Asymmetric x-ray diffraction by strained crystal wafers: 8×8-matrix dynamical theory

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Abstract
We have extended the matrix representation of the dynamical theory of x-ray diffraction to include Bragg planes that are oblique with respect to the surface of a crystal wafer. In place of the two independent 2×2 matrices of the Abelès method for planes parallel to the surface, we use a single 8×8 matrix. With such a matrix, rays may be skew with respect to the oblique Bragg planes and the wafer surface. Despite minor differences in approximations, computations by this method give results nearly identical to those from the Laue method in cases where either may be used. The new approach brings out the close analogy between the diffraction of visible light by blazed gratings and the diffraction of x rays by edges of oblique Bragg planes near the crystal surface. Matrix methods present no special problem in cases where the layers near the surface do not have the same spacing normal to the surface as those deeper down, resulting in curved oblique planes. Thus, epi- taxial layers of varying composition, crystals strained by ion implantation, and other orderly surface distortions can be treated as easily as uniform wafers, as long as distorted three-dimensional order remains. Another advantage of matrix methods is that refraction and external and internal surface reflections are included in the computation automatically. An additional set of diffracting Bragg planes parallel to the surfaces can be included with little complication, thus allowing investigation of double-diffraction effects at the intersection of two diffraction cones.

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