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

International Tables for Crystallography (2010). Vol. B, ch. 1.5, pp. 189-190   | 1 | 2 |

Section 1.5.5.4.1. Representation domains and asymmetric units

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

1.5.5.4.1. Representation domains and asymmetric units

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When the symmetry of the reciprocal lattice allows, the shape of the asymmetric unit may be chosen to be much simpler than that of the representation domain.

Examples

  • (1) Arithmetic crystal class [4/mmmI]. The parameter ranges for the special lines and planes of the asymmetric unit and for general k vectors of the reciprocal-space group [(F4/mmm)^*] [setting [(I4/mmm)^*]] are listed in Tables 1.5.5.3[link] and 1.5.5.4[link]. One can describe the corresponding conditions of the representation domain by the boundary plane [x,y,z] = [\{1+( c/ a)^2[1–2(x+ y)]\}/4] which for [c/a \,\lt \, 1] forms the triangle [[Z_0\,Z_2\,P]] in Fig. 1.5.5.3[link] but for [c/a\,\gt \, 1] the pentagon [[S_2\,R\,P\,G\,S]] in Fig. 1.5.5.4[link]. The inner points of this boundary plane are points of the general position GP with the exception of the line [Q = x,\textstyle{1 \over 2}-x,\textstyle{1 \over 4}], which is a twofold rotation axis. The boundary conditions for the representation domain depend on [c/a]; they are much more complicated than those, [x,y,z=\textstyle{1 \over 4}], for the asymmetric unit.

  • (2) Arithmetic crystal class [mm2F], see Figs. 1.5.5.5[link] to 1.5.5.7[link][link]. In the reciprocal-space group [(Imm2)^*] the lines [\Lambda] and LE belong to Wintgen position [2\ a\ mm2], as do the lines [Q,QA,\Lambda_1] and [\Lambda_3] if present. The lines [H] and [HA] belong to the Wintgen position [2\ b\ mm2]; as do the lines [G, GA,H_1] and [H_3] if present. The lines [\Sigma], [\Sigma_1], A, A1, C and U belong to the plane [x,0,z]; the lines [\Delta], B, B1 and D belong to the plane [0,y,z]. The decisive boundary plane of the representation domain is [x a^{*2} + yb^{*2} + zc^{*2} = d^{*2}/4], where [d^{*2} = a^{*2} + b^{*2} + c^{*2}]; it is a hexagon for Fig. 1.5.5.5[link] and a parallelogram for Figs. 1.5.5.6[link] and 1.5.5.7[link]. There is no relation of the lattice parameters for which all the above-mentioned lines are realized on the surface of the representation domain simultaneously; either two or three of them do not appear and the length of the others depends on the boundary plane, see Tables and Figs. 1.5.5.5 to 1.5.5.7.

    The boundary conditions for the asymmetric unit are independent of the lattice parameters and the boundary plane is always represented by the simple equation [x,y,\textstyle{1 \over 2}]: [0 \,\lt \, x,y \,\lt \, \textstyle{1 \over 2}]. By introducing flagpoles and wings, the description may become uni-arm.








































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