  We consider a fractal with a variable fractal dimension, which is a
generalization of the well known triadic Cantor set. In contrast with the usual
Cantor set, the fractal dimension is controlled using a scaling factor, and can
vary from zero to one in one dimension and from zero to three in three
dimensions. The intensity profile of small-angle scattering from the
generalized Cantor fractal in three dimensions is calculated. The system is
generated by a set of iterative rules, each iteration corresponding to a
certain fractal generation. Small-angle scattering is considered from
monodispersive sets, which are randomly oriented and placed. The scattering
intensities represent minima and maxima superimposed on a power law decay, with
the exponent equal to the fractal dimension of the scatterer, but the minima
and maxima are damped with increasing polydispersity of the fractal sets. It is
shown that for a finite generation of the fractal, the exponent changes at
sufficiently large wave vectors from the fractal dimension to four, the value
given by the usual Porod law. It is shown that the number of particles of which
the fractal is composed can be estimated from the value of the boundary between
the fractal and Porod regions. The radius of gyration of the fractal is
calculated analytically.
