Calculations on the e-H cross section by the correlation method using a modified potential

Volume 38A, number 5

CALCULATIONS

PHYSICS LETTERS

28 February 1972

ON T H E e - H C R O S S S E C T I O N BY THE CORRELATION METHOD USING A MODIFIED POTENTIAL M. M. FELDEN, J. C. BRAUN and M. RICHARD

Laboratoire de Physique des Milieux Ionis$s, 2 Rue de la Citadelle, 54 Nancy, France Received 11 January 1972 A modofieatJon on the e-H attractive potential is proposed and a study of its effect on the ls ~ 2s, l s --~ 2p and ionization cross-section calculations is made. Numerical results are compared with those previously obtained. In the c o r r e l a t i o n a p p r o x i m a t i o n used for the e - H c r o s s - s e c t i o n s c a l c u l a t i o n s [1], some a p p r o x i m a t i o n s a r e made to solve the SchrSdinger equation. In a p r e v i o u s p a p e r [2], we have proposed a m o d i f i c a t i o n to take into account the neglected t e r m by the a u t h o r s [1] and to obtain a solution s a t i s f y i n g the n e c e s s a r y b o u n d a r y conditions. To reduce the effect of the a t t r a c t i v e potential l / R , we i n t r o d u c e d a n u m e r i c a l s c r e e n i n g factor e 1 = ½ to the n u c l e a r charge. On the other hand, to i n c r e a s e the r e p u l s i v e potential between the incident and atomic e l e c t r o n s , we c o n s i d e r e d a second s c r e e n i n g factor E2 = ~" [The j u s t i f i c a t i o n being given in eqs. (7) and (8) of ref. [2].] However, t h e r e is a n o t h e r way to introduce the hypothesis R ~ ~ in the a t t r a c t i v e p o t e n t i a l ; it is to put: 2/

IR+~ol

~

lip

The t r a n s i t i o n amplitude r e l a t i v e to the wave function (3) is then:

Ton = 4Ngon(q) exp (-~O /2ko) q2 L ~ A

(5)

o2s

02

l/-

(2)

",. ~o5

~'~ 'l -

-

(3)

N e x p { i k o ' ( R + ~ ) ) } F(-i/k O, 1, i k o P - i k o.@) with

-i/ko

The peaking a p p r o x i m a t i o n is c o m p e n s a t e d as in ref. [2] in which the values of the factor 0 a r e done. The n u m e r i c a l c a l c u l a t i o n s a r e plotted in figs. 1-3 which r e s p e c t i v e l y r e p r e s e n t the e - H c r o s s - s e c t i o n s r e l a t i v e to t r a n s i t i o n s l s ~ 2s, l s ~ 2p and ionization. The r e s u l t s have been Obtained by u s i n g the

A solution having a s y m p t o t i c a l l y a c o r r e c t form is:

g(R,@) =

(4)

(I)

instead of I/R as in ref. [2]. With the hypothesis (1) and the modified potentials as in ref. [2] to take into consideration the neglected term, the SchrSdinger equation becomes:

(~VR1 2 + 2-1V2-p pl+k2o)g(R'P)=O

N = exp(-lr/2k o) r(1 + i / k o)

°"l""l'"'l l lll IIIIII I

2

5

10

I I iI

20

R.y.

50

Fig. 1. e-H cross-section calculations for the ls -'~2s excitation. 375

Volume 38A, number 5

PHYSICS LETTERS

28 February 1972

1.5

(b} / I

i" I

I

I

" % (b)

"

I-a I ' ~

'~

1

"

~.

(a'}

"

I I

%%

.,.

0.5

T

%

%'~%,

I.."" 05

0

']"",'"'1 1

, l I] 2

I 11111 5

10

,

I 20

] I I R.y.

50

Fig. 3. e-H cross-section calculations for ionization.

,,i,,,,f,,,,i 1 2

lJl 5iiifll

10

I 20I

I I 50I

R.y.

Fig. 2. e-H cross-section calculations for the ls -~ 2p excitation. t r a n s i t i o n a m p l i t u d e s given r e s p e c t i v e l y by eq. (5) (curves a) and ref. [2, eq. (15)] (curves a'). T h e s e s equations a r e unlike b e c a u s e the hypot h e s i s R ~ ~ (which is n e c e s s a r y so that in SchrSdinger equation, the v a r i a b l e s R and s e p a r a t e ) is i n t r o d u c e d differently. We notice that the fact to put R in the place of ~ (curves a') and ~ in the place R (curves a) in the

3?6

i n t e r a c t i o n potential incident e l e c t r o n - n u c l e u s give r i s e to a d i v e r g e n c e at low e n e r g i e s but this d i s c r e p a n c y is s m a l l and the r e s u l t s (a) and (a') a r e in good a g r e e m e n t with the e x p e r i m e n t a l r e s u l t s . The c u r v e s c o r r e s p o n d i n g to the B o r n a p p r o x i m a t i o n a r e indicated by (b). The e x p e r i m e n t a l data a r e done in ref. [2]. References

[1] L. Vainshtein, L. Presnyakov and I. Sobelman, Zh. Eksperim. i Teor. Fiz. 45 (1963) 2015, Sov. Phys. JETP 18 (1964) 1383. [2] M. M. Felden, M.A. Fe[den and J. C. Braun, Physica, to be published.