International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A, ch. 2.1, p. 172

Section 2.1.3.16. Crystallographic groups in one dimension

Th. Hahna and A. Looijenga-Vosb

2.1.3.16. Crystallographic groups in one dimension

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In one dimension, only one crystal family, one crystal system and one Bravais lattice exist. No name or common symbol is required for any of them. All one-dimensional lattices are primitive, which is symbolized by the script letter [{\scr p}]; cf. Table 2.1.1.1[link].

There occur two types of one-dimensional point groups, 1 and [m \equiv \bar{1}]. The latter contains reflections through a point (reflection point or mirror point). This operation can also be described as inversion through a point, thus [m \equiv \bar{1}] for one dimension; cf. Section 2.1.2[link].

Two types of line groups (one-dimensional space groups) exist, with Hermann–Mauguin symbols [{\scr p}1] and [{\scr p}m \equiv {\scr p}\bar{1}], which are illustrated in Fig. 2.1.3.13[link]. Line group [{\scr p}1], which consists of one-dimensional translations only, has merely one (general) position with coordinate x. Line group [{\scr p}m] consists of one-dimensional translations and reflections through points. It has one general and two special positions. The coordinates of the general position are x and [\bar{x}]; the coordinate of one special position is 0, that of the other [{1 \over 2}]. The site symmetries of both special positions are [m \equiv \bar{1}]. For [{\scr p}1], the origin is arbitrary, for [{\scr p}m] it is at a reflection point.

[Figure 2.1.3.13]

Figure 2.1.3.13 | top | pdf |

The two line groups (one-dimensional space groups). Small circles are reflection points; large circles represent the general position; in line group [{\scr p}1], the vertical bars are the origins of the unit cells.

The one-dimensional point groups are of interest as `edge symmetries' of two-dimensional `edge forms'; they are listed in Table 3.2.3.1[link] . The one-dimensional space groups occur as projection and section symmetries of crystal structures.

References

International Tables for Crystallography (2002). Vol. A, 5th ed., edited by Th. Hahn. Dordrecht: Kluwer Academic Publishers. [Abbreviated as IT A (2002).]
International Tables for Crystallography (2010). Vol. A1, 2nd ed., edited by H. Wondratschek & U. Müller. Chichester: Wiley. [Abbreviated as IT A1 (2010).]
International Tables for X-ray Crystallography (1952). Vol. I, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press. [Revised editions: 1965, 1969 and 1977. Abbreviated as IT (1952).]
Internationale Tabellen zur Bestimmung von Kristallstrukturen (1935). 1. Band, edited by C. Hermann. Berlin: Borntraeger. [Revised edition: Ann Arbor: Edwards (1944). Abbreviated as IT (1935).]
Wolff, P. M. de, Belov, N. V., Bertaut, E. F., Buerger, M. J., Donnay, J. D. H., Fischer, W., Hahn, Th., Koptsik, V. A., Mackay, A. L., Wondratschek, H., Wilson, A. J. C. & Abrahams, S. C. (1985). Nomenclature for crystal families, Bravais-lattice types and arithmetic classes. Report of the International Union of Crystallography Ad-hoc Committee on the Nomenclature of Symmetry. Acta Cryst. A41, 278–280.








































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