Electron-impact excitation of electric octupole transitions in positive ions: Asymptotic behavior of the sum over partial-collision strengths.

Abstract

An investigation has been made, for the case of octupole transitions, of the dependence of the partial-collision strength on the orbital angular momentum of the colliding electron. It is shown that, similar to the dipole and quadrupole transitions, the sum over partial-collision strengths is asymptotic to a geometric series of common ratio ${\mathit{E}}_{\mathit{j}}$/${\mathit{E}}_{\mathit{i}}$, where ${\mathit{E}}_{\mathit{i}}$ and ${\mathit{E}}_{\mathit{j}}$ are the initial and final energies of the colliding electron, respectively. For large incident energies (${\mathit{E}}_{\mathit{j}}$/${\mathit{E}}_{\mathit{i}}$\ensuremath{\sim}1) the convergence of the sum to the geometric series is rather slow, since the geometric-series method only starts to become valid for large values of angular momentum. This difficulty is overcome by developing an alternative method in which the approximation is made that ${\mathit{E}}_{\mathit{j}}$/${\mathit{E}}_{\mathit{i}}$=1. An analytic formula is then obtained to estimate the contribution to the total-collision strength from large values of angular momentum. Results of partial- and total-collision strengths are presented for direct electric octupole transitions in ${\mathrm{Ca}}^{+}$ and ${\mathrm{Sr}}^{+}$.