Small-angle scattering from fat fractals

A number of experimental small-angle scattering (SAS) data are characterized by a succession of power-law decays with arbitrarily decreasing values of scattering exponents. To describe such data, here we develop a new theoretical model based on 3D fat fractals (sets with fractal structure, but nonzero volume) and show how one can extract structural information about the underlying fractal structure. We calculate analytically the monodisperse and polydisperse SAS intensity (fractal form factor and structure factor) of a newly introduced model of fat fractals and study its properties in momentum space. The system is a 3D deterministic mass fractal built on an extension of the well-known Cantor fractal. The model allows us to explain a succession of power-law decays and respectively, of generalized power-law decays (superposition of maxima and minima on a power-law decay) with arbitrarily decreasing scattering exponents in the range from zero to three. We show that within the model, the present analysis allows us to obtain the edges of all the fractal regions in the momentum space, the number of fractal iteration and the fractal dimensions and scaling factors at each structural level in the fractal. We applied our model to calculate an analytical expression for the radius of gyration of the fractal. The obtained quantities characterizing the fat fractal are correlated to variation of scaling factor with the iteration number.Comment: The final publication is available at Springer via http://dx.doi.org/10.1140/epjb/e2014-41066-

Dense random packing with a power-law size distribution: the structure factor, mass-radius relation, and pair distribution function

We consider dense random packing of disks with a power-law distribution of radii and investigate their correlation properties. We study the corresponding structure factor, mass-radius relation and pair distribution function of the disk centers. A toy model of dense segments in one dimension (1d) is solved exactly. It is shown theoretically in 1d and numerically in 1d and 2d that such packing exhibits fractal properties. It is found that the exponent of the power-law distribution and the fractal dimension coincide. An approximate relation for the structure factor in arbitrary dimension is derived, which can be used as a fitting formula in small-angle scattering. The findings can be useful for understanding microstructural properties of various systems like ultra-high performance concrete, high-internal-phase ratio emulsions or biological systems.Comment: 8 pages, 8 figure