International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A, ch. 2.1, p. 148

Table 2.1.2.5 

Th. Hahna and M. I. Aroyoc

Table 2.1.2.5| top | pdf |
Graphical symbols of symmetry axes normal to the plane of projection and symmetry points in the plane of the figure

DescriptionAlphanumeric symbolGraphical symbolScrew vector of the defining operation of the screw axis (in units of the shortest lattice translation vector parallel to the axis)Symmetry elements represented by the graphical symbol
[\!\left.\matrix{\hbox{Twofold rotation axis}\hfill\cr \hbox{Twofold rotation point (two dimensions)}\cr}\right\}] 2 [Scheme scheme37] None 2
Twofold screw axis: `2 sub 1' 21 [Scheme scheme38] [{1 \over 2}] [2_{1}]
[\!\left.\matrix{\hbox{Threefold rotation axis}\hfill\cr \hbox{Threefold rotation point (two dimensions)}\cr}\right\}] 3 [Scheme scheme39] None 3
Threefold screw axis: `3 sub 1' 31 [Scheme scheme40] [{1 \over 3}] [3_{1}]
Threefold screw axis: `3 sub 2' 32 [Scheme scheme41] [{2 \over 3}] [3_{2}]
[\!\left.\openup3pt\matrix{\hbox{Fourfold rotation axis}\hfill\cr \hbox{Fourfold rotation point (two dimensions)}\cr}\right\}] 4 [Scheme scheme42] None 4
Fourfold screw axis: `4 sub 1' 41 [Scheme scheme43] [{1 \over 4}] [4_{1} ]
Fourfold screw axis: `4 sub 2' 42 [Scheme scheme44] [{1 \over 2}] [4_{2}]
Fourfold screw axis: `4 sub 3' 43 [Scheme scheme45] [{3 \over 4}] [4_{3} ]
[\!\left.\openup3pt\matrix{\hbox{Sixfold rotation axis}\hfill\cr \hbox{Sixfold rotation point (two dimensions)}\cr}\right\}] 6 [Scheme scheme46] None 6
Sixfold screw axis: `6 sub 1' 61 [Scheme scheme47] [{1 \over 6}] [6_{1}]
Sixfold screw axis: `6 sub 2' 62 [Scheme scheme48] [{1 \over 3}] [6_{2}]
Sixfold screw axis: `6 sub 3' 63 [Scheme scheme49] [{1 \over 2}] [6_{3}]
Sixfold screw axis: `6 sub 4' 64 [Scheme scheme50] [{2 \over 3}] [6_{4}]
Sixfold screw axis: `6 sub 5' 65 [Scheme scheme51] [{5 \over 6}] [6_{5}]
[\!\left.\openup3pt\matrix{\hbox{Centre of symmetry, inversion centre: `1 bar'}\hfill\cr\hbox{Reflection point, mirror point (one dimension)}\cr}\right\}] [\bar 1] [Scheme scheme52] None [\bar{1}]
Inversion axis: `3 bar' [\bar 3] [Scheme scheme53] None [\bar{3}, \bar{1}, 3]
Inversion axis: `4 bar' [\bar 4] [Scheme scheme54] None [\bar{4}, 2]
Inversion axis: `6 bar' [\bar 6] [Scheme scheme55] None [\bar{6}, 3]
Twofold rotation axis with centre of symmetry [2/m] [Scheme scheme56] None [2, \bar{1}]
Twofold screw axis with centre of symmetry [2_1/m] [Scheme scheme57] [{1 \over 2}] [2_{1}, \bar{1}]
Fourfold rotation axis with centre of symmetry [4/m] [Scheme scheme58] None [4, \bar{4}, \bar{1}]
`4 sub 2' screw axis with centre of symmetry [4_2/m] [Scheme scheme59] [{1 \over 2}] [4_{2}, \bar{4},\bar{1}]
Sixfold rotation axis with centre of symmetry [6/m] [Scheme scheme60] None [6,\bar{6},\bar{3},\bar{1}]
`6 sub 3' screw axis with centre of symmetry [6_3/m] [Scheme scheme61] [{1 \over 2}] [6_{3}, \bar{6},\bar{3},\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] (223): [\bar{1}]: [0, {1 \over 2}]; [\bar{4}]: [{1 \over 4}, {3 \over 4}].