Cross sections for attachment Wigner threshold law

Matsuzawa [45] has considered various kinds of collisions and predicts from the EFE model that, for those in which the Rydberg electron is transferred and attaches to the target, the rate constant for the transfer should equal the rate constant for attachment of free electrons of the same velocity. 

Experiments on attachment are performed at very low kinetic energy, since the resulting negative ions are fragile, which is why Rydberg electrons are very appropriate. The aim of the experiments is to measure the attachment cross section, or the attachment rate (which is essentially the product of the velocity and the cross section - see below) as a function of the incident energy. 

For a two-particle capture process, a general rule emerging from the work of Bethe and Wigner in nuclear physics, usually known as Wigner s threshold law describes the variation of the cross section close to the threshold, i.e. at very low kinetic energies, and leads to a cusp in the cross section. This law states that 

In particular, for i = 0 or 5-wave scattering, the cross section varies as 1 /E1/2 or 1 /v, which is at first sight a surprising result, since it differs from the tt 2d value, where Ad is the de Broglie wavelength. This arises because of the attractive nature of the binding force a polarisation potential yields a 1/E1/2 variation, while a Coulomb potential yields a l/E variation, so that most situations lie in between. 

The attachment rate at finite temperature is given by a thermal average  

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