Structural characterization of chaos game fractals using small-angle scattering analysis

Area of Science:

• Materials Science
• Condensed Matter Physics
• Complex Systems

Background:

• Spatial patterns with fractal characteristics are ubiquitous in nature and generated computationally.
• Understanding the nano and microstructural properties of these patterns is crucial for various applications.
• Small-angle scattering (SAS) is a powerful technique for probing structures at the nanoscale.

Purpose of the Study:

• To investigate the application of Small-angle Scattering (SAS) for characterizing spatial patterns generated via Chaos Game Representation (CGR).
• To determine if key geometrical and fractal properties of Sierpinski Gaskets (SG) can be extracted using SAS within the CGR framework.
• To develop and validate a theoretical model for the scattering structure factor of SGs.

Main Methods:

• Application of the Small-angle Scattering (SAS) technique.
• Utilizing a simplified Debye formula to calculate the scattering structure factor in momentum space.
• Analysis of randomly oriented, non-interacting 2D Sierpinski Gaskets (SG) generated by Chaos Game Representation (CGR).
• Development of an analytical theoretical model for the SG structure factor.

Main Results:

• The SAS technique, within the CGR approach, successfully recovers essential geometrical and fractal properties of 2D Sierpinski Gaskets (SG).
• Recovered properties include overall size, scaling factor, minimal inter-unit distance, fractal dimension, and the number of constituent units.
• Numerical results are confirmed by a developed theoretical model describing the SG structure factor analytically.

Conclusions:

• The Chaos Game Representation (CGR) combined with Small-angle Scattering (SAS) provides a robust method for characterizing fractal structures.
• The developed theoretical model accurately describes the scattering behavior of Sierpinski Gaskets (SG).
• This methodology is applicable to various fractal systems, including single-scale mass fractals and complex multiscale fractals like DNA sequences.