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Orthogonality in the adiabatic-exchange electron-atom scattering approximation
Volume 26A, number 7
PHYSICS LETTERS
ORTHOGONALITY ELECTRON-ATOM
26 February 1968
IN T H E A D I A B A T I C - E X C H A N G E SCATTERING APPROXIMATION
R. W. LABAHN Department of P h y s i c s and A s t r o n o m y , Louisiana State University, Baton Rouge, Louisiana, USA Received 9 January 1968
The effect of imposing orthogonality between the scattering state wave function and atomic wave functions is examined. It is found that imposing orthogonality has little effect on the scattering cross section so long as a proper representation of the adiabatic polarization interaction is used.
The u s u al a d i a b a t i c - e x c h a n g e a p p r o x i m a t i o n f o r e l e c t r o n - a t o m s c a t t e r i n g [1,2] y i e l d s s c a t t e r i n g wave functions which a r e not o r t h o g o n a l to the bound e l e c t r o n wave functions of the t a r g e t atom. This fact h as l i t t l e effect when only s c a t t e r i n g phase shifts a r e sought f r o m the a s y m p t o t i c f o r m of the s c a t t e r i n g wave function. H o w e v e r , if the s c a t t e r i n g wave functions a r e to be u s e d for calculating matrix elements for reaction p r o c e s s e s , p r o p e r l y orthogonal s t a t e s must be used. We have e x a m i n e d the e f f e c t s upon the total cross section for electron-helium scattering when o r t h o g o n al i t y between the f r e e and bound s t a t e s is r e q u i r e d . O r t h o g o n a l i t y b etw e e n the s c a t t e r i n g state and the bound a t o m i c s t a t e s can be i m p o s e d by e i t h e r p r o j e c t i o n o p e r a t o r t e c h n i q u e s [3] o r the u s e of L a g r a n g e u n d e t e r m i n e d m u l t i p l i e r s [4]. In e i t h e r c a s e , the equation f o r the e l e c t r o n - h e l i u m s c a t t e r i n g state function (Pk is [ - V 2 - ~ -4 + 2 V c ( R )
+ 2 I~(R) - k2] q~k(R) =
(1)
= [P + f¢ls(rl[ r - ~
(Pk(rldr] ~ls(R)
(2)
w h e r e Cls is the H a r t r e e - F o c k s i n g l e - e l e c t r o n 326
P =2
f~ls(r)Vp(r)gok(r)dr.
(3)
A c o n s i s t e n t choice for the adiabatic p o l a r i z a tion potential is given by H a r t r e e - F o c k p e r t u r b a tion t h e o r y [1,2]. T h i s is = f @ls(r)~r_~2R~ x ( r ; R ) d r
(4)
w h e r e × is the f i r s t - o r d e r p e r t u r b a t i o n c o r r e c t i o n to the l s bound st at e of h e l i m due to a point c h a r g e at R. H o w e v e r , n u m e r o u s o t h e r f o r m s have been u s e d f o r V_ with one of the most p o p u l ar iJ b ei n g the Buckingham potential [5]. Vp(R) = - [d 2 + R212
(5)
w h e r e a is the p o l a r i z a b i l i t y of the t a r g e t atom and d 2 is a c u t - o f f p a r a m e t e r . A r e a s o n a b l e c h o i c e for d 2 was given by M i t t l e m a n and Watson
as [6] d 2 : [½a z~l ~
w h e r e Vc(R) is the s e l f - c o n s i s t e n t e l e c t r o n i n t e r action and Vp is the adiabatic p o l a r i z a t i o n p o t e n tial. ~Pls is the H a r t r e e - F o c k wave function f o r the bound l s state and tz is a L a g r a n g e m u l t i p l i e r . In the u s u al a d i a b a t i c - e x c h a n g e a p p r o x i m a t i o n tt is r e p l a c e d by the t e r m j~ ~ ( e l s - k 2 ) f ~ l s ( r ) q ~ k ( r ) d r
e i g e n e n e r g y f o r helium. When o r t h o g o n al i t y is i m p o s e d , the i n t e g r a l in (2) v a n i s h e s . The L a g r a n g e m u l t i p l i e r to a c c o m p l i s h this is
(6)
w h e r e Z e is the n u c l e a r c h a r g e of the t a r g e t atom. We have c l c u l a t e d the s c a t t e r i n g of e l e c t r o n s f r o m h e l i u m through eq. (1) u si n g both f o r m s (2) and (3) f o r tL and both f o r m s (4) and (5) for the p o l a r i z a t i o n potential. T o t a l c r o s s s e c t i o n s d et e r m i n e d f r o m the f i r s t t h r e e p a r t i a l wave phase shifts of the s c a t t e r i n g state function q~k a r e shown in fig. 1. The c u r v e s l a b e l e d I and II in fig. 1 c o r r e s -
Volume 26A, number 7
PHYSICS LETTERS
i-
o
5
I0
15
20
25
26 February 1968
tion potential, eqs. (4) and (5), r e s p e c t i v e l y w e r e used. Th e s c a t t e r i n g lengths a p p r o p r i a t e to t h e s e c a l c u l a t i o n s a r e 1.067 a~ f o r c a s e I I I ~ n d 0.442 ao3 f o r c a s e IV. We see that e x p l i c i t l y r e q u i r i n g o r t h o g o n al i t y b et w een the s c a t t e r i n g and bound st at e functions m a k e s the s c a t t e r i n g state function highly d e p e n dent upon the choice of the adiabatic p o l a r i z a t i o n potential. H o w e v e r , if a f o r m a l l y c o n s i s t e n t c h o i c e is made f o r this potential, as was done above by H a r t r e e - F o c k p e r t u r b a t i o n t h e o r y , then the s c a t t e r i n g c r o s s sect i o n is only slightly affected when o r t h o g o n a l i t y is s p e c i f i c a l l y imposed.
ENERGY (eV)
Fig. 1. Electron-helium total scattering cross sections. I-Bethe, II-Buekingham polarization potentials by adiabatic-exchange equation without orthogonality. III-Bethe, IV-Buckingham polarization potentials with orthogonality. pond to using the u s u a l a d i a b a t i c - e x c h a n g e e q u a tion [~ of eq. (2)] and the a l t e r n a t i v e f o r m s (4) and (5), r e s p e c t i v e l y , f o r the p o l a r i z a t i o n p o t e n tial. The s c a t t e r i n g lengths a p p r o p r i a t e to t h e s e c a l c u l a t i o n s a r e 1.135 ao3for c a s e I and 1.136 ao3 f or c a s e II. The c u r v e s l a b e l e d HI and IV in fig. 1 r e s u l t e d when orthogonality was i m p o s e d [~ of eq. (3)] and the s a m e two f o r m s f o r the p o l a r i z a -
Th e author would like to acknowledge v a l u a b l e d i s c u s s i o n s with P r o f e s s o r J o s e p k Callaway and Dr. M a r v i n H. M i t t l e m a n
RefeT'ence8 1. A. Temkin and J. C. Lamkin, Phys. Rev. 121 (1961) 788. 2. R.W. LaBahn and J. Callaway, Phys. Rev. 135 (1964} A1539. 3. M.H. Mittleman, Advances in theoretical physics, Vol. 1, ed. K. A. Brueckner (Academic Press, New York, 1965), p. 283 - 315. 4. R.J.W. Henry, Phys. Rev. 162 (1967) 56. 5. R.A. Buckingham, Proc. Roy. Soc. (London} A160 (1937) 94. 6. M.H. Mittle man and K. M. Watson, Phys. Rev. 113 (1959) 198.
327
PHYSICS LETTERS
ORTHOGONALITY ELECTRON-ATOM
26 February 1968
IN T H E A D I A B A T I C - E X C H A N G E SCATTERING APPROXIMATION
R. W. LABAHN Department of P h y s i c s and A s t r o n o m y , Louisiana State University, Baton Rouge, Louisiana, USA Received 9 January 1968
The effect of imposing orthogonality between the scattering state wave function and atomic wave functions is examined. It is found that imposing orthogonality has little effect on the scattering cross section so long as a proper representation of the adiabatic polarization interaction is used.
The u s u al a d i a b a t i c - e x c h a n g e a p p r o x i m a t i o n f o r e l e c t r o n - a t o m s c a t t e r i n g [1,2] y i e l d s s c a t t e r i n g wave functions which a r e not o r t h o g o n a l to the bound e l e c t r o n wave functions of the t a r g e t atom. This fact h as l i t t l e effect when only s c a t t e r i n g phase shifts a r e sought f r o m the a s y m p t o t i c f o r m of the s c a t t e r i n g wave function. H o w e v e r , if the s c a t t e r i n g wave functions a r e to be u s e d for calculating matrix elements for reaction p r o c e s s e s , p r o p e r l y orthogonal s t a t e s must be used. We have e x a m i n e d the e f f e c t s upon the total cross section for electron-helium scattering when o r t h o g o n al i t y between the f r e e and bound s t a t e s is r e q u i r e d . O r t h o g o n a l i t y b etw e e n the s c a t t e r i n g state and the bound a t o m i c s t a t e s can be i m p o s e d by e i t h e r p r o j e c t i o n o p e r a t o r t e c h n i q u e s [3] o r the u s e of L a g r a n g e u n d e t e r m i n e d m u l t i p l i e r s [4]. In e i t h e r c a s e , the equation f o r the e l e c t r o n - h e l i u m s c a t t e r i n g state function (Pk is [ - V 2 - ~ -4 + 2 V c ( R )
+ 2 I~(R) - k2] q~k(R) =
(1)
= [P + f¢ls(rl[ r - ~
(Pk(rldr] ~ls(R)
(2)
w h e r e Cls is the H a r t r e e - F o c k s i n g l e - e l e c t r o n 326
P =2
f~ls(r)Vp(r)gok(r)dr.
(3)
A c o n s i s t e n t choice for the adiabatic p o l a r i z a tion potential is given by H a r t r e e - F o c k p e r t u r b a tion t h e o r y [1,2]. T h i s is = f @ls(r)~r_~2R~ x ( r ; R ) d r
(4)
w h e r e × is the f i r s t - o r d e r p e r t u r b a t i o n c o r r e c t i o n to the l s bound st at e of h e l i m due to a point c h a r g e at R. H o w e v e r , n u m e r o u s o t h e r f o r m s have been u s e d f o r V_ with one of the most p o p u l ar iJ b ei n g the Buckingham potential [5]. Vp(R) = - [d 2 + R212
(5)
w h e r e a is the p o l a r i z a b i l i t y of the t a r g e t atom and d 2 is a c u t - o f f p a r a m e t e r . A r e a s o n a b l e c h o i c e for d 2 was given by M i t t l e m a n and Watson
as [6] d 2 : [½a z~l ~
w h e r e Vc(R) is the s e l f - c o n s i s t e n t e l e c t r o n i n t e r action and Vp is the adiabatic p o l a r i z a t i o n p o t e n tial. ~Pls is the H a r t r e e - F o c k wave function f o r the bound l s state and tz is a L a g r a n g e m u l t i p l i e r . In the u s u al a d i a b a t i c - e x c h a n g e a p p r o x i m a t i o n tt is r e p l a c e d by the t e r m j~ ~ ( e l s - k 2 ) f ~ l s ( r ) q ~ k ( r ) d r
e i g e n e n e r g y f o r helium. When o r t h o g o n al i t y is i m p o s e d , the i n t e g r a l in (2) v a n i s h e s . The L a g r a n g e m u l t i p l i e r to a c c o m p l i s h this is
(6)
w h e r e Z e is the n u c l e a r c h a r g e of the t a r g e t atom. We have c l c u l a t e d the s c a t t e r i n g of e l e c t r o n s f r o m h e l i u m through eq. (1) u si n g both f o r m s (2) and (3) f o r tL and both f o r m s (4) and (5) for the p o l a r i z a t i o n potential. T o t a l c r o s s s e c t i o n s d et e r m i n e d f r o m the f i r s t t h r e e p a r t i a l wave phase shifts of the s c a t t e r i n g state function q~k a r e shown in fig. 1. The c u r v e s l a b e l e d I and II in fig. 1 c o r r e s -
Volume 26A, number 7
PHYSICS LETTERS
i-
o
5
I0
15
20
25
26 February 1968
tion potential, eqs. (4) and (5), r e s p e c t i v e l y w e r e used. Th e s c a t t e r i n g lengths a p p r o p r i a t e to t h e s e c a l c u l a t i o n s a r e 1.067 a~ f o r c a s e I I I ~ n d 0.442 ao3 f o r c a s e IV. We see that e x p l i c i t l y r e q u i r i n g o r t h o g o n al i t y b et w een the s c a t t e r i n g and bound st at e functions m a k e s the s c a t t e r i n g state function highly d e p e n dent upon the choice of the adiabatic p o l a r i z a t i o n potential. H o w e v e r , if a f o r m a l l y c o n s i s t e n t c h o i c e is made f o r this potential, as was done above by H a r t r e e - F o c k p e r t u r b a t i o n t h e o r y , then the s c a t t e r i n g c r o s s sect i o n is only slightly affected when o r t h o g o n a l i t y is s p e c i f i c a l l y imposed.
ENERGY (eV)
Fig. 1. Electron-helium total scattering cross sections. I-Bethe, II-Buekingham polarization potentials by adiabatic-exchange equation without orthogonality. III-Bethe, IV-Buckingham polarization potentials with orthogonality. pond to using the u s u a l a d i a b a t i c - e x c h a n g e e q u a tion [~ of eq. (2)] and the a l t e r n a t i v e f o r m s (4) and (5), r e s p e c t i v e l y , f o r the p o l a r i z a t i o n p o t e n tial. The s c a t t e r i n g lengths a p p r o p r i a t e to t h e s e c a l c u l a t i o n s a r e 1.135 ao3for c a s e I and 1.136 ao3 f or c a s e II. The c u r v e s l a b e l e d HI and IV in fig. 1 r e s u l t e d when orthogonality was i m p o s e d [~ of eq. (3)] and the s a m e two f o r m s f o r the p o l a r i z a -
Th e author would like to acknowledge v a l u a b l e d i s c u s s i o n s with P r o f e s s o r J o s e p k Callaway and Dr. M a r v i n H. M i t t l e m a n
RefeT'ence8 1. A. Temkin and J. C. Lamkin, Phys. Rev. 121 (1961) 788. 2. R.W. LaBahn and J. Callaway, Phys. Rev. 135 (1964} A1539. 3. M.H. Mittleman, Advances in theoretical physics, Vol. 1, ed. K. A. Brueckner (Academic Press, New York, 1965), p. 283 - 315. 4. R.J.W. Henry, Phys. Rev. 162 (1967) 56. 5. R.A. Buckingham, Proc. Roy. Soc. (London} A160 (1937) 94. 6. M.H. Mittle man and K. M. Watson, Phys. Rev. 113 (1959) 198.
327