International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

International Tables for Crystallography (2006). Vol. A, ch. 14.2, p. 849

Section 14.2.2.3. Weissenberg complexes

W. Fischera and E. Kocha*

aInstitut für Mineralogie, Petrologie und Kristallographie, Philipps-Universität, D-35032 Marburg, Germany
Correspondence e-mail:  kochelke@mailer.uni-marburg.de

14.2.2.3. Weissenberg complexes

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Depending on their site-symmetry groups, two kinds of Wyckoff position may be distinguished:

  • (i) The site-symmetry group of any point is a proper subgroup of another site-symmetry group from the same space group. Then, the Wyckoff position contains, among others, point configurations with the property that the distance between two suitable chosen points is shorter than any small number [\varepsilon \gt 0].

    Example

    Each point configuration of the lattice complex with the characteristic Wyckoff position [P4/mmm\ 4j\ m.2m\ xx0] may be imagined as squares of four points surrounding the points of a tetragonal primitive lattice. For [x \rightarrow 0], the squares become infinitesimally small. Point configurations with [x = 0] show site symmetry [4/mmm], their multiplicity is decreased from 4 to 1, and they belong to lattice complex [P4/mmm\ a].

  • (ii) The site-symmetry group of any point belonging to the regarded Wyckoff position is not a subgroup of any other site-symmetry group from the same space group.

    Example

    In Pmma, there does not exist a site-symmetry group that is a proper supergroup of mm2, the site-symmetry group of Wyckoff position [Pmma\ 2e\ {1 \over 4}0z]. As a consequence, the distance between any two symmetrically equivalent points belonging to Pmma e cannot become shorter than the minimum of [{1 \over 2}a, b] and c.

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)[link]]. These lattice complexes have been called Weissenberg complexes by Fischer et al. (1973)[link]. 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 [P2_{1}2_{1}2_{1}\ a]. 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, [{1 \over 4}00] refers to Pmmm a and [{1 \over 4}0{1 \over 4}] 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.








































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