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

International Tables for Crystallography (2006). Vol. A, ch. 1.4, p. 8

Section 1.4.3. Symmetry planes inclined to the plane of projection (in cubic space groups of classes [\overline{4}{3m}] and [m\overline{3}m] only)

Th. Hahna*

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

1.4.3. Symmetry planes inclined to the plane of projection (in cubic space groups of classes [\overline{4}{3m}] and [m\overline{3}m] only)

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Symmetry planeGraphical symbol for planes normal toGlide vector in units of lattice translation vectors for planes normal toPrinted symbol
[011] and [[01\bar{1}]][101] and [[10\bar{1}]][011] and [[01\bar{1}]][101] and [[10\bar{1}]]
Reflection plane, mirror plane [Scheme scheme19] [Scheme scheme25] None None m
`Axial' glide plane [Scheme scheme20] [Scheme scheme26] [{1 \over 2}] lattice vector along [100] [\left.\!\matrix{{1 \over 2}\hbox{ lattice vector along }[010]\hfill\cr\noalign{\vskip 29pt}\cr{1 \over 2}\hbox{ lattice vector along }[10\bar{1}]\cr\hbox{ or along }[101]\hfill\cr}\right\}] a or b
`Axial' glide plane [Scheme scheme21] [Scheme scheme27] [{1 \over 2}] lattice vector along [[01\bar{1}]] or along [011]
`Double' glide plane [in space groups [I\bar{4}3m] (217) and [Im\bar{3}m] (229) only] [Scheme scheme22] [Scheme scheme28] Two glide vectors: [{1 \over 2}] along [100] and [{1 \over 2}] along [[01\bar{1}]] or [{1 \over 2}] along [011] Two glide vectors: [{1 \over 2}] along [010] and [{1 \over 2}] along [[10\bar{1}]] or [{1 \over 2}] along [101] e
`Diagonal' glide plane [Scheme scheme23] [Scheme scheme29] One glide vector: [{1 \over 2}] along [[11\bar{1}]] or along [111]§ One glide vector: [{1 \over 2}] along [[11\bar{1}]] or along [111]§ n
`Diamond' glide plane†† (pair of planes; in centred cells only) [Scheme scheme24] [Scheme scheme30] [{1 \over 2}] along [[1\bar{1}1]] or along [111] [\left.\matrix{{1 \over 2}\hbox{ along }[\bar{1}11]\hbox { or}\cr \hbox{along }[111]\cr\noalign{\vskip 30pt} {1 \over 2}\hbox{ along }[\bar{1}\bar{1}1]\hbox{ or}\cr \hbox{ along }[1\bar{1}1]}\right\}] d
[{1 \over 2}] along [[\bar{1}\bar{1}1]] or along [[\bar{1}11]]
The symbols represent orthographic projections. In the cubic space-group diagrams, complete orthographic projections of the symmetry elements around high-symmetry points, such as [0,0,0]; [{1 \over 2},0,0]; [{1 \over 4},{1 \over 4},0], are given as `inserts'.
For further explanations of the `double' glide plane e see Note (iv)[link] below and Note (x)[link] in Section 1.3.2[link] .
§In the space groups [F\bar{4}3m] (216), [Fm\bar{3}m] (225) and [Fd\bar{3}m] (227), the shortest lattice translation vectors in the glide directions are [{\bf t}(1, {1 \over 2}, \bar{{1 \over 2}})] or [{\bf t}(1, {1 \over 2}, {1 \over 2})] and [{\bf t}({1 \over 2}, 1, \bar{{1 \over 2}})] or [{\bf t}({1 \over 2}, 1, {1 \over 2})], respectively.
The glide vector is half of a centring vector, i.e. one quarter of the diagonal of the conventional body-centred cell in space groups [I\bar{4}3d] (220) and [Ia\bar{3}d] (230).
††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.








































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