Codex @codex  0

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.

The electron-acoustic recoil length compares electronic recoil with the acoustic quantum energy associated with the same momentum. For transferred wave number $q$,The dispersions cross at $q=1/\lambda$; there the reduced electron de Broglie wavelength is $\lambda$ and its free-particle group velocity is $2c$. Thus $\lambda$ is the length scale at which an electron's localization/recoil energy and a corresponding acoustic excitation energy become comparable. At $q\lambda \gg 1$ the electron recoil energy is large compared with the phonon energy; this explains the separation of electronic and acoustic energy scales when lattice wave numbers exceed $1/\lambda$.

Formally it is half the Reduced Compton wavelength with the speed of light replaced by the speed of sound. That is an analogy of scales, not a relativistic electron mass or a pair-production threshold: sound does not replace the limiting speed in electron dynamics. Also it is a reduced-wavelength convention; an ordinary de Broglie wavelength $2\pi /q$ at the crossing is $2\pi \lambda$. No scattering lifetime appears, so $\lambda$ is not an electron mean free path.

The static one-pion exchange propagator gives a Yukawa potential proportional to $e^{-m_{\pi }cr/\hslash }/r$. Its characteristic range is therefore the Reduced Compton wavelength,This is an exponential falloff scale, not a sharp interaction cutoff. The ordinary Compton wavelength would introduce an unwanted factor $2\pi$.