Scattering from a Mass Fractal

If we consider a mass fractal object as a distribution of mass points, the normalized correlation function y(r) is the probability of finding a mass point a distance r apart from an arbitrary mass point selected within the object. We construct a spherical shell of radius r and thickness dr around the selected point. The number of mass points enclosed in the shell is proportional to Airrd l dr in view of Equation (5.114). Since the volume of the shell is equal to Artr2 dr, the correlation function is 

The range of validity of (5.118) is R r a, where R is the overall dimension of the object ( Rg), and a is the size of the basic building block of the structure, which could be as small as an atom or a molecule. On substituting (5.118) into (5.61) the intensity of scattering for an isotropic material is given by 

Evaluation of the integral in (5.119) requires care because of the limited range of validity of (5.118). Without dwelling on the intricacy of the problem, we may simply quote the result  

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