Elastic Scattering by Deterministic and Random Fractals: Self-Affinity of the Diffraction Spectrum

TL;DR

This paper calculates the diffraction spectrum of waves scattered from both deterministic and random fractals, revealing self-affinity and universal scaling laws related to fractal dimensions across various physical scattering phenomena.

Contribution

It provides an exact calculation of the diffraction spectrum for a broad class of fractals, including randomized versions, and establishes their self-affine nature in the large iteration limit.

Findings

Diffraction intensities follow a recursion relation.

Spectra become self-affine with a specific exponent.

Universal scaling applies across multiple scattering methods.

Abstract

The diffraction spectrum of coherent waves scattered from fractal supports is calculated exactly. The fractals considered are of the class generated iteratively by successive dilations and translations, and include generalizations of the Cantor set and Sierpinski carpet as special cases. Also randomized versions of these fractals are treated. The general result is that the diffraction intensities obey a strict recursion relation, and become self-affine in the limit of large iteration number, with a self-affinity exponent related directly to the fractal dimension of the scattering object. Applications include neutron scattering, x-rays, optical diffraction, magnetic resonance imaging, electron diffraction, and He scattering, which all display the same universal scaling.