Comments on the Dynamical Theory of X-Ray Diffraction in Crystals

Abstract

Abstract Von Laue reformulated Ewald's dynamical theory of X-ray diffraction in crystals by solving Maxwell's equations for an electric susceptibility which has the periodicity of the crystals lattice. Absorption was in the original paper not included. Kohler showed that in this case the electric susceptibility is directly proportional to the electron charge density within the crystal. Von Laue assumed that the divergence of the electric field vector E is equal to the electron charge density which is excited by the electromagnetic field, and this is in accordance with Lorentz' classical electron theory. Under this assumption E is in contrast to D not a transverse wave. This is the reason that von Laue derived the fundamental equations in terms of D, and in order to obtain a simple wave equation he used the approximation 4πP = ϰD. Here P is the vector of the electric polarization and ϰ is 4π times the electric susceptibility. This approximation is well justified, and von Laue's formulation is used in most publication on this subject. However some authors derived the fundamental equations in terms of fusing the correct relation 4πP = ϰE . In most publications they assumed that the longitudinal component of the vector E is small and neglected it. This point and the comparison with von Laue's formulation was discussed by Miyake and Ohtsuki. We show here that the divergence of E can be put zero if inelastic processes are excluded. This follows from Kohler's and Moliere's papers on the wave mechanical foundation of the dynamical theory of X-ray diffraction. However if higher perturbation terms are taken into account then the divergence of E differs from zero. In multiple beam cases the fundamental equations are much simpler if they are expressed in terms of E. In the region of anomalous dispersion, and here in particular in the region where the X-ray energy is close and above and crystal absorption edge, the wave equations should be expressed in terms of the electric field vector E.