Designation of symmetry operations

Hahn, Th.; Looijenga-Vos, A.

doi:10.1107/97809553602060000505

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

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International Tables for Crystallography (2006). Vol. A. ch. 2.2, pp. 26-27

Section 2.2.9.2. Designation of symmetry operations

Th. Hahn^a ^* and A. Looijenga-Vos^b ^‡

^a Institut für Kristallographie, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany, and ^bLaboratorium voor Chemische Fysica, Rijksuniversiteit Groningen, The Netherlands
Correspondence e-mail: hahn@xtl.rwth-aachen.de

2.2.9.2. Designation of symmetry operations

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An entry in the block Symmetry operations is characterized as follows.

(i) A symbol denoting the type of the symmetry operation (cf. Chapter 1.3 ), including its glide or screw part, if present. In most cases, the glide or screw part is given explicitly by fractional coordinates between parentheses. The sense of a rotation is indicated by the superscript or −. Abbreviated notations are used for the glide reflections $[a({1 \over 2},0,0) \equiv a]$ ; $[b(0,{1 \over 2},0) \equiv b]$ ; $[c(0,0,{1 \over 2}) \equiv c]$ . Glide reflections with complicated and unconventional glide parts are designated by the letter g, followed by the glide part between parentheses.
(ii) A coordinate triplet indicating the location and orientation of the symmetry element which corresponds to the symmetry operation. For rotoinversions, the location of the inversion point is given in addition.

Details of this symbolism are presented in Section 11.1.2 .

Examples

(1) $[a\ \ \ x,y,{1 \over 4}]$

Glide reflection with glide component $[({1 \over 2},0,0)]$ through the plane $[x,y,{1 \over 4}]$ , i.e. the plane parallel to (001) at $[z = {1 \over 4}]$ .
(2) $[\bar{4}^{+}\ \ \ {1 \over 4},{1 \over 4},z\semi \ \ {1 \over 4},{1 \over 4},{1 \over 4}]$

Fourfold rotoinversion, consisting of a counter clockwise rotation by 90° around the line $[{1 \over 4},{1 \over 4}, z]$ , followed by an inversion through the point $[{1 \over 4},{1 \over 4},{1 \over 4}]$ .
(3) $[g({1 \over 4},{1 \over 4},{1 \over 2})\ \ \ x,x,z]$

Glide reflection with glide component $[({1 \over 4},{1 \over 4},{1 \over 2})]$ through the plane x, x, z, i.e. the plane parallel to $[(1\bar{1}0)]$ containing the point 0, 0, 0.
(4) $[g({1 \over 3},{1 \over 6},{1 \over 6})\ \ \ 2x - {1 \over 2}, x,z]$ (hexagonal axes)

Glide reflection with glide component $[({1 \over 3},{1 \over 6},{1 \over 6})]$ through the plane $[2x-{1 \over 2}, x,z]$ , i.e. the plane parallel to $[(1\bar{2}10)]$ , which intersects the a axis at $[-{1 \over 2}]$ and the b axis at $[{1 \over 4}]$ ; this operation occurs in $[R\bar{3}c]$ (167, hexagonal axes).
(5) Symmetry operations in Ibca (73)

Under the subheading `For (0, 0, 0) set', the operation generating the coordinate triplet (2) $[\bar{x} + {1 \over 2},\bar{y},z + {1 \over 2}]$ from (1) x, y, z is symbolized by $[2 (0,0, {1 \over 2})\ \ {1 \over 4},0,z]$ . This indicates a twofold screw rotation with screw part $[(0,0,{1 \over 2})]$ for which the corresponding screw axis coincides with the line $[{1 \over 4},0,z]$ , i.e. runs parallel to [001] through the point $[{1 \over 4},0,0]$ . Under the subheading `For $[({1 \over 2},{1 \over 2},{1 \over 2}) +]$ set', the operation generating the coordinate triplet (2) $[\bar{x},\bar{y} + {1 \over 2}, z]$ from (1) x, y, z is symbolized by $[2\ \ 0, {1 \over 4}, z]$ . It is thus a twofold rotation (without screw part) around the line $[0,{1 \over 4},z]$ .