In the first part of the paper, cross sections for the electron-impact ionization of the $2s$ and the $2p$, $m=0, \pm 1$ states of atomic hydrogen have been calculated, using two forms, B. E. (i) and B. E. (ii), of the Born-exchange approximation. For the form B. E. (i), it is found that inclusion of exchange increases the cross section for the $2p(m=\pm 1)\to c$ transition, but decreases the cross sections for the $2s\to c$ and $2p(m=0\mathrm{})\mathrm{}\to c$ transitions. In the B. E. (ii) calculation, inclusion of exchange always decreases the cross sections. For the $2s\to c$ transition Prasad's Born-exchange calculations are verified. In the second part, the binary collision approximation of Vainshtein has been applied to the $2s$ and the $2p$ ionization, and it is found that the Vainshtein approximation with its proper choice of effective charge does not agree with the Born-exchange approximation. A new expression for the exchange amplitude has been derived in the Vainshtein approximation. We have concluded by comparing the results of the Born, the Born-exchange, the Born-Ochkur, and the Vainshtein's approximations for the ionization of the $1s$, $2s$, and the $2p$ states of atomic hydrogen.

• Received 1 April 1968

DOI:https://doi.org/10.1103/PhysRev.176.164