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.

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
[\left.\openup3pt\matrix{\hbox{Reflection plane, mirror plane}\hfill\cr \hbox{Reflection line, mirror line (two dimensions)}\cr}\right\}] [Scheme scheme8] None m
[\left.\openup3pt\matrix{\hbox{`Axial' glide plane}\hfill\cr\hbox{Glide line (two dimensions)}\cr}\right\}] [Scheme scheme9] [\!\openup2pt\matrix{{1 \over 2} \hbox{lattice vector along line in projection plane}\cr {1 \over 2} \hbox{lattice vector along line in figure plane}\hfill\cr}] [\!\matrix{a,\ b \hbox{ or } c\cr g\hfill\cr}]
`Axial' glide plane [Scheme scheme10] [{1 \over 2}] lattice vector normal to projection plane a, b or c
`Double' glide plane (in centred cells only) [Scheme scheme11] [\!\openup2pt\matrix{Two\hbox{ glide vectors:}\hfill\cr{1 \over 2}\hbox{ along line parallel to projection plane and}\cr{1 \over 2} \hbox{ normal to projection plane}\hfill}] e
`Diagonal' glide plane [Scheme scheme12] [\openup2pt\matrix{One\hbox{ glide vector with }two\hbox{ components:}\hfill\cr{1 \over 2}\hbox{ along line parallel to projection plane,}\hfill\cr{1 \over 2}\hbox{ normal to projection plane}\hfill}] n
`Diamond' glide plane (pair of planes; in centred cells only) [Scheme scheme13] [{1 \over 4}] along line parallel to projection plane, combined with [{1 \over 4}] 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)[link] below and Note (x)[link] in Section 1.3.2[link] .
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 [{1 \over 4}({\bf a} + {\bf b})] and [{1 \over 4}({\bf a} - {\bf b})]. 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 [Scheme scheme14] None m
`Axial' glide plane [Scheme scheme15] [{1 \over 2}] lattice vector in the direction of the arrow a, b or c
`Double' glide plane (in centred cells only) [Scheme scheme16] [\matrix{Two\hbox{ glide vectors:}\hfill\cr{1 \over 2}\hbox{ in either of the directions of the two arrows}}] e
`Diagonal' glide plane [Scheme scheme17] [\matrix{One\hbox{ glide vector with }two\hbox{ components}\cr{1 \over 2}\hbox{ in the direction of the arrow}\hfill}] n
`Diamond' glide plane§ (pair of planes; in centred cells only) [Scheme scheme18] [{1 \over 2}] 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 [h + {1 \over 2}] above the plane of projection; e.g. [{1 \over 8}] stands for [h = {1 \over 8}] and [{5 \over 8}]. No fraction means [h = 0] and [{1 \over 2}] (cf. Section 2.2.6[link] ).
For further explanations of the `double' glide plane e see Note (iv)[link] below and Note (x)[link] in Section 1.3.2[link] .
§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 [{1 \over 4}({\bf a} + {\bf b})] and [{1 \over 4}({\bf a} - {\bf b})]. 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 [\overline{4}{3m}] and [m\overline{3}m] 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 [[01\bar{1}]][101] and [[10\bar{1}]][011] and [[01\bar{1}]][101] and [[10\bar{1}]]
Reflection plane, mirror plane [Scheme scheme19] [Scheme scheme25] None None m
`Axial' glide plane [Scheme scheme20] [Scheme scheme26] [{1 \over 2}] lattice vector along [100] [\left.\!\matrix{{1 \over 2}\hbox{ lattice vector along }[010]\hfill\cr\noalign{\vskip 29pt}\cr{1 \over 2}\hbox{ lattice vector along }[10\bar{1}]\cr\hbox{ or along }[101]\hfill\cr}\right\}] a or b
`Axial' glide plane [Scheme scheme21] [Scheme scheme27] [{1 \over 2}] lattice vector along [[01\bar{1}]] or along [011]
`Double' glide plane [in space groups [I\bar{4}3m] (217) and [Im\bar{3}m] (229) only] [Scheme scheme22] [Scheme scheme28] Two glide vectors: [{1 \over 2}] along [100] and [{1 \over 2}] along [[01\bar{1}]] or [{1 \over 2}] along [011] Two glide vectors: [{1 \over 2}] along [010] and [{1 \over 2}] along [[10\bar{1}]] or [{1 \over 2}] along [101] e
`Diagonal' glide plane [Scheme scheme23] [Scheme scheme29] One glide vector: [{1 \over 2}] along [[11\bar{1}]] or along [111]§ One glide vector: [{1 \over 2}] along [[11\bar{1}]] or along [111]§ n
`Diamond' glide plane†† (pair of planes; in centred cells only) [Scheme scheme24] [Scheme scheme30] [{1 \over 2}] along [[1\bar{1}1]] or along [111] [\left.\matrix{{1 \over 2}\hbox{ along }[\bar{1}11]\hbox { or}\cr \hbox{along }[111]\cr\noalign{\vskip 30pt} {1 \over 2}\hbox{ along }[\bar{1}\bar{1}1]\hbox{ or}\cr \hbox{ along }[1\bar{1}1]}\right\}] d
[{1 \over 2}] along [[\bar{1}\bar{1}1]] or along [[\bar{1}11]]
The symbols represent orthographic projections. In the cubic space-group diagrams, complete orthographic projections of the symmetry elements around high-symmetry points, such as [0,0,0]; [{1 \over 2},0,0]; [{1 \over 4},{1 \over 4},0], are given as `inserts'.
For further explanations of the `double' glide plane e see Note (iv)[link] below and Note (x)[link] in Section 1.3.2[link] .
§In the space groups [F\bar{4}3m] (216), [Fm\bar{3}m] (225) and [Fd\bar{3}m] (227), the shortest lattice translation vectors in the glide directions are [{\bf t}(1, {1 \over 2}, \bar{{1 \over 2}})] or [{\bf t}(1, {1 \over 2}, {1 \over 2})] and [{\bf t}({1 \over 2}, 1, \bar{{1 \over 2}})] or [{\bf t}({1 \over 2}, 1, {1 \over 2})], 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 [I\bar{4}3d] (220) and [Ia\bar{3}d] (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 [{1 \over 4}({\bf a} + {\bf b})] and [{1 \over 4}({\bf a} - {\bf b})]. 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 [R3c] (161) and [R\bar{3}c] (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[link] 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[link] , but also for the further glide planes g which are mentioned in Section 1.3.2[link] (Note x[link] ) and listed in Table 4.3.2.1[link] ; they are explained in Sections 2.2.9[link] and 11.1.2[link] .

  • (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[link]); for details of e glide planes see Chapter 1.3[link] . Note that the graphical symbol [\downarrow\hskip -6pt\raise5pt\hbox{$\rightarrow$}] for e glide planes oriented `parallel' to the projection plane has already been used in IT (1935)[link] and IT (1952)[link].

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
[\!\left.\matrix{\hbox{Twofold rotation axis}\hfill\cr \hbox{Twofold rotation point (two dimensions)}\cr}\right\}] [Scheme scheme31] None 2
Twofold screw axis: `2 sub 1' [Scheme scheme32] [{1 \over 2}] [2_{1}]
[\!\left.\matrix{\hbox{Threefold rotation axis}\hfill\cr \hbox{Threefold rotation point (two dimensions)}\cr}\right\}] [Scheme scheme33] None 3
Threefold screw axis: `3 sub 1' [Scheme scheme34] [{1 \over 3}] [3_{1}]
Threefold screw axis: `3 sub 2' [Scheme scheme35] [{2 \over 3}] [3_{2}]
[\!\left.\openup3pt\matrix{\hbox{Fourfold rotation axis}\hfill\cr \hbox{Fourfold rotation point (two dimensions)}\cr}\right\}] [Scheme scheme36] None 4 (2)
Fourfold screw axis: `4 sub 1' [Scheme scheme37] [{1 \over 4}] [4_{1} ] [(2_{1})]
Fourfold screw axis: `4 sub 2' [Scheme scheme38] [{1 \over 2}] [4_{2}] [(2)]
Fourfold screw axis: `4 sub 3' [Scheme scheme39] [{3 \over 4}] [4_{3} ] [(2_{1})]
[\!\left.\openup3pt\matrix{\hbox{Sixfold rotation axis}\hfill\cr \hbox{Sixfold rotation point (two dimensions)}\cr}\right\}] [Scheme scheme40] None 6 (3,2)
Sixfold screw axis: `6 sub 1' [Scheme scheme41] [{1 \over 6}] [6_{1}] [ (3_{1},2_{1})]
Sixfold screw axis: `6 sub 2' [Scheme scheme42] [{1 \over 3}] [6_{2}] [ (3_{2},2)]
Sixfold screw axis: `6 sub 3' [Scheme scheme43] [{1 \over 2}] [6_{3} ] [(3,2_{1})]
Sixfold screw axis: `6 sub 4' [Scheme scheme44] [{2 \over 3}] [6_{4} ] [(3_{1},2)]
Sixfold screw axis: `6 sub 5' [Scheme scheme45] [{5 \over 6}] [6_{5} ] [(3_{2},2_{1})]
[\!\left.\openup3pt\matrix{\hbox{Centre of symmetry, inversion centre: `1 bar'}\hfill\cr\hbox{Reflection point, mirror point (one dimension)}\cr}\right\}] [Scheme scheme46] None [\bar{1}]
Inversion axis: `3 bar' [Scheme scheme47] None [\bar{3} ] [(3,\bar{1})]
Inversion axis: `4 bar' [Scheme scheme48] None [\bar{4} ] [(2)]
Inversion axis: `6 bar' [Scheme scheme49] None [\bar{6} \equiv 3/m]
Twofold rotation axis with centre of symmetry [Scheme scheme50] None [2/m ] [(\bar{1})]
Twofold screw axis with centre of symmetry [Scheme scheme51] [{1 \over 2}] [2_{1}/m ] [(\bar{1})]
Fourfold rotation axis with centre of symmetry [Scheme scheme52] None [4/m ] [(\bar{4},2,\bar{1})]
`4 sub 2' screw axis with centre of symmetry [Scheme scheme53] [{1 \over 2}] [4_{2}/m ] [(\bar{4},2,\bar{1})]
Sixfold rotation axis with centre of symmetry [Scheme scheme54] None [6/m] [ (\bar{6},\bar{3},3,2,\bar{1})]
`6 sub 3' screw axis with centre of symmetry [Scheme scheme55] [{1 \over 2}] [6_{3}/m ] [(\bar{6},\bar{3},3,2_{1},\bar{1})]

Notes on the `heights' h of symmetry points [\bar{1}], [\bar{3}], [\bar{4}] and [\bar{6}]:

  • (1) Centres of symmetry [\bar{1}] and [\bar{3}], as well as inversion points [\bar{4}] and [\bar{6}] on [\bar{4}] and [\bar{6}] axes parallel to [001], occur in pairs at `heights' h and [h + {1 \over 2}]. In the space-group diagrams, only one fraction h is given, e.g. [{1 \over 4}] stands for [h = {1 \over 4}] and [{3 \over 4}]. No fraction means [h = 0] and [{1 \over 2}]. In cubic space groups, however, because of their complexity, both fractions are given for vertical [\bar{4}] axes, including [h = 0] and [{1 \over 2}].

  • (2) Symmetries [4/m] and [6/m] contain vertical [\bar{4}] and [\bar{6}] axes; their [\bar{4}] and [\bar{6}] inversion points coincide with the centres of symmetry. This is not indicated in the space-group diagrams.

  • (3) Symmetries [4_{2}/m] and [6_{3}/m] also contain vertical [\bar{4}] and [\bar{6}] axes, but their [\bar{4}] and [\bar{6}] inversion points alternate with the centres of symmetry; i.e. [\bar{1}] points at h and [h + {1 \over 2}] interleave with [\bar{4}] or [\bar{6}] points at [h + {1 \over 4}] and [h + {3 \over 4}]. In the tetragonal and hexagonal space-group diagrams, only one fraction for [\bar{1}] and one for [\bar{4}] or [\bar{6}] is given. In the cubic diagrams, all four fractions are listed for [4_{2}/m]; e.g. [Pm\bar{3}n] (No. 223): [\bar{1}]: [0, {1 \over 2}]; [\bar{4}]: [{1 \over 4}, {3 \over 4}].


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 [Scheme scheme56] None 2
Twofold screw axis: `2 sub 1' [Scheme scheme57] [{1 \over 2}] [2_{1}]
Fourfold rotation axis [Scheme scheme58] [Scheme scheme64] None 4 (2)
Fourfold screw axis: `4 sub 1' [Scheme scheme59] [{1 \over 4}] [4_{1} ] [(2_{1})]
Fourfold screw axis: `4 sub 2' [Scheme scheme60] [{1 \over 2}] [4_{2} ] [(2)]
Fourfold screw axis: `4 sub 3' [Scheme scheme61] [{3 \over 4}] [4_{3} ] [(2_{1})]
Inversion axis: `4 bar' [Scheme scheme62] None [\bar{4} ] [(2)]
Inversion point on `4 bar'-axis [Scheme scheme63] [\bar{4}] 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 [h + {1 \over 2}] above the plane of projection; here, a fraction h attached to such a symbol indicates two axes with heights h and [h + {1 \over 2}]. No fraction stands for [h = 0] and [{1 \over 2}]. The rule of pairwise occurrence, however, is not valid for the horizontal fourfold axes in cubic space groups; here, all heights are given, including [h = 0] and [{1 \over 2}]. This applies also to the horizontal [\bar{4}] axes and the [\bar{4}] 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 [Scheme scheme65] [Scheme scheme71] None 2
Twofold screw axis: `2 sub 1' [Scheme scheme66] [{1 \over 2}] [2_{1}]
Threefold rotation axis [Scheme scheme67] [Scheme scheme72] None 3
Threefold screw axis: `3 sub 1' [Scheme scheme68] [{1 \over 3}] [3_{1}]
Threefold screw axis: `3 sub 2' [Scheme scheme69] [{2 \over 3}] [3_{2}]
Inversion axis: `3 bar' [Scheme scheme70] None [\bar{3} ] [(3,\bar{1})]
The dots mark the intersection points of axes with the plane at [h = 0]. In some cases, the intersection points are obscured by symbols of symmetry elements with height [h \geq 0]; examples: [Fd\bar{3}] (203), origin choice 2; [Pn\bar{3}n] (222), origin choice 2; [Pm\bar{3}n] (223); [Im\bar{3}m] (229); [Ia\bar{3}d] (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).]
International Tables for X-ray Crystallography (1952). Vol. I, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press. [Abbreviated as IT (1952).]
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.








































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