Electron-acoustic recoil length ($\lambda =\hslash /(2m_e{c}_s)$)

For free-electron recoil and a linear acoustic branch, $E_e(q)={\hslash }^2{q}^2/(2m_e)$ and $E_{\mathrm{ph}}(q)=\hslash c_{s}q$. Their ratio is $q\lambda$, so equality occurs at $q=1/\lambda$. This is a momentum/localization crossover scale, formally half a sound-speed analogue of the Reduced Compton wavelength. It does not replace the speed of light in electron relativity or determine a mean free path. A Fermi-surface electron emitting a phonon has fractional radial loss $\delta \simeq q/(2\lambda k_F^2)$ when the transferred energy is small.