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Heuristic theory of positron-helium scattering
Volume 37A, number 3
HEURISTIC
PHYSICS LETTERS
THEORY
OF
POSITRON-HELIUM
22 November 1971
SCATTERING
R. J. DRACHMAN Theoretical Studies Branch, Goddard Space Flight Center, Greenbelt, Maryland 20771, USA Received 23 September 1971 An L-dependent local potential is constructed to fit the well-known positron-hydrogen s, p and d-wave phase shifts below the rearrangement threshold. The same form of potential yields a positron-helium cross-section in agreement with a recent experimental measurement near threshold. T h e r e i s now w i d e s p r e a d i n t e r e s t in the exp e r i m e n t a l i n v e s t i g a t i o n of p o s i t r o n - h e l i u m s c a t t e r i n g and a n n i h i l a t i o n both by s w a r m e x p e r i m e n t s [1] and e n e r g y - r e s o l d e d b e a m e x p e r i m e n t s [2]. A c o n s i d e r a b l e amount of t h e o r e t i c a l work has been p e r f o r m e d [3,4], but it i s c l e a r that no definitive and r e l i a b l e p h a s e - s h i f t a n a l y s i s is yet available. On the other hand, the p o s i t r o n - h y d r o g e n e l a s t i c s c a t t e r i n g has been r a t h e r well analyzed, although the difficulty of producing monatomic hydrogen t a r g e t s has p r e c l u d e d e x p e r i m e n t a l v e r i f i c a t i o n . E s s e n t i a l l y exact s - w a v e , v e r y good p - w a v e and convincingly e x t r a p o l a t e d d-wave p h a s e - s h i f t s now exist [5-7]. A g e n e r a l effectiver a n g e f o r m u l a [8] should be adequate for L > 2, where no detailed c a l c u l a t i o n s exist. It would be d e s i r a b l e , then, for the s u c c e s s e s in hydrogen c a s e to be extended to h e l i u m even if only a p p r o x i m a t e l y ; such p r e d i c t i o n s could be useful for e x p e r i m e n t e r s and t h e o r e t i c i a n s alike. One such attempt, the 'modified a d i a b a t i c ' app r o x i m a t i o n [3, 9] has defects which we wish to confront here. In that a p p r o x i m a t i o n , one solves the p a r t i a l wave s c a t t e r i n g equation (in atomic u n i t s with e n e r g i e s in Rydbergs)
- u L + [ L ( L + 1)x -2 - k 2 ] u ~ + IV 1+ V2+(a
+l)V20]u L = 0
(1)
Here, V 1 and V 2 are first- and second-order adiabatic potentials; the latter includes all multipoles by the method of Dalgarno and Lynn [10]. The monopole potential V20 is considered to represent short-range effects that are evaluate, and the parameter a is caUed the monopole suppression parameter. For hydrogen it was found that ~ 0 gave good s-wave results at all energies
up to the p o s i t r o n i u m threshold, by c o m p a r i s o n with the ' e x a c t ' r e s u l t s [5]. P r e s u m a b l y due to a lack of sufficient v i r t u a l p o s i t r o n i u m in the adiabatic d e s c r i p t i o n , the phase shifts for L = 1 and 2 fall below t h e i r c o r r e c t v a l u e s [6, 7] as k i n c r e a s e s . More p r e c i s e l y , the e r r o r is due to a lack of g e n e r a l i t y in the form of the s h o r t - r a n g e c o r r e l a t i o n p a r t of the wave function for L > 0 i n h e r e n t in the adiabatic approximation. We p r o p o s e h e r e to c o r r e c t the defect h e u r i s t i c a l l y by allowing ~ to be L - d e p e n d e n t (~ * OeL). By choosing ot 1 = 3.49 and ot2 = 23 we can r e p r o d u c e the r e s u l t s of refs. [6] and [7]. All that is n e c e s s a r y , then, to make an i m p r o v e d p r e diction of the p o s i t r o n - h e l i u m p h a s e - s h i f t s and c r o s s - s e c t i o n is to scale the p o t e n t i a l s c o r r e c t l y . It is easily shown [3~ 11] that, if the h e l i u m ground state has the hydrogen form $ ( r l , r 2) : n - l z3 e x p [ - z ( r l + r2) ]
,
(2)
Tab|e 1 Phase shifts ~/L for e+- He scattering with 0t0 = 0, ot1 = 3.49 and 0t 2 = 23. E(eV)
?70
7/1
772
0 a 0,136 0.545 1.23 2.18 3.41 4.90 6.67 8.72 11.0
(-0.659) 0.050 0.072 0.071 0.056 0.032 0.002 -0.031 -0.066 -0.100
. . 0.003 0.010 0.024 0.038 0.055 0.072 0.088 0.103 0.114
.
. 0.0004 0.002 0.004 0.007 0.012 0.017 0.024 0.033 0.043
13.6 16.5
-0.134 -0.168
0.125 0.134
0.056 0.070
17.8
-0.183
0.137
0.077
aThe zero-energy entry is the scattering length. 187
PHYSICS L E T T E R S
Volume 37A, n u m b e r 3
section, obtained from the phase is plotted. For comparison, the r e f . [3], (in w h i c h ot L = 0 f o r a l l shown. The single experimental lies on the new curve and offers port for this heuristic analysis. between hydrogen and helium is f o r t h e p h a s e s h i f t s of t a b l e 1 to c l o s e to t h e t r u e o n e s .
POSITRON ENERGY k 2
0
0.2 [
'
0.4
0.6
0.8
1.0
1.2
I
[
I
I
I
~10" z
uJ
References
t
%
.
.
~
i
5.0
t~
10.0
i
i
i
I
i
,
i
15.0
POSITRON ENERGY IN ELECTRON VOLTS
Fig.1. Elastic p o s i t r o n - h e l i u m c r o s s - s e c t i o n in units of ?ra 2. The lower (adiabatic) curve is from ref. [3] with o~ "= 0, z = 1.5992, C = 1. The upper curve was computed from the phase shifts of table 1, using the effective range formula of ref. [8] for L > 2. The e x p e r i m e n t a l point at E = 16.5eV is from ref. [2].
then [VI(X)]H e =
2Z[Vl(ZX)]H
[V2(X)]He = 2[
V2(zx)]H
[V20(X)]He = 2[ V 2 0 ( z x ) ] H
(3)
i s t h e c o r r e c t f o r m of s c a l i n g , a n d to f i t t h e s t r e n g t h of t h e a s y m p t o t i c x - 4 p o t e n t i a l (the p o l a r i z a b i l i t y ) we c h o o s e z = 1.5992. T h e r e s u l t s a r e g i v e n i n t a b l e 1. In fig. 1. t h e t o t a l e l a s t i c s c a t t e r i n g c r o s s -
188
s h i f t s of t a b l e 1. r e s u l t s of L) are also p o i n t s h o w n [2] additional supThe analogy strong enough be considered
kc~ ~ 0 ~ _ _ .
co O e.9
5o.~
22 November 1971
[1] C.Y. Leung and D . A . L . Paul, J. Phys. B2 (1969) 1278; G. F. Lee, P. H. R. Orth and G. Jones, Phys. L e t t e r s 28A (1969) 674; S. J. Tao and T. M. Kelly, Phys. Rev. 185 (1969) 135. [2] D.G. Costello, D.E. Groce, D. F. H e r r i n g and J. W. MeGowan, p r e p r i n t (1971) ; Bull. Am. Phys. Soe. 13 (1968) 1397. Proe. VI Int. Conf. on Phys. Electron and Atomic Coll. (MIT P r e s s , Cambridge, 1969) p. 757. [3] R . J . Draehman, Phys. Rev. 144 (1966) 25. [4] N. R. Kestner, J. J o r t n e r , M.H. Cohen and S. A. Rize, Phys. Rev. 140 (1969) A56; M. Kraidy, PhD Thesis, Univ. of Western Ontario (1967) unpublished; S. K. Houston and R. J. Drachman, Phys. Rev. A 3 (1971) 1335; J. Callaway, R.W. LaBahn, R.T. Pu and W.M. Duxler, Phys. Rev. 168 (1968) 12; R. E. Montgomery and R. W. LaBahn, Can.J. Phys. 48 (1970) 1288. [5] A. K. Bhatia, A. Temkin, R.J. Drachman and H. Eiserike, Phys. Rev. A3 (1971) 1328. [6] R.L. Armstead, Phys. Rev. 171 (1968)91. [7] C.J. Kleinman, Y. Hahn and L. Sprueh, Phys. Rev. 140 (1965) A413. [8] T. F. O'Malley, L. Rosenberg and L. Spruch, Phys. Rev. 125 (1962) 1300. [9] R.J.Draehman, Phys. Rev. 138 (1965) A1582. [10] A. Dalgarno and N. Lynn, Proc. Phys. Soe. (London) A70 (1957) 223. [11] R.J. Drachman, Phys. Rev. 173 (1968) 190.
HEURISTIC
PHYSICS LETTERS
THEORY
OF
POSITRON-HELIUM
22 November 1971
SCATTERING
R. J. DRACHMAN Theoretical Studies Branch, Goddard Space Flight Center, Greenbelt, Maryland 20771, USA Received 23 September 1971 An L-dependent local potential is constructed to fit the well-known positron-hydrogen s, p and d-wave phase shifts below the rearrangement threshold. The same form of potential yields a positron-helium cross-section in agreement with a recent experimental measurement near threshold. T h e r e i s now w i d e s p r e a d i n t e r e s t in the exp e r i m e n t a l i n v e s t i g a t i o n of p o s i t r o n - h e l i u m s c a t t e r i n g and a n n i h i l a t i o n both by s w a r m e x p e r i m e n t s [1] and e n e r g y - r e s o l d e d b e a m e x p e r i m e n t s [2]. A c o n s i d e r a b l e amount of t h e o r e t i c a l work has been p e r f o r m e d [3,4], but it i s c l e a r that no definitive and r e l i a b l e p h a s e - s h i f t a n a l y s i s is yet available. On the other hand, the p o s i t r o n - h y d r o g e n e l a s t i c s c a t t e r i n g has been r a t h e r well analyzed, although the difficulty of producing monatomic hydrogen t a r g e t s has p r e c l u d e d e x p e r i m e n t a l v e r i f i c a t i o n . E s s e n t i a l l y exact s - w a v e , v e r y good p - w a v e and convincingly e x t r a p o l a t e d d-wave p h a s e - s h i f t s now exist [5-7]. A g e n e r a l effectiver a n g e f o r m u l a [8] should be adequate for L > 2, where no detailed c a l c u l a t i o n s exist. It would be d e s i r a b l e , then, for the s u c c e s s e s in hydrogen c a s e to be extended to h e l i u m even if only a p p r o x i m a t e l y ; such p r e d i c t i o n s could be useful for e x p e r i m e n t e r s and t h e o r e t i c i a n s alike. One such attempt, the 'modified a d i a b a t i c ' app r o x i m a t i o n [3, 9] has defects which we wish to confront here. In that a p p r o x i m a t i o n , one solves the p a r t i a l wave s c a t t e r i n g equation (in atomic u n i t s with e n e r g i e s in Rydbergs)
- u L + [ L ( L + 1)x -2 - k 2 ] u ~ + IV 1+ V2+(a
+l)V20]u L = 0
(1)
Here, V 1 and V 2 are first- and second-order adiabatic potentials; the latter includes all multipoles by the method of Dalgarno and Lynn [10]. The monopole potential V20 is considered to represent short-range effects that are evaluate, and the parameter a is caUed the monopole suppression parameter. For hydrogen it was found that ~ 0 gave good s-wave results at all energies
up to the p o s i t r o n i u m threshold, by c o m p a r i s o n with the ' e x a c t ' r e s u l t s [5]. P r e s u m a b l y due to a lack of sufficient v i r t u a l p o s i t r o n i u m in the adiabatic d e s c r i p t i o n , the phase shifts for L = 1 and 2 fall below t h e i r c o r r e c t v a l u e s [6, 7] as k i n c r e a s e s . More p r e c i s e l y , the e r r o r is due to a lack of g e n e r a l i t y in the form of the s h o r t - r a n g e c o r r e l a t i o n p a r t of the wave function for L > 0 i n h e r e n t in the adiabatic approximation. We p r o p o s e h e r e to c o r r e c t the defect h e u r i s t i c a l l y by allowing ~ to be L - d e p e n d e n t (~ * OeL). By choosing ot 1 = 3.49 and ot2 = 23 we can r e p r o d u c e the r e s u l t s of refs. [6] and [7]. All that is n e c e s s a r y , then, to make an i m p r o v e d p r e diction of the p o s i t r o n - h e l i u m p h a s e - s h i f t s and c r o s s - s e c t i o n is to scale the p o t e n t i a l s c o r r e c t l y . It is easily shown [3~ 11] that, if the h e l i u m ground state has the hydrogen form $ ( r l , r 2) : n - l z3 e x p [ - z ( r l + r2) ]
,
(2)
Tab|e 1 Phase shifts ~/L for e+- He scattering with 0t0 = 0, ot1 = 3.49 and 0t 2 = 23. E(eV)
?70
7/1
772
0 a 0,136 0.545 1.23 2.18 3.41 4.90 6.67 8.72 11.0
(-0.659) 0.050 0.072 0.071 0.056 0.032 0.002 -0.031 -0.066 -0.100
. . 0.003 0.010 0.024 0.038 0.055 0.072 0.088 0.103 0.114
.
. 0.0004 0.002 0.004 0.007 0.012 0.017 0.024 0.033 0.043
13.6 16.5
-0.134 -0.168
0.125 0.134
0.056 0.070
17.8
-0.183
0.137
0.077
aThe zero-energy entry is the scattering length. 187
PHYSICS L E T T E R S
Volume 37A, n u m b e r 3
section, obtained from the phase is plotted. For comparison, the r e f . [3], (in w h i c h ot L = 0 f o r a l l shown. The single experimental lies on the new curve and offers port for this heuristic analysis. between hydrogen and helium is f o r t h e p h a s e s h i f t s of t a b l e 1 to c l o s e to t h e t r u e o n e s .
POSITRON ENERGY k 2
0
0.2 [
'
0.4
0.6
0.8
1.0
1.2
I
[
I
I
I
~10" z
uJ
References
t
%
.
.
~
i
5.0
t~
10.0
i
i
i
I
i
,
i
15.0
POSITRON ENERGY IN ELECTRON VOLTS
Fig.1. Elastic p o s i t r o n - h e l i u m c r o s s - s e c t i o n in units of ?ra 2. The lower (adiabatic) curve is from ref. [3] with o~ "= 0, z = 1.5992, C = 1. The upper curve was computed from the phase shifts of table 1, using the effective range formula of ref. [8] for L > 2. The e x p e r i m e n t a l point at E = 16.5eV is from ref. [2].
then [VI(X)]H e =
2Z[Vl(ZX)]H
[V2(X)]He = 2[
V2(zx)]H
[V20(X)]He = 2[ V 2 0 ( z x ) ] H
(3)
i s t h e c o r r e c t f o r m of s c a l i n g , a n d to f i t t h e s t r e n g t h of t h e a s y m p t o t i c x - 4 p o t e n t i a l (the p o l a r i z a b i l i t y ) we c h o o s e z = 1.5992. T h e r e s u l t s a r e g i v e n i n t a b l e 1. In fig. 1. t h e t o t a l e l a s t i c s c a t t e r i n g c r o s s -
188
s h i f t s of t a b l e 1. r e s u l t s of L) are also p o i n t s h o w n [2] additional supThe analogy strong enough be considered
kc~ ~ 0 ~ _ _ .
co O e.9
5o.~
22 November 1971
[1] C.Y. Leung and D . A . L . Paul, J. Phys. B2 (1969) 1278; G. F. Lee, P. H. R. Orth and G. Jones, Phys. L e t t e r s 28A (1969) 674; S. J. Tao and T. M. Kelly, Phys. Rev. 185 (1969) 135. [2] D.G. Costello, D.E. Groce, D. F. H e r r i n g and J. W. MeGowan, p r e p r i n t (1971) ; Bull. Am. Phys. Soe. 13 (1968) 1397. Proe. VI Int. Conf. on Phys. Electron and Atomic Coll. (MIT P r e s s , Cambridge, 1969) p. 757. [3] R . J . Draehman, Phys. Rev. 144 (1966) 25. [4] N. R. Kestner, J. J o r t n e r , M.H. Cohen and S. A. Rize, Phys. Rev. 140 (1969) A56; M. Kraidy, PhD Thesis, Univ. of Western Ontario (1967) unpublished; S. K. Houston and R. J. Drachman, Phys. Rev. A 3 (1971) 1335; J. Callaway, R.W. LaBahn, R.T. Pu and W.M. Duxler, Phys. Rev. 168 (1968) 12; R. E. Montgomery and R. W. LaBahn, Can.J. Phys. 48 (1970) 1288. [5] A. K. Bhatia, A. Temkin, R.J. Drachman and H. Eiserike, Phys. Rev. A3 (1971) 1328. [6] R.L. Armstead, Phys. Rev. 171 (1968)91. [7] C.J. Kleinman, Y. Hahn and L. Sprueh, Phys. Rev. 140 (1965) A413. [8] T. F. O'Malley, L. Rosenberg and L. Spruch, Phys. Rev. 125 (1962) 1300. [9] R.J.Draehman, Phys. Rev. 138 (1965) A1582. [10] A. Dalgarno and N. Lynn, Proc. Phys. Soe. (London) A70 (1957) 223. [11] R.J. Drachman, Phys. Rev. 173 (1968) 190.