Small-angle scattering from Apollonian packings using Monte Carlo simulations

Analytic expressions for form factors of several simple deterministic fractals have been derived in the last years, including mass-, surface-, fat and multi-fractals. Such expressions are very useful for extracting various structural parameters such as the fractal dimensions, lower and upper limits of the fractal, iteration number or the scaling factor. However, for more complex structures such as Apollonian packings (AP), where the fractal symmetry can not be easily exploited, analytic expressions are not available and we have to resort to other methods. Here, we calculate the pair-distance distribution function and small-angle scattering intensity of 3D AP using Monte Carlo simulations, and show that the corresponding fractal dimension is D ≃ 2.46, which is very good agreement with theoretical predictions. Thus, Monte Carlo simulations can be used successfully to characterize complex deterministic systems at nano- and micro-scales.