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.

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.

We investigate theoretically the 75 keV proton-impact ionization of atomic helium. The convoluted quasi-Sturmian approach is extended to treat, on an equal footing, both the direct mechanism and the electron capture to the continuum. This is achieved by proposing an ansatz of the Green’s function of the three-body Coulomb system $(e^{-},{\mathrm{He}}^{+},p^{+})$ that is compatible with the well-known 3C correlated continuum wave function. The model that stems from this approximation, named $3\tilde{C}$, is tested numerically using parabolic Sturmian expansions. Calculations of fully differential cross sections are presented for different regimes of energy losses, namely for ejected electron energies below, nearly equal to, and above the cusp energy. Our results are compared with recent experimental measurements and other theoretical calculations. The proposed $3\tilde{C}$ model yields very encouraging results and paves the way towards a more advanced Lippmann–Schwinger approach based on the 3C model.