Revised scattering exponents for a power-law distribution of surface and mass fractals

TL;DR

This paper revises the understanding of scattering exponents in small-angle scattering from polydisperse surface and mass fractals, showing how power-law size distributions can alter fractal dimensions and affect scattering behavior.

Contribution

It introduces generalized formulas for scattering exponents of polydisperse fractals, extending Martin's formulas to account for size distribution effects.

Findings

Fractal dimension can change with the power-law exponent.

Scattering intensity at large momentum transfer is a sum of individual fractal intensities.

Restrictions on power-law exponents are established.

Abstract

We consider scattering exponents arising in small-angle scattering from power-law polydisperse surface and mass fractals. It is shown that a set of fractals, whose sizes are distributed according to a power-law, can change its fractal dimension when the power-law exponent is sufficiently big. As a result, the scattering exponent corresponding to this dimension appears due to the spatial correlations between positions of different fractals. For large values of the momentum transfer, the correlations do not play any role, and the resulting scattering intensity is given by a sum of intensities of all composing fractals. The restrictions imposed on the power-law exponents are found. The obtained results generalize Martin's formulas for the scattering exponents of the polydisperse fractals.