physics pp. 249-253
Weak correlation theory of electron hydrogen atom ionization collisions
J N DAS and K CHAKRABARTI
Department of Applied Mathematics, University College of Science, 92, Acharya Prafulla Chandra Road, Calcutta 700 009, India
MS received 30 April 1996; revised 24 June 1996
Abstract. A hyperspherical partial wave method of Das to calculate cross sections for ioniz- ation of hydrogen atoms by electrons has been applied for low energies. Here effect of coupling among different partial waves is neglected.
Keywords. Electron; hydrogen; ionization; partial waves.
PACS No. 34.80
1. Introduction
During the last few years considerable effort has been made in quest of a suitable theoretical method capable of giving accurate cross section results for atomic ioniz- ation problems over wide kinematic conditions. Distorted wave Born approximation (DWBA, Jones et al Eli), three-body distorted wave Born approximation (3DWBA, Jones et al [2]), dynamic screening of three two-body Coulomb interaction method (DS3C, Berakdar and Briggs [3]), BBK theory (Brauner et al l-4]), convergent close coupling calculation (CCC, Bray et al [5]), multiple scattering theory (MST, Das and Seal [6, 7]), are some, which may be mentioned in this context. Although most of these calculations give excellent cross section results for intermediate and high energies, none of these methods works well for low energies (Konovalov [8], R6sel et al [9]). At low energies, correlation in the final three particle state is so strong that any perturbation in the potentials in the final channel leads to unacceptable cross section results under many circumstances. Keeping this in view Das El0,11], treating all the particles on an equal footing, developed a formalism, using hyperspheri- cal coordinates, for electron hydrogen atom ionization collisions. This theory gives accurate cross section results and is expected to be suitable particularly for low energies, for which a small number of partial waves in the final three-particle state are important. Here we perform a simplified version of the calculation in which effect of coupling is neglected (in view of Lin's observation (Lin [12])) to gain some initial experience for a more complicated calculation. We report here some of these prelimi- nary results and informations which will be helpful in future for more accurate calculation along this line.
249
2. Theory
The (direct) T-matrix element for ionization of a hydrogen atom by an incident electron of energy E i and momentum Pl when the two outgoing electrons have m o m e n t u m Pl and P2 respectively is given by
T a
= (W~-)I Vi[(I)i ) , (1)where @i is the initial unperturbed state corresponding to a decomposition of the total Hamiltonian H, say
n = H i + Vi, (2a)
where
Vi(r 1 , r 2 ) = 1/r12 -
1/r 2,
(2b)r l, r 2 being position vectors of the atomic and the incident electron and W~-) is the exact final three particle scattering state with converging wave b o u n d a r y conditions. In hyperspherical coordinates ' ~ - ) may be represented as (see Das [10, 113, particularly for notations)
Here p =
PR,
R = x//~l 2 + r~, P = v ~ + p2, ~ = tan -l(r2/r 1
), % = tan- l(p 2,/pl )
andffoa(7, rl,rz)
= pt,l~(~t) l,. ^ -Yt,t~(rl,
r2), (3b)Ptn~t~(~ )
being the Jacobi polynomial.The radial waves F / s satisfy the coupled set of equations (see Das [10], Lin [12])
[dd~p2 z + 1
vz(v~+l)]p2 F~ +~2~'F~,=O.
(3c)Here 2 stands for the multiplet (n, l~, 12, l, m) or the eigenvalue
2n + l~ + l 2
depending on the context and v~ = 2 + 3/2 and( / 1 l ~ o s : ~ sin:~ , , , c o s : ~ - , 2 s i n ~ l l \ /
~t~a, = qS~ + C a / / P "
(3d)When the off-diagonal matrix elements :%~.,(2 ~ 2') are neglected, the solution ofeq. (3c) simply becomes C o u l o m b waves corresponding to charge parameter :~za ---=- ~a- In this approximation which we call
weak correlation approximation
the final three-particle scattering state becomesW~o ) =
Y]2(2n)5;2i'~e - i ' I ~ ~ ( ~ ) F t ° )
q~'(Z¢o,/~1,/~z) x q~(~,f~,f2). (4) F~ °) satisfies the b o u n d a r y conditions~ p ... , p - - , O
--- sin(p -
v~n/2 +
~ l n 2p + qz~, p ~ .30,where
~/2 = arg F(v2 + 1 - i~a).
Explicitly F ] °~ is given by
F~z°)(p) = e l / 2 ~ l F ( v ~ + 1 + i~)12~'p v~+ l e - i P
× 1F 1 ( i ~ + v~ + I, 2v z + 2, 2ip)/F(2va + 2).
Asymptotically W[ -) (and also qJ}o ~) behaves as a s u p e r p o s i t i o n of a distorted plane wave a n d incoming converging waves only.
Exchange a m p l i t u d e is o b t a i n e d after the substitution ~ ~ n / 2 - ~ a n d ~i ~ 2 - Differential cross section is then o b t a i n e d as usual.
In o u r present calculation we included terms in the e x p a n s i o n (4) for n, l, lt, 12 t a k i n g values f r o m zero to certain m a x i m u m values n . . . . I . . . . 11 . . . . lzm~ a n d set m = 0. F o r fixed values of other p a r a m e t e r s , convergence is practically o b t a i n e d with nm~ = 5 or 6 in all the calculations r e p o r t e d here. So we t o o k nm~ = 6 in this calculation. We varied Im~ ~ f r o m zero to a m a x i m u m value of six and for fixed llm~ we v a r i e d / l m ~ ( = 12m~ ) from zero to seven only. With these values convergence is practically obtained in the results. H e r e we have calculated triple differential cross sections in a plane configur- ation for incident electron energies of 17.6 eV, 27.2 eV a n d 54-4 eV in the low energy range for which there are certain experimental results. Results are presented a n d discussed in the next section.
3. Results
Three sets of results for triple differential cross section are presented in figure 1 for three different incident energies corresponding to the parameters (a) E i = 54.4 eV, E 1 = 5 eV, 02 = 4°; (b) E i = 27.2 eV, E 1 = E 2 = 6.8 eV, 02 = 15°; and (c) E i = 17.6 eV, E 1 = E 2 = 2eV, 02 = 140 °. Angles are m e a s u r e d positive or negative d e p e n d i n g on if the direction is anticlockwise or clockwise with reference to the incident electron m o m e n t u m direction.
In figure 1 (a) we c o m p a r e one set of o u r results for 54.4 eV energy with the theoretical results of convergent close coupling calculation (CCC, Bray et al [5]), B B K t h e o r y (Brauner et al [4]), t h r e e - b o d y distorted wave Born a p p r o x i m a t i o n (3DWBA, Jones et al [2]) a n d with the experimental results of E h r h a r d t et al [13] as normalized b y Bray et al. T h e e x p e r i m e n t a l results of E h r h a r d t a n d associates a r e n o t absolute. F o r a fixed energy a single multiplicative factor normalizes the results for all scattering angles. Since C C C calculation gives a good total ionization cross section, Bray et al [5]
n o r m a l i z e d these e x p e r i m e n t a l d a t a for each energy b y multiplying with a single factor which gives visually the best fit with all of their theoretical results. T h u s in normalizing the d a t a of E h r h a r d t et al for 54.4 eV, they considered all the four sets of results c o r r e s p o n d i n g to 02 = 4 °, 10 °, 16 ° a n d 23 ° during the visual fit (see also figure 2 in ref.
I-5]). N o w the c o m p a r i s o n with o u r results show that o u r results are really g o o d a n d are of the right order of magnitude. Inclusion of the effect of c o u p l i n g are expected to give still better results.
In figures 1 (b) a n d 1 (c) we c o m p a r e one set of o u r theoretical results for each of energies 27.2 eV a n d 17-6 eV with the experimental results of E h r h a r d t et al [ 13], again Pramana - J. Phys., Vol. 47, No. 3, September 1996 251
t Ei=54.4¢ v
"-:'- 3-0 "\. D
(a) EI=5eV
"c- 2.0
~ ~.o
I..)
180
/ ,,,
.',->'.
\ e2=A °,'/ ¢ /,.
• / ...
'.',,X", -' , / / "
-~\~ / \ #.-"
".k \\ i .,'i';':\ l"
"~_\\ -~..~ ,,-C.L~,~. \ ",~.#.- J ~.-.~"...x /.'"
i I i s~-'l - # i I , M i i
120 60 0 -60 -120 -180
O,(deg)
,~ 4.0
.~ 3.(]
g
~n o 2.0
1.0
(b)
",
E i =27.2ev El= E2=B.BeV
*
-60 -120 -180 120 60 0
Ol(deg )
(c) Ei=17.6eV
8.0 Ec'E~2eV
8 s.c
..-
o U
2.0
180 120 60 0 -60 -120 -180
el (de9)
Figure la, b, e. Triple differential cross sections for ionization of hydrogen atoms by electrons in a plane for different scattering angles (01) and for three different sets of kinematic conditions. Theory: continuous, present theory; dashed, BBK [-4];
dash-dotted, 3DWBA [2]; dotted, CCC [5]. Experiment: points Ehrhardt et al [ 13]
suitably normalized.
suitably n o r m a l i z i n g their data, since these are not absolute. H e r e we did not p r e s e n t results of o t h e r theories such as BBK, since these are n o t of c o m p a r a b l e accuracy. T h e c o m p a r i s o n qualitatively shows that the present results are not bad. Inclusion of coupling effect m a y be expected to i m p r o v e the results further. T h e results are also in conformity with the observations of M u r r a y et al [14] regarding contributing p a r t i a l waves.
The calculations presented here were done on a 486 P C with Linux as o p e r a t i n g system. F o r a single p o i n t (i.e. for fixed energy and angles) it t o o k a b o u t half an h o u r c o m p u t e r time for a d o u b l e precision calculation.
4. Conclusion
The present w e a k c o r r e l a t i o n theory is capable of representing gross features o f low energy cross section curves. F o r very low energies, effect of coupling a m o n g different partial waves is i m p o r t a n t and cannot be simply neglected. T h e calculation will then b e c o m e m u c h m o r e complicated. Such a calculation is n o w in progress.
A c k n o w l e d g e m e n t
O n e of the a u t h o r s (KC) is grateful to C S I R for p r o v i d i n g a research fellowship.
References
[1] S Jones, D H Madison and M K Srivastava, J. Phys. B25, 1899 (1992) [2] S Jones, D H Madison, A Franz and P L Altick, Phys. Rev. A48, R22 (1993) [-3] J Berakdar and J S Briggs, Phys. Rev. Lett. 72, 3799 (1994)
[4] M Brauner, J S Briggs and H Klar, J. Phys. B22, 2265 (1989)
[5] I Bray, D A Konovalov, I E McCarthy and A T Stelbovirs, Phys. Rev. A50, R2818 (1994) [6] J N Das and S Seal, Phys. Rev. A47, 2978 (1993)
[7] J N Das and S Seal, Aust. J. Phys. 47, 49 (1994) [8] D A Konovalov, J. Phys. B27, 5551 (1994)
1-9] T Rrsel, J Rrder, L Frost, K Jung and H Ehrhardt, Phys. Rev. A46, 2539 (1992) [10] J N Das, Aust. J. Phys. 47, 743 (1994)
U Fano and A R P Rau, Atomic collisions and spectra (Academic Press, New York, 1994) sec. 10
[11] J N Das, Invited talk, Euroconference, Colmar 7-9, December 1994 1-12] C D Lin, Phys. Rev. A10, 1986 (1974)
[13] M Brauner, J S Briggs, H Klar, J T Broad, T R/Ssel, K Jung and H Ehrhardt, J. Phys. B24, 657 (1991)
H Ehrhardt (1991) (private communication)
[14] A J Murray, F H Read, N J Bowring, Phys. Rev. A49, R3162 (1994)
Pramana - J. Phys., Vol. 47, No. 3, September 1996 253
