This paper is devoted to the analysis methodology of grazing-incidence small-angle x-ray scattering from dense assemblies of islands on a surface. To interpret the experimental data for increasing coverage, (i) the distorted wave Born approximation (DWBA) formalism has been extended to include the exact profile of electronic density, i.e., the fuzziness of the interface, and (ii) the scaling of the Voronoï cell size of each particle with its size has been introduced in the scattering formalism. Multiple scattering effects in the perpendicular direction, i.e., along the emergence angle, are treated within the DWBA using the full perpendicular profile of refraction index as a reference state of the perturbation formalism and not simply the bare substrate. Thus, the average along the surface of the perturbation due to the particles or the holes in between is zero. It is shown that the concept of island form factor is still valid and includes, in a continuous integral, scattering events from upward to downward propagating waves (and vice versa) in the graded interface. Contrary to the case of multiple scattering on the bare substrate, the shape of Yoneda’s peak as well as the location and sharpness of the perpendicular interference fringes depend on the coverage for monodisperse particles, or more generally, on the embedding profile of refraction index. In the parallel direction, i.e., along the surface plane, a one dimensional model based on the paracrystal is proposed to include correlations between the size and the spacing of the particles. In this model, the interplay between coherent and incoherent scattering can lead to a hollow of scattered intensity between the specular rod and the correlation peak, as experimentally observed. This size-spacing correlation approximation goes beyond the commonly used approximations of scattering from a dense collection of particles, i.e., (i) the decoupling approximation characterized by a random disorder and by a too intense incoherent scattering and (ii) the convenient local monodisperse approximation which relies on a description of the scattering system as a set of monodisperse domains, which is not physically sound in many cases. The three approximations are compared on a physically relevant case. In a following experimental paper [R. Lazzari et al., Phys. Rev. B 76, 125412 (2007)], these models are applied to the analysis of the $\mathrm{Au}∕\mathrm{Ti}{\mathrm{O}}_2\left(110\right)$ growth mode.

• Received 15 September 2006

DOI:https://doi.org/10.1103/PhysRevB.76.125411