Grazing incidence small angle x-ray scattering from free-standing nanostructures

We develop the theory for grazing incidence small-angle x-ray scattering (GISAXS) from nanometer-sized naked islands on a flat substrate in the framework of the distorted-wave Born approximation (DWBA). The scattered wave amplitude is composed of four terms, including all combinations of scattering from the islands and reflection from the substrate. We apply this theory to x-ray measurements on Ge islands grown on Si(111), and show that we can determine the full triangular symmetry of these islands. The results also show that the DWBA must be used for smooth substrates near the angle of total external reflection. We finally discuss the advantages of GISAXS as compared to transmission small angle x-ray scattering for determining the symmetry of nanostructures.