Scattering cross-section ($\sigma =R_{\mathrm{sc}}/j_{\mathrm{in}}$)

A scattering cross-section divides the scattered rate per target by the incident particle flux. Its differential form resolves that rate by outgoing direction. It has units of area for three-dimensional scattering and length for two-dimensional scattering. In two dimensions, the convention $\psi \sim e^{ikr\cos \theta }+e^{i\pi /4}f(\theta )e^{ikr}/\sqrt{kr}$ gives $d\sigma /d\theta =|f(\theta ){|}^2/k$ by comparing outgoing and incident probability currents.

The plane spanned by incoming and outgoing propagation directions when those directions are not parallel or antiparallel. Polarization perpendicular or parallel to it is useful when expressing a differential scattering cross-section. The plane is not unique at exact forward or backward scattering.