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.