Scaling functions for the finite-size effect in fractal aggregates

An exact scaling function for the finite-sized fractal aggregates sharply bounded by a sphere of radius R has been established by using the convolution square of the shape function of aggregates and the inhomogeneity function, which is introduced to take into account the presence of inhomogeneity in fractal aggregates. The scaling function for an inhomogeneous aggregate is mainly determined by the geometric shape of the aggregate but is also dependent upon the degree of inhomogeneity present in the aggregate. The differences between the scaling function reported in this paper and the commonly used ones, exp (−r/ξ) and exp [−(r/ξ)²], are discussed. The simulating calculations have shown that the use of different scaling functions will not only influence the cross-over behavior between the Guinier regime and the fractal regime, but also make the low-q scattering intensity converge to different values.