Symmetry directions

Burzlaff, H.; Zimmermann, H.

doi:10.1107/97809553602060000524

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. 12.1, p. 818

Section 12.1.4.1. Symmetry directions

H. Burzlaff^a and H. Zimmermann^b ^*

^a Universität Erlangen–Nürnberg, Robert-Koch-Strasse 4a, D-91080 Uttenreuth, Germany, and ^bInstitut für Angewandte Physik, Lehrstuhl für Kristallographie und Strukturphysik, Universität Erlangen–Nürnberg, Bismarckstrasse 10, D-91054 Erlangen, Germany
Correspondence e-mail: helmuth.zimmermann@krist.uni-erlangen.de

12.1.4.1. Symmetry directions

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The Hermann–Mauguin symbols for finite point groups make use of the fact that the symmetry elements, i.e. proper and improper rotation axes, have definite mutual orientations. If for each point group the symmetry directions are grouped into classes of symmetrical equivalence, at most three classes are obtained. These classes were called Blickrichtungssysteme (Heesch, 1929). If a class contains more than one direction, one of them is chosen as representative.

The Hermann–Mauguin symbols for the crystallographic point groups refer to the symmetry directions of the lattice point groups (holohedries, cf. Part 9 ) and use other representatives than chosen by Heesch [IT (1935), p. 13]. For instance, in the hexagonal case, the primary set of lattice symmetry directions consists of $[\{\hbox{[}001\hbox{]}, \hbox{[}00\overline{1}\hbox{]}\}]$ , representative is [001]; the secondary set of lattice symmetry directions consists of [100], [010], $[\hbox{[}\overline{1}{\hbox to .5pt{}}\overline{1}0\hbox{]}]$ and their counter-directions, representative is [100]; the tertiary set of lattice symmetry directions consists of $[\hbox{[}1\overline{1}0\hbox{]}, \hbox{[}120], \hbox{[}\overline{2}{\hbox to .5pt{}}\overline{1}0\hbox{]}]$ and their counter-directions, representative is $[\hbox{[}1\overline{1}0\hbox{]}]$ . The representatives for the sets of lattice symmetry directions for all lattice point groups are listed in Table 12.1.4.1. The directions are related to the conventional crystallographic basis of each lattice point group (cf. Part 9 ).

Table 12.1.4.1| top | pdf |
Representatives for the sets of lattice symmetry directions in the various crystal families

Crystal family		Anorthic (triclinic)	Monoclinic	Orthorhombic	Tetragonal	Hexagonal		Cubic
Lattice point group	Schoenflies	$[C_{i}]$	$[C_{2h}]$	$[D_{2h}]$	$[D_{4h}]$	$[D_{6h}]$	$[D_{3d}]$	$[O_{h}]$
Lattice point group	Hermann–Mauguin	$[\bar{1}]$	$[\displaystyle{2 \over m}]$	$[\displaystyle{2 \over m} {2 \over m} {2 \over m}]$	$[\displaystyle{4 \over m} {2 \over m} {2 \over m}]$	$[\displaystyle{6 \over m} {2 \over m} {2 \over m}]$	$[\displaystyle\bar{3} {2 \over m}]$ ^†	$[\displaystyle{4 \over m} \bar{3} {2 \over m}]$
Set of lattice symmetry directions	Primary	–	[010]
			b unique	[100]	[001]	[001]	[001]	[001]
			[001]
			c unique
	Secondary	–	–	[010]	[100]	[100]	[100]	[111]
	Tertiary	–	–	[001]	$[\hbox{[}1\bar{1}0\hbox{]}]$	$[\hbox{[}1\bar{1}0\hbox{]}]$	–	$[\hbox{[}1\bar{1}0\hbox{]}]$
	Tertiary	–	–	[001]	$[\hbox{[}1\bar{1}0\hbox{]}]$	$[\hbox{[}1\bar{1}0\hbox{]}]$	–	$[\hbox{[110]}]$ ^‡

^†In this table, the directions refer to the hexagonal description. The use of the primitive rhombohedral cell brings out the relations between cubic and rhombohedral groups: the primary set is represented by [111] and the secondary by [ $[1\bar 10]$ ].
^‡Only for $[\bar{4}3m]$ and 432 [for reasons see text].

The relation between the concept of lattice symmetry directions and group theory is evident. The maximal cyclic subgroups of the maximal rotation group contained in a lattice point group can be divided into, at most, three sets of conjugate subgroups. Each of these sets corresponds to one set of lattice symmetry directions.

References

Internationale Tabellen zur Bestimmung von Kristallstrukturen (1935). I. Band, edited by C. Hermann. Berlin: Borntraeger. [Reprint with corrections: Ann Arbor: Edwards (1944). Abbreviated as IT (1935).]Google Scholar

Heesch, H. (1929). Zur systematischen Strukturtheorie II. Z. Kristallogr. 72, 177–201.Google Scholar

International Tables for Crystallography (2006). Vol. A. ch. 12.1, p. 818