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

作者：Bernard Derrida, R. Orbach, K. W. Yu · 发表于：Physical review. B, Condensed matter · 年份：1984 · DOI：10.1103/physrevb.29.6645 · 被引用次数：100 · 研究领域：Theoretical and Computational Physics、Material Dynamics and Properties、Spectroscopy and Quantum Chemical Studies

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 $\stackrel{-}{\stackrel{-}{d}}=1$ for $2<d<4$, and have obtained explicit values for $\stackrel{-}{\stackrel{-}{d}}$ between $1<d<2$. We have calculated the vibrational density of states, $N(\ensuremath{\omega})$, for percolating networks within the EMA for $d$ in these ranges. We find at $2<d<4$ that a steep change in $N(\ensuremath{\omega})$ takes place between phonon, ${N}_{\mathrm{ph}}(\ensuremath{\omega})$, and fracton, ${N}_{\mathrm{fr}}(\ensuremath{\omega})$, excitation regimes at a critical frequency ${\ensuremath{\omega}}_{c}$ which scales as $p\ensuremath{-}{p}_{c}$. The ratio $\frac{{N}_{\mathrm{fr}}({\ensuremath{\omega}}_{c})}{{N}_{\mathrm{ph}}({\ensuremath{\omega}}_{c})}$ is found to scale as ${(p\ensuremath{-}{p}_{c})}^{1\ensuremath{-}\frac{d}{2}}$. 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 $d=2$, 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 ${(p\ensuremath{-}{p}_{c})}^{\frac{1}{2}}$, independent of $d$ for $2<d<4$. Finally, we have obtained within the...