Simulation and interpretation of 2D diffraction patterns from self-assembled nanostructured films at arbitrary angles of incidence: From grazing incidence (above the critical angle) to transmission perpendicular to the substrate

A method to calculate the location of all Bragg diffraction peaks from nanostructured thin films for arbitrary angles of incidence from just above the critical angle to transmission perpendicular to the film is reported. At grazing angles, the positions are calculated using the distorted wave Born approximation (DWBA), whereas for larger angles where the diffracted beams are transmitted though the substrate, the Born approximation ( BA) is used. This method has been incorporated into simulation code ( called NANOCELL) and may be used to overlay simulated spot patterns directly onto two-dimensional (2D) grazing angle of incidence small-angle X-ray scattering (GISAXS) patterns and 2D SAXS patterns. The GISAXS simulations are limited to the case where the angle of incidence is greater than the critical angle (alpha(i) \u3e alpha(c)) and the diffraction occurs above the critical angle (alpha(f) \u3e alpha(c)). For cases of surfactant self-assembled films, the limitations are not restrictive because, typically, the critical angle is around 0.2 degrees but the largest d spacings occur around 0.8 degrees 2 theta. For these materials, one finds that the DWBA predicts that the spot positions from the transmitted main beam deviate only slightly from the BA and only for diffraction peaks close the critical angle. Additional diffraction peaks from the reflected main beam are observed in GISAXS geometry but are much less intense. Using these simulations, 2D spot patterns may be used to identify space group, identify the orientation, and quantitatively fit the lattice constants for SAXS data from any angle of incidence. Characteristic patterns for 2D GISAXS and 2D low-angle transmission SAXS patterns are generated for the most common thin film structures, and as a result, GISAXS and SAXS patterns that were previously difficult to interpret are now relatively straightforward. The simulation code (NANOCELL) is written in Mathematica and is available from the author upon request