The Ioffe-Regel criterion and diffusion of vibrations in random lattices

TL;DR

This paper investigates vibrational diffusion in disordered 3D lattices, showing that above a certain frequency, vibrations behave like diffusons, affecting heat transfer and particle displacement diffusion, with implications for experimental interpretations.

Contribution

It demonstrates that vibrational excitations above the Ioffe-Regel crossover act as diffusons, providing a detailed analysis of their diffusivity and displacement diffusion, linking theory with experimental observations.

Findings

Diffusons dominate vibrational transport above w_IR.

Diffusivity D(w) remains nearly constant above w_IR.

Particle displacements also diffuse similarly to vibrational energy.

Abstract

We consider diffusion of vibrations in 3d harmonic lattices with strong force-constant disorder. Above some frequency w_IR, 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 w_IR is close to the position of the boson peak. Changing strength of disorder we can vary w_IR from zero value (when rigidity is zero and there are no phonons in the lattice) up to a typical frequency in the system. Above w_IR the energy in the lattice is transferred by means of diffusion of vibrational excitations. We calculated the diffusivity of the modes D(w) using both the direct numerical solution of Newton equations and the formula of Edwards and Thouless. It is nearly a constant above w_IR 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 w_IR a displacement structure factor S(q,w) coincides well with a structure factor of random walk on the lattice. As a result the vibrational line width Gamma(q)=D_u q^2 where D_u 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.