The Effect of Exchange on the Scattering of Slow Electrons from Atoms

Abstract

Reasons are given why the Born approximation is incapable of dealing with the scattering of slow electrons. Since this approximation assumes that the sine of the phase angle $\ensuremath{\delta}$ equals $\ensuremath{\delta}$, the results computed by the Born method become completely unreliable when $\ensuremath{\delta}$ is greater than $\frac{\ensuremath{\pi}}{2}$. Exact solutions then must be obtained. Equations are set up, including exchange effects, for the best possible wave function for an electron scattered from hydrogen or helium when the complete wave function is of the separable type usually used in atomic theory. These equations are solved on the differential analyzer to find the best possible curves for the $\ensuremath{\delta}'\mathrm{s}$, for the angle distribution of scattering and for the total cross section, for this type of wave function. The check with experiment for helium is good, the maximum discrepancy in any of the $\ensuremath{\delta}'\mathrm{s}$ being only ten degrees. No data for atomic hydrogen are available. The small error introduced by the use of separable wave functions (neglect of polarization) is discussed. The conclusions are that exchange effects are not important for electron energies greater than 30 volts, and below this energy have an appreciable effect only on the angle distribution curves, and not on the cross section curves. An analytic solution of the equations, valid for any atom having closed electronic shells, is obtained for a simplified form of atomic wave function and potential. The results confirm the above conclusions, and also show that exchange is less important in scattering from heavy atoms than from light ones.