Phonon scattering

Phonon scattering changes the occupation or propagation of lattice-vibration normal modes through interactions with boundaries, defects or other phonons. Anharmonic terms permit phonon-phonon scattering. Different processes need not relax heat flow equally: normal processes conserve total crystal momentum within the first Brillouin zone, whereas Umklapp scattering changes its representative by a nonzero reciprocal lattice vector.

Normal phonon collisions conserve the crystal momentum of the distribution, whereas Umklapp scattering transfers momentum by a reciprocal vector. A normal collision can redistribute heat-carrying modes without relaxing a drifting distribution in the same manner as a momentum-relaxing event. Boundaries, impurities and Umklapp processes can then set the resistance. Counting all collision rates as interchangeable inverse transport times can miss collective phonon flow and nontrivial current weighting.

The phonon mean free path is the typical distance traveled by a phonon between the relevant scattering events. For heat transport, the relevant relaxation time must describe relaxation of heat current, not merely count all collisions. An isotropic kinetic estimate is $\kappa \simeq c_V{v}_s\ell /3$, where $c_V$ is heat capacity per unit volume. Boundary-limited $\ell$ and the low-temperature Debye model give $\kappa \propto T^3$.

An Umklapp process conserves reduced crystal momentum modulo a nonzero reciprocal lattice vector. It can occur in electron-phonon scattering as well as phonon-phonon scattering; the lattice takes up the reciprocal-lattice momentum.

In a three-phonon Umklapp scattering process, the first-zone wavevectors obey the displayed equation for a nonzero reciprocal lattice vector $\mathbf{G}$, together with conservation of energy. The crystal momentum transferred to the lattice allows these processes to relax a phonon heat current. They are suppressed at sufficiently low temperature and often determine the approximate $1/T$ high-temperature thermal conductivity of a crystalline electrical insulator.