Multiple scattering

Multiple scattering occurs when a disturbance encounters more than one scatterer, or revisits scatterers after earlier deflections. Each object is driven by both the original incident wave and fields produced by other objects. Solving the coupled responses therefore differs from adding independent single-scattering intensities.

When wave phases remain correlated, alternative paths must be added as complex amplitudes before squaring, retaining their interference. If suitable randomization makes those cross terms negligible, transport may instead be approximated by a sequence of random deflections; the relevant mean free path compares the typical distance between encounters with sample size. A thick cloudy sample can transmit diffuse light even when its direct beam is weak. Under appropriate coherent conditions, repeated wave scattering also produces weak localization or Anderson localization, which cannot be inferred from independent path probabilities alone.

Multiple Coulomb scattering is the accumulated deflection of a charged particle passing through matter, mainly through repeated Rutherford scattering by atomic nuclei. Many small deflections broaden a beam or make a reconstructed particle track wander. A roughly Gaussian distribution describes the central angular core in a useful regime, while rarer large-angle encounters produce tails.^[ref]

The angular width increases with traversed material and decreases as particle momentum increases. It is commonly parameterized using material thickness relative to the radiation length. In a particle detector, adding measurement layers can improve position sampling while also increasing Multiple Coulomb scattering; track precision therefore depends on material as well as sensor accuracy. A width projected onto one measurement plane must be distinguished from the full spatial deflection angle.

For $d\sigma /d\Omega =K/[p^2{\sin }^4(\theta /2)]$ between angular cutoffs, the leading total cross section is $16\pi K/(p^2{\theta }_{\min }^2)$. Multiply its mean count ${\rho }_{n}x\sigma$ by the one-scatter second moment $2{\theta }_{\min }^2\ln ({\theta }_{\max }/{\theta }_{\min })$ to obtain the displayed total angular variance. A single-plane projected variance is half as large. Comparing the scattering angle to a magnetic bend explains the approximately momentum-independent scattering contribution to relative track uncertainty.