Space symmetry, Space groups

- Space groups are the product of possible combinations of symmetry operations including translations. - There exist 230 different space groups in 3-dimensional space - Comparing to point groups, space groups have 2 more symmetry operations (table 1). These operations include translations. Therefore they describe not only the lattice but also the crystal structure.

Table 1: Additional symmetry operations in space symmetry, their description, symmetry elements and Hermann-Mauguin symbols. symmetry H-M description symmetry element operation symbol screw 1. rotation by 360°/N screw axis NM (Schraubung) 2. translation along the axis (Schraubenachse) 1. reflection across the plane glide glide plane 2. translation parallel to the a,b,c,n,d (Gleitspiegelung) (Gleitspiegelebene) glide plane

Glide directions for the different gilde planes: a (b,c): translation along ½ a or ½ b or ½ c, respectively) (with ,, cba = vectors of the unit cell) n: translation e.g. along ½ (a + b) d: translation e.g. along ¼ (a + b), d from diamond, because this glide plane occurs in the diamond structure

Screw axis example: 21 (N = 2, M = 1) 1) rotation by 180° (= 360°/2) 2) translation parallel to the axis by ½ unit (M/N)

Only 21, (31, 32), (41, 43), 42, (61, 65) (62, 64), 63 screw axes exist in parenthesis: right, left hand screw axis

Space group symbol: The space group symbol begins with a capital letter (P: primitive; A, B, C: base centred I, body centred R rhombohedral, F face centred), which represents the Bravais lattice type, followed by the short form of symmetry elements, as known from the point group symbols. The symmetry elements are ordered according to the viewing directions in the respective crystal system (table 2). Some space groups have only one viewing direction, so their symbols are short (P4, P21/c...), others have up to 3 viewing directions and therefore a longer symbol (P4/mmm, I4/mcm, Pmmm…). Table 2: Viewing directions of the 7 crystal systems. position in the symbol triclinic monoclinic orthorhombic tetragonal trigonal hexagonal cubic 1. ― b a c c c ‹a› 2. ― b ‹a› ‹a› ‹a› ‹111› 3. ― c ‹110› ― ‹210› ‹110› ‹...›: ... and symmetry equivalent directions, respectively

Literature:

W. Borchardt-Ott, Kristallographie, 7th ed. Springer-Verlag, Berlin, Heidelberg, 2009. U. Müller, Anorganische Strukturchemie, 6th ed. Vieweg+Teubner, Wiesbaden, 2008. A. R. West, Basic Solid State Chemistry, 2nd ed. John Wiley & Sons LTD, Chichester, 2004. http://ruby.chemie.uni-freiburg.de/Vorlesung/Vorlagen/methoden_II_24.pdf Î (compact overview of space groups, bravais lattices and viewing directions, only 1 page) http://www.uni-leipzig.de/~minkrist/pdf/IMKM_Kristallographie_2.pdf http://chemkrist2009.uni-freiburg.de/Vortraege/englert_2.pdf http://img.chem.ucl.ac.uk/sgp/LARGE/sgp.htm http://cst-www.nrl.navy.mil/lattice/spcgrp/