View of AN ANALYTICAL APPROACH ON DIFFERENTIAL SCATTERING (DCS) FOR ELASTIC ELECTRON SCATTERING BY A CALCIUM ATOM PRESENT ENERGY REGIONS

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ACCENT JOURNAL OF ECONOMICS ECOLOGY & ENGINEERING

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Vol.04, Issue 09, September 2019 Available Online: www.ajeee.co.in/index.php/AJEEE

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AN ANALYTICAL APPROACH ON DIFFERENTIAL SCATTERING (DCS) FOR ELASTIC ELECTRON SCATTERING BY A CALCIUM ATOM PRESENT ENERGY REGIONS

Dr. Amrendra Singh Sengar, Assistant Professor, Physics Department,Veerangna Jhalkari Bai Girls College

Near IIITM College, Morena Link Road

Abstract:- Theoretical studies of elastic electron scattering by a calcium atom have been made by employing a model complex optical potential (composed of static, exchange, polarization and absorption terms). The complex optical potential, free from any adjustable parameter, is treated exactly in a variable phase approach to yield scattering complex phase shifts and the differential scattering cross section in the intermediate energy region. The calculations were performed at electron impact energies 10, 60, and 100 eV. We report on the elastic, inelastic, and momentum transfer cross sections up to 500 eV and compare them with other results. The present method is quite simple in nature and is able to reproduce fairly well the experimental differential cross sections (DCS) and other cross sections in the present energy regions.

1. INTRODUCTION

In recent years, there have been numerous studies of elastic electron scattering by metallic atoms. Although a considerable number of papers have been devoted to these processes, they still attract a lot of attention from both experimental and theoretical fields.

Differential cross sections (DCSs) for electron scattering are very important for the explanation and understanding of the electron interactions with different atoms and molecules and for determining the dynamics of the collision processes.

Elastic electron-calcium scattering has been studied both experimentally and theoretically, and several results are available regarding this process. The results of interest for our investigation reported by different authors. In the work by Gregory and Fink, based on solving the Dirac equation, the elastic electron- calcium DCSs have been reported at energies varying from 100 eV to 2 KeV, but these calculations excluded the electron exchange and polarization effects.

The other theoretical descriptions and calculations are mainly based on the optical potential approximation in which the main problem lies in exact determination, i.e., the direct calculation of the optical potential of the system, which is a complex, non-local, energy dependent, non-spherically symmetric potential. Therefore several approximations were performed where the optical potential was represented by different potentials. Khare et al. [4] used the optical model calculation for elastic

electron and positron scattering by a calcium atom. They obtained the differential and integral elastic cross sections, the critical points, the direct and spin- flip scattering amplitudes (f and g), and the spin polarization parameters S(_), T(_), and U(_) in the energy range from 10 to 500 eV. The many-body problem was reduced to a one-body problem, and the optical potential of the system was represented by a spherically symmetric, localized real potential which consists of static, exchange, and polarization potentials.

According to this, they performed results in the static-field (SF), static-field- polarization-exchange (SFPE), and static- field-polarization-exchange-spin-orbit (SFPESO) approximations. A calculation for the elastic scattering of a slow electron has been carried out by Nikolic and Tancc. They obtained the DCS in the Hartree Fock and random phase approximation with exchange for the polarization potential. The main motivation for the work by Yuan was to examine the influence of intra-atomic relativistic effects on electron spin polarization in low-energy (0.01–20 eV) electron scattering. In order to study the importance of these effects, he used three different kinds of atomic wave functions:

Dirac Fock (DF), quasi-relativistic Hartree Fock (QRHF), and non-relativistic Hartree Fock (HF) for bounded electrons, and obtained three sets of DCS and QI results.

The momentum transfer cross sections were obtained using only the DF atomic wave function.

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2 The study by Kelemen et al. was concerned with elastic and inelastic scattering in the energy range up to 200 eV. Using the complex optical potentials they calculated DCSs for elastic electron-calcium scattering at energies below the inelastic threshold and integral elastic cross sections in the whole energy region up to 200 eV. Buckman and Clark have summarized the extensive work done on observing resonance phenomena in electron-calcium scattering.

2. THEORY

All the major interactions of electron atom scattering can be represented by a complex, energy dependent, optical potential Vopt(r, k) as

(1) where Vst(r) is the static potential obtained from the DHFS function, Vex(r, k) is the exchange potential obtained from the FEG model, Vpol (r, k) is the polarization potential model, and Vabs (r, k) is the absorption potential that takes into account the loss of flux due to all energetically possible inelastic channels.

The only parameters required for the construction of the full optical potential are the first ionization potential and the dipole polarizability of the atom under consideration. The parameters used in our calculations are shown in Table I.

After generating the full optical potential of a given electron-atom system, we treat the generated potential exactly in a partial wave analysis by solving the following set of first order coupled differential equations for the real _1 and imaginary lm_l parts of the complex phase shift function under the variable phase approach.

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Table I: Summary of experimental and theoretical works on elastic electron calcium scattering. DCS: differential cross section, CS: Cross section, SF: static field approximation, SFPE: Static-field- polarization-exchange approximation, SFPESO: static-field-polarization- exchange-spin- orbit approximation, OP:

complex optical potential, SE: Static exchange, SEP: static exchange plus polarization calculations, SEa: Static exchange plus correlation-polarization potential, SEb: static exchange plus correlation-polarization potential (the screening, spin-polarization, and relativistic effects are accounted for), DF:

Dirac Fock atomic wave function, QRHF:

Cowan quasi-relativistic Hartree Fock atomic wave function, HF: non-relativistic Hartree Fock atomic wave function, CC:

close-coupling calculation using R-matrix method, PW: partial-wave expression.

TABLE I: Parameters used in the present calculations.

(3) Where

(4) (5)

jl(kr) and _l(kr) are the usual Riccati-Bessel functions. Equations (2) and (3) are integrated up to a sufficiently large r, different for different l and k values. Thus, the final S matrix is written as

(6) and the corresponding DCSs are defined as

(3)

3

(7) where Pl(cos _) is a Legendre polynomial of order l.

3. RESULTS AND DISCUSSION

Elastic electron scattering by a calcium atom has been studied theoretically at 10, 60, and 100 eV incident energies. We have compared our calculated results with the recent experimental work of Milisavljevic et al. The structure and change of the DCS curves are given in the following figures. The figures show how the shapes of the elastic DCS curves change as a function of electron impact energy.

Fig. 1: Differential cross sections for electron-Ca scattering at 10 eV.

In Figure 1, at 10 eV, we have shown our DCS results along with the theoretical results of Raj and Kumar, who also used an optical potential model. The experimental results are also displayed for comparison. We have good agreement with the observed results. There are three minima near 30_, 78_, and 120_. These minima are also detected in the experimental results. In contrast, the theoretical results are not even in qualitative agreement with the observation.

In Figure 2 we have depicted our DCS at 60 eV along with other work. We have excellent agreement of our DCS at scattering angles up to 30_ with the experiment, which implies that our model has correctly taken into account the effect of polarization effects. Beyond 90_ both of the theoretical results are lower than the experiment, which is due to the fact that the absorption potential employed is

overestimating the loss of flux to the electronic excited states for large scattering angles.

Fig. 2: Differential cross sections for electron-Ca scattering at 60 eV.

4. CONCLUSIONS

We have calculated the elastic electron scattering by a calcium atom at intermediate energies by employing a complex optical potential Vopt (r) approach. The potential Vopt (r) was constructed using only two parameters, namely the spherical dipole polarizability (_d) and the ionization potential energy (I.P.) of the ground state of the target atom. After generating the full optical potential of the scattering system, we treat it exactly in a partial wave analysis in terms of a set of first order coupled differential equations for the real and imaginary parts of the complex phase shift functions under the variable phase approach.

The present method is quite simple in nature and robust. To our best knowledge, the experimental DCS reported by Milisavljevic et al. are the first experimental data obtained so far in this energy range. The agreement between theory and experiment was observed in the general behavior, i.e., both in the shape and absolute nature of the angular distributions of the DCSs and energy dependence. There is a reasonable good agreement between the experimental and calculated DCSs; the results are encouraging. In general the optical potential model works best for

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4 symmetrical targets, like rare gases which have closed shells. However any target with singlet S symmetry is expected to yield reasonable values. The open shell targets that have doublet or triplet spin symmetries may suffer accuracy.

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