When a beam of particles or waves encounters a target, a cross section quantifies how strongly a specified interaction produces a specified outcome. It has dimensions of area, but it should not generally be interpreted as the literal geometric size of the target.

If an incident beam has flux $\Phi$—particles per unit area per unit time—and a thin target contains $N_t$ scattering centers per unit area, then the event rate for a process with total cross section $\sigma$ is approximately $$R=\Phi N_t\sigma.$$

Scattering is often directional. The differential cross section $$\frac{d\sigma}{d\Omega}$$ describes the effective cross section per unit solid angle around a particular outgoing direction. A small solid-angle element is $$d\Omega=\sin\theta,d\theta,d\phi,$$ and the total cross section is obtained by integrating over directions: $$\sigma=\int \frac{d\sigma}{d\Omega},d\Omega.$$

For example, if scattering is isotropic so that $d\sigma/d\Omega=C$ in every direction, then $$\sigma=C\int d\Omega=4\pi C.$$

Angular distributions carry more information than the total rate. A strong forward peak can indicate a long-range interaction or small momentum transfer, while resonant or interference structures can reveal internal dynamics.

Cross sections can depend strongly on incident energy, polarization, particle species and the final state being counted. They connect microscopic interaction amplitudes to experimentally measurable rates across atomic, nuclear and particle physics.