Second Born approximation with the Coulomb Green's function: Electron capture from a hydrogenlike ion by a bare ion

We have applied the second Born approximation with the Coulomb Green's function to the calculation of the cross section for electron capture from a hydrogenlike ion of high atomic number $Z_T$ by a bare ion of atomic number $Z_P$ moving with a high speed $v$. With the aid of a peaking approximation, which is valid when $\frac{\hslash v}{e^2}\gg Z_P$ and $Z_T\gg Z_P$, we reduced the second Born amplitude to a one-dimensional integral which was evaluated numerically for $1s\to 1s$ capture. It was found that by using the Coulomb Green's function, rather than the free Green's function, the second Born cross section is greatly reduced (and is comparable in size to the first Born cross section) when $\frac{\hslash v}{e^2}\lesssim Z_T$.