The scattering from generalized Cantor fractals

TL;DR

This paper analyzes the small-angle scattering from a generalized Cantor fractal with variable dimension, deriving intensity profiles, and identifying how scattering behavior transitions from fractal to Porod regimes.

Contribution

It introduces a generalized Cantor fractal with controllable dimension and provides analytical calculations of scattering intensity, including the transition from fractal to Porod behavior.

Findings

Scattering intensity shows minima and maxima superimposed on a power law decay.

The fractal dimension influences the scattering exponent.

The boundary between fractal and Porod regimes can estimate the number of particles.

Abstract

We consider a fractal with a variable fractal dimension, which is a generalization of the well known triadic Cantor set. In contrast with the usual Cantor set, the fractal dimension is controlled using a scaling factor, and can vary from zero to one in one dimension and from zero to three in three dimensions. The intensity profile of small-angle scattering from the generalized Cantor fractal in three dimensions is calculated. The system is generated by a set of iterative rules, each iteration corresponding to a certain fractal generation. Small-angle scattering is considered from monodispersive sets, which are randomly oriented and placed. The scattering intensities represent minima and maxima superimposed on a power law decay, with the exponent equal to the fractal dimension of the scatterer, but the minima and maxima are damped with increasing polydispersity of the fractal sets. It is shown that for a finite generation of the fractal, the exponent changes at sufficiently large wave vectors from the fractal dimension to four, the value given by the usual Porod law. It is shown that the number of particles of which the fractal is composed can be estimated from the value of the boundary between the fractal and Porod regions. The radius of gyration of the fractal is calculated analytically.