Fine structure constant expansions for electron capture

Formulae for the electron capture differential cross section given as a function of the momentum change by the first and second order OBK approximations as well as the eikonal approximation, are compared using a relativistic analysis and expanding in the fine structure constant α. This analysis is extended to the total electron capture cross section. The spin-flip as well as the non-spin-flip cases are examined. It is emphasized that the second order OBK approximation must be evaluated with the inclusion of only the leading order terms in α for satisfactory results to be obtained. The non-relativistic limiting case is also discussed with particular reference to proton-helium scattering. (author)