Second Born approximation for electron capture at ultrahigh relativistic impact energies

The relativistic second Born approximation is used to derive an approximate analytical expression for the scattering amplitude for electron capture by a nucleus of charge Z B e, moving with ultrahigh relativistic energy, from a target hydrogenic ion having nuclear charge Z A e. By expanding in powers of alpha Z A and alpha Z B , where alpha is the fine-structure constant, and retaining the leading terms it is shown that the second-order contribution to the capture cross section falls off as E -1 at ultrahigh relativistic impact energies E and is not negligible in comparison with the relativistic first-order OBK approximation which also has an E -1 decay.