Small-angle scattering from the Cantor surface fractal on the plane and the Koch snowflake

TL;DR

This paper analyzes small-angle scattering from the Cantor surface fractal and Koch snowflake, developing algorithms and relations to interpret scattering data and reveal fractal structural details.

Contribution

It introduces a recurrence relation for the scattering amplitude of the Koch snowflake and shows how to extract fractal parameters from SAS data.

Findings

Scattering intensity decays as $q^{D_s-4}$ with fractal dimension $D_s$

Log-periodic behavior of $I(q)q^{4-D_s}$ in the fractal region

Correlation effects are significant for Koch snowflake but negligible for Cantor fractal

Abstract

The small-angle scattering (SAS) from the Cantor surface fractal on the plane and Koch snowflake is considered. We develop the construction algorithm for the Koch snowflake, which makes possible the recurrence relation for the scattering amplitude. The surface fractals can be decomposed into a sum of surface mass fractals for arbitrary fractal iteration, which enables various approximations for the scattering intensity. It is shown that for the Cantor fractal, one can neglect with a good accuracy the correlations between the mass fractal amplitudes, while for the Koch snowflake, these correlations are important. It is shown that nevertheless, the correlations can be build in the mass fractal amplitudes, which explains the decay of the scattering intensity $I(q)\sim q^{D_{\mathrm{s}}-4}$ with $1 < D_{\mathrm{s}} < 2$ being the fractal dimension of the perimeter. The curve $I(q)q^{4-D_{\mathrm{s}}}$ is found to be log-periodic in the fractal region with the period equal to the scaling factor of the fractal. The log-periodicity arises from the self-similarity of sizes of basic structural units rather than from correlations between their distances. A recurrence relation is obtained for the radius of gyration of Koch snowflake, which is solved in the limit of infinite iterations. The present analysis allows us to obtain additional information from SAS data, such as the edges of the fractal regions, the fractal iteration number and the scaling factor.