Diffuse Reflexion of X-Rays

Abstract

IN a recent note Preston¹ has commented upon the agreement of his experimental results for the diffuse scattering of X-rays by single crystals of aluminium with the predictions of both the Preston–Bragg block hypothesis and a formula derived by Faxén. It seems necessary to correct the impression which might thereby be formed that the Preston–Bragg hypothesis is reconcilable with the Faxén–Waller theory. This is indeed not the case. It has been shown by Laue² that if the intensity of scattering from a finite crystal is plotted in the reciprocal lattice space as a function of the end-point of the vector (unit vectors in the direction of incident and reflected ray, wave-length) drawn from the origin of the reciprocal lattice, then the resulting distribution of scattering power is periodic with the periods of the reciprocal lattice, so that in particular there is the same distribution about every reciprocal lattice point. On the other hand, the Faxén formula discussed by Preston is obtained by specialization (see below) from a general expression derived by both Faxén³ and Waller⁴ for the first approximation to the coherent diffuse scattering from a monatomic cubical crystal. This general expression can be written in the form: where are unit (and orthogonal) vectors defining the directions of oscillation of those three acoustical vibrations the wave vector of which satisfies the Faxén-Waller vector condition (namely where is that reciprocal lattice vector the end-point of which lies nearest to that of ) and I₁, I₂, I₃ are the respective frequencies of these vibrations. Owing to the occurrence in this expression of the cosines of the angles between the vector and the vectors of oscillation, it is clear that the expression must lead to a different distribution of scattering power about each point of the reciprocal lattice. In a paper which I have in preparation, the general Faxén-Waller expression has been evaluated for various relative values of the three elastic constants c₁₁, c₁₂, c₄₄ of a cubic crystal, and the marked dependence of the scattering power upon the elastic anisotropy demonstrated. For example, for sodium single crystals, using the values of the elastic constants determined by Quimby and Siegel⁵, namely, c₁ = 0.760 c₁₁, c₄₄ = 0.914 c₁₁ (at 200° K.), with cubic anisotropy c₁₁-c₁₂’2c₄₄ = -1.59 c₁₁, the section of the contours of isodiffusion of the 200 and 110 reflexions in the 200,020 plane of the reciprocal lattice approach, in the limiting case of long elastic waves, that is, proximity to the Bragg spot, the shape shown in the accompanying diagram.