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

International Tables for Crystallography (2006). Vol. A. ch. 1.4, pp. 7-11
https://doi.org/10.1107/97809553602060000503

## Chapter 1.4. Graphical symbols for symmetry elements in one, two and three dimensions

Th. Hahna*

aInstitut für Kristallographie, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany
Correspondence e-mail: hahn@xtal.rwth-aachen.de

This chapter lists the graphical symbols for symmetry elements used throughout this volume. The lists are accompanied by notes and cross-references to recent IUCr nomenclature reports.

Keywords: symbols; crystallography; symmetry elements; symmetry planes; symmetry lines; symmetry axes.

### 1.4.1. Symmetry planes normal to the plane of projection (three dimensions) and symmetry lines in the plane of the figure (two dimensions)

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Symmetry plane or symmetry lineGraphical symbolGlide vector in units of lattice translation vectors parallel and normal to the projection planePrinted symbol
None m
Axial' glide plane lattice vector normal to projection plane a, b or c
Double' glide plane (in centred cells only) e
Diagonal' glide plane n
Diamond' glide plane (pair of planes; in centred cells only) along line parallel to projection plane, combined with normal to projection plane (arrow indicates direction parallel to the projection plane for which the normal component is positive) d
For further explanations of the double' glide plane e see Note (iv) below and Note (x) in Section 1.3.2 .
Glide planes d occur only in orthorhombic F space groups, in tetragonal I space groups, and in cubic I and F space groups. They always occur in pairs with alternating glide vectors, for instance and . The second power of a glide reflection d is a centring vector.

### 1.4.2. Symmetry planes parallel to the plane of projection

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Symmetry planeGraphical symbolGlide vector in units of lattice translation vectors parallel to the projection planePrinted symbol
Reflection plane, mirror plane None m
Axial' glide plane lattice vector in the direction of the arrow a, b or c
Double' glide plane (in centred cells only) e
Diagonal' glide plane n
Diamond' glide plane§ (pair of planes; in centred cells only) in the direction of the arrow; the glide vector is always half of a centring vector, i.e. one quarter of a diagonal of the conventional face-centred cell d
The symbols are given at the upper left corner of the space-group diagrams. A fraction h attached to a symbol indicates two symmetry planes with heights' h and above the plane of projection; e.g. stands for and . No fraction means and (cf. Section 2.2.6 ).
For further explanations of the double' glide plane e see Note (iv) below and Note (x) in Section 1.3.2 .
§Glide planes d occur only in orthorhombic F space groups, in tetragonal I space groups, and in cubic I and F space groups. They always occur in pairs with alternating glide vectors, for instance and . The second power of a glide reflection d is a centring vector.

### 1.4.3. Symmetry planes inclined to the plane of projection (in cubic space groups of classes and only)

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Symmetry planeGraphical symbol for planes normal toGlide vector in units of lattice translation vectors for planes normal toPrinted symbol
[011] and [101] and [011] and [101] and
Reflection plane, mirror plane None None m
Axial' glide plane lattice vector along [100] a or b
Axial' glide plane lattice vector along or along [011]
Double' glide plane [in space groups (217) and (229) only] Two glide vectors: along [100] and along or along [011] Two glide vectors: along [010] and along or along [101] e
Diagonal' glide plane One glide vector: along or along [111]§ One glide vector: along or along [111]§ n
Diamond' glide plane†† (pair of planes; in centred cells only) along or along [111] d
along or along
The symbols represent orthographic projections. In the cubic space-group diagrams, complete orthographic projections of the symmetry elements around high-symmetry points, such as ; ; , are given as inserts'.
For further explanations of the double' glide plane e see Note (iv) below and Note (x) in Section 1.3.2 .
§In the space groups (216), (225) and (227), the shortest lattice translation vectors in the glide directions are or and or , respectively.
The glide vector is half of a centring vector, i.e. one quarter of the diagonal of the conventional body-centred cell in space groups (220) and (230).
††Glide planes d occur only in orthorhombic F space groups, in tetragonal I space groups, and in cubic I and F space groups. They always occur in pairs with alternating glide vectors, for instance and . The second power of a glide reflection d is a centring vector.

### 1.4.4. Notes on graphical symbols of symmetry planes

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 (i) The graphical symbols and their explanations (columns 2 and 3) are independent of the projection direction and the labelling of the basis vectors. They are, therefore, applicable to any projection diagram of a space group. The printed symbols of glide planes (column 4), however, may change with a change of the basis vectors, as shown by the following example. In the rhombohedral space groups (161) and (167), the dotted line refers to a c glide when described with hexagonal axes' and projected along [001]; for a description with rhombohedral axes' and projection along [111], the same dotted glide plane would be called n. The dash-dotted n glide in the hexagonal description becomes an a, b or c glide in the rhombohedral description; cf. the first footnote in Section 1.3.1. (ii) The graphical symbols for glide planes in column 2 are not only used for the glide planes defined in Chapter 1.3 , but also for the further glide planes g which are mentioned in Section 1.3.2 (Note x ) and listed in Table 4.3.2.1 ; they are explained in Sections 2.2.9 and 11.1.2 . (iii) In monoclinic space groups, the parallel' glide vector of a glide plane may be along a lattice translation vector which is inclined to the projection plane. (iv) In 1992, the International Union of Crystallography introduced the double' glide plane e and the graphical symbol ..--..-- for e glide planes oriented normal' and inclined' to the plane of projection (de Wolff et al., 1992); for details of e glide planes see Chapter 1.3 . Note that the graphical symbol for e glide planes oriented parallel' to the projection plane has already been used in IT (1935) and IT (1952).

### 1.4.5. Symmetry axes normal to the plane of projection and symmetry points in the plane of the figure

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Symmetry axis or symmetry pointGraphical symbolScrew vector of a right-handed screw rotation in units of the shortest lattice translation vector parallel to the axisPrinted symbol (partial elements in parentheses)
Identity None None 1
None 2
Twofold screw axis: 2 sub 1'
None 3
Threefold screw axis: 3 sub 1'
Threefold screw axis: 3 sub 2'
None 4 (2)
Fourfold screw axis: 4 sub 1'
Fourfold screw axis: 4 sub 2'
Fourfold screw axis: 4 sub 3'
None 6 (3,2)
Sixfold screw axis: 6 sub 1'
Sixfold screw axis: 6 sub 2'
Sixfold screw axis: 6 sub 3'
Sixfold screw axis: 6 sub 4'
Sixfold screw axis: 6 sub 5'
None
Inversion axis: 3 bar' None
Inversion axis: 4 bar' None
Inversion axis: 6 bar' None
Twofold rotation axis with centre of symmetry None
Twofold screw axis with centre of symmetry
Fourfold rotation axis with centre of symmetry None
4 sub 2' screw axis with centre of symmetry
Sixfold rotation axis with centre of symmetry None
6 sub 3' screw axis with centre of symmetry

Notes on the heights' h of symmetry points , , and :

 (1) Centres of symmetry and , as well as inversion points and on and axes parallel to [001], occur in pairs at heights' h and . In the space-group diagrams, only one fraction h is given, e.g. stands for and . No fraction means and . In cubic space groups, however, because of their complexity, both fractions are given for vertical axes, including and . (2) Symmetries and contain vertical and axes; their and inversion points coincide with the centres of symmetry. This is not indicated in the space-group diagrams. (3) Symmetries and also contain vertical and axes, but their and inversion points alternate with the centres of symmetry; i.e. points at h and interleave with or points at and . In the tetragonal and hexagonal space-group diagrams, only one fraction for and one for or is given. In the cubic diagrams, all four fractions are listed for ; e.g. (No. 223): : ; : .

### 1.4.6. Symmetry axes parallel to the plane of projection

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Symmetry axisGraphical symbolScrew vector of a right-handed screw rotation in units of the shortest lattice translation vector parallel to the axisPrinted symbol (partial elements in parentheses)
Twofold rotation axis None 2
Twofold screw axis: 2 sub 1'
Fourfold rotation axis None 4 (2)
Fourfold screw axis: 4 sub 1'
Fourfold screw axis: 4 sub 2'
Fourfold screw axis: 4 sub 3'
Inversion axis: 4 bar' None
Inversion point on 4 bar'-axis point
The symbols for horizontal symmetry axes are given outside the unit cell of the space-group diagrams. Twofold axes always occur in pairs, at heights' h and above the plane of projection; here, a fraction h attached to such a symbol indicates two axes with heights h and . No fraction stands for and . The rule of pairwise occurrence, however, is not valid for the horizontal fourfold axes in cubic space groups; here, all heights are given, including and . This applies also to the horizontal axes and the inversion points located on these axes.

### 1.4.7. Symmetry axes inclined to the plane of projection (in cubic space groups only)

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Symmetry axisGraphical symbolScrew vector of a right-handed screw rotation in units of the shortest lattice translation vector parallel to the axisPrinted symbol (partial elements in parentheses)
Twofold rotation axis None 2
Twofold screw axis: 2 sub 1'
Threefold rotation axis None 3
Threefold screw axis: 3 sub 1'
Threefold screw axis: 3 sub 2'
Inversion axis: 3 bar' None
The dots mark the intersection points of axes with the plane at . In some cases, the intersection points are obscured by symbols of symmetry elements with height ; examples: (203), origin choice 2; (222), origin choice 2; (223); (229); (230).

### References

Internationale Tabellen zur Bestimmung von Kristallstrukturen (1935). I. Band, edited by C. Hermann. Berlin: Borntraeger. [Reprint with corrections: Ann Arbor: Edwards (1944). Abbreviated as IT (1935).]Google Scholar
International Tables for X-ray Crystallography (1952). Vol. I, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press. [Abbreviated as IT (1952).]Google Scholar
Wolff, P. M. de, Billiet, Y., Donnay, J. D. H., Fischer, W., Galiulin, R. B., Glazer, A. M., Hahn, Th., Senechal, M., Shoemaker, D. P., Wondratschek, H., Wilson, A. J. C. & Abrahams, S. C. (1992). Symbols for symmetry elements and symmetry operations. Final Report of the International Union of Crystallography Ad-hoc Committee on the Nomenclature of Symmetry. Acta Cryst. A48, 727–732.Google Scholar