A fully differential cross section for single ionization of helium induced by 1 MeV proton impact is calculated using the parabolic convoluted quasi-Sturmian (CQS) method. In the framework of this approach the transition amplitude is extracted directly from the asymptotic behavior of the solution of an inhomogeneous Schrödinger equation for the Coulomb three-body system $(e^{-},{\mathrm{H}\mathrm{e}}^{+},p^{+})$. The driven equation is solved numerically by expanding in convolutions of quasi-Sturmians for the two-body proton-He⁺ and electron-He⁺ systems. It is found, at least in the high energy limit, that the calculated cross sections within the proposed CQS method converge quickly as the number of terms in the expansions is increased, and are in reasonable agreement with experimental data and other theoretical results.

The parabolic quasi-Sturmian approach, recently introduced for the calculation of ion–atom ionizing collisions, is adapted and applied here to the single ionization of helium induced by an intermediate-energy proton impact. Within the method, the ionization amplitude is represented as the sum of the products of the basis amplitudes associated with the asymptotic behavior of the continuum states of the two noninteracting hydrogenic subsystems $(e^{-},{\mathrm{He}}^{+})$ and $(p^{+},{\mathrm{He}}^{+})$. The $p-e$ interaction is treated as a perturbation in the Lippmann–Schwinger-type (LS) equation for the three-body system $(e^{-},{\mathrm{He}}^{+},p^{+})$. This LS equation is solved numerically using separable expansions for the proton–electron potential. We examine the convergence behavior of the transition amplitude expansion as the number of terms in the representation of the $p-e$ interaction is increased and find that, for some kinematic regimes, the convergence is poor. This difficulty, which is absent for a higher proton energy impact, is solved by varying the momentum of the auxiliary proton plane wave introduced into the basis function. Fully differential cross-sections are calculated and compared with experimental data for 75 keV protons and the results obtained with the 3C model.