Crystal families

Wondratschek, H.

doi:10.1107/97809553602060000515

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

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International Tables for Crystallography (2006). Vol. A. ch. 8.2, pp. 729-730

Section 8.2.7. Crystal families

H. Wondratschek^a ^*

^a Institut für Kristallographie, Universität, D-76128 Karlsruhe, Germany
Correspondence e-mail: hans.wondratschek@physik.uni-karlsruhe.de

8.2.7. Crystal families

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Another classification of space groups, which is a classification of geometric crystal classes and Bravais flocks as well, is that into crystal families.

Definition: A crystal family³ is the smallest set of space groups containing, for any of its members, all space groups of the Bravais flock and all space groups of the geometric crystal class to which this member belongs.

Example

The space-group types R3 and $[P6_{1}]$ belong to the same crystal family because both R3 and P3 belong to the geometric crystal class 3, whereas both P3 and $[P6_{1}]$ are members of the same Bravais flock [6/mmmP] . In this example, P3 serves as a link between R3 and $[P6_{1}]$ .

There are four crystal families in $[E^{2}]$ (oblique m, rectangular o, square t and hexagonal h) and six crystal families in $[E^{3}]$ [triclinic (anorthic) a, monoclinic m, orthorhombic o, tetragonal t, hexagonal h and cubic c]; see Fig. 8.2.1.1.

The classification into crystal families is a rather universal crystallographic concept as it applies to many crystallographic objects: space groups, space-group types, arithmetic and geometric crystal classes of space groups, point groups (morphology of crystals), lattices and Bravais types of lattices.

Remark: In most cases of $[E^{2}]$ and $[E^{3}]$ , the lattices of a given crystal family of lattices have the same point symmetry (for the symbols, see Table 2.1.2.1 ): rectangular op and oc in $[E^{2}]$ ; monoclinic mP and mS, orthorhombic oP, oS, oF and oI, tetragonal tP and tI, cubic cP, cF and cI in $[E^{3}]$ . Only to the hexagonal crystal family in $[E^{3}]$ do lattices with two different point symmetries belong: the hexagonal lattice type hP with point symmetry [6/mmm] and the rhombohedral lattice type hR with point symmetry $[\bar{3}m]$ . In $[E^{4}]$ and higher dimensions, such cases are much more abundant.

Usually, the same type of coordinate system, the so-called `conventional coordinate system ', is used for all space groups of a crystal family, for instance `hexagonal axes' for both hexagonal and rhombohedral lattices; cf. Chapters 2.1 , 2.2 and 9.1 . Other coordinate systems, however, may be used when convenient. To avoid confusion, the use of unconventional coordinate systems should be stated explicitly.

References

International Tables for Crystallography (2006). Vol. A. ch. 8.2, pp. 729-730