Imaginary potential for electrons of CaF₂from a Bloch-wave analysis

Abstract

In a previous analysis of Bloch waves in electron diffraction experiments with a CaF: single-crystal wedge [Ishida, Johnson & Lehmpfuhl (1975). Z.Naturforsch. Tell A, 30, 1715Tell A, 30, -1729] absorption coefficients of the strong Bloch waves were determined. The experiments were performed with two types of crystal wedges, produced by (111) cleavage faces. One type had a wedge angle of 109 ° and the other type an angle of 71 ° . In the experiment with the blunt wedge the direction of incidence was close to [100] and in the sharp-wedge experiment close to [110]. The two sets of absorption coefficients were not consistent with an imaginary potential describing the absorption. From reexamination of the data a consistent absorption potential, expressed by an analytical formula, could be determined, which confirmed the absorption coefficients for the blunt wedge and modified the former values for the sharp wedge. Agreement between calculation and experiment could be achieved by convoluting the diffracted beams with the profile of the incident beam. This effect is important for the sharp wedge and negligible for the blunt wedge.The current density of electrons travelling through a crystal can be described by Bloch waves: The various Bloch waves show characteristic density distributions over the crystal unit cell, depending on the direction of the incident beam (Lehmpfuhl, 1973). Each of the Bloch waves is absorbed with its own characteristic absorption coefficient as predicted by van Laue (1953). For localized inelastic scattering processes this results in an orientation dependence of absorption as shown experimentally (Taft0 & Lehmpfuhl, 1982). The orientation-dependent absorption of the electrons can be described by introducing an imaginary part into the scattering potential, according to Moli~re (1939), in the form V(r) = V(r) real + iV(r) im.( 1 )