Angular interference integral for two point scatterers ($\int e^{ik\hat{\mathbf{r}}\cdot \mathbf{d}}d\Omega =4\pi \sin (kd)/(kd)$)

Taking the polar axis along the separation vector reduces the solid-angle integral to $2\pi {\int }_{-1}^1{e}^{ikdu}du$. Thus the total scattering cross-section contains the interference term $2\mathrm{Re}(c_1{c}_2^{\ast })\sin (kd)/(kd)$ as well as the two self terms. This is an outgoing-angle integral for fixed incident direction, rather than an average over the incoming beam.