Point-group symbols

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, pp. 818-820
https://doi.org/10.1107/97809553602060000524

Chapter 12.1. Point-group symbols

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

The three main kinds of point-group symbols in use today (Schoenflies symbols, Shubnikov symbols and Hermann–Mauguin symbols) are described and listed. International (Hermann–Mauguin) and Shubnikov symbols for symmetry elements and representatives for the lattice symmetry directions in the different crystal families are tabulated.

Keywords: point-group symbols; Schoenflies symbols; Shubnikov symbols; Hermann–Mauguin symbols; symmetry directions; generators; international space-group symbols.

12.1.1. Introduction

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For symbolizing space groups, or more correctly types of space groups, different notations have been proposed. The following three are the main ones in use today:

(i) the notation of Schoenflies (1891 , 1923 );
(ii) the notation of Shubnikov (Shubnikov & Koptsik, 1972 ), which is frequently used in the Russian literature;
(iii) the international notation of Hermann (1928) and Mauguin (1931). It was used in IT (1935) and was somewhat modified in IT (1952).

In all three notations, the space-group symbol is a modification of a point-group symbol.

Symmetry elements occur in lattices, and thus in crystals, only in distinct directions. Point-group symbols make use of these discrete directions and their mutual relations.

12.1.2. Schoenflies symbols

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Most Schoenflies symbols (Table 12.1.4.2 , column 1) consist of the basic parts C_n, D_n,¹ T or O, designating cyclic, dihedral, tetrahedral and octahedral rotation groups, respectively, with [n = 1, 2, 3, 4, 6] . The remaining point groups are described by additional symbols for mirror planes, if present. The subscripts h and v indicate mirror planes perpendicular and parallel to a main axis taken as vertical. For T, the three mutually perpendicular twofold axes and, for O, the three fourfold axes are considered to be the main axes. The index d is used for mirror planes that bisect the angle between two consecutive equivalent rotation axes, i.e. which are diagonal with respect to these axes. For the rotoinversion axes $[\overline{1}, \overline{2} \equiv m, \overline{3}]$ and $[\overline{4}]$ , which do not fit into the general Schoenflies concept of symbols, other symbols $[C_{i}, C_{s}, C_{3i}]$ and $[S_{4}]$ are in use. The rotoinversion axis $[\overline{6}]$ is equivalent to [3/m] and thus designated as $[C_{3h}]$ .

12.1.3. Shubnikov symbols

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The Shubnikov symbol is constructed from a minimal set of generators of a point group (for exceptions, see below). Thus, strictly speaking, the symbols represent types of symmetry operations . Since each symmetry operation is related to a symmetry element, the symbols also have a geometrical meaning. The Shubnikov symbols for symmetry operations differ slightly from the international symbols (Table 12.1.3.1 ). Note that Shubnikov, like Schoenflies, regards symmetry operations of the second kind as rotoreflections rather than as rotoinversions.

Table 12.1.3.1| top | pdf |
International (Hermann–Mauguin) and Shubnikov symbols for symmetry elements

The first power of a symmetry operation is often designated by the symmetry-element symbol without exponent 1, the other powers of the operation carry the appropriate exponent.

	Symmetry elements
	of the first kind	of the second kind
Hermann–Mauguin	1 2 3 4 6	$[\bar{1}\quad m\quad \bar{3}\quad \bar{4}\quad \bar{6}]$
Shubnikov^†	1 2 3 4 6	$[\tilde{2}\quad m\quad \tilde{6}\quad \tilde{4}\quad \tilde{3}]$

^†According to a private communication from J. D. H. Donnay, the symbols for elements of the second kind were proposed by M. J. Buerger. Koptsik (1966)

used them for the Shubnikov method.

If more than one generator is required, it is not sufficient to give only the types of the symmetry elements; their mutual orientations must be symbolized too. In the Shubnikov symbol, a colon (:), a dot (·) or a slash (/) is used to designate perpendicular, parallel or oblique arrangement of the symmetry elements. For a reflection, the orientation of the actual mirror plane is considered, not that of its normal. The exception mentioned above is the use of [3:m] instead of $[\tilde{3}]$ in the description of point groups.

12.1.4. Hermann–Mauguin symbols

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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.

12.1.4.2. Full Hermann–Mauguin symbols

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After the classification of the directions of rotation axes, the description of the seven maximal rotation subgroups of the lattice point groups is rather simple. For each representative direction, the rotational symmetry element is symbolized by an integer n for an n-fold axis, resulting in the symbols of the maximal rotation subgroups 1, 2, 222, 32, 422, 622, 432. The symbol 1 is used for the triclinic case. The complete lattice point group is constructed by multiplying the rotation group by the inversion $[\overline{1}]$ . For the even-fold axes, 2, 4 and 6, this multiplication results in a mirror plane perpendicular to the rotation axis yielding the symbols $[2n/m\ (n = 1, 2, 3)]$ . For the odd-fold axes 1 and 3, this product leads to the rotoinversion axes $[\overline{1}]$ and $[\overline{3}]$ . Thus, for each representative of a set of lattice symmetry directions, the symmetry forms a point group that can be generated by one, or at most two, symmetry operations. The resulting symbols are called full Hermann–Mauguin (or international) symbols. For the lattice point groups they are shown in Table 12.1.4.1 .

For the description of a point group of a crystal, we use its lattice symmetry directions. For the representative of each set of lattice symmetry directions, the remaining subgroup is symbolized; if only the primary symmetry direction contains symmetry higher than 1, the symbols `1' for the secondary and tertiary set (if present) can be omitted. For the cubic point groups T and $[T_{h}]$ , the representative of the tertiary set would be `1', which is omitted. For the rotoinversion groups $[\overline{1}]$ and $[\overline{3}]$ , the remaining subgroups can only be 1 and 3. If the supergroup is [2n/m] , five different types of subgroups can be derived: $[n/m,\ 2n,\ \overline{2n},\ n]$ and m. In the cubic system, for instance, $[4/m,\ 2/m]$ , $[\bar{4}]$ , 4 or 2 may occur in the primary set. In this case, the symbol m can only occur in the combinations [2/m] or [4/m] as can be seen from Table 12.1.4.2 .

Table 12.1.4.2| top | pdf |
Point-group symbols

Schoenflies	Shubnikov	International Tables, short symbol	International Tables, full symbol
$[C_{1}]$	1	1	1
$[C_{i}]$	$[\tilde{2}]$	$[\bar{1}]$	$[\bar{1}]$
$[C_{2}]$	2	2	2
$[C_{s}]$	m	m	m
$[C_{2h}]$
$[D_{2}]$		222	222
$[C_{2v}]$	$[2 \cdot m]$	mm2	mm2
$[D_{2h}]$	$[m \cdot 2:m]$	mmm	$[2/m\ 2/m\ 2/m]$
$[C_{4}]$	4	4	4
$[S_{4}]$	$[\widetilde{4}]$	$[\bar{4}]$	$[\bar{4}]$
$[C_{4h}]$
$[D_{4}]$		422	422
$[C_{4v}]$	$[4 \cdot m]$	4mm	4mm
$[D_{2d}]$	$[\widetilde{4}:2]$	$[\bar{4}2m]$ or $[\bar{4}m2]$	$[\bar{4}2m]$ or $[\bar{4}m2]$
$[D_{4h}]$	$[m \cdot 4:m]$		$[4/m\ 2/m\ 2/m]$
$[C_{3}]$	3	3	3
$[C_{3i}]$	$[\widetilde{6}]$	$[\bar{3}]$	$[\bar{3}]$
$[D_{3}]$		32 or 321 or 312	32 or 321 or 312
$[C_{3v}]$	$[3 \cdot m]$	3m or 3m1 or 31m	3m or 3m1 or 31m
$[D_{3d}]$	$[\widetilde{6} \cdot m]$	$[\bar{3}m]$ or $[\bar{3}m1]$ or $[\bar{3}1m]$	$[\bar{3}\; 2/m]$ or $[\bar{3}\; {2/m} 1]$ or $[\bar{3} 1 {2/m}]$
$[C_{6}]$	6	6	6
$[C_{3h}]$		$[\bar{6}]$	$[\bar{6}]$
$[C_{6h}]$
$[D_{6}]$		622	622
$[C_{6v}]$	$[6\cdot m]$	6mm	6mm
$[D_{3h}]$	$[m\cdot 3:m]$	$[\bar{6}m2]$ or $[\bar{6}2m]$	$[\bar{6}m2]$ or $[\bar{6}2m]$
$[D_{6h}]$	$[m\cdot 6:m]$		$[6/m\; 2/m\; 2/m]$
T		23	23
$[T_{h}]$	$[\widetilde{6}/2]$	$[m\bar{3}]$	$[{2/m} \bar{3}]$
O		432	432
$[T_{d}]$	$[3/\widetilde{4}]$	$[\bar{4}3m]$	$[\bar{4}3m]$
$[O_{h}]$	$[\widetilde{6}/4]$	$[m\bar{3}m]$	$[4/m\; \bar{3}\; 2/m]$

12.1.4.3. Short symbols and generators

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If the symbols are not only used for the identification of a group but also for its construction, the symbol must contain a list of generating operations and additional relations, if necessary. Following this aspect, the Hermann–Mauguin symbols can be shortened. The choice of generators is not unique; two proposals were presented by Mauguin (1931). In the first proposal, in almost all cases the generators are the same as those of the Shubnikov symbols. In the second proposal, which, apart from some exceptions (see Chapter 12.4 ), is used for the international symbols, Mauguin selected a set of generators and thus a list of short symbols in which reflections have priority (Table 12.1.4.2 , column 3). This selection makes the transition from the short point-group symbols to the space-group symbols fairly simple. These short symbols contain two kinds of notation components:

(i) components that represent the type of the generating operation, which are called generators;
(ii) components that are not used as generators but that serve to fix the directions of other symmetry elements (Hermann, 1931 ), and which are called indicators.

The generating matrices are uniquely defined by (i) and (ii), if it is assumed that they describe motions with counterclockwise rotational sense about the representative direction looked at end on by the observer. The symbols 2, 4, $[\bar{4}]$ , 6 and $[\bar{6}]$ referring to direction [001] are indicators when the point-group symbol uses three sets of lattice symmetry directions. For instance, in 4mm the indicator 4 fixes the directions of the mirrors normal to [100] and $[\hbox{[}1\bar{1}0\hbox{]}]$ .

Note : The generation of (a) point group 432 by a rotation 3 around [111] and a rotation 2 and (b) point group $[\bar{4}3m]$ by 3 around [111] and a reflection m is only possible if the representative direction of the tertiary set is changed from $[\hbox{[}1\bar{1}0\hbox{]}]$ to [110]; otherwise only the subgroup 32 or 3m of 432 or $[\bar{4}3m]$ will be generated.

References

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Hermann, C. (1928). Zur systematischen Strukturtheorie I. Eine neue Raumgruppensymbolik. Z. Kristallogr. 68, 257–287.Google Scholar

Hermann, C. (1931). Bemerkungen zu der vorstehenden Arbeit von Ch. Mauguin. Z. Kristallogr. 76, 559–561.Google Scholar

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International Tables for Crystallography (2006). Vol. A. ch. 12.1, pp. 818-820
https://doi.org/10.1107/97809553602060000524