Figure 1.5.5.2

Aroyo, M. I.; Wondratschek, H.

doi:10.1107/97809553602060000553

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

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International Tables for Crystallography (2006). Vol. B. ch. 1.5, p. 171 | 1 | 2 |

Figure 1.5.5.2

M. I. Aroyo^a ^* and H. Wondratschek^b

^a Faculty of Physics, University of Sofia, bulv. J. Boucher 5, 1164 Sofia, Bulgaria , and ^bInstitut für Kristallographie, Universität, D-76128 Karlsruhe, Germany
Correspondence e-mail: aroyo@phys.uni-sofia.bg

Figure 1.5.5.2

Symmorphic space group $[F\bar{3}m]$ (isomorphic to the reciprocal-space group $[{\cal G}^*]$ of $[m\bar{3}I]$ ). (a) The asymmetric unit (thick dashed edges) half imbedded in and half protruding from the Brillouin zone, which is a cubic rhombdodecahedron (as in Fig. 1.5.5.1). (b) The asymmetric unit $[\Gamma H\Gamma_{1}P]$ , IT A, p. 610. The representation domain of CDML is $[\Gamma HH_{3}P]$ . Both bodies have ΓHNP in common; $[H\Gamma_{1}NP]$ is mapped onto $[\Gamma NH_{3}P]$ by a twofold rotation around NP. The representation domain as the asymmetric unit would be the better choice because it is congruent to the asymmetric unit of IT A and is fully imbedded in the Brillouin zone. Coordinates of the points: $[\Gamma = 0, 0, 0 \sim \Gamma_{1} = {1 \over 2}, {1 \over 2}, 0]$ ; $[P = {1 \over 4}, {1 \over 4}, {1 \over 4}]$ ; $[H = {1 \over 2}, 0, 0 \sim H_{1} = 0, 0, {1 \over 2} \sim H_{2} = {1 \over 2}, {1 \over 2}, {1 \over 2} \sim H_{3} = 0, {1 \over 2}, 0]$ ; $[N = {1 \over 4}, {1 \over 4}, 0 ]$ $[\sim N_{1} ={1 \over 4}, {1 \over 4}, {1 \over 2}]$ ; the sign ∼ means symmetrically equivalent. Lines: $[\Lambda = \Gamma P = x, x, x \sim P \Gamma_{1} = x, x, {1 \over 2} - x]$ ; $[F = HP = {1 \over 2} - x, x, x \sim F_{1} = PH_{2} =]$ $[ x, x, x \sim F_{2} = PH_{1} = x, x, {1 \over 2} - x]$ ; $[\Delta = \Gamma H = x, 0, 0 \sim H \Gamma_{1} =]$ $[ {1 \over 2}, y, 0]$ ; $[D = PN = {1 \over 4}, {1 \over 4}, z]$ . ( [G = NH = x] , $[{1 \over 2} - x]$ , 0 and $[\Sigma = \Gamma N = x, x, 0 \sim N\Gamma_{1} = x, x, 0]$ are not special lines.) Planes: $[A = \Gamma HN =]$ [ x, y, 0] ; $[AA = \Gamma_{1} NH = x, y, 0]$ ; $[B = HNP = x, {1 \over 2} - x, z \sim PN_{1} H_{1} =]$ [ x, x, z] ; $[C = \Gamma NP = x, x, z]$ ; $[J = \Gamma HP = x, y, y \sim \Gamma PH_{1} = x, x, z]$ . (The boundary planes B, C and J are parts of the general position GP.) Large black circles: special points of the asymmetric unit; small black circle: special point $[\Gamma_{1} \sim \Gamma]$ ; small open circles: other special points; dashed lines: edges and special line D of the asymmetric unit. The edge $[\Gamma \Gamma_{1}]$ is not a special line but is part of the boundary plane $[A \cup AA]$ . For the parameter ranges see Table 1.5.5.2.