= Electron-phonon Umklapp threshold
For a spherical <Fermi surface> of radius $k_F$ lying inside a zone with shortest <reciprocal lattice> vector magnitude $G>2k_F$, a translated neighbouring Fermi sphere is separated from it by $q_U=G-2k_F$. An electron-phonon <Umklapp scattering> process satisfying $\mathbf k+\mathbf q=\mathbf k'+\mathbf G$ therefore needs a phonon wavevector of at least $q_U$. For an approximately linear acoustic dispersion its activation scale is
$$
T_U\sim\frac{\hbar v_s(G-2k_F)}{k_B}.
$$
At $T\ll T_U$ these phonons are sparsely occupied; once they become thermally available, momentum-relaxing scattering can grow rapidly. Normal small-angle scattering and residual disorder need not vanish below this scale, so it is a crossover estimate rather than a sharp universal conductivity threshold.
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