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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 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.

The phonon mean free path is the typical distance traveled by a phonon between the relevant scattering events. For heat transport, the relevant relaxation time must describe relaxation of heat current, not merely count all collisions. An isotropic kinetic estimate is $\kappa \simeq c_V{v}_s\ell /3$, where $c_V$ is heat capacity per unit volume. Boundary-limited $\ell$ and the low-temperature Debye model give $\kappa \propto T^3$.