Graphical symbols for symmetry elements in one, two and three dimensions

Hahn, Th.

doi:10.1107/97809553602060000503

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

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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. Hahn^a ^*

^a Institut 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 line	Glide vector in units of lattice translation vectors parallel and normal to the projection plane	Printed symbol
$[\left.\openup3pt\matrix{\hbox{Reflection plane, mirror plane}\hfill\cr \hbox{Reflection line, mirror line (two dimensions)}\cr}\right\}]$	None	m
$[\left.\openup3pt\matrix{\hbox{`Axial' glide plane}\hfill\cr\hbox{Glide line (two dimensions)}\cr}\right\}]$	$[\!\openup2pt\matrix{{1 \over 2} \hbox{lattice vector along line in projection plane}\cr {1 \over 2} \hbox{lattice vector along line in figure plane}\hfill\cr}]$	$[\!\matrix{a,\ b \hbox{ or } c\cr g\hfill\cr}]$
`Axial' glide plane	$[{1 \over 2}]$ lattice vector normal to projection plane	a , b or c
`Double' glide plane^† (in centred cells only)	$[\!\openup2pt\matrix{Two\hbox{ glide vectors:}\hfill\cr{1 \over 2}\hbox{ along line parallel to projection plane and}\cr{1 \over 2} \hbox{ normal to projection plane}\hfill}]$	e
`Diagonal' glide plane	$[\openup2pt\matrix{One\hbox{ glide vector with }two\hbox{ components:}\hfill\cr{1 \over 2}\hbox{ along line parallel to projection plane,}\hfill\cr{1 \over 2}\hbox{ normal to projection plane}\hfill}]$	n
`Diamond' glide plane^‡ (pair of planes; in centred cells only)	$[{1 \over 4}]$ 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)

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 $[{1 \over 4}({\bf a} + {\bf b})]$ 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 plane	Glide vector in units of lattice translation vectors parallel to the projection plane	Printed symbol
Reflection plane, mirror plane	None	m
`Axial' glide plane	$[{1 \over 2}]$ lattice vector in the direction of the arrow	a , b or c
`Double' glide plane^‡ (in centred cells only)	$[\matrix{Two\hbox{ glide vectors:}\hfill\cr{1 \over 2}\hbox{ in either of the directions of the two arrows}}]$	e
`Diagonal' glide plane	$[\matrix{One\hbox{ glide vector with }two\hbox{ components}\cr{1 \over 2}\hbox{ in the direction of the arrow}\hfill}]$	n
`Diamond' glide plane^§ (pair of planes; in centred cells only)	$[{1 \over 2}]$ 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

).
^‡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 $[{1 \over 4}({\bf a} + {\bf b})]$ 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 plane	Graphical symbol^† for planes normal to		Glide vector in units of lattice translation vectors for planes normal to		Printed symbol
Symmetry plane	[011] and $[[01\bar{1}]]$	[101] and $[[10\bar{1}]]$	[011] and $[[01\bar{1}]]$	[101] and $[[10\bar{1}]]$	Printed symbol
Reflection plane, mirror plane			None	None	m
`Axial' glide plane			$[{1 \over 2}]$ 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			$[{1 \over 2}]$ lattice vector along $[[01\bar{1}]]$ or along [011]		a or b
`Double' glide plane^‡ [in space groups $[I\bar{4}3m]$ (217) and $[Im\bar{3}m]$ (229) only]			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			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)			$[{1 \over 2}]$ 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 $[[\bar{1}11]]$ ^¶		d

^† 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)

below and Note (x)

in Section 1.3.2

.
^§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 (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 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 $[\downarrow\hskip -6pt\raise5pt\hbox{$\rightarrow$}]$ 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 point	Graphical symbol^†	Screw vector of a right-handed screw rotation in units of the shortest lattice translation vector parallel to the axis	Printed symbol (partial elements in parentheses)
Identity	None	None	1
$[\!\left.\matrix{\hbox{Twofold rotation axis}\hfill\cr \hbox{Twofold rotation point (two dimensions)}\cr}\right\}]$		None	2
Twofold screw axis: `2 sub 1'		$[{1 \over 2}]$	$[2_{1}]$
$[\!\left.\matrix{\hbox{Threefold rotation axis}\hfill\cr \hbox{Threefold rotation point (two dimensions)}\cr}\right\}]$		None	3
Threefold screw axis: `3 sub 1'		$[{1 \over 3}]$	$[3_{1}]$
Threefold screw axis: `3 sub 2'		$[{2 \over 3}]$	$[3_{2}]$
$[\!\left.\openup3pt\matrix{\hbox{Fourfold rotation axis}\hfill\cr \hbox{Fourfold rotation point (two dimensions)}\cr}\right\}]$		None	4 (2)
Fourfold screw axis: `4 sub 1'		$[{1 \over 4}]$	$[4_{1} ]$ $[(2_{1})]$
Fourfold screw axis: `4 sub 2'		$[{1 \over 2}]$	$[4_{2}]$
Fourfold screw axis: `4 sub 3'		$[{3 \over 4}]$	$[4_{3} ]$ $[(2_{1})]$
$[\!\left.\openup3pt\matrix{\hbox{Sixfold rotation axis}\hfill\cr \hbox{Sixfold rotation point (two dimensions)}\cr}\right\}]$		None	6 (3,2)
Sixfold screw axis: `6 sub 1'		$[{1 \over 6}]$	$[6_{1}]$ $[ (3_{1},2_{1})]$
Sixfold screw axis: `6 sub 2'		$[{1 \over 3}]$	$[6_{2}]$ $[ (3_{2},2)]$
Sixfold screw axis: `6 sub 3'		$[{1 \over 2}]$	$[6_{3} ]$ $[(3,2_{1})]$
Sixfold screw axis: `6 sub 4'		$[{2 \over 3}]$	$[6_{4} ]$ $[(3_{1},2)]$
Sixfold screw axis: `6 sub 5'		$[{5 \over 6}]$	$[6_{5} ]$ $[(3_{2},2_{1})]$
$[\!\left.\openup3pt\matrix{\hbox{Centre of symmetry, inversion centre: `1 bar'}\hfill\cr\hbox{Reflection point, mirror point (one dimension)}\cr}\right\}]$		None	$[\bar{1}]$
Inversion axis: `3 bar'		None	$[\bar{3} ]$ $[(3,\bar{1})]$
Inversion axis: `4 bar'		None	$[\bar{4} ]$
Inversion axis: `6 bar'		None	$[\bar{6} \equiv 3/m]$
Twofold rotation axis with centre of symmetry		None	$[(\bar{1})]$
Twofold screw axis with centre of symmetry		$[{1 \over 2}]$	$[2_{1}/m ]$ $[(\bar{1})]$
Fourfold rotation axis with centre of symmetry		None	$[(\bar{4},2,\bar{1})]$
`4 sub 2' screw axis with centre of symmetry		$[{1 \over 2}]$	$[4_{2}/m ]$ $[(\bar{4},2,\bar{1})]$
Sixfold rotation axis with centre of symmetry		None	$[ (\bar{6},\bar{3},3,2,\bar{1})]$
`6 sub 3' screw axis with centre of symmetry		$[{1 \over 2}]$	$[6_{3}/m ]$ $[(\bar{6},\bar{3},3,2_{1},\bar{1})]$

^†

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 and $[{1 \over 2}]$ . In cubic space groups, however, because of their complexity, both fractions are given for vertical $[\bar{4}]$ axes, including and $[{1 \over 2}]$ .
(2) Symmetries and 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 axis	Screw vector of a right-handed screw rotation in units of the shortest lattice translation vector parallel to the axis	Printed symbol (partial elements in parentheses)
Twofold rotation axis	None	2
Twofold screw axis: `2 sub 1'	$[{1 \over 2}]$	$[2_{1}]$
Fourfold rotation axis	None	4 (2)
Fourfold screw axis: `4 sub 1'	$[{1 \over 4}]$	$[4_{1} ]$ $[(2_{1})]$
Fourfold screw axis: `4 sub 2'	$[{1 \over 2}]$	$[4_{2} ]$
Fourfold screw axis: `4 sub 3'	$[{3 \over 4}]$	$[4_{3} ]$ $[(2_{1})]$
Inversion axis: `4 bar'	None	$[\bar{4} ]$
Inversion point on `4 bar'-axis	–	$[\bar{4}]$ 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 axis	Screw vector of a right-handed screw rotation in units of the shortest lattice translation vector parallel to the axis	Printed symbol (partial elements in parentheses)
Twofold rotation axis	None	2
Twofold screw axis: `2 sub 1'	$[{1 \over 2}]$	$[2_{1}]$
Threefold rotation axis	None	3
Threefold screw axis: `3 sub 1'	$[{1 \over 3}]$	$[3_{1}]$
Threefold screw axis: `3 sub 2'	$[{2 \over 3}]$	$[3_{2}]$
Inversion axis: `3 bar'	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).]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

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

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

1.4.2. Symmetry planes parallel to the plane of projection

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

1.4.4. Notes on graphical symbols of symmetry planes

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

1.4.6. Symmetry axes parallel to the plane of projection

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

References

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)