Sound Attenuation in Glasses

Calculation of the temperature and frequency dependent coefficient of sound attenuation in amorphous silicon can be found in J. Fabian and P. B. Allen, Theory of sound attenuation in glasses: The role of thermal vibrations, Phys. Rev. Lett. 82, 1478 (1999). 

Sound attenuation in glasses is a very complex phenomenon which is poorly understood. Different competing mechanisms lead to sound-wave damping, each relevant at different temperature regions and at different frequencies of the attenuated sound waves. At the lowest temperatures (below 1-5 K) sound attenuation in glasses is caused by the interaction of sound waves with hypothetical two-level systems. At higher temperatures one has to distinguish different frequency regimes. At the lowest frequencies (below about 1 MHz), sound attenuation is caused by thermally activated structural relaxation (a group of atoms jump over a barrier to a different metastable position). At high frequencies (above 10 GHz), sound waves lose their energy by a direct anharmonic interaction with other vibrational modes: a sound wave, which is an eigenstate of the harmonic potential, can decay to two or more other modes because of small anharmonic terms in the Hamiltonian. The most interesting behavior occurs at ultra- and hyper-sound frequencies, between 1 MHz and 100 GHz. This situation is interesting, because until recently we had no good mechanism at hand to explain the experimental data. In crystals, this regime is entirely dominated by the Akhiezer mechanism. Here phonons are considered to act as thermal bath for the sound wave. As the sound wave passes through the sample, it creates a strain field. Phonon frequencies change (the change is measured by the phonons' Gruneisen parameters), and so the phonon population finds itself out of equilibrium (the frequencies change but the occupation numbers not). As the phonon bath tries to get into the new equilibrium characterized by the sound-wave-induced strain, energy is dissipated. This mechanism was believed to be not relevant for glasses, since to explain the experiment, thermal vibrations in glasses would need giant Gruneisen parameters (10-100). Typical values for phonons are between 0 and 1. We have shown that some of the vibrational states in glasses indeed possess such giant Gruneisen parameters as a result of internal strain (see here). Those modes are the resonant modes and we have already shown that they are behind the observed sample dependence of thermal expansion in glasses (see here). The moral of our calculation is that, in contrast to previous views, thermal vibrations are responsible for the observed ultra and hyper-sound attenuation in glasses at temperatures above 5 K (above the realm of two level systems.

Above: Calculated coefficient of sound attenuation in amorphous silicon: solid (dashed) line is for longitudinal (transverse) sound wave; dash-point line is for longitudinal waves without taking into account internal strain (IS). Experimental values for longitudinal sound attenuation in silica (triangles and squares) suggest sample dependent sound attenuation. Crystalline values (for silicon) are denoted by circles. Temperature is 300 K. The plot is log-log, and the sound attenuation coefficient increases quadratic with increasing sound wave frequency, until the point where the sound wave frequency becomes comparable to the anharmonic decay rate of thermal vibrations (which is about 1 meV). Then the mechanism of sound attenuation changes and the calculation ceases to be valid.

Above: Calculated sound attenuation coefficient as a function of temperature, for different sound wave frequencies. At relatively low frequencies (1 MHz up to 1 GHz) there is a peak at low temperatures (50 K) which diminishes and moves to the right with increasing frequency, until it flattens out at about 40 GHz. The peak is our prediction which is yet to be confirmed experimentally, and its origin lies in giant Gruneisen parameters of resonant modes. This also makes the sound attenuation coefficient sample/model dependent, similarly to thermal expansion.