2sand2pElectron Impact Excitation in Atomic Hydrogen

Abstract

A numerical calculation has been carried out to evaluate the 3\ifmmode\times\else\texttimes\fi{}3 cross-section matrix involved in the electron impact excitation of the ground state of H atom to the $2s$ and $2p$ levels. The method of solution is that of atomic eigenstates expansion. In this paper, instead of the iterative technique used by other authors, the definite integral terms in the coupled radial differential equations are eliminated through some linear transformation of the radial functions, thus avoiding iteration of these equations. The accuracy of the numerical integration is tested by satisfying the equation of reciprocity and the equation of continuity of currents with an error-to-value ratio less than 1 per 1000 on the average; and the maximum of this ratio, except for a few cases, has been kept below 5%. The results are in agreement with the results of an interative technique. To evaluate the effect of the long range and the centrifugal potential, a simple perturbation theory is developed. The six cross sections $1s\ensuremath{\rightarrow}2s$, $1s\ensuremath{\rightarrow}2p$, $1s\ensuremath{\rightarrow}1s$, $2s\ensuremath{\rightarrow}2s$, $2s\ensuremath{\rightarrow}2p$, and $2p\ensuremath{\rightarrow}2p$ are tabulated elsewhere, only the $2s\ensuremath{\rightarrow}2p$ and the $2p\ensuremath{\rightarrow}2p$ cross sections are reported here. The $2p\ensuremath{\rightarrow}2p$ cross section requires the solution of the sets of differential equations with different parities. Assuming the validity of the eigenstates expansion, it is found by comparison with the eigenstates expansion calculation that the Born approximation, despite its simplicity, gives meaningful results for low and close-to-the-threshold energies of the bombarding electrons. The effect of the exchange potentials on the cross sections is also investigated. Finally, an interesting structure of the $1s\ensuremath{\rightarrow}2s$ excitation cross section above threshold is found.