The structural characterization of deterministic mass fractals at nano- and microscales is presented in this chapter using two complementary techniques in both reciprocal and real spaces. In the former case, fractal and geometrical features are obtained from the small-angle scattering (SAS) (neutrons, X-rays, light) spectrum in the reciprocal space. The lacunarity technique is considered to extract structural properties and differentiate textures of fractals in real space. We present and discuss various types of mass fractals, such as thin and fat fractals, as well as fractals generated with the Chaos game representation (CGR). We show how the main structural properties of the fractals, such as the fractal dimension, the iteration number, the scaling factor, the overall size of the fractal, and the size of the basic units of the fractal, can be extracted by using SAS and lacunarity techniques.
Part of the book: Chaos Theory
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 mono- and multi-fractal properties. We provide some useful relations between structural parameters of ACA that can be obtained experimentally from SAS.
Part of the book: Small Angle Scattering and Diffraction