Approximate Cross-Sections for Inelastic Collisions of Electrons with Atoms. I. Allowed Transitions.

The threshold and large energy properties of cross-sections are combined with the energy derivative of the cross-section at its maximum. The resulting formula relates the maximum cross-section to the energy 61 at this maximum. The multipole expansion then gives a simple formula within 10 per cent of the Born approximation maxima for some illustrative transitions of hydrogen induced by electron impact. Choosing El, the approximate formula gives the momentum cutoff in the Bethe approximation. The Bethe and threshold shape is then checked against electron plus hydrogen atom Born cross-sections for allowed transitions The simple formula gives (a) the Born maximum within a factor of 2 and (b) the Born approximation for all energies within a factor of 3. Thus, when the dipole moment is known from experiment or calculation, the approximate cross-section for any allowed atomic transition can be found at all energies by several minutes' calculation.