## research papers

## Symmetry Elements in Space Groups and Point Groups. Addenda to two IUCr Reports on the Nomenclature of Symmetry†

^{a}Laboratoire de Cristallographie, Université de Genève, 24 quai Ernest-Ansermet, CH-1211 Genève 4, Switzerland, ^{b}Institut für Kristallographie, Universität Karlsruhe, D-76128 Karlsruhe, Germany, ^{c}Institut für Kristallographie, RWTH Aachen, D-52056 Aachen, Germany, and ^{d}Physics Department, Southern Oregon University, Ashland, OR 97520, USA^{*}Correspondence e-mail: howard.flack@cryst.unige.ch

The definition of `symmetry element' given in the Report of the IUCr *Ad-Hoc* Committee on the Nomenclature of Symmetry by de Wolff *et al.* [*Acta Cryst.* (1989). A**45**, 494–499] is shown to contain an ambiguity in the case of space groups *P*6/*m*, *P*6/*mmm*, *P*6/*mcc* and point groups 6/*m* and 6/*mmm*. The ambiguity is removed by redefining the `geometric element' as a labelled geometric item in which the label is related to the rotation angle of the rotation or rotoinversion The complete set of different types of glide plane is shown to contain three more than the 15 that are illustrated in the 1992 Report by de Wolff *et al.* [*Acta Cryst.* (1992). A**48**, 727–732].

Keywords: symmetry.

### 1. Introduction

The IUCr *Ad-Hoc* Committee on the Nomenclature of Symmetry presented a Report in 1989 on the definition of symmetry elements in three dimensions (de Wolff *et al.*, 1989), and another in 1992 on the corresponding symbols for symmetry elements and symmetry operations (de Wolff *et al.*, 1992). Pacheco (1998) noticed that Fig. 3 in the 1992 Report, which aimed to present all possible glide-plane aspects, did not contain the diagram for an *n*-glide plane of the *tp* (tetragonal primitive) Bravais-lattice type. Flack (1998) pointed out that a strict application of the definitions in the 1989 Report to 6/*m* led to an undefined type of Both matters were referred to readily available members of the original Committee who agreed to investigate the two problems.

### 2. Definition of symmetry element

#### 2.1. Definition in 1989 Report

The term *symmetry element*, widely used over a long period in the crystallographic literature prior to 1989, had been subject to application in a variety of different ways. The Report of 1989 presented a careful definition of this term in order to clarify its meaning.

The 1989 Report first defined the *geometric element* by starting from a of a or A *reduced symmetry operation* may be formed from the For a screw rotation, this is the corresponding rotation; for a glide reflection, it is the corresponding reflection. A (and thus a symmetry element) is not defined for the identity mapping or the translations. The is thus the geometric item that allows the reduced to be located and oriented in space. This is a plane for reflections and glide reflections (mirror plane or glide plane), a line for rotations and screw rotations (rotation axis or screw axis), a point for inversions (centre of inversion), and a point on a line for rotoinversions (rotoinversion axis), see Tables 1 and 2 of the 1989 Report. These tables are repeated in the 1992 Report; an omitted comma after `Angle and sense of rotation', under `Additional parameters' for screw rotation operations in Table 1 of 1992, is noted to avoid the possibility of confusion.

Different symmetry operations of a symmetry group or crystal may have the same *element set of a geometric element* is defined as the set of *all* symmetry operations of a or which have the same see Tables 1 and 2 of the 1989 Report. The is the combination of the with the element set. Its name is formed by attaching the symbol of the element set to the for example, a twofold rotation axis, a threefold screw axis, a glide plane, a fourfold rotoinversion axis *etc.* The designation of a however, is derived from the complete element set of the *e.g.* the of a threefold screw rotation with axis through the origin in *P*3_{1} is a threefold screw axis, in *P*6_{1} it is a sixfold screw axis. It is noted that further specification of these symmetry elements is possible only if the (the space group) and thus its translation lattice is known. The combination of a counterclockwise rotation by 120° about the direction [111] with a lattice translation by **a**, *i.e.* the *z*+1, *x*, *y* [in the notation of *International Tables for Crystallography* (1996), Vol. A, hereafter *IT*A], of the cubic *P*23, is a threefold screw rotation. Its is a 3_{1} screw axis through the point , ;, 0 in *P*23, the of the same is a 3_{2} screw axis in *I*23. The of the glide reflection *x*+½, *y*, is an *a*-glide plane in *Pmma* but an *e*-glide plane in *Cmme* (formerly *Cmma*).

#### 2.2. Recommended definition

The definition of `symmetry element' in §2.1 breaks down in space groups *P*6/*m*, *P*6/*mmm*, *P*6/*mcc* and point groups 6/*m* and 6/*mmm*, as observed by Flack (1998). The of the rotoinversion in 6/*m* is a line with a point defined on it. However, not only and (the inverse of ) but also and belong to this The second sentence of the second paragraph in §5 of the 1989 Report: `Given that it is the element set (consisting of the operations sharing that geometric element) that determines the nature of (and, eventually, the symbol for) the There is always just one such set, so that no ambiguity can exist and it is only the symbol which may be open to discussion' hence does not hold in this example.

The *symmetry element* concept in the Report of 1989 is so clear and useful that it should be retained. However, the inconsistency in the last paragraph should be removed. This is easily accomplished by defining the *geometric element* not as a purely geometric item (point, line, plane, line + point on it) but as a *labelled geometric item*. The geometric item is marked by an appropriate designation of the original (defining) operation. The of the operation in 6/*m* is not line + point but, rather, is line + point coupled to . Thus defined, and do not belong to the element set of this because its element set is only and , see Tables 1 and 2 in the 1989 Report. The resulting ambiguity has been lifted by recalling the original (*defining*) For this reason, the definition of the is augmented by specifying a label related to the rotation angle of the rotation or rotoinversion symmetry operation.

It is noteworthy that such difficulties do not arise in space groups *P*6_{3}/*m*, *P*6_{3}/*mcm*, and *P*6_{3}/*mmc* due to the separation of invariant points in and .

### 3. Types of glide plane

The list of 15 types of glide plane in Fig. 3 of the 1992 Report is incomplete, as originally discovered by Pacheco (1998) who suggested the addition of a diagram such as (11) in the present Fig. 1. Two further omissions, displayed as diagrams (7) and (15) in the present Fig. 1, were discovered during this study by H. Wondratschek. Fig. 3 of the 1992 Report should hence be replaced by the present Fig. 1. It should be noted that the numbers of some diagrams in the present Fig. 1 are changed with respect to their numbering in Fig. 3 of the 1992 Report. This is necessary to place the new diagrams in the most appropriate sites.

It is also noted that the first term *oc* in the final column of Table 3 of the 1992 Report should be changed to *oc*, *tp*(*tc*). The special orientation of the tetragonal net should be indicated because the setting of the tetragonal net is centred in cubic space groups with lattice type *F*, see also the present Fig. 1.

### Acknowledgements

Appreciation is expressed to Mr J. V. Pacheco for bringing the omission in Fig. 3 of the 1992 Report to our attention. The help of Ralf Müller, Karlsruhe, Germany, in drawing Fig. 1 is gratefully acknowledged.

### References

Flack, H. D. (1998). Private communication. Google Scholar

*International Tables for Crystallography* (1996). Vol. A, *Space-Group Symmetry*, 4th revised ed., edited by Th. Hahn. Dordrecht: Kluwer Academic Publishers. Google Scholar

Pacheco, J. V. (1998). Private communication. Google Scholar

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). *Acta Cryst.* A**48**, 727–732. CrossRef Web of Science IUCr Journals Google Scholar

Wolff, P. M. de, Billiet, Y., Donnay, J. D. H., Fischer, W., Galiulin, R. B., Glazer, A. M., Senechal, M., Shoemaker, D. P., Wondratschek, H., Hahn, Th., Wilson, A. J. C. & Abrahams, S. C. (1989). *Acta Cryst.* A**45**, 494–499. CrossRef Web of Science IUCr Journals Google Scholar

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