Elastic scattering by deterministic and random fractals: Self-affinity of the diffraction spectrum

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.

Scattering from generalized Cantor fractals

“…A similar method for obtaining the form factor of non-random fractals was used in [15] (see also a generalization in [32]). The zeroth approximant is an initiator with a form factor F 0 (q).…”

“…The curves corresponding to the polydisperse cases approach the monodisperse ones as the distribution width σ r tends to zero. Smoothing of intensity curve [equation (32)] increases when the width of the distribution σ r becomes larger. The most interesting effect is changing the power law exponent from the fractal dimension D in the fractal region [equation (27)] to the usual Porod exponent of four beyond this region.…”

“…One can easily obtain an explicit analytical formula for the form factor [equation (6)] of the mth approximant. A similar method for obtaining the form factor of non-random fractals was used by Schmidt & Dacai (1986) [see also a generalization given by Hamburger-Lidar (1996)]. The zeroth approximant is an initiator with a form factor F 0 (q).…”

Fractal analysis of protein potential energy landscapes

“…(8)] if the curve is subjected to the Minkowski analysis of Eq.(9). 16,17 An example is the Weierstrass-Mandelbrot function:…”

Synthesis of fractal light pulses by quasi-direct space-to-time pulse shaping

“…In optics, the first scientific contributions on this topic were addressed to the analysis of light scattered and diffracted by fractal structures usually known as diffractals [2][3][4][5]. Here, it should be recalled that self-similarity means that, within a given distribution (i.e., spatial or frequency distribution), some of its parts have the same shape as the whole set.…”