A previous method for finding very simple approximate electron-atom cross-sections for allowed transitions is extended to forbidden transitions. From the high-energy properties of cross-sections and the form of the cross-section at its maximum, an expression is obtained for the momentum cutoff in terms of O , the energy at which the maximum occurs. The multipole expansion gives a simple approximate formula for the cross-section (T, in the region 0 o , in terms of the quadrupole moment determined from experiment or calculation; the threshold shape is adopted for e < E The curve for (T(E,) is found to cross the Born curves within 10 per cent of the Born maxima for several forbidden transitions in hydrogen With the particular method used for estimating o,, in the region 0 > o, the simple (T(o) formula satisfactorily reproduces the Born values for the An = 1, but not the An = 2, transitions considered. For use in this and other approximation methods, Born and Bethe cross-sections were calculated for seven forbidden and three allowed transitions in hydrogen, namely, 1s-2s, 3s; 2s-3s, 3p, 3d, 4s; 2p-3s, 3p, 3d, 4f.
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Scanlon et al. (1961) studied this question.