A standard Born approximation example is Rutherford Scattering, that is, Coulomb scattering of a particle of charge
screened Coulomb potential
The exponential represents the screening of the nuclear charge by atomic electrons.
Without screening, the total Coulomb scattering cross section is infinite because the range of the force in infinite.
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.
, 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.