structurefactor

The structure factor, in physics and materials science, describes how density variations in a system scatter radiation and provides a concise way to summarize spatial correlations among scattering centers. It is central to interpreting data from X-ray, neutron, and electron scattering experiments.

The static structure factor S(k) is the Fourier transform of the pair correlation function g(r) relative to an ideal gas: S(k) = 1 + ρ ∫ [g(r) − 1] e^{i k · r} d^3r, where ρ is the number density. Equivalently, for a system of N particles at positions r_j, S(k) = (1/N) ⟨∑_{j,l} e^{i k · (r_j − r_l)}⟩. In a single-component system, this reduces to a function of the magnitude k = |k| for isotropic liquids. In mixtures, the structure factor combines correlations between different species and includes their scattering form factors.

In crystals, the structure factor is related to Bragg scattering. The unit-cell structure factor F(hkl) = ∑_j f_j e^{i 2π (h x_j + k y_j + l z_j)} determines the intensity of reflections; the observed intensity is proportional to |F(hkl)|^2. The overall static structure factor describes how well the arrangement of scatterers matches a periodic lattice or a disordered configuration.

Dynamic structure factor S(k, ω) is the time Fourier transform of density-density correlations and is probed by inelastic scattering, providing information about excitations and dynamics. The static S(k) is the ω-integrated part of S(k, ω). Together, structure factors reveal order, phase behavior, and microscopic structure in complex materials.