M\1any arguments have been concerned with what these symbols mean, and how they

SOME DESIRABLE MODIFICATIONS OF THE INTERNATIONAL SYMMETRY SYMBOLS* BY MARTIN J. BUERGER MASSACHUSETTS INSTITUTE OF TECHNOLOGY Communicated August 21, 1967 With the publication of Hilton's Mathematical Crystallography' in 1903, Schoenflies' derivation of the space groups2' ' and his symbols for representing them4 became widely available. For those who understand Schoenflies' derivation of point groups, the symbols are self-explanatory. His symbolism for space groups, however, was less fortunate, for each of the several space groups isomorphic with a particular point group was identified by attaching a numerical superscript to the point-group symbol. This represented merely the order in which Schoenflies presented the derivation of this set of space groups. The arbitrary designation of a space group was unsatisfactory. In fact the arbitrariness probably led to the greater arbitrariness of numbering all space groups from 1 to 230 by Astbury and Yardley,5 a scheme which eventually found its way into the 1952 International Tables. Attempts to avoid the arbitrary aspect of spacegroup designation were made by C. Hermann,6 Schiebold,7 and Mauguin.8 Eventually the features of the Hermann and Mauguin schemes were incorporated into the first (1935) edition of the International Tables, and then came into common use. The symbols used in the International Tables were not entirely satisfactory, however, with respect to the choice of lattice symbol. For example, although trigonal symmetries can have only primitive Bravais lattice types, some space groups were described by the symbol H, others by C, the latter being referred to a double cell. For the tetragonal point group 42m, which can be placed in the cell of a simple or body-centered Bravais lattice so the pure rotation axis is parallel to a or to [1101, the two orientations were specified by P and C for the simple lattice, and by I and F for the body-centered lattice. Unfortunately the symbol C implies using a double cell for a simple lattice, while the symbol F implies using a quadruple cell for a body-centered lattice. A reaction against these unnecessarily doubled multiplicities set in, which was satisfied in the second (1952) edition of International Tables by designating any simple Bravais lattice by P (except for R) but indicating the orientation of the symmetry elements with respect to the lattice translation by inserting the symbol "1" in different places in the space-group sequence for trigonal crystals, or in reversing the order of the last two symbols for crystals of the classes 42m and 62m. The use of the symbol "1" to suggest orientation had already been used by Hermann6 and by Mauguin.8 In this author's opinion, the unfortunate choice of lattice symbols in the first edition of International Tables resulted from not recognizing exactly what function was to be served by that symbol. The change introduced by the second edition of International Tables patched up the trouble, in a way, but only at the expense of introducing symmetry symbols involving inconsistencies. As a consequence, the symmetry symbol then began to be interpreted, in part, as a symbol of orientation. M\1any arguments have been concerned with what these symbols mean, and how they 1768

VoL..58, 1967 PHYSICS: M. J. BUERGER 1769 can be used. The arguments still persist, and the confusion about the meanings of the symbols continues. In order to present a more consistent set of space-group symbols, some aspects of the Hermann-1\Iauguin symbols are discussed below. Possible Combinations of Rotations. Let 1, 2, 3, 4, and 6 each represent a possible operation of rotation about some axis, in this case an operation which is crystallographically permissible. Let a sequence like 62 represent a sequence (or combination) of rotations about two different axes. Each such combination is always equivalent to a third rotation, which, for crystals, must also be 1, 2, 3, 4, or 6. Such sets of three are therefore basic features of symmetry. It can be readily shown8 that the only possible crystallographic combinations involving pure rotations correspond to the following axial sequences: 222 32(2) 422 622 23(3) 432. These six combinations, together with the five symmetries generated by the five uncombined rotations 1, 2, 3, 4, and 6, represent the 11 basic kinds of pure rotational symmetries permissible in crystals. The numbers in parentheses in two of the sequences of three symbols, as given above, are normally omitted because the rotation resulting from combining the first two rotations is already represented in the second symbol when the group is completed. The rotoinversions combine in similar ways, and all permissible combinations of rotations and rotoinversion symmetries can be represented by the eleven basic symbol sequences noted above, modified (where necessary) by a bar over a numeral to indicate a rotoinversion axis, or by substituting a fraction for a numeral when both a pure rotation and a 2 (= m) axis are parallel. The sequence of one, two, or three numerals or fractions is a neat representation of the point-group symmetry, and each such sequence is the full Hermann-Mauguin symbol for one of the 32 possible point groups. A sequence of symmetry symbols cannot be tampered with and still retains its original neat representation of the point group. Specifically, if the numeral 1 is inserted in the sequence, it spoils the original meaning of the sequence. In particular, if, as in the second edition of the International Tables, 32 is modified to 312 and 321, in an attempt to imply orientation of the axis 2 with regard to a cell edge, then the consistency of the symbolism is destroyed, in the original meaning of a sequence, for 312 really implies that a rotation of 2Xr/3 about some axis, followed by a rotation of 2w7r/1 about another axis, is equal to a rotation of 27r/2 about some axis, a relation which cannot be satisfied by any axis. Another attempt to suggest orientation is provided by the symbol 121, which is sometimes used to imply that a 2-fold rotation is oriented parallel to the b axis. This implies A2,rTB2,/2 = C2,r, which is untrue. In general, then, the insertion of the symbol I into a pointgroup sequence produces an inconsistency. The standard point-group symmetry symbols, therefore, should not be modified by insertions. Neither should their substituents, which indicate screws and glides in the isomorphic space group, be modified. Similarities in Symmetry Symbols.-One of the features which characterize a science is that it considers similar things from the same viewpoint, which is related to the recognition of some kind of invariance. This desirable feature ought to appear in the science of crystallography, and should be present in symmetry symbols. Each sequence of point-group symbols not only indicates the point-group symmetry, but, in addition, is usually interpreted as representing the orientation of one

1770 PHYSICS: M. J. BUERGER PROC. N. A. S. axis of the symmetry with respect to a convenient reference system. Any single axis of rotation, or one axis of the sequence of two or three axes, can be conveniently set parallel to the same direction in space, say the X, or Y, or Z axis of a reference system. It is already common practice to place the axis represented by the first symbol of a sequence of one, two, or three symbols, parallel to the Z axis for all axial frames except that, for the orthorhombic sequence, it is now set parallel to X, and except that, some crystallographers prefer to set the axis corresponding to the symbol 2 (or 2 or 2/2), when it occurs alone, parallel to Y. Thus in only two of the eleven symbol sequences (or of ten, if the trivial sequence 1 is excluded) is an exception made to setting the axis corresponding to the first symbol parallel to Z. It is evident that to continue these two exceptions is to discard the only obvious similarity, or invariant aspect, of all the point groups. As a consequence, the following specific recommendation is made: For all point-group symbols (and for their substituents in isomorphic space groups) the first of a sequence of one, two, or three symbols is, by convention, the axis of the rotation (or its substituent) about Z. Symbols for Alternative Orientations of the Lattice.-In the first edition of International Tables the orientation of a Bravais lattice of end-centered type was symbolized by A, B, or C, according to which face of the parallelopiped was centered. These symbols have two functions: They reveal the general Bravais lattice type, and they reveal the relative orientation of lattice and symmetry axes. Let the symbol E be used to indicate an end-centered lattice without implying orientation; then it is seen that the permissible orientations of E for orthorhombic symmetries are A, B, and C. In a similar way, let a Bravais lattice type which is without points except on the cell vertices, be designated S ("simple") when it carries no orientation connotation, and let Z ("Zentriert") be a Bravais lattice type with an additional point at 1/2 1/2 '/2. For consistency, alternative symbols are needed for the two permissible orientations of S and two for Z in the tetragonal symmetries, and three for S in trigonal symmetries. The suggestion made here is that the alternative orientations for the simple lattice in tetragonal symmetries be designated TABLE 1 DEFINITION OF ORIENTATION SYMBOLS, FOR PLACEMENT OF SYMMETRY AXES WITH RESPECT TO CELL AXES OF GENERAL BRAVAIS LATTICE TYPES General Bravais Orientation Orientation of Axis of lattice type symbol First symbol Second symbol S (simple) P [011] in (100) D [001] f [110] orthogonal a[210] hexagonal R [111] [101] Z (body-centered) I [001] [100] J [001] [110] TABLE 2 LATTICE SYMBOLS Symbols for General Lattice Type, Implying Orientation Symbol for general lattice of Symmetry to Cell type, not involving Axial Diagonal Other orientation orientation orientation orientation S (simple) P D R E (end-centered) A, B, C Z (body-centered) I J F (face-centered) F

VOL. 58, 1967 PHYSICS: M. J. BUERGER 1771 TABLE 3 ORIENTATION SYMBOLS REQUIRED FOR THE VARIOUS POINT GROUPS Permissible Crystal system Point groups lattice orientation symbol Triclinic 1, 1 P, I (- A, B) Monoclinic 2, m, 2 P. A, B, C, I, F Tetragonal 4, 422 4, 4, 4mm PI M 4 2 2 m m m 4 2 m P, D, I, J Hexagonal 3, 3 P. R 32, 3m, 3 2 M 6, 622 ) Pi D, R 6,-, 6mm P 6 2 2 m m m 6 2 m P, D Isometric 23, 432 2-43m 4-2 P I - 3, -3- F M m ' 'm my P (for axial orientations) and D (for "diagonal" orientations). For the bodycentered lattice the symbols would be I (for axial orientation) and J ("justiert," for diagonal orientation). For trigonal crystals the three orientations of the simple cell would be P, D, and R. The more exact definitions of these orientations are given in Table 1. The distribution of these orientation symbols over the point groups is given in Table 2. The various lattice orientation symbols required for representing the space groups corresponding to each of the 32 point groups are listed in Table 3. New symbols are required only for the five point groups 42m, 62rn, 32, 3m, and 3-. m They actually appear in onlv 15 space groups, as indicated by the designation * in the last column of Table 4. The author urges the adoption of the new symbols for these space groups, together with the outlawing of the symbol "1" for point-group and space-group symbols, except id the point groups 1 and 1, and the space groups P1 and P1, where they behave in a way consistent with the meaning of sequences of symmetry symbols. The new symbols remove several inconsistencies in the older symbols, especially the lack of orientation indicated for any lattice type but the end-centered type. With these modifications the symmetry symbols have invariable meanings. In space-group sequences, the first symbol indicates the general Bravais lattice type and its mutual orientation with respect to the symmetry axes. It conveys no necessary indication of the coordinate system which an author wishes to use to describe his crystal. The next symbol corresponds to the generating operation about a primary axis, which is to be set parallel to the Z axis of the coordinate system. The third symbol indicates the generating operation about a secondary axis, which, when combined with that of the primary axis, gives the generating

1772 PHYSICS: M. J. BUERGER PROC. N. A. S. TABLE 4 Preferred Point Schoenflies 1935 I. T. 1952 I. T. group symbol symbol symbol Proposed 32 D31 H32 P312 D32 * D32 C32 P321 P32 D33 H312 P3J12 D312 * D34 C312 P3121 P312 D35 H322 P3212 D322 * D36 C322 P3221 P322 D37 R32 R32 R32 3m C3v 1 C3m P3ml P3m C3V2 H3m P31m D3m * CU'3 C3c P3c1 P3c C3.4 H3c P31c D3c * C3v5 R3m R3m R3m C3U6 R3c R3c R3c 2 DAd' H3m P31m D3m * m D d2 H3c P31c D3c * Ad3 C3m P3ml P3m DAd4 C3c P3c1 P3c D3d5 R3m R3m R3m DAd R3c R3c R3c 62m D3ah' C6m2 P6m2 D62m * D3h2 C6c2 P6c2 D62c * DAh3 H6m2 P62m Pf3m DMh4 H6c2 P62c P62c 42m D2d' P42m P42m P42m DAd2 P42c P42c P42c D2d3 P421m P42im P421m DAd 4 P421c P421c P421c D2d5 C42m P4m2 D42m * DAd6 C42c P4c2 D42c * D2d7 C42b P4b2 D42b * DAd8 C42n P4m2 D42n * Dad9 F42m 14m2 J42m * DAdP' F42c I4c2 J42c * D2d" I42m I42m I42m Ada 2 I42d 142d 142d operation about the tertiary axis. These symbols have the same meanings for all space groups. Every space group has the three last symbols in exactly the same sequence as the standard sequence for the corresponding point group. The distinctive features of a space group, and how it is derived from its point group, are very plain: Each point-group symbol has a substituent in the same place in all its space groups, an)d the permissible lattice type and its orientation with respect to the symmetry axes is obvious from the first symbol. The new symbolism should render less necessary the arbitrary numerical designation of a space group, as used in the 1952 edition of International Tables, because the meanings of the new symbols are more evident. They should also make it easier to teach space groups and their derivation. * This investigation was supported by a grant from the National Science Foundation. 1 Hilton, Harold, Mathematical Crystallography and the Theory of Groups of Movements (Oxford: Clarendon Press, 1903). 2 Schoenflies, Arthur, Krystallsysteme und Krystalstruktur (Leipzig: B. G. Teubner, 1891). 3 Schoenflies, Artur, Theorie der Kristallstruktur (Berlin: Gebruder Borntraeger, 1923). 4 For some unknown reason, chemists use symbols which look like the Schoenflies symbols, but which have a different meaning. Schoenflies notation for an operation is a symbol like A2,/3

VOL. 58, 1967 PHYSICS: M. J. BUERGER 1773 (which represents a rotation through 2r/3 about an axis A). The second power of this operation is A2,/32. The cyclical group generated by operation S2T/3 is C3. The second space group derived by Schoenflies as isomorphic with point group C3 is C32. But the chemists use the point-group symbol C3 to represent the operation A2,T/3. They thus use the crystallographic symbol which Schoenflies used for a space group to represent a power of an operation of a point group. This is not the Schoenflies notation. 5 Astbury, W. T., and Kathleen Yardley, "Tabulated data for the examination of the 230 spacegroups by homogeneous X-rays," Phil. Trans. Roy. Soc. London, Ser. A, 224, 221-257 (1924). 6 Hermann, C., "Zur systematischen Strukturtheorie I. Eine neue Raumgruppensymbolik," Z. Krist., 68, 257-287 (1928). 7 Schiebold, Ernst, "Uber eine neue Herleitung und Nomenklatur der 230 kristallographischen Raumgruppen," Abhandl. Saechs. Akad. Wiss., math.-physik. KM., 15, 204 (1929). 8 Mauguin, Ch., "Sur le symbolisme des groupes de repetition ou de symetrie des assemblages cristallins," Z. Krist., 76, 542-558 (1931). 9 Buerger, M. J., Elementary Crystallography (New York: John Wiley & Sons, 1956), esp. pp. 35-44.