Attenuation of Fast Electrons in Crystals and Anomalous Transmission

Abstract

The two-beam dynamical theory of electron diffraction (Laue case) with attenuation is treated using imaginary potentials and a damping vector α in the Schrödinger equation. Explicit formulas are obtained for centrosymmetric crystals when the Bragg condition is realized relating the damping vectors for the two wave fields with the mean imaginary inner potential P0 and the imaginary Fourier coefficient Pg. Using a simplified model, the imaginary Fourier coefficients are expressed in terms of the imaginary potential Pf for nearly free electrons (plasma) and Pc for bound atom cores. It is shown that the imaginary potentials can be directly related to the total inelastic collision cross sections and absolute numerical values so calculated. The values computed in this way are sufficiently near experimental results reported in the literature to warrant confidence in the approach. For 105-V electrons, the damping vectors α are around 10−3 Å−1, and the imaginary potential Pf≅0.6 V. The question of anomalous transmission or an electron ``Borrmann effect'' is considered on the basis of the results obtained. It is concluded at this stage that the effect will not be large for metals but will increase with atomic number and may become significant for Z>30. Uncertainties in both plasma and core electron cross sections make such predictions unreliable except for a few elements. A calculation of the contribution of those electrons that have lost energy to E1<E indicates that the E1 electrons also can produce an ``anomalous transmission.'' A velocity filter will be necessary to distinguish this E1 effect from a true anomalous transmission.