Classical Impulse Approximation for Inelastic Electron-Atom Collisions

Analytic expressions for the ionization and excitation cross sections of atoms by electrons are derived using the classical impulse approximation, i.e., by considering only the Coulomb interaction between the incident electron and one bound electron. The results obtained are slightly simpler and more self-consistent than those obtained in an earlier calculation by Gryzinski. The cross sections are found to be roughly as good as those obtained by the Born approximation except in the high-energy limit. The apparent superiority of Gryzinski's theory to quantum approximations arises from a subsidiary approximation made in averaging the cross section over the initial angular distribution rather than from the kinematic description of the bound electrons or the nature of the impulse approximation itself. The Coulomb cross section for transfer of energy $\Delta E$ between two particles of equal mass $m$, charge $e$, initial kinetic energies $E_1$ and $E_2$, relative velocity $V$, with an isotropic initial angular distribution is found to be $\frac{\mathrm{Vd}\sigma }{d\left(\Delta E\right)}=2^{\frac{1}{2}}\pi e^4{\left|\Delta E\right|}^{-2\mathrm{}}{\left(m{E}_1{E}_2\right)}^{\frac{-1\mathrm{}}{2}}\left(\frac{{\mathcal{E}}^{\frac{1}{2}}+\frac{4}{3}{\mathcal{E}}^{\frac{3}{2}}}{\left|\Delta E\right|}\right)$ where $\mathcal{E}$ is the smallest of the four initial and final kinetic energies. For single ionization this cross section is found to increase as the 3/2 power of the excess energy above threshold, reach a maximum at about 2½ times the threshold energy, and decrease as $E^{-1\mathrm{}}$ at high energies. For hydrogenic atoms in any state the cross section goes to 5/3 the classical Thomson ionization cross section in the high-energy limit.