For a mass fractal of dimension ${\mathit{d}}_{\mathit{m}}$ bounded by a surface fractal of dimension ${\mathit{d}}_{\mathit{s}}$, we show that the mass (M) versus radius (r) relationship is given by M∝${\mathit{r}}_{\mathit{m}}^{\mathit{d}}$[1-A(r/L${)}_{\mathit{s}}^{\mathit{d}\mathrm{-}\mathit{d}}$] for r/L<1, where d is the Euclidean dimension and ${\mathit{L}}^{\mathit{d}}$ 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)∝${\mathit{q}}_{\mathit{m}}^{\mathrm{-}\mathit{d}}$[1+A’(qL${)}_{\mathit{s}}^{\mathrm{-}(\mathit{d}\mathrm{-}\mathit{d}}$)] 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)∝${\mathit{r}}_{\mathit{m}}^{\mathit{d}}$-3${\mathit{e}}^{\mathrm{-}\mathrm{\alpha }(\mathit{r}/\mathit{L})\mathrm{\beta }}$, where β=d-${\mathit{d}}_{\mathit{s}}$.

• Received 9 September 1991

DOI:https://doi.org/10.1103/PhysRevB.45.7627