  Relativistic electrons experience very slight wave packet distortion and
negligible momentum recoil when interacting with nanometer-sized samples, as a
consequence of the ultra-short interaction time. Accordingly, modeling fast
electrons as classical point-charges provides extremely accurate theoretical
predictions of energy-loss spectra. Here we investigate the aloof interaction
of nanometer-sized electron beams of few keV with micron-sized samples, a
regime where the classical description generally fails due to significant
wavefunction broadening and momentum recoil. To cope with these effects, we use
macroscopic quantum electrodynamics to analytically derive a generalized
expression for the electron energy loss probability which accounts for recoil.
Quantum features of the interaction are shown to get dramatically strong as the
interaction length is increased and/or the electron kinetic energy is
decreased. Moreover, relatively large values of the energy loss probability are
found at higher energy losses and larger impact parameters, a marked quantum
effect which is classically forbidden by the evanescent profile of the field
produced by a moving point-charge.
