Born amplitude for a Yukawa potential

For $V(r)=A{e}^{-\kappa r}/r$, with real $A$ and $\kappa >0$, the first Born approximation gives the scattering amplitudeIndeed, orienting the polar axis along the momentum transfer gives the angular integral $4\pi \sin (qr)/(qr)$, and the remaining integral is ${\int }_0^{\infty }{e}^{-\kappa r}\sin (qr)dr=q/(q^2+{\kappa }^2)$. The differential scattering cross-section is $|f_B(q){|}^2$.

The limit $\kappa \to 0^{+}$ at fixed nonzero $q$ produces the Rutherford differential cross-section for a Coulomb potential. The unscreened interaction has long range, so this limiting calculation by itself does not establish the validity of a naive Born approximation to the full Rutherford scattering state. The exact Rutherford scattering amplitude has an additional phase, while its modulus agrees with the Rutherford differential cross-section.