Theory of electron capture from a hydrogenlike ion by a bare ion: Intermediate-state contributions to the amplitude

Abstract

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 $v$ has been studied using the strong-potential Born approximation to the amplitude. Under the conditions ${Z}_{P}\ensuremath{\ll}{Z}_{T}$ and ${Z}_{P}{e}^{2}\ensuremath{\ll}\ensuremath{\hbar}v$, it is shown that, in comparison with the impulse approximation, the correct weighting of the target spectrum of intermediate states in the strong-potential Born theory significantly alters the $1s\ensuremath{\rightarrow}1s$ cross section and at the same time makes peaking approximations to the amplitude more realistic, even for low $v$. The specific cases of ${Z}_{T}=6, 10,\mathrm{and} 18$ are treated over the velocity range $\frac{{Z}_{T}}{3}\ensuremath{\lesssim}\frac{\ensuremath{\hbar}v}{{e}^{2}}\ensuremath{\lesssim}\frac{{Z}_{T}}{(0.3)}$. Instituting a one-electron model, $K$-shell capture cross sections and probabilities for protons on carbon, neon, and argon are calculated and compared with experiment. The strong-potential Born theory is seen to give a good representation of the data. Total cross sections for $1s\ensuremath{\rightarrow}1s$ capture for ${Z}_{P}={Z}_{T}=1$ are also presented.