Small-angle scattering from generalized self-similar Vicsek fractals

An analytical approach for calculating the small-angle X-ray or neutron scattering (SAXS/SANS) from generalized self-similar Vicsek fractals (GSSVF) is presented; each fractal consists of spherical subunits. The system considered is a mass-fractal, generated iteratively from a regular 3D Vicsek fractal structure. Its fractal dimension is controllable and increases with increasing the value of the scaling factor. Small-angle scattering (SAS) intensity is determined from a set of non-interacting, randomly oriented and uniformly distributed GSSVF fractals. It is shown that in the fractal region, the curve I(q)q(D) is approximately log-periodic with the period equal to the scaling factor of fractal; here D and I(q) are the fractal dimension and the SAS intensity, respectively. In particular, the positions of deepest minima and highest maxima are log-periodic, and their number coincides with the number of fractal iterations. The log-periodicity of the scattering curves is a consequence of the self-similarity of GSSVF.