DIFFERENTIAL CROSS SECTIONS FOR ELASTIC ELECTRON SCATTERING BY A CALCIUM ATOM AT LOW ENERGY RANGE.

In this study the distorted wave method was applied in calculation of the differential cross sections (DCS) for elastic scattering of electron by a calcium atom at electron impact energies of 10, 15, 20, and 40 eV. At lower incident energies, 10, 15, and 20 eV the present DCS results are not in good agreement with other theoretical and experimental results. However, at 40 eV the present DCS results are in good agreement with other theoretical and experimental results.


INTRODUCTION
The studies of atomic collision processes have drawn immense attention since the introductory days of quantum mechanics. This has been occasioned by the fact that information about planetary processes is still insufficient. There is also need for detailed knowledge of atomic collisions in nuclear physics. To contribute to this, studies of atomic collisions have been carried out using different targets and projectiles. We have done a study on the lower energy range of collisions of calcium atom as the target and electrons. Elastic scattering of electron by a calcium atom has been studied both experimentally and theoretically, and several results are available regarding these processes. Milisavljevic et al. (2005) used a crossed electron-atom beam experimental technique to measure differential and integral cross sections for elastic scattering of electron by a calcium atom at electron-impact energies of 10,20,40,60 and 100 eV and for a range of scattering angles (Ө) from 10 0 up to 150 0 . Raj et al. (2007) used complex optical potential method to calculate DCS for elastic scattering of electrons by a calcium atom at electron impact energies 10-500 eV. Pandya et al. (2010) used a complex optical potential to calculate DCSs for elastic electron scattering by a calcium atom at electron impact energies 10, 60 and 100 eV. The two theoretical approaches results when compared with experimental results do not agree well. As a result, in this study differential and integral cross sections for elastic electron-calcium scattering was calculated using distorted wave method in the energy range from 10 -40 eV.

THEORY
Distorted wave method Total Hamiltonian, H, for electron-calcium elastic collision is given as (Singh, 2004;Madison and Bartschat, 1996) Where Ha is the Hamiltonian for the isolated atom, T is the kinetic energy operator for the projectile electron and V is the interaction between the projectile electron and the atom. The T-matrix for the electron-target collision in the two potential approach is given by Madison and Bartschat (1996), where is the initial plane wave given as, and  and  are respectively, the initial and final wave functions of the calcium atom.
When the Born series expansion is done for  , and considering the first term of it only, When the target is one electron atom or quasi one electron atom (when we consider only one electron of the atom in the collision process) and elastic collision is being considered, then the direct matrix (T d ) and exchange transition matrix (T ex ) corresponding to the above can be separated as, and since  =  (for elastic collision process) and = .
Here is the distorted wavefunction representing the projectile electron in the initial channel and is a solution to the wave equation, where is an arbitrarily chosen potential for the distortion of the initial state projectile electron and is the initial wave vector of the projectile electron. In this study = and it has been taken as the initial state static potential of calcium atom.

Distorted waves
The distorted waves and for the projectile in the initial and final states are expanded in terms of partial waves (Singh, 2005) and where are spherical harmonics.

Cross sections
The radial distorted wave equations for initial and final states will be solved using Numerov method and the differential cross sections will be obtained using the relation, Which when integrated yields the total cross section ( ) as,

Distortion potential
Since elastic scattering was being considered, both the initial and final distortion potentials were taken as the static potential of a calcium atom in the initial state that is, where and are initial and final distortion potentials respectively. At 15 eV, there are no available measured or calculated results to compare with. Figure 2 shows the present results with exchange potential (DWM WE) and present results without the exchange potential (DWM WOE). From 0° -120° DWM WE results are higher than DWM WOE results. This shows that at lower incident energies exchange effect is significant since interaction between projectile and the target is for a longer time.

RESULTS AND DISCUSSIONS
At 20 eV, figure 3 shows that the present results for the differential cross sections are not in agreement with the experimental results of Milisavljevic et al. Conclusions At lower incident energies (10-20 eV), energies just above ionization threshold, the present DCS results are not in good agreement with measured results. This can be attributed to the fact that first order distorted wave method does not give good results at lower incident energies. At 40eV, the present DCS results are in the better agreement with experimental results of Milisavljevic et al. (2005) compared to the calculated optical potential results of Khare et al. (1985), Raj and Kumar (2007) and Pandya et al. (2010).