Optical-potential approach to the electron-atom impact-ionization-threshold problem

The problem of the threshold law for electron-atom impact ionization is reconsidered as an analytic continuation of inelastic-scattering cross sections through the ionization threshold. The cross sections are evaluated from a distorted-wave matrix element, the final state of which describes the scattering from the Nth excited state of the target atom. The actual calculation is carried out for the $e-\mathrm{H}$ system, and a model is introduced in which the $r_{12}^{-1\mathrm{}}$ repulsion is replaced by ${(r_1+r_2)}^{-1\mathrm{}}$. This model is shown to preserve the essential properties of the problem while at the same time reducing the dimensionality of the Schrödinger equation. Nevertheless, the scattering equation is still very complex. It is dominated by the optical potential which is expanded in terms of the eigenspectrum of $\mathrm{QHQ}$. It is shown by actual calculation that the lower eigenvalues of this spectrum descend below the relevant inelastic-scattering thresholds; it follows rigorously that the optical potential contains repulsive terms. Analytical solutions of the final-state wave function are obtained with several approximations of the optical potential: (i) omission of the optical potential, (ii) inclusion of the lowest term and dominant pole term, and (iii) a closure approximation which depends on an effective energy ${\overline{\mathcal{E}}}_N$ for each threshold energy $E_N$. The threshold law in all these cases is obtained. In the closure approximation the law depends on the sign and $N$ dependence of $E_N-{\overline{\mathcal{E}}}_N$. However, the above phenomenon of eigenvalues descending below threshold suggests that $E_N-{\overline{\mathcal{E}}}_N$ is an oscillating function of $N$. In that case the derivative of the yield curve is an oscillating (but non-negative) function of the available energy $E$. A form of such a threshold law is given.