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

International Tables for Crystallography (2015). Vol. A, ch. 1.5, pp. 95-106

Section 1.5.4.3. Synoptic table of the space groups

B. Souvignier,c G. Chapuisd and H. Wondratscheka

1.5.4.3. Synoptic table of the space groups

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Table 1.5.4.4[link] gives a comprehensive listing of the possible space-group symbols for various settings and choices of the unit cell. The data are ordered according to the crystal systems. The extended Hermann–Mauguin symbols provide information on the additional symmetry operations generated by the compositions of the symmetry operations with lattice translations. An extended Hermann–Mauguin symbol is a complex multi-line symbol: (i) the first line contains those symmetry operations for which the coordinate triplets are explicitly printed under `Positions' in the space-group tables in this volume; (ii) the entries of the lines below indicate the additional symmetry operations generated by the compositions of the symmetry operations of the first line with lattice translations. For example, for A-, B-, C- and I-centred space groups, the entries of the second line of the two-line extended symbol denote the symmetry operations generated by combinations with the corresponding centring translations.3

In the triclinic system the corresponding symbols do not depend on any space direction. Therefore, only the two standard symbols P1 (1) and [P\bar{1}] (2) are listed. One should, however, bear in mind that in some circumstances it might be more appropriate to use a centred cell for comparison purposes, e.g. following a phase transition resulting from a temperature, pressure or composition change.

The monoclinic and orthorhombic systems present the largest number of alternatives owing to various settings and cell choices. In the monoclinic system, three choices of unique axis can occur, namely b, c and a. In each case, two permutations of the other axes are possible, thus yielding six possible settings given in terms of three pairs, namely [{\bf a\underline{b}c}] and [{\bf c\underline{\overline{b}}a}], [{\bf ab\underline{c}}] and [{\bf ba\underline{\overline{c}}}], [{\bf \underline{a}bc}] and [{\bf \underline{\overline{a}}cb}]. The unique axes are underlined and the negative sign, placed over the letter, maintains the correct handedness of the reference system. The three possible cell choices indicated in Fig. 1.5.3.1[link] increase the number of possible symbols by a factor of three, thus yielding 18 different cases for each monoclinic space group, except for five cases, namely P2 (3), [P2_{1}] (4), Pm (6), P2/m (10) and P21/m (11) with only six variants.

In monoclinic P lattices, the symmetry operations along the symmetry direction are always unique. Here again, as in the plane groups, the cell centrings give rise to additional entries in the extended Hermann–Mauguin symbols. Consider, for example, the data for monoclinic P12/m1 (10), C12/m1 (12) and C12/c1 (15) in Table 1.5.4.4[link]. For P12/m1 and its various settings there is only one line, which corresponds to the full Hermann–Mauguin symbols; these contain only rotations 2 and reflections m. The first line for C12/m1 is followed by a second line, the first entry of which is the symbol 21/a, because 21 screw rotations and a glide reflections also belong to this space group. Similarly, in C12/c1 rotations 2 and screw rotations 21 and c and n glide reflections alternate, and thus under the full symbol [C12/c1] one finds the entry [2_1/n].

In Table 1.5.4.4[link] the Hermann–Mauguin symbols of the orthorhombic space groups are listed in six different settings: the standard setting [{\bf abc}], and the settings [{\bf ba\overline{c}}], [{\bf cab}], [{\bf \overline{c}ba}], [{\bf bca}] and [{\bf a\overline{c}b}]. These six settings result from the possible permutations of the three axes. Let us compare for a few space groups the standard setting [{\bf abc}] with the [{\bf cab}] setting. For Pmm2 (25) the permutation yields the new setting P2mm, reflecting the fact that the twofold axes parallel to the c direction change to the a direction. The mirrors normal to a and b become normal to b and c, respectively.

The case of Cmm2 (35) is slightly more complex due to the centring. As a result of the permutation the C centring becomes an A centring. The changes in the twofold axes and mirrors are similar to those of the previous example and result in the A2mm setting of Cmm2.

The extended Hermann–Mauguin symbol of the centred space group Aem2 (39) reveals the nature of the e-glide plane (also called the `double' glide plane): among the set of glide reflections through the same (100) plane, there exist two glide reflections with glide components [ {\textstyle{1 \over 2}} {\bf b}] and [{\textstyle{1 \over 2}}{\bf c}] (for details of the e-glide notation the reader is referred to Section 1.2.3[link] , see also de Wolff et al., 1992[link]). In the [{\bf cab}] setting, the A centring changes to a B centring and the double glide plane is now normal to b and the glide reflections have glide components [ {\textstyle{1 \over 2}}{\bf a}] and [{\textstyle{1 \over 2}}{\bf c}]. The corresponding symbol is thus B2em. Note that in the cases of the five ortho­rhombic space groups whose Hermann–Mauguin symbols contain the e-glide symbol, namely Aem2 (39), Aea2 (41), Cmce (64), Cmme (67) and Ccce (68), the characters in the first lines of the extended symbols differ from the short symbols because the characters in the extended symbol represent symmetry operations, whereas those in the short and full symbol represent symmetry elements. In all these cases, the extended symbols listed in Table 1.5.4.4[link] are complemented by the short symbols, given in brackets.

The general discussion in Section 1.5.4.1[link] about the additional symmetry operations that occur as a result of combinations with lattice translations provides some rules for the construction of the extended Hermann–Mauguin symbols in the orthorhombic crystal system. In orthorhombic space groups with primitive lattices, the symmetry operations of any symmetry direction are always unique: either 2 or 21, either m or a or b or c or n. In C-centred lattices, owing to the possible combination of the original symmetry operations with the centring translations, the axes 2 along [100] and [010] alternate with axes 21. However, parallel to c there are either 2 or 21 axes because the combination of a rotation or screw rotation with a centring translation results in another operation of the same kind. Similarly, [m_{100}] alternates with [b_{100}], [m_{010}] with [a_{010}], [c_{100}] with [n_{100}] etc. The [m_{001}] reflection plane is simultaneously an [n_{001}] glide plane and an [a_{001}] glide plane is simultaneously a [b_{001}] glide plane. This latter plane with its double role is the [e_{001}] glide plane, as found for example in the full symbol of C2/m2/m2/e (67) and the corresponding short symbol Cmme. As another example, consider the space group C2/m2/c21/m (63). In Table 1.5.4.4[link], in the line of various settings for this space group the short Hermann–Mauguin symbols are listed, and the rotations or screw rotations do not appear. The [m_{100}], [c_{010}] and [m_{001}] reflections and glide reflections occur alternating with [b_{100}], [n_{010}] and [n_{001}] glide reflections, respectively. The entry under Cmcm is thus bnn.

F and I centring cause alternating symmetry operations for all three coordinate axes a, b and c. For these centrings, the permutation of the axes does not affect the symbol F or I of the centring type. However, the number of symmetry operations increases by a factor of four for F centrings and by a factor of two for I centrings when compared to those of a space group with a primitive lattice. In Fmm2 (42) for example, three additional lines appear in the extended symbol, namely ba2, [nc2_1] and [cn2_1]. These operations are obtained by combining successively the centring translations [t({\textstyle{1 \over 2}}, {\textstyle{1 \over 2}}, 0)], [t(0,{\textstyle{1 \over 2}}, {\textstyle{1 \over 2}})] and [t({\textstyle{1 \over 2}}, 0,{\textstyle{1 \over 2}})] with the symmetry operations of Pmm2. However, in space groups Fdd2 (43) and Fddd (70) the nature of the d planes is not altered by the translations of the F-centred lattice; for this reason, in Table 1.5.4.4[link] a two-line symbol for Fdd2 and a one-line symbol for Fddd are sufficient.

In tetragonal space groups with primitive lattices there are no alternating symmetry operations belonging to the symmetry directions [001] and [100]. However, for the symmetry direction [[1\overline{1}0]] the symmetry operations 2 and 21 alternate, as do the reflection m and the glide reflection g [g is the name for a glide reflection with a glide vector [({\textstyle{1 \over 2}},{\textstyle{1 \over 2}},0)]], and the glide reflections c and n. For example, the second line of the extended symbol of [P4_2/n\,2/b\,2/c] (133) contains the expression [2_1/n] under the expression [2/c].

For the space groups in the tetragonal system, the unique axis is always the c axis, thus reducing the number of settings and choices of the unit cell. Two additional multiple cells are considered in this system, namely the C and F cells obtained from the P and I cell by the following relations:[{\bf a'}={\bf a}\mp{\bf b}\semi\ {\bf b'}=\pm{\bf a}+{\bf b}\semi\ {\bf c'}={\bf c}.]The secondary [100] and tertiary [110] symmetry directions are interchanged in this cell transformation. As an example, consider P4/n (85) and its description with respect to a C-centred basis. Under the transformation [{\bf a}'={\bf a}+{\bf b}], [{\bf b}'=-{\bf a}+{\bf b}], [{\bf c}'= {\bf c}], the n glide [n({\textstyle{1 \over 2}}, {\textstyle{1 \over 2}}, 0)\ x, y, 0] is transformed to an a glide [a\ \, x, y, 0] while its coplanar equivalent glide [n(-{\textstyle{1 \over 2}}, {\textstyle{1 \over 2}}, 0)\ x, y, 0] is transformed to a b glide [b\ \, x, y, 0]. Thus, the extended symbol of the multiple-cell description of P4/n (85) shown in Table 1.5.4.4[link] is C4/a(b), while in accordance with the e-glide convention, the short Hermann–Mauguin symbol becomes C4/e.

In the case of I4/m (87), as a result of the I centring, screw rotations 42 and glide reflections n normal to 42 appear as additional symmetry operations and are shown in the second line of the extended symbol (cf. Table 1.5.4.4[link]). In the multiple-cell setting, the space group F4/m exhibits the additional fourfold screw axis [4_2] and owing to the new orientation of the [a'] and [b'] axes, which are rotated by 45° relative to the original axes a and b, the n glide of I4/m becomes an a glide in the extended Hermann–Mauguin symbol. The additional b glide obtained from a coplanar n glide is not given explicitly in the extended symbol.

The rhombohedral space groups are listed together with the trigonal space groups under the heading `Trigonal system'. For both representative symmetry directions [001]hex and [100]hex, rotations with screw rotations and reflections with glide reflections or different kinds of glide reflections alternate, so that additional symmetry operations always occur: rotations 3 or rotoinversions [\overline{3}] are accompanied by 31 and 32 screw rotations; 2 rotations alternate with 21 screw rotations and m reflections or c glide reflections alternate with additional glide reflections. As examples, under the full Hermann–Mauguin symbol R3 (146) one finds [3_{1,2}] and in the line under [R\,\overline{3}\, 2/c] (167) one finds [3_{1,2}\ 2_1/n].

The extended Hermann–Mauguin symbols for space groups of the hexagonal crystal system retain the symbol for the primary symmetry direction [001]. Along the secondary [\langle 100\rangle] and tertiary [\langle1\overline{1}0\rangle] symmetry directions every horizontal axis 2 is accompanied by a screw rotation 21, while the reflections and glide reflections, or different types of glide reflections, alternate.

The list of hexagonal and trigonal space-group symbols is completed by a multiple H cell, which is three times the volume of the corresponding P cell. The unit-cell transformation is obtained from the relation[{\bf a'}={\bf a}-{\bf b}\semi\ {\bf b'}={\bf a}+2{\bf b}\semi\ {\bf c'}={\bf c}]with centring points at 0, 0, 0; [\textstyle{{2}\over{3}}, \textstyle{{1}\over{3}},0] and [\textstyle{{1}\over{3}}, \textstyle{{2}\over{3}},0]. The new vectors [{\bf a'}] and [{\bf b'}] are rotated by −30° in the ab plane with respect to the old vectors [{\bf a}] and [{\bf b}]. There are altogether six possible such multiple cells rotated by ±30°, ±90° and ±150° (cf. Table 1.5.1.1[link] and Fig. 1.5.1.8[link]).

The hexagonal lattice is frequently referred to the ortho­hexagonal C-centred cell (cf. Table 1.5.1.1[link] and Fig. 1.5.1.7[link]). The volume of this centred cell is twice the volume of the primitive hexagonal cell and its basis vectors are mutually perpendicular.

In general, the space groups of the cubic system do not yield any additional orientations and only the short, full and extended symbols are given. The only exception to this general rule is the group [Pa\bar 3] (205) with its alternative setting [Pb\bar 3], whose basis vectors [{\bf a}',{\bf b}',{\bf c}'] are related by a rotation of 90° in the ab plane to the basis vectors [{\bf a},{\bf b},{\bf c}] of [Pa\bar 3]: [{\bf a}'={\bf b}, {\bf b}'=-{\bf a}, {\bf c}'={\bf c}]. The different general reflection conditions of [Pb\bar 3] in comparison to those of [Pa\bar 3] indicate its importance for diffraction studies (cf. Table 1.6.4.25[link] ). In some extended symbols of the cubic groups, we note the use of the g or [g_i] type of glide reflections as in, for example, [F\bar 43c] (219). The g glide is a generic form of a glide plane which is different from the usual glide planes denoted by a, b, c, n, d or e. The symbols g, [g_1] and [g_2] indicate specific glide components and orientations that are specified in the Note to Table 1.5.4.4[link].

References

International Tables for Crystallography (1995). Vol. A, Space-Group Symmetry. Edited by Th. Hahn, 4th revised ed. Dordrecht: Kluwer Academic Publishers.
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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