Effective-medium approximation for a percolation network: The structure factor and the Ioffe-Regel criterion

The effective-medium approximation for a percolation system is used to analyze the vibrational structure factor S(q,ω), measured in scattering experiments off self-similar samples. A sharp crossover to strong localization is found at a certain frequency ${\omega }_c$, corresponding to a crossover wave vector $q_c$. In particular, the linewidth ${\tau }^{\mathrm{-}1}$ obeys the Rayleigh law, ${\tau }^{\mathrm{-}1}$≃${\omega }^{d+1\mathrm{}}$, at frequencies lower than ${\omega }_c$ (d is the Euclidean dimensionality), and the Ioffe-Regel strong-scattering limit, ${\tau }^{\mathrm{-}1}$≃ω, at high frequencies, ω>${\omega }_c$. At wave vectors less than $q_c$, S(q,ω) is sharply peaked at ω=qc (where c is the sound velocity) and has a small structure at ${\omega }_c$. At wave vectors higher than $q_c$, the sharp peak is completely smeared out, while the small structure persists. The results are in qualitative agreement with experiments and with the fracton scaling model, regarding the frequency and wave-vector dependence. However, they fail to scale with ${\omega }_c$ alone in the low-frequency regime.