Scattering from Fractals

Abstract

AbstractThis chapter presents scattering from matters with fractal structures which exhibit characteristics such as self-similarity: the structures in real space are identical at all scales or approximately the same at different scales and fractal dimensions: the structures are characterized by the fractal dimensions greater than their topological dimensions. The mass fractal structure is characterized by the mass fractal dimension Dm (1 < Dm < 3), the lower cut-off length r0, the characteristic length of the structural units which build up the mass fractal structure, and the upper cut-off length Rg,MF, up to which the self-similarity persist with respect to the spatial distribution of the center of mass of the structural units with r0. The surface fractal structure is characterized by the surface fractal dimension Ds (2 < Ds < 3), the lower cut-off length l1, and the upper cut-off length l2, in between the lengths of which the self-similar rough surface of the scattering matters persists. The scattering formulae are derived for the mass fractal structure and surface fractal structure.KeywordsSelf-similarityMass fractal structureSurface fractal structureMass fractal dimension and surface fractal dimensionLower and upper cut-off lengthLower and upper cut-off wave numberStructure factor for mass fractal and surface fractal structure