Scattering curves of ordered mesoscopic materials.

Analytical expressions for the scattering functions of ordered mesoscopic materials are derived and compared to experimentally determined scattering curves. Ordered structures comprising spheres (fcc, bcc, hcp, sc), cylinders (hex, sq), and lamellar structures are considered. The expressions take into account particle size distributions and lattice point deviations, domain size, core/shell structures, as well as peak shapes varying analytically between Lorentzian and Gaussian functions. The expressions allow one to quantitatively describe high-resolution synchrotron small-angle X-ray (SAXS) and neutron scattering (SANS) curves from lipid and block copolymer lyotropic phases, core/shell nanoparticle superstructures, ordered nanocomposites, and ordered mesoporous materials. In addition, the diffuse out-of-plane scattering of grazing incidence GISAXS and GISANS experiments of laterally ordered thin films can be quantitatively analyzed.