Plasma Losses by Fast Electrons in Thin Films

TLDR

The authors derived the angle‑energy distribution of fast electrons in thin metallic foils by treating conduction electrons as a Fermi‑Dirac gas and applying Bloch hydrodynamics, showing that for very thin foils the energy loss can fall below the plasma energy and decreases further as thickness drops below \(v/\omega_p\). The derived distribution reveals both collective and individual interaction features, predicts that energy loss per unit thickness rises as foil thickness shrinks, and suggests that sub‑plasma‑energy losses observed experimentally arise from this effect.

Abstract

The angle-energy distribution of a fast electron losing energy to the conduction electrons in a thick metallic foil has been derived assuming that the conduction electrons constitute a Fermi-Dirac gas and that the fast electron undergoes only small fractional energy and momentum changes. This distribution exhibits both collective interaction characteristics and individual interaction characteristics, and is more general than the result obtained by other workers. Describing the conduction electrons by the hydro-dynamical equations of Bloch, it has been shown that for very thin idealized foils energy loss may occur at a value which is less than the plasma energy, while as the foil thickness decreases below $\ensuremath{\sim}\frac{v}{{\ensuremath{\omega}}_{p}}$ the loss at the plasma energy becomes less than that predicted by more conventional theories. The net result is an increase in the energy loss per unit thickness as the foil thickness is decreased. It is suggested that the predicted loss at subplasma energies may correspond to some of the low-lying energy losses which have been observed by experimenters using thin foils.