= Solution
For one conduction electron per atom, let $n_a$ be the atom density. <State counting in a Bloch band> gives the <Fermi wavevector> $k_F=(3\pi^2n_a)^{1/3}$. The <Debye model> preserves the three vibrational modes per atom by a spherical <Debye wavevector> cutoff $q_D=(6\pi^2n_a)^{1/3}$, so
$$
\boxed{q_D=2^{1/3}k_F\simeq1.26k_F.}
$$
Both wavevectors are of order the inverse atomic spacing. The numerical factor uses the spherical Debye approximation; a real zone boundary and the largest-frequency mode depend on direction and branch. Because $q_D<2k_F$, short-wavelength acoustic <phonons> can scatter <Fermi surface> electrons through substantial angles. For nearly elastic scattering within a spherical Fermi surface,
$$
q=2k_F\sin(\theta/2),\qquad1-\cos\theta=\frac{q^2}{2k_F^2}.
$$
Near room temperature, when a substantial fraction of the phonon spectrum is occupied, <electron-phonon scattering> is often the main source of <electrical resistance>. For $T\gg\Theta_D$, the phonon population and scattering rate are approximately proportional to $T$, giving $\rho\propto T$, $\sigma\propto T^{-1}$ and roughly constant $K_e$ when the <Wiedemann-Franz law> holds.
For <small-angle electron-phonon transport at low temperature>, thermally active acoustic modes instead have
$$
q_T\sim\frac{k_BT}{\hbar c_s}=q_D\frac{T}{\Theta_D}\ll k_F.
$$
They predominantly scatter forward. The factor $1-\cos\theta$ makes such collisions inefficient at relaxing an electrical current, while a phonon energy of order $k_BT$ can still substantially change an electron's thermal excitation. Heat-current relaxation therefore need not share the charge-current relaxation time. In the clean three-dimensional acoustic-phonon model this gives $\tau_Q^{-1}\propto T^3$ but $\tau_\sigma^{-1}\propto T^5$, so the phonon-limited electrical resistivity scales as $T^5$ and $K_e\propto T^{-2}$. These laws describe the regime before residual elastic scattering takes over; at sufficiently low temperature impurities give nearly constant residual resistivity and $K_e\propto T$, restoring the usual <Lorenz number>.https://home.itp.ac.ru/~grigorev/LectureSlides/MetalsLecture2.pdf[Electron-phonon scattering: distinct low-temperature thermal and electrical transport relaxation rates]{ref}
<Phonons> also transport heat themselves, with $K_{\rm ph}\simeq c_{\rm ph}c_s\ell_{\rm ph}/3$. At low temperature their population decreases and their <phonon mean free path> can become boundary- or defect-limited. At higher temperature <Umklapp scattering> can relax the phonon heat current. Such lattice conduction is an additional contribution, not the quantity constrained by the electronic <Wiedemann-Franz law>.
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