First Born Approximation of the Single Differential Cross Section (SDCS) for Electron Impact Ionization of H(3s)

TL;DR

This paper presents a detailed theoretical calculation of the Single Differential Cross Section for electron impact ionization of hydrogen in the 3s state using the First-Born Approximation, incorporating Coulomb effects and hypergeometric functions.

Contribution

It provides a rigorous theoretical framework and numerical results for SDCS in excited hydrogen, validating the First-Born Approximation against experimental data.

Findings

Distinct peak in ionization rate around 200 eV

Gradual decrease in ionization with increasing energy

Wave function diffuse structure influences ionization variations

Abstract

This investigation is a rigorous theoretical study of the Single Differential Cross Section (SDCS) for the ionization of hydrogen in the 3s state by electron impact computed by means of the First-Born Approximation. The transition matrix has been found by means of the integral process of Bethe-Lewis. The effect of the Coulomb attractive force and the continuum of outgoing radiation have been taken into account in conjunction with the hypergeometric function, which has been used to denote the states of collision. It has hence been possible to deduce the SDCS. for a considerable range of respective incoming electron energies (100 eV to 250 eV). The results show a very distinct marked peak in the rate of ionization taking place for these energies (about 200 eV) with a gradual fall-off after with an increase in energy. The diffuse structure of the wave function in the 3s state serves to a certain extent to make variations in the rate of ionization with that of the incoming electron increased. The final results have been arrived at by means of numerical integrations done so by means of a MATLAB computer, which has yielded very accurate numbers for the cross sections. The results show up very fatally, with experimental results existing at the present date and with theoretical deductions validating the procedure of the FBA. in giving the results of the ionization of excited hydrogen atoms within the field of ionizing processes of electron-atom impact systems. The work gives a most complete basis for the further study of those systems where the processes of ionization and the complexities of the scattering process take place in excited states.