Small-angle scattering from fat fractals

TL;DR

This paper introduces a new theoretical model based on 3D fat fractals to analyze small-angle scattering data, enabling extraction of detailed structural information about fractal structures with nonzero volume.

Contribution

The paper develops an analytical model for fat fractals in SAS, explaining power-law decays and extracting structural parameters from scattering data.

Findings

Model explains succession of power-law decays with decreasing exponents

Analytical expressions for fractal parameters and radius of gyration derived

Method allows determination of fractal dimensions and iteration levels

Abstract

A number of experimental small-angle scattering (SAS) data are characterized by a succession of power-law decays with arbitrarily decreasing values of scattering exponents. To describe such data, here we develop a new theoretical model based on 3D fat fractals (sets with fractal structure, but nonzero volume) and show how one can extract structural information about the underlying fractal structure. We calculate analytically the monodisperse and polydisperse SAS intensity (fractal form factor and structure factor) of a newly introduced model of fat fractals and study its properties in momentum space. The system is a 3D deterministic mass fractal built on an extension of the well-known Cantor fractal. The model allows us to explain a succession of power-law decays and respectively, of generalized power-law decays (superposition of maxima and minima on a power-law decay) with arbitrarily decreasing scattering exponents in the range from zero to three. We show that within the model, the present analysis allows us to obtain the edges of all the fractal regions in the momentum space, the number of fractal iteration and the fractal dimensions and scaling factors at each structural level in the fractal. We applied our model to calculate an analytical expression for the radius of gyration of the fractal. The obtained quantities characterizing the fat fractal are correlated to variation of scaling factor with the iteration number.