Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.5, p. 192   | 1 | 2 |
doi: 10.1107/97809553602060000762

Appendix A1.5.1. Reciprocal-space groups [({\cal G})^{*}]

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:

This table is based on Table 1 of Wintgen (1941)[link].

In order to obtain the Hermann–Mauguin symbol of [({\cal G})^{*}] from that of [{\cal G}], one replaces any screw rotations by rotations and any glide reflections by reflections. The result is the symmorphic space group [{\cal G}_{0}({\cal G})]. For most space groups [{\cal G}], the reciprocal-space group [({\cal G})^{*}] is isomorphic to [{\cal G}_{0}({\cal G})], i.e. [{\cal G}_{0}({\cal G})] and [({\cal G})^{*}] belong to the same arithmetic crystal class. In the following cases [({\cal G})^{*}] is isomorphic to a symmorphic space group [{\cal G}_0] which is different from [{\cal G}_{0}({\cal G})]. Thus the arithmetic crystal classes of [{\cal G}] and [({\cal G})^{*}] are different, i.e. [({\cal G})^{*}] can not be obtained in this simple way:

  • (1) If the lattice symbol of [{\cal G}] is F or I, it has to be replaced by I or F, e.g. [({\cal G})^{*}] is isomorphic to Imm2 for the arithmetic crystal class [{\cal G} = mm2F]. The tetragonal space groups form an exception to this rule; for these the symbol I persists.

  • (2) The other exceptions are listed in the following table (for the symbols of the arithmetic crystal classes see IT A, Section 8.2.3[link] ):[\eqalign{\hbox{Arithmetic }&\hbox{crystal class of } {\cal G}\quad\quad\hbox{Reciprocal-space group } ({\cal G})^{{*}}\cr &\overline{4}m2I\phantom{ithmetic crystal class of }\qquad \ I\overline{4}2m\cr &\overline{4}2mI\phantom{ithmetic crystal class of }\qquad \ I\overline{4}m2\cr &321P\phantom{ithmetic crystal class of }\qquad \ P312\cr &312P\phantom{ithmetic crystal class of }\qquad \ P321\cr &3m1P\phantom{ithmetic crystal class of }\qquad P31m\cr &31mP\phantom{ithmetic crystal class of }\qquad P3m1\cr &\overline{3}1mP\phantom{ithmetic crystal class of }\qquad P\overline{3}m1\cr &\overline{3}m1P\phantom{ithmetic crystal class of }\qquad P\overline{3}1m\cr &\overline{6}m2P\phantom{ithmetic crystal class of }\qquad P\overline{6}2m\cr &\overline{6}2mP\phantom{ithmetic crystal class of }\qquad P\overline{6}m2\cr}]


Wintgen, G. (1941). Zur Darstellungstheorie der Raumgruppen. Math. Ann. 118, 195–215.

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