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Small-angle neutron scattering at fractal objects

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Abstract

The calculation of the correlation function of an isotropic fractal particle with the finite size ξ and the dimension D is presented. It is shown that the correlation function γ(r) of volume and surface fractals is described by a generalized expression and is proportional to the Macdonald function (D–3)/2 of the second order multiplied by the power function r (D–3)/2. For volume and surface fractals, the asymptotics of the correlation function at the limit r/ξ < 1 coincides with the corresponding correlation functions of unlimited fractals. The one-dimensional correlation function G(z), which, for an isotropic fractal particle, is described by an analogous expression with a shift of the index of the Macdonald function and the exponent of the power function by 1/2, is measured using spin-echo small-angle neutron scattering. The boundary case of the transition from a volume to a surface fractal corresponding to the cubic dependence of the neutron scattering cross section Q −3 leads to an exact analytical expression for the one-dimensional correlation function G(z) = exp(−z/ξ), and the asymptotics of the correlation function in the range of fractal behavior for r/ξ < 1 is proportional to ln(ξ/r). This corresponds to a special type of self-similarity with the additive law of scaling rather than the multiplicative one, as in the case of a volume fractal.

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Authors and Affiliations

  1. St. Petersburg State University, St. Petersburg, 198504, Russia

    E. G. Iashina & S. V. Grigoriev

  2. Petersburg Nuclear Physics Institute, National Research Center “Kurchatov Institute”, Gatchina, St. Petersburg, 188300, Russia

    E. G. Iashina & S. V. Grigoriev

Authors
  1. E. G. Iashina
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  2. S. V. Grigoriev
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Correspondence to E. G. Iashina.

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Original Russian Text © E.G. Iashina, S.V. Grigoriev, 2017, published in Poverkhnost’, 2017, No. 9, pp. 5–16.

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Iashina, E.G., Grigoriev, S.V. Small-angle neutron scattering at fractal objects. J. Surf. Investig. 11, 897–907 (2017). https://doi.org/10.1134/S1027451017040334

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