Scattering from surface fractals in terms of composing mass fractals

TL;DR

This paper demonstrates that surface fractals can be modeled as sums of mass fractals with the same dimension, and provides methods to analyze small-angle scattering data to extract fractal properties.

Contribution

It introduces a novel approach to represent surface fractals as compositions of mass fractals and offers analytical tools for interpreting scattering data.

Findings

Surface fractals can be expressed as sums of mass fractals with identical fractal dimensions.

Scattering intensity follows a power-law decay related to the surface fractal dimension.

The model enables extraction of fractal parameters from scattering data.

Abstract

We argue that a finite iteration of any surface fractal can be composed of mass-fractal iterations of the same fractal dimension. Within this assertion, the scattering amplitude of surface fractal is shown to be a sum of the amplitudes of composing mass fractals. Various approximations for the scattering intensity of surface fractal are considered. 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), provided the distance between objects is much larger than their size for each composing mass fractal. 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 $d N(r) \propto r^{-\tau} dr$, 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.