Deterministic fractals: extracting additional information from small-angle scattering data

TL;DR

This paper analyzes deterministic mass fractals in small-angle scattering data, revealing log-periodic patterns in the scattering curves that relate to fractal properties and enabling extraction of detailed structural information.

Contribution

It introduces a method to extract fractal parameters from scattering data by analyzing log-periodicity, including fractal iteration number and scaling factor, which was not previously possible.

Findings

Log-periodic behavior of I(q)q^D in fractal region

Relation between scattering minima/maxima and pair distribution

Fractal parameters can be derived from scattering data

Abstract

The small-angle scattering curves of deterministic mass fractals are studied and analyzed in the momentum space. In the fractal region, the curve I(q)q^D is found to be log-periodic with a good accuracy, and the period is equal to the scaling factor of the fractal. Here D and I(q) are the fractal dimension and the scattering intensity, respectively. The number of periods of this curve coincides with the number of fractal iterations. We show that the log-periodicity of I(q)q^D in the momentum space is related to the log-periodicity of the quantity g(r)r^{3-D} in the real space, where g(r) is the pair distribution function. The minima and maxima positions of the scattering intensity are estimated explicitly by relating them to the pair distance distribution in the real space. It is shown that the minima and maxima are damped with increasing polydispersity of the fractal sets; however, they remain quite pronounced even at sufficiently large values of polydispersity. A generalized self-similar Vicsek fractal with controllable fractal dimension is introduced, and its scattering properties are studied to illustrate the above findings. In contrast with the usual methods, the present analysis allows us to obtain not only the fractal dimension and the edges of the fractal region, but also the fractal iteration number, the scaling factor, and the number of structural units from which the fractal is composed.