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 $\mathbf{k}+\mathbf{q}={\mathbf{k}}^{\prime }+\mathbf{G}$ therefore needs a phonon wavevector of at least $q_U$. For an approximately linear acoustic dispersion its activation scale isAt $T\ll T_U$ 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.