Scattering from surface fractals in terms of internal power-law polydispersity

A. Yu. Cherny, E. M. Anitas, V. A. Osipov, A. I. Kuklin

Published 2015-07-27Version 1

It is shown that small-angle scattering (SAS) from a surface fractal can be explained in terms of power-law distribution of sizes of objects composing the fractal (internal polydispersity). The power-law decay of the scattering intensity $I(q) \propto q^{D_{\mathrm{s}}-6}$, where $2 < D_{\mathrm{s}} < 3$ is the surface fractal dimension of the system, is realized as a non-coherent sum of scattering amplitudes of three-dimensional objects composing the fractal and obeying a power-law distribution $d N(r) \propto r^{-\tau}d r$, with $D_{\mathrm{s}}=\tau-1$. The distribution is continuous for random fractals and discrete for deterministic fractals. We suggest a model of surface deterministic fractal, the surface Cantor-like fractal, which is a sum of three-dimensional Cantor dusts at various iterations, and study its scattering properties. The present analysis allows us to extract additional information from SAS data, such us the edges of the fractal region, the fractal iteration number and the scaling factor.