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

International Tables for Crystallography (2016). Vol. A, ch. 1.2, p. 20

Table 1.2.3.1 

H. Wondratscheka and M. I. Aroyob

aLaboratorium für Applikationen der Synchrotronstrahlung (LAS), Universität Karlsruhe, Germany, and bDepartamento de Física de la Materia Condensada, Universidad del País Vasco (UPV/EHU), Bilbao, Spain

Table 1.2.3.1| top | pdf |
Symmetry elements in point and space groups

Name of symmetry elementGeometric elementDefining operation (d.o.)Operations in element set
Mirror plane Plane p Reflection through p D.o. and its coplanar equivalents
Glide plane Plane p Glide reflection through p; 2v (not v) a lattice-translation vector D.o. and its coplanar equivalents
Rotation axis Line l Rotation around l, angle 2π/N, N = 2, 3, 4 or 6 1st … (N − 1)th powers of d.o. and their coaxial equivalents
Screw axis Line l Screw rotation around l, angle 2π/N, u = j/N times shortest lattice translation along l, right-hand screw, N = 2, 3, 4 or 6, j = 1, …, (N − 1) 1st … (N − 1)th powers of d.o. and their coaxial equivalents
Rotoinversion axis Line l and point P on l Rotoinversion: rotation around l, angle 2π/N, followed by inversion through P, N = 3, 4 or 6 D.o. and its inverse
Centre Point P Inversion through P D.o. only
That is, all glide reflections through the same reflection plane, with glide vectors v differing from that of the d.o. (taken to be zero for reflections) by a lattice-translation vector. The glide planes a, b, c, n, d and e are distinguished (cf. Table 2.1.2.1[link] ).
That is, all rotations and screw rotations around the same axis l, with the same angle and sense of rotation and the same screw vector u (zero for rotation) up to a lattice-translation vector.