Ionization amplitudes in electron-hydrogen collisions

Recently Zatsarinny and Bartschat [Phys. Rev. Lett. 107, 023203 (2011)] have given an ansatz for extracting ionization amplitudes from close-coupling calculations of electron-impact ionization of atoms. They applied it with extraordinary success to a fully differential cross section of electron-helium single ionization leaving the residual ion in an $n=2$ state. By considering electron-impact ionization of atomic hydrogen we explain the origin of the ansatz and show that it forms an effective interpolation scheme for determining the amplitudes, so long as the pseudostate energy distribution is sufficiently dense.

Figure 1

Singly differential cross sections for 54.4 eV electron-impact ionization of atomic hydrogen in the S-wave model. The results are presented for six separate CCC($N$) calculations with $N=30,\cdots ,35$. The solid points arise from using Eq. (9), and the corresponding lines are obtained via the usage of Eq. (10).

Figure 2

Singly differential cross sections for 27.2 eV electron-impact ionization of atomic hydrogen, calculated with the $N_0=40$ and $l_{\max }=5$ CCC calculation of Bray [21]. Individual $l\le 5$ components are given together with the sum (bottom panel). The cubic spline interpolations over the presented solid points arising from Eq. (9) are compared with those obtained via the usage of Eq. (10). The step function estimate (as $N_0\to \infty$) [21] is also given in the bottom panel.

Figure 3

Fully differential cross sections for 27.2 eV electron-impact ionization of atomic hydrogen. The CCC calculations are the $N_0=40$ and $l_{\max }=5$ ones from Bray [21]. The original results utilizing Eq. (9) are compared with those obtained via the usage of Eq. (10). Experimental data are as presented by Berakdar et al. [22].