 Mass fractal dimension

The mass fractal dimension $d_f$ relates the particle count or mass in a self-similar aggregate to its spatial size. For primary particle radius $a$ and aggregate radius of gyration $R_g$, a common convention is $N\sim k_f(R_g/a{)}^{d_f}$, with a nonuniversal prefactor $k_f$. In three dimensions the mean occupied fraction scales as $N{a}^3/R_g^3\propto (R_g/a{)}^{d_f-3}$. A smaller $d_f$ therefore describes a more tenuous large cluster; a compact constant-density object has $d_f=3$. The scaling applies between the primary-particle and aggregate scales, not at all distances.