Quantum interaction of sub-relativistic aloof electrons with mesoscopic samples

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.