Non-orthogonality of atomic wavefunctions in electron capture at relativistic energies

A term g 1 which corrects for the lack of orthogonality between the atomic wavefunctions of the initial and final states occurring in the OBK scattering amplitude g OBK for electron capture from a hydrogenic ion by an incident nucleus, is evaluated using relativistic kinematics and Dirac atomic functions. A non-relativistic treatment using the same approach provides a one-dimensional integral for g 1 which yields good accordance with the non-relativistic calculations of McCarroll (1961) for electron capture from ground-state H atoms by protons having 0.5 and 1 MeV impact energies. Simple analytical expressions for g 1 and the capture cross sections are derived at relativistic as well as high but non-relativistic impact energies E. It is found that for small atomic numbers g 1 becomes negligible in comparison with g OBK before E is sufficiently high for relativistic effects to be significant, but that for large atomic numbers g 1 remains important at high energies. Moreover, in the ultra-high energy limit g 1 produces an E -1 (1n E) 2 decay of the capture cross sections which is more gradual than the E -1 decay of the relativistic OBK cross section.