The $d$-dimensional bond-percolating network has been examined with the use of the effective-medium approximation (EMA) of Odagaki and Lax and of Webman. We have found that the fracton dimensionality $\overline{\stackrel{―}{d}}=1\mathrm{}$ for $2<d<\mathrm{}4\mathrm{}$, and have obtained explicit values for $\overline{\stackrel{―}{d}}$ between $1<d<\mathrm{}2\mathrm{}$. We have calculated the vibrational density of states, $N\left(\omega \right)$, for percolating networks within the EMA for $d$ in these ranges. We find at $2<d<\mathrm{}4\mathrm{}$ that a steep change in $N\left(\omega \right)$ takes place between phonon, $N_{\mathrm{ph}}\left(\omega \right)$, and fracton, $N_{\mathrm{fr}}\left(\omega \right)$, excitation regimes at a critical frequency ${\omega }_c$ which scales as $p-p_c$. The ratio $\frac{N_{\mathrm{fr}}\left({\omega }_c\right)}{N_{\mathrm{ph}}\left({\omega }_c\right)}$ is found to scale as ${(p-p_c)}^{1-\frac{d}{2}}$. These results provide substantial support for the fracton interpretation of the thermal properties of epoxy resin, glasses, and neutron-irradiated quartz as hypothesized by Alexander, Laermans, Orbach, and Rosenberg. At $d=2\mathrm{}$, the transition between the two regimes is smoother (logarithmic), but a clearly defined phonon and fracton regime can be ascertained. The velocity of sound in the phonon regime scales as ${(p-p_c)}^{\frac{1}{2}}$, independent of $d$ for $2<d<\mathrm{}4\mathrm{}$. Finally, we have obtained within the EMA a closed expression for the mean-square diffusion length $〈R^2\mathrm{}\left(t\right)\mathrm{}〉$ for all times of order ${(p-p_c)}^{-2\mathrm{}}$. It is found to be a smooth function of time between the fractal and homogeneous diffusion regimes.

• Received 21 September 1983

DOI:https://doi.org/10.1103/PhysRevB.29.6645