= Solution
The <electron-acoustic recoil length> compares electronic recoil with the acoustic quantum <energy> associated with the same <momentum>. For transferred wave number $q$,
$$
E_{\rm recoil}(q)=\frac{\hbar^2q^2}{2m_e},\qquad
E_{\rm phonon}(q)=\hbar cq,\qquad
\boxed{\frac{E_{\rm recoil}}{E_{\rm phonon}}=q\lambda.}
$$
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 $q\lambda\gg1$ 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>.
\Image[/past-exam-of-the-condensed-matter-physics-course-of-the-university-of-oxford/2014-electron-acoustic-recoil-and-near-elasticity.png]
{title=<Electron>–acoustic recoil crossover and nearly conserved Fermi-wavevector magnitude}
{description=Original dimensionless dispersion comparison with $q\lambda$ on the horizontal axis. The separate right panel compares the exact radial loss $1-\sqrt{1-\epsilon/E_F}$ to its small-transfer approximation $\epsilon/(2E_F)$; the cutoff-phonon example is marked at $\epsilon/E_F=1/240$. Neither panel is a measured copper dispersion curve.}
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