Differential cross-section for scalar wave scattering ($d\sigma /d\Omega =|f_{\infty }{|}^2$)

If a unit incident plane wave has scattered field ${\psi }_s=e^{ikR}{f}_{\infty }(\hat{\mathbf{r}},{\hat{\mathbf{r}}}_0)/R+O(R^{-2})$, the differential wave scattering cross-section isIt is the far-field limit of $R^2$ times the scattered-to-incident energy flux ratio. The common physical amplitude normalization cancels when the incident and scattered waves are in the same background medium. The formula assumes that the $1/(4\pi )$ Green-function factor, if present, is already included in the far-field pattern.

In the Born approximation for scalar wave scattering, with $n=1+\mu W$ and zero-mean stationary unit-variance $W$, define $S_W(\mathbf{q})=\int C(\mathbf{s})e^{-i\mathbf{q}\cdot \mathbf{s}}{d}^{3}s$. The leading expectation of the differential cross-section for scalar wave scattering isFor a regular volume much larger than the covariance range, the overlap is approximately $|V|$, giving ${\mu }^2{k}^4|V|S_W({\mathbf{k}}_s)/(4{\pi }^2)$. Statistical isotropy, beyond statistical homogeneity, permits the scalar argument $|{\mathbf{k}}_s|=2k\sin (\theta /2)$. If the spectrum is defined for the index fluctuation rather than $W$, it already contains ${\mu }^2$.