Quantum defect theory VII. Analysis of resonance structures

Abstract

The scattering matrix for electron collisions with positive ions may be expressed in terms of a matrix R which varies slowly as a function of the energy. In the neighbourhood of resonances the scattering matrix varies rapidly. Resonances associated with a single closed channel, or with a group of degenerate closed channels, are analysed using a method similar to that published by Gailitis in 1963. It is shown that the poles of the scattering matrix are given by a Rydberg formula containing a complex quantum defect. The case of non-degenerate closed channels is also discussed. For electron collisions with highly ionized atoms the R matrix may be calculated using perturbation theory. It is shown that, in this limit, the resonant and non-resonant contributions to excitation rates are of comparable order of magnitude.