (63) - Cva(v) + v f(v)]y(v,t) I f some estimates can be made as to the i n i t i a l pos i tron v e l o c i t y d i s t r i b u t i o n y ( v , o ) , equation (63) can be solved by standard numerical techniques. The funct ion X(t) = / " y(v,t)v (v)dv / f"y(v,t)dv (64) O cl O y ie lds the ve loc i ty-averaged d i r e c t a n n i h i l a t i o n rate as a funct ion of time and i s the quantity which i s compared with the experiment-al ly-determined time-dependent d i r e c t a n n i h i l a t i o n ra te . Pre-l iminary resu l t s of th is type of c a l c u l a t i o n have been reported ( F a l k , Orth and Jones, 1965). However, the d i f f i c u l t y i n estimating a r e a l i s t i c y(v ,o) makes i t advisable to examine the equi l ibr ium solut ions to (63) which have no i m p l i c i t time dependence. 3 A . 3 « The case of no i m p l i c i t time dependence. Consider the case where the appl ied e l e c t r i c f i e l d i s zero and where the pos i tron a n n i h i l a t i o n rate i s n e g l i g i b l e . Then i t i s known that the i n i t i a l v e l o c i t y d i s t r i b u t i o n of positrons w i l l re lax af ter a time t' to a Maxwellian d i s t r i b u t i o n appropriate to the temperature T of the host gas. Since the ann ih i la t ions are n e g l i g i b l e i t can be assumed that they do not appreciably a f fec t the v e l o c i t y d i s t r i b u t i o n . Then the veloci ty-averaged a n n i h i l a t i o n rate at the time t 1 i s given by X(t') using (6*+). Because the shape of the v e l o c i t y d i s t r i b u t i o n remains Maxwellian a f ter t ' , -79-the average a n n i h i l a t i o n r a t e i s a constant t h e r e a f t e r . The p r o b a b i l i t y d e n s i t y of p o s i t r o n s , however, i s dependent on time, and can be seen to decrease e x p o n e n t i a l l y w i t h a r a t e g i v e n by x ( t * ) . . I t can be shown that n e i t h e r the a p p l i e d e l e c t r i c f i e l d nor the a n n i h i l a t i o n r a t e have to be n e g l i g i b l e f o r the e x p o n e n t i a l behaviour to be r e t a i n e d . I t i s o n l y necessary to r e q u i r e that a f t e r some time the v e l o c i t y d i s t r i b u t i o n has an e x p l i c i t dependence on time. Thus the shape of the v e l o c i t y d i s t r i b u t i o n i s unchanged f o r l a t e r times. P u t t i n g y ( v , t ) = T(t./Y(v) (65) s e p a r a t e s equation (63) i n t o d [ { a i + uv^KT }dY + , _ 2a£ _ 2y_^KT } = dv l3v, m Jdv 1 d 3v,v mv J d d > ( 6 6 ) [v + v- - X]Y a f and £ = - XT (67) at The s o l u t i o n to (65 J and (67), y ( v , t ) = Y ( v ) T ( o ) e i s c l e a r l y e x p o n e n t i a l i n t . The value of x i s found by i n t e g r a t i o n of (66) once w i t h r e s p e c t to v over a l l v e l o c i t i e s . Then [ { ai + y v d ^ } dY + , _ 2ai _ 2 ^ y ] ~ = 13v, m J dv 1 K d 3v ,v mv J o d d (68) f°° (v + - X)Ydv J o a 1 For p o t e n t i a l s of the type used here, the momentum-transfer- c r o s s -s e c t i o n tends to a constant value as the i n c i d e n t v e l o c i t y tends to zero (Landau and L i f s h i t z , 1965; p.500). Furthermore, the p r o b a b i l i t y d e n s i t y of p o s i t r o n s i n v e l o c i t y space, f ( v , t ) must be f i n i t e f o r v « 0 . I t f o l l o w s that Y(v) must tend to zero at -80-2 l e a s t as r a p i d l y as v nea r V P 0. For h i g h v e l o c i t i e s , i t i s expected the. momentum-transfer c r o s s - s e c t i o n remains f i n i t e . I n a d d i t i o n Y(v) must tend t o z e r o more r a p i d l y t h a n v"^ a t h i g h v e l o c i t i e s i n o r d e r t h a t the o v e r a l l energy a s s o c i a t e d w i t h the d i s t r i b u t i o n Y ( v ) , remain f i n i t e . Making use o f the l i m i t s e s t a b l i s h e d above, i t i s c l e a r t h a t the l e f t hand s i d e o f (68) v a n i s h e s , b o t h a t v = 0 and v=*». Hence x - 11 C v ( v ) + v~(v)]Y(v)dv / f" Y(v)dv (69) O a. ± O Thus when 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 becomes " s t a t i c " , t h e r e s u l t i n g a n n i h i l a t i o n r a t e i s a c o n s t a n t , and the p o s i t r o n p o p u l a t i o n decays e x p o n e n t i a l l y . T h i s e x p o n e n t i a l decay w i l l a l s o be e x h i b i t e d i f the a n n i h i l a t i o n c r o s s - s e c t i o n i s p r o p o r t i o n a l t o 1/v. The a n n i h i l a t i o n r a t e v & i s t h e n independent o f v e l o c i t y , and f r om (64) A(t) = v (70) a. 3 . 4 . 4 . S o l u t i o n o f the tim e - i n d e p e n d e n t e q u a t i o n . For a g i v e n s e t o f c r o s s - s e c t i o n s , the s o l u t i o n s t o (66) and (69) can be r e a d i l y o b t a i n e d u s i n g a d i g i t a l computer. C o n s i d e r the i n t e g r a l o f (66) from v'-O t o v'=-v. Then { a i +-yv^KT. }dY + { _ 2a^ _ ^ K T } y = l3v, m - Jdv l K d 3v,v mv J /r,n s d d (71) r ( v + v. - X)Ydv J o a r s i n c e t h e i n t e g r a l o f t h e l e f t hand s i d e o f (66) v a n i s h e s a t v=0 (s e e S e c t i o n 3 . 4 . 3 0 . I f the a n n i h i l a t i o n r a t e i s s m a l l compared w i t h the average s c a t t e r i n g r a t e , t h e n i t s e f f e c t on the v e l o c i t y -81-d i s t r i b u t i o n w i l l s i m p l y be a p e r t u r b a t i o n . I n Argon t h i s i s a 12 -1 good a p p r o x i m a t i o n s i n c e the s c a t t e r i n g r a t e i s about 10 sec a t 10 amagats f o r a s c a t t e r i n g c r o s s - s e c t i o n o f the o r d e r o f 2 i T a 0 w h i l e the a n n i h i l a t i o n r a t e i s e x p e r i m e n t a l l y about 6 - 1 5 x 10 sec f o r the same d e n s i t y (see T a b l e I V ) . Thus a f i r s t o r d e r s o l u t i o n t o Y i s o b t a i n e d by s o l v i n g (71) w i t h the r i g h t hand s i d e s e t e q u a l t o z e r o . T h i s y i e l d s Y Q = Cv^exp [- % yvv d / + H ^ ) d v ] ( 7 2 ) where C i s de t e r m i n e d by n o r m a l i z i n g t h e i n t e g r a l o f Y Q over a l l v e l o c i t i e s t o u n i t y . The f i r s t o r d e r s o l u t i o n t o * i s g i v e n by Xo = C ( v a + V Y d v ( 7 3 ) S u b s t i t u t i o n o f YQ, XQ i n t o the r i g h t hand s i d e o f (71) and s o l v i n g y i e l d s the second a p p r o x i m a t i o n t o Y. Y]_(v) = Y q ( V ) [Y + f 2 ( v » ) / Y Q(v')dv'] = Y Y q ( V ) + A Y 1 ( v ) (74) where f 2 ( v ) = Jl ( v a + v f - A O ) Y o ( v ' ) d V / ( ^ + H ^ ) (75) a l i m . . . . . S i n c e t h e V+Q ^ ( v ) / Y 0 ( v ) might be d i f f e r e n t from u n i t y the c o n s t a n t y i s i n c l u d e d , and can be c a l c u l a t e d by n u m e r i c a l i n t e -g r a t i o n o f (74) over a l l v e l o c i t i e s , and r e q u i r i n g t h a t i n t e g r a l s o f Y-|(v) and Y Q ( v ) b o t h be n o r m a l i z e d t o u n i t y . Thus any a p p r o x i -m a t i o n Y j i s thus o b t a i n e d i n terms of Y..^ u s i n g (74). The i t e r a t i v e p r o c e d u r e o u t l i n e d has been s u c c e s s f u l l y u s ed t o s o l v e e q u a t i o n (71). The i n i t i a l e s t i m a t e Y Q i s made u s i n g 8 - p o i n t G a u s s i a n i n t e g r a t i o n , and subsequent i n t e g r a t i o n s were made u s i n g Simpson's r u l e and l i n e a r i n t e r p o l a t i o n where -82-necessary. In. general , the procedure converged wi th in f i ve i t e r a t i o n s for a l l cross-sect ions used. The ef fect of the a n n i h i l a t i o n perturbat ion was to decrease the Y(v) r e l a t i v e to the o r i g i n a l Y Q (v) for those v e l o c i t i e s where the a n n i h i l a t i o n rates are large . Thus, for the monotonically decreasing a n n i h i l a t i o n rates considered, the v e l o c i t y d i s t r i b u t i o n was uniformly sh i f ted to a higher v e l o c i t y . In a d d i t i o n , for each v e l o c i t y d i s t r i b u t i o n that was c a l c u l a t e d , the average pos i tron v e l o c i t y v given by v = Y(v)vdv (76) was found. 3 - 5 - Resul t s . Pre l iminary resu l t s based on the procedure of the pre-ceding Sections have been reported (Jones, Fa lk and Orth , 1965; Jones and Orth , 1966 (a); Jones and Orth , 1966 (b) ) . In one case (Jones and Orth , 1966 (a ) ) , however, the published values of the ve loc i ty-averaged d i r e c t a n n i h i l a t i o n ra te , A a , are i n error to the extent that e l a s t i c - s c a t t e r i n g cross - sec t ions , rather than momentum-transfer cross-sect ions were used i n the v e l o c i t y d i s t r i b u t i o n c a l c u l a t i o n s , where only the Y Q were ca l cu la ted . 3-5-1 . Discussion of the potent ia l s used. Three representations of the positron-Argon i n t e r a c t i o n were s tudied. A l l exhibited a 1/R behaviour at large dis tances , but d i f f e r e d i n the s ize and type of the c u t - o f f employed and consequently i n the de ta i l ed shape of the'pos i tron-Argon i n t e r a c t i o n -83-i n the neighbourhood of the atom. The t o t a l i n t e r a c t i o n considered was the sum of the Hartree po tent ia l for Argon (Hartree and Hartree , 1936), VJJ, and the empir ica l p o l a r i z a t i o n p o t e n t i a l , Vp. Three types of V^ were s tudied: A. V p A * -5 .-5/(r2 + r D 2 ) 2 ; r 0 2 = 2.5 a 0 2 , B . V p B ^ -5.5 / ( r 2 •+ r Q 2 ) 2 ; r Q 2 = 0.62 a Q 2 , C. V p C = -5 .5r A(1 - e x p ( r / r 0 ) 8 ) ; r Q 8 = 14 a Q 8 . In each case, the numerator i s just the e l e c t r i c p o l a r i z i b i l i t y of Argon i n atomic units (Holtsmark, 1929). The form of the A B potent ia l s Vp , V^ has been used i n reproducing the observed Ramsauer ef fect ( K i v e l , 1959) i n Argon, and i s known i n the l i t e r a -ture as the Buckingham semi-empirical form (Labahn and Callaway, 1966) and sometimes as the Holtsmark p o t e n t i a l (Massey, et a l . , 1966). 3.5.2. Comparison with experiment. A 2 2 In V , r n = 2.5 a was selected for the cu t -o f f , p ' o " o ' s ince th i s was the value found to f i t low-energy e lectron scat ter ing i n Argon ( K i v e l , 1959). A s i m i l a r e f fec t ive i n t e r a c t i o n was employed by Massey, et a l . (1966) i n ca lcu la t ions of the momentum-transfer cross - sec t ion and Z e f f for positrons i n Argon, and the cut-2 of f parameter r Q chosen was also the value appropriate to e lectron s c a t t e r i n g . . Figures 15 and 16 show respec t ive ly the v e l o c i t y dependence of the Z e £ £ and momentum-transfer cross-sect ions r e s u l t i n g from the so lu t ion of the Schrodinger equation. The values obtained for C U ^ V B B " v * C v K v S C " V * > 100 — 5 0 LU > -EFFECT O ISI A. 5 , ^ * ^ A ^ ^ ^ ^ ^ """" 1 1 I 1 I 1 i 1 1 1 i 1 I 1 I 1 1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 k - POSITRON VELOCITY ( 0 Figure 15. Theoretical r e s u l t s f o r Z e f f as a function of positron wave number k. C U R V E A • cu Ave a • Cu Kve c • II V.' 2 I 1 1 1 I I I I I I I • I • I • 1 0.0 0.1 0.2 0 3 0.4 0.5 0.6 0.7 0.8 k — POSITRON VELOCITY (a0 H) Figure 16. Theoretical r e s u l t s f o r the momentum-transfer cross-section f o r positrons i n Argon as a function of k. -8*+-V^ a r e i n agreement w i t h those r e p o r t e d by Massey, e t a l . (1966). Jr Agreement w i t h r e s u l t s o b t a i n e d i n d e p e n d e n t l y i s n e c e s s a r y i n c a l c u l a t i o n s o f the type p r e s e n t e d h e r e , p r o v i d i n g an i m p o r t a n t check on the n u m e r i c a l t e c h n i q u e s employed. The r e s u l t s o f the v e l o c i t y - d i s t r i b u t i o n c a l c u l a t i o n s ( S e c t i o n 3AO a r e p r e s e n t e d i n F i g u r e 17' Comparison w i t h the dependence o f the a n n i h i l a t i o n r a t e on the a p p l i e d e l e c t r i c f i e l d o b t a i n e d e x p e r i m e n t a l l y shows t h a t t h e c a l c u l a t e d a n n i h i l a t i o n r a t e (A) i s f a r too s m a l l . F u r t h e r m o r e , the t h e o r e t i c a l c u r v e does n o t v a r y s u f f i c i e n t l y r a p i d l y w i t h a p p l i e d e l e c t r i c f i e l d . I n o r d e r t o re p r o d u c e a t l e a s t one a s p e c t o f the e x p e r i -m e n t a l r e s u l t s , the c u t - o f f parameter r 0 i n V was chosen t o be Jr 0.62 a Q so t h a t the c a l c u l a t e d a n n i h i l a t i o n r a t e a t z e r o e l e c t r i c f i e l d would approximate the e x p e r i m e n t a l r e s u l t . R e d u c t i o n o f t h e c u t - o f f parameter by such a l a r g e amount r e s u l t s i n an i n c r e a s e d p o s i t r o n a t t r a c t i o n , and t h i s i s r e f l e c t e d i n the g r e a t l y i n c r e a s e d momentum-transfer c r o s s - s e c t i o n ( F i g u r e 16). I n a d d i t i o n , t h e r e s u l t i n g i n c r e a s e i n the p o s i t r o n w a v e - f u c t i o n i n s i d e the atom i s i n d i c a t e d d i r e c t l y by the l a r g e r Z g f f ( F i g u r e 15). A l t h o u g h , by t h i s means, t h e magnitude o f t h e t h e o r e t i c a l a n n i h i l a t i o n r a t e i s made alm o s t e q u a l t o t h e e x p e r i m e n t a l v a l u e a t z e r o f i e l d , t he dependence o f the a n n i h i l a t i o n r a t e on a p p l i e d e l e c t r i c f i e l d i s too weak a t low e l e c t r i c f i e l d s ( F i g u r e 17). •g An i m p o r t a n t f e a t u r e f o r the c u r v e f o r Vp i n F i g u r e 17 i s the -1 -1 " b r e a k " w h i c h o c c u r s a t about 70 V cm amagat . F o r t h i s v a l u e o f E/P, t h e mean p o s i t r o n wave number as c a l c u l a t e d f rom E q u a t i o n Figure 17. Comparison of t h e o r e t i c a l and experimental a n n i h i l a t i o n rates as a funct ion of E/P. -85-(76), Sect ion 3-4.h. is.0.23 a Q . From Figure 16, th i s corres -2 ponds to a momentum-transfer cross - sec t ion of about 1 0 IT a Q . The s ign i f i cance of th i s cross - sec t ion at the break w i l l be d i s -cussed s h o r t l y . The po ten t ia l desribed by V c i s a lso a one parameter p o t e n t i a l , but follows more c l o s e l y the form of the p o l a r i z a t i o n p o t e n t i a l expected for inc ident electrons from a phys i ca l argu-ment (Lenander, 1966; Crown and Russek, 1965). This po ten t ia l rise's to a maximum near the surface of the atom and decreases r a p i d l y ins ide the atom. For pos i trons , however, the ef fect of e l ec tron-pos i t ron corre la t ions should r e s u l t i n an enhanced e f fec t ive p o t e n t i a l wi th in the atom as compared with the case of inc ident e lectrons . This i n t e r a c t i o n may be expected to be more important for low v e l o c i t y pos i trons , thus g iv ing r i s e to a velocity-dependent e f fec t ive p o t e n t i a l . This type of v e l o c i t y -dependent p o t e n t i a l i s not considered here, however. o o The s ize of the cu t -o f f parameter r D =. 1*+ a Q has been chosen so that the ZQff at zero v e l o c i t y coincides with the Zeff — 1 B at k =• 0 a Q ~ ca lcu lated using Vp (Figure 15). Again the momentum-transfer cross - sec t ion i s very large at k = 0 a 0 - 1 (Figure 16 ) , and f a l l s o f f r a p i d l y to a f a i r l y constant value . The dependence of a n n i h i l a t i o n rate on e l e c t r i c f i e l d (Figure 17) bears no resemblance to the experimental r e s u l t s . The p o s i t i o n of the break mentioned . ear l i er i s less wel l defined for th i s case, but can be taken to be i n the region of 120 V cm-^ amagat -''. The average pos i tron wave number corresponding to th i s f i e l d as -86-ca lcu la ted from equation (76), Sect ion 3 A A . i s 0.23 a . At th i s wave number the momentum-transfer cross - sec t ion i s once more 2 of the order of 10 * a . o 3«5'3« Discussion of the break i n the dependence of a n n i h i l a t i o n rate on e l e c t r i c f i e l d . The s ize of the momentum-transfer cross - sec t ion at the break appears to be independent.of the de ta i l ed shapes of the momentum-transfer cross - sec t ion and Zeff, but depends simply on the monotonic decrease of both the momentum-transfer cross - sec t ion and Zeff. This fac t can be demonstrated with the a id of equation ( 6 6 ) , Sect ion 3.*+.3. At low e l e c t r i c f i e l d s , the form of the pos i tron v e l o c i t y d i s t r i b u t i o n i s e s s e n t i a l l y independent of the e l e c t r i c f i e l d due to the fac t that the energy gain term, a /3v^ i s swamped by the e l a s t i c scat ter ing term, yv^KT/m. Once the e l e c t r i c f i e l d 2 i s large enough for a / 3 V £ to be comparable to uv^KT/m, however, a t r a n s i t i o n occurs where the shape of the v e l o c i t y d i s t r i b u t i o n becomes determined p r i m a r i l y by the value of the f i e l d term. Once the e l e c t r i c f i e l d i s increased beyond th is po int , the average pos i t ron v e l o c i t y increases more r a p i d l y with e l e c t r i c f i e l d , provided that there i s no accompanying increase i n the momentum-transfer cross - sec t ion . I t i s evident that a t r a n s i t i o n of th i s type w i l l lead to the break i n the e l e c t r i c - f i e l d dependence of ve loc i ty-averaged a n n i h i l a t i o n ra te , i f the Z i s a monotonically decreasing funct ion of v e l o c i t y . -87-In order, then, that the v e l o c i t y d i s t r i b u t i o n be mainly determined by the e l e c t r i c f i e l d term, the momentum-transfer cross-sect ion at the average wave number k appropriate to the v e l o c i t y d i s t r i b u t i o n must approximately s a t i s f y i n a host gas at 298°K. When the magnitude of the e l e c t r i c f i e l d - 1 - 1 — -1 (E/P = 70 V cm amagat ) and average wave number (k = 0.23 a 0 ) appropriate to the break i n curve B , Figure 17 are subst i tuted into (77), the i n e q u a l i t y i s < l 5 7 r a 0 2 . This i s i n agreement with the d iscuss ion i n Sect ion 3«5«2. where the value f o r o d at the average wave number corresponding to the break i n curve B was found from a de ta i l ed c a l c u l a t i o n . 3 . 5 A . Discuss ion of the experimental dependence of a n n i h i l a t i o n • rate on e l e c t r i c f i e l d . I t i s poss ib le to discuss the experimental resu l t s i n Figure 17 i n the l i g h t of the previous Sect ion . Compared with the t h e o r e t i c a l a n n i h i l a t i o n rates , the experimental values de-crease r a p i d l y at considerably smaller e l e c t r i c f i e l d s . Further-more, there i s evidence that the break s tarts to occur at E/P -1 -1 values less than 15 V cm amagat . The average pos i tron v e l o c i t y at the break must be of the order of thermal v e l o c i t i e s ( k ~ 0 . 0 5 a 0 ~ — -1 -1 or s l i g h t l y l a r g e r . Now for k = 0.05 a Q and E / P » 15 V cm amagat the expression (77) y i e l d s o ( j < '.^s-^, while larger k give r i s e to smaller upper l i m i t s for ° d - Thus i t i s reasonable to suppose that the momentum-transfer cross - sec t ion for positrons i n Argon -88-i s less than about \ ^ ^ B . 0 at thermal energies. I t i s in t ere s t ing to observe that the estimation of th i s upper l i m i t for the low-energy momentum-transfer cross - sec t ion i s independent of any assumptions regarding the nature of the positron-atom i n t e r a c t i o n , and i s a lso independent of a poss ible contr ibut ion to the observed a n n i h i l a t i o n rate from r a d i a t i v e capture processes. The momentum-transfer cross - sec t ion cannot r i s e appreciably at higher v e l o c i t i e s i n view of the increased positronium formation as a funct ion of e l e c t r i c f i e l d (Figure 9; see also Figure 8). For example, the amount of increased positronium formation at 100 V cm - 1amagat 1 indicates that an appreciable f r a c t i o n of the equ i l ibr ium d i s t r i b u t i o n i s at the threshold v e l o c i t y for pos i t ron-ium formation; that i s , at k ^ 0.8 a 0 ~ 1 . Further evidence for th i s i s to be found i n the f l a t t e n i n g o f f of the d i r e c t a n n i h i l a t i o n rate at high e l e c t r i c f i e l d s , which has been a t t r i b u t e d to increased positronium formation (Chapter 2, Sect ion 2.6.3.). For the e l e c t r i c f i e l d to dominate i n th i s region (as ind ica ted , i n f a c t , by the r a p i d dependence of positronium formation on e l e c t r i c f i e l d i n t h i s region) requires that < 5 1 t a 0 . 3.6. Conclusions. 3.6.1. Summary of experimental r e s u l t s . The d i r e c t component of pos i tron a n n i h i l a t i o n i n Argon has been measured as a funct ion of dens i ty . The a n n i h i l a t i o n rate appears to obey a l i n e a r dependence up to about 10 amagats, - 8 9 -with some i n d i c a t i o n of a n o n - l i n e a r i t y at higher d e n s i t i e s . The orthopositronium quenching rate has also been deter-mined as a funct ion of dens i ty . Again the resu l t s are consistent with a l i n e a r dependence up to about 17 amagats. The d i r e c t a n n i h i l a t i o n rate and r e l a t i v e orthopositronium production were both measured as a funct ion of appl ied e l e c t r i c f i e l d . These measurements have been used to provide an i n t e r n a l l y consistent p ic ture of the behaviour of the positrons i n Argon under the inf luence of an e l e c t r i c f i e l d . 3 . 6 . 2 . Theore t i ca l conclusions . The e l e c t r i c f i e l d resu l t s have been used as a test of the current models of in terac t ions of inc ident electrons and positrons with m u l t i - e l e c t r o n atoms. I t has been shown here that the simple one-parameter approximations for the e f f ec t ive atom-electron i n t e r a c t i o n , which have been found to y i e l d a reasonable f i t to the low-energy e l a s t i c scat ter ing data, are inadequate for the case of positron-Argon s c a t t e r i n g . The sharp i n i t i a l drop i n the observed d i r e c t a n n i h i l a t i o n rate as a funct ion of appl ied e l e c t r i c f i e l d i s not reproduced using the momentum-transfer cross-sect ions and a n n i h i l a t i o n rates derived using these p o t e n t i a l s . Furthermore, from the general arguments proposed i n Sect ion 3 ' 5 » 1 + . , i t appears that the momentum-transfer cross-sec t ion i n Argon i s less than 15 at thermal energies. In order to reproduce such a small momentum-transfer c r o s s - s e c t i o n , while requ ir ing a r e l a t i v e l y large a n n i h i l a t i o n - 9 0 -r a t e , i t seems that c e r t a i n effects neglected i n the simple model considered here should be taken into account. These effects inc lude the pos i t ron-e l ec tron c o r r e l a t i o n , e s p e c i a l l y i n the neighbourhood of the atom, and deformation of the Argon atom by the scat ter ing pos i t ron . The p o s s i b i l i t y of r a d i a t i v e capture of the pos i tron from the continuum into a bound Argon-posi tron system should a lso be expected to contribute to the observed d i r e c t a n n i h i l a t i o n r a t e . -91-REFERENCES. Alekseev, A . I . (1959). Soviet Phys. - JETP 2 , 1312. see also ( 1 9 5 8 ) . Soviet Phys. - JETP Z, 826. American Ins t i tu te of Physics Handbook, }+, 119. McGraw H i l l Book Co. , N.Y. Anderson, C D . ( 1 932 ) . Phys. Rev. }+]_, L+05. 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APPENDIX. MODIFICATIONS TO THE FAST-SLOW COINCIDENCE  CIRCUITRY USED BY FALK ( 1 9 6 5 ) . A block diagram of the fast-s low coincidence system i s shown i n Chapter 2, Figure 3 ' 1 . Photomult lp l ier c i r c u i t r y . The new 1+in. x 3 i n . diameter Na l (T l ) c r y s t a l s required replacement of the 2 i n . diameter RCA 6 8 1 0 photomult lp l ier tube used prev ious ly with the larger !+in. diameter RCA 701+6. The modified c i r c u i t for the dynode chain supplying the dynode potent ia l s i s shown i n Figure 1 8 . Since fast r iset imes were required at the anode the ef fect of inductance was minimized from stages 6 to 11 by use of heavy, short copper s t r i p as leads. Elsewhere, leads were kept as short as poss ib le . The photomulti-p l i e r high voltage was supplied by the commercial P r e c i s i o n Power Source, Model 122 B, manufactured by C a l i b r a t i o n Standards Inc. 2• Timing Pulse Generator. Both l i m i t e r s used by ,Fa lk have been replaced with t r a n s i s t o r i z e d equivalents . The c i r c u i t i s shown i n Figure 1 9 , and i s s i m i l a r to a type of c i r c u i t , employing tunnel diodes, that i s used for fas t coincidence work (Schwarzschild, 1963). Operation of the c i r c u i t i s as fol lows: the a r r i v a l of -95-an e lectron pulse at the p h o t o m u l t i p l l e r ' c o l l e c t o r (Figure 18) causes current to be conducted through the tunnel diode TD. S u f f i c i e n t current through the TD causes the tunnel diode to t r i g g e r . The threshold current required for t r igger ing i s set by the 10k ft potentiometer. Once the TD has t r iggered , the voltage at the base of t r a n s i s t o r T drops by about 0.2 V. The t r a n s i s t o r i s r a p i d l y dr iven to saturat ion and a pos i t ive voltage pulse of about 2.5 V appears at the 1 00ft r e s i s t o r . Hal f of the 25 mA produced i s dr iven through the 100ft cable into the input of the t imesorter . The maximum width of the output pulse i s defined by the time taken for the TD to return to i t s o r i g i n a l s tate . This time i s set by the LR network across the TD. The threshold l e v e l of the TD affects the re so lu t ion of the t imesorter to some degree. For each gamma-ray channel, the threshold was set so as to minimize the f u l l - w i d t h at h a l f -maximum of the prompt peak obtained with Na-22 i n aluminum. An advantage of th i s threshold method i s that noise pulses do not t r i g g e r the input to the t imesorter . The prompt re so lu t ion obtained with these timing pulse generators i s shown i n Figure 20 . Two curves are presented, one corresponding to those a n n i h i l a t i o n gamma rays which y i e l d pulses wi th in the 0.51 MeV fu l l - energy peak. The second was obtained for the same energy i n t e r v a l but inc luding only those pulses which occurred i n the "valley" region (Chapter 1, Section 2.2.1.). The prompt re so lu t ion had been previous ly optimized for the f i r s t case. There i s a small d i f ference i n the re so lu t ion -96-due mainly to time slewing, that i s , dependence of the t r igger ing time on the amplitude of the photomult lp l ier c o l l e c t o r pulse . 3. Ampl i f i ers and Single Channel Analyzers . The ampl i f i ers and s ingle channel analyzers used hy Falk have heen replaced with commercial t r a n s i s t o r i z e d units (Cosmic A m p l i f i e r , Model 901A and Single Channel Analyzer , Model 901 S . C . A . ) . The ampl i f i ers produce b ipo lar pulses which are fed in to the zero cross-over s ingle channel analyzers . Each s ingle channel analyzer has both a negative and a pos i t i ve output. The p o s i t i v e output was used to t r igger the coincidence gate (Chapter 2, Figure 3) described by F a l k . The negative pulses were counted by a sca ler (Nuclear Suppl ies , Model SA-250). Each sca ler was used to monitor the t o t a l number of counts admitted by the s ingle channel analyzers during a run. The number of counts i n the 1 .28 MeV gamma channel was used for normal izat ion from run to run. As the t y p i c a l number of counts encountered i n a run were of the 7 8 order of 10 - 10 , and the scalers were capable of sca l ing only to 10^ , a mechanical r e g i s t e r of counting capacity 10^  was connected to each sca ler count overflow. The c i r c u i t used for th i s purpose i s i l l u s t r a t e d i n Figure 21 . h. P i l e -up Re.iectors. The use of the Cosmic ampl i f i ers made i t poss ib le to feed the outputs of the ampl i f i ers d i r e c t l y into the p i l e - u p r e j e c t o r s , described i n F a l k ' s thesis (1965), without further a m p l i f i c a t i o n . The current ampl i f i er port ion of the detectors ) -97-cieveloped by F a l k was t h e r e f o r e n o t used. 5. ND 101 K i c k s o r t e r . The 100 c h a n n e l k i c k s o r t e r (Computing D e v i c e s o f Canada, Model AEP 2230) p r e v i o u s l y used was r e p l a c e d w i t h a 256 c h a n n e l a n a l y z e r ( N u c l e a r D a t a , Model 101). The d i f f e r e n c e i n the i n p u t r e q u i r e m e n t s t o t h e k i c k s o r t e r r e q u i r e d s l i g h t m o d i f i c a t i o n o f th e o u t p u t s t a g e s o f t h e t i m e s o r t e r v/hich i n no way a f f e c t e d t h e o p e r a t i o n o f the i n s t r u m e n t . 6. D i f f e r e n t i a l and i n t e g r a l l i n e a r i t y o f the t i m e s o r t e r . The d i f f e r e n t i a l and i n t e g r a l , l i n e a r i t y o f the o v e r a l l s ystem were o b t a i n e d r o u t i n e l y as r e p o r t e d p r e v i o u s l y (Jones and F a l k , 1965; F a l k , Jones and O r t h , 1965). and a r e shown i n F i g u r e 22' Two measurements o f t h e i n t e g r a l l i n e a r i t y were made, s e p a r a t e d by an i n t e r v a l o f 2 months. R e s u l t s o f t h e f i r s t measurement a r e shown i n F i g u r e 22. A l e a s t squares f i t t o t h i s s t r a i g h t l i n e y i e l d e d an average s l o p e o f 2.7*+-— 0.01 n s e c / c h a n n e l , . w h i l e t h e second measurement y i e l d e d 2.73 - 0.01 n s e c / c h a n n e l . The i n t e g r a l l i n e a r i t y measurement was used t o d e f i n e t h e average t i m e w i d t h p e r c h a n n e l ( t h a t i s , the time c a l i b r a t i o n o f t h e i n s t r u m e n t ) , w h i l e t h e d i f f e r e n t i a l l i n e a r i t y was used as a measure o f the a c t u a l i n d i v i d u a l w i d t h s . B o t h s e t s o f dat a were used f o r t h e e v a l u a t i o n o f t h e l i f e t i m e s ( C h a p t e r 2, Section2.5-)• 0 +• 2 8 7 5 V I 6 kV 10 UA TO TIMING PULSE CeNERATOR 39 KJl I zjl 500 pf 33 k A -( D 14 70 COSMIC A f W J R f i R 2 7 k^! s - H I — 100 c '0*1 A? x I TOTTA" < D 13 Cftio # 3 lOOpf 22 k i i 2 2 k J l 22 k J l 22 left. 22 kJZ. 22>KJ2 22 fc^< 2 2 kSl. £2 k i l 225 kA ( D 12 < D ll -( DiO -<D9 R C A 7 0 4 6 < D8 -(07 <D6 < D 5 < D 4 < D 3 < D Z -C D l GRID #2 j ( CATHODE Figure 18. Photomultiplierccircuit. 15V ! R $100/1 1N3716 TD Y~ To PM anode L HOOpH 1 JJ'H 3.3 kA 10 kXl 2N1195 !470il 100/1 . 0 1 L J F •o) 100 A cable Denotes heavy Ccrstrip F i g u r e 19. T i m i n g p u l s e g e n e r a t o r c i r c u i t . A-- -0.51 MeV peak position C-* 0.51 MeV valley position CHANNEL NUMBER F i g u r e 20. Prompt r e s o l u t i o n o f t h e e l e c t r o n i c system. The s p e c t r a were o b t a i n e d f o r Na-22 i n A l . Elect romechan ica l R e g i s t e r ! SODECO Model T C e B Z 5 E + 120 V 2 5 u F A A A " - K v w v — i I N 4 5 9 I X 2 N 3 4 4 0 2.2kJl: X Figure 2 1 . C i r c u i t f o r dr i v i n g electro-mechanical r e g i s t e r . RELATIVE CHANNEL WIDTHS (ARBITRARY UNITS) (Differential linearity) •* p •* t* TIME SEPARATION, BETWEEN PULSES (NSEC) ° (Integral linearity) 

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