Special directions in momentum space. III. Practical applications

Abstract

This paper complements two previous papers devoted to Special directions in momentum space. I. Cubic and II. 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.

How to estimate isotropic distributions and mean values in crystalline solids

“…Previous considerations [34,[36][37][38] and those presented in this paper for various symmetries led to similar conclusions, emphasising additionally that HSD are highly unprofitable and the traditional manner of calculating the isotropic average yields incomparably worse results than the use of SDs. Generally, of course, the quality of f 0 a ( p ) depends on the anisotropy as well as, in the case of experimental investigations, on the statistical precision of the data and the experimental resolution.…”

“…While there is no doubt that the number of terms used in equation ( 1) is important, the choice of sampling directions (orientation (Θ,ϕ)), which are used to determine functions f l,ν ( p ), is extremely important. As shown in figure 6 in [38], even a single direction, but a very particular single direction, could be better than three HSDs.…”

“…There are a few papers [28][29][30][31][32][33][34][35][36][37] devoted to determining the isotropic and anisotropic components of some quantities with applying SDs. Unfortunately, they are identified with some particular theoretical calculations and the standard way of determining the isotropic component is still based on papers [4] and [6], and usually employing Eq.…”

“…Bansil proposed sets of SDs, which would optimize calculations of the isotropic component, f 0 (p). In subsequent papers [29][30][31][32][33][34][35][36][37], in which (except for [32,33,36]) cubic structures were considered, the authors estimated such sets of SDs which also enabled the determination of anisotropic f l,ν (p). To the best of our knowledge, SDs were utilised in theoretical calculations only in a few papers [38][39][40][41][42][43][44][45][46][47][48].…”