Third Born approximation for electron capture at relativistic energies

Abstract

A relativistic generalisation of the third Born approximation is used to investigate the ground-state to ground-state capture of an electron from a hydrogenic ion A by a fully stripped positive ion B having atomic numbers Z_A and Z_B respectively. Formulae are obtained for the scattering amplitudes by expanding in powers of alpha Z_A and alpha Z_B and retaining the leading terms, alpha being the fine-structure constant. Closed analytical expressions for the relativistic electron capture cross sections for the cases without and with spin flip from the initial to the final states are derived making suitable approximations. In the limit of non-relativistic but high impact energies the authors obtain complete agreement with the electron capture cross section derived by Shakeshaft (1978) using the non-relativistic third Born approximation. At ultra-high relativistic impact energies E they find that the contributions to the total capture cross sections to leading order in alpha Z_A and alpha Z_B arising from the third Born correction terms fall off as E^-4 and E^-3 for the cases without and with spin flip respectively, which are both much more rapid than the E^-1 decay obtained with the second Born approximation.