= Thermal and electrical relaxation times in metals
{title2=$\kappa_e/(\sigma T)=L_0\tau_\kappa/\tau_\sigma$}
For a degenerate isotropic parabolic electron band, <electrical conductivity> is $\sigma=ne^2\tau_\sigma/m$ and electronic <thermal conductivity> is $\kappa_e=\pi^2nk_B^2T\tau_\kappa/(3m)$. Their ratio is $\kappa_e/(\sigma T)=L_0\tau_\kappa/\tau_\sigma$, where $L_0=(\pi^2/3)(k_B/e)^2$ is the <Lorenz number>. Nearly elastic scattering that relaxes charge and heat currents similarly gives $\tau_\kappa\simeq\tau_\sigma$ and the <Wiedemann-Franz law>. Inelastic scattering can relax <heat current> and directed <momentum> differently, invalidating the simplest identification even in a good <metal>.
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