Small-angle electron-phonon transport at low temperature

For acoustic modes ${\omega }_q\simeq c_{s}q$, thermally active wavevectors have $q_T\sim k_{B}T/(\hslash c_s)$. When $q_T\ll k_F$, a nearly elastic event on a spherical Fermi surface has $1-\cos \theta =q^2/(2k_F^2)$, suppressing momentum relaxation. In the clean three-dimensional acoustic model, the thermal relaxation rate scales as $T^3$ while the electrical transport rate gains two extra powers and scales as $T^5$. With electronic heat capacity proportional to $T$, this gives $K_e\propto T^{-2}$ and electrical resistivity proportional to $T^5$ in the phonon-limited regime. Residual elastic scattering cuts off these behaviours as temperature falls further.^[ref]