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

Abstract

This paper is a continuation of a previous one, Special directions in momentum space. I. Cubic symmetries [Kontrym‐Sznajd & Samsel‐Czekała (2011). J. Appl. Cryst.44, 1246–1254], where new sets of special directions (SDs), having the full symmetry of the Brillouin zone, were proposed for cubic lattices. In the present paper, such directions are derived for structures with unique six‐, four‐ and threefold axes, i.e. hexagonal, tetragonal and trigonal lattices, for both two‐ and three‐dimensional space. The SDs presented here allow for construction, in the whole space, of anisotropic quantities from the knowledge of such quantities along a limited number of SDs. The task at hand is to determine as many anisotropic components as the number of available sampling directions. Also discussed is a way of dealing with data when the number of anisotropic components is restricted by a non‐optimal set of SDs.

How to estimate isotropic distributions and mean values in crystalline solids

“…(1), which is analogous to 1D Gauss-Legendre quadrature. Other lattice harmonics containing cos(mϕ) are eliminated by the 1D quadrature for trigonometric polynomials cos(mϕ) [32,36]. Below we demonstrate this with the example when the first harmonic, omitted in Eq.…”

“…5 in Ref. [36]. As an example, 12-SDs in the hcp Brillouin zone (described in Table 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 …”

“…For structures with unique R-fold axes (R = 6, 4 and 3 for hcp, tetragonal and trigonal symmetry, respectively) the lattice harmonics with the full symmetry of the BZ have a very simple form [36]: (6) where P l |m| are associated, and a are the normalization constants. As is the case for cubic lattices, for these structures harmonics have the highest values along HSDs, particularly along…”

“…1 in Ref. [36]. Functions f 12,12 (p) is not presented because its absolute values are lower than 0.1% of f 0 (p=0), i.e.…”

“…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].…”