Calculation of scattering-patterns of ordered nano- and mesoscale materials

Analytical expressions for the scattering patterns of ordered nano- and mesoscopic materials are derived and compared to measured scattering patterns. Ordered structures comprising spheres (fcc, bcc, hcp, sc, and bct), cylinders (hex and sq), lamellae (lam) and vesicles, as well as bicontinuous cubic structures (Ia3d, Pn3m, and Im3m) are considered. The expressions take into account unit cell dimensions, particle sizes and size distributions, lattice point deviations, finite domain sizes, orientational distributions, core/shell-structures as well a variety of peak shapes. The expressions allow to quantitatively describe, model and even fit measured SAXS and SANS-patterns of ordered or oriented micellar solutions, lyotropic phases, block copolymers, colloidal solutions, nanocomposites, photonic crystals, as well as mesoporous materials.