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Collision of low-energy electrons by helium and neon atoms

S. A. ELKILANY(*)

Mathematics Department, Faculty of Education, Tanta University - Kafr Elsheikh, Egypt

(ricevuto il 21 Maggio 1996; approvato il 25 Marzo 1997)

Summary. — The variational-static approximation is used to investigate the collision

of low-energy electrons by helium and neon atoms. This method considers the mean static field of the atom and also considers the short-range forces experienced by the incident particle. Model potentials are used to describe the interaction between electrons and considered atom. Exchange potential is approximated by a local effective potential with two adjustable parameters. For each impact energy, the phase shifts of the first three partial waves are obtained using a polarization potential of Buckingham type. Also, total cross-sections are calculated. Results of phase shifts and total cross-sections show good agreement with other theoretical calculations as well as experimental measurements.

PACS 34.50 – Scattering of atoms, molecules, and ions.

1. – Introduction

The scattering of positrons and electrons from noble-gas atoms has been of continuous interest to experimental and theoretical physicists. Moreover, the importance of low-energy electron (positron)-atom scattering data in the different branches of physics is well known. It is very tedious to perform elaborate theoretical calculations for the complex systems. One must seek for some tractable method to investigate the complex systems. McEachran and Stauffer [1] formulated the elastic scattering of electrons from noble gases using a polarized-orbital method with the exchange-adiabatic approximation including only the dipole part of the polarization potential for electronic systems. A model-potential method has been used by Khan et

al. [2] to evaluate the elastic scattering of electrons and positrons by helium atoms at

low incident energies. The potential contains one parameter to include the effect of short-range correlation. Also, Nakanishi and Schrader [3] extended to neon and argon the method for constructing model polarization potentials for electron and positron-atom systems, previously presented for targets containing up two electrons.

(*) Present address: Faculty of Girls, Hafr Al-Baten, P.B. No: 31991, Kingdom of Saudi Arabia.

We plan to study the scattering of low-energy electrons from noble-gas atoms using the variational-static approximation introduced by Moussa [4]. This method considers the mean static field of the atom and also considers the short-range forces experienced by the incident particle on the target atom which are expressed as correlation terms. In case of the atom has a closed shell (or sub-shell), it may be approximately considered as a core. This method is applied successfully to the elastic scattering of positrons by helium, neon and argon atoms [5-7]. In case of electron-atom scattering the intistinguishability of electrons introduces a nonlocal term, the exchange potential, into the scattering equation. Therefore, a variety of local effective exchange potentials have been proposed to model the exchange effected by a simple free-electron-gas approximation, since the original work of Slater [8].

2. – The mean static approximation

In the mean static approximation, the collision problem is expressed theoretically by the differential equation:

d2Gl(r) dr2 1

y

z

where k is the momentum of incident electron, r is the electron nucleus distance, l is the total angular momentum and U(r) is a model potential given by

U(r) 42V(r) ,

(2) where

V(r) 4Vs(r) 1Vex(r) 1Vp(r) ,

with Vs(r) is the static potential of the system, Vex(r) is the local-exchange potential,

and Vp(r) is the polarization potential.

The solution of (1) that vanishes at the origin has the symptotic form

Gl(r) Asin

(

( 0 )

₎

(3)

where d( 0 )l represents the mean static phase shift.

3. – The static potential

The static potential Vs(r) due to the nucleus and orbital electrons is obtained by

averaging over the motion of the target electron [9]:

Vs(r) 4 Zep r 2i 41

!



where Z is the nuclear charge of the target atom. C(r1, r2, R , rz) is the antisymmetrized Hartree-Fock wave function of the target and is expanded in terms of

the Slater-type: Fopm(r) 4

!

with A(o , p , i) 4C(o, p, i)[2x(p, i) ]n(p , i) 11O2_{O] 2 n(p , i) !(}1 O2_{. The values of C(o , p , i),}

x(p , i) and n(p , i) are taken from Clementi-Roetti table for atom data [10]. Defining,

for convenience,

m 4n(p, i)1n(p, j ) , z 4x(p, i)1x(p, j ) , a 4A(o, p, i) A(o, p, j ) m! , s 4z2m 2 t_{, m 4 [1O(t11)!21O(t! m) ]Oz}m 2t_,

where i, j and t are integers, the static potential can be written as

Vs(r) 4ep

!

!

!

!

{

!

}

where ep is the projectile charge, N is the number of occupied shells in the atom, and

Nopis the number of electrons in the orbital (o , p).

4. – Local-exchange potential

In electron-atom scattering, the intistinguishability of electrons introduced a nonlocal term, the exchange potential, into the scattering equation. Therefore, a variety of local exchange potentials have been proposed to model the exchange effect by a simple free-electron-gas approximation, since the work of Slater [8].

The general form of the local exchange potential in free-electron-gas approximation may be expressed as

Vex(r) 42

2

pKF(r) F(h) ,

(7)

where KF(r) is the Fermi-electron momentum of the target atom and F(h) is the

Fermi-electron distribution function defined as

KF(r) 4 [3p2N(r) ]1 O3, F(h) 4 1 2 1 1 2h2 4 h ln

N

N

where N(r) is the electronic density of the target atom as calculated from Hartree-Fock atomic wave function [10], h is the momentum of the scattering electron normalized to the Fermi momentum, i.e. h 4KOKF, and K is the momentum of the

scattering electron which differs from the initial momentum k, where

K 4k2 1 2 Vion g 1lk2 1 K 2 F, (9)

where g 42.65, l46.0 for both helium and neon, this choice is made from comparison of our results of the phase shifts with that of [1-3]; and Vion is the first ionization

5. – Polarization potential

The effects of target polarization are included through a local effective potential

Vp(r). In the present calculations, we have used a polarization potential of Buckingham

type [9]:

Vp(r) 4

s2_r

2(r2_{1 d}2)2 , (10)

s being the dipole polarizability taken to be equal to 1.384 [11] for helium atom and

2.663 [12] for neon atom. d is an adjustable parameter for the wave of each considered atom obtained by fitting our results with the compared ones.

6. – The variational-static scattering approximation

The variational-static scattering approximation considers the mean static field of the atom, and also considers the short-range forces experienced by the incident particle on the target atom, which are expressed as correlation terms. In case the atom has a closed shell (or sub-shell), it may be approximately considered as a core. This will simplify the form of the correlation terms [4].

Thus, the trial wave function used here has the form Cl(r) 4Gl(r) 1

!

n 41 N

Cnle2ar( 1 2e2r) rn 21, (11)

where C1 l, C2 l, R , CNl are N variational parameters for each orbital l of the incident electron, a may be taken as an extra variational parameter, and Gl(r) is the mean static approximation radial wave function

(

)

The variational problem starts with the variational expression:

L 4



y

z

which should be stationary with respect to the variation of Cl(r). The Kohn variational method leads to the determination of the phase dlby the formula

tan dl4 ll ( 0 ) 1 ll ( 1 ) , (13) where ll ( 0 ) 4 tan dl ( 0 ) and ll ( 1 ) 4 1 2 k j 41

!



y

z

and Cjlare obtained by solving the equations

!



y

z

The term ( 1 2e2r_{) is introduced in the trial function C}

l(r) to avoid singularities in calculation of Rjland Mijl when l c 0 .

The total cross-section given by the formula

Q 4 4 p k2

!

will depend on the number of partial waves considered. However, since dl decreases with increasing l, we may neglect terms l E3.

7. – Results and discussion

The differential equation (1) has been solved numerically using Numerov’s method with an interval small enough in order to justify the condition that Gl(r) would be of the order rl 11 _{as r K0.}

A computer program for the calculation of V(r), Rjl, Mijl, Cjland hence the values of l( 1 )

l (i , j 410, 20, 40 and l40, 1, 2) has been developed and executed for different values of the incident electron energy.

TABLE I. – Partial waves and total cross-sections for electrons elastically scattered by helium

atoms. k (a.u.) s-wave (rad.) p-wave (rad.) d-wave (rad.) Total cross-section (pa0) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 2.95431 2.82398 2.68432 2.55507 2.43508 2.32689 2.23421 2.15512 2.08625 2.01511 1.93020 0.00751 0.02173 0.04151 0.06560 0.09381 0.12402 0.15811 0.19407 0.23245 0.26249 0.29426 0.00051 0.00243 0.00547 0.00923 0.01353 0.01821 0.02349 0.02982 0.03724 0.04643 0.06301 13.93485 9.89769 8.89957 7.99112 7.17904 6.40947 5.69723 5.07325 4.55864 4.11180 3.79662

Fig. 1. – s-wave phase shifts of the elastic scattering of electrons by helium atoms. —— present work, — P — ref. [1], —3— ref. [2],!ref. [13],mref. [14] andjref. [15].

7._{1. Electron-helium. – Our results for the partial-wave phase shifts (l E2) for}

incident electron momentum ranging from 0.1 to 1.1 a.u. are listed in table I along with the elastic total cross-sections. Figure 1 contains a comparison between the behaviour of the present s-wave phase shifts for the elastic scattering of electrons by helium atoms for the energies ranging from 0.1 to 1.1 a.u. with the available results calculated by McEachran and Stauffer [1] and Khan et al. [2]. The experimental values of Williams [13], Andrick and Bitsch [14] and Newell et al. [15] are also displayed in this figure. We notice that there is an overall agreement between our results and other theoretical and experimental ones, specially in energy range (k 40.4–0.9 a.u.). Also, in fig. 2, the present p-wave phase shifts are compared with those of McEachran and Stauffer [1] and Khan et al. [2]. The values obtained lie always above those of the compared values in the interval of energy (k 40.6–1.1 a.u.) and coincide with that of Khan et al. [2] for incident energy (k 40.6 a.u.). The present d-wave phase shifts are also displayed in fig. 3 with those of Khan et al. [2]. In this figure, we notice that

Fig. 2. – p-wave phase shifts of the elastic scattering of electrons by helium atoms. —— present work, — P — ref. [1], and —3— ref. [2].

our values deviate from the compared values for incident energy ranging from

k 40.2–0.5 a.u k40.8–1.0 a.u. and coincides for k40.675 a.u. We notice that all

values of the present s-, p- and d-waves are positive that means that the mean field is attractive, thus allowing for the polarization of the atom.

7._{2. Electron-neon. – The partial-wave phase shifts (l E2) for the collision of}

electrons by neon atoms for energy ranging from 0.1 to 1.1 a.u. are tabulated in table II along with the elastic total cross-sections. Our s-wave phase shifts for the collision of electrons by neon atoms are displayed in fig. 4 along with that of Nakanishi and Schrader [3]. We notice that all values are negative that means that the mean field is repulsive. Also, there is no peak in the present values of s-wave phase shifts and does not exist in the compared results. In fig. 5, the present values of p-wave phase shifts are compared with that of Nakanishi and Schrader [3]. Results show that the phase shift changes its sign (from positive to negative), and consequently the mean field changes from attractive to repulsive at k 40.261 a.u., whereas the change of sign of the mean field appears at k 40.294 a.u. in the results of Nakanishi and Schrader [3]. A peak exists at k 40.2 a.u. in our results which is the same as the results of Nakanishi and Schrader [3]. Figure 6 contains a comparison between the behaviour of the present

d-wave phase shifts for the considered range of energy with the available results

calculated by Nakanishi and Schrader [3]. The present values of the phases increase with k and are all positive that means that the mean field is attractive. Our values of s-,

Fig. 3. – d-wave phase shifts of the elastic scattering of electrons by helium atoms. —— present work, —3— ref. [2].

TABLE II. – Partial waves and total cross-sections for electrons elastically scattered by neon

atoms. k (a.u.) s-wave (a.u.) p-wave (a.u.) d-wave (a.u.) Total cross-section (pa0) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 2 0.04241 2 0.11053 2 0.19204 2 0.28920 2 0.39124 2 0.50092 2 0.62247 2 0.73032 2 0.83824 2 0.93039 2 1.05345 0.00121 0.00195 2 0.00124 2 0.01288 2 0.03625 2 0.07591 2 0.12202 2 0.17613 2 0.23702 2 0.31246 2 0.37701 0.00051 0.00243 0.00520 0.01261 0.02429 0.04153 0.06272 0.08197 0.10723 0.13415 0.16641 0.72143 1.22091 1.62527 2.06563 2.43684 2.84992 3.29812 3.56661 3.82925 4.06362 4.29477

Fig. 4. – s-wave phase shifts of the elastic scattering of electrons by neon atoms. —— present work, —1— ref. [3].

Fig. 5. – p-wave phase shifts of the elastic scattering of electrons by neon atoms. —— present work, —1— ref. [3].

Fig. 6. – d-wave phase shifts of the elastic scattering of electrons by neon atoms. —— present work, —1— ref. [3].

Fig. 7. – Total cross-section of the elastic scattering of electrons by neon atoms. —— present work,mref. [16],%ref. [17].

The present results of the total cross-sections are displayed in fig. 7 along with the measured values of Sinapius et al. [16] and Charlton et al. [17]. Our results show close agreement with the compared values.

Encouraged by our success in helium and neon we are pressing ahead with similar calculations for the collision of low-energy electrons by argon, krypton and xenon atoms.

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