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 $$\frac{{dR_{tot} \left( {\theta ,\phi ;\theta_{0} } \right)}}{d\Omega }$$ is obtained. The latter represents by itself a superposition of terms related to the GISAXS specular $$\frac{{dR_{spec} \left( {\theta ;\theta_{0} } \right)}}{d\Omega }$$ and diffuse $$\frac{{dR_{dif} \left( {\theta ,\phi ;\theta_{0} } \right)}}{d\Omega }$$ patterns, respectively. Hereafter, $$\theta$$ is the scattering meridian angle, $$\phi$$ is the scattering azimuth angle; $$\theta_{0}$$ 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 Rspec($$\theta_{0}$$) 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 χ2-fit fashion has been used to determine the corresponding values of the root-mean-square roughness σ from the specular GISAXS reflectivity data.