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

International Tables for Crystallography (2016). Vol. A, ch. 2.1, pp. 144-148

## Section 2.1.2. Symbols of symmetry elements

Th. Hahna and M. I. Aroyoc

### 2.1.2. Symbols of symmetry elements

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As already introduced in Section 1.2.3 , a symmetry element' (of a given structure or object) is defined as a concept with two components; it is the combination of a geometric element' (that allows the fixed points of a reduced symmetry operation to be located and oriented in space) with the set of symmetry operations having this geometric element in common (element set'). The element set of a symmetry element is represented by the so-called defining operation', which is the simplest symmetry operation from the element set that suffices to identify the geometric element. The alphanumeric and graphical symbols of symmetry elements and the related symmetry operations used throughout the tables of plane (Chapter 2.2 ) and space groups (Chapter 2.3 ) are listed in Tables 2.1.2.1 to 2.1.2.7. For detailed discussion of the definition and symbols of symmetry elements, cf. Section 1.2.3 , de Wolff et al. (1989, 1992) and Flack et al. (2000).

 Table 2.1.2.1| top | pdf | Symbols for symmetry elements and for the corresponding symmetry operations in one, two and three dimensions
SymbolSymmetry element and its orientationDefining symmetry operation with glide or screw vector
m Reflection plane, mirror plane Reflection through the plane
Reflection line, mirror line (two dimensions) Reflection through the line
Reflection point, mirror point (one dimension) Reflection through the point
a, b or c Axial' glide plane Glide reflection through the plane, with glide vector
a or
b or
c or
or
e Double' glide plane (in centred cells only) Two glide reflections through one plane, with perpendicular glide vectors
and
and
and
; and ; and
; and ; and
; and ; and
n Diagonal' glide plane Glide reflection through the plane, with glide vector
; ; ; ;
or or
; ; ; ;
d § Diamond' glide plane Glide reflection through the plane, with glide vector
; ; ; ;
; ; ; ;
; ; ; ;
g Glide line (two dimensions) Glide reflection through the line, with glide vector
; ;
1 None Identity
2, 3, 4, 6 n-fold rotation axis, n Counter-clockwise rotation of degrees around the axis
n-fold rotation point, n (two dimensions) Counter-clockwise rotation of degrees around the point
Centre of symmetry, inversion centre Inversion through the point
, Rotoinversion axis, , and inversion point on the axis †† Counter-clockwise rotation of degrees around the axis, followed by inversion through the point on the axis ††
n-fold screw axis, Right-handed screw rotation of degrees around the axis, with screw vector (pitch) () t; here t is the shortest lattice translation vector parallel to the axis in the direction of the screw
In the rhombohedral space-group symbols (161) and (167), the symbol c refers to the description with hexagonal axes'; i.e. the glide vector is , along [001]. In the description with rhombohedral axes', this glide vector is , along [111], i.e. the symbol of the glide plane would be n: cf. Table 1.5.4.4 .
Glide planes e' occur in orthorhombic A-, C- and F-centred space groups, tetragonal I-centred and cubic F- and I-centred space groups. The geometric element of an e-glide plane is a plane shared by glide reflections with perpendicular glide vectors, with at least one glide vector along a crystal axis [cf. Section 1.2.3 and de Wolff et al. (1992)].
§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.
Only the symbol m is used in the Hermann–Mauguin symbols, for both point groups and space groups.
††The inversion point is a centre of symmetry if n is odd.
 Table 2.1.2.2| top | pdf | Graphical symbols of symmetry planes normal to the plane of projection (three dimensions) and symmetry lines in the plane of the figure (two dimensions)
DescriptionGraphical symbolGlide vector(s) of the defining operation(s) of the glide plane (in units of the shortest lattice translation vectors parallel and normal to the projection plane)Symmetry element represented by the graphical symbol
None m
Axial' glide plane normal to projection plane a, b or c
Double' glide plane e
Diagonal' glide plane n
Diamond' glide plane (pair of planes) parallel to line in projection plane, combined with normal to projection plane (arrow indicates direction parallel to the projection plane for which the normal component is positive) d
The graphical symbols of the e'-glide planes are applied to the diagrams of seven orthorhombic A-, C- and F-centred space groups, five tetragonal I-centred space groups, and five cubic F- and I-centred space groups.
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.
 Table 2.1.2.3| top | pdf | Graphical symbols of symmetry planes parallel to the plane of projection
DescriptionGraphical symbolGlide vector(s) of the defining operation(s) of the glide plane (in units of the shortest lattice translation vectors parallel to the projection plane)Symmetry element represented by the graphical symbol
Reflection plane, mirror plane None m
Axial' glide plane in the direction of the arrow a, b or c
Double' glide plane e
Diagonal' glide plane n
Diamond' glide plane§ (pair of planes) 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 2.1.3.6).
The graphical symbols of the e'-glide planes are applied to the diagrams of seven orthorhombic A-, C- and F-centred space groups, five tetragonal I-centred space groups, and five cubic F- and I-centred space groups.
§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.
 Table 2.1.2.4| top | pdf | Graphical symbols of symmetry planes inclined to the plane of projection (in cubic space groups of classes and only)
DescriptionGraphical symbol for planes normal toGlide vector(s) (in units of the shortest lattice translation vectors) of the defining operation(s) of the glide plane normal toSymmetry element represented by the graphical symbol
[011] and [101] and [011] and [101] and
Reflection plane, mirror plane None None m
Axial' glide plane along [100] a or b
Axial' glide plane 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) 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'.
In the space groups (216), (225) and (227), the shortest lattice translation vectors in the glide directions are or and or , respectively.
§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.
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).
 Table 2.1.2.5| top | pdf | Graphical symbols of symmetry axes normal to the plane of projection and symmetry points in the plane of the figure
DescriptionAlphanumeric symbolGraphical symbolScrew vector of the defining operation of the screw axis (in units of the shortest lattice translation vector parallel to the axis)Symmetry elements represented by the graphical symbol
2 None 2
Twofold screw axis: 2 sub 1' 21
3 None 3
Threefold screw axis: 3 sub 1' 31
Threefold screw axis: 3 sub 2' 32
4 None 4
Fourfold screw axis: 4 sub 1' 41
Fourfold screw axis: 4 sub 2' 42
Fourfold screw axis: 4 sub 3' 43
6 None 6
Sixfold screw axis: 6 sub 1' 61
Sixfold screw axis: 6 sub 2' 62
Sixfold screw axis: 6 sub 3' 63
Sixfold screw axis: 6 sub 4' 64
Sixfold screw axis: 6 sub 5' 65
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. (223): : ; : .

 Table 2.1.2.6| top | pdf | Graphical symbols of symmetry axes parallel to the plane of projection
DescriptionGraphical symbolScrew vector of the defining operation of the screw axis (in units of the shortest lattice translation vector parallel to the axis)Symmetry elements represented by the graphical symbol
Twofold rotation axis None 2
Twofold screw axis: 2 sub 1'
Fourfold rotation axis None 4
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 None None
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.
 Table 2.1.2.7| top | pdf | Graphical symbols of symmetry axes inclined to the plane of projection (in cubic space groups only)
DescriptionGraphical symbolScrew vector of the defining operation of the screw axis (in units of the shortest lattice translation vector parallel to the axis)Symmetry elements represented by the graphical symbol
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).

The alphanumeric symbols shown in Table 2.1.2.1 correspond to those symmetry elements and symmetry operations which occur in the conventional Hermann–Mauguin symbols of point groups and space groups. Further so-called additional symmetry elements' are described in Sections 1.4.2.3 and 1.5.4.1 , and Tables 1.5.4.3 and 1.5.4.4 show additional symmetry operations that appear in the so-called extended Hermann–Mauguin symbols' (cf. Section 1.5.4 ). The symbols of symmetry elements (symmetry operations), except for glide planes (glide reflections), are independent of the choice and the labelling of the basis vectors and of the origin. The symbols of glide planes (glide reflections), however, may change with a change of the basis vectors. For this reason, the possible orientations of glide planes and the glide vectors of the corresponding operations are listed explicitly in columns 2 and 3 of Table 2.1.2.1.

In 1992, following a proposal of the Commission on Crystallographic Nomenclature (de Wolff et al., 1992), the International Union of Crystallography introduced the symbol e' and graphical symbols for the designation of the so-called double' glide planes. The double- or e-glide plane occurs only in centred cells and its geometric element is a plane shared by glide reflections with perpendicular glide vectors related by a centring translation (for details on e-glide planes, cf. Section 1.2.3 ). The introduction of the symbol e for the designation of double-glide planes (cf. de Wolff et al., 1992) results in the modification of the Hermann–Mauguin symbols of five ortho­rhombic groups:

 Space group No. 39 41 64 67 68 New symbol: Aem2 Aea2 Cmce Cmme Ccce Former symbol: Abm2 Aba2 Cmca Cmma Ccca

Since the introduction of its use in IT A (2002) the new symbol is the standard one; it is indicated in the headline of these space groups, while the former symbol is given underneath.

The graphical symbols of symmetry planes are shown in Tables 2.1.2.2 to 2.1.2.4. Like the alphanumeric symbols, 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 alphanumeric symbols of glide planes (column 4), however, may change with a change of the basis vectors. For example, the dash-dotted n glide in the hexagonal description becomes an a, b or c glide in the rhombohedral description. In monoclinic space groups, the parallel' vector of a glide plane may be along a lattice translation vector that is inclined to the projection plane.

The e'-glide graphical symbols are applied to the diagrams of seven orthorhombic A-, C- and F- centred space groups, five tetragonal I-centred space groups, and five cubic F- and I-centred space groups. The double-dotted-dash' symbol for e glides normal' and inclined' to the plane of projection was introduced in 1992 (de Wolff et al., 1992), while the double-arrowed' graphical symbol for e-glide planes oriented parallel' to the projection plane had already been used in IT (1935) and IT (1952).

The graphical symbols of symmetry axes and their descriptions are shown in Tables 2.1.2.5–2.1.2.7. The screw vectors of the defining operations of screw axes are given in units of the shortest lattice translation vectors parallel to the axes. The symbols in the last column of the tables indicate the symmetry elements that are represented by the graphical symbols in the symmetry-element diagrams of the space groups. Two main cases may be distinguished:

 (i) graphical symbols of symmetry elements that in the space-group diagrams represent just one symmetry element. Thus, the graphical symbol of a fourfold rotation axis or an inversion centre represent the symmetry element 4 or . Similarly, the graphical symbols of symmetry planes (Tables 2.1.2.2–2.1.2.4) represent just one symmetry element (namely, mirror or glide plane) in the space-group diagrams; (ii) graphical symbols of symmetry elements that in the space-group diagrams represent more than one symmetry element. For example, the graphical symbol described in Table 2.1.2.5 as Inversion axis: 3 bar' , represents in the diagrams the three different symmetry elements , 3, .

The last six entries of Table 2.1.2.5 are combinations of symbols of symmetry axes with that of a centre of inversion. When displayed on the space-group diagrams, the combined graphical symbols represent more than one symmetry element. For example, the symbol for a fourfold rotation axis with a centre of inversion (4/m), represents the symmetry elements , 4 and .

The meaning of a graphical symbol on the space-group dia­grams is often confused with the set of symmetry elements that constitute the site-symmetry group associated with the symmetry element displayed. As an example, consider the rotoinversion axis (described as Inversion axis: 6 bar' in Table 2.1.2.5). The site-symmetry group can be decomposed into three symmetry elements: , 3 and m (cf. de Wolff et al., 1989). However, the graphical symbol of in the diagrams represents the two symmetry elements and 3, as the symmetry element m' (that belongs' to ) is represented by a separate graphical symbol.

### 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 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).]
Flack, H. D., Wondratschek, H., Hahn, Th. & Abrahams, S. C. (2000). Symmetry elements in space groups and point groups. Addenda to two IUCr Reports on the Nomenclature of Symmetry. Acta Cryst. A56, 96–98.
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.
Wolff, P. M. de, Billiet, Y., Donnay, J. D. H., Fischer, W., Galiulin, R. B., Glazer, A. M., Senechal, M., Shoemaker, D. P., Wondratschek, H., Hahn, Th., Wilson, A. J. C. & Abrahams, S. C. (1989). Definition of symmetry elements in space groups and point groups. Report of the International Union of Crystallography Ad-hoc Committee on the Nomenclature of Symmetry. Acta Cryst. A45, 494–499.