Past exam of the quantum condensed matter course of the University of Cambridge / 2008 / Section B / B3 / b

For an acoustic phonon with ${\omega }_q\simeq v_{s}q$, energy and crystal momentum conservation require $E_{{\mathbf{k}}^{\prime }}-E_{\mathbf{k}}=\pm \hslash v_{s}q$ and ${\mathbf{k}}^{\prime }-\mathbf{k}=\pm \mathbf{q}+\mathbf{G}$. The sound speed is usually much smaller than the relevant electron velocity in a metal, so scattering can change the electron's direction substantially while changing its energy only slightly. It is approximately elastic for many transport calculations, though its finite energy transfer is essential for cooling.

The middle panel gives an allowed backscattering example with a small energy loss. In a degenerate metal, the Pauli exclusion principle blocks occupied final states, restricting low-temperature scattering to a thermal neighborhood of the Fermi surface. Phonon occupations also shrink on cooling. A normal process has $\mathbf{G}=0$; an Umklapp process transfers reciprocal-lattice momentum. In semiconductors, accessible valleys and their separation add further momentum constraints.