International
Tables for Crystallography Volume A Space-group symmetry Edited by Th. Hahn © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. A, ch. 14.2, p. 849
Section 14.2.2.3. Weissenberg complexes^{a}Institut für Mineralogie, Petrologie und Kristallographie, Philipps-Universität, D-35032 Marburg, Germany |
Depending on their site-symmetry groups, two kinds of Wyckoff position may be distinguished:
A lattice complex contains either Wyckoff positions exclusively of the first or exclusively of the second kind. Most lattice complexes are made up from Wyckoff positions of the first kind.
There exist, however, 67 lattice complexes that do not contain point configurations with infinitesimal short distances between symmetry-related points [cf. Hauptgitter (Weissenberg, 1925)]. These lattice complexes have been called Weissenberg complexes by Fischer et al. (1973). The 36 invariant lattice complexes are trivial examples of Weissenberg complexes. In addition, there exist 24 univariant (monoclinic 2, orthorhombic 5, tetragonal 7, hexagonal 5, cubic 5) and 6 bivariant Weissenberg complexes (monoclinic 1, orthorhombic 2, tetragonal 1, hexagonal 2). The only trivariant Weissenberg complex is . All Weissenberg complexes with degrees of freedom have the following common property: each Weissenberg complex contains at least two invariant limiting complexes belonging to the same crystal family.
Example
Pmma e is a comprehensive complex of Pmmm a and of Cmmm a. Within the characteristic Wyckoff position, refers to Pmmm a and to Cmmm a.
Except for the seven invariant plane lattice complexes, there exists only one further Weissenberg complex within the plane groups, namely the univariant rectangular complex p2mg c.
References
Fischer, W., Burzlaff, H., Hellner, E. & Donnay, J. D. H. (1973). Space groups and lattice complexes. NBS Monograph No. 134. Washington: National Bureau of Standards.Weissenberg, K. (1925). Der Aufbau der Kristalle. I. Mitteilung. Die Systematik der Symmetriegruppen von Punktlagen im Diskontinuum. Z. Kristallogr. 62, 13–51.