= Solution
An <optical phonon> often has approximately constant <energy> $\hbar\omega_0$ over the relevant <wavevector> range. Its absorption or emission therefore moves the <electron> between <energy> surfaces separated by this finite amount, with the <phonon> supplying the associated <crystal momentum> modulo a reciprocal vector. The last panel shows emission between two points on a parabolic <conduction band>.
Emission from a band with a lower edge requires at least $\hbar\omega_0$ of available kinetic <energy> and an unoccupied final state. For nondegenerate hot carriers this produces a threshold and can lead to rapid <energy> loss in successive <phonon> emissions. Absorption is weighted by the <Bose-Einstein distribution> $n_B(\omega_0)$, while emission has the factor $n_B+1$; absorption is strongly suppressed at $k_BT\ll\hbar\omega_0$. In a metal, Pauli blocking also prevents an <electron> near the Fermi <energy> from simply dropping into the occupied Fermi sea.
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