Quantum interaction of subrelativistic 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 ultrashort 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 a few keV with micron-sized samples, a regime where the classical description generally fails due to significant wave function 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.