Theory of electron capture from a hydrogenlike ion by a bare ion

Abstract

We have systematically developed a new approximation for the calculation of the cross section for electron capture from a hydrogenlike ion of large nuclear charge ${Z}_{T}e$ by a bare ion of charge ${Z}_{P}e$ moving with speed $v$. The amplitude in the wave treatment is obobtained through consistent expansion in the small parameters $\frac{{Z}_{P}}{{Z}_{T}}$ and $\frac{{Z}_{P}{e}^{2}}{\ensuremath{\hbar}v}$; however, the ratio $\frac{{Z}_{T}{e}^{2}}{\ensuremath{\hbar}v}$ is not assumed small. Following a careful analysis of the approach to the energy shell, an off-shell factor is seen to arise which does not appear in the impulse approximation. The effects of this factor on the capture amplitude are explored. Using a peaking approximation, we derive a closed-form expression for the $1s\ensuremath{\rightarrow}\mathrm{ns}$ capture amplitude which includes the effects of the off-shell factor and is accurate to order ${(\frac{{Z}_{P}{e}^{2}}{\ensuremath{\hbar}v})}^{2}$. We tabulate the peaking-approximation cross section. The derivation of the asymptotic form of the cross section when $\frac{{Z}_{P}}{{Z}_{T}}<<1$ is justified. Using a realistic one-electron model, $K$-shell capture cross sections are calculated for protons on carbon, neon, and argon. These are seen to represent the data generally better than do the impulse approximation cross sections.