Scattering from surface fractals in terms of composing mass fractals

• Published in: Journal of applied crystallography Vol. 50; no. 3; pp. 919 - 931
• Main Authors: Cherny, A. Yu, Anitas, E. M., Osipov, V. A., Kuklin, A. I.
• Format: Journal Article
• Language: English
• Published: 5 Abbey Square, Chester, Cheshire CH1 2HU, England International Union of Crystallography 01.06.2017; Blackwell Publishing Ltd
• Subjects: Aggregates; Amplitudes; Coherent scattering; Crystallography; Decay; Dilution; Fractal analysis; Fractals; Information processing; Iterative methods; mass fractals; Mathematical analysis; Polydispersity; power‐law polydispersity; Scaling; Scaling factors; Scattering amplitude; small‐angle scattering; surface fractals
• ISSN: 1600-5767; 0021-8898; 1600-5767
• DOI: 10.1107/S1600576717005696

It is argued 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 a surface fractal is shown to be a sum of the amplitudes of the composing mass fractals. Various approximations for the scattering intensity of surface fractals are considered. It is shown that small‐angle scattering (SAS) from a surface fractal can be explained in terms of a 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) ∝ , 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−τdr, with Ds = τ − 1. The distribution is continuous for random fractals and discrete for deterministic fractals. A model of the surface deterministic fractal is suggested, the surface Cantor‐like fractal, which is a sum of three‐dimensional Cantor dusts at various iterations, and its scattering properties are studied. The present analysis allows one to extract additional information from SAS intensity for dilute aggregates of single‐scaled surface fractals, such as the fractal iteration number and the scaling factor. The scattering properties of surface fractals are studied by means of their decomposition into a sum of mass fractals.