International
Tables for
Crystallography
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

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

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

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

 (1) If the lattice symbol of is F or I, it has to be replaced by I or F, e.g. is isomorphic to Imm2 for the arithmetic crystal class . 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 ):

### References

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