We consider diffusion of vibrations in 3d harmonic lattices with strong force-constant disorder. Above some frequency ωIR, corresponding to the Ioffe-Regel crossover, notion of phonons becomes ill defined. They cannot propagate through the system and transfer energy. Nevertheless most of the vibrations in this range are not localized. We show that they are similar to diffusons introduced by Allen, Feldman et al., Phil. Mag. B 79, 1715(1999 to describe heat transport in glasses. The crossover frequency ωIR is close to the position of the boson peak. Changing strength of disorder we can vary ωIR from zero value (when rigidity is zero and there are no phonons in the lattice) up to a typical frequency in the system. Above ωIR the energy in the lattice is transferred by means of diffusion of vibrational excitations. We calculated the diffusivity of the modes D(ω) using both the direct numerical solution of Newton equations and the formula of Edwards and Thouless. It is nearly a constant above ωIR and goes to zero at the localization threshold. We show that apart from the diffusion of energy, the diffusion of particle displacements in the lattice takes place as well. Above ωIR a displacement structure factor S(q, ω) coincides well with a structure factor of random walk on the lattice. As a result the vibrational line width Γ(q) = Duq 2 where Du is a diffusion coefficient of particle displacements. Our findings may have important consequence for the interpretation of experimental data on inelastic x-ray scattering and mechanisms of heat transfer in glasses.

The vibrational properties of model amorphous materials are studied by combining complete analysis of the vibration modes, dynamical structure factor and energy diffusivity with exact diagonalization of the dynamical matrix and the Kernel Polynomial Method which allows a study of very large system sizes. Different materials are studied that differ only by the bending rigidity of the interactions in a Stillinger-Weber modelization used to describe amorphous silicon. The local bending rigidity can thus be used as a control parameter, to tune the sound velocity together with local bonds directionality. It is shown that for all the systems studied, the upper limit of the Boson peak corresponds to the Ioffe-Regel criterion for transverse waves, as well as to a minimum of the diffusivity. The Boson peak is followed by a diffusivity's increase supported by longitudinal phonons. The Ioffe-Regel criterion for transverse waves corresponds to a common characteristic mean-free path of 5-7Å (which is slightly bigger for longitudinal phonons), while the fine structure of the vibrational density of states is shown to be sensitive to the local bending rigidity.

In amorphous solids, a non-negligible part of thermal conductivity results from phonon scattering on the structural disorder. The conversion of acoustic energy into thermal energy is often measured by the dynamical dtructure factor (DSF) thanks to inelastic neutron or x-ray scattering. The DSF is used to quantify the dispersion relation of phonons, together with their damping. However, the connection of the dynamical structure factor with dynamical attenuation of wave packets in glasses is still a matter of debate. We focus here on the analysis of wave-packet propagation in numerical models of amorphous silicon. We show that the damped harmonic oscillator model fits of the dynamical structure factors give a good estimate of the wave packets mean free path, only below the Ioffe-Regel frequency. Above the Ioffe-Regel frequency and below the mobility edge, a pure diffusive regime without a definite mean free path is observed. The high-frequency mobility edge is characteristic of a transition to localized vibrations. Below the Ioffe-Regel frequency, a mixed regime is evidenced at intermediate frequencies, with a coexistence of propagative and diffusive wave fronts. The transition between these different regimes is analyzed in detail and reveals a complex dynamics for energy transport, thus raising the question of the correct modeling of thermal transport in amorphous materials.