Impact Ionization in the Proton-H-Atom System: I. Theory

In a series of papers, of which this is the first, the calculation of the total cross section ${\sigma }_I\left(\epsilon \right)$ for the process ${\mathrm{H}}^{+}+\mathrm{H}\mathrm{}\left(1s\right)\mathrm{}\to 2{\mathrm{H}}^{+}\left(E^{\prime }\right)+e^{-}\left(\epsilon \right)$, for initial relative energies $E<\mathrm{}500\mathrm{}$ eV, is described and performed. Here $E^{\prime }$ is the relative energy of the final protons. This paper (I) describes the theory used, and paper II describes the calculation and results for the electronic part of the problem (wave functions and transition matrix elements). Work is being completed on the trajectory integrals which form the last part of the calculation. ${\sigma }_I\left(\epsilon \right)$ is a total cross section with respect to scattering angles, but is differential in the final electron energy. The principle upon which the calculation is based is the approximate separability of electron and heavy-particle dynamics for both initial and final states, due to the mass disparity. The theory is an extension of that developed earlier by Thorson for transitions to discrete states.