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

International Tables for Crystallography (2006). Vol. A, ch. 2.2, p. 40

Section 2.2.17. Crystallographic groups in one dimension

Th. Hahna* and A. Looijenga-Vosb

aInstitut für Kristallographie, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany, and bLaboratorium voor Chemische Fysica, Rijksuniversiteit Groningen, The Netherlands
Correspondence e-mail:  hahn@xtl.rwth-aachen.de

2.2.17. 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. Chapter 1.2[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. Chapters 1.3[link] and 1.4[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. 2.2.17.1[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 2.2.17.1]

Figure 2.2.17.1 | 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 10.1.2.1[link] . The one-dimensional space groups occur as projection and section symmetries of crystal structures.








































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