= Solution
For an <acoustic phonon> with $\omega_q\simeq v_sq$, <energy> and <crystal momentum> conservation require $E_{\boldsymbol k'}-E_{\boldsymbol k}=\pm\hbar v_sq$ and $\boldsymbol k'-\boldsymbol k=\pm\boldsymbol q+\boldsymbol G$. 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 $\boldsymbol G=0$; an <Umklapp process> transfers reciprocal-lattice momentum. In semiconductors, accessible valleys and their separation add further momentum constraints.
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