On the Bethe approximation

It is shown that in the limit of infinite impact energies the Coulomb-Bethe total collision cross section tends to twice the value given by the Born approximation. The discrepancy can be removed by introducing a lower cut-off in the partial-wave expansion in order to exclude close encounters for which the Coulomb-Bethe approximation is invalid. The approximation introduced by Bethe for electron impact excitation of a neutral target has been reformulated. The results generalised Bethe approximation is valid at all energies and involves a more precise definition of the cut-off than that suggested by Bethe. It also gives cross sections of correct form for transitions with vanishing oscillator strength.