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

International Tables for Crystallography (2006). Vol. A, ch. 1.3, p. 5

Section 1.3.1. Printed symbols for symmetry elements and for the corresponding symmetry operations in one, two and three dimensions

Th. Hahna*

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

1.3.1. Printed symbols for symmetry elements and for the corresponding symmetry operations in one, two and three dimensions

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For `reflection conditions', see Tables 2.2.13.2[link] and 2.2.13.3[link] .

Printed symbolSymmetry element and its orientationDefining symmetry operation with glide or screw vector
m [\Bigg\{] Reflection plane, mirror plane Reflection through the plane
Reflection line, mirror line (two dimensions) Reflection through the line
Reflection point, mirror point (one dimension) Reflection through the point
a, b or c `Axial' glide plane Glide reflection through the plane, with glide vector
a [\perp [010]] or [\perp [001]] [{1 \over 2}{\bf a}]
b [\perp [001]] or [\perp [100]] [{1 \over 2}{\bf b}]
c [\left\{\vbox to 22pt{}\right.] [\perp [100]] or [\perp [010]] [{1 \over 2}{\bf c}]
[\perp [1\bar{1}0]] or [\perp [110]] [{1 \over 2}{\bf c}]
[\!\openup1pt\matrix{\perp [100] \hbox{ or} \perp [010] \hbox{ or} \perp [\bar{1}\bar{1}0]\cr \perp [1\bar{1}0] \hbox{ or} \perp [120] \hbox{ or} \perp [\bar{2}\bar{1}0]\cr}] [\!\left.\openup1pt\matrix{{1 \over 2}{\bf c}\cr {1 \over 2}{\bf c}\cr}\right\} \hbox{ hexagonal coordinate system}]
e `Double' glide plane (in centred cells only) Two glide reflections through one plane, with perpendicular glide vectors
[\perp [001]] [{1 \over 2}{\bf a}] and [{1 \over 2}{\bf b}]
[\perp [100]] [{1 \over 2}{\bf b}] and [{1 \over 2}{\bf c}]
[\perp [010]] [{1 \over 2}{\bf a}] and [{1 \over 2}{\bf c}]
[\perp [1\bar{1}0]]; [\perp [110]] [{1 \over 2}({\bf a} + {\bf b})] and [{1 \over 2}{\bf c}]; [{1 \over 2}({\bf a} - {\bf b})] and [{1 \over 2}{\bf c}]
[\perp [01\bar{1}]]; [\perp [011]] [{1 \over 2}({\bf b} + {\bf c})] and [{1 \over 2}{\bf a}]; [{1 \over 2}({\bf b} - {\bf c})] and [{1 \over 2}{\bf a}]
[\perp [\bar{1}01]]; [\perp [101]] [{1 \over 2}({\bf a} + {\bf c})] and [{1 \over 2}{\bf b}]; [{1 \over 2}({\bf a} - {\bf c})] and [{1 \over 2}{\bf b}]
n `Diagonal' glide plane Glide reflection through the plane, with glide vector
[\perp [001]]; [\perp [100]]; [\perp [010]] [{1 \over 2}({\bf a} + {\bf b})]; [{1 \over 2}({\bf b} + {\bf c})]; [{1 \over 2}({\bf a} + {\bf c})]
[\perp [1\bar{1}0]] or [\perp [01\bar{1}]] or [\perp [\bar{1}01]] [{1 \over 2}({\bf a} + {\bf b} + {\bf c})]
[\perp [110]]; [\perp [011]]; [\perp [101]] [{1 \over 2}(- {\bf a} + {\bf b} + {\bf c})]; [{1 \over 2}({\bf a} - {\bf b} + {\bf c})]; [{1 \over 2}({\bf a} + {\bf b} - {\bf c})]
d § `Diamond' glide plane Glide reflection through the plane, with glide vector
[\perp [001]]; [\perp [100]]; [\perp [010]] [{1 \over 4}({\bf a} \pm {\bf b})]; [{1 \over 4}({\bf b} \pm {\bf c})]; [{1 \over 4}(\pm {\bf a} + {\bf c})]
[\perp [1\bar{1}0]]; [\perp [01\bar{1}]]; [\perp [\bar{1}01]] [{1 \over 4}({\bf a} + {\bf b} \pm {\bf c})]; [{1 \over 4}(\pm {\bf a} + {\bf b} + {\bf c})]; [{1 \over 4}({\bf a} \pm {\bf b} + {\bf c})]
[\perp [110]]; [\perp [011]]; [\perp [101]] [{1 \over 4}(- {\bf a} + {\bf b} \pm {\bf c})]; [{1 \over 4}(\pm {\bf a} - {\bf b} + {\bf c})]; [{1 \over 4}({\bf a} \pm {\bf b} - {\bf c})]
g Glide line (two dimensions) Glide reflection through the line, with glide vector
[\perp [01]]; [\perp [10]] [{1 \over 2}{\bf a}]; [{1 \over 2}{\bf b}]
1 None Identity
2, 3, 4, 6 [\left\{\vbox to 21pt{}\right.] n-fold rotation axis, n Counter-clockwise rotation of [360/n] degrees around the axis (see Note viii[link])
n-fold rotation point, n (two dimensions) Counter-clockwise rotation of [360/n] degrees around the point
[\bar{1}] Centre of symmetry, inversion centre Inversion through the point
[\bar{2} = m], [\bar{3},\bar{4},\bar{6}] Rotoinversion axis, [\bar{n}], and inversion point on the axis †† Counter-clockwise rotation of [360/n] degrees around the axis, followed by inversion through the point on the axis †† (see Note viii[link])
[2_{1}] n-fold screw axis, [n_{p}] Right-handed screw rotation of [360/n] degrees around the axis, with screw vector (pitch) ([p/n]) t; here t is the shortest lattice translation vector parallel to the axis in the direction of the screw
[3_{1}, 3_{2}]
[4_{1}, 4_{2}, 4_{3}]
[6_{1}, 6_{2}, 6_{3}, 6_{4}, 6_{5}]
In the rhombohedral space-group symbols [R3c] (161) and [R\bar{3}c] (167), the symbol c refers to the description with `hexagonal axes'; i.e. the glide vector is [{1 \over 2}{\bf c}], along [001]. In the description with `rhombohedral axes', this glide vector is [{1 \over 2}({\bf a} + {\bf b} + {\bf c})], along [111], i.e. the symbol of the glide plane would be n: cf. Section 4.3.5[link] .
For further explanations of the `double' glide plane e, see Note (x)[link] below.
§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.
Only the symbol m is used in the Hermann–Mauguin symbols, for both point groups and space groups.
††The inversion point is a centre of symmetry if n is odd.








































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