Small-Angle Scattering Analysis of Fractals Generated by Additive Cellular Automata

Abstract

Structural analysis of fractals generated using one-dimensional additive cellular automata (ACA) is presented in this chapter. ACA is a dynamical system that evolves in discrete steps and generates two-dimensional self-similar structures. We investigate the structure of M-state ACA Rule 90 and Rule 150 using small-angle scattering (SAS; X-rays, neutrons, light) technique and multi-fractal analysis. We show how the scattering data from ACA can provide information about the overall size of the system, the number of total units, the number of rows, the size of the basic fractal units, the scaling factor, and the fractal dimension. In this case, when a particular row number reproduces a complete structure of the fractals, we can also obtain the fractal iteration number. We show that subsets of different states of M-state ACA can manifest both monoand multi-fractal properties. We provide some useful relations between structural parameters of ACA that can be obtained experimentally from SAS.