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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    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$\frac{{dR_{tot} \left( {\theta ,\phi ;\theta_{0} } \right)}}{d\Omega }$$\end{document}d   R   totθ  ,  ϕ  ;   θ  0d  Ωis obtained. The latter represents by itself a superposition of terms related to the GISAXS specular    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$\frac{{dR_{spec} \left( {\theta ;\theta_{0} } \right)}}{d\Omega }$$\end{document}d   R   specθ  ;   θ  0d  Ωand diffuse    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$\frac{{dR_{dif} \left( {\theta ,\phi ;\theta_{0} } \right)}}{d\Omega }$$\end{document}d   R   difθ  ,  ϕ  ;   θ  0d  Ωpatterns, respectively. Hereafter,    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$\theta$$\end{document}   θis the scattering meridian angle,    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$\phi$$\end{document}   ϕis the scattering azimuth angle;    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$\theta_{0}$$\end{document}    θ  0is 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 (   \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$\theta_{0}$$\end{document}    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.

scientific reports

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