The absorption-correction factor of multifaceted crystals

A mathematical analysis of the evaluation of the absorption-correction factors in convex polyhedral crystals is made. Formulae are given that permit the exact evaluation of the factors and their derivatives and that do not depend on the subdivision of the crystal into tetrahedra. A new method of defining the Howells polyhedra as the included volume of a set of planes is also described. The method allows additional constraints to be introduced, which could be used to deal with inhomogeneities in either the crystal or the incident beam. Computation based on these ideas gives rapid and exact evaluation of both the absorption-correction factor and its derivatives, with negligible rounding-error problems. In addition, a formula is given for scattering by a general tetrahedron with a wide range of orientations that can be used as a test of computer programs.