Parabolic quasi-Sturmian approach to proton-impact ionization of helium

Abstract

A method for calculating ion-atom ionizing collisions is formulated and applied to single ionization of helium induced by energetic proton impact. Within the frozen-core model for the residual helium ion, the four-body problem in the exit channel is recast as an inhomogeneous Schr\"odinger equation for the Coulomb three-body system $({e}^{\ensuremath{-}},{\text{He}}^{+},{p}^{+})$. The asymptotic behavior of its solution contains the transition amplitude. We suggest to solve the driven equation in the representation of so-called parabolic convoluted quasi-Sturmian (CQS) basis functions which are constructed using quasi-Sturmians (QSs) for the $({e}^{\ensuremath{-}},{\text{He}}^{+})$ and $({p}^{+},{\text{He}}^{+})$ subsystems. By applying the corresponding Coulomb Green's function operators, these QSs are generated from orthogonal complements to square-integrable (${L}^{2}$) Sturmian functions in parabolic coordinates. In the proposed parabolic CQS approach explicit asymptotic expressions for the basis functions provide an expansion of the transition amplitude in terms of ``basis amplitudes,'' which we express analytically. The proton-electron interaction is treated as a perturbation and is approximated by a truncated Sturmian basis-set expansion. It is found that, at least in the high-energy limit, the ionization amplitude converges pretty fast as the number of terms in the separable expansion for the proton-electron potential is increased. The calculated fully differential cross sections for singly ionizing 1-MeV $p$-He collisions in several kinematical regimes in the scattering plane are found to be in reasonable agreement with experimental data and with the theoretical results obtained using the first Born approximation, the 3C model, and the wave-packet convergent close-coupling method. The theory-experiment discrepancy over the binary peak position observed by other authors remains unexplained.