For the single differential cross section (SDCS) for hydrogen ionization by electron impact ($e$-H problem), we propose a correction to the flux formula given by R. Peterkop [Theory of Ionization of Atoms by Electron Impact (Colorado Associated University Press, Boulder, 1977)]. The modification is based on an alternative way of defining the kinetic energy fraction, using Bohm's definition of velocities instead of the usual asymptotic kinematical, or geometrical, approximation. It turns out that the solution-dependent, modified energy fraction is equally related to the components of the probability flux. Compared to what is usually observed, the correction yields a finite and well-behaved SDCS value in the asymmetrical situation where one of the continuum electrons carries all the energy while the other has zero energy. We also discuss, within the $S$-wave model of the $e-\mathrm{H}$ ionization process, the continuity of the SDCS derivative at the equal energy sharing point, a property not so clearly observed in published benchmark results obtained with integral and $S$-matrix formulas with unequal final states.

• Received 1 July 2015

DOI:https://doi.org/10.1103/PhysRevA.92.062706