Crystal systems and crystal families

Wondratschek, H.

doi:10.1107/97809553602060000538

International
Tables for
Crystallography
Volume A1
Symmetry relations between space groups
Edited by Hans Wondratschek and Ulrich Müller

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International Tables for Crystallography (2006). Vol. A1. ch. 1.2, p. 16 | 1 | 2 |

Section 1.2.5.5. Crystal systems and crystal families

Hans Wondratschek^a ^*

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

1.2.5.5. Crystal systems and crystal families

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The example of [P1] mentioned above shows that the point group of the lattice may be systematically of higher order than that of its space group. There are obviously point groups and thus space groups that belong to a holohedral crystal class and those that do not. The latter can be assigned to a holohedral crystal class uniquely according to the following definition:⁸

Definition 1.2.5.5.1. A crystal class C of a space group $[{\cal G}]$ is either holohedral H or it can be assigned uniquely to H by the condition: any point group of C is a subgroup of a point group of H but not a subgroup of a holohedral crystal class $[{\bf H\,'}]$ of smaller order. The set of all crystal classes of space groups that are assigned to the same holohedral crystal class is called a crystal system of space groups.

The 32 crystal classes of space groups are classified into seven crystal systems which are called triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal and cubic. There are four crystal systems of plane groups: oblique, rectangular, square and hexagonal. Like the space groups, the crystal classes of point groups are classified into the seven crystal systems of point groups.

Apart from accidental lattice symmetries, the space groups of different crystal systems have lattices of different symmetry. As an exception, the hexagonal primitive lattice occurs in both hexagonal and trigonal space groups as the typical lattice. Therefore, the space groups of the trigonal and the hexagonal crystal systems are more related than space groups from other different crystal systems. Indeed, in different crystallographic schools the term `crystal system' was used for different objects. One sense of the term was the `crystal system' as defined above, while another sense of the old term `crystal system' is now called a `crystal family' according to the following definition [for definitions that are also valid in higher-dimensional spaces, see Brown et al. (1978) or IT A, Chapter 8.2 ]:

Definition 1.2.5.5.2. In three-dimensional space, the classification of the set of all space groups into crystal families is the same as that into crystal systems with the one exception that the trigonal and hexagonal crystal systems are united to form the hexagonal crystal family. There is no difference between crystal systems and crystal families in the plane.

The partition of the space groups into crystal families is the most universal one. The space groups and their types, their crystal classes and their crystal systems are classified by the crystal families. Analogously, the crystallographic point groups and their crystal classes and crystal systems are classified by the crystal families of point groups. Lattices, their Bravais types and lattice systems can also be classified into crystal families of lattices; cf. IT A, Chapter 8.2 .

References

Brown, H., Bülow, R., Neubüser, J., Wondratschek, H. & Zassenhaus, H. (1978). Crystallographic groups of four-dimensional space. New York: John Wiley & Sons.Google Scholar

International Tables for Crystallography (2006). Vol. A1. ch. 1.2, p. 16