Correlation function and structure factor for a mass fractal bounded by a surface fractal

For a mass fractal of dimension dₘ bounded by a surface fractal of dimension dₛ, we show that the mass (M) versus radius (r) relationship is given by M{∝}rₘᵈ[1-A(r/L)ₛ^d-d] for r/L1, where d is the Euclidean dimension and Lᵈ characterizes the volume enclosed by the surface. This prediction is borne out by computer simulations of Sierpinski carpets bounded by different fractal perimeters. It implies that the structure factor S(q) observed by small-angle scattering has the general form S(q){∝}qₘ^-d[1+A'(qL)ₛ^-(d-d)] for qL>1. Applications on real-space analyses of finite-size fractal objects are noted. We also conjecture that the geometric correlation function has the general form of g(r){∝}rₘᵈ-3e^-α(r/L)β, where {β}=d-dₛ.