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

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. Hahna*

aInstitut 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 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): : ; : .