Three-Beam Diffraction in Finite Perfect Crystals. I. Exploring Laue–Laue and Bragg–Laue Scattering Using a Series-Expansion Approach for the Solution of the Takagi–Taupin Equations

Using the boundary-value Green-function technique, analytical expressions in the form of finite series expansions are obtained for the relative change in the integrated power of the primary reflection due to the gradual excitation of a secondary reciprocal-lattice point on the Ewald sphere. Solutions are found for both a Laue–Laue and a Bragg–Laue case in finite shaped crystals confined by the scattering vectors. When the crystal sizes do not exceed the Pendellösung length of the involved reflections, the profiles exhibit the same qualitative features in the two cases. The solutions do however indicate a strong dependence on the outer crystal dimensions – which add a geometrical aspect to the interpretation of the Aufhellung and Umweganregung concepts.