Theory of Ionization Probability Near Threshold

The behavior of the cross section for ionization by electron impact is investigated in the vicinity of threshold by means of the quantum theory of inelastic scattering. A Coulomb-modified form of the Born approximation is used to calculate the $S$-wave ionization cross sections for ${\mathrm{H}}^{+}$, ${\mathrm{He}}^{+}$, and ${\mathrm{He}}^{++}$. The electron-nuclear interaction is included in the unperturbed Hamiltonian for the problem while the interaction between the incident and atomic electrons is considered as the perturbation. It can be shown that the limiting law for the ionization of a hydrogen atom has a linear dependence on the excess incident energy. The absolute cross sections evaluated are in good agreement with experiment. Generalization of the excess energy dependence of the threshold law to the case of multiple ($n$-fold) ionization is found to yield the forms ${E_{\mathrm{ex}}}^n$ for ionization by electron impact and ${E_{\mathrm{ex}}}^{n-1\mathrm{}}$ for photoionization.