Special directions in momentum space. II. Hexagonal, tetragonal and trigonal symmetries

How to estimate isotropic distributions and mean values in crystalline solids

The concept of special directions in the Brillouin zone and the applicability of Houston's formula (or its extended versions) to both theoretical and experimental investigations are discussed. We propose some expressions to describe the isotropic component in systems having both cubic and non-cubic symmetry. The results presented have implications for both experimentalists who want to obtain average properties from a small number of measurements on single crystals, and for theoretical calculations which are to be compared with isotropic experimental measurements, for example coming from investigations of polycrystalline or powder samples. As George Orwell might have put it: all directions are equal, but some directions are more equal than others.

Special directions in momentum space. III. Practical applications

This paper complements two previous papers devoted toSpecial directions in momentum space. I. CubicandII. Hexagonal, tetragonal and trigonal symmetries[Kontrym-Sznajd & Samsel-Czekala (2011).J. Appl. Cryst.44, 1246–1254; Kontrym-Sznajd & Samsel-Czekala (2012).J. Appl. Cryst.45, 1254–1260], in which sets of special directions (SDs) were proposed. Such directions, employing the full symmetry of the Brillouin zone, allow for constructing in the whole space anisotropic quantities from their known values along a few directions. SDs also define which spectra, measured in, for example, Compton scattering experiments, are the most efficient for reconstructing three-dimensional densities from their one-dimensional projections. This paper, in which new sets of special directions (SDs) for cubic structures are proposed, is devoted mainly to practical applications of SDs. Taking into account experimental uncertainty, an optimal strategy for experimental investigations is discussed.