Symmetric eikonal approximation for electron capture at relativistic energies

A symmetric eikonal approximation is used to obtain a symmetric approximate formula in closed form for the cross section for 1s-1s electron capture from a target hydrogenic ion with nucleus A having atomic number Z A by an incident bare positive ion B having atomic number Z B moving with relativistic velocity referred to A, assuming that alpha Z A and alpha Z B <<1 where alpha is the fine structure constant. The formula for the capture cross section yields satisfactory accordance with numerical calculations carried out by Deco and Rivarola (1986) using a similar approximation but produces cross sections which are significantly less than those given by the non-symmetric eikonal approximation formula derived by Eichler (1985) except at extremely high energies.