the scattered amplitude is:
where . using this:
then the amplitude reduce to
For Coulomb force:
which is Rutherford Scattering equation!
For electron scattering, the potential from finite size of the nucleus is
the is called the Form factor, which is due to the distribution of the charge.
the Form-factor is Fourier transform of the charge distribution alone. If reader familiarized with optics, the Form-factor is also the diffraction pattern. so, we can see, when the incoming particle wave has wavelength similar to the size of the charge distribution, then a diffraction occurred. here are some form of the Form factor.
for point charge:
for exponential :
Solid sphere :