Scattering by long-range and screened potentials

The scattering of particles by screened long- and short-range potential is discussed in the context of nonrelativistic quantum theory. Taylor's work, which established the relation between the screened and pure Coulomb scattering probabilities, is extended to other long-range potentials and to potentials made up of a Coulomb potential plus a spherically symmetric short-range part. The connection between the screened and unscreened Coulomb cross sections is then considered. It is shown, in Born approximation, that the cross section for a sharply cutoff Coulomb potential does not tend to the Rutherford cross section as the screening radius rho → infinity. On the other hand, for certain ''smooth'' screening functions, it is shown, again in Born approximation, that the screened Coulomb cross section does tend to the Rutherford cross section as rho → infinity. Finally, in the case of short-range potentials, it is shown exactly, not just in Born approximation, that the screened scattering probability and cross section both tend to the corresponding unscreened quantities as rho → infinity, whether the screening function is smooth or not