Excitation of Helium by Electrons and Protons

Abstract

The first part of this paper deals with generalized oscillator strengths $f\left(K\right)\mathrm{}$. On the basis of previous work of Lassettre, we obtain a suitable form for the power-series representations of the $f\left(K\right)\mathrm{}$ for transitions in which an electron jumps from a $1s$ to an $\mathrm{ns}$, $\mathrm{np}$, or $\mathrm{nd}$ orbital of an atom. We apply the method to give analytical representations of $f\left(K\right)\mathrm{}$ for the transitions $1{}_{}{}^1{}_{}{}^{}S\to 2{}_{}{}^1{}_{}{}^{}P$ and $1{}_{}{}^1{}_{}{}^{}S\to 3{}_{}{}^1{}_{}{}^{}P$ in helium. In the second part of this paper, we study the dependence of the "total" excitation cross sections ${\sigma }_E$ on the velocity $v$ of the incident electron or proton for various dipole-allowed and dipole-forbidden transitions. By expanding the theoretical ${\sigma }_E$ (direct and interference) in a series in inverse powers of $v^2$, we find a qualitative interpretation of the variation with $v$ of the experimental ${\sigma }_E$ of Moustafa and de Heer (electrons) and of Van den Bos, Winter, and de Heer (protons) for the transitions $1{}_{}{}^1{}_{}{}^{}S\to n{}_{}{}^1{}_{}{}^{}S, n{}_{}{}^1{}_{}{}^{}P, \mathrm{and} n{}_{}{}^1{}_{}{}^{}D$ in helium, with $n=3, 4, 5, \mathrm{and} 6$. From our analytical formula for $f\left(K\right)\mathrm{}$ in the transitions $1{}_{}{}^1{}_{}{}^{}S\to 2{}_{}{}^1{}_{}{}^{}P, 3{}_{}{}^1{}_{}{}^{}P$ in helium, we calculate the corresponding ${\sigma }_E$, and compare these values with some other theoretical and experimental ones. The influence of interference between direct and exchange scattering is taken into account by using the Ochkur approximation. It is shown that the Ochkur approximation can be obtained in a simple way from the Bethe and Mott formulas. In the final part of this paper we study the nature of $f\left(K\right)\mathrm{}$, for complex $K$, for ionization of atoms, and find the positions of the singularities (poles).