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

International Tables for Crystallography (2016). Vol. A, ch. 3.3, p. 790

Section 3.3.4. Changes introduced in space-group symbols since 1935

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

3.3.4. Changes introduced in space-group symbols since 1935

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Before the appearance of the first edition of International Tables in 1935, different notations for space groups were in use. A summary and comparative tables may be found in the introduction to that edition. The international notation was proposed by Hermann (1928a[link],b[link]) and Mauguin (1931[link]), who used the concept of lattice symmetry directions (see Section 3.3.1[link]) and gave preference to reflections or glide reflections as generators. Considerable changes to the original Hermann–Mauguin short symbols were made in IT (1952[link]).

The most important change refers to the symmetry directions. In the original Hermann–Mauguin symbols [IT (1935[link])], the distribution of symmetry elements is prescribed by the point-group symbol in the traditional setting, for example [\overline{4}2m] (not [\overline{4}m2]) but [\overline{6}m2] (not [\overline{6}2m]). This procedure sometimes implies the use of a larger unit cell than would be necessary. In IT (1952[link]) and in the present series, however, the lattice symmetry directions always refer to the conventional cell (cf. Chapter 3.1[link] ) of the Bravais lattice. The results of this change are (a) different symbols for centring types and (b) different sequences of the symbols referring to the point group. These differences occur only in some space groups that have a tetragonal or hexagonal lattice.

Thus, the two different space groups [D_{2d}^{1}] and [D_{2d}^{5}] were symbolized by [P\overline{4}2m] and [C\overline{4}2m] in IT (1935[link]) because in both cases the twofold axis had to be connected with the secondary set of symmetry directions. The new international symbols are [P\overline{4}2m] and [P\overline{4}m2]; since in the point group [{4/m}\ {2/m}\ {2/m}] of the Bravais lattice the secondary and tertiary set cannot be distinguished, the twofold axis in the subgroups [\overline{4}2m] and [\overline{4}m2] may occur in either the secondary or the tertiary set. Accordingly, the C-centred cell of [D_{2d}^{5}-C\overline{4}2m], used in IT (1935[link]), was transformed to a primitive one with the twofold axis along the tertiary set, resulting in the symbol [P\overline{4}m2].

The same considerations hold for [\overline{6}m2] and [\overline{6}2m] and for space groups with a hexagonal lattice belonging to the point groups 32, 3m and [\overline{3}m], which can be oriented in two ways with respect to the lattice.

For example, the point group 3m has two sets of symmetry directions. If the basis vector a is normal to the mirror plane m, two hexagonal cells with different centrings are possible:

  • (i) the hexagonal primitive cell, always described by C in IT (1935[link]), leads to [C3m = C_{3v}^{1}];

  • (ii) the hexagonal H-centred cell, with centring points in [{\textstyle{2\over 3}}, {\textstyle{1\over 3}}, 0] and [{\textstyle{1\over 3}}, {\textstyle{2\over 3}}, 0], leads to [H3m = C_{3v}^{2}] (cf. Chapter 3.1[link] ).

The latter can be transformed to a primitive cell in which the mirror plane is normal to the representative of the tertiary set of the hexagonal lattice. In IT (1952[link]) and the present editions, the primitive hexagonal cell is described by P. Thus, the above space groups receive the symbols [P3m1 = C_{3v}^{1}] and [P31m = C_{3v}^{2}].

Further changes are:

  • (i) In IT (1952[link]), symbols for space groups related to the point groups 422, 622 and 432 contain the twofold axis of the tertiary set. The advantage is that these groups can be generated by operations of the secondary and tertiary set. The symbol of the indicator is provided with the appropriate index to identify the screw part, thus fixing the intersection parameter.

  • (ii) Some standard settings are changed in the monoclinic system. In IT (1935[link]), only one setting (b unique, one cell choice) was tabulated for the monoclinic space groups. In IT (1952[link]), two choices were offered, b and c unique, each with one cell choice. In the present edition, the two choices (b and c unique) are retained but for each one three different cells are available. The standard short symbol, however, is that of IT (1935[link]) (b-unique setting).

  • (iii) In the short symbols of centrosymmetric space groups in the cubic system, [\bar{3}] is written instead of 3, e.g. Pm[\bar{3}] instead of Pm3 [as in IT (1935[link]) and IT (1952[link])].

  • (iv) Beginning with the fourth edition of this volume (1995), the following five orthorhombic space-group symbols have been modified by introducing the new glide-plane symbol e, according to a Nomenclature Report of the IUCr (de Wolff et al., 1992[link]).[\matrix{\hbox{Space group No.}&39\hfill&41\hfill&64\hfill&67\hfill&68\hfill\cr \hbox{Former symbol: } &Abm2&Aba2&Cmca&Cmma&Ccca\hfill\cr \hbox{New symbol: }\hfill&Aem2&Aea2&Cmce&Cmme&Ccce\hfill\cr}]The new symbol is indicated in the headline of these space groups. Further details are given in Section 2.1.2[link] .

    Difficulties arising from these changes are avoided by selecting the lexicographically first one of the two possible glide parts for the generating operation.

Example: [Aea2 \sim Aba2 = C_{2v}^{17}\ (41)]

The generators are[\displaylines{\openup-2pt\matrix{A \hbox{ centring: }\hfill&x,y+\textstyle{{1\over 2}},z+\textstyle{{1\over 2}}\hfill&(1)+(0, \textstyle{{1\over 2}}, \textstyle{{1\over 2}})\hfill\cr \hbox{glide reflection } b_{100}\!: \hfill&\overline{x},y+\textstyle{{1\over 2}},z \hfill&(4)\hfill\cr \hbox{or glide reflection } c_{100}\!:\hfill& \overline{x},y,z+\textstyle{{1\over 2}}\hfill&(4)+(0, \textstyle{{1\over 2}},\textstyle{ {1\over 2}})\hfill\cr \hbox{The first possibility is selected.}\hfill\cr \hbox{glide reflection } a_{010}\!:\hfill &x+\textstyle{{1\over 2}},\overline{y},z\hfill&(3).\hfill\cr}\hfill\cr \qquad\hbox{A shift of origin by } (-\textstyle{1\over 4},\! -\textstyle{{1\over 2}}, 0) \hbox{ is necessary.}\hfill\cr}]

The 1935 symbols and all the changes adopted in the present edition of International Tables can be seen in Table 3.3.3.1[link]. Differences in the symbols between IT (1952[link]) and the present edition may be found in the last column of this table; cf. also Section 2.1.3.4[link] .

References

International Tables for X-ray Crystallography (1952). Vol. I, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press. [Revised editions: 1965, 1969 and 1977. Abbreviated as IT (1952).]
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).]
Hermann, C. (1928a). Zur systematischen Strukturtheorie I. Eine neue Raumgruppensymbolik. Z. Kristallogr. 68, 257–287.
Hermann, C. (1928b). Zur systematischen Strukturtheorie II. Ableitung der Raumgruppen aus ihren Kennvektoren. Z. Kristallogr. 69, 226–249.
Koptsik, V. A. (1966). Shubnikov Groups. Moscow University Press. (In Russian.)
Mauguin, Ch. (1931). Sur le symbolisme des groupes de répetition ou de symétrie des assemblages cristallins. Z. Kristallogr. 76, 542–558.
Shubnikov, A. V. & Koptsik, V. A. (1972). Symmetry in Science and Art. Moscow: Nauka. (In Russian.) [Engl. transl: New York: Plenum (1974).]
Wolff, P. M. de, Billiet, Y., Donnay, J. D. H., Fischer, W., Galiulin, R. B., Glazer, A. M., Hahn, Th., Senechal, M., Shoemaker, D. P., Wondratschek, H., Wilson, A. J. C. & Abrahams, S. C. (1992). Symbols for symmetry elements and symmetry operations. Final Report of the International Union of Crystallography Ad-hoc Committee on the Nomenclature of Symmetry. Acta Cryst. A48, 727–732.








































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