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

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, p. 9

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

Th. Hahn^a ^*

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

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): $[\bar{1}]$ : $[0, {1 \over 2}]$ ; $[\bar{4}]$ : $[{1 \over 4}, {3 \over 4}]$ .

References

International Tables for Crystallography (2006). Vol. A. ch. 1.4, p. 9