Accurate amplitudes for electron-impact ionization

Abstract

We show that the ''two-potential'' formalism of conventional distorted-wave rearrangement theory, which is formally valid only for short-range interactions, can be used to evaluate amplitudes for the ionization of atomic hydrogen by electron impact. The triply differential cross sections calculated using this method validate earlier results obtained by extrapolating the quantum mechanical flux. Although it uses the same time-independent wave functions, this method offers significant advantages over flux extrapolation. It is more accurate, can be applied in practical calculations over a broader range of collision energies, and unlike the flux-extrapolation method, can be applied for arbitrary values of energy sharing between the ejected electrons. Since this rearrangement formalism provides the complete scattering amplitude for ionization, it can be used to calculate any differential cross section.

A complete numerical approach to electron–hydrogen collisions

“…We present TICS and spin asymmetry calculations for a broad range of energies in table 3, which compare extremely favourably with ECS calculations [23] where available. The number of L states required for convergence (L max ) increases significantly with energy.…”

“…There is no known analytic form for this condition when the integral is expanded in partial waves, and cannot be applied to (34). For the ionization of ground-state hydrogen, Baertschy et al [23] found that provided the same ρ was used for all partial waves, the phase ambiguities introduced by using uniform charges (Z 1 = Z 2 = 1) cancelled and the ionization cross sections were radially convergent. The Peterkop formulation and phase factor divergence problem have been recently addressed in a series of papers by Kadyrov et al [50,51].…”

“…We have included an explicit summation over parity in (23) and (25) to emphasize the contribution of both parity states, but as parity is determined by (−1) = (−1) l i +l 2 +L its inclusion is redundant. The total ICS is the sum of the cross sections for each spin state multiplied by the spin weighting factor (2S + 1)/4,…”

“…This proved to be inaccurate when the energy-sharing of the outgoing electrons was highly asymmetric, which at moderate collision energies is the region that has the highest contribution to the TICS. In later publications [23,24,28] they used the Peterkop [27] surface integral method, which overcame this problem and provided converged ionization cross sections. We will not repeat their derivations here, but simply give the relevant final equations, enhanced to include excited and charged hydrogenic targets.…”

“…Rescigno, Baertschy, McCurdy and coworkers [22][23][24] demonstrated that an exterior complex scaling (ECS) transformation [25,26], similar to that used for atomic resonance calculations, could be applied successfully to ionization problems. They obtained the first direct solutions for the scattering wavefunctions for e-H ionizing collisions in coordinate space, without explicit knowledge of the outer boundary conditions of their grid.…”