Relativistic second Born approximation for electron capture

Abstract

The capture of an electron from a hydrogenic ion by a fully stripped ion, having atomic numbers Z_A and Z_B respectively, is investigated using the relativistic second Born approximation. By expanding in powers of alpha Z_A and alpha Z_B and retaining the leading terms, alpha being the fine-structure constant, the authors have derived formulae for the scattering amplitude which are used to evaluate differential and total cross sections for electron capture between ground states. A closed formula for the relativistic total capture cross section, with full allowance for spin flips, is derived which tends to the non-relativistic second Born cross section first obtained by Drisco in the limit of non-relativistic but high impact energies. At ultra-high relativistic impact energies E it is found that the relativistic second Born capture cross section decays as E^-1 in accordance with the relativistic OBK approximation but with a reduced coefficient.