## teaching and education

## From patterns to space groups and the

of crystallographic orbits: a reinterpretation of some symmetry diagrams in IUCr Teaching Pamphlet No. 14^{a}Laboratorio de Cristalografía, Estado Sólido y Materiales (Cryssmat-Lab)/DETEMA, Facultad de Química, Universidad de la República, Avenida Gral Flores 2124, 11800 Montevideo, Uruguay, and ^{b}Université de Lorraine, Faculté des Sciences et Technologies, Institut Jean Barriol FR 2843, Cristallographie, Résonance Magnétique et Modélisations (CRM2), UMR – CNRS 7036, Boulevard des Aiguillettes, BP 70239, F54506 Vandoeuvre-lès-Nancy Cedex, France^{*}Correspondence e-mail: massimo.nespolo@crm2.uhp-nancy.fr

The

of a is the intersection group of the eigensymmetries of the crystallographic orbits corresponding to the occupied Wyckoff positions. Polar space groups without symmetry elements with glide or screw components smaller than 1/2 do not contain characteristic orbits and cannot be realized in patterns (structures) made by only one crystallographic type of object (atom). The space-group diagram of the general orbit for this type of group has an that corresponds to a special orbit in a centrosymmetric of the generating group. This fact is often overlooked, as shown in the proposed solution for Plates (i)–(vi) of IUCr Teaching Pamphlet No. 14, and an alternative interpretation is given.### 1. Introduction

Although crystallography is a centennial science, its presence in higher education is in jeopardy. Indeed, crystallography is very often nothing more than a chapter in introductory solid-state physics and chemistry books and in mineralogy textbooks, and therefore the treatment it receives in graduate-level courses is often only incidental. As a result of this lack of formal crystallographic education, many young crystallographers have acquired their knowledge in the field through a rather slow and sometimes tortuous self-education process, using some excellent books available on the different aspects and applications of crystallography, and attending schools and workshops covering basic and advanced aspects and new developments of crystallography at different levels. This continues to be the case nowadays, especially, but not only, in the developing world, where the lack of strong crystallographic societies or associations keeps crystallography as a very rarely taught topic at the undergraduate level and only for specific areas in graduate schools. Paradoxically, this difficulty of learning crystallography may also be at the root of its outstanding development in the past century, since modern crystallographers come from such different knowledge areas as physics, chemistry, materials science, mineralogy and biology, permanently enriching this already wide area of science.

This problem has been tackled by the IUCr at different times through different strategies, all aimed at compensating for the lack of formal education in crystallography. The creation of the IUCr Teaching Commission (IUCr-TC) in 1954 was one of these actions, and the systematic work of convincing academia to include crystallography as a separate subject in undergraduate and graduate-level courses has always been part of the work of the IUCr. Nowadays, in some universities in Europe specific graduate programmes on crystallography exist as a consequence of this push, but they are often isolated efforts by researchers in just a few universities. Being very aware of the differences in development of crystallographic teaching in different regions of the world, in the late 1970s the IUCr-TC undertook the task of providing academia with a series of short booklets or pamphlets directed at helping students to self-educate and teachers to introduce the basic concepts of crystallography to advanced undergraduate or graduate students. These so-called IUCr Teaching Pamphlets have become standard and widely used crystallographic teaching materials. Since the first series published in 1980, continued by the second series published in 1984, up to some recent additions, a total of 23 IUCr Teaching Pamphlets have been published and made available for free at the IUCr website (http://www.iucr.org/education/pamphlets ). These are in general high-level teaching materials checked carefully for errors and inconsistencies. Nevertheless, some topics that should form the common background of a crystallographer are not (yet?) included and their absence results in some inconsistencies, even in this professional series. Here, we point out the concepts of orbit and intersection symmetry, which are practically never presented even in graduate courses. Without them, serious oversights may occur, and indeed have occurred, as we will show.

### 2. Space groups as intersection groups of the eigensymmetries of crystallographic orbits

The operations of a *G* applied to an atomic position give rise to an infinite set of equivalent atoms called a or [for details of the difference between these terms, see Koch & Fischer (1985)]. Let the of the *i*th orbit *O*_{i} be *E*(*O*_{i}), or *E*_{i} for brevity. The relation between *E*_{i} and *G* gives rise to the following subdivision, where *T* is the of translations (Engel *et al.*, 1984):

(1) *E*_{i} = *G*: the orbit is called characteristic;

(2) *E*_{i} > *G*: the orbit is called noncharacteristic; it can be further subdivided depending on whether

(2.1) *T*(*E*_{i}) = *T*(*G*): the noncharacteristic orbit is non-extraordinary (term usually omitted);

(2.2) *T*(*E*_{i}) > *T*(*G*): the noncharacteristic orbit is extraordinary, the latter term taking priority over the former (an extraordinary orbit is always noncharacteristic, while the opposite is not true).

A *S* can be seen as the union (in the algebraic meaning) of all the crystallographic orbits *O* corresponding to the Wyckoff positions occupied by the atoms of the structure. The of the structure *G*(*S*) is, instead, the intersection of the eigensymmetries of these orbits. In fact, for each orbit, only the symmetry operations that are common to the other orbits are promoted to symmetry operations of the whole structure, the others being operations [for the meaning of a local operation, see Nespolo *et al.* (2008)];

A *S* is composed of only one type of atom, which, under the action of *G*, generates one orbit *O*, then necessarily *G*(*S*) = *E*(*O*), which requires that the orbit is characteristic. Space groups without characteristic orbits are typically pyroelectric groups without *d* mirrors, or screw axes with a screw component different from (*i.e.* containing only 2_{1}, 4_{2} and 6_{3} as screw axes). In these space groups, the of each orbit has an additional *q* perpendicular to the defining the polar direction(s): either a mirror perpendicular to the polar axis or a twofold axis perpendicular to the polar plane (in the absence of *P*1 is an exception because the triclinic metric is not compatible with a proper or improper rotation of order higher than 1). In fact, atoms in the orbit are in one of the following four situations: (i) on planes separated by full translations; (ii) on planes separated by half translations; (iii) along directions separated by full translations; and (iv) along directions separated by half translations. These atoms have *q* in their eigensymmetry, and the *s*(*q*) about *q* defines a *s*(*q*)*G* so that *G* ∪ *s*(*q*)*G* = *E* is the of the orbit. As a consequence, the general orbit in *G* corresponds to a special orbit in *E*, whose group is precisely defined by *q*. For these cases, the space-group diagrams in Volume A of *International Tables for Crystallography* (2011) do not indicate any additional symmetry elements, because for structures composed of more than one orbit these are local elements, although each diagram gives only one general orbit. However, when applying the opposite reasoning, from the orbit to the the implicit assumption that the orbit is general may result in the underestimation of the and thus of the as we will now show.

### 3. Missing symmetry elements in the IUCr Teaching Pamphlets

IUCr Teaching Pamphlet 14 (*Space Group Patterns*; Meier, 2001) is the continuation of IUCr Teaching Pamphlet 13 (*Symmetry*; Dent Glasser, 2001). It contains 15 plates showing groups of feet or hands (or, as we interpret them, footprints and handprints) periodically and symmetrically arranged to represent crystal patterns^{1} in each of the 230 types of in a particular setting. Pamphlet 14 is designed to put into practice the concepts of symmetry introduced in Pamphlet 13, and starts with an explanatory introduction where the symbols and rules for the use of the plates are outlined.

In general, two types of symbol are used for the symmetry patterns. Footprints are used for space groups containing only twofold symmetry operations [Plates (i)–(vi)], while handprints are used for space groups with rotations of higher order. The feet symbols are also used to exemplify planar groups that can be obtained as a projection of a

along the vertical axis. The difference between a hand and a foot may not be evident from examining real hands and feet, but in the plates footprints are only used to represent polar space groups, the polar direction being taken as the direction of projection, since footprints are always looked at from above. Feet differ, however, in their handedness (right or left). Handprints, instead, are shown both right and left and palm up or palm down.The polar space groups represented by footprint patterns are precisely the types without characteristic orbits: *Pma*2 (No. 28) [Plate (i)], *Pnc*2 (No. 30) [Plate (ii)], *Pbn*2_{1} (No. 33) [Plate (iii)], *Cc* (No. 9) [Plate (iv)], *Cmc*2_{1} (No. 36) [Plate (v)] and *Aea*2 (former symbol *Aba*2) (No. 41) [Plate (vi)]. The plates represent only the general orbit of these groups, so that *G*(*S*) = *E*(*O*). Because the of the orbit is higher than that of the generating group, the given in the text is systematically a of the corresponding to the plates. In other words, each plate shows a special orbit in a centrosymmetric while the text describes it as a general orbit in a polar group.

Let us examine Plate (i), reproduced in Fig. 1. This is a corresponding to the general orbit of a of type *Pma*2. The + symbol at the top right of the picture indicates that the *z* coordinate of the feet is located away from *z* = 0. A footprint has *m*. Because all feet are located at the same *z* coordinate, this mirror also occurs in the of the pattern, at *z* coordinates + and + with respect to the chosen origin. This mirror implies the existence of further symmetry elements, namely twofold screw axes parallel to [100], twofold axes parallel to [010] and inversion centres at the intersection of 2_{[001]} with *m*_{[001]} (Fig. 2). The of the pattern shown in Plate (i) is thus *Pmam* [standard symbol *Pmma* (No. 51) obtained by an transformation]. In this type of the footprints are no longer in a general position but in a special position with ..*m* (.*m*. in the standard setting). A shift of the origin is necessary to obtain the standard description. Once this shift is applied, it is possible to recognize that the orbit corresponds to 4*i* or 4*j*, depending on where the origin is placed with respect to the orbit. The of the pattern would only be of type *Pma*2 if the footprint did not possess *m*, *i.e.* if the top and the bottom of the footprint were different, as is the case for the handprints, for which palm up and palm down are shown.

The same argument applies to Plates (ii)–(vi), where the supposedly polar arrangements of footprints correspond not to a general orbit in *G* (polar) but to a special orbit in the centrosymmetric *E*. The correct space-group types are then *Pncm* [Plate (ii); standard symbol *Pmna* (No. 53)], *Pbnm* [Plate (iii); standard symbol *Pnma* (No. 62)], *C*2/*c* (No. 15) [Plate (iv)], *Cmcm* (No. 63) [Plate (v)] and *Aeam* (No. 64) (former symbol *Abam*) [Plate (vi); standard symbol *Cmce*].

### 4. Discussion

The usual way of introducing space groups in crystallography courses is *via* the application of space-group operations to objects in a general position to generate a The opposite approach, from pattern to is didactically more interesting, not only because a is indeed the *a posteriori* interpretation of a in terms of its symmetry, but also because it underlines several features that normally go unnoticed, namely (i) the of each orbit, (ii) the nature of a as the intersection group of these eigensymmetries, and (iii) the presence of operations, which are part of the of an orbit but not common to the other orbits. Adopting this approach in parallel with the more common way of introducing space-group symmetry avoids oversights like those present in Teaching Pamphlet No. 14 discussed in this article. This pamphlet is frequently downloaded from the IUCr web site, suggesting it is still in widespread use, making our reinterpretation and this discussion of some didactic value for teachers who use it.

### Footnotes

^{1}A is a generalization of a to a set of any objects or figures. A is a special case of a where the objects are atoms.

### Acknowledgements

LS thanks the anonymous student who first solved Plate (i) `incorrectly' by placing a mirror plane in the plane of the feet, drawing our attention to the possible misinterpretation that the use of footprints may lead to in the first six plates of pamphlet No. 14. The authors are also indebted to Brian McMahon from the Chester office of the IUCr for providing information on the download of IUCr Teaching Pamphlet No. 14 from http://www.iucr.org/education/pamphlets/14 . The critical remarks of two anonymous reviewers are gratefully acknowledged.

### References

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