Past exam of the theoretical physics II course of the University of Cambridge / 2014 / Paper 1 / 3 / d / Solution

Choose the polar axis along ${\mathbf{r}}_1-{\mathbf{r}}_2$. ThenUsing the angular interference integral for two point scatterers, integrating the two self terms and the interference terms givesFor $kd\ll 1$, the amplitudes add coherently and $\sigma \to 4\pi |c_1+c_2{|}^2/k^2$; when $kd\gg 1$ the angle-integrated interference is suppressed. These limits hold for fixed incoming direction; no incoming-orientation average has been silently imposed.