What is the relation between linear elastic theory and phonon transport?

Question

I am reading about Boltzmann equation and I am having a hard time making a link between elastodynamics and phonon equations. Clearly there should be a limit where both are one and the same, isn't it? How is the thermal conductivity or the time-relaxation rate related to the elastic coefficients of wave propagation in linear elasticity?

Or are both theories completely unrelated? Any hints on the limit in which microscopic theories link to acoustics would be great.

I am not looking on why the macroscopic model is wrong but under what limit one recovers this model from the microscopic model and how are the microscopic coefficients tied to the macroscopic ones.

Answer 1

Clearly there should be a limit where both are one and the same, isn't it?

I think the underlying contrast you want to make is the continuum model of a solid vs the atomistic model of a solid. The elastodynamic equations come from the continuum model, and the phonon equations come from the atomistic model. That also hints at where the two agree: in the limit of wavelengths much longer than the spacing of the atoms, the material can often be treated as a continuum, and the elastodynamic and phonon equations will converge --- at least for acoustic phonons.

However, optical phonons have no continuum equivalent; they exist due to having more than one atom in a unit cell, and continuum models cannot reproduce the effect.

How is the thermal conductivity or the time-relaxation rate related to the elastic coefficients of wave propagation in linear elasticity?

There's not a direct connection for a few reasons:

1. A continuum model allows for an arbitrarily small wavelength whereas the atomistic model does not. If you try to calculate thermal conductivity with the continuum model, those arbitrarily small wavelength waves will result in problems --- especially at high temperatures. This is a little like the ultraviolet catastrophe for light.

2. The continuum model leads to linear dispersion relations ($\omega \propto k$), which is correct in the limit of small $k$ but is otherwise wrong.

3. The continuum model cannot explain the relaxation time because a non-linear dispersion relation is required for phonon-phonon scattering. Linear dispersion relations allow waves to "pass though" each other without interacting, but an interaction is required for phonon-phonon scattering to exist.

4. The continuum model cannot explain some other sources of scattering (and the relaxation times they result in) such as mass-difference impurity scattering, which arises from some atoms having different masses than the rest of the crystal (e.g. most of the sample is Si 28, but other isotopes of Si are there in small quantities). The continuum model cannot account for this because it does not allow for atoms.

EDIT: For how to get elastodynamic coefficients (the elastic modulus tensor) from the phonon models, I'm going to cite two references:

1. Landau and Lifshitz "Theory of Elasticity" (volume 7 of their Course of Theoretical Physics)
2. M. J. P. Musgrave "Crystal Acoustics"

For crystals, Crystal Acoustics sections 18.7 and 18.8 shows how to get from the interatomic potential to the elastic modulus tensor. Doing this can be more or less complicated depending on the symmetry of the crystal (L&L section 10 for more information). Crystal Acoustics chapter 19 has examples for several flavors of cubic crystals (which are some of the simpler crystals).

For polycrystals, it is in general not possible to derive the elastic modulus tensor from the interatomic potential. L&L section 10 says more about this, but the short version is that it depends on the details of the boundaries between the many crystals, and that generally makes the problem intractable.

For amorphous materials, the situation is even more hopeless.