International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.5, p. 190   | 1 | 2 |

Section 1.5.5.4.3. k-vector types for non-holosymmetric space groups

M. I. Aroyoa* and H. Wondratschekb

aDepartamento de Fisíca de la Materia Condensada, Facultad de Cienca y Technología, Universidad del País Vasco, Apartado 644, 48080 Bilbao, Spain , and bInstitut für Kristallographie, Universität, D-76128 Karlsruhe, Germany
Correspondence e-mail:  wmpararm@lg.ehu.es

1.5.5.4.3. k-vector types for non-holosymmetric space groups

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The k-vector labels of CDML are primarily listed for the holosymmetric space groups. These lists are kept and supplemented for the non-holosymmetric space groups. In this way many superfluous k-vector labels are introduced.

Examples

  • (1) Arithmetic crystal class [m\overline{3}I]. In its reciprocal-space group [(Fm\overline{3})^*], the introduction of the plane [AA = [\Gamma \,H_2\,N]] is unnecessary because the plane [A = [\Gamma \,N\,H]] of Wintgen position 96 [j\ m..] of [(Fm\overline{3}m)^*] can be extended to [A \cup AA = [\Gamma \,H_2\,H]] in the reciprocal-space group [(Fm\overline{3})^*], cf. Fig. 1.5.5.2[link] and Table 1.5.5.2[link]. In [(Fm\overline{3})^*], both planes, A and AA, belong to Wintgen position [48\ h\ m..]. The parameter description is extended from [x,y,0]: [0 \,\lt \, x \,\lt \, y \,\lt \, \textstyle{1 \over 2} - x( \lt \,\textstyle{1 \over 4})] to [0 \,\lt \, y \,\lt \, \textstyle{1 \over 2} - x \,\lt \, \textstyle{1 \over 2}].

  • (2) In the previous example, during the transition from the group [(Fm\overline{3}m)^*] to the subgroup [(Fm\overline{3})^*] the order of the little co-group of the special k vectors of [(Fm\overline{3}m)^*] was not changed. In other cases, the little co-group may be reduced to a subgroup. Such k vectors may then be incorporated into a more general Wintgen position and described by an extension of the parameter range.

    Arithmetic crystal class [m\overline{3}mI], plane [[\Gamma \,H \,N] = x, y,0]. In [(Fm\overline{3}m)^*], see Fig. 1.5.5.1[link], all points (Γ, H, N) and lines (Δ, Σ, G) of the boundary of the asymmetric unit are special. In [(Fm\overline{3})^*], see Fig. 1.5.5.2[link], the lines Δ and [[\Gamma\,H_2]\sim\Delta] are special but Σ, G and [[N\,H_2]\sim G] belong to the plane ([A \cup AA]). The free parameter range on the line G is [0\,\lt\,y\,\lt\,\textstyle{1 \over 4}]. Therefore, the parameter ranges of ([A \cup AA\cup G \cup\Sigma]) in [x,y,0] can be taken as: [0 \,\lt \, x \,\lt \, \textstyle{1 \over 2} - y \,\lt \, \textstyle{1 \over 2}] for [A \cup AA \cup \Sigma] and [0 \,\lt \, y = \textstyle{1 \over 2} - x \,\lt \, \textstyle{1 \over 4}] for G.








































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