Replica field theory for anharmonic sound attenuation in glasses

A saddle-point treatment of interacting phonons in a disordered environment is developed. In contrast to crystalline solids, anharmonic attenuation of density fluctuations becomes important in the hydrodynamic regime, due to a broken momentum conservation. The variance of the shear modulus Δ² turns out to be the strength of the disorder enhanced phonon-phonon interaction. In the low-frequency regime (below the boson peak frequency) we obtain an Akhiezer-like sound attenuation law Γ ∝ Τω². Together with the usual Rayleigh scattering mechanism this yields a crossover of the Brillouin linewidth from a ω² to a ω⁴ regime. The crossover frequency ω_c is fully determined by the boson peak frequency and the temperature. For network glasses like SiO₂ at room temperature this crossover is predicted to be situated one order of magnitude below the boson peak frequency.