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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.

In an ordinary three-dimensional crystalline electrical insulator, heat is carried mainly by phonons. Kinetic theory gives the estimatewhere $c_V$ is heat capacity per unit volume, $v_s$ is a typical sound speed, and $\ell$ is the phonon mean free path. The thermal conductivity $\kappa$ has units $\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$.

At low temperature, $T\ll {\Theta }_D$, the Debye model gives $c_V\propto T^3$. With phonon-phonon collisions frozen out, diffuse scattering at the sample boundaries makes $\ell$ approximately constant, and $v_s$ is nearly constant. HenceAt high temperature, of order or above the Debye temperature, the Dulong-Petit law makes $c_V$ approximately constant. Anharmonic phonon scattering, especially resistive Umklapp scattering, has a collision rate approximately proportional to $T$. Thus $\ell \propto T^{-1}$ andThese two regimes usually give an intermediate maximum of thermal conductivity. The assumptions matter: glassy materials, strong defects or specular boundary reflection can change the low-temperature law; these are not universal laws for every electrical insulator.

For a lattice-periodic one-electron Hamiltonian, Bloch's theorem givesHere $\mathbf{R}$ is any lattice translation and $n$ labels an energy band. Under that translation the wavefunction gains phase $e^{i\mathbf{k}\cdot \mathbf{R}}$. The crystal momentum $\hslash \mathbf{k}$ labels this translation eigenvalue and is defined modulo $\hslash \mathbf{G}$ for a reciprocal lattice vector $\mathbf{G}$.

The mechanical momentum is different. In zero magnetic field its operator is $-i\hslash \nabla$, which acts asA general Bloch state is therefore not a mechanical-momentum eigenstate. Expanding its periodic factor gives plane-wave components with momenta $\hslash (\mathbf{k}+\mathbf{G})$, with probabilities determined by their Fourier coefficients. For a local Hamiltonian $p^2/(2m_e)+V(\mathbf{r})$, its mean momentum satisfiesnot generally $\hslash \mathbf{k}$. The lattice exchanges actual momentum with the electron; crystal-momentum conservation in scattering is instead modulo reciprocal-lattice vectors, as in Umklapp scattering.

For a spherical Fermi surface inside a primitive cubic lattice zone, the shortest reciprocal vector has magnitude $G=2\pi /a$. The nearest translated Fermi sphere is separated by the electron-phonon Umklapp thresholdA phonon of this wavevector can connect nearly opposite-velocity Fermi states using $\mathbf{k}+\mathbf{q}={\mathbf{k}}^{\prime }+\mathbf{G}$, transferring reciprocal-lattice momentum to the crystal and efficiently relaxing electrical current. With the supplied sound speed, its thermal scale isFor $T\ll T_U$ such phonons have exponentially small occupation. As they become populated, Umklapp scattering can make the electrical conductivity fall rapidly. Lower-energy normal scattering and residual disorder still give resistance below this scale; this is a crossover estimate, not a sharp transition. The numerical estimate extrapolates the linear acoustic dispersion to $q_U$ and assumes the spherical Fermi surface implicit in giving one $k_F$.

Using $2k_F$ alone as the phonon wavevector misses this mechanism: direct backscattering would need $1.4\pi /a$, beyond the first-zone edge along this direction, while the reciprocal-lattice transfer permits it with $0.6\pi /a$.

For one conduction electron per atom, let $n_a$ be the atom density. State counting in a Bloch band gives the Fermi wavevector $k_F=(3{\pi }^2{n}_a{)}^{1/3}$. The Debye model preserves the three vibrational modes per atom by a spherical Debye wavevector cutoff $q_D=(6{\pi }^2{n}_a{)}^{1/3}$, soBoth 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,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 haveThey predominantly scatter forward. The factor $1-\cos \theta$ makes such collisions inefficient at relaxing an electrical current, while a phonon energy of order $k_{B}T$ 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.^[ref]

Phonons also transport heat themselves, with $K_{\mathrm{ph}}\simeq c_{\mathrm{ph}}{c}_s{\ell }_{\mathrm{ph}}/3$. 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.

Phonon scattering changes the occupation or propagation of lattice-vibration normal modes through interactions with boundaries, defects or other phonons. Anharmonic terms permit phonon-phonon scattering. Different processes need not relax heat flow equally: normal processes conserve total crystal momentum within the first Brillouin zone, whereas Umklapp scattering changes its representative by a nonzero reciprocal lattice vector.

In a three-phonon Umklapp scattering process, the first-zone wavevectors obey the displayed equation for a nonzero reciprocal lattice vector $\mathbf{G}$, together with conservation of energy. The crystal momentum transferred to the lattice allows these processes to relax a phonon heat current. They are suppressed at sufficiently low temperature and often determine the approximate $1/T$ high-temperature thermal conductivity of a crystalline electrical insulator.