Ionization of Atomic Hydrogen by Electron Impact

The ionization problem is first formulated on the assumption of short-range potentials, particular attention being paid to the treatment of electron exchange. For the case of Coulomb potentials, the asymptotic form of the wave function is obtained for positive total energy and for zero energy. An exact integral expression is obtained for the scattering amplitude, and threshold laws are deduced. At threshold, the differential cross-section tends to a finite value for the case in which the total potential energy is negative in the asymptotic region. The total cross-section varies near threshold as a linear function of energy. Results of three experimental determinations of the cross-section are discussed. A number of calculations are made, all in the approximation of using a plane wave for the incident electron. The cross-section expression which has usually been employed in Born approximation ionization calculations is correct when the two electrons have opposite spins. An alternative form of the Born approximation for ionization, which is more closely analogous to that used for excitation, is shown to give improved results. The Born–Oppenheimer approximation gives poor results, but allowance for exchange in other approximations gives an improvement in the agreement with experiment.