small-angle scattering, surface fractals, mass fractals, power-law polydispersity
Publication Date
2017-06
Journal
JOURNAL OF APPLIED CRYSTALLOGRAPHY, v.50, no.3, pp.919 - 931
Publisher
WILEY-BLACKWELL
Abstract
We argue that a finite iteration of any surface fractal can be composed of mass-fractal iterations of the same
fractal dimension. Within this assertion, the scattering amplitude of surface fractal is shown to be a sum of the
amplitudes of composing mass fractals. Various approximations for the scattering intensity of surface fractal
are considered. It is shown that small-angle scattering (SAS) from a surface fractal can be explained in terms of
power-law distribution of sizes of objects composing the fractal (internal polydispersity), provided the distance
between objects is much larger than their size for each composing mass fractal. The power-law decay of the
scattering intensity I(q) ∝ qDs−6, where 2 < Ds < 3 is the surface fractal dimension of the system, is realized
as a non-coherent sum of scattering amplitudes of three-dimensional objects composing the fractal and obeying
a power-law distribution dN(r) ∝ r−dr, with Ds = −1. The distribution is continuous for random fractals
and discrete for deterministic fractals. We suggest a model of surface deterministic fractal, the surface Cantorlike
fractal, which is a sum of three-dimensional Cantor dusts at various iterations, and study its scattering
properties. The present analysis allows us to extract additional information from SAS data, such us the edges of
the fractal region, the fractal iteration number and the scaling factor.