Small-angle neutron scattering at fractal objects

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.