Phonon scattering and thermal transport

At thermal equilibrium, modes traveling in opposite directions carry equal average energy currents, giving zero net heat flow. A small temperature gradient shifts their populations unevenly. In the relaxation-time approximation, each mode's departure from local Bose equilibrium is estimated from its group velocity, the gradient, and a transport lifetime. Summing the resulting energy currents gives a conductivity tensor. The derivative of frequency with respect to angular wavevector is the group velocity; using phase velocity instead can badly misrepresent modes near a flat zone-edge dispersion.

For an isotropic gray approximation with one representative speed and lifetime, the angular average of a squared directional velocity supplies a factor one-third. Conductivity becomes one-third of volumetric heat capacity times speed times mean free path. The heat capacity must be per volume, not the total heat capacity of an arbitrary sample. The resulting units are watts per metre kelvin. This estimate is useful for scale analysis, but a real material usually has broad branch-dependent distributions of speed, occupation, and scattering lifetime.

Anharmonic interactions allow phonons to combine or decay subject to energy conservation and crystal wavevector conservation modulo a reciprocal vector. A three-phonon combination obeys omega1 plus omega2 equal to omega3 and q1 plus q2 equal to q3 plus G. A process with G zero is normal; one with nonzero G is Umklapp when all wavevectors are represented in the first zone. The lattice absorbs the reciprocal-vector bookkeeping. This does not violate total physical momentum conservation when the complete crystal and environment are included.

Normal collisions conserve total phonon crystal momentum and can drive a distribution toward a drifting local equilibrium rather than a motionless one. They redistribute energy and can influence heat current, but should not generally be inserted as independent resistive events in a scalar relaxation-time sum. Umklapp, defects, and boundaries can relax the relevant directed transport. When normal collisions dominate over resistive scattering, collective hydrodynamic behavior can require a coupled collision treatment instead of the naive picture that every collision independently destroys the same fraction of current.

Fourier's law relates heat flux to minus conductivity times temperature gradient. Combined with local energy conservation and constant coefficients, it gives a thermal diffusivity equal to conductivity divided by volumetric heat capacity. These are distinct quantities with different units. Local diffusion is appropriate when the spatial and temporal scales exceed relevant mean free paths and relaxation times. In a small sample or over short delays, ballistic propagation and boundary transmission can matter, making a bulk scalar conductivity insufficient to characterize the measured heat flow.

At low temperature boundary-limited mean free paths can be nearly temperature independent, so conductivity can inherit a bulk acoustic heat-capacity power law under additional assumptions. At higher temperature Umklapp scattering often lowers lifetimes, but an inverse-temperature conductivity is not a universal law. Strong disorder, mode localization, optical contributions, electrons, and explicit anharmonicity can change the behavior. The quasiparticle picture itself becomes doubtful when linewidths rival frequencies or paths are too short to define a clear wavevector. Distinguish a controlled model estimate from a measured material property or an exact transport solution.