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

Abstract

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 dRtot(θ,ϕ;θ0)dΩ is obtained. The latter represents by itself a superposition of terms related to the GISAXS specular dRspec(θ;θ0)dΩ and diffuse dRdif(θ,ϕ;θ0)dΩ patterns, respectively. Hereafter, θ is the scattering meridian angle, ϕ is the scattering azimuth angle; θ 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 Rspec(θ) 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 χ2-fit fashion has been used to determine the corresponding values of the root-mean-square roughness σ from the specular GISAXS reflectivity data.