Novel theoretical approach to the GISAXS issue: the Green function formalism using the q -Eigenwaves propagating through a twofold rough-surfaced medium

To describe the 1D and 2D patterns of the grazing-incidence small-angle X-ray scattering (GISAXS) from a rough fractal surface, the novel integral equations for the amplitudes of reflected and transmitted waves are derived. To be specific, the analytical expression for the 2D total intensity distribution dR_tot( θ ,ϕ ;θ_0)/dΩ is obtained. The latter represents by itself a superposition of terms related to the GISAXS specular dR_spec( θ ;θ_0)/dΩ and diffuse dR_dif( θ ,ϕ ;θ_0)/dΩ patterns, respectively. Hereafter, θ is the scattering meridian angle, ϕ is the scattering azimuth angle; θ_0 is the angle of incidence. By using the above analytical expressions, the 1D and 2D GISAXS patterns are numerically calculated. Some new experimental measurements of the specular reflectivity curves R spec ( θ_0 ) related to the fused quartz and crystal Si(111) samples have been carried out. Based on the theoretical approach developed, a direct least-squared procedure in a χ 2 -fit fashion has been used to determine the corresponding values of the root-mean-square roughness σ from the specular GISAXS reflectivity data.