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

International Tables for Crystallography (2016). Vol. A, ch. 3.3, pp. 777-779

Section 3.3.1.4. Hermann–Mauguin symbols

H. Burzlaffa and H. Zimmermannb*

aUniversität 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

3.3.1.4. Hermann–Mauguin symbols

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3.3.1.4.1. 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 symmetry equivalence, at most three classes are obtained. These classes were called Blickrichtungssysteme (Heesch, 1929[link]). 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. Sections 1.3.4.3[link] and 3.1.1.4[link] ) and use other representatives than chosen by Heesch [IT (1935[link]), 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 3.3.1.2[link]. The directions are related to the conventional crystallographic basis of each lattice point group (cf. Section 3.1.1.4[link] ).

Table 3.3.1.2| top | pdf |
Representatives for the sets of lattice symmetry directions in the various crystal families

Crystal familyAnorthic (triclinic)MonoclinicOrthorhombicTetragonalHexagonalCubic
Lattice point group Schoenflies [C_{i}] [C_{2h}] [D_{2h}] [D_{4h}] [D_{6h}] [D_{3d}] [O_{h}]
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{]}]
[\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.

3.3.1.4.2. 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 3.3.1.2[link].

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 Th, 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 3.3.1.3[link].

Table 3.3.1.3| top | pdf |
Point-group symbols

SchoenfliesShubnikovInternational Tables, short symbolInternational 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}] [2:m] [2/m] [2/m]
[D_{2}] [2:2] 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}] [4:m] [4/m] [4/m]
[D_{4}] [4:2] 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/mmm] [4/m\ 2/m\ 2/m]
[C_{3}] 3 3 3
[C_{3i}] [\widetilde{6}] [\bar{3}] [\bar{3}]
[D_{3}] [3:2] 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}] [3:m] [\bar{6}] [\bar{6}]
[C_{6h}] [6:m] [6/m] [6/m]
[D_{6}] [6:2] 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/mmm] [6/m\; 2/m\; 2/m]
T [3/2] 23 23
[T_{h}] [\widetilde{6}/2] [m\bar{3}] [{2/m} \bar{3}]
O [3/4] 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]

3.3.1.4.3. 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[link]). 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 Section 3.3.4[link]), 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 3.3.1.3[link], 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[link]), and which are called indicators.

The generating matrices are uniquely defined by (i)[link] and (ii)[link] 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

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).]
Heesch, H. (1929). Zur systematischen Strukturtheorie II. Z. Kristallogr. 72, 177–201.
Hermann, C. (1931). Bemerkungen zu der vorstehenden Arbeit von Ch. Mauguin. Z. Kristallogr. 76, 559–561.
Mauguin, Ch. (1931). Sur le symbolisme des groupes de répetition ou de symétrie des assemblages cristallins. Z. Kristallogr. 76, 542–558.








































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