Eikonal theory of charge exchange between arbitrary hydrogenic states of target and projectile

The approach to electron capture previously developed by Chan and Eichler is extended to describe charge exchange in fast collisions between arbitrary hydrogenic states $\mathrm{nl}$ and $n^{\prime }{l}^{\prime }$ of target and projectile. A closed formula for the cross section is derived. As an illustration, cross sections for the reactions ${\mathrm{H}}^{+}+\mathrm{H}\mathrm{}\left(2l\right)\mathrm{}\to \mathrm{H}\mathrm{}\left(3l\right)+{\mathrm{H}}^{+}$ and ${\mathrm{H}}^{+}+\mathrm{H}\mathrm{}\left(n\right)\mathrm{}\to \mathrm{H}+{\mathrm{H}}^{+}$ up to $n=10\mathrm{}$ are calculated. For the process ${\mathrm{He}}^{2+}+\mathrm{H}\mathrm{}(n=8,9\mathrm{})\mathrm{}\to {\mathrm{H}}^{+}+{\mathrm{He}}^{+}$, consistency with experimental results is obtained. Furthermore, a proposal is presented on how to apply the theory to multielectron targets by properly taking into account internal and external screening. As an example, Ar-$L$ capture by protons is considered. Finally, the post and prior versions of the theory are compared and it is argued that the prior form currently used is more adequate for electron capture. A comparison between experimental and theoretical total cross sections for $H\left(1s\right)\mathrm{}$ electrons is briefly updated.