Damped Electron Waves in Crystals

Abstract

Electrons in solids, as in electron diffraction, suffer inelastic impacts if their energy is great enough to exceed the resonance energy of the atoms of the crystal, resulting in strong damping of the electron beams, with consequent broadening of the reflected peaks about the Bragg scattering angles, and reduction of the reflection coefficient. The problem is discussed mathematically, on the basis of the conventional theory of electron diffraction, by introducing an empirical damping constant in the form of a pure imaginary term in the potential. It is shown that such a constant leads to damped waves of the required sort, and the equations for energy and reflection coefficient are set up. The results are qualitatively of the sort necessary to explain the observations, though no marked asymmetry in the reflected peaks is predicted. Presumably the observed asymmetries arise, as Harding has suggested, from irregular spacings of the atomic planes near the surface. One interesting result of the theory is that the sharp distinction between allowed energy bands and forbidden gaps, which is found in the theory of undamped electrons in periodic lattices, is lost, for all waves are more or less damped.