  We investigate the effects of crystal lattice vibrations on the dispersion of
plasmons. The loss function of the homogeneous electron gas (HEG) in two and
three dimensions is evaluated numerically in presence of electronic coupling to
an optical phonon mode. Our calculations are based on many-body perturbation
theory for the dielectric function as formulated by the Hedin-Baym equations in
the Fan-Migdal approximation. The coupling to phonons broadens the spectral
signatures of plasmons in the electron-energy loss spectrum (EELS) and it
induces the decay of plasmons on timescales shorter than 1 ps. Our results
further reveal the formation of a kink in the plasmon dispersion of the 2D HEG,
which marks the onset of plasmon-phonon scattering. Overall, these features
constitute a fingerprint of plasmon-phonon coupling in the EELS of simple
metals. It is shown that these effects may be accounted for by resorting to a
simplified treatment of the electron-phonon interaction which is amenable to
first-principles calculations.
