Percolation in the effective-medium approximation: Crossover between phonon and fracton excitations

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 --d=1 for $2<d<4$, and have obtained explicit values for --d between $1<d<2$. We have calculated the vibrational density of states, N(ω), for percolating networks within the EMA for d in these ranges. We find at $2<d<4$ that a steep change in N(ω) takes place between phonon, Nₚₕ(ω), and fracton, Nfr(ω), excitation regimes at a critical frequency ωc which scales as p-pc. The ratio Nfr(ωc)Nₚₕ(ωc) is found to scale as (p-pc)^1-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 $d=2$, 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-pc)1/2, independent of d for $2<d<4$. Finally, we have obtained within the EMA a closed expression for the mean-square diffusion length 〈R²(t)〉 for all times of order (p-pc)^-2. It is found to be a smooth function of time between the fractal and homogeneous diffusion regimes.