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

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 $Z_P\ll Z_T$ and $Z_P{e}^2\ll \hslash v$, 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 $1s\to 1s$ 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 $\frac{Z_T}{3}\lesssim \frac{\hslash v}{e^2}\lesssim \frac{Z_T}{\mathrm{}\left(0.3\right)\mathrm{}}$. 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 $1s\to 1s$ capture for $Z_P=Z_T=1\mathrm{}$ are also presented.