Differential cross sections for electron transfer, excitation, and elastic scattering in keV-energy collisions between protons and He$^+$ ions

Coupled-state differential cross sections are being determined for electron transfer, excitation, and elastic scattering in collisions between keV-energy protons and He$^{+}$ ions. Integrated cross sections for these and other collisions were recently reported.\footnote{T. G. Winter, Phys. Rev. A {\bf 87}, 032704 (2013).} Some differential cross sections were previously considered with smaller Sturmian bases as well as triple-center bases\footnote{T. G. Winter, Phys. Rev. A {\bf 49}, 1767 (1994).} using an eikonal approach.\footnote{L. Wilets and S. J. Wallace, Phys. Rev. {\bf 69}, 84 (1968).} The impact parameter $\rho$ is now expressed in terms of the scattering angle for potential scattering with an impact-parameter-dependent effective nuclear charge corresponding to partial screening by an electron in the He$^{+}$ ground state. Integrating over the scattering angle, this formal transformation does yield the same total cross sections for excitation and electron transfer as when integrating directly over $\rho$. Differential cross sections obtained in this way are likely to be valid at least for elastic scattering when this is the dominant channel.