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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 of PAUL HANS ROBERT ORTH B,Sc.,j  University of Capetown^ i960  B o Sc., (Hons) 9 University of Capetown,, 1961 M Sco 0  S  University of Capetown^ 1963  THURSDAY^ JANUARY 26,  I967  AT 2 s 30 P* M.  IN ROOM 301, HENNINGS BUILDING COMMITTEE IN CHARGE Chairmanz  I» McT. Cowan  G„ Jones L de Sobrino J , M Kennedy 0  0  F. W. Dalby J . H. Williamson D, C Frost  External Examiner; D„ A. L . Paul Department of Physics University of Toronto Toronto Research Supervisor:  G. Jones  THE ANNIHILATION OP POSITRONS IK ARGON . ABSTRACT The annihilation of positrons i n Argon has been investigated as a function of Argon density and j  applied electric f i e l d 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 i n g . Results obtained at zero electric f i e l d yielded a linear dependence on density for the direct amihi~ lation rate of (5.6 * O.l) x 10^ sec" amagat"""^ with 1  1  some evidence of. n o n ° l i n e a r i t y 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 1 0 sec" 6  1  in agree-  ment with the theoretical value of the free orthopositronium annihilation raite (7.2 x 10^ sec" ), 1  in  addition, an orthopositronium quenching rate of (0.29 * o.O"+) x 1 0 sec" araagat™ was obtained from 6  1  1  the linear dependence of the orthopositronium annihilation rate on density. The electric f i e l d 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 i n  a gas under the influence of an applied electric f i e l d . Furthermore3 these results have been compared with theo= r e t i c a l results for the direct annihilation rate obtained from one parameter representations of the effective positron=Argon atom interactions.  It is shown that, while  such potentials are successful i n describing the lowenergy elastic-scattering of electrons from noble gas atoms, they are inadequate for the case of positrons.  However  9  consideration of the way i n which the direct annihilation rate changes as a function of electric f i e l d leads to an 2 upper l i m i t of 157Ta  0  for the momentum-transfer cross-  section for positrons i n Argon at thermal energies.  Such  an estimate is shown to be independent of any assumption concerning the effective positron=Argon atom interaction.  GRADUATE STUDIES Positron Physics  F i e l d of Study %  Elementary Quantum Mechanics Nuclear Physics Special Relativity Theory Physics of Nuclear Reactions Theoretical Nuclear Physics Cosmic Rays and High Energy Physics Electronic Instrumentation  F, A Kaenrpffer Jo Bo Warren H Schmidt Bo L* White M McMillan J B, Warren F,K. Bowers Q  tt  0  6  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 )W Ro Falk,* PoH„Ro Orth and G, Jones. Effect of Electric Field on Positron Lifetimes i n Argon and Heliumo Phys* Rev* Letters 14, 507 (1965),. 0  :  G„ Jones, W» R* Falk and P^H.A,-Orth,,, Positron Annihilation i n the Noble Gases, p..-372, IVth International Conference on the Physics of Electronic and Atomic Collisions (abstracts of papers) Science 3  Bookcrafters Inc.,  N*Y  a  (1965)0  G Jones and P..H R,> Orth* The Annihilation of Positrons i n 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 ed« Academic Press e  0  s  (1966) N Yo a  Garth Jones and Robert Orth, Annihilation of Positrons i n Argon Gas. B u l l . Am* Phys* Soc. I I , 11, 7"+9 (1966),  THE ANNIHILATION OF POSITRONS I N ARGON by P a u l Hans R o b e r t B.Sc,  U n i v e r s i t y o f Cape Town,  B.Sc.( Hons.)., M.Sc,  Orth 1960  U n i v e r s i t y o f Cape Town, 1 961  U n i v e r s i t y o f Cape Town,  1963  A THESIS SUBMITTED I N P A R T I A L 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 required  t h e s i s as conforming  to the  standard  THE U N I V E R S I T Y OF B R I T I S H COLUMBIA December,  1966  In presenting for  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  an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h Columbia,, I a g r e e  t h a t t h e L i b r a r y s h a l l make i t f r e e l y study«  a v a i l a b l e f o r r e f e r e n c e and  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 o f 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 g r a n t e d by t h e Head o f my Department o r by h i s r e p r e s e n t a t i v e s .  I t i s understood that  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 w i t h o u t my w r i t t e n p e r m i s s i o n .  Department o f  pfr  The U n i v e r s i t y o f B r i t i s h C o l u m b i a Vancouver 8, Canada  copying  g a i n s h a l l n o t be a l l o w e d  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 A r g o n has been  g a t e d as a f u n c t i o n o f A r g o n d e n s i t y and a p p l i e d using  t h e t e c h n i q u e o f l i f e t i m e measurements.  were analyzed Results  obtained  , with  Lifetime  field spectra  thej maximum l i k e l i h o o d method o f c u r v e  a t zero  electric  field  yielded a linear  fitting.  dependence  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  t h a n 10 a m a g a t s .  ( 7 . 2 ioA)  x 10 s e c  free orthopositronium  i n d i c a t e d a zero  greater  density  intercept of  i n agreement w i t h  the t h e o r e t i c a l value  annihilation rate  6 —1 (7-2. x 10 s e c ) .  of the  In  q u e n c h i n g r a t e o f ( 0 2 9 - 0.Oh) x 1 0 ^  a d d i t i o n an orthopositronium -1  o  -1 amagat  positronium  was o b t a i n e d  field  r a t e and o r t h o p o s i t r o n i u m to provide positrons  from t h e l i n e a r dependence o f t h e o r t h o -  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  dependence o f t h e d i r e c t a n n i h i l a t i o n  formation  h a v e b e e n m e a s u r e d and a r e u s e d  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 t h e behaviour o f i n a gas u n d e r t h e i n f l u e n c e o f a n a p p l i e d e l e c t r i c  field.  F u r t h e r m o r e , t h e s e r e s u l t s h a v e b e e n 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 rate obtained meter r e p r e s e n t a t i o n s action. in  1  The d e n s i t y d e p e n d e n c e o f t h e l o n g - l i v e d com-  ponent o f the time s p e c t r a  sec  electric  f o r 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 ( 5 . 6 i 0.1) x 1 0 ^ s e c ~  on d e n s i t y amagat  using  investi-  f r o m one  of the e f f e c t i v e positron-Argon  I t i s shown t h a t , w h i l e  atom  parainter-  such p o t e n t i a l s a r e s u c c e s s f u l  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 t h e 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 t h e way i n w h i c h t h e 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 c h a n g e s as a f u n c t i o n o f e l e c t r i c 2 limit  o f l5 i ra Q  positrons  field  l e a d s t o an upper  f o r the momentum-transfer c r o s s - s e c t i o n f o r  i n Argon a t thermal  energies.  S u c h a n e s t i m a t e i s shown  t o be i n d e p e n d e n t o f a n y a s s u m p t i o n c o n c e r n i n g p o s i t r o n - A r g o n atom  interaction.  the  effective  - i v-  TABLE OF CONTENTS page ABSTRACT  i  i  L I S T OF TABLES  vlii  L I S T 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 g a s 2 1.2.1. I n t r o d u c t o r y r e m a r k s 1.2.2. D e s c r i p t i o n o f t h e a n n i h i l a t i o n t i m e s p e c t r u m 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 o n t h e slowing-down time 1,2.h. T h e 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 o n t h e 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 a n d m o m e n t u m - t r a n s f e r 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 -electron overlap 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 A r g o n 1-3-3- V e l o c i t y d e p e n d e n c e 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 1.3-^- E l e c t r i c f i e l d d e p e n d e n c e 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 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 Helium 1.h. P o s i t r o n i u m f o r m a t i o n a n d a n n i h i l a t i o n i n A r g o n 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 1 A . 3- P o s i t r o n i u m formation 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 formation, quenching and e l a s t i c - s c a t t e r i n g c r o s s sections 1.^.5.1. P o s i t r o n i u m formation 1A.5.2. P o s i t r o n i u m q u e n c h i n g a n d e l a s t i c scattering cross-sections 1.5- Summary o f w o r k c o n t a i n e d i n t h e 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. I n t r o d u c t i o n 21 2.2. E x p e r i m e n t a l method 23 2.2.1. L i f e t i m e measurements 2.2.2. V a l l e y - t o - p e a k r a t i o measurements 2.3- The e x p o n e n t i a l p o r t i o n s of the time s p e c t r a 27 2.3.1• The d i r e c t or f r e e a n n i h i l a t i o n s 2.3*2. 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 2.3*3. P a r a p o s i t r o n i u m a n n i h i l a t i o n s 2.3*'+. The observed spectrum i n the e x p o n e n t i a l region 2.*+. The v a l l e y - t o - p e a k r a t i o 29 2.5* A n a l y s i s of r e s u l t s 31 2.5.1. A n a l y s i s of time s p e c t r a 2.5*1*1. Maximum l i k e l i h o o d theory 2.5.1.2. I t e r a t i v e s o l u t i o n 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 f o u r parameters 2.5.1."+. E s t i m a t i o n of channel w i d t h s , w, , and random background B 2.5.1.5. E s t i m a t i o n of v a r i a n c e s 2.5*1*6. Goodness of f i t 2.5«2.Analysis of e x p e r i m e n t a l l y determined a n n i h i l a t i o n rates 2.6. E x p e r i m e n t a l r e s u l t s 38 2.6.1. C r i t e r i a f o r p r e s e n t a t i o n of data 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 : zero e l e c t r i c f i e l d results 2.6.2.1. R e s u l t s of f i t t i n g the a n n i h i l a t i o n r a t e to f u n c t i o n s of the Argon d e n s i t y 2.6.2.2. D i s c u s s i o n of the f i t s to the data 2.6.2.3* Comparison of the l i n e a r term w i t h previous r e s u l t s 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 v a l l e y - t o - p e a k ratio: electric f i e l d results 2.6.k. The shoulder i n the time s p e c t r a 2.6.-+.1. Width of the shoulder 2.6A.2. The l o g a r i t h m i c 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. 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 2.6.5.1. F i t t i n g of experimental data 2.6.5.2. D i s c u s s i o n of the l i n e a r d e n s i t y . 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 o r t h o p o s i t r o n i u m r e s u l t s  -vipage  2.6.6.  D i s c u s s i o n of e r r o r s not r e l a t e d to counting statistics 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 e c t r o n i c 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 i m e s o r t e r 2.6.6.3. Systematic e r r o r i n the a n n i h i l a t i o n rates 2.6.6.^-. A p p l i e d e l e c t r i c f i e l d 2.6.6.5. Measurement of gas d e n s i t y 2.6.6.6. U n c e r t a i n t y i n E / P 2.6.6.7. Gas composition  3.  THEORETICAL CONSIDERATIONS OF THE POSITRON-ARGON ATOM  6k  INTERACTION 3.1 • I n t r o d u c t i o n 6*+ 3-1.1. D i s c u s s i o n of the d i f f e r e n c e s between lowenergy p o s i t r o n and e l e c t r o n s c a t t e r i n g 3.1.2. O u t l i n e of procedure 3«2. The Schrodinger equation and i t s s o l u t i o n 69 3.2.1. The Schrodinger equation 3.2.2. C a l c u l a t i o n of phase s h i f t s and wave functions 3.2.2.1. Asymptotic s o l u t i o n f o r k^O 3.2.2.2. Asymptotic s o l u t i o n f o r k=0 3.3. C a l c u l a t i o n of Z 75 3-*+. 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 76 3A.1 . The modified W i l k i n s equation 3.k.2. General computer s o l u t i o n 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+. S o l u t i o n of the'time-independent equation 3-5- R e s u l t s 82 3.5.1. D i s c u s s i o n of the p o t e n t i a l s used 3.5.2. Comparison w i t h experiment 3.5.3. D i s c u s s i o n of the break i n the dependence of a n n i h i l a t i o n r a t e on e l e c t r i c f i e l d D i s c u s s i o n of the experimental dependence of a n n i h i l a t i o n r a t e on e l e c t r i c f i e l d 3.6.. C o n c l u s i o n s 88 3.6.1. Summary of experimental r e s u l t s 3-6.2. T h e o r e t i c a l c o n c l u s i o n s f  f  REFERENCES  91'  APPENDIX: M o d i f i c a t i o n s to the f a s t - s l o w c o i n c i d e n c e c i r c u i t r y of W. F a l k (1965).  9^  -viipage 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 Pile-up rejectors 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 t h e  timesorter  -viii-  LIST OF TABLES page Table I .  Gas p u r i t y a n a l y s i s  2*+  Table I I .  R e s u l t s of c h i - s q u a r e t e s t on the l i f e t i m e spectra  37  Table I I I .  Dependence  ^0  Table IV.  P u b l i s h e d values of 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  Table V .  Dependence  of  Table V I .  Dependence  of ~* on P  Table V I I .  Summary of p u b l i s h e d r e s u l t s f o r o r t h o p o s i t r o n i u m quenching i n Argon  57*  Table V I I I .  Dependence  58*  of A  on P  a  l2 2 T  o  n  e  ^-  e  c  ^ ^r  c  field  M-6  53  Q  of l i f e t i m e s  h2  on S . C . A .  * I n d i c a t e s page number preceding  table.  setting  -ix-  LIST OF FIGURES to  follow page 3  1.  A n n i h i l a t i o n mechanisms  2.  Representative i n Argon  3.  B l o c k diagram of e l e c t r o n i c s measurements  h.  Energy spectrum of 0.51 MeV gamma rays showing S . C . A . settings  25  5-  Representative i n Argon  26  6.  Dependence  7-  D i r e c 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 f i e l d as a f u n c t i o n of density'  39  8.  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 as a f u n c t i o n of E / P  hh  9-  F r a c t i o n of p o s i t r o n s of E / P  of p o s i t r o n s  i n gases  time spectrum of p o s i t r o n a n n i h i l a t i o n  2*+  used i n the l i f e t i m e  time spectrum of p o s i t r o n a n n i h i l a t i o n  of the l i k e l i h o o d f u n c t i o n on 1^,  I , 2  11  T 1  ,T  2  36  forming p o s i t r o n i u m as a f u n c t i o n  10.  Time spectrum f o r p o s i t r o n s  11 .  Comparison of d i r e c t - and ortho-enhanced time s p e c t r a obtained at P = h .9 amagats, E / P =0 V cm amagat"  *+8  Comparison of d i r e c t - and ortho-enhanced time s p e c t r a o b t a i n e d at P = 9.3 amagats, E / P =0 V cm" amagat"'  *+9  13-  Time spectrum f o r p o s i t r o n s  51  \h.  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 i n Argon as a f u n c t i o n of d e n s i t y  52  1 5-  T h e o r e t i c a l r e s u l t s f o r Z ff wave number k  83  16.  T h e o r e t i c a l r e s u l t s 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 as a f u n c t i o n of k  83  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 r a t e s as a f u n c t i o n of E / P  8^  18.  Photomultlplier c i r c u i t  97  i n Argon at s m a l l E / P -1  12.  1  e  hj  1  1  at h i g h E / P  as a f u n c t i o n of p o s i t r o n  -x-  to 19-  Timing  pulse generator  20.  Prompt r e s o l u t i o n  21.  Circuit  22.  Integral  for  circuit  of the e l e c t r o n i c  driving  97 system  electromechanical register  and d i f f e r e n t i a l  follow page  l i n e a r i t y of the timesorter  97 97 97  -xi-  ACKNOWLEDGMENTS. T h i s t h e s i s i s a r e p o r t of r e s e a r c h work performed under the aegis of D r . G. Jones. his  enthusiasm,  his  exacting  I am much indebted to him f o r  s t a n d a r d s , and h i s deep p h y s i c a l  insight. Thanks are a l s o due to the many members of the  Physics  Department and Computing C e n t r e , w i t h whom I had f r u i t f u l discussions  d u r i n g the course of t h i s  wish to acknowledge  project.  Furthermore, I  the a s s i s t a n c e of members of the  Workshop and Van de Graaff  technical  Physics  staff.  In a d d i t i o n , my s i n c e r e a p p r e c i a t i o n goes to my wife and  p a r e n t s , who have been a source of encouragement  throughout the past Finally,  and  incentive  years. I am indebted to the U n i v e r s i t y of  Columbia f o r two Graduate F e l l o w s h i p s , C o u n c i l of Canada f o r a S t u d e n t s h i p .  British  and to the N a t i o n a l Research  -1 -  1.  1.1.  POSITRONS AND THEIR INTERACTION WITH GAS ATOMS.  Introduction. Of a l l the p r e d i c t i o n s of modern r e l a t i v i s t i c quantum  mechanics,  s u r e l y one of the most s a t i s f y i n g  d u c t i o n of the e x i s t e n c e of the p o s i t r o n .  has been the  de-  T h i s achievement  is  due to D i r a c (1928), who proposed a r e l a t i v i s t i c wave equation for  the e l e c t r o n ,  characteristics  which, i n a d d i t i o n to p r e d i c t i n g such dynamical  as the s p i n of the e l e c t r o n ,  also predicted  the  e x i s t e n c e of a p o s i t i v e l y - c h a r g e d a n t i - p a r t i c l e ( D i r a c , 1931). This positive  particle,  the p o s i t r o n , was d i s c o v e r e d  experi-  m e n t a l l y by Anderson (1932), and was found to have a mass equal to t h a t of the The  electron. study of the i n t e r a c t i o n of slow p o s i t r o n s w i t h gas  atoms was f i r s t undertaken by Shearer and Deutsch (19^9), who s t u d i e d the slowing down of p o s i t r o n s d i r e c t r e s u l t of these experiments nium  i n v a r i o u s gases.  the e x i s t e n c e of a p o s i t r o -  atom p o s t u l a t e d by Ruark 0 9^5) was v e r i f i e d .  system c o n t a i n i n g a p o s i t r o n and an e l e c t r o n , s i m i l a r to hydrogen i n many of i t s  As a  properties.  A bound  positronium is However, i t  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  annihilate  differs each  other w i t h 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 p o s i t r o n s w i t h e l e c t r o n s b a s i s of most experiments of  positrons.  d e a l i n g w i t h the atomic  forms  the  interactions  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 t h e r the r e l a t i v e p o s i t r o n - e l e c t r o n v e l o c i t y at a n n i h i l a t i o n (Heinberg and Page, 1957; C e l i t a n s and Green, 196M-), or the time of the p o s i t r o n s i n a gas, The  l i q u i d or s o l i d ( F a l k ,  l a t t e r technique forms the experimental b a s i s Under c e r t a i n c o n d i t i o n s  ing  sections,  and p o s i t r o n i u m f o r m a t i o n .  of t h i s work.  to be made c l e a r i n the  this  follow-  f o r momentum t r a n s f e r , a n n i h i l a t i o n  It is  the t a s k of quantum mechanics  to g i v e p r e c i s e p r e d i c t i o n s of these c r o s s - s e c t i o n s ,  of  1965).  the l i f e t i m e of p o s i t r o n s i n a noble gas is. simply  connected to the c r o s s - s e c t i o n s  comparison,  life-  w i t h experiment can be made.  thesis is  i n order that  The t h e o r e t i c a l  aspect  concerned both w i t h the c a l c u l a t i o n of momentum  t r a n s f e r and a n n i h i l a t i o n c r o s s - s e c t i o n s  f o r some simple  potentials  d e s c r i b i n g the p o s i t r o n - a t o m i n t e r a c t i o n , and w i t h the r e d u c t i o n of  these c r o s s - s e c t i o n s  mental l i f e t i m e r e s u l t s .  i n order to make comparison w i t h e x p e r i I t i s necessary  to d e s c r i b e the p o s i t r o n -  atom i n t e r a c t i o n i n an approximate way, siLnce, although the c o r r e c t wave equation r e p r e s e n t i n g the i n t e r a c t i o n can be g i v e n , i t was not possible aspect.  to  make a complete a n a l y s i s because of the many p a r t i c l e  Thus approximate models must be used and j u s t i f i e d by  comparison w i t h experiment.  The c u r r e n t s i t u a t i o n r e g a r d i n g the  theory of p o s i t r o n - a t o m i n t e r a c t i o n s 1.2. 1.2.1.  is  enlarged upon i n Chapter 3-  The f a t e of p o s i t r o n s i n a gas. I n t r o d u c t o r y remarks. The  study of p o s i t r o n - a t o m i n t e r a c t i o n s has been l a r g e l y  c o n f i n e d to experiments u s i n g Na-22 or Cu-6 * as p o s i t r o n 1  sources.  -3-  The energy-spectrum of p o s i t r o n s from such sources Fermi d i s t r i b u t i o n . energy i s  the  continuous  In the case of Na-22, the maximum p o s i t r o n  5^2 keV, and the d i s t r i b u t i o n i s  (Macklin,e,t a l . , 1950). positrons  is  peaked at 170 keV  The method by which these h i g h energy  l o s e energy and subsequently  a n n i h i l a t e i n a gas  is  q u a l i t a t i v e l y understood. Figure 1 (Falk,  1965) i l l u s t r a t e s  involved i n this  energy l o s s .  isms,  is  discussion  the v a r i o u s  In order to d i s c u s s  mechanisms  these mechan-  confined to the case of Argon at a d e n s i t y  the order of .10 amagats.  (1 amagat= +. +589 x 10"^ m o l e s / c c . l  l e a n I n s t i t u t e of P h y s i c s Handbook.) out t h i s work and the d e n s i t y  is  Amer-  I  Argon i s  of  the gas used t h r o u g h -  such that the r a t e of the  cesses i n v o l v e d can be r e l a t e d to the time r e s o l u t i o n ,  pro-  7>k n s e c ,  c h a r a c t e r i s t i c of the i n s t r u m e n t a t i o n used. 1.2.2.  D e s c r i p t i o n of the a n n i h i l a t i o n time spectrum i n terms of a positron velocity 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 s p e c t r a of p o s i t r o n s  i n the f o l l o w i n g way. nucleus  is  necessary state.  The emission of a p o s i t r o n from a Na-22  f o l l o w e d w i t h i n 10~ ^ sees by a 1.28 MeV gamma r a y 1  to d e - e x c i t e the daughter nucleus Ne-22 to i t s  ground  T h i s gamma r a y provides the p o s i t r o n " b i r t h " s i g n a l .  a n n i h i l a t i o n of the p o s i t r o n w i t h an e l e c t r o n r e s u l t s of 1 .02 MeV of gamma r a d i a t i o n being e m i t t e d . this  are determined  gamma r a d i a t i o n s i g n i f i e s  in a total  The d e t e c t i o n  the "death" of a p o s i t r o n .  a n n i h i l a t i o n time-spectrum d i s p l a y s  The  of  An  the number of such events as  _  Ionization and Inelastic Collisions  Na-22 Positrons E : 5^2keV m a x  Rapid EnergyLoss V  Negligible Annihilation  Energy  Figured.  A n n i h i l a t i o n mechanisms o f p o s i t r o n s Ps-positronium.  i n gases.  -lf-  a f u n c t i o n of the l i f e t i m e of the p o s i t r o n . will  Such a time spectrum  i n g e n e r a l c o n t a i n components a r i s i n g from the a n n i h i l a t i o n  o f f r e e p o s i t r o n s ( d i r e c t a n n i h i l a t i o n ) , and of p a r a - and o r t h o positronium.  For a gas i n which there i s no p o s i t r o n i u m f o r m a t i o n ,  the time spectrum c o n s i s t s of d i r e c t a n n i h i l a t i o n s o n l y . In order to d i s c u s s such a gas,  i t is  the l i f e t i m e of a f r e e p o s i t r o n i n  convenient to c o n s i d e r the experiment as one i n  which a l l the p o s i t r o n s detected were emitted Furthermore the p o s i t r o n d e n s i t y i s  simultaneously.  thought of as so low t h a t no  one p o s i t r o n can i n t e r a c t w i t h another during i t s is  the case f o r the weak ( 1 0 / i C i )  case,  l i f e t i m e , which  sources used h e r e .  In such a  the swarm of p o s i t r o n s has some energy d i s t r i b u t i o n , which  i s m o d i f i e d as a f u n c t i o n of time by i n t e r a c t i o n of the w i t h the gas  positrons  atoms.  The l o g a r i t h m of the slope of a time spectrum at any time t i s  then I n v e r s e l y p r o p o r t i o n a l to the  velocity-dependent  a n n i h i l a t i o n r a t e averaged over the v e l o c i t y d i s t r i b u t i o n a p p r o p r i a t e to the time t .  Returning now to the a c t u a l c o n d i t i o n s under which  the experimental data i s  gathered i t  can be seen that there i s no  d i f f e r e n c e between these two d e s c r i p t i o n s . distribution y(v,t) is  consistent  w i t h e i t h e r of these d e s c r i p t i o n s  then the p r o b a b i l i t y t h a t a t a time t ,  between v and v t d v .  The p o s i t r o n v e l o c i t y  the p o s i t r o n has a v e l o c i t y  For the case of a swarm of p o s i t r o n s ,  positron velocity distribution is  the  e q u a l l y d e f i n e d as the f r a c t i o n  o f p o s i t r o n s at time t which have a v e l o c i t y between v and v f d v . The above d i s c u s s i o n a p p l i e s e q u a l l y to the case where  -5-  there i s  p o s i t r o n i u m f o r m a t i o n , except that p o s i t r o n i u m a n n i h i l a -  t i o n s modify the shape of the time  spectrum,somewhat.  1.2.3.  collisions  The i n f l u e n c e of i n e l a s t i c  on the  slowing-down  time. At 10 amagats the p o s i t r o n s from the continuous d i s t r i b u t i o n l o s e energy r a p i d l y i n the r e g i o n where c o l l i s i o n s w i t h the Argon atoms are p o s s i b l e . all  energy ( F a l k ,  1965).  The c a l c u l a t i o n of t h i s  on experimental and t h e o r e t i c a l values  inelastic  relies  energy  probably correct  l e v e l i n Argon at 11.6 eV i s  However, a r e a l i s t i c upper l i m i t to t h i s  be found by assuming t h a t a minimum of 11.6 eV i s elastic  time  The time i n v o l v e d f o r the remaining energy l o s s  from 10 keV to the l a s t less certain.  their  of dE/dx over the  range from 5^0 keV to 10 keV (Nelms, 1956), and i s to w i t h i n 20$.  inelastic  W i t h i n about 0.7 nsec  the p o s i t r o n s have l o s t a l l but about 1 0 keV o f  initial  energy  collision.  For an average i n e l a s t i c  time can  l o s t per i n -  c o l l i s i o n cross-  2 s e c t i o n of  ^ir a  Q  , where a -=Bohr r a d i u s , t h i s g i v e s an upper 0  l i m i t of ^ 5 x 1 0 ~ ^ seconds f o r a d e n s i t y of 10 amagats. it  is  clear that,  7 nsec,  g i v e n an experimental time r e s o l u t i o n of about  a l l a n n i h i l a t i o n s t h a t occur d u r i n g t h i s  initial  down p e r i o d occur w i t h i n the r e s o l u t i o n of the a p p a r a t u s . addition,  .the  p o s i t r o n a n n i h i l a t i o n during t h i s  time i s  4  (Gerhart,  Thus  et a l . , 195^5 K e n d a l l and Deutsch,  195*0-  slowingIn negligible  -6-  I, 2.h.  The i n f l u e n c e  of the e l a s t i c  collisions  on t h e s l o w i n g -  down t i m e . The  majority  of the positrons,  then, reach energies  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  only  collisions with  be s l o w e d down b y e l a s t i c  During t h i s slowing-down p e r i o d , be  depopulated by three  annihilate with the Ar-e  t h e A r g o n atoms.  the positron d i s t r i b u t i o n w i l l  main p r o c e s s e s .  an atomic e l e c t r o n during  F i r s t l y , a p o s i t r o n can a direct collision  atom, t h e 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  c o m p l e x may be f o r m e d 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 Section  1.3-  Thirdly, a  p o s i t r o n may c a p t u r e a n a t o m i c e l e c t r o n t o f o r m a a t o m 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 tic  spin-dependent l i f e t i m e s .  Section  1 ."+.  Scattering  of the p o s i t r o n s t e r i s e d only  will  This  one o f two c h a r a c t e r i s -  possibility  i s discussed i n  e l a s t i c a l l y only, the v e l o c i t y d i s t r i b u t i o n  by t h e temperature T o f t h e h o s t gas.  This e q u i l i b -  be M a x w e l l i a n e x c e p t f o r a p o s s i b l e  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 . occur i f the a n n i h i l a t i o n rate annihilation of a positron) For  positronium  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 -  rium d i s t r i b u t i o n w i l l  the  with  These  deviations  ( p r o b a b i l i t y per u n i t time f o r  is  velocity-dependent.  an a n n i h i l a t i o n r a t e w h i c h i s  r a t e o f removal o f positrons  velocity-independent  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 . that f o r this  small  I t follows  s p e c i a l c a s e 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 unaffected  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  Maxwellian  -7-  at equilibrium. al and  Since the a n n i h i l a t i o n rate i s d i r e c t l y  proportion-  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 t h e atom) the a n n i h i l a t i o n cross-section a t that v e l o c i t y , i ti s clear  that a velocity-independent annihilation cross-section  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 inversely proportional  to the positron  velocity. For decreasing  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  v e l o c i t y , positrons  with  a r e removed p r e f e r e n t i a l l y f r o m t h e  l o w - e n e r g y end o f t h e d i s t r i b u t i o n . A t a n y i n s t a n t t h e a v e r a g e p o s i t r o n energy w i l l  be h i g h e r  zero a n n i h i l a t i o n rate.  This  t h a n i n t h e ca-s-© o f a c o n s t a n t o r can only  occur i f the centre  of  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 The the  the Maxwellian d i s t r i b u t i o n . t i m e 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 d e p e n d s on  s h a p e 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 b e l o w '11.6 eV a n d  a l s o on t h e momentum-transfer c r o s s - s e c t i o n . m o m e n t u m - t r a n s f er- c r o s s - s e c t i o n a t i o n time i s greater  Typically, fora  o f t h e o r d e r o f 7Fa , t h e r e l a x 2  0  t h a n a b o u t 10 n s e c ( T a o , e t a l . , 1963),  which i s l a r g e r than the experimental time r e s o l u t i o n o f 7 nsec. 1.2.5.  Summary. For  a b o u t 10 a m a g a t s o f A r g o n , a n d a n 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 nsec, p o s i t r o n a n n i h i l a t i o n s during t h e initial  slowing-down p e r i o d  to 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 h a v e a n y o b s e r v a b l e l i f e t i m e . remaining positrons  can l o s e energy only  by e l a s t i c  The  collisions,  -8-  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 being and  positronium  p o s i t r o n has the  formation.  The  l a t t e r can  s u f f i c i e n t e n e r g y t o make up  i o n i z a t i o n p o t e n t i a l s o f A r g o n and  this  threshold  i s not  e n e r g y i s 8.9  possible.  The  to t h i s  o b s e r v e d and 1 .3'  o c c u r o n l y when  the  t h e d i f f e r e n c e between  positronium.  For  Argon  eV,below w h i c h p o s i t r o n i u m  formation  r e l a x a t i o n time of the d i s t r i b u t i o n to  l i b r i u m i s e x p e c t e d t o be E f f e c t s due  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  of the order  o f 10  nsec or  s l o w r e l a x a t i o n t i m e have been  are discussed  i n Section  1.3*  and  equi-  greater.  previously  i n Chapter  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  2.  cross-Sections  f o r : .positrons i n cArgon. 1-3.1.  Relationship  of a n n i h i l a t i o n r a t e to  positron-electron  overlap. D u r i n g t h 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 system, there  i s the p o s s i b i l i t y  t h a t the  positron w i l l  shown t h a t t h e  the  gas  positron annihilation rate i s proportional  d e n s i t y and  to the  e l e c t r o n d e n s i t y a t the  averaged over the p o s i t r o n p o s i t i o n . given  i n Chapter I f the  annihilate  F e r r e l l (1956)  w i t h an a t o m i c 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 . has  atomic  The  positron  d e t a i l e d expression  is  3' incoming p o s i t r o n i s described  by a p l a n e w a v e ,  the  i n t e g r a l i n v o l v i n g the 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 equal  the  t o t a l number, Z,  the d e p a r t u r e of the to the  to  of electrons  comprising  the atom.  Because  p o s i t r o n w a v e - f u n c t i o n f r o m a p l a n e wave  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 not  to  in  of  due  general  -9-  equal to Z .  The r e s u l t i n g e f f e c t i v e  number of e l e c t r o n s ,  Z ff, e  depends on the p a r t i c u l a r form of the p o s i t r o n - a t o m i n t e r a c t i o n assumed. An a l t e r n a t e 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  +  state is  possible.  A n n i h i l a t i o n s from such a system  would be i n d i s t i n g u i s h a b l e from the d i r e c t a n n i h i l a t i o n s by the experimental techniques  employed h e r e .  The l i f e t i m e  of t h i s  system  (10~ ^ s e e s ) .  w i l l be of the order, of the p a r a p o s i t r o n i u m l i f e t i m e  1  The observed a n n i h i l a t i o n r a t e c h a r a c t e r i s t i c of t h i s  process  is  thus the capture r a t e of p o s i t r o n s i n t o the bound Ar-e+ system and i s thus p r o p o r t i o n a l to  pressure.  No t h e o r e t i c a l d i s c u s s i o n of the a n n i h i l a t i o n r a t e so f a r has i n c l u d e d t h i s p o s s i b i l i t y . in this  The a n n i h i l a t i o n r a t e s  calculated  t h e s i s are those obtained by c o n s i d e r i n g o n l y the 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 an e l a s t i c  collision.  As y e t ,  there i s no  way of t e l l i n g which of the two processes predominates. c a l c u l a t i o n s presented i n t h i s  The  t h e s i s are based on the assumption  t h a t the c o n t r i b u t i o n to the observed a n n i h i l a t i o n r a t e from process  is negligible  this  compared w i t h the d i r e c t r a t e c o n s i d e r e d .  x. 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. Experimental r e s u l t s  and Green,  indicate  ^^6U••, Osmon, 1965; P a u l ,  (Falk,  196^) t h a t the 6  a n n i h i l a t i o n r a t e f o r Argon (about 5 x 10 sec a Z ff e  1965; T a o , B e l l , average  1  amagat  1  ) implies  of about 3 0 , s u b s t a n t i a l l y l a r g e r than the D i r a c Z f f e  of 18.  -10-  This indicates  that  the positron  must he a t t r a c t e d  t o t h e 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 t h e p o s i t r o n a t o m t o be s u b s t a n t i a l l y The  required  larger  than that  a t the  a p p r o p r i a t e t o a p l a n e wave.  a t t r a c t i o n c o u l d be d u e , a t l e a s t p a r t i a l l y ,  to the  standard long range p o l a r i z a t i o n term found necessary t o f i t low energy e l e c t r o n 1965).  scattering  f r o m n o b l e g a s atoms ( M o t t a n d M a s s e y ,  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  a s a/R *, 1  w h e r e R i s t h e p o s i t r o n - a t o m s e p a r a t i o n a n d <* i s t h e p o l a r i z a b i l i t y o f t h e A r g o n atom.  A further  contribution  to the high Z f f  could  e  a r i s e from d i s t o r t i o n o f the electron  c l o u d a r o u n d t h e A r g o n atom  by  tending t o increase the  the incoming p o s i t r o n ,  electron  density  this  at the positron  Two c a l c u l a t i o n s Argon have been r e p o r t e d Both use an e m p i r i c a l a/R  1+  position.  directed  specifically  1966).  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 asymptotic  behaviour, with a simple cutoff  i n order that  The c u t o f f s  are also  the potential chosen such  the Hartree-Fock p o t e n t i a l appropriate to the ground-state  a t o m d o m i n a t e s i n t h e i n t e r i o r o f t h e atom. is  t o t h e case o f  ( M a s s e y , e t a l . , 1966; J o n e s a n d O r t h ,  remain f i n i t e a t the o r i g i n . that  effect  A scattering  potential  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 a n d p o l a r i z a t i o n  potentials.  This type o f s c a t t e r i n g  which an a t t r a c t i o n i s p o s s i b l e ,  potential  occurring  i s the simplest f o r  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 Hartree-Fock p o t e n t i a l , which i s repulsive due  f o ra positron.  to electron  potential  I n both c a l c u l a t i o n s  d i s t o r t i o n are considered.  i s adequate f o r d e s c r i b i n g  r e p o r t e d no  Although this  t h e Ramsauer e f f e c t  effects  type of i n Argon  1  -11-  ( H o l t s m a r k , 1929; K i v e l , 1959; L a b a h n a n d 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 scattering 1.3•3•  case.  t o be i n a d e q u a t e f o r t h e p o s i t r o n - A r g o n i n C h a p t e r 3*  This point i sdiscussed further  Velocity  dependence o f t h e d i r e c t a n n i h i l a t i o n  That t h e d i r e c t a n n i h i l a t i o n  rate  rate  i n Argon.  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 t h e r e g i o n b e t w e e n 0 eV a n d a b o u t 10 eV follows  from the experimental observation that t h e d i r e c t  tion rate in  i s not exponential, but i scharacterised  thepositron  annihilation  time spectrum (Falk  annihila-  by a " s h o u l d e r " a n d J o n e s , 1964;  F a l k , 1965; T a o , B e l l a n d G r e e n , 1964; Osmon, 1965; P a u l ,  1964).  A t y p i c a l t i m e s p e c t r u m s h o w i n g 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 be  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 thedirect annihilation  (Falk,  1965).  rate  exponential which i s taken t o  f o rthe thermalized  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 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 C h a p t e r 2, S e c t i o n 1.3.4.  Electric field in  of the  2.1.  dependence o f t h e d i r e c t a n n i h i l a t i o n  use o f an applied  dependence o f t h e p o s i t r o n  electric field  annihilation  F a l k , O r t h a n d J o n e s (1965).  to study the v e l o c i t y  r a t e was i n t r o d u c e d b y  T h i s method r e l i e s o n t h e f a c t  velocity distribution of thepositrons at equilibrium  influenced  rate  Argon. The  the  positrons  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  S i n c e t h e p o s i t r o n s may g a i n e n e r g y f r o m t h e f i e l d , 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 . annihilation  that  c a n be  field.  t h e average  The o b s e r v e d  rate, which i s thevelocity-dependent  annihilation  o o o u~>'  §  I  III  I  58  +  P=8.9 E/P  o  r .0  2.74 nsec/channel  55CO  \ — o  §3 O U. O O •O  + Note: I n t h i s and a l l sub- + s e q u e n t t i m e s p e c t r a t h e random c o i n c i d e n c e b a c k g r o u n d h a s been subtracted.  ++  +  o o o  j 60.000  85.000  .  1  110.000  1  135.000  CHANNEL NUMBER  160.000  1 185.000  1  <  210.000  ++  r  +  +  235.000  , F i g u r e 2. R e p r e s e n t a t i v e t i m e s p e c t r u m p f p o s i t r o n a n n i h i l a t i o n i n A r g o n . P i s i n a m a g a t s , E/P i s i n V c m a m a g a t " ; I - p r o m p t p e a k ; 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 c o m p o n e n t . - 1  1  t  -12-  r a t e averaged over a l l p o s i t r o n 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 , then e l e c t r i c  f i e l d dependent.  The d e t a i l e d shape of the e q u i -  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 magnitude of the a p p l i e d e l e c t r i c  order by the  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 s c a t t e r i n g momentum-transfer c r o s s - s e c t i o n . order to t e s t a s p e c i f i c mental r e s u l t s ,  it  is  is  In  model of the i n t e r a c t i o n w i t h the e x p e r i -  t h e r e f o r e necessary  to compute both the  d i r e c t a n n i h i l a t i o n r a t e and the momentum-transfer  cross-section  f o r the model assumed. The f i r s t measurements  of the e l e c t r i c  of the d i r e c t a n n i h i l a t i o n i n Argon ( F a l k , Jones,  zero e l e c t r i c 8 0 V cm"  dependence Orth and  decreased w i t h  f i e l d , r e a c h i n g a value of about h a l f  that  at  f i e l d when the f i e l d reached a v a l u e of about -1  1  1.3«5-  1965; F a l k ,  1965) showed t h a t the a n n i h i l a t i o n rate'  increasing electric  field  amagat  D i r e c t a n n i h i l a t i o n r a t e of p o s i t r o n s  i n Helium.  In order to t e s t the g e n e r a l f e a t u r e s  of t h i s  picture  of the i n t e r a c t i o n of p o s i t r o n s w i t h atoms of the noble gases, is  intended to extend such measurements  o b t a i n e d noble gases (eg.  He, Ne, K r ) .  to the other r e a d i l y Indeed p r e l i m i n a r y measure-  ments being performed at present i n d i c a t e a s i g n i f i c a n t effect  it  shoulder  i n the a n n i h i l a t i o n time spectrum f o r Helium i n c o n t r a -  d i c t i o n to the e a r l i e r r e s u l t s Osmon (1965).  of F a l k ,  Orth and Jones  (1965), and  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 f o r the case of Argon. In f a c t ,  a complete  experimental  a n n i h i l a t i o n r a t e i n Helium i s there  of  of p a r t i c u l a r s i g n i f i c a n c e  appears to be c o n s i d e r a b l e  of t h i s problem at present  investigation  the  since  i n t e r e s t i n the t h e o r e t i c a l  aspects  (Drachman, 19665 K e s t n e r , et a l . , 1965;  Massey, et a l . , 1966). 1 .4.  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.  1.4.1.  S t r u c t u r e of p o s i t r o n i u m . As mentioned i n S e c t i o n  s t a t e of a p o s i t r o n and e l e c t r o n . energy l e v e l s i t reflecting  probabilities  Thus the i o n i z a t i o n  as  the average  The t r a n s i t i o n  The f i n e  distance  structure  c o n s i d e r a b l y from that of hydrogen due to the l a r g e  ence i n mass between the proton and p o s i t r o n . s t r u c t u r e term i s  energy l e v e l s i s  Details  of the  fine  nature  Only the gross s t r u c t u r e of  important i n t h i s work.  differ-  An a d d i t i o n a l  i n t r o d u c e d by the p a r t i c l e - a n t i p a r t i c l e  of the. e l e c t r o n and p o s i t r o n .  two  potential  positron-electron  twice that of the p r o t o n - e l e c t r o n d i s t a n c e .  differs  its  from the hydrogen atom by a f a c t o r of  6.78 eV i n s t e a d of 13*6 eV.  are h a l v e d  the bound  In the gross s t r u c t u r e of  a s m a l l e r reduced mass.  of p o s i t r o n i u m i s  is  differs  1.1., p o s i t r o n i u m i s  the  overall  s t r u c t u r e of p o s i t r o n i u m have been summarized by S e r i e s (1957). 1.4.2.  A n n i h i l a t i o n of p o s i t r o n i u m . P o s i t r o n i u m i n the ground s t a t e has two s p i n  Roughly s p e a k i n g ,  o r t h o p o s i t r o n i u m contains  e l e c t r o n w i t h spins  states.  the p o s i t r o n and  p a r a l l e l , whereas p a r a p o s i t r o n i u m has the  spins  -14-  anti-parallel. electron photons.  The p a r t i c l e - a n t i p a r t i c l e nature of the  system causes i t Because  to be u n s t a b l e  against  positron-  annihilation  into  of the s e l e c t i o n r u l e s governing the decay of a  s p i n 1 p a r t i c l e i n t o photons, an odd number of photons.  o r t h o p o s i t r o n i u m can only decay  By the same token,  o n l y decay i n t o an even number of photons. r e g a r d i n g the number of photons s t a t e and e x c i t e d  into  p a r a p o s i t r o n i u m can  The g e n e r a l  situation  f o r a n n i h i l a t i o n from the ground  s t a t e s of p o s i t r o n i u m has been summarized by  Kaempffer (1965). The l i f e t i m e momentum of the s t a t e .  of p o s i t r o n i u m depends on the t o t a l Orthopositronium i n i t s  a c a l c u l a t e d mean l i f e t i m e  of 1 .4 x 10 ' -  7  ground s t a t e has  seconds.  i n the ground s t a t e decays f a r more r a p i d l y , i t s  angular  Parapositronium mean  lifetime  —1 0 being c a l c u l a t e d to be 1.25 x 10 With r e f e r e n c e  to the experimental  quoted i n S e c t i o n 1.2.5. i t  is  seconds (Ore and P o w e l l , 1949). time r e s o l u t i o n of 7 x 10~9 sees  c l e a r that  the l i f e t i m e  of  ortho-  p o s i t r o n i u m w i l l be r e s o l v e d ,  whereas that of p a r a p o s i t r o n i u m w i l l  not be v i s i b l e .  of e x c i t e d  The l i f e t i m e  s t a t e s of p o s i t r o n i u m has  been c a l c u l a t e d f o r the S and P s t a t e s by Alekseev longer  (1959)  than f o r the ground s t a t e s w i t h the same s p i n  In p a r t i c u l a r the 2S l e v e l  and are a l l  •>  alignments.  should have a mean l i f e t i m e  of  about  _Q  1 x 10 excited  7  sec. level  Furthermore, i t appears that  this is  i n p o s i t r o n i u m from which i t  is  the  only  at a l l f e a s i b l e  d e t e c t a n n i h i l a t i o n , s i n c e t h i s s t a t e i s metastable a g a i n s t transitions  to the ground s t a t e .  For a l l other s t a t e s the  to  optical optical  t r a n s i t i o n r a t e s exceed the a n n i h i l a t i o n r a t e s by a f a c t o r of  at  least  1000.  1.h.3.  Positronium formation. Formation of p o s i t r o n i u m i n i t s  can occur i f the p o s i t r o n energy i s difference  ground s t a t e i n a gas  greater  than or equal to  between the i o n i z a t i o n p o t e n t i a l s  the p o s i t r o n i u m atom.  the  of the gas atom and  For Argon t h i s t h r e s h o l d energy ( E t h r )  8.9 eV.  The t h r e s h o l d energy f o r formation i n t o the f i r s t  state is  Ih.'l  is  excited  eV.  Because of the d i f f e r e n t  spins of ground s t a t e o r t h o -  and p a r a p o s i t r o n i u m , the r a t i o of the amounts formed i n each of these s t a t e s i s  expected to be the normal s t a t i s t i c a l  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 t a t e s .  ratio  Thus three  based  ortho-  p o s i t r o n i u m atoms should be formed f o r each p a r a p o s i t r o n i u m atom. The amount of n=2 p o s i t r o n i u m formed r e l a t i v e  to ground s t a t e  p o s i t r o n i u m w i l l be c o n s i d e r a b l y l e s s due to the expected smaller cross-section  f o r formation (Massey and Mohr, 195*+) and the  c o m p e t i t i o n from the i n e l a s t i c  collisions  Any p o s i t r o n i u m atom formed i n an e x c i t e d  w i t h the Argon atom. s t a t e (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 s t a t e annihilation. to s u f f e r hilate.  At 10 amagats of Argon, the 2S s t a t e i s more l i k e l y  c o l l i s i o n a l de-excitation  to the 2P s t a t e than to a n n i -  Furthermore, at these d e n s i t i e s ,  excitation  before  of any e x c i t e d  the c o l l i s i o n a l  de-  s t a t e i n t o the ground s t a t e i s probably  c o n s i d e r a b l y more r a p i d than the o p t i c a l t r a n s i t i o n r a t e of  8 10  1 sec"  (Wallace,  1955)•  about  T h i s might e x p l a i n to some extent  the  -16-  unsuccessful  attempts to d e t e c t the Lyman alpha s p e c t r a l l i n e  p o s i t r o n i u m (Brock and S t r e i b , 1958; Bennett,  et a l . ,  of  Duff  1961;  and Heymann, 1963)• 1.h.h.  Quenching of p o s i t r o n i u m The c o l l i s i o n s  also result  of  1  the most important type of  in collisions  w i t h noble gas atoms.  quenching  This e x t r a channel  a n n i h i l a t i o n of a p o s i t r o n i u m atom r e s u l t s i n a p r e s s u r e -  dependent for  the  This "pick-off "' quenching of the l i f e t i m e  the p o s i t r o n i u m atom i s  for  of a p o s i t r o n i u m atom w i t h a gas atom can  i n the a n n i h i l a t i o n of the p o s i t r o n w i t h one of  atomic e l e c t r o n s .  encountered  lifetimes.  annihilation rate.  The quenching r a t e has been measured  o r t h o p o s i t r o n i u m 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" 10 amagats,  the mean l i f e t i m e  of o r t h o p o s i t r o n i u m Is  to be reduced to about 1 .0 x 10'' sees. -  magnitude w i l l have a n e g l i g i b l e 1.^.5-  f o r Argon.  thus expected  A quenching r a t e of  7  effect  At  on p a r a p o s i t r o n i u m  this  lifetime.  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 elastic scattering  1 A.5*1•  cross-sections.  Positronium formation. Very l i t t l e  tion cross-section  work has been done on the p o s i t r o n i u m forma-  i n general.  C h e s h i r e 0 96'+) has  p o s i t r o n i u m formation by f a s t p o s i t r o n s the Born and impulse approximations.  considered  i n atomic hydrogen,  The c r o s s - s e c t i o n s so  using obtained  p  are of the order of ifa  Q  and decrease w i t h i n c r e a s i n g  energy,  in  -17-  agreement  to w i t h i n an order of magnitude w i t h those of Massey and  Mohr ( 1 9 5 ) ' l+  S i m i l a r c a l c u l a t i o n s u s i n g the Born approximation f o r  the case of Helium hy Massey and Moussa (1961) i n d i c a t e that cross-section  is  The c r o s s - s e c t i o n  of the order of 0.1^ao rises  the  or l e s s near t h r e s h o l d .  to a maximum of 0.4 T a  2 0  at 27 eV.  No  c a l c u l a t i o n s have been p u b l i s h e d f o r the case of many-electron atoms. 1.4.5.2.  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  cross-sections.  The g e n e r a l problem of low energy p o s i t r o n i u m s c a t t e r i n g from atomic Hydrogen has been i n v e s t i g a t e d u s i n g the Born a p p r o x i m a t i o n .  In t h i s  by Massey and Mohr (195^)  case, quenching of o r t h o -  p o s i t r o n i u m i s achieved by d i r e c t c o n v e r s i o n of o r t h o - i n t o p o s i t r o n i u m made p o s s i b l e by the exchange of the s i n g l e electron.  The c r o s s - s e c t i o n  for this  to be h i g h l y energy dependent.  para-  atomic  type of quenching was found  These c a l c u l a t i o n s have been r e -  peated i n more d e t a i l by F r a s e r (1961) who has c o n s i d e r e d the scattering  of o r t h o p o s i t r o n i u m from Helium atoms ( F r a s e r , 1962).  N e g l e c t i n g p o l a r i z a t i o n and e x c i t a t i o n , of e l e c t r o n exchange,  but i n c l u d i n g the  he has found an e l a s t i c  s e c t i o n f o r p o s i t r o n i u m w i t h Helium of 1 7 - 7 ^ a energy.  scattering 0  Again no c a l c u l a t i o n s  effects cross-  at zero k i n e t i c  The a p p r o p r i a t e p i c k - o f f quenching c r o s s - s e c t i o n  to be r e p o r t e d . atoms.  elastic  has  yet  e x i s t f o r other many-electron  -18-  1.5.  Summary of work contained i n the  1.5.1.  thesis.  Theoretical aspects. That a 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 i s  to g i v e r i s e  to the observed shoulder i n the time s p e c t r a has  shown by F a l k (1965). calculations  One purpose of t h i s  of such v e l o c i t y  dependent  thesis is  to  been  present  annihilation rates,  on a simple model of the p o s i t r o n - a t o m i n t e r a c t i o n . scattering  sufficient  based  Elastic—  momentum-transfer c r o s s - s e c t i o n s are a l s o d e r i v e d on  the b a s i s of these models.  The a p p r o p r i a t e v e l o c i t y  averaged  a n n i h i l a t i o n r a t e s at e q u i l i b r i u m as a f u n c t i o n of a p p l i e d f i e l d have a l s o been d e r i v e d i n order to compare w i t h The c a l c u l a t i o n of the v e l o c i t y dependent momentum t r a n s f e r c r o s s - s e c t i o n s ,  annihilation  experiment. rates,  and a p p r o p r i a t e v e l o c i t y  a n n i h i l a t i o n r a t e s as a f u n c t i o n of e l e c t r i c  electric  averaged  f i e l d constitute  an  o r i g i n a l c o n t r i b u t i o n i n that there are no p r e v i o u s l y p u b l i s h e d reports.  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 . L i m i t a t i o n s i n the i n s t r u m e n t a t i o n used by F a l k  prevented  the measurement of the 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  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 time-spectra.  Consequently,  of the l i f e t i m e were expected  there was some doubt as to the accuracy  of the l o n g - l i v e d component.  to have a n o t i c e a b l e  short-lived (direct)  component.  These  inaccuracies  e f f e c t on the e s t i m a t i o n of  the  These problems were f u r t h e r com-  pounded by e r r o r s i n the maximum l i k e l i h o o d method used i n  fitting  the experimental d a t a . The experimental r e s u l t s  v  contained i n t h i s  t h e s i s have  been obtained w i t h c o n s i d e r a b l y modified i n s t r u m e n t a t i o n , i n that the time s c a l e has been extended  i n order that the o r t h o p o s i t r o n i u m  component may be measured s i m u l t a n e o u s l y w i t h the d i r e c t  component.  Furthermore, the random background has been reduced by a f a c t o r of ten,yielding  increased s t a t i s t i c a l  accuracy.  The e r r o r s inherent i n the o r i g i n a l method of curve fitting  have been s u c c e s s f u l l y  removed and, i n a d d i t i o n , the  and a p p l i c a b i l i t y of the standard d e v i a t i o n s parameters are d i s c u s s e d .  of the  The o r i g i n a l s t a t i s t i c a l  resulting analysis  F a l k i n v o l v e d m o d i f i c a t i o n of the raw data to take i n t o instrumental effects.  In the present  case,  1 -5•3•  to  are the  to the raw d a t a .  Positronium formation. There i s  the d i s c u s s i o n dependence this  of  account  these e f f e c t s  taken i n t o account by making a p p r o p r i a t e m o d i f i c a t i o n s f u n c t i o n to be f i t t e d  size  some doubt a s - t o the i n t e r n a l c o n s i s t e n c y  of  i n F a l k ' s t h e s i s which r e l a t e s to the e l e c t r i c  of the p o s i t r o n i u m formation r a t e and to the e f f e c t  field of  p o s i t r o n i u m formation r a t e on the d i r e c t a n n i h i l a t i o n r a t e .  This arises  from the f a c t  that the measurements  f o r m a t i o n as a f u n c t i o n of e l e c t r i c  f i e l d due to Marder,  (1956) were used f o r that d i s c u s s i o n . such measurements lifetime  f o r the same gas  experiments  of p o s i t r o n i u m al.,  The present work contains  samples  presented h e r e .  et  as were used i n the  The r e s u l t s  differ  consider-  -20-  a b l y from those of Marder, et a l . doubt t h a t the Marder,  There i s ,  therefore,  reasonable  et a l . values were r e l e v a n t to the F a l k  experiments. 1.5•4.  Orthopositronium a n n i h i l a t i o n r a t e s . The pressure dependent quenching r a t e of o r t h o p o s i t r o n i u m  i n Argon i s d i s c u s s e d i n t h i s  t h e s i s i n some d e t a i l .  These measure-  ments are c o n s i d e r a b l y more accurate than any p r e v i o u s l y r e p o r t e d , a r e s u l t of i n c r e a s e d s t a t i s t i c a l accuracy and improved c u r v e fitting  techniques.  -21  2.  -  EXPERIMENTAL INVESTIGATION  OF  POSITRON L I F E T I M E S IN ARGON. 2.1.  Introduction. Recent i n v e s t i g a t i o n s  A r g o n h a v e shown t h a t be  d e s c r i b e d by  Paul, the  the  free  of the positron  a single exponential  196*+; Osmon, 1965;  Falk  and  by  a single  a r i s e from the  the  field  r e s u l t s as  amagat  series  Of  (Chapter  to It  l i f e t i m e of  has the  static  Typically, a  i s s u f f i c i e n t to decrease the two.  In view of  the  field  b e e n made w i t h  the  the  these  validity  of  further  improved i n s t r u -  statistical  v a l i d i t y of  direct  importance of  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  used to t e s t the  describing  the  e x p e r i m e n t a l t e s t of  to a g r e a t e r degree of  measurements are potentials  J o n e s , 1965)"  t h e s e m e a s u r e m e n t s has  m e n t a t i o n , and  the  of  shoulder  thermal v e l o c i t i e s .  a p p l i c a t i o n of a moderate  only available  models d e s c r i b i n g  Time s p e c t r a  —1  a f a c t o r of the  the  O r t h and  —1  l i f e t i m e by  cannot  G r e e n , 196*+;  evidence of a  s h o u l d e r i s r e m o v e d , and  (Falk,  o f a b o u t 80 V cm  in  e x p o n e n t i a l whose l i f e t i m e i s p r e s u m e d  e x p o n e n t i a l c h a n g e d , on electric  B e l l and  J o n e s , 196*+).  d i r e c t a n n i h i l a t i o n at  b e e n shown t h a t  annihilation rate  (Tao,  a n n i h i l a t i o n gamma r a y s show c l e a r  followed  l i f e t i m e of p o s i t r o n s  accuracy.  several  effective positron-Argon  These  empirical  interaction  3)Positrons  l o s e most o f Once p o s i t r o n s  e m i t t e d by  t h e i r e n e r g y by  a radioactive  inelstic  source i n t o a  c o l l i s i o n s with  the  gas gas  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  atoms. to  -22-  11.6 eV, the lowest e x c i t a t i o n energy of Argon, they can only l o s e 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 , p o s i t r o n i u m can be  formed u n t i l the t h r e s h o l d energy f o r p o s i t r o n i u m f o r m a t i o n , 8.9 eV, i s reached.  Those p o s i t r o n s which terminate t h e i r  rapid  slowing-down at k i n e t i c energies below 11.6 eV, can be d e s c r i b e d 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 - -  1965; F a l k , 1965).  f u n c t i o n ( F a l k , Orth and J o n e s ,  Once the p o s i t r o n s t h e r m a l i z e , the shape of the b u t i o n f u n c t i o n becomes time independent and  the d e p o p u l a t i o n by a n n i h i l a t i o n i s  exponential.  A single  distri-  ( e s s e n t i a l l y Maxwellian) c h a r a c t e r i s e d by a  single  exponential w i l l also occur, regardless  of  the shape of the p o s i t r o n d i s t r i b u t i o n f u n c t i o n , i f the a n n i h i l a t i o n r a t e i s v e l o c i t y independent  (Chapter 3» S e c t i o n 3 « ° ) ° ! ; f  The  observed shoulder i n the time spectrum from Argon, f o l l o w e d by a single is  exponential,  a l l superimposed on the o r t h o p o s i t r o n i u m decay,  t h e r e f o r e a c l e a r 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  r a t e of p o s i t r o n s  is velocity  dependent.  That the d i r e c t a n n i h i l a t i o n r a t e changes on the a p p l i c a t i o n of e l e c t r i c Orth and Jones,  f i e l d f u r t h e r confirms t h i s p o i n t of view  1965)°  An a p p l i e d e l e c t r i c  average energy of the p o s i t r o n s ,  f i e l d increases  (Falk, the  the e q u i l i b r i u m d i s t r i b u t i o n  being to f i r s t order a f u n c t i o n of the p o s i t r o n - A r g o n e l a s t i c s c a t t e r i n g momentum-transfer c r o s s - s e c t i o n , The observed a n n i h i l a t i o n r a t e i s dependent a n n i h i l a t i o n r a t e i s velocity distribution.  and e l e c t r i c  thus changed s i n c e  the  field. velocity-  averaged over a d i f f e r e n t e q u i l i b r i u m  The experimental r e s u l t s  contained h e r e i n  -23-  are measurements  of the a n n i h i l a t i o n r a t e s  for this equilibrium  distribution. 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 a l s o been made by monitoring the and  "peak" count r a t e s  et a l . , 1956).  of the 0.51 MeV gamma-ray s p e c t r a  "valley"  (Marder,  T h i s r a t i o i s d i r e c t l y r e l a t e d to the f r a c t i o n of  pqsiftrons forming p o s i t r o n i u m i n Argon. E x p e r i m e n t a l method.  2.2. 2.2.1.  L i f e t i m e measurements. A 10 ;uCi Na-22 source d e p o s i t e d on 30 / l i n c h aluminum  foil  served as the p o s i t r o n source.  The Instrumentation f o r  the  r e c o r d i n g of the time s p e c t r a i s p r i m a r i l y that used p r e v i o u s l y by F a l k (1965) (see 1965).  a l s o F a l k and Jones,  196^-5 F a l k , Orth and Jones, •  To improve r e l i a b i l i t y , the l i m i t e r s and slow  s e c t i o n s were r e p l a c e d by t r a n s i s t o r i z e d e q u i v a l e n t s . these m o d i f i c a t i o n s are given i n the Appendix. the s t a t i s t i c a l  accuracy of the measurements,  background was reduced  (relative  coincidence Details  of  In order to improve the random c o i n c i d e n c e  to the measurements  of F a l k ) by  a f a c t o r of ten by reducing the p o s i t r o n source s t r e n g t h by the ' same f a c t o r .  In o r d e r , however,  c o i n c i d e n c e count r a t e , and  detectors  s o l i d angle were r e q u i r e d .  used by F a l k ,  consisting  to m a i n t a i n the same o v e r a l l t r u e of s i g n i f i c a n t l y g r e a t e r  To t h i s  end the gamma ray  efficiency detectors  of 2 i n . x 2-g- i n . diameter N a l ( T l )  c r y s t a l s mounted on R . C . A .  6810 p h o t o m u l t i p l i e r s were r e p l a c e d by  h i n . x 3 i n . diameter N a l ( T l )  c r y s t a l s mounted on R . C . A . 70^6  -24photomultipliers. I n a d d i t i o n , use it  p o s s i b l e to extend the  of a  The  gas  decay, thus f a c i l i t a t i n g  chamber u s e d i s t h e  in this  laboratory  gas  purified  was  1965;  (Falk,  as b e f o r e  by  by The  M a t h e s o n Co.,  same as  made  (ND101)  the  include  analysis  t h a t used' p r e v i o u s l y  1965).  The  continuous r e c i r c u l a t i o n over a  hot  1952).  the  F a l k , O r t h and  CaMg e u t e c t i c m i x t u r e ( C o l l i and gas  channel analyzer  t i m e range o f the measurements t o  the whole o r t h o p o s i t r o n i u m of data.  256  Fachini,  I n c . N.J.  Jones,  Analysis  of  i n d i c a t e s t h a t the main  impurity  4 p r e s e n t was  N 2 a t a b o u t one  p a r t i n 10  .  The  M a t h e s o n a n a l y s i s on b o t h t h e b o t t l e gas p u r i f i c a t i o n ) are  and  r e s u l t s of chamber gas  the (after  shown i n T a b l e I . TABLE I . Gas  Impurity gas N  Bottle gas  190  2  ppm  ppm  3  ppm  co  <4  ppm  <4  ppm  23  ppm  32  ppm  1 53  ppm  2  2  e f f e c t i v e n e s s of the  diagram of the  shown i n F i g u r e 3.  <100  ppm  p u r i f i e r i n at l e a s t maintaining  p u r i t y l e v e l of the Argon i s q u i t e  is  119  ppm  5  He  A block  Chamber gas  °2 H  The  Purity Analysis.  the  apparent.  electronic instrumentation  A t y p i c a l run  extended over a p e r i o d  used of  two  1.28 MeV CHANNEL (PROMPT)  0 . 5 \ MeV X CHANNEL (DELAYED)  DYNODE PULSE  TIME SORTER  INVERTERDRIVER  TS OUTPUT  DyNODE PULSE  TS SIGNAL  > CATG  NEGATIVE TIME SIMM. ELIMINATOR COlNCIDENCt  256 CHANNEL KICKSORTER  Ik  GATE PULSE  PILE-UP RETECTOR NO. I  ANTICOINCIDENCE CIRCUIT  PILE-UP REJECTOR NO. 2  GATE PULSE SINGLE  CHANNEL  ANALYZER MO. I  F i g u r e 3.  GATE PULSE GENERATOR SLOW COINCIDENCE CIRCUIT  Block diagram o f e l e c t r o n i c s used ments .  SINGLE CHANNEL  ANALYZER Ho. 2  i n lifetime  measure-  -25d a y s i n w h i c h t i m e some 2 x 10 c o u n t e d i n t h e photo peak. was  1.28  MeV n u c l e a r  The number o f t h e s e  gamma r a y s gamma r a y s  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  difficulties  caused by t h e source  analyzer d r i f t s . by u s i n g  t h e two i n d e p e n d e n t o u t p u t s  amplifier.  The s i n g l e c h a n n e l  analyzer  counted  to avoid  decay and t h e s i n g l e  Each s i n g l e channel  were  channel  s e t t i n g was o b t a i n e d  of the appropriate  linear  analyzer associated with the anni-  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 t h e p h o t o p e a k o f t h e 0.51 gamma r a y . width  H o w e v e r , many r u n s w e r e t a k e n w i t h t h e same  MeV  channel  b u t w i t h t h e 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 t h e r e g i o n  b e t w e e n t h e p h o t o p e a k a n d t h e Compton e d g e , t h e s o - c a l l e d region.  At this  setting  valley  t h e r a t i o o f t h r e e p h o t o n t o two p h o t o n  e v e n t s was e n h a n c e d r e l a t i v e  to theusual setting,  since the three  photon count r a t e i n t h e 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 p e a k r e g i o n ( O r e  and P o w e l l ,  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 . much g r e a t e r a c c u r a c y s p e c t r u m o f t h e 0.51  t h e two  This  allowed  for the orthopositronium  setting lifetime.  analyzers  a r e shown i n F i g u r e h>  o v e r a l l t i m e 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  channels, using a  The e n e r g y  MeV gamma r a d i a t i o n t o g e t h e r w i t h t h e two  s e t t i n g s o f t h e s i n g l e channel The  1949)', w h i l e  o r 7.k n s e c s  ( f u l l width  2.7  a t h a l f maximum), m e a s u r e d  Na-22 s o u r c e i n a l u m i n u m ( s e e  Appendix).  A r e p r e s e n t a t i v e t i m e s p e c t r u m i s shown i n F i g u r e 5« The  p e a k i s due t o a n n i h i l a t i o n  chamber a n d i n t h e s o u r c e the i n i t i a l  of positrons i nthewalls of the  holder, of parapositronium  formed  during  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 5.0  t  o LU  i4.0  $3.0 (J  01 LU CL  • •••  2.0  to I—  \ \  9.0 keV/channel  V  B  0.0 1 to  20  30  CHANNEL  ts  40  v 1 1  t^t -tr c  JO JO  NUMBER  60  J]  70  *••«...  F i g u r e h. Energy spectrum o f 0.51 MeV gamma r a y s showing S.G.A. s e t t 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 ~ d i r e c t - e n h a n c e d s e t t i n g ; 2 ~ ortho-enhanced setting; A B - valley-to-peak r a t i o setting. 2  80  -26-  the gas w i t h i n about 7 nsecs of emission  from the source.  o f the shoulder i s marked by a r e l a t i v e l y component, give r i s e is  the d i r e c t component.  short-lived  The o r t h o p o s i t r o n i u m  to the l o n g - l i v e d e x p o n e n t i a l .  the f l a t  The end  exponential annihilations  Omitted from the  random background r e g i o n which occupies  o f the k i c k s o r t e r up to the prompt peak.  that  figure  portion  The spectrum i n t h i s  r e g i o n was obtained by r e c o r d i n g events where a 0.51 MeV gamma-ray is  d e t e c t e d before a 1.28 MeV gamma r a y .  i n t h i s p a r t of the time spectrum, t h e n , background counts per c h a n n e l . w i t h t h i s random c o i n c i d e n c e 2.2.2.  Valley-to-peak ratio  The counts per channel correspond to the random  The spectrum i n F i g u r e 5 i s  shown  background s u b t r a c t e d . measurements.  The energy spectrum of the a n n i h i l a t i o n gamma rays as a f u n c t i o n of the e l e c t r i c  f i e l d was obtained u s i n g 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 single  channel a n a l y z e r was used to gate the k i c k s o r t e r , and was set that  so  only pulses between the Compton edge f o r 0.51 MeV gamma rays  and the h i g h energy s i d e of the P.51 MeV photopeak would be analyzed by the k i c k s o r t e r ( F i g u r e >+). h-5 minutes  i n which some 2 x 1 0  Each run l a s t e d and 8 x 1 0  approximately  counts per channel  were obtained i n the v a l l e y and peak p o s i t i o n s  respectively.  8 o  i  I  60,000  I  85.000  110.000  I  135.000  I L60.000  CHANNEL NUMBER  I  165.000  1 210.000  "I  1  I 23S.00O  F i g u r e 5» R e p r e s e n t a t i v e t i m e s p e c t r u m o f p o s i t r o n a n n i h i l a t i o n i n A r g o n . P i s i n a m a g a t s , : E / P . i s - i n - V cm" a m a g a t - I o - ' - i p r b m p t p e a k ; 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 : component"; I V ; - • o r t h o p o s i t r o n i u m c o m p o n e n t . 1  -  1  -27-  2.3-  The e x p o n e n t i a l p o r t i o n s of the time The appearance of a s i n g l e  spectra.  e x p o n e n t i a l superimposed on  the o r t h o p o s i t r o n i u m component i n the time s p e c t r a i s  evidence  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 of the p o s i t r o n s has become Chapter 1 , S e c t i o n 1 . 3 . and Chapter 3 , S e c t i o n 3 . " . ) .  (see is,  1 1  the time dependence  by a s i n g l e is  "static'"  of the v e l o c i t y d i s t r i b u t i o n i s  exponential.  The time spectrum under these  composed of the f o l l o w i n g  2.3.1-  The d i r e c t or f r e e  described conditions  components:  annihilations.  The p o p u l a t i o n of f r e e p o s i t r o n s at any time t i s by  = -C x  where X  d  is  Xf i s  f  + x  d  That  ] N(t)  the v e l o c i t y - a v e r a g e d  given (l)  p o s i t r o n i u m f o r m a t i o n r a t e , and  the v e l o c i t y - a v e r a g e d d i r e c t a n n i h i l a t i o n r a t e .  Assuming  t h a t the d i r e c t a n n i h i l a t i o n s are a l l by two photon decay whose efficiency rate  for detection  is  e • the observed d i r e c t a n n i h i l a t i o n 2  is: R (t) = e X N(0)e" f (X  d  2.3.2.  2  +  d  X  d  } t  sec"  (2)  1  Orthopositronium a n n i h i l a t i o n s . The o r t h o p o s i t r o n i u m p o p u l a t i o n i s  g i v e n by  ^ M i i = -XoNo(t) + |* N(t)  (3)  f  where and  x  Q  i  s  the sum of the f r e e  o r t h o p o s i t r o n i u m decay r a t e  the o r t h o p o s i t r o n i u m quenching r a t e  X  q  .  It is  the f r a c t i o n of p o s i t r o n i u m atoms which are formed as tronium i s  the s t a t i s t i c a l  r a t i o 3: +« 1  assumed  x  0  that  orthoposi-  The p o p u l a t i o n at any time  -28-  t is  then  where N ( 0 ) i s o  t=0.  The observed o r t h o p o s i t r o n i u m decay r a t e ,  X~ N ( t ) 0  the number of o r t h o p o s i t r o n i u m atoms present  corresponding to  is:  0  .(5)'  R (t) = A e N (t) Q  where  £3 i s  that t h i s  0  3  0  the three photon d e t e c t i o n  contains  efficiency.  two terms, one, R ^ ( t ) , 0  It is  F Q  (t),  r e p r e s e n t i n g a n n i h i l a t i o n from the "delayed"  formation.  apparent  r e p r e s e n t i n g the decay  of the o r t h o p o s i t r o n i u m atoms present at time t=0, the R  at  1  other,  positronium  Thus_  Rj(t) = A ^ N o C C D e " ^ R  o^> = *o*£•  It  T ^ p ^  NCO) [ e" ot _ e - ( M ^ )t j f  becomes a p p r e c i a b l e under the i n f l u e n c e x  d  + x  f  »T  6  )  (  7  )  r  should be noted t h a t i f the p o s i t r o n i u m formation r a t e  and i f  (  of the e l e c t r i c  X  f  field,  , the observed o r t h o p o s i t r o n i u m spectrum w i l l  Q  e x h i b i t a c h a r a c t e r i s t i c e x p o n e n t i a l growth, f o l l o w e d by a slow e x p o n e n t i a l decay (of 2.3-3•  lifetime  l/T  ).  Parapositronium a n n i h i l a t i o n s . Since p a r a p o s i t r o n i u m has a mean l i f e  (^ 1 0~ ^sec) 1  c o n s i d e r a b l y s h o r t e r than the experimental time r e s o l u t i o n a v a i l a b l e ( 7 A nsec) i t s  c o n t r i b u t i o n to the spectrum i s governed  by the p a r a p o s i t r o n i u m p o p u l a t i o n present a t any time t . the decay r a t e i s Rp(t) =• e  entirely Thus  given by 2  [ X N (t) + £f. N(t) ] q  G  (8)  -29-  The  effect  of o r t h o p o s i t r o n i u m quenching i s  here n o t i n g that t h i s  is  account  e q u i v a l e n t to p a r a p o s i t r o n i u m formation  a t a r a t e given by the quenching r a t e of  taken i n t o  A  q  .  The separate  formation  p a r a p o s i t r o n i u m from f r e e p o s i t r o n s i s a l s o taken i n t o  T h i s component has the same d e t e c t i o n annihilations since  efficiency  account.  as the d i r e c t  the a n n i h i l a t i o n i s by two photons o n l y (Yang,  1950). 2.3A.  The observed spectrum i n the e x p o n e n t i a l r e g i o n . The  components,  t o t a l observed spectrum i s  the sum of these  three  viz.,  R(t) = I e " x  ( X  d f n  } t  +  (9)  where  and  i  2  It  - [.,*„ is  •  V  q  ] c N o) • i x e<  fT  ^  s  -]  m  ,  c l e a r that although there may be a growth i n the o r t h o -  p o s i t r o n i u m component the observed spectrum i s always a simple sum of  the two exponentials  u n l e s s I-j happens to be a n e g a t i v e .  Such  a s i t u a t i o n has not yet been observed. 2.4.  The v a l l e y - t o - p e a k r a t i o . The  energy spectrum of t h r e e - p h o t o n a n n i h i l a t i o n i s  c o n t i n u o u s , w i t h a maximum at 0.51 MeV (Ore and P o w e l l , 1949). In  the a n n i h i l a t i o n spectrum of p o s i t r o n s i n Argon, t h e r e f o r e ,  the  r a t i o of counts i n the v a l l e y r e g i o n (between the 0.51 MeV peak and  Compton edge) to the counts  i n the 0.51 MeV peak i n c r e a s e s  as  -30-  a f u n c t i o n of p o s i t r o n i u m f o r m a t i o n . Consider a 0.51 MeV gamma-ray spectrum obtained with no t h r e e - p h o t o n events.  The v a l l e y - t o - p e a k r a t i o , R , w i l l be l e s s  than t h a t i n a,spectrum where three-photon events were a l s o counted. If,  i n a spectrum of the l a t t e r t y p e ,  r a t e s are C p , C , r e s p e c t i v e l y ,  C  3  = C  then C ^ , where  - ( C - R3C3) R  v  the peak and v a l l e y count  p  (12)  Q  w i l l be the count r a t e due to the three-photon events o n l y , i n the valley region.  The c o e f f i c i e n t  of R  is  Q  the count r a t e Cp i n the  0.51 MeV photopeak w i t h the c o n t r i b u t i o n due to three-photon events i n t h i s r e g i o n , R3C3, s u b t r a c t e d .  In t h 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 a n n i h i l a t i o n s (Ore and P o w e l l , 194-9), and the efficiencies  of the c r y s t a l f o r counting gamma rays whose energies  correspond to the peak and v a l l e y regions r e s p e c t i v e l y , determined. form:  relative  R^ can be  E q u a t i o n (12) can be more u s e f u l l y r e w r i t t e n i n the  C3 = ( C  v  - C R ) / p  Q  (1 - R 3 R ) .  (13)  0  In the case of p o s i t r o n s a n n i h i l a t i n g i n Argon under the influence  of an a p p l i e d e l e c t r i c f i e l d , where i t  is possible  to  change o n l y the r e l a t i v e number of three-photon o r t h o p o s i t r o n i u m and two-photon decays,  C^ w i l l be d i r e c t l y p r o p o r t i o n a l to  f r a c t i o n of p o s i t r o n s forming o r t h o p o s i t r o n i u m . f = kC where f i s  3  the  Thus (14)  the f r a c t i o n of p o s i t r o n s forming p o s i t r o n i u m 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 if  independent of e l e c t r i c  field,  the f r a c t i o n of o r t h o p o s i t r o n i u m atoms which are quenched  independent of e l e c t r i c  field.  In p r a c t i c e the peak count  is  rate  has to be reduced by an a d d i t i o n a l f a c t o r of 1-W, where W represents the f r a c t i o n of p o s i t r o n s which a n n i h i l a t e i n the w a l l s chamber.  of the  gas  Thus f i n a l l y  f  =  k  [C  -  v  Cp  (1  -  W)  R Q ]  /  [ 1  -  R  3  R  Q  ]  (15)  The p r o p o r t i o n a l i t y constant k i s determined i n S e c t i o n 2.6.3. from a knowledge of the f r a c t i o n of p o s i t r o n s forming p o s i t r o n i u m a t zero f i e l d ( F a l k and Jones, stopped i n the gas (C ) y  196*+), the f r a c t i o n ( 1 - W ) of  1965), and from the peak ( C ) and v a l l e y  (Falk,  p  count r a t e s at zero a p p l i e d e l e c t r i c  (15) i s  s i m i l a r to t h a t deduced by Marder,  field.  A n a l y s i s of  2.5-1• 2.5-1-1-  The e x p r e s s i o n  et a l . ,  s i m p l e r i n that no magnetic f i e l d quenching i s 2.5.  positrons  (1956), but  present i n t h i s  is case.  results.  A n a l y s i s of time  spectra.  Maximum l i k e l i h o o d t h e o r y . In order to f i t  exponentials  (as  the experimental data to a sum of two  expected a c c o r d i n g to the d i s c u s s i o n of  Section  2.3-), a computer programme was d e v i s e d u t i l i z i n g m a x i m u m - l i k e l i hood theory (Orear, constant  1958).  For f i n i t e  channel w i d t h s , w^, and a  random background B per u n i t c h a n n e l ,  spectrum shape  is  the  theoretical  -32W  + W K  0  yK = / o  C i'• expC-t/x ) + I e x p ( - t / T ) ] dt + w B  (16)  I  2  K  2  w  where w  is  the sum of the channel widths up to the s t a r t  channel k.  S i n c e the counts  distributed, p  i n i n d i v i d u a l channels are P o i s s o n  the p r o b a b i l i t y of observing  k = tyy^expt-y*) I k N  of  counts  i n channel k i s  <>  !  17  w h i l e f o r the whole e x p o n e n t i a l r e g i o n 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 f u n c t i o n L =.j!P  (18)  k  The aim i s  to determine the f o u r paramenters, 1-j , l ^ i  which t h i s  probability L is  a maximum.  T  1 • 2> ^ T  I t i s more convenient  o  r  to  d e a l w i t h the l o g a r i t h m i c p r o b a b i l i t y W = I lnP k  = I N lny, - y, - ln(N !) k  k  k  (18a)  R  The programme which maximizes t h i s p r o b a b i l i t y f o l l o w s lines  as t h a t of F a l k (1965).  which W i s a maximum i s 9  W  -  n  8  3TJ " °'  W  n  3  31, = °'  2.5.1.2.  given by the set n  W  9^  Iterative  The v a l u e s  = Q ;  ™  the  same  of the paramenters f o r of four  equations  n'  A  JT = °'  O  N  ( 1 9 )  2  s o l u t i o n of the maximum l i k e l i h o o d problem.  Since the equations  (19).are extremely n o n l i n e a r ,  the  method f o l l o w e d was an i t e r a t i v e procedure i n v o l v i n g 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  expansion of each of the p a r t i a l d e r i v a t i v e s initial  estimates.  a^, a^_, r e p e c t i v e l y , reduce to a set  series  (19) about these  I f the f o u r parameters are denoted a-| , &21 then the f o u r simultaneous  of four simultaneous  equations  l i n e a r equations  above  represented  -33-  (20)  CA=V. The h x h matrix C c o n t a i n s  the elements (21)  The v e c t o r s A and V are  respectively (22)  (23) S o l u t i o n of the f o u r simultaneous  equations y i e l d s  A a i which are to be added to the i n i t i a l  estimates  a-± so obtained are then i n s e r t e d i n the set the process  repeated.  estimates are s u f f i c i e n t l y  The new (20) and  once the  exponentials initial  good t h a t none of the parameters become  Should the l a t t e r occur i t was found convenient  the problem i s  i n three unknowns.  u s i n g the best values as the i n i t i a l  Once convergence under t h i s the f o u r parameter  fix  three three  iterations,  obtained from the three parameter  estimate,  to  In t h i s way,  reduced to a problem i n v o l v i n g s o l u t i o n of  parameter v a r i a t i o n was a c h i e v e d ,  2.5-1-3-  a^.  c f equations  one of the parameters, u s u a l l y the long l i f e t i m e .  equations  increments  Convergence f o r the case of two  g e n e r a l l y r e q u i r e s l e s s than s i x i t e r a t i o n s ,  negative.  the  solution  u s u a l l y converged.  I n i t i a l estimates of the f o u r parameters. The i n i t i a l  estimates were made using a v e r y f a s t  procedure rased on the l e a s t squares method. d i v i d e d i n t o three r e g i o n s .  iterative  The spectrum i s  The f i r s t r e g i o n i s  the range over  i.  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 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 these  over  i s the  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 t h e 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 m a k i n g a s t r a i g h t  l i n e f i t t o the logarithm of the channel region.  This f i t i s extrapolated i n t o the f i r s t  contribution to the f i r s t of the r e s u l t i n g curve straight and  counts i n t h e second  line.  This  r e g i o n , and i t s  r e g i o n subtracted out.  i nthe f i r s t  The l o g a r i t h m  r e g i o n i s then f i t t e d  to a  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 ,  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 t h e r e m a i n d e r i n t h e s e c o n d r e g i o n i s o n c e more to  a s t r a i g h t l i n e and t h e whole procedure  2.5-1•^•  E s t i m a t i o n o f channel The  using  r e l a t i v e channel  widths  repeated.  w^ a n d r a n d o m b a c k g r o u n d B.  w i d t h s , w^, h a v e b e e n m e a s u r e d  t h e random t i m e g e n e r a t o r  ( s e e S e c t i o n 2.6.6.2.).The r e s u l t s  o f t h e m e a s u r e m e n t a r e shown i n t h e A p p e n d i x . random b a c k g r o u n d c o u n t s ,  fitted  B per u n i t channel,  Estimation of the f o l l o w s the procedure  u s e d b y F a l k (1965) a n d i n v o l v e s t h e m e a s u r e m e n t o f t h e random coincidence 2.5-1 • 5'•  r a t e d i r e c t l y f r o m t h e 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.)  Estimation of variances. *K  Once t h e most p r o b a b l e have been o b t a i n e d , in  each o f these  that a matrix  an estimate  values  j|e  s(c  s e t o f v a l u e s : a-j, a , a ^ , a ^ , 2  of the s t a t i s t i c a l  i sdesired.  o f t h e type  J|C  uncertainty  I t h a s b e e n shown ( O r e a r ,  1958)  •35-  when i n v e r t e d , y i e l d s (H  = ""(a -  _1  i  where  is  ai  ")  the s o - c a l l e d e r r o r matrix  (aj- aj")  (25)  the most probable v a l u e of the parameter a^.  If i t  i s assumed that the l i k e l i h o o d f u n c t i o n L i s  Gaussian w i t h r e s p e c t  to each of the parameters a^, and that  parameters are independent of each o t h e r ,  then  L - r[ exp C- i ( i * - X ) ] i °i a  a  (26)  2  2  where o |  is  the v a r i a n c e of the G a u s s i a n .  W= I - i ( i l) i i a  2  a  the  It follows  + constant  that (27)  0  and 3W 2  - 1/ai  2  whence  Thus the d i a g o n a l elements of H are TT  3^  _  "ii "  " Sip  In t h i s  case,  (29) s i n c e H^^ i s a d i a g o n a l m a t r i x ,  ( H - l ) ^ = (Hii)-l SO  oz = ( ±  H i i  )-i  y i e l d the v a r i a n c e of the parameters i f L i s Gaussian i n shape. I f H i s not d i a g o n a l , i n v e r s i o n of the e n t i r e m a t r i x H  -36-  allows  the c o r r e l a t i o n between d i f f e r e n t  parameters to be taken  i n t o account i n the e s t i m a t i o n of the v a r i a n c e s Should any of the o f f - d i a g o n a l  ( O r e a r , 1958).  elements be n e g l i g i b l e  compared with  the ma'in d i a g o n a l elements, then the r e l e v a n t parameters have an a p p r o p r i a t e l y s m a l l degree of c o r r e l a t i o n . In order to check the assumption that the  likelihood  f u n c t i o n d e s c r i b i n g t y p i c a l experimental r e s u l t s  i s near to Gaussian  i n shape,  Each of  the f o l l o w i n g a n a l y s i s was performed.  parameters i n t u r n was set on e i t h e r  a t one and then two standard  deviations  s i d e of the best v a l u e obtained u s i n g the maximum  l i k e l i h o o d programme.  The remaining parameters were then v a r i e d  to maximize the l i k e l i h o o d once more. relative  the  The r e s u l t i n g  likelihood  to the best maximum l i k e l i h o o d was then p l o t t e d as a  f u n c t i o n of the parameter concerned. shape of the l i k e l i h o o d f u n c t i o n i s  Figure 6 indicates  that  the  indeed approximately Gaussian f o r  each of the f o u r parameters. 2.5.1.6.  Goodness of  fit.  In order to d i s c u s s fit, since  the goodness - or otherwise  the u s u a l c h i - s q u a r e t e s t was made on each spectrum.  - of  the  However,  the c h i - s q u a r e t e s t i s d e f i n e d i n terms of the n o r m a l i z e d  p r o b a b i l i t y of g e t t i n g  a worse f i t  a c c o r d i n g to Gaussian s t a t i s t i c s 1965) i t  i s not s u f f i c i e n t  computation i n the present number of counts,  for quantities  distributed  ( O r e a r , 1958; Mathews and Walker,  to make a s t r a i g h t  channel"-by-channel  case.  because of the  This arises  u s u a l l y t h i r t y to f o r t y ,  i n the t a i l of the  low time  —  VALUES OF PARAMETER F i g u r e 6.  Fix^p  a  Dependence o f t h e l i k e l i h o o d f u n c t i o n on ll , I » , p« The v a l u e s o f t h e s e p a r a m e t e r s 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 t h e 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 o m 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 are denoted-Gaussian i n the legend. °*' T  2  x  1  -37-  spectrum ( F i g u r e  . . Although the v a r i a n c e of these P o i s s o n -  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 d e v i a t i o n about the mean s i n c e tion is counts  the P o i s s o n d i s t r i b u -  skew about the mean f o r counts below about 1 0 0 . lower than the mean are more p r o b a b l e .  becomes a p p l i c a b l e .  is  above about 1 0 0 ,  Table  II  non-Gaussian d i s t r i b u t e d counts,  of  the c h i - s q u a r e t e s t  annihilation rates,  method was employed. method l i e s s o l e l y  In order to f i t  the  d i r e c t or o r t h o p o s i t r o n i u m decays,  p a r t i c u l a r polynomial dependence  results  on d e n s i t y ,  the l e a s t  rates.  i n a gas  observed to a squares  The j u s t i f i c a t i o n f o r u s i n g the l e a s t  i n the f a c t  for  test.  In g e n e r a l the a n n i h i l a t i o n r a t e s of p o s i t r o n s on the gas d e n s i t y .  test  the  A n a l y s i s of e x p e r i m e n t a l l y - d e t e r m i n e d a n n i h i l a t i o n  are dependent  the  chi-square  Table II f u r t h e r c o n t a i n s  f o r the c h a n n e l - b y - c h a n n e l c h i - s q u a r e  2.5-2.  such that  In order to r e i n f o r c e  point concerning the n o n - a p p l i c a b i l i t y :bf  the  then the c h i - s q u a r e  shows the r e s u l t s  t e s t s on a l l the s p e c t r a p r e s e n t e d .  fact,  However, i f  counts are summed over a c e r t a i n number of channels, t o t a l number of counts  In  squares  that the l i k e l i h o o d f u n c t i o n s  for  the t y p i c a l time s p e c t r a r e p o r t e d here have been demonstrated be n e a r l y Gaussian i n shape  (Section 2 . 5 * 1 * 5 • ) •  That they are  shows t h a t 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 I , 1  I  , T , T^,  i s nearly Gaussian.  1  lifetimes,  it  is  Thus,  to so  parameters  i n order to analyze  the  c o n s i s t e n t 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 , method of curve f i t t i n g  which g i v e s r i s e to the l e a s t  (Orear,  1958).  squares  Furthermore, the standard  TABLE I I . RESULTS OF CHI-SQUARE TEST ON LIFETIME SPECTRA. Q is  to be i n t e r p e t e d as the n o r m a l i z e d (to  getting Q  B  Q  A  is  a worse f i t  1) p r o b a b i l i t y of  should the experiment be r e p e a t e d .  the r e s u l t of the c h a n n e l - b y - c h a n n e l computation, w h i l e results  from the a l t e r n a t i v e i n t e g r a l method o u t l i n e d  in Section 2 . 5 . 1 - 6 .  -fiA-  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  0A5  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  <10  - 8  0.93  0.03  0.25  0.02  0.56  0.45  0.97  0.73  0.34  0.004  -38-  chi-square test is relevant i n this  case ( S e c t i o n 2.5.1.6.).  The polynomial parameters were obtained using a computer s o l u t i o n f o r the set  of a n a l y t i c equations p e r t i n e n t to the  squares problem f o r a f u n c t i o n l i n e a r i n these parameters  1958; Rose, 1953).  The f u n c t i o n which i s  a^ (to be determined)  least  (Orear,  l i n e a r i n the parameters  is  M y(x) = I a- fi(x) i=l  (30)  where the f-^(x) are any f u n c t i o n s  of x o n l y .  p experimental values  have been obtained as a f u n c t i o n  N ( X J ) ± aj  In a s i t u a t i o n where  of  the p data p o i n t s X j , the l e a s t squares s o l u t i o n s MD ' a.* = H ^ j l ^ i j l ( I T M W k,j=l J where  f o r the aj_ are (31)  0  = I a^k>4ji2kl  k=l k The e r r o r m a t r i x i s J  (32)  G  ( H ) ^ = (ai-a^) - 1  g i v e n as u s u a l by  (aj-aj )  (33)  ft  The c h i - s q u a r e t e s t was a l s o performed on each  fit.  Experimental r e s u l t s .  2.6. 2.6.1.  C r i t e r i a f o r p r e s e n t a t i o n of d a t a . The data presented here r e p r e s e n t the r e s u l t s  made w i t h f o u r d i f f e r e n t Argon gas samples. fitted  to two exponentials  ( S e c t i o n 2.5-1•)• ing  conditions:  of runs  A l l the s p e c t r a were  by the maximum-likelihood technique  The r e s u l t s  presented here s a t i s f i e d  the  follow-  -39( a ) C o n v e r g e n c e was ly  obtained v a r y i n g a i l  i n the maximum-likelihood  (b) No h i g h v o l t a g e  I t was found  could  to .significant  were a s c r i b e d  t o the  simultaneous-  programme.  breakdowns o c c u r r e d d u r i n g  question. lead  four parameters  that  a c t u a l run  in  t h e o c c u r r e n c e o f such breakdowns  deviations  effect,  the  i n the  results.  o f contaminating  breakdown p r i o r t o t h e a b s o r p t i o n  These  deviations  gases l i b e r a t e d by the  o f s u c h c o n t a m i n a n t s by the  purifier. ( c ) The r e s u l t s o f a l l t h e probability Out  alone, two  o f g e t t i n g a w o r s e f i t was g r e a t e r  of a total  criteria.  o f 33 r u n s ,  s i n c e a poor r e s u l t here s i g n i f i e d  exponentials  time spectra this  was p o o r f o r t h e  starting  2.6.2.1.  Results  Figure r a t e a t zero the  7 shows t h e  electric  functions  assumption o f This  fault  c h a n n e l number o f t h e The c h i - s q u a r e t e s t a good  the  electric  field  results.  a n n i h i l a t i o n rate t o functions of  density.  field  results of fitting  simple  test  analyses.  of - f i t t i n g  the A r g o n  the  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  D i r e c t a n n i h i l a t i o n r a t e : zero  2.6.2.  the  a t w h i c h a n a l y s i s was s t a r t e d .  p o i n t f o r the  chi-square  o f the that  0.1 (10$).  basis o f these  case considered.  be r e m e d i e d b y a l t e r i n g  case, thus serves  than  6 were r e j e c t e d on the  None was r e j e c t e d o n t h e b a s i s  could normally  in  chi-square tests indicated that the  o f the  the  dependence o f the on Argon d e n s i t y .  direct annihilation T a b l e I I I shows  r e s u l t s shown i n F i g u r e  density.  Ther e s u l t s ,  7 to various  Q, o f a  chi-square  4 6 8 10 12 14 P - DENSITY (AMAGATS) F i g u r e 7. 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 a t zero e l e c t r i c f i e l d as a f u n c t i o n o f d e n s i t y . The s t r a i g h t l i n e r e p r e s e n t s V » 5 . 6 P x 10°sec-1.  -ko-  c a l c u l a t i o n are a l s o g i v e n .  The v a l u e o f Q i s  as being the p r o b a b i l i t y of g e t t i n g of experiments  be r e p e a t e d .  were made to a l l the p o i n t s  to be i n t e r p r e t e d  a worse f i t  Unless otherwise  should the  stated,  the  series fits  shown i n F i g u r e 7-  On the b a s i s of the c h i - s q u a r e t e s t alone i t would appear that form VI i s  the best f i t  to the experimental d a t a .  However, there i s no reason to suppose a n n i h i l a t i o n rate is  different  that the observed  from  zero at zero Argon  u n l e s s a model i n v o l v i n g an A r g o n - p o s i t r o n complex i s A c c o r d i n g to f i t  direct density,  invoked.  VI such a system would have to have a  half-life  _Q  o f about *t2 x 1 0  s e e s , which i s  c o n s i d e r a b l y longer than that  -9  which would be expected  ( 1 0 sec),  because of the h i g h e r  electron  d e n s i t y at the p o s i t r o n p o s i t i o n compared w i t h the p a r a p o s i t r o n i u m atom (see  Chapter 1 , S e c t i o n 1 A . 2 . ) .  ment f i t s V and VI can be r e j e c t e d .  On the b a s i s of t h i s  F u r t h e r examination of Table  shows t h a t below 1 0 amagats a l i n e a r f i t  IV to the data i s  I n c l u s i o n of the data obtained at h i g h e r d e n s i t i e s r e s u l t s s i g n i f i c a n t l y worse for f i t s  linear f i t  1  Some improvement i s  c o n t a i n i n g a n o n - l i n e a r dependence  the l i n e a r term ( f i t s for a  I.  I I and I I I ) .  between 5-5 and 5-7 x 1 0 sec  arguIII  adequate. in a  obtained  on d e n s i t y as w e l l  as  I t i s apparent that a value amagat  describes  well  the  l i n e a r dependence  on Argon d e n s i t y of the d i r e c t a n n i h i l a t i o n r a t e .  Such a v a l u e f o r a  1  is  i n keeping w i t h the o b s e r v a t i o n that  l i n e a r term i n f i t s  I to IV i s  somewhat  independent  of the  n a t u r e of the f i t .  Regarding the n o n - l i n e a r i t y at h i g h e r  the detailed densities,  the tendency i s f o r the a n n i h i l a t i o n r a t e to be reduced r e l a t i v e  to  TABLE I I I . DEPENDENCE OF A Summary o f t h e r e s u l t s  of f i t t i n g  Tvne o f f i t  I. II. III. IV.  A  a  X X  X  =  a  =  a  1 & 1  a  7.  -  sec  -1  .-1 amagat  Q  a /10  a3/l0°"  6  2  sec  -1  . -2  amagat  s e c ^ amagat -  5.42 i 0.05  P  a  =  sec ^  - 1  P  a = 1 a  af/106  6  0  6  the curve i n F i g u r e  Results a /10 -  10 s e c  ON P.  a  + a P  * 3 a  p  1  5.71 i 0.12  2  2  0.02 -0.027  31  5.59 - 0.08  p 3  P  0.08  0.011 0.0013 - 0.0004  0.13  5.53 - 0.06  0.16  5.28 1 0.11  0.02  P < 1 0 amagats V.  X  a  =  a  VI.  X  a  =  a  o  +  o  +  1  a  a  2  P  p 2  11  .44±  0.94  23 .85 i 0.52  0.250 - 0.006  Note: U n l e s s o t h e r w i s e s t a t e d , made t o a l l t h e p o i n t s  the f i t s  i n Figure  7-  were  0.21  the l i n e a r f i t of  acceptable  at lower d e n s i t i e s .  The d e t a i l e d  t h i s n o n - l i n e a r i t y remains undetermined by these  as examination of Q f o r f i t s  II and I I I  D i s c u s s i o n of the f i t s  2 . 6 . 2 . 2 .  The Table I ,  estimated  Section  nature  experiments,  shows.  to the  data.  p u r i t y of the Argon gas has been given  in  The s m a l l d e v i a t i o n from l i n e a r i t y of  2 . 2 . 1 .  the  a n n i h i l a t i o n r a t e as a f u n c t i o n of d e n s i t y might be due i n some way to the N i t r o g e n p r e s e n t .  However, as  rate i n Nitrogen i s  expected to the f i r s t  with density,  difficult  or  it  is  any other i m p u r i t y gas  annihilation It  the'direct  annihilation  order to v a r y l i n e a r l y  to see how the presence of N i t r o g e n  could a f f e c t  the l i n e a r i t y of the  rate. is  f a r more reasonable  to suppose t h a t the  non-linear-  i t y at h i g h d e n s i t i e s a r i s e s from the i n t e r a c t i o n of the w i t h more than one Argon atom at a time. an  direct  positron  The p o s s i b i l i t y  of  e f f e c t has been r a i s e d by Tao, B e l l and Green ( 1 9 6 + ) ( s e e  also  l  Kivel,  1 9 5 9 ) .  At  1  5  amagats,  the average  interatomic  such  distance  - 7  is  about 1 0 c m ,  wavelength  which i s  of the same order as the de B r o g l i e  of a t h e r m a l i z e d p o s i t r o n .  It is  also possible  i n t e r a t o m i c d i s t a n c e s of t h i s magnitude could r e s u l t screening and  i n some  of the f i e l d of the p o s i t r o n at the s c a t t e r i n g  thus reduce the magnitude of the v e r y important  polarization potential. by f u r t h e r experiments  Clearly,  atom,  attractive  t h i s problem can o n l y be  at h i g h p r e s s u r e s ,  that  resolved  and by c a l c u l a t i o n s  take i n t o account the presence of more than one s c a t t e r i n g  that  atom.  -h2-  From the r e s u l t s  g i v e n here i t  seems that the d e v i a t i o n 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 r a t e on d e n s i t y  is  l e s s than 10$ at about 17 amagats. 2.6.2.3.  Comparison of the l i n e a r term w i t h previous  results.  The magnitude of the d i r e c t a n n i h i l a t i o n r a t e per u n i t amagat i s  compared i n Table IV w i t h the values  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 i n d i c a t e d i n S e c t i o n . 2 . 6 . 2 . 1 .  e r r o r a s s o c i a t e d w i t h the v a l u e r e f l e c t s and  also  obtained by other  the s t a t i s t i c a l  The  errors  the u n c e r t a i n t y i n the exact n o n - l i n e a r behaviour of  annihilation rate.  The systematic  e r r o r s (Sec;  w i t h t h i s measurement are of the order of \%.  2.6.6.3}  the  associated  This estimate  is  not i n c l u d e d i n the v a l u e given i n Table I V . TABLE IV. P u b l i s h e d values  of the D i r e c t A n n i h i l a t i o n R a t e .  D i r ,e c t a ntnei h i l a t i o n ra 1 0 sec" amagat" 1  The v a l u e s  Author  1  5-90*0.23  Falk  (1965)  ^.96  Tao, B e l l and Green  5.18  Duff and Heymann (1962)  3.0M-  Osmon (1965)  5.78  Paul  5.6*0.1  Present work  (196*0'  (196*0  presented f o r other workers d i f f e r from the p u b l i s h e d  298 273.  v a l u e s where n e c e s s a r y by t h e f a c t o r which results  This i s t h e f a c t o r by  o b t a i n e d a t room t e m p e r a t u r e (25°C) h a v e t o be -1  -1 multiplied  i n o r d e r t o e x p r e s s them i n u n i t s The  by  amagat  l a c k o f agreement between a l l t h e r e s u l t s  thus f a r a r i s e s and  o f sec  published  mainly from the presence o f i m p u r i t i e s  inadequate analysis  of results.  Osmon (1965) i s s i g n i f i c a n t l y  p r o b a b l y because t h e l o n g - l i v e d  i n the gas,  F o r example, t h e r e s u l t  smaller than the other  values  o r t h o p o s i t r o n i u m component was  completely ignored i n the l i f e t i m e analysis.  I f such a  long-lived  component i s n o t t a k e n i n t o a c c o u n t , i t c a n h a p p e n t h a t contribution  will  raise  the t a i l  of the short-lived  r e l a t i v e t o t h e r e s t o f t h e spectrum. single  exponential then r e s u l t s  short-lived The  given  its  component  Subsequent a n a l y s i s  as a  i n an i n c r e a s e d l i f e t i m e f o r  component, and hence i n a r e d u c e d a n n i h i l a t i o n  this  rate.  A r g o n u s e d b y T a o , B e l l a n d G r e e n (196*+) h a s s u b s e q u e n t l y  b e e n shown ( T a o a n d B e l l , The  1966) t o c o n t a i n s i g n i f i c a n t  impurities.  agreement between d i f f e r e n t workers w i l l  only  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 g a s e s c o n t a i n i n g t h a n a t o t a l o f 1 ppm o f i m p u r i t y .  less  I n t h i s sense t h e r e s u l t s  q u o t e d i n T a b l e I V s h o u l d be assumed t o be t h o s e f o r Impure A r g o n . The  annihilation rate  presented here i s that to  since  1  (Falk,  the annihilation rate  1965),  1  impurities  S i n c e t h e main i m p u r i t y  the direct a n n i h i l a t i o n rate  5.5 x 10 s e c " ' a m a g a t " that  6  a p p r o p r i a t e t o Argon containing  the extent given i n Table I .  and  o f (5.6*0.1) x 1 0 s e c " a m a g a t "  i s Nitrogen,  i n N i t r o g e n i s about  i t can hopefully  given here i s very l i t t l e  be assumed different  from  -44-  the 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 electric  field  Figure 8 p e r u n i t amagat on  the v a l l e y - t o - p e a k  ratio:  results.  illustrates  the  e f f e c t of a p p l i e d e l e c t r i c  the d i r e c t a n n i h i l a t i o n r a t e .  For  small  the a n n i h i l a t i o n rate begins to decrease r a p i d l y reaching  a  constant  1  value  o f a b o u t 2.8  x 1O^sec amagat - 1  1  field  a t 90 V  E/P fairly  cm~ amagat~  A comparison of the c u r r e n t r e s u l t s w i t h those of F a l k  1  (1965)  i n d i c a t e s good agreement. The as d i s c u s s e d  measurements o f t h e v a l l e y - t o - p e a k ( S e c t i o n 2.4.)  as a f u n c t i o n o f E/P  the i n c r e a s e d  ratio ( indicating  formation  of  positronium)  a r e p l o t t e d on t h e same g r a p h f o r c o m p a r i s o n .  T h e s e d i f f e r f r o m t h e m e a s u r e m e n t s o f M a r d e r , e t a l . , (1956) i n t h a t the r a t e of r i s e of the v a l l e y - t o - p e a k applied electric  field  I n v e s t i g a t i o n of the contamination  o f up  r a t e o f r i s e can  i s considerably  ratio with  l e s s i n the present  s h a p e o f t h e c u r v e as a f u n c t i o n o f t o 1$  of the  Argon.  i n c r e a s e somewhat w i t h t h i s gas  b e c a u s e some o t h e r unknown i m p u r i t y was T h a t t h e r e was  case.  Nitrogen  t o t a l density indicated that  However, the r e s u l t s o f Marder, e t a l . were n o t probably  increasing  as  an  the  impurity.  reproduced, present  i n that  a t e n d e n c y 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 i m e i s c o n s i s t e n t w i t h the  evolution  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 is  an  i n c r e a s e i n the  i s increased,  t h r e e - p h o t o n component a t t h e  expense of  there the  A / P - D I R E C T . ANNIHILATION a  VALLEY-TO-PEAK (Normalized  to  1 at  RATE  (1cfsec*amagatT )  RATIO E / P = 0)  l  t w o - p h o t o n component.  Thus i t i s r e a s o n a b l e  of the f l a t t e n i n g o f f o f the creased  positronium  *  a  formation.  t o s u p p o s e t h a t some  v s E/P c u r v e a r i s e s f r o m t h e i n I n t h i s case, positronium  production  thus c o n t r i b u t e s a n e x t r a c h a n n e l by w h i c h the 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 c a ndecay. from the field,  equilibrium d i s t r i b u t i o n increases with applied  electric  9) w h i c h r e s u l t s f r o m t h e  averaged positronium  formation  of p o s i t r o n s forming  positronium  Section 2 . T h e  rate.  Figure  shows t h e  fraction 15,  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  p o s i t r o n s i n A r g o n ( F a l k , 1965).  a s s u m i n g t h a t 2,7% o f t h e p o s i t r o n s tronium  '9  velocity-  as c a l c u l a t e d from Equation  o f t h e chamber was c a l c u l a t e d f r o m d a t a  a t zero f i e l d  t h a t f o r the h i g h e s t form  formation  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  for  As p o s i t r o n i u m  d e r i v e d f r o m the dE/dx  The c o n s t a n t  data  k was d e t e r m i n e d  stopped i n the gas form p o s i -  ( F a l k a n d J o n e s , 196*+). electric  walls  I t i s quite  apparent  f i e l d s used a m a j o r i t y o f the  positrons  positronium. T a b l e V shows t h e d e p e n d e n c e o f t h e o v e r a l l number o f  c o u n t s i n t h e l o n g - l i y e d component a s a f u n c t i o n o f e l e c t r i c  field.  Two  with  sets o f data  are given corresponding  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  t o spectra obtained  s e t a t the peak ( " d i r e c t -  e n h a n c e d " ) a n d 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 (see S e c t i o n 2,2.1.). the  The p r o d u c t I T  long-lived exponential  t =0 t o t=», Ip and x  2  2  I exp( r-t/t ) 2  a n d was c a l c u l a t e d u s i n g  2  i s found by i n t e g r a t i n g ( s e e S e c t i o n 2.3.*+.) f r o m  the appropriate  found from maximum-likelihood f i t s t o the  values  lifetime  for spectra.  -1+6-  In t h e case of the data presented i n Table V t h e zero been a r b i t r a r i l y d e f i n e d point  positronium  as  this  field  from Table V  d e p e n d e n c e o f t h e t o t a l number o f o r t h o -  a n n i h i l a t i o n s r e c o r d e d by t h e t i m e s o r t e r .  i s that the s t a t i s t i c a l uncertainty  i n ^2 2  The r e a s o n ^  X  s  a  ^  -*-  l a r g e as t h e s i z e o f t h e 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  .9)?  evan a t t h e h i g h e s t To  as  i s n o t p o s s i b l e t o draw a n y c o n c l u s i o n s  to the electric  for  a s c h a n n e l 89 o f t h e k i c k s o r t e r , a t w h i c h  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. It  as  o f time has  fields  e a s t  8 and  used.  summarize, t h e r a p i d decrease o f a n n i h i l a t i o n r a t e  a function of applied  electric  field  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 a s t h e p o s i t r o n v e l o c i t y increases. the  The e x p l i c i t v e l o c i t y d e p e n d e n c e c a n n o t be f o u n d  experimental  transfer  data unless  the velocity-dependent  c r o s s - s e c t i o n i s known.  2.6.h. l  i n the time  196^;  displayed, the usual  spectra.  applied  shoulder  electric  !  5. ) •  Since  field,  a l l the time  s t r u c t u r e (Tao, B e l l  F a l k a n d J o n e s , 196 +; Osmon, 1965; P a u l ,  example F i g u r e direct  are discussed i n  Width of the shoulder. In t h e case o f zero  spectra  field  3«5«  The s h o u l d e r  2.6. +.1.  momentum-  Some i m p l i c a t i o n s o f t h e s h a p e  o f . a n n i h i l a t i o n r a t e d e p e n d e n c e on- e l e c t r i c C h a p t e r 3, S e c t i o n  from  and Green,  196*+) ( s e e f o r  t h e form o f t h e time spectrum o f t h e  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 T 2  C  ON E L E C T R I C  Ortho-enhanced I  4  10  T  77  counts  I T 2  4--1  1  _  V cm  0.3  6.0*  Direct-enhanced E/P  2- 2  FIELD.  10  amagat 0  E/P  2  counts 4.7  - 0.3  i  V era  amagat 0  ± 0.3  14.4  4.4  0.4  14.0  5.6 ± 0 . 4  34.7  4.9 - 0 . 5  3^.7  6.3 t 0.5  50.7  5.1 t-0.7  48.9  ± 0.8  70.7  4.5 ±0.8  70.7  5.8  6.1  125.1  6.5 - 1.1  Note:  The r e s u l t s  g i v e n above are f o r Argon at an average  d e n s i t y of 9 . 0 amagats. and The  The terms '"ortho-enhanced"  "direct-enhanced" are d e f i n e d i n S e c t i o n  3  2.6.3.  e r r o r s quoted were o b t a i n e d by simply compounding  the e r r o r s i n I  2  and t  2  maximum l i k e l i h o o d f i t s .  t h a t were found from the Each spectrum was normalized  to the same t o t a l number of 1.28 MeV counts as in Section  2.2.1.  indicated  _L _ 7  exponential,  the end of the shoulder s i g n i f i e s  either a thermalized  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 , or t h a t the a n n i h i l a t i o n r a t e becomes v e l o c i t y o f these two. reduces  independent at low v e l o c i t i e s or some combination  The a p p l i c a t i o n of a s m a l l e l e c t r i c  however,  the a n n i h i l a t i o n r a t e s i g n i f i c a n t l y without p e r t u r b i n g the  shoulder markedly. electric  field,  This i s  f i e l d increases  the d i r e c t l i f e t i m e  shown i n F i g u r e 10.  Since the a p p l i e d  the average p o s i t r o n energy at  equilibrium,  corresponds to a v e l o c i t y - d e p e n d e n t  annihilation  r a t e averaged over v e l o c i t i e s higher than t h e r m a l . seems c o n s i s t e n t to i n t e r p r e t the s i n g l e  Therefore,  exponential after  the.  shoulder i n terms of a t h e r m a l i z e d p o s i t r o n d i s t r i b u t i o n . width of the s h o u l d e r , t h e n ,  is  it  The time  the time taken f o r a p o s i t r o n to  t h e r m a l i z e from energies corresponding to E ^ . ^ (8.9 e V ) .  It  has  a l r e a d y been shown (Chapter 1, S e c t i o n 1.2.3*) t h a t the time taken f o r t h e p o s i t r o n to reach such energies (about 10 eV) i s  l e s s than  the time r e s o l u t i o n o f the apparatus used. The shoulder width i s  i n v e r s e l y p r o p o r t i o n a l to  density.  The w i d t h - d e n s i t y product was measured to be about 3*+0 nsec-amagat, i n agreement w i t h previous r e s u l t s 196k).  ( F a l k and Jones,  196*f; P a u l ,  A narrower shoulder width would i n d i c a t e a more r a p i d  slowing down of the p o s i t r o n s , which would occur through the of f o r e i g n gas atoms. low-lying excitation significantly,  Thus the presence  agency  of f o r e i g n molecules  l e v e l s , would be expected  to narrow the  simply by i n c r e a s i n g the energy l o s s per  with  shoulder  collision  and thereby the r a t e of t h e r m a l i z a t i o n . A positron scattered  by the host atoms, w i t h momentum-  o o o IO  56 o o o  P = 9.0  + ++  E/P o o  -  1 3 . 9  2.74 nsec/channel  CO  f-o  Z ° 3 ~ O N O  U_ O o -o  8 o  1  r 60.000  85.000  110.000  1  135.000  CHANNEL NUMBER  1—•  160.000  T  T  T  185.000  210.000  235.000  F i g u r e 1 0 . Time s p e c t r u m f o r p o s i t r o n s i n A r g o n a t s m a l l E / P . P i s i n a m a g a t s , E/P i s i n V cm" ' a m a g a t . 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 the points. -1  -1+8-  transfer cross-sections 11  10  of the order of n a  , will  2 0  suffer  about  1 2  -10  collisions  sec  i  h  positron suffers  10 -10 J  at 1 0 amagats.  - 1  collisions.  Thus i n 3 0 nsec  the.  An i m p u r i t y c o n c e n t r a t i o n  L.  as low as a few p a r t s i n 1 0  c o u l d be expected  cant e f f e c t on the shape of the s h o u l d e r . experiments  to have a s i g n i f i -  Throughout these  the shoulder width remained e s s e n t i a l l y  showing t h a t here there was no s i g n i f i c a n t  constant  change i n gas  composition  from run to r u n , e i t h e r due to e v o l u t i o n of i m p u r i t i e s from the walls,  or due to the d i f f e r e n t gas samples The l o g a r i t h m i c slope of the  2.6.*+.2.  used.  shoulder.  A second f e a t u r e of the shoulder i s constant  logarithmic slope.  its  reasonably  Furthermore i t appears t h a t the  of the time spectrum i n the r e g i o n of the shoulder i s  shape  somewhat  independent of the shape of the o r t h o p o s i t r o n i u m component i n that region.  This i s demonstrated by comparing the two time  spectra  i n F i g u r e 11 which were obtained w i t h the 0 . 5 1 MeV s i n g l e a n a l y z e r set enhanced)  at the peak ( d i r e c t - e n h a n c e d )  positions  i n t u r n (see  Section 2 . 2 . 1 . ) .  o f the shoulder there i s v e r y l i t t l e despite  the f a c t  and v a l l e y  difference  that the r e l a t i v e i n t e n s i t i e s  channel  (ortho-  In the r e g i o n  i n the time s p e c t r a , of the d i r e c t  and o r t h o p o s i t r o n i u m components d i f f e r i n the two c a s e s .  Such  a s i t u a t i o n c o u l d a r i s e i f the o r t h o p o s i t r o n i u m component underl y i n g the shoulder has a l o g a r i t h m i c slope  little  different  from  the l o g a r i t h m i c slope of the d i r e c t a n n i h i l a t i o n c o n t r i b u t i o n to the s h o u l d e r .  T h i s indeed seems to be the case f o r Argon at  1  60-000  85.000  i  r  110.OOO  135.OOO  CHANNEL NUMBER  160.000  185.000  210.000  235.0001  F i g u r e 1 1 . C o m p a r i s o n o f d i r e c t - and o r t h o - e n h a n c e d t i m e s p e c t r a o b t a i n e d a t P=H-.9 a m a g a t s , E/P=0 V cm'Tamagat" . 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 - direct-enhanced; C - ortho-enhanced. 1  -49-  (Figure 1 V ) .  a d e n s i t y of 4.9 amagats  For higher d e n s i t i e s  l o g a r i t h m i c slopes should d i f f e r to a g r e a t e r the d i f f e r e n t  pressure dependence,  e x t e n t , because of  and hence the shoulder i n the  d i r e c t - e n h a n c e d time spectrum should be somewhat d i f f e r e n t t h a t i n the ortho-enhanced time spectrum.  much narrower i n t h i s  any c o n c l u s i o n s here If  it  case, i t  it  i s assumed, however,  is  Because the  field)  possible  argument.  that a l l the p o s i t r o n i u m i s  at the time corresponding to  to e x t r a p o l a t e  the  the o r t h o p o s i t r o n i u m  component i n the r e g i o n of the two exponentials r e g i o n of the s h o u l d e r .  shoulder  c l e a r l y d i f f i c u l t to draw  concerning the previous  formed ( f o r zero e l e c t r i c prompt peak,  is  from  F i g u r e 12 shows such  a p a i r of s p e c t r a obtained at 9«3 amagats. is  these  back i n t o  the  S u b t r a c t i o n of the o r t h o p o s i t r o n i u m  component i n the shoulder r e g i o n i n t h i s r e s u l t i n g curve g r a p h i c a l l y to a s i n g l e  f a s h i o n and f i t t i n g exponential y i e l d s  the  an  average a n n i h i l a t i o n r a t e i n the shoulder r e g i o n of about  6 - 1 1.5 x 10  sec  -1 amagat  f o r the two s p e c t r a i n F i g u r e 11.  the two s p e c t r a i n F i g u r e 12 the r e s u l t  is v K O  6  The s i z e of t h i s  - 1  amagat  6  1  sec"  g i v e n by Osmon (1965) ( t h i s v a l u e has been c o r r e c t e d as i n S e c t i o n 2.6.2.3.).  - 1  x 10 sec  T h i s a g r e e s • f a i r l y w e l l w i t n the f i g u r e of 1.2 x 10  For  —1  amagat  indicated  ""shoulder a n n i h i l a t i o n r a t e " ,  in  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 r a t e s observed as a f u n c t i o n  of  electric  field,  is  i n d i c a t e d i n F i g u r e 8.  A reasonably constant  l o g a r i t h m i c s l o p s of the  direct  a n n i h i l a t i o n c o n t r i b u t i o n to the shoulder cannot be i n t e r p r e t e d simply i n terms of a changing a n n i h i l a t i o n r a t e as a f u n c t i o n of  o o o •. +  51  o o o  P=9.3 E/P  8  o •o  = .0  2.74 nsec/channel  +  o o o 60.000  i 85.000  1  1  110.000  135.000  — i 160.000  CHANNEL NUMBER  T 185.000  T 210.000  T 235.000  F i g u r e 12. Comparison o f d i r e c t - and ortho-enhanced time s p e c t r a o b t a i n e d a t P-9.3 a m a g a t s , E/P»0 V - c m " a m a g a t " 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 t h e p o i n t s . 1  -50-  time.  In order that a constant  l o g a r i t h m i c slope appear i n the  d i r e c t component of the time s p e c t r a , d i s t r i b u t i o n must be at e q u i l i b r i u m , r a t e must be e s s e n t i a l l y range of the s h o u l d e r . S e c t i o n 3-4-.  either  the p o s i t r o n v e l o c i t y  or the d i r e c t a n n i h i l a t i o n  v e l o c i t y independent over the a p p r o p r i a t e T h i s i s d i s c u s s e d i n d e t a i l i n Chapter 3?  I t has a l r e a d y been p o i n t e d out that the  single  e x p o n e n t i a l f o l l o w i n g the shoulder corresponds to a n n i h i l a t i o n from the e q u i 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 . the constant  is  clear  that  l o g a r i t h m i c slope i n the shoulder r e g i o n can only a r i s e  from a reasonably v e l o c i t y velocities  Thus i t  independent a n n i h i l a t i o n r a t e  at  somewhat h i g h e r than t h e r m a l . Therefore,  it  appears t h a t i n a v e l o c i t y range a p p r o p r i a t e  to the time span d e f i n e d by the s h o u l d e r , the d i r e c t a n n i h i l a t i o n r a t e 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 p o s i t r o n i u m f o r m a t i o n occurs at the time c o r r e s ponding to the prompt peak. should be taken as evidence  The s m a l l shoulder a n n i h i l a t i o n r a t e that the v e l o c i t y dependent  6  direct  -1  a n n i h i l a t i o n r a t e can become as s m a l l as ^ 1 . 5 x 10 sec Furthermore,  since  the shoulder has a constant  over the g r e a t e r p a r t of i t s velocity-dependent  extent,  it  threshold) 2.6.4-.3-  and 1A0  eV (thermal  E f f e c t of the e l e c t r i c  amagat  logarithmic  seems l i k e l y that  d i r e c t a n n i h i l a t i o n rate is  of the energy range between 8.9 eV (the  -1  constant  slope the  over some  positronium formation  energy). f i e l d on the  shoulder.  I t has been suggested i n the previous s u b s e c t i o n  that  x  the v e l o c i t y  averaged a n n i h i l a t i o n r a t e corresponding to the  i n t e r v a l occupied by the shoulder i s independent  due to a somewhat  time  velocity  a n n i h i l a t i o n r a t e at v e l o c i t i e s higher than t h e r m a l .  Furthermore, t h i s shoulder a n n i h i l a t i o n r a t e has been shown to be c o n s i d e r a b l y l e s s than the a n n i h i l a t i o n r a t e at thermal v e l o c i t i e s Reference  to F i g u r e 8 shows that the a p p l i c a t i o n of an  f i e l d reduces not f a l l  electric  the d i r e c t a n n i h i l a t i o n r a t e , but t h a t t h i s does  below the shoulder a n n i h i l a t i o n r a t e .  This i s  so p r e -  sumably because of the onset of i n c r e a s e d p o s i t r o n i u m formation as the e l e c t r i c  f i e l d is  the shoulder disappears  applied.  However, i t  as the e l e c t r i c  is  f i e l d is  evident increased  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 r a t e averaged the v e l o c i t y  d i s t r i b u t i o n at h i g h E / P tends towards  annihilation  rate.  Lack of knowledge c h a r a c t e r i z i n g the v e l o c i t y below 1 1 . 6 eV (see detailed  increasing  over  shoulder  distribution  S e c t i o n 1 . 2 A . ) prevents  a more  of the way i n which the shoulder changes as  a f u n c t i o n of e l e c t r i c  dependent  (Fig. 1 '  d i s t r i b u t i o n of p o s i t r o n s w i t h energie  Chapter 1 ,  discussion  observations  of the i n i t i a l v e l o c i t y  the  that  field.  are e n t i r e l y  I t can o n l y be s t a t e d t h a t  c o n s i s t e n t w i t h the view of a  the  velocity-  a n n i h i l a t i o n r a t e which decreases as a f u n c t i o n of velocity.  o o o  55 +  o o  P = 9.3 E/P z 70.6 2.74 nsec/channel  to f-o O  o •o o o + ++  o o o  + 60.000  T  85.000  1  1  110.000  135.000  1  160.000  T  185.000  -+-t-  210.000  •+-  1-  235.000  CHANNEL NUMBER  F i g u r e 13* Time spectrum f o r p o s i t r o n s at h i g h 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 p o i n t s . P i s i n amagats, E / P i s i n V cm" amagat" . T  1  2.6.5.  Orthopositronium a n n i h i l a t i o n r a t e . F i t t i n g of experimental d a t a .  2.6.5.1.  F i g u r e 14 shows the A r g o n - d e n s i t y dependence  of  the  a n n i h i l a t i o n r a t e a p p r o p r i a t e to the l o n g - l i v e d component of time s p e c t r a .  Only those r e s u l t s  the c r i t e r i a d i s c u s s e d  are presented which  the  satisfied  i n S e c t i o n 2.6.1. and are thus f o r the  s p e c t r a as are the d i r e c t a n n i h i l a t i o n r a t e s  discussed  same  i n Section  2.6.2. The r e s u l t s functions is  of f i t t i n g  the data i n F i g u r e 14 to  of the d e n s i t y are shown i n Table V I .  the same as i n S e c t i o n 2.6.2.1.  Unless  were made to a l l the experimental values 27 p o i n t s p r e s e n t e d , field.  i n d i c a t e d , the  i n F i g u r e 14.  E i t h e r the parameter a  I to V I I ) or i t  is  electric  is  Q  ( f i t s V I I I to X) the  6 cal  free orthopositronium a n n i h i l a t i o n rate  fits  Of the  to which the data have been f i t t e d  d i v i d e d i n t o two c a t e g o r i e s . by the data ( f i t s  The meaning of Q  11 were obtained w i t h an a p p l i e d  The f u n c t i o n s  simple  (7.2 x 10  can be determined theoreti-  1 sec" ;  see Chapter 1, S e c t i o n 1.4.2.). The r e l a t i v e l y l a r g e s c a t t e r is  reflected  i n the poor f i t s v  some systematic dependent  terms,  10 amagats, the e l e c t r i c a  Q  and a  1  trend i s is  of the r e s u l t s  obtained i n Table V I .  apparent.  w i t h E/P=0, a good l i n e a r f i t the f i t  However,  F i t I , which c o n t a i n s no d e n s i t y  c l e a r l y unacceptable.  field results,  i n F i g u r e 14  VI i s  For d e n s i t i e s l e s s II i s  obtained.  obtained.  are r e l a t i v e l y unchanged but the f i t  is  than  Including  The parameters  judged worse by  6» 0  '  i 1 i I 2 4 6 8 10 12 14 16 P — DENSITY (AMAGATS) l  '  I  »  I  i  I  i  I  i  I  »  I  18  F i g u r e 14. 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 i n Argon as a function of density. The s t r a i g h t l i n e r e p r e s e n t s the f u n c t i o n X » ( 7 . 2 * .0.29 P) x-10°sec"1.  -53-  the c h i - s q u a r e t e s t .  A s i m i l a r s i t u a t i o n holds f o r  l e s s than 20 amagats where the corresponding f i t s As d i s c u s s e d  previously,  densities  are I I I and V I I .  the l o n g - l i v e d component  expected to be due to 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 , while change i n a n n i h i l a t i o n r a t e as a f u n c t i o n of gas d e n s i t y a s s o c i a t e d w i t h the quenching of the f r e e  is the  is  orthopositronium  life-  time  (Chapter 1, S e c t i o n 1 A A . ) .  free  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 measured by these  experiments.  The parameter a , 0  In view of the t h e o r e t i c a l  then,  is  p r e d i c t i o n of t h i s  the  life-  time (Chapter 1, S e c t i o n 1A.2.) and the a s s o c i a t e d a n n i h i l a t i o n  6 rate  (7.2 x 10  The v a l u e of a  -1 sec  0  ),  f i t V has l i t t l e  for this f i t ,  dependence on d e n s i t y ,  is  which contains  to the d a t a .  where a l i n e a r term i s a b l e Q, and the f a c t to the t h e o r e t i c a l  if  justification.  only a quadratic  considerably different  orthopositronium a n n i h i l a t i o n r a t e , the l i n e a r f i t s  physical  from the  compared w i t h the a  A s i m i l a r argument holds  included.  theoretical  f o r f i t IV  However, i n view of the  that the v a l u e f o r a  Q  is  from  0  reason-  significantly  orthopositronium a n n i h i l a t i o n r a t e ,  the  closer possi-  b i l i t y of a q u a d r a t i c d e n s i t y dependence i n a d d i t i o n to the term can not be r u l e d out by the set  of data  presented.  In the f i t s V I I I to X the parameter a  6 v a l u e of 7.2 x 10 time.  a  0  Q  0  is  given  sec  , the t h e o r e t i c a l  orthopositronium  was not f i x e d .  This i s  so s i n c e the v a l u e  the best f i t  for  to the d a t a , and  6 0  life-  are c o n s i d e r a b l y worse than those  from any of the l a t t e r y i e l d s  these values f o r a  the  1  As expected the f i t s  o b t a i n e d where a  linear  were not equal to the 7.2 x 10  —1 sec  .  The  TABLE V I . DEPENDENCE OF  ON P.  Summary o f t h e r e s u l t s o f f i t t i n g  t h e c u r v e i n F i g u r e 14. Q_  Results  a /10  6  a i  Q  »  sec  -1  sec ^  o = a  0  E/P«  o ,  r  a  0  =.  sec  1  a /10  6  4.-1 amagat  6  2  sec  -1  -2 amagat  10.57* o .05  ^o A  "  -  /10  o  -» a.,P  <10  6.83+0.50  0.339 t 0.059  0.88  18  0 . 2 4 0 * 0.013  0.16  P < 10 amagats + a P  7 . 5 5 1 0 .  E/P = 0 r  o  - o a  + a ^ + a P  7-97* 0. 5 9  + a P  8.90±0. 11  2  0.164* 0.102  0.0030* 0.0041  0.15  0.0095* 0.0005  0.10  E/P = 0 r  _  a  • -o 0 0 E/P =  2  2  -6  TABLE VI (continued) DEPENDENCE OF ^  OJJ P .  Type of f i t  Results a /10 0  10 VI.  VII. VIII.  IX. X.  . X = a 0  o  sec"  6  + a  l  sec  1  l P  & 1  = 7.2 + a P o x" = 7 . 2 + a.,P t a o X  -1 sec  /1 0  £_ a /10  6  2  ^-1 amagat  o.305±  0.056  0.2h2±  0.011  6  sec ^ amagat ^ -  0.02  amagats  x~ = a + a . 0 X = 7.2 + P 0 P < 1 0 amagats Q  a i  - 1  7.07^0.1+8  P  P <10  6  7.5^-0.15  0.00*+  0.29*+ ± 0 . 0 0 8  0.017  0.267±  0.000*+  0.00*+  0.303 t 0.016 Note: Unless otherwise s t a t e d ,  -0.002*+- 0.0010  the f i t s  to a l l the points i n F i g u r e 1 * + .  were made The e l e c t r i c  f i e l d E / P i s i n u n i t s of V cm" amagat 1  -1  0.0012  -54-  values the  f o r t h e p a r a m e t e r a-j i n f i t s  a-| o b t a i n e d  2.6.5.2.  i n the  Discussion  fits  of  the  orthopositronium As density  linear  i s e x p e c t e d t o be This  as  density  was  Q  the  the  free orthopositronium  The  On. t h e  basis  the d e n s i t y dependence of the be  gas  annihilation  1.4,4.).  The  be v e l o c i t y d e p e n d e n t ,  A ^ i s thus  quenching r a t e of the  gas  also  d i s c u s s i o n p r e s e n t e d here,,  q u e n c h i n g r a t e w o u l d be  expected  to  linear. The  l a r g e s c a t t e r of the  to r e f l e c t  slight differences  impurities  i n t h e A r g o n gas  quenched o r t h o p o s i t r o n i u m amagats o f A r g o n . of the  shoulder  I n a d d i t i o n , the  This  experimental  i n the  from run  points  concentration  to run.  On  i s considerably  observed i n the  longer  l o n g - r a n g e 1/R^  nsec i n  van  der  S e c t i o n 3*1 •) as  10  2.6.4.).  orthoposi-  l e s s than that f o r f r e e p o s i t r o n s  5  the  width  (Section  elastic-scatteringcross-section for  1.3*2.; C h a p t e r 3  of  the average  than the  same t i m e s p e c t r a  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  Section  is interpreted  t y p e and  atom e x i s t s f o r a b o u t 100  t r o n i u m i s e x p e c t e d t o be the  the  quenching r a t e p r o p o r t i o n a l to the  velocity-averaged  d e n s i t y dependent.  vary,  by p i c k - o f f q u e n c h i n g  ( s e e C h a p t e r 1, S e c t i o n  velocity-dependent  density.  from  the a n n i h i l a t i o n r a t e at zero  quenching c r o s s - s e c t i o n v / i l l i n general and  to  d e n s i t y dependence of  a n n i h i l a t i o n rate increases increases  allowed  little  annihilation rate.  mentioned e a r l i e r ,  rate. the  where a  V I I I to X d i f f e r  (Chapter  compared w i t h  Waals a t t r a c t i o n e x p e c t e d f o r t h e  where 1, the  positron-  -55-  ium  case.  For an e l a s t i c s c a t t e r i n g  cross-section  of the order  p of 10  Ta  the o r t h o p o s i t r o n i u m atom would make of the order of  0  collisions/sec  1 1  a t 1 0 amagats. k  Thus the o r t h o p o s i t r o n i u m  atom makes of the order of 1 0  collisions  Should one of these c o l l i s i o n s  i n v o l v e an i m p u r i t y atom w i t h a  l a r g e quenching c r o s s - s e c t i o n ,  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  be c o n s i d e r a b l y shortened. ium  a n n i h i l a t i o n rate is  before a n n i h i l a t i n g .  will  I t thus appears that the o r t h o p o s i t r o n -  at l e a s t as s e n s i t i v e to i m p u r i t i e s as  the shoulder width i n the Argon time s p e c t r a .  The l e v e l  of  N i t r o g e n contamination found i n many of the Argon samples  used  in  explain  these experiments  these  (Table I)  is  probably h i g h enough to  discrepancies.  2.6.5.3*  Influence  of the e l e c t r i c  field.  Although there i s no p h y s i c a l b a s i s f o r expecting electric  f i e l d dependence  i n 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 ,  the data has been checked f o r such an e f f e c t . the data i n Table VI shows no evidence dependence.  The i n c r e a s e d s c a t t e r  when the e l e c t r i c  f i e l d results  An examination of  f o r any e l e c t r i c  are i n c l u d e d ( r e f l e c t e d  gases l i b e r a t e d by undetected h i g h - v o l t a g e is  expected  field  of the a n n i h i l a t i o n d a t a ,  s m a l l e r v a l u e f o r Q) i s most probably due to the e f f e c t  It  any  i n the of  foreign  breakdowns d u r i n g a r u n .  that f o r e i g n gases added to the Argon i n t h i s manner  would not be immediately removed by the p u r i f i e r , and c o u l d thus affect  the measured 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  i n Section  2.6.5.2.  discussed  -56-  2.6.5. .  Summary of the o r t h o p o s i t r o n i u m r e s u l t s .  k  From Table V I , F i g u r e 1*+, and the above d i s c u s s i o n i t  is  c l e a r that some 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 .-. annihilation  rate  is~  necessary.  A s t a t i s t i c a l a n a l y s i s of the  data i n d i c a t e s that a l i n e a r dependence on d e n s i t y i s if  the e l e c t r i c f i e l d r e s u l t s are excluded.  electric  f i e l d r e s u l t s worsens the f i t  sufficient  I n c l u s i o n of  the  but does not change the  v a l u e 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 c o n t a i n a  l i n e a r d e n s i t y dependence and a zero d e n s i t y i n t e r c e p t a , 0  —1  6 i n d i c a t e s a value for a r e s u l t s of f i t .for  a  Q  between 6.8 and 7.6 x 10  IV are n e g l e c t e d  (Section 2.6.5.1.).  takes i n t o account the systematic  Q  sec  , if  the  Such a range  e r r o r i n t r o d u c e d by the  e l e c t r i c f i e l d as d i s c u s s e d i n Section2.6.5.3*  On t h i s  basis 6 -1 a 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 of ( 7 . 2 0 . ) x 10 sec i s g i v e n by these experiments, i n good agreement w i t h 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 The 0.3  k  sec  values f o r a  x 10° sec amagat" -1  i n Table VI a l l l i e between 0 . 2 and L  1  i f once a g a i n f i t  Furthermore, the values f o r a-| i n f i t s limits.  neglected.  V I I I to X l i e w i t h i n these  e r r o r i n t r o d u c e d by the e l e c t r i c f i e l d .  quenching r a t e X q  IV i s  F i n a l l y ^ t h i s range of values f o r a^ a l s o allows f o r the  systematic  X  L  = (0.29  q  The l i n e a r  g i v e n as a r e s u l t of these experiments i s  * 0.05)  x 10  6  then  sec" amagat" . 1  1  I t should be emphasized that the quenching r a t e has been obtained w i t h about 100 ppm of N i t r o g e n 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 A r g o n c o n t a i n i n g l e s s t h a n 1 ppm of 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 published values.  results with previously  Both of the quadratic f i t s  ( C e l i t a n s , Tao a n d  G r e e n , 1964; C e l i t a n s a n d G r e e n , 1964) w e r e o b t a i n e d where t h e s h o u l d e r  from  data  w i d t h was a b o u t 90 n s e c - a m a g a t s , w h i c h i s a t  l e a s t a f a c t o r o f three smaller than the c u r r e n t l y accepted This  smaller shoulder  width  i s a result of a relatively  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 , arguments i n S e c t i o n 2 . 6 . 5 . 2 . , were p r o b a b l y  the orthopositronium  were o b t a i n e d  from  extent.  t h e s i s w h i c h were o b t a i n e d  2.6.6.  The r e s u l t s o f  the standard  from l i f e t i m e measurements.  stated otherwise,  of counting  only.  r o l e o f errors Introduced  statistics.  d e v i a t i o n s quoted thus f a r , except  where i t i s e x p l i c i t l y statistics  MeVi.  presented i n  Discussion o f errors not r e l a t e d to counting All  lifetimes  a n a l y s i s o f 0..51  gamma-ray s p e c t r a , a n d a g r e e w e l l w i t h t h e v a l u e s this  large  1965), a n d f o l l o w i n g t h e  affected to a considerable  Heymann e t a l . ( 1 9 6 l )  value.  a r e based on t h e e f f e c t  I t i s necessary  by i n s t a b i l i t i e s  also to discuss the  i n thee l e c t r o n i c  i n s t r u m e n t a t i o n , b y 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 t h e i n t e g r a l and d i f f e r e n t i a l the a p p l i e d e l e c t r i c t h e gas c o m p o s i t i o n  field  linearities  of the timesorter,  and t h e gas d e n s i t y , and by changes i n  from r u n t o r u n .  TABLE V I I . SUMMARY OF PUBLISHED RESULTS FOR ORTHOPOSITRONIUM QUENCHING IN ARGON. Type of  density an/1O  dependence  6 10 A  =  o  r  -1  a-i/10  -1 sec  -1  a /1 0  6  £  -1  amagat  sec  0  ar  4  A  -=7.0  A  —a  4 a.P  0.277 *  7.2-* 0.4  -1  -2  amagat  0.017 - 0.002  Celitans,  0.015 - 0.002  C e l i t a n s and Green (1964).  0  2  Reference  2  7.0 + a P ' 2.  O  o  sec  sec  =7.0  _°  Results 6  0.005  Tao and Green (1964).  Heymann, et a l . (1961). Present work.  0.29 ± 0.04  + a P o  1  Note: The r e s u l t s (see  due to other workers have been c o r r e c t e d by the f a c t o r 298/273  S e c t i o n 2.6.2.3.) on the assumption t h a t the atmosphere u n i t s g i v e n by o  them are at 25 C .  The value of 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  assumed by the other workers i s not i n accordance w i t h the c a l c u l a t e d (Ore  and P o w e l l , 1949; A l e k s e e v ,  1959).  values  -582.6.6.1.  Effect of i n s t a b i l i t i e s The d i s c u s s i o n  can of  be d i v i d e d  i n the electronic  instrumentation.  of the effect of electronic  i n t o two s e c t i o n s .  instability  I n t h i s subsection the effects  c h a n g e s b r o u g h t a b o u t i n t h e prompt r e s o l u t i o n a n d 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 . with the uncertainties  The f o l l o w i n g  subsection deals  i n t h e time c a l i b r a t i o n o f t h e  timesorter.  F i g u r e 20 ( A p p e n d i x ) , shows t h e 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 setting.  c h a n g e i n t h e 0.51 MeV s i n g l e c h a n n e l  analyzer  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 p h o t o p e a k , t h e o t h e r w i t h t h e s i n g l e analyzer set a t the v a l l e y position The r e s o l u t i o n Table V I I I  (see S e c t i o n 2.2.1.  channel and F i g .  5).  i s somewhat d i f f e r e n t f o r t h e two s e t t i n g s .  shows t h e r e s u l t s o f l i f e t i m e m e a s u r e m e n t s f o r some  representative  pairs  o f t i m e s p e c t r a where t h e main  difference  was 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 t i n g . T a k i n g ; n o t e o f . t h e . s t a n d a r d deviations trend .  in'the To f i r s t  r e s u l t s , there i s l i t t l e  e v i d e n c e f o r a marked  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  modify the time r e s o l u t i o n  t o the extent indicated  i n Figure 20,  h a v e 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 m e a s u r e m e n t s .  The  e f f e c t i s e x 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 are  o f t h e o r d e r o f , o r l e s s t h a n , t h e prompt t i m e  resolution.  D u r i n g t h e c o u r s e o f t h e e x p e r i m e n t s t h e prompt peak (see  F i g u r e 6) o f t h e t i m e s p e c t r a was o b s e r 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 t h e prompt peak o f t h i s  m a g n i t u d e d u r i n g a r u n w i l l h a v e no e f f e c t o n t h e l i f e t i m e o f t h e e x p o n e n t i a l s b u t does m o d i f y t h e i n t e n s i t i e s .  I n f a c t , any non-  TABLE V I I I .  DEPENDENCE OF LIFETIME RESULTS ON S.C.A. SETTING.  Direct-enhanced T  1  2  P  nsec  amagats  T  nsec  0 r tho - enhanc ed 1  T  2  P  nsec  amagats  T  nsec  1 -7  97.8 * 2.6  9.3  19.4  1.7  90.3 * 4.9  9.0  38.7 + 3.4  27.9 ± 0.8  95.0 ± 4.0  9.0  29.3  1.5  0.5  107.0 t 4.0  9.0  24.2  20.7 ± 0 . 5  104.3.± 3.0  8.9  18.6 + 1 . 0 100.5  34.3 ± 1.4  90.3 * 4-.3  9.0  33-1  13-9 * 0.4  95.8 * 2.0  13.5  14.2  35.2 * 1.5  1 1 2 . 9 * 8.9  5.3  35.6  18.3 * 37.8  22.0  t t  0.7  +  101 .2 * 2.1 101 . 0  t  9.3  5.0  9.0  105.5 * 3.3  9.0  1 . 0 108.5 * 2.4  9.0  * 2.0  8.8  2.4  98.8 t 3 A  8.7  0.6  95.2  3-8  t  1 .4  13.5  114.8 ± 5.9  5.3  -59-  exponential region  " p o r t i o n o f the time spectrum  i s m o d i f i e d by a s h i f t Other  have an  effect  integral  and  i n t h e prompt  instabilities considerably  differential  s u c h as peak.  i n the e l e c t r o n i c s  less  the shoulder  are expected  than the accuracy to which  linearities  to  the  o f t h e t i m e s o r t e r have  been  measured.  2.6.6.2.  The  integral  and  differential  The  integral  linearity  linearities  o f the  o f t h e t i m e s o r t e r was  by J o n e s  t h e method d e s c r i b e d  error  a s s o c i a t e d w i t h t h e measurement o f t h e a v e r a g e  p e r c h a n n e l g i v e n by t h e i n t e g r a l t h a n 1$,  arising  m a i n l y from  between p u l s e s f r o m  and  (1965).  using  Falk  linearity  The  integral  to  that  time w i d t h per channel. the  first  (see  by two  the of  The  months was  width  i n order  changes i n t h e  \% o f t h e f i r s t  time  measure-  average  s e c o n d measurement, s e p a r a t e d within  less  from  measurement  Appendix). The  is  significant  overall  time  linearity  r e p e a t e d d u r i n g the course o f the experiment t h e r e were no  The  i n r e a d i n g the  ment was check  measured  i s e s t i m a t e d t o be  the u n c e r t a i n t y  the p u l s e r .  timesorter.  differential  m e a s u r e d by  the r e l a t i v e  linearity  number o f c o u n t s  t i m e s o r t e r , when t h e i n p u t pulse pairs  ( F a l k , Jones directly channel.  and  s e p a r a t e d by Orth,  proportional The  (or r e l a t i v e  1965).  i n each  to the t i m e s o r t e r time i n t e r v a l s The  relative  to the r e l a t i v e  channel  channel  is a  o f a random  length  channel widths  linearity  of  source  number o f c o u n t s  a c c u r a c y of the d i f f e r e n t i a l  width)  are  i n each  .  measurement  -60-  is  thus governed by the counting s t a t i s t i c s ,  and s i n c e  about  1000 counts per channel were r e c o r d e d , the r e l a t i v e channel widths are a c c u r a t e 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  linearities  o f the t i m e s o r t e r are shown i n F i g u r e 22 (Appendix). It  is  c l e a r 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  S e c t i o n 2.5.1.)  a l r e a d y c o n s t i t u t e s a f i r s t order c o r r e c t i o n 'to the e s t i m a t i o n of the l i f e t i m e s .  Any u n c e r t a i n t y i n the r e l a t i v e channel widths  must then be c o n s i d e r e d as  affecting  this  c o r r e c t i o n to  the  l i f e t i m e measurements  (when compared w i t h the u n c e r t a i n t y i n the  integral linearity).  From t h i s p o i n t of view i t  is  reasonable  to suppose t h a t the t o t a l u n c e r t a i n t y i n the l i f e t i m e s c a l i b r a t i o n of the t i m e s o r t e r i s  due to  of the order of 1$, t h i s  r e p r e s e n t i n g the maximum systematic  the  figure  e r r o r 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 e r r o r i n the a n n i h i l a t i o n r a t e s . Taking i n t o account the above d i s c u s s i o n ,  t h a t the systematic rates is  is  of the order of 1$, and a r i s e s  Applied e l e c t r i c  from the u n c e r t a i n t y timesorter.  field.  S i n c e much of the a n a l y s i s of the experiments  relies  on the assumption of a s p a t i a l l y uniform e l e c t r i c f i e l d , n e c e s s a r y to d i s c u s s , of t h i s  clear  e r r o r i n the measurement of the a n n i h i l a t i o n  i n the a b s o l u t e time c a l i b r a t i o n of the  2.6.6.k.  it  it  is  not o n l y the u n c e r t a i n t y i n the magnitude  electric field,  but a l s o the degree to which t h i s  electric  -61-  field  i s uniform w i t h i n the  chamber.  The v o l t a g e a p p l i e d to the e l e c t r i c found by measuring the c u r r e n t flowing resistor  connected  f i e l d r i n g s was  through a s e l e c t e d 500 M ft  i n p a r a l l e l w i t h the e l e c t r i c  field  rings.  The v a l u e of the r e s i s t o r was measured to ^% at 2 0 . 0 kV and 0.5$ at 3 kV, and was found to have a n e g l i g i b l e  voltage  coefficient.  The Avometer used to measure the c u r r e n t through the r e s i s t o r a c c u r a t e to \% f o r a f u l l highest f i e l d s  used.  scale deflection  The d i s t a n c e  corresponding to  was  the  between the ground p l a t e and  the h i g h v o l t a g e r i n g has been p r e v i o u s l y r e p o r t e d by F a l k (1965) and i s known  to  less  than \%.  magnitude of the e l e c t r i c is  The o v e r a l l u n c e r t a i n t y i n  f i e l d t a k i n g i n t o account  of the order of 3% f o r a l l e l e c t r i c  fields  analogue  of the g r i d s t r u c t u r e .  factors  measured.  The s p a t i a l u n i f o r m i t y of the e l e c t r i c v e s t i g a t e d by F a l k (1965) u s i n g a l o w - v o l t a g e  these  the  f i e l d was i n -  two-dimensional  The r e s u l t s i n d i c a t e d t h a t  the  n o n - u n i f o r m i t i e s were confined to the r e g i o n immediately s u r r o u n d i n g the e l e c t r i c enclosed  f i e l d r i n g s , and occupied about 8% of the  by these r i n g s .  The e f f e c t  of these  w i l l be l e s s marked at h i g h Argon d e n s i t i e s , reduced p o s i t r o n range, points  non-uniformities because of  compared w i t h lower d e n s i t i e s .  i n F i g u r e 9 have been obtained f o r a v a r i e t y of  and l i e  on a continuous is  c u r v e , depending on the E / P .  of t h i s ,  it  c o n s i d e r e d that  electric  f i e l d are unimportant.  volume  the The densities,  In view  the s m a l l n o n - u n i f o r m i t i e s  i n the  -62-  2.6.6.5.  Measurement of gas d e n s i t y . The d e n s i t y of the gas was found u s i n g the p e r f e c t  law.  D e v i a t i o n s from t h i s  equations  are n e g l i g i b l e  law as expressed  by the van der Waals  compared w i t h the u n c e r t a i n t y i n the  This u n c e r t a i n t y \s  p r e s s u r e measurement.  a s s o c i a t e d w i t h the  c a l i b r a t i o n of the h i g h pressure gauge and i s (Falk,  1965).  gas  The measurement of the absolute  estimated  to be 2%  temperature of  the  gas was done u s i n g a mercury thermometer w i t h the bulb p l a c e d against  the chamber w a l l .  There i s  a p o s s i b i l i t y t h a t the tempera-  t u r e of the gas was h i g h e r than that of the chamber w a l l s ,  due to  the h e a t i n g a c t i o n of the p u r i f i e r .  evidence  However, there was no  t h a t the temperature of the chamber w a l l s  i n c r e a s e d when the  was turned on, i n d i c a t i n g that most of the gas i s  purifier  i n thermal  e q u i l i b r i u m w i t h the chamber w a l l s . . The average a b s o l u t e  temperature of the gas was known  i n t h i s way to w i t h i n \% during a r u n . d e n s i t y measurement i s account u n c e r t a i n t i e s  The u n c e r t a i n t y i n the  thus of the order of 3%, t a k i n g  into  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 d i s c u s s i o n s  subsections,  the u n c e r t a i n t y i n the measurement of E / P i s  order of 6% which i s i n E and P.  i n the previous  obtained by compounding the  two of  the  uncertainties  2 . 6 . 6 . 7 .  Gas  composition.  I n experiments of the t o be  f a i r l y c e r t a i n t h a t t h e gas  over the The  type reported  l e n g t h of time t h a t the  composition  here, i t i s desirah1 remained  e x p e r i m e n t s were  e f f e c t i v e n e s s o f t h e Ca-Mg e u t e c t i c p u r i f i e r  t h e gas  composition  Section 2.2.1.  The  o v e r a p e r i o d o f t i m e has possibility  c o n c e n t r a t i o n have a f f e c t e d the Section  2.6.2.3.  and  Section  performed. in  maintaining  been d i s c u s s e d  t h a t s m a l l changes i n lifetime results  2.6.5.2.  constant  in  impurity  i s discussed  in  3-  THEORETICAL CONSIDERATIONS  OF THE POSITRON-ARGON ATOM INTERACTION.  3.1.  Introduction. The  s u b j e c t of low-energy p o s i t r o n - a t o m i n t e r a c t i o n s  has r e c e i v e d c o n s i d e r a b l e a t t e n t i o n r e c e n t l y (Massey, et a l . , 1966;  Drachman, 1966) due to the development of experimental  techniques which make comparison between theory and experiment possible  ( F a l k , Orth and Jones,  1965; P a u 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 H a m i l t o n i a n H f o r a system of electrons H  =  lET^N N  bound to a n u c l e u s , and an unbound p o s i t r o n i s g i v e n by ~ S T I - i " 2m~ lp~l |r.-rU e 1 e 1 '—l -N V  +  ,  2  7 Z  e  1  -  I  2 I  e  + 1  1  I 1  l^]rwv  3  —;  i#j (3Ha)  where the s u b s c r i p t N denotes the n u c l e u s , p denotes the p o s i t r o n and  summation over i r e f e r s  to the Z e l e c t r o n s  Mott and Massey, 1965? p. 287).  i n the atom  (see  The wave equation d e s c r i b i n g  the complete system i s g i v e n by Hr = EY. E is  (34b)  the t o t a l energy of the The  system.  s o l u t i o n to a system such as (3*+a) and (3*+b) i s beyond  the scope o f the present day t e c h n i q u e s , fications tractable.  and r a t h e r d r a s t i c s i m p l i -  have to made i n order that the problem become at a l l  The s i m p l e s t involves  approximation to t h i s m a n y - p a r t i c l e problem  the use of an e f f e c t i v e two-body i n t e r a c t i o n .  f i e l d of the low energy e l e c t r o n s c a t t e r i n g  In the  from noble gas  atoms,  such a s i m p l i f i c a t i o n has met with c o n s i d e r a b l e success (Holtsmark, 1929; K i v e l ,  1959; Labahn and C a l l a w a y , 1966).  In t h i s model  the incoming e l e c t r o n s c a t t e r s from the unperturbed atom, 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  the  electronic  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 incoming e l e c t r o n ,  expected  (Crown and Russek,  1965), i s  to the atomic p o t e n t i a l ,  p o l a r i z a t i o n of the atom by the  from f i r s t  order p e r t u r b a t i o n theory  taken i n t o account by the a d d i t i o n ,  of an a t t r a c t i v e  ct/R^ f o r l a r g e e l e c t r o n - a t o m s e p a r a t i o n s chosen to be the c l a s s i c a l  electric  term which behaves as R.  The parameter ct i s  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 consisting  of the two p o t e n t i a l s  reproduce the Ramsauer e f f e c t some d e t a i l .  interaction  i n d i c a t e d above was a b l e  to  i n Argon and other noble gases i n  Much of the recent work i n the s c a t t e r i n g  from noble gases has confirmed t h i s  of  electrons  p o i n t of view ( K i v e l , 1959;  Labahn and C a l l a w a y , 1966). 3.1.1.  D i s c u s s i o n of the d i f f e r e n c e s and e l e c t r o n  between low-energy  scattering.  In a recent review of the s u b j e c t zation potentials Callaway,  positron  of approximate  polari-  f o r the e l e c t r o n - H e l i u m i n t e r a c t i o n (Labahn and  1966), the o v e r a l l l a c k of s e n s i t i v i t y  of the  velocity-  -66-  dependence of the t o t a l s c a t t e r i n g the e f f e c t i v e  c r o s s - s e c t i o n s to d e t a i l s  scattering potential is  of  c l e a r l y demonstrated.  agreement w i t h experiment i s obtained f o r a v a r i e t y of  Good  potentials  as long as they e x h i b i t a 1 / R dependence f o r l a r g e 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. chapter c o n t a i n s  the r e s u l t s  of c a l c u l a t i o n s which, when compared  w i t h the experimental r e s u l t s of positron-Argon s c a t t e r i n g It  is  the s c a t t e r i n g  possible  This  of Chapter 2 i n d i c a t e that the  case  i  is  f a r l e s s ambiguous.  i n such a c a l c u l a t i o n to compute not only  cross-section  as a f u n c t i o n of energy but a l s o  positron-electron annihilation rates.  the  The l a t t e r depend s o l e l y on  the o v e r l a p of p o s i t r o n and e l e c t r o n d e n s i t y i n the atom. experiment which depends on both the s c a t t e r i n g  An  cross-sections  and a n n i h i l a t i o n r a t e s as a f u n c t i o n of v e l o c i t y should thus impose more s t r i n g e n t  c o n d i t i o n s on the choice of  than experiments which depend on one Such a s i t u a t i o n i s positrons  elastically  is  field.  then determined to f i r s t  s e c t i o n and e l e c t r i c  field.  rate is  i n Argon gas under the  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 cross-  The observed a n n i h i l a t i o n r a t e  is  a n n i h i l a t i o n r a t e averaged over  Thus, i n g e n e r a l ,  a f u n c t i o n of the a p p l i e d e l e c t r i c Further,  influence  order by the momentum-transfer  g i v e n by the v e l o c i t y - d e p e n d e n t this velocity distribution.  alone.  p r o v i d e d i n the a n n i h i l a t i o n of  scattering  of an a p p l i e d dc e l e c t r i c  potential  the a n n i h i l a t i o n  field.  the s c a t t e r i n g of p o s i t r o n s from atoms  s i g n i f i c a n t l y from t h a t f o r e l e c t r o n s  i n that the p o s i t r o n  differs is  -67d i s t i n g u i s h a b l e 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 f u n c t i o n o v e r l a p w i t h the e l e c t r o n s atom.  of  the  For t h i s r e a s o n ,  short-range p o s i t r o n - e l e c t r o n c o r r e l a t i o n  e f f e c t s may be expected  to p l a y a more important r o l e than i n the  corresponding case f o r e l e c t r o n s c a t t e r i n g . i n p a r t i c u l a r , should r e f l e c t  the d e t a i l e d extent to which both the  atomic and p o s i t r o n wave f u n c t i o n s Finally,  The a n n i h i l a t i o n r a t e ,  are d i s t o r t e d by the i n t e r a c t i o n .  i t may be expected t h a t both long-range p o l a r i z a t i o n of  the atom and s h o r t - r a n g e c o r r e l a t i o n of the p o s i t r o n - e l e c t r o n wave f u n c t i o n s w i l l p l a y an important p a r t i n the e f f e c t i v e  interaction.  As mentioned i n S e c t i o n 3»1.» the problem of p o s i t r o n atom s c a t t e r i n g  is  complicated by the many p a r t i c l e aspect.-  It  been p o i n t e d out that a v e r y simple two-body approximation has p a r t i c u l a r l y u s e f u l i n the e l e c t r o n s c a t t e r i n g case. then,  It is  has been  desirable,  to know the extent to which such two-body approximations apply  to the case of p o s i t r o n s c a t t e r i n g .  Furthermore, i t  to e s t a b l i s h whether the ambiguity which a r i s e s i n g occurs f o r p o s i t r o n s . of this  Accordingly,  is  of  i n electron  the t h e o r e t i c a l  interest scatter-  results  Chapter were obtained u s i n g the e m p i r i c a l p o l a r i z a t i o n  potentials  of the type used i n e l e c t r o n s c a t t e r i n g .  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  (see  1/R  a l s o Moussa, 1959).  Some c u t - o f f  parameter i s always used i n order t h a t 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 . is  taken to be the sum o f V  the H a r t r e e - F o c k s e l f  p  The e f f e c t i v e  interaction  and the p o t e n t i a l V  consistent  H  V  characterizing  f i e l d f o r Argon i n the ground  -68-  s t a t e (Hartree and H a r t r e e , 1936).  The r e s u l t s i n d i c a t e  c l e a r l y t h a t these simple two-body p o t e n t i a l s account f o r the experimental  3.1.2.  quite  are inadequate  to  results.  O u t l i n e of procedure. In order to compute the e f f e c t  on the a n n i h i l a t i o n r a t e of p o s i t r o n s dependent  annihilation rate  cross-section  a  d( )  of' the e l e c t r i c  i n a gas both the  \>a(v) and the  positron-atom i n t e r a c t i o n ,  velocity  momentum-transfer  have to be known. For a s p e c i f i c  v  field  model of  these can be found by o b t a i n i n g  the  the  u s u a l \ p a r t i a l wave s o l u t i o n to the a p p r o p r i a t e Schrodinger equation.  It  should be emphasized that the a n a l y s i s  of  this  S e c t i o n n e g l e c t s any c o n t r i b u t i o n to the a n n i h i l a t i o n r a t e from radiative;  capture of a p o s i t r o n i n t o a bound A r - e  +  system.  N e i t h e r the e x i s t e n c e nor p r o b a b i l i t y of formation of such bound s t a t e s has been d i s c u s s e d Should i t  exist,  it  is  i n any q u a n t i t a t i v e way i n the  expected t h a t the l i f e t i m e  order of the p a r a p o s i t r o n i u m l i f e t i m e rate is  comparable, t h e n ,  the p o s i t r o n - e l e c t r o n  (10~  10  sec).  literature.  would be of I f the  the  capture  to the d i r e c t a n n i h i l a t i o n r a t e due  overlap during e l a s t i c s c a t t e r i n g ,  from t h i s channel c o u l d r e p r e s e n t  a significant  to  competition  contribution  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 r a t e . The v e l o c i t y of  an e l e c t r i c  u s i n g the  ;  0  d i s t r i b u t i o n of p o s i t r o n s under the  f i e l d i n a gas at temperature T i s  J ( V ) and u  ^  ( ) a v  discussed  equation d e s c r i b i n g t h i s s i t u a t i o n  is  above.  influence  then determined  The d i f f e r e n t i a l  s i m i l a r to the W i l k i n s  -69-  equation used to d e s c r i b e capture i s  thermal neutron d i f f u s i o n where neutron  important (Sobrino and C l a r k ,  1961), except t h a t ,  t h i s c a s e , there i s an a d d i t i o n a l term due to the a p p l i e d field.  In a d d i t i o n , the p o s i t r o n s  L o r e n t z gas  in  electric  i n the gas approximate a  to a v e r y h i g h degree by v i r t u e of t h e i r s m a l l mass  and low d e n s i t y .  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  r e p o r t e d p r e v i o u s l y i n p r e l i m i n a r y work done on t h i s problem  1965; F a l k , Orth and J o n e s , 1965).  (Falk,  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 f o r a p a r t i c u o ( 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 r a t e d a  lar  J  is  X  found by averaging the v e l o c i t y - d e p e n d e n t  r a t e over the e n t i r e v e l o c i t y 3.2.  distribution.  The two-body Schrodinger equation and i t s  3.2.1.  The Schrodinger  annihilation  solution.  equation.  Because of the s p h e r i c a l symmetry of 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 assumed, to s o l v i n g This i s ,  effective  the problem i s  the r a d i a l p a r t of the r e l e v a n t Schrodinger  i n atomic u n i t s  (Wu and Ohmura,  reduced equation.  1962)  2 C  dW + k  where is  2  -  }  - 2V(r)] X = 0  (35)  £  V(r) = h£±  - V  p  «  the e f f e c t i v e i n t e r a c t i o n p o t e n t i a l at a d i s t a n c e  (35a) r from the  2 origin,  and k  is  the k i n e t i c  energy of the i n c i d e n t  positron  ( i n Rydberg u n i t s ) The term Z „ ( r ) takes i n t o account the s c r e e n i n g  of  the  -70-  nucleus by the atomic e l e c t r o n s . Zp(0) =18 and  ZpC  0 0  )  Thus, f o r Argon  a " , 1  0  =0  a  , where a  1 Q  is  Q  For a l l the p o t e n t i a l s , used h e r e ,  the Bohr r a d i u s . the H a r t r e e - F o c k p a r t of  the  i n t e r a c t i o n ( Z p ( r ) / r ) , a s c r i b e d to the unperturbed Argon atom, is  due to H a r t r e e and H a r t r e e  3.2.2.  (1936).  C a l c u l a t i o n of phase s h i f t s The s o l u t i o n of ( 3 5 )  and wave f u n c t i o n s .  was undertaken u s i n g the Runge-  K u t t a method f o r s o l v i n g d i f f e r e n t i a l e q u a t i o n s . solution consisted  The method of  of i n t e g r a t i n g the d i f f e r e n t i a l  n u m e r i c a l l y from a p o i n t near the o r i g i n u n t i l the form f o r  x^? which i s  known a n a l y t i c a l l y ,  to b e g i n the i n t e g r a t i o n ,  the s l o p e of  x , A  is  For non-zero A , - a n d kr<<A  v a l u e of  C  asymptotic  obtained.  In order  d x ^ / d r , was set  some a r b i t r a r y i n i t i a l value at a s m a l l d i s t a n c e origin.  equation  to  r away from the  , the corresponding i n i t i a l  obtained by s o l v i n g  dr^  ]  x  * "  0  (  3  6  )  Is g i v e n by ( 3 7 )  T h i s ensures that  x  A  has the c o r r e c t 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 r Z ( r ) <,<. 1 ,' the i n i t i a l v a l u e of x p  is  n u  g i v e n by x  0  •  r  <38  >  -71-  where x  is  Q  the s o l u t i o n  d 2ZL(r)-| dP" ~ - ^ X 2  r  [  ]  Q  _  n  =  0  to  «  (39)  i n the r e g i o n where Z ( r ) =Z. p  combined to _  v  for  r  dx  kr «  E x p r e s s i o n s (37) and (38) can be  give  0  £ if  * >0, and r Z p ( r ) «  1 i f l- 0.  The a b s o l u t e magnitude of the n u m e r i c a l s o l u t i o n f o r x expression for x .  A  was determined by matching  to the a p p r o p r i a t e asymptotic  The v a l u e o f the asymptotic x  £  £  was found by _2  n o r m a l i z i n g the incoming p o s i t r o n f l u x to 1 p o s i t r o n cm i n the u s u a l manner (Wu and Ohmura,  3.2.2.1.  sec  1962).  the asymptotic  solution for  i n the r e g i o n where the V ( r ) term i s n e g l i g i b l e  x  £  compared w i t h  the  i n (35) i s  X ( r ) = kr CA j (kr) - B n (kr)] A  £  where j ^ k r ) ,  Jl  £  n^Ckr)  and second k i n d .  are s p h e r i c a l B e s s e l f u n c t i o n s  For s t i l l  ^  — £ 2 —  term i s n e g l i g i b l e  Ohmura, X (r) = £  (41)  £  iU+i)  l a r g e r r where,  of the  in addition,  first  the  2 compared w i t h k , t h i s becomes (Wu and  1962) sin (kr - | ^ + 6 ) £  (42)  where tan  i  Asymptotic s o l u t i o n f o r k i 0.  For k unequal to z e r o ,  others  _  h=  (43)  -72and Cj = Aj + Bj . Thus i t of  r  (44)  i s u n n e c e s s a r y to compute the wave f u n c t i o n up  f o r which i t becomes  to.values  s i n u s o i d a l , s i n c e the phase s h i f t and  n o r m a l i z a t i o n constant can be c a l c u l a t e d when the wave f u n c t i o n satisfies (V(r) x  (41).  is negligible),  and  &  F o r any r s a t i s f y i n g  .  the constants  the asymptotic requirement •  Ajt, % can be c a l c u l a t e d from  D i f f e r e n t i a t i n g (41 ) w i t h r e s p e c t  jj*£ = (t+1) x / r - k r CA^j^Ckr) - B n £  £  £+1  to r g i v e s  (kr)]  (45)  where use has been made of the r e l a t i o n  where f for  is  t  any s p h e r i c a l B e s s e l f u n c t i o n .  A £ , and B^ y i e l d s  A. = x . [ ( W ) ^ t o ) B  =x  £  S o l v i n g (41 ) and (45)  A  to (kr)] l+1  CU+1) j ( k r ) - k r j 4  £ + 1  -^ ( k r ).  ^  (kr)] - rSJ^Ckr)  In p r a c t i c e the r a t i o B jt/Ae, was c o n t i n u o u s l y monitored as differential converged,  equation (35) was s o l v e d .  Once the r a t i o  respectively.  The phase s h i f t s  (Bowe,  1960)  o  I  0  £  (4+1)  sin2(6  £  - 6  £+1  £  shift  (43) and (44)  were then used to c a l c u l a t e the p  t o t a l momentum-transfer c r o s s - s e c t i o n s  = p  B /A  the c a l c u l a t i o n was .terminated, and the phase  and n o r m a l i z a t i o n constant Cj_ found u s i n g equations  d  the  )  In p r a c t i c e , a l l the phase s h i f t s  o  d  ( i n u n i t s of i r a  Q  ) from  (47)  f o r p a r t i a l waves up to and  -73-  including  4 =  5 were c a l c u l a t e d ,  the h i g h e r order phase s h i f t s  being  negligible.  3.2.2.2.  Asymptotic s o l u t i o n f o r k 0. =  For the case of very s m a l l i n c i d e n t v e l o c i t y , r  2  (that i s ,  k  and V ( r ) <•<  (35) s i m p l i f i e s r  d  4U+D-,  2  Cgpr- — r T "  This free X  A  =C  the Schrodinger equation  to _  *£ ~  ]  ,. .  n  lQ  (48)  0  p a r t i c l e equation has as i t s  general  solution  + C /r .  1 + 1  (49)  l  i r  2  However, i t  is  4=0 f o r t h i s in  2  4(4+1)/r ) ,  and l a r g e  o n l y necessary  special  comparison w i t h  to compute the wave f u n c t i o n  case as a l l the other phase s h i f t s  $ . Q  for  are s m a l l  T h i s occurs o n l y because a l l the  potentials  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  is  a p p r o p r i a t e l y normalized i f  at  large r X  Q  =r +C  (50)  2  The constant  C-j has been set  to u n i t y s i n c e a s y m p t o t i c a l l y  the  wave f u n c t i o n R =x/r has u n i t y a m p l i t u d e . The momentum-transfer c r o s s - s e c t i o n the t o t a l  elastic  scattering  cross-section,  expressed  by (Landau and L i f s h i t z ,  i s now i d e n t i c a l  the l a t t e r  being  1965; p.500)  a = 4(0^)2 If  to  the wave f u n c t i o n is n o r m a l i z e d ( t h a t i s ,  (51) 0^1), t h i s  reduces  to  -74-  o - 4C  (&2>  2 2  LL  For  smaller  values  of  r,  f o r  which  the  a/r  term  i n  V(r)  i s  2 s i n i f l e a n t  and  Schrodinger  is  k  term  equation  s t i l l A=  f o r  n e g l i g i b l e ,  the  solution  by  the  (53)  Q  given  to  0  X '» 0  |p  C  the  (Landau  and  1965;  L i f s h i t z ,  p.504-)  where Y  For  /2a  S  large  form  r,  given  by  putting  i n  terms  of  this  i n  The  The  =  x , Q  wave  function  by  i n t e g r a t i n g  of  the  form  (see  o  * (r)r. l>  f o r  very  (35)  given  (5*+)  •  the  r a t i o  converged,  the  calculation was  normalized  setting  could  be  by  C /C' 2  found  calculated.  C-  small  k=0,  monitoring  cross-section  ( r )  tends  to  a d d i t i o n  (5 *)  and  C  can  2  3.2.2.1.)  the  1  asymptotic  i s  also and  normalized be  this  expressed yields  l(  ( f o r by  Q  Section  - £  0  x  i f i n  constants  iix /dr  %<¥  f o r  (50),  equation  C^ 1.  <=! ' ^  expression  was  using  C.*1,  and  This as  i n  incident v e l o c i t y .  •••fc«0)  was  given  numerically  checked by  (55)'  terminated (51)« order  was  The that  by  wave the  the  u n t i l  x  found was  Q  continuously  Once and  thus  the  r a t i o  momentum-transfer  f u n c t i o n was appropriate  also Z  e  f f  -75-  3.3*  Calculation of Z ff. ?  As  m e n t i o n e d i n C h a p t e r 1, S e c t i o n  annihilation rate of positrons  i n a gas 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 over the p o s i t r o n p o s i t i o n .  1.3.1. t h e d i r e c t  This  I* ! a t t h e p o s i t r o n -  2  averaged  a n n i h i l a t i o n r a t e , i s given by  ( F e r r e l l , 1956)  v = rrr cnJd3x|iT| |V"| 2  2  O  a,  where r  S  (56)  2  ™"  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  Q  s  i s t h e gas d e n s i t y  i n atoms cm , a n d c i s t h e v e l o c i t y o f l i g h t . J  E q u a t i o n ( 5 6 ) may b e r e w r i t t e n i n t h e f o r m v (k) = i r r c n Z 2  a  o  where Z ^ e  s  e f f  (k)  (57)  i s the value o f thei n t e g r a l i n (56).  f  wave r e p r e s e n t a t i o n  o f t h e p o s i t r o n wave <J< t h e v a l u e o f f  j u s t t h e a t o m i c number Z o f t h e atom. i n t e r a c t i o n i s taken i n t o account, relevant x  * i s given  Zggv i s  When t h e p o s i t r o n - a t o m  'l' must b e f o u n d b y s o l v i n g t h e +  S c h r o d i n g e r e q u a t i o n ( s e e S e c t i o n 3.2.).  o f S e c t i o n 3.2.,  a  F o ra plane  I n terms o f t h e  b y ( f o r k*0)  +  00  *  +  = 1  $£ »  where t h e  P  (  c  o  s  9)  (58)  a r e Legendre polynomials.  over angles i n the expression wave c o n t r i b u t e s tion  e  t h e n shows t h a t e a c h  t o t h et o t a l Z f-r, t h e 4 e  t h  p a r t i a l wave  partial  contribu-  being  Wi^CI*"' ^ 2  The  f o r Z ff  Performing the i n t e g r a t i o n  electron density  1 1  ^*  (59)  I*""! 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 h e r e 2  -76-  is  that appropriate  t o t h e A r g o n atom i n t h e 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 The  numerical  (1936).  calculation of ( Z f f ) e  i n v o l v e s an  £  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 found s u f f i c i e n t l y accurate. r=7a  a t which point  0  overlap  i n t e g r a t i o n was  The i n t e g r a t i o n was c a r r i e d o u t t o  the electron density  k~|  2  i s less  than  -5 10  o f i t s maximum v a l u e For  Z  e f f .= £  w i t h i n t h e atom.  k=0, t h e Z' +- was c a l c u l a t e d f r o m ef  i n ' ^ d r  (60)  w h e r e x ( r ) was c a l c u l a t e d a s i n d i c a t e d i n S e c t i o n 3'2.2.2. Q  3- '  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 .  l+  3.4.1.  The m o d i f i e d The  Wilkins  differential  d i s t r i b u t i o n of positrons only  elastic  collisions  equation.  equation  i n a gas i n t h e energy range where  c a n t a k e p l a c e h a s b e e n shown t o b e  ( F a l k , 1965; F a l k , O r t h a n d J o n e s , ^  =  v  -  !  ^  describing the v e l o c i t y  1965)  ^ (61)  D  - Cv (v)-+ v ( v ) ] f ( v , t ) a  f  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 t h e 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 electric  field;  e  i s the positron  charge;  E  i s the applied  m  i s t h e p o s i t r o n mass;  electric  fieldj  -77-  v  d<  v )  = Vd  «jj i s  (  v  )  5  v  the momentum-transfer c r o s s - s e c t i o n  for  positron-gas  atom c o l l i s i o n s ; n  is  the d e n s i t y o f s c a t t e r i n g  v^(v) = ur *cn Z ~.(v)j ct  r  O  o  c  S  (see  atoms;  Equation  57)  Si*  is  the c l a s s i c a l  electron radius;  is  the v e l o c i t y of  light;  v (v) = n a (v)v; f  f  g  f  ! .p(v) a  is  the c r o s s - s e c t i o n  for positronium formation;  v »m/M; M  is  the mass o f s c a t t e r i n g  T  is  the temperature of the host g a s . i n ° K ;  K  i s Boltzmann's  The v e l o c i t y v i s the f i n e  atoms; ,  constantj  r e l a t e d to wave" number k by v=k<*c, where ot i s  structure constant.  The f u n c t i o n y ( v , t ) - v f ( v , t ) 2  the v e l o c i t y d i s t r i b u t i o n of p o s i t r o n s  per u n i t v e l o c i t y  is  interval.  By analogy w i t h the Maxwellian and Druyvesteyn d i s t r i b u t i o n s and from p h y s i c a l c o n s i d e r a t i o n s , conditions  for y(v,t)  it  is  expected  that the boundary  are  y(o,t)*0 (62) y(-,t) = 0 f o r a l l p h y s i c a l l y meaningful momentum-transfer, a n n i h i l a t i o n and formation  cross-sections.  -78-  3. .2. k  General computer s o l u t i o n of the d i f f e r e n t i a l The  at  equation.  equation (61) can be transformed i n t o  9v 3v,  m  L1  9v  1  .  t y v v  d  3v,v  d  - mv  jy » < w  x  J  (63)  d  - Cv (v) + v (v)]y(v,t) a  If  some estimates can be made as to the i n i t i a l p o s i t r o n v e l o c i t y  distribution y(v,o),  equation (63) can be s o l v e d by standard  numerical techniques. X(t)  The f u n c t i o n  = / " y(v,t)v (v)dv / f"y(v,t)dv cl  O yields of  f  O  the v e l o c i t y - a v e r a g e d d i r e c t 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  time and i s  the q u a n t i t y which i s  a l l y - d e t e r m i n e d time-dependent liminary results (Falk,  (64)  of t h i s  Orth and Jones,  compared w i t h the  experiment-  direct annihilation rate.  Pre-  type of c a l c u l a t i o n have been r e p o r t e d  1965).  However, the d i f f i c u l t y i n e s t i m a t i n g  a r e a l i s t i c y ( v , o ) makes i t a d v i s a b l e to examine the e q u i l i b r i u m solutions 3A.3«  to (63) which have no i m p l i c i t time  The case of no i m p l i c i t time  dependence.  dependence.  Consider the case where the a p p l i e d e l e c t r i c zero and where 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 Then i t  field  negligible.  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  w i l l r e l a x a f t e r a time t '  it  positrons  to a Maxwellian d i s t r i b u t i o n a p p r o p r i a t e  to the temperature T of the host gas. negligible  is  Since the a n n i h i l a t i o n s are  can be assumed that they do not a p p r e c i a b l y a f f e c t  the v e l o c i t y d i s t r i b u t i o n . r a t e at the time t  g i v e n by  X(t') u s i n g (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 t e r  t',  1  is  Then the v e l o c i t y - a v e r a g e d a n n i h i l a t i o n  -79the average  annihilation  probability  d e n s i t y o f p o s i t r o n s , however, i s d e p e n d e n t on  and  can  be  x(t*).  .  seen  It nor  to decrease  can  be  after on  I t i s only necessary  the v e l o c i t y  Thus t h e  later  times.  The  d i s t r i b u t i o n has  shape o f t h e v e l o c i t y  f o r the  d ai dv 3 v , [ {  }  field  to r e q u i r e that  an  explicit  distribution  dependence  i s unchanged  Putting  equation  uv^KT dY m dv  +  (65)  (63)  ,  +  J  l  by  exponential  y ( v , t ) = T(t./Y(v) separates  time,  exponentially with a rate given  r a t e have t o be n e g l i g i b l e  retained.  some t i m e  time.  for  t o be  thereafter.  shown t h a t n e i t h e r t h e a p p l i e d e l e c t r i c  the a n n i h i l a t i o n  behaviour  rate i s a constant  into  _  2a£ _ 2y_^KT 3v,v mv d  d  1  d  }  = J  > ( 6 6 )  [v + v- - X]Y  a  f  and £ = - XT at The  (67)  solution  exponential (66) [{ 1  to  i n t.  once w i t h  ai  +  3v,  yvd^ m  (65 J and  J  value of x  The  r e s p e c t to v over  dY dv  }  (67), y ( v , t ) = Y(v)T(o)e  +  , 1 K  d  d  _  2ai _ 2  y ] J  potentials  of the  section  tends  to  ( L a n d a u and  zero  probability be  =  (68)  finite  type used  f°° (v o  here,  t o a c o n s t a n t v a l u e as Lifshitz,  1965;  a  +  1  - X)Ydv  t h e momentum-transfer- c r o s s the  incident velocity  p.500).  density of positrons i n v e l o c i t y  for v « 0 .  of  Then  d  J  For  ~ o  clearly  by i n t e g r a t i o n  a l l velocities.  ^ mv  3v ,v  i s found  is  I t f o l l o w s t h a t Y(v)  Furthermore, space,  must t e n d  tends the  f(v,t) to zero  must at  -80l e a s t a s r a p i d l y a s v 2 n e a r V P 0. For cross-section  high  velocities,  i t i s e x p e c t e d the. m o m e n t u m - t r a n s f e r  remains f i n i t e .  I n a d d i t i o n Y(v)  must t e n d 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 energy a s s o c i a t e d  with  the d i s t r i b u t i o n Y ( v ) , remain  Making use o f t h e l i m i t s that  t h e l e f t h a n d s i d e o f (68)  established  vanishes,  the o v e r a l l  finite.  above, i t i s c l e a r  b o t h a t v = 0 a n d v=*».  Hence - 11 C v ( v )  x  + v~(v)]Y(v)dv / f" Y(v)dv  a.  O  (69)  O  ±  Thus when 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 becomes the  r e s u l t i n g a n n i h i l a t i o n rate i s a constant,  population  decays e x p o n e n t i a l l y .  This  and t h e p o s i t r o n  exponential  decay  a l s o be e x h i b i t e d i f t h e a n n i h i l a t i o n c r o s s - s e c t i o n t o 1/v.  The a n n i h i l a t i o n r a t e v  and  (64)  from  "static",  will  i s proportional  i s then independent o f v e l o c i t y ,  &  A(t) = v  (70)  a.  3.4.4.  S o l u t i o n o f the time-independent For  and  (69)  Consider { l  a given  +  set of cross-sections,  c a n be r e a d i l y o b t a i n e d t h e i n t e g r a l o f (66)  a i -yv^KT. dY 3v, m - dv d }  +  J  { l K  _ d  using  the solutions  a digital  f r o m v'-O t o v'=-v.  2a^ _ ^ K T 3v,v mv d  } y  Then  = /r  r(v o  a  t o (66)  computer.  J  J  since  equation.  , (71) ns  + v . - X)Ydv r  t h e i n t e g r a l o f t h e l e f t h a n d s i d e o f (66) v a n i s h e s a t v=0 3.4.30.  (see  Section  I f the a n n i h i l a t i o n rate i s small  compared  with  t h e 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 t h e v e l o c i t y  -81-  distribution will  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  good a p p r o x i m a t i o n s i n c e at  the scattering  10 amagats f o r a s c a t t e r i n g  rate  cross-section  i s a b o u t 10  -1  sec  of the order of  2 iTa  while the a n n i h i l a t i o n rate  0  i s experimentally  about  6 - 1  5 x 10 s e c  f o r t h e same d e n s i t y  Thus a f i r s t (71) Y  yvv  /  d  +  s e t equal t o zero.  H ^ )  d  v  X  to unity.  o C a V =  ( v  +  yields ( 7 2 )  the integral of Y  order s o l u t i o n to * i  Y d v  Substitution yields  The f i r s t  This  ]  where C i s determined by n o r m a l i z i n g velocities  IV).  order s o l u t i o n t o Y i s obtained by s o l v i n g  w i t h t h e r i g h t hand s i d e = Cv^exp [- %  Q  (see Table  over a l l  Q  g i v e n by  s  ( 7 3 )  of Y,  X  Q  i n t o t h e r i g h t hand s i d e  Q  o f (71) a n d s o l v i n g  t h e s e c o n d a p p r o x i m a t i o n t o Y.  Y _(v) = Y ( V ) ]  q  [  f ( v » ) / Y (v')dv'] = YY (V) + AY (v)  +  Y  2  Q  q  (74)  1  where f (v) 2  = Jl  (v  v  a+  lim S i n c e t h e +Q V  - A )Y (v')dV  f  O  o  . ..  ..  of  +  a  H ^ )  (75)  ^ ( v ) / Y ( v ) m i g h t be d i f f e r e n t f r o m u n i t y t h e 0  constant y i s included, gration  / ( ^  a n d c a n be c a l c u l a t e d  by n u m e r i c a l  o f (74) o v e r a l l v e l o c i t i e s , a n d r e q u i r i n g  Y-|(v) a n d Y ( v ) Q  b o t h be n o r m a l i z e d  to unity.  that  inte-  integrals  Thus a n y  approxi-  m a t i o n Y j i s t h u s o b t a i n e d i n t e r m s o f Y..^ u s i n g ( 7 4 ) . The used t o solve using  i t e r a t i v e procedure o u t l i n e d equation  (71).  has been  successfully  The i n i t i a l e s t i m a t e Y  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  Q  i s made  integrations  w e r e made u s i n g S i m p s o n ' s r u l e a n d l i n e a r i n t e r p o l a t i o n w h e r e  -82-  necessary.  In. g e n e r a l ,  iterations  the procedure converged w i t h i n  for a l l cross-sections The e f f e c t  used.  of the a n n i h i l a t i o n p e r t u r b a t i o n was  decrease the Y(v) r e l a t i v e  Q  Thus, f o r the  a n n i h i l a t i o n rates considered,  was u n i f o r m l y s h i f t e d  to a h i g h e r  v =  the v e l o c i t y  monotonically distribution  velocity.  In a d d i t i o n , f o r each v e l o c i t y calculated,  to  to the o r i g i n a l Y ( v ) f o r 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 r a t e s are l a r g e . decreasing  five  d i s t r i b u t i o n that was  the average p o s i t r o n v e l o c i t y v given by  Y(v)vdv  (76)  was found.  3-5-  Results. Preliminary results  based on the procedure of the p r e -  ceding S e c t i o n s have been r e p o r t e d (Jones, Jones and O r t h ,  1966 ( a ) ;  (Jones and O r t h , velocity-averaged  Jones and O r t h , 1966 ( b ) ) .  1966 ( a ) ) ,  however,  cross-sections,  c r o s s - s e c t i o n s were used i n the where only the Y  D i s c u s s i o n of the p o t e n t i a l s Three r e p r e s e n t a t i o n s  were s t u d i e d .  the p u b l i s h e d values of a  distribution calculations, 3-5-1.  Q  to  r a t h e r than velocity  used.  of the p o s i t r o n - A r g o n i n t e r a c t i o n  A l l e x h i b i t e d a 1/R  i n the d e t a i l e d  the  were c a l c u l a t e d .  behaviour at l a r g e  but d i f f e r e d i n the s i z e and type of the c u t - o f f consequently  In one case  d i r e c t a n n i h i l a t i o n r a t e , A , are i n e r r o r  the extent that e l a s t i c - s c a t t e r i n g momentum-transfer  F a l k and O r t h , 1 9 6 5 ;  distances,  employed and  shape of t h 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  -83-  in  the neighbourhood of the atom.  The t o t a l i n t e r a c t i o n c o n s i d e r e d  was the sum of the H a r t r e e p o t e n t i a l  f o r Argon (Hartree and  1936), VJJ, and the 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 , Vp.  Hartree,  Three types of V^ were  studied:  A.  V  * -5 .-5/(r  B.  V p ^ -5.5/(r  C.  V  A p  B  =  C p  2  2  + r  D  •+ r  Q  2  )  2  ; r  0  ; r  Q  = 2.5 a  2  = 0.62  2  8  A  0  of Argon i n atomic u n i t s  potentials  2  )  -5 . 5 r ( 1 - e x p ( r / r ) ) ; r  In each case, the numerator i s  A  2  8 Q  2 0  a  , 2  Q  ,  = 14 a  8 Q  .  j u s t the e l e c t r i c p o l a r i z i b i l i t y  (Holtsmark, 1929).  The form of  the  B  Vp , V^  Ramsauer e f f e c t  has been used i n reproducing the  (Kivel,  observed  1959) i n Argon, and i s known i n the l i t e r a -  t u r e as the Buckingham s e m i - e m p i r i c a l 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 w i t h experiment. A  In V , r p ' o n  2 = 2.5 a 2 was s e l e c t e d f o r the " o  s i n c e t h i s was the v a l u e found to f i t i n Argon ( K i v e l ,  1959).  transfer cross-section  low-energy e l e c t r o n  A similar effective  employed by Massey, et a l .  cut-off, ' scattering  i n t e r a c t i o n was  (1966) i n c a l c u l a t i o n s of the momentum-  and Z  e  f  f  f o r p o s i t r o n s i n Argon, and the  cut-  2 off  parameter r  scattering.  Q  chosen was a l s o the v a l u e a p p r o p r i a t e to .  F i g u r e s 15 and 16 show r e s p e c t i v e l y of  electron  the v e l o c i t y  the Z £ £ and momentum-transfer c r o s s - s e c t i o n s e  the s o l u t i o n of the Schrodinger e q u a t i o n .  dependence  r e s u l t i n g from  The values  obtained f o r  CU^VB B  "  CvKvS C  "  *  v  V  *  >  100 —  50  O  -EFFECT  LU >  ISI  A.  5  ,  1 0.0  ^*^A  1  I  0.1  1  k -  ^  I  0.2  1  i  0.3  1  ^  1  0.4  POSITRON  ^  1  i  0.5  ^  1  VELOCITY  ^  I  """"  0.6  1  I  0.7  1  1  0.8  (0  F i g u r e 15. T h e o r e t i c a l r e s u l t s f o r Z f f as a f u n c t i o n o f p o s i t r o n wave number k. e  CURVE A •  cu Ave a • Cu Kve c •  2  I 0.0  1  1  1  0.1 k —  I  0.2  I  I  I  03  POSITRON  I 0.4  II  V.'  I  I • I 0.5 0.6  •  I  0.7  •  1  0.8  VELOCITY (a ) 0  H  F i g u r e 1 6 . T h e o r e t i c a l r e s u l t s f o r the momentum-transfer s e c t i o n f o r p o s i t r o n s i n Argon as a f u n c t i o n o f k.  cross-  -8*+-  V^  a r e i n a g r e e m e n t w i t h t h o s e r e p o r t e d by M a s s e y , e t a l . ( 1 9 6 6 ) .  Jr  Agreement w i t h r e s u l t s  obtained independently i s necessary i n  c a l c u l a t i o n s o f t h e t y p e p r e s e n t e d h e r e , p r o v i d i n g an c h e c k on t h e 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 t h e v e l o c i t y - d i s t r i b u t i o n ( S e c t i o n 3AO  are presented  important  i n F i g u r e 17'  calculations  Comparison w i t h the  d e p e n d e n c e o f t h e a n n i h i l a t i o n r a t e on t h e a p p l i e d e l e c t r i c  field  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 small.  Furthermore,  the t h e o r e t i c a l curve  not vary 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 In mental  order t o reproduce  results,  does  field.  a t l e a s t one a s p e c t o f t h e e x p e r i -  the c u t - o f f parameter r  0  inV  was c h o s e n t o be Jr  0.62  a  Q  so 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 t z e r o  f i e l d would approximate  the experimental r e s u l t .  Reduction of  t h e c u t - o f f p a r a m e t e r b y s u c h a l a r g e amount r e s u l t s p o s i t r o n a t t r a c t i o n , and t h i s  is  i n d i c a t e d d i r e c t l y by t h e l a r g e r Z  zero f i e l d ,  g f f  (Figure  15).  b y t h i s means, t h e m a g n i t u d e 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 a l m o s t  electric  In addition, the  i n c r e a s e i n t h e 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 t h e atom  Although,  at  i n an i n c r e a s e d  i s r e f l e c t e d i n the g r e a t l y increased  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 ( F i g u r e 16). resulting  electric  equal to the experimental  t h e dependence o f t h e a n n i h i l a t i o n  value  r a t e on a p p l i e d  field  i s t o o weak a t l o w e l e c t r i c f i e l d s ( F i g u r e 17). •g A n i m p o r t a n t f e a t u r e f o r t h e c u r v e f o r Vp i n F i g u r e 17 i s t h e  -1 "break" which of  o c c u r s a t a b o u t 70 V cm  -1 amagat  .  For this  E / P , t h e mean p o s i t r o n wave number a s c a l c u l a t e d f r o m  value  Equation  F i g u r e 1 7 . 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 r a t e s as a f u n c t i o n o f E/P.  -85-  (76),  S e c t i o n 3-4.h. is.0.23 a  .  Q  From F i g u r e 16, t h i s  corres-  2 ponds to a momentum-transfer c r o s s - s e c t i o n The  significance  cussed  of t h i s  cross-section  of about 1 0 IT a  Q  .  at the break w i l l be d i s -  shortly. The  potential, potential  p o t e n t i a l d e s r i b e d by V  but f o l l o w s expected  more c l o s e l y  i s a l s o a one parameter  c  the form of the p o l a r i z a t i o n  for incident electrons  ment (Lenander, 1966; Crown and Russek,  from a p h y s i c a l a r g u -  1965).  This  potential  rise's to a maximum near the s u r f a c e of the atom and decreases r a p i d l y i n s i d e the atom.  For p o s i t r o n s ,  electron-positron correlations  however,  should r e s u l t  the e f f e c t  of  i n an enhanced  e f f e c t i v e p o t e n t i a l w i t h i n the atom as compared w i t h the case of incident electrons.  T h i s i n t e r a c t i o n may be expected  important f o r low v e l o c i t y velocity-dependent dependent  positrons,  to be more  thus g i v i n g r i s e to a  effective potential.  This type of  p o t e n t i a l i s not considered h e r e ,  velocity-  however. o  o  The s i z e of the c u t - o f f parameter r =. 1*+ a has been chosen so that the ZQff at zero v e l o c i t y c o i n c i d e s w i t h the Zeff —1 B a t k =• 0 a ~ c a l c u l a t e d u s i n g Vp ( F i g u r e 15). Again the D  Q  Q  momentum-transfer c r o s s - s e c t i o n ( F i g u r e 16 ) , The  and f a l l s  dependence  o f f r a p i d l y to a f a i r l y constant  of a n n i h i l a t i o n r a t e on e l e c t r i c  bears no resemblance of  i s very l a r g e at k = 0 a  field  to the experimental r e s u l t s .  the break mentioned . e a r l i e r i s  - 1 0  value.  ( F i g u r e 17)  The p o s i t i o n  l e s s w e l l defined f o r t h i s  but can be taken to be i n the r e g i o n of 120 V cm ^ amagat ''. -  average p o s i t r o n wave number corresponding to t h i s  -  field  as  case, The  -86-  c a l c u l a t e d from equation ( 7 6 ) , S e c t i o n 3 A A . i s 0.23 a t h i s wave number the momentum-transfer c r o s s - s e c t i o n of  .  is  At  once more  2  the order of 10 * a  . o  3«5'3«  D i s c u s s i o n of the break i n the dependence of a n n i h i l a t i o n r a t e on e l e c t r i c The  field.  s i z e of the momentum-transfer c r o s s - s e c t i o n  break appears to be i n d e p e n d e n t . o f momentum-transfer c r o s s - s e c t i o n the monotonic decrease and  and Z ff, e  e  the  of both the momentum-transfer  cross-section equation  S e c t i o n 3.*+.3. At  low e l e c t r i c  fields,  the form of the p o s i t r o n v e l o c i t y  distribution is  essentially  due to the f a c t  that the energy g a i n term, a /3v^ i s  the e l a s t i c is  the  but depends simply on  T h i s f a c t can be demonstrated w i t h the a i d of  Z ff.  (66),  the d e t a i l e d shapes of  at  independent of the e l e c t r i c  s c a t t e r i n g term, 2  yv^KT/m.  field  swamped by  Once the e l e c t r i c  l a r g e enough f o r a / 3 £ to be comparable to V  uv^KT/m,  field 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 v a l u e of the f i e l d term. Once the e l e c t r i c  f i e l d is  p o s i t r o n v e l o c i t y increases  i n c r e a s e d beyond t h i s p o i n t , more r a p i d l y w i t h e l e c t r i c  the  average  field,  p r o v i d e d that there i s no accompanying i n c r e a s e i n the momentumtransfer cross-section. type w i l l  It is  evident  that a t r a n s i t i o n of  l e a d to the break i n the e l e c t r i c - f i e l d dependence  velocity-averaged annihilation rate, d e c r e a s i n g f u n c t i o n of  velocity.  if  the Z  is  this of  a monotonically  -87-  In o r d e r , t h e n ,  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  s e c t i o n at the average wave number k a p p r o p r i a t e to the d i s t r i b u t i o n must approximately  i n a host gas at 298°K. - 1 - 1 ( E / P = 70 V cm amagat  velocity  satisfy  When the magnitude of the e l e c t r i c  field  — -1 ) and average wave number (k = 0.23 a ) 0  a p p r o p r i a t e to the break i n curve B , F i g u r e 17 are into  cross-  ( 7 7 ) , the i n e q u a l i t y i s  <l5  7 r  a  2 0  .  This i s  substituted i n agreement  the d i s c u s s i o n i n S e c t i o n 3«5«2. where the v a l u e f o r o  d  at  with  the  average wave number corresponding to the break i n curve B was found from a d e t a i l e d 3.5A.  calculation.  D i s c u s s i o n of the experimental dependence r a t e on e l e c t r i c It is  possible  field. to d i s c u s s  the experimental r e s u l t s  F i g u r e 17 i n the l i g h t of the p r e v i o u s S e c t i o n . the 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 s ,  values  evidence  that the break s t a r t s  -1 -1 l e s s than 15 V cm amagat .  fields.  the e x p r e s s i o n  de-  Further-  to occur at E / P  The average p o s i t r o n v e l o c i t y  a t the break must be of the order of thermal v e l o c i t i e s or s l i g h t l y l a r g e r .  in  Compared w i t h  the experimental values  crease r a p i d l y at c o n s i d e r a b l y s m a l l e r e l e c t r i c more, there i s  of a n n i h i l a t i o n •  (k~0.05 a ~ 0  — -1 -1 Now f o r k = 0.05 a and E / P » 15 V cm amagat  (77) y i e l d s  Q  o j  <  (  to s m a l l e r upper l i m i t s f o r ° d  '.^s-^, w h i l e l a r g e r k g i v e Thus i t  t h a t the momentum-transfer c r o s s - s e c t i o n  is  reasonable  for positrons  to  rise suppose  i n Argon  -88-  is  l e s s than about It  is  \ ^ ^ B .  at thermal e n e r g i e s .  0  interesting  to observe  that  the e s t i m a t i o n  t h i s upper l i m i t f o r the low-energy momentum-transfer is  independent  of any assumptions  positron-atom i n t e r a c t i o n ,  cross-section  r e g a r d i n g the nature of  and i s  a l s o independent  of  of a  the  possible  c o n t r i b u t i o n to the observed a n n i h i l a t i o n r a t e from r a d i a t i v e capture  processes. The momentum-transfer  at higher v e l o c i t i e s  field  1  indicates  equilibrium distribution is ium f o r m a t i o n ; t h a t i s , this is  ( F i g u r e 9; see  also Figure 8).  that an a p p r e c i a b l e f r a c t i o n of at the t h r e s h o l d v e l o c i t y  at k ^ 0.8 a ~ . 1  0  to be found i n the f l a t t e n i n g  r a t e at h i g h e l e c t r i c  fields,  o f f of the d i r e c t  3.6. 3.6.1.  that  annihilation  indicated,  For the  in fact,  r a p i d dependence of p o s i t r o n i u m formation on e l e c t r i c requires  positron-  which has been a t t r i b u t e d to  to dominate i n t h i s r e g i o n (as  t h i s region)  for  <5 a 1 t  0  the  F u r t h e r evidence f o r  p o s i t r o n i u m formation (Chapter 2, S e c t i o n 2.6.3.). field  appreciably  the amount of i n c r e a s e d p o s i t r o n i u m formation at  100 V cm amagat -1  cannot r i s e  i n view of the i n c r e a s e d p o s i t r o n i u m formation  as a f u n c t i o n of e l e c t r i c For example,  cross-section  increased electric  by the  field  in  .  Conclusions. Summary of experimental  results.  The d i r e c t component of p o s i t r o n a n n i h i l a t i o n i n Argon has been measured as a f u n c t i o n of d e n s i t y .  The a n n i h i l a t i o n  r a t e appears to obey a l i n e a r dependence up to about 10 amagats,  - 8 9 -  w i t h 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  densities.  The o r t h o p o s i t r o n i u m quenching r a t e has a l s o been d e t e r mined as a f u n c t i o n of d e n s i t y .  Again the r e s u l t s  w i t h a l i n e a r dependence up to about 1 7 amagats.  are  consistent  The d i r e c t  a n n i h i l a t i o n r a t e and r e l a t i v e o r t h o p o s i t r o n i u m p r o d u c t i o n were both measured as a f u n c t i o n of a p p l i e d e l e c t r i c  field.  measurements have been used to p r o v i d e an i n t e r n a l l y  These consistent  p i c t u r e of the behaviour of the p o s i t r o n s i n Argon under the i n f l u e n c e of an e l e c t r i c 3.6.2.  field.  Theoretical conclusions. The e l e c t r i c  f i e l d results  the c u r r e n t models of i n t e r a c t i o n s  have been used as a t e s t of  of i n c i d e n t e l e c t r o n s  p o s i t r o n s w i t h m u l t i - e l e c t r o n atoms.  I t has been shown here  the simple one-parameter approximations f o r the atom-electron i n t e r a c t i o n , reasonable f i t  that  effective  which have been found to y i e l d a  to the low-energy e l a s t i c  scattering  d a t a , are  inadequate f o r the case of p o s i t r o n - A r g o n s c a t t e r i n g . initial  and  The sharp  drop i n the observed d i r e c t a n n i h i l a t i o n r a t e as a f u n c t i o n  of a p p l i e d e l e c t r i c  f i e l d i s not reproduced u s i n g the momentum-  transfer cross-sections  and a n n i h i l a t i o n r a t e s  derived using  these p o t e n t i a l s .  Furthermore, from the g e n e r a l arguments proposed  i n Section 3 ' 5 »  i t appears that the momentum-transfer  section  1 +  .,  i n Argon i s  l e s s than 1 5  at thermal  cross-  energies.  In order to reproduce such a s m a l l momentum-transfer cross-section,  while r e q u i r i n g a r e l a t i v e l y large a n n i h i l a t i o n  -90-  rate,  it  seems that c e r t a i n e f f e c t s n e g l e c t e d i n the simple model  c o n s i d e r e d here should be taken i n t o account. i n c l u d e the p o s i t r o n - e l e c t r o n  These  effects  correlation, especially  in  the  neighbourhood of the atom, and deformation of the Argon atom by the s c a t t e r i n g  positron.  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 p o s i t r o n from the continuum i n t o a bound A r g o n - p o s i t r o n system should a l s o be expected to c o n t r i b u t e to the direct annihilation  rate.  observed  -91REFERENCES. Alekseev, see  A . I . (1959).  S o v i e t Phys.  - JETP 2 , 1312.  (1958).  S o v i e t Phys.  - JETP Z,  also  Handbook, }+, 119.  American I n s t i t u t e of Physics Book Co. , N . Y . Anderson, C D . ( 1 9 3 2 ) .  McGraw H i l l  Phys. Rev. }+]_, +05. L  B e n n e t t , W.R. , Thomas,W., Hughes, B u l l . Am. Phys. Soc. 6 , *+9. Bowe, J . C ( 1 9 6 0 ) .  826.  Phys. Rev. 1V7_, 1 *+16.  B r o c k , R . L . and S t r e i b , J . F . ( 1 9 5 8 ) .  Phys. Rev. 1 0 2 , 399-  Celitans,  G . J . and Green, J . H . ( 1 9 6 ) .  Celitans, Soc.  G . J . , Tao, S . J . and Green, J . H . ( 1 9 6 ) .  L  P r o c . Phys. Soc. L  83., 833.  C h e s h i r e , I . M . (196*+). P r o c . Phys. Soc. Colli,  (1961)  V.W. and Wu, C S .  Phys.  6Q, 227.  L . and F a c c h i n i , U . ( 1 9 5 2 ) . Rev. S c i .  Crown, J . C and Russek,  Proc.  8J,, 823.  Instr.  2 J , 39-  A. ( 1 9 6 5 ) - Phys. Rev. 1 3 8 , A 6 6 9 .  D i r a c , P . A . M . ( 1 9 2 8 ) . P r o c . Roy. Soc. A117. 6 1 0 , P r o c . Roy. A 1 1 8 . 351 • Dirac, P.A.M.  (1931)•  Soc.  P r o c . Roy. Soc. A126, 36O.  Drachman, R . J . ( 1 9 6 6 ) . Phys. Rev. 1J>0, 1 1 . D u f f , R . G . and Heymann, F . F . ( 1 9 6 3 ) . P r o c . Roy. Soc. - A272, 363. F a l k , W.R. ( 1 9 6 5 ) . P h . D . T h e s i s , . U n i v e r s i t y (Unpublished) F a l k , W.R. and Jones, F a l k , W . , Jones,  21, 3^5.  F a l k , W.R., Orth, 1j+, ¥ f 7 .  G. 0 9 6 ) . L  of B r i t i s h Columbia.  Can. J . Phys.  hg,.1751.  G. and O r t h , R. ( 1 9 6 5 ) . N u c l . I n s t r . P.H.R.  and Jones,  (London)  and Meth.  G. ( 1 9 6 5 ) . Phys. Rev.  Letters  -92-  F e r r e l l , R.A.  0956).  Rev. Mod. Phys. 28, 308.  F r a s e r , P.A. (1961). Proc. Phys. Soc. 28, 329F r a s e r , P.A. (1961). Proc. Phys. Soc. 22, 721. G e r h a r t , J.B. , C a r l s o n , B.C. and S h e r r , R. 9j+, 917.  (195*0-  Phys. Rev.  H a r t r e e , D.R. and H a r t r e e , W. ( 1 9 3 6 ) . Proc. Roy. Soc. A166.  450.  Heinberg, M. and Page, L.A. ( 1 9 5 7 ) . Phys. Rev. 102, 1 5 8 9 Heymann, F.F., Osmon, P.E., V e l t , J . J . and W i l l i a m s , W.F. Proc. Phys. Soc. 28, 1 0 3 8 . Holtsmark,  (1961).  J . (1929). Z. P h y s i k £5, 437.  Jones $ G. and F a l k , W.R. ( 1 9 6 5 ) . N u c l . I n s t r . and Meth. 32, 2 2 . Jones, G., F a l k , W.R. and Orth, P.H.R. ( 1 9 6 5 ) . Paper presented a t the IVth I n t e r n a t i o n a l Conference o f the P h y s i c s o f E l e c t r o n i c and Atomic C o l l i s i o n s ( 1 9 6 5 ) . Jones, G. and Orth, P.H.R. ( 1 9 6 6 ( a ) ) . " P o s i t r o n A n n i h i l a t i o n " . (Proceedings o f the Conference on P o s i t r o n A n n i h i l a t i o n h e l d a t Wayne S t a t e U n i v e r s i t y , D e t r o i t , Michigan, 1 9 6 5 A.T. Stewart and L.O. R o e l l i g , ed.) Academic P r e s s , N.Y.  Jones, G. and Orth, P.H.R. (1966(h)). B u l l . Am. Phys. S o c . I I , J M , 749. Kaempffer, F.A. ( 1 9 6 5 ) . "Concepts i n Quantum Academic P r e s s , N.Y. K e n d a l l , H.W. and Deutsch,  Mechanics".  M. ( 1 9 5 4 ) . Phys. Rev. j £ , 932.  K e s t n e r , N.R., J o r t n e r , J . , Cohen, M.H. and R i c e , S.A. (1965). Phys. Rev. lJ40, A 5 6 . K i v e l , B. ( 1 9 5 9 ) . Phys. Rev. U6,  4.  Labahn, R.W. and Callaway, J . 0 9 6 6 ) . Bull.Am. Phys. Soc. J J . , 3 0 7 . Lenander, C.J. ( 1 9 6 6 ) . Phys. Rev. 142. 1. M a c k l i n , P.A., L i d o f s k y , L . J . and Wu, C S . ( 1 9 5 0 ) . Phys. Rev. 28, 3 1 8 . Marder, S., Wu, C S . and Bennett, W. ( 1 9 5 6 ) . Phys. Rev. 1 C £ , 1 2 5 8 . Massey, H.S.W. and Mohr, C.B.O. 0 954-). Proc. Phys. Soc. A, 62, 6 9 5 .  -93-  M a s s e y , H.S.W. a n d Moussa, A . H . (1961). P r o c . P h y s . S o c . 22? M a s s e y , H.S.W., Lawson, communication.  J . a n d Thompson,  811.  D.G. (1966). P r i v a t e  M a t h e w s , J . a n d W a l k e r , R . L . (1965). " M a t h e m a t i c a l M e t h o d s P h y s i c s . " W.A. B e n j a m i n I n c . , N.Y.  of  0  M o t t , N . F . a n d M a s s e y , H.S.W. (1965). " T h e T h e o r y o f A t o m i c C o l l i s i o n s . " 3rd E d . O x f o r d . C l a r e n d o n P r e s s . M o u s s a , A . H . (1959). P r o c . P h y s . S o c . 2it, Nelms,  A . T . (1956). N a t . B u r . o f S t a n d a r d s ,  101. Circular  O r e , A . a n d P o w e l l , J . L . (19^9)• P h y s . R e v . 2£« O r e a r , J . (1958). " N o t e s o n S t a t i s t i c s Report, Berkeley, C a l i f o r n i a . Osmon,  577-  1696.  for Physicists,"'  UCRL - 84-17  P . E . (1965). F h y s . R e v . 1^8, B 216.  P a u l , D . A . L . (1964). P r o c . P h y s . S o c . 8 4 , 563R o s e , M . E . (1953). P h y s . R e v . 2±, 610. Ruark,  E.. (194-5). P h y s . R e v . 68, 278.  Schwarzschild,  A. (1963). N u c l . I n s t r .  a n d M e t h . 21_, 1.  S e r i e s , G.W. (1957). " T h e S p e c t r u m o f A t o m i c H y d r o g e n . " University Press. Shearer,  J.W. and D e u t s c h ,  de S o b r i n o ,  M.  0  Oxford  94-9). P h y s . R e v . 76, 4-62.  L. a n d C l a r k , M. (1961). N u c l e a r S c i . a n d E n g . 10, 377.  T a o , S . J . , ' G r e e n , J . H . a n d C e l i t a n s , G . T . (1963). P r o c . P h y s . S o c .  81, 1091..  Tao, S . J . , B e l l ,  83, 453.  J . a n d G r e e n , J . H . (1964). P r o c . P h y s . S o c .  Tao, S . J . and B e l l , W a l l a c e , P.R.  J . (1966). P r i v a t e  (1955). P h y s . R e v . 100,  communication. 738.  Wu, T . a n d Ohmura, T . (1962). '"Quantum T h e o r y o f S c a t t e r i n g " . P r e n t i c e - H a l l I n c . , N.Y. Yang,  C . N . (1950). P h y s . R e v . 22?  242.  APPENDIX.  MODIFICATIONS TO THE FAST-SLOW COINCIDENCE CIRCUITRY USED BY FALK ( 1 9 6 5 ) . A b l o c k diagram of the f a s t - s l o w is  shown i n Chapter 2, F i g u r e 3 '  1.  Photomultlplier c i r c u i t r y .  coincidence  The new 1+in. x 3 i n . diameter N a l ( T l )  system  crystals  required  replacement of the 2 i n . diameter RCA 6 8 1 0 p h o t o m u l t l p l i e r tube used p r e v i o u s l y w i t h the l a r g e r !+in. diameter RCA 701+6.  The  m o d i f i e d c i r c u i t f o r the dynode c h a i n s u p p l y i n g the dynode potentials  is  shown i n F i g u r e 1 8 . S i n c e f a s t  risetimes  were  r e q u i r e d at the anode the e f f e c t of inductance was minimized from stages 6 to 11 by use of heavy, Elsewhere,  s h o r t copper s t r i p as  leads were kept as short as p o s s i b l e .  leads.  The p h o t o m u l t i -  p l i e r h i g h v o l t a g e was s u p p l i e d by the commercial P r e c i s i o n Power S o u r c e , Model 122 B , manufactured by C a l i b r a t i o n Standards I n c . 2•  Timing P u l s e G e n e r a t o r . Both l i m i t e r s used b y , F a l k have been r e p l a c e d w i t h  transistorized equivalents. and i s  The c i r c u i t i s  s i m i l a r to a type of c i r c u i t ,  t h a t i s used f o r f a s t  shown i n F i g u r e 1 9 ,  employing t u n n e l d i o d e s ,  c o i n c i d e n c e work ( S c h w a r z s c h i l d , 1963).  Operation of the c i r c u i t i s as f o l l o w s :  the a r r i v a l  of  -95an e l e c t r o n 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 ( F i g u r e 18) causes c u r r e n t to be conducted through the t u n n e l diode TD. Sufficient trigger.  c u r r e n t through the TD causes the t u n n e l diode The t h r e s h o l d c u r r e n t r e q u i r e d f o r t r i g g e r i n g i s  by the 10k ft potentiometer. a t the base of t r a n s i s t o r is  Once the TD has t r i g g e r e d , T drops by about 0.2 V .  r a p i d l y d r i v e n to s a t u r a t i o n and a p o s i t i v e  2.5 V  of about  produced i s  appears at the  d r i v e n through the  timesorter.  1 00ft 100ft  set  Half  of t h i s  the  is  This  resolution  For each gamma-ray c h a n n e l ,  t h r e s h o l d method i s  half-  Na-22 i n aluminum. An  that n o i s e pulses do not  timesorter.  The prompt r e s o l u t i o n obtained w i t h these t i m i n g generators  the  defined by  so as to minimize the f u l l - w i d t h at  t r i g g e r the i n p u t to the  mA  the TD.  maximum of the prompt peak obtained w i t h advantage  25  of the  original state.  The t h r e s h o l d l e v e l of the TD a f f e c t s  the t h r e s h o l d was set  pulse  cable i n t o the i n p u t of  by the LR network across  to some degree.  voltage  The t r a n s i s t o r  The maximum width of the output pulse i s  of the t i m e s o r t e r  set  the  voltage  resistor.  the time taken f o r the TD to r e t u r n to i t s time i s  to  shown i n F i g u r e 20 .  Two curves are  pulse  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 w i t h i n the 0.51  MeV f u l l - e n e r g y peak.  The second was  o b t a i n e d f o r the same energy i n t e r v a l but i n c l u d i n g o n l y those pulses  which o c c u r r e d  2.2.1.).  i n the " v a l l e y " r e g i o n (Chapter 1,  Section  The prompt r e s o l u t i o n had been p r e v i o u s l y optimized  f o r the f i r s t  case.  There i s  a small difference  i n the  resolution  -96due mainly to time s l e w i n g ,  that i s ,  dependence of the t r i g g e r i n g  time on the amplitude of the p h o t o m u l t l p l i e r c o l l e c t o r 3.  pulse.  A m p l i f i e r s and S i n g l e Channel A n a l y z e r s . The a m p l i f i e r s and s i n g l e  channel a n a l y z e r s used hy  F a l k have heen r e p l a c e d w i t h commercial t r a n s i s t o r i z e d u n i t s (Cosmic A m p l i f i e r , 901 S . C . A . ) .  Model 901A and S i n g l e Channel A n a l y z e r , Model  The a m p l i f i e r s produce b i p o l a r pulses which are fed  i n t o the zero c r o s s - o v e r s i n g l e channel a n a l y z e r has positive  channel a n a l y z e r s .  Each  single  both a n e g a t i v e and a p o s i t i v e o u t p u t .  The  output was used to t r i g g e r the c o i n c i d e n c e gate (Chapter  2, F i g u r e 3) d e s c r i b e d by F a l k . by a s c a l e r  The n e g a t i v e  (Nuclear S u p p l i e s , Model  SA-250).  pulses were counted Each s c a l e r was  used to monitor the t o t a l number of counts admitted by the channel a n a l y z e r s during a r u n .  single  The number of counts i n the  1 .28 MeV gamma channel was used f o r n o r m a l i z a t i o n from run to r u n . As the t y p i c a l number of counts encountered i n a run were of  7 order of 10  the  8 - 10 , and the s c a l e r s were capable of s c a l i n g o n l y  to 10^, a mechanical r e g i s t e r of counting c a p a c i t y 10^ was connected to each s c a l e r count o v e r f l o w . this h.  purpose i s  The c i r c u i t used f o r  i l l u s t r a t e d i n F i g u r e 21 .  P i l e - u p Re.iectors. The use of the Cosmic a m p l i f i e r s made i t  possible  to  feed the outputs of the a m p l i f i e r s d i r e c t l y i n t o the p i l e - u p rejectors,  described i n Falk's  amplification.  thesis  (1965),  without f u r t h e r  The c u r r e n t a m p l i f i e r p o r t i o n of the )  detectors  -97c i e v e l o p e d b y F a l k was t h e r e f o r e n o t u s e d . 5.  ND 101  Kicksorter. The 100  k i c k s o r t e r (Computing D e v i c e s o f Canada,  2230) p r e v i o u s l y  M o d e l AEP analyzer  channel  u s e d was r e p l a c e d w i t h a  ( N u c l e a r D a t a , M o d e l 101).  256  channel  The d i f f e r e n c e i n t h e i n p u t  requirements t o the k i c k s o r t e r required s l i g h t m o d i f i c a t i o n of the  output stages  operation 6.  o f t h e t i m e s o r t e r v/hich i n n o way a f f e c t e d t h e  of the instrument.  D i f f e r e n t i a l and i n t e g r a l  linearity  of the timesorter.  The d i f f e r e n t i a l a n d i n t e g r a l , l i n e a r i t y system were o b t a i n e d Falk,  1965;  r o u t i n e l y as r e p o r t e d  F a l k , Jones and O r t h ,  Two m e a s u r e m e n t s o f t h e i n t e g r a l by are  an i n t e r v a l o f 2 months. shown i n F i g u r e 2 2 .  a n d a r e shown i n F i g u r e  of the f i r s t  22'  separated  measurement  A l e a s t squares f i t t o t h i s s t r a i g h t  2.7*+-— 0.01  s e c o n d m e a s u r e m e n t y i e l d e d 2.73  integral  p r e v i o u s l y (Jones and  l i n e a r i t y w e r e made,  Results  l i n e y i e l d e d an average s l o p e o f the  1965).  of the o v e r a l l  - 0.01  nsec/channel,.  nsec/channel.  while The  l i n e a r i t y m e a s u r e m e n t was u s e d t o d e f i n e t h e a v e r a g e  time width  per channel  instrument), while  ( t h a t i s , the time c a l i b r a t i o n o f the  the d i f f e r e n t i a l  l i n e a r i t y was u s e d a s a  measure o f t h e a c t u a l i n d i v i d u a l w i d t h s . used f o r the e v a l u a t i o n of the l i f e t i m e s  B o t h s e t s o f dat a were (Chapter  2, S e c t i o n 2 . 5 - ) •  0 +• 2 8 7 5 V UA  10  I  TO TIMING PULSE  6 kV  CeNERATOR  -( D 14 3 9 KJl  I  zjl 500 pf  x  33 k A 70 COSMIC AfWJRfiR  '0*1 A?  <  D 13  Cftio # 3  ( D 12  lOOpf  2 7 k^!  s-HI— 100 c  I  TOTTA"  <  D ll  22 k i i  RCA -(  22 k J l 22  DiO  7046  -<D9  kJl  < 22 left. 22 kJZ.  D8  -(07 <D6  22>KJ2 <D5 22  fc^< <D4  22  kSl.  £2  kil  <D3  <DZ-  C Dl GRID #2 225 kA j ( CATHODE  Figure 18.  Photomultiplierccircuit.  15V !  R $100/1  1N3716 TD  .01LJF L  Y~  HOOpH  2N1195  To PM anode 1 JJ'H  !470il •o) 100 A cable  3.3 kA 100/1 10 kXl  Denotes heavy Ccrstrip  F i g u r e 19.  Timing  pulse generator  circuit.  A---0.51 MeV peak position C-* 0.51 MeV valley position  CHANNEL  NUMBER  F i g u r e 20. P r o m p t 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 s p e c t r a w e r e o b t a i n e d f o r Na-22 i n A l .  system.  The  + 120 V  i  Electromechanical Register! SODECO Model T C e B Z 5 E  I N  459  25uF AAA  v w v —  "-K  2N3440 IX  2.2kJl:  X  Figure 2 1 .  C i r c u i t for driving  electro-mechanical  register.  RELATIVE CHANNEL WIDTHS (ARBITRARY UNITS) (Differential linearity) •* p •* t*  TIME SEPARATION, BETWEEN PULSES (NSEC) ° (Integral linearity)  

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