The potential energy then is

We need to calculate its Fourier Transform.

Since the potential has spherical symmetry, we can choose to be in the direction and proceed with the integral.

Since , we have . For elastic scattering, . The differential cross section is

In the last step we have used the non-relativistic formula for energy and .

The screened Coulomb potential gives a finite total cross section.
It corresponds well with the experiment Rutherford did in which
particles were scattered from atoms in a foil.
If we scatter from a bare charge where there is no screening, we can take the limit in which
.

The total cross section diverges in due to the region around zero scattering angle.

Jim Branson 2013-04-22