Small-angle neutron scattering at fractal objects


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Resumo

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.

Sobre autores

E. Iashina

St. Petersburg State University; Petersburg Nuclear Physics Institute

Autor responsável pela correspondência
Email: yashina_91@inbox.ru
Rússia, St. Petersburg, 198504; Gatchina, St. Petersburg, 188300

S. Grigoriev

St. Petersburg State University; Petersburg Nuclear Physics Institute

Email: yashina_91@inbox.ru
Rússia, St. Petersburg, 198504; Gatchina, St. Petersburg, 188300

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