Relativistic second Born approximation for electron capture

The capture of an electron from a hydrogenic ion by a fully stripped ion, having atomic numbers Zsub(A) and Zsub(B) respectively, is investigated using the relativistic second Born approximation. By expanding in powers of αZsub(A) and αZsub(B), α 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 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. (author)