Electron scattering from excited states of hydrogen: Implications for the ionization threshold law

Abstract

The elastic scattering wave function for electrons scattered from the $N$th excited state of hydrogen is the final state of the matrix element for excitation of that state. This paper deals with the solution of that problem primarily in the context of the Temkin-Poet (TP) model [A. Temkin, Phys. Rev. 126, 130 (1962); R. Poet, J. Phys. B 11, 3081 (1978)], wherein only the radial parts of the interaction are included. The relevant potential for the outer electron is dominated by the Hartree potential, ${V}_{N}^{\mathrm{H}}(r)$. In the first part of the paper, ${V}_{N}^{\mathrm{H}}(r)$ is approximated by a potential ${W}_{N}^{}(r)$, for which the scattering equation can be analytically solved. The results allow formal analytical continuation of $N$ into the continuum, so that the ionization threshold law can be deduced. Because the analytic continuation involves going from $N$ to an imaginary function of the momentum of the inner electron, the threshold law turns out to be an exponentially damped function of the available energy $E$, in qualitative accord with the result of Macek and Ihra [J. H. Macek and W. Ihra, Phys. Rev. A 55, 2024 (1997)] for the TP model. Thereafter, the scattering equation for the Hartree potential ${V}_{N}^{\mathrm{H}}(r)$ is solved numerically. The numerical aspects of these calculations have proven to be challenging and required several developments for the difficulties to be overcome. The results for ${V}_{N}^{\mathrm{H}}(r)$ show only a simple energy-dependent shift from the approximate potential ${W}_{N}^{}(r)$, which therefore does not change the analytic continuation and the form of the threshold law. It is concluded that the relevant optical potential must be included in order to compare directly with the analytic result of Macek and Ihra. The paper concludes with discussions of (a) a quantum mechanical interpretation of the result, and (b) the outlook of this approach for the complete problem.