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
[\!\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}].









































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