Slow positron processes in gases

Volume 16, number 3

PHYSICS L E T T E R S

SLOW

POSITRON

PROCESSES

1June 1965

IN G A S E S

P. E. OSMON Westfield College, University of London Received 6 May 1965

T h e r e is at p r e s e n t no overlap of e x p e r i m e n tal and t h e o r e t i c a l s t u d i e s of slow p o s i t r o n - a t o m collision p r o c e s s e s . T h i s is unfortunate when p r o g r e s s has b e e n a c c e l e r a t i n g f r o m both d i r e c tions. On the e x p e r i m e n t a l side, fine s t r u c t u r e has been o b s e r v e d in the l i f e t i m e s p e c t r a of p o s i t r o n s in the noble g a s e s [1 - 4 ] , and r e c e n t l y the f r e e p o s i t r o n a n n i h i l a t i o n rate in argon has been found to d e c r e a s e c o n s i d e r a b l y with an applied e l e c t r i c field [5]. Both of t h e s e effects a r e to be i n t e r p r e t e d as m e a s u r e m e n t s of the free p o s i t r o n a n n i h i l a t i o n r a t e , as a function of energy, in the r a n g e z e r o to t h r e s h o l d for p o s i t r o n i u m production (~ 10 eV). On the t h e o r e t i c a l side, the concept of s c a t t e r i n g length has b e e n applied to the g e n e r a l p r o b l e m of low e n e r g y e l e c t r o n - atom c o l l i s i o n s . In this connection e x t r e m a p r i n c i p l e s have been e s t a b l i s h e d m a k i n g p o s s i b l e a c o m p a r i s o n of the m e r i t of different a p p r o x i m a t e solutions without r e c o u r s e to e x p e r i m e n t [6]. Recently, extensive c a l c u l a t i o n s have b e e n done f o r slow p o s i t r o n s on atomic hydrogen [7], and h e l i u m [8]. The quantity c a l c u l a t e d has always been a cross-section, whereas annihilation rates are the m e a s u r e d q u a n t i t i e s . They a r e only i n d i r e c t l y r e l a t e d . It is the p u r p o s e of this l e t t e r to draw attention to the p o s s i b i l i t y of c a l c u l a t i n g a n n i h i l a tion r a t e s . (Until slow p o s i t r o n b e a m s a r e a v a i l a ble it will continue to be i m p o s s i b l e to m e a s u r e c r o s s - s e c t i o n s . The only c r o s s - s e c t i o n data [10] has been i n f e r r e d i n d i r e c t l y . ) The s p i n - a v e r a g e d c r o s s - s e c t i o n for twophoton a n n i h i l a t i o n s of a slow p o s i t r o n is ,,

=

/v

(1)

r 0 is the c l a s s i c a l e l e c t r o n r a d i u s , v is the r e l a tive velocity of e l e c t r o n and p o s i t r o n . The a n n i h i l a t i o n r a t e for a b e a m of p o s i t r o n s of unit intensity is t h e r e f o r e = pcrv

(2)

where p is the e l e c t r o n density in the path of the positrons. In p r a c t i c e the p o s i t r o n s a r e d i s t r i b u t e d in a gas so that 61 = p_p+ o~v6T

(3)

6~ is the a n n i h i l a t i o n r a t e f r o m a volume e l e m e n t 6v, with local e l e c t r o n and p o s i t r o n d e n s i t i e s p_, p+. If the gas is at p r e s s u r e p a t m o s p h e r e s , and L is L o s c h m i d t ' s n u m b e r , the f r e e p o s i t r o n a n n i h i l a t i o n r a t e is ~. = Irr02CLp

fp_ p+dr

(4)

the i n t e g r a l being taken over the volume of a s i n g l e atom. It is i n t e r e s t i n g that this equation does not contain v explicitly. T h e r e a r e v a r i o u s a p p r o x i m a t i o n s of i n t e r e s t : High energy limit. ( P o s i t r o n velocity >>o r b i t a l velocity of atomic e l e c t r o n s . ) This is the s i t u a tion d u r i n g the e a r l y h i s t o r y of a p o s i t r o n in a gas, say, d u r i n g the f i r s t one or two n a n o s e c o n d a t m o s p h e r e s . The p o s i t r o n is d e s c r i b e d by a plane wave, and t h e r e f o r e X = ~ro2cLpZ

(5)

where Z is the atomic n u m b e r . Static f i e l d approximation. The atomic e l e c t r o n s a r e a s s u m e d to be moving so much f a s t e r then the p o s i t r o n that it e x p e r i e n c e s a steady e l e c t r i c field due to the atomic charge cloud. This is the situation in p r a c t i c e a f t e r the p o s i t r o n has lived for about 100 n a n o s e c o n d - a t m o s p h e r e s in a noble gas. Neglecting d i s t o r t i o n of this cloud by the p o s i t r o n , the p o s i t r o n e x p e r i e n c e s r e p u l s i o n . B e c a u s e of this r e p u l s i o n ~. d e c r e a s e s smoothly with d e c r e a s i n g energy, f r o m the high energy l i m i t , p_ is s h e r i c a l for atomic hydrogen and the noble g a s e s so that t h e r e a r e no c r o s s - t e r m s f r o m the p a r t i a l wave expansion c o n t r i b u t i n g to the i n t e g r a l ( a s s u m i n g that one extended this app r o x i m a t i o n to high enough e n e r g i e s f o r t h e r e to be any c o n t r i b u t i o n other than S - wave. )

271

Volume 16, number 3

PHYSICS LETTERS

D i s t o r t e d static f i e l d . O b s e r v e d annihilation r a t e s [4] a r e g r e a t e r than the high e n e r g y l i m i t so one must assume considerable atomic distortion. As a p o s i t r o n a p p r o a c h e s an atom f r o m a g r e a t d i s t a n c e , the f i r s t i n t e r a c t i o n to take e f f e c t is via the e l e c t r i c dipole m o m e n t induced in the atom. The i n t e r a c t i o n e n e r g y is U = e2~/r 4 ,

(6)

w h e r e ~ is the p o l a r i z a b i l i t y of the a to m . T h e r e is e x p e r i m e n t a l e v i d e n c e f o r the i m p o r t a n c e of this long r a n g e dipole i n t e r a c t i o n : o v e r a wide r a n g e of a t o m i c and m o l e c u l a r g a s e s a c o r r e l a t i o n has been o b s e r v e d b e t w e e n annihilation r a t e and dipole p o l a r i z a b i l i t y [9]. As the p o s i t r o n gets c l o s e r , h i g h e r m u l t i poles become important. A stationary state t r e a t m e n t e x a g g e r a t e s the d i s t o r t i o n b e c a u s e it t a k e s no acco u n t of the ~nertia of the e l e c t r o n cloud, but is c o r r e c t f o r v e r y low e n e r g i e s . It is p o s s i b l e that a t t r a c t i o n of the p o s i t r o n to the atom as a r e s u l t , at f i r s t , of the long r a n g e f o r c e could end in the p o s i t r o n a tt a c h i n g to the atom. And t h e r e is s o m e e v i d e n c e that this effect has been o b s e r v e d [4] at v e r y low p o s i t r o n e n e r g i e s . Annihilation of an attached p o s i t r o n would be n e a r l y i n s t a n t a n e o u s , so that h e r e it is the c r o s s - s e c t i o n that should be c a lc u l a t e d . V i r t u a l a t t a c h m e n t and v i r t u a l p o s i t r o n i u m f o r m a t i o n a r e ways of d e s c r i b i n g the s h o r t r a n g e a t o m i c d i s t o r t i o n s . The f o r m e r m a y be s u i t a b l e f o r v e r y low e n e r g i e s , the l a t t e r n e a r p o s i t r o n i -

FOURIER-TRANSFORM WITHOUT COMPUTING

1June 1965

um production t h r e s h o l d . Again a s t a t i o n a r y c a l culation of the annihilation r a t e would p r o v i d e i n f o r m a t i o n f o r the e x p e r i m e n t a l l y i n t e r e s t i n g c a s e of v e r y low e n e r g y . F o r a c o m p l e t e i n t e r p r e t a t i o n of e x p e r i m e n t a l data, which is in the f o r m of l i f e t i m e s p e c t r a , one n eed s the c r o s s - s e c t i o n and the annihilation r a t e as a function of e n e r g y . And w h e r e a s p r e v i ously c r o s s - s e c t i o n s alone have been computed, t h e r e would s e e m to be no g r e a t difficulty in obtaining the r a t e s as well.

References 1) S.J. Tao, J. Bell and J. H. Green, Proc. Phys. Soc. (Izondoa) 83 (1964) 453. 2) W. R. Falk and G. Jones, Can. J. Phys. 42 (1964) 1751. 3) D.A.L. Paul, Proc. Phys. Soc. (London) 84 (1964) 563. 4) P. E. Osmon, Phys. Rev. 138 (1965) B216. 5) W. R. Falk, P.H.R. Orth and G. Jones, Phys. Rev. Letters 14 (1965) 447. 6) P. G. Burke and Kenneth Smith, Rev. Mod. Phys. 34 (1962) 458. 7) W. J. Cody, Joan Lawson, Sir Harrie Massey and K. Smith, Proc. Roy. Soc. (London) 278A (1964) 479. 8) K. Smith (private communication). 9) P. E. Osmon (unpublished). 10) W. B. Teutsch and V.W. Hughes, Phys. Rev. 103 (1956) 1261.

SPECTROSCOPY USING HOLOGRAPHIC IMAGING AND WITH STATIONARY INTERFEROMETERS

G. W. STROKE and A. T. FUNKHOUSER The University of Michigan, Ann Arbor, Michigan Received 6 May 1965

The r e m a r k a b l e p r o p e r t i e s and the i m p o r t a n t a d v a n t a g e s of s e v e r a l f o r m s of ' F o u r i e r - t r a n s f o r m s p e c t r o s c o p y ' have now been e s t a b l i s h e d [1-6], following initial w o r k by J a c q u i n o t [4], F e l l g e t t [5], Strong [6] and o t h e r s [1, 3, 6]. A m o n g the p r i n c i p a l a d v a n t a g e s a r e a s i m u l t a n e ous r e c o r d i n g of al l s p e c t r a l e l e m e n t s ( r e c o r d i n g - t i m e independent of the s p e c t r a l width) and high l u m i n o s i t y . The m e t h o d r e q u i r e s v e r y a c c u r a t e ( r u l i n g - e n g i n e quality) m o v i n g - m i r r o r m o t i o n (or scanning) and computation (by F o u 272

r i e r t r a n s f o r m a t i o n ) of the s p e c t r u m f r o m the photoelectrically recorded interferogram. Recent a d v a n c e s in h o l o g r a p h i c ( w a v e f r o n t - r e c o n struction) i m ag i n g [7-11], and in its F o u r i e r t r a n s f o r m f o r m u l a t i o n [7, 10, 11], m a k e it r e a sonable to i n v e s t i g a t e p o s s i b l e s i m p l i f i c a t i o n s which m i g h t r e s u l t f r o m e x t e n s i o n s of holography to such a p p l i c a t i o n s a s s p e c t r o s c o p y and a s t r o n o my. In this l e t t e r we d e m o n s t r a t e the t h e o r e t i c a l p r i n c i p l e s and e x p e r i m e n t a l v e r i f i c a t i o n s of a