= Small-angle electron-phonon transport at low temperature
For acoustic modes $\omega_q\simeq c_s q$, thermally active wavevectors have $q_T\sim k_BT/(\hbar c_s)$. 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.https://home.itp.ac.ru/~grigorev/LectureSlides/MetalsLecture2.pdf[Low-temperature electron-phonon scattering: thermal lifetime versus transport lifetime]{ref}
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