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

© International Union of Crystallography 2006
International Tables for Crystallography (2006). Vol. B. ch. 1.5, p. 173   | 1 | 2 |

Figure 1.5.5.3 

M. I. Aroyoa* and H. Wondratschekb

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 1.5.5.3]
Figure 1.5.5.3

(a), (b) Symmorphic space group [I4/mmm] (isomorphic to the reciprocal-space group [{\cal G}^*] of [4/mmmI]). Diagrams for [a \;\gt\; c], i.e. [c^{*} \;\gt\; a^{*}]. In the figures [a = 1.25c], i.e. [c^{*} = 1.25a^{*}]. (a) Representation domain (thick lines) and asymmetric unit (thick dashed lines, partly protruding) imbedded in the Brillouin zone, which is a tetragonal elongated rhombdodecahedron. (b) Representation domain [\Gamma MXZ_{1}PZ_{0}] and asymmetric unit [\Gamma MXTT_{1}P] of [I4/mmm], IT A, p. 468. The part [\Gamma MXTNZ_{1}P] is common to both bodies; the part [TNPZ_{0}] is equivalent to the part [NZ_{1}PT_{1}] by a twofold rotation around the axis [Q = NP]. Coordinates of the points: [\Gamma = 0, 0, 0]; [X = 0, {1 \over 2}, 0]; [M = {1 \over 2}, {1 \over 2}, 0]; [P = 0, {1 \over 2}, {1 \over 4}]; [N = {1 \over 4}, {1 \over 4}, {1 \over 4}]; [T = 0, 0, {1 \over 4} \sim T_{1} = {1 \over 2}, {1 \over 2}, {1 \over 4}]; [Z_{0} = 0, 0, z_{0} \sim Z_{1} = {1 \over 2}, {1 \over 2}, z_{1}] with [z_{0} = [1 + (c/a)^{2}]/4]; [z_{1} = {1 \over 2} - z_{0}]; the sign ∼ means symmetrically equivalent. Lines: [\Lambda = \Gamma Z_{0} = 0, 0, z]; [V = Z_{1}M = {1 \over 2}, {1 \over 2}, z]; [W = XP = 0, {1 \over 2}, z]; [\Sigma = \Gamma M = x, x, 0]; [\Delta = \Gamma X = 0, y, 0]; [Y = XM = x, {1 \over 2}, 0]; [Q = PN = x, {1 \over 2} - x, {1 \over 4}]. The lines [Z_{0}Z_{1}], [Z_{1}P] and [PZ_{0}] have no special symmetry but belong to special planes. Planes: [C = \Gamma MX = x, y, 0]; [B = \Gamma Z_{0}Z_{1}M = x, x, z]; [A = \Gamma XPZ_{0} = 0, y, z]; [E = MXPZ_{1} = x, {1 \over 2}, z]. The plane [Z_{0}Z_{1}P] belongs to the general position GP. Large black circles: special points belonging to the representation domain; small open circles: [T \sim T_{1}] and [Z_{0} \sim Z_{1}] belonging to special lines; thick lines: edges of the representation domain and special line [Q = NP]; dashed lines: edges of the asymmetric unit. For the parameter ranges see Table 1.5.5.3[link]. (c), (d) Symmorphic space group [I4/mmm] (isomorphic to the reciprocal-space group [{\cal G}^*] of [4/mmmI]). Diagrams for [c \;\gt\; a], i.e. [a^{*} \;\gt\; c^{*}]. In the figures [c = 1.25a], i.e. [a^{*} = 1.25c^{*}]. (c) Representation domain (thick lines) and asymmetric unit (dashed lines, partly protruding) imbedded in the Brillouin zone, which is a tetragonal cuboctahedron. (d) Representation domain [\Gamma S_{1}RXPMSG] and asymmetric unit [\Gamma M_{2}XTT_{1}P] of [I4/mmm], IT A, p. 468. The part [\Gamma S_{1}RXTNP] is common to both bodies; the part TNPMSG is equivalent to the part [T_{1}NPM_{2}S_{1}R] by a twofold rotation around the axis [Q = NP]. Coordinates of the points: [\Gamma = 0, 0, 0]; [X = 0, {1 \over 2}, 0]; [N = {1 \over 4}, {1 \over 4}, {1 \over 4}]; [M = 0, 0, {1 \over 2} \sim M_{2} = {1 \over 2}, {1 \over 2}, 0]; [T = 0, 0, {1 \over4} \sim T_{1} = {1 \over 2}, {1 \over 2}, {1 \over 4}]; [P = 0, {1 \over 2}, {1 \over 4}]; [S = s, s, {1 \over 2} \sim S_{1} = s_{1}, s_{1}, 0] with [s = [1 - (a/c)^{2}]/4]; [s_{1} = {1 \over 2} - s]; [R = r, {1 \over 2}, 0 \sim G = 0, g, {1 \over 2}] with [r = (a/c)^{2}/2]; [g = {1 \over 2} - r]; the sign ∼ means symmetrically equivalent. Lines: [\Lambda = \Gamma M = 0, 0, z]; [W = XP = 0, {1 \over 2}, z]; [\Sigma = \Gamma S_{1} = x, x, 0]; [F = MS = x, x, {1 \over 2}]; [\Delta = \Gamma X = 0, y, 0]; [Y = XR = x, {1 \over 2}, 0]; [U = MG = 0, y, {1 \over 2}]; [Q = PN = x, {1 \over 2} - x, {1 \over 4}]. The lines [GS \sim S_{1} R], [SN \sim NS_{1}] and [GP \sim PR] have no special symmetry but belong to special planes. Planes: [C = \Gamma S_{1} RX = x, y, 0]; [D = MSG = x, y, {1 \over 2}]; [B = \Gamma S_{1} SM = x, x, z]; [A = \Gamma XPGM = 0, y, z]; [E = RXP = x, {1 \over 2}, z]. The plane [S_{1} RPGS] belongs to the general position GP. Large black circles: special points belonging to the representation domain; small open circles: [M_{2} \sim M]; the points [T \sim T_{1}], [S \sim S_{1}] and [G \sim R] belong to special lines; thick lines: edges of the representation domain and special line [Q = NP]; dashed lines: edges of the asymmetric unit. For the parameter ranges see Table 1.5.5.3[link].