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I am reading "Supercollision cooling in undoped graphene." There the authors write: ``Above $T_{BG}$ (the Bloch-Gruneisen temperature), only a fraction of acoustic phonons with wave vector $q\le 2k_{F}$ can scatter off electrons."

I assume the condition $q\le 2k_{F}$ for electron-phonon scattering is a basic result, but I couldn't find it proved anywhere. Could anyone point me in the right direction, or state the proof here? Thanks.

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The electron-phonon scattering is a process that takes the electron across its Fermi surface (from an occupied state to an empty state, or vice versa) by absorbing or emitting a phonon. Compared to the electron energy in most solid state materials, the phonon energy is neglectable, such that the electron will (almost) not change its energy when scattering with a phonon. Therefore the phonon can only scatter off the electrons on the Fermi surface. However the greatest possible momentum transfer on the Fermi surface is $2k_F$ (assuming spherical Fermi surface with radius $k_F$). So all scattered phonon must have a momentum $q\leq 2k_F$.

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