Past exam of the theoretical physics II course of the University of Cambridge / 2018 / 2 / b / Solution

In the first Born approximation, with incident and outgoing wave vectors $\mathbf{k},{\mathbf{k}}^{\prime }$, put $\mathbf{q}={\mathbf{k}}^{\prime }-\mathbf{k}$. For elastic scattering,Choose the polar axis along $\mathbf{q}$. The angular integral isThe radial Dirac delta distribution then givesThus, introducing the dimensionless shell strength $g=2m{V}_{0}R/{\hslash }^2$,The forward value is the continuous limit $f_B(0)=-gR$. This is the delta-shell Born amplitude, with its angular dependence expressed through the momentum transfer.