Mass Dependence of the Angular Distribution of Charged-Particle Emission from Crystals: Transition to the Classical Limit

Abstract

In recent years, the emission of charged particles from crystals has received considerable attention, both experimentally and theoretically. The customary interpretation of the emission partterns has been derived from classical mechanics, which predicts patterns very close to those observed for heavy charged particles (protons, $\alpha$ particles) but only qualitatively similar to those exhibited by electrons and positrons. In the present paper, it is shown that at least one consequence of the classical model, namely, mass independence of the angular distribution at constant energy, is violated dramatically by the quantum-mechanical calculation of the emission pattern. Specifically, the emission patterns for electrons and positrons are shown to differ from the emission patterns for much-higher-mass particles by factors of up to 4 in half-width and intensity. The condition under which the classical limit is regained is that the lattice spacing be large, not compared with the particle de Broglie wavelength, but compared, rather, with ${\lambda }_p=\frac{\hslash }{2}{\left(\mathrm{m}\overline{V}\right)}^{\frac{1}{2}}$, where $\overline{V}$ is an average value for the interaction potential inside the unit cell, and $m$ is the particle mass.