= Born amplitude for a Yukawa potential
{c}
For $V(r)=Ae^{-\kappa r}/r$, with real $A$ and $\kappa>0$, the <first Born approximation> gives the <scattering amplitude>
$$
f_B(q)=-\frac{m}{2\pi\hbar^2}\int_{\mathbb R^3}e^{-i\mathbf q\cdot\mathbf r}V(r)\,d^3r
=-\frac{2mA}{\hbar^2(q^2+\kappa^2)}.
$$
Indeed, 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 $\kappa\to0^+$ 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>.
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